Surveys in Differential Geometry

Vol. 1: Lectures given in 1990 edited by S.-T. Yau and H. Blaine Lawson Vol. 2: Lectures given in 1993 edited by C.C. Hsiung and S.-T. Yau Vol. 3: Lectures given in 1996 edited by C.C. Hsiung and S.-T. Yau Vol. 4: Integrable systems edited by Chuu Lian Terng and Karen Uhlenbeck Vol. 5: Differential geometry inspired by string theory edited by S.-T. Yau Vol. 6: Essays on Einstein manifolds edited by Claude LeBrun and McKenzie Wang Vol. 7: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer edited by S.-T. Yau Vol. 8: Papers in honor of Calabi, Lawson, Siu, and Uhlenbeck edited by S.-T. Yau Vol. 9: Eigenvalues of Laplacians and other geometric operators edited by A. Grigor’yan and S-T. Yau Vol. 10: Essays in geometry in memory of S.-S. Chern edited by S.-T. Yau Vol. 11: Metric and comparison geometry edited by Jeffrey Cheeger and Karsten Grove Vol. 12: Geometric flows edited by Huai-Dong Cao and S.-T. Yau Vol. 13: Geometry, analysis, and algebraic geometry edited by Huai-Dong Cao and S.-T.Yau Vol. 14: Geometry of Riemann surfaces and their moduli spaces edited by Lizhen Ji, Scott A. Wolpert, and S.-T. Yau Vol. 15: Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday edited by Tomasz Mrowka and S.-T. Yau Volume XV

Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday

edited by Tomasz Mrowka and Shing-Tung Yau

International Press www.intlpress.com Series Editor: Shing-Tung Yau

Surveys in Differential Geometry, Vol. 15 (2010) Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday

Volume Editors: Tomasz Mrowka (Massachusetts Institute of Technology) Shing-Tung Yau (Harvard University)

2010 Mathematics Subject Classification. 00A79, 00Bxx, 14H52, 20-XX, 53-XX, 53D05, 81T13, 83-XX.

All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Repub- lication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author. (Copyright ownership is indicated in the notice on the first page of each article.)

ISBN 978-1-57146-145-2

Printed in the United States of America.

15 14 13 12 11 1 2 3 4 5 6 7 8 9 Surveys in Diﬀerential Geometry XV

Preface

This volume honors the 85th birthday of our friend and teacher Isadore Singer. We organized a conference to honor this event in May of 2009. The lectures were given at the Massachusetts Institute of Technology and at Har- vard University. Included herein are papers by many of the speakers, as well as contributions from friends of Is. The breadth and depth of these papers reflect the many areas of mathematics and physics that Is has influenced. Over the past 60 years, Singer’s work has transformed many areas of mathematics and physics. Singer is a major force in transforming work in geometry from a local to a global point of view, as well as pioneering the modern interactions between mathematics and physics. After receiving his PhD with Irving Segal at the University of Chicago in 1950, his early work was in operator algebras and Riemannian geometry. Two results from this period are the famous Ambrose-Singer holonomy theorem, and the Kadison-Singer problem (which remains open to this day and is now known to be equivalent to important questions in harmonic analysis and wavelet theory). In the early 1960s, Singer began his long collaboration with Sir Michael Atiyah with their legendary work on index theory. There were several proofs of the index theorem: the original cobordism proof, the K-theoretic proof, and, finally, the heat equation proof. McKean and Singer calculated the asymptotic expansion of the solution of heat equation on a manifold. Heat equation methods led to several important works of Singer: the Atiyah-Patodi-Singer index theorem for manifolds with boundary and the introduction of the η- invariant, as well as the work with Ray on analytic torsion. In the 1970s, Is began his long- running effort to bring mathematics and modern physics closer together. The use of the index theorem to compute the dimension of the moduli space of self-dual connections on a four-manifold, and the explanation of the Gribov ambiguity, marked new a level of serious modern mathematics being applied to the current work of the physicists. Since then, Singer has worked on many aspects of the relations between mathematics and physics, with collaborators including Alexrod, Alvarez, Bealieu, Hop- kins, and Ramadas. These efforts are reflected strongly in the topics covered in this volume.

v vi PREFACE

The volume consists of twelve papers: “A shifted view of fundamental physics” by Michael Atiyah and Gregory W. Moore; “Subgroups of depth three” by Sebastian Burciu and Lars Kadison; “Yukawa couplings in F-theory and non-commutative geometry” by Sergio Cecotti, Miranda C.N. Cheng, Jonathan J. Heckman, and Cumrun Vafa; “Spin structures and superstrings” by Jacques Distler, Daniel S. Freed, and Gregory W. Moore; “Operator traces and holography” by Michael R. Douglas; “A loop of SU(2) gauge fields stable under the Yang-Mills flow” by Daniel Friedan; “Automor- phisms of graded super symplectic manifolds” by Joshua Leslie; “The signature of the Seiberg-Witten surface” by Andreas Malmendier; “Eta forms and the odd pseudodifferential families index” by Richard Melrose and Frédéric Rochon; “Anomaly constraints and string/F-theory geometry in 6D quantum gravity” by Washington Taylor; “A new look at the path integral of quantum mechanics” by Edward Witten; and “Quasi-local mass in general relativity” by Shing-Tung Yau. We would like to thank the Simons Foundation, the mathematics depart- ments at Harvard University and the Massachusetts Institute of Technology, the National Science Foundation, and Rosemarie Singer, for all their help in various ways in making the conference and this volume a success. We hope this volume will be remembered as a gift from his friends and students. We appreciate his friendship, his teaching, and his contributions to science.

Tomasz S. Mrowka Massachusetts Institute of Technology Shing-Tung Yau Harvard University

This material is based upon work supported by the National Science Foundation under Grant No. 0928515. Surveys in Diﬀerential Geometry XV

Contents

Preface ...... v A Shifted View of Fundamental Physics MichaelAtiyahandGregoryW.Moore...... 1 Subgroups of Depth Three SebastianBurciuandLarsKadison...... 17 Yukawa Couplings in F-theory and Non-Commutative Geometry Sergio Cecotti, Miranda C. N. Cheng, Jonathan J. Heckman, 37 andCumrunVafa...... Spin Structures and Superstrings JacquesDistler,DanielS.Freed,andGregoryW.Moore...... 99 Operator Traces and Holography MichaelR.Douglas...... 131 ALoopofSU (2) Gauge Fields Stable under the Yang- Mills Flow DanielFriedan...... 163 Automorphisms of Graded Super Symplectic Manifolds JoshuaLeslie...... 237 The Signature of the Seiberg-Witten Surface AndreasMalmendier...... 255 Eta Forms and the Odd Pseudodifferential Families Index Richard Melrose and FrédéricRochon...... 279 Anomaly Constraints and String/F-theory Geometry in 6D Quantum Gravity WashingtonTaylor...... 323

vii viii CONTENTS

A New Look at the Path Integral of Quantum Mechanics EdwardWitten...... 345 Quasi-Local Mass in General Relativity Shing-TungYau...... 421 Surveys in Diﬀerential Geometry XV

A Shifted View of Fundamental Physics

Michael Atiyah and Gregory W. Moore

Abstract. We speculate on the role of relativistic versions of delayed diﬀerential equations in fundamental physics. Relativis- tic invariance implies that we must consider both advanced and retarded terms in the equations, so we refer to them as shifted equations. The shifted Dirac equation has some novel properties. A tentative formulation of shifted Einstein-Maxwell equations naturally incorporates a small but nonzero cosmological constant.

1. Prologue by Michael Atiyah The past three or four decades have seen a remarkable fusion of theoretical physics with mathematics. Much of the impetus for this is due to Is Singer whose 85th birthday is being celebrated at this meeting. He introduced me1 to many of the physical ideas and instructed me in the areas of mathematics that had interacted strongly with physics in the first part of the 20th century. These were differential geometry, flourishing in the wake of Einstein’s theory of general relativity and functional analysis, which provided the rigorous background for quantum mechanics as laid down by von Neumann. By contrast my own mathematical background has centered round algebraic geometry and topology, the areas of mathematics which have played an increasingly important part in the developments in physics of the last quarter of the 20th century and beyond. I think Is and I were fortunate to be at the right place at the right time, working on the kind of mathematics which became the new focus of interest in theoretical physics of gauge theories and string theory. Looking to the past the great classical era began with Newton, took a giant stride with Maxwell and culminated in Einstein’s spectacular theory of general relativity.

1This joint article is based on the lecture by the ﬁrst author at Singer 85.

c 2011 International Press

1 2 M. ATIYAH AND G. W. MOORE

But then came quantum mechanics and quantum field theory with a totally new point of view, rather far removed from the geometric ideas of the past. The major problem of our time for physicists is how to combine the two great themes of GR, that governs the large scale universe, and QM that deals with the very small scale. At the present time we have string theory, or perhaps “M-theory,” which is a beautiful rich mathematical story and will certainly play an important role in the future of both mathematics and physics. Already the applications of these ideas to mathematics have been spectacular. To name just a few, we have 1. Results on the moduli spaces of Reimann surfaces. 2. The Jones polynomials of knots and their extension by Witten to “quantum invariants” of 3-manifolds. 3. Donaldson theory of 4-manifolds and the subsequent emergence of Seiberg-Witten theory. 4. Mirror symmetry between holomorphic and symplectic geometry. Although string theory or M-theory are thought by many to be the ultimate theory combining QM and GR no-one knows what M-theory really is. String theory is recognized only as a perturbative theory, but the full theory is still a mystery (one of the roles of the letter M). Some claim that the final theory is close at hand – we are almost there. But perhaps this is misplaced optimism and we await a new resolution based on radical new ideas. There are, after all, some major challenges posed by astronomical observations. • Dark Matter • Dark Energy (with a very small cosmological constant) Moreover, the direct linkage between the rarified mathematics of string theory and the world of experimental physics is, as yet, very slender. A friend of mine, a retired Professor of Physics, commentating on a string theory lecture, said it was ”pure poetry”! This can be taken both as criticism and as a tribute. In the same way science fiction cannot compete with the modern mysteries of the quantum vacuum. But, if we need new ideas, where will they come from? Youth is the traditional source of radical thoughts, but only a genius or a fool would risk their whole future career on the gamble of some revolutionary new point of view. The weight of orthodoxy is too heavy to be challenged by a PhD student. So it is left to the older generation like me to speculate. The same friend who likened string theory to poetry encouraged me to have wild ideas, saying “you have nothing to lose!” That is true, I have my PhD. I do not need employment and all I can lose is a bit of my reputation. But then allowances are made for old-age, as in the case of Einstein when he persistently refused to concede defeat in his battle with Niels Bohr. A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 3

So my birthday present to Is is to tell him that we senior citizens can indulge in wild speculations!

2. An exploration The idea we2 want to explore is the use of retarded (or advanced) differential equations in fundamental physics. These equations, also known as “functional differential equations,” or “delayed differential equations” have been much studied by engineers and mathematicians, but the applications in fundamental physics have been limited. This idea has a number of different origins: (i) Such retarded differential equations occur in Feynman’s thesis [5]. (ii) In the introduction to Bjorken and Drell [4] it is suggested that, if space-time at very small scales is “granular,” then one would have to use such equations. (iii) Their use in the context of quantum mechanics has been advocated by C.K. Raju [7]. We will discuss the general idea briefly before going on to explain how to develop a more scientific version. This one of us worked on at an earlier stage as reported in the Solvay Conference [2]. Let us begin by looking at the simplified example of a linear retarded differential equation for a function x(t): (2.1) x˙(t)+kx(t − r)=0 where the positive number r is the retardation parameter and k is a constant. A rescaling of the time variable shows that the equation really only depends on a single dimensionless parameter μ = kr. Moreover, the initial data for such an equation is an arbitrary function g(t) over the interval [0,r]. Successive integration then allows us to extend the function for all t ≥ 0, while successive differentiation (for smooth initial data g(t)) enables us to extend to negative t. A second way to discuss the solutions is to note −zt/r z that the functions x(t)=x0e solve (2.1) provided z = μe . The latter transcendental equation has an infinite set of roots tending to z = ∞.(In general, all the roots z have nonzero real part, and hence the solutions have an unphysical divergence in the far past or future.) From either approach, we note that equation (2.1), like the equations of quantum mechanics, has an infinite-dimensional space of initial data: It can be taken to be the Hilbert space L2[0,r]. We will take that as an encouraging sign and, without pursuing further the parallel with quantum theory at present we can ask whether retarded differential equations make any sense in a relativistic framework where there is no distinguished time direction in which to retard. In fact this can be done and there is a natural and essentially unique way to carry this out. The first observation is that 2 The work of G.M. is supported by DOE grant DE-FG02-96ER40959. He would like to thank T. Banks and S. Thomas for discussions. 4 M. ATIYAH AND G. W. MOORE

d the translation t → t − r has the infinitesimal generator −r dt , so that the d translation is formally just exp(−r dt ). In Minkowski space we need a rela- d tivistically invariant version of dt , i.e. a relativistically invariant first order differential operator. But this is just what Dirac was looking for when he invented the Dirac Operator D. The important point is that D acts not on scalar functions but on spinor fields. Thus, we can write down a relativistic analogue of (2.1) which is a retarded version of the usual Dirac equation

(2.2) {iD − mc + ik exp(−rD)}ψ =0 where D is the Dirac operator and ψ is a spinor field. Note that r has dimensions of length L whilst k (slightly different from that appearing in (2.1)) now has the physical dimension MLT−1. Both k and r are required to be real for physical reasons which will be clarified shortly (see (2.6) and (2.12) below). This appears to be rather a formal equation and one can question whether it makes any sense. In fact (2.2) makes sense for all physical fields ψ, i.e. those which propagate at velocities less than the velocity of light. Any such ψ is a linear combination of plane-waves and the operator exp(−rD) applied to such a plane-wave component just retards it by r in its own time-direction. For waves which travel with velocity c mathematical arguments based on continuity, or physical arguments using clocks, require that there be no retardation. Although we have said that (2.2) is a retarded equation the fact that spinors have both positive and negative frequencies implies that it is also an advanced equation. Perhaps we should use a neutral word such as “shifted” instead of advanced or retarded. Having said that we may consider several variants of the shifted Dirac equation where we replace

→ D k −rD (2.3a) D + := D + e → D − k rD (2.3b) D − := D e → D k (2.3c) D s := D + sinh(rD) → D k (2.3d) D c := D + i cosh(rD)

For brevity we will focus on the modiﬁcation (2.3a) in what follows. It is instructive to examine the plane-wave solutions of the modiﬁed Dirac equations in Minkowski space. We take ψ = s(p)e−ip·x/ where s(p)is 2 μ 2 2 a constant spinor and there is a dispersion relation p = pμp = E0 /c ,with E0 > 0 representing the inertial rest energy of a particle. Acting on such a E0 wavefunction Dψ = c γ · pψˆ where γ · pˆ squares to 1. Let s±(p) denote the eigenspinors. The planewave solutions of the shifted Dirac equation

(2.4) (iD+ − mc)ψ =0 A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 5

−ip·x/ ip·x/ are then s+(p)e and s−(p)e provided E0 irE0 (2.5) − mc + ik exp =0. c c Since the first two terms are real we derive a quantization condition rE0 (2.6) =(n +1/2)π with n integral c Note that, thanks to the half-integer shift, the value r = 0 is excluded in (2.6).3 This is related to and explains the factor i in (2.2) (k being real). Replacing n by n − 1 in (2.6) is equivalent to changing the sign of k in (2.2). The two signs of k are on an equal footing and so we should consider both. A similar discussion with a massless dispersion relation, i.e. E0 =0,shows that there are no solutions: It is not possible to retard a massless stable fermion. There are two ways to interpret the quantization condition (2.6). First, it is useful to rewrite it by recalling that the Compton wavelength of a particle 2πc 0 c of rest energy E is λ = E0 .Thuswehave 2n +1 (2.7) r = λc. 4 The first interpretation of this equation is that it demands that different fermions have different retardation parameters, given by (2.7). We might expect this to become problematic when, say, electrons interact with protons, neutrons, or neutrinos. A second interpretation declares that there is a universal retardation time in Nature, denoted r.Inthisnotewewilladopt the second point of view. The question arises as to the magnitude of r and n. The hypothesis we are entertaining is that the modified Dirac equation (2.4) should apply to stable fermions whose propagation in vacuum would ordinarily be described (to good approximation) by a standard Dirac equation. Thus we are led to consider protons, neutrons, electrons, and neutrinos. The Compton wavelength e −12 of the electron is λc ∼ 10 m, while for the proton and neutron we have ν p,n e −3 5 λc 9 λc /λc ∼ 10 , whilst the lightest neutrino probably has 10 < e < 10 λc [6, 9]. Equation (2.7) shows that r is bounded below by λc/4. Optimistically taking the smallest nonzero neutrino mass ∼1 eV we have (2.8) r 10−5cm. This scale is uncomfortably large and we hence take the corresponding integer for the lightest neutrino to be of order one.4 3 Given the speculative connections to quantum mechanics mentioned above it is natural to wonder if this is related to the zero point energy of the harmonic oscillator. 4 It is possible that the neutrino is a Majorana particle, and that the above analysis does not hold for a Majorana fermion. We defer investigation of this possibility to another occasion. If we disregard the neutrino then the electron is the lightest stable fermion and the bound (2.8) is weakened to r 10−10cm. 6 M. ATIYAH AND G. W. MOORE

Returning to (2.5) we have: 2 n (2.9) E0 = mc +(−1) kc, This is not a deviation from Einstein’s formula relating rest energy to mass, but simply the relation of the inertial rest energy to the parameters of the modified Dirac equation. Of course, the modified Dirac equations will lead to deviations from standard physical results and hence we expect k and r to be small. One interesting deviation is in the equivalence principle. The modified Dirac equation can be derived from an action principle (2.10) vol ψ¯ (iD+ − mc) ψ from which one can derive the energy-momentum tensor Tμν. For our purposes it will suffice to consider the action in a weak gravitational field (2.11) ds2 = −(1 − 2Φ)dt2 + dxidxi, where Φ is the Newtonian gravitational potential, and extract the coefficient of Φ to obtain the gravitational rest energy. In the curved metric (2.11) the 0 1 Dirac operator D = γ · ∂ + S + ··· with S =Φγ ∂0 + 2 γi∂iΦ Making this substitution, and using on-shell spinor wavefunctions for a particle at rest we find n (2.12) T00 = E0 (1 − μ(−1) ) where kr (2.13) μ := is a dimensionless number. We thus find a deviation from the equivalence principle T00 − E0 (2.14) =(−1)n+1μ. E0 Consequently, the parameter μ will have to be very small (less than about 10−13) not to contradict observational evidence [10]. Since r is uncomfortably large according to (2.8) it must be that k/ is small, something which will prove to be interesting in Section 3.2 below. Indeed

k 5 −1 −8 −1 (2.15) < 10 μcm < 10 cm . The parameter μ is a fundamental constant of our “theory” and links together the two key dimensionful parameters r, the shift, and k, the coeﬃcient that measures the magnitude of the term that shifts the Dirac operator. More- over (2.13) involves Planck’s constant, reﬂecting the quantum character of the parameter kr. All these parameters are embodied in our basic choice of the shifted Dirac operator. We close this section with three sets of remarks. A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 7

2.1. Remarks regarding the formal modification of the Dirac operator. 1. Let us comment on the modified Dirac equation for the other choices D−, Ds, Dc. Using D− instead of D+ we obtain the same results by replacing particles with antiparticles. For the modified Dirac equations using Dc and Ds there is no quantization condition such as (2.6), whilst (2.9) is replaced by a transcendental relation between E0,m,k,r. The equation with Ds and m = 0 is compatible with a massless dispersion relation but that with Dc is not. The violation of the equivalence principle (2.12) is similar in all four cases. 2. One may ask how to define the shifted version of the Klein-Gordon equation for scalar fields. The usual relation D2 = −∇2 is more complicated for modified Dirac operators (2.4). One simple possi- 2 bility is to replace D → D+D−. The latter can be interpreted as a power series in D2: 2 2 2 sinh rD k (2.16) D+D− = D − 2μD − rD 2

Replacing D2 →−∇2 in this expression produces a candidate modified Klein-Gordon operator. 3. We can extend, or couple, the Dirac operator to other fields. Thus coupling to spinors again we get the operator d + δ acting on all differential forms (where δ is the Hodge adjoint of d). We can expo- nentiate this operator but since it does not preserve the degrees of forms, we cannot just restrict it say to 2-forms. Treating Maxwell’s equations requires more care and we will come back to this later in Section 4. 4. Following Dirac, our search for relativistically shifted equations inevitably led us to spinors. An interesting question is how such shifted equations interact with supersymmetry. Perhaps one should shift field equations in superspace.

2.2. Further remarks on advanced and delayed equations in general. 1. Returning to (2.1), we note that it is an unusual equation since it involves a sum of a skew adjoint operator with a unitary operator. Thus it involves an element of an “affine operator group.” This group is in turn a degenerate form of a semi-simple group, reminiscent of the Wigner contraction of semi-simple groups to inhomoge- neous orthogonal groups such as the Poinaré group. 2. Shifting a differential equation in a way that involves both retarded and advanced terms drastically alters its nature, even in the simple 8 M. ATIYAH AND G. W. MOORE

1-dimensional case when (2.1) is replaced by (2.17)x ˙(t)+k{x(t − r)+x(t + r)} =0. This is no longer a simple evolution equation and the theory of such equations has hardly been developed. However, once again, there is an inﬁnite-dimensional space of exponential solutions x(t)=e−zt/r where z now satisﬁes the equation (2.18) z = μ(ez + e−z).

As before, there is an infinite set of roots zn which tend to infinity for n →±∞. An infinite-dimensional space of solutions is given by the linear combinations xn exp(−znt/r). n 3. The roots of (2.18) always have nonzero real parts so once again there is unacceptable behavior in the past or future. However, it should be noted that in the limit μ →∞ these real parts tend to zero. This is particularly clear if one takes k →∞ in (2.17), since the solutions plainly become anti-periodic with period 2r.Thus,for large μ there is a finite-dimensional space of solutions with acceptable oscillatory behavior, at least in a restricted time domain. Of course our interest is in the opposite case of very small k, but this might conceivably be related by some kind of duality. The parallel interchange of small and large values of r arises in Fourier theory. In fact Feynman in [5] studies an equation similar to (2.17) where r represents the distance between a particle and a virtual image. He shows that, surprisingly, there is a conserved energy. (In fact his system is equivalent to a standard physical one without retardation.) For Feynman r is large while for us r is small. 4. For our shifted Dirac operator we expect that mathematical difficulties of the sort encountered in the previous two remarks arise from the mixing of the positive and negative frequencies in the presence of external forces. In standard quantum field theory such mixing of states forces the introduction of Fock space in quantum field theory. In other words our shifted Dirac operator may be easy to define but its mathematical and physical implications can be profound. A link to quantum theory would not therefore be too surprising.

2.3. Some more phenomenological remarks. 1. We have not attempted to investigate the modified Dirac equation in nontrivial electromagnetic backgrounds. Moreover, we have not attempted to include the effects of quantum interactions and thus explore the effects on the standard successes of QED such as Bhaba A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 9

scattering, Compton scattering, the Lamb shift, the anomalous magnetic moment etc. This would be a logical next step if our present speculations are to be taken further. The existence of nontrivially interacting string ﬁeld theories gives some hope that such interactions can be sensibly included. 2. Since the modiﬁed Dirac equation (2.4) is not compatible with massless fermions it is not at all obvious how to include important properties of the standard model such as chiral representations of the gauge group and the Higgs mechanism into our framework. 3. It follows from (2.7) that all particles have a Compton wavelength (and hence mass) related by a ratio of two odd integers. Of course any real number admits such an approximation, but the numerator and or denominator in such an approximation cannot become too large without making r too large. For the electron and proton these integers must be large, but this is not obviously so for neutrinos. Perhaps this idea can be tested experimentally as our knowledge of neutrino masses improves.

3. The shifted Einstein equations 3.1. Definition. Even if we do not know how shifted equations may relate to quantum mechanics we can still ask if there is any way of producing a shifted version of the Einstein equation of General Relativity. We begin by recalling some standard differential geometry, initially in the positive-definite Riemannian version. Recall there are two natural second order differential operators active on the space of 1-forms, both generalizing the Laplacian of flat space, (i) the Hodge Laplacian = (d + δ)2 =Δ (ii) the Bochner Laplacian = ∇∗∇ where ∇ :Ω1 → Ω1 ⊗ Ω1 is the covariant derivative. The Weitzenbock formula asserts that

(3.1) Hodge − Bochner = Ricci This, on a compact manifold, was originally used by Bochner to prove that, if the Ricci tensor was positive definite, then there were no harmonic 1-forms and so the first Betti number was zero. Formula (3.1) continues to hold for a Lorentzian manifold, though sign conventions have to be carefully checked. We replace ∇∗∇ with −∇2 and define δ using the Hodge star. More fundamentally, we view D = d + δ as a Dirac operator coupled to the spin bundle. Then, we have (3.2) D2 + ∇2 = Ricci 10 M. ATIYAH AND G. W. MOORE as an operator equation on 1-forms. We are indebted to Ben Mares for his careful calculations to confirm signs. Equation (3.2) says that the Einstein vacuum equations, Ricci = 0, asserts the equality of the two Laplacians on 1-forms. This indicates how we might define a shifted Hodge Laplacian on 1-forms: We replace D by one of the modified Dirac operators (2.3). There are 10 ways to do this, but we will focus on only three: 2 2 (3.3a) Ricci++ := c(D+) + ∇ 2 (3.3b) Ricci+− := D+D− + ∇ 2 2 (3.3c) Ricciss := Ds + ∇

In equation (3.3a) we cannot simply square the operator D+ because the resulting operator does not preserve the degree of forms. However, we can then compress it on 1-forms by taking the composite operator 2 2 (3.4) c(D+):=P (D+) I where I is the inclusion Ω1 → Ω∗ and P is the projection Ω∗ → Ω1. The other two operators do not need compression. The shifted Einstein equations should then be given by equating Ricci to zero. Written out, these are the equations 2 − 2k k (3.5a) Ricci D sinh(rD)+2 cosh(2rD)=0 2 − 2k − k (3.5b) Ricci D sinh(rD) 2 =0 2 2k k 2 (3.5c) Ricci + D sinh(rD)+2 sinh (rD)=0 It is not at all clear how to interpret these operator equations. We will comment further on this point in Section 3.3 below. One interpretation is to regard the operator as an expansion in D2, then, using (3.2), convert this to an expansion in ∇2. Then we can view higher order terms in ∇2 as an expansion in low energy and small momenta.

3.2. The cosmological constant. If we adopt the viewpoint that the shifted Einstein equations (3.5) can be understood as an expansion in low energies then the leading term is a well-deﬁned equation on the metric given by replacing D2 → Ricci in (3.5). Using the fact that μ must be small the modiﬁed Einstein equations become approximately 2 −k (3.6a) Ricci = 2 k2 (3.6b) Ricci = 2 (3.6c) Ricci = 0 A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 11

In other words the ﬁrst correction to the Einstein equation produced by ± k2 our shifted operators is to introduce a cosmological constant Λ = 2 .This relates our parameter k to cosmology. Evidently, the theory can accommodate all three cases of positive, negative, and zero cosmological constant, and hence the correct sign of the cosmological constant is hardly a triumph for our approach. The observational data support the case (3.6b). Any logical consideration which distinguishes or rules out some of the possible shifted Ricci tensors would be most interesting. The magnitude of Λ is also interesting. As we have seen, observed bounds on violations of the equivalence principle imply that k/ is small and hence Λ is small. In fact, the observational evidence5 gives an order of magnitude for ∼ 4 8πG Λ corresponding to an energy density ρvac (1eV ) and hence Λ = c4 ρvac of order (3.7) Λ ∼ 10−56cm−2 and hence k ∼ −28 −1 (3.8) 10 cm , much smaller than the upper bound (2.15). If we now take the order of magnitude of r derived from our quantization condition then it follows from (2.8) that (3.9) μ 10−33.

3.3. Further comments on interpretation. In Section 3.1 we showed formally how to shift the Ricci tensor, giving formula (3.3), and we focused on the lowest order correction which gave the cosmological constant. We then noted the order of magnitudes that emerged. Notably the estimate (3.9) for our dimensionless parameter μ = kr/. But, as we noted, it is not at all clear how to interpret the operator equation (acting on 1-forms) (3.5). We will now discuss this question. In the ﬁrst place, although D does not preserve the degrees of forms, a power series in the dimensionless quantity r2D2 = r2Δ should make sense (even non-perturbatively) on the space of 1-forms. Second it seems reasonable to restrict (3.5) to 1-forms which are locally solutions of the wave equation (3.10) −∇2φ =0 and hence determined by initial conditions along a space-like 3-space. In view of (3.1), on such 1-forms φ,wehave Δφ = Ricci (φ) so that Δ = D2 acts tensorially on such φ. Unfortunately it does not preserve the solutions of (3.10), since Δ need not commute with the Ricci operator. 5 See, for example, [8]. 12 M. ATIYAH AND G. W. MOORE

However if we are looking for a perturbation expansion in powers of μ this procedure might be made to work. An alternative idea is to regard the equations (3.5) as equations on an 1 infinite-dimensional Hilbert space, namely Ω (M4). That is we consider the equation 2 (3.11) (D+D− + ∇ )φ =0 as an equation for the pair (g, φ) where g is the metric tensor and φ is a 1-form.6 This needs detailed investigation and again it could first be looked at perturbatively in powers of μ. However it should be emphasized that (3.11) makes sense non-perturbatively. Finally, we remark that in string field theory one very naturally runs into exponentials exp[α∇2] acting on fields. This is usually not considered to be too disturbing since at energy-momenta small compared to the string scale such terms are close to 1, and at energy-momenta on the order of the string scale the usual notion of commutative space and time might be breaking down in any case. Certainly, topology changing effects in string theory take place at that scale. Witten has in the past speculated that string theory would necessarily lead to a revision of quantum mechanics. (His main reason being that there is no dilaton-field in 11-dimensional supergravity.) There is thus some remote connection here to Witten’s speculation.

4. The shifted Einstein-Maxwell equations In addition to shifting the Dirac operator and the Ricci operator we should also shift the Maxwell operator ∗ ∗ (4.1) d FA = d dA. One way to do this is to use Kaluza-Klein reduction and the shifted Einstein equations in ﬁve-dimensional spacetime. That is, we take the 5-dimensional space M5 to be a principal circle bundle over the Lorentzian spacetime M4 equipped with a connection Θ and a metric: (4.2) ds2 = e2σΘ2 + ds¯2 2 where ds¯ is the pullback of a metric on the four-manifold M4 and σ is the dilaton ﬁeld, again pulled back from M4. The shifted Einstein equations for such a metric produce the shifted Einstein-Maxwell equations in four dimensions. In addition to the shifted Einstein-Maxwell equations the equation of motion for the scalar σ is amusing. When FA = 0 it is simply 2 −∇¯ 2 σ k σ (4.3) e = 2 e

6 This is reminiscent of Einstein æther theory. See, for example, [1]. However, a minimal requirement would appear to be that the restriction to any x ∈ M4 of the φ’s solving ∗ (3.11) should span Tx M4. A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 13 where = ±1, so the radius eσ of the KK circle is an eigenfunction of the Laplacian on the four-dimensional spacetime. If we take the metric to be deSitter space, and consider spatially homogeneous solutions to (4.3) then the general solution is √ √ σ kc t − kc t (4.4) e = a+e + a−e

Thus, in general the circle radius grows exponentially in the past or the future. This looks potentially devastating since the circle starts growing or shrinking exponentially on a time-scale ∼ kc. However, if we take k/ ∼ 10−28cm−1 then this timescale is approximately (3π)−1 × 1011 years. For comparison the age of the universe is approximately 1.4 × 1010 years! The √1 observation that Λc is the same order of magnitude as the age of the universe is a version of the famous “coincidence problem” in modern cosmology. It means that we just happen to live at an era in the history of the universe when the effects of the cosmological constant change from being negligible to being dominant. In our discussion the coincidence “problem” turns out to be a blessing, not a curse.7 Of course, once the exponential behavior starts to become important one should surely no longer use the zero-mode approximation (i.e. neglecting the O(D2) corrections). We close with three remarks 1. Clearly the above procedure could be extended to include nonabelian Yang-Mills equations by considering Kaluza-Klein reduction on spaces with nonabelian isometry groups. 2. This discussion shows that the idea of shifting geometric operators is a natural process, akin to geometric quantization. It would be fascinating to find any relation between the two processes especially since our shifting process includes the gravitational field, a target not yet achieved in quantization. 3. The fact that our treatment rests fundamentally on spinors, the Dirac operator, and Kaluza-Klein theory suggests possible connections to super-gravity and string theory.

5. Conclusion In this paper we have, very tentatively, put forward a speculative new idea that seems worth exploring. The idea is to introduce in a natural geometric way operators which shift (i.e. retard or advance) the basic operators of mathematical physics. This includes the Dirac, Maxwell and Ricci operators (occurring in the Einstein equations of GR). The shifting involves just

7 Still, the existence of a massless scalar with gravitational strength couplings is problematic in view of fifth-force experiments [10]. 14 M. ATIYAH AND G. W. MOORE two key physical parameters (5.1) r 10−5cm k ∼ −28 −1 (5.2) 10 cm . Here r measures the timeshift and k measures the magnitude of the shift. There is a natural quantization condition 2n +1 (5.3) r = λc 4 where λc is the Compton wavelength of a stable fermion. Of course, this has a quantum-mechanical aspect and involves Planck’s constant . The constant k is at first sight arbitrary (except that it clearly must be very small). However once we introduce our shifted Ricci operator we find that k2/2 is related to the cosmological constant. Using the observed value of this gives the estimate (5.2). Thus the two key constants r, k are determined by physical observations at the atomic and cosmological scales respectively. This is a satisfactory situation. It is also reminiscent of some of the ideas of T. Banks [3]. Note that our approach has a dimensionless fundamental constant linking r, k and (5.4) μ = kr/ μ 10−33. We leave the reader with three key questions 1. What is the correct interpretation of equation (3.5)? 2. Can the above ideas can be made into a coherent model of physics, compatible with the successes of the Standard Models of particle physics and modern cosmology? 3. How is this idea of shifted equations related to quantum mechanics? We leave this to others and to the future. There are tantalizing hints of possible connections, not least the philosophical and mathematical difficulties on both sides!

References [1] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, “Effective field theory for massive gravitons and gravity in theory space,” Annals Phys. 305, 96 (2003) [arXiv:hep- th/0210184]. [2] M.F. Atiyah, Topology and physics, Proc. of the XVIIIth Solvay Conference at Univ. of Texas, Austin, Texas, North-Holland Physics Pub. Amsterdam, 203-206 (1984). [3] T. Banks, “TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory,” arXiv:1007.4001 [hep-th]. [4] J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill Book Co., New York (1965), Section 11.1, p.3. [5] R.P. Feynman, Feynman’s Thesis - A new approach to quantum theory, edited by Laurie M. Brown (Northwestern University, USA), World Scientific Publishing Co. Pte. Ltd. (2005). A SHIFTED VIEW OF FUNDAMENTAL PHYSICS 15

[6] Particle Data Group, http://pdg.lbl.gov/2010/reviews/rpp2010-rev-neutrino- mixing.pdf. [7] C.K. Raju, Towards a consistent theory of time, Kluwer Academic Publishers, Dordrecht, Netherlands (1994). [8] M. Tegmark et al. [SDSS Collaboration], “Cosmological parameters from SDSS and WMAP,” Phys. Rev. D 69, 103501 (2004) [arXiv:astro-ph/0310723]. [9] P. Vogel and A. Piepke, “Neutrinoless Double-β Decay,” SPIRES entry. [10] C. M. Will, “Resource Letter PTG-1: Precision Tests of Gravity,” arXiv:1008.0296 [gr-qc].

School of Mathematics, JCMB, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK

NHETC, Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA

Surveys in Diﬀerential Geometry XV

Subgroups of Depth Three

Sebastian Burciu and Lars Kadison

Abstract. A subalgebra pair of semisimple complex algebras B ⊆ A with inclusion matrix M is depth two iﬀ MMtM ≤ nM for some positive integer n and all corresponding entries. If A and B are the group algebras of ﬁnite group-subgroup pair H3 by successive right multiplications of this inequality with alternately M and M t. We show that a Frobenius complement in a Frobenius group is a nontrivial class of examples of depth three subgroups. A tower of Hopf algebras A ⊇ B ⊇ C is shown to be depth-3 if C ⊆ core(B); and this is also a necessary condition if A, B and C are group algebras.

Introduction Induction of characters from a subgroup to a group is a useful technique for completing character tables [8] found by nineteenth century algebraists. At about the same time, Frobenius discovered reciprocity, which in modern terms states that induction is naturally isomorphic to coinduction of G-modules, either forming an adjoint pair with the restriction functor, and applies to any Frobenius extension of algebras. Finite index subfactors are a certain type of Frobenius extension, where an analytic notion of finite depth was discovered in connection with classification, with depth two being part of a remarkable type of Galois theory of paragroups. The notion of finite depth was eventually made algebraic and applied to Frobenius extensions; later, depth two and its Galois theory of quantum groupoids and Hopf algebroids were exposed in simplest terms for ring extensions (see [18] for an application to J. Roberts field algebra construction [16]). It was noted in [10] that the notion of depth two applies to characters of a finite group and subgroup pair via complex group algebras: a subgroup

c 2011 International Press

17 18 S. BURCIU AND L. KADISON is depth two if no new constituents arise when inducing-restricting-inducing a character as compared with inducing just one time. By means of general theory in one direction and Mackey theory in the other, depth two subgroup is shown to be precisely a normal subgroup [10]. (A similar statement is true for semisimple Hopf C -subalgebras [4].) In this paper we generalize this approach to depth two subgroup to a semisimple subalgebra pair, giving a condition in terms of inclusion matrix [7], which is the same as a induction- restriction table [1] up to a permutation change of basis. The depth two condition is essentially that the cube of the inclusion matrix is less entrywise than a multiple of the inclusion matrix, noted more precisely in the abstract and Proposition 1.2 below. In [11] it was shown that finite depth Frobenius extension has a simplified definition in terms of a generalization of depth two to a tower of three algebras in the Jones tower. In this paper we extend a particular case of an embedding theorem in [11] to characterization of certain finite depth separable Frobenius extension in terms of depth two extension in Jones tower (see Theorems 2.1 and 2.5 below). Then one may check that a subgroup is depth three or more by comparing cube of symmetric matrix S of inner products of induced irreducible characters with multiples of S (see Prop. 2.2). In somewhat the same spirit, Corollary 2.9 below implies that a subgroup is depth three if no new constituents arise from applying restriction-induction one extra time to a character. Although amusing to test for depth three property from character tables of groups and non-normal subgroup, it is not clear from this definition what precisely a depth three subgroup is. A number of proposals to remedy this are given below: depth three quasi-bases are given in Theorem 2.10, a characterization of certain depth three Frobenius extension in terms of similar bimodules, tensor-square and overalgebra in Theorem 2.7, and a class of examples in Section 3, a Frobenius group and its Frobenius complement. Even the notion of depth-3 tower of algebras may be viewed as an alternative to defining finite depth in terms of iterated endomorphism algebra extensions (perhaps applied instead to an iteration of another useful construction). Depth-3 towers of finite group algebras are completely classified in Theorem 1.1 following the spirit of [11]. Depth-3 towers of Hopf algebras are also considered at the end of the second section. A tower of Hopf algebras A ⊇ B ⊇ C is depth-3 if C ⊆ core(B) (see subsection 1.6 for the definition of the core of a Hopf subalgebra). Using then notion of kernel of a module introduced in [3] we formulate a conjecture on the core of a Hopf subalgebra. This conjecture would imply that the condition C ⊆ core(B) is also a necessary condition for the Hopf algebra tower A ⊇ B ⊇ C to be depth-3 (which is true for group algebras by the Theorem 1.1 below). Although our algebras are often over the complex numbers, we have tried to write this paper in a change-of-characteristic-friendly way. SUBGROUPS OF DEPTH THREE 19

1. Preliminaries on depth two extensions All algebras in this paper are associative algebras (not necessarily commutative) over a ﬁeld k. Given an (A, A)-bimodule M, we let M A denote the A-central elements {m ∈ M |∀a ∈ A, am = ma}. Two r × s integer matrices M and N satisfy M ≤ N if each of the coeﬃcients mij ≤ nij: this property is independent of permutation of bases. Note that if X is a third q × r matrix of non-negative integers, then XM ≤ XN;ifXiss×q, then MX ≤ NX.WesayM is strictly positive if all entries mij > 0.

1.1. Frobenius extensions. A Frobenius extension A | B is an extension of associative algebras where the natural bimodule BAA is isomorphic to the (B,A)-bimodule Hom (AB,BB) (of right B-module homomorphisms) given by (b · f · a)(x)=bf(ax)fora, x ∈ A, b ∈ B,f ∈ Hom (AB,BB). This is equivalent to the existence of a mapping F ∈ Hom (BAB,BBB)withdual n n n n bases {xi}i=1 and {yi}i=1 such that i=1 F (axi)yi = a and i=1 xiF (yia)= a for all a ∈ A: we call the data system F a Frobenius homomorphism with dual bases {xi}, {yi}. For example, a group algebra A = k[G] is a Frobenius extension of any subgroup algebra B = k[H], where H ≤ G is a subgroup of ﬁnite index [G : n H]=n.Forif{gi}i=1 denotes left coset representatives of H in G, where −1 g1 =1G, a Frobenius system is given by xi = gi , yi = gi with bimodule projection given by (nhgi ∈ k) n

(1) F nhgi hgi = nhh, i=1 h∈H h∈H a routine exercise. A Frobenius extension A | B enjoys isomorphic tensor-square and endo- ∼ morphism ring as (A, A)-bimodules. We note that A ⊗B A = End AB via ∼ x ⊗B y → λ(x) ◦ F ◦ λ(y). Also A ⊗B A = End BA via x ⊗ y → ρ(y) ◦ F ◦ ρ(x)[9]. Composing the two isomorphisms we obtain an anti-isomorphism End AB → End BA given by f → i F (−f(xi))yi, which restricts to an anti- automorphism on the subring End BAB, and plays the role of antipode in case of depth two Frobenius extension deﬁned below.

1.2. Separable extensions. If the characteristic of the ground ﬁeld k is coprime to [G : H]=n, then the extension of group algebras A | B noted above is a separable extension: i.e. the multiplication map μ : A ⊗B A → A is a split (A, A)-epimorphism. The image of 1A under a section A → n A ⊗B A is a separability element e = i=1 ei ⊗B fi satisfying ae = ea for all n a ∈ A and μ(e)= i=1 eifi =1A, which characterizes separable extension. 20 S. BURCIU AND L. KADISON

Notice that n 1 −1 (2) gi ⊗B gi [G : H] i=1 is a separability element for the group algebras A over B. In the situation that C ⊇ A ⊇ B is a tower of algebras and A | B is a separable extension, the canonical epi C ⊗B C → C ⊗A C given by c1 ⊗B c2 → c1 ⊗A c2 splits. A section for this mapping is of course given by c1 ⊗A c2 → n i=1 c1ei ⊗B fic2.

1.3. Depth-3 towers of algebras. A tower of three algebras A ⊇ B ⊇ C, where C is a unital subalgebra of B which is in turn unital subalgebra of A, is said to be right depth-3, or right d-3, if there is a complementary (A, C)-bimodule P and n ∈ N such that ∼ n (3) A ⊗B A ⊕ P = A as natural (A, C)-bimodules. Equivalently, there is a split (A, C)-bimodule epimorphism from a finite direct sum of A with itself to A ⊗B A (P is the kernel of such an epi). Left d-3 towers are defined oppositely, so that A ⊇ B ⊇ C is left d-3 iff the tower of opposite algebras Aop ⊇ Bop ⊇ Cop is right d-3. It has been noted in [11, 5] that if A | B is a Frobenius, or quasi-Frobenius (QF, where isomorphisms above are replaced by similarity of bimodules) extension, then left d-3 is equivalent with right d-3 extension. For example, a subnormal series of subgroups GNHwith corresponding group algebras A ⊇ B ⊇ C (over any ground field) is a depth- 3 tower, since the normal closure HG ⊆ N G = N and [11, Theorem 3.1] applies. 1.3.1. Depth-3 towers of semisimple algebras. Suppose a tower A ⊇ B ⊇ C of semisimple finite dimensional k-algebras is right d-3. Tensoring 3 by −⊗C M this implies that the following inequality: A A A A M ↑C ↓B↑B,Q≤n M ↑C ,Q holds for any simple C-module M and any simple A-module Q. Using this relation a necessary and sufficient condition for a tower of groups to be depth-3 will be given in the next theorem. For H a subgroup of G let g coreG(H)=∩g∈G H be the largest subgroup of H which is normal in G. (Here gH = gHg−1.) Let G ⊇ N ⊇ H be a tower of groups. Since HG is the subgroup of G gen- −1 erated by the elements ghg with g ∈ G and h ∈ H note that H ⊆ coreG(N) if and only if HG ⊆ N.

Theorem 1.1. AtowerG ⊇ N ⊇ H of groups is depth three if and only if H ⊂ coreG(N). SUBGROUPS OF DEPTH THREE 21

G Proof. If H ⊂ coreG(N) then H ⊆ N and the proof of Theorem 3.1 from [11] applies. Suppose now that the tower is depth-3. The above argument for the tower kG ⊇ kN ⊇ kH of semisimple algebras implies that there is n ∈ N such that G G G G α ↑H ↓N ↑N ,μ≤n α ↑H ,μ for any characters α ∈ Irr(H)andμ ∈ Irr(G). ↑G Put μ =1G , the trivial character in the above inequality. Since α H ↑G ↓G ↑G , 1G = α, 1H it follows that α H N N , 1G =0ifα =1H . By Frobenius ↑G ↓G reciprocity this implies that α H N , 1N =0ifα =1H . On the other hand applying Mackey’s theorem one has: G G g gH N 0= α ↑H ↓N , 1N = α ↓N∩ gH ↑N∩ gH , 1N NgH∈N\G/H g ↓gH = α N∩ gH , 1N∩ gH NgH∈N\G/H H = α ↓g−1 , 1 −1 N∩ H g N∩ H NgH∈N\G/H H = α, 1 −1 ↑g−1 g N∩ H N∩ H NgH∈N\G/H On the other hand using Frobenius reciprocity again one has H 1H , 1 −1 ↑g−1 = 1 −1 , 1 −1 =1 g N∩ H N∩ H g N∩ H g N∩ H Thus H 1 −1 ↑g−1N =1H g N∩ H ∩ H −1 −1 which implies that H = g N ∩ H or H ⊂ g N = g−1Ng.ThusH ⊂ coreG(N).

1.4. Depth two algebra extensions. An algebra extension A ⊇ B is defined to be right depth two (equivalently, subalgebra B ⊆ A is rD2) if the partially trivial tower A ⊇ B ⊇ B is right d-3; similarly we define left D2 in terms of partially trivial left d-3 tower. It is obvious that a finite dimensional algebra A is a depth two extension ∼ n of its unit subalgebra B = k1A:ifdimk A = n, then of course AA⊗k A = AA . Similarly, we may show that if C is a finite dimensional dimensional algebra, the tensor algebra A = C ⊗ B is a depth two extension of its subalgebra B =1C ⊗ B. The main examples in the literature of depth two extension are Hopf- Galois extensions as well as its classical, weakened and pseudo- variants. The defining Condition (3), with B = C, for right depth two extension is similar to the characterization of projective module as isomorphic to a direct summand of a free module. Like the derivation of projective bases for 22 S. BURCIU AND L. KADISON a projective module, we may derive from this condition right D2 quasi-bases for the right D2 extension A | B as follows. For any ring extension, using ∼ the hom-tensor relation, note that Hom (AA ⊗B AB, AAB) = End BAB.By ∼ B evaluation at 1A note that Hom (AAB, AA ⊗B AB) = (A ⊗B A) . n Then the split epi from π : A → A ⊗B A satisfies an equation π ◦ σ = n → idA⊗BA.Wehaven standard split epis A A, which compose with π and σ n ◦ ∈ ⊗ to give the equation i=1 fi gi =idA⊗BA, where fi Hom (A, A B A)and gi ∈ Hom (A ⊗B A, A), to which we apply the simplifications noted above. B Suppose fi → ui ∈ (A⊗B A) , while gi → γi ∈ End BAB for each i =1,...,n. As a consequence, we obtain for any x, y ∈ A the identity n (4) x ⊗B y = xγi(y)ui i=1 B Note that an extension A | B having elements ui ∈ (A ⊗B A) and endo- morphisms γi ∈ End BAB satisfying this identity, eq. (4), also implies that n A | B is right D2, since A → A ⊗B A given by (a1,...,an) → i aiui is an (A, B)-epimorphism with section given by x ⊗B y → (xγ1(y),...,xγn(y)). For example, a normal subgroup N of index n in any group G (over any −1 ground ring) is depth two with D2 quasi-bases given by ui = gi ⊗ gi and −1 γi(g)=F (ggi )gi for coset representatives {g1 = e, g2,...,gn}.

1.5. When inclusion matrix is depth two. Let the ground field k = C be the complex numbers when we consider semisimple algebras, which consequently become multi-matrix algebras (or split semisimple algebras). Suppose B ⊆ A is a subalgebra pair of semisimple algebras. As one constructs an induction-restriction table for a subgroup H in a finite group G [1,p. 166], we briefly review the procedure for generalizing to any pair of semisimple algebras (such as finite dimensional complex group algebras). Label the simples of A by V1,...,Vs and the simple modules of B by W1,...,Wr.To obtain the i’th column restrict the i’th simple A-module Vi to a B-module and express in terms of direct sum of simples ∼ r (5) Vj ↓B = ⊕i=1mijWi

We let M be the r × s-matrix, or table, with entries mij: M =(mij). By a well-known generalization of Frobenius reciprocity, the rows give induction of the B-simples: A A s (6) Wi ↑ = Wi = ⊕j=1mijVj A ∼ A since Wj = Wj ⊗B A and Vi ↓B = Hom (AB,Vi); i.e. if [Wj ,Vi] denotes the A number of constituents in Wj isomorphic to Vi, Frobenius reciprocity is given by A (7) [Wi ,Vj]=mij =[Wi,Vj ↓B] The matrix M is also known as the inclusion matrix of B in A [7]. SUBGROUPS OF DEPTH THREE 23

For example, the induction-restriction table (based on Frobenius reci- G procity (ψi ,χj)G =(ψi,χj ↓H )H ) for the standard embedding of permutation groups S2 ≤ S3 is given by

1 2 1 • • • S2 ≤ S3 χ1 χ2 χ3 101 ψ1 101 M = \ / \ / 011 ψ2 011 ◦ ◦ 1 1 where ψ1 =1H , χ1 =1G denote the trivial characters, ψ2, χ2 the sign characters, and χ3 the two-dimensional irreducible character of S3. Note too the inclusion diagram or Bratteli diagram, a bicolored weighted multigraph [7]. G G For example, 1H = χ1 + χ3 and 1H ↓H =2· 1H + ψ2. Burciu [3]notes G that a subgroup H is normal in G if and only if 1H ↓H =[G : H]1H .In[10] it is established that the notion of depth two subalgebra for subalgebra pair of complex group algebras is equivalent to the notion of normal subgroup.

Proposition 1.2. The inclusion matrix M of a subalgebra pair of semisimple complex algebras B ⊆ A satisﬁes

(8) MMtM ≤ nM for some positive integer n ifandonlyifB is depth two subalgebra of A.

∼ n Proof. (⇐) The depth two condition A ⊗B A ⊕ P = A as natural B-A-bimodules, becomes

A A A (9) [Wi ↓B↑ ,Vj] ≤ n[Wi ,Vj]=nmij

A for all i =1,...,r and j =1,...,s.ButWi is given by row i of M,or eiM, where ei denotes row matrix with all zeroes except 1 in i’th column. A t t t A A Then Wi ↓B is given by M(eiM) = MM ei. Finally Wi ↓B↑ is given by t t t t (MM ei) M, i.e. row i of MM M. (⇒) If the inclusion matrix M of semisimple subalgebra pair B ⊆ A satis- t A A A fies MM M ≤ nM for some n ∈ Z +, then [IndBResBIndBWi,Vj] ≤ A n[IndBWi,Vj] for all B-simples Wi and A-simples Vj(fix these orderings). Via unique module decomposition into simples, we find a monic natural A A A A transformation IndBResBIndB → nIndB from category B-Mod into A-Mod. Now B, A and so Bop ⊗ A are separable C -algebras, so as in [[10], Theorem 2.1(6), pp. 3107–3108], we apply the natural monic to the right regular module BB, apply the natural transformation property to all left multiplications n λb (b ∈ B), and note that A ⊗B A → A splits by Maschke as B-A-bimodule monic. Hence A is depth two over its subalgebra B. 24 S. BURCIU AND L. KADISON

1.6. Depth-3 tower of of Hopf algebras. For B ⊂ A an extension of ﬁnite dimensional Hopf algebras, deﬁne core(B) to be the largest Hopf subalgebra of B which is normal in A. It is easy to see that core(A)always exists (see also [3]). If H ⊂ G is a group inclusion with A = kG and B = kH note that core(B)=kcoreG(H).

Theorem 1.3. Suppose that A ⊇ B ⊇ C is a tower of semisimple Hopf algebras. If C ⊂ core(B) then the tower is depth three.

Proof. Since core(B) is a normal Hopf subalgebra of A it follows that the extension core(B) ⊂ A is D2 and therefore A ⊗core(B) A is a direct sum- n mand of the bimodule A(A )core(B).ThusA ⊗core(B) A is also a direct sum- n mand of the A − C bimodule A(A )C since C ⊂ core(B). Since core(B) ⊂ B the canonical map

A ⊗core(B) A → A ⊗B A is a surjective morphism of A − A-bimodules, in particular of A − C bimodules. Since the category of A ⊗ Cop-modules is semisimple it follows that n A ⊗B A is a direct summand in A(A )C . 1.6.1. Kernel of a module. Let A be a semisimple Hopf algebra over an algebraically closed ﬁeld k.ThenA is also cosemisimple and S2 =Id(see

[13]). Let ΛA be the idempotent integral of A. Denote by Irr(A) the set of irreducible A-characters and let C(A) be the character ring of A with basis Irr(A). There is an involution “ ∗ ”onC(A) determined by the antipode.

Remark 1.4. and If X ⊂ C(A∗) is closed under multiplication then it generates a subbialgebra of A denoted by AX [15]. Moreover if X is also ∗ closed under “ ” it follows from the same paper that AX is a Hopf subalgebra. Since A is ﬁnite dimensional any subbialgebra is a Hopf subalgebra and therefore any subset X closed under multiplication is also closed under “ ∗ ”.

Let M be an A-module with character χ. Deﬁne kerM to be the set of simple subcoalgebras C of A such that cm = ε(c)m for all c ∈ C.Itcan ∗ be proven that the set kerM is closed under multiplication and “ ”and therefore from [15] it generates a Hopf subalgebra AM (or Aχ )ofA [3]. One ⊕ has AM = C∈kerM C. Remark 1.5. ↓A 1. Aχ is the largest subbialgebra B of A such that χ B= χ(1)εB .

Equivalently, Aχ is the largest subbialgebra B of A such that + AB A ⊂ AnnA(M).

2. If A = kG is a group algebra then Aχ = k[ker χ] where ker χ is the kernel of the character χ.

3. It is not known if Aχ is a normal Hopf subalgebra of A.In[3]it ∗ was proven that Aχ is normal in A if χ ∈ Z(A ). SUBGROUPS OF DEPTH THREE 25

4. If N is a submodule or a quotient of M then clearly AM ⊂ AN . (since A is semisimple.)

A Notation: If B is a Hopf subalgebra of A thenwedenotebyε ↑B the ↑A character εB B.

Proposition 1.6. Suppose B and C are Hopf subalgebras of a ﬁnite dimensional semisimple Hopf algebra A.If ∼ n (10) A(A ⊗B ⊗A)C ⊕∗=A A C as A − C- bimodules then ⊂ C Aε↑A . B

Proof. As in subsection 1.3.1 it follows that that A A A A M ↑C ↓B↑B,P≤n M ↑C ,P for any simple C-module M and any simple A-module P . In terms of the characters this can be written as A A A A (11) mA (α ↑C ↓B↑B,χ) ≤ nmA (α ↑C ,χ) for any irreducible character α of C and any irreducible character χ of A.

Here mA is the usual multiplication form on the character ring C(A). Put ↑A χ = εA , the trivial A-character, in the above inequality. Since mA (α C ,εA ) ↑A↓A↑A = mC (α, εC )itfollowsthatmA (α C B B,εA )=0 if α = εC .By ↑A↓A Frobenius reciprocity this implies that mB (α C B,εB )=0 if α = εC . Adding over all irreducible characters α ∈ Irr(C) it follows that ⎛⎛ ⎞ ⎞ ⎝⎝ ⎠ A A ⎠ A A (12) mB α(1)α ↑C ↓B,εB = mB (ε ↑C ↓B,εB ) α∈Irr(C) Since α(1)α is the regular character of C (see [14]) it follows α∈Irr(C) A A |A| that ( α∈Irr(C) α(1)α) ↑C ↓B is the regular character of B multiplied by |C| . ↑A↓A |A| ↑A↓A Thus mB (ε C B,εB )= |B| . Frobenius reciprocity implies that mC (ε B C , |A| ↑A↓A |A| εC )= |B| . A dimension argument now shows that ε B C = |B| εC and ﬁrst ⊂ item of Remark 1.5 implies that C Aε↑A . B The above Proposition and Theorem 1.3 suggest the following conjecture:

Conjecture 1. For any Hopf subalgebra B of a semisimple Hopf algebra A one has:

(13) core(B)=Aε↑A . B 26 S. BURCIU AND L. KADISON

The next Proposition gives a description of core(B) in terms of kernels ⊆ and shows the inclusion core(B) Aε↑A . In order to prove it we need the B following lemmas.

Lemma 1.7. Let K and L be two Hopf subalgebras of a semisimple Hopf algebra A.IfΛK ΛL =ΛL then K ⊂ L.

Proof. By Corollary 2.5 of [2] there is a coset decomposition for A

(14) A = ⊕C/∼CL. where ∼ is an equivalence relation on the set of simple subcoalgebras of A given by C ∼ C if and only if CL = C L.In[2] this equivalence relation is A denoted by rk, L . The equality ΛK ΛL =ΛL shows that any subcoalgebra of K is equivalent to k1 and therefore it is contained in L.

Lemma 1.8. Suppose that B is a Hopf subalgebra of a semisimple ﬁnite ⊂ dimensional Hopf algebra. Then Aε↑A B. Equality holds if and only if B B is normal in A.

Proof. + Let K = Aε↑A . By the deﬁnition of K it follows that AK anni- B ∼ + hilates A ⊗B k. On the other hand A ⊗B k = A/AB as A-modules and + + + therefore AK ⊂ AB .Thus1− ΛK ∈ AB which implies ΛK ΛB =ΛB . The above Lemma implies that K ⊂ B. The second statement of the lemma is Corollary 2.5 from [3].

Proposition 1.9. Suppose that B is a Hopf subalgebra of a semisimple ﬁnite dimensional Hopf algebra A. Deﬁne inductively

0 r+1 B = B, B = Aε↑A . Br Then B1 ⊇ B2 ⊇···⊇Bs ⊇ Bs+1 ⊇··· .

If Bs = Bs+1 then Bs =core(B).

Proof. The above proposition implies that Bi ⊇ Bi+1 for any i. Since B is ﬁnite dimensional there is s such that Bs = Bs+1 = Bs+2 = ···.Thus s B = Aε↑A and the above lemma implies that B is normal in A.Wehaveto Bs show that core(B)=Bs. Suppose that K is normal in A and that K ⊆ B. It is enough to show K ⊆ Bs. Clearly K ⊆ B0.IfK ⊆ Bi then there is a + → + ⊂ canonical surjection of A-modules A/AK A/ABi .ThusAε↑A Aε↑A K Bi by the last item of Remark 1.5. On the other hand Aε↑A = K since K is K normal. Therefore K ⊆ Bi+1. SUBGROUPS OF DEPTH THREE 27

1.6.2. The correspondent of conjugate Hopf subalgebras. Let A be a semisimple Hopf algebra over an algebraically closed field k and let A ∗ be the set of simple subcoalgebras of A. Since A is cosemisimple note that A ∗ can be identified with Irr(A∗)[12]. Let B be a Hopf subalgebra of A and C a simple subcoalgebra of A. Define { ∈ ∗ | ∈ ∈ } XC B = D A dcΛB = ε(d)cΛB for all c C, d D Proposition . ∗ 1.10 The set XC B is closed under multiplication and “ ” and it generates a Hopf subalgebra C B of A.

Proof. By Remark 1.4 it is enough to show that the above set is closed under multiplication. Suppose that D and D are subcoalgebras in XC B .

If E is a simple subcoalgebra of DD then any e ∈ E can be written as s ∈ ∈ i=1 didi with di D and di D .ThenecΛB = ε(e)cΛB which show that ∈ E XC B . Notation: C B will also be denoted with cB if c is the irreducible character of A∗ corresponding to C.

Example 1.11. Let A = kG and B = kN where N is a subgroup of ∈ 1 G. The simple subcoalgebras of A are kg with g G and ΛB = |N| n∈N n. g −1 ∈ { ∈ | } { ∈ Then B = gBg for all g G. Indeed XgB = h G hgΛB = gΛB = h G | hgN = gN} = gNg−1

Proposition 1.12. Let B be a Hopf subalgebra of A and g ∈ G(A) be a grouplike element of A.ThengB = gBg−1.

Proof. First note that 1B = B. Clearly B ⊂ 1B. On the other hand 1 the deﬁnition of B implies that Λ 1B ΛB =ΛB . Then Lemma 1.7 implies 1B ⊂ B. g Let now C be a simple subcoalgebra of B.ThencgΛB = ε(c)gΛB for all −1 −1 1 c ∈ C.Thusg cgΛB = ε(c)ΛB which shows that g Cg ⊂ B = B. There- fore C ⊂ gBg−1 which shows that gB ⊂ gBg−1. A direct computations shows that gBg−1 ⊂g B.ThusgB = gBg−1.

Proposition 1.13. Let B be a Hopf subalgebra of A.Then C ∩ ∗ Aε↑A = C∈A B. B

Proof. Recall the coset decomposition

(15) A = ⊕C/∼CB. form Corollary 2.5 of [2]. If k is the trivial B-module then A k ↑B= ⊕C/∼CB ⊗B k. 28 S. BURCIU AND L. KADISON

C From the deﬁnition of B it follows that CB ⊗B k is trivial as left C C ∩ ∗ ⊂ B-module. Therefore C∈A B Aε↑A . B A ∼ Note that k ↑ = A ⊗B k = AΛ as left A-modules via aΛ → a ⊗B Λ . B B ∼ B B The decomposition 15 implies that CB⊗B k = CΛB under the above isomor- ↑A phism. Any simple subcoalgebra of Aε↑A acts trivially on k B and therefore B C on each CB ⊗B k. This implies that any such coalgebra is contained in B. ⊂C ∈ ∗ Thus Aε↑A B for any simple coalgebra C A . B

Corollary 1.14. Let B be a Hopf subalgebra of A.ThenB isanormal Hopf subalgebra if and only if ∩C B = C∈A ∗ B.

Proof. ∩C Since Aε↑A = ∗ B this is Corollary 2.5 of [3]. B C∈A

Remark 1.15. ⊂ ∩C 1. Theorem 1.9 implies that core(B) Aε↑A = ∗ B. This can also B C∈A be seen directly as follows. Fix C ∈ A ∗. For any x ∈ core(B)andc ∈

C one has that xcΛB = c1(S(c2)xc3)ΛB = c1ε(x)ε(c2)ΛB = ε(x)cΛB since core(B) is normal in A. Thus core(B) ⊂C B.

2. If Aχ is normal Hopf algebra for any χ ∈ Irr(A) then Proposition 1.9 implies the above conjecture on the core of a Hopf subalgebra.

2. Depth three Frobenius extension A Frobenius extension A | B is defined to be depth three if the following tower of subalgebras in the endomorphism ring E =EndAB is right or left depth-3: via the algebra monomorphism, left multiplication λ : A → E given by λ(a)(x)=ax (x, a ∈ A) we obtain the (ascending) tower, λ(B) ⊆ λ(A) ⊆ E.By[11, Theorem 3.1] the given tower is left d-3 if and only if the tower is right d-3. The definitions and first properties of depth two and three extensions are introduced in detail in [11]. There it is determined that a tower of three group algebras corresponding to the subgroup chain G ≥ H ≥ K is depth three if the normal closure KG (of K in G) is contained in H.In [10] it is shown that, with k = C and G a finite group, the group algebra A of G is depth two over subgroup algebra B of H if and only if H is a normal subgroup of G. This normality result for depth two subalgebras is extended to semisimple Hopf algebras over an algebraically closed field of characteristic zero in [4]. The following is a characterization of depth three for a separable, Frobe- nius extension in terms of the more familiar depth two property. The following is true more generally for QF-extensions [5, Theorem 3.8]. SUBGROUPS OF DEPTH THREE 29

Theorem 2.1. Suppose A | B is a separable extension and Frobenius extension. Let E denote End AB and λ : A → E be understood as the extension E | A.TheA | B is depth three if and only if the composite extension E | B is depth two.

Sketch of Proof. (⇒) This direction does not apply separability. By ∼ the Frobenius extension property, we noted above that E = A⊗B A as (A, A)- ∼ bimodules. Then E ⊗A E ⊗A E = E ⊗B E as natural (E,E)-bimodules. By deﬁnition of right D3 extension, E ⊗A E is isomorphic to direct summand n ∼ of E as natural (E,B)-bimodules for some n ∈ N , whence E ⊗A E ⊗A E = n E ⊗B E is (E,B)-isomorphic to a direct summand of E ⊗A E , which in turn 2 is isomorphic to a direct summand of En by the right D3 property. Hence ∼ n2 E | B is right D2, since E ⊗B E ⊕∗= E as natural (E,B)-bimodules. n (⇐) There is a split (E,B)-epimorphism from E → E⊗B E for some n ∈ N. In addition, there is a split (E,E)-epimorphism from E ⊗B E → E ⊗A E by the separability property of the extension A | B. Composing the two split n epis we obtain a split epi E → E ⊗A E showing A | B is right D3.

Proposition 2.2. Let M be the inclusion matrix of a subalgebra pair of semisimple complex algebras B ⊆ A,andS = MMt. The symmetric matrix S satisﬁes (16) S3 ≤ nS for some positive integer n ifandonlyifB is a depth three subalgebra of A.

Proof. Let Mm(C )=EndC A = E where m =dimA, which contains both A and B via left regular representation. It is shown in [7, 2.3.5] that the centralizers EA ⊆EB have transpose inclusion matrix; i.e. inclusion matrix t of A → End AB via a → λa is M . It is not hard to show from transitivity of induction that matrix multiplication yields new inclusion matrix of two successive subalgebra pairs. Hence, inclusion matrix of B → E via b → λb (b ∈ B) is given by MMt. The algebra A is separable, whence separable extension over B. The extension A ⊇ B is a split Frobenius extension by application of [7, Goodman-De la Harpe-Jones, ch. 2], very faithful conditional expectations. Then AB is a progenerator since B is semisimple and B → A is split B-module monic, so E and B are Morita equivalent semisimple algebras. By the theorem above, B ⊆ A is depth three iﬀ B → E is depth two, and we may apply Proposition 1.2 to the composite inclusion matrix S = MMt. In general for any subgroup H in ﬁnite group G with inclusion matrix M, if the irreducible characters of H are given by {ψ1,...,ψr} = Irr(H), note that the matrix S = MMt is given by ⎛ ⎞ G G G G ψ1 |ψ1 ... ψ1 |ψr (17) S = ⎝ ...... ⎠ . G G G G ψr |ψ1 ... ψr |ψr 30 S. BURCIU AND L. KADISON

For example, we revisit the inclusion S2

Corollary 2.3. The subgroup H is depth three in G if symmetric matrix S is strictly positive.

Another example: the standard inclusion of full permutation group algebras B = C [S3] → C [S4]=A has inclusion matrix (computed from character tables in e.g. [6]) and symmetric matrix: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10010 201 15 7 21 M = ⎝ 01001⎠ S = ⎝ 021⎠ S3 = ⎝ 71521⎠ 00111 113 21 21 43

It is clear that there is no positive integer n for which S3 ≤ nS, since 3 S has zero entries but S is strictly positive. We conclude that S3 is not a depth three subgroup of S4. (Using the next theorem one computes that S3 is a depth ﬁve subgroup of S4.)

2.1. Higher depth. Recall from [11] that depth n>2 is deﬁned as follows. Begin with a Frobenius extension (or QF extension [5]) B = A−1 ⊆ A = A0.LetA1 =EndAB and inductively An =End(An−1)A .Bythe ∼n−2 Frobenius hypothesis and its endomorphism ring theorem, An = A⊗B ···⊗B A (n +1 times A). Embedding An → An+1 via left regular representation λ, we obtain a Jones tower of algebras,

B → A → A1 →··· → An → An+1 →···

The subalgebra B in A is depth n if An−2 ⊇ An−3 ⊇ B is a depth-3 tower defined above; infinite depth if there is no such positive integer n. Of course, this agrees with the definition of depth three subalgebra above. If B and A are semisimple complex algebras, A ⊇ B becomes a split, separable Frobe- nius extension via the construction of a very faithful conditional expection [7]. This type of extension has an endomorphism ring theorem [9], and enjoys transitivity, so that all extensions in this Jones tower are split, separable Frobenius extensions, and all algebras are semisimple by Morita’s theorem (or Serre’s theorem on global dimension). Indeed, all the odd An’s are Morita equivalent to B, while all the even An’s are Morita equivalent to A. The proof of the lemma below is similar to that of Prop. 1.2 and therefore omitted. A ∼ A B (One notes that IndC = IndBIndC is given by the rows of matrix NM.) SUBGROUPS OF DEPTH THREE 31

Lemma 2.4. Suppose C ⊆ B ⊆ A is a tower of semisimple algebras with inclusion matrices N and M respectively. Then the tower is depth-3 if and only if there is a positive integer n such that (19) NMMtM ≤ nNM

Notice that Prop. 1.2 follows from letting B = C and N equal the identity matrix of rank dim Z(B). Conversely, if A ⊇ B is depth two, and C any subalgebra of B, then the lemma follows in this special case from Prop. 1.2 by multiplying the inequality there from the left by the inclusion matrix N of C ⊆ B. Let n, m and q denote positive integers below.

Theorem 2.5. Suppose B ⊆ A is a subalgebra pair of semisimple algebras. Let M be the inclusion matrix and S = MMt.Ifn =2m+1 then A ⊇ B is depth n ifandonlyifSm+1 ≤ qSm for some q.Ifn =2m,thenA ⊇ B is depth n ifandonlyifSmM ≤ qSm−1M for some q.

Proof. The proof follows from noting that if M is the inclusion matrix t of B ⊆ A, then M is the inclusion matrix of A → A1,andS is the inclusion matrix of their composite B → A1. The proof now follows from applying the last lemma to the depth-3 tower B → An−3 → An−2 in the even and odd case. It is worth emphasizing that a depth n algebra extension is also depth n + 1 (so one might denote this as depth ≥ n); in the special case of the theorem, this is seen by multiplying the given inequality from the right by the inclusion matrix M or M t. Of course one should strive to use the least depth to one’s knowledge. Let m be a positive integer and G a ﬁnite group in the next result on subgroups of ﬁnite depth.

Corollary 2.6. Suppose H

Proof. Applying the theorem we see Sm+1 ≤ qSm for some positive integer q since Sm is a strictly positive matrix. 2 For example, while S3 0 such that the p-power of each diagonal block is a positive 32 S. BURCIU AND L. KADISON matrix. Applying Theorem 2.5 this implies that the extension B ⊆ A is of depth at least 2p + 1; in other words, all semisimple subalgebra pairs are of ﬁnite depth.

2.2. Simpliﬁed condition for depth three. Again let A ⊇ B be an algebra extension. In case AB is a generator, such as when the extension is free or right split, there is a particularly simpliﬁed condition for when a Frobenius extension is depth three.

Theorem 2.7. Suppose A ⊇ B is a Frobenius extension where the natural module AB is a generator. Then A ⊇ B is depth three if and only if there is a B-B-bimodule P and positive integer n such that ∼ n (20) BA ⊗B AB ⊕ P = BAB

Proof. (⇒)LetE =EndAB. By the Frobenius extension hypothesis ∼ on A ⊇ B,asE-A-bimodules E = A⊗B A via the mapping in subsection 1.1. Recall that A ⊇ B is depth three if B ⊆ A → E is depth three tower, i.e. ∼ n EE ⊗A EB ⊕ Q = EEB for some E-B-bimodule Q and positive integer n. Then by substitution ∼ n (21) EA ⊗B A ⊗B AB ⊕ Q = EA ⊗B AB.

But AB is a progenerator by hypothesis, whence B and E are Morita equivalent algebras. The context bimodule are BHom (AB,BB)E (the right B- ∗ dual of A denoted by (AB) )andEAB with B-B-bimodule isomorphism =∼ Hom (AB,BB) ⊗E A −→ B given by evaluation. Now tensor all components ∗ of eq. (21) by B(AB) ⊗E − and cancel B⊗B to obtain eq. (20), where of ∗ course P =(AB) ⊗E Q. (⇐) Tensor all components of eq. (20) from the left by the natural bimodule EAB given by f · a · b = f(a)b, obtain eq. (21), and reverse the argument above it. This direction of proof does not make use of generator hypothesis. ∼ Remark 2.8. Since BAB ⊕ Ω = BA ⊗B AB is always the case for some B-B-bimodule Ω, the B-B-bimodules A ⊗B A and A are similar or H- equivalent under the conditions of the theorem: thus their endomorphism algebras are Morita equivalent. By Theorem 2.1, left multiplication B → E is depth two. There is a general Galois theory of depth two extensions which in this case specializes to total algebra End BA ⊗B AB and base algebra B ∼ E = End BAB as parts of a bialgebroid. It is interesting to note that base and total algebras in this case are Morita equivalent.

G Let Res = ResH denote restriction of G-modules to H-modules in the G corollary below, and Ind = IndH denote induction of H-modules to G-modules. SUBGROUPS OF DEPTH THREE 33

Corollary 2.9. A subgroup H of a ﬁnite group G is depth three if (22) ResIndResIndψ | χ≤n ResIndψ | χ for all irreducible characters ψ, χ of H. Proof. Note that the corresponding complex group algebras A ⊇ B satisfy the conditions of the theorem. One arrives at the condition on inner products of characters by tensoring a simple B-module V by the components in eq. (20). Of course, whatever simple B-module components of BA⊗B A⊗ n BV has, also BA ⊗B V has.

For example, from the character tables of the permutation groups S4 and S5 [6] we compute the induction-restriction table by restricting irreducible characters on S5, given below in matrix form (with ﬁrst column and row corresponding to trivial characters): ⎛ ⎞ 1010000 ⎜ ⎟ ⎜ 0101000⎟ ⎜ ⎟ M = ⎜ 0000011⎟ ⎝ 0010110⎠ 0001101 G Let ηi ∈ Irr(H)andχi ∈ Irr(G)(i =1,...,5). Then from row 1, η1 = G G G χ1 + χ3, η1 ↓H =2η1 + η4 and ﬁnally η1 ↓H ↑ ↓H =5η1 + η3 +5η4 + η5. Note G G G η1 ↓H ↑ ↓H |η3 =1≤ n η1 ↓H |η3 = 0 for all positive integers n, whence S4 is not a D3 subgroup in S5. Computing the 5×5 matrix S = MMt, we may compute that the matrix 3 4 S and S are strictly positive, so that S4 is a depth seven subgroup in S5 by Theorem 2.5 and its corollary. (Observing the pattern, we might conjecture at this point that the canonical subgroup Sn

2.3. Depth three quasi-bases. The condition (20) for a depth three extension has an interpretation in terms of split epis, including the canonical split epis of a product. This should give us depth three condition in terms of quasi-bases somewhat similar to dual bases for projective modules. Meanwhile the Frobenius hypothesis on extension A ⊇ B is needed to reduce the quasi-bases to simplest terms. Suppose F is a Frobenius homomorphism A → B with dual bases {xi} and {yi} in A.

Theorem 2.10. Suppose A ⊇ B is a Frobenius extension where AB is a generator. Then A ⊇ B is a depth three extension if and only if there are B elements ui,ti ∈ (A ⊗B A ⊗B A) such that for all x, y ∈ A, n 1 2 3 1 2 3 (23) x ⊗B y = ti ⊗B ti F (ti ui F (ui F (ui x)y)) i=1 where u = u1 ⊗ u2 ⊗ u3 is Sweedler notation that suppresses a possible summation over simple tensors. 34 S. BURCIU AND L. KADISON

Proof. (⇒) First note from eq. (20) that there are mappings fi ∈ Hom (BAB, BA ⊗B AB)andgi ∈ Hom (BA ⊗B AB, BAB) such that n ◦ fi gi =idA⊗BA. i=1 ∼ Next recall that for any B-module M,CoIndM = IndM for a Frobe- nius extension A over B [9]; I.e., there is a natural A-module isomor- ∼ phism Hom (AB,MB) = M ⊗B A via f → f(xi)⊗yi with inverse m⊗a → ∼ mF (a−). Applied to M = A⊗B A, this restricts to Hom (BAB, BA⊗B AB) = B (A ⊗B A ⊗B A) via f → i f(xi) ⊗ yi with inverse (24) t −→ t1 ⊗ t2F (t3−). Next apply the hom-tensor relation and the Frobenius isomorphism between endomorphism ring and tensor-square of extension: ∼ B Hom (BA ⊗B AB, BAB) = Hom (AB,EB) ∼ ∼ B = Hom (BAB, BA ⊗B AB) = (A ⊗B A ⊗B A) .

Following the isomorphisms, the forward composite mapping is given by g → i,j g(xi ⊗ xj) ⊗ yj ⊗ yi with inverse given by

(25) u −→ (x ⊗ y → u1F (u2F (u3x)y)) B for all u ∈ (A ⊗B A ⊗B A) , x, y ∈ A. Now suppose the mappings we begin with fi → ti and gi → ui in (A ⊗B B A ⊗B A) via isomorphisms displayed above. Then eq. (23) results. n (⇐) Deﬁne a split B-B-bimodule epimorphism A → A⊗B A by (a1,..., n 1 2 3 n an) → i=1 ti ⊗ ti F (ti ai) with section A ⊗B A → A given by x ⊗ y → 1 2 3 (ui F (ui F (ui x)y))i=1,...,n. B For example, a left depth two quasi-bases ti ∈ (A ⊗B A) and βi ∈ ⊇ ⊗ n ∈ End BAB for A B satisfy x y = i=1 tiβi(x)y for all x, yA.IfA is ∼ B Frobenius extension of B, then End BAB = (A⊗B A) via α → i α(xi)⊗yi 1 2 B 1 2 with inverse t → t F (t −). Let ui ∈ (A ⊗B A) satisfy ui F (ui −)=βi.Then ⎧ ⎫ ⎧ ⎫ ⎨ ⎬ ⎨ ⎬ 1 ⊗ 2 ⊗ ⊗ 1 ⊗ 2 (26) ⎩ ti B ti xj B yj⎭ ⎩ xj B yjui B ui ⎭ j j i=1,...,n i=1,...,n are D3 quasi-bases, because 1 2 1 2 1 2 1 2 ti ⊗ ti xjF (yjxkF (ykui F (ui x)y)) = ti ⊗ ti xkF (ykui F (ui x)y) i,j,k i,k 1 2 1 2 = ti ⊗ ti ui F (ui x)y = x ⊗ y. i SUBGROUPS OF DEPTH THREE 35

3. Hall subgroup in Frobenius group is depth three A Frobenius group is a ﬁnite group G with nontrivial normal subgroup M (called the Frobenius kernel) which contains the centralizer of each of its ∗ nonzero elements: CG({x}) ⊆ M for each x ∈ M [8, 17]. This is equivalent to G having a Hall subgroup, or Frobenius complement, H such that G = MH, M ∩ H = {e}, H ∩ Hx = {e} where Hx = x−1Hx for any x ∈ G − H; −1 ∗ in addition, M = G −∪x∈Gx H x. The Hall subgroup H is not normal in G (and therefore not depth two in the terms of this paper). We will see below that H

Theorem 3.1. Let G be a Frobenius group with Hall subgroup H.Then H is depth three subgroup of G.

Proof. From the deﬁning condition (22), we easily ﬁnd a positive integer n if ResIndψ|χ > 0 for all irreducible characters ψ, χ of H. We compute using Mackey subgroup theorem [8, p. 74] and Frobenius reciprocity, where T denotes a set of n double coset representative {e = g1,g2,...,gn}: G t H t ψ ↓H | χ = (ψ ↓Ht∩H )↑ ,χ = ψ ↓Ht∩H | χ↓Ht∩H ≥n − 1 t∈T t∈T

t since H ∩ H = {e} for each t = g1. Indeed it is easy to check that

G ψ ↓H |χ =(n − 1)(deg ψ)(deg χ) if ψ = χ and equals 1 + (n − 1)(deg ψ)2 if χ = ψ.

For example the subgroup S2 in S3 has two double coset reprenta- G tives, both irreducible characters are linear, and the values ψ ↓H |χ >= ψG|χG are 1 on the oﬀ-diagonal and 2 on the diagonal, the coeﬃcients of the matrix S in eq. (18). The proof of the theorem also follows from eq. (17), Corollary 2.3 and Mackey’s theorem.

Acknowledgments The second author is grateful to David Harbater and Gestur Olafsson for discussions related to this paper. The first author was supported by the strategic grant POSDRU/89/1.5/S/58852, Project ”Postdoctoral pro- gramme for training scientific researchers” cofinanced by the European Social Found within the Sectorial Operational Program Human Resources Devel- opment 2007–2013. 36 S. BURCIU AND L. KADISON

References [1] J. L. Alperin with R. B. Bell, Groups and Representations,GTM162, Springer, New York, 1995. [2] S. Burciu, Coset decomposition for semisimple Hopf Algebras, Comm. Alg., 37, 10, (2009), 3573–3585. [3] S. Burciu, Normal Hopf subalgebras of a semisimple Hopf algebra, Proc. A. M. S., 12 (2009), 3969–3979. [4] S. Burciu and L. Kadison, Depth two Hopf subalgebras of a semisimple Hopf algebra J. Algebra, 322, (2009), 162–176. [5] F. Castano Iglesias and L. Kadison, Similarity, codepth two bicomodules and QF bimodules, preprint arXiv:0712.4362. [6] W. Fulton and J. Harris, Representation Theory, GTM 129, Springer, 1991. [7] F. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, M.S.R.I. Publ. 14, Springer, Heidelberg, 1989. [8]I.M.Isaacs,Character Theory of Finite Groups, Dover, New York, 1976. [9] L. Kadison, New Examples of Frobenius Extensions,Univ.Lect.Ser.14, A.M.S., 1999. [10] L. Kadison and B. Külshammer,Depth two, normality and a trace ideal condition for Frobenius extensions, Comm. Alg. 34 (2006), 3103–3122. [11] L. Kadison, Finite depth and Jacobson-Bourbaki correspondence, J. Pure & Applied Alg. 212 (2008), 1822–1839. [12] R. G. Larson, Characters of Hopf algebras, J. Algebra 17 (1971), 352–368. [13] R. G. Larson, and D. E. Radford, Finite dimensional cosemisimple Hopf Algebras in characteristic zero are semisimple, J. Algebra 117 (1988), 267–289. [14] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math. Vol. 82, AMS, Providence, 1993. [15] W. D. Nichols and M. B. Richmond, The Grothendieck group of a Hopf algebra, I, Comm. Alg. 26 (1998), 1081–1095. [16] M. Rieffel, Category theory and quantum field theory, in: Noncommutative Rings(Berkeley, CA, 1989), 115–129, Math. Sci. Res. Inst. Publ. 24, Springer, New York, 1992. [17] W. R. Scott, Group Theory, Dover, New York, 1987. [18] K. Szlachanyi, On field algebra construction, arXiv preprint 0806.0041.

Inst. of Math. “Simion Stoilow” of the Romanian Academy P.O. Box 1-764, RO-014700, Bucharest, Romania

University of Bucharest, Faculty of Mathematics and Computer Science, 14 Academiei St., RO-010014, Bucharest 1, Romania E-mail address: [email protected]

Department of Mathematics, University of Pennsylvania, David Ritten- house Lab, 209 S. 33rd St. Philadelphia, PA 19104 E-mail address: [email protected] Surveys in Diﬀerential Geometry XV

Yukawa Couplings in F-theory and Non-Commutative Geometry

Sergio Cecotti, Miranda C. N. Cheng, Jonathan J. Heckman, andCumrunVafa

Abstract. We consider Yukawa couplings generated by a configuration of intersecting seven-branes in F-theory. In configurations with a single interaction point and no fluxes turned on, the Yukawa matrices have rank one. This is no longer true when the three-form H-flux is turned on, which is generically the case for F-theory compactifications on Calabi-Yau fourfolds. In the presence of H-fluxes, the Yukawa coupling is computed using a non-commutative deformation of holomorphic Chern-Simons theory (and its reduction to seven-branes) and subsequently the rank of the Yukawa matrix changes. Such fluxes give rise to a hierarchical structure in the Yukawa matrix in F-theory GUTs of the type which has recently been proposed as a resolution of the flavor hierarchy problem.

Contents 1. Introduction 38 2. F-theory Yukawas and Seven-Branes 42 3. Yukawas in Commutative Geometry 44 3.1. The Partially Twisted Seven-Brane Theory 45 3.2. Matter Curves and Localized Zero Modes 47 3.2.1. Enhancement Loci and Matter Curves 47 3.2.2. Zero Modes Localized on Curves 49 3.2.3. The Local Cohomology Theory of Chiral Matter 52 3.3. The Yukawa Coupling 55 3.3.1. Two Examples 56 3.3.2. Yukawas from Residues 60 3.3.3. A Rank Theorem 62 4. H -flux and Non-Commutative F-terms 62 4.1. HR-flux and the Myers Effect 64 c 2011 International Press

37 38 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

4.2. H -flux and the B-model 65 4.3. Poisson Bi-Vectors and Generalized KählerGeometry 67 5. Yukawas in Non-Commutative Geometry 68 5.1. Deforming the Field Theory 69 5.2. Matter from Non-Commutative Geometry 71 5.2.1. Background Field Configurations 71 5.2.2. Fuzzy Matter Curves and Localized Zero Modes 72 5.3. The Yukawa Coupling 75 5.3.1. A Quantum Residue Formula and a Selection Rule 76 5.3.2. An Explicit Example 79 6. Applications 81 6.1. Review of FLX and DER Expansions 81 6.2. Comparison with the Non-Commutative Gauge Theory 82 6.3. Scaling Estimates 83 6.4. Matching to the Non-Commutative Gauge Theory 85 6.5. Masses and Mixing Angles with λ and λΦ 87 7. Conclusions 88 Acknowledgements 89 Appendix A. Calculation of the Commutative Residue 89 Appendix B. The and Product 91 Appendix C. Proof of Commutativity Upon Integration 93 References 94

1. Introduction It has been known for a long time that the existence of Yukawa couplings is crucial in understanding the masses and mixing angles for quarks and leptons. Indeed, Yukawa matrices in flavor space determine the structure of the corresponding mass matrices. These parameters exhibit striking hierarchies which are not explained by the Standard Model. Turning this around, the existence of hierarchical structure in the Yukawa couplings can serve as a hint for physics beyond the Standard Model. It is thus natural for string theory to use this hint as a way to connect to particle phenomenology. This idea is very natural in the context of N = 1 supersymmetric theories, since Yukawa couplings are holomorphic and as such are often exactly computable.1 There are various contexts in which Yukawa couplings appear in string theory, and in all cases these computations are quasi-topological in nature. Assuming that supersymmetry is a part of Nature at some scale, this suggests a link between hierarchical structures in the internal geometry, and those of the Standard Model. In particular, it is well-known that up-type quarks have a very hierarchical mass structure. The top quark is very heavy, with a mass of ∼170 GeV, 1 Of course the physical Yukawa couplings also depend on the normalization of fields which are not holomorphic quantities, and are therefore hard to compute exactly. YUKAWA COUPLINGS IN F-THEORY 39 which is close to that of the weak scale, while the up and charm are far lighter. It is therefore natural to search for a topological scenario in string theory where we start with one massive quark and two massless ones. In other words, it is natural to approximate the Yukawa matrix for the (u,c,t) quarks as a rank one matrix which is then deformed by higher order hierarchical corrections. Recent work has shown that a class of Grand Unified Theories in F- theory known as F-theory GUTs provide a potentially promising starting point for making detailed contact between string theory and the Standard Model. See for example [1–35] for recent work on F-theory GUTs. In these theories, matter and Yukawa interactions respectively localize on Riemann surfaces and at points of an intersecting seven-brane configuration. Yukawas are controlled by the overlap of matter field wave-functions and are concentrated at the point of maximal overlap. As observed in [2](seealso[19]), minimal geometries with a single Yukawa point generate rank one Yukawa matrices, and adding more points simply produces higher rank structures. Building on this observation, it was proposed in [14] that starting from a minimal geometry with a single Yukawa point, subleading corrections to this geometric structure could generate hierarchical flavor structure for the remaining generations. Subleading corrections can in principle be generated by the spread of the matter field wave-functions in the internal geometry. Hierarchical structures are then expected from small flux induced distor- tions to the matter field wave-functions. Furthermore, the U(1) symmetries induced from local rotations of the complex internal space at the intersection point constrain the structure of the hierarchical corrections to the Yukawas, implementing a mechanism rather similar to the Froggatt-Nielsen mechanism [35]. A crucial element of this proposal is that the leading order deformation to the Yukawas originates from a three tensor index structure, such as the gradient of the gauge field strength ∇¯iFjk¯ or ∇iF¯jk¯. It was argued in [14] that the constant two-form flux F on the seven-brane alone will not induce any correction to the Yukawa couplings. Using crude estimates for the expected form of flux-induced deformations, it was found that hierarchies in the quark masses and mixing angles could naturally be generated. One of the primary aims of this paper is to evaluate this proposal. Recall that there are essentially two types of fluxes in F-theory backgrounds. These correspond to gauge field fluxes F threading the worldvolume of a seven-brane, and to bulk three-form H-fluxes which can be decom- posed as H = HR + τHNS in perturbative IIB language. In a dual M-theory description, these two contributions both descend from a four-form G-flux.2 When only F is activated, we find that no change can occur in the rank of

2 It is also sometimes possible to consider F-theory vacua with background five-form fluxes activated. Such fluxes lie somewhat outside the realm of the local model defined by a configuration of seven-branes, and so we will not consider this case in this paper. 40 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA the Yukawa matrix away from the minimal form expected from geometry. We shall sometimes refer to this result as the “rank theorem” for Yukawa couplings in commutative geometry. This result also implies that more general F-theory backgrounds must be considered to deform the structure of the Yukawa couplings. When H-flux also participates, however, we find that the rank generically does increase, and that the form of these subleading corrections generate hierarchical structures which are almost identical to those proposed in [14]. Indeed, in simplified situations, the presence of the three- form flux H can be viewed as a mild shift in the gradient of the background gauge field strength ∇F = −∇B. At a pragmatic level, it is possible to work in “physical gauge”, by specifying a choice of background Kählermetric, computing the precise profile of the matter field wave-functions in this background and then evaluating the explicit overlap integrals in a local neighborhood containing the Yukawa enhancement point. This was the point of view advocated for example in [14]. On the other hand, in a given overlap integral, there will typically be non-trivial cancellations from distinct terms. The primary advantage of working in terms of a manifestly holomorphic formulation is that such cancellations are immediate from the start of the computation. Indeed, when only the gauge flux F is present, we find highly non-trivial cancellations in the form of the overlap integral which are quite subtle in physical gauge. In the present paper we have performed the relevant overlap integrals in both holomorphic and physical wave-function bases as a check on the expressions found. The holomorphic formulation of Yukawa couplings is based on the observation made in [2] that the Yukawa couplings in F-theory are computed using the reduction of holomorphic Chern-Simons theory to the seven-brane worldvolume. In other words, the matter fields correspond to first order deformation moduli of the holomorphic Chern-Simons theory, and the Yukawa couplings measure the obstruction to extending these moduli to higher order. We show how the computation of local Yukawa couplings can be simplified and recast in the form of a residue formula. This residue formula for the Yukawas turns out to be powerful in establishing the rank theorem when H =0. From the perspective of the quasi-topological theory which computes the coupling, the effects of H = 0 can be captured in terms of a non-commutative deformation of holomorphic Chern-Simons theory and its reduction to a configuration of seven-branes.3 In the non-commutative geometry, the holomorphic coordinates Xi and Xj satisfy the relation:

[Xi,Xj]=θij(X)

3 For other applications of non-commutative Chern-Simons theory, see for example [36]. YUKAWA COUPLINGS IN F-THEORY 41 where θij is a holomorphic bi-vector. It follows from the Jacobi identity of this commutator that θij is a Poisson bi-vector:

i jk θ ∂θ + (antisymmetrization in i, j, k)=0.

As we will argue, the non-commutativity parameter θij is related to the H-flux by: ij ij ∇kθ = Hk to leading order in the weak field expansion. This kind of non-commutativity is encountered in the context of topological strings [37–39]. Here we show that the natural setting in the superstring context involves the H-flux. It is well-known that this kind of non- commutativity does not typically arise in perturbative string compactifications because the H-flux satisfies a Gauss’ law type constraint.4 On the other hand, it is also well known that F-theory vacua typically do have H-flux in compact geometries [40]. Indeed, F-theory vacua generically evade such no- go theorems because the SL(2, Z) duality group acts non-trivially on H.Ata practical level, what this means is that far from being an exotic element of a compactification, background H-fluxes will generically deform the structure of F-theory Yukawas in potentially phenomenologically interesting ways. The resulting hierarchical structures we find are quite similar to those initially proposed in [14]. Indeed, another benefit of a manifestly holomorphic formulation is that it is possible to give more reliable estimates on the size of the “order one” coefficients expected in each entry of the Yukawa matrix. We find that in all examples we consider, the magnitudes of the coefficients multiplying each hierarchical suppression factor are indeed order one numbers, in accord with crude numerological expectations. The organization of the rest of this paper is as follows. In section 2 we review aspects of F-theory in connection with Yukawa couplings. In section 3 we show how one can compute the Yukawa couplings for seven-branes locally in terms of a residue formula. We also derive the rank theorem which is valid in the absence of H-fluxes in this section. In section 4 we show why H-fluxes are generic in F-theory compactifications on Calabi-Yau fourfolds and why the F-terms are computed by non-commutative holomorphic Chern-Simons theory. In section 5 we consider non-commutative holomorphic Chern-Simons theory and its reduction to seven-branes. In particular, we compute the Yukawa couplings in this setup, finding hierarchical corrections to the leading order result. In section 6 we apply our results to the study of flavor hierarchy and recover structures very similar to those found in [14]. Section 7 contains our conclusions and future directions of investigation. Appendix A is devoted to further elaboration on the residue formulation of Yukawas in commutative geometries, and Appendices B and 4 Compactifications on hyperkähler manifolds and non-compact Calabi-Yau threefolds provide exceptions to this general point, but also serve to illustrate just how strong this constraint is. 42 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

C contain some technical aspects of the non-commutative deformation and its application to deformed F-terms.

2. F-theory Yukawas and Seven-Branes The primary aim of this paper is to study the superpotential of a configuration of seven-branes in the presence of non-trivial background fluxes in F-theory. In this section we recall the main ingredients entering into the computation of Yukawa couplings in F-theory vacua. F-theory can be defined as a strongly coupled formulation of type IIB string theory in which the axion-dilaton is allowed to vary over points of the ten-dimensional spacetime. This information can be conveniently packaged in terms of a twelve-dimensional geometry. F-theory compactified on an elliptically fibered Calabi-Yau fourfold with a section preserves N =1 supersymmetry in the four uncompactified dimensions. The discriminant locus of the elliptic fibration defines the location of seven-branes wrapping complex surfaces in the threefold base X of the elliptic fibration. Locally, the seven-brane wrapping a complex surface S can be described as a singularity of ADE type fibered over S. Six-dimensional chiral matter fields correspond to the enhancements of the singularity type to higher rank along complex curves, and Yukawa interactions correspond to further enhancements of the singularity type at points in the geometry. In addition to the geometric elements of an F-theory compactification, there can also be additional contributions from background fluxes. Such fluxes can either descend from two-form fluxes, such as gauge field fluxes on the various seven-branes, or from three-form NSNS and RR fluxes (in type IIB language). Upon further compactification on a circle down to three dimensions, we have the dual M-theory description on the same Calabi-Yau fourfold, where the Kähler class of the elliptic fiber is the inverse of the radius. It is well-known that in this context, one generically has a (2,2) G4-flux turned on [41]. Lifting it back up to F-theory, this corresponds to turning on H-flux [40,42]:

(2.1) G4 = H ∧ dz + H¯ ∧ dz,¯ where z is the holomorphic coordinate along the elliptic fiber of the Calabi- Yau fourfold, and H = HR + τHNS is a (1,2)-form. As reviewed for example in [43], abelian gauge field fluxes threading the seven-brane correspond to an appropriate localization of the G4-fluxes on the components of the discriminant locus, while the remaining degrees of freedom of the G4-fluxes correspond to the bulk three-form fluxes H of interest to us in the present paper. The background fluxes of an F-theory compactification will in general distort the profile of matter field wave-functions of the compactification. At an abstract level, these chiral matter fields correspond to representatives of elements in an appropriately defined cohomology theory. Indeed, the zero YUKAWA COUPLINGS IN F-THEORY 43 mode wave-functions Ψ(i) localized on a matter curve Σ are defined in terms of the Dirac equation:

(2.2) D¯Ψ(i) =0 where D¯ is an appropriate Dolbeault operator defined in terms of the background fluxes of the compactification, and we have labelled the distinct zero modes by the index i =1,...,n for n zero modes localized on the curve Σ. Yukawa couplings between three matter fields in the four-dimensional effective theory are given by triple overlaps of wave-functions in the internal directions of the geometry: ijk (i) (j) (k) (2.3) λ = ΨΣ1 ΨΣ2 ΨΣ3 .

For phenomenological applications, the case of primary interest is where one of the matter field wave-functions corresponds to a Higgs field of the MSSM. For this reason, we shall be primarily interested in cases where the Yukawa couplings reduce to a g × g matrix for a g-generation model. Although this might suggest that it is necessary to specify the explicit form of the matter wave-functions, much of this data is unnecessary in a computation of F-terms due to holomorphy considerations. Specifically, note that for any appropriately defined theory of Yukawa couplings, the corresponding overlap integral should be independent of the particular representative chosen for the matter wave-function. Indeed, this is a necessary property of the superpotential in order for it to have a formulation which is independent of the D-term data, such as the Kähler form of the complex surface.5 A more pragmatic benefit of having a holomorphic formulation of Yukawa couplings is that it also provides a useful check that there are no non-trivial cancellations between different contributions to the wave-function overlap integrals. Indeed, although such cancellations can appear to be rather mys- terious and non-trivial in “physical gauge”, holomorphy considerations typically make such effects more transparent. In the context of perturbative type IIB string theory, the evaluation of Yukawas reduces to a disc diagram computation in the topological B-model. Phrased in this way, our primary task is to determine the deformation theory of the Yukawa couplings in the topological B-model. Once these deformations have been determined, we next establish a link to the physical theory. In other words, it is then enough to determine how various background field configurations in the physical theory such as fluxes translate into parameters in the topological string.

5 Here we are neglecting possible instanton type contributions to the superpotential which contain terms of the form exp(−Vol(S)). 44 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

Recall that in the topological B-model on a Calabi-Yau threefold X, the closed string moduli space is:

(2.4) ⊕Hp(ΛqT 1,0) p,q where T 1,0 denotes the holomorphic tangent bundle. This moduli space is known as the “extended moduli space” in the sense of Witten [44](see also [39]). The more familiar class of deformations satisfying p + q = 2 correspond to marginal deformations of the worldsheet theory. For example, p = q = 1 correspond to the usual complex deformations typically studied in the context of the B-model. Here, the case of p =2 and q = 0 corresponds to “gerby” deformations, and p =0 and q = 2 corresponds to deformations which induce the Kontsevich -product between open string states [39]. On the other hand, one of the most attractive features of F-theory compactifications is that some Yukawas which cannot be realized perturbatively are naturally generated from E-type singularity structures in the fibration of the elliptic Calabi-Yau fourfold. Even so, it is still possible to retain computational control over some aspects of the seven-brane theory by appealing to the formulation of the partially twisted eight-dimensional gauge theory [2]. Since Yukawa couplings correspond to points of the geometry where the singularity type enhances to Gp, in a sufficiently small neighborhood of this point, the physical system defined by this geometry is given by a gauge theory with gauge group Gp which is Higgsed down to GΣ along matter curves, and GS in the bulk of the complex surface wrapped by the seven-brane. This provides a topological formulation of Yukawa couplings in a patch containing the enhancement point, and this is the approach we shall follow in this paper. Furthermore, extending the relationship between deformations of the topological theory by type IIB field configurations to F-theory, we will be able to study the Yukawa couplings in the partially twisted eight- dimensional gauge theory deformed by the presence of F-theory fluxes.

3. Yukawas in Commutative Geometry Our main aim in this section will be to derive a quasi-topological formulation of Yukawa couplings among the localized zero modes of the theory. The formulation of Yukawa couplings we present will be based on the local profile of the background fields in a sufficiently small neighborhood of the Yukawa interaction point. For this reason, it is often sufficient to study the gauge theory as if it were defined in a patch of C2. Within this patch, we shall model the system as an eight-dimensional gauge theory with gauge group Gp which away from the Yukawa point has been Higgsed down to GΣ along matter curves, and GS in the bulk. A quasi-topological formulation of Yukawas helps to illustrate two important features. First, the reduction of the computation to a residue integral demonstrates that small changes in the shape of the patch cannot alter the YUKAWA COUPLINGS IN F-THEORY 45 value of the Yukawa coupling, and hence the Yukawa coupling is a well- defined observable of the quasi-topological theory. Second, in this manifestly holomorphic formulation, it is straightforward to demonstrate that background gauge field fluxes do not alter the structure of the Yukawa couplings. This result may appear surprising at first sight since background gauge fields do distort the profile of the matter field wave-functions. As a check on these results, we have also computed in some explicit examples presented in section 3.3.1 the same overlap integrals by using wave-functions which satisfy both the F-term and D-term equations of motion. In this “physical gauge”, the holomorphy of the superpotential is more obscure, but the actual profile of the matter field wave-functions is easier to track. The absence of corrections to the Yukawa couplings can be traced to subtle cancellations between different contributions to the wave-function overlap integrals. It follows from these computations that additional ingredients must be included in the seven-brane theory if we are to realize flavor hierarchies in F-theory GUTs. In subsequent sections we shall study the hierarchies generated by one such deformation. The rest of this section is organized as follows. After a quick review of the seven-brane gauge theory, we discuss the notion of matter curves and zero modes localized on them. We then show how, for a given background, the gauge-inequivalent localized zero modes are captured by a cohomology theory. From there we derive a residue formula for the Yukawa triple coupling of the localized zero modes and arrive at a theorem on the rank of the Yukawa matrix. As checks of the formula, we also present two explicit examples in which we calculate the zero mode wave-functions for a given choice of Kähler form and background gauge flux. We find that these computations agree with the results of the residue formula.

3.1. The Partially Twisted Seven-Brane Theory. In this subsection we will briefly review the partially twisted supersymmetric Yang-Mills theory describing seven-branes filling the Minkowski spacetime R3,1 and wrapping a Kähler surface S [2](seealso[1]). To motivate this theory, recall that the superpotential of the field theory describing a stack of D9-branes with gauge group G wrapping a Calabi-Yau threefold X is determined by the holomorphic Chern-Simons Lagrangian [45] 2 (3.1) Ω3,0 ∧ Tr A¯ ∧ ∂¯A¯ + A¯ ∧ A¯ ∧ A¯ X 3 with A¯ the connection for G. Here and in what follows we have suppressed the non-compact R3,1 directions which do not play any role in our discussions. Given a background gauge field configuration A¯, zero mode fluctuations δA¯ about this background correspond to massless excitations of the four-dimensional effective theory. 46 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

The superpotential of the seven-brane gauge theory can be defined by dimensional reduction of the holomorphic Chern-Simons theory. Schemat- ically, denoting by A¯⊥¯ the component of the gauge field transverse to the seven-brane world-volume and replacing Ωij⊥A¯⊥¯ by ϕij, the superpotential for a seven-brane with gauge group G on R3,1 × S is given by the supersym- metrization of: (3.2) W = Tr ϕ ∧ F (0,2) , S where F (0,2) denotes the (0, 2)-component of the gauge field strength:

F (0,2) = ∂¯A¯ + A¯ ∧ A.¯

When the context is clear, we shall drop the label indicating the Hodge type of the form. In addition, we shall not distinguish between chiral superfields and their bosonic components given that there is little room for con- fusion. Notice from the above prescription that we can naturally identify ϕ as a (2, 0)-form on the complex surface S. This is consistent with the result obtained from the unique twisting of N = 1 supersymmetric Yang-Mills theory on R3,1 ×S when S isaKähler surface [2]. Let us note here that although we have motivated this superpotential using a formulation based on dimensional reduction and compactification on a Calabi-Yau threefold, all that is really required to derive the superpotential of the seven-brane theory is the existence of N = 1 supersymmetry in R3,1. Indeed, this is the original approach used in [2]. This approach is also useful in cases which cannot be realized in perturbative D-brane constructions. From the above superpotential it is straightforward to derive the F-term equations of motion

(3.3) F (0,2) = ∂¯A¯ + A¯ ∧ A¯ =0

(3.4) ∂¯Aϕ =0.

2 The ﬁrst equation ensures ∂¯A = 0 and hence the existence of a complex structure on the gauge bundle, while the second equation states that the adjoint-valued two-form ﬁeld ϕ is holomorphic with respect to this complex structure. The superpotential of the seven-brane theory is invariant under the gauge transformations:

(3.5) A¯ → g−1Ag¯ + g−1∂g,¯ ϕ → g−1ϕg.

Note that in the present theory the action is invariant not only under real gauge transformations, but also complex (or holomorphic) gauge transformations. As we will see later, this gauge invariance will allow us to develop a cohomology theory for the zero mode solutions and will also greatly simplify our computation of the Yukawa couplings between chiral matter ﬁelds. YUKAWA COUPLINGS IN F-THEORY 47

Up to this point, our discussion has focussed solely on the superpotential of the seven-brane theory. The D-term equation of motion for the partially twisted seven-brane is given by [2]: i (3.6) ω ∧ F + [ϕ, ϕ¯]=0, 2 where ω is the K¨ahlerform on S. This D-term constraint constitutes non- holomorphic data, and so is not expected to play a crucial role in a holomorphic formulation of the Yukawa couplings. As we will explain later, in our case this can be explicitly seen from the gauge invariance of the superpotential.

3.2. Matter Curves and Localized Zero Modes. In this subsection we will first review part of [2], showing how the intersection of other seven- branes with the seven-brane on S on which our gauge theory is defined can be modelled by a specific background field configuration, and how chiral matter fields localized on the intersection locus are related to fluctuations around this background. After this we will see how the chiral modes can be described by a cohomology theory. In subsequent sections this will make manifest the quasi-topological nature of the Yukawa coupling. 3.2.1. Enhancement Loci and Matter Curves. Following [2,46], we now discuss how matter curves are specified by a background field configuration of the seven-brane theory. This amounts to a choice of supersymmetric background field configuration A¯(0),ϕ(0). Since the main object of interest in the present paper is the triple coupling of chiral zero modes localized on three “matter curves” meeting at one point, we can work in a local neighborhood U of this point. ¯2 In a local patch, the BPS equation ∂A(0) = 0 ensures we can further simplify our discussion by working in the holomorphic gauge

(3.7) A¯(0) =0.

As guaranteed by the gauge invariance of the theory, no generality is lost by making such a choice. To discuss the background for the two-form ﬁeld ϕ(0), recall that it is a section of the canonical bundle of the surface S. Adopting local holomorphic coordinates x and y in the patch U, we can simply write dx ∧ dy as a basis for the (2, 0)-form. Writing

ϕ(0) = φ(0)dx ∧ dy, let us now focus on the simplest kind of background in which ϕ(0) takes values in the Cartan subalgebra h of the Lie algebra g of the gauge group Gp:

(3.8) φ(0) ∈ h. 48 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

In this case, the D-term equation (3.6) reduces to the usual instanton equation ω ∧ F = 0. The non-zero value for ϕ(0) specifies a breaking pattern for Gp. A justification for this choice of ϕ(0) comes from the physical consideration that we are interested in modelling a situation in which other seven- branes intersect the surface S along some curves Σ and the gauge symmetry on S is enhanced along the loci of intersection. For φ(0) valued in the Car- tan subalgebra h, the gauge group Gp is broken to its maximal torus at a generic point on S, as is the case for intersecting branes. At a generic point, no gauge symmetry will be left when φ(0) is not in the Cartan.6 The general configuration of matter curves is conveniently packaged in terms of the root space decomposition of the Lie algebra g of Gp so that:

(3.9) g = h ⊕α gα where α denotes the roots of the algebra. For φ(0) ∈ h we can write

(0) (0) [φ ,Xα]=α(φ )Xα, where Xα is the generator in the Lie algebra corresponding to the root α. From this expression, it is now immediate that the curve Σα deﬁned by α(φ(0)) = 0 for some positive root α corresponds to a locus of gauge enhancement and will hence be identiﬁed as the curve of intersection between seven-branes.

An example To illustrate the main ideas discussed above, we now consider a system with gauge group U(1) at a generic point on the surface S, and a matter curve where the singularity type enhances to SU(2). In this case, we can choose the gauge and the coordinates such that our background conﬁgura- tion reads (0) (0) x 0 A¯ =0,ϕ= dx ∧ dy, 0 −x where x, y denote local coordinates of the patch. The locus of gauge enhancement in these coordinates is the complex line x = 0. Note that in more general situations where ϕ(0) embedsasa2× 2 sub-block of a larger (traceless) matrix, we can also consider conﬁgurations of the form: (0) (0) x +Φ 0 A¯ =0,ϕ= dx ∧ dy 0 −x +Φ

6 It would be interesting to extend the analysis presented here to more general choices for φ(0),suchascaseswhereφ(0) is nilpotent. See for example, [47] for a study of the spectrum of B-branes in a similar setup. YUKAWA COUPLINGS IN F-THEORY 49 for Φ = Φ(x, y). In this case, the matter curve defined as the gauge enhancement locus is still given by the complex line x =0. 3.2.2. Zero Modes Localized on Curves. In order to study the chiral matter in a specific background, we first determine the available deformations of a given supersymmetric background field configuration. The massless fluctuations about this background will then define the zero modes of the system. Separating the fields ϕ and A¯ into a background value and fluctuations about this background, we write:

(3.10) ϕ = ϕ(0) + ϕ(1), A¯ = A¯(0) + A¯(1).

Expanding the F-term equations (3.3)–(3.4) to linearized order in ﬂuctua- tions ϕ(1) and A¯(1), we obtain the equations of motion to be satisﬁed by the zero modes

(1) (3.11) ∂¯A(0) A¯ =0 (1) (1) (0) (3.12) ∂¯A(0) ϕ +[A¯ ,ϕ ]=0.

One can check that these are indeed the equations of motion for the fermionic zero modes in the supersymmetric Yang-Mills theory [2]. Similarly, expanding the D-term equation (3.6) we get the following equation for the zero modes7

(1) i (0) (1) (3.13) ω ∧ ∂ (0) A¯ + [¯ϕ ,ϕ ]=0. A 2 Let us now examine the properties one can expect from a zero mode satisfying the above BPS equations. In a local patch, and for any given zero (1) mode solutions, we can integrate (3.11) to write A¯ = ∂¯A(0) ξ for some regular function ξ with values in the adjoint of Gp. Combined with the other zero mode equation, we can express any zero mode solution as

(1) (1) (0) (3.14) A¯ = ∂¯A(0) ξ, ϕ =[ϕ ,ξ]+h for some adjoint-valued (2, 0)-form h which is holomorphic with respect to ∂¯A(0) . Again using the local basis for (2, 0)-forms to write

ϕ(1) = φ(1)dx ∧ dy and decomposing the zero modes in the basis g = h ⊕α gα by writing

(1) (1) (1) (1) A¯ = A¯α Xα,φ= φα Xα etc.,

7 Technically speaking, expanding about the holomorphic and anti-holomorphic background yields a single equation of motion for the zero modes and their complex conjugates. Compatibility with the holomorphic structure speciﬁed by the F-terms then leads to equation (3.13). 50 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA in the background discussed in (3.7)–(3.8) the zero modes can now be expressed in component form as

(1) (1) (0) (3.15) A¯α = ∂ξ¯ α,φα = α(φ )ξα + hα, with some regular function ξα and a holomorphic section hα of the canonical bundle of the surface S satisfying ∂h¯ α =0. This expression illustrates that it is possible to have zero modes with different supports on our local patch. Our primary interest will be in configu- (0) rations where these zero modes localize on matter curves Σα = {α(φ )=0} for positive roots α.8 To define a localized chiral zero mode, let us rewrite the above equation as

(1) (1) φα − hα (1) (3.16) A¯α = ∂¯ ,φα |α(φ(0))=0= hα is holomorphic. α(φ(0))

By deﬁnition, a zero mode localized at the matter curve Σα is given by the (1) above formula with φα vanishing rapidly away from the curve. The class of physically distinct zero mode wave-functions are those spec- iﬁed by solutions to the F-term equations of motion, modulo gauge equiva- (1) (1) lences. For example, all zero modes φα which satisfy φα |α(φ(0))=0= hα for (1) (0) a given hα with (φα − hα)/α(φ ) smooth everywhere on the patch are in fact gauge equivalent. To see this, consider two sets of such zero mode (1) (1) (1) (1) wave-functions A¯α ,φα and A¯α ,φα . Since:

(1) (1) (1) (1) φα − φα φα − hα φα − hα δξα = = − α(φ(0)) α(φ(0)) α(φ(0)) is a function which is everywhere smooth on the patch, the diﬀerence between the two “diﬀerent” zero modes is: (1) (1) (1) (1) (0) (3.17) A¯α − A¯α = ∂¯(δξα),φα − φα = α(φ ) δξα .

By inspection, the diﬀerence between these two solutions is an inﬁnitesimal version of the gauge transformation (3.5). Note that a similar argument holds for a change of the holomorphic section hα by

(0) hα → hα + α(φ )h˜α for some holomorphic function h˜α. Combining the above two observations, we conclude that in our local patch the space of gauge inequivalent classes of zero mode solutions localized (0) (0) on the curve Σα = {α(φ )=0} is isomorphic to the space OU /(α(φ )),

8 Note that negative roots of the form −α for α positive deﬁne the same matter curve. YUKAWA COUPLINGS IN F-THEORY 51 namely the space of holomorphic functions on the patch with the equivalence relation α(φ(0)) ∼ 0.9

An example As alluded to earlier, the gauge invariance of the physically distinct zero mode solutions is consistent with the fact that the superpotential W is independent of the Kähler data. To illustrate this point, we now work in “physical gauge” and present an example of zero mode solutions in a geometry where the non-holomorphic data of the system is as simple as possible. To this end, consider a background field configuration where the background gauge field strength is zero, and the Kähler metric is diagonal and flat. The corresponding values of the gauge field A(0) and Kähler form ω can then be written in local coordinates x and y as: i A(0) =0,ω= g dx ∧ dx¯ + dy ∧ dy¯ . 2 Next consider zero mode solutions localized on the matter curve α(φ(0))= −x. Plugging in the form of the localized zero modes dictated by equation (3.16) and writing hα = hx(x, y), the D-term equation (3.13) for the zero modes reads (1) φα − hx (1) g∂x∂x¯ =¯xφα . x As a brief aside, although x appears in the denominator of an expression which is acted on by the Laplacian, the resulting zero mode solutions will nevertheless be regular at the origin x = y = 0. This is because the numerator (1) φα − hx will vanish at the origin as well. As can be checked, this equation admits solutions of the form:

− √1 |x|2 (1) g φα = e hx(y) where hx(y) is a holomorphic function of the coordinate y on the curve {x =0}, on which the zero mode is localized. The D-term constraint specifies a particular representative for the wave- function, specified by a Gaussian falloff away from the matter curve. Com- paring the profile of the matter field wave-functions for two different values (1) (1) of g which we refer to as φα (g1)andφα (g2), it can be shown that there exists a δξα of the form given by equation (3.17). In other words, a change in the Kähler metric simply corresponds to a gauge transformation of the

9 Recall that the hα’s are actually holomorphic sections of the restriction of the canonical bundle of S to U, KS |U , a fact consistent with the above equivalence relation since (0) φ is also a holomorphic section of KS |U . They only reduce to holomorphic functions upon local trivialization of the bundle KS . Since we mostly work in a local patch in the present paper, we will sometimes abuse terminology and refer to them as “holomorphic functions” when the context is clear. 52 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA wave-function. Moreover, the gauge inequivalent classes of zero modes are distinguished by different holomorphic functions hx(y). In particular, we will often find it convenient to use the basis {1,y,y2,...} to label the zero modes localized on the curve {x =0}. This example also shows that the triple coupling of the zero modes localized on three matter curves is localized at the point of intersection. To see this, first observe that the freedom to choose the gauge, or equivalently the Kähler data, allows us to take the limit g → 0. In this limit the zero mode wave-functions become sharply peaked along the matter curve:

− √1 |x|2 (1) 1 g A¯α = lim √ e hx(y)dx¯ = πδ(x)hx(y)dx¯ g→0 g (3.18) − √1 |x|2 (1) g φα = lim e hx(y)=2 1 − H(|x|) hx(y) g→0 where here, δ(x)√ is the two-dimensional delta function so that upon writing x =Re(x)+ −1Im(x), δ(x)=δ(Re(x))δ(Im(x)) = δ(|x|)/|x|. Further, H(|x|) is the Heaviside step function, which satisfies: d (3.19) H(|x|)=δ(|x|)=π|x|δ(x). d|x| By an appropriate choice of Kählermetric, the profile of the matter field wave-functions can be made to vanish away from the matter curves. In particular, the evaluation of the Yukawa couplings is concentrated near the point of mutual overlap between the zero mode wave-functions. This localization of the matter field wave-functions near the Yukawa point is the primary justification for our approach of working in a local neighborhood containing this point. Furthermore, the δ-function form of the matter field wave-functions also suggests that the triple overlap integral should be zero unless all holomorphic functions hα are constants. 3.2.3. The Local Cohomology Theory of Chiral Matter. It turns out that there is a more formal and economical way to package the information about the equivalence classes of the localized zero modes by defining an appropriate cohomology theory. See [23] for a related discussion. This way of encoding the zero mode solutions helps to make the topological nature of our twisted supersymmetric Yang-Mills theory, and in particular the Yukawa coupling of the theory, manifest. To define this cohomology, let us start by considering the total space of the canonical bundle of the Kähler surface S, KS → S which we will refer to as KS. The complex threefold KS is a non–compact Calabi–Yau threefold with holomorphic (3, 0) form given locally as:

(3.20) Ω = dx ∧ dy ∧ dz, where x, y are holomorphic local coordinates on S and z is the local coordinate along the ﬁber. YUKAWA COUPLINGS IN F-THEORY 53

Using the isomorphism between the canonical bundle and normal bundle:

(3.21) KS NS/KS we can map a (2, 0)-form on S toa(0, 1)-form on KS by

(3.22) φdx∧ dy → κˆ ∧ φ where 1¯ı 2¯j κˆ = g g Ω¯ıj¯z¯ dz.¯ In the above equation gij¯ is the K¨ahlermetric on S andwehavedenotedthe coordinates by x = x1,y= x2. From the fact that the holomorphic three-form is covariantly constant and that a K¨ahlermetric respects the holomorphic structure, we have

i¯ı jj¯ ∇¯ κˆ = ∂¯κˆ = ∂¯(g g Ω¯ıj¯z¯ dz¯)=0, a fact that will prove important later in order for our Dolbeault operator to correctly capture the zero mode equations. Notice that the non-compact Calabi-Yau KS should really be thought of as a book-keeping device in the present construction rather than literally as the physical geometric space. Extending the above map (3.22), we can map (2,k− 1)-forms on S to (0,k)-forms on KS by

(3.23) F dx ∧ dy → κˆ ∧ F ,

(0,k−1) where F = F is a KS-valued (0,k− 1)-form on S. In this way a pair of (0,k)-form and (2,k− 1)-forms on S is mapped to a (0,k)-form on KS as

(3.24) (F, F dx ∧ dy) → F +ˆκ ∧ F .

By this construction, we have unified the two kinds of forms on S in the standard Hodge decomposition by introducing a fictitious extra dimension. For this reason we will from now on refer to both (0,k)-forms and (2,k− 1)- forms on S as “twisted k-forms”. There is also a natural way to define the integral of a twisted three-form on the patch S. Writing a twisted 3-form as

μ =ˆκ ∧ μˆ whereμ ˆ is an KS-valued (0, 2)-form on S, the integral of μ on the surface is simply given by (3.25) μ ≡ μdxˆ ∧ dy S S as follows from the isomorphism (3.21). 54 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

Now we are ready to deﬁne the relevant cohomology. For a given background A¯(0),φ(0), deﬁne the Dolbeault operator

(0) (3.26) D¯ = ∂¯A(0) +ˆκ ∧ adj(φ ) where adj(φ(0))=[φ(0), · ] is the adjoint action of the background field φ(0). It is easy to verify the following properties of the Dolbeault operator D¯: First, we observe that D¯ takes twisted k-forms to twisted (k + 1)-forms and that D¯ 2 =0 as a consequence of the F-term equations (3.3)–(3.4) satisfied by the background fields A¯(0) and φ(0). Second, D¯ satisfies the Leibniz rule:

(3.27) D¯(Ψ ∧ Ψ)=D¯Ψ ∧ Ψ +(−1)degΨΨ ∧ D¯Ψ .

Third, we have D¯ = ∂¯ when acting on gauge singlets. The ﬁrst property shows it is meaningful to talk about a cohomology with respect to this operator. As we will see in more detail later, the last two properties ensure that the Yukawas are invariant under a change of zero mode wave-functions by a D¯-exact piece, given that some mild convergence properties are satisﬁed by these wave-functions. Finally we are ready to establish the connection between this cohomology and the zero mode solutions discussed in section 3.2.2. When acting on a twisted one-form

(3.28) Ψ(1) = A¯(1) +ˆκ ∧ φ(1), we have (1) (1) (1) (1) (0) (3.29) D¯Ψ = ∂¯A(0) A¯ − κˆ ∧ ∂¯A(0) φ +[A¯ ,φ ] . The F-term equation on the zero modes is hence equivalent to the closure of the corresponding twisted one-form under the Dolbeault operator D¯.Fur- thermore, what we get when acting on a scalar

(0) (3.30) Df¯ = ∂¯A¯(0) f +ˆκ ∧ [φ ,f] is exactly an inﬁnitesimal gauge transformation. To sum up, the deﬁnition of equivalence classes of the zero modes can be written as

(3.31) D¯Ψ(1) =0, Ψ(1) ∼ Ψ(1) + Df¯ for any smooth adjoint-valued scalar f vanishing near the boundary of our patch. From here we can conclude that the space of inequivalent zero mode ¯ 1 P solutions is isomorphic to the space of D cohomology HD¯ (U, adj( )), where P is the principal Gp-bundle. YUKAWA COUPLINGS IN F-THEORY 55

Having speciﬁed the formal aspects of the D¯ cohomology theory, we now turn again to zero modes which are localized on the matter curve (0) Σα = {α(φ )=0} of gauge enhancement. This means we have the following expression for the localized zero modes as a twisted one-form (1) (1) φα − hα (1) (3.32) Ψα Xα = D¯ Xα +ˆκ ∧ hαXα,φα |α(φ(0))=0 = hα, α(φ(0))

(1) where φα vanishes rapidly oﬀ the curve Σα. Of course, this expression reduces to our old expression (3.16) when written out in components. Notice (1) that the ﬁrst term is not a pure gauge piece because the expression (φα − (0) hα)/α(φ ) is not localized in the patch. Furthermore, the freedom to add a D¯-exact (local) piece shows that the zero modes localized on the curve Σα (0) are indeed given by OU /(α(φ )), as we have seen in section 3.2.2. The Dolbeault operator D¯ also has a natural role in interpreting the D-term condition (3.13). Using the metric

2 i j¯ −1 ds = gij¯ dx dx +det(g) dz dz¯ on the threefold KS, we see that the D-term equation (3.13) is nothing but the statement that D¯ †Ψ(1) = 0. Hence the physical wave-function is really the harmonic representative of the D¯-cohomology. Nevertheless, as we have argued earlier and as we will see explicitly later, the topological nature of the Yukawa coupling in our theory guarantees that the answer does not depend on which representative, harmonic or not, we pick in the cohomology for the zero mode wave-function.

3.3. The Yukawa Coupling. Finally we are ready to compute the Yukawa coupling of interest. Putting (3.10) into the superpotential (3.2) and using the BPS equations satisfied by the background and the zero modes, we get the following expression for the superpotential (1) (1) (1) (3.33) WYuk = Tr(A¯ ∧ A¯ ∧ ϕ ) . U In the language of the twisted one-forms of section 3.2.3, using equations (3.25) and (3.28) the above equation can be written in a more symmetric- looking form as: (1) (1) (1) (3.34) WYuk = Tr(Ψ ∧ Ψ ∧ Ψ ) . U As explained in the last subsection, the gauge invariance of the superpotential ensures that WYuk is well-defined in the sense that it only depends on the D¯-cohomology of the wave-function Ψ(1). Indeed, the chiral superfields localized on the matter curve Σ have the cohomological interpretation as 56 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA elements of the group Ext1(E, E) where E, E are the coherent sheaves associated with the holomorphic gauge bundles along the seven-branes S and S which intersect on the curve Σ. The Yukawa coupling is in this case given by the Yoneda pairing of the Ext-groups:

Ext1(E, E) ⊗ Ext1(E, E) ⊗ Ext1(E, E) → C .

See for example [48] for a discussion of the Yoneda pairing, and [49]fora discussion in the context of superpotentials for D-branes. In the context of F-theory GUTs, a formulation of Yukawas in terms of Yoneda pairings has been discussed for example in [23,28]. Let us now turn to the physical meaning of this triple coupling. While the given zero modes satisfy the linearized equation of motion (and hence the absence of mass terms), there is no guarantee that the corresponding deformation remains unobstructed beyond the linearized level. In fact, a non-zero value for this triple coupling means the deformation is obstructed. The Yoneda pairing formulation also demonstrates that the gauge bundle data is already taken into account in computing the form of the Yukawa couplings. This last point is somewhat at odds with the expectation of [14] rein- forced in [32] that because the gauge field fluxes distort the profile of the matter wave-functions, the Yukawa couplings should also be distorted. Indeed, much of this distortion can be ascribed to the D-term equations of motion for the matter field wave-functions. Since the contribution to the Yukawa coupling are independent of non-holomorphic data, it follows that only the F-term equations of motion (3.11)–(3.12) are relevant in discussions of the Yukawa coupling. On the other hand, in a local patch, compatibility of the complex structure of the surface S and the gauge bundle through the ¯2 equation ∂A(0) = 0 ensures that we can pass to a holomorphic gauge in which A¯(0) = 0. In particular, this suggests that it is always possible to recast our computation in terms of a configuration where the local profile of the fluxes are absent. Indeed, in the remainder of this section we will show that gauge field fluxes alone do not alter the form of the Yukawa couplings. To reach this conclusion, we first study two representative examples in “physical gauge” where we solve for wave-functions which satisfy both the F- and D-term equations of motion. In physical gauge, the absence of a change in the structure of the Yukawas can be ascribed to a non-trivial cancellation between various contributions to the Yukawa overlap integrals. To provide further evidence for this general point, we next present a residue formula which makes the quasi-topological nature of the Yukawa coupling manifest. 3.3.1. Two Examples. We now present two examples of Yukawa couplings between zero modes, first in the absence of a background gauge field flux, and then in the presence of a gauge field flux with non-trivial first order gradients. To be concrete, we work in physical gauge by specifying YUKAWA COUPLINGS IN F-THEORY 57 zero mode wave-functions which satisfy both the F- and D-term equations of motion. For simplicity, we consider a geometry with a single point of Gp = SU(3) enhancement and choose a flat and diagonal Kähler form: i (3.35) ω = g dx ∧ dx¯ + dy ∧ dy¯ . 2 (0) We consider a background value for ϕ such that Gp = SU(3) is broken down to its maximal torus U(1)2 at generic points of the patch through the vev: ⎛ ⎞ −2x + y 00 1 (3.36) ϕ(0) = φ(0)dx ∧ dy = ⎝ 0 x + y 0 ⎠ dx ∧ dy. 3 00x − 2y

Indeed, from the F-term equation (3.4), we see that φ(0) ∈ h forces the gauge background A¯(0) to be in the Cartan subalgebra at generic points as well. To be entirely explicit we work in the gauge in which the gauge ﬁeld is real. To set notation, let α1 and α2 denote the two roots of SU(3) such that

Xα1 and Xα2 are given by 3×3 matrices with a single non-zero entry respectively in the 1-2 and 2-3 entry. In this notation, the relevant zero modes are labelled by the 1-2, 2-3 and 3-1 entries of a 3 × 3 matrix which respectively correspond to the roots α1, α2 and α3 ≡−α1 − α2. We will solve for the zero mode wave-functions in the following way: From the expression for the localized zero modes (3.16), or equivalently (3.32), we see that the zero modes satisfy (1) (1) φα (3.37) Ψα Xα = D¯ Xα , on U \ Σα α(φ(0)) everywhere away from the matter curve Σα. Then the D-term equation implies (1) ¯ † ¯ φα \ (3.38) D D (0) Xα =0, on U Σα. φα

The ‘Laplacian’ operator D¯ †D¯ is positive definite, and with our Ansatz for the Kähler form it reads † (0) 2 (D¯ D¯)α = −g (∂x + iAα,x)(∂x¯ + iAα,x¯)+(∂y + iAα,y)(∂y¯ + iAα,y¯) + |φα | when acting on the Xα component of the zero mode. The reason that the equation (3.38) can have a solution is that it does not hold on the matter curve. Using this equation, we next specify the behavior of the zero mode on (1) the matter curve through the relation hα = φα |α(φ(0))=0. This then determines the full profile of the zero mode solution. 58 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

First example: Without fluxes Having discussed the general set-up, we now compute the Yukawa couplings in a configuration with all gauge field fluxes turned off. In particular, we choose the gauge (0) (0) Aα1 = Aα2 =0. In this case the zero mode solutions are of the Gaussian form we have seen earlier in (3.18). Explicitly, they are

− √1 |x|2 − √1 |x|2 (1) 1 g n (1) g n A¯α = √ e y dx,¯ ϕα = e y dx ∧ dy 1 g 1 1 − √1 |y|2 − √1 |y|2 ¯(1) −√ g m (1) g m ∧ Aα2 = e x dy,¯ ϕα2 = e x dx dy (3.39) g − √1 |x−y|2 (1) 1 2g A¯α = −√ e (x + y) (dx¯ − dy¯), 3 2g (1) − √1 |x−y|2 2g ∧ ϕα3 = e (x + y) dx dy

The Yukawa coupling (3.33) in this case is: (1) (1) (1) WYuk = Tr(A¯ ∧ A¯ ∧ ϕ ) S √ √ 1+ 2 − √1 (|x|2+|y|2+|x−y|2/ 2) = e g ynxm (x + y)dx ∧ dx¯ ∧ dy ∧ dy¯ g

It follows from this expression that the integrand admits a natural rephasing symmetry under the U(1) action

x → eiθx, y → eiθy.

In particular, this implies that the Yukawa coupling vanishes between all zero modes, except when = m = n = 0. Further note that the answer is indeed independent of the K¨ahlerdata g as we expected. In particular we are free to take the zero-volume limit g → 0 in which the wave-functions converge to δ-functions (3.18). An explicit evaluation of the integral yields 1 mn 1 = m = n =0 (3.40) WYuk = . (2π)2 0 otherwise

Second example: Non-constant fluxes We now study the effects (or lack thereof) of gauge field fluxes on Yukawa couplings. With φ(0) as before, we next consider a background gauge field which takes values in the Cartan subalgebra of the Lie algebra YUKAWA COUPLINGS IN F-THEORY 59

of Gp = SU(3). The projection of this direction in the Cartan onto the roots α1 and α2 of su(3) are speciﬁed as:

(0) (0) 2 2 3 3 α1(A )=α2(A )=i1(¯y dx − x¯ dy + 2(¯y dx +¯x dy) ) + c.c. which describes a background gauge ﬁeld strength of the form: (0) (0) α1(F )=α2(F )=i1 2(¯xdy ∧ dx¯ − ydx¯ ∧ dy¯) 2 2 − 32(¯x dy ∧ dx¯ +¯y dx ∧ dy¯) + c.c.

Note in particular that this gauge field strength has non-trivial gradients, and so will produce three tensor index objects of the type required by the flavor hierarchy argument of [14]. As we now show, however, such fluxes do not alter the structure of the Yukawa couplings. We begin by solving for the zero modes in this background. The solution to the D-term equation (3.38) is to first order in the expansion of small flux given by: − √1 |y|2 (1) g − 2 3 φα2 (x, x,¯ y, y¯)=e hα2 (x)+1 x¯ y + 2x¯ y hα2 (x) 2 √ + −2¯xy +32x¯ y gh (x)+ −2y +62xy¯ gh (x) α2 α2 3/2 +62yg hα2 (x) √ 2 − 3 1 2 1 3 +¯1 x y¯ ¯2x y¯ hα2 (x)+ y + ¯2y ghα2 (x) 2 3 + O(F 2) .

The symmetry of the fluxes we chose implies that, for the given holomorphic (1) function hα1 (y), the zero mode φα1 (x, x,¯ y, y¯) is obtained by exchanging x ↔−y in the above formula. To simplify our analysis, we now restrict to the (1) | (0) Yukawas in the special case where φα3 α3(φ )=0= 1. This is an appropriate description for models with a single MSSM Higgs field localized on the curve (0) {α3(φ )=0}, with some number of generations on the other matter curves. The corresponding profile for this zero mode is then given by: − √1 |x−y|2 (1) 2g 1 2 1 2 φα (x, x,¯ y, y¯)=e 1+1 − (x − y)(¯x − y¯) − (x − y)(¯x +¯y) 3 6 2 2 1 2 3 2 + 2|x − y| (¯x − y¯) + (¯x +¯y) 16 8 2 (x − y)2(¯x − y¯) (x + y)2(¯x − y¯) +¯1 2g (x − y)+ + 3 6 2 60 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

√ 2 3 2g (x − y) 1 3 − ¯2 + (x − y) (¯x − y¯) 16 16 3 1 + (x − y)(x + y)2(¯x − y¯) − (x + y)3(¯x +¯y) + O(F 2) , 8 4 and the A¯(1) wave-functions are given by the above expression and the zero mode formula (3.16). To reliably estimate the 2-2 entry of the Yukawa matrix, it is actually necessary to include second order corrections from the fluxes. These expressions are not particularly illuminating, and so for the sake of brevity we shall only present the explicit expressions for the first order corrections to the wave-functions. From the form of the zero mode wave-functions, it follows that there is an effective U(1) rotational symmetry which is broken by the spurion parameters 1 and 2. Indeed, under the diagonal U(1) which acts as

x → eiθx, y → eiθy, the flux parameters 1 and 2 both carry charge one. This is in accord with the analysis of [14] that fluxes can generate an effective Froggatt-Nielsen mechanism. See also [25] for further discussion on a discussion of the associated effective field theory. Although consistent with this selection rule, in the actual computation of wave-function overlaps, we find that the resulting Yukawa structure remains unchanged by the background gauge field flux. To illustrate this point, we consider a two generation model in which the wave-functions on the matter curves x =0andy = 0 are specified by the associated hα as:

m n hα1 (y)=y ,hα2 (x)=x for m, n =0, 1, while that of the “Higgs” wave-function is speciﬁed as:

hα3 =1.

In this case, we ﬁnd that the resulting Yukawa is given by: 1 mn 1 m = n =0 (3.41) WYuk = . (2π)2 0 otherwise

In other words, comparing this result with that of equation (3.40), we find that there is no difference in the two Yukawa structures. This result is due to a number of non-trivial cancellations between contributions to the overlap integral, and in the remainder of this section, our aim will be to make such cancellations more transparent. 3.3.2. Yukawas from Residues. The examples presented in the previous subsection strongly suggest that gauge field fluxes do not distort the structure of the Yukawa couplings. In this subsection we generalize this result by YUKAWA COUPLINGS IN F-THEORY 61 showing how Yukawa couplings can be formulated in terms of a residue formula defined in a neighborhood containing the Yukawa enhancement point Gp. This quasi-topological formulation of Yukawa couplings will allow us to prove a more general “Rank Theorem” that when only gauge field fluxes are activated, the rank of the Yukawa matrix is completely determined by the intersection theory of the matter curves, and so in particular, is independent of the gauge field flux. To obtain a useful formula for this triple coupling, let us focus on the (0) (0) point of intersection of the two matter curves α1(φ )=0andα2(φ )=0. Here we assume that these two curves intersect transversely. Notice that (0) this necessarily implies the presence of the third matter curve α3(φ )=0, as long as α3 = −α1 − α2 is in the root system. Our starting point for the Yukawa coupling is the overlap integral for three twisted one-forms: (1) (1) (1) WYuk = Tr(Ψ ∧ Ψ ∧ Ψ ) U (1) − ∧ ∧ ¯ φα3 hα3 ∧ ∧ ∧ = Ψα1 Ψα2 D (0) + hα3 Ψα1 Ψα2 dx dy U α3(φ ) U (1) − (1) − ¯ φα1 hα1 ∧ ¯ φα2 hα2 ∧ (3.42) = hα3 ∂A(0) (0) ∂A(0) (0) dx dy U α1(φ ) α2(φ ) where in the above formula we have dropped the first term in the second line by first rewriting it as a boundary term and then using the property that Ψα1 ∧ Ψα2 is localized around the intersection point of the two matter curves Σα1 and Σα2 . Let us now focus on the remaining term in the integral. From the form it is rather clear that it can be rewritten a residue integral. See Appendix 7 for the actual calculation. The final answer reads

1 hα1 hα2 hα3 (3.43) 2 WYuk =Res (0) (0) . (2πi) α1(φ )α2(φ )

This formula makes immediately obvious the two major conclusions drawn from our previous explicit calculations of lines (3.40) and (3.41). (1) The ﬁrst is that although the actual wave-functions given by φα are different, the Yukawa coupling has to be the same since it only depends on (1) the holomorphic section hα = φα |α(φ(0))=0 of the canonical bundle, namely the wave-function evaluated on the matter curve. Second, as long as our matter curves are “simple” and hence can be described by x =0andy =0 locally near the intersection point, the only set of three zero modes having non-vanishing Yukawa coupling is just

hα1 = hα2 = hα3 =1. 62 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

3.3.3. A Rank Theorem. As we mentioned at the end of section 3.3.2, the residue expression for the Yukawa coupling (3.43) makes manifest that the rank of the Yukawa matrices in the examples we computed earlier is one. Now we will sharpen this statement in the following way. In the theory described by the superpotential (3.2), if the matter curves are smooth, reduced, and intersect transversely at point p, then the rank of the Yukawa coupling associated with the intersection point p is at most one. To simplify the formal analysis, let us ﬁx the zero mode on the third matter curve to be hα3 = 1, motivated by the phenomenological application to the MSSM and the fact that there is only one Higgs mode present in a given class of Yukawas. In other words, we associate the curve Σα3 to the

Higgs and the other two curves Σα1 ,Σα2 to chiral generations.

Let hα1,m, hα2,n (m, n =1, 2,...) be the holomorphic sections corresponding to the various generations. From the residue formula (3.43) we see that the Yukawa matrix is proportional to mn hα1,mhα2,n (3.44) WYuk =Res (0) (0) α1(φ )α2(φ ) The local duality theorem (see ref. [50] pages 659 and 693) states that the bi-linear pairing defined by the above residue fg f, g =Resp (0) (0) α1(φ )α2(φ ) is a perfect pairing in the ring (0) (0) (3.45) O/J≡O/(α1(φ ),α2(φ )) . In particular, all localized modes h ∈J decouple from the cubic Yukawa interaction, again a fact reflecting the gauge invariance of the theory. We therefore obtain: (0) (0) (3.46) rank·, · =dimC O/(α1(φ ),α2(φ )) ≡ (Σα1 · Σα2 )p where (Σα1 · Σα2 )p stands for the intersection number of the curves (divisors) Σα1 ,Σα2 at the point p. Indeed, the last equality in equation (3.46) is precisely the definition of the intersection number of two distinct curves on a complex surface. Comparing equations (3.44) and (3.46), we get the following result Rank theorem. In the theory with superpotential given by (3.2), the mn rank of the Yukawa matrix WYuk associated with a given intersection point p is equal to or less than the intersection degree (Σα1 · Σα2 )p of the matter curves at p.

4. H -flux and Non-Commutative F-terms In the previous section we studied the Yukawa coupling of the seven- brane field theory with background gauge fluxes turned on, and concluded YUKAWA COUPLINGS IN F-THEORY 63 that the Yukawa coupling does not change in the presence of the gauge field fluxes. In a certain sense, however, this analysis is incomplete because there are more general fluxes present in generic F-theory compactifications. In this section, we focus on the three-form fluxes H = HR + τHNS in F-theory and ask the question of how their presence deforms the superpotential of the seven-brane gauge theory. Indeed, from the tadpole condition of F-theory compactified on a Calabi-Yau fourfold Z [42] χ(Z) HR ∧ HNS + ND3 = , X 24 where X denotes the base of the fourfold, ND3 is the number of D3-branes and χ(Z) is the Euler characteristic of the fourfold, we see that the F-theory three-form fluxes will generically be present. The (1, 2) component of the H-flux which we denote by H = HR +τHNS is the object which enters the four-dimensional N = 1 superpotential as a chiral superfield. For example, in compactifications of type IIB string theory, H enters the closed string superpotential as [40]: (4.1) W = Ω ∧ H where Ω is the holomorphic three-form on a Calabi-Yau threefold. This expression generalizes to F-theory, and so it is appropriate to treat H in the same way there as well. From this perspective, it is therefore quite natural to expect H to enter into the superpotential for the open string sector as well. By analyzing the superpotential of probe branes in the presence of background fluxes, we will find that in the presence of H-fluxes, the F-term equations of motion are modified. This induces a non-commutative deformation of the topological theory so that for holomorphic coordinates Xi,wehave: [Xi,Xj]=θij. Here, the field theory parameter θij parameterizing the non-commutative deformation satisfies the relation: jk jk (4.2) ∇iθ = Hi , to leading order in H. In the above, jk jj¯ kk¯ Hi = g g Hij¯k¯. Our strategy for establishing this result is as follows. Since we are mainly interested in local questions, we take the “internal directions” of the com- 3 pactification to be C and ignore the effects of monodromy on τIIB by an SL(2, Z) transformation for simplicity. Indeed the relation (4.2) would suggest that θ should be a section of a bundle. To further simplify the discussion we take τ = τIIB to be constant. We first show that a non-commutative 64 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

deformation of the topological theory is generated for non-zero HR as a consequence of the Myers effect [51]. Furthermore since only chiral fields can appear in the topological B-model sector, and since H is a chiral field, this will show that the relation also holds when τHNS is added to HR. Next, we interpret the presence of H-flux in terms of the topological B-model. As a final check on this relation, we provide an independent argument for this relation by studying realizations of the Poisson bi-vector θ in the context of generalized complex geometry. In the next section we will study how this non-commutative deformation affects the field theory and in particular its Yukawa couplings.

4.1. HR-flux and the Myers Effect. In this section we will show how HR-flux leads to superpotential terms for D3-branes. Since the D3-brane corresponds to a probe of the topological B-model, this will allow us to deduce the presence of a non-commutative deformation of the seven-brane F-terms. Indeed, as we show in the next subsection, the D3-branes can also be described by defects in the non-commutative deformation of holomorphic Chern-Simons theory. First let us briefly recall the Myers effect [51]. Consider type IIA string theory with four-form G-fluxes turned on. A stack of N D0-branes feels a potential due to this G-flux given to leading order in constant weak G by

I J K VD0 =Tr(X X X )G0IJK + ··· where 0 denotes the time direction and XI denotes a U(N) valued field of the D0-brane theory whose eigenvalues indicate the positions of the D0-branes in the nine spatial directions. This configuration admits a T-dual description. Focussing on fluxes with spatial indices in C3, if we compactify the three remaining spatial directions on a T 3, applying three T-dualities to the D0-brane configuration yields a D3-brane probe in type IIB in the presence of HR flux (given by the Hodge dual of the seven-form field strength) with potential:

I J K LMN VD3 =Tr(X X X )IJK HLMN + ··· where I,...,N =1,...,6 are indices labelling the six transverse directions to the stack of D3-branes. As we have explained in section 2, HR is a (1,2) form in the case of interest to us. In terms of the local complex coordinates Xi in the normal direction to the D3-brane (viewed as the internal geometry of type IIB string), we can thus write the potential as

i ¯j k¯ VD3 =Tr(X X X )Hi¯jk¯ + ···

Moreover, in the case of interest to us, N = 1 supersymmetry is preserved. Assuming a canonical K¨ahlerpotential, the potential for the stack of probe YUKAWA COUPLINGS IN F-THEORY 65

D3-branes can then be written in terms of a superpotential as: VD3 = (∂iW )(∂¯iW ).

For a stack of N D3-branes, this implies that the leading terms of the superpotential in the H expansion are

ijk i j ij k ∂kW =[X ,X ] − Hk X + ··· , which indicates that the F-term moduli space for the stack of D3-branes is non-commutative. Indeed, the F-term equation of motion [Xi,Xj]=0fora probe of commutative space is now replaced by:

i j ij ij k (4.3) [X ,X ]=θ = Hk X + ··· .

This shows that to leading order in weak H we have:

ij ij ∇kθ = Hk

In a local patch, this can be solved by

ij i¯i j¯j θ = g g B¯i¯j + const., where B is the RR (0, 2)-form gauge potential. The constant can be absorbed into a gauge choice for B and we will choose it such that B vanishes when H vanishes. This non-commutativity is reminiscent of what is found in the D-brane sector of the superstring in the presence of BNS [52,53]. However, there are diﬀerences from the present case. First, we have non-commutativity even when we only have BR. Secondly, the non-commutativity is only in the topological sector, rather than the full superstring theory. Even though this formula was derived for HR =0 and HNS = 0, since W is holomorphic with respect to H = HR + τHNS, it follows that the non- commutative deformation of the F-term equation of motion for the D3-brane theory extends to more general H (which is consistent with our notation above, where we occasionally omitted the subscript from H).

4.2. H -ﬂux and the B-model. Having motivated equation (4.2), we now explain why this means that the topological B-model which computes such superpotential terms lives in non-commutative space. In [37,39] the deformation of the topological B-model by turning on an element of H0(Λ2T 1,0) is considered. From the viewpoint of the Kodaira-Spencer theory [54], these correspond to giving vevs to the ghost ﬁelds (more generally one could consider deformations given by Hp(∧qT 1,0)), and a deformation of the complex structure by

ij ∂¯ → ∂¯ = ∂j¯ + θ ∂j. 66 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

More explicitly, in local coordinates the deformed ∂¯ acts on ⊕m,n Hm(ΛnT 1,0)as

¯ i1...in → i1...in ij i1...in ∂ :Ωj¯1...j¯m ∂j¯Ωj¯1...j¯m + θ ∂jΩj¯1...j¯m with all uncontracted indices anti-symmetrized. The integrability condition ∂¯2 = 0 can be written as two conditions: i jk ∂θ¯ =0,θ∂θ + (anti-symmetrization of ijk)=0. This shows that θij has to be a holomorphic Poisson bi-vector. As has been shown in [39], this deformation of the topological string makes all the open string sectors non-commutative. In other words, we obtain the non-commutative deformation of the holomorphic Chern-Simons theory living on the six internal directions of D9-branes wrapping a Calabi- Yau manifold. More precisely, the three complex dimensions now have non- commutative coordinates satisfying [Xi,Xj]=θij. Similarly, by turning on fluxes on the D9-brane we obtain its reduction to D7-, D5- and D3-branes. There is yet another way to see this. As was discussed in [40], turning on fluxes gives rise to a superpotential term W = Ω ∧ H, where Ω is the background value of the KS field A. Giving a vev to θij corresponds to a deformation of the form Ω → Ω+θ where θ is a one-form given by ij θk = θ Ωijk. Moreover, the totality of RR fields unify to a sum over all odd-dimensional field strengths. It is therefore natural to generalize the above superpotential to all such fields. Given the above deformation of Ω by θ wegetanaddi- tional contribution from the five-form field strength of the D3-brane gauge potential. This implies that if we have a stack of D3-branes and move it from one internal point to another, we get an additional contribution to the superpotential (in addition to the usual contribution from the N = 4 theory, i j k ijkTr([X ,X ]X )) given by X ΔW = W (X) − W (X0)=− θ X0 In other words, we have

∂W i j ij =([X ,X ] − θ )Ωijk, ∂Xk YUKAWA COUPLINGS IN F-THEORY 67 which agrees with what we have obtained by identifying the topological string deformation parameter θ with the B field. Having analyzed the behavior of a stack of probe D3-branes in the presence of a background H-flux, let us now briefly mention supersymmetric configurations for other probe branes. It would be interesting to analyze whether non-commutative deformations could be used to engineer novel examples of supersymmetry breaking. Returning to equation (4.3), in the case of a single D3-brane, the usual commutator of the N = 4 theory vanishes, and we are left with the condition θij(X) = 0. In the case of a five-brane which wraps a one complex dimensional submanifold of the internal geometry, the worldvolume coordinate clearly commutes with itself. The two complex coordinates in directions normal to the five-brane will not in general commute, and so just as for the D3-brane theory, a supersymmetric configuration corresponds to the special case where it wraps a locus along which θ vanishes. For example, if the five-brane wraps a complex curve defined by the intersection of the two divisors {f1 =0} and {f2 =0}, then as shown in [37], we have the condition θ(df 1,df2) = 0, where here we are using the fact that the holomorphic bi-vector acts naturally on pairs of holomorphic one-forms. Next consider the case of seven-branes. In this case, there is only a single direction normal to the seven-brane, so the condition that it commutes with itself is trivially satisfied. On the other hand, the internal worldvolume of the seven-brane is two complex-dimensional, and so will now be non-commutative. The theory on a nine-brane is similar but of less interest for F-theory. Since the topological B-model captures F-terms of the seven-brane theory, we conclude that the effects of the H-fluxes on F-terms such as the Yukawa couplings are computed by a non-commutative deformation of the partially twisted seven-brane theory.

4.3. Poisson Bi-Vectors and Generalized KählerGeometry. In the previous subsections we have argued that in the weak field limit, the presence of a background H-flux induces a non-commutative deformation of the seven-brane F-terms. To further motivate equation (4.2), we now derive the same relation between the Poisson bi-vector θ and H-flux in the context of generalized Kähler geometry. We begin by first reviewing a few facts about generalized Kählergeom- etry. The manifold with data (I±,g,H) was first introduced by the authors of [55] as the target manifold of N =(2, 2) sigma-models with worldsheet coupling to a B-field such that dB = H locally. The two complex structures I± and the bi-hermitian metric g satisfy

−1 (4.4) ∇±I± =0, ∇± = ∇±g H, where to avoid overloading the notation, we have suppressed the tensor indices. As shown in [57], this set of data is equivalent to a pair of commuting generalized complex structures (I, J ), which satisfy the further property 68 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA that −I · J is positive deﬁnite as a metric on T ⊕ T ∗ and we shall refer to the manifold equipped with this data as generalized K¨ahler. This type of geometry enters the discussion of topological string theory in the following way: In [37, 38], it was argued that a deformation of the B-model by an element of H0(Λ2T 1,0) corresponds to having a block-upper- triangular structure generalized complex structure I. In particular, writing the corresponding pair of complex structures as I± = I ± δ,wehave[57] −1 I δ · g I = ,I= I + O(δ). 0 −It

As we have seen in the previous subsection, the upper-right corner is given by a holomorphic Poisson bi-vector of type (2, 0) with respect to the complex structure I. The fact that this corresponds to a non-commutative deformation can be seen from the open string sector [39, 56], with the corresponding star product given by the Poisson vector θ = δ · g−1 · I = δ · g−1 · I + O(δ2). Using ∇δ = − g−1 · H · I, ∇I = − g−1 · H · δ, it is now straightforward to see that

∇θ ∼ g−1 · H · g−1 + O(δ2), which is of course nothing but the same relation (4.2) upon proper antisymmetrization of the indices. This provides further evidence that in the weak ﬁeld limit, the relation we propose (4.2) should be true for general three-form ﬂuxes H.

5. Yukawas in Non-Commutative Geometry In the previous section we argued that the presence of H-flux in F-theory induces a non-commutative deformation in the open topological string sector. We now study the effect of this non-commutative deformation on the eight-dimensional theory defined in a neighborhood containing the Yukawa enhancement point of a configuration of seven-branes. As in section 3, we first study the profile of the matter field zero modes, and then compute the triple overlap integral of these wave-functions. In the presence of the non-commutative deformation, the matter fields are no longer strictly localized on holomorphic curves. Recall that in the commutative case, the matter curves are identified with the loci of gauge enhancement. When the non-commutative deformation is turned on, this relation only holds to leading order in an expansion in the deformation parameter of the theory. Indeed, although the zero modes still exhibit a Gaussian profile, YUKAWA COUPLINGS IN F-THEORY 69 as we will see in section 5.2.2, the wave-functions now acquire more complicated extra dependence on both the holomorphic and anti-holomorphic coordinates of the local patch. Similar wave-function distortion can also be induced by gauge field fluxes, and it is therefore important to determine whether this distortion also filters through to the Yukawas. We find that the non-commutative deformation generates hierarchical corrections to the structure of the Yukawa couplings. To establish this, we first derive a residue-like formula which makes the quasi-topological nature of the Yukawa coupling manifest. This general formalism also allows us to deduce the selection rule determined by the local rephasing symmetry of the geometry, which makes the structure of such hierarchical corrections manifest. As a check on this result, we next present an explicit example in which the Yukawa matrix as a perturbative expansion is computed. The rest of this section is organized much as in section 3. First we discuss localized zero modes and then establish a cohomology theory for them. After this, we obtain a residue-like formula for these Yukawa couplings, and then provide an explicit example which confirms the presence of hierarchical Yukawa structures.

5.1. Deforming the Field Theory. As shown in section 4, the presence of background H-ﬂux in F-theory descends to a non-commutative deformation of the seven-brane superpotential so that the local coordinates x and y satisfy the relation:

(5.1) xy− yx= θθ.

Here, θ is to be viewed as a quantum expansion parameter, and the two index anti-symmetric tensor θ = θxy corresponds to a local presentation of the holomorphic Poisson bi-vector:

θˆ = θ(x, y) ∂x ∧ ∂y.

For constant θ and holomorphic functions f and g, the product fgcorre- sponds to the usual Moyal product:

jk fg= exp(θ θ ∂ζj ∂ξk )f(z + ζ)g(z + ξ)|ζ=ξ=0 jk 2 (5.2) = fg + θ θ ∂jf∂kg + O(θ).

For general, non-constant θ, the -product is given in (B.3). To keep our discussion as general as possible, we shall consider θ non-constant. In particular, we shall later see in section 6 that the case where θ vanishes at the classical Yukawa point leads to phenomenologically interesting Yukawa structures. To write down the non-commutative version of the superpotential, it is necessary to extend the action of to diﬀerential forms as well. We shall refer to the non-commutative deformation of the wedge product as . Utilizing 70 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA the properties of the product detailed in Appendix B, it is possible to show that the -product is the only available deformation of the ∧ product which satisﬁes the following properties: 1. Associativity 2. The ∂¯-chain rule

∂¯(F F )=∂F¯ F +(−1)degF F ∂F¯ for (p, q)-forms F, F

3. Commutativity Upon Integration (5.3) Tr F F = Tr F F , S S for forms F and F which satisfy an appropriate notion of localization. As discussed in [53], these properties are all crucial in order for the corresponding theory to be gauge invariant. We refer to Appendix 7 for the proof of property (5.3). The non-commutative deformation of the seven-brane superpotential for gauge group G is then given by: (5.4) W = Tr ∂¯A¯ + A¯ A¯ ϕ . S Here, it is important to note that in the presence of the non-commutative deformation and for a general gauge group G, the fields A¯ and ϕ will not in general take values in the adjoint representation of G, but will instead take values in the universal enveloping algebra of G [59, 60]. Due to the associative structure of the group U(N), it follows that in this special case, this subtlety is absent. To simplify our analysis, we will therefore confine our discussion to this choice of gauge group. Nevertheless, much of the general methodology discussed throughout this paper is likely also applicable to general gauge groups, and in particular to E-type gauge groups which have proven to be quite important in the context of F-theory GUTs [31]. With the above definition of the non-commutative -product, the superpotential (5.4) is indeed a deformation of the original theory (3.2) which reduces to the original one when the deformation parameter θ vanishes. As promised, this deformed superpotential is invariant under the gauge transformation:

(5.5) A¯ → g−1 A¯ g + g−1 ∂g,¯ ϕ → g−1 ϕ g, where g−1 is the inverse of g under :

g−1 g = g g−1 =1.

−1 Here, the formal inverse g is deﬁned in terms of an expansion in θ.So long as g−1 exists, it can be shown that a unique g−1 can be constructed recursively. YUKAWA COUPLINGS IN F-THEORY 71

For later use we also write down the inﬁnitesimal version of the above gauge transformations:

(5.6) δA¯ = ∂f¯ +[A,¯ f]∗,δϕ=[ϕ, f]∗, where we have introduced the notation

[F, F ]∗ = F F − F F for later convenience. Varying the superpotential of equation (5.4) with respect to A¯ and ϕ yields the deformed F-term equations of motion:

(0,2) F = ∂¯A¯ + A¯ A¯ =0 (5.7) ∂ϕ¯ +[A,¯ ϕ]∗ =0 which are given by replacing all wedge products by -products in the undeformed equations of motion. From the gauge invariance of (5.5), we see that it is again possible to choose the holomorphic gauge A¯ = 0 in the deformed theory for a supersymmetric ﬁeld conﬁguration without any loss of generality.

5.2. Matter from Non-Commutative Geometry. In this subsection we repeat our analysis of localized zero modes done in the case of the commutative theory, but now in the presence of the non-commutative deformation. To this end, we first study solutions to the background field F-term equations of motion, which reduce in the limit θ → 0 to those of the commutative theory. Next, we study the profile of zero modes which are localized on “fuzzy” matter curves. Just as in the undeformed case, the gauge equivalence of the localized zero modes is captured by an appropriately defined cohomology, a fact which will play an important role in our analysis of the Yukawa coupling between these localized zero modes. 5.2.1. Background Field Configurations. First we will specify the supersymmetric background configuration A¯(0),ϕ(0) of the non-commutative topological field theory in which we would like to study the “quantum Yukawa coupling” of the non-commutative theory. As mentioned earlier, to keep our analysis of the non-commutative theory as simple as possible, we shall confine our attention to the case of a U(n) non-commutative gauge theory so that the background fields, and the corresponding zero modes are given by u(n)-valued differential forms. It will often be useful to think of them as n×n matrices and denote different modes as distinct entries in this matrix. Much as in the commutative theory, in the non-commutative theory we can choose a holomorphic gauge with

(5.8) A¯(0) =0. 72 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

In this case the other F-term equation (5.7) again implies that ϕ(0) is a matrix with holomorphic entries. As before, the most natural choice for such a background is one which satisﬁes: (0) (0) (0) (0) [ϕ , ϕ¯ ]∗ =[ϕ , ϕ¯ ]=0.

For this reason, we shall again focus on backgrounds where ϕ(0) takes values in the Cartan of u(n) and can be represented in matrix form as:

(0) (0) (0) (0) (5.9) ϕ = φ dx ∧ dy = diag(φ1 ,...,φn ) dx ∧ dy,

(0) where the φi (x, y)’s are some holomorphic sections of the bundle KS|U . An important feature of the above definition of φ(0) is that the overall (0) (0) (0) trace φ1 + ···+ φn is not subject to any constraint, since the matrix φ takes values in u(n). Indeed, as we will show later in this section, this overall trace generates an additional expansion for hierarchical Yukawa structures. It might at first appear that the absence of the trace condition is in tension with the expectation that the Yukawa enhancement point defines a gauge theory of ADE type. Indeed, in the commutative case, the associated matrices φ(0) are typically traceless with respect to the Lie algebra for the gauge group defined at the Yukawa enhancement point. We have already mentioned that in the non-commutative gauge theory, the fields take values in the universal enveloping algebra, and so will typically not be traceless. Aside from this formal expectation, there is another more direct reason to expect the corresponding trace to be important in many cases of interest. Returning to the undeformed field theory, note that although φ(0) will satisfy an overall trace condition, any m×m sub-block for 1 ≤ m ≤ n of φ(0) need not satisfy such a constraint. In particular, the overall trace of a sub-block can then enter into the Yukawa overlap integral. This is especially natural in the context of higher rank Yukawa enhancement points, as has been advocated for example in [14,24,31]. 5.2.2. Fuzzy Matter Curves and Localized Zero Modes. Having specified a class of background field configurations which closely mimic those of the commutative gauge theory, we now study localized zero mode solutions in this background. In the limit where θ → 0, we recover the expected loci of gauge enhancement corresponding to the matter curves Σab:

(0) (0) Σab =Σba = {φa − φb =0},a,b=1,...,n.

For this reason we will sometimes refer to the Σab as the “classical” matter curves. Including the eﬀects of the non-commutative deformation, the precise notion of where the zero modes are localized becomes “fuzzy”. The equations of motion the zero modes satisfy are obtained by separating the ﬁelds into the background (A¯(0),ϕ(0)) and the deformation (A¯(1),ϕ(1)) part as before and expanding the deformed F- and D-term equations of YUKAWA COUPLINGS IN F-THEORY 73 motion to linearized order. The resulting equations are again simply given by replacing all wedge products in the original zero mode equations (3.11– 3.12) by -products and read

∂¯A¯(1) + A¯(0) A¯(1) + A¯(1) A¯(0) =0 (5.10) (1) (0) (1) (1) (0) ∂ϕ¯ +[A¯ ,ϕ ]∗ +[A¯ ,ϕ ]∗ =0. for the F-term equations of motion. The D-term equations given by the analogue of equation (3.13) read as:10 (1) (0) (1) (1) (0) i (0) (1) (5.11) ω ∂¯A¯ + A A¯ + A¯ A + [¯ϕ ,ϕ ]∗ =0. 2 As before we shall refer to zero modes which satisfy both the F- and D-term equations of motion as the “physical” zero mode wave-functions. We now solve for the zero modes in the presence of the background field configuration defined in section 5.2.1 where ϕ(0) takes values in the Cartan subalgebra of u(n). The analysis closely parallels the undeformed case presented in equation (3.15). The solutions to the zero mode equation (5.10) can be written in the following form

(1) (1) (0) (5.12) A¯ = ∂ξ,¯ ϕ =[ϕ ,ξ]∗ + h with some matrix-valued regular function ξ and holomorphic section h of the canonical bundle. In matrix components they read

¯(1) ¯ (1) (0) (0) (5.13) Aab = ∂ξab,φab = φa ξab − ξab φb + hab no sum over a, b, where, as we recall from the deﬁnition of (B.5) (0) (0) −1 φa ξab = (θφa ) ξab θ .

As opposed to the commutative case, note that the holomorphic section (1) hab and the zero mode wave-function φab will in general only agree on the classical matter curve Σab in the strict limit θ → 0. We are interested in zero mode solutions which vanish rapidly away from the classical matter curves Σab. With the gauge equivalence (5.6) taken into account, we conclude that, on our patch, the space of gauge inequivalent

10 Although we have argued in section 4 that H-fluxes induce a non-commutative deformation of the F-term equations of motion, the D-term equations of motion are strictly speaking outside the realm of the topological B-model, and so cannot be captured in a similar fashion. Note, however, that D-terms play a secondary role in the present case, because we shall mainly be interested in deformations of the superpotential. It is therefore enough to specify the Kählerdata such that it is in principle compatible with the gauge invariance of the superpotential. For example, our definition of the product acts trivially on anti-holomorphic coordinates. 74 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA classes of localized zero mode solutions is isomorphic to the space of holomorphic sections, with a now deformed equivalence relation

(0) ˜ ˜ (0) hab ∼ hab + φa hab − hab φb , for any functions h˜ab. Stated in another way, for a given holomorphic section hab one can ﬁnd holomorphic functions on the patch U \ Σab such that

(0) (0) (5.14) hab = φa ζab − ζab φb on U \ Σab.

The smooth function ξab reduces to −ζab far away from the matter curve (1) Σab due to the localization of the wave-function φab . The two zero modes given by the holomorphic functions hab and hab are gauge equivalent if and only if the corresponding ζab and ζab diﬀer by a function that can be holomorphically extended to the entire patch U. Furthermore, we can deﬁne a deformed version of the D¯ operator introduced in section 3.2.3

(0) (5.15) D¯ ∗ = ∂¯ +ˆκ ∧ [φ , ·]∗.

In the matrix notation, where we deﬁne Eab, a, b =1,...,n to be the n × n matrix with 1 in the a-b-th entry and zero for all other entries, it reads ¯ ¯ ¯ (0) (5.16) D∗ (Fab +ˆκ ∧ Gab) Eab = ∂Fab − κˆ ∧ ∂Gab + Fab φb (0) − φa Fab Eab, where Fab is a (0,k)-form and Gab a KS-valued (0,k−1)-form on the surface S as before. It is not diﬃcult to check that the deformed D¯ ∗ operator still has the following properties:

2 D¯ ∗ =0 (5.17) degΨ D¯∗(Ψ Ψ )=D¯ ∗Ψ Ψ +(−1) Ψ D¯ ∗Ψ , and that the zero mode equation and the gauge invariance of the theory (5.4) can now be written in the cohomological form

(5.18) D¯ ∗Ψ=0, Ψ ∼ Ψ+D¯∗ f for any matrix-valued, smooth function f subject to some support properties. Finally, for future use we note that the general form of the zero mode solutions (5.12) can be conveniently rewritten as

(1) (1) (1) (5.19) Ψ = A¯ +ˆκ ∧ φ = D¯ ∗ ξ +ˆκ ∧ h, ∂h¯ =0. YUKAWA COUPLINGS IN F-THEORY 75

An example To illustrate the effects of the non-commutative deformation on the profile of the zero mode wave-functions, we now present an example in which we solve the F- and D-term equations of motion. For simplicity, we choose a canonical Kählerform, so that in local coordinates: i ω = g dx ∧ dx¯ + dy ∧ dy¯ , 2 and assume that the Poisson bi-vector θˆ is constant on the patch. We consider a configuration in which the background gauge field strength is switched off A(0) =0, and in which φ(0) is diagonal. In this case the F-term (5.7) and D-term (5.11) equations, in matrix notation, become ¯ ¯(1) ∂Aab =0 ¯ (1) ¯(1) (0) (0) ¯(1) (5.20) ∂φab + Aab φb − φa Aab =0 ¯(1) ¯(1) ¯(0) ¯(0) (1) g ∂xAab,x¯ + ∂yAab,y¯ − (φa − φb ) φab =0.

Now take φ(0) in the Cartan of u(2) so that: (0) 1 x + C 0 ϕ = − dx ∧ dy, 2 0 −x + C where C is a complex constant. The zero mode solution in the 1-2 entry of the matrix is

− √1 x¯(x+C) − √1 x¯(x−C) (1) 1 2 g 2 g A¯12 = √ e hx(y) e dx¯ (5.21) g − √1 x¯(x+C) − √1 x¯(x−C) (1) 2 g 2 g φ12 = e hx(y) e , where hx(y) is a holomorphic function of the coordinate y. In this setup, the wave-functions are simply the “normal-ordered” versions of the wave-functions (3.39) of the undeformed theory. Furthermore, as long as the non-commutativity parameter θ does not vanish, the wave- functions cannot be deformed to a δ-function anymore, no matter how we tune the K¨ahler data g. This is to be contrasted with the commutative case (3.18) and suggests that the triple coupling does not take place just strictly at a point anymore but rather in a “fuzzy neighborhood” of the classical Yukawa point. We now show that this allows the non-commutative deformation to violate the rank theorem of section 3.3.3.

5.3. The Yukawa Coupling. In this sectioned we study the effects of the non-commutative deformation on the structure of the Yukawa. We find 76 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA that on the one hand, a very similar residue-like formula can be obtained and in particular, the deformed Yukawa coupling is still a topological quantity in the sense that it does not depend on the representative in the D¯∗-cohomology one picks for the zero mode wave-functions. On the other hand, we also find that the rank theorem discussed in section 3.3.3 for the undeformed theory does not hold when the non-commutative deformation is activated. In particular, we find that the non-commutative deformation generates hierarchical corrections to the Yukawas. 5.3.1. A Quantum Residue Formula and a Selection Rule. In the deformed superpotential (5.4), substituting A¯ = A¯(0) + A¯(1) and ϕ = ϕ(0) + ϕ(1) we obtain the Yukawa coupling (1) (1) (1) (5.22) WYuk = Tr A¯ A¯ ϕ U as the only potentially non-vanishing piece of the superpotential. In the language of the twisted one-forms (5.19) we again obtain the equivalent formula (1) (1) (1) (5.23) WYuk = Tr Ψ Ψ Ψ U where the rule of integration is given by (3.25) as before. To make the discussion more explicit and without loss of generality, we focus on the contribution: (1) (1) (1) WYuk,123 = Ψ12 Ψ23 Ψ31 U to the above Yukawa coupling. It is clear that such a term can be discussed independently from other possible terms. We will hence drop the subscript ‘123’ in WYuk,123. To evaluate this overlap integral, we follow the same general strategy outlined in section 3.3.2 and Appendix A. To be explicit, we shall take the patch U to be the bi-disk:

U = Dx × Dy = {x : |x|≤R}×{y : |y|≤R}, with the coordinates chosen such that the classical matter curves are given (0) (0) (0) (0) by φ1 − φ2 = −x =0 and φ2 − φ3 = y = 0. From here we immediately (0) (0) see that automatically there is a third classical matter curve φ3 − φ1 = x − y = 0 passing through the same intersection point. In other words, to study the Yukawa coupling associated to a specific intersection point, we will work with a U(3) gauge group with the background ⎛ ⎞ −2x + y +Φ 0 0 1 (5.24) ϕ(0) = ⎝ 0 x + y +Φ 0 ⎠ dx ∧ dy, 3 00x − 2y +Φ YUKAWA COUPLINGS IN F-THEORY 77 where Φ = Φ(x, y) is a holomorphic function. Working on the patch U and following the analysis in section 3.3.2 and Appendix A, we obtain: (1) (1) (1) (1) WYuk = Ψ12 Ψ23 D¯ ∗ ξ31 + Ψ12 Ψ23 h31 dx ∧ dy U U = ∂ξ¯ 12 ∂ξ¯ 23 h31 dx ∧ dy U (5.25) = ξ12 ξ23 h31 dx ∧ dy , |x|=R |y|=R where we have dropped the first term in the first line because it can be written as a boundary term: (1) (1) (1) (1) Ψ12 Ψ23 D¯ ∗ ξ31 = Ψ12 Ψ23 ξ31 U ∂U which corresponds to an exponentially suppressed contribution because (1) (1) Ψ12 Ψ23 vanishes rapidly away from the origin, assuming again that the matter curves Σ12 and Σ23 intersect transversely. Finally, from the fact that the functions ξab reduce to the holomorphic functions −ζab away from the corresponding matter curves due to the localization of the wave-functions (5.14), for large R we can further rewrite the above expression as (5.26) WYuk = ζ12 ζ23 h31 dx ∧ dy , |x|=R |y|=R

(0) with ζab given by the holomorphic section hab by the relation hab = φa (0) ζab − ζab φb which holds away from the matter curve. Again ﬁxing the zero mode on the third matter curve to be given by h31 = 1, we get the following expression for the Yukawa coupling as the “quantum bilinear pairing” of the two curves h12 h23 ∧ (5.27) WYuk = (0) (0) dx dy , |x|=R |y|=R [φ , ·]12,∗ [φ , ·]23,∗ where we have introduced the notation

hab ⇔ (0) − (0) ζab = (0) hab = φa ζab ζab φb [φ , ·]ab,∗

(0) (0) for convenience. Note that on both sides of the equation, hab and φa , φb are all sections of the canonical bundle KS, and hence it is easy to see that the quantity ζab is indeed a holomorphic function away from the matter curve. This equation can be thought of as the quantized version of the residue 78 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA formula (3.44) for the undeformed theory and indeed reduces to the residue formula when the limit θ → 0istaken. To see whether the rank theorem also applies to the deformed theory, we now take a closer look at the object ζ12 which enters the quantum residue formula (5.27). Define the bi-differential operator k,k=0, 1,...as the coefficients of the θ-expansion of the operator , such that ∞ k f g = θ (f k g ), k=0 ∞ k then ζ12 = k=0 θ ζ12,k is given by the following recursive relation

h12 −h12 ζ12,0 = (0) (0) = , φ1 − φ2 x n (0) (0) φ1 k ζ12,n−k − ζ12,n−k k φ2 =0 foralln>0. k=0 At next order, the second equation gives 1 (0) (0) h12 ζ12,1 = − (φ1 + φ2 ) 1 ( ) x x 1 (0) (0) h12 (0) (0) h12 −1 (5.28) = − V1(θφ + θφ ) V2( ) − V2(θφ + θφ ) V1( ) θ x 1 2 x 1 2 x where V1,2 are the diﬀerential operator deﬁned by the Poisson bi-vector (B.2). And the same structure also holds for ζ23 of course. From the above expression and the quantum residue formula (5.27) we can immediately see a U(1) selection rule which is very reminiscent of the Froggatt-Nielsen mechanism. A necessary condition for a term in the integrand in the Yukawa quantum residue formula (5.27) to contribute is that it has total charge zero under the geometric U(1)

x → eiθx, y → eiθy.

Expanding θ(x, y) as: m1 m2 (5.29) θ(x, y)= θm1,m2 x y m1+m2≥ for some ≥ 0, it follows that each θm1,m2 can be viewed as having a charge under this U(1) rephasing symmetry. In particular, θm1,m2 has charge m1 + m2 − 2. The order of vanishing of θ can therefore have a signiﬁcant impact on the possible Yukawa textures expected. Similar considerations apply to Φ(x, y)=Tr φ(0) : (0) m1 m2 (5.30) Tr φ =Φ(x, y)=Φ0 + Φm1,m2 x y . m1+m2≥0 YUKAWA COUPLINGS IN F-THEORY 79

In this case, we ﬁnd that the terms Φm1,m2 have charge m1 + m2 − 1 relative (0) (0) to φa − φb under the geometric U(1). In particular, the only term of Φ (0) (0) which carries a negative charge relative to φa − φb is Φ0, with charge −1. From this analysis, we obtain the conclusion that, when using the mono- m n mials y , x as a basis for the holomorphic functions h12(y), h23(x) labelling the zero modes, a necessary condition for the Yukawa coupling m n mn y x ∧ WYuk = (0) (0) dx dy |x|=R |y|=R [φ , ·]12,∗ [φ , ·]23,∗ k1 k2 = Tk1,k2 θ Φ0 k1,k2∈Z+,k1≥k2 k1(2−)+k2≥m+n

k1 k2 to contain a term of the form θ Φ0 is

k1 ≥ k2,k1(2 − )+k2 ≥ m + n, where is the minimal degree of the polynomial θ in holomorphic coordinates x, y.

An example Above we have derived a geometric U(1) selection rule which naturally implements the Froggatt-Nielsen mechanism that renders a hierarchical structure in the Yukawa matrix. To illustrate it, let us take = 1, namely let us choose our Poisson bi-vector to have the expansion

θ(x, y)=θx x + θy y + higher power and similarly (0) Tr φ =Φ(x, y)dx ∧ dy =(Φ0 + higher power)dx ∧ dy. For example, in a three-generation model, the Yukawa coupling matrix can now be expressed in terms of a series expansion in θ and Φ0 of the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 1 θ θ 00θ 00 0 ⎝ 2 3⎠ ⎝ 2⎠ 2 ⎝ ⎠ (5.31) WYuk ∼ θ θ θ +Φ0 0 θ θ +Φ0 00 0 2 3 4 2 3 2 θ θ θ θ θ θ 00θ for the scaling of the leading terms in the expansion in θ, Φ0 1. 5.3.2. An Explicit Example. To illustrate the structure of the deformed Yukawa coupling, we now present a simple explicit example. In the background ⎛ ⎞ −2x + y +Φ0 00 (0) (0) 1 ⎝ ⎠ (5.32) A =0,ϕ = 0 x + y +Φ0 0 3 00x − 2y +Φ0 × dx ∧ dy, Φ0 ∈ C, 80 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA take the Poisson bi-vector to be

θˆ =2(x − y) ∂x ∧ ∂y = z∂z ∧ ∂w, where z = x − y, w = x + y, then the solution to (0) (0) (0) (0) φ1 ζ12 − ζ12 φ2 = h12,φ2 ζ23 − ζ23 φ3 = h23 is given by ∞ n n −2 −2 1 (12) ζ12 = θ ζ12,n,ζ12,0 = h12,ζ12,n = Dm ζ12,n−m n=0 z + w z + w m=1 m! ∞ − − n n 2 2 1 (23) ζ23 = θ ζ23,n,ζ23,0 = − h23,ζ23,n = − Dm ζ23,n−m n=0 z w z w m=1 m! m (12) 1 (−1) m m−1 m Dm = + w∂w − mz∂z∂w +2 z∂w 6 3 Φ0 m m + − 1+(−1) ∂w 3 m (23) (−1) 1 m m−1 m Dm = − + w∂w − mz∂z∂w − (−2) z∂w 6 3 Φ0 m m + − 1+(−1) ∂w 3 Taking the zero modes to be given by ym xn h12,m = ,h23,n = , m! n! for m, n =0, 1, 2, the leading order behavior of the Yukawa matrix we obtain by evaluating (5.27) is: ⎛ ⎞ 4 112 ⎜ 1 θ θ ⎟ ⎜ 3 9 ⎟ 1 ⎜ ⎟ ⎜−4 −262 −1073⎟ 2 WYuk = ⎜ θ θ θ⎟ (2π) ⎝ 3 9 27 ⎠ 11 2 107 3 1225 4 θ θ θ 9 ⎛ 27 162 ⎞ 1 ⎛ ⎞ ⎜ 00− θ⎟ ⎜ 6 ⎟ 00 0 ⎜ 2 2 ⎟ 2 ⎜00 0 ⎟ +Φ0 ⎜ 00 θ ⎟ +Φ0 ⎝ ⎠ , ⎜ 9 ⎟ 1 2 ⎝ ⎠ 00− θ 1 2 2 θ 0 36 6 9 θ which is indeed of the form (5.31) with order one coeﬃcients. YUKAWA COUPLINGS IN F-THEORY 81

6. Applications In this section we make contact between the formal Yukawa structures found in previous sections, and more phenomenological applications. To this end, we ﬁrst review the proposal of [14] that background ﬂuxes can distort the structure of the Yukawa couplings. We next show how quite similar expressions to those estimated in [14] appear in the non-commutative deformation of the partially twisted seven-brane theory.

6.1. Review of FLX and DER Expansions. We now brieﬂy review the proposal of [14] that background ﬂuxes can generate hierarchical corrections to the structure of the Yukawa couplings. Letting zΣi denote a local coordinate along a matter curve Σi such that zΣ1 = zΣ2 = zΣ3 = 0 denotes the interaction point, there exists a basis of wave-functions organized by the order of vanishing near the interaction point so that:

(i) i i+1 (6.1) ΨΣ ∼ (zΣ) + O((zΣ) ), where the localized matter fields exhibit an exponential falloff in directions normal to the matter curve. The leading order profile of the wave-functions exhibits a rephasing symmetry due to the local action of the internal Lorentz symmetries on the holomorphic coordinates zΣ. This overall rephasing symmetry implies that overlap integrals of the form: ijk i j k (6.2) λ ∼ (zΣ1 ) (zΣ2 ) (zΣ3 ) × Gauss will vanish unless i = j = k = 0. Here, “Gauss” is shorthand for the rephasing invariant exponential falloff of the wave-functions, with the precise form set by the behavior of the Kähler form near the intersection point. The rephasing symmetry of the holomorphic coordinates determines a selection rule for the Yukawa matrices derived from the geometry. As an example, in the up type Yukawa coupling defined by the interaction term:

ij i j (6.3) WMSSM ⊃ λu Q U Hu,

ij the 3 × 3 matrix λu has rank one when the U(1) selection rule is exactly obeyed. This would correspond to a situation with one massive generation, and two exactly massless generations. In any semi-realistic theory of ﬂa- vor, this structure must be corrected to incorporate hierarchical masses and mixing angles. Subleading corrections to the form of the Yukawa coupling correspond to violations of this rephasing symmetry. In [14], it was proposed that dis- tortions of the integrand by additional powers of the anti-holomorphic coor- dinatesz ¯Σ could induce higher order corrections to the structure of the Yukawas. The ﬁrst order correction to the form of the integrand can occur 82 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA from objects with one holomorphic and two anti-holomorphic tensor indices of the form:

(6.4) OF = Fi¯jk¯ziz¯¯jz¯k¯, so that locally F behaves as a (1, 2) form. Examples of such tensor structures are gradients of the gauge ﬁeld strength of the form ∇iF¯jk¯ and ∇¯iFjk¯,as well as the (1, 2) component of the background H-ﬂux. As we shall argue later, these contributions are actually closely linked in the context of the non-commutative deformation of the superpotential. On general grounds, the rephasing symmetry can be violated by higher order terms either of the form:

M i j k (6.5) (OF ) · (zΣ1 ) (zΣ2 ) (zΣ3 ) × Gauss or: ¯M M i j k (6.6) (∂ OF )¯z · (zΣ1 ) (zΣ2 ) (zΣ3 ) × Gauss.

The ﬁrst expansion corresponds to additional contributions in powers of the ﬂux, and was referred to as the “FLX expansion” in [14]. The second expansion corresponds to including higher order gradients of OF and was referred to as the “DER expansion”. Counting the overall U(1) charge of each integrand, the resulting structure is similar to that of the Froggatt- Nielsen mechanism [35]. The resulting form of the two expansions leads to Yukawa matrices of the form [14]: ⎛ ⎞ ⎛ ⎞ 1 ε2 ε4 1 ε2 ε3 ⎝ 2 4 6 ⎠ ⎝ 2 3 4 ⎠ (6.7) λFLX ∼ ε ε ε ,λDER ∼ ε ε ε . ε4 ε6 ε8 ε3 ε4 ε5

As estimated in [14], crude√ scaling arguments relate ε to the GUT ﬁne structure constant as ε ∼ αGUT . One of our aims in this section will be to see how similar structures appear in the non-commutative theory we have studied. To summarize, counting the number of powers of z andz ¯ in a physical basis for the wave-functions constrains the form of possible higher order corrections to the structure of the Yukawa couplings. Note, however, that counting powers of z andz ¯ certainly obscures the overall holomorphy of the superpotential. Indeed, one of the aims of this paper has been to develop a manifestly holomorphic formulation of Yukawa couplings.

6.2. Comparison with the Non-Commutative Gauge Theory. In the previous section we reviewed the argument that a generic background flux will alter the internal profile of matter field wave-functions, and thus the Yukawa interactions as well. A more precise analysis reveals, however, that background gauge field fluxes alone do not disort the structure of the YUKAWA COUPLINGS IN F-THEORY 83

Yukawas. On the other hand, we have seen in this paper that background H- fluxes do distort the structure of the Yukawas. Indeed, at the level of matching tensor indices, both H-fluxes and gradients of the gauge field strength with one holomorphic and two anti-holomorphic indices given by:

(6.8) Hi¯jk¯, ∇iF¯jk¯, ∇¯iFjk¯ are of the general form required for the FLX and DER expansions. The contributions from the H-flux and the gauge field strength are actually closely related. For example, in the simplified case of perturbative type IIB vacua, the DBI action for a single D7-brane contains the gauge field strength F in tandem with the NS B-field through the combination F + B. Indeed, in this simplified case, one of the equations of motion for the D7- brane theory is:

(6.9) F¯jk¯ + B¯jk¯ =0 for the (0, 2) component of the gauge field strength and the background B- field. Taking the gradient of this equation, we then obtain a relation between the gauge field strength and the H-flux in directions along the D7-brane:

(6.10) ∇iF¯jk¯ + Hi¯jk¯ =0.

From this perspective, it is now immediate that the presence of the background H-ﬂux can simply be viewed as a deformation of the ordinary Her- mitian Yang-Mills equations by higher dimension operators. Although a full discussion is beyond the scope of this paper, it would be interesting to study in more general terms higher dimension operator contributions to the superpotential. In particular, such an analysis likely admits an interpretation in terms of Massey products.

6.3. Scaling Estimates. The structure of Yukawa matrices found in section 5 are characterized in terms of θ, the quantum expansion parameter of the non-commutative deformation and Φ0, the constant part of the trace of the (2, 0) form in a sub-block of the matrix φ(0). We now estimate the scaling behavior of these parameters. In the next subsection we shall use these estimates to compare the hierarchical expansions found in previous sections with those of the FLX and DER expansions. To estimate the scaling behavior of θ, we first recall the primary mass scales which enter into F-theory GUTs. First, there is the characteristic scale of the complex surface S, which is given by a mass scale MGUT such that 4 MGUT ∼ 1/Vol(S). In addition, there is a characteristic mass scale associated with stringy modes, which we shall refer to as M∗. For example, the units of 2 gauge field flux naturally scale as F ∼ MGUT , while the three-form H-flux which is not localized on a particular seven-brane more naturally scales as 3 H ∼ M∗ . The fine structure constant of the GUT is given in terms of MGUT 84 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

and M∗ as [6]:

4 MGUT ∼ ∼ 1 (6.11) 4 αGUT . M∗ 25 Using these two basic mass scales, we now proceed to estimate the scaling of θ and Φ0. First consider the quantum expansion parameter θ. In units where θ is set to 1, an estimate for the size of the non-commutative deformation can be deduced by computing the overall scaling of the Poisson bi-vector θ. Since a pointlike probe D3-brane can detect the presence of this parameter, it follows that the overall scaling of θ is fully determined by the index structure of θ.In particular, it follows that although θ will typically be a non-trivial function of the internal coordinates of the complex surface, this further functional dependence can effectively be viewed as a dimensionless contribution. Thus, it is enough to evaluate the overall size of the parameter θ based on its index structure. As shown in section 4, the parameter θ is defined by its relation to the background H-flux through the equation:

jk jk jj kk (6.12) ∇iθ = Hi = g g Hijk.

Now, although the H-ﬂux is naturally quantized in units suited to M∗, the metric on the complex surface will instead scale as an appropriate power of MGUT . Moreover, because the volume of the complex surface is given as: √ (6.13) Vol(S)= g, S

2 it follows that each inverse power of the metric contributes a factor of MGUT to the overall scaling of θ. Finally, because the gradient ∇i is again measured in closed string units, we conclude that the only MGUT scaling is, in dimensionless units: 4 jk ∼ MGUT ∼ ∼ 2 (6.14) θ 4 αGUT ε , M∗ where in the final estimate we have introduced an expansion parameter ε defined as in [14]. Next consider the scaling of the parameter Φ0, the constant part of the trace of a sub-block of the (2, 0) form, or more abstractly, by its trace in the universal enveloping algebra. It might at first appear that Φ0 should scale as one power of MGUT , associated with the scaling of a single field. Writing 2 ϕ = φdx ∧ dy on a patch, since ||φ|| scales inversely with the volume, Φ0 2 scales as MGUT . This is also in accord with the fact that ϕ has two indices along the GUT seven-brane, scaling in the same way as the gauge field flux YUKAWA COUPLINGS IN F-THEORY 85

2 in the internal directions of the seven-brane theory, namely, as MGUT .Asa unitless expansion parameter, Φ0 therefore scales as: 2 √ ∼ MGUT ∼ ∼ (6.15) Φ0 2 αGUT ε. M∗ Summarizing the analysis just presented, the expansion parameters of the non-commutative gauge theory are: 2 (6.16) θ ∼ ε , Φ0 ∼ ε.

6.4. Matching to the Non-Commutative Gauge Theory. We now compare the hierarchical Yukawa matrices found in section 5 with those of the FLX and DER expansions. We find that the most phenomenologically attractive Yukawa matrices originate from configurations where the holomorphic bi-vector θ vanishes to first order at the interaction point. In this specific case, the types of Yukawa structures for a g generation model are of the form: ⎛ ⎞ ⎛ ⎞ 2 3 2 1 θ θ θ ... 00θ θ ... ⎜ 2 3 4 ⎟ ⎜ 2 3 ⎟ ⎜ θ θ θ θ ... ⎟ ⎜ 0 θ θ θ ... ⎟ ⎜ 2 3 4 5 ⎟ ⎜ 2 3 4 ⎟ λNC ∼ ⎜ θ θ θ θ ... ⎟ +Φ0 ⎜ θ θ θ θ ... ⎟ ⎝ 3 4 5 6 ⎠ ⎝ 2 3 4 5 ⎠ θ θ θ θ ... θ θ θ θ ...... ⎛ ⎞ 0000... ⎜ 2 ⎟ ⎜ 000θ ... ⎟ 2 ⎜ 2 3 ⎟ (6.17) +Φ0 ⎜ 00θ θ ... ⎟ + ··· . ⎝ 2 3 4 ⎠ 0 θ θ θ ......

By inspection of the above matrices, we conclude that the exact form of the FLX expansion is reproduced by the Φ0 independent contribution to 2 equation (6.17). Indeed, setting θ ∼ ε yields: ⎛ ⎞ 1 ε2 ε4 ε6 ... ⎜ 2 4 6 8 ⎟ ⎜ ε ε ε ε ... ⎟ ≡ | ∼ ∼ ⎜ 4 6 8 10 ⎟ (6.18) λ λNC Φ0=0 λFLX ⎜ ε ε ε ε ... ⎟ . ⎝ ε6 ε8 ε10 ε12 ... ⎠ ......

The form of the FLX expansion requires that terms independent of Φ0 dominate over those which have some Φ0 dependence. Indeed, as argued in [14], the FLX expansion is expected to dominate in Yukawas involving matter fields with large couplings to background fluxes. The DER expansion is expected to dominate in those Yukawas involving matter fields with smaller hypercharges. In the present context, it is natural to expect that this milder type of coupling is captured through the power 86 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

series in Φ0. Working to leading order in this expansion, the form of λNC given by equation (6.17) is:

(6.19)⎛ ⎞ ⎛ ⎞ 2 2 3 5 1 θ Φ0 · θ Φ0 · θ ... 1 ε ε ε ... ⎜ 2 2 2 ⎟ ⎜ 2 3 5 6 ⎟ ⎜ θ Φ0 · θ Φ0 · θ Φ0 · θ ... ⎟ ⎜ ε ε ε ε ... ⎟ ⎜ 2 2 2 2 3 ⎟ ⎜ 3 5 6 8 ⎟ λΦ ∼ ⎜ Φ0 · θ Φ0 · θ Φ0 · θ Φ0 · θ ... ⎟ ∼ ⎜ ε ε ε ε ... ⎟ . ⎝ 2 2 2 2 3 3 3 ⎠ ⎝ 5 6 8 9 ⎠ Φ0 · θ Φ0 · θ Φ0 · θ Φ0 · θ ... ε ε ε ε ......

Although similar, the precise structure of the expansion λΦ deviates from the DER expansion by a small amount. Indeed, restricting to the case of a three generation model, we see that by comparison, the three entries in the lower righthand corner of the 3 × 3 matrix diﬀers by a factor of ε: ⎛ ⎞ ⎛ ⎞ 1 ε2 ε3 1 ε2 ε3 ⎝ 2 3 5 ⎠ ⎝ 2 3 4 ⎠ (6.20) λΦ ∼ ε ε ε , λDER ∼ ε ε ε . ε3 ε5 ε6 ε3 ε4 ε5

Let us stress that there could in principle be additional contributions to the Yukawa matrices which might reproduce exactly the form of the DER expansion. As an example, one might consider more elaborate contributions from higher order terms in the series expansions: m1 m2 θ(x, y)= θm1,m2 x y , m1+m2>0 and: m1 m2 Φ(x, y)=Φ0 + Φm1,m2 x y . m1+m2>0 Our aim here has been to explore the most generic situation consistent with a θ which vanishes to first order. It would be interesting to see how much “fine-tuning” of higher order corrections is necessary to reproduce precisely the form of the DER expansion just from the non-commutative deformation already studied. Indeed, given the quite general arguments provided in [14], it is reasonable to expect that additional corrections to the structure of the Yukawa will likely reproduce precisely the structure of the DER expansion. For example, there will typically be higher order corrections to the relation ∇θ = H found for weak H-fluxes. In the effective field theory, such corrections constitute the presence of additional higher dimension operators, which might lead to a milder hierarchy, in line with the estimates of [14]. There is also a broader class of non-commutative deformations which one might consider. Indeed, besides the non-commutative deformation in the local coordinates x and y, one might also consider a non-commutative YUKAWA COUPLINGS IN F-THEORY 87 deformation of the form: xy yz zx [x, y]=Sθ , [y, z]=⊥θ , [z,x]=⊥θ where z denotes a local coordinate parameterizing directions normal to the seven-brane, and we have distinguished the expansion parameters S and ⊥ for Poisson bi-vectors with respectively both legs, or one leg along the seven-brane wrapping the complex surface S. Although it is beyond the scope of the present paper, we now briefly sketch in quite heuristic terms how a computation of Yukawas in this setting might go. The effects of this deformation are most readily analyzed as a non-commutative deformation of holomorphic Chern-Simons theory. In this language, the configuration of seven-branes and matter curves are specified by expanding about a background field configuration which satisfies the BPS equations of motion. Note that just as the location of the matter curves in the seven-brane theory is no longer sharply defined once = 0, so too will the location of the seven- branes become fuzzy. This is in accord with the co-isotropic condition for seven-branes studied in [37]. Even so, just as for the seven-brane theory we have analyzed, the theory still contains localized zero modes, and so an appropriate cohomology theory can be developed in this case as well. The holomorphic dependence of the zero modes will now contain dependence on the local coordinates of the seven-brane, x and y, as well as z, the coordinate normal to the seven-brane. It is therefore quite tempting to simply take n n the formal structure of the Yukawa matrix λ, and replace θ by either S n−1 or S · ⊥ . Assuming that the number of indices along the GUT seven- 2 brane dictates the overall scaling of the Poisson bi-vector, setting S ∼ ε and ⊥ ∼ ε would now appear to provide an exact match respectively to the FLX and DER expansions. It would be interesting to perform a more precise computation and estimate of the resulting Yukawas, along similar lines to what has been developed in this paper.

6.5. Masses and Mixing Angles with λ and λΦ. Restricting now to the Yukawa structures λ and λΦ found in this paper, we now ask whether these structures reproduce the main features of the estimates for the mixing angles and masses given in [14]. To compare with these estimates, we switch to a basis of states ordered from lightest to heaviest in mass, so that for example, the (3, 3) entry of 5 λDER is of order one, while the (1, 1) entry is of order ε .In[14], the up-type quark Yukawa was identiﬁed with the FLX expansion, while the down-type Yukawa was matched to the DER expansion. Identifying the up- type quark Yukawa matrix with the λ matrix and that of the down-type Yukawa matrix by λΦ, the three generation CKM matrix is of the form: ⎛ ⎞ 1 εε3 ⎝ 2 ⎠ (6.21) VCKM ∼ ε 1 ε . ε3 ε2 1 88 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

As in [14], it is interesting to estimate the form of the CKM matrix in the case of a g-generation model for g ≥ 3 in which the three generations of the Standard Model are the lightest. We ﬁnd that the CKM matrix of the Stan- dard Model alternates between the structure for odd and even generations: ⎛ ⎞ ⎛ ⎞ 1 εε3 1 ε2 ε3 g odd ⎝ 2 ⎠ g even ⎝ 2 ⎠ (6.22) VCKM ∼ ε 1 ε , VCKM ∼ ε 1 ε . ε3 ε2 1 ε3 ε 1

In other words, the expected hierarchy in a three generation model is in better agreement with observation than what is expected in the four generation model. Next consider the masses of associated with these Yukawa structures. Since the up-type quarks and charged leptons both descend from the FLX expansion, which is reproduced by λ, the estimates for these masses are the same as in [14]. On the other hand, the masses from λΦ are slightly diﬀerent from those of the DER expansion:

6 3 (6.23) NC : md : ms : mb ∼ εd : εd :1 5 3 (6.24) DER : md : ms : mb ∼ εd : εd :1.

Fitting to the masses of the down type quarks, this leads to a slightly higher estimate for the value of εd when compared to what is expected from the DER√ expansion. Nevertheless, the resulting value of εd is still on the order of αGUT ∼ 0.2.

7. Conclusions Yukawa couplings provide an important link between the physics of the Standard Model and that of possible ultraviolet completions. In this paper, we have seen that computations of Yukawa couplings in the context of F-theory with background gauge field and H-fluxes activated admit a rather interesting local geometric formulation near the intersection point of seven-branes. As we have shown in this paper, Yukawa couplings map to a residue integral computation defined in a neighborhood containing the Yukawa enhancement point. Although gauge field fluxes alone do not alter the structure of the Yukawas, background H-fluxes do deform the structure of the Yukawa couplings. Moreover, when H-fluxes are turned on, there is a natural formulation of the effective superpotential in terms of a non- commutative space with non-commutativity fixed by the strength of the H-flux. We have also found evidence that background fluxes can generate flavor hierarchies in F-theory GUTs of the type proposed in [14]provided we identify the local profile of the three-form field with the three-form flux H. There are a number of issues which need further study. On the theoretical side, it would be important to better understand the non-commutative YUKAWA COUPLINGS IN F-THEORY 89 theory for E-type groups. Even though in principle this can be done along the lines suggested in [60] by allowing all fields to take values in the universal envelopping algebra, from the viewpoint of F-theory it is clear that there are some subtleties to be resolved. For example, in general F-theory compactifications, H and consequently θ as well will undergo SL(2, Z) monodromies near such enhancement points. It would be important to better understand this feature, especially in view of the fact that F-theory GUTs demand E-type enhancement points. In more realistic settings, the configuration of seven-branes will contain monodromies. It is clearly important to study the effects of such monodromies on the form of the superpotential, and the interplay between such geometric ingredients and fluxes. For example, in order to obtain the requi- site flavor hierarchy from H-flux, we assumed that θ should vanish to first order at the Yukawa enhancement point. It would be interesting to understand how this may be related to seven-brane monodromies. The study of non-commutative deformations of topological strings related to the extended moduli space is in its infancy. We have already uncovered a very rich structure in the disk amplitudes for a deformation in this generalized moduli space. It would be very interesting to study this issue more systematically, as well as in a more general setup. Along these lines, it would be interesting to study the structure of Yukawa couplings expected directly in terms of the non-commutative deformation of holomorphic Chern-Simons theory. Finally, from a phenomenological point of view, the contribution from the non-commutative deformation of the F-terms can be viewed as a specific class of higher dimension operators. It would be interesting to determine the most general class of such higher dimension operators, and the resulting flavor hierarchies expected.

Acknowledgements We thank A. Tavanfar for discussions and collaboration at an early stage of this project. We also thank F. Denef, O. Lunin, A. Neitzke, V. Pestun, M. Rocek, A. Tomasiello and M. Wijnholt for helpful discussions. MC, JJH and CV would also like to thank the Seventh Simons Workshop in Mathematics and Physics for hospitality while some of this work was performed. The work of MC is supported by the Netherlands Science Organisation (NWO). The work of JJH and CV was supported in part by NSF grant PHY-0244821. The work of JJH is also supported by NSF grant PHY-0503584.

Appendix A. Calculation of the Commutative Residue In this Appendix we compute the Yukawa coupling in the commutative theory between localized zero modes of the form (3.32) and show that the 90 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA answer is given by the residue formula (3.43). Starting from (1) (1) (1) WYuk = Tr(Ψ ∧ Ψ ∧ Ψ ) U (1) − ∧ ∧ ¯ φα3 hα3 ∧ ∧ ∧ (A.1) = Ψα1 Ψα2 D (0) + hα3 Ψα1 Ψα2 dx dy, U α3(φ ) U we will ﬁrst show that we can discard the ﬁrst term. Using the Leibniz rule (3.27) and the fact that D¯ = ∂¯ on a gauge singlet to write it as a boundary term (1) − ∧ ∧ φα3 hα3 Ψα1 Ψα2 (0) . ∂U α3(φ )

From the fact that Ψα1 and Ψα2 are localized on two distinct, transversely intersecting matter curves which only intersect at the point our patch U is chosen to enclose and in particular do not intersect at the boundary of the patch, we conclude that this surface term does not contribute to the Yukawa integral. This statement can be made more precise in the following way. As we argued in section 3.2.2, the superpotential and in particular the Yukawa coupling should not depend on which gauge orbit, or equivalently which representative in the cohomology, one chooses for the wave-function. So let us now choose the zero modes

(1) (1) φα − hα (1) (A.2) Ψα Xα = D¯ Xα +ˆκ ∧ hαXα,φα |α(φ(0))=0 = hα, α(φ(0))

(1) with the function φα localized in a tube of radius ρα surrounding the matter curve Σα:

(1) (0) (A.3) φα =0 when|α(φ )| >ρα.

Let K be any convex compact set in C2 with smooth boundary containing the triple intersection point and let K(R)beK rescaled by R.Takeour patch to be U = K(R ρα). Then from the fact that Tα1 ∩ Tα2 ∩ ∂U = ∅, where Tα is the tube centering around the matter curve Σα with radius ρα, we see that this boundary term indeed vanishes. Let us now compute the remaining term

WYuk = hα3 Ψα1 ∧ Ψα2 ∧ dx ∧ dy U (1) − (1) − ¯ φα1 hα1 ∧ ¯ φα2 hα2 ∧ (A.4) = hα3 ∂A(0) (0) ∂A(0) (0) dx dy, U α1(φ ) α2(φ ) YUKAWA COUPLINGS IN F-THEORY 91

Since hα is holomorphic, we can also write this term as a boundary term (1) − (1) − φα1 hα1 ∧ ¯ φα2 hα2 ∧ ∧ hα3 (0) ∂A(0) (0) dx dy ∂U α1(φ ) α2(φ ) (1) − (1) − φα1 hα1 ∧ ¯ φα2 hα2 ∧ = hα3 (0) ∂A(0) (0) dx dy 1 2 ∂U∩Tα2 α (φ ) α (φ ) (1) − − hα1 hα3 ∧ ¯ φα2 hα2 ∧ (A.5) = (0) ∂A(0) (0) dx dy, 1 2 ∂U∩Tα2 α (φ ) α (φ )

(1) (1) where we have again used the support property of φα1 and φα2 . (0) From the holomorphicity of hα and φ we can now perform another integration by parts and write hα1 hα2 hα3 ∧ (A.6) WYuk = (0) (0) dx dy 1 2 ∂U∩∂Tα2 α (φ )α (φ )

This is a residue integral, and in particular ∂U ∩ ∂Tα2 is diﬀeomorphic to a product of two circles surrounding the intersection point. Hence the ﬁnal answer can be written as equation (3.43).

Appendix B. The and Product The precise definition of is given as a natural extension of the product for functions to the case of differential forms. Both operations are defined in terms of an expansion in successive powers of θ. When the Poisson bi- vector is constant, the -product between two functions is simply the Moyal product (B.1) ∞ n n (θθ) θ θ∂x∧∂y ⊗ − k n−k k k n−k fg= e f g := − ( 1) ∂x ∂y f ∂x∂y g . n=0 k=0 k!(n k)!

The form of the product is more involved when the Poisson bi-vector is not constant. To illustrate this point, now consider the form of in a neighborhood U of the classical Yukawa point, which we take to lie at x = y = 0, in local coordinates x and y of U.Whenθ does not vanish at the origin, the Darboux theorem ensures we can ﬁnd local coordinates X, Y such that θˆ = ∂X ∧ ∂Y . The situation is more complicated when θ does vanish at the origin. Nevertheless we still have the following result: in the neighborhood U of the origin, denoting D ≡ (θ) as the divisor of vanishing θ, then on U \ D there exist two holomorphic vector ﬁelds V1, V2 such that

(B.2) θˆ = θ(x, y)∂x ∧ ∂y = V1 ∧ V2, [V1,V2]=0, 92 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA given that either (1)θ(0, 0) = 0, or (2) the divisor D has support on a smooth curve passing through the origin, or (3) the divisor D has support on two smooth component curves meeting transversely at the origin. The simplest way to convince oneself of the above statement is to observe that the above three classes of situations correspond respectively to the following three classes of Poisson bi-vectors (1) V1 = ∂x,V2 = ∂y,(2)V1 = x∂x,V2 = ∂y,(3) V1 = x∂x,V2 = y∂y. The -product, in this case, is the simple generalization of the Moyal product (B.1):

∞ n − kn θV1∧V2 ( 1) θ n−k k k n−k (B.3) fg= e = − V1 V2 f V1 V2 g . n=0 k=0 k!(n k)!

Recall that a theorem by Kontsevich [58] states that the requirement for the -product to be associative and having the leading behavior

2 fg= fg + θ θ (∂xf∂yg − ∂yf∂xg)+O(θ) determines the -product as a formal series in θ uniquely for a given Poisson bi-vector θ. And hence the -product (B.3) we defined above is in fact unique. We next define the -product, the non-commutative deformation of the ∧-product, acting on forms. Since we are concerned with non-commutativity in holomorphic coordinates only, the -products between a (n1,m1)- and a(n2,m2)-form take the same form for all m1,m2 =0, 1,.... The natural generalization of the non-commutative product from the space of functions (p,q) to that of all forms ⊕(p,q)Λ (U) on the neighborhood U is to “covariantize” the product by writing forms in the basis 1,2 which are meromorphic closed one-forms dual to the vectors V1,V2 defined above, and then taking the -product between the coefficients. This basis for one-forms and vectors can be thought of as the analogue of the “vielbein” basis but now defined by the Poisson structure. To discuss the deformation of the field theory with superpotential given by (3.2), we need to define the -products among (0, ∗)- and (2, ∗)-forms. For (0, ∗)-forms f and g,wehave:

(B.4) f g = fg.

Next consider the product between a (2, ∗) form, Ψ = ψdx∧dy and a (0, ∗) form f. To follow the covariantization procedure described above, note that when Ψ is for example a (2, 0) form, ψ is a section of the canonical bundle 2 ∗ KS =(Λ TS) . More generally, note that although ψ and θ both transform non-trivially under a change of coordinates, their product θψ can be treated YUKAWA COUPLINGS IN F-THEORY 93 asa(0, ∗)-form. The product is therefore given as: dx ∧ dy f (ψdx∧ dy)=f(θψ) θ dx ∧ dy (ψdx∧ dy) f =(θψ) f . θ We shall also sometimes write: 1 f ψ =(f(θψ)) θ 1 ψ f =((θψ) f) . θ The -product between diﬀerential forms satisﬁes the following important properties: 1. Associativity 2. The ∂¯-chain rule

∂¯(F F )=∂F¯ F +(−1)degF F ∂F¯ for (p, q)-forms F, F

3. Commutativity Upon Integration (B.5) Tr F F = Tr F F , S S for forms F and F which satisfy an appropriate notion of localization. These properties are in turn crucial in order for the non-commutative deformation of the seven-brane superpotential to be well-deﬁned.

Appendix C. Proof of Commutativity Upon Integration In this Appendix we prove the claim mentioned in section 5, line (5.3) and in Appendix B, line (B.5), namely that the -product we deﬁned in (B.5) satisﬁes the property: (C.1) Tr F F = Tr F F , S S for forms F and F which satisfy an appropriate notion of localization. Note that for our present purposes, it is enough to show that for a (0, 2)-form f and a (2, 0)-form Ψ = ψdx∧ dy which satisfy the localization condition:

m1 m2 n1 n2 ∂x ∂y f∂x ∂y ψ → 0 for all m1,m2,n1,n2 > 0 at the boundary of the patch U, that the following property holds: (C.2) f ψdx∧ dy = ψdx∧ dy f. U U 94 S. CECOTTI, M. C. N. CHENG, J. J. HECKMAN, AND C. VAFA

To prove this, ﬁrst deﬁne

Vˆi = Vi + Vi log θ, i =1, and then write f ψdx∧ dy − ψdx∧ dy f ∞ 1+2n − k − 1+2n ( 1) k 1+2n−k ˆ 1+2n−k ˆ k ∧ = 2 θ − V1 V2 f V1 V2 ψ dx dy n=0 k=0 k!(1 + 2n k)! ∞ 1+2n 2n − θ − k (2n)! 2n−k k ˆ ˆ k ˆ 2n−k = 2 ( 1) − V2(V1 V2 f)V1(V1 V2 ψ) n=0 (2n +1)! k!(2n k)! k=0 2n−k k k 2n−k − V1(V1 V2 f)Vˆ2(Vˆ1 Vˆ2 ψ) dx ∧ dy.

Note that each summand in the above expression can be written as

(V2f˜Vˆ1ψ˜ − V1f˜Vˆ2ψ˜) dx ∧ dy for some f˜ and ψ˜. It is therefore enough to show that each of these terms corresponds to an exact diﬀerential form. To this end, note that:

(V2f˜Vˆ1ψ˜ − V1f˜Vˆ2ψ˜) dx ∧ dy ψ˜ = − (V1fV˜ 2ψ˜ − V2fV˜ 1ψ˜)+ (V1fV˜ 2θ − V2fV˜ 1θ) dx ∧ dy θ = − ∂xf∂˜ y(θψ˜) − ∂yf∂˜ x(θg˜) dx ∧ dy = −df˜∧ d(ψθ˜ ).

This proves our claim.

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Scuola Internazionale Superiore di Studi Avanzati via Beirut 2-4 I-34100 Trieste, Italy E-mail address: [email protected] Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA E-mail address: [email protected] E-mail address: [email protected] School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA E-mail address: [email protected]

Surveys in Diﬀerential Geometry XV

Spin Structures and Superstrings

Jacques Distler, Daniel S. Freed, and Gregory W. Moore

Abstract. In superstring theory spin structures are present on both the 2-dimensional worldsheet and 10-dimensional spacetime. We present a new proposal for the B-ﬁeld in superstring theory and demonstrate its interaction with worldsheet spin structures. Our formulation generalizes to orientifolds, where various twistings appear. A special case of the orientifold worldsheet B-ﬁeld amplitude is a KO-theoretic construction of the Z/8Z-valued Kervaire invariant on pin− surfaces.

The Type II superstring in the NSR formulation is a theory of maps from a closed surface Σ—the worldsheet—to a 10-manifold X—spacetime. The spin structures of the title are present on both the worldsheet and the spacetime. Their roles have been explored in many works; a sampling of references includes [GSO1,GSO2,SS1,SS2,R,SW,DH,AgMV, AgGMV, AW]. In this paper we identify several new phenomena which are intimately related to a new Dirac quantization condition for the B-field (Proposal 1.4). For example, in our approach the B-field amplitude depends on the worldsheet spin structure. In particular, the distinction between Types IIB and IIA is encoded in the B-field and the worldsheet B-field amplitude includes the usual signs in the sum over spin structures. In another direction we answer the question: How does the spacetime spin structure impact the worldsheet theory in the lagrangian formulation? It turns up in

To Isadore Singer on the occasion of his 85th birthday Date: July 17, 2010 Report #: UTTG-07-10. The work of J.D. is supported by the National Science Foundation under grant PHY- 0455649 and a grant from the US-Israel Binational Science Foundation. The work of D.S.F. is supported by the National Science Foundation under grant DMS-0603964. The work of G.W.M. is supported by the DOE under grant DE-FG02-96ER40949. We also thank the Aspen Center for Physics for providing a stimulating environment for many discussions related to this paper.

c 2011 International Press

99 100 J. DISTLER, D. S. FREED, AND G. W. MOORE the definition of the partition function of worldsheet fermions, i.e., in computing the pfaffian of the Dirac operator on Σ. For orientifolds of the Type II superstring, including the Type I superstring, there are several new features. For example, we define precisely the twisted notions of spin structure needed onΣandonX. We also consider the worldsheet B-field amplitude and the partition function of worldsheet fermions. It turns out that each is anomalous and that these anomalies cancel. That anomaly cancellation is the subject of a future paper [DFM2]; here we are content to motivate that work and consider some special cases. Evidently, these spin structure considerations are closely tied to the B-field βˇ,withwhichwebeginin§1. Quite generally, Dirac quantization of charges and fluxes is implemented by generalized cohomology theories. For the oriented bosonic string the B-field has a flux quantized by H3(X; Z). We locate the superstring B-field quantization condition in a generalized cohomology theory R which is a truncation of connective KO-theory. Then the B-field is modeled in the differential cohomology group Rˇ−1(X)using the general development of differential cohomology in [HS]. In §2wetake up the integral of φ∗βˇ on the worldsheet Σ for maps φ:Σ→ X. The presence of KO-theory suggests the dependence on worldsheet spin structures. We show how the standard Z/2Z-valued quadratic function on spin structures [A1] is embedded in the B-field amplitude, leading to the distinction between Types IIB and IIA. A generalization of the Scherk-Schwarz construction [SS1, SS2]isalsopartofourB-field amplitude. Orbifolds (in the sense of string theory) and orientifolds are introduced in §3. To accommodate the former we allow X to be an orbifold (in the sense of differential geometry); the orientifold is encoded in a double cover π : Xw → X of orbifolds. The B-field βˇ is now quantized by the R-cohomology of the Borel construction applied to X, with local coefficients determined by the double cover π (Proposal 3.7). The integral of φ∗βˇ is taken up in §4. We posit a spin structure on the orientation double coverπ ˆ : Σ → Σ of the worldsheet. In case this refines and is refined to a pin− structure the integral of φ∗βˇ may be easily defined. For a certain universal B-field this yields a KO-theoretic construction of the Z/8Z-valued Kervaire invariant on pin− surfaces [Bro], [KT, §3]. For a general (non-pin−) spin structure on Σ the B-field amplitude is anomalous (4.13); its definition is postponed to [DFM2]. In §5weproveaformula for the pfaffian line of the Dirac operator in a related one-dimensional supersymmetric quantum mechanical model, the one which computes the index of the Dirac operator. That formula is a categorified index theorem in low dimensions. We see explicitly how the spin structure on spacetime enters. This result is included here as motivation for [DFM2], where we take up the analogous problem on the two-dimensional worldsheet. The precise nature of the spin structure on spacetime for orientifolds is the subject of §6. It is a twisted version of the usual notion of spin structure, where the twisting depends on the orientifold double cover π : Xw → X as well as the B-field βˇ. SPIN STRUCTURES AND SUPERSTRINGS 101

The telegraphic précis [DFM1] outlines many aspects of orientifold theory. This is the first of several papers which expatiate on this résumé. These papers provide motivation, give precise definitions, develop some background mathematics, state and prove the main theorems, and give applications to physics. The geometry of the B-field is further developed in subsequent papers. In [DFM2] we build a geometric model of Rˇ−1(X). The geometric model is used in [DFM3]totwistK-theory and its cousins, thus defining the home of the Ramond-Ramond field on X.TheB-field is a twisting of K-theory. This relation to twistings of K-theory is one of the main motiva- tions for the choice of Dirac quantization condition for the B-field. The ideas here touch on many mathematical works of Isadore Singer: among others his recent paper [HS] on quadratic forms and generalized differential cohomology, his many contributions to index theory and the geometry of Dirac operators, and even his use of frame bundles to express geometric structures on manifolds [S]. Beyond that his prescient recognition 30 years ago of the role that theoretical high energy physics would play in late 20th century and early 21st century mathematics has had enormous influence on the entire field. We thank Andrew Blumberg, Mike Hopkins, Isadore Singer, and Edward Witten for helpful discussions.

1. B-fields and generalized differential cohomology In classical physics an abelian gauge field is determined by its field strength F , a closed differential form on spacetime X. The archetype is the Maxwell electromagnetic field, a closed 2-form in 4 spacetime dimensions.1 Abelian gauge theories include an electric current j, which in Maxwell theory is a closed 3-form with compact support on spacelike hypersurfaces. The de Rham cohomology class of F is called the classical flux 2 and the de Rham cohomology class of j the classical charge. (The latter is taken with compact supports in spatial directions.) In quantum theories Dirac’s quantization principle constrains these classical fluxes and charges to full lattices inside the appropriate de Rham cohomology groups. For example, the quantum Maxwell electromagnetic flux is constrained to the image 2 2 ∼ 2 of H (X; Z)inH (X; R) = HdR(X). It is natural to refine the flux to the abelian group H2(X; Z). Indeed, in the quantum theory the Maxwell electromagnetic field is modeled as a connection on a principal circle bundle P → X, and the flux is the topological equivalence class of P . The electric charge is then refined to H3(X; Z) (with appropriate supports), and there is a magnetic charge in the quantum theory as well. This leads to the notion that for any abelian gauge field, charges and fluxes lie in abelian groups which are cohomology groups of spacetime. It is a relatively recent discovery

1 2 ∼ The word ‘gauge’ in ‘classical gauge theory’ applies when we identify Ω (X)exact = 1 1 Ω (X)/Ω (X)closed. 2 Our usage of ‘flux’ is not entirely standard. 102 J. DISTLER, D. S. FREED, AND G. W. MOORE that generalized cohomology groups may occur. Spacetime anomaly cancellation [GHM, MM] led to the proposal, further elaborated in [W2], that the Ramond-Ramond charges in superstring theory are properly quantized by K-theory, at least in the large distance and weak coupling limit. Similarly, the fluxes are also quantized by K-theory [FH, MW]. In general, to quantize a classical abelian gauge field one must choose a generalized cohomology group which reproduces the appropriate de Rham cohomology vector space after tensoring over the reals. The choice of cohomology theory is an input. There are many physical considerations which motivate the choice and can be used to justify it. See [F1,Part3],[W3, OS, M] for leisurely expositions of these ideas, including some examples. In string theory, spacetime X is a smooth manifold whose dimension is 26 for the bosonic string and 10 for the superstring.3 In each case there is an abelian gauge field—the “B-field”—whose field strength is a closed 3-form H ∈ Ω3(X). Dirac’s principle applies and we must locate the quantum flux in a cohomology group. The most natural choice applies a simple degree shift to the Maxwell case. Supposition 1.1. The flux of the oriented bosonic string B-field lies in H3(X; Z). This supposition is certainly well-established [RW]. In this section we make a new proposal for the oriented superstring.

1.1. The cohomology theory R. Let ko denote connective KO- theory. One construction [Se] starts with the symmetric monoidal category of real vector spaces and applies a de-looping machine to construct an inﬁnite loop structure on its classifying space. More concretely, ko is the real version of K-theory developed in [A2] before inverting the Bott element; for any space M the abelian groups koq(M) vanish for q>0and −q ∼ −q 4 ko (M) = KO (M)forq ≥ 0. Deﬁne the Postnikov truncation (1.2) R := ko0 ···4. Then R is a generalized multiplicative cohomology theory, more precisely an E∞-ring spectrum. Its nonzero homotopy groups are ∼ (1.3) {π0,π1,π2,π3,π4}(R) = {Z, Z/2Z, Z/2Z, 0, Z}, a truncated Bott song. These are also the nonzero R-cohomology groups of −q a point and they occur in nonpositive degrees, as R (pt) = πq(R). If we represent the theory as a (loop) spectrum {Rp}p∈Z, so that for any space M

3 We use ‘superstring’ as a shorthand for ‘Type II superstring’ in a sigma model formulation. 4 ∞ We use the version of Postnikov truncation for connective E -ring spectra [B]. The notation ‘R’ for a multiplicative spectrum is generic, ergo uninformative, but it would be cumbersome to use ‘ko0 ···4’ instead. SPIN STRUCTURES AND SUPERSTRINGS 103

−q and q ≥ 0 we compute R (M)=[M,R−q] as the abelian group of homotopy classes of maps into the space R−q, then (1.3) are the homotopy groups of the space R0. Here is our new proposal for the B-ﬁeld in superstring theory. Let X be a smooth 10-dimensional manifold which plays the role of spacetime in the superstring.

Proposal 1.4. The ﬂux of the oriented superstring B-ﬁeld βˇ lies in R−1(X).

As a first check we note that the nonzero homotopy groups of the space R−1 are ∼ (1.5) {π0,π1,π2,π3}(R−1) = {Z/2Z, Z/2Z, 0, Z}, so after tensoring with the reals we obtain the Eilenberg-MacLane space K(R, 3) which computes real cohomology in degree 3. This is as it should be: the classical fluxes of the classical field H lie in the degree 3 de Rham cohomology of the manifold X. We explore some physical consequences of the nonzero torsion homotopy groups in §2. We record the exact sequence of abelian groups (1.6) (t,a) 0 −→ H3(M; Z) −→ R−1(M) −−−→ H0(M; Z/2Z) × H1(M; Z/2Z) −→ 0 which follows from the Postnikov tower (see (1.5)) and holds for any space M. There is not a corresponding exact sequence of cohomology theories; the k-invariant between the bottom two homotopy groups is nonzero. The quotient group in (1.6) is more properly regarded as the group of equivalence classes of Z/2Z-graded real line bundles (equivalently: Z/2Z-graded double covers) over M. The exact sequence (1.6) immediately implies −1 ∼ (1.7) R (pt) = Z/2Z, −1 ∼ and we can identify a generator with the nonzero element η ∈ ko (pt) = −1 ∼ KO (pt) = Z/2Z. There is a natural splitting of (1.6) as sets (not as abelian groups). To construct it we interpret the quotient group as the group of Z/2Z-graded real line bundles and apply the following lemma.

Lemma 1.8. Let V → M be a real vector bundle over a space M and [V ] ∈ R0(M) its equivalence class under the map ko0(M) → R0(M). Then for η[V ] ∈ R−1(M) we have (1.9) t η[V ] =rank(V )(mod2) (1.10) a η[V ] = w1(V ), where rank(V ): π0M → Z is the rank. 104 J. DISTLER, D. S. FREED, AND G. W. MOORE

Proof. The map t in (1.6) is determined on the 0-skeleton M 0 of M, and V is equivalent to rank(V )inko0(M 0). This reduces (1.9) to the assertion t(η) = 1, which is essentially the isomorphism (1.7). The map a in (1.6) is determined on the 1-skeleton, and as a(η) = 0 we can replace V by its reduced determinant line bundle Det V −1, which is equivalent to V −rank V 0 in the reduced group ko (M 1). Hence it suffices to prove (1.10) for the uni- univ ∞ −1 ∼ 0 1 versal real line bundle L → RP .Identifyko (pt) = ko (RP )and represent η by the reduced Möbiusline bundle (H − 1) → RP1.Then η[Luniv] is represented by the external tensor product (H−1)⊗Luniv → RP1× RP∞. To compute the a-component in (1.6) we restrict to the 1-skeleton RP1 ⊂ RP∞,overwhichLuniv is identified with H. Again since a(η)=0 we may pass to (H − 1) ⊗ (H − 1) → RP1 × RP1, and this represents 2 −2 η ∈ ko (pt), which is nonzero. This proves η[H − 1] is the nonzero class in −1 1 0 ∼ 1 1 0 R (RP /RP ) = H (RP /RP ; Z/2Z). Therefore a η[H − 1] , hence also a η[Luniv] , is nonzero.

1.2. Generalized differential cohomology and superstring B-fields. Semi-classical models of abelian gauge fields, which appear as background fields or as inputs to a functional integral, combine the local information of the classical field strength with the integrality of the quantum flux. As mentioned earlier the model for the Maxwell field is a circle bundle with connection: its curvature is the classical field strength and its Chern class the quantum flux. Notice that there are nontrivial connections for which both of these vanish. In other words, the combination of classical field strength and quantum flux do not determine the semi-classical gauge field. Equivalence classes of Maxwell fields, thus of circle connections, on any smooth manifold M form an infinite dimensional abelian Lie group Pic∇(M), a differential-geometric analog of the Picard group in algebraic geometry. Its group of path components is ∼ 2 (1.11) π0 Pic∇(M) = H (M; Z) the group of equivalence classes of circle bundles. The map Pic∇(M) → 1 π0 Pic∇(M) forgets the connection. The torus H (M; Z)⊗R/Z of equivalence classes of flat connections on the trivial circle bundle acts freely on the 0 identity component Pic∇(M) by tensor product, and the quotient 0 2 (1.12) Pic∇(M) → Ωexact(M) is the vector space of exact 2-forms. Other components of Pic∇(M)are also total spaces of principal H1(M; Z) ⊗ R/Z bundles; the bases are affine 2 translates of Ωexact(M) in the topological vector space of closed 2-forms, affine spaces of closed forms with a fixed de Rham cohomology class in the lattice Image H2(M; Z) → H2(M; R) . Cheeger-Simons [CS] introduced abelian Lie groups Hˇ q(M) for all inte- 2 ∼ 1 gers q which generalize Hˇ (M) = Pic∇(M). The group Hˇ (M) is the group SPIN STRUCTURES AND SUPERSTRINGS 105 of smooth maps M → T into the circle group. The group Hˇ 3(M)may be modeled as equivalence classes of T-gerbes with connection or bundle gerbes [Br, Hi, Mu]. The original definition of Hˇ q(M)isintermsof the integral over smooth singular (q − 1)-cycles, generalizing the holonomy of a T-connection around a loop. There is an alternative approach using sheaves, modeled after a construction of Deligne in algebraic geometry. Hopkins-Singer [HS] provide two important supplements. First, they define differential cohomology groups hˇ•(M) for any cohomology theory h. Sec- 5 ∼ p ond, they define spaces hˇp(M) such that π0hˇp(M) = hˇ (M). Thus points of hˇp(M) may be considered as geometric objects whose equivalence class lies in hˇp(M), just as a circle bundle with connection has an equivalence class in Pic∇(M). For the specific cohomology theory R in (1.2) fix a singu- 3 lar cocycle ι ∈ C (R−1; R) whose cohomology class is a normalized generator 3 of H (R−1; R). Then a point of degree −1 is a triple (c, h, ω), where

c: M −→ R−1 2 (1.13) h ∈ C (M; R) ω ∈ Ω3(M) and h satisfies δh = ω − c∗ι. (It follows that dω =0.)WegiveRˇp(M) the structure of an abelian Lie group for which p ∼ p (1.14) π0Rˇ (M) = R (M) and each component is a principal Rp−1(M; R/Z)-bundle over an affine space of closed differential forms. The preceding discussion leads to corollaries of Supposition 1.1 and Pro- posal 1.4:

(1.15) The oriented bosonic string B-ﬁeld βˇ is a point in Hˇ3(X).

(1.16) The oriented superstring B-ﬁeld βˇ is a point in Rˇ−1(X).

In [DFM2] we give a concrete differential-geometric model of the superstring B-field, whereas the model in terms of the spaces Rˇp(X) is more homotopy-theoretic. In any case for the purposes of this paper we only need the equivalence class [βˇ] ∈ Rˇ−1(X)ofβˇ.Weremarkthatβˇ determines −1 β ∈ R−1 whose equivalence class is [β] ∈ R (X); see (1.19) below. Then using (1.6) we define (1.17) t(βˇ),a(βˇ) ∈ H0(X; Z/2Z) × H1(X; Z/2Z).

The physical significance of (1.17) is explained in subsequent sections. We record the following exact sequences, which are specializations to the case at hand of general facts about differential cohomology and hold for any 5 In fact, they define simplicial sets. We use the moniker ‘points’ for its 0-simplices. 106 J. DISTLER, D. S. FREED, AND G. W. MOORE smooth manifold M: −(q+1) −q 4−q (1.18) 0 −→ R (M; R/Z) −→ Rˇ (M) −→ ΩZ (M) −→ 0 3−q 3−q −q −q (1.19) 0 −→ Ω (M)/ΩZ (M) −→ Rˇ (M) −→ R (M) −→ 0 4−q Here q =1, 2, 3andΩZ (M) denotes the space of closed forms with integral periods. In particular, it follows from these sequences and (1.7) that −2 ∼ −1 ∼ −1 ∼ (1.20) R (pt; R/Z) = Rˇ (pt) = R (pt) = Z/2Z. The nonzero elementη ˇ of (1.20) pulls back to any M and is a special B-field in oriented superstring theory. It may be identified with the generator of −2 ∼ −2 ∼ ko (pt; R/Z) = KO (pt; R/Z) = Z/2Z. Of course,η ˇ maps to η under the Bockstein homomorphism R−2(pt; R/Z) → R−1(pt; Z). Any real line bundle L → M determines (1.21) ηˇ[L] ∈ R−2(M; R/Z) −→ Rˇ−1(M) with t ηˇ[L] =1anda ηˇ[L] = w1(L); see Lemma 1.8. Remark 1.22. An oriented superstring spacetime X10 is endowed with a spin structure κ. (See §2.1 for a review of spin structures. The twisted notion of spin structure for superstring orientifold spacetimes is the subject of §6.) Now the B-field βˇ may be written (Lemma 1.8) as a sum of an 3 object βˇ0 in Hˇ (X)anda Z/2Z-graded double cover K → X, the latter with characteristic class t(βˇ),a(βˇ) ∈ H0(X; Z/2Z)×H1(X; Z/2Z). We can shuffle the data: Define two spin structures κ = κ, κr = κ+K on spacetime and consider the B-field to be βˇ0. The two spin structures then correlate with the two spin structures α,αr on the worldsheet; see Definition 2.4 below. This splitting into ‘left’ and ‘right’ does not generalize to orientifolds.

2. The B-field amplitude and worldsheet spin structures The spacetime for oriented bosonic string theory is a smooth 26-manifold X, and the B-field βˇ has an equivalence class in Hˇ 3(X); see Supposition 1.1. The worldsheet in oriented bosonic string theory is a closed 2-manifold Σ with orientation o and a smooth map φ:Σ→ X. (It represents the propagation of closed strings; for open strings Σ may have a boundary.) One factor in the exponentiated action of the worldsheet theory is (2.1) exp 2πi φ∗βˇ ; Σ it only depends on the equivalence class [βˇ] ∈ Hˇ 3(X) and is defined using the pushforward in ordinary differential cohomology: φ∗[βˇ] ∈ Hˇ 3(Σ) and the orientation o on Σ determines a pushforward map [HS, §3.5] 3 1 ∼ (2.2) : Hˇ (Σ) −→ Hˇ (pt) = R/Z. (Σ,o) SPIN STRUCTURES AND SUPERSTRINGS 107

In this section we deﬁne the analog for the superstring and explore some consequences.

2.1. Spin structures on superstring worldsheets. As a preliminary we quickly review spin structures. Recall that the intrinsic geometry of a smooth n-manifold M is encoded in its principal GLnR-bundle of frames n B(M) → M.ApointofB(M) is a linear isomorphism R → TmM for some m ∈ M. Choose a Riemannian metric on M, equivalently, a reduction to an On-bundle of frames BO(M) → M. The spin group

(2.3) ρ: Spinn −→ On is the double cover of the index two subgroup SOn ⊂ On.Aspin structure on M is a principal Spinn-bundle BSpin → M together with an isomorphism of the associated On-bundle with BO(M). It induces an orientation on M via the cover Spinn → SOn. The space of Riemannian metrics is contractible, so a spin structure is a topological choice and can alternatively be described in terms of a double cover of an index two subgroup of GLnR. An isomorphism of spin structures is a map BSpin →BSpin such that the induced map on On-bundles commutes with the isomorphisms to BO(M). The opposite spin structure to BSpin → M is the complement of BSpin − − in the principal Pinn -bundle associated to the inclusion Spinn → Pinn ; see [KT, Lemma 1.9] for more elaboration.6 If M admits spin structures, then the collection of spin structures forms a groupoid whose set of equivalence classes S(M) is a torsor for H0(M; Z/2Z) × H1(M; Z/2Z); the action 0 of a function δ : π0M → Z/2Z in H (M; Z/2Z) sends a spin structure to its opposite on components where δ = 1 is the nonzero element. The automorphism group of any spin structure is isomorphic to H0(M; Z/2Z); a function δ : π0M → Z/2Z acts by the central element of Spinn on components where δ = 1. The manifold M admits spin structures if and only if the Stiefel-Whitney classes w1(M),w2(M)vanish. A superstring worldsheet (Σ, o) is oriented and is equipped with a pair of 7 spin structures α,αr which induce opposite orientations at each point. Our convention is that the left spin structure α induces the chosen orientation o. Observe that a spin structure is local and can be considered as a field in the sense of physics. It is a discrete field, in fact a finite field on a compact manifold: there are only finitely many spin structures up to isomorphism. As with gauge fields, spin structures have automorphisms so there is a groupoid of fields rather than a space of fields.

Definition 2.4. The topological data on an oriented superstring worldsheet (Σ, o) is a discrete ﬁeld α which on each connected orientable open

6 ± ± 2 Recall that Pinn sits in the Clifford algebra Cliffn whose generators satisfy γ = ±1. Either sign can be used to construct the opposite spin structure. 7 ‘’ and ‘r’ stand for ‘left’ and ‘right’. 108 J. DISTLER, D. S. FREED, AND G. W. MOORE set U ⊂ Σ is a pair of spin structures which induce opposite orientations of U. In more detail, this is the indicated data on each connected orientable open set, isomorphisms of the spin structures on intersections of such open sets, and a coherence condition among the isomorphisms on triple intersections. The global orientation o is used to construct from α a global spin structure α which induces o and a spin structure αr which induces the opposite orientation −o. The global spin structures α,αr need not be opposites (as defined in the previous paragraph). For orientifold models (§3) the worldsheet does not have a global orientation, indeed may be nonorientable, but it retains the discrete field α; see Definition 4.8. In string theory one integrates over α, i.e., sums over the spin structures. Remark 2.5. We could, of course, replace α in Definition 2.4 with the pair of spin structures α,αr. Our formulation emphasizes both the local nature of the spin structure and that the same local field is present on worldsheets in orientifold superstring theories.

2.2. Superstring B-field amplitudes. Let X be a 10-manifold—a superstring spacetime—and βˇ a B-field on X as defined in (1.16). We define the oriented superstring B-field amplitude (2.1), which only depends on the equivalence class [βˇ] ∈ Rˇ−1(X). To do so we replace (2.2) with a pushforward in differential R-theory. The main point is that the cohomology theory R is Spin-oriented, that is, there is a pushforward in topological R-theory on spin manifolds. It is the Postnikov truncation of the pushforward in ko-theory defined from the spin structure (which by the Atiyah-Singer index theorem has an interpretation as an index of a Dirac operator). In fact, because we are in sufficiently low dimensions we can identify it exactly with the pushforward in ko, a fact which is useful in the proof of the Theorem 2.9 below. Combining with integration of differential forms we obtain a pushforward [HS, §4.10] −1 −3 ∼ (2.6) : Rˇ (Σ) −→ Rˇ (pt) = R/Z Σ,α in differential R-theory defined using the spin structure α on Σ. (Use (1.18) −3 ∼ to see the isomorphism Rˇ (pt) = R/Z.) This completes the definition of the B-field amplitude. In the remainder of this section we investigate special cases which go beyond the B-field amplitude for the oriented bosonic string. Let (Σ, o) be a closed oriented surface and S(Σ, o) the set of equivalence classes of spin structures which refine the given orientation. Note S(Σ, o)is a torsor for H1(Σ; Z/2Z). Let (2.7) q : S(Σ, o) −→ Z/2Z be the affine quadratic function which distinguishes even and odd spin structures. It dates back to Riemann and is the Kervaire invariant in dimension SPIN STRUCTURES AND SUPERSTRINGS 109 two; see [HS, §1] for some history. The characteristic property of the quadratic function q is

q(α + a1 + a2) − q(α + a1) − q(α + a2)+q(α)=a1 · a2, 1 (2.8) α ∈ S(Σ, o),a1,a2 ∈ H (Σ; Z/2Z), where a1 · a2 ∈ Z/2Z is the mod 2 intersection pairing.

Theorem 2.9. Let ηˇ be the nonzero universal B-ﬁeld in (1.20). For any q(α ) superstring worldsheet φ:Σ→ X,theB-ﬁeld amplitude is (−1) .

This demonstrates that the B-ﬁeld amplitude (2.1) is sensitive to the worldsheet spin structure.

α 0 −2 Proof. Let p:Σ→ pt and p∗ : ko (Σ; Z) → ko (pt; Z) the pushforward (2.6) deﬁned using the spin structure α. . Since [HS, §4.10] pushforward is compatible with the exact sequence (1.18), we use push-pull to compute the integral in (2.1) as

α ∗ α (2.10) p∗ p ηˇ =ˇηp∗ (1).

α 2 2 −2 The main theorem in [A1] states that p∗ (1) = q(α)η , where η ∈ ko (pt; ∼ 2 −4 ∼ Z) = Z/2Z is the generator. Finally,η ˇ · η ∈ ko (pt; R/Z) = R/Z is the nonzero element 1/2 of order two [FMS, Proposition B.4]. The space of fields F in the worldsheet formulation has many components, distinguished by the equivalence class of the spin structures α, the homotopy class of φ:Σ→ X,etc.Ifβˇ is any B-field on X, then Theorem 2.9 q(α ) implies that the theory with B-field βˇ +ˇη differs only by the sign (−1) ˇ ˇ on components of F with spin structure α. Note that t(β +ˇη)=t(β)+1. Recall the notation in (1.17).

Definition 2.11. An oriented superstring has Type IIB on components of X on which t(βˇ): π0X → Z/2Z vanishes and has Type IIA on components of X on which t(βˇ) is nonzero.

Remark 2.12. In the Hamiltonian formulation the distinction between Type IIA and Type IIB is a sign in the GSO projection. In the Lagrangian formulation this sign is manifested by the sign (−1)q(α) in the sum over spin structures [SW]. Also, since the set of isomorphism classes of B-ﬁelds is an abelian group there is a distinguished element, namely zero. In this sense our approach favors Type IIB as more “fundamental” than Type IIA.

Next, we consider the worldsheet amplitude for the special flat B-fields defined in (1.21). 110 J. DISTLER, D. S. FREED, AND G. W. MOORE

Theorem 2.13. Let L → X beareallinebundleandηLˇ the corresponding B-ﬁeld. For a superstring worldsheet φ:Σ→ X,theB-ﬁeld amplitude q(α +φ∗L) is (−1) .

Proof. We proceed as in the proof of Theorem 2.9. The right hand ∗ side of (2.10) is nowηp ˇ ∗[φ L]. Conclude by observing that the pushforward ∗ of [φ L] in the spin structure α is equal to the pushforward of 1 in the spin ∗ structure α + φ L. Lemma 1.8 implies that t ηˇ[L] =1anda ηˇ[L] = w1(L). We can consider instead the B-ﬁeldη ˇ(L − 1) for which t =0anda is as before; then combine Theorem 2.9 and Theorem 2.13 to compute the B-ﬁeld amplitude

∗ (2.14) (−1)q(α+φ L)−q(α) for the B-ﬁeldη ˇ(L − 1).

3. Orbifolds and orientifolds In this section we take up two important variations of the basic Type II superstring. First, suppose a finite group Γ acts on a smooth 10-manifold Y . Then there is a superstring theory—the orbifold—whose spacetime is constructed from the pair (Y,Γ) by “gauging” the symmetry group Γ. The main new feature is the inclusion of twisted sectors [DHVW]: in addition to strings φ: S1 → Y one considers for each γ ∈ Γ maps φ: R → Y such that φ(s +1)=γ · φ(s) for all s ∈ R. The analog for surfaces is a bit more complicated. Twisted sectors are labeled by a principal Γ-bundle P → Σovera superstring worldsheet Σ, and then a map to spacetime is a Γ-equivariant map φ˜: P → Y .Ifφ˜ : P → Y is another orbifold worldsheet, then a morphism φ˜ → φ˜ is an isomorphism P → P of principal Γ-bundles which inter- twines φ,˜ φ˜. The space of these fields is an infinite-dimensional groupoid. Points of Y connected by elements of Γ represent the same points of spacetime—Γ is a gauge symmetry—so it is natural to take spacetime as the quotient Y//Γ. We keep track of isotropy subgroups, due to non-identity elements γ ∈ Γandy ∈ Y with γ · y = y. Now an old construction in differential geometry [Sa], also dubbed [Th] ‘orbifold’, does exactly that. Furthermore, we can admit as spacetimes orbifolds X which are not global quotients by finite groups, thus widening the collection of models introduced in the previous paragraph. Orbifolds are presented by a particular class of groupoids8 [ALR], a special case being the presentation of a global quotient X = Y//Γ by the pair (Y,Γ). We take up groupoid presentations in subsequent papers, but here simply work directly with X. A worldsheet is then a map φ:Σ→ X of orbifolds, and the infinite-dimensional orbifold of

8 We could write ‘orbifold’=‘smooth Deligne-Mumford stack’, smooth understood as in ‘smooth manifold’. SPIN STRUCTURES AND SUPERSTRINGS 111 such maps includes twisted sectors. The reader unfamiliar with diﬀerential- geometric orbifolds may prefer to consider only global quotients Y//Γand work equivariantly on Y .

3.1. Equivariant cohomology and orbifold B-fields. There are many extensions of a given cohomology theory h to an equivariant cohomology theory for spaces Y with the action of a compact Lie group G. The simplest is the Borel construction. It attaches to (Y,G) the space YG = EG×GY , where EG is a contractible space with a free G-action. Then one defines the Borel equivariant h-cohomology as hG(Y ):=h(YG). This is not a new cohomology theory, but rather the nonequivariant theory applied to the Borel construction, a functor from G-spaces to spaces. That functor generalizes to orbifolds which are not necessarily global quotients—the functor is geometric realization—and so leads to a notion of “Borel cohomology” theories on orbifolds. But usually h has other extensions to an equivariant theory. For example, the Atiyah-Segal geometric version of equivariant K-theory, defined in terms of equivariant vector bundles, is more delicate: Borel equivariant K-theory appears as a certain completion [AS]. The Atiyah-Segal theory is extended to orbifolds, in fact to “local quotient groupoids”, in [FHT]. We recalled at the beginning of §1 that the charges and fluxes associated to an abelian gauge field in a quantum gauge theory lie in generalized cohomology groups. When we pass to theories formulated on orbifolds we must additionally specify a flavor of equivariant cohomology to locate the charges and fluxes. For example, the Ramond-Ramond field in superstring theory has charges and fluxes in K-theory. In the corresponding orbifold theory they are in Atiyah-Segal equivariant K-theory. This choice has consequences even locally, at the level of differential forms: it is consistent with extra Ramond-Ramond fields in twisted sectors. We hope to elaborate in a future paper. Here we limit consideration to B-fields on orbifolds. Let M be a 26-dimensional orbifold. We posit the following generalization of Supposition 1.1.

Supposition 3.1. For the oriented bosonic orbifold the ﬂux of the B-ﬁeld βˇ lies in the Borel cohomology H3(X; Z).

Furthermore, there is a generalization of differential cohomology to orbifolds [LU, G]. So an immediate reformulation locates the B-field itself in orbifold differential cohomology (see (1.16)). Supposition 3.1 is implicit in the literature, for example in [Sh, GSW]. The B-field amplitude (2.1) is defined as before; the integration is still over a smooth manifold, the worldsheet Σ. For the superstring case we also posit Borel cohomology for the B-field. Let X be a 10-dimensional orbifold.

Proposal 3.2. For the superstring orbifold the ﬂux of the B-ﬁeld βˇ lies in the Borel cohomology R−1(X). 112 J. DISTLER, D. S. FREED, AND G. W. MOORE

We are not aware of any general equivariant version of generalized differential cohomology, much less a version for orbifolds. In [DFM2] we develop a geometric model of Rˇ−1(X) for a local quotient groupoid X and locate the B-field there. The pullback to a worldsheet then lives in the differential R-theory as in the non-orbifold case, and the amplitude (2.1) is defined as before.

3.2. Orientifolds and B-fields. The orientifold construction applies to both the bosonic string and the superstring. In its simplest incarnation the construction involves a pair (Y,σ) of a smooth manifold Y and an involution σ : Y → Y . Fields on Y have a definite transformation law under σ.For example, the metric is invariant whereas the 3-form field strength H of the B-field is anti-invariant: σ∗H = −H. We combine the orbifold and this simple orientifold by starting with a triple (Y,Γ,υ) consisting of a finite group Γ, a smooth Γ-manifold Y , and a surjective homomorphism υ :Γ→ Z/2Z. Then fields on Y transform under Γ: e.g., the 3-form field strength of the B-field satisfies

(3.3) γ∗H =(−1)υ(γ)H, γ ∈ Γ.

As before Γ acts as a gauge symmetry and the physical points of spacetime lie in the quotient. Therefore, we arrive at a more general model in a geometric formulation.

Definition 3.4. The spacetime of an orientifold string model is an orbifold X equipped with a double cover of orbifolds π : Xw → X.

The equivalence class w ∈ H1(X; Z/2Z) of the double cover lies in the Borel cohomology of X. For the triple (Y,Γ,υ) the double cover is π : Y// ker υ → 1 Y//Γ with characteristic class in HΓ(Y ; Z/2Z). Deﬁnition 3.4 applies to both the bosonic string and the superstring. There is a particular special case of the orientifold construction which goes back to the early superstring theory literature.

Definition 3.5. The Type I superstring on a smooth 10-manifold Y is the orientifold with spacetime X = Y × pt // (Z/2Z), the orbifold quotient of the trivial involution on Y .

We next generalize Supposition 3.1 and Proposal 3.2 to bosonic and superstring orientifolds. First, recall that if M is any space and A → M a fiber bundle of discrete abelian groups—a local system—then we can define twisted ordinary cohomology H•(M; A) with coefficients in A. In particular, if Mw → M is a double cover, then we form the associated bundle Aw → M of free abelian groups of rank one, defined by the action of {±1} on Z.We denote the associated twisted cohomology by Hw+•(M; Z). It has a concrete manifestation in terms of cochain complexes: the deck transformation of SPIN STRUCTURES AND SUPERSTRINGS 113

• the double cover Mw → M acts on the cochain complex C (Mw; Z), and Hw+•(M; Z) is the cohomology of the anti-invariant subcomplex. If M is a smooth manifold there is a corresponding twisted version Hˇ w+•(M)of diﬀerential cohomology. We use the model of diﬀerential cohomology as a • • cochain complex of triples (c, h, ω), where c ∈ C (Mw; Z), ω ∈ Ω (Mw), and •+1 h ∈ C (Mw; R)(see[DF, §6.3], [HS, §2.3]), and take the anti-invariant subcomplex.

Supposition 3.6. Let Xw → X be a double cover of 26-dimensional orbifolds and suppose X is the spacetime of a bosonic orientifold. Then the ﬂux of the B-ﬁeld βˇ lies in the twisted Borel cohomology Hw+3(X; Z).

This appears in the literature using a different model of twisted degree three cohomology [GSW]. The equivalence class of the B-field lies in the twisted differential cohomology group Hˇ w+3(X), consistent with the transformation law (3.3). The B-field quantization law for the superstring orientifold is expressed in terms of twisted R-cohomology. The following discussion applies to any cohomology theory h.LetMw → M be a double cover of a space M with deck transformation σ, and as after (1.3) let {hp}p∈Z denote a spectrum representing h-cohomology. Recall that hp(M) is the abelian group of homotopy classes of maps M → hp.Letip : hp → hp be a map which represents the additive inverse on cohomology classes, and we may assume ip ◦ ip =idhp . Define a w-twisted h-cocycle of degree p on M to be a pair (c, η) of a map ∗ c: Mw → hp and a homotopy η from σ c to ipc. A homotopy of w-twisted h-cocycles is a w-twisted h-cocycle on Δ1 × M, where Δ1 is the 1-simplex. Then hw+p(M) is defined as the group of homotopy classes of w-twisted h-cocycles of degree p. A small elaboration using triples as in (1.13) defines w-twisted hˇ-cohomology if M is a smooth manifold. In [DFM2] we develop a differential-geometric model for Rˇw−1(M).

Proposal 3.7. Let Xw → X be a double cover of 10-dimensional orbifolds and suppose X is the spacetime of a superstring orientifold. Then the ﬂux of the B-ﬁeld βˇ lies in the twisted Borel cohomology Rw−1(X).

Remark 3.8. There is an important restriction on the B-field flux which we will derive in §6. Namely, a superstring orientifold spacetime X carries a suitably twisted spin structure defined in terms of the B-field, and its existence leads to the constraints (6.9), (6.10).

3.3. Universal B-ﬁelds on orientifolds. Let BZ/2Z =pt// (Z/2Z) and π0 :pt→ BZ/2Z the universal double cover, which we denote w0. The geometric realization of BZ/2Z is RP∞, so the Borel R-cohomology of BZ/2Z is the R-cohomology of RP∞. For orientifolds there are universal B-ﬁelds pulled back from the classifying map X → BZ/2Z of the orientifold 114 J. DISTLER, D. S. FREED, AND G. W. MOORE

double cover Xw → X. For the bosonic orientifold we ﬁrst apply the exact sequence analogous to (1.18),

w+2 w+3 w+3 (3.9) 0 −→ H (M; R/Z) −→ Hˇ (M) −→ ΩZ (M) −→ 0, w +3 ∼ w +2 to M = BZ/2Z and deduce Hˇ 0 (BZ/2Z) = H 0 (BZ/2Z; R/Z). Now the twisted chain complex of the geometric realization RP∞, starting in degree zero, is

2 0 2 0 (3.10) ZZZ Z Z ···

Apply Hom(−, R/Z) to compute

w0+3 ∼ w0+2 ∼ (3.11) Hˇ (BZ/2Z; Z) = H (BZ/2Z; R/Z) = Z/2Z.

This is the universal group of B-ﬁelds on bosonic orientifolds.

Remark 3.12. The Bockstein map Hw0+2(BZ/2Z; R/Z) → Hw0+3 (BZ/2Z; Z) is an isomorphism, as follows easily from the long exact sequence associated to Z → R → R/Z. This is also obvious from the geometric picture of diﬀerential cohomology given around (1.14) since in this case Hˇ w0+3(BZ/2Z) is ﬁnite, hence equal to its group of components Hw0+3 (BZ/2Z; Z).

For superstring orientifolds we also have a ﬁnite group of universal twistings. w −1 ∼ w −2 ∼ Theorem 3.13. The group Rˇ 0 (BZ/2Z) = R 0 (BZ/2Z; R/Z) = Rw0−1(BZ/2Z; Z) is cyclic of order 8. For any generator θˇ we can identify 4θˇ with the nonzero element in (3.11). Furthermore, the pullback of θˇ under π0 :pt→ BZ/2Z is ηˇ.

Recall thatη ˇ is the nonzero class in (1.20). In [DFM3] we interpret Rw0−1 (BZ/2Z; Z) as a group of universal twistings of KO-theory (modulo Bott periodicity), which may be identified with the super Brouwer group [Wa, p. 195], [De, Proposition 3.6]. Proof. All cohomology groups in this proof have Z coefficients. We first show

w0−1 w−1 ∞ ∼ w−1 4 ∼ w−1 4 (3.14) R (BZ/2Z):=R (RP ) = R (RP ) = ko (RP ), where ‘w’ denotes the nontrivial double cover of projective space. The first equality is the definition of (twisted) Borel cohomology. The second group is computed as the space of sections of a twisted bundle of spectra over RP∞ whose fiber is R−1;see[ABGHR, MS]. The second isomorphism follows from elementary obstruction theory since R−1 has vanishing homotopy groups above degree 3; see (1.5). Finally, the (−1)-space of the ko-spectrum SPIN STRUCTURES AND SUPERSTRINGS 115

and R−1 have the same 5-skeleton, which justiﬁes the ﬁnal isomorphism in (3.14). w−1 4 ∼ w0−1 4 Write ko (RP ) = koZ/2Z (S ). Here we use the Atiyah-Segal equivariant ko-theory for the antipodal action on the sphere; the equivariant double cover w0 is pulled back from a point. Next, we claim

w0−1 ∼ 0 (3.15) koZ/2Z (pt) = ko (pt).

For in the Atiyah-Bott-Shapiro (ABS) model with Clifford algebras [ABS], the left hand side is the K-group of a category of Z/2Z-graded real modules for the Z/2Z-graded algebra A generated by odd elements γ,α with γ2 = −1, α2 =1,andαγ = −γα. (That the generator α of Z/2Z is odd reflects the twisting w0; the Clifford generator γ is always odd.) But A is isomorphic to the Z/2Z-graded matrix algebra End(R1|1), and so the category of A-modules is Morita equivalent to the category of Z/2Z-graded real 9 −1 w0−1 vector spaces. Let ξ denote the element in koZ/2Z (pt) which corresponds to 1 ∈ ko0(pt) under the isomorphism (3.15). In the ABS model ξ−1 is represented by −1 1|1 0 −1 01 (3.16) ξ : R with γ = ,α= . 10 10

−1 0 4 ∼ w0−1 Then multiplication by ξ induces an isomorphism ko Z/2Z(S ) = koZ/2Z 4 0 4 ∼ 0 (S ), where the tilde denotes reduced ko-theory. Now ko Z/2Z(S ) = ko (RP4)andko 0(RP4) is cyclic of order 8 generated by H −1, where H → RP4 0 4 is the nontrivial (Hopf) real line bundle: the order of ko (RP ) is bounded by 8 by the Atiyah-Hirzebruch spectral sequence, and because w4 4(H − 1) = 0 we conclude 4(H − 1) =0. The assertion about 4θˇ follows from the twisted version of the exact sequence (1.6) on BZ/2Z: the kernel group Hw0+3(BZ/2Z; Z) is (3.11). To prove the last statement we observe that the argument in the previous para- w0−1 4 −1 graph identiﬁes the generator of koZ/2Z (S ) as the pullback of ξ under the Z/2Z-equivariant map h: S4 → pt. Let i:pt → S4 be the (nonequivariant) ∗ ∗ −1 inclusion of a point. Then π0(θˇ) is the image of h ξ under the composition ∗ w0−1 4 −1 4 i −1 −1 koZ/2Z (S ) → ko (S ) −→ ko (pt), which is evidently the image of ξ w0−1 −1 4 under koZ/2Z (pt) → ko (pt). (We choose orientations of pt and S to trivialize the pullback of w0 under π0.) Finally, in the ABS model this pullback simply drops the action of α, and what remains of (3.16) is the generator η −1 ∼ of ko (pt) = Z/2Z.

9 We reserve the notation ‘ξ’fortheinverseclassintwistedperiodic KO-theory. It is the KO-Euler class of the real line with involution −1, viewed as an equivariant line bundle over a point. It has many beautiful properties, some of which we exploit in [DFM3]. 116 J. DISTLER, D. S. FREED, AND G. W. MOORE

4. The B-field amplitude for orientifolds A worldsheet in an orientifold string theory has several fields [DFM1, Definition 5]. For the bosonic case they all appear in Definition 4.1; for the superstring there are additional fields articulated in Definition 4.8 and Definition 5.1.

4.1. Bosonic orientifold worldsheets. As a preliminary recall that a smooth n-manifold M has a canonical orientation double coverπ ˆ : M → M + + deﬁned as the quotient M := B(M)/GLn R, where GLn R is the group of orientation-preserving automorphisms of Rn. The manifold M is canonically oriented. It is natural to denote the double coverπ ˆ : M → M as ‘w1(M)’.

Definition 4.1. Let π : Xw → X be the spacetime of an orientifold string theory. An orientifold worldsheet is a triple (Σ,φ,φ˜) consisting of a compact 2-manifold Σ, a smooth map φ:Σ→ X, and an equivariant lift φ˜: Σ → Xw of φ to the orientation double cover of Σ.

In theories with open strings Σ may have nonempty boundary. The surface Σ is not oriented and need not be orientable. In fact, the existence of the equivariant lift implies a constraint involving its first Stiefel-Whitney class: ∗ (4.2) φ w = w1(Σ); the equivariant lift φ˜ is an isomorphism of the double covers in (4.2).10 For an orientifold spacetime defined by a triple (Y,Γ,υ) as above, Defini- tion 4.1 unpacks to a principal Γ-bundle P → Σ, an orientation on P ,anda Γ-equivariant map P → Y . There is a constraint: if υ(γ) = 0, then the action of γ on P preserves the orientation; if υ(γ) = 1, then γ reverses the orientation. There is an obvious notion of equivalence of triples (Σ,φ,φ˜), and the collection of such triples forms a groupoid presentation of an infinite dimensional orbifold.

Remark 4.3. Definition 4.1 applied to a single string clarifies the nature of twisted sectors in orientifold theories. Namely, if φ: S1 → X is a string, then the constraint implies that φ∗w = 0, since the circle is orientable. Thus φ lifts to the double cover Xw. Put differently, the homotopy class of φ does not detect a nontrivial double cover, so does not sense the orientifold. Now the “twisting” in a twisted sector for a global quotient orbifold X = Y//Γ measures the extent to which a string S1 → X fails to lift to a string S1 → Y .SoifX = Y//Γ is a global quotient with υ :Γ→ Z/2Z specifying the orientifold, then φ lifts to Xw = Y// ker υ and the twisted sectors are labeled by conjugacy classes in ker υ. In case Xw = Y is a smooth manifold and X the orbifold quotient by an involution, then any string φ: S1 → X 10 In our ambiguous notation ‘w’and‘w1(M)’ denote both a double cover and its equivalence class. SPIN STRUCTURES AND SUPERSTRINGS 117 lifts to a loop S1 → Y . Hence there are no twisted sectors in a “pure” orientifold.

4.2. B-field amplitudes for bosonic orientifolds. Recall that if M is a smooth compact n-manifold then integration of differential forms (4.4) :Ωn(M) −→ R M,o is only defined after choosing an orientation o. Absent an orientation one may integrate densities, which in our current notation are w1-twisted differential forms: forms on the orientation double cover M which are odd under the deck transformation. Integration of densities is a homomorphism (4.5) :Ωw1(M)+n(M) −→ R M which lifts to integration in twisted differential cohomology: w1(M)+n+1 1 ∼ (4.6) : Hˇ (M) −→ Hˇ (pt) = R/Z. M

To define (4.6) one may follow [HS, §3.4] working in the model with smooth singular cochains. That understood, the definition of the B-field amplitude (2.1) for bosonic orientifolds is straightforward. Let βˇ be a bosonic orientifold B-field as in Supposition 3.6; its equivalence class is [βˇ] ∈ Hˇ w+3(X). Then for an orientifold worldsheet as in Definition 4.1 the isomorphism (4.2) (defined by φ˜ in Definition 4.1) places the pullback φ∗[βˇ] in the group Hˇ w1(Σ)+3(Σ). The B-field amplitude is then computed using a twisted integration (4.6) in place of (2.2). This bosonic orientifold B-field amplitude is described using a particular model for Hˇ w+3(X)in[GSW]. The universal B-field amplitude is easy to compute.

Proposition 4.7. Let βˇ be the nonzero universal B-ﬁeld in (3.11). Then for any bosonic orientifold worldsheet the B-ﬁeld amplitude (2.1) is (−1)Euler(Σ),whereEuler(Σ) is the Euler number of the closed surface Σ.

Proof. If φ:Σ→ X is the worldsheet map, then we can identify φ∗[βˇ] ∈ Hw1(Σ)+2(Σ; R/Z) as the pullback of x2 ∈ H2(RP∞; Z/2Z) via the map ∞ w1 :Σ→ RP which classiﬁes w1(Σ). The latter pulls back the generator 1 ∞ ∗ 2 2 x ∈ H (RP ; Z/2Z)tow1(Σ), so φ [βˇ]=w1(Σ) .Noww1(Σ) = w2(Σ) since the diﬀerence of the two sides is the second Wu class, which vanishes on manifolds of dimension less than four. Finally, w2(Σ) is the mod 2 reduction of the Euler class (which in general lives in twisted integral cohomology). 118 J. DISTLER, D. S. FREED, AND G. W. MOORE

4.3. Spin structures on superstring orientifold worldsheets. Turning to the worldsheet in a superstring orientifold theory we begin by specifying the appropriate notion of spin structure. We could not ﬁnd this deﬁnition in the string theory literature, even for the Type I superstring.

Definition 4.8. The topological data on a superstring orientifold worldsheet Σ is a discrete ﬁeld α which on each connected orientable open set U ⊂ Σ is a pair of spin structures which induce opposite orientations of U.

Definition 4.8 is identical to Definition 2.4 except for the omission of the global orientation. Although α is locally a pair of spin structures, there is no global spin structure on Σ. Rather, the local pair of spin structures with opposite underlying orientation glue to a global spin structure on the orientation double cover Σ. The global description is equivalent to the local Definition 4.8, and we use ‘α’ to denote the spin structure on Σ as well as the local field in Definition 4.8. Letσ ˆ denote the involution on Σ. If the spin structures are locally opposite consistent with gluing—more simply, if the pullbackσ ˆ∗α of the global spin structure on Σ is the opposite (−α)—then a refinement to a pin− structure on Σ may be possible, but is additional data.

Remark 4.9. The oriented double cover S2 of RP2 has a unique spin ∼ structure (up to =) compatible with the orientation. It refines to two inequivalent pin− structures on RP2. On the other hand, the oriented double cover S1 × S1 of the Klein bottle K has 4 inequivalent spin structures compatible with the orientation. Two of them each refine in two inequivalent ways to give four inequivalent pin− structures on the Klein bottle; the other two each refine in two inequivalent ways to give four inequivalent pin+ structures on the Klein bottle.

Remark 4.10. It is important to emphasize that for general α there is no refinement to a pin− structure. (Indeed, if α refines to a pin− structure then the pullback to the orientation double cover is a spin structure which is invariant under the deck transformation.) This has important ram- ifications for the physics. Consider a connected open set U ⊂ Σwiththe topology of a cylinder. On U there are four choices of a pair of spin structures: each spin structure can be either bounding or non-bounding when restricted to the circle. In the case where one spin structure bounds, and the other does not, it is impossible to refine α to a pin− structure since the pullback of the pair to the oriented double cover of U is not invariant under the deck transformation. From the physical viewpoint, it is clear from the Hamiltonian formulation of the string theory that this mixed choice of spin structures occurs for Feynman diagrams in which spacetime fermions propagate along an internal line corresponding to U. Conversely, restricting attention to only those α which do refine to a pin− structure misses all of the sectors of the worldsheet theory in which space-time fermions propagate along that channel. SPIN STRUCTURES AND SUPERSTRINGS 119

Remark 4.11. Consider an orientifold theory in which the orientifold double cover π : Xw → X is trivial and trivialized. Then Definition 4.8 reduces to Definition 2.4. For the trivialization may be modeled as a section of π. Then for an orientifold worldsheet (Definition 4.1) φ:Σ→ X the ∗ ∼ equivariant lift φ˜ identifies φ (Xw → X) = (Σ → Σ), and so the section of π pulls back to a section ofπ ˆ : Σ → Σ. But the latter is precisely a global orientation o of Σ.

4.4. B-field amplitudes for superstring orbifolds. To describe the B-field amplitude (2.1) for the superstring we need the analog of (4.6) in differential R-theory. A complete definition involves twistings of cohomology theories beyond twists by double covers (see the discussion preceding Proposal 3.7) and is deferred to [DFM2]. For now recall that R is Spin- oriented and there is a pushforward (2.6) on spin manifolds. More generally, the obstruction to a spin structure on an n-manifold M determines a twisting τ R(M)ofR-theory, so too of differential R-theory, and a twisted pushforward τ R(M)−3 −3 ∼ (4.12) : Rˇ (M) −→ Rˇ (pt) = R/Z. M

The twisting τ R(M) includes the dimension of M, as well as the Stiefel- Whitney classes w1(M),w2(M). A spin structure produces an isomorphism n → τ R(M) and so reduces the pushforward (4.12) to a pushforward on untwisted differential R-theory, as in (2.6). Now suppose π : Xw → X is the orientifold double cover of a 10-dimensional superstring spacetime X with B-field βˇ. Given a worldsheet as in Definitions 4.1 and 4.8 the pullback φ∗[βˇ] of the equivalence class of the B-field lies in Rˇw1(Σ)−1(Σ). It seems, then, that to push forward to a point R using (4.12) we need an isomorphism w1(Σ) → τ (Σ) − 2 of twistings of R-theory. However, the local spin structures α on Σ—equivalently global spin structure on Σ—do not give such an isomorphism. This puzzle stymied the authors for a long period. The resolution is that the B-field amplitude in general is not a number, but rather an element in a complex line:

(4.13) The B-ﬁeld amplitude for a superstring orientifold is anomalous.11

R There is one case in which there is an isomorphism w1(Σ) → τ (Σ) − 2, namely when α is refined to a pin− structure on Σ. Then the B-field amplitude may be defined as a number. Notice that on a pin− worldsheet the two local spin structures α are opposites. The anomaly measures the extent to which that fails for general α.

11 We refer to a term in an (eﬀective) action as anomalous if it takes values in a (noncanoncially trivialized) complex line rather than the complex numbers. 120 J. DISTLER, D. S. FREED, AND G. W. MOORE

Remark 4.14. To illustrate, suppose that the superstring orientifold worldsheet Σ is diffeomorphic to a 2-dimensional torus. Even though Σ is orientable, the fields do not include an orientation. The field α is equivalent to a pair of spin structures α,α on Σ with opposite underlying orientations. Up to isomorphism there are 4 choices for each of α,α, so 16 possibilities in total. Of those 4 refine uniquely to pin− structures on Σ. The B-field amplitudes for the remaining 12 are anomalous.

Recall from Theorem 2.9 that in the oriented case the universal B-ﬁeld amplitude for the superstring computes the well-known Z/2Z-valued quadratic form on spin structures. We now investigate the analogous amplitude in the orientifold case for pin−worldsheets. Let Σ be a closed 2-manifold and P−(Σ) the H1(Σ; Z/2Z)-torsor of equivalence classes of pin− structures. Let θˇ be a generator of the cyclic group Rw0−2(BZ/2Z; R/Z); see Theorem 3.13. Now the orientation double cover determines a map h:Σ→ BZ/2Z and so a class h∗θˇ ∈ Rw1(Σ)−2(Σ; R/Z). Let p:Σ→ pt. Then a pin− structure α− on Σ determines a pushforward map

− α w1(Σ)−2 −4 ∼ (4.15) p∗ : R (Σ; R/Z) −→ R (pt; R/Z) = R/Z. Deﬁne q− : P−(Σ) −→ R/Z (4.16) − α− ∗ α −→ p∗ (h θˇ)

We can replace the R-cohomology groups in (4.15) with ko-groups or even periodic KO-groups. − 1 ∼ Theorem 4.17. The function q takes values in 8 Z/Z = Z/8Z,isa quadratic reﬁnement of the intersection pairing, and its reduction modulo two is congruent to the Euler number Euler(Σ).

Proof. The ﬁrst statement follows since 8θˇ = 0. We must show that 1 for a1,a2 ∈ H (Σ; Z/2Z), (4.18) − − − − − − − − 1 − − q (α +a1+a2)−q (α +a1)−q (α +a2)+q (α )= a1·a2,α∈ P (Σ). 2 The argument of [A1, p. 53] applies verbatim through Lemma (2.3), which 2 we replace with the following assertion. Let i:pt → Σandu = i∗(η ) ∈ ko0(Σ; Z); then

α− ∗ (4.19) p∗ (h θˇ · u)=1/2.

To prove this we note that u is supported in a neighborhood of a point in Σ, so by excision we can compute the left side on a sphere S2.Fixan 2 orientation of S , which is a section of the orientation double cover w1(Σ). SPIN STRUCTURES AND SUPERSTRINGS 121

2 This lifts h: S → BZ/2Z to π0 :pt→ BZ/2Z. Then since by Theorem 3.13 ∗ 2 we have π0θˇ =ˇη, we reduce (4.19) to p∗(ˇη · u), which by push-pull isη ˇ · η . As in the proof of Theorem 2.9 this is nonzero. The last statement follows from Proposition 4.7 since 4θˇ is the nonzero element of (3.11); see Theorem 3.13. − Pin− Recall [KT, §3] that the pin bordism group Ω2 is cyclic of order eight and the Kervaire invariant is an isomorphism.

Corollary 4.20. With an appropriate choice of generator θˇ in Theo- rem 3.13, the quadratic form (4.16) is the Kervaire invariant.

For oriented surfaces the Z/2Z-valued Kervaire invariant (2.7) has a well- known KO-theoretic interpretation [A1]. Corollary 4.20 provides a similar KO-theoretic interpretation in the unoriented case. Proof. The deﬁnition (4.16) of q− is evidently a bordism invariant. The 2 − Pin− real projective plane RP has two pin structures; either generates Ω2 . Since RP2 has odd Euler number, the value of q− on either pin− structure is a generator of Z/8Z. The four possible choices of θˇ in the deﬁnition of q− give the four generators of Z/8Z, so we can choose the one which matches the standard Kervaire invariant on RP2, hence on all pin− surfaces.

5. Worldsheet fermions and spacetime spin structures A fermionic functional integral is, by definition, the pfaffian of a Dirac operator. It is naturally an element of a line, so in a family of bosonic fields a section of a line bundle over the parameter space [F1, Part 2]. For an orientifold superstring worldsheet the B-field amplitude is also anomalous (4.13). The main result of [DFM2] is that the product of these anomalies is trivializable, and furthermore the correct notion of spin structure on spacetime (§6) leads to a trivialization. In this section, after identifying the fermionic fields in the 2-dimensional worldsheet theory, we work out an analogous phenomenon in a familiar 1-dimensional theory: the “spinning particle”. Namely, in Theorem 5.11 we identify the pfaffian line of the Dirac operator on a circle in terms of the frame bundle of spacetime, and show how a spin structure on spacetime leads to a trivialization.

5.1. Fermions on orientifold superstring worldsheets. This is the last in the triad of definitions (see Definitions 4.1 and 4.8) specifying the fields on an orientifold superstring worldsheet [DFM1, Definition 5].

Definition 5.1. An orientifold superstring worldsheet consists of (Σ,φ, φ,˜ α) as in Definitions 4.1 and 4.8 together with a positive chirality spinor field ψ on Σ with coefficients in πˆ∗φ∗TX and a negative chirality spinor field χ on Σ with coefficients in T ∗Σ. 122 J. DISTLER, D. S. FREED, AND G. W. MOORE

The notion of chirality is defined by the canonical orientation on the orientation double cover Σ; the spinors use the spin structure α.Bothψ (the “matter fermion”) and χ (the “gravitino”) should be regarded as local fields on Σ, but the global description on Σ is more transparent; the action is local on Σ. The crucial factor in the functional integral over ψ, χ for fixed φ and α is the pfaffian of a Dirac operator on Σ, which may be written ∗ ∗ − (5.2) pfaff DΣ,α πˆ φ TX T Σ . The pfaffian line bundle is local, so we can heuristically analyze it on a small contractible open set U ⊂ Σ. Now π−1U ⊂ Σ is the disjoint union of two oppositely oriented open sets diffeomorphic to U with spin structures α,α refining the underlying orientations. The pfaffian (5.2) is anomalous on each component of π−1U. If the spin structures α,α are opposite, then the product of the anomalies is trivializable; an isomorphism of α with the opposite of α trivializes the anomaly. So we see that the anomaly measures the failure of α and α to be opposites, just as for the B-field.12 (See the text leading to Remark 4.14.) For the oriented superstring a global argument for the triviality of the pfaffian line bundle—the anomaly in the fermionic functional integral (5.2)— is given in [FW, §4]. In the non-orientifold case there is no anomaly in the B-field amplitude (see (2.6)). The argument in [FW] only proves the triviality; it does not provide a trivialization so does not determine a definition of (5.2) as a function. (This additional data is sometimes termed a ‘setting of the quantum integrand’.) In fact, the superstring data does determine a trivialization: it is the spacetime spin structure which is critical. We explore this two-dimensional anomaly problem in [DFM2] and show that the trivialization varies under a change of spacetime spin structure. Remark 5.3. For an oriented superstring worldsheet (Definition 2.4), the dependence is as follows. Suppose a ∈ H1(X; Z/2Z) is a change of space- 1 time spin structure and b = αl − αr ∈ H (Σ; Z/2Z) the difference of the two global worldsheet spin structures. Then the trivialization for a worldsheet φ:Σ→ X multiplies by

∗ (5.4) (−1)φ a,b where −, − is the Z/2Z-valued pairing on H1(Σ; Z/2Z). Combining this factor with (2.14) one sees that our formulation of the oriented superstring has the expected left-right symmetry. See (5.10) for a 1-dimensional analog of (5.4). Equation (5.4) is consistent with [AW].

5.2. A supersymmetric quantum mechanical theory. Here we illustrate the impact of the spacetime spin structure on the worldsheet pfaf- ﬁan in a simpler quantum ﬁeld theory: the 1-dimensional supersymmetric

12 ∗ The anomaly also depends on the topology of φ (TX). SPIN STRUCTURES AND SUPERSTRINGS 123 quantum mechanical system whose partition function computes the index of the Dirac operator [Ag, FW, W1]. In this theory spacetime X is a Rie- mannian manifold of arbitrary dimension n. For the classical theory it does not have a spin structure or even an orientation. However, to simplify we assume that X is oriented. The worldsheet of superstring theory is replaced by a 1-dimensional manifold S with a map φ: S → X. The manifold S is endowed with a single spin structure. The fermionic fields of Definition 5.1 are replaced by a single spinor field ψ on S with coefficients in φ∗TX. Consider S = S1 with the nonbounding spin structure α. The first step in computing the partition function is to compute the fermionic functional integral over ψ for a fixed loop φ: S1 → X, which is the pfaffian

∗ (5.5) pfaﬀ DS1,α(φ TX).

As the Dirac operator on the circle is real, the square of its pfaffian line bundle is canonically trivial and so the square of (5.5) is a well-defined function. There is a standard regularization and the result (see [A3], for example) is ∗ 2 (5.6) pfaff DS1,α(φ TX) =det 1 − hol(φ) , where hol(φ) ∈ SOn is the holonomy, well-defined up to conjugacy. We may as well assume that n is even, or else (5.6) vanishes identically. Now the function g → det(1 − g)onSOn does not have a smooth square root. However, its lift to Spinn does have a square root f, the difference of the characters of the half-spin representations: n/2 − ∈ (5.7) f(˜g)=i χΔ+ (˜g) χΔ− (˜g) , g˜ Spinn . Hence given a spin structure on X we can lift the holonomy function hol: LX → SOn on the loop space of X to a function hol: LX → Spinn,andso define (5.5) as ∗ (5.8) pfaff DS1,α(φ TX):=f hol(φ) .

The right hand side of (5.8) manifestly uses the spin structure on spacetime X. Note that we can equally replace the function f by its negative; the overall sign is not determined by this argument.

Remark 5.9. If we change the spin structure on X by a class a ∈ H1(X; Z/2Z), then it follows immediately from (5.8) that the pfaﬃan multiplies by

∗ 1 (5.10) (−1)φ (a)[S ].

The pfaﬃan is more naturally an element of a line and for the analogy with the 2-dimensional worldsheet theory it is more illuminating to analyze 124 J. DISTLER, D. S. FREED, AND G. W. MOORE

∗ the pfaffian line Pfaff DS1,α(φ TX) directly. (See [F2, §3] for the definition of the pfaffian line of a Dirac operator.) Write E → S1 for the oriented ∗ vector bundle φ TX. The Dirac operator DS1,α is the covariant derivative ∇ 1 acting on sections of E → S . It is real and skew-adjoint, so its pfaffian line is real. Replace a real line L by the Z/2Z-torsor π0 L\{0} , so obtain 1 the pfaffian torsor Pfaff ∇.LetBSO(E) → S denote the bundle of oriented orthonormal frames of E. It is trivializable since SOn is connected. The space ∼ of sections Γ has two components and is naturally a torsor for π1(SOn) = 1 Z/2Z. Furthermore, a spin structure BSpin →BSO(E) → S trivializes the torsor π0Γ: there is a distinguished component of sections which lift to BSpin. ∼ Theorem 5.11. There is a canonical isomorphism Pfaff ∇ = π0Γ. Therefore, a spin structure on E determines a trivialization of Pfaff ∇. Suppose Z is any manifold and E → Z × S1 an oriented bundle with covariant derivative. Then the Pfaffian torsors vary smoothly in z ∈ Z so form a double cover of Z. Its characteristic class may be computed from the 1 Atiyah-Singer index theorem as the slant product w2(E)/[S ]; see [FW, (5.22)]. Theorem 5.11 is a “categorification” of this topological result—an isomorphism of line bundles rather than simply an equality of their isomorphism classes—necessary in order to discuss trivializations. We remark that more sophisticated categorifications of the Atiyah-Singer index theorem are needed for anomaly problems in higher dimensions, such as [DFM2]; see [Bu] for a recent result in dimension two. Proof. Fix a Riemannian metric on S1 of total length 1. The covariant 1 derivative of a framing e ∈ Γ is a function ∇(e): S → son. Using parallel transport choose√e so that ∇(e) is a constant skew-symmetric matrix A whose eigenvalues a −1 satisfy −π

6. The twisted spin structure on a superstring orientifold spacetime The spacetime X of an oriented superstring theory has a spin structure. There is a modiﬁcation for orientifolds in superstring theory: the notion SPIN STRUCTURES AND SUPERSTRINGS 125

of spin structure is twisted by both the orientifold double cover π : Xw → X and the B-field. In this section we describe this twisted notion of spin structure in concrete differential-geometric terms. Recall quite generally that if ρ: G → G is a homomorphism of Lie groups and P → M a principal G-bundle over a space M, then there is an associated principal G-bundle ρ(P ) → M, defined by the “mixing construction” ρ(P )=P ×G G . Conversely, if Q → M is a principal G -bundle, then a reduction to G along ρ isapair(P, ϕ) consisting of a principal G-bundle P → M and an isomorphism ϕ: ρ(P ) → Q.IfM n is a smooth manifold and ρ: G → GLnR, then a reduction of the GLnR frame bundle B(M)toG along ρ is called a G-structure on M. We defined orientations in these terms in §4.1 and spin structures in these terms in §2.1; for convenience we used a metric and so a homomorphism (2.3) into the orthogonal group. A principal G-bundle is classified by a map13 M → BG whose homotopy class is an invariant of P → M. The topological classification of reductions along ρ: G → G may be analyzed as a lifting problem:

(6.1) BG P Bρ Q M BG

Two particular cases are of interest here: (i) ρ is the inclusion of an index two subgroup, in which case Bρ: BG → BG is a double cover and the obstruction to (6.1) lies in H1(M; Z/2Z); and (ii) ρ is a surjective double cover, in which case Bρ: BG → BG is a principal K(Z/2Z, 1)-bundle14 and the obstruction to (6.1) lies in H2(M; Z/2Z). The spin group (2.3) is a double cover of an index two subgroup of On.

We now deﬁne groups G0,G1 which bear the same relation to On := On × Z/2Z × Z/2Z via homomorphisms

(6.2) ρi : Gi −→ On,i=1, 2

which factor through an index two subgroup Gi ⊂ On. First, let D4 → Z/2Z × Z/2Z be the dihedral double cover in which the generators of the Z/2Z factors lift to anticommuting elements of order two. Deﬁne G0,G0 as the ﬁrst two groups in

(6.3) ρ0 : (Spinn ×D4)/{±1}−→SOn × Z/2Z × Z/2Z −→ On,

13 More precisely, a classifying map for P → M is a G-equivariant map P → EG for EG → BG a universal G-bundle. 14 K(Z/2Z, 1) is an Eilenberg-MacLane space; a topological group model is the group of projective linear transformations of an inﬁnite dimensional real Hilbert space. 126 J. DISTLER, D. S. FREED, AND G. W. MOORE

where −1 ∈{±1} is the product of the central elements of Spinn and D4. For G1 we ﬁrst deﬁne the surjective homomorphism

On −→ Z/2Z (6.4) (g, a, b) −→ c + a, det(g)=(−1)c,

and let G1 be the kernel. Then G1 is the inverse image of G1 under − (6.5) (Pinn ×D4)/{±1}−→On. Suppose X is a superstring spacetime—a 10-dimensional orbifold—and Xw → X an orientifold double cover. Proposal 3.7 implies that a B-ﬁeld βˇ is a geometric object whose equivalence class [βˇ] lies in Rˇw−1(X). As in (1.17) there are topological invariants t(βˇ): π0X → Z/2Z and a double cover Xa(βˇ) → X.

Definition 6.6. Let Xw → X be the orientifold double cover of a Rie- mannian orbifold X which represents a superstring spacetime. Let βˇ be a B-ﬁeld on X.Thenatwisted spin structure is a reduction of the principal

O10-bundle

(6.7) BO(X) ×X Xw ×X Xa(βˇ) → X along ρ: Gi → O10,wherei ∈ Z/2Z is chosen on each component of X according to the value of t(βˇ).

Deﬁnition 2.11 expresses the two types in more familiar terms as Type IIB for t(βˇ)=0andTypeIIAfort(βˇ) = 1. Typically spacetime is connected and only one of these occurs. Equivalence classes of twisted spin structures, if they exist, form a torsor for H0(X; Z/2Z) × H1(X; Z/2Z). The existence is settled by the following.

Proposition 6.8. Let Xw → X and βˇ be as in Deﬁnition 6.6. Then a twisted spin structure exists if and only if

(6.9) w1(X)=t(βˇ)w 2 (6.10) w2(X)=a(βˇ)w + t(βˇ)w

These equations live in the Borel cohomology of the orbifold X. Proof. Equation (6.9) is the condition to reduce the structure group of (6.7) along the inclusion Gi → O10.ForG0 = SO10 × Z/2Z × Z/2Z it is the condition w1(X) = 0 for an orientation. For t(βˇ) = 1 the homomorphism (6.4) induces a map BO10 → BZ/2Z which pulls the generator 1 1 1 of H (BZ/2Z; Z/2Z)backtow1 + x, where H (BO10; Z/2Z)=H (BO10; 1 1 Z/2Z)×H (BZ/2Z; Z/2Z)×H (BZ/2Z; Z/2Z) has generators w1,x,y.Then (6.9) follows by pullback along the classifying map of (6.7). SPIN STRUCTURES AND SUPERSTRINGS 127

For (6.10) we first observe that the double cover D4 → Z/2Z × Z/2Z is classified by xy ∈ H2(BZ/2Z × BZ/2Z; Z/2Z). Then the first homomorphism in (6.3) induces a principal K(Z/2Z, 1)-bundle BG0 → BG0 classified by w2 + xy, from which (6.10) follows on components with t(βˇ)= 0. For components with t(βˇ) = 1 we first recall [KT, Lemma 1.3] that − 2 the universal K(Z/2Z, 1)-bundle B Pin10 → BO10 is classified by w1 + 2 w2 ∈ H (BO10; Z/2Z). Then the definition (6.5) of G1 shows that BG1 → 2 BG1 is classified by w1 + w2 + xy; equation (6.10) now follows from this and (6.9).

Remark 6.11. The occurrence of D4 in our deﬁnition of a twisted spin 15 structure is closely related to the D4 symmetry group which appears in Hamiltonian treatments of orientifolds in the physics literature. We hope to elaborate on this elsewhere.

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Theory Group, Department of Physics, and Texas Cosmology Center, Uni- versity of Texas, 1 University Station C1600, Austin, TX 78712-0264 E-mail address: [email protected] Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257 E-mail address: [email protected] NHETC and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855–0849 E-mail address: [email protected] Surveys in Diﬀerential Geometry XV

Operator Traces and Holography

Michael R. Douglas

Abstract. We describe a version of AdS/CFT duality proposed by Sezgin, Sundell, Klebanov and Polyakov, between the O(N) model in two dimensions and Fradkin-Vasiliev higher spin gravity in three dimensions, and begin to reformulate it as a mathematical conjecture expressing traces of differential and pseudodifferential operators in terms of a classical boundary value problem in an infinite-component Chern-Simons theory.

Contents 1. Introduction 132 2. Dualities between quantum ﬁeld theory and gravity 135 2.1. Matrix model/noncritical string duality 137 2.2. AdS/CFT and the conformal boundary 138 2.3. Global symmetries and gauge symmetries 140 3. Free ﬁeld-higher spin gravity duality 142 3.1. Higher spin gravity 143 3.2. Duality between correlation functions 145 4. Why should gauge-gravity duality work? 146 4.1. AdS1/CFT0 duality 147 4.2. Determinants of operators 150 4.3. Higher derivative operators 151 4.4. Multiplicative anomaly 153 4.5. Relation to the RG 154 5. Chern-Simons gravity and generalized Liouville theory 155 5.1. Liouville and Chern-Simons gravity 155 5.2. Generalized Liouville theory 157 References 159

Dedicated to Is Singer on the occasion of his 85th birthday.

c 2011 International Press

131 132 M. R. DOUGLAS

1. Introduction One of the most striking and unexpected discoveries of the 1994-98 “second superstring revolution” was the AdS/CFT correspondence [33,41,65], according to which N = 4 supersymmetric Yang-Mills theory in four dimen- 5 sions is dual to type IIb superstring theory on AdS5 × S . This finally made precise the physics intuition, developed in the 1970’s by ’t Hooft, Polyakov, Migdal, Witten and others [49, 58], that a four-dimensional gauge theory should have a string description in the large N limit. Since then, the correspondence has been much generalized [1] and has found many applications, especially in providing simple solvable models exhibiting nonperturbative physical phenomena such as confinement, dissipation and quantum phase transitions [56]. However, despite a good deal of work, the microscopic workings of the duality are still not well understood. Various ideas have been developed, such as the holographic renormalization group [18], and integrability [44], which has led to very impressive computations and comparisons at finite gauge coupling [9, 31]. But many basic questions, such as the class of field theories with gravity duals, or the mapping between the variables on the two sides, seem to not yet have precise answers. Although dualities can be understood in depth in simple models [16], usually the complexity of the two theories being related puts most precise statements out of reach. For example, just to define the N =4 super Yang-Mills correlation functions requires renormalization – although the beta function is zero, all nontrivial gauge invariant operators are composite operators, which must be renormalized. The usual way this renormalization is phrased in physics requires making choices, at the very least a basis for Hilbert space, and usually much more structure. This dependence plausi- bly corresponds to some scheme dependence in the dual superstring theory, including and generalizing coordinate transformations, field redefinitions, and perhaps other symmetries. But to make the duality precise for correlation functions of generic operators, one must get these prescriptions to match, which is a formidable technical problem. Another symptom of this complexity is that, compared to other topics in string theory, so far AdS/CFT has inspired relatively few mathematical developments. While there are some very interesting works along these lines, such as [42,66], more typically what is discussed is rather intricate, and often simpler to understand in other terms. From this point of view, we should start with simpler versions of the duality, still involving field theory in higher dimensions, to make more fundamental contact with mathematics. Probably the simplest field theory-gravity duality, after the “old matrix models” [22] which are dual to linear dilaton backgrounds rather than AdS, is the conjecture of Sezgin and Sundell [53] and Klebanov and Polyakov [38], which states that the so-called O(N) model in D = 3 dimensions is dual to a D + 1 = 4 dimensional theory of higher spin gravity developed by Fradkin OPERATOR TRACES AND HOLOGRAPHY 133 and Vasiliev, and many others [59]. Although less discussed, both the O(N) models and higher spin gravity make sense in arbitrary dimension D [60], and one can make analogous duality conjectures for any D. The O(N) model is exactly solvable in the large N limit, because (as we review shortly) it is simply related to a quantum theory of N free bosons. This theory has an infinite series of higher spin conserved currents, and by physics arguments we will explain, the dual gravitational theory must have a corresponding series of massless fields with spin greater than two. It turns out to be difficult to find unitary quantum field theories of this type, and the consistency of the Fradkin-Vasiliev type theory is nontrivial. It turns out to make sense only in AdS backgrounds, which is part of the motivation for the duality conjecture. Since the conjecture was made, it received support from several comparisons of correlation functions, however since there are significant differences with the standard AdS/CFT conjectures (for example, the O(N) model is not a gauge theory) its status remained unclear. But recently, a complete check at cubic order has been done by Giombi and Yin [28], which seems to us fairly compelling evidence. Physically, although the O(N) model itself is now considered a bit trivial, in principle the duality extends to a much larger class of theories, with arbitrary kinetic terms and thus arbitrary dispersion relations for free bosons or fermions, which are of interest in condensed matter physics. The problem of finding the gravity dual of a free field theory has also attracted attention as a starting point for understanding AdS/CFT in more depth [29]. From a mathematical point of view, the partition function of N free bosons is simply the determinant of a Schrödingeroperator (raised to the power −N/2) on a D-dimensional Riemannian manifold, while its generalization to the generating functional of correlation functions is the determinant of a general multi-parameter family of partial differential operators. Thus, the mathematical statement of this duality is an exact formula for such determinants,1 in terms of critical points of an action functional of D + 1-dimensional fields evaluated with specified boundary conditions, corresponding to the choice of operator. This sounds rather interesting and surprising, as determinants and traces of partial differential operators have been studied very extensively and, at least to the author, no such general mathematical results come to mind. One goal of the present note is simply to explain this duality in a way that we hope a mathematician can more easily follow. Besides the usual language barriers, which by now are not that major (especially for Is), there are a number of significant differences in how these topics are approached in math and in physics, which need to be overcome to make a satisfactory and precise mathematical statement. We will also try to explain these differences to physics readers.

1 At least for manifolds which can appear as a conformal boundary, such as the sphere. 134 M. R. DOUGLAS

In section 2, we briefly review gauge-gravity dualities, their origins in the large N limit, and the development of the gravitational point of view. A particularly important point for us, which was actually discovered before the rest of the picture [14], is the relation between symmetries of the gauge theory, and asymptotic symmetries of gravity. A good mathematical introduction to AdS/CFT is [3], and many points which we skim over are explained in more detail there. In section 3, we describe the physics duality conjecture relating the O(N) model with Vasiliev theory, and explain why this is essentially equivalent to a duality conjecture involving a free bosonic theory. We then discuss some of the physics background behind the conjecture. In section 4, we survey a few of the questions we should answer in explaining how such dualities work. We then turn to a simple but so far as we know unexplored special case, that of D = 0, in which the mathematics reduces to the need to take the inverse and the trace of the logarithm of a matrix. We find that, as one might hope, a simple 1d bulk theory can work in this case. Since the D ≥ 1 theories involve analogous mathematics, with matrices replaced by differential operators, a reasonable way to proceed might be to carefully make this replacement, which might enable a simple and explicit proof of the duality. In section 5, we begin to turn this idea into a mathematical conjecture about determinants of families of pseudo-differential operators. While the physics arguments suggest that there will be a similar conjecture in any dimension D, here we will only cover the case D = 2, where it generalizes a known

Fact 1. The logarithm of the determinant of the scalar Laplacian Δ on a two-dimensional manifold Σ, as a functional of the metric on Σ,is equal to the action of a particular solution of an SL(2, R)×SL(2, R) Chern- ∼ Simons theory on a three-dimensional manifold M with boundary ∂M = Σ, with boundary conditions determined by the metric on Σ.

This follows by combining the Polyakov formula for these determinants [2, 48] with results in 2 + 1 gravity [7, 17, 43, 55]. Conceptually, this is an example of a general relationship between anomalies in D-dimensional QFT (here the conformal anomaly) and Chern-Simons terms in D+1-dimensions. Similarly, by combining the proposed O(N) model duality with results on higher spin gravity, we make

Conjecture 1. The logarithm of the determinant of a pseudodifferential operator O on a two-dimensional manifold Σ , or a family of operators O(t), is equal to the action of a particular solution of an G×G Chern-Simons theory on a three-dimensional manifold M with boundary ∼ ∂M = Σ, and boundary conditions determined by the choice of O(t).HereG is an infinite dimensional Lie algebra called hs(1, 1) anddefinedin§3.1. OPERATOR TRACES AND HOLOGRAPHY 135

One realization of hs(1, 1) is the subalgebra of area-preserving vector ﬁelds on the two-dimensional hyperboloid a2 − b2 − c2 = 1, which are odd under inversion. To begin explaining why this conjecture extends the theorem, we note that since SL(2, R) acts on this hyperboloid by isometries, we have SL(2, R) ⊂G. Thus, the case O(t) = Δ corresponds to a subset of boundary conditions in which the G×G solution actually lives in SL(2, R)× SL(2, R). Of course, we will need to say far more to motivate a conjecture of this generality. It is a pleasure to thank Luca Mazzucato, Leonardo Rastelli and Shlomo Razamat for discussions on these topics, and Is Singer for encouraging me over the years to think more mathematically about large N limits.

2. Dualities between quantum field theory and gravity Let us start by reviewing the physical framework of large N limits. We begin by defining a quantum field theory partition function ZQF T,N for a sequence of QFTs labelled by N ∈ Z+, schematically 2 i −N t Oi[Φ] (1) ZQF T,N = [dΦ] e i

Here (∂M,γ) is a manifold with Riemannian metric γ, and Φ is a map from ∂M to a ‘configuration space’ – this could be a U(N) connection, a map to a vector space with a linear U(N) action, a direct sum of these, perhaps with some fermionic (odd) variables, and so on. We denote the space of CN ≥ such maps as .Thus,fordimM 1, the integral CN [dΦ] is a functional integral, requiring regularization, renormalization, and the whole works. The Oi are real or complex valued functionals on CN , usually called ‘observables’ or ‘operators.’ They appear in Eq. (1) weighed by parameters i (or ‘couplings’) t . One way to think about this is to regard ZQF T,N as a generating function for expectation values under the integral, as in ∂ (2) log ZQF T,N = E[Oi] ≡Oi, ∂ti and analogous n’th order derivatives, the n-point correlation functions. To some extent, the metric γ can also be thought of as one of the parameters t, and its corresponding observable Oγ is called the ‘stress-energy tensor.’ However, in the standard discussions, since almost all of the other observables Oi depend on the metric, typically in a nonlinear way, the metric dependence is more complicated than that suggested by Eq. (2). And this is the tip of an iceberg, as once one gets into regularization and renormalization, the exponentiated linear dependence on the ti which we wrote in Eq. (1) turns out to be a great oversimplification. The defining property of the sequences of QFTs we consider is that the CN have a uniform definition in terms of a space C of maps into an algebra A, or related structures such as a module for A and a connection over it. We 136 M. R. DOUGLAS will also need a trace on A, and might also need an involution on A or other structure. The concrete physics model ZQF T,N is then obtained by taking N N A = MatN ≡ Hom(C , C ). Thus CN admits an action of U(N) inherited from the linear action on CN . We require that this is a symmetry; in other words U(N)actsonCN in a way which preserves the measure [dΦ] and the observables Oi. Although normally one discusses gauge theories in which C includes a connection over ∂M, the focus in this paper will be on the simpler case ∼ N CN = Map(∂M,C ), in more abstract terms C isamoduleforA. Having defined the sequence of QFTs, a basic claim about the N →∞ limit is

Claim 1. (’t Hooft, Witten): There is a list of observables Oi which have good limits, obtained by taking a normalized trace of elements of A, such that the limiting partition function is a sum over saddle points in the following sense: 2 N SQG[∂M,γ,Oi] (3) lim ZQF T,N [∂M,γ,t] = lim e . N→∞ N→∞ ∂SQG/∂Oi=0

In words, the large N limit of the logarithm of ZQF T (the free energy or effective action) is given by evaluating a functional SQG, which is universal for a particular QFT, at its critical point(s) dSQG =0. Let us sketch a proof of this in the simplest case, of a matrix integral, following [12]. We take ∂M to be a point, and C to be the hermitian elements Φ=Φ∗ of A. We then consider the integral i −N tiTr Φ (4) ZMM,N[g]= [dΦ] e i≥1 , C where [dΦ] is the product of Lebesgue measure for each matrix component. Since the integrand is invariant under the conjugation Φ → U †ΦU with U ∈ U(N), it is a function of the N eigenvalues of Φ, or equivalently the first N moments Tr Φi. One can then change variables in the integral from the components of Φ, to a choice of group element U ∈ U(N) and a choice of moments. This brings in a Jacobian, which is the volume of a U(N) orbit with specified moments. It has a simple closed form expression, which can be found in [12]. To heuristically justify the large N limit, we make the following two observations. First, for a ‘typical’ orbit in which Φ has order 1 eigenvalues, the moments Tr Φi ∝ N, so the exponent in Eq. (6) is of order N 2.Thuswe define the observables

1 i (5) Oi ≡ Tr Φ , N OPERATOR TRACES AND HOLOGRAPHY 137 and write 2 i −N t Oi (6) ZMM,N[g]= dOi Vol (Oi) e i≥1 , i where Vol (Oi) is the volume of a U(N) orbit. Then, the volume of a typical orbit has the large N asymptotics

2 (7) Vol (Oi) ∼ exp N Seff (Oi), in terms of an ‘entropy functional’ Seff , the ‘free entropy’ of Voiculescu [61]. Thus, in the limit, Eq. (6) is dominated by its saddle points, and we recover Eq. (3) with i (8) SSG = Seff (Oi) − t Oi. i

It is natural to try to explain this by finding an algebra with trace A, such that the values of Oi at the saddle point are obtained by evaluating Eq. (5) for a particular Φ ∈A, the ‘master field.’ In this example, this can be done using the free probability theory of Voiculescu [62]. Equally explicit treatments of the limit have been made for other simple quantum theories, such as matrix quantum mechanics, and D = 2 Yang-Mills [54]. It is tempting to believe that someday this will be made precise for more complicated quantum field theories. There are also probabilistic definitions of the limit of the integral Eq. (6), based on large deviation principles [34].

2.1. Matrix model/noncritical string duality. So far, while we have motivated the claim of Eq. (3), we have not explained why SSG should have any relation to string theory or quantum gravity in D + 1 dimensions. Let us brieﬂy recall how this works in the simpler matrix model examples, developed in the late 1980’s. In these theories, the relation to string theory is precisely the one postulated by ’t Hooft, that a planar diagram is a discrete approximation to a string world-sheet, and that string theory is obtained by taking a continuum limit. The simplest example or “pure gravity” is deﬁned in terms of the matrix integral Eq. (4) as a “double scaling limit” [13,23,32] taking the couplings ti to a critical point as N →∞. Here D = 0 and the dual theory is the so-called “c = 0 string,” a non-critical bosonic string with a one-dimensional target space. In terms of the Φ variable, one formulates observables depending on a new coordinate r,as

−rΦ (9) Or ≡ Tr e or some functional transform of this, and rewrites Eq. (8) in terms of these. ∼ + The coordinate r then parameterizes a one dimensional manifold M = R , whose boundary at r = 0 might be identiﬁed with the point ∂M. 138 M. R. DOUGLAS

The idea that the D = 1 theory should be geometric and thus is some theory of gravity is motivated by the general arguments that closed string theory contains gravity. Of course, readers who have heard that bosonic string theory requires D = 26 may be wondering how a one dimensional string theory can make sense. The answer [49] is that one can turn on a world-sheet coupling which shifts the conformal central charge. Such a coupling arises naturally upon considering a two-dimensional world-sheet metric, so that the new coordinate r then corresponds (loosely) to a distance scale on the world-sheet. This suggests that fixing the couplings t to local observables on the world-sheet, corresponds to putting boundary conditions at r → 0, leading to a string theory space-time picture. This picture became clearer in the “c = 1 model,” for which the QFT is the D = 1 quantum mechanics of a hermitian matrix M, with a potential V = −tr Φ2. Now we can take the field theoretic dimension to be ‘time’ and the extra r dimension of Eq. (9) to be ‘space,’ so that the dual D +1=2- dimensional theory naturally lives on a Lorentzian space-time. In this case, the equations of motion dSSG = 0 following from the counterpart of Eq. (8) are a second order hyperbolic PDE [19, 47], one can develop a scattering theory, according to which waves (small perturbations) sent in at r ∼ 0enter the system, bounce off a ‘wall’ at large r, and come back to r ∼ 0, defining an S-matrix. Again, Eq. (3) holds with a QFT coupling–boundary data relation. The heuristic physics explanation for this relation is the ‘UV/IR correspondence,’ according to which the extra dimension r in the quantum gravity corresponds to a ‘renormalization group scale’ parameter, here acting on the world-sheet. The boundary condition corresponds to fixing the QFT action at short distances (the UV), while the evolution equation dSSG = 0 some- how corresponds to incorporating quantum fluctuations at larger distances. Finally, the large r boundary condition or ‘wall’ is a regularity condition, corresponding to the fact that the renormalization group removes (and does not create) degrees of freedom. Thus many elements of gauge-gravity duality were visible in these models; on the other hand supersymmetry played no role, and the extra dimension looked very different from the field theory dimensions, giving little guidance for how to go to D>1.

2.2. AdS/CFT and the conformal boundary. Before AdS/CFT, the search for string duals of gauge theory had generally focused on modifying the world-sheet action for the string, to avoid the following paradox: In quantum field theory, local observables, for example a correlation function of a set of operators at distinct points, are easy to define. On the other hand, in quantum gravity, diffeomorphism invariant local observables are nearly impossible to define, because they are highly nonlinear in the metric, a fluctuating quantum variable. And in string theory, the extended nature of the string makes this problem even worse. OPERATOR TRACES AND HOLOGRAPHY 139

The now accepted solution of this paradox seems to have first appeared in [50], and consists of taking the string to propagate in a D+1-dimensional metric of the form 2 (10) ds2 = dr2 + γ , r2 where r is an extra ‘radial’ coordinate, and is a (fixed) curvature length. In words, we introduce a family of space-time metrics parameterized by an extra coordinate r, related by an overall rescaling. For γ the Minkowski metric on space-time, the metric Eq. (10) covers part of a constant negative curvature metric with isometry group SO(D, 2), the anti-de Sitter (AdS) space-time. The metric Eq. (10) first appeared this way in string theory by looking at the geometry near the horizon of a group of D3-branes; it is the direct product of Eq. (10) with a round S5. Since the D3-branes can also be described (at low energies) by N = 4 super Yang-Mills theory, there is a direct physical argument for the duality in this case [41]. The relation of correlation functions to boundary conditions was then proposed in [33,65]. N = 4 super Yang-Mills, here denoted MSYM4,isaD = 4 Yang-Mills theory with various scalar and fermionic fields and an action chosen to realize the maximal supersymmetry possible in D = 4 in a non-gravitational theory. By doing quantum perturbation theory, one learns that it is a superconformal field theory, so the Yang-Mills coupling is a parameter of the quantum theory. Doing the functional integral over fields which live on a D- dimensional Riemannian manifold (∂M,γ), one obtains ZQF T,N of Eq. (3). The right-hand side of Eq. (3), at finite N, is a quantum gravity or string theory partition function. For MSYM, it is type IIb superstring theory compactified on S5,togetaD + 1 = 5-dimensional quantum gravity. Heuristically, one also thinks of this partition function as defined by a functional integral, now over a fluctuating metric g and various fields φ. While this intuition has never been made precise for quantum gravity or string theory, even to physics standards, in Eq. (3) we sidestep this issue by arguing that in the large N limit, the quantum gravity partition function reduces to a sum over critical points of the supergravity action SQG. Thus, granting the picture of the previous subsection, we obtain a concrete form of Eq. (3). Now, to explain the solution of the paradox, let us grant that the problem with local observables in quantum gravity only arises when we try to localize on length scales shorter than some fixed length scale L, which characterizes the fluctuations of the metric (the Planck scale), the size of a string (the string scale), or other quantum gravity fluctuations. This is physically reasonable: after all, despite the underlying quantum nature of gravity, experimental physicists can work with very short distances d using the ordinary classical picture of space-time, because L ∼ 10−33 or so (the Planck scale in our universe) and d L. 140 M. R. DOUGLAS

Applying these ideas to quantum gravity in AdS, since L is a physical length scale, it relates to distances defined using the metric Eq. (10). But, a fixed space-time distance d measured in the metric γ corresponds to an AdS metric distance d/r,whichasr → 0 becomes much greater than L. Thus, by associating QFT correlation functions with boundary conditions at r → 0, we can localize them on arbitrarily short scales d, and avoid the paradox. 2 It is convenient that the metric Eq. (10) is conformal to a metric ds˜ = r2ds2 which is non-singular at r = 0. Indeed, since r = 0 is at finite distance in this metric, we can think of it as the boundary ∂M of a closed manifold with metric (M,g˜). In mathematical terms, (M,g˜) is the conformal compactification of (M,g). In this way, we can replace the physics idea of scattering boundary conditions at r ∼ 0, with local boundary conditions on ∂M. Thus we have taken another step towards making Eq. (3) precise. A familiar variation is to take γ with Riemannian signature, defining a ‘statistical field theory.’ This corresponds to Riemannian γ in Eq. (10). For example, taking (∂M,γ) to be the round D-sphere, the metric Eq. (10) will be D + 1-dimensional hyperbolic space. One then expects the equations dSSG = 0 to be elliptic, and the corresponding boundary value problem to have a unique solution. For example, a metric on M satisfying the Einstein equation with negative cosmological constant (constant negative Ricci curvature), should be determined by the conformal class of its restriction to ∂M. As discussed in [3], the nonlin- earity of the Einstein equations makes such a claim highly nontrivial, and false without restrictive assumptions. But, as shown by Fefferman and Gra- ham [27], it is generally true in the sense of an asymptotic expansion: given γ on ∂M, one can solve for the higher order terms γ(k) in

2 (11) ds2 = dr2 + γ + r2γ(2) + ··· , r2 if D is odd to all orders. This can be used to deﬁne correlation functions of the metric as in Eq. (2) (with t → γ). A simpler example is a scalar ﬁeld φ whose equation of motion ∂SSG/∂φ = 0 is simply the Laplace equation on M. By taking Dirichlet boundary conditions, one gets a well posed problem, whose solution is given by a “boundary-to-bulk Green function,” (12) φ(x, r)= dDx φ(x, 0)G(x; x, r). ∂M

One can then treat nonlinear terms in ∂SSG/∂φ by the usual perturbative approach. Since one only needs a classical solution, the series expansion for an n-point function will have a ﬁnite number of terms, and there are many such calculations for low n in the literature.

2.3. Global symmetries and gauge symmetries. It is rather magi- cal that the large N action SSG of Eq. (3), which according to the arguments OPERATOR TRACES AND HOLOGRAPHY 141 so far might be too complicated or abstract to work with, ever takes such a simple and explicit form as a supergravity action. While this follows from the string theory intuition, apriorithis argument is very closely tied to the specific theories which come out of string and M theory, leaving the question of what class of QFTs might admit gravity duals completely open. Is it only these theories, or theories which can be obtained from these by adding operators? If it is a larger class, what characterizes it – gauge symmetry? conformal invariance? supersymmetry? In some cases such as MSYM4, one has integrability. Each of these properties have been suggested to be important. Perhaps the best clue we have at present is the relation between global symmetries of the QFT, and gauge symmetries of the gravity theory. For example, MSYM4 has an SU(4) R-symmetry (a global symmetry under which the supercharges transform nontrivially), while IIb supergravity compactified on S5 leads to a five-dimensional gauged supergravity with SU(4) gauge fields. The most far reaching case is the relation between the stress- tensor of QFT, a spin two operator which generates diffeomorphisms acting on (∂M,γ), and the metric g of the dual gravity theory. Clearly this relation lies at the heart of the matter. There are various physical arguments for it. In the context of theories which can be obtained from branes in string theory, such as MSYM4 and D3- branes, one has the general comment that any gauge symmetry of the string theory, must couple to some current (operator) on the brane which generates a corresponding global symmetry [8]. One also has a general relation between global anomalies on the brane, and corresponding Chern-Simons terms in the bulk [30,65]. Granting the gauge-gravity duality, one can reverse this logic by observing that global symmetries of QFT are generated by operators (currents) with spin (i.e., which transform nontrivially under the Lorentz group). Such an operator will correspond to a massless field with spin in the bulk theory. But massless fields with spin only make sense in a unitary quantum theory if there is a corresponding gauge invariance. Without gauge invariance, the Hilbert space for a particle with spin would decompose into finite dimensional linear representations of the Lorentz group, but this is noncompact and has no finite dimensional unitary representations. This paradox is evaded by embedding the Lorentz group into a larger group, whose representations can be unitary after quotienting by the gauge group (this is usually done by BRST quantization, i.e. taking equivariant cohomology). The same logic can be applied to translational symmetry, which is generated by a spin two current (the stress tensor), to explain the need for general covariance and derive the Einstein-Hilbert action [21]. A final reversal of this argument would be to derive the global symmetry of the QFT from the bulk theory, purely in the context of gauge-gravity duality (as opposed to string theory or branes). In fact such an argument in a sense precedes the others, as it amounts to deriving the group of symmetries 142 M. R. DOUGLAS which act on the asymptotic region of a space-time in gravity. This is an old question: for example, how does one see geometrically that the Poincaré group acts on an asymptotically flat solution, without referring to explicit coordinates? How does one define the corresponding conserved quantities? These are subtle problems, about which entire reviews have been written [57], and are clearly central to this subject. Recent works on the asymptotic symmetries of higher spin gravity include [15,36].

3. Free field-higher spin gravity duality Although the usual discussion involves gauge theory, there is a simpler and older class of field theories with large N limits, the so-called vector models. The basic example is a theory of N complex scalar fields denoted ψ, with the action √ ij 2 2 2 (13) Sft = γγ ∂i¯ψ · ∂jψ + t|ψ| + λ(|ψ| ) . ∂M

Here a·b is the usual hermitian inner product on CN , |a|2 = a·a, λ ∈ R,and in addition we allow a real-valued function t : M → R (in physics terms, a position-dependent mass). It is not hard to obtain a large N action of the form Eq. (8) for this theory, as done for example in [20]. It is also not hard to compute the partition function directly. Let us start with the special case λ = 0. Since the action is then quadratic in ψ, a straightforward application of the theory of Gaussian functional integrals tells us that

−NF[g] −N ZQF T,N [γ,t] ≡ e =det(Δγ + t) (14) = exp −N Tr log (Δγ + t) ,

ij where Δγ = −γ ∇i∇j is the scalar Laplacian. This depends on N, but in a trivial way. Of course, this will require regularization to make sense. We could take det to be the zeta function regularized determinant [51]. ∂ (15) det A = exp lim Tr As. s→0 ∂s We will return to this point in §4.4. Having warmed up with λ = 0, to handle the general case, we note that 2 √ t +t|ψ|2 −λ(ψ|2)2 (16) dte 2λ = 2πλ e .

Thus, we can eliminate the quartic term in Eq. (13) in favor of a quadratic term, at the cost of doing another functional integral over t.Ifwetake OPERATOR TRACES AND HOLOGRAPHY 143

λ ∝ 1/N , this integral becomes Nt2 (17) dte 2λ ZQF T,N [γ,t]

2 N t +Tr log(Δ +t) (18) = dte 2λ γ and in the large N limit, it will be dominated by a saddle point. Thus, the case λ = 0 can also be solved in the large N limit, if we can get a sufficiently explicit expression for Eq. (14). In the usual physics case of γ the Euclidean metric and constant λ,it is easy to get such an explicit expression by diagonalizing Δγ, and thus the O(N) model is exactly solvable. However, if we consider more general metrics, or position-dependent t, this is not so easy. One can of course treat the case of perturbations around this solvable case, but an explicit formula for Eq. (14) is not known. Since the relation between Eq. (13) and Eq. (14) was simply that of Gaussian functional integration, at least formally it generalizes to an arbitrary linear operator in the action, (19) dψ e−(¯ψ,Oψ) =(const.) exp −N Tr log O, with i ij ijk (20) O = t + γ1∂i + γ2 ∂i∂j + γ3 ∂i∂j∂k + ···. We could treat this expansion in two ways. One is to try to define a functional of all of these coefficients, say Tr log O[t, γ1,γ2,...]. Less ambitiously, we could regard Eq. (19) as the generating functional of correlation functions of general local operators, obtained by taking derivatives with respect to the γn at γ = 0. This allows us to make contact with computations such as that of [28], so for this section let us be satisfied with this definition.

3.1. Higher spin gravity. Although there are several approaches to higher spin gravity, perhaps the simplest is to define it as a gauge theory, along the general lines of Ashtekar’s approach to 3 + 1 gravity [4] and Wit- ten’s approach to 2 + 1 gravity [64], but with an infinite dimensional gauge group. In general terms, instead of the basic field being the metric tensor, one takes as fields the frame (or D-bein) and spin connection, and reinter- prets these as components of a gauge connection. Just as it turns out that the Einstein-Hilbert action can be re-expressed as a gauge theory action, so too can the higher spin gravity action. Thus, let us first review the analogous formulation of standard 2 + 1- dimensional gravity. In these dimensions, the simplest gauge theory action is the Chern-Simons action, and Einstein gravity was reformulated in these terms in [64]. One starts with a three dimensional manifold M, and a vector 144 M. R. DOUGLAS bundle V over M with structure group SO(2, 1). One then introduces a frame e ∈ Hom(TM,V) and a connection ω on V . One then embeds SO(2, 1) in a larger group G whose additional generators correspond to translations, and identifies (e, ω) with components of a G-connection. Chern-Simons gravity is then simply Chern-Simons theory on M with group G, so a classical solution isaflatG connection. For 2 + 1 gravity with a negative cosmological constant, so that a solu- 2 ∼ tion has constant negative Ricci curvature − , one takes G = SL(2, R) × SL(2, R), with connections A and A˜ and the identification

1 1 (21) A = ω + e; A˜ = ω − e.

The action is the difference of two Chern-Simons actions, 2 3 (22) SQG = SCS[A] − SCS[A˜]; SCS[A]=tr AdA + A . 3 Its simplest classical solution is the group manifold of SL(2, R) itself, taking its left- and right-invariant connections as A and A˜ respectively. Its univer- ∼ 1 sal cover is AdS3. It is conformally compact, with boundary ∂M = R × S carrying the flat (Minkowski) metric. The duality conjecture Eq. (3) thus relates this gravitational theory on AdS3 to a two-dimensional QFT on the cylinder. The formulation of higher spin gravity in these terms is attributed in [36] to [10]. We again take the action Eq. (22), but A and A˜ are now connections on M each taking values in the “higher spin algebra hs(1, 1),” an infinite dimensional Lie algebra containing an sl(2, R) subalgebra. Thus, one again has AdS3 as a solution, by embedding the previous connection into this larger algebra. There are various definitions of hs(1, 1) in the physics literature, almost all in terms of generators and relations, or else oscillators. In [11], it is identified with the odd area preserving vector fields acting on a 2d hyperboloid with sl(2, R) symmetry (i.e., the hypersurface a2 − b2 − c2 =1 in R3,and ‘odd’ means odd under inversion). Another suggested geometric interpretation of hs(1, 1)⊕hs(1, 1), and its generalizations to arbitrary D, is the Lie algebra contained in the associative algebra of symmetries of the Laplacian.

Definition 1. (Eastwood, [25]) A symmetry of the Laplacian Δ is a linear diﬀerential operator D such that

(23) ΔD = δΔ for some linear diﬀerential operator δ. The algebra of symmetries of the Laplacian is the algebra of such operators with the natural product, with the OPERATOR TRACES AND HOLOGRAPHY 145 equivalence relation ∼ (24) D = D + PΔ with P a linear diﬀerential operator.

Eastwood then argues that symmetries of the Laplacian correspond to conformal Killing tensors, which are also the natural symmetries which arise from the point of view discussed in §2.3. In the particular case of the Laplacian on RD (with the Euclidean metric), their algebra is the universal enveloping algebra of SO(D +1, 1), modulo a two-sided ideal generated by the Killing form (schematically ∼ 1). The relation to the algebras used in deﬁning higher spin gravity is discussed in [60]. For present purposes, however, let us simply cite a concrete deﬁnition of hs(1, 1). It is the space of even polynomials in two variables ξ1,ξ2 modulo constants, with the bracket derived from the associative product ∗ ≡ ∂ ∂ − ∂ ∂ (25) (f g)(ξ) exp i 1 2 2 1 f(ξ)g(η) . ∂ξ ∂η ∂ξ ∂η ξ=η

The quadratic polynomials generate an sl(2, R) subalgebra, say

(26) J + ≡ ξ1ξi; J 3 ≡ ξ1ξ2; J − ≡ ξ2ξ2.

3.2. Duality between correlation functions. It is easy to develop a perturbative expansion for Eq. (19), using

(−1)n (27) Tr log(O + δO)=TrlogO + Tr (O−1δO)n. n≥1 n

Let us look at the three-point correlation function, discussed for D =3 in Giombi and Yin [28]. We take z to be a coordinate on ∂M and O to be the Laplacian, so that O−1 is the standard ﬁeld theoretic Green function. For ∼ D ∂M = R , 2−D (28) ψ¯(z1) ψ(z2)≡F (z1,z2)=const. ·|z1 − z2| .

An n-point function of operators ψ¯Oiψ is then a sum of (n − 1)! terms of the form 2 2 (29) A = d z1 ...d zn O1F (z1,z2)O2F (z2,z3) ...OnF (zn,z1).

For deﬁniteness, consider D =3 and n = 3, with three operators Ji =: ψ¯(zi)∂iψ(zi) :, one gets expressions like ∂ ∂ ∂ 1 (30) +(2↔ 3). i1 i2 i3 | − || − || − | ∂z1 ∂z2 ∂z3 z1 z2 z2 z3 z3 z1 146 M. R. DOUGLAS

The dual expression is found by following the approach outlined in §2.2; one ﬁnds (SG) 2 A 1 2 3 1 2 3 (31) k1,k2,k3 (z ,z ,z )= d zdr Gk1 (z ; z,r)Gk2 (z ; z,r)Gk3 (z ; z,r),

where Gki (zi; z,r) is the bulk-to-boundary Green function Eq. (12) for a massless spin 1 field (Eq. (2.26) in [65]). At this general level, there is little resemblance between the two expressions. Actually, given a metric on ∂M with enough conformal isometries, such as the standard cases of RD or SD, symmetry forces the three-point functions to have the same functional dependence, so the nontrivial pre- diction of the duality is the overall normalizations. However this argument quickly peters out for higher point functions, and loses all strength for general metrics and operators O. The best studied examples come with their own special simplifications. For example, D = 4 MSYM has a large superconformal symmetry, which forces many relations between correlation functions. In our primary example of D = 2, one has holomorphic factorization. In the Chern-Simons dual, this is reflected in the form G×Gfor the gauge group. There has been a fair amount of work trying to give more general arguments for this equality, which could work for any of the proposed dualities. One of the more interesting ones appears in [29] and involves a relation between the proper time parameterizations ∞ (32) O−1 = dτe−τO 0 for the QFT Green function Eq. (28) and the bulk-to-boundary Green function Eq. (12), motivated by the physics intuition relating the two sides of Eq. (3) to open and closed strings. This argument is quite simple for the three-point function in a symmetric background, but it is not clear how to take it farther. In any case, the equality of Eq. (30) and Eq. (31), as demonstrated for D =3 in[28], looks sufficiently nontrivial and convinces us to take O(N)- higher spin gravity duality seriously.

4. Why should gauge-gravity duality work?

Most of the work on this question focuses on the AdS5/MSYM4 duality. This is a very rich subject which we will not try to do justice to here, instead raising a few questions which the author ﬁnds interesting. First, compared to previous ideas about large N limits and Eq. (3), a lot of the power of AdS/CFT comes from the statement that one can take a second limit, of strong (large) Yang-Mills coupling, in which SSG becomes a local ﬁeld theory, such as type IIb supergravity. OPERATOR TRACES AND HOLOGRAPHY 147

Now there is a simple physics argument for this – it follows from the mapping of parameters (N,gYM) to the dual string theory coupling and string scale ls, which translates the limit gYM →∞into the limit ls → 0. In this limit, the higher dimension operators of string theory go off to infinite dimension, leaving only the fields of supergravity. While this argument sounds quite reasonable, it remains surprising from any field theoretic point of view. Are we sure that it is true? After all, there is another scale in the problem, the AdS curvature radius. There exist gravity theories which are nonlocal on this scale – in fact the higher spin gravity we discussed in section 3 is an example. One might entertain a different conjecture that, even at large Yang-Mills coupling, gravity duals can have similar nonlocality. For the case of MSYM4, results from integrability are believed to show that all non-supergravity operators do in fact go off to infinite dimension at strong coupling, eliminating this loophole. Granting this, for what class of theories does this work? Integrability is very special, and almost all theories of physical interest are not integrable; the principle which makes this work should be more general. Second, where does the extra dimension come from? The standard intuition involves the renormalization group. While very believable, it seems fair to say that this has not yet been made as precise as one would like. This question is discussed further in [24]. Third, as we commented in the introduction, making a precise statement away from strong coupling probably requires matching up all of the operators on both sides of the duality. How do we organize all the higher derivative operators? This is surely a question of geometry – what geometry underlies the duality?

4.1. AdS1/CFT0 duality. Let us return to discuss the duality between free QFT and higher spin gravity. The simplest point of view would be that this discussion depends so little on details, that even a schematic expression like Eq. (19) could have a gravity dual. Since readers with some familiarity with higher spin gravity or other detailed physics discussions may be skeptical at this point, let us explore this idea in the simplest possible context. Thus, let us consider the D = 0 case, in which the operators O[t]are N × N hermitian matrices, i.e. elements of the Lie algebra u(N ). The reason we write N is that, in the standard discussions, one restricts attention to gauge theory or at least to u(N) singlet operators, whereas we want to consider general operators. Of course, one can consider the configuration space CN ⊗ CN , and tensor the operator O[t] of Eq. (19) with the identity acting on CN , so in a free theory this is a distinction without a difference. Thus for ease of notation we simply take CN as the configuration space in the following, and consider general operators. 148 M. R. DOUGLAS

Let ea be a basis for u(N) and write a (33) O[t] ≡ t ea; a then the partition function Eq. (14) becomes ¯ Z[t]= d2N ψe −ψO[t] ψ (34) = exp −Tr log O[t].

The observables of Eq. (2) are

∂ −1 (35) ua = log Z =TrO[t] ea. ∂ta If we think of t and u as matrices, we have u = t−1. Now, one may ask, how can the simple results Eq. (34) and Eq. (35) be obtained by introducing an extra dimension? Since the D = 0 theory admits a global symmetry O[t] → g−1O[t]g,we expect the bulk theory to be a u(N)orevengl(N) gauge theory,. Thus, we introduce a covariant derivative Dr with connection Ar. Of course, the connection can be gauged to zero in one dimension, so this is not the key point. We take the couplings t to be boundary conditions for a ﬁeld g(r)which is an adjoint of u(N). Since we know we will need to take the logarithm, we deﬁne φ(r)tobe

(36) g = exp φ

Let us try postulating the simplest possible second order equation for φ, ∂2 (37) φ =0, ∂r2 as would follow from the action −1 2 (38) S = dr (g Dr g) in the gauge Ar = 0. Its general solution is φ = Ar + B. According to the general philosophy of gauge-gravity duality, we should use the couplings t as a Dirichlet boundary condition, in other words as the leading or non- normalizable solution. Thus we take A =logt. We then, as in the matrix model examples, postulate a wall at r = 1. At the wall, we impose the boundary condition φ(r = 1) = 0, so B = −A. This is consistent with the variational equation for the gauge connection, which is [φ,φ] = 0 which forces [A, B] = 0. Finally, the normalizable mode determining u is B = −A,and the corresponding part of g is exp − log t = t−1 = u. OPERATOR TRACES AND HOLOGRAPHY 149

Thus, it is easy to get the relation Eq. (35) from a D + 1-dimensional dual theory. Of course, we postulated all of its ingredients, so the significance of this can only be judged by going on to cases with more structure. Still, we are not done with D = 0 yet, as we need to check that the value of the action on this solution reproduces Eq. (34) in some sense. This is Tr A, but the action Eq. (38) gives us the (divergent) volume dr multiplied by Tr A2. Even if we could argue that this divergent term were meaningful, it would not be correct. Thus, we need to add additional terms and fields to the action to cancel it. Since the additional term is negative, adding it will lead to problems of stability when we go on to higher dimensions. This is a sign that we need to bring features of gravity into the discussion. In particular, the action Eq. (38) does not have r-reparameterization invariance. Thus, we introduce a metric ds2 = e(r)dr2 and consider the reparameterization invariant action −1 −1 2 e −1 (39) S = dr e Tr (g Drg) + + αTr (g Drg). 2 Starting with α = 0 and taking constant e gauge, then the constraint from varying e determines 2 2 −1 2 (40) e = (g Drg) (41) = 2Tr A2 given our boundary conditions. Thus, the divergent part of the action is now e 1√ (42) S ∼ dr ∼ dr Tr A2. 2 This is also not what we want. In fact, the term Tr A which would match −1 Eq. (34), would arise from the αTr (g Drg) term. Since it is a total derivative, it does not change the equation of motion, so we can add it, but we still need to subtract the term Eq. (42). To write an action which does this, we drop the term and instead work μ in first order formalism with a canonical pair of variables (φ , Πν), and write μ 1 μν 1 (43) S = dr Tr ΠμDrφ + eη Tr ΠμΠν +TrDrφ . 2 with a constant metric ημν. Now the equations of motion and constraint are μ μν μν (44) Drφ = −eη Πν; DrΠμ =0; η ΠμΠν =0.

To solve the constraint with Π = 0, there must be more than one component in (φ, Π), and the metric ημν must have indeﬁnite signature, say η00 = −1 11 1 −1 and η = 1. We take one component, say Drφ to be g Drg as above, thus 1 we control its boundary condition. Then Π1 = A as above and φ = Ar + B. 150 M. R. DOUGLAS

2 2 We also need Tr Π0 =TrA so the second sector just cancels the unwanted term in the action. The term Tr A we want comes out by taking the total derivative term to only see φ1. While we would not make overly strong claims for the importance of this toy model, it is a very simple illustration of how the dual of a free theory can work. What did the extra dimension buy us? In terms of the (φ, Π) variables, the dynamics is linear, and the transformation A → B = −A is rather trivial. It becomes more nontrivial when we formulate the dynamics in terms of g = eφ. The equations of motion are now

−1 (45) g Drg = −eΠ

−1 (46) DrΠ=[Π, (g Drg)].

The second is still solved by constant Π, while the ﬁrst gives a local evolution role for g, solved (taking Ar =0)by − r dre(r)Π(r) (47) g = g0 · e .

The “wall” boundary condition g = 1 then determines Π = log g0, and the dynamics propagates it back to the boundary. In higher dimensions, one can hope that the ‘wall’ boundary condition will emerge in a less artiﬁcial way, as a consequence of continuity in the interior of AdS. Thus, we can take the logarithm of O[t] using a local 1d bulk theory. It exhibits two other features which we might look for in higher dimensions. First, the indeﬁnite metric and constraint in Eq. (43) is suggestive of the role of the conformal factor in the Hamiltonian constraint for gravity. Second, −1 the term αTr (g Drg) which reproduced the quantum free energy is a one- dimensional analog of the Wess-Zumino term of higher dimensions, which will play an important role in section 5.

4.2. Determinants of operators. From an abstract point of view, linear differential operators are not so different from matrices, so the D ≥ 1 case could be treated the same way. Several questions present themselves: • What is the analog of the group U(N) in the D = 0 discussion? In other words, what are the natural symmetry algebras and groups associated to a space of differential operators acting on a D-dimensional manifold ∂M? • To define a determinant such as Eq. (14), we need to regulate the theory. This is no surprise and can be done in many ways, but regularization will spoil some of the formal properties used in our arguments, such as cyclicity of the trace Tr AB = BA. Can we either show that all the properties we need are true, or else characterize their anomalies and fix up the arguments to take these into account? OPERATOR TRACES AND HOLOGRAPHY 151

• Assuming this can be done, and that we can substitute the appropriate analog of U(N) into Eq. (43), can we explicitly write done the resulting effective action in D + 1 dimensions? if so, will it be local? In any case, why would there be any symmetry relating the extra dimension to the original D dimensions? The physics word ‘local’ usually means that one can write the action as an integral over a functional of the fields and finitely many of their derivatives. Unfortunately, this definition becomes meaningless in a theory with an infinite number of fields. Probably, higher spin gravity should not be regarded as local. Naively, the analog of U(N)inD ≥ 1 is a group of differential operators, for example the operators −Δ+t and everything we can get by taking arbitrary combinations of these. Of course, these are unbounded operators and, although we can add and multiply them, our ability to do anything else is severely limited. And since the inverse of a differential operator is not a differential operator, they do not by themselves form a group. The basic operation in §4.1 turned out to be Eq. (45), the evolution of g by right action of a semigroup evolving it to the identity. If we can make sense of the semigroup action exp −rΠ, then a prescription with boundary conditions along the lines we just gave, could naturally lead to the relation Π=logg. But while a semigroup action is easier to define mathematically, the need to work with logarithms of differential operators again forces us to generalize our space of operators.

4.3. Higher derivative operators. To turn Eq. (3) into a satisfactory mathematical conjecture, we should also revisit the meaning of operators like: ψ¯Oψ involving higher derivatives, as in Eq. (20),

i ij ijk O = t + γ1∂i + γ2 ∂i∂j + γ3 ∂i∂j∂k + ···.

In section 3, we regarded Eq. (1) as a formal power series in the couplings γn. Of course, it would be more attractive to think of Tr log O as a well- defined real valued function of the couplings γn. However, one cannot sum these Taylor expansions around zero; if one starts with γn =0forn>nmax (say nmax = 2 as in Eq. (13)) the γn Taylor series is divergent. This is to say that the higher order derivatives are singular perturbations, on general grounds and in QFT because they change the high energy behavior of O and thus require changing the regularization prescription. Although one can take various approaches to dealing with this inconve- nience, conceptually the cleanest is to put an explicit cutoff into the definition of O. If one takes the couplings γn to be constant, this is easily done in momentum (Fourier transform) space; for example we could take p2 → p2 exp −αp2. One can then add similarly regulated operators pn exp −αp2,to get regular perturbations. The precise details tend to depend on the problem 152 M. R. DOUGLAS at hand, because all such regulators complicate the explicit expressions, and one has to take advantage of specifics to get tractable results. In any case, this approach requires us to consider operators O which are non-polynomial in derivatives. In fact, there are many other reasons to do this, which in mathematics led to the development of the theories of pseudo-differential operators (PDOs) and Fourier integral operators. While we cannot provide an introduction here (see e.g. [52]), the starting point for this is to consider operators O(t) acting on functions f : RN → R of the form (48) (O(t) f)(x) ≡ dDξt(x, ξ) dDyei(x−y)·ξf(y).

For example, a linear differential operator with constant coefficients is given by taking t(x, ξ) to be a polynomial in ξ. However, we do not require t(x, ξ) to be polynomial; rather we impose conditions on how it behaves at infinity, such as t(x, ξ) ∼ ξm for an m’th order operator (this is not very precise, see [52] for the actual definitions). The operators satisfying this condition then form a linear space denoted Sm. One can also define operator classes with logarithmic growth. One then has multiplication laws such as

(49) O(t) O(u)=O(t ∗ u) with 1 (50) (t ∗ u)(x, ξ)= e−i(x−y)·(ξ−η)t(x, η)u(y, ξ)dDydDη (2π)n ∼ 1 I (51) | | ∂ξ tDI,xu, I I ! where ∼ means up to exponentially small corrections. As one might expect, the product of an operator in Sm with an operator in Sn will be an operator in Sm+n. Thus the inverse of an operator in Sn can be a PDO in S−n,and one can define groups of PDOs. This suggests that we deal with the problem of singular perturbations which we raised at the beginning of the section, by restricting O[t]tobe in a particular class of operators Sm,withm = 2 for the standard theory obtained by perturbing the Laplacian. Since we can use non-polynomial symbols, we can still use a family of operators as general as Eq. (20), in principle more general since the symbols need not be analytic. Another style of definition, which would fit better with Eq. (45), would be to choose a reference operator O[0], say the Laplacian, and consider the family of operators

i i 0 (52) exp t Oi ·O[0]; O ∈S . OPERATOR TRACES AND HOLOGRAPHY 153

Either way, we are led to another question: while the physics duality conjecture is made for m =2;doesitmakesenseforotherm? Presumably, the relevant background solution of gravity for m = 2 would not be AdSD+1, but something else. On the other hand, the Laplacian does seem to play a distinguished role. As discussed in §3.1, the gauge algebra in D + 1-dimensional higher spin gravity appears to be the symmetry algebra of the Laplacian on RD, which is a small subalgebra of the algebra of diﬀerential operators. The corresponding gauge group would be functions from M to the corresponding symmetry group. While this group and the group of invertible elements of S0 can both be written formally in terms of star products, the relation between them is not immediately obvious.

4.4. Multiplicative anomaly. A well studied example of Eq. (52) is i to take O[0] to be a 2d Laplacian, and t Oi to be a general function σ,in other words the family of operators ∂2 (53) O[σ]=e2σ − . ∂z∂z¯

In D = 2, there is a well-known formula [2, 48] for the variation of det Δγ with respect to the conformal factor σ, 1 2 √ (54) δ (− log det Δγ)= d z γ0 (−Δγ0 σ + R[γ0]) δσ, 6π ∂M where R[γ] is the curvature scalar. While the scalar Laplacian is conformally invariant in D = 2, the regulator needed to define the determinant is not, leading to an anomaly. However the variation is a local functional of the background curvature and conformal factor. It can be integrated to define the Liouville action Eq. (65), a functional on metrics within a given conformal class. To directly apply the finite dimensional formulas of §4.1, we would presumably want to have a relation such as

(55) det A · B =detA · det B.

Of course, this is false for these operators. But there are corrected versions of these relations [40, 46] which take regularization into account, and can lead to anomalies such as Eq. (54). The basic idea is that the functional

(56) F (A, B) = log det A · B − log det A − log det B is local, because its second variation δ1δ2F (A, B) can be computed along the lines of Eq. (27), and vanishes for variations with disjoint support. Thus one can get a finite series expansion for it, in terms of local functionals of the coefficients of A and B. In fact it can be expressed as a residue trace [67], a 154 M. R. DOUGLAS natural trace on the pseudo-differential operators which (in favorable cases) can be computed purely from the symbol of the operator, and agrees with the log trace. The Liouville action was computed this way from Eq. (53) in Appen- dix A of [46]. In principle, the formulas there could be used to compute the analog of Eq. (54) for an arbitrary operator O[0] in any dimension D. More generally, since the relation Eq. (55) holds up to a computable local correction, one could try to follow the approach of §4.1 making this single modification.

4.5. Relation to the RG. The usual physics intuition is that the extra dimension is related to renormalization group flow. We can see whether this has any analog in the toy model, by looking at the following simplified version of the RG. One of the standard formulations of the RG is to cut off the functional integral, in other words replace an integral over ‘all’ modes of a field φ(x), with an integral only over ‘long wavelength’ modes, say those satisfying Δψ<Λ2ψ for some cutoff Λ. One then derives a formula for the variation of the functional integral with respect to Λ. Let us consider a decomposition of the configuration space into orthonormal subspaces (57) C = Cλ, λ with projectors dP (λ)ontoCλ satisfying (58) 1 = dP (λ)

Λ (59) = dP (λ)+ dP (λ). Λ At fixed Λ, this defines a splitting of C into the λ<Λandλ>Λ subspaces. Parameterize these subspaces as u and v, then we could integrate over the λ>Λ subspace using a formula like 2 † † AB u (60) log [d v] exp − u v CD v (61) =TrlogD + u†(A + B · D−1 · C)u to get an expression for the result of integrating out the modes v, as a partition function and a ‘RG transformed’ action A + B · D−1 · C. Differentiating this with respect to Λ will define an RG flow. In fact, we can look at Eq. (47) as a formally simpler way to accomplish the same thing, in which the variation of the cutoff is determined by the operator Π(r). In this picture, we have some freedom in the specific choice of Π(r)ateachr, subject to the overall constraint that the evolution reach g = 1 at the wall, corresponding to a freedom in how we vary the cutoff. OPERATOR TRACES AND HOLOGRAPHY 155

Let us compare this D = 0 discussion, with the RG in QFT. One very important difference is that since C is now infinite dimensional, we can choose isomorphisms between the subspaces Cλ<Λ for different Λ. This is what is done in the standard RG, by the device of rescaling space-time after integrating over CΛ. Of course, the operation of rescaling is only canonically defined for very special metrics; more generally one needs to accept that these isomorphisms are not canonical, and work with this ambiguity. This is one of the many arguments that a full understanding of the RG must be more geometrical than the existing discussions. Perhaps the gauge group in the dual higher spin gravity is related to this ambiguity; one might even conjecture that the Chern-Simons higher spin gravity of §3.1 is related to a multiplicative anomaly in the symmetry algebra of the Laplacian also discussed there. Leaving these ideas for future work, we recall the exact renormalization group equation near a Gaussian fixed point [45],

2 −S[φ] D D δ −S[φ] (62) δΛ e = d xd yC(x, y) e , δφ(x)δφ(y) where (63) S[φ]= dDx(∂φ(x))2 − V (φ(x)) + ··· is an action containing all possible operators, and δC(x, y) is the variation of the covariance with respect to the cutoff Λ. Conceptually, this is an infinite-dimensional heat equation, and the mathematical treatments justify this picture. Thus, renormalization group evolution smooths out the integration measure e−S, and corresponds to parabolic evolution. This looks quite different from the elliptic or hyperbolic Einstein equation, and the problem of how to reconcile these two pictures has been around since the matrix model days. Perhaps the very recent [35] will shed light on this point. For the O(N) model and free QFT, one can make a complete analysis of Eq. (62), and an attempt to derive gauge-gravity duality from this starting point will appear in [24].

5. Chern-Simons gravity and generalized Liouville theory The proposed duality of free QFT and the O(N) model to higher spin gravity should give us some concrete answers to the questions of the previous section. It generalizes a fairly well-studied relation which we now review.

5.1. Liouville and Chern-Simons gravity. We begin by reconsid- ering the simplest case of Eq. (14), the determinant of the D = 2 scalar ∼ 1 Laplacian with metric γ and t = 0, on a cylinder ∂M = R × S . For any γ, 156 M. R. DOUGLAS we can find a complex coordinate z, in terms of which 2σ 2σ 2 (64) γ = e γ0 ≡ e |dz| . As discussed in §4, the log determinant is then given by the Liouville action 1 2 2 2 2σ (65) SL = − d z |∂σ| + λ e . 24π ∂M More generally, one can consider varying the conformal class of metric. In a fixed complex coordinate z, this could be expressed in terms of a Beltrami differential (66) γ = e2σ|dz + μdz¯|2.

Variations of log det Δγ with respect to μ then define correlation functions, zz 2 Eq. (29) with O = γ ∂z . All of these results can be obtained from SL(2, R) × SL(2, R) Chern- Simons theory in an AdS3 background. This goes back to the 1980’s work on 2d quantum gravity [39], and was related to 2 + 1 Chern-Simons gravity in [17]. The point that this relation is a simple version of AdS/CFT has been made in many works such as [43,55]. The problem is to calculate the value of the Chern-Simons action on a flat connection (A, A˜) with specified boundary conditions. In general terms, this can be reduced to a calculation on ∂M by writing (67) A = g−1 · dg; A˜ = dg˜ · g˜−1 and expressing the action as a total derivative in r. This leads to a chiral Wess-Zumino-Witten action on ∂M for each of the SL(2, R) factors, and after taking boundary conditions corresponding to Eq. (64), to the Liouville action [7,17]. One of the difficulties in carrying this out explicitly is that whereas some components of (A, A˜) correspond to the metric on ∂M, others must be determined by solving the flatness conditions. These can be simplified by a clever parameterization of g, g˜. Another subtlety is the need to add boundary terms to Eq. (22) which correspond to the choice of boundary conditions. To explain, the variational equations for a first order action (68) A ∧ dB M will include boundary terms unless we take variations with A|∂M = 0 (Dirich- let) or (ιndB)|∂M = 0 (Neumann; here n is a normal to ∂M). We can choose the other boundary conditions (say, Dirichlet for B) at the cost of adding a boundary term ∂M AB. Following [7,17], one can write the flat connections Eq. (21) in the form

(69) A = rJ3 (d + α)r−J3 ; A˜ = r−J3 (d +˜α)rJ3 , OPERATOR TRACES AND HOLOGRAPHY 157 where α andα ˜ are flat SL(2, R) connections on a surface of fixed r,and (J +,J3,J−) is a basis of the Lie algebra sl(2, R). Taking the surface at r → 0, we can use (α, α˜) to specify the metric (actually the frame or zweibein) on ∂M. Expanding Eq. (69) gives us 1 − dr 3 3 + (70) A = α J− + − + α J + rα J+ r r 1 + dr 3 3 − (71) A˜ = α˜ J+ + +˜α J + rα˜ J−, r r so they correspond to a frame and metric on ∂M 1 − + (72) e = α J− − α˜ J+ ; 2 1 dr (73) ds2 = dr2 − α−α˜+ + · (α3 − α˜3)+O(r2). r2 r The cross term dr · α does not belong in an expansion of the form Eq. (11), but it can be canceled by taking a connection with α3 =˜α3. This implicitly tells us what type of boundary terms must be added to Eq. (22); see [7]for the details. Evaluating the resulting action on Eq. (69), one finds 1 3 3 2 2 (74) SSG = − Tr (α) − (˜α) +3((α +˜α )J3 dr 3 M 1 (75) + r−2α−α˜+ − α3α˜3 + r2α+α˜−. ∂M 2 −2 − + This contains a quadratically divergent term r ∂M α α˜ , and a logarith- mically divergent term dr.... Both are expected divergences in the QFT, the first proportional to the volume, and the second reproducing the trace anomaly. The rest of the action is finite, and reproduces the Liouville action Eq. (65) (shown in [7, 17]), as well as the generating functional of stress- tensor correlators defined using the metric Eq. (66) (shown in [7]). Combin- ing this with the physics derivations of these results for the 2d free boson, or the more general approach of §4.4, we have explained ‘Fact 1’ of the introduction.

5.2. Generalized Liouville theory. We can now explain ‘Conjecture 1’ of the introduction. It is a generalization of the above, starting from higher spin Chern-Simons gravity, fixing boundary conditions to obtain a generalization of the SL(2, R) Wess-Zumino-Witten or Liouville actions. The duality relation Eq. (3) then states that this is equal to the logarithm of the 158 M. R. DOUGLAS determinant of a general 2d linear pseudo-differential operator. Similar conjectures should hold in any dimension D with a higher spin gravity theory, as in D = 3 where it follows from the physics conjecture of [38,53]. Although the conceptual parallel with the SL(2, R) × SL(2, R) analysis seems clear, we have not yet worked out the details. Some steps in this direction are taken in [15, 36]. The starting point is to generalize Eq. (69), and identify the leading coefficients in the expansion of α, α˜ in powers of r, as coefficients in the expansion Eq. (20). Thus we identify the generators of SL(2, R) ⊂ hs(1, 1) as in Eq. (26), and generalize Eq. (70) to dr 1 (76) A = − ξ1ξ2 + eσdzξ1ξ1 + γ(4)ξ1ξ1ξ1ξ1 + γ(6)(ξ1)6 + ··· r r + O(r0)

(resp. A˜), where the subleading terms are determined by the flatness conditions, perhaps using a generalization of the Bruhat decomposition of A = g−1dg used in [7]. Here we have written γ(2) = e2σ and generalized the higher order coefficients in like fashion. Of course, there are many further components of the tensors γ(k) in Eq. (20), but as these can be simplified into terms involving the Laplacian, they only lead to local effects. We should distinguish the 2d conjecture under discussion from another statement about determinants. This is based on the well known physics relation between 2d free QFTs (bosons or fermions) coupled to gauge connections, and Wess-Zumino-Witten theory [63]. For free fermions, this states that −1 (77) Tr log DA = SWZW(G)+ tr A·G dG ∂M where DA is a connection on a rank K vector bundle E over ∂M, G is a map ∂M → GL(K), and we evaluate the right hand side at a critical point in G. This can also be expressed in terms of the Chern-Simons action [26], and is another example of the relation between anomalies (here in chiral GL(K) invariance) and Chern-Simons. Given this relation, and then relating higher order differential operators to systems of first order operators as K (78) det γ(k)∂k ↔ det (∂ · 1 + A) , k one gets an alternate ‘holographic’ description of these determinants, in terms of a gauge theory with a finite number of fields. However, no analog of this second description is known for dimensions D>2. Perhaps it would be worth searching for one, but physicists have looked hard for bosonization OPERATOR TRACES AND HOLOGRAPHY 159 formulas in D>2, with limited success. The present conjecture is quite different in using an infinite number of fields. If Conjecture 1 holds, as do analogous conjectures in other dimensions D, since the picture is so general, it seems very likely to us that it will have a simple proof, perhaps along the lines sketched in section 4.

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Simons Center for Geometry and Physics and IHES

Surveys in Diﬀerential Geometry XV

ALoopofSU (2) Gauge Fields Stable under the Yang-Mills Flow

Daniel Friedan

Abstract. The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial SU(2) gauge transformation. Local stability is demonstrated by calculating the flow equations to leading order in perturbations of the loop. The stable loop might play a role in physics as a classical winding mode of the lambda model, a 2-d quantum field theory that was proposed as a mechanism for generating spacetime quantum field theory. We also present evidence for 2-manifolds of SU(3) and SU(2) gauge fields that are stable under the Yang-Mills flow. These might provide 2-d instanton corrections in the lambda model. For Isidore M. Singer in celebration of his eighty-fifth birthday.

Contents 1. Introduction 165 2. Summary of the result 168 2.1. BPST instantons 168 2.2. Twisted pairs 169 2.3. Conformal symmetry 169 2.4. The Y-M ﬂow near the twisted pairs 170 3. Preliminaries 173 3.1. Parametrization of S3 173 3.2. Parametrization of S4 173 3.3. Round metric on S4 173 3.4. Action of U(2) on S3 and S4 173 3.5. S3 identiﬁed with SU(2) 174 c 2011 International Press

163 164 D. FRIEDAN

3.6. U(2)-invariant su(2)-valued 1-forms on S3 174 3.7. The Maurer-Cartan form ω on SU(2) 174 3.8. U(2)-invariant connections on S4 174 3.9. The Yang-Mills action 175 3.10. Hodge duality 176 3.11. Hodge duality for U(2)-invariant connections 176 3.12. Connections dω − f(x)ω 177 3.13. The basic instanton 177 3.14. The basic anti-instanton 178 3.15. Twisted (anti-)instantons 178 4 3.16. Nontrivial U(2)-invariant maps φh : S → SU(2) 179 4. Computer calculation 180 4.1. Rationale 180 4.2. Numerical results 181 5. Twisted pairs 182 5.1. The U(2)-invariant twisted pairs 183 5.2. The nontrivial loop of twisted pairs 183 6. The slow manifold 184 7. The Y-M flow equation on the slow manifold 185 7.1. The flow equation in each open hemisphere 186 7.2. Continuity conditions at the equator 190 7.3. Summary: the Y-M flow equation on the slow manifold 193 8. The gradient formula and SYM on the slow manifold 194 9. The metric on the slow manifold in A/G 196 10. Long time behavior of the flow 196 11. The outgoing trajectory 197 12. Stable 2-manifolds of SU(2) and SU(3) gauge fields 198 12.1. SU(3) 198 12.2. SU(2) 201 13. Questions and comments 218 13.1. Does the outgoing trajectory end at the flat connection? 218 13.2. Asymptotic behavior of the outgoing trajectory? 218 13.3. The lambda model 219 Appendices 222 A. U(2)-invariant connections on S4 222 A.1. S3, SU(2), U(2), SO(4), S4 222 A.2. U(2)-invariant su(2)-valued 1-forms on S3 223 A.3. U(2)-invariant su(2)-valued 2-forms on S3 223 A.4. U(2)-invariant su(2)-valued forms on S4 223 A.5. Hodge ∗ 224 A.6. F, F± 224 A.7. L± 225 A.8. Products of 1-forms and 2-forms 225 A.9. Inner products 225 A.10. New basis for the U(2)-invariant forms 226 A LOOP OF SU(2) GAUGE FIELDS 165

A.11. Formulas in the new basis 226 A.12. Instanton covariant derivatives 227 A.13. The instanton laplacian 227 B. SU(2)→SU(3)→S5 pulled back along [−1, 1]×S4→S5 229 5 B.1. SU(3)/SU(2) = S 229 B.2. U(2) acts on SU(3) → S5 230 B.3. A map [−1, 1]×S4 → S5 230 B.4. Trivialize 230 B.5. A formula for ΔA(s) 231 B.6. Discrete symmetries of the loop 231 B.7. Action of the discrete symmetries 232 B.8. Nontriviality of the loop 234 Acknowledgments 235 References 235

1. Introduction We are interested in the long time behavior of the Yang-Mills flow acting on topologically nontrivial loops and 2-spheres of SU(2) and SU(3) gauge 4 fields on S .Singer[1] noted that the homotopy groups πn(A/G)ofthe space A/G of gauge fields on S4 modulo gauge equivalence are given by the homotopy groups of the gauge group G,

(1) πn(A/G)=πn−1G = πn+3G.

There are nontrivial loops of SU(2) gauge fields becausen π4SU(2) = Z2 [2, 3]. For SU(3), there are no nontrivial loops because π4SU(3) = 0 [4]. There are topologically nontrivial 2-spheres of gauge fields for both SU(2) and SU(3) because π5SU(3) = Z [4]andπ5SU(2) = Z2 [5,6]. Our motivation is a hypothetical effect in a speculative theory of physics. The lambda model [7] is a 2-dimensional nonlinear model whose target space is the manifold of spacetime fields. The short distance fluctuations in a 2-d nonlinear model generate a measure on its target manifold, called the apri- ori measure. In the lambda model, the apriorimeasure is a measure on the manifold of spacetime fields: a quantum field theory. The apriorimeasure of the lambda model is generated by a diffusion process in the loop space of the target manifold, driven by the gradient flow of the classical spacetime action. We are pursuing the possibility that the quantum field theory generated by the lambda model will be different from the canonically quantized field theory because of nonperturbative 2-dimensional effects. The dominant nonpertubative effects at weak coupling will be due to winding modes, which are associated with topologically nontrivial loops in the target manifold, and instantons, which are associated with topologically nontrivial 2-spheres in the target manifold. Winding modes in the lambda model might give rise to 166 D. FRIEDAN non-canonical physical states in the spacetime quantum field theory. Instan- tons in the lambda model might produce non-canonical interactions. We are motivated by these possibilities to investigate the concentration points of the gradient flow of the Yang-Mills action as it acts on loops and on 2-spheres of SU(2) and SU(3) gauge fields on R4. We replace R4 by its conformal compactification S4, studying gauge fields in topologically trivial bundles over S4. As it turns out, our results will be applicable to gauge fields on R4 because they will concern fixed points of the Yang-Mills flow, i.e., critical points of the Yang-Mills action, which is conformally invariant. We find that the Yang-Mills flow concentrates on a nontrivial loop of singular SU(2) gauge fields made out of a zero-size instanton and a zero- size anti-instanton. We find evidence that the Yang-Mills flow concentrates on a nontrivial 2-sphere of singular SU(3) gauge fields also made from a zero-size instanton and a zero-size anti-instanton. We find evidence that the flow concentrates on a nontrivial 2-sphere of SU(2) gauge fields made from configurations of two zero-size instantons and two zero-size anti-instantons. These singular gauge fields live in the boundary of the manifold of gauge fields. The natural metric on the manifold of gauge fields degenerates at the boundary, so the stable loop of gauge fields has zero length and the pre- sumptive stable 2-spheres have zero area. This keeps alive the hope that they might have observable effects at low energy in the quantum field theory. Loops or 2-spheres of nonsingular gauge fields, with nonzero length or area, would make contributions in the lambda model only visible at extreme small distance in spacetime. Let A be the space of connections (gauge fields) in the trivial SU(2) principle bundle over S4. A connection is described by its corresponding covariant derivative D = d + A, where A is an su(2)-valued 1-form on S4. The curvature 2-form is

(2) F = D2 = dA + A2.

The group of gauge transformations, G, is the group of maps φ : S4 → SU(2) acting on connections by

(3) d + A → φ(d + A)φ−1 A → φd(φ−1)+φAφ−1.

A/G is the space of gauge equivalence classes of connections. The Yang-Mills (Y-M) action is 1 − ∗ (4) SYM (A)= 2 tr( F F ) 8π S4 where ∗ is the Hodge operator, which takes k-forms to (4 − k)-forms and 2 k satisfies ∗ =(−1) . The action SYM is normalized so that the BPST instanton [8] has action 1. The Yang-Mills flow on the space of connections is the A LOOP OF SU(2) GAUGE FIELDS 167 gradient flow of the Y-M action [9–13], dA (5) = −∇SYM = ∗ D ∗ F = ∗ (d ∗ F +[A, ∗F ]) . dt The sign is such that SYM decreases along the flow. The gradient is taken with respect to the L2 metric on variations δA of A, 2 1 − ∗ (6) (ds)A = 2 tr( δA δA). 4π S4 The Y-M flow is gauge invariant (commutes with gauge transformations), so it acts on the gauge equivalence classes A/G. The Y-M flow acts pointwise on parametrized loops in A/G, acting simul- taneously on each connection along the loop. This action on parametrized loops is invariant under reparametrizations of the loop, so the Y-M flow acts on the unparametrized loops Maps(S1 →A/G)/Diff (S1). We are interested in the long time behavior of the Y-M flow acting on the unparametrized loops in A/G. We expect that each connected component of the loop space contains a stable loop that is the generic attractor for the Y-M flow. There is an obvious stable attractor among the topologically trivial loops: the constant loop at the flat connection. All nearby connections are driven to the flat connection, so all nearby loops are driven to the constant loop. The connected components of the loop space are the elements of the fundamental group π1(A/G). As Singer [1] pointed out, the long exact sequence of homotopy groups implies πn(A/G)=πn−1G, since A is a contractible space. In particular,

(7) π1(A/G)=π0G = π4SU(2) = Z2. The loop space of A/G thus has two connected components: the trivial (contractible) loops and the nontrivial (non-contractible) loops. The nontrivial loops in A/G lift to paths in A whose endpoints are gauge equivalent under a nontrivial gauge transformation, i.e., one that belongs to the nontrivial connected component of G. Heuristically, we expect a stable nontrivial loop to be associated with an index 1 fixed point — a fixed point whose unstable manifold is one- dimensional. The unstable manifold will consist of two outgoing branches. We expect each of the two branches to flow to a flat connection, the two flat connections being gauge equivalent under a nontrivial gauge transformation. The unstable manifold will thus form a nontrivial loop in A/G.Thisloop will be locally stable because any nearby loop will intersect the codimension 1 stable manifold of the fixed point. Here, we use elementary methods to find a locally stable attractor among the nontrivial loops. We start out completely ignorant of the long time fate of a generic nontrivial loop of connections under the Y-M flow. In hope of relieving our ignorance, we pick a particular nontrivial loop of connections, derived from the homogeneous space SU(3)/SU(2) = S5, then try 168 D. FRIEDAN by numerical calculation to discover its long time behavior under the Y-M flow. The numerical results suggest the existence of an index 1 fixed point lying within the space of singular connections that consist of a zero-size instanton at one point and a zero-size anti-instanton at a second point and are flat everywhere else. A nontrivial loop of such singular connections is written explicitly. The loop is parametrized by the angle σ that measures the relative rotation between the instanton and the anti-instanton. The Y-M flow is calculated asymptotically near these twisted pairs. The stable loop of twisted pairs is found by examining the flow lines. Sibner, Sibner and Uhlenbeck [14] study a related problem. They consider the submanifold (A/G)inv ⊂A/G consisting of the SU(2) connections on S4 invariant under a certain U(1) symmetry group. The submanifold (A/G)inv separates into a series of connected components, indexed by m ≥ 1. Each connected component has nontrivial π1.Foreachm, they write a nontrivial loop consisting of a zero-size m-instanton at one pole in S4 glued to a zero-size m-anti-instanton at the other pole. For m = 1, their loop is exactly the loop of twisted pairs considered here. They point to [15] for references on the nontriviality of such loops. They apply a min-max procedure: minimizing the maximum value of SYM along the loop, over all nontrivial loops in (A/G)inv that belong to the same homotopy class. For m ≥ 2, they are able to make a small perturbation of the loop of singular connections to obtain a loop of nonsingular connections that has SYM < 2m everywhere on the loop. They then prove that the min-max connection provides a non-singular critical point of the Y-M action that is neither self-dual nor anti-self-dual — the first examples of such in 4 dimensions. Their min-max connections should have index 1 within the submanifold (A/G)inv and should correspond to globally stable loops under the Y-M flow acting on (A/G)inv . Here, we treat a much more elementary question: the local stability of the loop of twisted pairs (their m = 1 loop) within the full A/G. We present a summary of our results on the stable loop of SU(2) gauge fields, then some preliminaries on notation and basic formulas, then the computer calculation, then the explicit loop of twisted pairs and its nontriviality, then the calculation of the flow asymptotically nearby and the demonstration of local stability. We present evidence of stable 2-manifolds for the gauge groups SU(3) and SU(2). For SU(3) we expect this to be a stable 2-sphere. For SU(2) we expect either a 2-torus or 2-sphere. At the end, we raise some mathematical questions and make some very preliminary remarks about possible effects in the lambda model.

2. Summary of the result 2.1. BPST instantons. The BPST instanton [8] is the self-dual gauge field, ∗F = F , in the SU(2) bundle of Pontryagin index +1 over S4.The anti-instanton is the anti-self-dual gauge field, ∗F = −F , in the bundle of Pontryagin index −1. Explicit formulas are given in section 3.13 below. The A LOOP OF SU(2) GAUGE FIELDS 169 instantons are parametrized by a point in S4 — the location of the instanton — and by a nonnegative real number — the size of the instanton — and by an element in SU(2)/{±1} — the orientation of the instanton. Strictly speaking, all the orientations of an isolated instanton are gauge equivalent. The orientation becomes significant when instantons are combined, the relative orientations being gauge invariant. In the limit where the size of the instanton goes to 0, the instanton becomes a singular connection whose action density (8π2)−1tr(−F ∗ F )is a Dirac delta-function concentrated at the location of the instanton, while F = 0 everywhere else.

2.2. Twisted pairs. A twisted pair is a singular gauge field in the trivial bundle consisting of a zero-size instanton at one point in S4 and a zero-size anti-instanton at a second point. The zero size limit is taken with the ratio of the sizes held fixed. A twisted pair is everywhere either self-dual or anti-self-dual or flat, so each twisted pair is a fixed point of the flow. The 4 twisted pairs are parametrized by the location x+ ∈ S of the instanton, by 4 the location x− ∈ S of the anti-instanton, by the ratio ρ+/ρ− of the size of the instanton to the size of the anti-instanton, and by the relative orientation or twist, gtw ∈ SU(2)/{±1}. Of the two orientations, the instanton’s and the anti-instanton’s, one is eliminated by a gauge transformation, leaving only the relative orientation to parametrize the twisted pairs. We establish by an explicit calculation that a loop of twisted pairs is nontrivial in A/G if gtw traverses a nontrivial loop in the space SU(2)/{±1} of relative orientations.

2.3. Conformal symmetry. The Hodge ∗-operator acting on 2-forms is conformally invariant in four dimensions, so the conformal symmetry group of S4,whichisSO(1, 5), acts on the space of critical points of the Yang-Mills action, in particular on the space of twisted pairs. Using conformal transformations, we can move the zero-size instanton to the south pole in S4 and the zero-size anti-instanton to the north pole. We can make the sizes of the instanton and the anti-instanton equal. The remaining subgroup of the conformal group is SO(4), which acts on the twist gtw ∈ SU(2)/{±1} by conjugations. So we can diagonalize gtw . The twisted pair is invariant under the remaining U(2) subgroup of SO(4). The conformal equivalence classes of twisted pairs form a one parameter family labelled by the conjugacy classes of SU(2)/{±1}. Each twisted pair has a U(2) symmetry. The metric on A is not conformally invariant, so the Y-M flow is not conformally invariant away from the fixed points. Near the fixed points, the conformal group acts merely by rescaling parameters, so the qualitative behavior of the flow in the neighborhood of the fixed points is conformally invariant. It is enough to study the Y-M flow near a slice of the conformal equivalence classes, consisting of a representative in each conformal equivalence class of twisted pairs. 170 D. FRIEDAN

Figure 1. The ﬂow on the slow manifold. The twisted pairs are represented by the horizontal axis. The vertical axes at σ =0andσ =2π are identiﬁed under a nontrivial gauge transformation.

2.4. The Y-M flow near the twisted pairs. Instantons and anti- instantons are individually stable under the Y-M flow, so the Y-M flow very near the twisted pairs reduces to a flow in a slow manifold parametrized by an asymptotically small instanton and an asymptotically small anti- instanton. We represent the conformal equivalence classes by puting the instanton at the south pole and the anti-instanton at the north pole, by making their sizes equal, ρ+ = ρ− = ρ ≈ 0, and by diagonalizing the twist, 1 iσ e 2 0 ∈ (8) gtw = − 1 iσ ,σ[0, 2π]. 0 e 2

The slow manifold is represented by a two dimensional space of connections parametrized by ρ ≈ 0andbyσ. The twisted pairs are at ρ =0. We calculate the Y-M flow equations to leading order in ρ, dρ dσ (9) = ρ3(1 + 2 cos σ)+O(ρ5), = −8ρ2 sin σ + O(ρ4). dt dt The flow lines follow the curves (10) ρ8(1 − cos σ)sinσ = C. as pictured in Figure 1. The twisted pairs lie on the horizontal axis, ρ =0. The vertical axes at σ =0andatσ =2π are identified by a nontrivial gauge A LOOP OF SU(2) GAUGE FIELDS 171 transformation φ : S4 → SU(2). The maximally twisted pairs are those with tr(gtw ) = 0, represented in Figure 1 by the point σ = π on the horizontal axis. The attracting (stable) manifold of the maximally twisted pairs is represented in Figure 1 by the vertical line σ = π. It has codimension 1 in A/G. If a loop of connections intersects it, the Y-M flow drives the intersection point to a maximally twisted pair. An infinitesimal neighborhood within the loop around the intersection point is driven to the unstable manifold of that maximally twisted pair, which is represented in Figure 1 by the three axes: the horizontal axis ρ = 0 and the vertical axes σ =0andσ =2π.The unstable manifold of the maximally twisted pair is one dimensional. One outgoing branch consists of the segment of the horizontal axis going from σ = π to σ = 0, followed by the outgoing trajectory along the vertical axis at σ = 0. The other branch consists of the segment of the horizontal axis going from σ = π to σ =2π, followed by the outgoing trajectory along the vertical axis at σ =2π. The unstable manifold of the maximally twisted pair has to be constructed asymptotically in the limit ρ → 0. In the limiting unstable manifold, the first segment of each branch — on the horizontal axis in Figure 1 — is in fact a line of fixed points. Effectively, the maximally twisted pairs are fixed points of index 1. The stable loops are indexed by the maximally twisted pairs. The stable loop passing through a general maximally twisted pair tr(gtw )=0is obtained from the stable loop in Figure 1 by the inverting the conjugation that diagonalized gtw . The segment of the stable loop lying within the twisted pairs consists of the shortest geodesic loop in SU(2)/{±1} that starts and ends at ±1 and that passes through ±gtw . This segment of fixed points is preceeded and followed by the outgoing trajectory leaving from the twisted pair at σ =0, 2π,withtwist±1, the untwisted pair. The twisted pairs look more literally like fixed points of index 1 when pictured in the riemannian geometry of the space of gauge fields. To leading order in ρ, the metric on the slow manifold in A/G is

2 slow 2 2 2 (11) (ds )A/G = 16(dρ) + ρ (dσ) .

Geometrically, the space of connections is a cone, as pictured in Figure 2. The loop of twisted pairs — the horizontal axis in Figure 1 — collapses to the vertex of the cone. The vertex of the cone looks like an index 1 fixed point lying on the boundary of A/G. The outgoing trajectories at σ =0andσ =2π, i.e., at gtw =+1and gtw = −1, are gauge equivalent. The orientations of the small instanton and the small anti-instanton are lined up within SU(2). Under the Y-M flow, the instanton and anti-instanton grow larger, presumably merging together and annihilating, flowing eventually to the flat connection. It remains to be proved that this does in fact happen in general, that the outgoing trajectory from the untwisted pair, gtw = ±1, ends at the flat connection, and not at some other fixed point with SYM > 0. Here, we prove this only for the 172 D. FRIEDAN

Figure 2. Conical geometry of the slow manifold. The twisted pairs are represented by the vertex. The two axes are identiﬁed under a nontrivial gauge transformation.

special case where the instanton and anti-instanton of the untwisted pair are located at opposite poles in S4. Then the entire outgoing trajectory is SO(4)-invariant, and we can show that there are no SO(4)-invariant fixed points besides the flat connection and the untwisted pair itself. 2.4.1. Global stability. We have only established the local stability of the loop of twisted pairs. We can argue for global stability based on a theorem of Taubes [16] which states that, for connections in the trivial SU(2) bundle 4 over S , the hessian of SYM must have at least 2 negative eigenvalues at any smooth solution of the Yang-Mills equation (fixed point of the Y-M flow). It follows that there are no smooth fixed points with unstable manifolds of dimension 0 or 1. Any index 1 fixed point must be completely singular, so it must be a twisted pair or must have SYM ≥ 4. Therefore any loop of gauge fields with SYM < 4 must flow to the stable loop of twisted pairs. This argument does not work for stable 2-spheres, since Taubes’ theorem allows smooth fixed points of Morse index 2. In retrospect, Taubes’ theorem and/or the paper of Uhlenbeck, Sibner, and Sibner could have made our numerical explorations unnecessary, leading directly to consideration of the loop of twisted pairs. A LOOP OF SU(2) GAUGE FIELDS 173

3. Preliminaries We will be doing elementary, explicit calculations with the U(2)-invariant connections over S4. In this section, we establish notation and collect some basic formulas. More detail is given in Appendix A.

3.1. Parametrization of S 3. We realize S3 as the unit sphere in C2, parametrized by the unit vectors z1 † (12) z = , z z =¯z1z1 +¯z2z2 =1. z2

We write the complementary projection matrices

(13) P (z)=zz†,Q(z)=1− P (z).

ThevolumeformonS3 is 1 † † 2 (14) dvolS3 = − (z dz)(dz dz), dvolS3 =2π . 2 S3

3.2. Parametrization of S 4. We realize S4 as the unit sphere in R ⊕ C2, parametrized by the unit vectors

2 † (15) y =(y0, y) y0 + y y =1.

In polar coordinates,

(16) y =(cosθ, z sin θ), z ∈ S3.

Most often, we use coordinates (x, z) where θ (17) x =lntan , −∞ ≤ x ≤∞. 2

The north pole of S4 is at θ =0,x = −∞. The south pole is at θ = π, x = ∞.

3.3. Round metric on S 4. The round metric on S4 is 2 2 2 † 2 −2 (18) (ds)S4 = R (x) (dx) + dz dz ,R(x) = (cosh x) .

3.4. Action of U (2) on S 3 and S 4. U ∈ U(2) acts on S3 ⊂ C2 by 4 2 z → Uz and acts on S ⊂ R ⊕ C by (y0, y) → (y0,Uy)or(x, z) → (x, Uz). 174 D. FRIEDAN

3.5. S 3 identiﬁed with SU (2). S3 is identiﬁed with SU(2) by 1 z1 −z¯2 (19) g(z)e = z, e = ,g(z)= . 0 z2 z¯1

The action of U(2) on S3 becomes 10 (20) g(Uz)=Ug(z) . 0 (det U)−1

The rotation group SO(4) is identiﬁed with SU(2) × SU(2)/Z2 via

−1 (21) g(O(z)) = gLg(z)gR ,O∈ SO(4), (gL,gR) ∈ SU(2) × SU(2).

3.6. U (2)-invariant su(2)-valued 1-forms on S 3. The general U(2)-invariant su(2)-valued 1-form on S3 is

† (22) fη − fη¯ + f3η3,f3 ∈ R,f ∈ C where

(23) η = −PdP = −(z†dz)zz† − zdz† (24) η† = −dP P =(z†dz)zz† − dzz† † † † (25) η3 =(z dz)(P − Q)=(z dz)(2zz − 1) is a natural basis that diagonalizes the U(1) generated by i(P − Q),

† † (26) [P − Q, η]=2η, [P − Q, η ]=−2η , [P − Q, η3]=0.

3.7. The Maurer-Cartan form ω on SU (2). The Maurer-Cartan form ω on SU(2) is

−1 † (27) ω = gd(g )=−η + η − η3, satisfying

2 1 3 (28) dω + ω =0, − ω =dvolS3 1. 6

3.8. U (2)-invariant connections on S 4. A connection in the trivial SU(2) bundle over S4 is described by its covariant derivative

(29) D = d + A(x, z) where A(x, z)isansu(2)-valued 1-form on S4. Regularity at the poles requires

(30) A(±∞, z)=0. A LOOP OF SU(2) GAUGE FIELDS 175

Invariance under U(2) is the condition

(31) A(x, Uz)=UA(x, z)U −1 U ∈ U(2).

Deﬁne

−1 (32) dω = gdg = d + ω which satisﬁes

2 (33) dω =0, [dω,P− Q]=0.

The U(1) generated by i(P − Q) thus leaves dω invariant. It is convenient to write the U(2)-invariant connections in the U(1)-covariant form

(34) D = d + A = dω +ΔA

† (35) ΔA = A0(x)dx i(P − Q)+f(x)η − f¯(x)η + f3(x)η3 with

(36) A0(x),f3(x) ∈ R,f(x) ∈ C.

The U(2)-invariant gauge transformations act by

iϕ(x)(P −Q) −iϕ(x)(P −Q) D → e De = dω − (∂xϕdx) i(P − Q) (37) + eiϕ(x)(P −Q)ΔAe−iϕ(x)(P −Q) iϕ(x)(P −Q) −iϕ(x)(P −Q) 2iϕ(x) e ΔAe = A0(x)dx i(P − Q)+e f(x)η −2iϕ(x) † (38) − e f¯(x)η + f3(x)η3.

Connections with A0(x)=0aresaidtobeinA0 = 0 gauge. Any connection can be brought to A0 = 0 gauge by the gauge transformation with ∂xϕ = A0(x), perhaps at the cost of introducing singularities at the poles x = ±∞.

3.9. The Yang-Mills action. The curvature 2-form is

(39) F = D2 = dA + A2.

The Yang-Mills action is 1 1 − ∗ (40) SYM = 2 tr( F F ) 2π S4 4 where ∗ is the Hodge operator taking k-forms to (4−k)-forms and satisfying ∗2 =(−1)k. 176 D. FRIEDAN

For U(2)-invariant connections, the integrand is an invariant volume form 1 (41) tr(−F ∗F )=dx LYM (x)dvolS3 4 and the Y-M action is ∞ (42) SYM = dx LYM (x). −∞

3.10. Hodge duality. The (anti-)self-dual curvature is 1 (43) F± = (F ±∗F ) 2 The Y-M action splits into contributions of the two chiralities, 1 1 (44) SYM = S+ + S− S± = tr(−F±∗F±). 2π2 4 The integer instanton number is

(45) Stop = S+ − S−.

The instanton number vanishes for connections in the trivial bundle over S4.

3.11. Hodge duality for U (2)-invariant connections. For U(2)- invariant connections,

(46) LYM (x)=L+(x)+L−(x)

1 (47) tr(−F±∗F±)=dx L±(x)dvolS3 4

∞ (48) S± = dx L±(x) −∞

∞ ∞ (49) SYM = dx [L+(x)+L−(x)] ,Stop = dx [L+(x) − L−(x)] . −∞ −∞

For a U(2)-invariant connection in A0 = 0 gauge,

† (50) D = dω +ΔA, ΔA = f(x)η − f¯(x)η + f3(x)η3,

1 2 2 1 2 (51) L± = ∂xf3 ± 2 f3 −|f| + |∂xf ± 2(1− f3) f)| 4 2 A LOOP OF SU(2) GAUGE FIELDS 177

1 2 2 2 2 2 2 (52) LYM = (∂xf3) + |∂xf| +2 f3 −|f| +4(1− f3) |f| 2 2 2 2 (53) L+ − L− = ∂x f3 +2|f| − 2f3 |f| , x=∞ 2 2 2 Stop = f3 +2|f| − 2f3 |f| . x=−∞

3.12. Connections d ω − f(x)ω. The U(2)-invariant connections that are invariant under the full O(4) = SU(2) × SU(2)/{±1} are of the form

† (54) D = dω − f(x)ω = dω + f(x)(η − η + η3).

Substituting in equations 51 and 53,

3 2 (55) L±(x)= [∂xf ± 2f(1 − f)] . 4 2 − 3 x=∞ (56) Stop = 3f 2f x=−∞ .

3.13. The basic instanton. The BPST instanton [8] is the self-dual connection, F = F+, of instanton number 1. For us, the basic instanton is the self-dual U(2)-invariant connection of the form

(57) D+ = dω − f+(x)ω.

The self-duality equation F− = 0 becomes the ordinary diﬀerential equation

(58) ∂xf+ =2f+(1 − f+).

The general solution is 1 1 (59) f+(x)= = . −2(x−x+) −2 −2x 1+e 1+ρ+ e

−x The parameter ρ+ = e + is the size of the instanton. The Y-M action density is 3 (60) LYM (x)=L+(x)= 4 4 cosh (x − x+) and the Y-M action is

(61) SYM = S+ = Stop =1.

The basic instanton is regular at the south pole (x = ∞), because D+ → d −1 there. Near the north pole, D+ → dω = gdg , so the instanton lives in the nontrivial bundle formed from trivial bundles over the two hemispheres, 178 D. FRIEDAN patched together at the equator using the index +1 map z → g(z)fromS3 to SU(2). When the instanton size goes to zero, when x+ →∞, the action density becomes a delta-function concentrated at the south pole, at x = ∞. We say that the basic instanton is located at the south pole.

3.14. The basic anti-instanton. The anti-instanton is the anti-self- dual connection of instanton number −1. Our basic anti-instanton is

(62) D− = dω − f−(x)ω

(63) ∂xf− = −2f−(1 − f−)

1 1 (64) f−(x)= = . 2(x−x−) −2 2x 1+e 1+ρ− e x The parameter ρ− = e − is the size of the anti-instanton. The basic instanton and anti-instanton are related by the orientation reversing map x →−x, ρ+ ↔ ρ−. The Y-M action is 3 (65) LYM (x)=L−(x)= 4 ,SYM = S− = −Stop =1. 4 cosh (x − x−)

The basic anti-instanton is regular at the south pole (x = −∞), because −1 D+ → d. Near the north pole, D+ → gdg , so the anti-instanton lives in the nontrivial bundle formed from trivial bundles on the hemispheres patched together at the equator using the index −1 map z → g(z)−1.When the anti-instanton size goes to zero, when x− →−∞, the action density becomes a delta-function concentrated at the north pole, at x = −∞.The basic anti-instanton is located at the north pole.

3.15. Twisted (anti-)instantons. The U(2)-invariant twisted instanton with twist angle σ+ ∈ [0, 2π]is

iα+(x)(P −Q) −iα+(x)(P −Q) (66) e D+e where α(x) satisﬁes 1 (67) α+(−∞)= σ+,α+(∞)=0. 2 and the U(2)-invariant twisted (anti-)instanton with twist angle σ− is

iα−(x)(P −Q) −iα−(x)(P −Q) (68) e D−e

1 (69) α−(−∞)=0,α−(∞)= σ−. 2 A LOOP OF SU(2) GAUGE FIELDS 179

These are of course merely gauge transforms of the basic (anti-)instanton. The twist angle σ± is gauge invariant only if we restrict our notion of gauge equivalence to the group of pointed gauge transformations, that act as the identity at a base-point in S4, here the gauge transformations eiϕ(x)(P −Q) with ϕ(±∞)=0. The (anti-)instanton twisted by a general element gtw ∈ SU(2)/{±1} is

−1 (70) D±(gtw )=φ(gtw )D±φ(gtw ) where −1 (71) φ(gtw )(x, z) → g(z)gtw g(z) ,x→±∞.

The U(2)-invariant twisted (anti-)instanton corresponds to 1 iσ± 2 −1 i 1 σ (P −Q) e 0 2 ± (72) gtw = − 1 iσ ,g(z)gtw g(z) = e 0 e 2 ±

Rotations O =(gL,gR)inSO(4) = SU(2)×SU(2)/Z2 transform the twisted (anti-)instanton by

−1 −1 (73) D±(gtw )(x, Oz)=gLD±(gR gtw gR)(x, z)gL .

The gR act by conjugation on the twist gtw , so every twisted instanton can be taken to a U(2)-invariant one by a rotation in SO(4). The gL are symmetries, as are the gR that commute with gtw.

4 3.16. Nontrivial U (2)-invariant maps φh : S → SU (2). The Hopf ﬁbration [2] is the map h : S3 → S2 ⊂ R ⊕ C,

2 2 (74) h(z)=(|z1| −|z2| , 2¯z1z2).

The nontrivial element in π4SU(2) = Z2 is represented by the suspension, Sh : S4 → S3 = SU(2), of the Hopf ﬁbration [3]. In particular, the U(2)- 4 invariant maps φh : S → SU(2) of the form iϕ (x) iϕ (x)(P −Q) e h 0 −1 (75) φh(x, z)=e h = g(z) g(z) 0 e−iϕh(x) with

(76) ϕh(−∞)=π, ϕh(∞)=0 represent the nontrivial element in π4SU(2) = Z2 [17]. Explictly,

(77) 2 2 10 |z1| −|z2| 2z1z¯2 φh(x, z)=cosϕh(x) + i sin ϕh(x) 2 2 . 01 2¯z1z2 −|z1| + |z2| 180 D. FRIEDAN

4. Computer calculation We start with a numerical calculation, looking for a clue to the long term behavior of the Y-M ﬂow on the nontrivial loops. We pick a particular nontrivial loop and try to discover what it ﬂows to. The calculation is sketched here. Details are given in Appendix B.

4.1. Rationale. We use the homogeneous space SU(3)/SU(2) = S5 to construct a nontrivial loop of connections on S4.TheSU(2) bundles over S5 are classified topologically by π4SU(2), since they are made by gluing two trivial bundles along the equator in S5 by a map from the equator, S4,to SU(2). The bundle SU(2) → SU(3) → S5 represents the nontrivial element in π4SU(2) [4]. 5 There is a canonical invariant connection Dinv in SU(2) → SU(3) → S . 4 5 We pull back Dinv along a certain map [−1, 1] × S → S to obtain a one parameter family D(s) of connections over S4. The map is chosen so that the endpoint connections D(±1) are both flat, so s → D(s) forms a closed loop in A/G. The nontriviality of the loop is verified explicitly in Appendix B. The map [−1, 1] × S4 → S5 preserves a U(2) subgroup of the symmetries of 4 Dinv ,soeachD(s)isaU(2)-invariant connection over S . We want to see what happens to this particular nontrivial loop under the Y-M flow. The Y-M flow preserves symmetry, so loop will remain within the U(2)-invariant connections on S4. Two additional discrete symmetries of Dinv are likewise preserved by our construction, one taking each D(s)to itself, the other taking D(s)toD(−s). The connection D(0) at the midpoint of the loop thus has an extra discrete symmetry. Again, the Y-M flow preserves these discrete symmetries. In order to simplify the computational problem, we assume a plausible- seeming scenario. We assume that the midpoint D(0) of the initial loop will flow to a fixed point in A/G of Morse index 1, while the rest of the loop will flow to the one dimensional unstable manifold of the fixed point. The discrete symmetry that takes D(s)toD(−s) will exchange the two outgoing branches of the unstable manifold. Now we do not need to run the Y-M flow on the entire loop, but only on the single connection D(0). We simplify still further by assuming that D(0) will flow to a connection that minimizes SYM among all the U(2)-invariant conections with the same two discrete symmetries as D(0). Assuming this scenario, there is no need to run the Y-M flow at all. We need only minimize SYM on this class of invariant connections, which is quite easy to do numerically. There is a fairly extensive literature on minimizing SYM over connections with specific prescribed symmetries [13, 14, 18–27], 2 but seemingly not the U(2) × Z2 symmetry of interest here. The closest seemstobe[25], which studies U(2) invariant connections on non-round S4 and finds a solution of the Yang-Mills equation which degenerates, in the round limit, to a zero-size instanton/anti-instanton pair. A LOOP OF SU(2) GAUGE FIELDS 181

The purpose of the numerical calculation is only heuristic. The simplify- ing assumptions are justiﬁed by the clue that emerges from the computation. It could have turned out otherwise. In particular, it could have turned out that an initial loop with such special symmetries would not detect generic properties of the Y-M ﬂow acting on loops.

4.2. Numerical results. The midpoint connection D(0) = dω+ΔA(0) of the initial loop is calculated in Appendix B, † 1 2 (78) ΔA(0) = cos θ (η − η )+ 1 − sin θ η3. 2 1 It is convenient to use the polar angle θ here, rather than x =lntan2 θ which we use elsewhere. The two discrete symmetries of ΔA(0) are derived in Appendix B. The general U(2)-invariant connection with these two additional discrete symmetries has the form † (79) ΔA = f(θ)(η − η )+f3(θ)η3 with

(80) f = ff¯ (π − θ)=−f(θ) f3(π − θ)=f3(θ). Regularity at the poles requires the boundary conditions

(81) f = f3 =1 atθ =0. We change variables again, to (82) t =cosθ The Y-M action is given by equation 52, 1 2 −1 (83) SYM = dt (1 − t ) LYM −1

1 2 2 2 2 2 2 2 2 2 2 (84) LYM = (1−t ) (∂tf3) +2(f3 −f ) +(1−t ) (∂tf) +4(1−f3) f . 2 The initial connection D(0) has

3 2 4 (85) LYM = (1 − t )(1 − t ),SYM =2.4. 2

To minimize SYM numerically, we use a ﬁnite mode approximation [23]. We write f3 and f as polynomials in t obeying the symmetry and boundary conditions, N/2 N/2 2n 2n+1 (86) f3 =1+ (t − 1)a2n f = t + (t − t)a2n−1 n=1 n=1 182 D. FRIEDAN

Table 1. Results of numerical minimization of SYM on aﬃne subspaces of A of dimension N.

N min(SYM ) 2 2.15627 12 2.00723 22 2.00286 4 2.06011 14 2.00504 24 2.00251 6 2.03019 16 2.00368 26 2.00202 8 2.01735 18 2.00346 28 2.00186 10 2.01086 20 2.00313 30 2.00147

where N is an even number. The N real variables ak parametrize an affine subspace of A of dimension N. We are approximating A by an increasing family of finite dimensional affine subspaces. On each subspace, SYM evaluates to a quartic polynomial in the ak, which is minimized numerically using mathematical software such as Sage [28]. Typical results are shown in Table 1. The numerical results suggest that there is a global minimum with SYM = 2. The possibility of an integer global minimum motivates examining the self-dual and anti-self-dual action densities L±(x) of the approximate minima obtained from the computer calculations. Figure 3 plots the evolution of L±(x)asN increases. It looks like the global minimum is a connection that consists of a zero-size instanton at the south pole and a zero-size anti- instanton at the north pole, and is otherwise flat. Closer inspection suggests that the minimum is attained at the connection given by 1 (87) f3 = f = f− = ,x<0 1+e2(x−x−) 1 (88) f3 = −f = f+ = ,x>0 1+e−2(x−x+) in the limit x+ →∞, x− →−∞. This is the zero-size basic anti-instanton at the north pole combined with a twisted zero-size instanton at the south pole, twisted by π.

5. Twisted pairs Motivated by the numerical calculation, we investigate the long time behavior of the Y-M ﬂow near the singular connections that consist of a zero- size instanton and a zero-size anti-instanton patched together on a 3-sphere separating their locations. We are calling such connections twisted pairs.The general twisted pair is parametrized by the locations of the instanton and anti-instanton and by their relative twist gtw .TheU(2)-invariant twisted pair has the instanton at the south pole and the anti-instanton at the north pole and has diagonal gtw , so is parametrized by the twist angle σ ∈ [0, 2π]. We write the U(2)-invariant twisted pair explicitly in the next section. The general twisted pair is obtained by making a conformal transformation of S4. A LOOP OF SU(2) GAUGE FIELDS 183

Figure 3. Plots of L±(x) for the connections numerically minimizing SYM for N ≤ 30. The curves move away from the origin as N increases. For comparison, the rightmost curve is −x L+(x) for the instanton of size ρ+ = e + , x+ =4.

5.1. The U(2)-invariant twisted pairs. A U(2)-invariant twisted pair combines a U(2)-invariant twisted instanton of small size ρ+ at the south pole with a U(2)-invariant twisted anti-instanton of the same size ρ− at the north pole, in the limit ρ± → 0, iα (x)(P −Q) −iα (x)(P −Q) e + D−e + x>0 (89) Dtw (α+,α−) = lim ρ →0 iα−(x)(P −Q) −iα−(x)(P −Q) ± e D+e x<0.

The functions α±(x) should vanish fast enough at the poles to ensure that the connection is regular there,

∓2x (90) α±(x)=O(e ),x→±∞.

The connection Dtw (α+,α−) and its curvature are discontinuous at the equator as long as ρ± > 0, but the discontinuities disappear in the zero-size limit. The relative twist is

(91) σ =2α+(0) − 2α−(0).

iϕ(x)(P −Q) The U(2)-invariant gauge transformations e act by α±(x) → α±(x) − ϕ(x), so the relative twist σ is gauge invariant. Any two U(2)- invariant twisted pairs with the same twist σ are gauge equivalent.

5.2. The nontrivial loop of twisted pairs. The U(2)-invariant twisted pairs with twist σ ∈ [0, 2π] form a nontrivial closed loop in A/G (for references on the nontriviality of such loops, reference [14] refers to reference [15]). To see this explicitly, let Dtw (2π)=Dtw (α+,α−)beany 184 D. FRIEDAN

twisted pair with σ =2α+(0) − 2α−(0) = 2π, and let Dtw (0) = Dtw (0, 0), which has twist 0. Then −1 (92) Dtw (2π)=φhDtw (0)φh 4 where φh is one of the nontrivial maps S → SU(2) described in section 3.16 above, iϕ (x)(P −Q) α+(x) x>0 (93) φh(x, z)=e h ,ϕh(x)= π + α−(x) x<0.

6. The slow manifold A twisted pair is everywhere self-dual or anti-self-dual or flat, so the twisted pairs are all fixed points under the Y-M flow. Instantons in iso- lation are stable under the Y-M flow, as are anti-instantons. All perturbations of the instanton transverse to the space of instantons are driven rapidly to zero under the Y-M flow. Therefore, the Y-M flow, acting on a small neighborhood of the twisted pairs, rapidly compresses the neighborhood down to a space of approximate fixed points, the slow manifold,which is parametrized by an asymptotically small instanton and an asymptotically small anti-instanton. The long time behavior of the Y-M flow near the twisted pairs is determined by the flow on the slow manifold, which can be represented as a flow on the parameter space of the instanton-anti-instanton pair. To find the long time behavior, it will be enough to calculate the asymptotic expansion of the flow equation on the slow manifold to leading order in the sizes of the instanton and anti-instanton, at least if the leading order flow is robust against small perturbations. The slow manifold is parametrized by the location, size and twist of the instanton and by the location, size and twist of the anti-instanton. A gauge transformation eliminates one of the twists, leaving the relative twist. The slow manifold is thus parametrized by the two locations, the two sizes and the relative twist gtw . The Y-M action SYM is invariant under the 15 parameter conformal group SO(1, 5), so SYM on the slow manifold, as a function of the parameters of the instanton-anti-instanton pair, is invariant under SO(1, 5). Therefore it suffices to calculate the generator of the Y-M flow, which is the gradient of SYM , on a representative slice through the orbits of the conformal group. We move the instanton to the south pole in S4 using a conformal transformation, and the anti-instanton to the north pole using another. The remaining subgroup of SO(1, 5) consists of the rotation group SO(4) and the translations in x (which are the dilations of R4 in the stereographic projection). A rotation in SO(4) diagonalizes the relative twist gtw .Wenowhavea U(2)-invariant instanton and a U(2)-invariant anti-instanton. A translation −a a x → x + a takes ρ+ to e ρ+ and ρ− to e ρ−, so we can use a translation in x to set ρ+ = ρ− = ρ. We now have a representative slice of the slow A LOOP OF SU(2) GAUGE FIELDS 185 manifold parametrized by the U(2)-invariant instanton-anti-instanton pairs of equal size ρ and relative twist σ.

7. The Y-M ﬂow equation on the slow manifold To calculate the asymptotic expansion of the Y-M ﬂow equation on the slow manifold, we start with a larger than necessary slice of the slow manifold: all the U(2)-invariant instanton-anti-instanton pairs, parametrized by their asymptotically small sizes ρ± and their twist angles σ± ∈ [0, 2π]. This slice of the slow manifold is a family of U(2)-invariant connections iα+(x)(P −Q) −iα+(x)(P −Q) slow e (D+ + δA+) e x>0 (94) D = iα (x)(P −Q) −iα (x)(P −Q) e − (D− + δA−) e − x<0,

1 (95) α±(0) = σ±. 2

The δA± are asymptotically small perturbations of the instanton and anti- instanton, to be determined by the condition that the family of connections slow D , parametrized by ρ± and σ±, is preserved under the Y-M flow. There must be velocity vector fields d d d (96) ρ± =˙ρ±, α±(x)=α ˙±(x), σ± =˙σ± =2˙α±(0) dt dt dt such that the Y-M flow equation is satisfied on the slow manifold d (97) Dslow = ∗Dslow ∗F slow dt where F slow is the curvature of Dslow . We solve for the velocities in two steps: 1. First we solve equation 97, the Y-M flow equation, separately in each open hemisphere. The general solution in each hemisphere depends on several undetermined parameters, includingρ ˙± andσ ˙ ±. 2. Then we require Dslow and F slow to be continuous at the equator, x = 0, so that the flow equation holds there as well. At this stage, to simplify the calculation, we specialize to the subfamily where there is an x →−x symmetry, where 1 (98) ρ+ = ρ− = ρ, α+(x)=−α−(−x)=α(x),σ+ = −σ− = σ. 2 The symmetric twisted pairs still represent every orbit of the conformal group. The continuity conditions at x = 0 fix all parameters in the separate solutions on the two hemispheres, thereby determining the velocity vectorsρ ˙,˙α(x), andσ ˙ . 186 D. FRIEDAN

All calculations are to leading order in ρ. With more work, the method would produce the velocity vectors on the slow manifold to all orders in ρ.

7.1. The flow equation in each open hemisphere. In this section, we solve the flow equation, equation 97, in each hemisphere separately, to leading order in ρ±. The calculation is the same in each hemisphere, so, for the sake of legibility, we temporarily write ρ instead of ρ± and f instead of f±. The lhs of equation 97 is, to leading order, d slow iα±(x)(P −Q) ∂ (99) D = e ρ˙ D± +[D±, −iα˙ ±(x)(P − Q)] dt ∂ρ × e−iα±(x)(P −Q) We only need the rhs of equation 97 expanded to first order in δA. To this 2 order, the connection D = D± + δA has curvature F = D± + D±δA where 2 D± is the curvature of the (anti-)instanton. So 2 2 (100) ∗D∗F = ∗[δA, D±]+∗D±∗D±δA = ∓∗D±δA + ∗D±∗D±δA = ∗D±(∗∓1)D±δA so the rhs of equation 97 becomes

slow slow iα±(x)(P −Q) −iα±(x)(P −Q) (101) ∗D ∗F = e [∗D±(∗∓1)D±δA] e . The ﬂow equation, equation 97,isnow ∂ (102)ρ ˙ D± +[D±, −iα˙ ±(x)(P − Q)] = ∗D±(∗∓1)D±δA. ∂ρ This takes a particularly simple form if we make a change of basis † † † (103) ω = −η + η − η3,ω1 = η − η − 2η3,ω2 = −i(η + η ). Recall that 1 1 (104) f = , − 1=ρ−2e∓2x 1+ρ−2e∓2x f so ∂ ∂ (105) ρ D± = ρ (dω − fω)=−2f(1 − f)ω. ∂ρ ∂ρ The U(2)-invariant inﬁnitesimal gauge transformations of the (anti-) instanton are

(106) [D±, −iϕ(x)(P − Q)] = −∂xϕ(x)dx i(P − Q) − ϕ(x)[−fω, i(P − Q)] (107) = −∂xϕ(x)dx i(P − Q) − 2fϕ(x)ω2. A LOOP OF SU(2) GAUGE FIELDS 187

Using this formula with ϕ =˙α±, equation 102 becomes

−1 (108) −2ρ ρf˙ (1−f)ω −∂xα˙ ±dx i(P −Q)−2fα˙ ±ω2 = ∗D±(∗∓1)D±δA.

We expand in the new basis

(109) δA = δA0(x)dx i(P − Q)+δf(x)ω + δA1(x)ω1 + δA2(x)ω2.

The laplacian, derived in Appendix A.13, is diagonal in this basis,

(110) ∗D±(∗∓1)D±δA 2 −1 2 2 −1 −1 2 = −8R (x) f δA˜0idx(P − Q) − 2R (x) f ∂x f δA˜0 ω2 2 −1 −1 −1 2 2 −1 −1 (111) + R (x) f (1 − f) ∂xf (1 − f) ∂xf (1 − f) δf ω 2 −1 −1 2 2 −4 −1 2 (112) + R (x) f (1 − f) ∂xf (1 − f) ∂xf (1 − f) δA1 ω1 where R2(x) = (cosh x)−2 is the conformal factor in the round metric on S4 as written in equation 18 and where

1 −1 (113) δA˜0 = δA0 − ∂x(f δA2). 2

Note that δA˜0 vanishes for perturbations of the form δA0 = −∂xϕ, δA2 = −2fϕ, which are the infinitesimal gauge transformations of the (anti-) instanton as given in equation 107. So the laplacian annihilates the infinitesimal gauge transformations, as it should. We take advantage of the infinitesimal gauge transformations to set δA0 = 0, keeping the perturbation δA in A0 = 0 gauge. Then

1 −1 (114) δA˜0 = − ∂x(f δA2). 2 The ﬂow equation, equation 108, is now four ordinary equations

2 −1 2 (115) −∂xα˙ ± = −8R (x) f δA˜0 2 −1 −1 2 (116) −2fα˙ ± = −2R (x) f ∂x f δA˜0 −1 2 −1 −1 −1 2 2 (117) −2ρ ρf˙ (1 − f)=R (x) f (1 − f) ∂xf (1 − f) −1 −1 × ∂xf (1 − f) δf 2 −1 −1 2 2 −4 (118) 0=R (x) f (1 − f) ∂xf (1 − f) . −1 2 × ∂xf (1 − f) δA1

Since we are solving the ﬂow equation only to leading order, we expand

2 −2 4f(1 − f) 2 −1 4 (119) R (x) = (cosh x) = 2 =4ρ f (1 − f)+O(ρ ) [ρ−1f + ρ(1 − f)] 188 D. FRIEDAN and keep only the leading order term. This approximation expresses the fact that, in the limit ρ → 0, only the metric at the location of the (anti-) instanton enters into the solution of the ﬂow equation. The four equations 115–118 become −2 3 −1 (120) ∂xα˙ ± =2ρ f (1 − f) δA˜0 1 −2 −1 −1 2 (121) α˙ ± = ρ f (1 − f) ∂x f δA˜0 4 −1 1 −2 −1 −3 2 2 −1 −1 (122) ρ ρ˙ = − ρ f (1 − f) ∂xf (1 − f) ∂xf (1 − f) δf 8 2 −4 −1 2 (123) 0=∂xf (1 − f) ∂xf (1 − f) δA1.

For Dslow to be regular at the pole, the perturbations must vanish at x = ±∞, ∓2x (124) δf = δA1 = δA2 = O(e )=O(1 − f) x →±∞,f→ 1.

7.1.1. The first two flow equations. The first of the four flow equations, equation 120, is trivially solved to give

1 2 −3 (125) δA˜0 = ρ f (1 − f)∂xα˙ ±. 2 Then the second of the four ﬂow equations, equation 121, becomes an equation onα ˙ ±(x), −1 (126) 8f(1 − f)˙α± = ∂x f (1 − f)∂xα˙ ± .

If we change independent variable from x to f(x),

(127) df = ±2f(1 − f)dx, this becomes 2 (128) 2α ˙ ± = ∂f (1 − f) ∂f α˙ ± which has two independent solutions, 1 − f and (1 − f)−2. The latter is singular at the pole x = ±∞,sowemusthave

(129) α˙ ± = Cα±(1 − f).

At x = 0, this is

Cα± (130) α˙ ±(0) = 1+ρ2 so, to leading order, 1 1 (131) Cα± = σ˙ ±, α˙ ± = σ˙ ±(1 − f). 2 2 A LOOP OF SU(2) GAUGE FIELDS 189

Using equations 131 and 114 in equation 125,weget

−1 1 2 −2 2 (132) ∂x(f δA2)=− ρ σ˙ ±∂x f (1 − f) . 4

The unique solution that goes to zero at the pole, where f =1,is

1 2 −1 2 (133) δA2 = − ρ σ˙ ±f (1 − f) . 4

We now have the general solution of the ﬁrst two equations,

1 2 −1 2 1 (134) δA2 = − ρ σ˙ ±f (1 − f) , α˙ ± = σ˙ ±(1 − f). 4 2

7.1.2. The last two ﬂow equations. The last two of the four ﬂow equations, equations 122 and 123, become, after the change of independent variable from x to f(x),

2 3 3 −1 −1 (135) −2ρρ˙(1 − f) = ∂f f (1 − f) ∂f f (1 − f) δf 3 −3 −1 2 (136) 0=∂f f (1 − f) ∂f f (1 − f) δA1.

Integrating once, we get

2 −3 −1 −1 (137) ρρf˙ = ∂f f (1 − f) δf 3 2 −3 3 −1 2 (138) −2C1ρ f (1 − f) = ∂f f (1 − f) δA1.

The integration constant in the ﬁrst equation is ﬁxed by the boundary condition that δf should go to zero at f = 1. We write the integration constant 2 in the second equation as −2C1ρ for later convenience. Integrating again, we get

1 −1 (139) δf = − ρρf˙ (1 − f)+Cf f(1 − f) 3 2 −1 −1 −2 (140) δA1 = C1ρ f +2− 6(1 − f) − 6f(1 − f) ln f .

The integration constant Cf can be absorbed into a redefinition of ρ,sowe set Cf = 0. The new integration constant in the second equation is fixed by the boundary condition at f =1. 7.1.3. Summary: the general solution in each hemisphere. Now we restore the ± subscripts to ρ and f, indicating the hemisphere in which they obtain. The general solution to the flow equation in each hemisphere, 190 D. FRIEDAN

with the gauge ﬁxing condition δA0± =0,is

1 −1 (141) δf± = − ρ±ρ˙±f± (1 − f±) 3 2 −1 −1 −2 (142) δA1± = C1±ρ± f± +2− 6(1 − f±) − 6f±(1 − f±) ln f± 1 2 −1 2 (143) δA2± = − ρ±σ˙ ±f± (1 − f±) 4 1 (144) α˙ ±(x)= σ˙ ±(1 − f±). 2

The solution in each hemisphere is parametrized by three quantities,r ˙±, σ˙ ± and C1±, which are to be determined by the continuity equations at the equator.

7.2. Continuity conditions at the equator. Now we specialize to the subfamily of connections Dslow with

1 (145) ρ+ = ρ− = ρ, σ+ = −σ− = σ, α+(x)=−α−(−x)=α(x). 2

These connections Dslow have an x →−x symmetry that simpliﬁes the calculations. The general solution to the ﬂow equation in each hemisphere is now

(146) δA0± =0

1 −1 (147) δf± = − ρρf˙ ± (1 − f±) 3 2 −1 −1 −2 (148) δA1± = C1±ρ f± +2− 6(1 − f±) − 6f±(1 − f±) ln f± 1 2 −1 2 (149) δA2± = ∓ ρ σf˙ ± (1 − f±) 8 1 (150) α˙ ± = ± σ˙ (1 − f±) 4 1 (151) α±(0) = ± σ 4

We will need, at x = 0, the values

1 −1 1 (152) δf±(0) = − ρ ρ,˙ δA1±(0) = C1±,δA2±(0) = ∓ σ,˙ 3 8 and the ﬁrst derivatives

2 −1 1 (153) ∂xδf±(0) = ± ρ ρ,˙ ∂xδA1±(0) = ∓2C1±,∂xδA2±(0) = σ.˙ 3 4 A LOOP OF SU(2) GAUGE FIELDS 191

7.2.1. Continuity of Dslow . The continuity of Dslow at the equator is the condition, at x =0,

iα+(x)(P −Q) −iα+(x)(P −Q) (154) e (D+ + δA+) e

iα−(x)(P −Q) −iα−(x)(P −Q) = e (D− + δA−) e .

This is equivalent to

1 iσ(P −Q) 2 − 1 iσ(P −Q) (155) e 4 (−ρ ω + δA+) e 4 − 1 iσ(P −Q) 2 1 iσ(P −Q) = e 4 (−ρ ω + δA−) e 4 since D± = dω − f±ω and ∂xα+(0) = ∂xα−(0) by the symmetry α−(x)= −α+(−x)and

2 4 (156) f+(0) = f−(0) = ρ + O(ρ ).

At x =0, 2 2 1 −1 1 (157) −ρ ω + δA± = −ρ − ρ ρ˙ ω + C1±ω1 ∓ σω˙ 2 3 8 2 1 −1 1 (158) = ρ + ρ ρ˙ + C1± ± iσ˙ η 3 8 2 1 −1 1 † − ρ + ρ ρ˙ + C1± ∓ iσ˙ η 3 8 2 1 −1 (159) + ρ + ρ ρ˙ − 2C1± η3 3 so the continuity condition becomes the two equations

(160) 1 iσ 2 1 −1 1 − 1 iσ 2 1 −1 1 e 2 ρ + ρ ρ˙ + C1+ + iσ˙ = e 2 ρ + ρ ρ˙ + C1− − iσ˙ 3 8 3 8 2 1 −1 2 1 −1 (161) ρ + ρ ρ˙ − 2C1+ = ρ + ρ ρ˙ − 2C1− 3 3 which are equivalent to the two equations

(162) C1+ = C1− and 2 1 −1 1 1 1 (163) ρ + ρ ρ˙ + C1+ sin σ = − σ˙ cos σ. 3 2 8 2 192 D. FRIEDAN

7.2.2. Continuity of F slow . Continuity of F slow at x =0is

1 iσ(P −Q) 2 − 1 iσ(P −Q) e 4 (D+ + D+δA+) e 4 − 1 iσ(P −Q) 2 1 iσ(P −Q) (164) = e 4 (D− + D−δA−) e 4

The self-dual part of this condition is

1 iσ(P −Q) 2 − 1 iσ(P −Q) (165) e 4 (2D+ +(∗ +1)D+δA+) e 4 − 1 iσ(P −Q) 1 iσ(P −Q) = e 4 (∗ +1)D−δA− e 4 while the anti-self-dual part is

1 iσ(P −Q) − 1 iσ(P −Q) (166) e 4 (∗−1)D+δA+ e 4 − 1 iσ(P −Q) 2 1 iσ(P −Q) = e 4 (−2D− +(∗−1)D−δA−) e 4 .

The self-dual and anti-self-dual continuity equations are equivalent under the x →−x symmetry. From Appendix A.6, the (anti-)instanton curvature is

2 (167) D± = −2f±(1 − f±)(∗±1)dxω which is, at x =0,

2 2 (168) D± = −2ρ (∗±1)dxω.

From Appendix A.13,

(169) D±δf±ω = ∂xδf±dxω + λδf±∗dxω (170) D±δA1±ω1 = ∂xδA1±dxω1 + λ1δA1±∗dxω1 (171) D±δA2±ω2 = ∂xδA2±dxω2 + λ2δA2±∗dxω2 where, to leading order,

(172) λ = λ1 = λ2 =2.

At x = 0, using the values collected in equations 152 and 153,

2 −1 (173) D±δf±ω = − ρ ρ˙(∗∓1)dxω 3 (174) D±δA1±ω1 =2C1±(∗∓1)dxω1 1 (175) D±δA2±ω2 = ∓ σ˙ (∗∓1)dxω2 4 so 2 −1 1 (176) D±δA± =(∗∓1)dx − ρ ρω˙ +2C1±ω1 ∓ σω˙ 2 . 3 4 A LOOP OF SU(2) GAUGE FIELDS 193

Equation 165, the self-dual continuity condition, becomes

1 iσ(P −Q) 2 − 1 iσ(P −Q) (177) e 4 ρ ωe 4 − 1 iσ(P −Q) 1 −1 1 1 iσ(P −Q) = e 4 ρ ρω˙ − C1±ω1 − σω˙ 2 e 4 . 3 8

Equation 166, the anti-self-dual continuity condition, becomes the equivalent equation 1 iσ(P −Q) 1 −1 1 − 1 iσ(P −Q) (178) e 4 ρ ρω˙ − C1±ω1 + σω˙ 2 e 4 3 8 − 1 iσ(P −Q) 2 1 iσ(P −Q) = e 4 ρ ωe4 .

Their solution is

1 2 (179) C1+ = ρ (cos σ − 1) 3 (180) ρ−1ρ˙ = ρ2(1 + 2 cos σ) (181) σ˙ = −8ρ2 sin σ.

slow where we have used C1+ = C1− which was required for continuity of D . Finally, we check that the remaining continuity condition on Dslow , equation 163, is now also satisﬁed.

7.3. Summary: the Y-M ﬂow equation on the slow manifold. The slow manifold is represented by the family of U(2)-invariant connections iα+(x)(P −Q) −iα+(x)(P −Q) slow e (D+ + δA+) e x>0 (182) D = iα (x)(P −Q) −iα (x)(P −Q) e − (D− + δA−) e − x<0. obeying the symmetry condition

(183) ρ+ = ρ− = ρα+(x)=−α−(−x)=α(x).

The relative twist of the instanton and anti-instanton is

(184) σ =4α(0).

The slow manifold is parametrized by the instanton size ρ, the relative twist 1 σ, and by the gauge function α(x), α(0) = 4 σ. The gauge transformations

(185) Dslow → eiϕ(x)(P −Q)Dslow e−iϕ(x)(P −Q),ϕ(x)=−ϕ(−x) act on the slow manifold by

(186) α(x) → α(x)+ϕ(x) 194 D. FRIEDAN so the slow manifold in A/G is parametrized by ρ and σ alone. The Y-M ﬂow equations on the slow manifold are

(187) ρ˙ = ρ3(1 + 2 cos σ)+O(ρ5) (188) σ˙ = −8ρ2 sin σ + O(ρ4)

1 4 (189) α˙ = σ˙ (1 − f+)+O(ρ ). 4

The perturbation δA± of the (anti-)instanton is

(190) δA± = δf±(x)ω + δA1±(x)ω1 + δA2±(x)ω2 where

1 4 −1 6 (191) δf± = − ρ (1 + 2 cos σ)f± (1 − f±)+O(ρ ) 3 1 4 −1 −1 (192) δA1± = ρ (cos σ − 1)[f± +2− 6(1 − f±) 3 −2 6 − 6f±(1 − f±) ln f±]+O(ρ ) 4 −1 2 6 (193) δA2± =2ρ sin σf± (1 − f±) + O(ρ ). which indeed is a small perturbation of the (anti-)instanton D± everywhere on S4.

8. The gradient formula and S YM on the slow manifold We check that the Y-M flow on the slow manifold is a gradient flow with respect to the metric induced from the space of connections A. Let dDslow be an infinitesimal variation in the slow manifold, corresponding to variations dρ and dα(x) of the parameters. In the metric on A, given by equation 6, the length-squared of the variation is 2 slow 1 − slow ∗ slow (194) (ds ) = 2 tr dD dD 4π S4 In each hemisphere, to leading order, slow iα±(x)(P −Q) ∂ (195) dD = e dρ± D± +[D±, −idα±(x)(P − Q)] ∂ρ± × e−iα±(x)(P −Q) with

∂ −1 (196) dρ± D± = −2ρ± dρ±f±(1 − f±)ω ∂ρ± (197) [D±, −idα±(x)(P − Q)] = −∂xdα±dx i(P − Q) − 2f±dα±ω2. A LOOP OF SU(2) GAUGE FIELDS 195

Using the inner-product formulas given in Appendix A.11, equation 430, we get ∞ 2 slow 2 −2 2 2 2 (198) (ds ) = dx R (x)[12ρ± (dρ±) f±(1 − f±) −∞ 2 2 2 +(∂xdα±) +8f±(dα±) ] We replace the conformal factor R2(x) by its leading order approximation, equation 119, getting ∞ 2 slow 2 −1 −2 2 2 2 (199) (ds ) = dx 4ρ±f± (1 − f±)[12ρ± (dρ±) f±(1 − f±) −∞ 2 2 2 +(∂xdα±) +8f±(dα±) ] Specializing to the x ↔−x symmetric subfamily, and again writing f for f+,wehave (200) ∞ (ds2)slow =2 dx 4ρ2f −1(1 − f)[12ρ−2(dρ)2f 2(1 − f)2 0 2 2 2 +(∂xdα) +8f (dα) ] 1 2 2 2 2 2 2 2 (201) =16 df [3(dρ) (1 − f) + ρ (1 − f) (∂f dα) +2ρ (dα) ] 0 1 2 2 2 2 2 (202) = 16(dρ) +16ρ df [(1 − f) (∂f dα) +2(dα) ] 0 The generator of the Y-M ﬂow,ρ ˙,˙α, has inner product with a general variation

(203) 1 2 slow 2 2 (ds ) (˙ρ, α˙ ; dρ, dα)=16˙ρdρ +16ρ df (1 − f) ∂f α∂˙ f dα +2˙αdα 0 1 2 2 (204) =16˙ρdρ +4˙σρ df ∂ f −(1 − f) dα 0 (205) =16˙ρdρ +˙σρ2dσ (206) =16ρ3(1 + 2 cos σ)dρ − 8ρ2 sin σρ2dσ so 2 slow (207) (ds ) (˙ρ, α˙ ; dρ, dα)=−dSYM with 4 6 (208) SYM =2− 4ρ (1 + 2 cos σ)+O(ρ ).

This is the gradient formula, equation 5. The additive constant in SYM is ﬁxed because SYM = 2 for the twisted pairs at ρ =0. 196 D. FRIEDAN

9. The metric on the slow manifold in A/G The metric on the slow manifold in A is given by equation 202.To find the metric on the slow manifold in A/G, we need to project on the horizontal subspace of the tangent space of A — the variations orthogonal to the infinitesimal gauge transformations. The infinitesimal gauge transformations are the perturbations dαV (x) with dαV (0) = 0. A variation dαH is perpendicular to the gauge transformations iff, for all dαV (x)withdαV (0) = 0, 1 2 (209) df (1 − f) (∂f dαV )(∂f dαH )+2(dαV )(dαH ) =0 0 whichistosaythatdαH satisfies the ordinary differential equation 2 (210) −∂f (1 − f) ∂f +2 dαH =0.

The only solution that vanishes at f =1is 1 (211) dαH = dσ(1 − f). 4 Substituting in equation 202, we get the metric on the slow manifold in A/G,

2 slow 2 2 2 (212) (ds )A/G = 16(dρ) + ρ (dσ) .

The gradient formula of course holds here as well,

2 slow (213) (ds )A/G (˙ρ, σ˙ ; dρ, dσ)=−dSYM .

10. Long time behavior of the flow The Y-M flow on the slow manifold, dρ dσ (214) = ρ3(1 + 2 cos σ), = −8ρ2 sin σ, dt dt has flow lines given by dρ ρ 1+2cosσ (215) = − dσ 8 sin σ which integrates to

(216) ρ8(1 − cos σ)sinσ =4C.

Changing variable from σ to σ (217) s =cos ,s∈ [−1, 1], 2 A LOOP OF SU(2) GAUGE FIELDS 197 the ﬂow is dρ ds (218) = ρ3(4s2 − 1), =8ρ2s(1 − s2), dt dt and the ﬂow lines are

(219) ρ8s(1 − s2)3/2 = C.

On a ﬂow line, say on the side s ≥ 0 where C ≥ 0, the ﬂow equation is ds (220) =8C1/4(1 − s2)5/8s3/4. dt which integrates to

1 1 1/4 4 2 4 2 (221) 2C t = st F (st ) − s0 F (s0) where H(s2) is the hypergeometric function 1 5 9 1 3 (222) F (z)=F , ; ; z ,F(0) = 1,F(1 − )=F (1) − 8 + O(). 8 8 8 3 Substituting for C,weget

1 4 2 2 3/8 st 2 2 (223) 2ρ0(1 − s0) t = F (st ) − F (s0) s0

1 4 2 2 3/8 2 s0 2 (224) 2ρt (1 − st ) t = F (st ) − F (s0) st

Suppose t large. If we hold ρ0 ﬁxed and letting s0 vary near 0, we see explicitly from these formulas that the trajectory moves ﬁrst towards ρ = 0,s = 0, then along the s-axis to the neighborhood of ρ =0,s = 1, then outward to increasing ρ with s near 1.

11. The outgoing trajectory For the symmetric twisted pair, where the instanton and anti-instanton are located at opposite poles in the round S4, we can show that the outgoing trajectory at σ =0, 2π ends at the ﬂat connection. The argument does not work for other twisted pairs, whose outgoing trajectories have less symmetry. The perturbations δA1,2, equations 192 and 193 vanish for σ =0, 2π, so the outgoing trajectory has the full SO(4) symmetry of the aligned instanton-anti-instanton and of the round geometry on S4. The connections on the outgoing trajectory are therefore all of the form

(225) D = dω − fω, f(±∞)=1. 198 D. FRIEDAN

From equation 55, the Y-M action is 3 2 2 2 (226) SYM = dx (∂xf) +4f (1 − f) . 2 From Appendix A, 1 (227) ∗F± = − [∂xf ± 2f(1 − f)] (∗±1)dxω 2 1 2 −1 (228) ∗D∗F± = − R (x) [∂x ± 2(2f − 1)] [∂xf ± 2f(1 − f)] ω 2 so the Y-M flow equation is df 2 −1 2 (229) = R (x) ∂xf +4f(1 − f)(2f − 1) . dt Let us assume that the flow ends at a fixed point. The fixed point equation is

2 (230) ∂xf +4f(1 − f)(2f − 1) = 0.

For any solution f of the ﬁxed point equation, the quantity

2 2 2 (231) A =(∂xf) − 4f (1 − f) is constant, ∂xA = 0, and must vanish because SYM < ∞.So,forallx,

(232) ∂xf = ±2f(1 − f).

The only solution of this equation compatible with the boundary conditions f(±∞) = 1, besides the twisted pair, is f = 1, the ﬂat connection. There is no other ﬁxed point where the outgoing trajectory can end.

12. Stable 2-manifolds of SU (2) and SU (3) gauge fields Nontrivial stable 2-spheres of gauge fields might give 2-d instanton corrections to the space-time quantum field theory in the lambda model (discussed in section 13.3 below). Nontrivial 2-spheres of gauge fields are classified by π2(A/G), which is π5 of the gauge group. Potentially interesting examples are π5SU(2) = Z2 and π5SU(3) = Z. We describe some partial results towards constructing stable 2-spheres for SU(3) and for SU(2) gauge groups.

12.1. SU (3). Numerical evidence suggests that there is a stable 2-sphere of SU(3) connections on S4 consisting again of zero-size instanton-anti-instanton twisted pairs [29]. The numerical calculation is analogous to the SU(2) calculation reported above (and was actually done 6 ﬁrst). The SU(3) principle bundles over S are classiﬁed by π5SU(3) = Z. 6 The homogeneous space SU(3) → G2 → S represents a generator of A LOOP OF SU(2) GAUGE FIELDS 199

2 4 6 π5SU(3) [30]. Pulling back along a suitably chosen map S × S → S gives a nontrivial 2-sphere of connections in the trivial SU(3) bundle over 4 4 S , representing a generator of π2(A/G). The south pole of S is mapped to the flat connection. Some of the G2 symmetry survives, so that all of the connections on S4 are SU(2)-invariant. An additional U(1) symmetry acts on the 2-sphere family of connections, rotating the 2-sphere around its poles. The north pole of the 2-sphere is left fixed, so the connection at the north pole has an additional U(1) symmetry. It also has a discrete symmetry exchanging x →−x. It seems plausible that this connection flows to an index 2 fixed point whose two dimensional unstable manifold is a stable 2-sphere. It also seems plausible that this connection flows to the connection that minimizes SYM among all connections with the same symmetries. Car- rying out this minimization of SYM numerically, we find strong indications that the minimum value is SYM = 2, realized by a twisted pair. All SU(3) instantons on S4 of instanton number ±1 are reducible [31]. That is, they are SU(2) instantons embedded in SU(3). We identify SU(2) with the upper-left 2 × 2 block in SU(3), identifying an element g ∈ SU(2) with the block matrix g 0 (233) g ∈ SU(2) ≡ ∈ SU(3). 0 1

The basic SU(2) instanton D+ is now an SU(3) instanton. The general −1 SU(3) instanton — of given size and location — is GD+G for G ∈ SU(3), up to the equivalence G ∼ GK(θ), for K(θ)intheU(1) subgroup of SU(3) of elements that commute with SU(2), which take the block matrix form eiθ 0 (234) K(θ)= . 0 e−2iθ

The space of orientations of the SU(3) instanton is thus SU(3)/U(1). The SU(3) space of relative twists of a twisted pair of SU(3) instantons is Mtw = U(1)\SU(3)/U(1), which consists of the individual orientations of the instanton and anti-instanton, SU(3)/U(1)×SU(3)/U(1), modulo the global SU(3) SU(3) SU(3) gauge transformations. Mtw contains nontrivial 2-spheres, π2Mtw ⊃ Z, that can represent π2(A/G). To write a concrete nontrivial 2-sphere of relative twists, it is convenient to parametrize SU(3) as SU(2) × D2 × SU(2),

−1 (235) G(g−,u,g+)=g− G1(u)g+ where ⎛ ⎞ 10 0 ⎝ 2⎠ (236) G1(u)= 0 u − 1 −|u| , |u|≤1. 0 1 −|u|2 u¯ 200 D. FRIEDAN

The parametrization is faithful for |u| < 1, while at the boundary of the 2-disk, |u| = 1, it gives a redundant parametrization of the subgroup SU(2)× U(1) ⊂ SU(3). The U(1) subgroup of SU(3) acts on the left and right by (237) −1 2iθ−2iθ K(θ)G(g−,u,g+)K(θ ) = G(h(θ +2θ )g−,e u, h(2θ + θ )g+) where eiθ 0 (238) h(θ)= . 0 e−iθ

The O(4) = SU(2) × SU(2)/{±1} group of rotations around the poles of S4 −1 acts on the relative twists by (gL,gR):G → gR GgR, the gL all leaving the twisted pair invariant. In our parametrization of SU(3), the symmetries act by

(239) G(g−,u,g+) → G(g−gR,u,g+gR)

We represent the symmetry classes of twists by the G(1,u,g+) subject to the gauge equivalence −1 2iθ 2θ θ (240) G(1,u,g+) ≡ G(1,e u, h(θ)g+)=K G(1,u,g+)K − 3 3 and a remaining U(1) symmetry −1 (241) G(1,u,g+) → G(1,u,h(θ)g+h(θ) ).

The symmetry classes of twisted pairs with an additional U(1) invariance are the G(1,u,1) and also G(1, 0,g0)with 01 (242) g0 = . −10

The latter, G(1, 0,g0), is the U(2)-invariant twisted pair indicated by the computer calculation. A 2-sphere family of twisted pairs invariant under U(1) acting by rotation around the poles of S2 is given by 2 (243) G(w)=G(1,w ,g1(w)), |w|≤1 where w − 1 −|w|2 (244) g1(w)= . 1 −|w|2 w¯

The U(1) symmetry is

(245) G(w) → G(eiθw). A LOOP OF SU(2) GAUGE FIELDS 201

The twisted pair at the north pole of S2, w = 0, is the U(2)-invariant G(1, 0,g0). At |w| =1,

(246) G(w)=K(w) ≡ 1 so |w| = 1 can be identified to the the south pole in S2, which is mapped to the aligned twisted pair, G =1. It should be straightforward to check directly that w → G(w) represents a generator of π2(A/G), by the same argument used above to check the nontriviality of the loop of SU(2) twisted pairs. We leave A0 = 0 gauge, making −1 4 G(w)D+G(w) non-singular at the south pole of S by a gauge transformation φ(w, x) ∈ SU(3). The twisted pairs at |w| = 1 will all be gauge equivalent, giving a loop in the gauge group, a map S1 × S4 → SU(3). This will factor through a map S5 → SU(3), which we can check is a generator 5 of π5SU(3) by composing with SU(3) → S = SU(3)/SU(2) to get a map S5 → S5 whose index should be ±1[4]. A quicker way to check the nontriviality of the 2-sphere w → G(w)isto evaluate the family index [32] of the Dirac operator on S4 acting on spinors tensored with the defining representation, 3,ofSU(3). The chiral zero-modes of the Dirac operator of each handedness are localized respectively in the instanton and and the anti-instanton. It is a simple calculation to show that the left-handed zero mode forms a line bundle of Chern number 1 over the 2-sphere of twisted pairs, which must then necessarily be a generator of π2(A/G)=Z. The Y-M flow on the slow manifold remains to be calculated in order to check that that the 2-sphere w → G(w), or some deformation, is locally stable under the flow. The calculation is the same, in principle, as for the SU(2) twisted pairs. For SU(3), the symmetry classes of twists are described by 3 parameters, analogous to the twist angle σ for SU(2) twists. Unfortunately, there does seem to be any symmetry that singles out a distinguished set of representatives of the symmetry classes, closed under the flow, analogous to the U(2) symmetry for SU(2) twisted pairs. It might be possible to find a perpendicular slice through the symmetry classes, which would be closed under the gradient flow. Otherwise, it will be necessary to parametrize the slow manifold by the full 6 parameter space of SU(3) twists, in addition to the instanton size ρ. Inverting the instanton laplacian will be considerably more work than in the SU(2) case. In any case, the calculation of the Y-M flow on the slow manifold and the check of local stability remain to be done.

12.2. SU (2). Since π5SU(2) = Z2, there should be a nontrivial stable 2-sphere of SU(2) gauge fields on S4. We do not know of a homogeneous realization of the generator of π5SU(2) analogous to the bundles SU(2) → 5 6 SU(3) → S for π4SU(2) and SU(3) → G2 → S for π5SU(3), but there is available a realization with enough symmetry to reduce the problem to minimizing SYM on the space of connections with a certain fixed symmetry 202 D. FRIEDAN group, as in the other two cases. In this case, the symmetry group is large enough that numerical minimization is (barely) practical. We construct a nontrivial 2-sphere family of SU(2) bundles over S4, each having the symmetry group (U(1) × U(1)/Z2) × Z2. The bundle at 2 the north pole in S has an extra Z2 × Z2 symmetry. We attempt to minimize SYM numerically over connections with the enhanced symmetry group 3 (U(1)×U(1)/Z2)×(Z2) , again approximating the space of such connections by finite dimensional affine subspaces. We find min(SYM ) < 4.0053. The numerical computations are more expensive in processing time and memory than the previous ones because the two continuous symmetries reduce S4 to a 2-dimensional domain, instead of the 1-dimensional domain of the previous calculations. We have to minimize SYM over connections that are polynomials in two variables. The numerical results suggest that, at the enhanced symmetry point in the 2-sphere family, min(SYM ) is realized by a fixed point of the Y-M flow that consists of two zero-size instantons and two zero-size anti-instantons, arranged along the x axis in the order II¯ II¯ . Writing the sizes of the instan- −x x x tons r+1 = e +1 , r+2 = e +2 and the sizes of the anti-instantons r−1 = e −1 , −x r−2 = e −2 , the zero-size limit is taken with

(247) x−1 x+2 0 x−2 x+1.

Each pair of neighbors in the sequence is maximally twisted. It seems plausible that repulsion between neighbors will drive such a configuration of finite- size instantons and anti-instantons to this zero-size limit. In the limit, there is an an instanton/anti-instanton pair at each of the poles. The evidence for min(SYM ) = 4 at the enhanced symmetry point, realized by the twisted quadruplet of zero-size (anti-)instantons, is good, though perhaps not as compelling as in the previous calculations. The twisted quadruplet has SYM =4,soSYM ≤ 4 is a rigorous upper bound at the enhanced symmetry point. The continuous U(1) × U(1)/Z2 symmetry restricts the relative twists of the instantons to the diagonal SU(2) matrices. We write explicitly a 2-parameter family of twisted quadruplet connections, in the 2-parameter family of bundles. This family of connections forms a 2-torus, not a 2-sphere. It remains to calculate the Y-M flow in the slow modes, to check first that the twisted quadruplet connection at the enhanced symmetry point has a 2- dimensional unstable manifold, and then to find the global structure of that unstable manifold, presumably either a 2-torus of zero area or a 2-sphere of nonzero area. The first possibility would be of interest for the lambda model. 12.2.1. A nontrivial 2-sphere of SU(2) bundles over S4. The nontrivial element in π5SU(2) was originally realized as the suspension map S(h◦Sh): S5 → S3, where h : S3 → S2 is the Hopf fibration, and Sh : S4 → S3 is its A LOOP OF SU(2) GAUGE FIELDS 203 suspension [6]. We write explicitly

(248) S(h ◦ Sh):[0,π]2 × SU(2) → SU(2) −1 −1 −1 (249) S(h ◦ Sh)(β1,β2,g)= ghβ2 g hβ1 ghβ2 g where eiβ 0 (250) hβ = . 0 e−iβ

We make a topologically insignificant modification, defining 2 (251) Φ2 :[0,π] × SU(2) → SU(2) −1 −1 −1 −1 −1 2 1 2 ◦ (252) Φ (β ,β )(g)=hβ1 S(h Sh)=hβ1 ghβ2 g hβ1 ghβ2 g , which satisfies

(253) Φ2(β1, 0,g)=Φ2(β1,π,g)=Φ2(0,β2,g)=Φ2(π, β2,g)=1, so the boundary of the square [0,π]2 can be identiﬁed to a point, the square 2 becoming a 2-sphere, and Φ2 becoming a nontrivial map S × SU(2) → 2 4 SU(2). For each (β1,β2) ∈ S , we construct an SU(2) bundle over S using 4 g → Φ2(β1,β2,g) as the gluing map at the equator in S .ThusΦ2 deﬁnes a nontrivial 2-sphere of trivial SU(2) bundles over S4. The group SO(4) = SU(2) × SU(2)/{±1} of rotations of S4 around the polar axis acts by −1 −1 −1 −1 −1 −1 (254) Φ2(β1,β2,gLggR )=gL h1 gh2g h1 gh2g gL where −1 −1 (255) h1 = gL hβ1 gL,h2 = gR hβ2 gR.

If gL and gR are both diagonal,

(256) gL = hα ,gR = hα, then −1 −1 (257) Φ2(β1,β2,hα ghα )=hα Φ2(β1,β2,g)hα so each of the SU(2) bundles over S4 is invariant under the U(1)×U(1)/{±1} subgroup of diagonal matrices (hα ,hα) modulo (−1, −1). In addition, the entire 2-sphere family of bundles is invariant under the Z2 × Z2 subgroup generated by (gL,gR)=(μ2,μ2)and(μ1,μ3) where 0 i 01 i 0 (258) μ1 = ,μ2 = ,μ3 = . i 0 −10 0 −i 204 D. FRIEDAN

This Z2 × Z2 acts on the family of bundles by

−1 −1 (259) Φ2(β1,β2,μ2gμ2 )=μ2Φ2(π − β1,π− β2,g)μ2 −1 −1 (260) Φ2(β1,β2,μ1gμ3 )=μ1Φ2(π − β1,β2,g)μ1 −1 (261) Φ2(β1,β2,μ3gμ1)=μ3Φ2(β1,π− β2,g)μ3 .

Finally, there is a Z2 symmetry

−1 −1 2 1 2 2 − 1 2 (262) Φ (β ,β ,g) = hβ1 Φ (π β ,β ,g)hβ1 that acts by reﬂecting S4 in the equator, taking θ → π − θ, x →−x. −1 Combining with the discrete symmetry g → μ1gμ3 , we get a reﬂection symmetry of each bundle in the family,

−1 −1 −1 −1 2 1 2 2 1 2 1 1 (263) Φ (β ,β ,g) = hβ1 μ1 Φ (β ,β ,μ gμ3 )μ hβ1 so each connection has symmetry group (U(1) × U(1)/{±1}) × Z2. π The SU(2) bundle at the midpoint β1 = β2 = 2 thus has an extra Z2×Z2 symmetry. If we were to choose a 2-sphere family of connections in this 2-sphere family of SU(2) bundles, respecting the symmetries of the bundles, then run the Y-M flow on the family of connections, we might expect that π the connection at the midpoint β1 = β2 = 2 would flow to a fixed point with effective Morse index 2 that minimizes SYM among all connections with the π enhanced symmetry of the bundle at β1 = β2 = 2 . With this scenario in mind, we attempt to minimize SYM among the connections invariant under 3 this (U(1) × U(1)/{±1}) × (Z2) group. 12.2.2. Reduction to 2-dimensions. The continuous symmetry group U(1) × U(1)/{±1} acts on S4 by iα−iα iα+iα z1 −z¯2 −1 z1e −z¯2e (264) g = → hα ghα = −iα−iα −iα+iα z2 z¯1 z2e z¯1e iα −iα z1e 1 −z¯2e 2 = iα −iα z2e 2 z¯1e 1

1 1 (265) α = (α1 − α2),α= (−α1 − α2). 2 2 We write

iθ1 iθ2 1 1 (266) z1 = r1e ,z2 = r2e ,r1 =cos ψ, r2 =sin ψ, ψ ∈ [0,π] 2 2 4 and use θ, ψ, θ1,θ2 as coordinates on S . We can write the coordinate map

−1 1 (267) g = h (θ1−θ2)g(ψ)h 1 (−θ −θ ) 2 2 1 2 A LOOP OF SU(2) GAUGE FIELDS 205 where r1 −r2 − 1 ψμ (268) g(ψ)= = e 2 2 . r2 r1

The coordinate map is redundant at the poles θ =0,π and at ψ =0,π. All of S4 is covered when ψ ranges over [0,π], but it is useful to think of ψ taking any real value, the coordinate map being many-to-one. The slice θ1 = θ2 = 0 contains one representative in each symmetry class (except at the poles θ =0,π). A connection on S4 invariant under the continuous symmetry will reduce to a connection on the slice, the 2- dimensional domain parametrized by θ and ψ. A connection in the bundle deﬁned by the patching map Φ2(β1,β2)consists of a connection in each hemisphere, D± = d+A±, related on the overlap of the hemispheres by

−1 (269) D− =Φ2(β1,β2)D+Φ2(β1,β2) .

Writing

−1 (270) Φ2(β1,β2)=Φ− Φ+ with

−1 −1 (271) Φ+ = hβ1 ghβ2 g , Φ− = ghβ2 g hβ1 , the patching formula becomes

−1 −1 (272) Φ−D−Φ− =Φ+D+Φ+ .

The continuous symmetry group (U(1) × U(1)/{±1})actsby

−1 −1 (273) D±(θ, hα ghα )=hα D±(θ, g)hα or, equivalently,

−1 −1 (274) A±(θ, hα ghα )=hα A±(θ, g)hα .

We eliminate the dependence on θ1,2 by a gauge transformation

−1 ˜± ˜± 1 ± (275) D = d + A = h (−θ1+θ2)D h 1 (−θ +θ ) 2 2 1 2

−1 1 ˜± 1 ± 1 − 2 3 (276) A = h (−θ1+θ2)A h 1 (−θ +θ ) + (dθ dθ )μ 2 2 1 2 2

˜ ˜ (277) ∂θ1 A± = ∂θ2 A± =0. 206 D. FRIEDAN

The patching formula now becomes

−1 −1 (278) Φ˜ −D˜−Φ˜ − = Φ˜ +D˜+Φ˜ + where

−1 ˜ ± 1 ± (279) Φ = h (−θ1+θ2)Φ h 1 (−θ +θ ) 2 2 1 2 also do not depend on θ1,2,

˜ −1 ˜ −1 (280) Φ+ = hβ1 g(ψ)hβ2 g(ψ) , Φ− = g(ψ)hβ2 g(ψ) hβ1 .

Finally, we deﬁne

−1 −1 (281) D = d + A = Φ˜ −D˜−Φ˜ − = Φ˜ +D˜+Φ˜ + which is regular everywhere on S4 except at the poles, and which is independent of θ1,2. At the poles, 1 −1 (282) D → Φ˜ − d + (dθ1 − dθ2)μ3 Φ˜ − at the north pole, θ =0. 2 1 −1 (283) D → Φ˜ + d + (dθ1 − dθ2)μ3 Φ˜ + at the south pole, θ = π. 2

We have traded the patching condition at the equator and the dependence on θ1,2 for boundary conditions at θ =0,π. We now can write

(284) A = Aθ(θ, ψ)dθ + Aψ(θ, ψ)dψ + Aθ1 (θ, ψ)dθ1 + Aθ2 (θ, ψ)dθ2.

π 12.2.3. Reduction from SU(2) to U(1) by a Z2 symmetry at β1 = β2 = 2 . π The extra Z2 symmetry at β1 = β2 = 2 , π π −1 π π −1 (285) Φ2 , ,μ2gμ = μ2Φ2 , ,g μ 2 2 2 2 2 2 becomes, on the slice,

−1 (286) A(θ, ψ, −θ1, −θ2)=μ2A(θ, ψ, θ1,θ2)μ2 because

−1 −1 (287) μ2g(ψ)μ2 = g(ψ),μ2h 1 (−θ +θ )μ2 = h 1 (θ −θ ), 2 1 2 2 1 2 and, at the enhanced symmetry point,

1 1 −1 − ψμ2 ψμ2 −ψμ2 ˜ − π π 2 3 2 3 − (288) Φ = g(ψ)h 2 g(ψ) h 2 = e μ e μ = e 1 1 −1 − ψμ2 + ψμ2 ψμ2 ˜ + π π 3 2 2 3 − (289) Φ = h 2 g(ψ)h 2 g(ψ) = μ e e μ = e . A LOOP OF SU(2) GAUGE FIELDS 207

The connection on the slice therefore takes the form † † (290) A = aθμ2dθ + aψμ2dψ +(v1μ+ − v¯1μ+)dθ1 +(v2μ+ − v¯2μ+)dθ2 where

1 −1 (291) μ+ = (μ3 + iμ1),μ2μ+μ = −μ+, [μ2,μ+]=2iμ+ 2 2 and where the components aθ, aψ,andv1,2 are functions only of θ and ψ. Thus the invariant SU(2) connection reduces to a U(1) connection on the slice, plus the two additional ﬁelds v1,2. We write the U(1) connection as

(292) Dr = d + iAr = dθDr,θ + dψDr,ψ,Ar = aθdθ + aψdψ.

Its curvature 2-form is

(293) Fr = dAr = Fr,θψdθdψ, Fr,θψ = ∂θaψ − ∂ψaθ

12.2.4. SYM. The curvature 2-form of D is

(294) F =[Fr + i(v1v¯2 − v¯1v2)dθ1dθ2] μ2 +(Drv1dθ1 + Drv2dθ2) μ+ † − Drv1dθ1 + Drv2dθ2 μ+ where the covariant derivatives of the ﬁelds v1,2 are given by

(295) Drvk =(d +2iAr)vk,Dr,θvk =(∂θ +2iaθ)vk, Dr,ψvk =(∂ψ +2iaψ)vk.

The round metric on S4 is

2 2 2 1 2 2 1 2 (296) (ds) 4 =(dθ) +(sinθ) (dψ) +(sinθ) (1 + cos ψ)(dθ1) S 4 2 2 1 2 +(sinθ) (1 − cos ψ)(dθ2) . 2 The volume element of S4 reduced to the 2-dimensional domain is 2π 2π 1 1 3 (297) 2 dθ1dθ2 dvolS4 = dθdψ (sin θ) sin ψ. 8π 0 0 8 The Yang-Mills action is most neatly written in terms of a certain metric on the 2-dimensional domain 2 −1 −2 2 2 (298) (ds)2 =(2sinψ) 4(sin θ) (dθ) +(dψ) =(2sinψ)−1[4(dx)2 +(dψ)2] whose area element is −1 −1 (299) dvol2 = dθdψ (sin θ sin ψ) = dxdψ (sin ψ) 208 D. FRIEDAN and in terms of hermitian forms on the line bundles (300) 2 2 2 2 1 1 11¯ v12 = h1¯1|v1| , v22 = h2¯2|v2| ,h1¯1 =tan ψ, h2¯2 =cot ψ = h . 2 2

That is, v1 is a section of a line bundle L1 with hermitian form h,andv2 is −1 a section of L2 = L¯1 .ThenSYM is given by the covariant formula 1 2 2 2 2 (301) SYM = dvol2 F 2 + v1v¯2 − v¯1v22 + Drv12 + Drv22 2 where 2 ab 2 2 (302) F 2 = Fr,abFr , v1v¯2 − v¯1v22 = h1¯1h2¯2|v1v¯2 − v¯1v2| , 2 a Drvk2 = hkk¯ Dr,avkDr vk. 12.2.5. Discrete symmetries and boundary conditions. The remaining Z2 × Z2 symmetries are:

(303) aθ(π − θ, ψ)=aθ(θ, ψ) v1(π − θ, ψ)=¯v1(θ, ψ) (304) aψ(π − θ, ψ)=−aψ(θ, ψ) v2(π − θ, ψ)=¯v2(θ, ψ)

(305) aθ(θ, π − ψ)=−aθ(θ, ψ) v1(θ, π − ψ)=−v¯2(θ, ψ) (306) aψ(θ, π − ψ)=aψ(θ, ψ) v2(θ, π − ψ)=−v¯1(θ, ψ)

In addition, if we regard the connection as a function of ψ ∈ R,wehave

(307) aθ(θ, −ψ)=−aθ(θ, ψ) v1(θ, −ψ)=¯v1(θ, ψ) (308) aψ(θ, −ψ)=aψ(θ, ψ) v2(θ, −ψ)=¯v2(θ, ψ).

Combined with the ψ → π − ψ symmetry, this gives

(309) aθ(θ, ψ + π)=aθ(θ, ψ) v1(θ, ψ + π)=−v2(θ, ψ) (310) aψ(θ, ψ + π)=aψ(θ, ψ) v2(θ, ψ + π)=−v1(θ, ψ) so the U(1) connection Dr lives on the 2-sphere parametrized by θ, ψ with the identiﬁcation ψ ∼ ψ + π. The ﬁelds v1,2 live on a 2-sheeted covering of this 2-sphere. The boundary conditions at θ =0,π are: (311) 1 −2iψ 1 −2iψ aθ(0,ψ)=0,aψ(0,ψ)=1,v1(0,ψ)= e ,v2(0,ψ)=− e , 2 2 (312) 1 2iψ 1 2iψ aθ(π, ψ)=0 aψ(π, ψ)=−1 v1(π, ψ)= e v2(π, ψ)=− e . 2 2 A LOOP OF SU(2) GAUGE FIELDS 209

At ψ =0andatψ = π, the U(1) × U(1)/{±1} continuous symmetry degenerates to U(1), giving rise to boundary conditions at ψ =0,π (313) 1 aθ(θ, 0) = 0,v1(θ, 0) =v ¯1(θ, 0),v2(θ, 0) = − ,Dr,ψv2(θ, 0) = 0, 2 (314) 1 aθ(θ, π)=0 v2(θ, π)=¯v2(θ, π) v1(θ, π)= Dr,ψv1(θ, π)=0. 2 Because of the degeneration of the continuous symmetry group, the U(1) gauge transformations must act trivially at ψ =0,π. The boundary conditions on v1,2 at ψ =0,π are gauge invariant for this restricted group of gauge transformations. Over the 2-sphere ψ ∼ ψ + π, 1 (315) 2Fr = −2 2π so the fields v1,2 live in the U(1) bundle of Chern number −2. 12.2.6. Numerical computations. We try to minimize SYM numerically in this two dimensional setting by the same technique as in the previous one dimensional problems, approximating the space of connections by increasing finite dimensional affine subspaces of polynomial connections. We let the fields be polynomials of finite degree, whose coefficients are real variables. If there are N of these real variables, we are approximating the space of connections by an affine subspace of dimension N. We use mathematical software [28]toevaluateSYM as a quartic polynomial in these N real variables, and then to minimize it. First, we design the polynomial approximation so that the evaluation of SYM requires only multiplication of polynomials (to conserve computational resources). We use as coordinates (316) t =cosθ, s =cosψ, and write 2 (317) aθ = Qθ(t, s)sinθ sin ψ, aψ = t +(1− t )Qψ(t, s)

(318) v1 = v11 + iv˜12 sin ψv2 = v21 + iv˜22 sin ψ

1 2 2 (319) v11 = − + s +(1− t )(1 + s)P11(t, s) 2 1 2 2 (320) v21 = − s − (1 − t )(1 − s)P11(−t, −s) 2 2 (321) v˜12 = −ts +(1− t )P12(t, s) 2 (322) v˜22 = ts − (1 − t )P12(−t, −s) 210 D. FRIEDAN

Table 2. Numerical minimization of SYM for the reduced 2-dimensional U(1) system.

nt ns N min(SYM ) 2 2 16 5.23 3 3 36 4.91 4 4 64 4.73 5 5 100 4.60 7 3 84 4.48 where Qθ, Qψ, P11,andP12 are polynomials in t and s obeying the symmetry conditions

(323) Qt(t, s)=−Qt(t, −s)=Qt(−t, s)

(324) Qψ(t, s)=Qψ(t, −s)=−Qψ(−t, s)

(325) P11(t, s)=P11(−t, s),P12(t, s)=−P12(−t, s).

All of the discrete symmetries are automatically satisﬁed, as are all of the boundary conditions except the boundary conditions on Dr,ψv1,2. These last conditions are solved by

(326) P12(t, s)=Qψ(t, s)+(1+s)Qv(t, −s) where Qv is a new polynomial with symmetry

(327) Qv(t, s)=−Qv(−t, s).

The connection is now specified by the polynomials P11, Qθ, Qψ,andQv, which obey the various symmetries written above, but are otherwise arbitrary. For simplicity in the computer program, each of the four polynomials is written so as to contain the first nt powers of t and the first ns powers of s consistent with the symmetries, so each polynomial contains ntns coefficients, so the total number of coefficients is N =4ntnu. We are approximating the space of connections by an affine subspace of dimension N. Numerical results are shown in Table 2. They were obtained using the Sage mathematics software [28]. The calculations became too time- consuming for N>100. Nothing is especially suggested by the values of min(SYM ), besides insufficiency of the computing resources. Slightly more suggestive are the graphs of the chiral action density 1 1 (328) tr −F (1 + ∗)F = dtds L+(t, s) 8π2 2 A LOOP OF SU(2) GAUGE FIELDS 211

Figure 4. Plots of L+(x) for the connections numerically minimizing SYM . The bumps at positive x move away from the origin as min(SYM ) decreases (becomes a better upper bound). The graphs of L−(x) are given by reﬂecting x →−x. or rather, of its projection onto the t or x coordinate 1 (329) dx L+(x)=dt ds L+(t, s). −1 Recall that 1 (330) x =lntan θ, t =cosθ = − tanh x. 2 The graphs are shown in Figure 4. There appears to be a separation into four lumps of alternating topological charges (the two lumps of positive charge are shown in the graphs), though there is no indication that the topological charges are quantized. Still, we can guess that the outer lumps will travel to x = ±∞, the lump going to x = ∞ resolving into a zero-size instanton and the lump going to x = −∞ resolving into a zero-size anti-instanton, the argument being that there seems to be nothing to stop this happening. A better method of approximation is needed that could give more convincing numerical evidence in support of this extrapolation. 12.2.7. Assume zero-size (anti-)instantons at the poles. We now assume that a zero-size instanton has gone to the south pole and a zero-size instanton to the north pole. These provide new boundary conditions at θ =0,π for the connection away from the poles. We minimize SYM numerically using the new boundary conditions. −x An instanton of size r+ = e + at the south pole is the connection

−1 1 (331) D+ = d +(1− f+)ω, ω = gd(g ),f+(x)= . 1 − e−2(x−x+) In the limit of zero size, the instanton becomes −1 (332) D+ = d + ω = gdg 212 D. FRIEDAN which will provide the new boundary condition at the south pole, θ = π, x = ∞. Going to the slice θ1,2 = 0, the zero-size instanton becomes

−1 ˜ + 1 + (333) D = h (−θ1+θ2)D h 1 (−θ +θ ) 2 2 1 2 −1 −1 1 (334) = g(ψ)h 1 (−θ −θ )dh (−θ1−θ2)g(ψ) 2 1 2 2 1 −1 (335) = g(ψ) d − (dθ1 + dθ2)μ3 g(ψ) 2 1 1 − ψμ2 1 ψμ2 (336) = e 2 d − (dθ1 + dθ2)μ3 e 2 2

−1 (337) D = Φ˜ +D˜+Φ˜ +

ψμ2 −ψμ2 (338) = e D˜+e 1 1 ψμ2 1 − ψμ2 (339) = e 2 d − (dθ1 + dθ2)μ3 e 2 2 1 1 iψ −iψ † (340) = d − dψμ2 − (dθ1 + dθ2)(e μ+ − e μ+). 2 2 At the north pole we put the reﬂected connection, respecting the θ → π − θ symmetry:

1 1 −iψ iψ † (341) D = d + dψμ2 − (dθ1 + dθ2)(e μ+ − e μ+). 2 2 Working backwards, 1 1 − ψμ2 1 ψμ2 (342) D = e 2 d − (dθ1 + dθ2)μ3 e 2 2 −1 1 −1 −1 −1 (343) = Φ˜ −μ3g(ψ)μ d − (dθ1 + dθ2)μ3 μ3g(ψ) μ Φ˜ − 3 2 3 so the zero-size anti-instanton is given by −1 −1 (344) D− = μ3(gdg )μ3 which is the maximally twisted zero-size anti-instanton. The new boundary conditions at θ =0,π are: (345) 1 1 −iψ 1 −iψ aθ(0,ψ)=0,aψ(0,ψ)= ,v1(0,ψ)=− e ,v2(0,ψ)=− e , 2 2 2 (346) 1 1 iψ 1 iψ aθ(π, ψ)=0 aψ(π, ψ)=− v1(π, ψ)=− e v2(π, ψ)=− e . 2 2 2

The ﬁelds v1,2 now live in the U(1) bundle of Chern number −1. A LOOP OF SU(2) GAUGE FIELDS 213

Table 3. Numerical minimization of SYM for the reduced 2-dimensional U(1) system, with a zero-size instanton at the south pole and a zero-size anti-instanton at the north pole. SYM here does not include the contribution of 2 units from the instantons. N is the dimension of the aﬃne subspace of connections on which SYM is minimized.

nt ns N min(SYM ) 4 4 64 2.0174 5 5 100 2.0109 6 3 72 2.0073 7 3 84 2.0053

Figure 5. Plots of L+(x) for the connections numerically minimizing SYM , with a zero-size instanton at the south pole and a zero-size anti-instanton at the north pole. The bumps move away from the origin as min(SYM ) decreases. The graphs of L−(x) are given by the reﬂection x →−x.

12.2.8. Numerical calculations with the new boundary conditions. We use the same technique to minimize SYM with the zero-size instanton at the south pole and the zero-size anti-instanton at the north pole, using the same Sage program, making only the changes needed to implement the new boundary conditions. The numerical results are shown in Table 3.Counting the two units of action from the instantons, we now have an upper bound min(SYM ) < 4.0053. Graphs of L+(x) are shown in Figure 5. It seems clear that an instanton is moving towards x = ∞ and an anti-instanton towards x = −∞. Given a widely separated sequence of instantons, II¯ II¯ , the symmetry conditions at the enhanced symmetry point force all three of the neighboring pairs to be maximally twisted. The zero-size limit of such a sequence, II¯ II¯ , of instantons and anti- instantons, widely separated in x, seems a plausible candidate for the 214 D. FRIEDAN

Table 4. Numerical minimization of SYM with coordinate re-scaling x → x/x1, choosing x1 to obtain the best minimum (roughly).

nt ns Nx1 min(SYM ) 3 5 60 3.0 4.34 4 2 32 3.0 4.17 5 2 40 3.5 4.13 5 5 100 4.0 4.13 6 2 48 4.5 4.08 8 2 64 3.5 4.05 10 2 80 4.0 4.04 12 2 96 4.0 4.04 enhanced symmetry fixed point connection with effective index 2. It seems at least worth trying to check by calculating the Yang-Mills flow on the slow manifold. It is worrisome that the first numerical minimization did not get closer to min(SYM ) = 4. We would naively expect the sequence of four separated (anti-)instantons to appear quickly, leaving only the sizes as slow modes. Perhaps there is a competing process. Or perhaps there is an error in the computer program. Most likely, the polynomials are not of high enough degree in t to sufficiently resolve the region near t = ±1. There is no compelling evidence from the numerical calculations that min(SYM ) = 4 at the enhanced symmetry point. There could still be a smooth fixed point with SYM < 4, or a hybrid connection containing a zero- size instanton and a zero-size anti-instanton plus a smooth part, with total action 2

Figure 6. L+(x) for the connection minimizing SYM at 4.04 with the improved resolution, the last run in Table 4.For comparison, the curve centered at x =0isL+(x) for an instanton.

Figure 7. L(x) for the connection minimizing SYM at 4.04 with the improved resolution, the last run in Table 4.The separation into four small (anti-)instantons is more apparent.

Figure 7 shows the full action density L(x)=L+(x)+L−(x). The evidence is stronger for separation into a quadruplet of zero-size (anti-)instantons at the minimum. 12.2.10. A 2-torus family of twisted quadruplets. We write an explicit 2-parameter family of twisted quadruplets of zero-size (anti-)instantons living in the 2-parameter family of bundles constructed in Section 12.2.1 above, 2 parametrized by (β1,β2) ∈ [0,π] . Although the family of bundles forms a 2-sphere, the boundary of the square being identiﬁed to a point, the 2-parameter family of connections forms a 2-torus, opposite sides of the square being identiﬁed with each other. 216 D. FRIEDAN

Write the basic instanton as 1 (347) DI (x − x+)=d + ω − f+ω, f+ = . 1+e−2(x−x+) The basic anti-instanton is

(348) DI¯(x − x−)=DI (−x + x−).

The twisted quadruplet is constructed from an instanton DI (x−x+), an anti- instanton DI¯(x − x−), and their reﬂections under x →−x, the instanton DI (x + x−)=DI¯(−x − x−), and the anti-instanton DI¯(x + x+)=DI (−x − x+), in the limit

(349) x− →∞,x+ − x− →∞ twisted as follows, ⎧ −1 −1 1 ⎪Φ−h D ¯(x + x+)hβ Φ x<− (x+ + x−) ⎨⎪ β1 I 1 − 2 D x x − 1 x x

Recall that the patching map for the bundle is

−1 −1 −1 (351) Φ2(β1,β2)=Φ− Φ+, Φ+ = hβ1 ghβ2 g , Φ− = ghβ2 g hβ1 .

In the limit, the instantons and anti-instantons agree at the junctions, taking the values ⎧ −1 1 ⎨ gdg x = − 2 (x+ + x−) (352) D = dx=0 ⎩ −1 −1 1 hβ1 gdg hβ1 x = 2 (x+ + x−)

At the north pole, x = −∞,

−1 −1 −1 (353) D =Φ−hβ1 dhβ1 Φ− =Φ−dΦ− and at the south pole, x = ∞,

−1 (354) D =Φ+dΦ+ so D satisﬁes the boundary conditions deﬁning the bundle. Alternatively, we have connections on the two hemispheres, nonsingular at the poles,

(355) −1 1 ¯ + − + − hβ1 DI (x + x )hβ1 x< 2 (x + x ) D− = −1 1 Φ− DI (x + x−)Φ− − 2 (x+ + x−)

(356) −1 −1 1 ¯ − Φ+ hβ1 DI (x x−)hβ1 Φ+ 0

−1 −1 −1 −1 − − + + + (357) D =Φ− dΦ ,D=Φ+ hβ1 dhβ1 Φ =Φ+ dΦ which are related by the patching map deﬁning the bundle

−1 (358) D− =Φ2(β1,β2)D+Φ2(β1,β2) .

So the connection D(β1,β2) lives in the bundle deﬁned by the patching map Φ2(β1,β2). Moreover, it can be checked that all the symmetry conditions of the family of bundles are satisﬁed by the family of connections D(β1,β2). On the boundary of the square, β1,2 =0,π, the patching map is trivial, Φ2(β1,β2) = 1, but the connections D(β1,β2) are not all gauge equivalent on the boundary. At β1 =0,π,

(359) ⎧ D x x x<− 1 x x ⎪ I¯( + +) 2 ( + + −) ⎨⎪ −1 gh g−1 D x x gh g−1 − 1 x x

At β2 =0,π,

⎧ −1 1 ⎪ h D¯(x + x+)hβ x<− (x+ + x−) ⎨⎪ β1 I 1 2 −1 − 1 hβ1 DI (x + x−)hβ1 2 (x+ + x−)

The boundary of the square cannot be identified to a single point to give a 2-sphere family of connections. Rather, the opposite sides of the square are identified, giving a 2-torus family of connections, as might have been expected from the symmetry conditions on the family. It remains to calculate the Y-M flow near this family of connections, first π to check that the connection at the enhanced symmetry point β1 = β2 = 2 has a two-dimensional unstable manifold, then to trace the global shape of that 2-manifold. We see two possibilities, depending on details of the flow on the slow manifold, yet to be calculated. The 2-torus of twisted quadruplets could be connected to the flat connection by a single outgoing trajectory leaving from the distinguished point (β1,β2)=(0, 0) (identified with the other 3 corners of the square). This outgoing trajectory in A/G would lift to a 2-cylinder in A, each point on the trajectory in A/G lifting to a nontrivial loop in the group of gauge transformations, G. In this scenario, the Y-M flow would have a stable 2-torus of zero area which might, in the lambda model, 218 D. FRIEDAN produce non-canonical low energy interactions in SU(2) gauge theory. A second, less attractive, scenario would have outgoing trajectories leaving from each point on the boundary of the square, travelling to the flat connection. The unstable manifold would form a 2-sphere stable under the Y-M flow, with non-zero area.

13. Questions and comments 13.1. Does the outgoing trajectory end at the flat connection? The topology of the Y-M flow is the same whatever the locations of the twisted pair in S4 and whatever the geometry on S4. There is always an outgoing trajectory from the aligned twisted pair. Does that outgoing trajectory always end at the flat connection? Euclidean R4 is the setting of interest for the possible physics application (discussed in section 13.3 below). The twisted pair in euclidean R4 can be obtained as the limit of twisted pairs in S4 in which the instanton and anti-instanton are brought together while the metric on S4 is scaled so that the distance of separation remains constant. Scaling the metric on S4 is equivalent to scaling the Y-M flow time, so the outgoing trajectory for twisted pairs in R4 is the same as the limiting trajectory for twisted pairs in S4 as the instanton and anti-instanton approach each other. Even if the outgoing trajectory ends at the flat connection for all twisted pairs in S4, there would still remain the possibility of cross-over to another fixed point as the locations of the instanton and anti-instanton approach each other, which would govern the outgoing trajectory for twisted pairs in R4.Does such a cross-over take place? or does the outgoing trajectory end at the flat connection for twisted pairs in euclidean R4? Whatever the fixed point at the end of the outgoing trajectory, it will have SYM < 2, so it cannot be singular. If it is not the flat connection, then Taubes’ theorem [16] says it must have at least a two dimensional unstable manifold. It would seem extraordinary for the outgoing trajectory from the twisted pairs to end exactly on a twice unstable fixed point.

13.2. Asymptotic behavior of the outgoing trajectory? For the possible application to physics, we would like to know how the outgoing trajectory approaches the flat connection at large time, especially for twisted pairs in euclidean R4 (presuming the outgoing trajectory does end at the flat connection). Near its end, the trajectory At will approach the flat connection as a decaying perturbation whose Fourier transform in R4 takes the form

−tp2 (361) A˜t(p)=e A˜0(p).

The amplitudes A˜0(p) will depend on the locations and twist of the twisted pair and will presumably control the observable properties of the hypothetical physical states associated with the twisted pair. A LOOP OF SU(2) GAUGE FIELDS 219

We might explore for clues to the outgoing trajectories by looking at the outgoing trajectory for the U(2)-invariant twisted pair on S4, given by equation 229, df 2 −1 2 = R (x) ∂xf +4f(1 − f)(2f − 1) dt 2 2 (362) = (cosh x) ∂xf +4f(1 − f)(2f − 1) . The ﬂow equation for the outgoing trajectory in the slow manifold is given by equation 214 at σ =0orσ =2π, dρ (363) =3ρ3. dt It determines the asymptotic initial conditions in the far past, t →−∞,for the outgoing trajectory, (364) 1 1 −1 ft(x) → f+(|x|),f+(x)= ,x+(t)= ln(−6t)+O(t ). 1+e−2(x−x+(t)) 2 If, instead of the round metric on S4, we were to use the cylindrical metric, R2(x) = 1, the outgoing trajectory would be given by

df 2 (365) = ∂xf +4f(1 − f)(2f − 1) dt which is a nonlinear diffusion or reaction-diffusion equation known as the Newell-Whitehead-Segel equation, a special case of the Kolmogorov- Petrovsky-Piskounov/FitzHugh-Nagumo equation. It is apparently not integrable, but some exact solutions are known (see for example [33]). Given the asymptotic t →−∞conditions we need for the outgoing trajectory, the methods by which the exact solutions were produced do not seem applicable [34]. Still, the possibility of an exact solution for the outgoing trajectory is tantalizing. For twisted pairs in R4, the only symmetry of the outgoing trajectory is the SO(3) group of rotations around the axis that passes through the locations of the instanton and anti-instanton. The unstable trajectory is given by a set of nonlinear diffusion equations in two spatial dimensions. Numerical integration might be the only way to find its long time asymptotic behavior.

13.3. The lambda model. The lambda model [7] is a two-dimensional nonlinear model whose target space is the manifold of spacetime fields: gauge fields, fermion fields, scalar fields, and the spacetime metric of general relativity. The functional integral of the lambda model is 2 1 i j (366) Dλ exp − d z Gij(λ)∂λ ∂λ¯ g2 220 D. FRIEDAN

(367) Dλ = dρ(λ(z,z¯)). z,z¯

The field λ(z,z¯) maps a two-dimensional domain, parametrized by a complex coordinate z, to the manifold of spacetime fields. The λi are coordinates on the manifold of spacetime fields (e.g., their momentum modes), i j Gij(λ)dλ dλ is the natural metric on the manifold of spacetime fields, g is the coupling constant, and dρ(λ) is a measure on the spacetime fields, the apriorimeasure of the nonlinear model. The lambda model differs from the standard two dimensional nonlinear model in that the fields λi(z,z¯)are not precisely dimensionless, but change with the two-dimensional scale Λ according to the gradient flow ∂ ∂ 1 (368) Λ λi = −∇iS(λ), ∇iS = g2Gij S ∂Λ ∂λj g2

1 where g2 S is the classical action functional of the spacetime field theory. For small fluctuations, the dimension of the mode λi is p(i)2 where p(i) is the spacetime momentum of the mode. The apriorimeasure is produced by the fluctuations of the lambda fields at short 2-d distances, acting in combination with the gradient flow. The mechanism of production is expressed by the renormalization group equation for the apriorimeasure, which at leading order is the driven diffusion equation ∂ 2 ij 1 (369) Λ dρ = ∇i g G ∇j + ∂j S dρ ∂Λ g2 which equilibrates at

− 1 S (370) dρ = dλ e g2 , dλ being the metric volume element. We recognize the apriorimeasure produced by the small fluctuations as the functional measure of the canonically quantized spacetime quantum field theory. The exact apriorimeasure produced by the lambda model is a quantum field theory in spacetime that might not be identical to the canonically quantized field theory. We are pursuing the possibility that non-canonical corrections to the canonical quantum field theory might be produced by large two-dimensional fluctuations in the lambda model. At weak coupling, large fluctuations in a nonlinear model show up as winding modes, associated with π1 of the target manifold, and as 2-d instantons, associated with π2 of the target manifold. The winding modes might provide weakly interacting states not present in the canonical quantum SU(2) gauge field theory. The 2-d instantons might provide interactions not present in the canonical quantum SU(2) and SU(3) gauge field theories. A LOOP OF SU(2) GAUGE FIELDS 221

The evolution of the nonlinear model in the two dimensional scale can be represented by the radial quantization, in which the wave functions live on the loop space of the target manifold and the hamiltonian is the 2-d dilation operator. The dilation operator of the lambda model is the ordinary dilation operator of the nonlinear model combined with the gradient flow. The winding modes are wave functions on the nontrivial component(s) of the loop space. In a normal nonlinear model, the low-lying winding modes are concentrated on the nontrivial loops of minimal length. In the lambda model, the gradient flow attempts to concentrate the wave function on the stable nontrivial loops. The two processes compete, in principle, so searching only for loops stable under the gradient flow was ill-conceived. Finding a stable loop of zero length was pure luck. It remains to quantize the stable loop of SU(2) gauge fields. The ground state or states will be concentrated on the nontrivial loop at the tip of the cone in Figure 2. Regarding the cone as an orbifold of the plane, the states of the stable loop are the twist states for the orbifold. The classical ground state energy is zero because the length of the loop is zero. But, if the stable loop is to provide any low energy states, the quantum corrections to the ground state energy must also vanish, at least to many orders in the coupling constant. We assume a high fundamental spacetime energy scale in the lambda model. Fermion zero-modes localized in the zero-size instanton and anti-instanton of the stable loop will provide degenerate ground states that offers at least a possibility of canceling the quantum corrections to the ground state energy, but to get cancellation to many orders or to all orders, we will probably need perturbative spacetime supersymmetry. The hypothetical non-canonical SU(2) gauge theory states would only be visible at low energy in theories with perturbative supersymmetry. It also remains to figure out the spacetime interpretation of the states of the stable loop. Do they appear as additional fields in the quantum field theory, or as extra states, in addition to the quantum field theory? In any case, they will presumably be bi-local objects, depending on the two instanton locations parametrizing the loop of twisted pairs. For consistency in the 2-d quantum field theory, twist fields have to be accompanied by a projection that eliminates all states with nontrivial monodromy around the twist field. Here, nontrivial monodromy means change of sign under nontrivial SU(2) gauge transformations. Such states arise in canonically quantized gauge theories that have a global SU(2) anomaly [35]. By projecting out the anomalous states, and adding new non-canonical states, the lambda model might produce a non-anomalous quantization of such gauge theories. We expect that interactions with the ordinary modes of the gauge field theory will be determined by the outgoing trajectory that leads from the twisted pairs to the flat connection. We picture two twist fields — two loops of twisted pairs — merging in the two dimensional domain. Any closed curve surrounding them will be a topologically trivial loop in the twisted pairs. In 222 D. FRIEDAN the radial quantization, it will be driven down the outgoing trajectory to the flat connection, where it can join to the rest of the two dimensional domain where there are only small fluctuations around the flat connection. The interactions of the new states associated with the stable loop should then be determined by how the outgoing trajectory approaches the flat connection. We picture a 2-d instanton in the lambda model to consist of a point-like core that is the nontrivial 2-sphere of twisted pairs of SU(3) gauge fields (or twisted quadruplets of SU(2) gauge fields), evolving outwards in the radial quantization, down the outgoing trajectory towards the flat connection. Again, the effects on the ordinary states will be determined by how the outgoing trajectory approaches the flat connection. On the two dimensional domain, the instanton will look like a defect, around which the nearly flat gauge field winds by a nontrivial loop in the group of gauge transformations. So far, we have only such vague speculations. The task now is to figure out how to calculate in the lambda model with the stable loop and the stable 2-spheres of gauge fields. Appendices

A. U (2)-invariant connections on S 4 A.1. S3, SU (2), U (2), SO(4), S 4. S 3: z1 † † (371) z = , z z =¯z1z1+¯z2z2 =1,P(z)=zz ,Q(z)=1−P (z), z2 1 † † 2 (372) dvolS3 = − (z dz)(dz dz), dvolS3 =2π . 2 S3 SU(2): z1 −z¯2 (373) g(z)= . z2 z¯1 U(2): 10 (374) g(Uz)=Ug(z) . 0 (det U)−1

SO(4) = SU(2) × SU(2)/Z2: −1 (375) g(O(z)) = gLg(z)gR . S4: θ (376) y =(cosθ, z sin θ),x=lntan , −∞ ≤ x ≤∞. 2 A LOOP OF SU(2) GAUGE FIELDS 223

2 2 2 † 2 −2 (377) (ds)S4 = R (x) (dx) + dz dz ,R(x) = (cosh x) .

A.2. U (2)-invariant su(2)-valued 1-forms on S 3.

(378) η = −PdP = −(z†dz)zz† − zdz† (379) η† = −dP P =(z†dz)zz† − dzz† † † † (380) η3 =(z dz)(P − Q)=(z dz)(2zz − 1)

0 −dz¯2 † 00 dz1 0 (381) η(e)= η (e)= η3(e)= 00 −dz2 0 0 −dz1

2 † 2 2 (382) η =(η ) = η3 =0

2 (383) dωω = dω + {ω, ω} = ω

† † † (384) dω(−η + η − η3)={η3,η}−{η3,η }−{η, η } so, by U(1)-covariance,

† † † (385) dωη = −{η3,η} dωη = −{η3,η } dωη3 = {η, η }

A.3. U (2)-invariant su(2)-valued 2-forms on S 3. Deﬁne

1 1 † † (386) σ = dωη = − {η3,η} = z dzPdP = −z dz η 2 2 † 1 † 1 † † † † (387) σ = dωη = − {η3,η} = −z dzdP P = z dz η 2 2 1 1 † 1 2 1 † (388) σ3 = dωη3 = {η, η } = (dP ) = dz dz(P − Q) 2 2 2 2

(389) 0 dz1dz¯2 † 00 1 10 σ(e)= σ (e)= σ3(e)= dz¯2dz2 00 dz¯1dz2 0 2 0 −1

A.4. U (2)-invariant su(2)-valued forms on S 4. The volume form on S3 is

(390) 1 1 † † 2 dvolS3 (e)=− dz1dz¯2dz2 dvolS3 = − (z dz)(dz dz) dvolS3 =2π . 2 2 S3 224 D. FRIEDAN

In S4,ateachx, we have a basis for the U(2)-invariant su(2)-valued forms

(391) 0-forms: i(P − Q), † (392) 1-forms: idx(P − Q),η,η ,η3, † † (393) 2-forms: σ, σ ,σ3,dxη,dxη ,dxη3, † (394) 3-forms: idvolS3 (P − Q),dxσ,dxσ ,dxσ3.

A.5. Hodge ∗. At (x, e) ∈ S4, the Hodge ∗ operator acts on 1-forms and 3-forms: 2 2 2 (395) ∗ = −1 ∗dx = R (x)dvolS3 ∗dz1 = −R (x)dxdz¯2dz2 2 2 (396) ∗dz2 = −R (x)dxdz1dz2 ∗dz¯2 = R (x)dxdz1dz¯2

2-forms: (397) 2 1 ∗ =1 ∗(dxdz1)= dz¯2dz2 ∗(dxdz2)=dz1dz2 ∗(dxdz¯2)=−dz1dz¯2 2 So Hodge ∗ acts on the U(2)-invariant su(2)-valued forms by: 1-forms and 3-forms: 2 (398) ∗idx(P − Q)=R (x)dvolS3 i(P − Q)

2 † 2 † 2 (399) ∗η = −R (x)dxσ ∗η = −R (x)dxσ ∗η3 = −R (x)dxσ3

2-forms: † † (400) ∗(dxη)=σ ∗(dxη )=σ ∗(dxη3)=σ3

A.6. F, F ±. Recall

(401) d + A = dω +ΔA

† (402) ΔA = f(x)η − f¯(x)η + f3(x)η3,f3 = f¯3

2 2 (403) F =(dω +ΔA) = dωΔA +(ΔA)

1 (404) F± = (1 ±∗)F. 2 Calculate † † (405) F = ∂xfdxη − ∂xfdxη¯ + ∂xf3dxη3 +2fσ − 2fσ¯ +2f3σ3 − 2f3fσ † +2f3fσ¯ − 2ffσ¯ 3 A LOOP OF SU(2) GAUGE FIELDS 225

1 (406) F± =[∂xf ± 2(1 − f3)f)] (1 ±∗)dxη 2 1 † − [∂xf¯± 2(1 − f3)f¯)] (1 ±∗)dxη 2 1 (407) +[∂xf3 ± 2(f3 − ff¯)] (1 ±∗)dxη3 2

A.7. L±.

1 1 2 (408) tr(−F±∗F±)=∓ tr(F±)=L±(x)dx dvolS3 4 4

A.8. Products of 1-forms and 2-forms. The nonzero products of 1-forms and 2-forms are (412) † † † † ση = ησ =2dvolS3 Pση = η σ =2dvolS3 Qη3σ3 = σ3η3 = −dvolS3 1

† (413) [ω, σ]=[ω, σ ]=−2dvolS3 (P − Q)[ω, σ3]=0

A.9. Inner products. The non-zero inner products are: 1-forms: (414) tr [−idx(P − Q)∗idx(P − Q)] = tr η∗η† =tr η†∗η 2 =tr(−η3∗η3)=2R (x)dxdvolS3

2-forms:

† † (415) tr(−dxη3σ3)=tr(dxησ )=tr(dxη σ)=2dx dvolS3 226 D. FRIEDAN

A.10. New basis for the U (2)-invariant forms. Change basis for the U(2)-invariant su(2)-valued 1-forms and 2-forms on S3 to

† † † (416) ω = −η + η − η3 ω1 = η − η − 2η3 ω2 = −i(η + η ) † † † (417) χ = −σ + σ − σ3 χ1 = σ − σ − 2σ3 χ2 = −i(σ + σ )

Correspondingly, on S4,

(418) 1-forms: idx(P − Q),ω,ω1,ω2 (419) 2-forms: χ, χ1,χ2, (420) dx ω, dx ω1,dxω2

(421) 3-forms: idvolS3 (P − Q),dxχ,dxχ1,dxχ2.

A.11. Formulas in the new basis.

(422) dωω =2χdωω1 =2χ1 dωω2 =2χ2 (423) {ω, ω} =4χ {ω, ω1} = −2χ1 {ω, ω2} =2χ2 (424) D±ω =(2− 4f±)χD±ω1 =(2+2f±)χ1 D±ω2 =(2− 2f±)χ2

(425) [ω, χ]=[ω, χ1]=0 [ω, χ2]=4idvolS3 (P − Q)

Hodge ∗ on 1-forms:

2 (426) ∗idx(P − Q)=R (x)dvolS3 i(P − Q)

2 2 2 (427) ∗ω = −R (x)dxχ ∗ω1 = −R (x)dxχ1 ∗ω2 = −R (x)dxχ2

Hodge ∗ on 2-forms:

(428) ∗χ = dxω ∗χ1 = dxω1 ∗χ2 = dxω2

Non-zero inner products of 1-forms:

2 (429) tr [−idx(P − Q)∗idx(P − Q)] = 2R (x)dxdvolS3

1 1 1 2 (430) tr(−ω∗ω)= tr(−ω1∗ω1)= tr(−ω2∗ω2)=R (x)dxdvolS3 6 8 4 Non-zero inner products of 2-forms:

(431) 1 1 1 tr(−dxω∗dxω)= tr(−dxω1∗dxω1)= tr(−dxω2∗dxω2)=dxdvolS3 6 8 4 A LOOP OF SU(2) GAUGE FIELDS 227

A.12. Instanton covariant derivatives. Using the formulas in Appendix A.2, we calculate the instanton covariant derivatives of the 1-forms (432) D±idx(P − Q)=−2f±dxω2 D±ω = λχ D±ω1 = λ1χ1 D±ω2 = λ2χ2 where

(433) λ =2(1− 2f±) λ1 =2(1+f±) λ2 =2(1− f±).

The covariant derivatives of the 2-forms are (434) D±(dx ω)=−λdx χ D±(dx ω1)=−λ1dx χ1 D±(dx ω2)=−λ2dx χ2

(435) D±χ =0 D±χ1 =0 D±χ2 = −4f±dvolS3 i(P − Q)

Their Hodge duals are

2 −1 2 −1 (436) ∗D±(dx ω)=−R (x) λω ∗D±(dx ω1)=−R (x) λ1ω1

2 −1 (437) ∗D±(dx ω2)=−R (x) λ2ω2

2 −1 (438) ∗D±χ =0 ∗D±χ1 =0 ∗D±χ2 =4R (x) f±idx(P − Q)

A.13. The instanton laplacian.

(439) D±δA0±(x)idx(P − Q)=−δA0±2f±dxω2 (440) (∗∓1)D±δA0±(x)idx(P − Q)=−2f±δA0±(χ2 ∓ dxω2) (441) D±(∗∓1)D±δA0±(x)idx(P − Q)=−2∂x(f±δA0±)dxχ2 − 2f±δA0±(D±χ2 ∓ D±dxω2) (442) 2 −1 ∗D±(∗∓1)D±δA0±(x)idx(P − Q)=−2∂x(f±δA0±)R (x) ω2 2 −1 (443) − 2f±δA0± 4R (x) f±idx(P − Q) 2 −1 (444) ∓(−λ2)R (x) ω2 2 −1 (445) = −2R (x) (∂x ± λ2)(f±δA0±)ω2 2 −1 2 (446) − 8R (x) f±δA0±idx(P − Q) 228 D. FRIEDAN

(447) D±δf±(x)ω = ∂xδf±dxω + δf±λχ (448) (∗∓1)D±δf±(x)ω =(∂x ∓ λ)δf±(χ ∓ dxω) (449) D±(∗∓1)D±δf±(x)ω = ∂x(∂x ∓ λ)δf±dxχ +(∂x ∓ λ)δf±(D±χ ∓ D±dxω) 2 −1 (450) ∗D±(∗∓1)D±δf±(x)ω = R (x) (∂x ± λ)(∂x ∓ λ)δf±ω

(451) D±δA1±(x)ω1 = ∂xδA1±dxω1 + δA1±λ1χ1 (452) (∗∓1)D±δA1±(x)ω1 =(∂x ∓ λ1)δA1±(χ1 ∓ dxω1) (453) D±(∗∓1)D±δA1±(x)ω1 = ∂x(∂x ∓ λ1)δA1±dxχ1 +(∂x ∓ λ1)δA1±(D±χ1 ∓ D±dxω1) 2 −1 (454) ∗D±(∗∓1)D±δA1±(x)ω1 = R (x) (∂x ± λ1)(∂x ∓ λ1)δA1±ω1

(455) D±δA2±(x)ω2 = ∂xδA2±dxω2 + δA2±λ2χ2 (456) (∗∓1)D±δA2±(x)ω2 =(∂x ∓ λ2)δA2±(χ2 ∓ dxω2) (457) D±(∗∓1)D±δA2±(x)ω2 = ∂x(∂x ∓ λ2)δA2±dxχ2 +(∂x ∓ λ2)δA2±(D±χ2 ∓ D±dxω2) (458) 2 −1 ∗D±(∗∓1)D±δA2±(x)ω2 = ∂x(∂x ∓ λ2)δA2±R (x) ω2 2 −1 +(∂x ∓ λ2)δA2±(4R (x) f±idx(P − Q) 2 −1 ± R (x) λ2ω2) (459) 2 −1 ∗D±(∗∓1)D±δA2±(x)ω2 = R (x) (∂x ± λ2)(∂x ∓ λ2)δA2±ω2 2 −1 +4R (x) f±(∂x ∓ λ2)δA2±idx(P − Q)

4 We expand δA± in a basis of U(2) invariant su(2)-valued 1-forms on S ,

(460) δA± = δA±0(x)idx(P − Q)+δf±(x) ω + δA±1(x) ω1 + δA±2(x) ω2. From Appendix A.13, (461) 2 −1 ∗D±(∗∓1)D±δA± = R (x) 4f±[−2f±δA0± +(∂x ∓ λ2)δA2±]idx(P − Q) 2 −1 + R (x) (∂x ± λ)(∂x ∓ λ)δf±ω 2 −1 + R (x) (∂x ± λ1)(∂x ∓ λ1)δA1±ω1 2 −1 + R (x) (∂x ± λ2)[−2f±δA0± +(∂x ∓ λ2)δA2±]ω2 where

(462) λ =2(1− 2f±) λ1 =2(1+f±) λ2 =2(1− f±). A LOOP OF SU(2) GAUGE FIELDS 229

−1 −1 (463) (∂x ± λ)=f± (1 − f±) ∂xf±(1 − f±) −1 −1 (∂x ∓ λ)=f±(1 − f±)∂xf± (1 − f±) −1 2 −2 (464) (∂x ± λ1)=f± (1 − f±) ∂xf±(1 − f±) −2 −1 2 (∂x ∓ λ1)=f±(1 − f±) ∂xf± (1 − f±) −1 (465) (∂x ± λ2)=f± ∂xf± −1 (∂x ∓ λ2)=f±∂xf± so

(466) 2 −1 2 −1 ∗D±(∗∓1)D±δA± = R (x) 4f±[−2δA0± + ∂x(f± δA2±)]idx(P − Q) 2 −1 −1 −1 2 2 + R (x) f± (1 − f±) ∂xf±(1 − f±) −1 −1 × ∂xf± (1 − f±) δf±ω 2 −1 −1 2 2 −4 + R (x) f± (1 − f±) ∂xf±(1 − f±) −1 2 × ∂xf (1 − f±) δA1±ω1 ± 2 −1 −1 2 −1 + R (x) f± ∂x f±[−2δA0± + ∂x(f± δA2±)] ω2

B. SU (2)→SU (3)→S 5 pulled back along [−1, 1]×S 4→S 5 5 B.1. SU (3)/SU (2)= S . S5 is represented as the unit sphere in C⊕ C2. The unit vectors are written w0 2 † (467) w = |w0| + w w =1 w

The north pole in S5 is 1 (468) n = 0

SU(3) acts by block matrices on C ⊕ C2. SU(2) is identiﬁed with the subgroup of SU(3) leaving n ﬁxed, the block matrices of the form 1 0† (469) . 0 g

We will write this SU(3) matrix simply as g. w ∈ S5 is identiﬁed with the SU(2) coset

(470) {G ∈ SU(3) : Gn = w}. 230 D. FRIEDAN

B.2. U (2) acts on SU (3) → S 5. U ∈ U(2)actsasasymmetryof the bundle SU(3) → S5 by 1 0† 1 0† (471) G → G 0 U 0 (det U)−1/2 1 w0 w0 (472) → w Uw where the sign of (det U)−1/2 is immaterial, because −1 acts trivially on connections, being in the center of SU(2).

B.3. A map [−1, 1]×S 4 → S 5. We construct a nontrivial loop of connections by pulling back along a map [−1, 1] × S4 → S5, (473) (s, y) → w(s, y) cos θ + is sin θ (474) w(s, y)= √ ,y =(cosθ, z sin θ). z sin θ 1 − s2 This map is manifestly U(2)-invariant so, for each s ∈ [−1, 1], the pulled back connection A(s)onS4 is U(2) invariant. At s = ±1, the pulled back connection over S4 is ﬂat, so we get a closed loop in A/G.

B.4. Trivialize. We ﬁnd a formula for A(s) by trivializing SU(3) → S5 over a convenient region of S5, giving, for each of the SU(2)-bundles in the loop, a trivialization over S4\south pole. Recall w0 w0 † (475) w = = 2 ,P= zz ,Q=1− P. w z 1 −|w0| Deﬁne a partial section S5 → SU(3) † w0 −w (476) w → G(w)= . w w¯0P + Q 5 which is regular on S except where |w0| =1,w0 =1.Foreach s except s ± 1, it is regular on S4 away from the south pole θ = π. The invariant connection in SU(2) → SU(3) → S5 takes the explicit form −1 (477) d + Ainv (w)=dω +ΔAinv (w)=d + PV G(w) dG(w) where PV is the invariant projection on su(2) ⊂ su(3). We calculate, 1 2 † (478) ΔAinv (w)= (w0dw¯0 − w¯0dw0) − (1 −|w0| )z dz 2 1 × (P − Q)+w ¯0dP P − w0PdP. 2 A LOOP OF SU(2) GAUGE FIELDS 231

B.5. A formula for ΔA(s). We substitute w0 = cosθ + is sin θ and, for each s, restrict the connection to S4 1 (479) ΔA(s)=− isdθ(P − Q)+(cosθ − is sin θ)η− +(cosθ + is sin θ)η+ 2 1 2 2 + 1 − (1 − s )sin θ η3 2 At the midpoint of the loop, s =0, 1 2 (480) ΔA(0) = cos θ (η+ + η−)+ 1 − sin θ η3 2

B.6. Discrete symmetries of the loop. In addition to the U(2) symmetry, there are two discrete symmetries. B.6.1. Reﬂection symmetry. The bundle SU(2) → SU(3) → S5 has a discrete symmetry −10 10 (481) G → G 0 1 0 i1 which acts on S5 by −w0 (482) w → Rw = w so on [−1, 1] × S4 by

(483) s →−s, (y0, y) → (−y0, y). B.6.2. Conjugation symmetry. Complex conjugation acts on SU(2) by −1 01 (484)g ¯ = g1gg g1 = . 1 −10

Deﬁne, for G ∈ SU(3),

(485) Gc = Gg¯ 1. Then, for g ∈ SU(2),

(486) (Gg)c = G¯gg¯ 1 = Gg¯ 1g = Gcg 5 so G → Gc is a symmetry of the bundle SU(2) → SU(3) → S .Itactson S5 by (487) w → w¯ so on I × S4 by

(488) s →−s, (y0, y) → (y0, y¯). 232 D. FRIEDAN

B.7. Action of the discrete symmetries. B.7.1. Action of the reﬂection symmetry. −1 (489) G(Rw)=G(w)rgr(w) † † −w0 −w −w0 iw −1 (490) = gr(w) w −w¯0P + Q w i(¯w0P + Q)

(491) † † −1 −w¯0 w −w0 −w 10 gr(w) = = −iw −i(w0P + Q) w −w¯0P + Q 0 i(P − Q)

(492) gr = −i(P − Q)

−1 (493) dω +ΔA(Rw)=gr(w)(dω +ΔA(w))gr (w) −1 (494) = dω + gr(w)ΔA(w)gr (w) since

(495) (d + ω)P = gdg−1gP(e)g−1 = P (d + ω) so −1 (496) ΔA(Rw)=grΔAA(w)gr We have −1 † −1 † −1 (497) grηgr = −ηgrη gr = −η grη3gr = η3 so, writing † (498) ΔA(s)=A0(s, θ)dθ i(P − Q)+f(s, θ)η − f¯(s, θ)η + f3(s, θ)η3

Then the reﬂection symmetry is

(499) A0(s, θ)=−A0(−s, π − θ) (500) f(s, θ)=−f(−s, π − θ) (501) f3(s, θ)=f3(−s, π − θ)

B.7.2. Action of the conjugation symmetry. G(¯w)andG(w)c belong to the same SU(2) coset, so there is gc ∈ SU(2) such that

−1 −1 −1 (502) G(¯w)=G(w)cgc(w) = G(w)g1gc(w) = G(¯w)g1gc(w) so 01 (503) gc = g1 = . −10 A LOOP OF SU(2) GAUGE FIELDS 233

G → Gc is a symmetry, so

−1 (504) d + A(¯w)=gc(d + A(w))gc or

−1 (505) A(w)=gcA(¯w)gc .

Complex conjugation acting on su(2) gives

−1 −1 † † −1 (506) gcη¯3gc = η3 gcηg¯ c = −η gcη¯ gc = −η

−1 In particular, gcωg¯ c = ω, so the conjugation symmetry is

−1 (507) ΔA(w)=gcΔA(¯w)gc

(508) A0(s, θ)=−A0(−s, θ) (509) f(s, θ)=f¯(−s, θ) (510) f3(s, θ)=f3(−s, θ).

B.7.3. Summary of the actions of the discrete symmetries. Each connection A(s) along the loop has the discrete symmetry that combines reﬂection and conjugation,

(511) A0(s, θ)=A0(s, π − θ) (512) f(s, θ)=−f¯(s, π − θ) (513) f3(s, θ)=f3(s, π − θ).

In addition, there is a discrete symmetry that reﬂects the loop, leaving the midpoint s = 0 ﬁxed,

(514) A0(s, θ)=−A0(−s, θ) (515) f(s, θ)=f¯(−s, θ) (516) f3(s, θ)=f3(−s, θ).

B.7.4. Discrete symmetries at the midpoint s =0. The two discrete symmetries of the loop leave ﬁxed the midpoint, s = 0, so they are symmetries of the connection A(0). The conjugation symmetry is

(517) A0(0,θ)=0,f(0,θ)=f¯(0,θ).

The reﬂection symmetry is

(518) f(0,θ)=−f(0,π− θ),f3(0,θ)=f3(0,π− θ).

Both can be checked in equation 480. 234 D. FRIEDAN

B.8. Nontriviality of the loop. B.8.1. Trivialize over S4\north pole. Deﬁne a second partial section which is nonsingular on S4 except at the north pole, † − w0 −w 10 (519) G (w)= = G(w) . w w¯0P − Q 0 P − Q The pulled back connection is − (520) dω +ΔA (s)=(P − Q)(dω +ΔA(s))(P − Q) (521) = dω +(P − Q)ΔA(s)(P − Q) (522) ΔA−(s)=(P − Q)ΔA(s)(P − Q)

B.8.2. Transform A(s) to A0 =0gauge. Gauge transform A(s)toA0 = 0 gauge, using a gauge transformation that is regular at θ = 0, the north pole,

− 1 isθ(P −Q) 1 isθ(P −Q) (523) dω +ΔA˜(s)=e 2 (dω +ΔA(s))e 2

1 − 1 isθ(P −Q) 1 isθ(P −Q) (524) = dω + isdθ(P − Q)+e 2 ΔA(s)e 2 2 so, in A0 = 0 gauge, the loop is (525) ΔA˜(s)=(cosθ + is sin θ)e−isθη − (cos θ − is sin θ)eisθη† 1 2 2 + 1 − (1 − s )sin θ η3 2 − − B.8.3. Transform A (s) to A0 =0gauge. Gauge transform A (s)to A0 = 0 gauge, using a gauge transformation that is regular at θ = π, the south pole,

− − 1 is(θ−π)(P −Q) − 1 is(θ−π)(P −Q) (526) dω +ΔA˜ (s)=e 2 (dω +ΔA (s))e 2 (527) = φ(s)(D +ΔA˜(s))φ(s)−1 with − 1 is(θ−π)(P −Q) −1 1 isθ(P −Q) 1 i(s−1)π(P −Q) (528) φ(s)=e 2 i (P − Q)e 2 = e 2 B.8.4. Patching map at the equator is the suspension of the Hopf ﬁbra- tion. Now we have a loop of bundles over S4, made from trivial bundles over the two hemispheres patched together by the loop of gauge transformations φ(s), which we can write 1 iπ(s−1) e 2 0 −1 (529) φ(s)=g(z) − 1 iπ(s−1) g(z) 0 e 2 which is the suspension of the Hopf ﬁbration. So the loop of connections is nontrivial. A LOOP OF SU(2) GAUGE FIELDS 235

Acknowledgments I thank V. Calian of the Natural Science Institute, University of Ice- land, for discussions of exact solutions of nonlinear diﬀusion equations. I am grateful to T. Mrowka for pointing out references on solving the Yang-Mills equations with symmetry and on the Yang-Mills ﬂow.

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Department of Physics and Astronomy, Rutgers, The State University of New Jersey Piscataway, New Jersey, 08854-8019 USA

Natural Science Institute, The University of Iceland, Reykjavik, Iceland Surveys in Diﬀerential Geometry XV

Automorphisms of Graded Super Symplectic Manifolds

Joshua Leslie

Abstract. In section 1 we present the language of differential analysis in graded locally convex infinite dimensional vector spaces that was studied in [5]. In section 2 we give results on diffeological Lie groups and Lie algebras and formulate diffeological versions of the fundamental theorems of Lie. In section 3 we introduce the notion of graded supermanifolds and we indicate a method of constructing non-trivial compact graded supermanifolds with non-degenerate graded differential forms which generalize symplectic and contact structures to the graded case. In section 4 we study automorphisms of generalizations of some of the classical geometric structures to compact graded supermanifold, M, and show that these automorphism groups are diffeological Lie subgroups of the Lie group of superdiffeomorphisms of M.

Introduction We shall show how to construct compact supermanifolds having geometric structures determined by certain elements of ∧p ⊗ ⊗q Γ (TΓM) Γ0 Γ0 (TΓM) ∧p ⊗ ⊗q where Γ (TΓM) Γ0 Γ0 (TΓM) represents the smooth sections of the bundle ∧p ⊗ ⊗q → ΠM : (TΓM) Γ0 Γ0 (TΓM) M, where by abuse of notation the “Γ” in TΓM and TΓM represents a finitely generated Grassmann algebra. The geometric automorphisms of these geometric structures will be seen to be diffeological Lie subgroups of the infinite c 2011 International Press

237 238 J. LESLIE dimensional Lie group of G∞ diffeomorphisms of an a non trivial compact G∞ supermanifold. Underlying the infinite dimensional Lie algebras of Cartan are geometric structures of differential geometry; such as contact structures, symplectic structures, and foliated structures. It is known [4] in the the compact case that the automorphism groups of these geometric structures are Lie subgroups of the Lie group of diffeomorphisms of the underlying differentiable manifold. We show this is also the case for the generalized contact and symplectic structures on graded G∞ super manifolds.

1. Analysis in graded infinite dimensional topological vector spaces We designate by N the natural numbers. Let Γ be the N graded Grass- mann algebra over the reals, R, of super numbers generated by an arbitrary set X = {ξi}i∈I , deg(ξi)=1),when X is infinite we suppose that Γ is fur- nished with the topology given by the inductive limit of ΓK ,forK ∈ J, where J is the collection of finite subsets of I ordered by inclusion and ΓK i i the finite dimensional subspaces of Γ generated by ξ 1 ,...,ξ nK . With this topology Γ is a complete locally convex topological vector space. Γ is a Z2 graded commuatative (i.e. ab =(−1)|a||b|ba algebra, where |a| designates the parity of a. Γ with this topology will be used as our base ring when X is infinite. Note that Γ is a bigraded ((N,Z2)) ring, graded over the non-negative integers N by the the number of generators, ξ ∈ X in the expression of an element and by the Z2-parity of that number. When we consider Γ as a right Γ module there are two possible right Γ module structures potentially of interest for us. Let Γ:Γ→ R be the unique R − algebra homomorphism such that Γ(k1Γ)=k, k ∈ R,andsetI(Γ) = ker(), we have Γ = K1˙ Γ + I(Γ). |γ||γ| Define a right Γ module structure on Γ noted ΓR by (k, γ)˙γ =(−1) k˙(γ),γγ˙ ), where k ∈ K, γ ∈ I(Γ), and γ ∈ Γ. Note that with this right Γ module structure ΓR remains a right bigraded Γ module. When the right Γ module structure on Γ is given by right Γ multiplication by abuse of notation we shall simply designate by Γ this right Γ module. Let Γ0 (resp.Γ1) be the even (resp. odd) Γ0-submodule of Γ. Let V and W be topological graded modules over Γ0, a continuous mapping f : V ×···×V → W is said to be an n − multimorphism when f is n-multilinear with repect to the ground field K and

f(e1,...,eiγ,ei+1,...,en)=f(e1,...,ei,γei+1,...,en),γ∈ Γ0 and f(e1,...,enγ)=f(e1,...,en)γ, γ ∈ Γ0. In case Γ is ﬁnite dimensional a Γ- multimorphism is called regular when there exists a countably generated inﬁnite dimensional algebra Λ containing AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 239

Γ as a proper unitary subalgebra such that the map

fΛ0 : VΛ0 ×···×VΛ0 → WΛ0 isaΛ0- multimorphism extension of f, where VΛ0 = V Γ0 Λ0, deﬁned by fΛ0 (v1 ⊗ λ1,...,vn ⊗ λn)=f(v1,...,vn) ⊗ λ1 ...λn.

Definition 1.1. Let V and W be topological graded modules over Γ0, given U ⊆ V open, a function f : U → W will be called super smooth when for every n ≥ 1 there exists continuous maps, which are regular k-multimorphisms in the k−terminal variables, for x ∈ U ﬁxed: Dkf(x; ...): U × V ×···×V → W , k ≤ n, such that

1 1 k Fk(h)=f(x + h) − f(x) − Df(x, h) −···− D f(x,h,...,h), 1 ≤ k ≤ n, 1! k! satisﬁes the property that k Fk(th)/t ,t=0 (1.1) Gk(t, h)= 0,t=0 is continuous at (0, h) ∀h ∈ V .

Proposition 1.1. If a function f : U → W is smooth and Df(x; αγ)= Df(x; α)γ,thenf(x) is super smooth.

k Proof. In this proof we suppose that D f(x; α1,...αk, ):U ×V ×···× n V → W symmetric in α1,...αk It suﬃces to prove for n ≥ 2 that a C n−1 n k function f : U → W which is G is G . It is known that Dt=0D f(x + th,h...,h)=Dk+1f(x + th,h...,h). For k ≥ 2, we have Dkf(x + th,h..., k−1 k−1 k hγ)=Dt=0(D f(x + th,h...,h)γ)=Dt=0D f(x + th,h...,h)γ = D f(x + th,h...,h)γ. Super smooth functions are smooth and therefore satisfy the classical theorems such as the chain rule. Note that multiplication deﬁnes a super smooth map Γ × Γ → Γ.

Given a well-ordering, ≺,ofI, we obtain an associated basis of Γ, {ξi1 ,...,

ξin } of Γ, where ik ≺ ik+1; we shall call this basis a canonical basis of Γ. To each of these basis elements is associated an unique 1 − morphism {ξ ,...,ξ } → p i1 in :Γ Γ, deﬁned by

p{ξ ...ξ }(aj1...jn ξ ...ξ +···+a ξ ...ξ +···+a hξ ...ξ ) i1 in 1 i1 jn1 l1...ln l1 ln k1...kn2 k1 kn2

= ai1,...,in ξi1 ,...,ξin It is straightforward to verify that as in the classical case

Proposition 1.2. If f : V → W is a continuous Γ0 homomorphism, then f(x) is a supersmooth function such that Dxf = f ∀x ∈ V . 240 J. LESLIE

∞ GivenanopensubsetUofΓ.x0 ∈ U, C (U, Γ) has the structure of a ∞ ∞ graded module over the ground ﬁeld K determined by C (U, Γ)0 = C (U, ∞ ∞ Γ0)andC (U, Γ)1 = C (U, Γ1). n n Designate by Γ0 (resp. Γ1 ) the n-fold direct sum of the Γ0 modules n m Γ0 (resp. Γ1) and suppose U ⊂ Γ0 ⊕ Γ1 is a non-empty open subset, set n m ηi =(0,...,0, 1Γ, 0,...,0) ∈ Γ0 ⊕ Γ1 , where 1Γ is in the ith position, i ≤ n. Given a smooth function f : U → Γset∂f/∂xi(x)=Dt=of(x + tηi). { } → Given a system of generators ξi of Γ deﬁne a linear map Δξi :Γ n − (k−1) ΓbyΔξi (ξi1 ...ξk ...ξin )= k=1 δiik ( 1) (ξi1 ...ξik−1 ξik+1 ...ξin ), and

Δξj (1Γ)=0.

Lemma 1.1. Δξi :Γ→ Γ is a 1-morphism, and a graded odd derivation; |γ1| that is, Δξi (γ1.γ2)=Δξi (γ1).γ2 +(−1) γ1.Δξi (γ2),“|·|” designates “the parity of”(0 for even, 1 for odd). Δξi is odd in the sense that Δξi (Γi) ⊂ Γi+1.

Proof. By definition Δξi is a 1-morphism. To see that Δξi is a derivation it suffices to consider Δξi (γ1.γ2) and to do an induction on the Z-grading of γ1. Definition 1.2. Let X ⊂ Γ be a set of generators of the Grassmanian algebra Γ. given an open subset U of Γ. x0 ∈ U, a generator ξ ∈ X ⊂ Γ, and a smooth function f : U → Γset∂f/∂ξ(x)=Δξ ◦ Dt=0f(x + tξ). We have the following important proposition which is immediate from the definitions: ∞ ∞ Proposition 1.3. Let x ∈ U, ∂f/∂ξ(x):C (U, Γi) → C (U, Γi+1) defines a smooth odd derivation for the C∞ topology.

Designate by Ln(Γ, Γ) the vector space of n-multilinear n-morphisms.

Proposition 1.4. Suppose that in L1(Γ, Γ) we take set theoretic composition as a product. Then L1(Γ, Γ) is a Z2-graded algebra, where L1(Γ, Γ)0 = {T ∈L1(Γ, Γ) : T (Γ0) ⊂ Γ0 and T (Γ1) ⊂ Γ1},andL1(Γ, Γ)1 = {T ∈L1(Γ, Γ): T (Γ0) ⊂ Γ1 and T (Γ1) ⊂ Γ0}.

Proposition 1.5. Given an open subset U of Γ, x0 ∈ U, a generator ξ ∈ X ⊂ Γ, and a super smooth function f : U → R ⊂ Γ.Then∂f/∂ξ(x)=0. Proof. Recall that Df(x, .) extends to countably infinite generated Λ where Γ ⊂ Λ. ∂f/∂ξ(x) =0.Therealnumber Dt=0f(x+tξ) satisfies for every element η ∈ Λ that 0 = Df(x; ξξη)=Df(x; ξ)ξη. Therefore, ∂f/∂ξ(x)=Δξ◦ Dt=0f(x + tξ)=0. n m Definition 1.3. GivenanopensubsetU ⊂ Γ0 ⊕Γ1 , a graded derivation ∞ ∞ D = D0 + D1 : G (U, Γ) → G (U, Γ), is a Γ homomorphism, where Γ is ∞ ∞ acting from the right, such that D0 : G (U, Γ) → G (U, Γ) is a derivation, ∞ ∞ |f| and D1 : G (U, Γi) → G (U, Γi + 1) satisfies D1(f ·g)=D1(f)·g+(−1) f · D1(g), where |f| designates the parity of f. AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 241

Using the fact that a G∞ super smooth functions f : U → ΓisaC∞ smooth function we deduce from the Taylor series with Lagrange remainder that

Proposition . ∞ → ∞ 1.6 Every smooth derivation D : G (U, Γ) G (U, Γ) n N ∞ is of the form D = i=1 gi(x)∂/∂ξi + j=1 fj(x)∂/∂xj,wherefj,gi ∈ C (U, Γ).

2. Diffeological lie groups and lie algebras The details for this section can be found in [7]. Given the category M whose objects are manifolds modelled on the open subsets of complete Hausdorff locally convex topological vector spaces, and whose morphisms are C∞ functions. A diffeological space is a set S together with a contravariant subfunctor on M FS(N) ⊆ Homset(N,S) such that constant maps are in F (N) for each object N of M and each x ∈ S and such that F restricted to the subcategory of open subsets of a fixed manifold, N, whose morphisms are the canonical injection of open subsets of N into each other, F satisfies the axioms of a set valued sheaf. Given an open U ⊂ E, E a graded complete Hausdorff locally convex topological vector space. An element f ∈ FS(U)is called a plot of S. Given diffeological spaces (X1,F1)and(X2,F2) a function f : X1 → X2 ∞ is called smooth or C if for each g ∈ F1,wehavef◦g ∈ F2. We shall say that f is locally smooth at x0 ∈ X1 when given any smooth map, g ∈ F1,from a neighborhood, U,of0∈ E,where f(0) = x0 there exists a neighborhood, U0 ⊂ U of 0 such that f ◦ (g|U0) ∈ F2. When S is a C∞ manifold we shall suppose without explicit mention to the contrary that it has its underlying diffeology given by F (C) being defined to be the set of C∞ maps with domain N and values in S.Weshall call this diffeology the canonical diffeology on the C∞ manifold S. Given a manifold M and a diffeological space (S, F ), a mapping, f, from a subset C ⊆ M to S is called smooth, when there exists an open neighborhood, U,ofC and a smooth extension of f to U, f˜: U → S. Given any collection of diffeological structures on a set S, Fi,wehave that ∩Fi is a diffeological structure, thus any assignment of functions, GS(U) ⊂ Homset(U, S),U ⊂ E, E a Hausdorff locally convex topological vector space, generates a diffeology; namely, the smallest or finest diffeology containing the GS(U). For the diffeology so generated we shall call GS(U) a system of generators. A useful notion for diffeological structures is that of the pull − back: given a diffeological stucture on a set T ,(T,G), and a function, f : S → T , define f G(U)={g ∈ Homsets(U, S):f ◦ g ∈ G(U)}. It is straightforward to verify that f G is a diffeological stucture on S. Given a subset S1 ⊂ S2, where (S2,F2) is a diffeological space, there is a diffeological structure induced on S1 by F1(C)={f ∈ F2(C):f(C) ⊂ S1}. 242 J. LESLIE

Note that F1 = i (F2), where i : S1 → S2 is the canonical inclusion. In the rest of this paper when we consider a subset of a diffeological space as a diffeological space it will be with the above described structure unless there is an explicit mention to the contrary. When C is an open subset of a graded Hausdorff locally convex topologoical vector space we shall call f of FS(C) a plot of the diffeological structure at a point s = f(x), x ∈ C. Given diffeological spaces (X1,F1)and(X2,F2),andanopensubset U of an complete Hausdorff locally convex topological vector space E a ∞ mapping f : U → C (X1,X2)issmooth when F : U × X1 → X2, given by ∞ F (u, x)=f(u)(x) is smooth. The finest diffeology on C (X1,X2) admitting these functions as generating plots will be called the function space diffeology ∞ ∞ on C (X1,X2) or the canonical diffeology on C (X1,X2). It follows from the definitions that

Proposition 2.1. Given compact C∞ manifolds M and N a function f : N → C∞(M,Rn) is smooth for the C∞ topology on C∞(M,Rn) if and only if the induced map f˜: N × M → Rn is C∞.

Corollary 2.1. Given a compact C∞ manifold M and an open subset U ⊂ Rn a function f : U → Diff∞(M) is smooth for the C∞ topology on Diff∞(M) if and only if the induced map f˜: U × M → M is C∞.

Definition 2.1. We shall call a diﬀeological structure lattice or L type when given two plots f : M1 → S, g : M2 → S at a point t = f(x)=g(y) ∈ S there always exists a third plot through which the germs of f at x and g at y factor; that is, there exists a plot h : N → S such that f = h ◦ φ, g = h ◦ γ, where φ : M˜1 → N,γ : M˜2 → N are smooth functions such that φ(x)=γ(y), where M˜i ⊂ Mi is a neighborhood of x (resp. y).

Suppose that (S, F ) is a diffeological space of L − type; consider the equivalence relation generated between germs of one dimensional plots at s ∈ S as follows: let f and g be one dimensional plots at s from domains open intervals C1 and C2 containing 0, we write f0 ≡ g0,whenf(0) = g(0) = s and there exists a plot k:U→ S through which the germs of f and g factor at 0 andwehavethatD0(h ◦ f)=D0(h ◦ g), where h is any smooth real valued function defined on k(U). The equivalence classes will be called tangent vectors at s, we shall designate the set of tangents at s by TsS. It is immediate that a smooth map f : S → W defines a function Tf : TsS → Tf(s)W . When M is a manifold modelled on a locally convex topological vector space, this definition is equivalent to the classical one for the canonical diffeology associated to the manifold structure. Given a plot f : U → S,we use Tf : TU → TS to define plots on TS and thus a diffeology on TS,in what follows TS will be considered as a diffeological space with the finest diffeology admitting such Tfs as plots. AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 243

Given diffeological spaces (X1,F1)and(X2,F2) the Cartesian product diffeology F1 × F2 on X1 × X2 is defined by f ∈ F1 × F2 if and only if pr1 ◦ f ∈ F1 and pr2 ◦ f ∈ F2. The cartesian product when considered as a diffeological space will be considered to have this cartesian product diffeology unless there is an explicit mention to the contrary. AgroupG with a diffeology F on its underlyling set will be called a diffeological group when multiplication and inversion define smooth maps G × G → G and G → G. One readily verifies that each diffeological group is of L − type. In a similar vein we define other diffeological algebraic structures. Given a plot at the identity, e ∈ G, f : U → G, x ∈ U, U ⊂ E, E a locally convex topological vector space, U open, define Df(x; α)=[f(x + tα]t=0, x ∈ U. In [7]weproved

Proposition 2.2. If G is a diﬀeological group, then TeG is a diﬀeological vector space.

Remark 2.1. Let {G, G} be a diffeological group and N a normal subgroup of G, then one has that G/N is a diffeological group, where the quotient diffeology on G/N is the diffeology generated by the sets of functions {p◦g, p : G → G/N canonical map, g ∈G(U), U an open subset of a complete Hausdorff locally convex topological vector space}.

We shall call a collection of plots S coﬁnal for a diﬀeological space {X, F} when every plot of X, factors germwise smoothly through a map of S; that is, given a plot of X, u : U → X and u(x0)=y0∃ an open neighborhood of x0 in U, U0 and a smooth map φ : V0 → X ∈S such that u|U0 = φ ◦ v, where v : U0 → V0 is smooth.

Definition 2.2. A diffeological group G will be called a diffeological Lie group when the tangent space at the identity, TeG, is a diffeological vector space which (i) admits for every non zero α ∈ TeG a smooth real valued linear map T : TeG → R such that T (α) =0,and (ii) the linear plots of TeG are cofinal.

Example 2.1. A Lie group, G modeled on a C∞ super manifold (possibly infinite dimensional) admits the structure of a diffeological Lie group with the canonical diffeology.

Theorem 2.1. Let G be a diffeological Lie group, then TeG is a diffeological vector space which admits the structure of a Lie algebra such that the bracket operation defines a smooth linear map X : TeG → TeG,where

X (Y )=[X, Y ]. 244 J. LESLIE

Let’s recall a diffeological Lie group G is called regular when the logarithmic derivative f (t)f(t)−1 defines a diffeological isomorphism from the diffeological set of smooth mappings from the unit interval into G which ∞ map zero to the identity, e ∈ G, C0 (I,G), to the diffeological set of smooth ∞ ∞ mappings from the unit interval into TeG, χ : C0 (I,G) → C (I,TeG). Note that a map from the unit interval into a diffeological space is called smooth when there exists an open interval containing the unit interval to which the map can be smoothly extended. It is straightforward to verify

Proposition 2.3. Given two regular diﬀeological Lie groups G and H, and a smooth homorphism F : G → H,thenHom(I,dF)◦χ = χ◦Hom(I,F)

Theorem 2.2. Let G be a regular diﬀeological Lie group, then there exists a smooth function exp : TeG → G such that exp((t + s)) = exp(tξ) × exp(sξ) and Dt[exp(tξ)] = Rexp(tξ)(ξ).

Corollary 2.2. If G is a regular diﬀeological Lie group, then TeG admits the structure of a diﬀeological Lie algebra.

Definition 2.3. A diffeological vector space E will be called integral ∞ when there exists a smooth linear map : C (I,E) → E, such that given any smooth real valued linear function H : E → R,wehaveH( (f)) = ∈ I H(f(t))dt and such that given any v E we have (f(t)v)=( I f(t)dt)v. We shall sometimes for the sake of clarity use the notation f(t))dt especially when f takes its values in a function space. Given an integral diffeological vector space, E,f ∈ C∞(I,E),t∈ I define t (f)= (f(ts)t)ds A subspace, K, of an integral diffeological vector space, E, will be called closed when (C∞(I,K)) ⊆ K.

Definition 2.4. Let K{x1,...,xn+2} be the free associative algebra over the field K generated by x1,...,xn+2, K{x1,...,xn+2} is isomomorphic to the tensor algebra generated by an n + 2 dimensional vector space. Definition 2.5. Let L be an integral diffeological Lie algebra such that the linear plots of L are cofinal. A closed subalgebra, H, will be called strongly integrable when s s (i) defines a smooth map : C∞(I,L) → C∞(I,L), and ∞ (ii) when for each sequence ai(t) ∈ C (I,H), i =1,...,n; g(t) ∈ ∞ C (I,H)andP (x1,...,xn+2) ∈ K{x1,...,xn+2} the non homogeneous linear differential equation, ( )y = g(t)+P (ada (t),...,ada s 1 n (y), ) admits an unique smooth flow, Φ(a(t),s,l) such that Ds Φ(a(t),s,l,g(s)) = g(s)+[a(s), Φ(a(t),s,l)], Φ(a(t), 0,l)=l and such that Φ defines a smooth map (iii) Φ : C∞(I,H) × I ×H×C∞(I,H) →H, such that AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 245

(iv) Φ(a(t),s,˙,g):H →H is a diﬀeomorphism which induces a diﬀeo- morphism Φ:C∞(I, I) → C∞(I, I), where Φ(f)(t)=Φ(a, t, f(t))

Example 2.2. Let M be a compact C∞ manifold and let X (M) be the Lie algebra of C∞ vector ﬁelds on M.

It results from the classical theory of diﬀerential equations on a compact manifold that the canonical diﬀeology associated to the C∞ topology on X (M) renders a closed sub algebra H⊆X(M) a strongly integrable Lie algebra. We proved in [7]:

Theorem 2.3. Let G be a simply connected, regular, diffeological Lie ∞ ∞ group with canonical diffeomorphism χ : C0 (I,G) → C (I,TeG) and suppose that H is a connected normal subgroup of G such that there exists an integrable Lie ideal, H⊆TeH ⊆ TeG = G with a diffeological vector space complement, K in G, satisfying −1 ∞ ∞ (i) χ (C (I,H)) ⊆ C0 (I,H); (ii) given h ∈ H suppose there exists a smooth path f :[0, 1] → H such that f(0) = e, f(1) = h,andχ(f) ∈ C∞(I,H); (iii) given any k =0 ∈G/H, suppose there exists a smooth Lie algebra homomorpism φ : G/H →S,whereS is the Lie algebra of a regular diffeological Lie group S such that φ(k) =0 . Then H and G/H are regular diffeological Lie groups with Lie algebra H (resp. G/H.

Theorem 2.4. Suppose that G is a strongly integrable diffeological Lie algebra such that [a, f(t)] = [a, f(t)],wheref(t) ∈ C∞(I,G) and a ∈G. Then C∞(I,G) with its canonical diffeology is a regular diffeological Lie group with a group product given by ( )(u·v)(s)=χ−1 (χ(u)(s)) + Φ (χ(v),s,χ(u)(s))) .

Proof. ( )[7] provides C∞(I,G) with a group strructure. Since χ is a diffeomorphism of diffeological spaces it follows that C∞(I,G) is a diffeological group. It’s Lie algebra is again C∞(I,G)with Lie product given by t t {f,g}(t):=[f(t), g(s)] + [ f(s),g(t)]

∞ To see that C (I,G) is an integral Lie algebra, suppose given a smooth F : ∞ ∞ I → C (I,G), deﬁne F ∈ C (I,G)by F (s)= F (t)(s)dt. From the definitions and utilizing the equality ( f(x, y)dx)dy = ( f(x, y)dy)dx we obtain that this new : C∞(I,C∞(I,G)) → C∞(I,G) determines an integral 246 J. LESLIE

Lie algebra stucture on C∞(I,G). The above integral structure on C∞(I,G) together with the strong integrability of G) implies the strong integrability of C∞(I,G).

Definition 2.6. Given two smooth paths f,g : I → L into a diﬀeolog- ical Lie algebra, L, we say that f,g : I → L are Lie homotopic when there exists smooth maps from the square V,W : I × I → L such that V (t, 0) = f,V (t, 1) = g, W(0,s) ≡ W (1,s) ≡ 0,Vs − Wt =[V,W]

It has been shown [7] that the subset N of C∞(I,G) consisting of ∞ G { ∈ paths Lie homotopic to 0 is a normal subgroup of C (I, ). Let Ω = γ ∞ 1 C (I,G): 0 γ(s)ds =0}, it follows from of the proof of theorem 2.4 that Ω is a Lie ideal of C∞(I,G). From [7]wehave

Theorem 2.5. Suppose that G is a strongly integrable integral Lie algebra, then there exists a simply connected regular diﬀeological Lie group G ∼ such that TeG = G.

Under these conditions we can now formulate a version of Lie’s second fundamental theorem:

Theorem 2.6. Let G be the integral Lie algebra of a simply connected regular diﬀfeological Lie group and suppose that H⊆G is a closed strongly integrable Lie subalgebra of G. Then there exists a connected subgroup H ⊆ G such that TeH H.

Proof. Let ρ : C∞(I,G) → C∞(I,G)/N = G be the canonical Lie homomorphism of groups. The the smooth Lie group homomorphism of C∞(I,H) into C∞(I,G) induced by the canonical smooth Lie algebra homomorphism of H into G composed with ρ gives a map onto a subgroup H of G which satisﬁes the conditions of the theorem.

3. A method of construction of non-trivial graded compact supermanifolds with non degenerate graded differential forms For the rest of this paper we suppose that Γ is the finitely generated Grassmann algebra with generators: ξ1 ...ξN , and suppose that C is the canonical basis of Γ associated to system of generators {ξi}.LetM be the category whose objects are super manifolds modelled on the open subsets of n m ∞ ∞ the graded Γ0 module V =Γ0 ⊕Γ1 having G smooth maps of G smooth super manifolds as its morphisms. As in the classical case we may define the tangent bundle to a supermanifolds M to obtain a smooth vector bundle

ΠM : TM → M where a vector α ∈ TxM is deﬁned to be an equivalence classes of smooth curves through x, a(t), parametrized by an open interval containing 0 such AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 247

that a(0) = x with the equivalence relation a ≡ a if and only if Dt=0(f ◦ a)(t)=Dt=0(f ◦a )(t) for every germ of a smooth function f : U → Γ defined on a neighborhood U of x ∈ M. This definition is equivalent to the choice of a vector in the modeling space αi ∈ Ei, Ui ⊂ Ei and Uj ⊂ Ej,andφi : Ui → M models M with the equivalence relation at x ∈ φi(Ui) ∩ φj(Uj) given by: (Ui,αi)) ∼ (Uj,αj)if −1 −1 and only if D(φi ◦ φj)(φj (x),αj)=αi, which implies ∼ n m Proposition 3.1. TxM = Γ0 ⊕ Γ1 = V . Definition 3.1. A supermanifold will be called pre-graded when the tangent bundle can be reduced to the group G of Γ0- automorphisms of n m n n m m Γ0 ⊕ Γ1 such that g ∈ G ⇔g(Γ0 )=Γ0 and g(Γ1 )=Γ1 . Proposition 3.2. Given a pre-graded supermanifold M, the even (resp. odd) vectors Ex(resp.Ox) ⊂ TxM form a sub vector space of TxM. A pre-graded supermanifold is said to be graded when there exists a system of charts U = {Ui},φi : Ui → M for its supermanifold structure such −1 that Dx(φi ◦ φj)(x) ∈ G, ∀x ∈ Ui ∩ Uj. It follows from the definitions that Proposition 3.3. If M is a graded supermanifold, then there exists a subbundle E,(resp.O) ⊂ TM such that α ∈ Ex(resp.Ox) is vector tangent to −1 n m acurveC through x with φi (C) ⊂ Γ0 (resp.Γ1 such that TM = E + O. ∞ n m Define δ(ξi)(s, x0) ∈ G (U, Γ0 ⊕ Γ1 )by

δ(ξi)(s, x0)=x0+[(sξi⊕...⊕0)⊕...⊕(0⊕...⊕sξi,...,0)⊕...⊕(0⊕...⊕sξi)], for |s| > 0 sufficiently small. Now define ∞ ∞ ∂f/∂ξ(x0):G (U, Γ)i → G (U, Γ)i+1 by ∂f/∂ξ(x0)=Δξ ◦ Ds=0f(δ(ξ)(s, x0)), we recall Δξ : Given a system of { } → generators ξi of Γ define a linear map Δξi :Γ ΓbyΔξi (ξi1 ...ξk ...ξin )= n − (k−1) k=1 δiik ( 1) (ξi1 ...ξik−1 ξik+1 ...ξin ), and Δξj (1Γ)=0. n m GivenanopensubsetU ⊂ Γ0 ⊕ Γ1 , we define a graded right Γ-module ∞ ∞ ∞ ∞ structure on G (U, Γ) by setting G (U, Γ)0 = G (U, Γ0)andG (U, Γ)1 = ∞ G (U, Γ1), and setting (fγ)(x)=f(x)γ. As in the classical case we identify ∞ a vector αi ∈ Ei with the class of G curves in the direction of that vector and write ∂/∂xi(resp.∂/∂ξi) for the vector in the direction of ηi (resp. δ(ξi)(1,x0)). For the rest of this section we shall assume that our manifolds are graded supermanifolds and therefore that the fibers of ΠM : TM → M have a canonically assigned gradation. The associated vector bundle with fiber Γ ⊗Γ0 V

ΠM Γ : TΓM → M inherits a graded structure by the convention |γ ⊗Γ0 v| = |γ| + |v|. 248 J. LESLIE

It is straightforward to prove that

Proposition 3.4. The G∞ smooth graded derivations of G∞(U, Γ) are ﬁnite sums of the form fi(x, ξ)∂/∂ξi + gi(x, ξ)∂/∂xi + hi(x, ξ)∂/∂ξi + ki(x, ξ)∂/∂xi,

n m ∞ ∞ where (x, ξ) ∈ U ⊂ Γ0 ⊕ Γ1 , fi,ki ∈ G (U, Γ)0,andgi,hi ∈ G (U, Γ)1, des- ∞ n m ignate by D(U) the space of derivations of G (U, Γ), where U ⊂ Γ0 × Γ1 , x ∈ Rn. D(U) can be given the structure of a left G∞(U, Γ) module by (γd)(f) (x)=γ(x)(d(f)(x))

Now let ΠM : TΓM → M ∼ be the associated bundle with ﬁber HomΓ(Γ ⊗Γ0 V,Γ) = HomΓ0 (V,Γ). Its ﬁber now is assigned a Z2 grading by

HomΓ(Γ ⊗Γ0 V,Γ)0 = HomΓ0 (Vi, Γi)andHomΓ(Γ ⊗Γ0 V,Γ)1

= HomΓ0 (Vi, Γi+1).

Given a trivializing chart at x ∈ U ⊂ M for ΠM : TM → M Suppose γi ∈ Γ,γ0 =1, k =1,...,N,j=1,...,n, is a basis of Γ over R, then a basis of TΓxM over R can be given by dxjγi,dξkγi and dxj, where dξj ∈ HomΓ0 (V,Γ) are deﬁned by {∂/∂xi|dxj} = δjl = {∂/∂ξi|dξj} and {∂/ ∂ξi|dxj} =0={∂/∂xi|dξj}.

Definition 3.2. A bilinear map ω : D(U) ×D(U) → G∞(U, Γ) is called a2− form when

|X1||f| {X1,fX2|ω} =(−1) f{X1,X2,ω}

|X1||f| ∞ =(−1) {fX1,X2,ω},f ∈ G (U, Γ).

We place a Z2 gradation on 2 − forms such that |ω| = |{X1,X2|ω}| + |X1| + |X2|(mod2) In the Z2-graded case we now deﬁne wedge algebra of a graded vector space V as follows: suppose that R is the graded ideal generated by α ⊗ β − (−1)|α||β|β ⊗ α in the doubly graded tensor algebra ∞ ⊗∞V = ⊗nV, n=0 where αand β are homogeneous elements of V . ∞ ⊗∞ ≡ ∞ n n ∩⊗n Set V = V/R n=0 V , where V = V/R V, ∞ ∞ n n On ⊗ V = n=0 ⊗ V ‘, a Z2-gradation is placed on ⊗ V by the convention |α1 ⊗···⊗αn| = |α1| + ···+ |αn|. AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 249

n As in the classical case we may consider a smooth vector bundle T n M → M and smooth sections, ω of the linear bundle T M → M, where M is a supermanifold. Rogers [9] p. 75 implies

2 Proposition 3.5. Given a trivializing chart U of ΠM |∧ TΓU → U, there is a 1−1 parity preserving correspondence between sections of ΠM |ΠM | 2 ∧ TΓU → U and 2-forms on U determined by

|X2||ω1| {X1,X2|ω1 ∧ ω2} =(−1) {X1, |ω1}{X2|ω2}

|ω1|||ω2|+|X1||ω2| − (−1) {X1, |ω2}{X2|ω1}

Remark 3.1. Derivations are are given the structure of left G∞(U, Γ) modules and we give two forms the structure of right G∞(U, Γ) module structure by the convention {X1,X2|ωf} = {X1,X2|ω}f.

f: Suppose that A is an n × n upper triangular matrices of entries from (ΓN )0,with1s down the diagonal; and suppose that B is an n × m matrix with odd entries from (ΓN )1,andIm an m × m identity matrix. The matrices of the form AB 0 Im form a connected nilpotent Lie group G of dimension 2L−2n(2m + n − 1).

Lemma 3.1. Let H ⊂ G be a the subgroup whose entries are elements of (ΓN )0 and (ΓN )1 with integer coeﬃcients of the canonical Grassmann ≤ generators ξi1 ...ξik ...ξin ; ik ik+1. Then H is a lattice subgroup of G.

Proof. G is clearly nilpotent, we have only to establish that G/H is compact. Firstly order the entries of the matrix lexicographically; that is, so that the entries are simply ordered by (i, j) ≺ (k, l) if and only if i

Now observe that the matrix (bij)=B = PklT has the property that |bij|≤1for(ij) (k, l), since bkl = tkl − ι(tkl)andbij = tij for ij ≺ kl. The above establishes that given any matrix (tij)=T ∈ G there exists a ij ∈ ij ij matrix P H such that PT = B =(b ) has all of its entries b = k ai1,...,ik ij i i | |≤ ξ 1 ,...,ξ k with real coefficients ai1,...,ik 1 with respect to th canonical basis ξi1 ,...,ξik ,...,ξin . The above implies that G/H is compact, and therefore that H ⊂ G is a discrete lattice subgroup of the connected nilpotent Lie group G. The homotopy sequence of the fibration H → G → G/H implies that ∼ Π1(G/H) = H. Corollary 3.1. G/H is a connected compact supermanifold modeled n(n−1)/2 mn ∼ on ((ΓN )0) ⊕ ((ΓN )1) , such that Π1(G/H, [e]) = H. Example 3.1. (Odd Compact Graded Symplectic Manifold): Consider the Abelian super Lie group G1 ⊂ G consisting of matrices of the form AB M = 01 where m =1 and the first row of A is a vector of the form α =(1,a12,..., a1n+1),wherea1j ∈ (Γn)0 and B is of the form ⎛ ⎞ b ⎜ ⎟ ⎜ 0 ⎟ ⎜ . ⎟ ⎝ . ⎠ 0 where b ∈ (Γn)1. Further, we suppose that eliminating the first row and column of M, M becomes an n × n identity matrix.

The construction of Example 3.1 provides a torus T = G1/H ∩G1 of real n−1 n dimension 2 (n + 1) modeled on ((Γn)0) ⊕ (Γn)1). Now suppose that G1 is modeled at the identity by the trivializing chart n (a12,...,a1n+1),b) → ((Γn)0) ⊕ (Γn)1). Let dxj,dξk ∈ (TΓ)eG be the elements determined by this trivialization at the identity. The right invariant 2 Maurer Cartan form ω ∈ Γ( TΓT ) such that n Π (ω(identity)) = dxidξi i=1 determines the odd graded super-symplectic manifold structure on T = G1/H ∩ G1. Example 3.2. (even symplectic compact graded manifold): With the notation of the above example suppose that α =(1,a12,...,a12s+1) and suppose that Γ is the Grassmanian algebra generated by ξ1 ...ξq. Now the con- 2s struction of Example 3.1 gives a torus T = G1/H∩G1 modeled on ((Γq)0) ⊕ AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 251

q−1 ∈ 2 (Γq)1) of real dimension 2 (2s +1). Consider the unique ω Γ( TΓT ) s q such that Π (ω(identity)) = 2 i=1 dxi ∧ dxi+s + i=1 dξi ∧ dξq−i.

The super symplectic structures of Examples 3.1 and 3.2 are nondegenerate in the sense that: n given any X = i=1 fi∂/∂ξi +gi(X, Ξ)∂/∂xi +hi∂/∂ξi +ki(X, Ξ)f∂/∂xi, n where fi,gi ∈ Γ there exists Y = i=1 hi∂/∂ξi + ki∂/∂xi such that ω(X, Y ) =0

Example 3.3. (compact graded contact super manifold) With the notation of the above example suppose that α =(1,a12,..., a12s+2) and suppose as above that Γ is the Grassmanian algebra generated by ξ1 ...ξq. Now the construction of Example 3.1 gives a torus T = G1/H ∩ G1 2s+1 q−1 modeled on ((Γq)0) ⊕(Γq)1) of real dimension 2 (2s+2). Consider the s unique 1-form ω ∈ Γ(TΓT ) such that Π (ω(identity)) = 2 i=1(xidxi+s − q xs+idxi)+ i=1 ξidξq−i+1.

Example 3.4. (graded compact odd contact super manifold) In Example 3.3 we suppose Γ is generated by ξ1 ...ξn+1 and that T = G/H is the torus n n+1 modeled on ((Γ)0) ⊕(Γ)1 of real dimension 2 (n+1). Consider the unique 1-form ω ∈ Γ(T T ) Γ n such that Π (ω(identity)) = dξn+1 + i=1(xidξi + ξidxi).

4. Automorphisms of graded compact manifolds It is known [5] that

Theorem 4.1. Suppose that M is a compact G∞ manifold modeled on n m Γ0 ⊕Γ1 ,whereΓ is the algebra generated by the ﬁnite sequence ξ1,...,ξk,..., ξN . The tangent space TxM has canonically Γ0 module structure such that TM is a G∞ manifold and such that the canonical projection TM → M is a G∞ bundle.

∞ Let Diff∞(M)andDiffG (M) be respectively the group of C∞ (resp. ∞ G∞) diffeomorphisms of M, Diff∞(M)andDiffG (M) are regular diffeological Lie groups with Lie algebras X and D, where X represent the C∞ vector fields on M and D is the closed Lie subalgebra of X of G∞ vector fields on M with the canonical Γ0 module structure. Further X and D are strongly integrable Lie algebras in the sense of section 2.

∞ Remark 4.1. The modeling of DiffG (M) is done by the restriction of ∞ the exponential map of a G − connection on M, expG : X → Diff (M)to ∞ G∞ n D and deﬁnes a G map (expG|D):D→ Diff (M), where G ⊂ GL(Γ0 ⊕ m N−1 Γ1 )=GL(2 (m + n),R), is the subgroup of of Γ0 automorphisms of n m Γ0 × Γ1 . 252 J. LESLIE

∞ DiffG (M) is the group of automorphisms of this G structure. The ∞ manifold structure on DiffG (M) is the canonical closed submanifold structure induced by the C∞ manifold structure on Diff∞(M). Now we recall that the Lie exponential exp : X → Diff∞(M) is the unique solution of the differential equation on the compact manifold M determined by ∞ ( )d/dt(exp(t, X)=X(exp(t, X)),exp(0,X)=idM ∈ Diff (M). This differential equation restricted to D determines a G∞ differential equation on the compact G∞ manifold M so that exp detemines locally a G∞ ∞ mapping exp : D→ DiffG (M).

Corollary 4.1. Given X ∈D the Lie derivative deﬁnes a Γ0 pcon- ∧p ⊗ ⊗q → travariant (q covariant) homorphism LX :Γ( (TΓM Γ0 Γ0 TΓM)) ∧p ⊗ ⊗q ∧p ⊗ ⊗q Γ( (TΓM Γ0 Γ0 TΓM)),whereΓ( (TΓM) Γ0 Γ0 TΓM) represents the smooth sections of the bundle ∧p ⊗ ⊗q → ΠM : TΓM Γ0 Γ0 TΓM M, ∞ → D× ∧p ⊗ ⊗q → and induces a G map (X, ω) LX (ω), Γ( (TΓM) Γ0 Γ0 TΓM) ∧p ⊗ ⊗q Γ( (TΓM) Γ0 Γ0 TΓM).

Theorem 4.2. Suppose that M is a compact G∞ manifold modeled on n m Γ0 ⊕ Γ1 ,whereΓ is the Grassmann algebra generated by the ﬁnite sequence ∈ ∧p ⊗ ⊗q { ∈D ξ1,...,ξk,...,ξN . Given ω Γ( (TΓM) Γ0 Γ0 TΓM),LetW = X : LX (ω)=0}.If W is Lie sub algebra of D, then W is a strongly integrable Lie sub algebra of D.

Idea of Proof: W ⊂D⊂X is a closed Lie sub algebra for the C∞ topology, and therefore the differential equation ( )y = g(t)+P (ada (t),..., s 1 adan (y), ) of definition 2.4 induces a smooth differential equation on the compact manifold M, the existence and uniqueness theorem for compact manifolds imply that admits a solution in W with the required properties.

References [1] Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups and representations of reductive groups, Ann. of Math. Stud. 94 Princeton University Press, Princeton, NJ, 1980. [2]Cantarini,N.andKac,V.,Infinite-dimensional Primitive Linearly Compact Lie Superalgebras, Adv. in Math, (2006), 328–419. [3] Iglesias, P., Thesis, L’Universite de Provence, 1985. [4] Leslie, J., On the Lie Subgroups of Infinite Dimensional Lie Groups, Bull. of AMS Jan.(1987). [5] Leslie, J., On a Super Lie Group Structure for the Group of G∞ Diffeomorphisms of acompactG∞ supermanifold Geometry & Physics. (1997). [6] Leslie, J., Lie’s Third Theorem in Supergeometry, Algebras, Groups, and Geometries. 14:4(1997), 359–406. AUTOMORPHISMS OF GRADED SUPER SYMPLECTIC MANIFOLDS 253

[7] Leslie, J., On a Diffeological Group Realization of Certain Generalized Symmetrizable Kac-Moody Lie Algebras, Journal of Lie Theory 13 (2003), 427–442. [8] Milnor, J., Remarks on Infinite Dimensional Lie Groups,Proc.ofSummerSchoolon Quantum Gravity (1983). [9] Rogers, A., Supermanifolds Theory and Applications, World Scientific (2008). [10] Singer, I. M. and Sternberg, S., On the infinite groups of Lie and Cartan, I. Ann Inst, Fourier (Grenoble) 15 (1965), 1–114. [11] Souriau, J. M., in “Feuilletages et Quantification geometrique”, Travaux en Cours, Hermann, 1984, 365–398.

Surveys in Diﬀerential Geometry XV

The Signature of the Seiberg-Witten Surface

Andreas Malmendier

Abstract. The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface Z over CP1. We show that for the ∂¯-operator along the fiber the logarithm of the regularized deter- 1 ¯∗ ¯ minant − 2 log det (∂ ∂) satisfies the anomaly equation of the one- loop topological string amplitude derived in Kodaira-Spencer theory. We also show that not only the determinant line bundle with the Quillen metric but also the ∂¯-operator itself extends across the nodal fibers of Z. The extension introduces current contributions to the curvature of the determinant line bundle at the points where the fibration develops nodal fibers. The global anomaly of the determinant line bundle then determines the signature of Z which equals minus the number of hypermultiplets.

1. Introduction and Statement of results There has been a continuing interest in the non-perturbative properties of the supersymmetric Yang-Mills theory on four-dimensional manifolds. One of the results of the work of Seiberg and Witten [19] is that the moduli space of the topological SU(2)-Yang-Mills theory on a four- dimensional manifold decomposes into two branches, the Coulomb branch and the Seiberg-Witten branch. The branches are interpreted as the moduli spaces of simpler physical theories on the four-manifold. The Coulomb branch, also called the Seiberg-Witten family of curves, is the moduli space of a topological U(1)-gauge theory, called the low energy effective U(1)- gauge theory. We investigate the geometry and topology of the Coulomb branch as it is fundamental for the definition and the understanding of the N = 2 supersymmetric, low energy effective field theory by using the results and techniques developed in [2, 18, 5, 6]. This article is structured as follows: In Section 2 we explain how the Seiberg-Witten families of elliptic curves for the N = 2 supersymmetric SU(2)-gauge theory with Nf =0, 1, 2, 3, 4 additional fields, called c 2011 International Press

255 256 A. MALMENDIER hypermultiplets, define four-dimensional, Jacobian rational elliptic surfaces Z → CP1 with singular fibers. We describe the correspondence between the rational elliptic surfaces and the constraints on the masses of the hypermultiplets. Unless noted otherwise, we will then always assume that the masses are generic so that any singular fiber over [u∗ :1]∈ CP1 for finite u∗ is a node and does not give rise to a surface singularity. Only the singular fiber over [1 : 0] is a cusp giving rise to a surface singularity. Thus, after removing the singular fiber over [1 : 0], we obtain a smooth elliptic surface Z with three-dimensional boundary and elliptically fibered over a bounded disc in CP1, called the u-plane UP ⊂ CP1. In Section 3 we review the construction of the regularized determinant of the ∂¯-operator on an elliptic curve. When the elliptic curve Eu is varied in the Seiberg-Witten family over the u-plane, we obtain the determinant line bundle DET ∂¯ → UP of the ∂¯-operator along the fiber of Z → UP. The regularized determinant det(∂¯∗∂¯) of the Laplacian along the fiber becomes a smooth function on UP which we will later use in the Quillen construction to define a metric and connection on DET ∂¯. In Section 4, we show that the 1 ¯∗ ¯ logarithm of the regularized determinant of the Laplacian − 2 ln det (∂ ∂) satisfies the anomaly equation for the one-loop topological string amplitude of Kodaira-Spencer theory derived in [4]. In Section 5 we show that the local anomaly of DET ∂¯ vanishes and determine the non-trivial global anomaly as holonomy of the determinant section. We use the results of Bismut and Bost [5, 6] to show that the determinant line bundle with the Quillen metric extends smoothly across the nodal fibers. Because of the non-trivial holonomy the extension of the determinant line bundle introduces current contributions to the curvature over the points in the u-plane where the fiber develops a node. In the case that the singular fibers are nodes Seeley and Singer [18] showed that there is a ∂¯-operator that is defined on the nodal fiber as well so that in a neighborhood U ⊂ UP the family of operators {∂¯u}u∈U is a continuous family. However, their procedure of obtaining the determinant line bundle is different from Bismut and Bost [5, 6] because their Laplacian is different. In Section 6 we explain the connection between the two pro- cedures by a local change in the conformal gauge of the fiber metric in a neighborhood of the nodal fiber. In Section 7 and Section 8 we discuss the elliptic operators connected to the signature of the elliptic surface Z → UP. First, for the signature operator D along the fiber of Z → UP we compute the global anomaly. We show that there is a canonical trivialization of the determinant line bundle (DET D)⊗6, and the well-defined logarithmic monodromies of the canonical section of DET D determine the signature of Z → UP. We interpret the determinant line bundle as a solution to a Riemann-Hilbert problem on CP1. On the other hand, the generalization of Hirzebruch’s signature theorem for manifolds with boundary by Atiyah, Patodi, Singer (APS) [3] shows that the elliptic signature operator on the four-dimensional surface Z has an analytic index THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 257 if one imposes APS boundary conditions on ∂Z. We show that this analytic index equals minus the number Nf of hypermultiplets.

2. The Jacobian elliptic surfaces for N = 2 Yang-Mills theory An elliptic curve E in the Weierstrass form can be written as 2 3 (2.1) y =4x − g2 x − g3, 3 where g2 and g3 are numbers such that the discriminant ΔΔ=g2 − 27g3 does not vanish. In homogeneous coordinates [X : Y : W ] Eq. (2.1) becomes 2 3 2 3 WY =4X − g2 XW − g3 W . One can check that the point P with coordinates [0 : 1 : 0] is a smooth point of the curve. We consider P the base point of the elliptic curve and the origin of the group law on E. The two types of singularities that can occur as Weierstrass cubic are a rational curve with a node, which appears when the discriminant vanishes and g2,g3 =0,oracuspwhen g2 = g3 =0. Next, we look at a family of cubic curves over CP1. The family is parametrized by the base space CP1 and a line bundle N→CP1.Thequan- ⊗4 ⊗6 tities g2 and g3 are promoted to global sections of N and N respectively; the discriminant becomes a section of N ⊗12. If the sections are generic enough so that they do not always lie in the discriminant locus, we obtain a Weierstrass fibration π : Z → CP1 with a section, called a Jacobian elliptic fibration. Each fiber comes equipped with the base point P that defines a section S of the elliptic fibration which does not pass through the nodes or cusps. We will always assume that N = OCP1 (−1). In addition, we will assume from now on that in the coordinate chart 1 [u :1]∈ CP , the discriminant Δ is a polynomial of degree 2 ≤ Nf +2≤ 6in u where 0 ≤ Nf ≤ 4, and g2 and g3 are polynomials in u of degree at most 2 and 3 respectively. The space of all such Weierstrass elliptic surfaces has Nf + 1 moduli. To see this consider first the case where Nf = 4. From the seven parameters defining g2 and g3 two can be eliminated by scaling and a shift in u. Furthermore, we can arrange the coefficient of g2 of degree two and the coefficient of g3 of degree three to be the modular invariants of an elliptic curve with periods 1 and τ0. The remaining four coefficients can be expressed in terms of four complex parameters. Following the convention of [19] we will denote the parameters by m1,...,mNf . In physics, they are called the masses of the hypermultiplets. A non-trivial elliptic fibration has to develop singular fibers. The classification of the singular fibers is part of Kodaira’s classification theorem of all possible singular fibers of an elliptic fibration [9]. For generic values ∗ ∗ of the masses, the polynomial ΔΔhasNf + 2 simple zeros u1,...,uNf +2 for ∗ ∗ |u| < ∞ with g2(ui ),g3(ui ) =0 for i =1,...,Nf + 2. From Kodaira’s classification theorem of singular fibers [9] it follows that the elliptic fibration develops the nodes, i.e., a singular fibers of Kodaira type I1 over the points 258 A. MALMENDIER

∗ ∗ ∗ 1 u1,u2,...,uNf +2. For special values for m ,...,mNf , several singular fibers of Kodaira type I1 can coalesce and form singular fibers of Kodaira type Ik with k ≥ 2, where the discriminant has a zero of order k. The second chart over the base space is [1 : v] ∈ CP1. The intersection of the two charts is given by u =1/v with v = 0. The Weierstrass coordinates transform accord- 2 3 4 6 ing to x → v x and y → v y; since g2 and g3 are sections of N and N 4 6 respectively, they transform according to g2 → v g2 and g3 → v g3.The 12 discriminant Δ → v Δ becomes a polynomial in v of degree 10 − Nf .From Kodaira’s classification theorem it follows that the singular fiber E∞ over ∞ ∗ u = (v = 0) is a cusp, a singular fiber of Kodaira type I4−Nf . Remark 1. Under the change of the coordinate chart from [u :1]to [1 : v]onCP1 by u = −1/v with v = 0 the holomorphic one-form dz = dx/y transforms as dxu dxv dzu = = − v = − vdzv. yu yv It follows that (dz)⊗2 has the same transformation under a coordinate −1 N 2 ∼ −1 change as (du) whence = ωCP1 where ωCP1 is the canonical bundle on CP1. The elliptic surface Z is a hyper-surface in the variables (u, [X : Y : W ]). Z has surface singular points whenever all partial derivatives in u, x, y simul- taneously vanish. Singular fibers of Kodaira type I1 do not give rise to surface singularities, whereas all singular fibers of Kodaira type In,withn ≥ 2, and ∗ In,withn ≥ 0, do. The monodromy around singular fibers of type In or ∗ In is parabolic [9]. It is known [12, Sec. 4.6] that a Weierstrass fibration 2 is rational (i.e., birational to CP ), if g2 and g3 are polynomials in u of degree at most 4 and 6 respectively. The minimal resolution Z of Z is the blow-up of CP2 in nine points, and therefore has Picard number 10. Con- versely, by contracting every component of the fiber which does not meet S, we obtain back the normal surface Z. The section S and smooth fiber use up two dimensions, and so the number of components of any singular fiber is at most eight since the components are always independent in the Neron-Severi group. However, as we will see below not all configurations of singular fibers exist. For later use we also introduce the following notation: we will denote by UP the base curve CP1 minus a small disc around u = ∞, and the restrition ◦ of the elliptic fibration to UP will be denoted by Z → UP. Similarly, UP will denote the base curve CP1 with small open discs around all points with ◦ singular fibers removed, and the restriction of the elliptic fibration to UP ◦ ◦ will be denoted by Z → UP.

Remark 2. We denote by ωZ the canonical bundle of Z. The space 0 H (ωZ) is the space of global holomorphic two-forms of dimension THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 259

2,0 ∗ −1 pg = h . The bundle ωZ/CP1 = ωZ ⊗ (π ωCP1 ) restricts to the canonical line bundle Ku = ωEu on each smooth fiber Eu. It is well-know that for any rational elliptic surface we have h2,0 = h1,0 =0[12, Thm. 2.10] and the 2 canonical class is minus the fiber class. Hence, we have c1(Z) = 0. It also follows that the first Chern class c1(ωZ/CP1 ) is a pullback from the base 2 manifold and c1(ωZ/CP1 )=0.

The above discussion motivates the following deﬁnition:

Definition 2.1. A Seiberg-Witten curve for Nf hypermultiplets is a Jacobian rational elliptic surface with one singular ﬁber of Kodaira type ∗ ∗ n I4−Nf and singular ﬁbers of Kodaira type I and In only.

Using the explicit Weierstrass parametrization given in [19]for0≤ Nf ≤ 4 it is easy to compose a list of the possible configurations of singular fibers that appear as Seiberg-Witten curves. In Table 1, we list the constraints on the moduli, which substituted into the Weierstrass presentation in [19] realize the configuration of singular fibers, the structure of the ∗ ∗ singular fibers Eun over finite un,andE∞ for the rational elliptic surface 1 Z → CP . Here, r is given by r =8− ν(mν − 1) where the sum in ν runs over all singular fibers of the elliptic surface, and mν denotes the number of irreducible components in the singular fiber.

Table 1

∗ Nf rE∞ singular ﬁbers Eun mass constraints ∗ 44I0 6I1 - ∗ 43I0 I2, 4I1 m3 = m4 =0 ∗ 42I0 I3, 3I1 m2 = m3 = m4 =0 ∗ 42I0 2I2, 2I1 m3 = m4 =0 ∗ 41I0 I4, 2I1 m2 = m3 = m4 =0 ∗ 41I0 3I2 m1 = m2,m3 = m4 =0 ∗ ∗ 40I0 I0 m1 = m2 = m3 = m4 =0 ∗ 33I1 5I1 - ∗ 32I1 I2, 3I1 m2 = m3 ∗ 31I1 I3, 2I1 m1 = m2 = m3 =0 ∗ 31I1 2I2,I1 m1 = m2 =0 ∗ 30I1 I4,I1 m1 = m2 = m3 =0 ∗ 22I2 4I1 - ∗ 21I2 I2, 2I1 m1 = m2 =0 ∗ 20I2 2I2 m1 = m2 =0 ∗ 11I3 3I1 - ∗ 00I4 2I1 - 260 A. MALMENDIER

Remark. The completeness of Table 1 follows by comparing the list with the list of impossible conﬁgurations in [11]. It follows from [14] that among the Seiberg-Witten curves the ones with r = 0 are the only modular elliptic surfaces.

3. The regularized determinant on an elliptic curve We consider an elliptic curve E with periods 2ω and 2ω, modular ω 1 2 parameter τ = ω , and complex coordinate z.Letξ = ξ + iξ be the complex coordinate on the normalized torus with periods 1 and τ such that z ξ = 2 ω .Forn1,n2 ∈ N, a complex function ϕ on the normalized torus with the periodicities ϕ(ξ1 +1,ξ2)=−eπiν1 ϕ(ξ1,ξ2), ϕ(ξ1 +Reτ,ξ2 +Imτ)=−eπiν2 ϕ(ξ1,ξ2), is given by 1 2 1 − ν1 1 ϕn ,n (ξ ,ξ ) = exp 2πi n1 + ξ 1 2 2 1 1 − ν2 1 − ν1 2 + n2 + − Re τ n1 + ξ . Im τ 2 2

In fact, the set of functions {ϕn1,n2 } constitutes a complete system of eigenfunctions for the Laplace operator −4∂ξ∂¯ξ where 2∂¯ξ = ∂ξ1 + i∂ξ2 . Their eigenvalues under 2∂¯ξ are 2π 1 − ν1 1 − ν2 (3.1) n1 + τ − n2 + . Im τ 2 2 ¯ ¯ 1 ¯ ¯ Because of 2∂ =2∂z = ω ∂ξ the functions ϕn1,n2 are also eigenfunctions of 2∂ for the eigenvalues π 1 − ν1 1 − ν2 n1 + τ − n2 + . Im τ ω 2 2 The holomorphic line bundle of positive spinors on an elliptic curve E can also be interpreted as a holomorphic square root K1/2 of the bundle of holomorphic (1, 0)-forms K =Ω1,0(E). The chiral Dirac operators are + ∂/ = ∂¯ : C∞(K1/2) → C∞(K1/2 ⊗ K), − ∂/ = −∂ : C∞(K1/2) → C∞(K1/2 ⊗ K). Equivalently, we can view the situation as follows: we choose the unique even 1/2 1/2 spin structure on E as a reference square root K0 · K0 is the preferred spin bundle for the chosen homology basis, its divisor κ =1/2+τ/2 is the + vector of Riemann constants. If we twist the Dirac operator ∂/ by a ﬂat THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 261

ˇ − ν2 − ν1 holomorphic line bundle W(ν1,ν2) of order two with divisor ξ = 2 2 τ and ν1,ν2 ∈{0, 1}, the twisted chiral Dirac operator becomes

(3.2) + ¯ ∞ 1/2 ⊗ → ∞ 1/2 ⊗ ⊗ ∂/(ν1,ν2) = ∂(ν1,ν2) : C K0 W(ν1,ν2) C K0 W(ν1,ν2) K .

In other words, the twisted chiral Dirac operator is the ∂¯-operator coupled 1/2 ⊗ to the holomorphic line bundle L = K0 W(ν1,ν2). The functions √ + ∈ ∞ 1/2 ⊗ ϕn1,n2 (z)= dz ϕn1,n2 (z) C K0 W(ν1,ν2)

− ¯ form a complete system of eigenfunctions for the operator ( 4∂∂)(ν1,ν2) with the eigenvalues 2 2 π 1 − ν1 1 − ν2 n1 + τ − n2 + . Im τ |ω| 2 2 √ √ Using the Kähler form one can identify dz ⊗ dz¯ with dz¯.Theζ-function (3.3) 1 ζ(ν1,ν2)(s)= 2 2 s 1−ν1 2 1−ν1 1−ν2 n1,n2 n1 + 2 Im τ + n1 + 2 Re τ − n2 + 2 is absolutely convergent for Re s>1. When (ν1,ν2)=(1, 1) it is understood that the summation does not include n1 = n2 = 0. The function ζ(ν1,ν2) is well-defined and has a meromorphic extension to C and 0 is not a pole. The regularized determinant of −4 ∂∂¯ is defined by setting ⎡ ⎤ ⎢ ⎥ − ¯ − 1 ln det( 4∂∂)(ν1,ν2) := ⎣ 2s ζ(ν1,ν2)(s)⎦ . π Im τ | ω|

It follows that 2 π ln det(−4∂∂¯)(ν ,ν ) = −ζ (0) + ln ζ(ν ,ν )(0). 1 2 (ν1,ν2) Im τ |ω| 1 2

It was shown in [17] that ζ(0) = 0 for (ν1,ν2) =(1 , 1), and ζ(0) = −1for (ν1,ν2)=(1, 1). It follows that

2 2 4Im (τ) |ω| 4 (3.4a) det (−4∂∂¯)(1,1) = |η(τ)| , (2π)2 2 2 2π 2 ˇ ϑν1ν2 (τ) − Im ξˇ ϑ ξ τ − ¯ Im τ ( ) (3.4b) det( 4∂∂)(ν1,ν2) = = e , η(τ) η(τ) 262 A. MALMENDIER

where the Dedekind η-function and the Jacobi ϑ-function ϑ(v|τ)=ϑ00(v|τ) are given by ∞ πi τ 2πin τ (3.5a) η(τ)=e 12 1 − e , n=1 a 2 a b (3.5b) ϑab(v|τ)= exp iπ n + τ +2πi n + v + . n∈Z 2 2 2

4. The topological one-loop string amplitude

For each smooth fiber Eu of the fibration Z → UP we have dim 1 H (Eu) = 2. Since a base point is given in each fiber by the section S,wecan choose a symplectic basis {αu,βu} of the homology H1(Eu) with respect to the intersection form, called a homological marking consisting of the A-cycle and B-cycle. We cannot define αu,βu globally over UP. The cycles are transformed by monodromies around the points with singular fibers. However, we can define globally an analytical marking. An analytical marking is a choice of a non-zero one-form on each smooth fiber Eu. We choose the analytical marking that identifies the canonical differential dx/y (where (x, y) are the Weierstrass coordinates in Eq. (2.1) on the fiber Eu) with the holomorphic one-form dz. Given the elliptic surface Z → UP and the analytic marking we associate to it a holomorphic symplectic two-form [8], given by dx (4.1) λ = du ∧ . y Using the two-form λ the period integrals of the elliptic fiber Eu over u with periods 2ωω, 2ω can be written as follows λ =2ω du, λ =2ω du. αu βu Then, there is a globally defined, real closed non-vanishing two-form ◦ form ΩΩonUP (4.2) ΩΩ= λ ∧ λ =8i Im τ |ω|2 du ∧ du.¯ Eu −1 The ∂¯-operator along the fiber Eu = π (u) is the operator 0,0 0,1 (4.3) ∂¯ :Ω (Eu) → Ω (Eu). Its adjoint will be denoted by ∂¯∗. We have the following lemma: Lemma 4.1. The regularized determinant det(∂¯∗∂¯) of the Laplace operator along the fiber is a smooth function on UP given by 2 2 ∗ vol(Eu) 1 (4.4) det (∂¯ ∂¯)= det Δ= Δ 12 (2π)4 where Δ is the modular discriminant of the elliptic fiber Eu. THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 263

Proof. It follows from Eq. (3.4) that 2 2 2 ∗ vol(Eu) η (τ) det (∂¯ ∂¯)= det Δ= . (2π)2 2ω

12 η24(τ) The discriminant Δ of the elliptic curve Eu is given by ΔΔ=(2π) (2ω)12 . 2 Finally, we have vol(Eu)=4Imτ |ω| . We write da = ωωdu and daD = ω du such that ω daD τ = = . ω da This notation should not suggest that da is integrable, i.e., that there is a globally defined function a. On every open set U ⊂ UP, we can integrate and da find holomorphic functions (a, aD) such that on U we have ω = du and ω = daD du .TheKähler metric (4.2) becomes ΩΩ=8i Im τda ∧ da. The following lemma was proved in [8]:

◦ Lemma 4.2. The Levi-Civita connection ∇LC on UP is given by ∂ i ∂ ∂ i ∂ (4.5) ∇LC = − dτ ⊗ , ∇LC = dτ¯ ⊗ . ∂a 2Imτ ∂a ∂¯a 2Imτ ∂¯a The scalar curvature of the Levi-Civita connection is 2 1 ∂τ (4.6) S = . 8Im3τ ∂a

(1,0) (1,0) Proof. Let π ∈ Ω (TCUP) be the projection onto the (1, 0) part of the complexified tangent bundle. π(1,0) is a one-form with values in the ◦ tangent bundle T (UP). The Levi-Civita connection ∇LC on UP is defined by ∂ ∂ ∂ ∂ ∂ ∂ d Ω , =ΩΩ ∇LC , − Ω , ∇LC . ∂a ∂¯a ∂a ∂¯a ∂a ∂¯a It is the unique connection which is compatible with the metric and the complex structure. It follows that the Levi-Civita connection satisfies Ω π(1,0), ∇LCπ(1,0) = 0. Eqns. (4.5) follow. We find LC ∂ i 1 ∂ i ∂ (4.7) d∇ ∇LC = − d ∧ dτ ⊗ − dτ ∧∇LC ∂a 2 Im τ ∂a 2Imτ ∂a 1 ∂ (4.8) = − dτ ∧ dτ¯ ⊗ . 4Im2τ ∂a The Riemannian curvature R of the Kähler metric ΩΩis a Raaa¯ 0 R = a¯ da ∧ da 0 Raa¯ a¯ 264 A. MALMENDIER where 2 a ∇LC LC ∂ 1 ∂τ Raaa¯ = da d ∇ (∂a,∂a) = − . ∂a 4Im2τ ∂a

Pulling down the summation index using the metric ΩΩ=ihaa¯ da ∧ da we ﬁnd 2 a 2 ∂τ Raaa¯ a¯ = haa¯ Raaa¯ = − . Im τ ∂a The scalar curvature is obtained by contracting the summation indices using the inverse K¨ahlermetric Ω. We obtain 2 aa¯ 2 1 ∂τ S = −4 h Raaa¯ a¯ = . 8Im3τ ∂a

We obtain the following proposition:

(1) 1 Proposition 4.3. The function F = − 2 ln det Δ is a smooth func- ◦ tion on UP and satisfies the equation (1) (4.9) ΔUP F = S, where Δ is the Laplace operator along the fiber of Z → UP, S is the scalar ◦ curvature of the Kähler metric Ω on UP,andΔUP is the Laplace-Beltrami operator 1 (4.10) ΔUP = ∂a∂a¯. Im τ Proof. The proof follows from Lemma 4.1 and Lemma 4.2.

Remark. Eq. (4.9) is the anomaly equation of the one-loop topological string amplitude derived in [4], i.e., 2 (1) 1 ∂τ (4.11) ∂a∂a¯ F = . 8Im2τ ∂a

5. The vertical ∂¯-operator on Z → UP In Section 3, we computed the regularized determinant of the ∂¯-operator on an elliptic curve. When the elliptic curve Eu is varied in an elliptic surface, we obtain the determinant line bundle DET ∂¯ → UP of the ∂¯-operator along the ﬁber of Z → UP. The determinant line bundle is the holomorphic line bundle (5.1) ◦ ¯ → ¯ 0,0 C −1 ⊗ 0,1 C DET ∂ UP with ﬁbers DET ∂ u = H (Eu, ) H (Eu, ). THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 265

There is a factorization of the determinant line bundle L = DET ∂ as the tensor product of L and H, corresponding to the non-zero and zero eigenvalues respectively. The line bundle DET ∂¯ can be identified with H since 0,0 ∼ 0,1 ∼ ∗ H (Eu, C) = ker ∂,¯ H (Eu, C) = ker ∂¯ . The bundle L has a holomorphic section det 2∂¯ that determines an iso- ∼ morphism DET ∂¯ = H [2]. The isomorphism does not preserve the metric or connection. Bismut and Freed defined the smooth metric 2 (5.2) .Q := (2π) det Δ .L2 on DET ∂¯ and determined its unitary connection. Since ωZ/UP is equipped with a Hermitian C∞-metric, the Quillen metric is a Hermitian C∞-metric on the holomorphic fibers (DET ∂¯)u. The curvature of this connection is called the local anomaly in physics. It follows:

◦ Lemma 5.1. The determinant line bundle DET ∂¯ → UP is ﬂat. σ =(dz)−1 is a non-vanishing holomorphic section of DET ∂¯ with

1 σQ = |Δ| 12 . σ∗ = dz is a non-vanishing holomorphic section of the dual bundle ◦ ∗ ∗ − 1 (DET ∂¯) → UP with σ Q∗ = |Δ| 12 .

Proof. The ﬂatness follows from the curvature formula of Bismut and Freed ([7]or[2, Prop. 5.14]): 2 c1 ωZ/UP c1 DET ∂¯ = − . Eu 6 2 It follows from Remark 2 that c1(ωZ/UP)=0. 0,1 1,0 ∼ 1,0 H (Eu)andH (Eu) are Serre duals. Thus, we have H = [H (Eu) ⊗ 0,0 −1 2 H (Eu)] . The kernel consists of the constant function φ =1withφ = ∗ vol(Eu). By Serre duality we identify the cokernel ker ∂¯ with the dual of the space of holomorphic one-forms. Thus, the cokernel is spanned by the section −1 2 (dz) and dz =vol(Eu). Using Eq. (3.4) we obtain for the Quillen norm of the section (dz)−1

2 ¯ −1 4 det(−4∂∂)(1,1) 2 (dz) =(2π) = |Δ| 12 . Q φ2 dz2 It is possible to factorize the right hand side holomorphically in τ. We use the Quillen metric to obtain a smooth section σ# of the dual bundle (DET ∂¯)∗ by setting # 1 σ = gQ(σ, •)=|Δ| 6 dz, # 1/12 and σ Q∗ = |Δ| by deﬁnition. The claim follows. 266 A. MALMENDIER

Remark. Under the change of the coordinate chart from [u :1]to[1:v] 1 dxu C − u on P by u = 1/v the holomorphic diﬀerential transforms as dz = yu = dxv − − v v yv = vdz . The Quillen metric is compatible with a change of coordinates since 2 2 1 2 1 2 −1 2 −1 2 12 12 (dzv) = |v| (dzu) = |v| Δu = Δv . Q Q

◦ Lemma 5.2. The Bismut-Freed connection on DET ∂¯ → UP is ﬂat and given by 1 ∂Δ ∇BFσ = ⊗ σ, 12 Δ and ∇BF (0,1) = ∂¯.

Proof. It follows for the Quillen metric in the coordinate chart [u :1]∈ CP1 2 dσQ 1 ∂ lnΔΔ 1 ∂ lnΔ¯ 2 = du + du.¯ σQ 12 ∂u 12 ∂u¯

Although DET ∂¯ is ﬂat and hence the local anomaly vanishes, there is a global anomaly arising from monodromy around the non-contractible closed Nf +2 loops. These are the non-contractible loops (γn)n=1 encircling the nodes at ∗ Nf +2 (un)n=1 clockwise, and γ∞ encircling the cusp at inﬁnity counterclockwise.

Lemma 5.3. There exist constants in R+ such that ∗ 1 ∗ σQ ∼ cn |u − un| 12 (u → un), 10−Nf σQ ∼ c∞ |v| 12 (v → 0).

Proof. The proof follows from Lemma 5.1, Remark 5, and the fact that Δ has a simple zero at each node un and is a polynomial of degree Nf +2 in u. We denote the holonomy of the determinant section of DET ∂¯ on the iπ boundary circle γn around un by exp(− 2 η∂¯[γn]). The following lemma is an immediate consequence of Lemma 5.3: Lemma . − πi 5.4 The holonomy of the section σ on a cycle γi is exp 2 η∂¯[γi] with

1 10 − Nf η ¯[γn] ≡− mod 4,η¯[γ∞] ≡− mod 4. ∂ 3 ∂ 3 THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 267

Since we have restricted ourselves to the case where the fibration Z → UP −1 ∗ has no surface singularities, the singular fibers Eu = π (u)atu = u1,..., ∗ − ∗ → uNf +2 are nodal curves. A current δ(u un)onZ UP is defined by saying that for every differential form α on Z with compact support, the equality ∗ δ(u − un) du ∧ du¯ ∧ α = α Z E ∗ un holds. The first Chern class c1(DET ∂,¯ .Q) is defined as a current of type (1, 1) by 1 2 (5.3) c1 DET ∂,¯ .Q := ∂∂¯log σQ 2πi where σ is a local non-vanishing holomorphic section.

Proposition 5.5. The determinant line bundle with the Quillen metric .Q of the ∂¯-operator along the ﬁber of Z → UP extends to a holomorphic line bundle DET ∂¯ → UP with curvature Nf +2 1 ∗ (5.4) c1 DET ∂¯ = − δ(u − un) du ∧ du.¯ n=1 12

Proof. The definition of the holomorphic determinant line bundle in Eq. (5.1) can be extended across the singular fibers of an elliptic fibration using the results of Knudsen and Mumford [5]. The Quillen metric is smooth by Lemma 4.4. Let f be a function that is differentiable in the disc D with |u| <. Suppose further that f and its derivatives with respect tou ¯ are bounded on the disc. Let T denote the function 1 dw dw¯ (5.5) T f (u, u¯)= f(w, w¯) . 2πi |w|< u − w

It is well-known [15] that the linear operator T is diﬀerentiable and admis- sible on D and satisﬁes ∂¯(T f)=f. In this sense, we write 1 ∂ 1 = δ(u − w). 2πi ∂u¯ u − w Since ∂∂¯ + ∂∂¯ = 0, Eq. (5.4) follows from Eq. (5.3) and the application of Lemma 5.3.

Remark. Corollary 5.5 shows that we can include the nodal fibers of Z → UP when considering the determinant line bundle of the ∂¯-operator along the fiber. The contributions η∂¯[γn] to the global anomaly of the determinant line bundle around the nodal fibers are then viewed as current contributions of type (1, 1) to the first Chern class of the extended determinant line bundle. 268 A. MALMENDIER

The results in [5,Thm.2.1]and[6] describe the more general situation of the ∂¯-operator coupled to a holomorphic vector bundle V → Z. There, the authors prove that c1 DET ∂¯V , .Q = − ch(V ) ∧ Todd(ωZ/UP, .) π (4) (5.6) rank(V ) ∗ − δ(u − un) du ∧ du.¯ n 12

2 For Todd(ωZ/UP, .)=1+c1(ωZ/UP)/2(sincec1(ωZ/UP)=0),andch(V )= 1weobtainEq.(5.4).

6. Extending ∂¯ to nodal curves Let us restrict the fibration Z → UP to a small neighborhood U of a point u∗ whose singular fiber is a nodal curve. We identify any smooth elliptic fiber Eu of the fibration Z → U with the complex plane (with complex coordinate z) modulo the action of the lattice generated by the periods 2ωω, 2ω.For ω τ = ω we set q = exp 2πiτ. After a suitable SL(2, Z) transformation we can assume that as we approach the node for u → u∗ we have Im τ →∞, q → 0. By making the neighborhood U smaller if necessary, we assume that |q| < 1 uniformly in U. Next, we consider the annulus annann(r1,r2)inC (with complex coordinate W ) with inner radius 0

2 |ω| (6.1) g = dz. dz¯ = dW · dW. π|W | The following lemma computes the regularized determinant for the Laplace- Beltrami operator on the annulus with respect to the metric (6.1) and with Dirichlet boundary conditions, i.e., for eigenfunctions vanishing on the outer and inner radius:

Lemma 6.1. The regularized determinant for the Laplace operator Δ= −4∂z∂¯z on the annulus annann(r1,r2) with Dirichlet boundary conditions is THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 269 given by

Im τ 2 (6.2) detD Δ= |η(τ)| . 2π

1 2 Proof. The eigenfunctions ψn1,n2 (ξ ,ξ ) of the Laplace-Beltrami operator with Dirichlet boundary conditions are similar to the eigenfunctions in section 3. In the case (ν1,ν2)=(1, 1) we have 1 2 1 Re τ 2 2πn2 2 ϕn1,n2 (ξ ,ξ ) = exp 2πin1 ξ − ξ cos ξ Im τ Im τ 2πn2 + i sin ξ2 . Im τ

1 1 The Dirichlet boundary conditions are ψn1,n2 (ξ , 0) = ψn1,n2 (ξ , Im τ)=0. Hence, we have n2 > 0and 1 2 1 Re τ 2 2πn2 2 ψn ,n (ξ ,ξ ) = exp 2πin1 ξ − ξ sin ξ . 1 2 Im τ Im τ

Therefore, the zeta-function ζD(s) for the Laplace operator with Dirichlet boundary conditions is 1 1 − 2 ζD(s)= s = ζ(1,1)(s) 2s ζ(2s) , 2 2 2 2 |τ| n1,n2>0 n1Im τ +(n1Re τ − n2) ∞ −2s where ζ(1,1)(s) was deﬁned in Eq. (3.3) and ζ(2s)= n=1 n is the Rie- mann zeta function. Thus, we have 1 1 ζD(0) = ζ(1,1)(0) − 2ζ(0) ,ζD(0) = ζ (0) + 4 ln |τ|ζ(0) − 4ζ (0) . 2 2 (1,1) Eq. (6.2) then follows from ζ(0) = −1/2, ζ(0) = − ln (2π)/2, and Eq. (3.4).

Instead of the Laplacian Δ = −4∂z∂¯z Seeley and Singer [18] use the Laplace-Beltrami operator for the ﬂat metric on the annulus. The ﬂat metric g on the annulus is obtained from the metric g in Eq. (6.1) by a change in the conformal gauge, i.e.,

(6.3) g = dW. dW = e2Φ g with Φ = ln (π |W |/|ω|). It follows that the Laplace-Beltrami operator for the ﬂat metric is given by −2Φ (6.4) Δ=−4 ∂W ∂¯W = e Δ.

We use the results of [21] to derive the following lemma: 270 A. MALMENDIER

Lemma 6.2. The regularized determinant of the Laplace operator Δ on the annulus annann(r1,r2) with Dirichlet boundary conditions is given by Im τ 2 1 (6.5) detD Δ= |η(τ)| |q| 6 . 2π Proof. The Gaussian curvature as well as the geodesic curvature on the boundary vanish for g. Similarly, the Gaussian curvature and the average geodesic curvature on the boundary vanish for g. Then, [21, Eq. (3)] implies L (6.6) detD Δ= detD Δ exp 6π 1 ab g a b where L = 2 ann(r1,r2) vol g (∂ Φ) (∂ Φ). A calculation shows that i dW ∧ dW r2 2 L = | |2 = π ln =2π Im τ 4 ann(r1,r2) W r1 and exp L/6π = |q|(−1/6). We obtain − 1 (6.7) detD Δ= detD Δ |q| 6 . The application of the result of Seeley and Singer [18] yields the following proposition:

Proposition 6.3. In a small neighborhood U ⊂ UP of a point u∗ with ∗ nodal fiber Eu∗ such that q = exp(2πiτ) → 0 as u → u the family of operators {∂¯W,u}u∈U is a continuous family and the operator ∂¯W is well-defined on the singular fiber Eu∗ .

Remark. The limiting Laplace operator of [18] is the Laplace operator −4 ∂W ∂¯W on C (with complex coordinate W ). The eigenfunction for an 2 eigenvalue λ with λ>0 satisfying Dirichlet boundary conditions is Jn(λr) exp(inθ)withW = r exp(iθ)andn ∈ N.Jn(λr) is the√ Bessel function of the ﬁrst kind that is regular at r = 0 and decays as 1/ r for r →∞.

7. The vertical signature operator on Z → UP

Using its complex structure the signature operator on each fiber Eu can be identified with the operator 0,0 1,0 0,1 1,1 D=∂ + ∂1 :Ω(Eu) ⊕ Ω (Eu) → Ω (Eu) ⊕ Ω (Eu). Again, there is a factorization of the determinant line bundle L =DETDas the tensor product of L and H, corresponding to the non-zero and zero eigenvalues respectively [2]. The bundle L has the holomorphic section det D. The fiber of the line bundle H is −1 ∼ 0,0 1,0 0,1 1,1 H = H (Eu) ⊗ H (Eu) ⊗ H (Eu) ⊗ H (Eu) . THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 271

0,0 1,1 The bundles H (Eu)andH (Eu) can be identified by duality. Similarly, 0,1 1,0 H (Eu)andH (Eu) are Serre duals. On each fiber Eu multiplication by ∼ 1,0 −2 dz converts ∂ into ∂1. Thus, we have H = [H (Eu)] and the determinant line bundle of the operator D is isomorphic to the two-fold tensor product of DET ∂. We have the following lemma: Lemma 7.1. −2 (1) σD =(dz) is a non-vanishing holomorphic section of DET D → ◦ UP with 1 σDQ = |Δ| 6 . ∗ 2 ∗ σD =(dz) is a holomorphic section of the dual bundle (DET D) → ◦ ∗ − 1 UP with σDQ∗ = |Δ| 6 . ◦ (2) The flat Bismut-Freed connection on DET D → UP is given by

BF 1 ∂Δ ∇ σD = ⊗ σD, 6 Δ and ∇BF (0,1) = ∂¯. (3) The determinant line bundle with the Quillen metric extends to a holomorphic line bundle DET D → UP. The curvature is a current with Nf +2 1 ∗ c1 (DET D) = − δ(u − un) du ∧ du.¯ n=1 6 Proof. The proof is the same as for Lemma 5.2. ◦ Lemma 7.2. The line bundle (DET D)∗ 6 → UP is canonically trivial. ◦ Proof. The section σ =ΔΔ1/6 (dz)2 of (DET D)∗ → UP satisﬁes ◦ σQ∗ = 1 and is invariant under the action of π1(UP)uptoasixthrootof unity. The trivializing, holomorphic, non-vanishing section of (DET ◦ D)∗⊗6 → UP is σ6. ∼ ∗ It follows from Remark 1 that H = T UP. Consequently, we can obtain ◦ well-deﬁned logarithmic monodromies for the bundle (DET D)∗ → UP. We 0 denote this distinguished choice for the monodromy by ηD[γ] as opposed to ηD[γ] which appeared in Lemma 5.4 and was only determined modulo 4. Lemma 7.3. The logarithmic monodromies of the bundle (DET D)∗ → ◦ UP are 0 2 0 2(10 − Nf ) ηD[γn]=− ,ηD[γ∞]=− . 3 3 272 A. MALMENDIER

∼ ∗ 2 Proof. Under the isomorphism H = T UP the form (dz) is identiﬁed ◦ with du−1. Thus, the bundle (DET D)∗ ⊗ T ∗UP → UP has a standard trivi- ◦ alization on UP given by (dz)2 ⊗du−1. In this trivialization the holomorphic section is Δ1/6. The claim follows.

Lemma 7.4. The signature of the elliptic surface Z → UP is

Nf +2 0 1 0 sign(Z) = ηD[γn] − ηD[γ∞]+2. n=1 2

It follows sign(Z) = −Nf .

Proof. The signature of the rational elliptic surface Z → CP1 in terms of its Chern classes c1,c2 is

2 c − 2c2 sign(Z) = 1 . 3 2 The canonical class is minus the fiber class so that c1 = 0 while c2 is the sum of the exceptional fibers of the fibration whence

Nf +2 2 2 − ∗ − sign(Z) = e(Eun ) e(E∞). 3 n=1 3

◦ ◦ The elliptic surface Z → UP is obtained by cutting out all singular fibers ◦ −∪ ∗ − → Z=Z Eun E∞; the elliptic surface Z UP is obtained by cutting out only the singular fiber at infinity whence Z = Z−E∞. Since the singular ∗ ∗ fibers at u = u1,...,uNf +2 are nodes they do not contribute to the signature. Hence, we have

◦ sign(Z) = sign(Z) = sign(Z) − sign(E∞), where E∞ is the singular ﬁber of Z at inﬁnity. Thus, we obtain

Nf +2 2 2 − ∗ − − sign(Z) = e(Eun ) e(E∞) sign(E∞). 3 n=1 3

∗ The Euler number e(Eun ) is equal to the degree of the zero the discriminant ∗ 2 0 2 − ∗ − assumes at u = un. Therefore, it follows 3 e(Eun )=ηD[γn]and 3 e(E∞)= 0 ηD[γ∞]. By Kodaira’s classiﬁcation result it follows that singularities which are not of type Ik satisfy sign(E∞)=2−e(E∞). Lemma 7.3 yields sign(Z) = −Nf . THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 273

7.1. Regular singularities and the Riemann-Hilbert problem. The definition of the holomorphic determinant line bundle in Eq. (5.1) can be extended across the singular fibers of an elliptic fibration using the results of Knudsen and Mumford [5] to include the higher-rank singularities of Kodaira ∗ type Ik and Ik . In this section, we allow any Jacobian rational elliptic sur- ∗ ∈ C 1 face Z with singular fibers of Kodaira type Ikn over [un :1] P (with 1 ≤ n ≤ K such that kn = Nf + 2) and a singular fiber of Kodaira type ∗ ∞ I4−Nf over u = . The following solution to the Riemann-Hilbert problem on CP1 was given by Röhrl in terms of differential equations with regular singular points [10]:

Fact 7.5. (1) The functor mapping all conjugate classes of one-dimensional rep- ◦ resentations of π1(UP) to the set of isomorphism classes of ﬂat line ◦ bundles over UP is an equivalence of categories. ◦ (2) For a ﬂat line bundle E → UP together with the natural connection ◦ d on UP, there exists a holomorphic line bundle L → CP1 together ∗ with an integrable connection ∇ with regular singular points u1, ∗ u2,..., such that the restriction of L is an isomorphism i : L → E with

d ◦ i | ◦ =(i ⊗ 1) ◦∇| ◦ . UP UP (3) The holomorphic line bundle L admits a global meromorphic section σ,soL is meromorphically trivial and the connection ∇ coincides with a homomorphism deﬁned by a global meromorphic Pfaﬃan system (d − θ)ρ =0 where θ is a global meromorphic one-form on CP1.Sinceθ = dρ/ρ the curvature dθ vanishes. Then, dθ can be prolonged to the whole CP1.

This implies the following result:

−1 Proposition 7.6. On the holomorphic anti-canonical line bundle ωCP1 , there exists an integrable meromorphic connection ∇ with regular singular ◦ ∗ ∗ ∞ −1 points u1,u2,..., such that the restriction of ωCP1 to UP is isomorphic to ◦ the determinant line bundle (DET D)∗ → UP of the signature operator along ◦ the ﬁber of Z → UP. The curvature Ω of ∇ equals i Ω kn ∗ 10 − Nf = δ(u − un) du ∧ du¯ + δ(v) dv ∧ dv¯ 2π n 6 6 whence CP1 i Ω/(2π)=2. 274 A. MALMENDIER

◦ Proof. The determinant line bundle (DET D)∗ → UP of the signature operator along the fiber takes the place of the holomorphic flat line bundle E in Fact 7.5. The bundle E has the meromorphic section σ =ΔΔ1/6 (dz)2 with σQ∗ = 1. Hence, the Bismut-Freed connection acts on σ simply as the exterior derivative d. Outside the set of singular points, the determi- −1 ∈ C 1 nant line bundle is isomorphic to L = ωCP1 . In the chart [u :1] P , the 1/6 −1 isomorphism is given by multiplication with ρu =ΔΔu and identifying du 2 −1 2 with (dzu) such that i(du )=ρu (dz) . The bundle L carries the integrable meromorphic connection dρu ∇du−1 = − ⊗ du−1 ρu ∗ with regular singular points un. In particular, the form θu = dρu/ρu has ∗ simple poles at every regular singular point un and the counterclockwise contour integral evaluates to 1 dρu kn = . ∗ 2πi un ρu 6 Under a change of the coordinate chart from [u :1]to[1:v]onCP1 by u = −1/v the holomorphic differential transforms as dzu = −vdzv. The iso- 1/6 2 morphism is given by multiplication with ρv =ΔΔv = v ρu and identify- −1 2 −1 2 ing dv with (dzv) such that i(dv )=ρv (dzv) . In particular, the form θv = dρv/ρv has simple pole at v = 0 and the counterclockwise contour integral evaluates to 1 dρv 10 − Nf = . 2πi v=0 ρv 6 The connection one-forms θu and θv patch together to give a meromorphic connection on CP1: on the intersection of the two charts we have dv−1 = u2 du−1 and du 2 ρu − dρu ∇dv−1 =2udu⊗ du−1 + u2 ∇du−1 = u ⊗ u2 du−1 (7.1) ρu dρv = − ⊗ dv−1. ρv

It follows that d ◦ i | ◦ =(i ⊗ 1) ◦∇| ◦ . The curvature vanishes on all open UP UP sets. The curvature of the line bundle extended across the singular points is given by ∗ 2 − 1 2 (7.2) Ω=∂∂¯ log σ Q∗ = ∂∂¯ log |Δ 6 | . We obtain K Ω kn ∗ 10 − Nf = − δ(u − un) du ∧ du¯ − δ(v) dv ∧ dv.¯ 2πi n=1 6 6 THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 275

The equality CP1 i Ω/(2π) = 2 then follows from n kn = Nf +2.

Remark. Under the functor of Fact 7.5 the isomorphism class of ◦ −1 ∇ (ωCP1 , ) corresponds to the monodromy representation of π1(UP)onthe ﬂat line bundle DET D.

8. The signature of Z In the case that the elliptic fibration Z → UP has no surface singularities −1 ∗ ∗ u and the singular fibers E = π (u)atu1,...,uNf +2 are nodal curves, the manifold Z is a smooth four-dimensional manifold with boundary ∂Z. The generalization of Hirzebruch’s signature theorem for manifolds with boundary by Atiyah, Patodi, Singer [3] shows that the elliptic signature operator A on Z has an analytic index if one imposes APS boundary conditions on ∂Z. The operator A on Z is of the form ∂ A=σ + D ∂|v| near the boundary, where |v| is the inward normal coordinate, σ a certain bundle isomorphism, and D is the selfadjoint signature operator on the boundary ∂Z[2]. For the boundary circle γ∞ around u = ∞, one obtains a three-dimensional manifold W∞ = ∂Z fibered over a circle. On the boundary component W∞ the selfadjoint signature operator D on the differential forms of even degree is ∞ 2 ∞ 2 D = ∗d − d∗ : C (W∞) ⊕ Ω (W∞) → C (W∞) ⊕ Ω (W∞).

The eigenvalues of the operator D can be positive λj or negative −μj.If we set −s −s −s −s ζ|D|(s)= λj + μj ,ηD(s)= λj − μj , j j j j −2s −2s ζD2 (s)= λj + μj , j j we obtain ζD2 (s)=ζ|D|(2s). The logarithm of the regularized determinant ln det D should equal [20] ⎛ ⎞ d ⎝ −s s −s⎠ − |s=0 λj +(−1) μj . ds j j

Making the choice (−1)s = eiπs we obtain d ζ|D| + ηD iπs ln det D = − |s=0 + e (ζ|D| − ηD) ds 2 iπ = −ζ (0) − ζ|D|(0) − ηD(0) . |D| 2 276 A. MALMENDIER

It follows that ln det |D| = −ζ|D|(0) = −ζD2 (2s)/2. Since D is a self adjoint operator on the odd-dimensional manifold W∞ it follows that ζ|D|(0) = 0. We obtain det D iπ = exp − ηD(0) . det |D| 2 It follows:

Corollary 8.1. The elliptic surface Z → UP satisﬁes

sign (Z) = −ηD (0) = −Nf .

Proof. For the elliptic surface Z the canonical class is minus the ﬁber 2 class. It follows that c1(Z) = 0. The main theorem of [3] when applied to 2 the elliptic surface Z → UP with c1(Z) = 0 yields 2 c1 (Z) sign (Z) = − ηD (0) = −ηD (0) . Z 3

On the other hand, the application of Lemma 7.4 yields sign (Z) = −Nf .

9. Conclusion and outlook We have shown that the Seiberg-Witten family of elliptic curves defines a four-dimensional, Jacobian elliptic surface Z → UP with boundary. The signature of Z is the analytic index of the signature operator on Z if we impose APS boundary conditions on ∂Z. On the other hand, we can compute the index from the logarithmic monodromy of the canonical section of the flat determinant line bundle DET D → UP of the signature operator along the fiber of Z → UP. The signature of Z coincides with the number Nf of hypermultiplets in gauge theory. The identification of the hypermultiplets in the N = 2 supersymmetric low energy SU(2)-Yang-Mills theory with the zero modes of the signature operator on the Jacobian elliptic surface defined by the Seiberg-Witten curve is interesting in the context of string theory. String theory predicts that N = 2 supersymmetric SU(2)-gauge theory in four dimensions emerges from the compactification the type IIB string on a certain K3-fibration X3 → 1 CP . The Calabi-Yau three-fold X3 is determined by the gauge bundle in the heterotic string theory, and in the large base limit becomes C × Z. Thus, we conclude that after the compactification the hypermultiplets must arise from the string fields on the internal manifold which are the zero modes of the signature operator.

Acknowledgments I would like to thank Isadore Singer and David Morrison for many helpful discussions and a lot of encouragement. THE SIGNATURE OF THE SEIBERG-WITTEN SURFACE 277

References [1] L. Alvarez-Gaumé, G. Moore, C. Vafa, “Theta functions, modular invariance, and strings.” Comm. Math. Phys. 106 (1986), no. 1, 1–40. [2] M. F. Atiyah, “The Logarithm of the Dedekind η-Function.” Math. Ann. 278 (1987), 335–380. [3]M.F.Atiyah,V.K.Patodi,I.M.Singer,“SpectralasymmetryandRiemannian geometry. I.” Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. [4] M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes” Comm. Math. Phys. 165 (1994), no. 2, 311–427. [5] J.-M. Bismut, J.-B. Bost, “Fibrés déterminants, métriques de Quillen et dé- générescence des courbes.” C. R. Acad. Sci. Paris. Sér.I Math. 307 (1988), no. 7, 317–320. [6] J.-M. Bismut, J.-B. Bost, “Fibrés déterminants, métriques de Quillen et dé- générescence des courbes.” Acta Math. 165 (1990), no. 1-2, 1–103. [7] J. M. Bismut, D. S. Freed, “The analysis of elliptic families, I, Metrics and connections on determinant bundles.” Comm. Math. Phys. 107 (1986), 103–163. [8] D. Freed, “Special Kählermanifolds.” Comm. Math. Phys. 203 (1999), no. 1, 31–52. [9] K. Kodaira, “On compact complex analyric surfaces. I, II, and III.” Ann. Math. 71(1960), 111-152, 77(1963), 563-626, 78 (1963), 1-40. [10] H. Majima, “Asymptotic analysis for integrable connections with irregular singular points.” Lecture Notes in Mathematics, 1075. Springer-Verlag, Berlin, 1984. [11] R. Miranda, “Persson’s list of singular fibers for a rational elliptic surface.” Math. Z. 205 (1990), 191–211. [12] R. Miranda, “An Overview of Algebraic surfaces.” Algebraic geometry (Ankara, 1995), 157–217, Lecture Notes in Pure and Appl. Math., 193, Dekker, New York, 1997. [13] R. Miranda, U. Persson, “On extremal rational elliptic surfaces.” Math. Z. 193 (1986), 537–558. [14] M. Nori, “On certain elliptic surfaces with maximal Picard number.” Topology 24 (1985), 175–186. [15] H. K. Nickerson, “On the complex form of the Poincaré lemma.” Proc. Amer. Math. Soc. 9 (1958), 183–188. [16] D. Quillen, “Determinants of Cauchy-Riemann operators on Riemann surfaces.” Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96. [17] D. B. Ray, I. M. Singer, “Analytic torsion for complex manifolds.” Ann. of Math. (2) 98 (1973), 154–177. [18] R. Seeley, I. M. Singer, “Extending ∂ to singular Riemann surfaces.” J. Geom. Phys. 5 (1988), no. 1, 121–136. [19] N. Seiberg, E. Witten, “Monopoles, duality and chiral symmetry breaking in N =2 supersymmetric QCD.” Nucl. Phys. B 431 (1994), 484–550. [20] I. M. Singer, “Families of Dirac operators with applications to physics. The mathematical heritage of Elie´ Cartan.” Astérisque1985, Numero Hors Serie, 323–340. [21] W. I. Weisberger, “Conformal Invariants for Determinants of Laplacians on Rie- mann Surfaces.” Comm. Math. Phys. 112 (1987), no. 4, 633–638.

Department of Mathematics, Colby College, Waterville, ME 04901 E-mail address: [email protected]

Surveys in Diﬀerential Geometry XV

Eta Forms and the Odd Pseudodiﬀerential Families Index

Richard Melrose and Fr´ed´eric Rochon

Abstract. Let A(t) be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration φ with base Y. The standard example is A+it where A is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and t ∈ R is the ‘suspending’ parameter. Let πA : A(φ) −→ Y be the infinite-dimensional bundle with fibre at y ∈ Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t)+q(t) invertible for all t ∈ R. The total eta form, ηA, as described here, is an even form on A(φ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: ∗ odd (*) dηA = πAγA, Ch(ind(A)) = [γA] ∈ H (Y ). The 1-form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family A with (*) giving the ‘curving’ as the 3-form part of the Chern character of the index. We also give ‘universal’ versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate ηA with the Bismut-Cheeger eta form.

Introduction Eta forms, starting with the eta invariant itself, appear as the boundary terms in the index formula for Dirac operators [2], [5], [4], [21], [22]. One aim of the present paper is to show that, with the freedom gained by

The research of the ﬁrst author was partially supported by the National Science Foundation under grant DMS-0408993; the second author was supported by a NSERC discovery grant. c 2011 International Press

279 280 R. MELROSE AND F. ROCHON working in the more general context of families of pseudodifferential operators, these forms appear as universal transgression, or connection, forms for the cohomology class of the index. That these forms arise in the treatment of boundary problems corresponds to the fact that boundary conditions amount to the explicit inversion of a suspended (or model) problem on the boundary. The odd index of the boundary family is trivial and the eta form is an explicit trivialization of it in cohomology. To keep the discussion within bounds we work here primarily in the ‘odd’ setting of a family of self-adjoint elliptic pseudodifferential operators, taken to be of order 1, on the fibres of a fibration of compact manifolds

(1) Z M 1 ∗ φ A ∈ Ψ (M/Y ; E),A = A, elliptic Y, where a smooth, positive, fibre density on M and a Hermitian inner product on the bundle have been chosen to define the adjoint. A similar discussion is possible in the more usual ‘even’ case. From the fibration and pseudodifferential family an infinite-dimensional bundle of principal spaces, A(φ), of invertible perturbations on each fibre, can be constructed: −∞ A //1,1 (2) Gsus (φ; EL ) (φ) Ψ9ps9(φ; E) LL ss LLL ss LL pA ss qA LL ss [A+it] L%%ss Y. The vertical map here does not correspond to a principal bundle in the conventional sense since it has a non-constant bundle of structure groups, −∞ Gsus (φ; E), with fibre consisting of the invertible suspended smoothing perturbations of the identity on the corresponding fibre of φ. The individual groups in this bundle are flat pointed loop groups and hence are classifying for even K-theory. Despite the twisting by fibre diffeomorphisms, the −∞ −∞ homotopy group Π0(Gsus (M/Y ; E)), where Gsus (M/Y ; E) is the space of −∞ 0 global smooth sections of Gsus (φ; E), is canonically identified with K (Y ), −∞ see for instance [30]. In this sense Gsus (φ; E) is a ‘classifying bundle’ for the K-theory of Y . Throughout this paper we use notation such as B(φ) for the total space of a bundle over Y associated with a given fibration (1) and B(M/Y )for the corresponding space of global sections of the bundle. Thus, on the right 1,1 in (2), Ψps (φ; E) is the space of product-type suspended pseudodifferential operators on the fibres of φ (and acting on sections of the bundle E)– 1,1 an element of Ψps (φ, E) is thus a family of pseudodifferential operators acting on smooth sections of E on the fibre above a point y ∈ Y where the parameter in the family is in R (the suspension variable t) with ‘product ETA FORMS AND FAMILIES INDEX 281 symbolic’ dependence on this parameter (as indicated by the suffix ps). In the diagram above, A + it is such a family, although we consider a more general situation in the body of the paper. On the total space of the structure bundle in (2) there is a deRham form, Cheven, representing the even Chern character, i.e. which pulls back under any section to a representative of the Chern character of the K-class defined by that section. The eta form, ηA, in this setting is a form on A(φ), defined by regularization of the formula for the Chern character on the structure bundle (see (6.4) and (6.6)). Under the action of a section of the structure bundle, this eta form shifts by the pull back of the Chern character up to an exact term (3) −∞ ∗ ∗ ∗ α : Gsus (M/Y ; E) ×Y A(φ) −→ A (φ),αηA =pr2 ηA +pr1 Cheven +dγ, −∞ γ a smooth form on Gsus (M/Y ; E) ×Y A(φ), −∞ −∞ where pr1 : Gsus (M/Y ; E) ×Y A(φ) −→ Gsus (M/Y ; E)and −∞ pr2 : Gsus (M/Y ; E) ×Y A(φ) −→ A (φ) are the natural projections. The central result below is:

Theorem 1. The eta form, ηA, on A(φ) has basic diﬀerential representing the odd Chern character of the index bundle of the given family A in (1) (4) ∗ ∞ odd odd dηA = pAγA,γA ∈C (Y ;Λ ),dγA =0, Chodd(ind(A)) = [γA] ∈ H (Y ).

Once a choice of connection is made, the form γA, which can be written explicitly in terms of the formal trace of [20] (see (6.13) below), gives a representative of the Chern character of the index class. For the particular case of families of Dirac operators associated to a pseudodiﬀerential bundle, Paycha and Mickelsson, in [28], obtained a related representative of the Chern class using the Wodzicki residue instead of the formal trace. To prove Theorem 1 we use the smooth delooping sequence for the ﬁbra- tion, which is the top row in the diagram

−∞ //−∞ //−∞ (5) Gsus (φ; E) G˜sus (φ; E) G (φ; E) pp77?? ppp ppp ppp p // A(φ) O A(φ) OO WW ind( A) OOO A˜ OOO OOO OO'' Y. 282 R. MELROSE AND F. ROCHON

Here A(φ) is an extension of A(φ) to a bundle of principal spaces (in the −∞ same sense as for A) with bundle of structure groups, G˜sus (φ; E), the half- open (smooth-flat) loop group bundle. This has contractible fibres and hence A(φ) has a section A as indicated in (5). Taking the quotient by the original structure group, this projects to a section, ind( A), of G−∞(φ; E) with homotopy class giving (minus) the index in K1(Y ) of the family. Ultimately, (4) follows from the fact that there is a corresponding multiplicativity formula linking ηA to an analogous form, η. Thus η is a universal transgression form for the delooping sequence, in that it restricts to the Chern character on −∞ Gsus (φ; E)anddη is basic; it is the pull-back of the odd Chern character −∞ Chodd on G (φ; E). In §1 the smooth delooping sequence for K-theory is described. The universal Chern forms on the odd and even classifying spaces are constructed in §2; the regularization to a universal eta form on the half-open loop group is carried out in §3. The constructions of Chern forms is extended to the classifying bundle given by a fibration in §4. The bundle of invertible perturbations for a self-adjoint elliptic family, or more generally an elliptic family of product-type suspended operators, is introduced in §5andin§6 the eta forms are generalized to this case and further extended in §7. The index formula, Theorem 1, is proved in §8. The realization of the exponentiated eta invariant, the τ-invariant, as a determinant is discussed in §9 and the adiabatic determinant of a doubly-suspended family is discussed in §10. This is used to construct a smooth and primitive form of the determinant line bundle over the even classifying space in §11. The K-theory gerbe is realized as a bundle gerbe in the sense of Murray [26]in§12 and the geometric version of this gerbe for an elliptic family is described in §13. Finally, in §14, we discuss the relationship between the eta forms as introduced here and the eta forms of Bismut-Cheeger [3].

1. Delooping sequence We first consider the ‘universal’ case with constructions directly over classifying spaces. Despite the infinite-dimensional base, this setting is a little simpler than the geometric case of a fibration since there is no twisting by diffeomorphisms. Let Z be a compact manifold with dim Z>0and let E −→ Z be a complex vector bundle over it. The algebra of smoothing operators on sections of E is −∞ ∞ 2 Ψ (Z; E)=C (Z ; Hom(E) ⊗ ΩR) ∗ where ΩR = πRΩ is the pull-back of the density bundle by the projection ∗ ∗ πR : Z × Z −→ Z onto the right factor and Hom(E)=πRE ⊗ πLE with πL : Z × Z −→ Z the projection onto the left factor. The product is given by the integral (1.1) (A ◦ B)(z,z)= A(z,z)B(z,z). Z ETA FORMS AND FAMILIES INDEX 283

The topological group (1.2) G−∞(Z; E)= A ∈ Ψ−∞(Z; E); ∃ (Id +A)−1 =Id+B, B ∈ Ψ−∞(Z; E) is an open dense subset and is classifying for odd K-theory. The ‘suspended’ (or flat-pointed loop) group −∞ 2 −∞ (1.3) Gsus (Z; E)= A ∈S(Rτ × Z ; Hom(E) ⊗ ΩR); A(τ) ∈ G (Z; E) is therefore classifying for even K-theory. It is an open (and dense) subspace of the Schwartz functions on R with values in Ψ−∞(Z; E). Thus, for any other manifold X, the sets of equivalence classes of (smooth) maps reducing to the identity outside a compact set under (smooth) homotopy through such maps are the K-groups: K1(X)= f ∈C∞(X; G−∞(Z; E)); f =IdonX \ K, K X / ∼, c 0 ∞ −∞ Kc(X)= f ∈C (X; Gsus (Z; E)); f =IdonX \ K, K X / ∼ . By definition, Schwartz functions are ‘flat at infinity’ and we introduce a larger space of functions which are Schwartz at −∞ but more generally ‘flat to a constant’ at +∞ and the corresponding group −∞ ∞ 2 G˜ (Z; E)= A ∈C (Rτ × Z ; Hom(E) ⊗ ΩR); lim A(τ)=0, sus τ→−∞

dA(τ) 2 ∈S(Rτ × Z ; Hom(E) ⊗ ΩR), dτ (1.4) A(τ) ∈ G−∞(Z; E) ∀ τ ∈ [−∞, ∞] .

Thus A can be recovered from its derivative, τ dA(s) (1.5) A(τ)= ds. −∞ ds Moreover, there is a well-deﬁned map ‘restriction to τ = ∞’, −∞ −∞ (1.6) R∞ : G˜sus (Z; E) −→ G (Z; E) which is surjective since G−∞(Z; E) is connected and a general curve between two points can be smoothed and ﬂattened at the ends. The delooping sequence in the present context is the short exact sequence of groups

−∞ ι //−∞ R∞ //−∞ (1.7) Gsus (Z; E) G˜sus (Z; E) G (Z; E) where the map to the quotient group is explicitly given by (1.6) and the ﬂatness of the paths at +∞ ensures exactness in the middle.

−∞ Lemma 1. The group G˜sus (Z; E) is contractible. 284 R. MELROSE AND F. ROCHON

−∞ Proof. It is only the ‘flatness at infinity’ of the elements of G˜sus (Z; E) that distinguishes this result from the standard contractibility, by shortening the curve, of the pointed path space of a group. To maintain this condition during the contraction, first identify (−∞, ∞) by radial compactification with the interior of [0, 1]. Since the singularities in the compactification are swamped by the rapid vanishing of the derivatives at the end points, this gives the alternative description of the group as −∞ ∞ −∞ da ∞ G˜sus (Z; E)= a ∈C ([0, 1]x; G (Z; E)); ∈ C˙ ([0, 1]; dx (1.8) Ψ−∞(Z; E)),a(0) = 0 , where C˙∞([0, 1]; Ψ−∞(Z; E)) is the space of smooth functions vanishing together with all their derivatives at x =0andx =1. Now, let ρ :[0, 1] −→ [0, 1] be a smooth function with ρ(0) = 0 and ρ(x)=1nearx = 1 and consider the homotopy ⎧ ⎪ 1 ⎨⎪2tρ(x),t∈ 0, , 2 (1.9) ψt(x)= ⎪ 1 ⎩⎪ρ(x)+(2t − 1)(x − ρ(x)),t∈ , 1 , 2 between the constant map, ψ0(x), and the identity map ψ1(x)=x. Note 1 1 that ψt(1) = 1 for 2 ≤ t ≤ 1andψt is flat at 1 for 0 ≤ t ≤ 2 . It follows that ∞ the composite f(ψt(x)) with f ∈C ([0, 1]) is flat at x =1forallt if f is flat at x =1. Thus composition −∞ −∞ (1.10) Ψt : G˜sus (Z; E) a −→ a ◦ ψt ∈ G˜sus (Z; E) gives the desired deformation retraction to the identity element. This argument is not limited to this particular group and holds in greater generality.

2. Universal Chern forms The group G−∞(Z; E), identified as an open dense subset of Ψ−∞(Z; E), is an infinite dimensional manifold modelled on the Fréchet space C∞(Z2; −∞ Hom(E) ⊗ ΩR). We shall fix the space of smooth functions on G (Z; E) and more generally the smooth sections of form bundles and other tensor bundles. First, it is natural to identify the tangent space at any point with the linear space in which the group is embedded. Then the cotangent space can be −∞ 2 identified with its dual, C (Z ; Hom(E )⊗ΩL), the space of distributional sections, where ΩL is the left density bundle. Thus ∗ −∞ −∞ 2 (2.1) Ta G (Z; E)=C (Z ; Hom(E ) ⊗ ΩL) ETA FORMS AND FAMILIES INDEX 285 with the duality between smooth tangent and cotangent fibres given by distributional pairing. This can be written formally as bundle pairing followed by integration

∗ −∞ −∞ T G (Z; E) × TaG (Z; E) (α, B) −→ α · B ∈ C, a (2.2) α · B = α(z,z)B(z,z). Z2

Having defined the tangent and cotangent fibres at each point, the fibres of the cotensor bundles are interpreted as completed tensor products. Thus ⎛ ⎞ ∗ ⊗k −∞ ⎝ 2k ∗ ⎠ (2.3) (T )a = C Z ; πj Hom(E ) ⊗ ΩkL j where ΩkL is the tensor product of the (trivial) real line bundles on each left factor of all the pairs and the homomorphism bundle is lifted from each pair of factors. Since G−∞(Z; E) is a metric space with the topology induced from Ψ−∞(Z; E), continuity for functions is immediately defined. More generally, continuity for sections of any of the tensor bundles is defined by insisting that a k-cotensor field should be a continuous map from the metric space G−∞(Z; E) (or indeed any subset of it) into the distributional space (2.3) in the strong sense that it should map locally into some fixed Sobolev, hence Hilbert, space ⎛ ⎞ −N ⎝ 2k ∗ ⎠ H Z ; πj Hom(E ) ⊗ ΩkL j and should be continuous for the metric topologies. The meaning of direc- tional derivatives is then clear. For a map to be C1, we insist that all direc- tional derivatives exist at each point, that they are jointly defined by an element of the next higher tensor space, i.e. distribution in two more variables, and that the resulting section of this tensor bundle is also continuous. Then infinite differentiability is defined by iteration. The form bundles are defined, as usual, as the totally antisymmetric parts of the corresponding cotensor bundles. Smoothness as a form is smoothness as a cotensor field. The deRham differential is the map from smooth k-forms to smooth (k + 1)-forms given in the usual way by differentiation followed by antisymmetrization. If F : G−∞(Z; E) −→ C is smooth and b ∈ G−∞(Z; E) then L(b)∗F (a)= F (ba) is also smooth, as is R(b)∗F defined by R(b)∗F (a)=F (ab−1). Thus G−∞(Z; E) acts on its space of smooth functions, as in the setting of finite dimensional Lie groups. These actions extend to cotensor fields and hence to forms. 286 R. MELROSE AND F. ROCHON

The universal odd Chern character is given by a slight reinterpretation of the standard finite-dimensional formula 1 −1 2 1 −1 t(1 − t)(a da) (2.4) Chodd(a)= Tr a da exp dt 2πi 0 2πi ∈C∞(G−∞(Z; E); Λodd). Namely expanding out the exponential in formal power series and carrying out the resulting integrals reduces this to an infinite sum ∞ −1 2k+1 1 k! (2.5) Chodd(a)= ck Tr (a da) ,ck = k+1 . k=0 (2πi) (2k +1)! Here, each da is the identification of the tangent space at a with Ψ−∞(Z; E) – so can be thought of as the differential of the identity. Thus, for any 2k +1 −∞ ⊗(2k+1) elements bj ∈ Ψ (Z; E), the evaluation on (Ta) of an individual term is −1 2k+1 (2.6) Tr (a da) (b1,...,b2k+1) sign(σ) −1 −1 −1 = (−1) Tr(a bσ(1)a bσ(2) ...a bσ(2k+1)). σ The trace is well defined since the product is an element of Ψ−∞(Z; E). It is also defined at each point by a distribution, as required above, and the same is true of all derivatives. Namely at each point the distribution defining this form is just the total antisymmetrization (of variables in pairs) of (2.7) A(z2k+1,z1)A(z1,z2)A(z2,z3) ...A(z2k,z2k+1) where A is the Schwartz kernel of a−1. Note that while this is smooth in the sense described above, the kernel representing the form at a given point is not a smooth function because of the presence of the identity factors in the operators. Due to the identity d dat (2.8) a−1 = −a−1 a−1 dt t dt differentiation gives a similar form, but with less symmetrization, with respect to parameters. Thus (2.6) defines a form in each odd degree. As a result of antisymmetrization the form corresponding to (2.6) for an even power is identically zero. Moreover the computation of the deRham differential, based on the identities (2.8), d2a =0andd(a−1daa−1) = 0 yields −1 2k+1 −1 2k+2 (2.9) d Tr (a da) = − Tr (a da) =0=⇒ d Chodd =0 globally on G−∞(Z; E). The Chern character (2.4) is universal in the sense that if f : X −→ G−∞(Z; E) is any smooth map from a compact manifold X, then ∗ odd (2.10) [f Chodd]=Chodd([f]) ∈ H (X; C) ETA FORMS AND FAMILIES INDEX 287 represents the odd Chern character of the K-class defined by the homotopy class [f]off. The abelian group structure on K1(X) is derived from the non-abelian group structure on G−∞(Z; E) and in particular the linearity of the odd Chern character is a consequence of the following result. Here we say that a form on a product of two (infinite-dimensional) manifolds M1 × M2 ‘has no pure terms’ if it vanishes when restricted to {p1}×M2 or M1 ×{p2} for any points p1 ∈ M1 or p2 ∈ M2.

−∞ −∞ Proposition 1. There is a smooth form δeven on G (Z; E) × G (Z; E) of even degree which has no pure terms, vanishes when pulled back to the ‘product diagonal’ {(a, a−1)} and is such that in terms of pull-back under the product map and two projections:

−∞ (2.11) G (OZO; E) m

G−∞(Z; E) × G−∞(Z; E) jjj TTTT jjjj TTTT jjjj TTTT ujjujj πL πR TTT)) G−∞(Z; E) G−∞(Z; E) the form in (2.4) satisﬁes

∗ ∗ ∗ (2.12) m Chodd = πL Chodd + πR Chodd + dδeven.

Proof. For any two bundles the group G−∞(Z; E) ⊕ G−∞(Z; F )can be identiﬁed as the diagonal subgroup of G−∞(Z; E ⊕ F ) and the Chern form restricted to this subgroup clearly splits as the direct sum. So, to prove (2.12) we work on E ⊕ E and take a homotopy which connects ab acting on the left factor of E, so as ab ⊕ Id on E ⊕ E, with a ⊕ b acting on E ⊕ E. This can be constructed in terms of a rotation between the two factors. Thus cos t sin t (2.13) M(t)= ,t∈ [0,π/2] − sin t cos t is such that −1 b 0 b 0 (2.14) B(t)=M (t) M(t) satisﬁes B(0) = , 0Id 0Id Id 0 B(π/2) = . 0 b

Using this family, consider the map

(2.15) H :[0, 1]×G−∞(Z; E)×G−∞(Z; E) −→ A(0)B(t) ∈ G−∞(Z; E⊕E). 288 R. MELROSE AND F. ROCHON

∗ It follows that the form α = H Chodd is a closed form on the product and hence decomposing in terms of the factor [0, 1],

(2.16) α = dt ∧ α1(t)+α2(t) −∞ where the αi are smooth 1-parameter families of forms on G (Z; E) × G−∞(Z; E), ∂ (2.17) dα2 =0,dα1 = α2 ∂t where d is now the deRham diﬀerential on G−∞(Z; E) × G−∞(Z; E). Thus, setting π/2 (2.18) δeven = − α1(t)dt, 0 (2.12) follows. ∗ Now, if a is held constant, H Chodd is independent of a and reduces to the Chern character for B(t). It follows that the individual terms in α1 are multiples of Id 0 ∂M(t) −1 −1 2k (2.19) Tr M (t)((db)b ) 00 ∂t Id 0 −1 ∂M(t) −1 2k − M (t) (b db) . 00 ∂t

∂M(t) −1 −1 ∂M(t) Since ∂t M (t)andM (t) ∂t are off-diagonal this vanishes. A similar argument applies if b is held constant, so δeven in (2.18) is without pure terms. −1 Under inversion, a −→ a , Chodd simply changes sign, so under the involution I :(a, b) −→ (b−1,a−1) both the left side and the two Chern terms, together, on the right change sign. Thus δ = δeven can be replaced by its odd 1 ∗ part under this involution, 2 (δ − I δ), which ensures that it vanishes when pulled back to the submanifold left invariant by I, namely {b = a−1}. It still is without pure terms so the proposition is proved. −∞ The discussion of the suspended group Gsus (Z; E) is similar. Namely the tangent space is the space of Schwartz sections S(R × Z2; Hom(E) ⊗ ΩR) which can be identified, by radial compactification of the line, with ∞ 2 ∞ 2 C˙ ([−1, 1]×Z ; Hom(E)⊗ΩR) ⊂C ([−1, 1]×Z ; Hom(E)⊗ΩR), consisting of the space of smooth sections on this manifold with boundary, vanishing to infinite order at both boundaries. The dual space is then the space of 2 Schwartz distributions S (R × Z ; Hom(E ) ⊗ ΩL), or in the compactified picture the space of extendible distributional sections. Apart from these minor alterations, the discussion proceeds as before and the even Chern forms, defined by pull-back and integration are ∗ ∞ −∞ even (2.20) Cheven = p∗(ev Chodd) ∈C (Gsus (Z; E); Λ ), ETA FORMS AND FAMILIES INDEX 289 where −∞ −∞ (2.21) ev : R × Gsus (Z; E) (s, A) −→ A(s) ∈ G (Z; E) is the evaluation map and p∗ is the pushforward map along the fibres of −∞ −∞ the projection p : R × Gsus (Z; E) −→ Gsus (Z; E) on the right factor. So, at least formally, 1 −1 2 1 −1 t(1 − t)(a da) (2.22) Cheven = Tr (a da) exp dt, 2πi R 0 2πi where the outer integral mean integration with respect to τ of the coefficient of dτ. The analogue of Proposition 1 for the even Chern character follows from that result. Namely if we consider the corresponding product map, pointwise in the parameter, and projections: −∞ (2.23) Gsus (OZO; E) m

−∞ −∞ Gsus (Z; E) × Gsus (Z; E) jjj TTTT jjjj TTTT jjjj TTTT ujjujj πL πR TTT)) −∞ −∞ Gsus (Z; E) Gsus (Z; E) then there is a smooth form δodd on the product group such that ∗ ∗ ∗ (2.24) m Cheven = πL Cheven +πR Cheven +dδodd.

This odd form can be constructed from δeven using the pull-back and pushforward operations for the (product) evaluation map −∞ −∞ (2.25) Ev : R × Gsus (Z; E) × Gsus (Z; E) (τ,a,b) −→ (a(τ),b(τ)) ∈ G−∞(Z; E) × G−∞(Z; E) as ∗ (2.26) δodd = − δ (τ)dτ, Ev δeven = dτ ∧ δ (τ)+δ (τ). R Since the forms are Schwartz in the evaluation parameter, the additional term ∂ (2.27) δ(τ)=0 R ∂τ and (2.24) follows; note that it does not follow from the fact that δeven has no pure terms that this is true of δodd – and it is not! Smoothness of forms in the sense discussed above certainly implies that the pull-back of such a form to a ﬁnite dimensional manifold, under a smooth map Y −→ G−∞(Z; E) is smooth on Y and closed if the form on G−∞(Z; E) is closed. Thus if f : Y −→ G−∞(Z; E) is a representative of [f] ∈ K1(Y ) then 290 R. MELROSE AND F. ROCHON

∗ f Chodd is a sum of closed odd-degree forms on Y. The cohomology class is constant under homotopy of the map. Indeed, a homotopy between f0 and −∞ f1 is a map F :[0, 1]r × Y −→ G (Z; E). The fact that the Chern form ∗ pulls back to be closed shows that F Chodd is of the form dA(r) A(r)+dr ∧ B(r),dY A(r)=0, = dY B(r) dr 1 (2.28) =⇒ A(1) − A(0) = d B(r)dr. 0 Thus cohomology classes in the even case are also homotopy invariant and these universal Chern forms deﬁne a map from K-theory to cohomology. This is the Chern character. The theorem of Atiyah and Hirzebruch shows that the combined even and odd Chern characters give a multiplicative isomorphism Ch : K0(X) ⊕ K1(X) ⊗ C −→ H*(X; C).

3. Universal eta form As a link between the odd and even universal Chern characters deﬁned above on the end groups in (1.7), we consider the corresponding eta form on −∞ G˜sus (Z; E). It has the same formal deﬁnition as the even Chern character but now lifted to the larger (and contractible) group. This consists of paths in G−∞(Z; E) so there is still an evaluation map −∞ −∞ (3.1) Ev : R × G˜sus (Z; E) (s, A) −→ A(s) ∈ G (Z; E) just as in (2.21).

−∞ Definition 1. The universal eta form on G˜sus (Z; E)isdeﬁnedasin −∞ (2.20) but interpreted on the group Gsus (Z; E) with the evaluation map (3.1) ∗ ∞ −∞ even (3.2)η ˜ =˜p∗ Ev Chodd ∈C (Gsus (Z; E); Λ ) and withp ˜∗ the push-forward map corresponding to integration along the −∞ −∞ ﬁbres of the projectionp ˜ : R × Gsus (Z; E) −→ Gsus (Z; E).

Integration on the fibres here is well defined since, in the integrand – which is the contraction with ∂/∂τ – necessarily one of the terms is differentiated with respect to the suspension parameter, which has the effect of removing the constant term at infinity. Thus the integral in (3.2) still converges rapidly. If X is a compact smooth manifold and if a : X −→ G−∞(Z; E)isa smooth map, the associated eta form is (3.3) η(a)=a∗η.˜ ETA FORMS AND FAMILIES INDEX 291

Now, consider the diagram analogous to (2.23) but for the extended group, and hence with an additional map corresponding to restriction to t = ∞ in each factor:

˜−∞ (3.4) Gsus (OZO; E) m

−∞ −∞ G˜sus (Z; E) × G˜sus (Z; E) kkk TTTT kkkk TTTT kkkk TTTT kukukk πL πR TTT)) −∞ R ×R −∞ G˜sus (Z; E) ∞ ∞ G˜sus (Z; E)

G−∞(Z; E) × G−∞(Z; E).

−∞ Proposition 2. The eta form in (3.2) restricts to Cheven on Gsus (Z; E), satisﬁes the identity

∗ ∗ ∗ ∗ (3.5) m η˜ = πLη˜ + πRη˜ + d(δ˜odd)+(R∞ × R∞) δeven

−∞ −∞ where δ˜odd is a smooth form on G˜sus (Z; E) × G˜sus (Z; E) which restricts to −∞ −∞ δodd on Gsus (Z; E) × Gsus (Z; E) and moreover η˜ has basic diﬀerential

∗ (3.6) dη˜ = R∞ Chodd where R∞ is the quotient map in (1.7).

Proof. To compute the diﬀerential of the eta form, write the pull-back under Ev as in (2.28):

∗ dA(τ) (3.7) Ev Chodd = A(τ)+dτ ∧ B(τ)=⇒ dB(τ)= . dτ

Since dA(τ) ∗ η˜ = B(τ),dη˜ = dB(τ)dτ = dτ = A(∞)=R∞ Chodd . R R R dτ

This proves (3.6). Similarly, as in the proof of (2.24), pulling back the corresponding addi- tivity formula, (2.12), for the odd Chern character gives (3.5) with the additional term arising from the integral which vanishes as in (2.27) on the suspended subgroup. 292 R. MELROSE AND F. ROCHON

4. Geometric Chern forms Next we pass to a discussion of the ‘geometric case’. Fix a connection on the fibration (1). That is, choose a smooth splitting (4.1) TM = T H M ⊕ T (M/Y ) where the subbundle T H M is necessarily isomorphic to φ∗TY. Also choose a connection ∇E on the complex vector bundle E −→ M. Consider the infinite- dimensional bundle (4.2) C∞(φ; E) −→ Y ∞ −1 which has fibre C (Zy; Ey),Zy = φ (y),Ey = E at y ∈ Y and space of Zy smooth global sections written C∞(M/Y ; E), which is canonically identified with C∞(M; E). The choice of connections induces a connection on C∞(φ; E) through the covariant differential ∇φ,E ∇E C∞ ∈C∞ (4.3) X u = XH u,˜ (M/Y ; E) u =˜u (M; E), ∞ where XH is the horizontal lift of X ∈C (Y ; TY). The curvature of this connection is a 2-form on the base with values in the first-order differential operators on sections of E on the fibres φ,E 2 2 1 (4.4) ω =(∇ ) ∈ Λ Y ⊗C∞(Y ) Diff (M/Y ; E). This covariant differential can be extended to the bundle Ψm(φ; E), for m each m including m = −∞, which has fibre Ψ (Zy,Ey)aty, and space of global smooth sections Ψm(M/Y ; E) through its action on C∞(M; E): (4.5) ∇φ,EQ =[∇φ,E,Q],Q∈ Ψm(M/Y ; E). The curvature of the induced connection is given by the commutator action of the curvature (4.6) (∇φ,E)2 =[ω, ·]. Let π : G−∞(φ; E) −→ Y be the infinite-dimensional bundle over Y with fibre (4.7) −∞ −∞ G (Zy; Ey)= IdEy +Q; Q ∈ Ψ (Zy; Ey), IdEy +Q is invertible . This is naturally identified with an open subbundle of Ψ−∞(φ; E) ⊂ Ψm (φ; E) and as such has an induced covariant differential. If σ ∈ G−∞ (M/Y ; E) is a global section, the corresponding odd Chern character is

(4.8) 1 φ,E φ,E 1 −1 φ,E w(s, σ, ∇ ) Chodd(σ, ∇ )= Tr (σ ∇ σ) exp ds , 2πi 0 2πi where w(s, σ, ∇)=s(1 − s)(σ−1∇σ)(σ−1∇σ)+(s − 1)ω − sσ−1ωσ. ETA FORMS AND FAMILIES INDEX 293

Even though the curvature ω from (4.4) is not of trace class, the term σ−1∇φ,Eσ is a 1-form with values in smoothing operators, the identity being annihilated by the covariant diﬀerential, so the argument of Tr is a smoothing operator. The form in (4.8) is the pull-back under the section σ of a ‘universal’ odd Chern character on the total space of the bundle. To see this, ﬁrst pull the bundle back to its own total space

(4.9) π∗G−∞(φ; E) −→ G−∞(φ; E).

This has a tautological section

(4.10) a : G−∞(φ; E) −→ π∗G−∞(φ; E) and carries the pulled back covariant diﬀerential ∇˜ φ,E = π∗∇φ,E. The geometric odd Chern character on G−∞(φ; E)is (4.11) 1 φ,E φ,E 1 −1 φ,E w(s, a, ∇˜ ) Chodd(∇˜ )= Tr a ∇˜ a exp ds , where 2πi 0 2πi w(s, a, ∇˜ φ,E)=s(1 − s)(a−1∇˜ φ,Ea)(a−1∇˜ φ,Ea)+(s − 1)˜ω − sa−1ωa˜ ; hereω ˜ = π∗ω is the pull-back of the curvature. This clearly has the desired universal property for smooth sections:

φ,E ∗ φ,E (4.12) Chodd(σ, ∇ )=σ Chodd(∇˜ ).

The basic properties of the geometric Chern character are well known and discussed, for example, in [1]. In particular of course, the forms are closed. This follows from identities for the forms w = w(s, a, ∇˜ φ,E)andθ = a−1∇˜ φ,Ea in (4.11) which will be used below. Namely the Bianchi identity for the connection implies (cf. (3.5) in [1]) that

∇˜ φ,Ew = s[w, θ] and hence w dw w w ∇˜ φ,E θ exp = − exp − s θ exp ,θ , 2πi ds 2πi 2πi 1 d w (1−r)w dw rw = −2πi exp + e 2πi , e 2πi dr ds 2πi 0 ds w (4.13) − s θ exp ,θ . 2πi 294 R. MELROSE AND F. ROCHON

All the commutators have vanishing trace so 1 φ,E 1 φ,E −1 φ,E w(s, a, ∇˜ ) (4.14) d Chodd = Tr ∇˜ a ∇˜ a exp ds 2πi 0 2πi 1 d w(s, a, ∇˜ φ,E) = − Tr exp ds =0 0 ds 2πi w(0) w(1) since Tr e 2πi − e 2πi =0.

Lemma 2. Under the inversion map I : G−∞(φ, E) σ −→ σ−1 ∈ G−∞ ∗ (φ; E) the Chern character pulls back to its negative I Chodd = − Chodd .

Proof. This follows directly from (4.8) since (4.15) w(s, σ−1, ∇)=σw(1 − s, σ, ∇)σ−1,σ∇φ,Eσ−1 = −(∇φ,Eσ)σ−1 and the conjugation invariance of the trace.

Furthermore, the cohomology class deﬁned by Chodd is additive, in the sense that

(4.16) Chodd(σ1σ2)=Chodd(σ1)+Chodd(σ2)+dF. Again it is useful to give a universal version of such a multiplicativity formula. Let (G−∞(φ, E))[2] be the ﬁbre product of G−∞(φ, E) with itself as a bundle over Y . Then there are the usual three maps −∞ (4.17) G (Oφ,O E) m

−∞ [2] (G (φ, E))P nnn PPP πSnnn PPπPF nnn PPP nvnvn PP(( G−∞(φ, E) G−∞(φ, E), where πF (a, b)=b, πS(a, b)=a and m(a, b)=ab.

Proposition 3. Thereisasmoothformμ on (G−∞(φ, E))[2] such that ∗ ∗ ∗ (4.18) m Chodd = πF Chodd +πS Chodd +dμ.

Proof. This follows from essentially the same argument as used in the proof of Proposition 1. Thus, consider the bundle of groups with E replaced by E ⊕ E in which the original is embedded as acting on the ﬁrst copy and as the identity on the second copy. As in (2.15), this action on E ⊕ E is homotopic under a family of 2 × 2 absolute rotations, i.e. not depending on the space variables, to the action on the second copy with the identity ETA FORMS AND FAMILIES INDEX 295

∗ on the ﬁrst. Now, m Chodd is realized through the product, i.e. diagonal action on the ﬁrst factor. Applying the rotations but just in the second term of this product action, the map m is homotopic to πS ⊕ πF . Since the Chern character is closed, the same argument as in Proposition 1 constructs the transgression form μ as the integral of the variation along the homotopy.

5. Odd elliptic families Now we turn to the consideration of a given family of self-adjoint elliptic pseudodifferential operators, A ∈ Ψ1(M/Y ; E). In fact it is not self- adjointness that we need here, but rather the consequence that A+it, which 1,1 is a product-type family in the space Ψps (M/Y ; E), should be fully elliptic and hence invertible for large real t, |t| >T.See [25]and[24] for a discussion of product-suspended pseudodifferential operators. More generally we may simply start with an elliptic family in this sense, possibly of different order, m,l A ∈ Ψps (M/Y ; E). It follows from the assumed full ellipticity that for each value of the parameter y ∈ Y the set of invertible perturbations −∞ Ay = A + it + q(t); q ∈ Ψ (Zy; Ey), sus −1 −1 (A + it + q(t)) ∈ Ψ (Zy; Ey) ∀ t ∈ R or (5.1) −∞ Ay = A(t)+q(t); q ∈ Ψ (Zy; Ey), sus −1 −m (A(t)+q(t)) ∈ Ψ (Zy; Ey) ∀ t ∈ R is non-empty. This is discussed in the proof below.

Proposition 4. If A ∈ Ψ1(M/Y ; E) is an elliptic and self-adjoint fam- m,l ily or A ∈ Ψps (M/Y ; E) is a fully elliptic product-type family then (5.1) m,l defines a smooth (infinite dimensional) Fréchet subbundle A(φ) ⊂ Ψps (φ, E) (where m = l =1in the standard case) over Y with fibres which are principal −∞ spaces for the action of the bundle of groups Gsus (φ; E).

Proof. The non-emptiness of the fibre at any point follows from standard results for the even index. Namely at each point in the base, the family Ay(t) is elliptic and invertible for large |t| as a consequence of the assumed 0 full ellipticity. Thus the index of this family is an element of Kc(R)and hence vanishes. For such a family there is a compactly supported, in the parameter, family q(t) of smoothing operators on the fibre which is such that A(t)+q(t) is invertible for all t ∈ R. −∞ Now the fact that the fibre is a principal space for the group Gsus (Zy; Ey) follows directly, since for two such perturbations qi,i=1, 2, −∞ (5.2) A(t)+q1(t)=(IdEy +q12(t))(A(t)+q2(t)),q12 ∈ Ψsus (Zy; Ey) and conversely. The local triviality of this bundle follows from the fact that invertibility persists under small perturbations. 296 R. MELROSE AND F. ROCHON

It is this bundle, A(φ), which we think of as the index bundle since the existence of a global section is equivalent to the vanishing, in odd K-theory, of the index of the original family. Note that since we permit the orders m m,l and l of a fully elliptic family in Ψps (φ, E) to take values other than 1, the index is additive in the sense that the index bundles of two elliptic families (acting on the same bundle) compose in the obvious way.

6. Eta forms for an odd family To define the geometric eta form, recall that it is shown above that a covariant derivative is induced on the bundle Ψm(φ; E) −→ Y from the connection on φ and the connection on E. Consider the subbundle of elliptic and invertible pseudodifferential operators, Gm(φ; E). Since product-suspended operators can be seen as one-parameter families of pseudodifferential operators, there is an evaluation map m (6.1) ev : Rτ ×A(φ) (τ,a) −→ a(τ) ∈ G (φ; E) compatible with the bundle structure. On Gm(φ; E), consider as before the tautological bundle (6.2) π∗Gm(φ; E) −→ Gm(φ; E) obtained by pulling back the bundle to its own total space. With a : Gm (φ; E) −→ π∗Gm(φ; E) the tautological section consider the odd form (6.3) 1 1 s(1 − s)(a−1∇˜ a)(a−1∇˜ a)+(s − 1)˜ω − sa−1ωa˜ λ = a−1∇˜ a exp ds 2πi 0 2πi taking values in π∗Ψ0(φ; E). Here ∇˜ = π∗∇ andω ˜ = π∗ω with ω defined in (4.4). This is formally the same as the argument of the trace functional in (4.11), except of course that now the section a is no longer a perturbation of the identity by a smoothing operator, but an invertible operator of order m. Nevertheless, the identities in (4.13) still hold, since they are based on the Bianchi identity. Writing the pull-back under the evaluation map as ∗ t n n (6.4) ev (λ)=λ + λ ∧ dτ, λ = ι∂τλ both the tangential and normal parts are form-valued sections of the bundle ∗ 0,0 (6.5) πAΨps (φ; E) −→ A 0,0 obtained by pulling back π :Ψps (φ; E) −→ Y to A.

Definition 2. On the total space of the bundle A the (even) geometric eta form is n (6.6) ηA =Trsus(λ ) ETA FORMS AND FAMILIES INDEX 297

where Trsus is the regularized trace of [20] taken ﬁbrewise in the ﬁbres of the bundle (6.5).

The bundle (6.5) does not have a tautological section, but

∗ m,l (6.7) πAGps (φ; E) −→ A

m,l m,l does, where Gps (φ; E) ⊂ Ψps (φ; E) is the subbundle of elliptic invertible elements, with product-type pseudodiﬀerential inverses. Denote this section ∗ m,l by αA : A−→πAGps (φ; E). Then consider also the odd form 1 1 −1 γ˜A = Tr αA ∇ˆ αA 2πi 0 −1 2 −1 s(1 − s)(α ∇ˆ αA) +(s − 1)ˆω − sα ωαˆ A (6.8) × exp A A ds 2πi

∗ ∗ where Tr is the formal trace from [20]and∇ˆ = πA∇,ˆω = πAω.

Proposition 5. For an odd elliptic family of ﬁrst order (so either self- adjoint or directly of product type), the exterior derivative of the geometric eta form ∗ (6.9) dηA =˜γA = πAγA

∞ odd is the pull-back of a closed form on the base γA ∈C (Y ;Λ ).

Proof. Recall ﬁrst that the regularized trace is deﬁned in [20] by taking the constant term in the asymptotic expansion of τ τp τ1 p ··· ∂ t ··· (6.10) Tr p λ (r) drdτ1 dτp −τ 0 0 ∂r

∂p t as τ → +∞. Here p ∈ N is chosen large enough so that ∂τp λ (τ) is of trace class – the product-suspended property implies that high τ derivatives are of correspondingly low order in both senses. This is a trace, i.e. vanishes on ∂A commutators, but is not exact in the sense that Tr(A)=Trsus( ∂τ )doesnot necessarily vanish, but is determined by the asymptotic expansions of A(τ) as ±τ →∞since it does vanish for smoothing Schwartz perturbations of A. As noted above, the identities (4.13) hold for λ. For the pull-back under the evaluation map this means that modulo commutators ∂ (6.11) ∇˜ λn ≡ λt. ∂τ Now, taking p further derivatives with respect to τ gives the same identity modulo commutators where the sum of the orders of the terms becomes low 298 R. MELROSE AND F. ROCHON as p increase. So, applying the trace functional the commutator terms vanish and it follows that ∂p ∂p ∂ (6.12) Tr ∇˜ λn(τ) ∧ dτ =Tr λt(r) ∧ dτ. ∂τp ∂τp ∂τ Therefore, n ∂ t dηA =Trsus(∇˜ λ )=Trsus λ ∂τ t (6.13) = Tr(λ )=˜γA by definition of the formal trace. As already noted, the formal trace vanishes on low order perturbations soγ Ã is basic, i.e. is actually the pull-back of a well-defined form γA on Y depending only on the initial family A.

The index formula (4) therefore amounts to showing that the form γA represents the (odd) Chern character of the index of the family A in cohomology. This is difficult to approach directly, computationally, so instead we show how the index bundle A(φ) can be ‘trivialized’ by extending the bundle of structure groups. m,l If A ∈ Ψps (M/Y ; E) is a fully elliptic family then it has an ‘inverse’ family which, whilst not completely well-defined, is determined up to smoothing terms. Namely the bundle A is locally trivial over Y and in particular has local sections. Taking a partition of unity ψi on Y subordinate to a cover by open set Ui over each of which there is a section Ai the inverse family can be taken to be −1 −m,−l (6.14) ψi(y)Ai ∈ Ψps (M/Y ; E). i It is fully elliptic and, essentially by definition, the corresponding bundle of invertible perturbations is naturally identified with the bundle A−1(φ) ⊂ −m,−l Ψps (φ; E) consisting of the inverses of the elements of A(φ).

−1 Lemma 3. Under the inversion map A(φ) −→ A (φ) the eta form ηA−1 −1 on A (φ) associated with the inverse family (6.14) pulls back to −ηA.

Proof. The proof is similar to the one of Lemma 2, namely since the regularized trace vanishes on commutators, the result follows from the analog of (4.15) for the bundles A(φ)andA−1(φ). We also need a variant of the multiplicative formula (4.18). For this, consider the ﬁbre product A[2](φ) of two copies of the bundle A(φ) and the product map given by inversion in the second map [2] −1 −∞ (6.15)m ˜ : A (φ) (A ,A) −→ A A ∈ Gsus (φ; E). ETA FORMS AND FAMILIES INDEX 299

Proposition 6. Under pull-back under the three maps

−∞ (6.16) Gsus (Oφ,O E) m˜

A[2](φ) s LL π s LL π Ssss LLF ss LLL syssy L%% A(φ) A(φ), ∗ ∗ ∗ (6.17) m˜ Cheven −πSηA + πF ηA = dδA

[2] for a smooth form δA on A (φ).

Proof. We perform the same deformation as in Proposition 3 and its earlier variants. Taking into account Proposition 5 and Lemma 2, it follows that the same conclusion holds except that extra terms may appear from τ →±∞. These give a basic form so in place of the desired identity (6.17) we find instead that ∗ ∗ ∗ ∗ (6.18)m ˜ Cheven −πSηA + πF ηA = dδA + π μ [2] where δA is smooth form on A (φ)andμ is smooth form on the base, with π : A[2](φ) −→ Y. However, under exchange of the two factors, the left side of (6.18) changes sign, while the final, basic, term is unchanged. Thus taking the odd part of (6.18) gives (6.17). It is also of interest to see how the eta form transforms under a change of connections for the fibration φ : M → Y and the bundle E → M.

Lemma 4. If ηA and ηA are eta forms associated to the self-adjoint elliptic family A ∈ Ψ1(M/Y ; E) with respect to two diﬀerent choices of connections for the ﬁbration φ : M → Y and the vector bundle E → M,then there exist a form β ∈ Ωeven(Y ) and a form α ∈ Ωodd(A) such that ∗ ηA − ηA = πAβ + dα.

Proof. Consider the new ﬁbration φ×Id : M ×[0, 1]t → Y ×[0, 1]t with family pr∗ A where pr : M × [0, 1] → M is the projection on the left factor. Clearly, we can choose connections for the ﬁbration φ × Id and the bundle ∗ pr E → M × [0, 1]t such that if ηpr∗ A is the corresponding eta form, then ∗ ∗ ι0ηpr∗ A = ηA,ι1ηpr∗ A = ηA, ∼ ∗ where for i ∈ [0, 1], ιi : A→A×{i}⊂A×[0, 1]t = pr A is the natural inclusion. The form ηpr∗ A can be written as (6.19) ∗ ∗ even odd ηpr∗ A =prA ω0(t)+prA ω1(t) ∧ dt, ω0(t) ∈ Ω (A),ω1(t) ∈ Ω (A), 300 R. MELROSE AND F. ROCHON

∗ ∼ where prA :pr A = A×[0, 1] →A is the natural projection. This suggests to deﬁne α ∈ Ωeven(A)by 1 α = (prA)∗(ηpr∗ A)= ω1(t)dt. 0 Then we have that dα = (prA)∗(dηpr∗ A)+ηA − ηA ∗ (6.20) = πA pr∗(γpr∗ A)+ηA − ηA. 1 Thus, the result follows by taking β = − pr∗(γpr∗ A)=− 0 v1(t)dt where ∗ ∗ odd even γpr∗ A =pr v0(t)+pr v1(t) ∧ dt, v0(t) ∈ Ω (Y ),v1(t) ∈ Ω (Y ), and where pr also denotes the natural projection pr : Y × [0, 1]t → Y .

7. Extended eta invariant The bundle A(φ) in (5.1) is a bundle of principal spaces for the action −∞ of the fibres of Gsus (φ; E). The fibres can be enlarged to give an action of the central, contractible, group in (1.7) by setting −∞ (7.1) Ay = G˜ (Zy; E) ·Ay. This is not so easily characterized additively but is the image of the quotient map on the fibre product −∞ (7.2) p∼ : Gsus (φ; E) ×Y A(φ) −→ A(φ),p∼(˜g, A)=˜gA. In particular there is an exact and fibrewise delooping sequence coming from (1.7):

R˜ // ∞ //−∞ (7.3) A(φ) A(φ) Gsus (φ; E). The quotient map here can be defined in the fibre Ay by −1 (7.4) R˜∞(A ) = lim A A y τ→∞ y y where Ay ∈Ay is any point in the fibre of A(φ) over the same basepoint. Clearly the result does not depend on this choice of Ay. The construction above of the eta form on A(φ) extends to A(φ). Thus, the same form (6.3) pulls back under the evaluation map m (7.5) ev : Rτ × A(φ) −→ G (φ; E) to give ∗ (7.6) ev (λ)=λt + λn ∧ dτ. Then, extending Definition 2, set n (7.7) ηA =Trsus(λ ). ETA FORMS AND FAMILIES INDEX 301

We use this extended bundle and eta form to analyse the invariance properties of ηA. Consider the ﬁbre product with projections and quotient map (7.8) A(φ) R OO RRR RRR R˜ p RRR∞ ∼ RRR RR(( −∞ R //−∞ G (φ; E) ×Y A(φ) G (φ; E) sus RR OO nn RR π πA nn RR G˜ nnn RRR R nnn RRR ∞ wnnwn R(( −∞ A(φ) Gsus (φ; E)

Proposition 7. The diagram (7.8) commutes and there are smooth −∞ forms δÃ and μÃ respectively on the fibre product and G (φ; E) such that the three eta forms pull back to satisfy ∗ ∗ ∗ ˜ ∗ (7.9) p∼ηA = πAηA + πG˜η˜ + dδA + R μÃ. Proof. The commutativity of the parallelogram on the right is discussed above and defines the diagonal map, R. The formula (7.9) is a generalization of that of Proposition 3 and the proof proceeds along the same lines. Consider the odd Chern character on E ⊕ E. Thus, from the fibre product there are two evaluation maps, Ev and ev and we may combine these using the bundle rotation as in (2.13). This −∞ gives the two-parameter family of maps from Gsus (φ; E) ×Y A(φ): −∞ [0, 1]t × R × Gsus (φ; E) ×Y A(φ) (t, τ, g,˜ A) −1 g˜y(τ)0 Id 0 m (7.10) −→ M (t) M(t) ∈ G (φ, E ⊕ E). 0Id 0 Ay(τ) Pulling back the form λ of (6.3) under this map and ‘integrating’ over [0, 1]t×Rτ gives the identity (7.9), where the τ integral is to be interpreted as part of the regularized trace. Since the form λ is closed modulo commutators, if the product decomposition of its pull-back is (7.11) dt ∧ dτ ∧ μ + dt ∧ λt + dτ ∧ λτ + λ then ∂λt ∂λτ (7.12) dμ − + ≡ 0 ∂τ ∂t again modulo commutators. The regularized trace and integral of the last term gives the difference of the three pulled-back eta forms and μ defines the term δÃ on the fibre product. Thus it remains to analyse the second term in (7.12). The exterior dif- −∞ ferentials in (4.11) each fall on either a factor from G˜sus (φ; E)oronA. The terms involving no derivative of the first type, so the ‘pure A part’, makes no 302 R. MELROSE AND F. ROCHON contribution, since as discussed earlier, the rotation factor M(t) disappears. Thus, only terms with at least one derivative falling on the first three factors of (7.10) need to be considered. This results in a smoothing operator and the regularization of the trace functional is not necessary. Then the τ integral reduces to the value of λt at τ = ∞ which depends only on the leading term in A as τ →∞, which is to say the corresponding term in A itself, and R∞(˜g). This leads to the additional termμ Ã in (7.9). The left action of the groups on the fibres of the index bundle induces a contraction map −∞ L : Gsus (φ; E) ×Y A(φ) −→ A (φ) (7.13) −∞ L(s, u)=su, ∀ s ∈ G (Zy; Ey),u∈Ay ∀ y ∈ Y.

−∞ Corollary 1. There is a smooth odd form δ˜ on Gsus (φ; E) ×Y A(φ) such that ∗ ∗ ∗ (7.14) L ηA = πAηA + πG Cheven +dδ.˜

−∞ Proof. Restricting to the subbundle Gsus (φ; E) in (7.8) gives a diagram which includes into it and on which R∞ and R are trivial: A (7.15) (Oφ)O p

−∞ × A Gsus (φ; E) Y R (φ) nnn RRR πA nn RRRπG nn RRR nnn RRR nvnvn R)) −∞ A(φ) Gsus (φ; E).

Thus (7.9) restricts to this diagram withμ ˜A vanishing andη ˜ reducing to −∞ Cheven on Gsus (φ; E). Thus (7.14) follows.

−∞ Remark 1. If s : U −→ Gsus (φ; E)andα : U −→ A are sections over an open subset U ⊂ Y, then corollary 1 shows that ∗ (7.16) η(sα) − η(α)=Cheven(s)+dα δ.˜

8. Index formula: Proof of Theorem 1 The bundle A(φ) has contractible ﬁbres and hence has a global continuous section A˜ : Y −→ A(φ); this section is easily made smooth. The inverse image of the range of this section under the vertical map, p∼, in (7.8) is −∞ a submanifold F⊂Gsus (φ; E) ×Y A(φ). Indeed for each y ∈ Y and each −∞ By ∈Ay there is a unique Qy ∈ Gsus (Zy; E) such that

Q(τ)By = A˜y(τ) ETA FORMS AND FAMILIES INDEX 303

is the value of the section at that point. Thus, πA restricts to an isomorphism from F to A. Using the section A˜ : Y −→ A˜(φ), we can identify p∼(F)withY so that restricting (7.8) to F gives the commutative diagram

(8.1) J YOOJJ JJγ JJ π JJ J%% γ //−∞ F II G (φ; E) || II OO πA || II | II R∞ || π ˜ I ~||~ G I$$ −∞ A(φ) Gsus (φ; E) where γ is the restriction of R to F and γ is the classifying map deﬁned to make this diagram commutes. Restricted to F the identity (7.9) becomes modulo exact forms ∗ ∗ ∗ ˜∗ − ∗ ∈C∞ even (8.2) πAηA + πG˜η˜ = π βA˜,βA˜ = A ηA˜ γ μ˜A (Y ;Λ ). From (3.6) ∗ ∗ (8.3) πG˜dη˜ = γ (Chodd) so pulling back to A under the isomorphism πA gives the index formula (4): ∗ ∗ (8.4) dηA = −π γ (Chodd)+dβA˜. Since the homotopy class of the section γ represents minus the index class, this shows that γA in (6.9) represents the Chern character of the index.

9. Determinant of an odd elliptic family The eta invariant, interpreted here as the degree zero part in the eta form (there is a factor of 2 compared to the original normalization of Atiyah, Patodi and Singer) is a normalized log-determinant. In the universal case, for 0 −∞ the classifying spaces,η ˜ is a well-deﬁned function on G˜sus (Z; E) and then (9.1) det(g) = exp(2πiη˜0) is the Freholm determinant on G−∞(Z; E). In the geometric case essentially the same result is true.

m,k Proposition 8. For A ∈ Ψps (M/Y ; E) a fully elliptic family of product-type operators on the fibres of a fibration, 0 ∞ ∗ (9.2) τ(A) = exp(2πiηA) ∈C (Y ; C ), 0 where ηA is the degree zero part in (6.6), is a multiplicative function on fully elliptic operators on a fixed bundle, m,k m,k (9.3) τ(AB)=τ(A)τ(B),A∈ Ψps (M/Y ; E),B∈ Ψps (M/Y ; E) 304 R. MELROSE AND F. ROCHON which is constant under smoothing perturbation and which represents the class associated to ind(A) ∈ K1(B) in H1(Y ; Z).

0 Proof. Theorem 1 shows that dηA defined in principle on A, the bundle of invertible perturbations of a given fully elliptic family A, is basic and represents the first odd Chern class of the index. For the zero form part, −∞ (7.14) implies true multiplicativity under the action of Gsus (φ; E), with the zero form part of Cheven being the numerical index. Thus indeed the tau invariant in (9.2) is a well defined function on the base which represents the first odd Chern class in integral cohomology. Full multiplicativity follows as in [20].

10. Doubly suspended determinant −∞ Let Gsus(2)(Z; E) be the double flat-smooth loop group. Thus, its elements are Schwartz functions a : R2 −→ Ψ−∞(Z; E) such that Id +a(t, τ)is invertible for each (t, τ) ∈ R2. Let −∞ 2 −∞ 2 −∞ (10.1) Gsus(2)(Z; E)[/ ]=Gsus(2)(Z; E) ⊕S(R ;Ψ (Z; E)) be the group with the truncated ∗ (or Moyal) product obtained as in [25]by adiabatic limit from the isotropic product on S(R2;Ψ−∞(Z; E)) and then passing to the quotient by terms of order 2. Explicitly this product is (10.2) 1 ∂a0 ∂b0 ∂a0 ∂b0 (a0 + a1)[∗](b0 + b1)=a0b0 + a0b1 + a1b0 − − 2i ∂t ∂τ ∂τ ∂t where the underlying product is in Ψ−∞(Z; E). As shown in [25], in the adiabatic limit the Fredholm determinant, for operators on Z × R, induces the ‘adiabatic determinant’ (10.3) −∞ 2 ∗ detad : Gsus(2)(Z; E)[/ ] −→ C , detad(g1g2)=detad(g1)detad(g2), which, as in the unsuspended case, generates the 1-dimensional integral −∞ 2 cohomology of Gsus(2)(Z; E)[/ ] – which is classifying for odd K-theory. Note that there is no such multliplicative function on the leading group, without the first order (in ) ‘correction’ terms in (10.2). To define the adiabatic determinant, one needs to consider the adiabatic −∞ 2 trace on Ψsus(2)(Z; E)[/ ] defined by (10.4) 1 −∞ 2 Trad(a)= TrZ (a1(t, τ))dtdτ, a = a0 + a1 ∈ Ψsus(2)(Z; E)[/ ] 2π R2 −∞ with a0,a1 ∈ Ψsus(2)(Z; E). A special case of Lemma 5 below shows that this is a trace functional −∞ 2 (10.5) Trad(a ∗ b − b ∗ a)=0, ∀ a, b ∈ Ψsus(2)(Z; E)[/ ]. ETA FORMS AND FAMILIES INDEX 305

Consider the 1-form

−1 −∞ 2 (10.6) α(a)=Trad(a ∗ da)onG (Z; E)[/ ],

−∞ 2 where the inverse of a ∈ Gsus(2)(Z; E)[/ ] is with respect to the the truncated ∗-product −1 −1 −1 1 ∂a0 −1 ∂a0 ∂a0 −1 ∂a0 −1 (10.7) a = a − a a1 + a − a a . 0 0 2i ∂t 0 ∂τ ∂τ 0 ∂t 0 The adiabatic determinant is then deﬁned by ∗ (10.8) detad(g) = exp γ α [0,1]

−∞ 2 where γ;[0, 1] −→ Gsus(2)(Z; E)[/ ] is any smooth path from the identity to g. Since the integral of α along a loop gives an integer multiple of 2πi (see for instance proposition 4.4 in [24]), this deﬁnition does not depend on the choice of γ. From (10.5),

−1 −1 −1 (10.9) α(ab)=Trad((a ∗ b) d(a ∗ b)) = Trad(a ∗ da)+Trad(b ∗ db), and hence

∗ ∗ ∗ (10.10) m α = πLα + πRα where

−∞ 2 −∞ 2 (10.11) m : Gsus(2)(Z; E)[/ ] × Gsus(2)(Z; E)[/ ] (a, b) −∞ 2 −→ a[∗]b ∈ Gsus(2)(Z; E)[/ ]. is the composition given by the truncated ∗-product while πL and πR are the projections on the left and right factor. The multiplicativity of the adiabatic determinant follows directly from (10.10).

11. The determinant line bundle We next describe the construction, and especially primitivity, of the determinant line bundle over a smooth classifying group for even K-theory.

Definition 3. A primitive line bundle over a (Fr´echet-Lie) group

(11.1) L

G 306 R. MELROSE AND F. ROCHON is a smooth, and locally trivial, line bundle equipped with an isomorphism of the line bundles ∗ ∗ ∗ πLL⊗πRL −→ m L over G×G, (11.2) πL : G×G (a, b) −→ a ∈G,πR : G×G (a, b) −→ b ∈G, m : G×G (a, b) −→ ab ∈G which is associative in the sense that for any three elements, a, b, c ∈G, the two induced isomorphisms

(11.3) Lab ⊗Lc n77 JJ nnn JJ nn JJ nnn JJ nnn J%% ______// La ⊗L ⊗Lc L b PP t99abc PPP tt PP tt PPP tt PP'' tt La ⊗Lbc are the same.

−∞ For the reduced classifying group, Gsus,ind=0(Z; E), a construction of the determinant line bundle, with this primitivity property, was given in [25], although only in the ‘geometric case’. A variant of the construction there, also depending heavily on the properties of the suspended determinant but using instead the ‘dressed’ delooping sequence (for the loop group)

−∞ 2 /˜/−∞ 2 //−∞ (11.4) Gsus(2)(Z; E)[/ ] Gsus(2)(Z; E)[/ ] Gsus,ind=0(Z; E) again constructs the determinant line bundle, with primitivity condition, over the component of the identity in the loop group. In this section, by modifying an idea from the book of Pressley and Segal, [29], we show how to extend this primitive line bundle to the whole of the classifying group. In (11.4) the central, contractible, group is based on the half-open but smooth-ﬂat loop group:

(11.5) G˜−∞ (Z; E)= a˜ : R2 −→ Ψ−∞(Z; E); lim a˜(t, τ)=0, sus(2) (t,τ) t→−∞ ∂a˜ ∈S(R2;Ψ−∞(Z; E)), a˜(t, τ), a˜(∞,τ) ∈ G−∞(Z; E) ∀ t, τ ∈ R . ∂t

Note that automatically, limτ→∞ a˜(t, τ) = 0 for all t ∈ [−∞, ∞]. This group has an extension with the product having the same ‘correction term’ given by the Poisson bracket on R2 as in (10.1): ˜−∞ 2 ˜−∞ −∞ (11.6) Gsus(2)(Z; E)[/ ]=Gsus(2)(Z; E) ⊕ Ψsus(2)(Z; E) ETA FORMS AND FAMILIES INDEX 307 where the additional terms at level are just Schwartz functions valued in the smoothing operators without any additional invertibility. Note that the term in the product involving the Poisson bracket always leads to a Schwartz function on R2, since one factor is differentiated with respect to ˜−∞ 2 t. Thus Gsus(2)(Z; E)[/ ] is again contractible, with just the addition of a lower order ‘affine’ term. To expand the quotient group to the whole classifying group, choose one −∞ j element s ∈ Gsus (Z; E)ofindex1. Then s is in the component of index j −∞ so each element a ∈ Gsus (Z; E) can be connected by a curve, and hence a flat-smooth loop, to sj for precisely one j. The group in (11.5) may then be further enlarged to (11.7) −∞ 2 −∞ ∂a˜ 2 −∞ Dsus(2)(Z; E)= a˜ : R(t,τ) −→ Ψ (Z; E); ∈S(R ;Ψ (Z; E)), ∂t lim a˜(t, τ)=sj for some j, a˜(t, τ), a˜(∞,τ) ∈ G−∞(Z; E) ∀ t, τ ∈ R . t→−∞ This expanded group has countably many components, labelled by j, and the −∞ restriction map to t = ∞ is a surjection to Gsus (Z; E). Thus, after adding the same affine lower order terms, (11.4) is replaced by the new short exact sequence ˜ −∞ 2 /D/−∞ 2 R∞ //−∞ (11.8) Gsus(2)(Z; E)[/ ] sus(2)(Z; E)[/ ] Gsus (Z; E). The central group is no longer contractible, although each of its connected component is. However the 1-form α in (10.6) can be extended to give a −∞ 2 smooth 1-form on Dsus(2)(Z; E)[/ ]. Indeed, the adiabatic trace has an obvious extension to a functional on −∞ 2 −∞ −∞ (11.9) Ψsus(2)(Z; E)[/ ]=Ψsus(2)(Z; E) ⊕ Ψsus(2)(Z; E) where −∞ ∞ 2 −∞ ∂a 2 −∞ Ψsus(2)(Z; E)= a ∈C (R ;Ψ (Z; E); ∈S(R ;Ψ (Z; E)), ∂t (11.10) lim a(t, τ)=0 . t→−∞

Namely 1 −∞ 2 (11.11) Trad(a)= TrZ (a1)dtdτ, a = a0 + a1 ∈ Ψsus(2)(Z; E)[/ ] 2π R2 −∞ 2 Thus, on Dsus(2)(Z; E)[/ ], one can consider the smooth 1-form

1 −1 −1 (11.12) α(a)= Trad(a ∗ da + da ∗ a ) 2 −∞ 2 which restricts to α on Gsus(2)(Z; E)[/ ]. 308 R. MELROSE AND F. ROCHON

−∞ 2 Lemma 5. For a = a0 + a1 and b = b0 + b1 in Ψsus(2)(Z; E)[/ ], 1 ∂a0 Trad(a ∗ b − b ∗ a)= TrZ (∞,τ)b0(∞,τ) dτ. 2πi R ∂τ −∞ 2 In particular, this trace-defect vanishes if a, b ∈ Ψsus(2)(Z; E)[/ ].

Proof. By deﬁnition of the truncated ∗-product and using the trace property of TrZ , 1 ∂a0 ∂b0 ∂a0 ∂b0 (11.13) Trad(a ∗ b − b ∗ a)=− TrZ − dtdτ. 2πi R2 ∂t ∂τ ∂τ ∂t Integrating by parts the ﬁrst term on the right, 2 ∂a0 ∂b0 ∂ a0 TrZ dtdτ = − TrZ b0 dtdτ 2 ∂t ∂τ 2 ∂τ∂t R R ∂ ∂a0 = − TrZ b0 dtdτ 2 ∂t ∂τ R ∂a0 ∂b0 (11.14) + TrZ dtdτ. R2 ∂τ ∂t Thus, 1 ∂ ∂a0 Trad(a ∗ b − b ∗ a)= TrZ b0 dtdτ, 2πi 2 ∂t ∂τ R 1 ∂a0 (11.15) = TrZ (∞,τ)b0(∞,τ) dτ. 2πi R ∂τ

Proposition 9. Under the maps on the product

−∞ 2 (11.16) D (Z; E)[/ ] sus(2) OO m

−∞ 2 −∞ 2 Dsus(2)(Z; E)[/ ] ×Dsus(2)(Z; E)[/ ] m QQ mmm QQQ mmm QQQ mmm QQQ mvmvm πL πR QQ(( −∞ 2 ˜ ˜ −∞ 2 Dsus(2)(Z; E)[/ ] R∞×R∞ Dsus(2)(Z; E)[/ ]

−∞ 2 (Gsus (Z, E)) the 1-form α˜ in (11.12) satisﬁes ∗ ∗ ∗ ∗ (11.17) m α˜ = πLα˜ + πRα˜ +(R˜∞ × R˜∞) δ, ETA FORMS AND FAMILIES INDEX 309 with 1 −1 ∂b −1 −1 −1 ∂a δ(a, b)=− TrZ a (da) b − (db)b a dτ 4πi R ∂τ ∂τ −∞ −∞ on Gsus (Z; E) × Gsus (Z; E).

−∞ 2 Proof. If a, b ∈Dsus(2)(Z; E)[/ ], the trace-defect formula of Lemma 5 gives −1 Trad((a ∗ b) ∗ d(a ∗ b)) −1 −1 −1 = Trad(b ∗ a ∗ da ∗ b + b ∗ db) −1 −1 = Trad(a ∗ da)+Trad(b ∗ db) −1 −1 −1 −1 + Trad(b ∗ (a ∗ da ∗ b) − (a ∗ da ∗ b) ∗ b ) −1 −1 = Trad(a ∗ da)+Trad(b ∗ db) −1 1 ∂b0 −1 + TrZ (∞,τ)(a0 da0b0)(∞,τ) dτ 2πi R ∂τ −1 −1 = Trad(a ∗ da)+Trad(b ∗ db) 1 ∂b0 −1 −1 (11.18) − TrZ (∞,τ)b0 (∞,τ)a0 (∞,τ)da0(∞,τ) dτ. 2πi R ∂τ Similarly, −1 Trad(d(a ∗ b) ∗ (a ∗ b) ) −1 −1 = Trad(da ∗ a )+Trad(db ∗ b ) 1 −1 −1 ∂a0 (11.19) + TrZ db0(∞,τ)b (∞,τ)a (∞,τ) (∞,τ) dτ. 2πi R ∂τ Combining these two computations, the result follows.

Proposition 10. The adiabatic determinant on the normal subgroup in (11.8) induces the determinant line bundle, L, which is primitive over the quotient and α˜ in (11.12) deﬁnes a connection ∇ad on L with curvature form the 2-form part of the universal even Chern character of (2.22).

−∞ Proof. The formα ˜ in (11.12) restricts to α in (10.6) on Gsus(2)(Z; E) 2 [/ ]. The latter is the differential of the logarithm of detad . As a special case −∞ of Proposition 9 above, the first factor may be restricted to Gsus(2)(Z; E), and then δ in (11.17) vanishes since a ≡ 0. This shows that as a connection −∞ 2 on the trivial bundle over Dsus(2)(Z; E)[/ ],d− α˜ is invariant under the −∞ 2 left action of Gsus(2)(Z; E)[/ ], acting through the adiabatic determinant on the fibres. Thus d − α˜ projects to a connection ∇ad on the determinant −∞ line bundle over Gsus (Z; E) defined as the quotient by this action, i.e. as the line bundle induced by detad as a representation of the structure group. 310 R. MELROSE AND F. ROCHON

To compute the curvature we simply need to compute the diﬀerential of α.˜ Using the trace-defect formula of Lemma 5, −1 dTrad(a ∗ da) −1 −1 1 −1 −1 = −Trad(a ∗ da ∗ a ∗ da)=− Trad([a ∗ da, a ∗ da]), 2 1 ∂ −1 −1 = − TrZ (σ dσ) σ dσ dτ, with σ = a0(∞,τ), 4πi ∂τ R 1 −1 ∂σ −1 2 −1 ∂σ −1 (11.20) = TrZ σ (σ dσ) − σ d σ dσ dτ. 4πi R ∂τ ∂τ Similarly, we compute that (11.21) −1 dTrad(da ∗ a ) −1 −1 1 −1 −1 = Trad(da ∗ a ∗ da ∗ a )= Trad([da ∗ a ,da∗ a ]), 2 1 ∂ −1 −1 = TrZ (dσσ ) dσσ dτ, 4πi ∂τ R 1 −1 ∂σ −1 −1 ∂σ −1 −1 = − TrZ dσσ σ dσσ − d σ dσσ dτ. 4πi R ∂τ ∂τ −∞ Recall that the 2-form part of the universal even Chern character on Gsus (Z; E) is given by (cf. formula (3.7) in [23]) 1 −1 ∂σ −1 2 (11.22) (Cheven)[2] = 2 TrZ σ (σ dσ) dτ. 2(2πi) R ∂τ Thus, combining (11.20) and (11.21), 1 −1 ∂σ −1 2 ∗ (11.23) dα˜ = TrZ σ (σ dσ) dτ =2πiR˜∞((Cheven)[2]), 4πi R ∂τ that is,

i 2 ∗ (11.24) ∇ad = R˜∞((Cheven)[2]). 2π Next, this construction of the determinant bundle is extended to the geometric case. The sequence, (11.8), being natural, extends to give smooth bundles over the ﬁbres of (1): ˜ −∞ 2 /D/−∞ 2 R∞ //−∞ (11.25) Gsus(2)(φ; E)[/ ] sus(2)(φ; E)[/ ] Gsus (φ; E).

Furthermore, using the connection chosen earlier, the formα ˜ in (11.12) can φ,E be replaced byα ˜φ by substituting ∇ for d throughout. The resulting −∞ 1-form is well-deﬁned on Dsus(2)(φ; E). Moreover the proof of proposition 9 only depends on the derivation property of d so extend directly toα ˜φ. In ETA FORMS AND FAMILIES INDEX 311 particular (11.17) carries over to the ﬁbre products. This leads to the following geometric version of Proposition 10.

Proposition 11. The adiabatic determinant on the ﬁbres of the struc- −∞ ture bundle in (11.8) induces the determinant line bundle, L, over Gsus (φ; E); the 1-form α˜φ deﬁnes a connection ∇φ on L with curvature the −∞ 2-form part of the even Chern character on Gsus (φ; E).

Proof. What is slightly diﬀerent in the geometric case is the computation of the curvature of ∇φ. Taking into account (4.6), the analogue of (11.20) and (11.21) is 1 −1 ∂σ −1 φ,E 2 dα˜φ = TrZ σ (σ ∇ σ) dτ 4πi R ∂τ 1 −1 −1 (11.26) + Trad a ∗ ([ω, a]) + ([ω, a]) ∗ a 2

−1 where σ = a0(∞,τ). To compute the second term, we use the identity a ∗ (aω)=(ωa) ∗ a−1 = ω to rewrite it as

1 −1 −1 1 −1 −1 Trad a ∗ ([ω, a]) + ([ω, a]) ∗ a = Trad a ∗ (ωa) − (ωa) ∗ a 2 2 1 −1 −1 (11.27) + Trad a ∗ (aω) − (aω) ∗ a . 2

Using the trace-defect formula of lemma 5, this gives 1 −1 ∂σ −1 φ,E 2 −1 dα˜φ = TrZ σ (σ ∇ σ) − σ ωσ − ω dτ 4πi R ∂τ ∗ φ,E (11.28) =2πiR˜∞((Cheven(∇ ))[2]), from which the result follows.

12. The K-theory gerbe First we consider the universal K-theory gerbe, i.e. the gerbe over the classifying space G−∞(Z; E) for odd K-theory. Such a gerbe was originally introduced by Carey and Mickelsson [9], [10] over a slightly diﬀerent classifying space for odd K-theory, namely the space of unitary operators which are perturbations of the identity by operators of trace class. We propose a diﬀerent construction of the universal K-theory gerbe using the determinant line bundle of proposition 10. 312 R. MELROSE AND F. ROCHON

Recall that the delooping sequence (5) for a single manifold

(12.1) L

π −∞ //−∞ Gsus (Z; E) G˜sus (Z; E)

R∞ G−∞(Z; E) is a classifying sequence for K-theory, the normal subgroup is classifying for even K-theory, the central group is contractible and the quotient is classifying for odd K-theory. Moreover, in the preceding section, we have −∞ constructed the smooth primitive determinant line bundle over Gsus (Z; E) with connection ∇ad given in Proposition 10. This induces the K-theory gerbe over the classifying space, as a line bundle over the ﬁbre product of two copies of the ﬁbration R∞ :

(12.2) m˜ ∗L L π π πL −∞ oo −∞ [2] m˜ //−∞ G˜sus (Z; E) (G˜sus (Z; E)) Gsus (Z; E) PP πR PPP PP [2] PPP (R∞) R∞ PP(( G−∞(Z; E).

−∞ [2] −∞ Herem ˜ :(G˜sus (Z; E)) −→ Gsus (Z; E) is the fibre-shift mapm ˜ (a, b)= −1 ab , where R∞(a)=R∞(b), by definition of the fibre product, som ˜ (a, b) ∈ −∞ Gsus (Z; E) by the exactness of (12.1), as indicated.

∗ Theorem 2. There is a connection ∇˜ ad on m˜ L with curvature

i ∗ ∗ −∞ [2] (12.3) F ˜ = πLη˜2 − πRη˜2 on (G˜sus (Z; E)) 2π ∇ad where the B-ﬁeld, η˜2, is the 2-form part of the eta form in (3.2) which has basic diﬀerential the 3-form part of the odd Chern character on G−∞(Z; E), as shown by (3.6).

−∞ Proof. The connection ∇ad on L as a bundle over Gsus (Z; E) given ∗ ∗ by Proposition 10 pulls back to a connectionm ˜ ∇ad onm ˜ L. The curvature −∞ is just the pull-back of the curvature on Gsus (Z; E) and again by Propo- sition 10 this is the 2-form part of the Chern character. By Proposition 2, −∞ the 2-form part of the eta form on G˜sus (Z; E) pulls back under the product map as in (3.5). To apply this result here we need to invert the right factor, ETA FORMS AND FAMILIES INDEX 313 to change from m tom, ˜ which also has the effect of changing the sign of the eta form from that factor leading to ∗ ∗ ∗ ∗ (12.4)m ˜ η˜ = πLη˜ − πRη˜ + d(δõdd)+(R∞ × R∞) δeven where the primes indicate that the forms are first pulled back under inversion in the second variable. Now, restricting (12.4) to the fibre diagonal gives ∗ ∗ ∗ (12.5)m ˜ Cheven = πLη˜ − πRη˜ + d(δõdd) since the last term now factors through the constant map to the identity. This corresponds to the middle row of (12.2). In particular, if the connection is modified by the 1-form part of δodd ∗ (12.6) ∇˜ ad =˜m ∇ad +2πi(δõdd)1 then it has curvature as claimed i 2 ∗ ∗ ∗ (12.7) (∇˜ ad) =˜m (Cheven)2 − d(δodd)1 = πLη˜2 − πRη˜2 2π which is precisely the statement thatη ˜2 is a B-field for the gerbe. The curving of the gerbe is then the basic form of which the differential of the B-field is the pull-back and from (3.6) ∗ (12.8) dη˜2 = R∞(Chodd)3.

The bundle gerbe (12.2) with connection given by theorem 2 is universal in the sense that given an odd K-theory class [g] ∈ K1(Y ) represented by a smooth map g : Y → G−∞(Z; E), the pull-back of (12.2) to Y by g gives a bundle gerbe (with connection) on Y whose curving 3-form is given by ∗ g Chodd (cf. Theorem 5.1 in [13]). Since 1 −1 3 (12.9) (Chodd)3 = Tr((σ dσ) ) 6(2πi)2 3 −∞ 3 −∞ ∼ is the image in H (G (Z; E); C) of the generator of H (G (Z; E); Z) = Z, we also note that the bundle gerbe (12.2) is basic for the group G−∞ (Z; E). By considering an n-dimensional subspace of L2(Z; E)withnorm defined by a choice of metric on Z and of Hermitian metric on E,weget a natural inclusion U(n) ⊂ G−∞(Z; E). Pulling back (12.2) to U(n) via this inclusion gives the basic bundle gerbe U(n). This is a ‘smooth’ construction in any reasonable sense although it is infinite dimensional in nature. Infinite dimensional constructions of the basic bundle gerbe of a Lie group first appeared in the book of Brylinski [6] and later in [7]. The tautological bundle gerbe of Murray [26] for 2-connected manifolds also provides such a construction for simply connected Lie groups (see also [12]). More recently, finite dimensional constructions of the basic gerbe were obtained by Gawedzki and Reis [14]forSU(n) and shortly after by Meinrenken [19] 314 R. MELROSE AND F. ROCHON for simple simply connected Lie groups. The construction of Meinrenken was subsequently generalized to non-simply connected Lie groups in [15] and [27].

13. Geometric gerbe for an odd elliptic family As pointed out in [11], gerbes are intimately related to index theory. In our case, we have the following construction of the index gerbe (with connection) associated to a family of self-adjoint elliptic pseudodiﬀerential operators and more generally to a product-type family of fully elliptic operators.

Theorem 3. Let A ∈ Ψ1(M/Y ; E) be a self-adjoint elliptic family as in m,l Section 6, or a product-type fully elliptic family A(t) ∈ Ψps (M/Y ; E) then the determinant bundle induces a bundle-gerbe

(13.1) S∗L L π π πL S oo [2] //−∞ A B A Gsus (φ; E) BπR BB B [2] p BB p B!! Y.

The connection on S∗L

∗ (13.2) ∇A = S ∇ + γ, γ =2πi(δ˜A)1, given by the 1-form part of the form in (6.17), is primitive in the sense that the curvature on A[2] splits

i 2 ∗ ∗ (13.3) (∇A) = πLηA,2 − πRηA,2 2π showing that ηA,2 is a B-ﬁeld and that the gerbe has curving 3-form

∗ (13.4) dηA,2 = p ChA,3 the 3-form part of the Chern character of the index bundle of the family.

Proof. This result follows by an argument parallel to the preceding one, given (6.17), Proposition 10, Proposition 11 and the 3-form part of (8.4).

This result can be seen as a pseudodiﬀerential generalization of a result of Lott (Theorem 1 in [18]). See also [8] for a diﬀerent treatment of the index gerbe and [13] for a generalization of ([18], Theorem 1) to families of Dirac operators on odd dimensional manifolds with boudary. ETA FORMS AND FAMILIES INDEX 315

14. Relation with the Bismut-Cheeger eta form Amongst the most important geometric examples of self-adjoint elliptic operators are the Dirac-type operators on odd dimensional manifolds. Sup- pose now that the fibres of the fibration (1) are odd dimensional. Let g ∈ C∞(M; 2T ∗(M/Y )) be a family of fibrewise metrics and let C(T (M/Y )) be the associated bundle of Clifford algebras for the vertical tangent bundle T (M/Y ). Let E → M be a Clifford module with respect to C(T (M/Y )) with Clifford action c : C(T (M/Y )) → End(E). Let also ∇E be a family of fibrewise Clifford connections, that is a family of unitary connections such that ∇E ∇LC ∈C∞ E ∀ ∈C∞ [ X1 ,c(X2)] = c( X1 X2) (M;End( )), X1,X2 (M; T (M/Y )), where ∇LC is the fibrewise Levi-Civita connection associated to the family of metrics g. This data allows us to define a family of Dirac-type operators by (14.1) ð = c ◦∇E. For invertible families of this type, Bismut and Cheeger introduced in [3] an eta form on the base. Their construction was subsequently generalized in [21] to situations where the family is not invertible, but admits a perturbation by a family Q ∈ Ψ−∞(M/Y ; E) of self-ajoint smoothing operators such that ð + Q is invertible. The odd families index of the family ð is precisely the obstruction to the existence of such a family of perturbations; for the boundary operators of a family of Dirac-type operators on a fibration of manifolds with boundary, this index obstruction vanishes by the cobordism invariance of the index, so that invertible perturbations exist in this case. When the family ð is invertible, the Bismut-Cheeger eta form is given by ∞ 1 dBt −B2 even (14.2) ηBC(ð)=√ STrC (1) e t dt ∈ Ω (Y ), π 0 dt where Bt is the rescaled Bismut superconnection associated to ð (see (10.8) and (13.7) in [21] for a detailed discussion). For a family perturbed to be invertible, ð + Q, the definition is slightly modified to ∞ 1 dBt −B2 even (14.3) ηBC(ð + Q)=√ STrC (1) e t dt ∈ Ω (Y ), π 0 dt 1 where Bt = Bt + t 2 χ(t)Qσ,withσ ∈ C(1) a generator of C(1) such that σ2 =1 and χ ∈C∞(R) is a non-negative function with χ(t)=0 for t<1 and χ(t)=1 for t>2. The Bismut-Cheeger eta form satisfies the following transgression formula.

Proposition 12. The exterior differential of ηBC(ð+Q) does not depend on the choice of perturbation Q and is given by − n+1 dηBC(ð + Q)=(2πi) 2 Aˆ(Rg)Ch(E), M/Y 316 R. MELROSE AND F. ROCHON where n is the dimension of the fibres of the fibration φ : M → Y and Ch(E) is the twisting Chern character of E.

Proof. Because of the cut-oﬀ function χ we have that 1 −B2 1 −B2 lim √ STrC (1) e t = lim √ STrC (1) e t t→0 π t→0 π (14.4) 1 ˆ = n+1 A(Rg)Ch(E). (2πi) 2 M/Y On the other hand, because the family ð + Q is invertible, 1 −B2 (14.5) lim √ STrC (1) e t =0 t→∞ π exponentially fast. Thus, the result follows by combining (14.4) and (14.5) with d −B2 dBt −B2 STrC (1) e t = −dY STrC (1) e t dt dt and integrating in t.

If we consider the rescaled version of the Bismut-Cheeger eta form, ∞ −j (14.6) ηBC(ð + Q)= (2πi) ηBC,[2j](ð + Q), j=0 where ηBC,[2j] is the part of ηBC of degree 2j, then Proposition 12 shows that the Chern character of the family index is trivial in cohomology, which is consistent with the fact the odd index of ð must vanish for the invertible perturbation ð + Q to exist. It also follows from Proposition 12 that if Q1 and Q2 are two perturbations giving invertible families, then ηBC(ð + Q1) − ηBC(ð + Q2) is a closed form. Moreover, using Proposition 12 again, it can be seen that the coho- molgy class represented by the form ηBC(ð+Q1)−ηBC(ð+Q2) only depends on the homotopy classes of Q1 and Q2 in the space of such perturbations. This cohomology class can be identiﬁed usign the notion of spectral sections introduced in [21].

Definition 4. A spectral section for the family of self-adjoint Dirac- type operators ð of (14.1) is a family of self-adjoint projections P ∈ Ψ0(M/Y ; E) such that for some smooth function R : Y → R+ (depending on P )and ∈ every y Y, Pyu = u, if λ>R(y), ðyu = λu =⇒ Pyu =0, if λ<−R(y). ETA FORMS AND FAMILIES INDEX 317

If P1 and P2 are spectral sections for the family ð, then as shown in [21], 0 their formal difference defines a K-class [P1−P2] ∈ K (Y ). If P1P2 = P2, then [P1 −P2] is represented by the vector bundle given by the range of the family of the finite rank projections (Id −P2)P1. In general, one can reduce to this case by choosing a third spectral section R such that P1R = R, P2R = R and setting 0 (14.7) [P1 − P2]=[P1 − R] − [P2 − R] ∈ K (Y ). It is shown in [21] that such a spectral section R always exists and that the definition of the K-class [P1 − P2] does not depend on the choice of R. To obtain a spectral section from an invertible self-adjoint perturbation Q,we need an extra assumption.

Definition 5. A family Q ∈ Ψ−∞(M/Y ; E) of self-adjoint operators is spectrally ﬁnite with respect to the family of Dirac-type operators ð if there exists a smooth function R : Y → R+ such that for every y ∈ Y ,

ðyu = λu =⇒ Qyu =0if|λ| >R(y).

If Q ∈ Ψ−∞(M/Y ; E) is an invertible self-adjoint perturbation of the family ð which is not spectrally finite, then using an approximation argument, it is shown in [21] that it is possible to deform it through invertible self-adjoint perturbations to one which is spectrally finite. Suppose now that Q ∈ Ψ−∞(M/Y ; E) is a spectrally finite invertible self- adjoint perturbation of the family ð. Then there is a corresponding spectral 0 section PQ ∈ Ψ (M/Y ; E)with(PQ)y defined to be the projection onto the positive eigenspace of ðy + Qy. The following relative index theorem was proved in [21].

Proposition 13. If Q1 and Q2 are two spectrally ﬁnite invertible self- ajoint perturbations of the family ð, then the Chern character Ch([PQ1 − 0 PQ2 ]) of the K-class [PQ1 − PQ2 ] ∈ K (Y ) is represented by the closed form

ηBC(ð + Q1) − ηBC(ð + Q2).

If the odd index of the family ð does not vanish, it is still possible to define a version of the Bismut-Cheeger eta form, but now over an infinite dimensional bundle : ASA −→ Y defined in terms of self-adjoint smoothing perturbations. Namely the fibre at y ∈ Y is

SA −∞ ∗ (14.8) Ay = {ðy + Qy; Qy ∈ Ψ (Zy; E),Qy = Qy, ðy + Qy is invertible}. ∗ SA SA The pull-back A of A to itself has a tautological section σASA : ASA −→ ∗ASA which can be used to deﬁne a form on the total space of ASA via formula (14.3), even SA (14.9) ηBC ∈ Ω (A ). 318 R. MELROSE AND F. ROCHON

It is well-deﬁned since the space ASA has a natural structure of smooth Fr´echet manifold. As in (14.6), we can also consider its rescaled version even SA ηBC ∈ Ω (A ). Proposition 12 then extends as follows.

Proposition 14. The exterior diﬀerential of the Bismut-Cheeger eta form ηBC in (14.9) is the basic form ∗ − n+1 dηBC = (2πi) 2 Aˆ(Rg)Ch(E) . M/Y

Let ρ ∈S(R) be a a choice of Schwartz function such that ρ(0) = 1. Let also A be the infinite dimensional bundle of (5.1) with A = ð.The function ρ can be used to define an injective bundle map ι : ASA →Adefined fibrewise by SA (14.10) ιy : Ay ðy + Qy −→ ðy + it + Qyρ(t) ∈Ay. The definition of ι clearly depends on the choice of ρ, but since the space of Schwartz functions equal to one at the origin is convex, the homotopy class of the map ι does not depend on the choice of ρ.

SA Proposition 15. For each y ∈ Y, the map ιy : Ay −→ A y isaweak SA homotopy equivalence; in particular, Ay is a classifying space for even K-theory and ι : ASA −→ A is also an homotopy equivalence.

−∞ Proof. Let W ∈ Ψ (Zy; Ey) be a fixed choice of spectrally finite invertible self-adjoint perturbation of ðy. Let B be a smooth closed mani- SA fold. Given a smooth map f : B −→ A y with f(b)=ðy + Qb, it is always possible to deform it through self-adjoint invertible perturbations so that −∞ the family b −→ Qb ∈ Ψ (Zy; Ey) becomes spectrally finite with respect to ðy. This means there is a well-defined map SA 0 (14.11) μ :[B; Ay ] [f] −→ [Pf − PW ] ∈ K (B) where Pf is the spectral section associated to the spectrally finite invertible self-adjoint perturbation b → Qb and PW is the spectral section associated to W seen as a spectrally finite invertible perturbation for the trivial family SA b → ðy over B. The map μ is easily seen to be bijective, so that Ay is a classifying space for even K-theory. On the other hand, under the identification ∼ −∞ (14.12) Ay = Gsus (Zy; Ey), −1 given by composing on the right with (ιy(ðy + W )) , we see the fibre Ay is also a classifying space for even K-theory. In fact, this identification induces a map −∞ ∼ −2 ν :[B; Ay] −→ [B; G (Zy; Ey)] = K (B) ETA FORMS AND FAMILIES INDEX 319 and a corresponding commutative diagram SA ι∗ // (14.13) [B; Ay ] [B; Ay]

μ ν p K0(B) /K/−2(B) where the bottom map p is Bott periodicity. In particular, this shows the SA map ι : Ay −→ A y is a weak homotopy equivalence. The proof of the previous proposition also gives the following result.

SA SA Corollary 2. Suppose that σ1 : Y →A and σ2 : Y →A are two sections of the bundle ASA. Then the closed forms ∗ ∗ ∗ (σ1ηBC − σ2ηBC) and ((ι ◦ σ1) ηA − (ι ◦ σ2)ηA) even represent the same cohomology class in HdR (Y ).

Proof. Using section σ1 instead W to deﬁne a map μ as in (14.11), we get a commutative diagram analogous to (14.13), namely,

ι∗ (14.14) [Y ; ASA] /[Y/; A]

μ ν p K0(B) /K/−2(B). ∗ ∗ Thanks to Proposition 13, we then see that the forms (σ1ηBC − σ2ηBC)and ∗ ((ι ◦ σ1) ηA − (ι ◦ σ2)ηA) represent the Chern character of the same K-class, from which the result follows. Since Proposition 15 shows that the bundles ASA and A are homotopy equivalent, the map ι allows ηA to be compared with the Bismut-Cheeger eta form ηBC. The definitions of ηA and ηBC involve different regularizations of the underlying Dirac family. Assuming choices of regularization and choices of connections affect the eta form in a similar way, it is natural in light of Lemma 4 to expect the following relation between the two types of eta forms.

Conjecture 1. There exist forms β ∈ Ωeven(Y ) and α ∈ Ωodd(ASA) such that ∗ ∗ ι ηA − ηBC = β + dα.

As an indication that this conjecture might be true, we will prove it in a particular case.

Theorem 4. The conjecture is true when the odd families index of the family ð ∈ Ψ1(M/Y ; E) vanishes. 320 R. MELROSE AND F. ROCHON

Proof. Since ind(ð)=0 in K1(Y ), we know ð admits an invertible −∞ self-adjoint perturbation ð + Q0 with Q0 ∈ Ψ (M/Y ; E). Without loss of generality, we can assume Q0 is spectrally finite. The perturbation Q0 defines SA a section σ : Y −→ A with σ(y)=ðy +(Q0)y. There is also an induced section ι ◦ σ : Y −→ A for the bundle A−→Y. The form on the base is then taken to be ∗ ∗ β = σ ηBC − (ι ◦ σ) ηA. − ∗ − ∗ The form ω = ηBC ι ηA πASA β can then be written as the difference of two closed forms, ω = ωBC − ωA with ∗ ∗ ∗ ∗ ∗ ωBC = ηBC − πASA σ ηBC,ωA = ι ηA − πASA (ι ◦ σ) ηA. Let B be a closed manifold and f : B −→ A SA a smooth map. By perturbing f in its homotopy class as necessary, it can be arranged that f induces a spectrally finite invertible self-adjoint pertrubation of the fam- ∗ ∗ ∗ ily f ð parametrized by B. By Corollary 2, both f ωBC and f ωA repre- ∗ sents the Chern character of [Pf − ((πASA ◦ f) PQ], where Pf is the spectral section associated to the invertible family over B defined by the map f ∗ and (πASA ◦ f) PQ is the spectral section associated to the invertible family obtained by pulling back the invertible family ð + Q under the map πASA ◦ f : B −→ Y. Since B and f are arbitrary, this means ω is trivial in the singular cohomology of ASA. From the infinite dimensional version of the de Rham theorem in this context (see Lemma 6 below), it follows that there exists α ∈ Ωodd(ASA) such that dα = ω, from which the result follows.

Lemma 6 (de Rham theorem). The de Rham theorem holds for the inﬁ- nite dimensional space ASA.

Proof. According to Theorem 16.10 and 34.7 in [17] or the discussion on p.25 of [29], it suffices to show that ASA satisfies the following two properties: (i) the Fréchet space F on which ASA is locally modelled has enough smooth functions, which means that for each open set U in F, there is a nonvanishing real-valued smooth function which vanishes outside U; (ii) The manifold ASA is Lindelöf, which means each open covering has a countable refinement. −∞ To show property (i), fix y ∈ Y and consider the Fréchet space Ψ (Zy; Ey). Then the closed subspace −∞ −∞ ∗ ΨSA (Zy; Ey)={Q ∈ Ψ (Zy; Ey); Q = Q} ETA FORMS AND FAMILIES INDEX 321 is also naturally a Fréchet space. If k =dimY , then our local model for ASA can be taken to be the Fréchet space −∞ k F =ΨSA (Zy; Ey) × R . −∞ Since Ψ (Zy; Ey) is a nuclear Fréchet space, so is F (see Corollary 21.6.4 and Corollary 21.2.3 in [16]). Thus, by Proposition 14.4 in [17], F has enough smooth functions. −∞ To prove property (ii), notice that the Fréchet space Ψ (Zy; Ey)is −∞ separable, so it is in particular second-countable. This implies ΨSA (Zy; Ey) SA SA and Ay are second-countable, and more generally, that A is second- countable, which means in particular it is Lindelöf.

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Department of Mathematics, Massachusetts Institute of Technology E-mail address: [email protected]

Department of Mathematics, Australian National University E-mail address: [email protected] Surveys in Diﬀerential Geometry XV

Anomaly Constraints and String/F-theory Geometry in 6D Quantum Gravity

Washington Taylor

Abstract. Quantum anomalies, determined by the Atiyah-Singer index theorem, place strong constraints on the space of quantum gravity theories in six dimensions with minimal supersymmetry. The conjecture of “string universality” states that all such theories which do not have anomalies or other quantum inconsistencies are realized in string theory. This paper describes this conjecture and recent work by Kumar, Morrison, and the author towards devel- oping a global picture of the space of consistent 6D supergravities and their realization in string theory via F-theory constructions. We focus on the discrete data for each model associated with the gauge symmetry group and the representation of this group on matter fields. The 6D anomaly structure determines an integral lattice for each gravity theory, which is related to the geometry of an elliptically fibered Calabi-Yau three-fold in an F-theory construction. Possible exceptions to the string universality conjecture suggest novel constraints on low-energy gravity theories which may be identified from the structure of F-theory geometry.

1. Introduction String theory is not yet a mathematically well-deﬁned system. Nonethe- less, the components of this theory which have been pieced together so far suggest a framework in which quantum physics and general relativity are uniﬁed. In certain limits it seems that the theory can be described geometrically and gives rise to 4D “vacuum solutions”, in which space-time has four macroscopic dimensions and additional microscopic dimensions are curled

The author would like to thank collaborators Vijay Kumar and David Morrison, with whom the work described in this paper was carried out. The author is immensely grateful to IMS for regular stimulating discussions and interactions over the last two decades, and for his optimism and constant support over the years. This research was supported by the DOE under contract #DE-FC02-94ER40818. c 2011 International Press

323 324 W. TAYLOR up into a compact manifold. Each 4D string vacuum solution gives rise to a low-energy physical theory in four dimensions which generally is a theory of gravity coupled to Yang-Mills theory with matter fields transforming under the Yang-Mills symmetry group. The vast landscape of possible four- dimensional string theory vacua presents a major challenge for physicists who hope to derive predictions for observable physics from string theory. String theorists have struggled with this problem since the early days of the subject, when it was realized that ten-dimensional string theory could be compactified to four dimensions on any Calabi-Yau threefold, giving a wide variety of low-energy theories. Efforts to remove unphysical massless scalar fields associated with Calabi-Yau moduli, such as through inclusion of generalized magnetic fluxes, have sharpened the problem further by generating exponentially large (or infinite) classes of vacua associated with each Calabi-Yau geometry (for a review of recent work in this direction see e.g. [1]). Beyond the enormous range of string vacua which can be constructed using this and other known mechanisms, it may be that even larger classes of string vacua can be constructed using less well-understood non-geometric compactifications. Despite the wide range of possible string vacuum constructions, the range of low-energy theories which exhibit no known quantum inconsistencies when coupled to gravity seems to be far vaster still. The term “swampland” was coined by Vafa to characterize the set of low-energy gravity theories with no known inconsistencies which cannot be realized in string theory [2]. At present, the swampland for four dimensions appears to be very large, and we lack tools powerful enough to make strong global statements about the space of consistent quantum gravity theories. In six space-time dimensions, however, a global understanding of the space of supersymmetric string vacua may be within reach. In 4k +2 dimensions, quantum anomalies can lead to a violation of diffeomorphism or Yang- Mills invariance. The condition that such anomalies are absent strongly constrains the range of possible quantum-consistent supergravity theories in six dimensions. The following conjecture was presented at the conference “Perspectives in Mathematics and Physics,” in honor of I. M. Singer, and stated more precisely in the paper [3]. conjecture 1.1 (“String Universality in six dimensions”). All N = (1, 0) supersymmetric theories in 6D with one gravity and one tensor multiplet which are free of anomalies or other quantum inconsistencies admit a string construction. In the statement of this conjecture, N =(1, 0) supersymmetry is the minimal possible amount of supersymmetry in six dimensions. Supersymmetry is a graded symmetry relating bosonic (commuting) to fermionic (anticommuting) fields. The gravity and tensor multiplets are combinations of fields transforming under irreducible representations of the supersymmetry algebra. We describe some relevant features of these multiplets in more detail in the following section. Over the last year, further work with V. Kumar [4] ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 325 and with V. Kumar and D. Morrison [5, 6] has led to a global picture of the space of constraints on these low-energy supergravity theories, and an explicit correspondence with the F-theory description of string vacua, not only for theories with one tensor multiplet, but also for a much richer class of theories with multiple tensor multiplets. We have proven that when the number of tensor multiplets is less than 9, there are only a finite number of distinct gauge groups and matter representations which can appear in consistent low-energy theories [4, 6]. This analysis has also, however, produced some apparent counterexamples to the conjecture. Either these counterexamples represent new classes of string vacua, which cannot be realized using conventional F-theory, or there must be additional stringy constraints on the space of possible low-energy theories. In the latter case, if the conjecture is correct, it may be the case that these constraints can be identified from the low-energy data describing the theory as obstructions to a consistent quantum UV completion. The analogue of Conjecture 1.1 in 10 dimensions was recently proven by Adams, DeWolfe, and the author by demonstrating the quantum inconsistency of the two models with gauge groups U(1)496 248 and E8 × U(1) which were previously thought to lie in the swampland [7]. Further progress towards proving or disproving the string universality conjecture in 6 dimensions will require achieving a better mathematical or physical understanding of the set of possible string constructions and low- energy constraints. In this paper we describe some of the context and motivation for the 6D string universality conjecture, and give a somewhat self-contained presentation of some of the core mathematical arguments underlying recent progress in understanding the relevant space of 6D supergravity theories. The group- theoretic structure of the anomaly cancellation conditions is reviewed in Section 2 and used in Section 3 to place bounds on the set of gauge groups and matter representations which can be realized in N = 1 supersymmetric 6D theories of gravity coupled to Yang-Mills theory. From the anomaly conditions, an integral lattice Λ can be associated with each such low-energy supergravity theory. In Section 4, we describe the connection with F-theory, where Λ is a sublattice of the lattice of integral second homology classes on a complex surface which is the base of an elliptic fibration. In Section 5 we give some explicit examples of 6D theories and describe some apparent counterexamples to the conjecture. This leads to a discussion in the concluding section 6 of possible new constraints on low-energy supergravity theories.

2. 6D supergravity: gauge groups, matter representations, and anomalies In this section we describe some of the key mathematical structures in 6D supergravity theories. In particular, we show how the set of possible theories is constrained by using group theory to analyze quantum anomaly constraints. Our focus here is on topological, rather than analytic aspects of 326 W. TAYLOR these theories. Thus, we are primarily interested in classifying the discrete data associated with the set of quantum ﬁelds in the theory, in particular the Yang-Mills symmetry group and associated representations. The following abbreviated outline of the structure of 6D supergravity theories highlights this discrete structure, and elides many of the details of the physics of these theories. For a more detailed description of 6D N =(1, 0) supergravity theories, see the original papers [10, 11, 12].

2.1. Gauge groups and matter ﬁelds. Any 6D supergravity theory with N =(1, 0) supersymmetry is characterized by the following discrete data

TT∈ Z,T ≥ 0 [number of anti-self-dual tensor fields] G compact finite-dimensional Lie group [gauge group] M finite-dimensional representation of G [matter content]

This data describes the set of dynamical quantum fields in the supergravity theory. As mentioned above, the fields in the theory live in supersymmetry multiplets–that is, in representations of the supersymmetry algebra of the theory. There are four types of supersymmetry multiplets appearing in N =(1, 0) theories. Each theory contains a single “gravity” multiplet. The gravity multiplet includes among its bosonic components a symmetric tensor field gμν, which describes the metric on the 6D space-time manifold M,and 0 an antisymmetric two-form field Bμν, which satisfies a self-dual condition ∼ dB0 = ∗dB0. T denotes the number of “tensor” multiplets, which each i i include an anti-self-dual two-form field Bμν and a scalar field φ ,i=1,...T. Each model has some some number of “vector” multiplets, which contain one-form gauge fields Aμ describing a connection on a principal G-bundle over M. Each model also has a set of “hyper” multiplets, each containing four scalar fields ϕa. These scalar fields parameterize a quaternionic Kähler manifold, and transform under a (generally reducible) representation M of the gauge group G. Each multiplet, in addition to the bosonic fields just mentioned, also contains fermionic (Grassmann) fields related to the bosonic fields through supersymmetry. We will not be concerned with many details of the field structure in the analysis here; the principal data we are concerned about are the data T, G, M characterizing the number of tensor multiplets, the gauge group, and the matter representation. Furthermore, we will focus on the nonabelian structure of the gauge group, taking G to be semisimple and ignoring U(1) factors. We write G as a product of simple factors Hi

(2.1) G = H1 × H2 ×···×Hk.

We denote the number of vector multiplets associated with generators of G by V =dimG, and the number of hypermultiplets transforming (trivially or nontrivially) under G by H =dimM. ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 327

2.2. Anomalies. Not all possible combinations of discrete data T, G, M can be associated with consistent supersymmetric quantum theories of gravity. Quantum anomalies impose strong constraints on the set of possible theories. In fact, the condition that a theory be free of anomalies, along with a simple physical sign constraint on the structure of the gauge fields, is sufficient to prove that when T<9 there are only a finite number of possible nonabelian gauge groups G and matter representations M possible for a consistent quantum supergravity theory. This was proven for T =1 in [4], and for T<9in[6]. After summarizing the anomaly conditions in this subsection, we present the key arguments in the proof of finiteness in the following subsection. A quantum anomaly is a breakdown at the quantum level of a classical symmetry. One way to understand anomalies (Fujikawa [13]) is in terms of the path integral formulation of a quantum theory. A quantum theory containing fermion fields ψ, ψ¯ can be defined through a path integral ¯ (2.2) Z = DψDψe¯ iS(ψ,ψ), where the action S(ψ, ψ¯)=ψ¯Dψ/ + ··· contains a Dirac operator D/ and is invariant under local gauge transformations ψ → gψ. For chiral fermionic fields, nontrivial transformation of the measure factor DψDψ¯ leads to a quantum violation of the classical gauge symmetry. In terms of Feynman diagrams, such quantum anomalies appear in one-loop computations, and cannot be removed by any choice of the regularization prescription used to remove infinities from the quantum calculation. For gravitational theories, chiral fermions and self-dual/anti-self-dual antisymmetric tensor fields lead to similar anomalies associated with violations of local diffeomorphism invariance [14]. As was shown in [15, 16], gauge and gravitational anomalies are determined by characteristic classes associated with the index of the Dirac operator coupled to the appropriate vector potential. The anomaly in D dimensions is associated with an index in D +2 dimensions. Thus, in particular, gauge, gravitational, and mixed gauge-gravitational anomalies in six dimensions are associated with an 8-form I8 containing terms built from the geometric and Yang-Mills curvatures, such as trR4, trF 4, trR2trF 2,.... It was shown by Green and Schwarz [17] that the anomaly in some 10- dimensional supergravity theories can be cancelled by classical (tree-level) terms associated with exchange of quanta associated with antisymmetric two-form fields. This mechanism was extended to six dimensions in [18]for theories with one tensor multiplet, and in [19] for theories with any number of tensor multiplets. Anomalies in six-dimensional supergravity theories with a single supersymmetry can be cancelled by the Green-Schwarz-Sagnotti mechanism if the anomaly polynomial I8 can be written in the form

1 α β (2.3) I8(R, F )= ΩαβX4 X 2 4 328 W. TAYLOR

α where X4 is a 4-form taking the form α 1 α 2 α 2 2 (2.4) X4 = a trR + bi trFi . 2 i λi α α 1,T Here, a and bi are vectors in the space R carrying a symmetric bilinear form Ωαβ. The normalization factor λi is chosen depending on the gauge factor Hi such that minimal instantons have compatible normalization factors.

For example, λSU(N) =1, while λE8 = 60. α β The anomaly polynomial does not specify the vectors a ,bi , but only 2 constrains the SO(1,T) invariant quantities a ,a· bi,bi · bj, where the inner product is taken with respect to the form Ω. A detailed calculation of the anomaly arising from all chiral fields in the theory (see [18, 20, 21]for details) gives the form of the complete anomaly polynomial as a function I(R, F ) of the gravitational and Yang-Mills curvature two-forms R and F . In order for the anomaly to factorize as in (2.3), I8 cannot contain terms proportional to trR4, which gives the condition (2.5) H − V = 273 − 29T. The trF 4 contribution to the total anomaly must also cancel for each gauge group factor, giving the condition i i i (2.6) BAdj = xRBR R where we use the group theory coefficients AR,BR,CR defined through 2 2 (2.7) trRF = ARtrF 4 4 2 2 (2.8) trRF = BRtrF + CR(trF ) i and where xR denotes the number of hypermultiplets transforming in irreducible representation R under gauge group factor Hi. The remaining conditions that the anomaly factorize relate inner products between the vectors a, bi to group theory coefficients and the representations of matter fields (2.9) a · a =9− T 1 i i i (2.10) a · bi = λi Aadj − xRAR 6 R 1 2 i i i (2.11) bi · bi = − λi Cadj − xRCR 3 R ij i j (2.12) bi · bj = λiλj xRSARAS RS ij where xRS denotes the number of hypermultiplets transforming in irreducible representations R, S under the gauge group factors Hi,Hj. ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 329

3. Constraints from anomalies The conditions (2.5-2.12) place strong constraints on the range of possible theories compatible with anomaly cancellation, particularly for small values of T . For one thing, these conditions imply that the inner products (1,T ) a · a, a · bi,bi · bj are all integral and hence that the vectors a, bi ∈ R span an integral lattice Λ (which may be degenerate, for example if several bi are equal). For gauge group factors Hi with an irreducible quartic invariant (nonzero B’s), the integrality of the inner products follows from group theory 2 i identities which show that, e.g.,3|λi CR for all representations R. For group factors with no irreducible quartic invariant, such as SU(2),SU(3),E8,integrality of the inner products depends upon cancellation of additional global anomalies. Details of the proof of integrality are given in [6]. Beyond the integral lattice structure imposed by (2.9-2.12), these conditions also provide strong bounds on the set of possible gauge groups and matter content. To give a ﬁnite bound to the set of allowed G, M for small T we need to impose an additional condition, associated with the constraint that each gauge group component has a sensible physical description without instabilities. This condition amounts to the constraint that there exists a vector j ∈ R1,T such that 2 (3.1) j · bi > 0 ∀i, j =1. The vector j in this equation corresponds to the scalar ﬁelds in the T tensor multiplets of the supergravity theory. Given these conditions, we can prove

Theorem 3.1. For T<9, there are a ﬁnite number of distinct nonabelian gauge groups G and matter representations M satisfying (2.5–2.12), such that (3.1) holds for some j.

This theorem is proven in [4]forT =1,andin[6]forT<9. The proof essentially relies upon geometry in the (1,T) signature space containing the vectors a, bi,j. We present here the main arguments in the proof. Proof. The proof proceeds by contradiction. We assume that there is an infinite family of models with nonabelian gauge groups {Gγ}. For any given model in the family we decompose the gauge group into a product of simple factors Gγ = H1×H2×···×Hk. There are a finite number of groups G with dimension below any fixed bound. For each fixed G, there are a finite number of representations whose dimension is below the bound (2.5) on the number of hypermultiplets. Thus, any infinite family {Gγ} must include gauge groups of arbitrarily large dimension. We divide the possibilities into two cases. (1) The dimension of the simple factors in the groups Gγ is bounded across all γ. In this case, the number of simple factors is unbounded over the family. 330 W. TAYLOR

(2) The dimension of at least one simple factor in Gγ is unbounded. For example, the gauge group is of the form Gγ = SU(Nγ) × G˜γ, where Nγ →∞. Case 1: In this case we can rule out infinite families with bounded values of T . The dimension of each factor Hi is bounded, say by dim Hi ≤ D. Assume that we have an infinite sequence of models whose gauge groups have Nγ factors, with Nγ unbounded. Consider one model in this infinite sequence, with N factors. We divide the factors Hi into 3 classes, depending 2 upon the sign of bi : 2 (1) Type Z: bi =0 2 (2) Type N: bi < 0 2 (3) Type P: bi > 0 Since the dimension of each factor is bounded, the contribution to H − V from −V is bounded below by −ND. For fixed T the total number of hypermultiplets is then bounded by (3.2) H ≤ 273 − 29T + ND ≡ B ∼O(N). This means that the dimension of any irreducible component of M is bounded by the same value B. The number of gauge group factors λ under which any matter field can transform nontrivially is then bounded by 2λ ≤ B, so λ ≤O(ln N). Now, consider the different types of factors. Denote the number of type N, Z, P factors by NN,Z,P , where

(3.3) N = NN + NZ + NP .

We can write the bi’s in a (not necessarily integral) basis where the inner product matrix takes the form Ω = diag(+1, −1, −1,...)as

(3.4) bi =(xi,yi).

For any type P factor, |xi| > |yi|,sobi · bj > 0 for any pair of type P factors. Thus, there are hypermultiplets charged under both gauge groups for every pair of type P factors. A hypermultiplet charged under λ ≥ 2 gauge group factors appears in λ(λ − 1) (ordered) pairs (i, j)withbi · bj > 0, and contributes at least 2λ to the total number of hypermultiplets H. Each ordered pair under which this hypermultiplet is charged then contributes at least 2λ (3.5) ≥ 1 λ(λ − 1) to the total number of hypermultiplets H. It follows that the NP (NP − 1) pairs under which at least one hypermultiplet is charged contribute at least NP (NP − 1) to H,so

(3.6) NP (NP − 1) ≤ B ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 331

Thus, √ √ (3.7) NP < B +1≤O( N) which is much smaller than N for large N. So most of the bis associated with gauge group factors in any infinite family must be type Z or type N. Now consider type N factors. Any set of r mutually orthogonal type N vectors defines an r-dimensional negative-definite subspace of R1,T .This means, in particular, that we cannot have T +1 mutually orthogonal type N vectors. If we have NN type N vectors, we can define a graph whose nodes are the type N vectors, where an edge connects every two nodes associated with perpendicular vectors. Turán’stheorem [22] states that the maximum number of edges on any graph with n vertices which does not contain a subset of T + 1 completely connected vertices is (3.8) (1 − 1/T )n2/2 where the total number of possible edges is n(n − 1)/2. Thus, applying this theorem to the graph described above on nodes associated with type N vectors, the number of ordered pairs with charged hypermultiplets must be at least 2 NN (3.9) − NN . T It then follows that, assuming T is fixed, √ √ (3.10) NN ≤ TB + T ∼O( N).

Finally, consider type Z factors. Vectors bi,bj of the form (3.4) associated with two type Z factors each have |xi| = |yi| and have a positive inner product unless they are parallel, in which case bi · bj =0.Denotebyμ the size of the largest collection of parallel type Z vectors. Each type Z vector is perpendicular to fewer than μ other type Z vectors, so there are at least NZ (NZ − μ) ordered pairs of type Z factors under which there are charged hypers. We must then have

(3.11) NZ (NZ − μ)=(NZ − μ)(N − NP − NN ) ≤ B.

But from (3.7, 3.10) this means that NZ − μ is of order at most O(1) (and is bounded by D as N →∞), while NZ is of order O(N). Thus, all but a fraction of order 1/N of the type Z factors have vectors in a common parallel direction. In [4], we carried out a case-by-case analysis demonstrating that all group + matter conﬁgurations which give type Z factors have a positive value for H − V . All of the μ factors associated with the largest collection of parallel type Z vectors thus contribute positively to H − V ,counting matter charged under these group factors only once. The total contribution to H − V is then bounded by

(3.12) H − V>μ− [(NZ − μ)+NP + NN ](D) ∼O(N) 332 W. TAYLOR

Table 1. Allowed charged matter for an inﬁnite family of models with gauge group H(N). The last two columns give 2 the values of a · b, b in the anomaly polynomial I8.

Group Matter content H − Va· bb2 SU(N)2N N 2 +1 0 −2 1 2 15 (N +8) +1 2 N + 2 N +1 1 −1 1 2 15 (N − 8) +1 2 N − 2 N +1 −1 −1 16 +2 15N +1 2 0 1 2 7 SO(N)(N − 8) 2 N − 2 N −1 −1 1 2 7 Sp(N/2) (N +8) 2 N + 2 N 1 −1 16 +1 15N − 120 which exceeds the bound H − V ≤ 273 − 29T for sufficiently large N.Thus, we have ruled out case 1 by contradiction for any fixed T>0, and in consequence ruled out any infinite family of the type of case 1 with bounded values of T .

Case 2: We now consider the possibility of infinite families with factors of unbounded size. We assume that we have an infinite family of models with gauge groups of the form Gγ = Hγ × G˜γ where dim Hγ →∞.ForSU(N) the F 4 anomaly cancellation condition (3.13) BAdj =2N = xRBR R can only be satisfied at large N when the number of multiplets xR vanishes in all representations other than the fundamental, adjoint, and two-index antisymmetric and symmetric representations (we consider representations and their conjugates to be equivalent for the purposes of this analysis). This follows from the fact that for all other representations, BR grows faster than N. For the representations just listed, indexed in that order, (3.13) becomes

(3.14) 2N = x1 +2Nx2 +(N − 8)x3 +(N +8)x4. The solutions to this equation at large N, along with the corresponding solutions for the other classical groups SO(N),Sp(N) are easy to tabulate, and are listed in Table 1. We discard solutions (x1,x2,x3,x4)=(0, 1, 0, 0) 2 and (0, 0, 1, 1), where a · bi = bi = 0 since for T<9 these relations combined 2 with a > 0 imply that bi = 0 and therefore that the kinetic term for the gauge field is identically zero. The contribution to H − V from each of the group and matter combinations in Table 1 diverges as N →∞. This cannot be cancelled by contributions to −V from an infinite number of factors, for the same reasons which rule out case 1. Thus, any infinite family must have an infinite sub-family, with gauge group of the form Hˆ (M)×H(N)× G˜M,N, ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 333 with both M,N →∞. For any factors Hi,Hj with a · bi,a ·√bj =0,ina (non-integral) basis where Ω = diag(+1, −1, −1,...), and a =( a2, 0, 0,...) writing

(3.15) bi =(xi,yi) √ 2 with xi = a · bi/ a we have 2 (3.16) xixj =(a · bi)(a · bj)/a ≥ bi · bj = xRSARAS. R,S

Since xixj is constant for any infinite family of pairs Hˆ (M),H(N), while AR grows for all representations besides the fundamental, the only possible fields charged under more than one of the infinite factors in Table 1 are bifundamentals. We now consider all possible infinite families built from products of groups and representations in Table 1 which have bifundamental fields and bounded H −V . There are 5 such combinations with two factors. These combinations were enumerated in [4], and are listed in Table 4 in that paper. These combinations include two infinite families shown to satisfy anomaly factorization by Schwarz [21], as well as three other similar families. For example, the simplest such family consists of an infinite sequence of models with gauge group SU(N) × SU(N) and two matter fields transforming in the bifundamental (N,N) representation. This model has H =2N 2,V = 2(N 2−1) so H−V = 2 for each model in the infinite family. In addition to the infinite families built from products of two factors with unbounded dimension, when T>1 there are several infinite families arising from products of three factors with unbounded dimension. For example, there is such a family of models with gauge group G = SU(N −8)×SU(N)×SU(N +8). The other possibilities are described in [6]. For T<9, the models in all of these infinite families with unbounded factors are unacceptable because the gauge kinetic term for one of the factors is always unphysical. This can be shown as follows: For each two-factor infinite family we have two vectors b1,b2 which 2 2 satisfy a · (b1 + b2)=0and(b1 + b2) =0.Butwhena > 0 these conditions imply b1 +b2 = 0, so that j·b1 and j·b2 cannot both be positive. For example, for the theory found by Schwarz with gauge group SU(N) × SU(N)with 2 2 two bifundamental fields, we have a · b1 = a · b2 =0,b1 = b2 = −2,b1 · b2 =2, from which it follows that b1 = −b2. Essentially the same argument works for each three-factor infinite family, where a · (b1 + b2 + b3) = 0. This method of proof breaks down when T>8, where a2 ≤ 0, since then a · b = b2 =0 is not sufficient to prove b = 0. In the following section we give an example of an infinite family of models with no clear inconsistency at T =9. This proves case 2 of the analysis. So we have proven that for T<9 there are a finite number of distinct gauge groups and matter content which satisfy anomaly cancellation with physical kinetic terms for all gauge field factors. We have ruled out infinite families with unbounded numbers of 334 W. TAYLOR gauge group factors at any finite T , though infinite families with unbounded numbers of factors satisfying the anomaly and group theory sign constraints can exist when T is unbounded, as discussed in [6]. We have not ruled out infinite families with a finite number of gauge group factors which become unbounded at finite T>8. Indeed, we give an explicit construction of such a family in Section 5.

Theorem (3.1) strongly constrains the range of possible supergravity theories, at least when T<9. Furthermore, the analysis in the proof of the theorem can be used to give explicit algorithms for systematically enumer- ating all the possible gauge groups and matter content which can appear in acceptable theories, as discussed in [4, 5].

4. 6D supergravities from F-theory 4.1. F-theory models from elliptically fibered Calabi-Yau threefolds. There are many ways in which string theory can be used to construct supergravity theories in various dimensions. In general, such constructions involve starting with a 10-dimensional string theory, and “compactifying” the theory by wrapping 10 − D dimensions of the space-time on a compact (10 − D)-dimensional manifold to give a theory in D dimensions which behaves as a supergravity theory at low energies. At this point in time, we do not have any fundamental “background-independent” definition of string theory, in which all the different compactifications of the theory arise on equal footing. Rather, string theory consists of an assemblage of tools, including low-energy supergravity, perturbative string theory, and nonperturbative structures such as D-branes and duality symmetries, which give an apparently consistent way of constructing and relating the various approaches to string compactification. One of the most general approaches to string compactification is “F-theory” [8, 9]. Technically, F-theory describes certain limits of string compactifications in which certain information (such as Kählermoduli) is lost. Mathematically, an F-theory compactification to D dimensions is characterized by an elliptically fibered Calabi-Yau manifold of dimension 12−D. The data needed to define an F-theory compactification to six dimensions consist of a complex surface B which acts as the base of the fibration, and an elliptic fibration with section over B. The elliptic fibration may have singularities, as long as such singularities can be resolved to give a total space which is Calabi-Yau. The structure of such an elliptic fibration is generally described by a Weierstrass model

(4.1) y2 = x3 + f(u, v)xz4 + g(u, v)z6 where u, v are coordinates on the base B. The functions f,g are sections of the line bundles −4K and −6K respectively, where K is the canonical bundle of B. ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 335

The gauge group and matter structure of the low-energy theory associated with an F-theory compactification on an elliptically fibered Calabi-Yau threefold are determined by the singularity structure of the fibration. Simple nonabelian gauge group factors Hi in the associated 6D theory are associated with codimension one loci in the base where the fiber degenerates. Such codimension one singularities were classified by Kodaira [23]. These singularities can be associated with A-D-E Dynkin diagrams and give rise to the corresponding simply-laced gauge group factors. Nontrivial monodromies about codimension 2 singularities expand the possible gauge group factors to include the non-simply laced groups, Sp(N),F4 and G2 [25]. Given a Weier- strass form for the elliptic fibration, the singularity types can be determined in terms of the orders of vanishing of f,g, and the discriminant locus (4.2) Δ = 4f 3 +27g2, which is a section of −12K. The resulting gauge group can be then determined via Tate’s algorithm [24]. For example, a singularity of Kodaira type In, corresponding to the Dynkin diagram An−1, arises on a codimension one singularity locus where neither f nor g vanishes, but Δ vanishes to order n. Such a singularity gives a gauge group factor SU(n) when the base of the fibration is (complex) one-dimensional, and SU(n)orSp(n/2) in the low-energy 6D theory when the base is 2 dimensional. A complete list of the possible singularity types and associated orders of vanishing of f,g,Δ can be found in, for example,[9, 25]. Each nonabelian gauge group factor Hi is associated with a singularity locus on an effective irreducible divisor ξi ∈ H2(B; Z). Matter fields (hypermultiplets) in the low-energy 6D theory are associated with codimension two singularities in the elliptic fibration. Generally such singularities arise at intersections of divisors associated with codimension one singularities. For example, an An−1 singularity intersecting an Am−1 singularity gives rise to a codimension two An+m−1 singularity, where Δ vanishes to order n+m. The resulting matter fields transform in the fundamental of the associated SU(n) gauge group factor and the (anti)-fundamental of the associated SU(m) gauge group factor. (This can be thought of as splitting the adjoint representation of SU(n + m) as a representation of SU(n) × SU(m)). The singularities associated with possible matter representations which can arise in this fashion have not been fully classified, but an analysis of many such representations appears in [27, 28]. As discussed in [9], it was shown by Kodaira for a base of complex dimension 1, and subsequently generalized to higher dimension by other authors, that the total space of the elliptic fibration can be resolved to a Calabi-Yau manifold when −12K = Δ, where Δ is the total divisor class of the singular (discriminant) locus. The component of the discriminant locus for each nonabelian factor is given by an effective irreducible divisor ξi with a multiplicity νi. For example, for an An−1 singularity, νi = n.The residual singularity locus, which is not associated with nonabelian gauge 336 W. TAYLOR group structure, is associated with another effective divisor Y , so that the Kodaira relation is (4.3) −12K =Δ= νiξi + Y. i In this fashion, a wide range of 6D supergravity theories can be constructed from particular choices of elliptically fibered Calabi-Yau threefolds. In all such situations, the resulting gauge group and hypermultiplet matter content automatically satisfy the anomaly cancellation conditions [26, 27]. We now discuss the connection between the data of the low-energy theory and that of the F-theory compactification.

4.2. Mapping 6D supergravities to F-theory. To identify the set of low-energy 6D supergravity models which are compatible with F-theory, we would like to have a systematic approach to constructing an F-theory model which realizes a particular low-energy gauge group G and matter representation M. In fact, as shown in [26, 27], the anomaly cancellation conditions give a close correspondence between the data of a geometric F-theory construction and the corresponding 6D supergravity theory. This correspondence characterizes a map from low-energy data to topological F-theory data [5, 6]; identifying those cases where such a map cannot be consistently deﬁned allows us to determine which G, M cannot be found through any F-theory construction. The connection between the 6D supergravity theory and F-theory data is given by the following correspondences [5, 6, 26, 27]: the number of tensor multiplets T in the 6D theory is equal to h1,1 − 1 of the base B of the F-theory elliptic ﬁbration. The lattice Λ determined by the anomaly cancellation conditions through (2.10–2.12) can be embedded into the integral second homology of B

(4.4) Λ → H2(B; Z) in such a way that

(4.5) a → K (4.6) bi → ξi

Thus, for example, we have

(4.7) a · a =9− T =10− h1,1 = K2.

For an elliptic ﬁbration which resolves to a smooth Calabi-Yau threefold, 2 the possible smooth bases B consist of P and the Hirzebruch surfaces Fm (m ≤ 12) and blow-ups of these bases, as well as the somewhat trivial cases of K3 and the Enriques surface [9]. For example, at T = 1, the set of possible F-theory bases are Fm,m≤ 12. In this case, a basis for H2(B; Z) is given by ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 337

2 2 Dv,Ds where Dv = −m, Dv · Ds =1,Ds =0.Asshownin[5], in this case, the vector b associated with a given gauge group factor maps to the divisor α m α˜ (4.8) b → ξ = Dv + Ds + Ds, 2 2 2 where α, α˜ satisfy −a · b = α +˜α, b2 = αα/˜ 2. In the following section we give some explicit examples of 6D supergravity theories and the corresponding F-theory constructions. In some cases, there is no map of the form (4.4) where a, bi map to effective divisors in any B with the correct properties for an F-theory compactification. For these models, which are apparently consistent from the low-energy.of view and yet have no F-theory construction, we can identify which aspect of the F-theory structure breaks down. We include some examples of this type in the following section. In some other cases, there are multiple possible F-theory realizations of the specified 6D gauge group G and matter representation M. Thus, the correspondence does not give a uniquely defined map from low-energy data to F-theory. It does, however, give us a framework for analyzing which low-energy theories admit an F-theory realization. (Note that in those cases where the gauge group and matter content are not sufficient to uniquely determine the F-theory construction, further information about the low-energy theory may single out a particular F-theory model.)

5. Examples In this section we consider some examples of possible gauge groups and matter representations for 6D supergravities. We begin with two examples of theories which seem to admit consistent embeddings in F-theory, and then consider two speciﬁc examples and one inﬁnite family of theories which cannot be realized using standard F-theory techniques (or any other known string construction).

Example 1. T =0,G= SU(N),M =3× +(24− N) ×

A simple class of examples involve theories with no tensor multiplets (T = 0). In this case, a2 = 9, and the space R1,T = R1 is just one-dimensional Euclidean space, so −a = (3) in an integral basis. Any SU(N) factor is associated with a vector b =(k). For simplicity, we assume that the only matter consists of F hypermultiplets transforming in the fundamental representation and A hypermultiplets in the two-index antisymmetric representation ( ). We can use the group theory coeﬃcients

Af =1,Bf =1,Cf =0 (5.1) Aa = N − 2,Ba = N − 8,Ca =3 AAdj =2N, BAdj =2N, CAdj =6 338 W. TAYLOR for SU(N) to write the anomaly cancellation equations: 0=2N − F − A(N − 8) (5.2) −a · b =3k = −(2N − F − A(N − 2))/6=A b2 = k2 =(−6+3A)/3=A − 2 For k = 1, we have solutions for A =3,with F =24− N for N ≤ 24. This gives a set of models with simple gauge group G = SU(N) which satisfies all anomaly cancellation conditions. The gauge kinetic term condition is satisfied for any positive j =(j0). For the corresponding F-theory construction, we have P2 as the only possible base with h1,1 = T + 1 = 1. The second homology of P2 is generated by the cycle H, corresponding to the hyperplane divisor, satisfying H·H =1. The map from Λ → H2(B; Z) is then given by the trivial map (1) → H,so that (5.3) −a =(3)→ 3H = −K b =(1)→ H. This gives the topological data associated with the corresponding F-theory construction. To verify that there exists an F-theory model from an elliptic fibration associated with this data one must construct a Weierstrass model with singularity type AN−1 on the divisor H. We discuss explicit construction of Weierstrass models in the following section. One can similarly construct a variety of anomaly-free models with gauge group of the form G = i SU(Ni) and matter in fundamental, antisymmetric, and bifundamental representations. We explicitly constructed all such models for T =1 in [5] and found that all appear to correspond to acceptable topological data for F-theory constructions.

Example 2. T =1,G= E8 × E7

This gauge group has two simple factors, and thus two vectors b8,b7 1,1 in R .ForE8 and E7 we have the normalization factors λ8 =60,λ7 = 12. Neither E8 or E7 has a fourth order invariant, so (2.6) is automatic. Assume we have F matter ﬁelds in the fundamental representation of E7,withno matter transforming under the E8 (the fundamental representation for E8 is equivalent to the adjoint). The E8 anomaly equations state that 2 (5.4) −a · b8 = −10,b8 = −12 2 where a = 8. In a basis where 01 (5.5) Ωαβ = 10 we can choose −a =(2, 2), and b8 =(1, −6). The anomaly equations for E7, 2 2 including the trF7 trF8 condition, state that 2 (5.6) −a · b7 =2F − 6,b7 =2F − 8,b7 · b8 =0. ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 339

These equations have the unique solution b7 =(1, 6), realized when F =10. Thus, the anomaly cancellation conditions for this class of theory are only realized for this value of F . In this case we can read the corresponding F-theory model directly from (4.8): 1 (5.7) b8 = (α8, α˜8) → ξ8 = Dv +(m/2 − 6)Ds 2 1 (5.8) b7 = (α7, α˜7) → ξ7 = Dv +(m/2+6)Ds 2 which gives eﬀective irreducible divisors precisely when m = 12. In this case we have an acceptable F-theory construction on the base manifold F12 [9], with ξ8 = Dv,ξ7 = Du = Dv +12Ds. This model was identiﬁed in the corresponding heterotic string formulation by Seiberg and Witten in [29]. We now present several models which seem consistent from the point of view of anomalies and gauge kinetic terms, but which have no known string theory realization. Example 3. T =1,G= SU(4),M =Adj+10× +40×

For this model the anomaly cancellation conditions give a2 −a · b 810 (5.9) Λ = = −a · bb2 10 10 This lattice cannot be embedded in any unimodular lattice [6]. Since the intersection form on H2(B; Z) must give a unimodular lattice by Poincaré duality, this model cannot be embedded in F-theory. Example 4. T =1,G= SU(8),M = Using the coefficients A =10,B=16,C= 3 for the symmetric representation, we find that the anomaly conditions give a2 −a · b 8 −1 (5.10) Λ = = . −a · bb2 −1 −1

While this lattice can be embedded in H2(B; Z) for some F-theory bases, the resulting divisor b is not an eﬀective irreducible divisor. For example, we could choose −a →−K =2Dv +3Ds on F1 giving b →−Dv. But since the divisors ξi carrying the singularity locus on the F-theory base must be irreducible eﬀective divisors, this will not work for constructing an F-theory realization of this model using existing methods. Example 5. T =9,G= SU(N) × SU(N),M =2× ( , ) For this gauge group and matter representations, at T = 9 we need vectors −a, b1,b2 with inner product matrix ⎛ ⎞ 00 0 (5.11) Λ = ⎝ 0 −22⎠ . 02−2 340 W. TAYLOR

In a basis with Ω = diag(+1, −1, −1,...), this can be realized through the vectors

−a =(3, −1, −1, −1, −1, −1, −1, −1, −1, −1) b1 =(1, −1, −1, −1, 0, 0, 0, 0, 0, 0) b2 =(2, 0, 0, 0, −1, −1, −1, −1, −1, −1)

This satisﬁes the correct gauge kinetic term sign conditions j · bi > 0for j =(1, 0, 0,...). Thus, this inﬁnite family of models cannot be ruled out by the low-energy constraints we have imposed here. These models are not valid in F-theory, however, when N>12, since in that case

(5.12) Y = −12K − N(ξ1 + ξ2)=(N − 12)K which cannot be eﬀective for N>12 since −K is eﬀective.

6. String universality? The results presented so far demonstrate that, although anomaly cancellation and other known constraints provide substantial limits on the range of gauge groups and matter representations that can be realized in supersymmetric six-dimensional quantum theories of gravity, not all models which satisfy these constraints can at this time be realized in string theory. Some of the examples presented in the previous section seem like counterexamples to the string universality conjecture stated in the introduction. At this time, however, our understanding of the situation is still incomplete. We do not have a complete deﬁnition of string theory that allows us to determine with certainty which low-energy models can or cannot be realized within string theory. It is also quite possible that string theory imposes additional constraints, like the sign condition on the gauge kinetic terms, which may be seen as quantum consistency conditions given the data of the low-energy theory. To prove or disprove the conjecture of string universality in six dimensions, it seems that further progress is needed in understanding both string theory and low-energy supergravity. There are also some purely mathematical questions whose solutions may contribute signiﬁcantly to making progress in this direction. In this concluding section, we summarize some of the most relevant open questions, both mathematical and physical, in this regard. Most of these questions are discussed in further detail in [6].

6.1. Mathematics questions. There are a number of concrete mathematical questions related to F-theory whose solutions would help clarify the precise set of theories which can be realized in string theory via F-theory. We brieﬂy summarize here a few of these questions that are most closely related to the discussion in this paper. ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 341

Weierstrass models We have argued that the correspondence between the low-energy data associated with the anomaly cancellation conditions and the mathematical structure of F-theory leads to a map (4.4) from the anomaly lattice Λ to the second homology lattice of the F-theory base B. In many cases this map associates with each nonabelian factor in the gauge group an effective divisor on B. In order to show that the model can be realized in F-theory, however, it is necessary to construct an explicit Weierstrass model with the desired singularity locus associated with these divisors. In many cases it seems that this can be done. For a class of models with T = 1 and gauge group SU(N), we showed in [5] that there is a precise correspondence between the number of degrees of freedom needed to fix the desired singularity structures on the proper divisor locus and the number of degrees of freedom encoded in the scalar matter fields through H−V . It may be possible to give a mathematical proof that the existence of a Weierstrass model follows given some necessary topological conditions on the divisor locus. Such a proof would be a useful step forward in identifying a large class of the anomaly-free models which have F-theory realizations.

Classifying matter singularities As mentioned above, while codimension one singularities in elliptic fibrations are well-understood, codimension two singularities arising at the intersection of codimension one singularities are not completely understood. Such singularities give rise in F-theory compactifications to a variety of different representation structures for matter fields. In studying the set of anomaly-free supergravity theories in six dimensions, we have encountered some novel matter representations which may or may not be associated with valid F-theory compactifications. For example, in [5], we found some models with an SU(N) gauge group and matter in a four-index antisymmetric representation. While some F-theory constructions are known which give rise to three-index antisymmetric representations of SU(N)[28] there is as yet no understanding of a singularity structure which would give a four-index antisymmetric representation. A complete classification of codimension two singularities in elliptic fibrations of Calabi-Yau manifolds would be of great assistance in systematically understanding the set of allowed F-theory compactifications.

Classification of elliptically fibered Calabi-Yau manifolds It has been shown by Gross [30] that the number of distinct topological types of elliptically fibered Calabi-Yau threefolds is finite up to birational equivalence. Finiteness of the set of topologically distinct elliptically-fibered Calabi-Yau threefolds is shown in [6] using minimal surface theory and the fact that the Weierstrass form for an elliptic fibration over a fixed base has a finite number of possible distinct singularity structures. These arguments, however, do not give a clear picture of how such compactifications 342 W. TAYLOR can be systematically classified. A complete mathematical classification of elliptically fibered Calabi-Yau threefolds would be helpful in understanding the range of F-theory compactifications. The analogue of this question for four dimensions, while probably much more difficult, would be of even greater interest, since at this time we have very little handle on the scope of the space of four dimensional supergravity theories which can be realized through F-theory compactifications on Calabi-Yau fourfolds.

6.2. Physics questions. Some of the most difficult questions which need to be answered to prove or disprove the conjecture of string universality in six dimensions are essentially physical in nature. The key questions amount to A) Are there as-yet unknown string compactifications that extend the range of 6D theories beyond those attainable using known F-theory constructions? B) Can we identify new quantum consistency constraints on low-energy theories that constrain the space of allowed models to match more closely the set which can be realized through string compactifications? While A) is certainly an important question, it seems that the most likely opportunity for narrowing the gap between the space of allowed theories and the space of string constructions is by identifying new constraints on low-energy theories. If string universality is even approximately correct, it is probably necessary to find a low-energy quantum consistency condition which rules out the infinite families of theories compatible with anomaly cancellation but which cannot be realized in F-theory. One place to look for such constraints is in the set of conditions imposed by F-theory. Considering the examples 3-5 in the previous section which cannot be realized in F-theory, we can identify some possible constraints that may be needed in addition to anomaly cancellation and the gauge kinetic term sign constraint. It seems plausible that some of these constraints may in fact be needed for quantum consistency of any low-energy theory. We briefly mention here a few of these constraints, and make a few comments on how they may be realized.

Unimodular lattice embedding As we see from example 3, a necessary condition for a given low-energy theory to be realizable through F-theory is that the lattice Λ associated with the anomaly cancellation constraints must be embeddable in a unimodular lattice. Associated with any of the supersymmetric six dimensional gravity theories we are considering here, there is a lattice of dyonic strings charged under the anti-symmetric tensor ﬁelds of the theory, carrying a natural inner product of signature (1,T). Some aspects of this lattice are discussed in [6]. We do not know any reason why this lattice must be unimodular for quantum consistency, but it is possible that this may be necessary for unitary of the theory, or for some other consistency reason. Note that an ANOMALY CONSTRAINTS IN 6D QUANTUM GRAVITY 343 analogous unimodularity constraint follows from modular invariance for the heterotic fundamental string construction. Demonstrating that the string dyonic lattice in a general 6D theory needs to be unimodular for a quantum theory to exist would help in ruling out some of the apparently consistent theories which we do not know how to realize in F-theory.

Kodaira constraint and effective divisors As we noted in examples 4 and 5, some apparently consistent theories cannot be realized in F-theory because the topological data for the F-theory model realized through (4.4) leads to certain divisors in F-theory not satisfying an effectiveness condition which is needed for the F-theory construction. In particular, the divisors −K, ξi associated with the vectors −a, bi must all be effective (inside the Mori cone), as must the residual divisor locus Y associated with the vector 12a − i νibi. This condition has no obvious counterpart in the low-energy theory. It seems plausible, however, that some of these conditions may be associated with sign constraints in the low-energy theory analogous to the sign constraint on the gauge kinetic term. For example, a term proportional to a·j trR2 must appear in the action of the theory by supersymmetry. Arguments analogous to those in [31] may suggest that this term must have a particular sign for consistency with causality in field theory. It is less clear how the condition on the residual divisor locus can be seen from the low-energy theory, but it seems possible that such a condition would arise from consistency with supersymmetry, since this is the ultimate origin of the Kodaira/Calabi-Yau condition in F-theory. To conclude, we have not yet proven or disproven the conjecture of string universality in six dimensions. Simply pursuing this conjecture, however, has given new insights into the global structure of the space of 6D supergravity theories and string compactifications, and has suggested some intriguing avenues for future progress, both in physics, and in mathematics.

References [1] M. R. Douglas and S. Kachru, “Flux compactiﬁcation,” Rev. Mod. Phys. 79, 733 (2007) arXiv:hep-th/0610102. [2] C. Vafa, “The string landscape and the swampland,” arXiv:hep-th/0509212. [3] V. Kumar and W. Taylor, “String Universality in Six Dimensions,” arXiv:0906.0987 [hep-th]. [4] V. Kumar and W. Taylor, “A bound on 6D N = 1 supergravities,” JHEP 0912, 050 (2009) arXiv:0910.1586 [hep-th]. [5] D. R. Morrison, V. Kumar and W. Taylor, “Mapping 6D N = 1 supergravity to F-theory,” arXiv:0911.3393 [hep-th]. [6] D. R. Morrison, V. Kumar and W. Taylor, “Global aspects of the space of 6D N =1 supergravities,” arXiv:1008.1062 [hep-th]. [7] A. Adams, O. DeWolfe, and W. Taylor, “String universality in ten dimensions,” arXiv:1006.1352 [hep-th]. [8] C. Vafa, “Evidence for F-Theory,” Nucl. Phys. B 469, 403 (1996) arXiv:hep- th/9602022. [9] D. R. Morrison and C. Vafa, “Compactiﬁcations of F-Theory on Calabi–Yau Three- folds – I,” Nucl. Phys. B 473, 74 (1996) arXiv:hep-th/9602114. 344 W. TAYLOR

D. R. Morrison and C. Vafa, “Compactifications of F-Theory on Calabi–Yau Three- folds – II,” Nucl. Phys. B 476, 437 (1996) arXiv:hep-th/9603161. [10] H. Nishino and E. Sezgin, “Matter And Gauge Couplings Of N=2 Supergravity In Six-Dimensions,” Phys. Lett. B 144, 187 (1984). H. Nishino and E. Sezgin, “The Complete N=2, D = 6 Supergravity With Matter And Yang-Mills Couplings,” Nucl. Phys. B 278, 353 (1986). [11] L. J. Romans, “Self-duality for interacting fields: covariant field equations for six dimensional chiral supergravities,” Nucl. Phys. B 276, 71 (1986). [12] M. B. Green, J. H. Schwarz and P. C. West, “Anomaly Free Chiral Theories In Six- Dimensions,” Nucl. Phys. B 254, 327 (1985). [13] K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D21, 2848 (1980). [14] L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,” Nucl. Phys. B 234, 269 (1984). [15] M. F. Atiyah and I. M. Singer, “Dirac operators coupled to vector potentials,” Proc. Natl. Acad. Sci. 81 2597-600 (1984). [16] L. Alvarez-Gaumé and P. Ginsparg, Ann. Phys. 161 423 (1985) [17] M. B. Green and J. H. Schwarz, “Anomaly Cancellation In Supersymmetric D=10 Gauge Theory And Superstring Theory,” Phys. Lett. B 149, 117 (1984). [18] M. B. Green, J. H. Schwarz and P. C. West, “Anomaly Free Chiral Theories In Six- Dimensions,” Nucl. Phys. B 254, 327 (1985). [19] A. Sagnotti, “A Note on the Green-Schwarz mechanism in open string theories,” Phys. Lett. B 294, 196 (1992) arXiv:hep-th/9210127. [20] J. Erler, “Anomaly Cancellation In Six-Dimensions,” J. Math. Phys. 35, 1819 (1994) arXiv:hep-th/9304104. [21] J. H. Schwarz, “Anomaly-Free Supersymmetric Models in Six Dimensions,” Phys. Lett. B 371, 223 (1996) arXiv:hep-th/9512053. [22] P. Turán . “On an extremal problem in graph theory,” Matematikai és Fizikai Lapok 48: 436-452 (1941). [23] K. Kodaira, Ann. Math. 77 563 (1963); Ann. Math. 78 1 (1963). [24] J. Tate, in: Modular functions of one variable IV, Lecture Notes in Math, vol. 476, Springer-Verlag, Berlin (1975). [25] M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov and C. Vafa, “Geometric singularities and enhanced gauge symmetries,” Nucl. Phys. B 481, 215 (1996) arXiv:hep-th/9605200. [26] V. Sadov, “Generalized Green-Schwarz mechanism in F theory,” Phys. Lett. B 388, 45 (1996) arXiv:hep-th/9606008. [27] A. Grassi, D. R. Morrison, “Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds”, J. Algebraic Geom. 12, 321-356 (2003) arXiv:math/0005196. [28] A. Grassi, D. R. Morrison, “Anomalies and the Euler characteristic of elliptically fibered Calabi-Yau threefolds,” to appear. [29] N. Seiberg and E. Witten, “Comments on String Dynamics in Six Dimensions,” Nucl. Phys. B 471, 121 (1996) arXiv:hep-th/9603003. [30] M. Gross, “A finiteness theorem for elliptic Calabi-Yau threefolds,” Duke Math. J. 74, 2, 271-299 (1994) arXiv:alg-geom/9305002. [31] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, “Causal- ity, analyticity and an IR obstruction to UV completion,” JHEP 0610, 014 (2006) arXiv:hep-th/0602178.

Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology; 77 Massachusetts Avenue, Cambridge, MA 02139, USA E-mail address: [email protected] Surveys in Diﬀerential Geometry XV

A New Look at the Path Integral of Quantum Mechanics

Edward Witten

Abstract. The Feynman path integral of ordinary quantum mechanics is complexiﬁed and it is shown that possible integration cycles for this complexiﬁed integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern- Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N = 4 super Yang- Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N = 4 path integral in four dimensions.

Contents 1. Introduction 346 2. Integration cycles for quantum mechanics 350 2.1. Preliminaries 350 2.2. The basic Feynman integral 351 2.3. Analytic continuation 353 2.4. The simplest integration cycles 355 2.5. Review of Morse theory 357 2.6. A new integration cycle for the Feynman integral 360 2.7. I and the A-model 362 2.8. Interpretation in sigma-model language 364 2.9. The physical model 368 2.9.1. The boundary condition and localization 370 2.10. Recovering the Hilbert space 372 3. Hamiltonians 374 3.1. Rerunning the story with a Hamiltonian 374 c 2011 International Press

345 346 E. WITTEN

3.2. Hamiltonians and superpotentials 377 3.2.1. Hyper-Kahler symmetries 379 3.2.2. An example 380 3.2.3. General analysis 381 4. Running the story in reverse 382 4.1. Sigma-models in one dimension 382 4.2. Sigma-models in two dimensions 388 5. Analogs with gauge ﬁelds 390 5.1. Supersymmetric quantum mechanics with gauge ﬁelds 391 5.1.1. Construction of the model 391 5.1.2. Geometrical interpretation 395 5.1.3. Gauge-invariant integration cycles 402 5.2. Application to Chern-Simons gauge theory 405 5.2.1. The Chern-Simons form as a superpotential 405 5.2.2. The Chern-Simons path integral from four dimensions 408 5.2.3. Comparison to a sigma-model 411 5.2.4. Elliptic boundary conditions 411 5.3. Quantization with constraints 412 Appendix A. Details on the four-dimensional boundary condition 413 A.1. Ellipticity 415 A.2. A generalization 417 Acknowledgements 418 References 418

1. Introduction The Feynman path integral in Lorentz signature is schematically of the form (1.1) DΦ exp (iI(Φ)) , where Φ are some fields and I(Φ) is the action. Frequently, I(Φ) is a real- valued polynomial function of Φ and its derivatives (the polynomial nature of I(Φ) is not really necessary in what follows, though it simplifies things). There is also a Euclidean version of the path integral, schematically (1.2) DΦ exp (−I(Φ)) , where now I(Φ) is a polynomial whose real part is positive definite, and which is complex-conjugated under a reversal of the spacetime orientation.1 1 General Relativity departs from this framework in a conspicuous way: in Euclidean signature, the real part of its action is not positive definite. This has indeed motivated the proposal [1] that the Euclidean path integral of General Relativity must be carried out THE PATH INTEGRAL OF QUANTUM MECHANICS 347

The analogy between the Feynman path integral and an ordinary ﬁnite- dimensional integral has often been exploited. For example, as a prototype for the Euclidean version of the Feynman integral, we might consider a one- dimensional integral ∞ (1.3) I = dx exp(S(x)), −∞ where S(x) is a suitable polynomial, such as

(1.4) S(x)=−x4/4+ax, with a a parameter. One thing which we can do with such an integral is to analytically continue the integrand to a holomorphic function of z = x + iy and carry out the integral over a possibly diﬀerent integration cycle in the complex z-plane:

(1.5) IΓ = dz exp(S(z)). Γ The integral over a closed contour Γ will vanish, as the integrand is an entire function. Instead, we take Γ to connect two distinct regions at infinity in which Re S(z) →−∞. In the case at hand, there are four such regions (with the argument of z close to kπ/2, k =0, 1, 2, 3) and hence there are essentially three integration cycles Γr, r =0, 1, 2. (The cycle Γr interpolates between k = r and k = r+1, as shown in the figure.) In general, as reviewed in [2], the integration cycles take values in a certain relative homology group. In the case at hand, the relative homology is of rank three, generated by Γ0, Γ1, Γ2. Given a similar integral in n dimensions, (1.6) dx1 dx2 ...dxn exp (S(x1,...,xn)) Rn again with a suitable polynomial S, one can analytically continue from real variables xi to complex variables zi = xi + iyi and replace (1.6) with an integral over a suitable integration cycle Γ ⊂ Cn: (1.7) dz1 dz2 ...dzn exp (S(z1,...,zn)) . Γ The appropriate integration cycles are n-cycles, simply because what we are trying to integration is the n-form dz1 dz2 ...dzn exp(S(z1,...,zn)). Of course, the differential form that we are trying to integrate is middle- dimensional simply because, in analytically continuing from Rn to Cn,we have doubled the dimension of the space in which we are integrating. So what over an integration cycle different from the usual space of real fields. The four-dimensional case is difficult, but in three dimensions, the possible integration cycles can be understood rather explicitly [2]. 348 E. WITTEN

Γ1 Γ0

Γ2 Γ3

Figure 1. Integration cycles for the integral IΓ of eqn. (1.5). The four cycles obey one relation Γ1 +Γ1 +Γ2 +Γ3 = 0, and any three of them give a basis for the space of possible integration cycles.

began in the real case as a differential form of top dimension has become middle-dimensional upon analytic continuation. In [2], it was shown that, at least in the case of three-dimensional Chern- Simons gauge theory, these concepts can be effectively applied in the infinite- dimensional case of a Feynman integral. But what do we learn when we do this? When one constructs different integration cycles for the same integral – or the same path integral – how are the resulting integrals related? For one answer, return to the original example IΓ. Regardless of Γ (using only the facts that it is a cycle, without boundary, that begins and ends in regions where the integrand is rapidly decaying, so that one can integrate by parts), IΓ obeys the differential equation d3 (1.8) − a IΓ =0. da3

Indeed, IΓ where Γ runs over a choice of three independent integration cycles gives a basis of the three-dimensional space of solutions of this third-order differential equation. For quantum field theory, the analog of (1.8) are the Ward identities obeyed by the correlation functions. Like (1.8), they are proved by integration by parts in function space, and do not depend on the choice of the integration cycle. One might think that different integration cycles would correspond to different vacuum states in the same quantum theory, but this is not always right. In some cases, as we will explain in section 2.4 with an explicit example, different integration cycles correspond to different quantum systems that have the same algebra of observables. In other cases, the interpretation is more exotic. THE PATH INTEGRAL OF QUANTUM MECHANICS 349

The ﬁrst goal of the present paper is to apply these ideas to a particularly basic case of the Feynman path integral. This is the phase space path integral of nonrelativistic quantum mechanics with coordinates and momenta q and p: (1.9) Dp(t) Dq(t) exp i (p dq − H(p, q)dt) .

(H(p, q) is the Hamiltonian and plays a secondary role from our point of view.) Integration cycles for this integral are analyzed in section 2. There are some fairly standard integration cycles, such as the original one assumed by Feynman, with p(t)andq(t) being real. The main new idea in this paper is that by restricting the integral to complex-valued paths p(t),q(t) that are boundary values of pseudoholomorphic maps – in a not obvious sense – we can get a new type of integration cycle in the path integral of quantum mechanics. Moreover, this cycle has a natural interpretation in a two- dimensional quantum field theory – a sigma-model in which the target is the complexification of the original classical phase space. The sigma-model is in fact a topologically twisted A-model, and the integration cycle can be described using an exotic type of A-brane known as a coisotropic brane [3]. It has been known from various points of view [4–9] that there is a relationship between the A-model and quantization. In the present paper, we make a new and particularly direct proposal for what the key relation is: the most basic coisotropic A-brane gives a new integration cycle in the Feynman integral of quantum mechanics. The fact that boundary values of pseudoholomorphic maps give a middle- dimensional cycle in the loop space of a symplectic manifold (or classical phase space) is one of the main ideas in Floer cohomology [10]. For an investigation from the standpoint of field theory, see [11–13]. The middle- dimensional cycles of Floer theory are not usually interpreted as integration cycles, because there typically are no natural middle-dimensional forms that can be integrated over these cycles. In the present paper, we first double the dimension by complexifying the classical phase space, whereupon the integrand of the usual Feynman integral becomes a middle-dimensional form that can be integrated over the cycles given by Floer theory of the complexified phase space. The relationship we describe between theories in dimensions one and two has an analog in dimensions three and four. Here the three-dimensional theory is Chern-Simons gauge theory, with a compact gauge group G,and the four-dimensional theory is N = 4 super Yang-Mills theory, with the same gauge group. To some extent, the link between the two was made in [2]. It was shown that to define an integration cycle in three-dimensional Chern- Simons theory, it is useful to add a fourth variable and solve certain partial differential equations that are related to N = 4 super Yang-Mills theory. Here we go farther and show exactly how a quantum path integral in N =4 super Yang-Mills theory on a four-manifold with boundary can reproduce 350 E. WITTEN the Chern-Simons path integral on the boundary, with a certain integration cycle. This has an application which will be described elsewhere [14]. The application involves a new way to understand the link [15] between BPS states of branes and Khovanov homology [16] of knots. In sections 2 and 3 of this paper, we begin with the standard Feynman integral of quantum mechanics and motivate its relation to a twisted supersymmetric theory in one dimension more. In section 4, we run the same story in reverse, starting in the higher dimension and deducing the relation to a standard Feynman integral in one dimension less. Some readers might prefer this second explanation. Section 5 generalizes this approach to gauge fields and contains the application to Chern-Simons theory. What is the physical interpretation of the Feynman integral with an exotic integration cycle? In the present paper, we make no claims about this, except that it links one and two (or three and four) dimensional information in an interesting way.

2. Integration cycles for quantum mechanics 2.1. Preliminaries. A classical mechanical system is described by a 2n-dimensional phase space M, which is endowed with a symplectic structure. The symplectic structure is described by a two-form f that is closed

(2.1) df =0, and also nondegenerate, meaning that the matrix fab deﬁned in local coordi- a a b ab nates x ,a=1,...,2n by f = a

The Poisson bracket of two functions u, v on M is deﬁned by { } ab ∂u ∂v ∂u ∂v − ∂u ∂v (2.3) u, v = f a b = s s . ∂x ∂x s ∂q ∂ps ∂ps ∂q

In quantization, one aims to associate to that data a Hilbert space H and an algebra R of observables. H will be ﬁnite-dimensional if and only if M has ﬁnite volume. In that case, to elements U1,...,Un of R,onecan associate the trace

(2.4) TrHU1U2 ...Un, which describes R and its action on H, up to unitary equivalence. In most physical applications, there is an element H of R known as the Hamiltonian, THE PATH INTEGRAL OF QUANTUM MECHANICS 351 and one is particularly interested in traces

(2.5) TrHU1(t1)U2(t2) ...Un(tn) exp(−iHt), where for U ∈R, U(t)isdefinedasexp(−iHt)U exp(iHt). However, the basic problem of quantization2 is to associate to M a Hilbert space H and algebra R, and what we will say about this problem in the present paper mostly has nothing to do with the choice of H.WewillomitH (that is, set H = 0) except in section 3. A more incisive approach to quantum mechanics is to consider not only a trace as in (2.5) but matrix elements ψf |U1U2 ...Un|ψi between initial and final quantum states |ψi, |ψf . (This also avoids the technical difficulty that if M has infinite volume, H is infinite-dimensional and the traces in (2.4) and (2.5) may diverge, depending on the Ui and H.) However, the approach to Feynman integrals in the present paper is most easily explained if we begin with traces rather than matrix elements. The additional steps involved in describing matrix elements between initial and final states are sketched in section 2.10.

2.2. The basic Feynman integral. The basic Feynman integral represents the trace (2.4) as an integral over maps from S1 to M: r s (2.6) TrH U1U2 ...Un = Dpr(t) Dq (t) exp i psdq U × u1(t1)u2(t2) ...un(tn).

r Here pr(t)andq (t) are periodic with a period of, say, 2π, so they deﬁne a map T : S1 →M, or in other words a point in the free loop space U of M. The uα are functions on M that upon quantization will correspond to the operators Uα. We have assigned a time tα to each function uα, but as we have taken the Hamiltonian to vanish, all that matters about the tα is their cyclic ordering on S1.Asisusual,u(t) is an abbreviation for u(T (t)). As written, the Feynman integral depends on a choice of canonical coor- r dinates q and momenta pr. We could, for example, make a canonical trans- s s formation from p, q to −q, p, replacing psdq with − q dps. This amounts s to adding to the exponent of the path integral a term d(−psq ) = 0, where the vanishing holds because the functions are periodic in time. Rather than picking a particular set of canonical coordinates, a more intrinsic approach is to observe that, as the two-form f is closed, it can be

2 Many elementary aspects of this problem are not as well-known as they might be. For example, the passage from classical mechanics to quantum mechanics does not map Poisson brackets to commutators, except in leading order in (which we set to 1 in the present paper) and certain special cases such as quadratic functions on M. A review can be found in [9]. 352 E. WITTEN regarded as the curvature of an abelian gauge field b: 2n a (2.7) f =db, b = ba dx . a=1 Then we can write (2.6) more intrinsically as a a (2.8) TrH U1U2 ...Un = Dx (t) exp i badx U × u1(t1)u2(t2) ...un(tn). We say that the Dirac condition is obeyed if the periods of f are integer multiples of 2π. In that case, a unitary line bundle L →Mwith a connection of curvature f exists and we take b to be that connection; its structure group is U(1). L is called a prequantum line bundle. If H1(M,Z) = 0, there are inequivalent choices of L and quantization depends on such a choice. When a L exists, the factor exp i badx is the holonomy of the connection b on L (pulled back to S1 via the map T : S1 →M). If the Dirac condition is not obeyed, the Feynman integral with the a usual integration cycle (real phase space coordinates x ) does not make a sense since the factor exp i badx in the path integral is not well-defined. We can make this factor well-defined by replacing U by its universal cover –orbyanycoverU ∗ on which the integrand of the path integral is single- valued. Once we do this, the integral over the usual integration cycle of the Feynman integral is not interesting because all integrals (2.6) vanish. (They transform with a non-trivial phase under the deck transformations of the cover U ∗ →U.) However, as analyzed in [2], and as we will see below, after analytic continuation, there may be sensible integration cycles (related to deformation quantization, which does not require the Dirac condition, rather than quantization, which does). So we do not want to assume that the Dirac condition is obeyed. In what follows, it might be helpful to have an example in mind. A simple example is the case that M = S2, defined by an equation 2 2 2 2 (2.9) x1 + x2 + x3 = j , for some constant j.Wetake

ijkxi dxj ∧ dxk dx1 ∧ dx2 (2.10) f = 2 = . 3!R x3 The first formula for f makes SO(3) invariance manifest, and the second is convenient for computation. One can verify that S2 f =4πj, so that the Dirac condition becomes j ∈ Z/2. However, as already explained, we do not necessarily want to assume this condition. We can think of f as the magnetic field due to a magnetic monopole (of magnetic charge 2j) located at the center of the sphere; b is the gauge connection for this monopole field. THE PATH INTEGRAL OF QUANTUM MECHANICS 353

2.3. Analytic continuation. The Feynman integral, as we have formulated it so far, is an integral over the free loop space U of M, or possibly a cover of this on which the integrand of the Feynman integral is well-defined. Our first step, as suggested in the introduction, is to analytically continue from U to a suitable complexification U . We simply pick a complexification M of M and let U be the free loop space of M (or, if necessary, an appropriate cover of this). What do we mean by a complexification of M? At a minimum, M should be a complex manifold with an antiholomorphic involution3 τ such that M is a component of the fixed point set of τ. Moreover, we would like M to be a complex symplectic manifold endowed with a holomorphic two-form Ω that is closed and nondegenerate and has the property that its imaginary part, when restricted to M, coincides with f. We introduce the real and imaginary parts of Ω by

(2.11) Ω = ω + if.

The fact that Ω is closed and nondegenerate implies that ω and f are each closed and nondegenerate. And we assume that under τ, Ω is mapped to −Ω, so in other words

(2.12) τ ∗(ω)=−ω, τ∗(f)=f.

It follows that on the ﬁxed point set M, ω must vanish:

(2.13) ω|M =0.

We denote local complex coordinates on M as Xa, and their complex conjugates as Xa, and we denote a complete set of local real coordinates (for example, the real and imaginary parts of the Xa)asY A. Since Ω is a closed form, we can write it as the curvature of a complex-valued abelian gauge ﬁeld: A (2.14) Ω = dΛ, Λ= ΛAdY . A

If the original model obeyed the Dirac condition and the complexification introduces no new topology, we can regard Λ as a gauge field with structure group C∗ (the complexification of U(1)). In general, however, as stated in section 2.2, we do not assume this to be the case, and instead we replace U by a suitable cover on which the integrand of the integral (2.20) introduced below is well-defined. It is convenient to introduce the real and imaginary parts of Λ just as we have done for Ω. So we write

(2.15) Λ = c + ib,

3 An involution is simply a symmetry whose square is the identity. 354 E. WITTEN where b and c are real-valued connections. Thus

(2.16) dc = ω, db = f.

We will slightly sharpen (2.13) and assume that there is a gauge with

(2.17) c|M =0.

Let us describe what these definitions mean for our example with M = S2.WedefineM by the same equation (2.9) that we used to define M, 4 except we regard it as an equation for complex variables Xi rather than real variables xi:

2 2 2 2 (2.18) X1 + X2 + X3 = j .

And we deﬁne Ω by the same formula as before except for a factor of i:

ijkXi dXj ∧ dXk dX1 ∧ dX2 (2.19) Ω = i 2 = i . 3!R X3 (The factor of i in the definition of Ω is a minor convenience; the formulas that follow are slightly more elegant if we take f to be the imaginary part of Ω – restricted to M – rather than the real part.) In addition to the conditions that we have already stated, M must have one additional property. Some condition of completeness of M must be desireable, since certainly we do not expect to get a nice theory if we omit from M a randomly chosen τ-invariant closed set that is disjoint from M.It is not obvious aprioriwhat the right condition should be, but as we will find (and as found in [9] in another way) the appropriate condition is that M , regarded as a real symplectic manifold with symplectic structure ω,must have a well-defined A-model. For noncompact symplectic manifolds, this is a non-trivial though in general not well understood condition. Our example of the complexification of S2 certainly has a good A-model, since in fact this manifold admits a complete hyper-Kahler metric – the Eguchi-Hansen metric. As for the functions uα that appear in the path integral (2.6), to make sense of them in the context of an analytic continuation of the Feynman integral, they must have an analytic continuation to holomorphic functions on M (which we still denote as uα). Moreover, so as not to affect questions involving convergence of the path integral, the analytically continued functions should not grow too fast at infinity. For example, in the case of 2 S , a good class of functions are the polynomial functions u(x1,x2,x3). The analytic continuation of such a polynomial is simply the corresponding polynomial u(X1,X2,X3). These are the best observables to consider, because

4 For a critical discussion of the sense in which this analytic continuation is or is not natural, as well as a discussion of the class of observables considered below, see [9]. THE PATH INTEGRAL OF QUANTUM MECHANICS 355 they are the (nonconstant) holomorphic functions with the slowest growth at inﬁnity. Having analytically continued the loop space U of M to the corresponding complexiﬁed loop space U of M , and having similarly analytically continued the symplectic structure and the observables, we can formally write down the Feynman integral over an arbitrary integration cycle Γ ⊂ U : A A (2.20) DY (t) exp ΛAdY u1(t1) ...un(tn). Γ

Γ is any middle-dimensional cycle in U on which the integral converges. Eqn. (2.17) ensures that if we take Γ to be the original integration cycle U of the Feynman integral, then (2.20) does coincide with the original Feynman integral.

2.4. The simplest integration cycles. The reason that there is some delicacy in choosing Γ is that the real part of the exponent in (2.20) is not bounded above. The troublesome factor in (2.20) comes from the real part of Λ. (We assume that the observables uα do not grow so rapidly as to aﬀect the following discussion.) The integration cycle Γ must be chosen so that the dangerous factor A A (2.21) exp cAdY = exp Re ΛA dY does not make the integral diverge. The reason that this is troublesome is that the exponent A (2.22) h = cAdY is unbounded above and below. For example, the map

(2.23) Y A(t) → Y A(t)=Y A(nt) multiplies h by an arbitrary integer n, which can be positive or negative. Leaving aside the question of convergence, how can we ﬁnd a middle- dimensional cycle Γ ⊂ U ? The most elementary approach is to deﬁne Γ by a local-in-time condition. We pick a middle-dimensional submanifold M0 ⊂

M and take Γ ⊂ U to be the free loop space of M0. In other words, Γ 1 parametrizes maps T : S → M whose image lies in M0 for all time. This type of choice cannot be wrong, since if we take M0 = M,weget back to the original Feynman integral. Now let us ask for what other choices of M0 we get a suitable integration cycle. When restricted to the loop space of M0, the function h must be identically zero, or the argument around eqn. (2.23) will again show that it is 356 E. WITTEN not bounded above or below. The variation of h under a small change in a map T : S1 → M is A B (2.24) δh = ωABδY dY .

For this to vanish identically when evaluated at any loop in M0, we require 5 that ω restricted to M0 must vanish. The requirement, in other words, is that M0 must be a Lagrangian submanifold with respect to ω. This is a familiar condition in the context of the two-dimensional A-model: it is a classical approximation to the condition for M0 (endowed with a trivial Chan-Paton bundle) to be the support of an A-brane. This is no coincidence, but a first hint that the possible integration cycles for the path integral are related to the A-model. Upon picking M0 so that the real part of the exponent of the path integral is bounded above, we are not home free: to make sense of the infinite- dimensional path integral, the phase factor that comes from the imaginary part of the exponent must be nondegenerate. For this, we want f to be nondegenerate when restricted to M0. In other words, M0 should have some of the basic properties of M: when restricted to M0, ω vanishes and f is nondegenerate.6 Under these conditions, the Feynman integral (2.20) for Γ equal to the loop space of M0 is simply the original Feynman integral (2.8) with M0 replacing M. In other words, it is the usual Feynman integral associated with quantization of M0. In this situation, different quantum mechanics problems with different choices of M0 correspond to different integration cycles in the same complexified Feynman integral. For a concrete example of how this can happen, let us return to the familiar example, with M defined by the equation 2 2 2 2 (2.25) X1 + X2 + X3 = j .

The condition on M0 is that Re Ω|M0 =0,orinotherwords dX1 ∧ dX2 (2.26) Im =0. X3 M0

A sufficient condition for this is that X1,X2,X3 are all real. This brings us back to the original phase space M. To get another example, we define M by requiring that X1 is real and positive while X2 and X3 are imaginary. 5 This is enough to ensure that h is constant in each connected component of the free loop space of M0. For it to vanish identically, we need in addition that c is a pure gauge when restricted to M0. 6 We imposed one more condition on M: it is a component of the fixed point set of an antiholomorphic involution τ. As explained in [9], this is needed so that quantization of M in the sense of the A-model admits a hermitian structure. So if M0 does not obey this condition, its “quantization” via the A-model – this operation is reviewed in section 2.10 – is not unitary. THE PATH INTEGRAL OF QUANTUM MECHANICS 357

It is not hard to see that while M is a two-sphere S2, M is a copy of the upper half plane H2. In quantization of either M or M, the observables are the same, namely polynomials u(X1,X2,X3) (modulo the relation (2.25)), restricted to M or M. Moreover, any identities in correlation functions of these observables (apart from (2.25)) arise as Ward identities in the path integral and are proved by integration by parts in field space. So just like the differential equation (1.8), the same identities hold regardless of which integration cycle we pick. So we have arrived at a pair of quantum mechanical systems – associated to quantization of M and M – that have the same algebra of quantum observables, though with inequivalent representations. Finally, we make a few remarks about this example that are more fully explained in [9] and are not really needed for the present paper. Concretely, at the classical level, the only relation that the Xi obey, apart from (2.25), is that they commute. Quantum mechanically, commutativity of the Xi is deformed to the sl(2) relations [X1,X2]=X3 and cyclic permutations thereof. (The constant j2 in (2.25) is also modified quantum mechanically.) This means that the algebra of observables is the universal enveloping algebra of sl(2) – whether we quantize M or M. This gives a concrete explanation of why the two systems have the same algebra of observables. An important detail is that the equivalence between the algebra of observables in quantizing M with that in quantizing M does not map hermitian operators to hermitian operators. Most simply, this is because the polynomials u(X1,X2,X3) that are real when restricted to M are not the same as the ones that are real when restricted to M. In footnote 6, we noted that in general M0 may fail to obey one condition that we imposed on the original M: there may not be an antiholomorphic involution with M0 as a component of its fixed point set. However, in the case of M , there is such an involution τ , acting by X1,X2,X3 → X1, −X2, −X3. Hence M actually obeys all of the conditions satisfied by M. The relation between the two is completely symmetrical. We may consider M to arise by analytic continuation from either M or M.

2.5. Review of Morse theory. To construct more interesting integration cycles, we will use Morse theory and steepest descent. A detailed review of the relevant ideas is given in [2]. Here we give a brief synopsis to keep this paper self-contained. Let Z be an m-dimensional manifold with local coordinates wi,i= 1,...,m and a Morse function h. A Morse function is simply a real-valued function whose critical points are nondegenerate. A critical point of h is a point p at which its derivatives all vanish. p is called a nondegenerate critical point of h if the matrix of second derivatives ∂2h/∂wi∂wj is invertible at p. If so, the number of negative eigenvalues of this matrix is called the Morse index of h at p;wedenoteitasip. i j Pick a Riemannian metric gijdw dw on Z. Introducing a “time” coordinate s (we reserve the name t for the time in the original quantum mechanics 358 E. WITTEN problem, to which we return later), we define the Morse theory flow equation: dwi ∂h (2.27) = −gij . ds ∂wj The first property of the flow equation is that the Morse function is always decreasing along any nonconstant flow, since dh ∂h ∂h (2.28) = −gij . ds ∂wi ∂wj The right hand side is negative unless ∂h/∂wi = 0, in which case the flow sits at a critical point for all s. In this statement, of course, we rely on i j positivity of the metric gijdw dw . The fact that h decreases along the flow is the reason that the flow equation will be useful. Let us look at the flows in the neighborhood of a critical point p. After diagonalizing the matrix of second derivatives, we can find a system of Rie- i m 2 mann normal coordinates w centered at p in which h = h0 + i=1 eiwi + 3 2 O(w ), gij = δij + O(w ), with constants h0, ei. The flow equations become

i dw i (2.29) = −eiw , ds with the solution i i (2.30) w = r exp(−eis), with constants ri. A solution of the flow equation, if it does not sit identically at p (that is, at wi = 0) for all s, can only reach the point p at s = ±∞.Let us focus on solutions that flow from the critical point p at s = −∞.From i (2.30), clearly, the condition for this is that r must vanish whenever ei > 0. This leaves ip undetermined parameters, so the solutions that start at p at s = −∞ form a family of dimension ip. We define an ip-dimensional subspace Cp of Z that consists of the values at s = 0 of solutions of the flow equations that originate at p at s = −∞.Thepointp itself lies in Cp, since it is the value at s = 0 of the trivial flow that lies at p for all s. Since h is strictly decreasing along any non-constant flow, the maximum value of h in Cp is its value at p. In favorable situations, the closures of the Cp are homology cycles that generate the homology of Z. As reviewed in [2], a very favorable case is that Z is a complex manifold, say of complex dimension n and real dimension m =2n, of a type that admits many holomorphic functions, and h is the real part of a generic holomorphic function S . The local form of h near a n 2 nondegenerate critical point p is h = h0 +Re i=1 zi , with local complex 2 coordinates zi. Setting zi = xi +iyi,withxi, yi real, and noting that Re zi = 2 2 xi − yi , we see that stable and unstable directions for h are paired. As a result, the Morse index ip of any such p always equals n = m/2, and the THE PATH INTEGRAL OF QUANTUM MECHANICS 359

corresponding Cp is middle-dimensional. In this situation, the Cp are closed (but not compact) for generic7 S and furnish a basis of the appropriate relative homology group, which classifies cycles on which h is bounded above and goes to −∞ at infinity. Now let us take Z = Cn and consider an exponential integral of the sort described in the introduction: (2.31) dz1 dz2 ...dzn exp (S(z1,...,zn)) . Γ Setting h =ReS, the main problem with convergence of the integral comes from the fact that the integrand has modulus exp(h). Convergence is assured if h →−∞ at infinity along Γ, so the cycles Cp just described give a basis for the space of reasonable integration cycles. For instance, let us consider the one-dimensional integral that was discussed in the introduction: (2.32) dz exp(S(z)),S(z)=−z4/4+az. Γ

The equation for a critical point of h =ReS is the cubic equation dS/dz =0. This equation has three roots in the complex z-plane, in accord with the fact that the space of possible integration cycles has rank three, as is evident in fig. 1 of the introduction. In our application, our Morse function h will be the real part of a holomorphic function S, but its critical points will not be isolated. The above discussion then needs some small changes. Let N be a component of the critical point set. N will be a complex submanifold of Z, say of complex dimension r;westilltakeZ to have complex dimension n. In this case, there are n − r complex dimensions or 2(n − r) real dimensions normal to N .We assume that h is nondegenerate in the directions normal to N , meaning that the matrix of second derivatives of h evaluated at a point on N has 2n − 2r n−r 2 nonzero eigenvalues. In that case, the local form of h is h = h0 +Re i=1 zi , and the matrix of second derivatives of h has precisely n − r negative eigenvalues. The space CN of solutions of the flow equation that begin on N at s = −∞ will have real dimension 2r +(n − r)=n + r, where 2r parameters determine a point on N at which the flow begins and n − r parameters arise because the flow has n − r unstable directions. Thus, CN is a cycle that is above the middle dimension. To get a middle-dimensional cycle, we have to impose r conditions, by requiring the flow to begin on a middle-dimensional cycle V ⊂N. The values at s = 0 of solutions of the flow equation on the

7 A sufficient criterion, as explained in [2], is that there are no flows between distinct critical points. Since Im S is conserved along the flow lines, this is the case if distinct critical points (or distinct components of the critical set, if the critical points are not isolated) have different values of Im S. The exceptional case with flows between critical points leads to Stokes phenomena, which were important in [2], but will not be important in the present paper. 360 E. WITTEN

n half-line (−∞, 0] that begin on V form a cycle CV ⊂ C that is of middle dimension. Let us consider a simple example. With Z = C3,wetake 2 2 2 2 2 (2.33) S =(z1 + z2 + z3 − j ) . Then S has an isolated nondegenerate critical point at the origin. In addition 2 2 2 2 it has a family N of critical points given by z1 +z2 +z3 = j . N has complex dimension two, and h =Re(S) is nondegenerate in the directions normal to N . N happens to be equivalent to the complex manifold M that was introduced in eqn. (2.18), so for examples of middle-dimensional cycles in N , we can take our friends M and M, deﬁned respectively by setting the zi to be real, or by taking z1 to be real and positive while z2 and z3 are imaginary.

2.6. A new integration cycle for the Feynman integral. Hope- fully it is clear that in attempting to describe integration cycles for the Feynman integral, we are in the situation just described. The exponent of A the Feynman integral is a holomorphic function ΛAdY on the complexified free loop space U . We want to take its real part, namely A A (2.34) h =Re ΛAdY = cAdY as a Morse function and use the flow equations to generate an integration cycle on which h is bounded above. The first step is to find the critical points of h. This is easily done. We have A B (2.35) δh = δY dY ωAB.

Since ω is nondegenerate, the condition for δh to vanish for any δY A is that dY B = 0. In other words, a critical point is a constant map T : S1 → M . This should be no surprise. Since we have taken the Hamiltonian to vanish, Hamilton’s equations say that the coordinates and momenta are independent of time. The space of critical points is thus a copy of M , embedded in its free loop space U as the space of constant maps. Let us write M∗ for this copy of M . As explained in section 2.5, to get an integration cycle, we pick a middle-dimensional cycle V ⊂ M∗ and consider all solutions of the flow equation on a half-line that start at V . The difference from the practice case discussed in section 2.5 is that the flow equations will be two-dimensional. The objects that are flowing are functions Y A(t), describing a map T : S1 → M . To describe a flow we have to introduce a second coordinate s, the flow variable, which will take values in (−∞, 0], and consider functions Y A(s, t) that describe a map T : C → M . 1 Here C is the cylinder C = S × R+, where R+ is the half-line s ≤ 0. THE PATH INTEGRAL OF QUANTUM MECHANICS 361

The flow equations will not automatically be differential equations on C, but this will happen with a convenient choice of metric on U .Wepick A B an ordinary metric gABdY dY on M. In principle, any metric will do, but we will soon find that choosing a certain type of metric leads to a nice simplification. We also pick a specific angular variable t on S1 with dt =2π. (Until this point, since we have taken the Hamiltonian to vanish, all formulas have been invariant under reparametrization of the time.) Then we define a metric on U by 2 A B (2.36) |δY | = dtgAB(Y (t))δY δY .

With the metric that we have picked, the flow equation becomes A A ∂Y (s, t) AB ∂Y (s, t) (2.37) = −g ωBC . ∂s ∂t The boundary condition at s →−∞ is that Y A(s, t) approaches a limit, independent of t, that lies in the subspace V of the critical point set M∗. There is no restriction on what the solution does at s = 0 (except that it must be regular, that is well-defined). For any choice of metric gAB, the space of solutions of the flow equation on the cylinder C with these conditions gives an integration cycle for the path integral. If we change gAB a little, we get a slightly different but homologically equivalent integration cycle. However, something nice happens if we pick gAB judiciously. Let A AB (2.38) I C = g ωBC.

We can think of I as an endomorphism (linear transformation) of the tangent bundle of M . The ﬂow equation is A B ∂Y A ∂Y (2.39) = −I B . ∂s ∂t Now suppose we pick g so that I obeys

(2.40) I2 = −1, A B A or more explicitly B I BI C = −δ C .(Thespaceofg’s that has this property is always nonempty and contractible; the last statement means that there is no information of topological significance in the choice of g.) This condition means that I defines an almost complex structure on M . When that is the case, the flow equation is invariant under conformal transformations of w = s + it. Indeed, since I is real-valued and obeys I2 = −1, it is a direct sum of 2 × 2 blocks of the form 0 −1 (2.41) . 10 362 E. WITTEN

In each such 2 × 2 block, the ﬂow equations look like ∂u ∂v ∂v ∂u (2.42) = , = − . ∂s ∂t ∂s ∂t

These are Cauchy-Riemann equations saying that (∂s + i∂t)(u + iv)=0 or in other words that u + iv is a holomorphic function of w = s + it. Their invariance under conformal mappings is familiar. Actually, to literally interpret the ﬂow equation as saying that the map T : C → M is holomorphic, we need I to be an integrable complex structure. If I is a nonintegrable almost complex structure, then (2.39) is known as the equation for a pseudoholomorphic map (or an I-pseudoholomorphic map if one wishes to be more precise). This is a well-behaved, elliptic, and conformally invariant equation whether I is integrable or not. Thus, as soon as I2 = −1, the ﬂow equations are invariant under conformal mappings of w. There is in fact a very convenient conformal mapping: we set z = exp(w), mapping the cylinder C to the punctured unit disc 0 < |z|≤1. Generically, a map from the punctured unit disc to M would not extend continuously over the point z = 0. In this case, however, the boundary condition that Y A(s, t) is a constant independent of t for s →−∞ precisely means that Y A,when regarded as a function of z, does have a continuous extension across z =0. Moreover, this extended map is still pseudoholomorphic, by the removeable singularities theorem for pseudoholomorphic maps. Thus, we arrive at a convenient description of an integration cycle CV for the Feynman integral of quantum mechanics. CV consists of the boundary values of I-pseudoholomorphic maps T : D → M , where D is the unit disc |z|≤1, and T maps the point z = 0 to the prescribed subspace V ⊂ M .

2.7. I and the A-model. What sort of complex or almost complex structure is I? M is by definition a complex manifold; it was introduced as a complexification of M. The defining conditions on M were that it is a complex symplectic manifold, with a complex structure that was previously unnamed and which we will now call J, and with a holomorphic two-form Ω whose imaginary part, when restricted to the original classical phase space M⊂M , coincides with the original symplectic form f of M. (There were some additional conditions that we do not need right now.) For example, in the familiar case M = S2, J is the complex structure in which the coordinates X1,X2,X3 of eqn. (2.18) are holomorphic. Since we already know about one complex structure on M , namely J,one might wonder if we can pick the metric g so that I = J. This is actually not possible. Since ω = Re Ω, where Ω is of type (2, 0) with respect to J, it follows AB that ω is of type (2, 0) ⊕ (0, 2) with respect to J. Therefore, for g ωBC to A coincide with J C , gAB wouldalsohavetobeoftype(2, 0) ⊕ (0, 2) with respect to J. But this would contradict the fact that g is supposed to be a positive-definite Riemannian metric. (See the discussion of eqn. (2.28), THE PATH INTEGRAL OF QUANTUM MECHANICS 363 where this positivity was a key ingredient in the Morse theory construction of appropriate integration cycles.) So I will have to be something new, that is not something that was introduced along with M. On the other hand, the conditions obeyed by gAB A and I B are famlliar in one branch of two-dimensional quantum field theory. This is the A-model, in fact in the present case the A-model obtained by twisting a two-dimensional sigma-model in which the target space is M and the symplectic structure is ω. We have already encountered this A-model in a naive way in section 2.4, and it will now enter our story in a more interesting fashion. In defining the A-model of a symplectic manifold X –suchasX = M – with a given symplectic structure ω, one introduces an almost complex structure I on X with respect to which ω is of type (1, 1) and positive. A AB Positivity means that the metric g defined by I B = g ωBC is in fact a positive-definite Riemannian metric on X. The nicest case is that one can choose I to be an integrable complex structure. In that case, the metric g is Kahler. In general, one cannot pick I to be integrable and one can define the A-model for any almost complex structure I such that ω is of type (1, 1) and positive. Indeed, to make sense of the A-model, one only needs the equation for an I-pseudoholomorphic map. (The non-integrable case was important in the early mathematical applications of the A-model [10,17] as well as many more recent ones and was described from a quantum field theory perspective in [18].) In general, A-model computations are localized on such maps. Usually, one encounters finite-dimensional families of I-pseudoholomorphic maps. In the A-model with target X on a Riemann surface Σ without boundary, one encounters the moduli spaces of I-pseudoholomorphic maps T :Σ→ X; these are finite-dimensional. If Σ has a boundary, we usually consider boundary conditions associated with Lagrangian A-branes, and in this case the moduli spaces of I-pseudoholomorphic maps are again finite- dimensional. What may be unfamiliar about the present problem from the point of view of the A-model is that our integration cycle CV is actually an infinite-dimensional space of I-pseudoholomorphic maps. The relation of this cycle to the A-model is explained more fully in sections 2.8 and 2.9. In the meanwhile, let us give a concrete example of what I can be, given that it cannot coincide with the complex structure J by which M was defined. We return first to the familiar example in which M is defined, in 2 2 2 2 complex structure J, by the equation X1 + X2 + X3 = j . In fact, this complex manifold admits a complete hyper-Kahler metric, the Eguchi-Hansen metric. The original complex structure J and the holomorphic two-form Ω are part of the hyper-Kahler structure of M . Indeed, a hyper-Kahler manifold has a triple of complex structure I,J,K obeying the quaternion relations I2 = J 2 = K2 = IJK = −1. It also has a triple of real symplectic structures ωI ,ωJ ,ωK , where ωI is of type (1,1) and positive with respect to I,and 364 E. WITTEN

similarly for ωJ and ωK . Finally, ΩI = ωJ + iωK is a holomorphic two-form with respect to I, and similarly with cyclic permutations of indices I,J,K. Now, in our example, deﬁne Ω by eqn. (2.19) and normalize the hyper-Kahler structure of M so that Ω = ωI − iωK (this is −i(ωK + iωI ), so it is holomorphic with respect to J). So ω = Re Ω is equal to ωI , and is of type (1, 1) and positive with respect to I. In other words, in this example, we can take the metric g on M to be the Eguchi-Hansen hyper-Kahler metric, and I to be one of the complex structures for which that metric is Kahler. The other real symplectic structure of M in our original description is f =ImΩ=−ωK . We have used no special property of M except that its complex symplectic structure J, Ω extends to a hyper-Kahler structure. Whenever this is so, the corresponding hyper-Kahler metric on M is a very convenient choice. (It can happen that the extension of J, Ω to a hyper-Kahler structure is not unique; varying it in a continuous fashion will give a family of equivalent and convenient integration cycles.) Since M is complex symplectic, its real dimension is always divisible by four, but it may not admit a hyper-Kahler structure that extends its complex symplectic structure. While we cannot necessarily pick I to be integrable, we can always pick it so that IJ = −JI.In this case, deﬁning K = IJ and ωJ = Jg, we arrive at what one might call an almost hyper-Kahler structure. The three almost complex structures I,J,K and the three two-forms ωI ,ωJ ,ωK obey all the usual algebraic relations. J is integrable and ωI and ωK are closed; I and K may not be integrable and ωJ may not be closed.

2.8. Interpretation in sigma-model language. At this point, we are supposed to do a Feynman integral A A (2.43) DY (t) exp ΛAdY uα(tα) CV α

A where Y (t) is a one-dimensional field, but the integration cycle CV is described in two-dimensional terms, in terms of boundary values of I-pseudoholomorphic maps. This is a hybrid-sounding recipe. A natural idea is to try to reformulate (2.43) as a two-dimensional path integral, with a field Y A(s, t) that describes a map from D to M , and certain additional fields that we will need to introduce along the way. The first step is obvious – after extending Y to a function defined on D, we introduce another field T that will be a Lagrange multiplier enforcing the desired equation (2.39) of an I-pseudoholomorphic map. We can write (2.39) in the form U A = 0 where

A A A B (2.44) U =dY + I BdY ; THE PATH INTEGRAL OF QUANTUM MECHANICS 365 here is the Hodge star operator on D (deﬁned so that ds =dt, dt = −ds). We view U asaone-formonD with values in the pullback of the tangent bundle of M . U obeys the identity

A A B (2.45) U = I BU .

We introduce a Lagrange multiplier ﬁeld TA that is a one-form on D with values in the pullback of the cotangent bundle of M , and further obeys the dual relation

A (2.46) TB = TAI B.

Then if we include a term in the two-dimensional action of the form D TA ∧ A A U , the integral over TA will give a delta function setting U =0. So a rough approximation to the two-dimensional path integral that we want would be B A (2.47) DTA(s, t) DY (s, t) exp i TA ∧ U (···) , D where the ellipses refer to the integrand in (2.43). After integrating out T , this expression will lead to an integral over Y that is supported on the cycle CV , but it is not the integral we want. In fact, the integral (2.47) will depend on the details of the almost complex structure A I that is used in defining U . The problem is that the integral over TA(s, t) indeed generates a delta function setting U A(s, t) equal to zero, but this delta function multiplies an unwanted determinant 1/| det(δU/δY )|. (An analog of the appearance of this determinant in the case of an ordinary integral is that, if f(x) is a function that vanishes precisely at x = a, then the integral dλ/2π exp(iλf(x)) equals not δ(x − a) but δ(x − a)/|f (a)|.) To cancel this determinant, we add fermions with a kinetic energy that is precisely the linearization of the equation U = 0. As a result, the fermion determinant will cancel the boson determinant up to sign. In the present problem, the sign will not do anything essential; this is because CV is connected, and we can pick the sign of the fermion measure so that the sign is +1. In a more general A-model problem, fermion and boson determinants cancel only up to sign and contributions of I-pseudoholomorphic curves are weighted by the sign of the fermion determinant (the boson determinant is always positive). The fermions will also carry a new U(1) quantum number (“fermion number”) that we call F . We need fermions ψA of F = 1 that take values in the pullback to D of the tangent bundle of M . And we need fermions χA of F = −1 that have the same quantum numbers as the bosons TA: they are a one-form on D with values in the pullback of the cotangent bundle of A M, and they obey a constraint χB = χAI B. We take the fermion action A to be i D χADψ , where the operator D is defined as the linearization of 366 E. WITTEN the equation U A = 0. This means that if we vary Y A around a solution of U A = 0, we have to first order δUA = DδY A. Concretely, if M is Kahler (so that I is covariantly constant, which makes the formulas look more familiar), then A B A Dψ A Dψ (2.48) Dψ = + I B , Ds Dt where D/Ds and D/Dt are defined using the pullback to D of the Rie- mannian connection on the tangent bundle to M . We now consider a two- dimensional path integral A A (2.49) DT DY Dχ Dψ exp i (TA ∧ U − χA ∧Dψ ) D A × exp ΛAdY uα(tα) OV (0). α

Here we have included explicitly all the factors from the original path integral (2.43). We have also included an operator OV (0) – the details of which we will describe presently – that is supposed to incorporate the constraint that Y A(z) lies in V at the point z =0. The two-dimensional action that we have arrived at has a fermionic symmetry that squares to zero. It is invariant under

(2.50) δY A = ψA,δψA =0, together with, roughly speaking,8

(2.51) δχA = TA,δTA =0.

Clearly δ2 = 0. The exponent in (2.49) is

A A A (2.52) i TA ∧ U − χA ∧Dψ = δ iχA ∧ U , D D which makes clear its invariance under δ. This also makes it clear that the path integral (2.49) is invariant under deformations of the almost complex 8 We are engaging here in a small sleight of hand and omitting terms of higher order in fermions. As χ is a section of the pullback of the cotangent bundle of M , to keep it “constant” when Y is varied, we must transport it using some connection; we will use the Riemannian connection. So we measure δχ relative to parallel transport by the Riemannian connection. With this understanding, we should write the first part of eqn. (2.51) in the C B C B B form δχA − ΓABδY χC = TA,whereΓAB are the Christoffel symbols. Since δY = ψ , C B 2 we may write this as δχA − ΓABψ χC = TA. To ensure that δ = 0, we must take δTA = C B C B C D B −δ(ΓABψ χC ). This is equivalent to δTA − ΓABδY TC = −(1/2)RADB ψ ψ χC ,where C RADB is the Riemann tensor. In deriving eqn. (2.53) below, the terms proportional to Γ cancel because the connection is metric-compatible, leaving a four-fermi term proportional to R. THE PATH INTEGRAL OF QUANTUM MECHANICS 367 structure I (or equivalently of the metric g). Indeed, I appears only in terms in the action of the form δ(···); varying such a term does not change the value of the path integral. Without changing anything essential, we can add to the action a further exact term AB AB (2.53) δ g χA TB = g TA ∧ TB 2 D 2 D CA D B + RDBχC ∧ χAψ ψ , 4 D with an arbitrary parameter . (The origin of the four-fermi term is explained in footnote 8.) After performing the Gaussian integral over T ,wegetan equivalent path integral for the other fields 1 A B A (2.54) DY Dχ Dψ exp − gABU ∧ U − i χADψ 2 D D CA D B A − RDBχC ∧ χAψ ψ exp ΛAdY 4 D × uα(tα) OV (0). α

In fact, the fermionic symmetry that was introduced in eqns. (2.50), (2.51) is the usual topological supersymmetry of the A-model with target M . The symmetry generated by δ is usually denoted as Q (and called the BRST operator or topological supercharge). The action is a standard A- model action, as we discuss in section 2.9. When the model is interpreted as an A-model, the operator OV (0) that imposes the constraint that the point z = 0 is mapped to V is a conventional closed string observable of the A-model. This comes about in a standard way, which we sketch for completeness. Operators S(Y (z0),ψ(z0)) that depend on Y and ψ only (evaluated at some point z = z0) are naturally associated to differential forms on M . Here one simply thinks of ψA as the one-form dY A.An arbitrary such S is a finite linear combination of expressions of the form S A1 A2 Ak A1A2...Ak (Y )ψ ψ ...ψ ; we associate such an expression to the differ- S A1 A2 Ak A A ential form A1A2...Ak (Y )dY dY ...dY . The relations [Q, Y ]=ψ , {Q, ψA} = 0 imply that Q acts on differential forms as the exterior derivative d. To be more precise, if γ is a differential form on Y and Sγ(Y,ψ)is the corresponding quantum field operator, then [Q, Sγ]=Sdγ.SoQ-invariant local operators of the A-model of this type correspond to closed differential forms on M . (Similarly, cohomology classes of Q acting on local operators of this type correspond to the de Rham cohomology of M .) Now, given any submanifold V ⊂ M , let V be a differential form (of degree equal to the codimension of V ) that is Poincaré dual to V . (This notion is explained in eqn. 368 E. WITTEN

(4.13).) Then V is a closed diﬀerential form with delta function support on V , and the operator OV that we need in (2.54) is simply SV . So it is indeed a standard Q-invariant local operator of the A-model.

2.9. The physical model. Let us explicitly evaluate the bosonic 1 A B kinetic energy term K = 2 D gAB U U in (2.54), using the deﬁnition (2.44) of U. We ﬁnd 1 dY A dY B dY A dY B (2.55) K = ds dtgAB + ds ds dt dt D 2 dY A dY B + ds dtωAB . ds dt

With ωAB = ∂AcB − ∂BcA, the last term in (2.55) is 2 dY A dY B 2 ∂ ∂Y B (2.56) ds dtωAB = ds dt cB ds dt ∂s ∂t D B ∂ ∂Y 2 B 2h − cB = cB dY = . ∂t ∂s 

Now let us examine the purely bosonic factors in the integrand of the path integral (2.54), ignoring the insertions of operators ui(ti)andOV (0) (which will not aﬀect the convergence of the path integral). Those factors are A A (2.57) exp(−K) exp ΛAdY = exp(−K) exp (cA + ibA)dY A = exp −K + h + i bAdY .

The terms proportional to h in the exponent cancel out if we eliminate K using eqns. (2.55), (2.56) and also set = 2. Then the bosonic factors become A B A B 1 dY dY dY dY A (2.58) exp − ds dtgAB + + i bAdY . 2 D ds ds dt dt ∂D

At this particular value of , our path integral is essentially that of an ordinary supersymmetric nonlinear sigma-model with a particular boundary condition that is compatible with unitarity. The bulk integral in the exponent in (2.58) is the ordinary bulk bosonic action of a sigma-model: 1 dY A dY B dY A dY B (2.59) ds dtgAB + . 2 D ds ds dt dt THE PATH INTEGRAL OF QUANTUM MECHANICS 369

The boundary contribution to the exponent is imaginary precisely at this value of , where it reduces to A (2.60) i bAdY ∂D

Being imaginary means that this contribution can be interpreted as the coupling of the sigma-model to a rank one unitary Chan-Paton bundle L →

M. The connection on this bundle is b and its curvature is f =db. As for the A fermion kinetic energy −i χA ∧Dψ , it is not the standard fermion kinetic energy of the usual supersymmetric sigma-model, but it is the standard fermion kinetic energy in the A-twisted version of this model. Finally, the four-fermion coupling in (2.54) is a standard part of the supersymmetric nonlinear sigma-model, again written in A-twisted notation. In short, at this value of , it must be possible to interpret the boundary interaction that we have at s = 0 as a supersymmetric brane in the usual supersymmetric sigma-model with target M . (Twisting does not affect the classification of branes, because on a flat worldsheet, the twisting is only a matter of notation; near the boundary of a Riemann surface Σ, we can always consider Σ to be flat.) More specifically, what we have at s =0 isa brane in the usual sigma-model that preserves A-type supersymmetry. Which brane is it? The support of the brane is all of M (since the bosonic fields are locally allowed to take any values at s = 0) and the curvature of its Chan-Paton bundle is f. These properties uniquely identify this brane: it is the most simple coisotropic A-brane constructed in the original paper on that subject by Kapustin and Orlov [3]. Their presentation contains more or less the same ingredients as in our derivation, but arranged quite differently. Their starting point was the A-model of a symplectic manifold X with symplectic structure ω. To define the A-model, an almost complex structure I is chosen such that ω is of type (1, 1) and positive. The question asked was then what branes of rank 1 are possible in this A-model. Such a brane is characterized by its support and by the curvature of the Chan- Paton line bundle that it carries. For brevity, let us state the answer in [3] only for the case of a rank one brane B whose support is all of X.The answer turned out to be that the Chan-Paton curvature f of such a brane must have the property that J = ω−1f is an integrable complex structure on X (which will necessarily be different from I,whichmayormaynotbe integrable). This was quite a surprising answer at the time; previously the only known branes of the A-model were Lagrangian A-branes, supported on a middle-dimensional submanifold of X. Saying that J = ω−1f is equivalent to saying that Ω = ω + if is a holomorphic two-form with respect to J. So whenever we study the A-model of a symplectic manifold X –suchasX = M – endowed with a symplectic form ω that is the real part of a holomorphic two-form Ω, in some complex structure J, there is a canonical way to obey the Kapustin-Orlov conditions. We 370 E. WITTEN can define a rank one A-brane whose support is all of X by taking the Chan- Paton curvature to be f = Im Ω. Since this is the simplest way to satisfy the relevant conditions, and also the brane constructed this way seems to arise in many applications, this brane has been called the9 canonical coisotropic A-brane and denoted Bcc. In our formulation in the present paper, we have arrived at the same structure from a different end. We started with a classical symplectic manifold M with symplectic form f. Seeking to analytically continue the Feynman integral that arises in quantizing M, we replaced M by a complexification M and analytically continued from f to Ω = ω + if. Then we found an integration cycle in the complexified Feynman integral that has a natural interpretation via a path integral on the unit disc D in the complex z-plane. From the standpoint of [3], what we have constructed is the path integral of the A-model on D with a boundary condition set by the A-brane Bcc, with boundary insertions of open string vertex operators ui(ti), and with a closed string A-model vertex operator OV inserted at z = 0. The closed string insertion is needed since otherwise, because of an anomaly in the fermionic quantum number F , the path integral on the disc would vanish. As for the open string vertex operators uα(tα) that are inserted on the boundary of the disc, it is a result of [3] that the (Bcc, Bcc) strings correspond to holomorphic functions in complex structure J. The reader may want to compare this discussion to the analysis in [11– 13] of localization of sigma-model path integrals on a middle-dimensional subspace of the loop space of a target manifold. (The target space was not assumed to be a complex symplectic manifold, so the middle-dimensional cycles were not interpreted as integration cycles.) The approach was the reverse of what we have explained here – and more like what we will explain in section 4 – in the sense that the starting point was taken to be a two- dimensional supersymmetric sigma-model, rather than the problem of finding an integration cycle for a path integral in dimension one. 2.9.1. The boundary condition and localization. Here we will add a few remarks on the boundary condition obeyed by the field Y A at the boundary of the punctured disc D. (We reconsider the boundary conditions in a related problem and include the fermions in section 4.1.) The purpose is to clarify the meaning of A-model localization in the presence of coisotropic branes. We read off from eqn. (2.57) that the bosonic part of the integrand of the path integral is exp(−K ), where the “action” K , with boundary contributions included, is A (2.61) K = K − h − i bAdY .

9 Calling this brane “the” canonical coisotropic A-brane is perhaps a little too cavalier, since it depends on the choice of J. Depending on the context, a distinguished J may or may not present itself. THE PATH INTEGRAL OF QUANTUM MECHANICS 371

When we vary K with respect to Y A,weget 2 B 2 B 2 A D Y D Y (2.62) δK = − ds dtδY gAB + Ds2 Dt2 D B B A 2 dY dY + dtδY gAB + ωAB ds dt ∂D dY B −(ωAB + ifAB) . dt Setting this to zero, we ﬁnd that the bulk equation of motion is the equation D2Y B/Ds2 + D2Y B/Dt2 = 0 for a harmonic map (this equation is satisﬁed for I-pseudoholomorphic maps). Moreover, as we wish to place no local restriction on the boundary values of δY A, to set the boundary term to zero, the boundary condition on Y A must be 2 dY B dY B dY B (2.63) gAB + ωAB − (ωAB + ifAB) =0. ds dt dt If we set = 2, the boundary condition becomes

dY B dY B (2.64) gAB − ifAB =0, ds dt which is the boundary condition10 of the physical sigma-model with a coisotropic brane as presented in [3]. On the other hand, to get the simplest topological field theory description, we take → 0. Precisely in the limit = 0, the boundary condition merely says that the equation for a pseudoholomorphic map should be obeyed on ∂D. This is not really a boundary condition at all, since at = 0, the equation for a pseudoholomorphic map is obeyed everywhere. The best way to understand what happens precisely at =0 istogo back to the form (2.49) of the path integral with an auxiliary field T .When we repeat the above exercise, assuming no constraint on the boundary values A of δY or δTA, we find a boundary condition for T but none for Y .(This happens because the boundary contribution to the variation of the action has a TδY term but no YδT term.) So in this form of the theory, Y obeys no boundary condition at all. The only condition on Y is the equation for an I-pseudoholomorphic map, which is enforced when we do the T integral. The question “on what class of I-pseudoholomorphic maps does the A- model localize in the presence of a coisotropic brane?” does not seem to have 10 The meaning of this boundary condition in quantum theory is a little subtle, because of the factor of i multiplying the second term, while classically the field Y B is real. This subtlety is a standard phenomenon in Euclidean field theory and has nothing to do with issues considered in the present paper. If we replace D by a two-manifold of Lorentz signature, the factor of i disappears and the interpretation of the boundary condition becomes straightforward. 372 E. WITTEN been answered in the literature. The answer for the case that the coisotropic brane is the canonical one considered here is that localization occurs in and only in the limit → 0 and the localization is on the infinite-dimensional space of all I-pseudoholomorphic maps. For any = 0, we can do a path integral that gives the same results for all A-model observables, but it arrives at these results in a more complex way, not by a simple localization on a space of I-pseudoholomorphic maps. Another way to describe the localization that occurs at → 0 is that the localization is on quantum mechanics – the two-dimensional A-model path integral localizes on a new integration cycle for a path integral of ordinary quantum mechanics.

2.10. Recovering the Hilbert space. We have defined an integration cycle by flowing in s over a semi-infinite interval (−∞, 0]. An obvious question is to ask what we would get if we flow in s for a finite interval only, say the interval [−s0, 0]. In other words, what happens if we replace the semi- 1 1 infinite cylinder C = S × R+ with a compact cylinder Cs0 = S × [−s0, 0]? We now need a different type of boundary condition at s = s0. We cannot start the flow from a critical point, since as we observed in section 2.5, a nonconstant flow can only start from or leave a critical point at s = ±∞. Let us approach the problem from the standpoint of the A-model. Since

Cs0 has a second boundary component at s = −s0, we need a boundary condition there. This boundary condition will have to correspond to a second A-brane. Though we could consider the case that the second brane is a coisotropic one (like the canonical coisotropic brane Bcc that we will continue to use at s = 0), let us consider instead the case that the second brane is a Lagrangian A-brane – supported on a submanifold L⊂M that is Lagrangian with respect to ω. We denote the second brane as BL. The boundary condition associated to BL requires the boundary values of the map T : Cs0 → M at s = −s0 to lie in L. Those boundary values deﬁne an arbitrary11 point in the free loop space of L. Then we solve the

flow equations for a “time” s = s0. The resulting integration cycle Γs0 in the free loop space of X consists of all possible boundary values at s = 0 of the solution of the flow equation. As s0 is varied, we get a one-parameter family of integration cycles. The value of s0 does not matter, since in general the integral of a holomorphic differential form of top dimension – in this case the integration form of the Feynman integral – over a middle-dimensional cycle is invariant under continuous deformations of that cycle. (Alternatively, the value of s0 does not matter, since the metric of Cs0 is immaterial in the A-model.) We can try to evaluate the integral in the limit s0 → 0. The limit is particularly simple if L, while Lagrangian for ω = Re Ω, is symplectic from

11 This is a slight simpliﬁcation. The boundary values at s = −s0 must be initial values of a solution of the ﬂow equation that is regular at least up to s =0. THE PATH INTEGRAL OF QUANTUM MECHANICS 373

2 2 the standpoint of f =ImΩ.(IfM is the familiar example given by X1 +X2 + 2 2 X3 = j , then M and M as described in section 2.4 are possible choices of L.) In the limit that s0 → 0, the flow equation does not do anything (since no time is available for the flow) and the limit of Γs0 is simply the free loop space of L. The integral over this free loop space is simply the original Feynman integral (2.8), with L replacing M. This path integral is associated with quantization of L. So the result is that the A-model path integral for strings stretched between the brane Bcc and a Lagrangian brane BL (with f nondegenerate on L) is associated with quantization of L (with respect to symplectic structure f). The corresponding statement from a Hamiltonian point of view is that the space of (Bcc, BL) strings is the Hilbert space that arises in quantization of L. As is usual, the path integral on the cylinder can be interpreted in terms of a trace in this Hilbert space. If rather than a trace, we wish to consider specific initial and final quantum states, we can consider a path integral on adiscD with its boundary ∂D divided into an interval ∂D1 labeled by Bcc and a second interval ∂D2 labeled by BL. (The associated A-model path integral will localize on I-pseudoholomorphic maps that send ∂D2 to L and whose boundary values on ∂D1 are unconstrained. The space of such maps is infinite-dimensional.) At the two points where ∂D1 and ∂D2 meet,wemust insert (Bcc, BL)and(BL, Bcc) vertex operators, which correspond to initial and final quantum states in the quantization of L. Insertions of (Bcc, Bcc) strings on ∂D1 will give matrix elements of quantum observables between initial and final quantum states. None of these assertions really require the analysis in the present paper, and indeed fuller and more direct explanations have been given in [9], following a variety of earlier clues and examples [4–8]. In our brief and somewhat cavalier explanation here, we have omitted some key details (involving the conditions for the space of (Bcc, BL) strings to have a hermitian structure, the role of the flat Chan-Paton line bundle of BL, etc.) that can be found in [9]. If f when restricted to L is degenerate, then the integral (2.8) (with L replacing M) is not well-defined. The limit s0 → 0 needs to be taken more carefully; higher order bosonic terms in the Lagrangian cannot be omitted, and fermion fields cannot be dropped as they have zero modes. The extreme case that f restricted to L is zero was treated in [6]; it leads to D-modules rather than to quantization. We can also consider the opposite limit of s0 →∞. In this limit, the finite cylinder Cs0 approaches the semi-infinite cylinder C, whose compactification isadiscD. It is natural to compare the path integral on the cylinder Cs0 with boundary conditions set by BL at s = −s0 to a path integral on D with an insertion of the closed string vertex operator OL at the center. For s0 →∞, the path integral on Cs0 converges to the corresponding path integral on D, but some information is lost. This is related to the fact that the path integral on D makes sense for any middle-dimensional cycle L, while the path integral 374 E. WITTEN

on Cs0 can only be deﬁned if L is Lagrangian. The path integral on either Cs0 or D with boundary insertions at s = 0 gives a cyclically symmetric trace- like function on the noncommutative algebra R that arises by deformation quantization of the ring of holomorphic functions on M . (A trace-like function is a family of cyclically symmetric functions fn(u1,u2,...,un), deﬁned for a cyclically ordered n-plet of elements u1,...,un ∈R, for any n ∈ N,and obeying fn−1(u1u2,u3,...,un)=fn(u1,u2,...,un).) The path integral on

Cs0 gives a trace-like function that actually can be interpreted as a trace in an R-module (namely the R-module obtained by quantizing the (Bcc, BL) strings), while the path integral on D, in the general case that L is not Lagrangian, gives a trace-like function that is not necessarily the trace in any R-module.

3. Hamiltonians Once we associate a Hilbert space H and an algebra of observables R to a classical phase space M, we can take an element of R and call it the Hamiltonian. However, one may ask if there is some useful way to include a Hamiltonian in the analysis from the beginning. Here we will approach this question in two ways. In section 3.1, we rerun the analysis of section 2 in a straightforward fashion with a Hamiltonian in place from the beginning. The analysis is not diﬃcult, and makes sense for any Hamiltonian, but is probably not very enlightening. In section 3.2, we do something that is probably more useful. We ask whether, in the construction of section 2, it is possible to relate the Hamiltonian of a one-dimensional description to the superpotential of a two-dimensional description. This is interesting when it is possible, though it is not usually possible.

3.1. Rerunning the story with a Hamiltonian. In the presence of a Hamiltonian H(p, q), the integrand in the quantum path integral (2.6) is i i modiﬁed in the familiar fashion: the integral pi dq is replaced by pi dq − H(p, q)dt . (We now take the t coordinate to have period τ; this parameter is meaningful when the Hamiltonian is nonzero.) The path integral with the Hamiltonian included is then

i i (3.1) Dpi(t) Dq (t) exp i pi dq − H(p, q)dt U × u1(t1)u2(t2) ...un(tn).

Actually, this is the real time version of the path integral, related to the evaluation of exp(−iτH). The imaginary time version of the path integral, related instead to exp(−τH), diﬀers only by dropping the factor of i in front THE PATH INTEGRAL OF QUANTUM MECHANICS 375 of −H(p, q)dt:

i i (3.2) Dpi(t) Dq (t) exp ipi dq − H(p, q)dt U × u1(t1)u2(t2) ...un(tn).

Most of our considerations in this section apply equally to the real or imaginary time version of the path integral. For deﬁniteness, we consider the real time version until section 3.2.2, where it is useful to consider both cases. As in section 2.3, we begin by analytically continuing the classical phase space M to a complex symplectic phase space M . We will certainly not be able to incorporate a Hamiltonian in the discussion unless it too can be analytically continued, so we assume that H can be analytically continued to a holomorphic function H on M . It is convenient to introduce the real and imaginary parts of H.Wewrite

(3.3) H = H + iG, with H, G real. The analog of the analytically continued path integral (2.20) is 

A A (3.4) DY (t) exp ΛAdY − iHdt u1(t1)u2(t2) ...un(tn). Γ

The dangerous real exponential factor in the path integral is exp(h) where now 

A (3.5) h = cAdY + G dt .

Just as in section 2, we need to pick the integration cycle Γ such that this exponential factor does not cause the path integral to diverge. The most obvious type of integration cycle is given by the sort of local- in-time condition considered in section 2.4. We take Γ to be the loop space of a middle-dimensional submanifold M ⊂ M. We can practically borrow the A analysis of section 2.4. Under the transformation 2.23, the term cAdY in h is multiplied by an arbitrary integer n, while the term Gdt is invariant. A To ensure that h is bounded above, a necessary condition is that cAdY should vanish identically (for paths in M). Just as in section (2.4), this means that M should be Lagrangian with respect to ω. The exponent in (3.5) then reduces to G dt, which is bounded above precisely if G is bounded above when restricted to M. So that is our answer: M must be a Lagrangian submanifold (with respect to ω)onwhichG is bounded above. We can also imitate the construction of integration cycles associated to flow equations. The only real differences are that we have to include the Hamiltonian in the condition for a critical point and in the flow equations. 376 E. WITTEN

Let us ﬁrst write the conditions for a critical point. Prior to analytic con- i tinuation, requiring the functional pi dq − H dt to be stationary gives Hamilton’s equations:

dqi ∂H (3.6) = dt ∂pi dpi ∂H = − . dt ∂qi

So critical points correspond to arbitrary periodic solutions of Hamilton’s equations. Of course, it does not really suﬃce to consider only the real Hamilton equations. Integration cycles in the analytically continued path integral can be derived from all critical points of the function h on the free loop space of M , real or not. The general critical point is a periodic solution of the complexiﬁed Hamilton equations, which we can write

dY B ∂H (3.7) −iΩAB = . dt ∂Y A To every critical point, one associates an integration cycle in the Feynman integral. It is deﬁned by solving the ﬂow equations

A B ∂Y AB δh A ∂Y AB ∂G (3.8) = −g = −I B − g ∂s δY B(s, t) ∂t ∂Y B

(for some choice of metric g on M ) for flows that start at s = −∞ at the given critical point. Moreover, by using some more information about solutions of the flow equations [2], the original integration cycle of the Feynman integral – with real p’s and q’s – can be expressed as a linear combination of these critical point cycles. So if one asks, “Can the path integral of a quantum system be expressed in terms of properties of the classical orbits?” then this procedure gives an answer of sorts. There are two problems which will tend to make this answer unuseful. First, it requires an unrealistic degree of knowledge about the classical system. Except for an integrable system, we cannot even describe the generic periodic solutions of Hamilton’s equations, even when we restrict to real p’s and q’s. And it will also be hard to say a lot about the solutions of the flow equations. What is more, it is not clear what one can say about the Feynman integral evaluated on a cycle associated to a critical point (except that perturbation theory for this integral is likely to be Borel-summable). Second, for a generic Hamiltonian, the flow equations lack the two- dimensional symmetry which was the main reason for the power of the analysis in section 2. THE PATH INTEGRAL OF QUANTUM MECHANICS 377

But is there a class of special Hamiltonians for which the ﬂow equations will once again have two-dimensional symmetry? We consider this question next.

3.2. Hamiltonians and superpotentials. To incorporate a Hamil- tonian while preserving two-dimensional symmetry, we will need to consider the A-model with a superpotential. As usual, we consider the A-model with target X and symplectic structure ω. (In our application, X is a complex symplectic manifold M and ω is the real part of a holomorphic symplectic form Ω on M .) To include a superpotential in the A-model is most natural when X is actually Kahler, so we henceforth assume that the metric g of X is Kahler and in particular the almost complex structure I = g−1ω is integrable. We let W : X → C be a holomorphic function12 that we call the superpotential. Consider the A-model on a Riemann surface Σ with local complex coordinate w. In the absence of a superpotential, the A-model localizes on holomorphic maps T :Σ→ X. In the presence of a superpotential, the condition for a holomorphic map is perturbed and becomes

∂xi ∂W (3.9) + gij =0, ∂w ∂xj where xi are local I-holomorphic functions on X. For the case of a single chiral superfield, this equation was studied in [19]. The more general case was studied from a physical point of view in [20]. For mathematical studies of this equation, with an elucidation of some important points, see [21,22]. The derivation of this equation will be sketched in section 5.1.1. To make sense of eqn. (3.9) globally along Σ, we cannot simply interpret it as an equation for a map from Σ to X. To explore this point, consider the case of a single chiral superfield x with Kahler metric |dx|2 and superpotential W (x)=xn/n. The equation is ∂x (3.10) + xn−1 =0. ∂w If x is regarded as a scalar function, then the first term in this equation, namely ∂x/∂w,isa(0, 1)-form on Σ, while the second term, namely xn−1, is a scalar function. So with that interpretation the equation does not make sense globally. To make sense of (3.10) globally, while preserving two-dimensional symmetry, we need a line bundle L → Σ with an isomor- n ∼ phism L = K, where K is the canonical bundle of Σ. We also need a Kahler metric on Σ, compatible with its complex structure; this gives an isomorphism between K (the space of (0, 1)-forms on Σ) and K−1. We interpret x 12 One can define the equation for a holomorphic function on any almost complex manifold. However, this equation is overdetermined and has no nonconstant solutions on a generic almost complex manifold. This is why we assume that I is integrable. 378 E. WITTEN as a section of L. Given this structure, xn−1 and ∂x/∂w are both sections of L−n+1, so eqn. (3.10) makes sense. The choice of Kahler metric on Σ is inessential in the A-model, in the same sense that the Kahler metric on X is inessential: a choice is needed to define the A-model, but the results are independent of the choice. By contrast, the choice of nth root L of the canonical bundle is an important part of the structure. The isomorphism of Ln with K is allowed to have certain singularities at insertion points of vertex operators, as described in [19]; there is a constraint on the genus of Σ and the choices of vertex operators such that L exists globally. In the general case with several chiral superfields, one proceeds similarly. One needs an action on X of U(1) (or in general of a covering group of U(1)) that preserves its Kahler structure and under which W transforms with charge 1; that is, under the element eiθ of U(1), W maps to eiθW . GivenanactionofU(1) on X, let K1 be the subbundle of K consisting of unit vectors (with respect to the metric on Σ), and define a fiber bundle Y→Σ, whose fiber is isomorphic to X,by

(3.11) Y = K1 ×U(1) X.

(Thus an element of the fiber is a pair (k, x) ∈ K1 × X with an equivalence ∼ −1 relation (k, x) = (ka ,ax)fora ∈ U(1).) If the group that acts on X is actually an n-fold cover U(1)n of U(1), for some integer n, then as above n ∼ we pick a line bundle L → Σ with an isomorphism L = K, and denote its subbundle of unit vectors as L1. The action of U(1) on K1 lifts to an action Y × of U(1)n on L1,and is defined as L1 U(1)n X.TheA-model is defined for sections of the bundle Y. 1 In our application, Σ is actually the cylinder C = S × R+. Its canonical bundle is naturally trivial, so the details of the last two paragraphs will not play an important role. We have explained them to make clear in what sense the eqn. (3.9) that we will be using does have two-dimensional symmetry. 1 With w = s + it, so that ∂/∂w = 2 (∂s + i∂t), eqn. (3.9) can be written ∂xk ∂xk ∂ (3.12) = −i − 2gkk W + W . ∂s ∂t ∂xk After combining xk and xk to real coordinates Y A and replacing the complex number i with the complex structure I,wecanwrite A B ∂Y A ∂Y AB (3.13) = −I B − g ∂B (4 Re W ) . ∂s ∂t (The part of this equation that is of type (1, 0) with respect to I coincides with (3.12), and the (0, 1) part is the complex conjugate.) Comparing this to (3.8), we see that the supersymmetry condition of the A-model with a superpotential coincides with the flow equation for quantum mechanics with a Hamiltonian precisely if (3.14) G =4ReW. THE PATH INTEGRAL OF QUANTUM MECHANICS 379

In other words, G, which was defined as the imaginary part of the J- holomorphic function H, must also be the real part of the I-holomorphic function 4W . 3.2.1. Hyper-Kahler symmetries. This condition might sound impossi- bly restrictive, but it actually leads to something relatively nice. We can analyze the condition fully in case the metric g on X = M is actually hyper- Kahler; as described in section 2.7, this is the most elegant case of our construction. We begin by considering a special situation, and show in section 3.2.3 that this special situation actually is typical. Let us suppose that X admits a Killing vector field V that preserves its hyper-Kahler structure. This means in particular that the corresponding Lie derivative LV annihilates the three symplectic structures ωI ,ωJ ,ωK .Thisis equivalent to saying that, with ιV the operation of contraction with V , the one-forms ιV ωI , ιV ωJ , ιV ωK are closed. Integrating those closed one-forms, 13 we find functions μI ,μJ ,μK that up to inessential additive constants are defined by

dμI = ιV ωI (3.15) dμJ = ιV ωJ dμK = ιV ωK .

The triple of functions μ =(μI ,μJ ,μK ) deﬁne the hyper-Kahler moment map; their main properties are described in [23]. Of particular importance to us, νI = μJ + iμK is I-holomorphic and νJ = μK + iμI is J-holomorphic. (These statements have an obvious analog in complex structure K and indeed in any complex structure that is a linear combination of I,J, and K.) So if we set

(3.16) H = iνJ , or in other words

(3.17) H = −μI ,G= μK , then G is indeed the real part of an I-holomorphic function. In fact, G = 4ReW with iνI (3.18) W = − . 4 Finally, when can we ﬁnd a U(1) symmetry of X that preserves the A-model complex structure I but rotates W =(μK − iμJ )/4? Such a U(1) symmetry does not preserve complex structures J or K; rather, in IJK space, it acts by a rotation around the I axis. Many hyper-Kahler manifolds admit such a symmetry. 13 If necessary, we replace M by a cover on which these functions are single-valued. The logic is the same as it was in section 2.2 where we chose not to impose the Dirac condition. 380 E. WITTEN

3.2.2. An example. An example is the familiar case of the Eguchi- 2 2 2 2 Hansen manifold M, deﬁned in complex structure J by X1 + X2 + X3 = j . This manifold, for any value of its hyper-Kahler moduli, has an SU(2) group of isometries that preserves its hyper-Kahler structure. We take V to be the vector ﬁeld that generates a one-parameter subgroup of this SU(2). In addition, for generic moduli, M has a U(1) symmetry that perserves one complex structure and rotates the other two. Which complex structure is preserved depends on the values of the hyper-Kahler moduli of M . For our purposes, we want to pick the moduli so that M admits a U(1) symmetry that acts by rotation about the I axis. (Concretely, M has three real moduli, namely the real and imaginary parts of j and a third real modulus that is a Kahler parameter from the point of view of J; we set to zero Im j and the third modulus. In this case, in complex structure I, M is a blowup rather than deformation of the A1 singularity and has the desired symmetry.) The U(1) symmetry that rotates about the I axis does not preserve complex structure J, so it is not easily visible in that complex structure. A typical choice of V is

∂ ∂ (3.19) V = X1 − X2 + complex conjugate ∂X2 ∂X1 (we add the complex conjugate because V is supposed to generate an isometry). According to eqn. (2.19) and section 2.7, the holomorphic two-form Ω =

ωI − iωK of M is idX1 ∧ dX2/X3. So we compute that ιV Ω=−i(X1dX1 + X2dX2)/X3 = idX3.ThusμI − iμK = iX3,soμK = −Re X3, μI = −Im X3. In view of (3.17), we then have

(3.20) H =ImX3,G= −Re X3.

Now let us interpret the Hamiltonian H in terms of real quantum mechanics. In doing this, we want to interpret M as the complexification of an “underlying real classical phase space” M0,andH as the analytic continuation to M of a real Hamiltonian H on M0. The candidates for M0 that we will consider are the two that have been discussed throughout this paper: M0 may be M, defined by X1,X2,X3 real, or M , defined by X1 real and positive, and X2,X3 imaginary. For H to be the analytic continuation to M of a real function H on M0, we at least need H = H on M0.Thus,a necessary condition is that G =ImH vanishes on M0. Otherwise, the path integral (3.4) that we have investigated is not an analytic continuation of the ordinary Feynman integral (3.1). Looking at (3.20), we see that this condition is obeyed by M and not by M. (In fact, H vanishes identically on M so H is not very interesting as a Hamiltonian on M.) So for the case at hand, we should take the mechanical system of interest to be the one with phase space M. M has SL(2, R) symmetry, and H is one of the generators of SL(2, R). THE PATH INTEGRAL OF QUANTUM MECHANICS 381

If we want an example well-adapted to M, we must do the Feynman integral in imaginary time rather than real time. Changing from real time to imaginary time in (3.4) means that we replace i dt by dt. This is equivalent to replacing H by −iH, so we can repeat the whole analysis for imaginary time by making that simple replacement. Then (3.16) becomes H = νJ and (3.17) becomes H = μK , G = μI . In our example, we now have H =ReX3, G =ImX3. Now the vanishing of G on M0 allows M0 to be M but not M . M is a two-sphere, and H generates its rotations around the X3 axis. We motivated the condition that G = 0 when restricted to M0 by ask- ing that the space of solutions of the flow equations can be intepreted as an integration cycle for the Feynman integral of M0 with Hamiltonian H.How- ever, this condition has another interpretation [24]: it is the condition for M0 to be the support of a Lagrangian A-brane in the presence of the super- 1 potential W . Consider the path integral on a finite cylinder S × [−s0, 0], with boundary conditions set at s = −s0 by a Lagrangian brane of support M0 and at s = 0 by a coisotropic brane (the coisotropic brane is defined by allowing any solution of the flow equations near s = 0, as in sections 2.8, 2.9). This path integral computes Tr exp(−itH), with the trace taken in the quantum Hilbert space of M0. The argument is the same as it was in 1 section 2.10. The path integral on S × [−s0, 0] with the indicated boundary conditions is independent of s0, and for s0 → 0 it goes over to the standard Feynman integral representation of the trace. 3.2.3. General analysis. Now we would like to show that, for M hyper- Kahler, the construction that we have described using a vector field V that preserves the hyper-Kahler structure is the most general possibility. First of all, the assertion that H + iG is J-holomorphic is equivalent to14 (1 + iJ t)d(H + iG) = 0, since 1 + iJ t projects onto the (0, 1) part of a one-form. Equivalently, we can write

(3.21) dH = J tdG.

Similarly, the fact that G is the real part of an I-holomorphic function means that there is a function U such that

(3.22) dG = ItdU.

t The fact that ωK + iωI is J-holomorphic means that J (ωK + iωI )= i(ωK + iωI ), or

t t (3.23) J ωK = −ωI ,JωI = ωK .

Similarly,

t t (3.24) I ωJ = −ωK ,IωK = ωJ .

14 We regard a complex structure such as J as a linear transformation that acts on the tangent bundle; its transpose J t acts on the cotangent bundle. 382 E. WITTEN

The assertion that a vector ﬁeld V preserves the hyper-Kahler structure of M is equivalent to saying that V preserves the three symplectic structures ωI , ωJ , ωK . Indeed, using (3.23) and (3.24), we can compute the complex structures from the symplectic structures:

t −1 t −1 (3.25) J = −ωI ωK ,I= ωJ ωK ,K= IJ.

t t t And the metric is g = I ωI = J ωJ = K ωK . So I,J,K and g are all V - invariant if the three symplectic forms are. In general, a vector field V preserves a symplectic structure ω if there −1 is a function Q (called the moment map) with V = ω dQ. Given functions G, H, U obeying (3.21), (3.22), we will find a vector field V that preserves the three symplectic structures and has G, H, U as moment maps. −1 We simply define V = ωK dG,soV certainly preserves ωK .Thenwe −1 −1 t −1 compute using the above formulas that ωI dH = ωI J dG = −ωI −1 −1 −1 t ωI ωK dG = −V .SoV preserves ωI also. Finally, ωJ dU = −ωJ I dG = −1 −1 −ωJ ωJ ωK dG = −V . So again V preserves ωI . Finally, the relations V = −1 −1 −1 ωK dG = −ωI dH = −ωJ dU that we have just found imply that H, G, U are the moment maps for the action of V : H = −μI , U = −μJ , G = μK .

4. Running the story in reverse So far, our point of view has been to start with a hopefully natural question – ﬁnd a new integration cycle for the Feynman integral of quantum mechanics – and express the answer in terms of a sigma-model with an extra spacetime dimension. In the present section, we will run the story in reverse. We start with a sigma-model and pick boundary conditions in the sigma-model so that the sigma-model path integral has an interesting interpretation as an integral over boundary data. In section 4.1, we consider a one-dimensional sigma- model. So the boundary is a point and the integration over boundary data is an ordinary ﬁnite-dimensional integral. In section 4.2, we consider sigma- models in two dimensions, so that the boundary integral is a quantum mechanical path integral, such as we considered so far. We will hopefully emerge from this analysis not just with a better understanding of why the constructions of section 2 and 3 work, but a better understanding of why the machinery in those sections is necessary – that is, why some simpler tries do not work.

4.1. Sigma-models in one dimension. To begin with, we consider supersymmetric quantum mechanics – a sigma-model in one dimension with a target space Z, endowed with a Riemannian metric g. The model is a supersymmetric theory of maps T : L → Z, where L is a one-manifold, parametrized by a “time” coordinate s.(WetakeL to have Euclidean signature.) We describe T by bosonic fields xI (s) that correspond to local THE PATH INTEGRAL OF QUANTUM MECHANICS 383 coordinates xI on Z. The fermi fields are two sections ψI (s)andχI (s)of T ∗(TZ), the pullback to L of the tangent bundle of Z. The model has a “fermion number” symmetry F ; ψ and χ respectively have F =1 and F = −1. There is also a “superpotential” h, which is a real-valued function on Z. In superspace, xI ,ψI ,χI and an auxiliary field can be combined to a superspace field XI (s, θ, θ) and the action can be written concisely

1 I J (4.1) I = ds dθ dθ gIJDX DX + h , 2 with superspace derivatives D, D. For our purposes, we will simply describe the theory in terms of the component ﬁelds, emphasizing its relation to Morse theory [25]. There are two supersymmetry operators Q and Q, of respectively F =1 and F = −1: I ∂ − ∂h Q = ψ I I ∂x ∂x ∂ ∂h (4.2) Q = χI − − . ∂xI ∂xI 2 They obey Q2 = Q =0, {Q, Q} =2H, where H is the Hamiltonian. For our purposes, we focus on Q and consider path integrals with Q-invariant boundary conditions. The commutation relations generated by Q are [Q, xI ]=ψI (4.3) {Q, ψI } =0 dxI ∂h {Q, χI } = − + gIJ . ds ∂xJ I J IJ J I To get these relations, we use {χ ,ψ } = g and gIJ∂x /∂s = −∂/∂x (the last formula depends on the fact that in Euclidean signature, the usual relation p = −i∂/∂x becomes p = −∂/∂x). One of the main observations I I in [25] is that, if we identify ψ with the one-form dx , we can identify Q I I with a conjugated version of the exterior derivative d = I dx ∂/∂x :

(4.4) Q =dh = exp(h)d exp(−h). To derive this result, one simply observes that it reproduces the commutation relations (4.3). In particular, the form of Q ensures that the condition for a map T : L → Z to be Q-invariant – which is that {Q, χI } must vanish – is the ﬂow equation of Morse theory: dxI ∂h (4.5) = −gIJ . ds ∂xJ 384 E. WITTEN

We take L to be the half-line s ≤ 0. We assume that h is a Morse function (its critical points are isolated and nondegenerate) and we let p be one of the critical points. We consider a path integral on L with the boundary condition that xi(s) → p for s →−∞. We are going to assume that all flow lines starting at p flow to infinity (rather than to another critical point), in which case, as reviewed in section 2.5, the possible boundary values xi(0) of the flow form a cycle Cp ⊂ Z. The dimension of this cycle is ip, the Morse index of h at p. A typical example is that Z is a noncompact complex manifold that admits lots of holomorphic functions (for instance, Z = Cn for some n)andh is the real part of a generic holomorphic function on Z. One may consider two points of view about the path integral. Regarded as a function of the boundary values of the fields at s = 0, the path integral on L computes a Q-invariant physical state Υ. Alternatively, if we integrate over the boundary values (with some boundary condition), the path integral computes a number. We first explore the first point of view. Exactly what the state Υ will turn out to be depends on exactly what action we take. In the most physically natural version of the theory, the bosonic part of the action is 1 I J IJ ∂h ∂h (4.6) Iphys = ds gIJx˙ x˙ + g , 2 ∂xI ∂xJ with a parameter . From a topological field theory perspective, it is natural to use a slightly different action that differs from Iphys by a boundary term: I 2 1 dx IJ ∂h 0 (4.7) Itop = ds + g = Iphys +(h/)| . 2 ds ∂xJ s=−∞ Finally, there is an equivalent action with an auxiliary field T : I − dx IJ ∂h IJ (4.8) Iaux = i dsTI + g J + dsg TI TJ . L ds ∂x 2 L Integrating out T brings us back to (4.7). The parameter is inessential (and has been set to 1 in formulas such as (4.2) above), because it can be absorbed in rescaling g → g/, h → h/. But it is convenient to make this parameter explicit, since localization of the path integral on the solutions of the flow equations occurs for → 0. Henceforth, we write h for h/. The path integral on the half-line (−∞, 0] gives a Q-invariant state15 in the Hilbert space associated to the boundary. In the physical formalism with the action Iphys, the path integral gives a state Υphys that is annihilated by Q = eh de−h . −h (4.9) d e Υphys =0.

15 This assertion can fail if there are ﬂow lines that interpolate from p to some other critical point q. We assume that there are none, for instance because h istherealpartof a generic holomorphic function on a complex manifold. THE PATH INTEGRAL OF QUANTUM MECHANICS 385

Since the relation between path integrals in the physical and topological actions comes from (4.7) or

(4.10) exp(−Itop) = exp(−Iphys) exp(−h(x(0)) + h(x(−∞))), the result of replacing Iphys by Itop is simply to multiply Υphys by the function exp(−h + h(p)). Thus the output of the topological path integral is Υtop = exp(−h + h(p))Υphys, and it obeys simply

(4.11) dΥtop =0.

The state Υtop depends on , but only by Q-exact terms. It is easiest to compute Υtop if we use the version (4.8) of the path integral with an auxiliary field T . In the limit → 0, the path integral over T I dx IJ ∂h (4.12) DT exp i TI + g J L ds ∂x gives a delta function supported on the solutions of the flow equations. This means that the wavefunction Υtop has (for = 0) delta function support on Cp, the locus of boundary values of solutions of the flow equations. The closed differential form supported on Cp with the smallest possible degree (and the only one that can be defined without more information) is known as the Poincaré dual to Cp. This concept is defined more systematically in [26], but 1 2 m in brief, if Cp is defined locally by equations x = x = ···= x = 0 (where m is the codimension of Cp), then the Poincaré dual is locally

(4.13) δ(x1)δ(x2) ···δ(xm)ψ1ψ2 ...ψm.

Since, for a fermionic variable ψ, a delta function is simply a linear function δ(ψ)=ψ, we can equally well write the Poincar´e dual as

(4.14) δ(x1)δ(x2) ···δ(xm)δ(ψ1)δ(ψ2) ···δ(ψm).

To get the fermionic delta functions, simply do the integral over χ,which takes the form I (4.15) Dχ exp i χI Dψ , L where D is the linearization of the flow equations. This gives a delta function setting Dψ = 0. When restricted to s = 0, this delta function forces ψI (0) to 1 2 m be tangent to Cp, so it sets to zero precisely the modes ψ ,ψ ,...,ψ of ψ that are valued in the normal bundle to Cp. This explains why Υtop is the Poincaré dual to Cp. (For more on this type of integral, see the discussion of eqn. (5.2) in [11].) So far, we have identified the state Υtop that emerges when the path integral (with the action Itop) is computed as a function of the boundary 386 E. WITTEN values. Now we would like to integrate over the boundary values, perhaps with the help of some additional boundary condition or boundary couplings, to get a number. If Cp has positive dimension ip, its Poincarédual has the wrong degree to be integrated. We will choose boundary couplings that involve a choice of closed form Λ of degree ip, and we will aim to give a path integral recipe for computing (4.16) Υtop ∧ Λ. Z Actually, we will take Z to be a noncompact Calabi-Yau manifold of complex dimension n, endowed with a holomorphic volume form Ω. (We take the metric of Z to be hermitian, possibly Kahler.) For example, we might take Z = Cn, with complex coordinates z1,...,zn,andΩ=dz1dz2 ···dzn.We take (4.17) Λ = Ω exp(S), where S is a holomorphic function on Z. Then the finite-dimensional integral that we will represent by a path integral on L =(−∞, 0] is (4.18) Υtop ∧ Λ= Λ= Ω exp(S). Z Cp Cp

(We have reduced an integral over Z to one over Cp using the fact that Υp has delta function support on Cp.) Since Ω is an n-form, the integral vanishes unless Cp is n-dimensional. In addition, of course, the Morse function h must be related to S in such as way that the integral converges. A simple way to ensure this is to set (4.19) h =ReS, as discussed in [2] and in section 2.5. But actually, any h that is close enough to Re S will work just as well. In the example discussed in the introduction, with n =1andS(z)=−z4 + az, with a constant a,wecouldtake (4.20) h =Re(−z4 + az), with some other constant a; changing h by subleading terms does not aﬀect the convergence of the integral. To represent (4.18) as a path integral, we simply write the usual Feynman integral for this supersymmetric system, but with a boundary insertion of i1 i2 exp(S)Ω. Of course, Ω is represented in the path integral as Ωi1i2...in ψ ψ ... ψin . Thus, we consider the path integral

(4.21) DX Dψ Dχ exp(−Itop) exp(S(x(0))

i1 i2 in × Ωi1i2...in ψ (0)ψ (0) ...ψ (0). THE PATH INTEGRAL OF QUANTUM MECHANICS 387

Here Itop is the supersymmetric completion of Itop defined in (4.7), including fermionic terms. We would like to verify directly that the path integral in (4.21) is supersymmetric, that is, Q-invariant. It is reasonable to expect this, since this path integral equals the period Cp Ω exp(S), which has a topological meaning, that is, it is invariant under small deformations of the integration cycle Cp. First, let us just directly verify Q-invariance. Since [Q, Itop]=0,weonly need to worry about Q-invariance of the boundary insertion exp(S(x(0)) i1 i2 in Ωi1i2...in ψ (0)ψ (0) ...ψ (0). Since {Q, ψ} = 0, we only need to worry about [Q, xI ]=ψI , or in terms of local complex coordinates xi,[Q, xi]=ψi, [Q, xi]=ψi. Actually, [Q, xi] will not enter, since S and Ω are holomorphic. And the contributions that come from [Q, xi] all vanish by fermi statistics, since the boundary insertion is already proportional to the product of all n fermions of type (1, 0), namely ψi1 ψi2 ···ψin . So the boundary condition is Q-invariant. Now we will obtain the same result in a longer but illuminating way. Let us work out the boundary conditions on bosons and fermions that arise naturally from the above path integral. Because the boundary insertion is proportional to the product of all fermion fields ψi of type (1, 0), and a fermion field ψi is equivalent to a delta function δ(ψi), the boundary condition sets the (1, 0) part of ψ to zero at s =0:

(1,0) (4.22) ψ |s=0 =0.

What about χ? The proper boundary condition on χ is determined by setting to zero the boundary contribution in the equations of motion. The fermion 0 I kinetic energy is If = i s=−∞ dsχI Dψ , and when we vary it, we get bulk equations of motion Dψ = Dχ = 0. After imposing these equations, we still ﬁnd a nonzero boundary contribution to the variation of If ,andwemust pick the boundary conditions to set this to zero. The boundary contribution I i is χI δψ |s=0. Since the only restriction on δψ is that its (1, 0) part δψ is equal to zero, the boundary condition on χ must be vanishing of the (0, 1) part χi. We can analyze the bosonic boundary conditions in the same way. The purely bosonic part of the exponent of the path integral is 0 I K − 1 dx IJ ∂h dx KL ∂h | (4.23) dsgIK + g J + g L + S(x) s=0. 2 −∞ ds ∂x ds ∂x

Upon varying this and imposing the Euler-Lagrange equations to set the bulk part of the variation to zero, we are left with a boundary variation at s = 0, which takes the form J −1 dx ∂h ∂S I (4.24) gIJ + J + I δx . dx ∂x ∂x s=0 388 E. WITTEN

Assuming that we want free boundary conditions, so that δxI is unconstrained, the boundary condition must be 1 dxJ ∂h ∂S (4.25) − gIJ + + =0. ds ∂xI ∂xI Let us verify that this condition, combined with the fermionic boundary conditions, is in fact supersymmetric. In general, supersymmetry of a boundary condition means that the component of the supercurrent normal to the boundary vanishes. But here, as we are in spacetime dimension one, the normal component of the super- J I currrent is simply the supercharge Q = gIJdx /ds + ∂I h ψ . Using the I bosonic boundary condition (4.25), this is equivalent at s =0 to ∂I Sψ . i Because S is holomorphic, this is equivalent to ∂iSψ , and vanishes at s =0 since the (1, 0) part of ψ vanishes at s = 0. So again we have established the Q-invariance of the boundary condition. This experience will stand us in good stead in more complicated examples.

4.2. Sigma-models in two dimensions. The sigma-model studied in section 4.1 can be regarded as a reduction to one dimension of a model defined in two (or even more) dimensions. Let us consider the two- dimensional case. For brevity, rather than considering unintegrable structures, we will assume that the target space Z is a Kahler manifold, with complex structure I and Kahler metric g. Then a sigma-model with target Z has four supercharges, and admits a topologically twisted A-model. Just as in one dimension, the bosonic fields xI describe a map T :Σ→ Z,for some Riemann surface Σ, and the fermi fields ψI of F = 1 are sections of the pullback to Σ of TZ, the tangent bundle of Z. The fermi fields of F = −1are a one-form χ on Σ with values in the pullback of T ∗Z (the cotangent bundle K of Z); as in section 2.8, χ obeys an algebraic constraint χJ = χK I J . Let us ask whether we can naively imitate the construction of section 4.1 and add to the bosonic part of the action a boundary coupling (4.26) dtS(x(t)), ∂Σ where S is an I-holomorphic function on Z,anddt isaone-formon∂Σ. Whether we can add this term depends on what supersymmetry we want to preserve. The two-dimensional sigma-model with target Z has two topologically twisted versions – the A-model and the B-model. In the B-model, there is no problem in adding the boundary interaction (4.26). However, the B-model localizes on constant maps to Z, rather than on the nontrivial integration cycles associated to Morse theory that are of interest in the present paper. Hence the B-model with the boundary coupling (4.26) is not a good generalization to two dimensions of what we have said in section 4.1. THE PATH INTEGRAL OF QUANTUM MECHANICS 389

The A-model does lead to interesting integration cycles related to Morse theory in loop space. However, the boundary coupling (4.26) does not work in the A-model. The reason is that because of the A-model transformation law {Q, xI } = ψI , invariance of the boundary coupling requires us to set ψ(1,0) to zero on the boundary of Σ, just as we did in section 4.1. The problem is that in two dimensions, unlike one dimension, this is not a satisfactory boundary condition. A quick explanation of why is that, in one dimension, setting the boundary values of ψ(1,0) to zero amounts to a boundary insertion i1 i2 in Ωi1i2···in ψ ψ ···ψ that has finite fermion number F . In two dimensions, the analog would have to be formally i1 i2 in (4.27) Ωi1i2···in ψ (t)ψ (t) ···ψ (t), t∈∂Σ now carrying an infinite F , as a result of which all correlation functions would vanish. Here is a better explanation. The fermion kinetic energy of the A-model is D (1,0) D (0,1) (4.28) i χ(1,0) ψ + χ(0,1) ψ . Σ Dz Dz When we vary this kinetic energy, the boundary contributions are (1,0) (0,1) (4.29) i χ(1,0)δψ + χ(0,1)δψ . ∂Σ In general, any boundary condition sets to zero a middle-dimensional subspace of the fermion boundary values in such a way that the boundary variations vanish. To obey these conditions, a boundary condition setting (1,0) ψ to zero must leave the boundary values of χ(1,0) unconstrained. But this is not an elliptic boundary condition, and concretely, as the equation of motion of χ(1,0) is Dχ(1,0) = 0, if we leave its boundary values unconstrained, χ(1,0) will have infinitely many zero modes, and all correlation functions will vanish. To avoid this, all A-branes, both Lagrangian ones [27] and coisotropic ones [3], involve a boundary condition that sets to zero a linear combination (1,0) (0,1) of ψ and ψ (and similarly a linear combination of χ(1,0) and χ(0,1)). From the standpoint of the present paper, the cure for the problem is that S in eqn. (4.26) should be holomorphic not in complex structure I, but in some other complex structure J which obeys IJ = −JI. Thus, assuming that I and J are both integrable, the Kahler structure of Z should be extended to a hyper-Kahler structure and the sigma-model has eight supercharges rather than four. (We can get by with an almost hyper-Kahler structure, as described at the end of section 2.7.) If S is holomorphic in complex structure J, then to ensure invariance of the boundary coupling (4.26), we require vanishing of the boundary values of ψ(1,0;J), that is the (1, 0) part of ψ with 390 E. WITTEN respect to J.AsJ anticommutes with I, there is no component of ψ that is of type (1, 0) with respect to both I and J, and the problem encountered above disappears. The rest of the analysis of the boundary conditions given in the one-dimensional case in section 4.1 has a straightforward extension to the present two-dimensional case. The boundary condition on χ must set to I zero the boundary values of χ(0,1;J).Onx , we take free boundary conditions (no restriction on the boundary values of δxI ). The vanishing of the normal component of the topological supercurrent is demonstrated in the same way as in section 4.1. By taking S to be holomorphic in a complex structure that anticommutes with I, and constructing a boundary condition as just described, we ensure that the boundary coupling (4.26) is compatible with the Q-invariance of the A-model. But we are still not out of the woods: we need to ask whether the A-model path integral converges. The bulk equations for supersymmetry of the A-model are flow equations of the general form ∂xI /∂t = −gIJδh/δxJ , for a suitable functional h which is determined by the bulk action of the A-model. To ensure convergence of the path integral, we must pick S so that its real part has the same asymptotic behavior as h (they may differ by a correction that grows too slowly to be problematical). The last condition is hard to obey, since h is determined entirely by the A-model defined with complex structure I and is supposed to be related to the real part of a holomorphic function in some other complex structure J. The constructions in sections 2 and 3.2 are based on cases in which this can actually happen. These cases, assuming that I and J are both integrable, involve a hyper-Kahler target space and a doubling of supersymmetry relative to the one-dimensional analysis of section 4.1. The special nature of the target space was explicit in sections 2 and 3. And in section 3.2, the Hamil- tonians that work are related to the twisted masses [28] which are precisely the potentials that preserve all the supersymmetry of a sigma-model with a hyper-Kahler target. A similar story holds in section 5.3 where we include gauge fields.

5. Analogs with gauge fields In the present section, we repeat the analysis of section 4, this time in the presence of gauge interactions. In section 5.1, we add gauge fields to the one-dimensional supersymmetric sigma-models studied in section 4.1, taking the target space of the sigma-model to be of finite dimension. In section 5.2, we consider the case that the target space of the sigma- model is the infinite-dimensional space of gauge-connections on a three- manifold. This leads to one of the main insights of the present paper: under certain conditions, the path integral of N = 4 super Yang-Mills theory on a half-space reduces to the path integral of three-dimensional Chern-Simons gauge theory on the boundary of the half-space. THE PATH INTEGRAL OF QUANTUM MECHANICS 391

Finally, in section 5.3, we consider two-dimensional supersymmetric gauge theories with gauge ﬁelds. This generalization of the analysis of section 2 leads to the construction of a new integration cycle for quantum mechanics with constraints.

5.1. Supersymmetric quantum mechanics with gauge fields. 5.1.1. Construction of the model. Though the preliminary setup did not require this, our main application in section 4 required the target space Z of the sigma-model to be a complex manifold. In the present discussion, we will want Z to be a complex manifold for much the same reason, and we want Z to be endowed with a compatible symplectic structure so that we can define the moment map for a group action on Z. So we will take Z to be a Kahler manifold to begin with. The sigma-model with a Kahler target space has four supersymmetries. Here we will consider a gauged version of the sigma-model, still with four supercharges. We assume that Z has a compact connected group H of symmetries, and we will consider the combined theory of a map Φ : L → Z together with vector multiplets that gauge the H symmetry. This theory can arise by dimensional reduction from four dimensions, and to understand some of its properties, it is convenient to start there. We formulate the theory on R4 with Euclidean signature and spacetime μ coordinates y , μ =0, 1, 2, 3. The only propagating bosonic fields are the 3 μ 4 gauge field A = μ=0 Aμ dy and the map Φ : R → Z, which we describe i in terms of complex-valued fields x , i =1,...,dimC Z that correspond to local complex coordinates on Z. The four supercharges are a spinor field Qα, α =1, 2, of positive four-dimensional chirality and another spinor field Qα˙ , α˙ =1, 2 of negative chirality. The algebra they generate is 3 μ (5.1) {Qα, Qα˙ } = σαα˙ Pμ, μ=0 μ where σαα˙ are the Dirac matrices written in a chiral basis, and Pμ are the momentum generators. Now we dimensionally reduce to two dimensions, taking the fields to be 2 3 independent of y and y . The components A0,A1 of the four-dimensional gauge field survive as a two-dimensional gauge field, but the components A2, A3 become scalar fields with values in the adjoint representation. The supersymmetry algebra still takes the form (5.1), but the momentum components P2 and P3 are now simply the commutators with A2 and A3, respectively. It is convenient to combine A2 and A3 to a complex scalar field σ = A2 − iA3 with values in the adjoint representation. We write [σ, ·] for the infinitesimal gauge symmetry generated by σ. Upon dimensional reduction to two dimensions, the four-dimensional rotation group SO(4) reduces to SO(2) × SO(2), or equivalently U(1) × U(1), where the first factor rotates y0,y1, and the second rotates y2,y3. 392 E. WITTEN

A spinor of positive chirality, such as Qα, has components of U(1) × U(1) charges ±1/2, ±1/2, and a spinor of negative chirality, such as Qα˙ , has components of charges ±1/2, ∓1/2. We denote the components of Q and Q with these quantum numbers as Q±± and Q±∓. If the superpotential vanishes (or more generally if it is quasihomogeneous), there is a third U(1) symmetry, present already in four dimensions before dimensional reduction. This is the U(1)R symmetry under which Q and Q have respective charges −1and1. (U(1)R may be anomalous in four dimensions, but not in the dimensional reduction to two dimensions.) The usual supercharge of the two-dimensional A-model is (5.2) Q = Q++ + Q−+.

If U(1)R is a symmetry, then upon suitably twisting the theory and restricting to Q-invariant observables and correlation functions, one gets a two- dimensional topological field theory that can be formulated on any oriented two-manifold Σ. (One has to pick a metric on Σ to define the theory, but the results are independent of this metric.) The starting point in twisting is that Q is invariant under a certain modified rotation symmetry; if J is the generator of U(1) and R is the generator of U(1)R, then Q commutes with a linear combination J = J +R/2. In the twisted theory, the spins of all fields are their J eigenvalues rather than their J eigenvalues. Moreover, since Q is J -invariant, its square does not generate a translation along R2 (those translation generators have J = ±1). Rather, the supersymmetry algebra (5.1) implies that

(5.3) Q2 =[σ, ·].

In particular, this means that on gauge-invariant fields and states, Q2 = 0 and one can define the cohomology of Q. Upon restricting states and operators to Q-invariant ones and projecting to the cohomology of Q,one obtains (in the R-symmetric case) a two-dimensional topological field theory known as the A-model; it was analyzed in [29], where the following formulas are obtained and described in more detail (with some minor differences in notation). The A-model is Z-graded by U(1). It is convenient to introduce a generator F of U(1) that is normalized so that Q has F =1 and σ has F = 2, while the fermions have F = ±1. An important property of the model is that

(5.4) [Q, σ]=0.

This reflects the fact that there is no elementary fermion of F =3,andis consistent with Q2 =[σ, ·] since [σ, σ]=0. After twisting, the fermionic fields in the vector multiplet are an adjoint- valued one-form λ of F = 1 and adjoint-valued zero-forms η, ρ of F = −1. Under the action of Q, the gauge field is contained in a multiplet that takes THE PATH INTEGRAL OF QUANTUM MECHANICS 393 the form

(5.5) [Q, Aμ]=λμ, {Q, λμ} = −Dμσ.

The rest of the vector multiplet becomes

(5.6) [Q, σ]=η, {Q, η} =[σ, σ], and

(5.7) {Q, ρ} = D, [Q, D]=[σ, ρ].

All these formulas are consistent with Q2 =[σ, ·]. In (5.7), D is an auxiliary ﬁeld with values in the adjoint representation; if we take the minimal form of the sigma-model action, the equation of motion of motion of D is D = F +μ, where F is the curvature dA+A∧A, is the Hodge star, and μ is the moment map for the action of H on Z. The moment map is deﬁned in the usual way by

∂μa j ∂μa i (5.8) i = Va ωji, = Va ωij. ∂x ∂xj Here ω is the Kahler form of Z and Va, a =1,...,dim H are the vector a fields that generate the action of H on Z. We will write V (σ)= a σ Va for the vector field corresponding to σ. After eliminating D by its equation of motion, the first equation in (5.7) becomes

(5.9) {Q, ρ} = F + μ.

As for the chiral multiplets, in the twisted theory the F = 1 fermions are zero-forms ψi, ψi with values in Φ∗(T (1,0)Z)andΦ∗(T (0,1)Z) (that is, the pullbacks to R2 of the (1, 0) and (0, 1) parts of the tangent bundle of Z). These ﬁelds obey

(5.10) [Q, xi]=ψi, {Q, ψi} = V i(σ) [Q, xi]=ψi, {Q, ψi} = V i(σ).

The F = −1 fermi ﬁelds are a (0, 1)-form χi valued in Φ∗(T (1,0)Z), and a (1, 0)-form χi valued in Φ∗(T (0,1)Z). They transform in the familiar sort of i i i i i multiplet {Q, χ } = F ,[Q, F ]=[σ, χ ] where F is an auxiliary ﬁeld; there j is a complex conjugate multiplet φj,χ , F j. After eliminating F, F by their equations of motion, we get

i i ij ∂W (5.11) {Q, χ } = ∂Ax + g ∂xj i i ij ∂W {Q, χ } = ∂Ax + g , ∂xj 394 E. WITTEN where g is the Kahler metric of Z and we make the usual decomposition dA = ∂A +∂A for the gauge-covariant exterior derivative dA on the Riemann q surface Σ. (Explicitly if w ,q=1, 2 are local coordinates on Σ, then dA is i q i a i defined by dAx =dw (∂qx + Aq Va ).) The equations for a supersymmetric configuration of the bosonic fields are {Q, Λ} = 0 for every fermionic field Λ. In view of the above formulas, the supersymmetry conditions that involve σ are

(5.12) Dμσ = V (σ)=[σ, σ]=0.

These equations say that the gauge transformation generated by σ leaves fixed A, x,andσ, so it is a symmetry of the whole configuration. These conditions are very restrictive, and in many applications they force σ =0. The supersymmetry conditions for the other fields are

(5.13) F + μ =0

i ij ∂W ∂Ax + g =0, ∂xj along with the complex conjugate of the second equation.16 As we have already explained in section 3.2, the second equation in (5.13) has two-dimensional symmetry only if W is quasihomogeneous. (A generic W violates the U(1)R symmetry that was assumed in the construction of the twisted A-model.) In the present section, we wish to consider a generic W , so we will have no two-dimensional symmetry. For this reason among others,17 we make a further dimensional reduction to one dimension, where the above equations are natural for any W . In this reduction, the gauge field splits up into a one-dimensional gauge field A0 and an adjoint-valued real scalar field A1. (We already generated two such fields A2 and A3 in the first stage of reduction; one might think that there should be an SO(3) symmetry rotating these three fields, but this symmetry is spoiled by the choice of Q.) We will return to the two-dimensional case in section 5.3. The equations (5.13) are flow equations in the y0 direction, up to a gauge transformation. To agree with our previous notation, we write s for y0.In

16 i i In [29], instead of the second equation, two separate equations ∂Ax =0=∂W/∂x were given. This is because separate invariance was imposed under Q++ and Q−+.Inthe present paper, we will consider boundary conditions that conserve not Q++ or Q−+ but only their sum Q = Q++ + Q−+, so we add the two equations. 17 There is another diﬃculty in continuing the analysis in two dimensions: unless we double the supersymmetry (as we will do in section 5.3), we will run into diﬃculties analogous to those that were described in section 4.2. THE PATH INTEGRAL OF QUANTUM MECHANICS 395

the gauge A0 = 0, the equations (5.13) can be written

dA1 ∂h (5.14) = −μ = − ds ∂A1 i dx a i ij ∂W ij ∂h = −iA1Va − g = −g , ds ∂xj ∂xj with a (5.15) h = A1μa +2ReW. (To verify that the gradient of h is as claimed, one needs the usual relation gij = −iωij between the metric and the Kahler form.) So as usual the equations for supersymmetry can be written as flow equations for a Morse function h. 5.1.2. Geometrical interpretation. Before trying to interpret the above formulas in differential geometry, we need a little background. First consider a general manifold Y with action of a connected Lie group H.LetΩ∗(Y )be the space of differential forms on Y , graded in the usual way by the degree of a form. Introduce a variable σ, taking values in the Lie algebra h of H,and considered to be of degree 2. Let Sym∗(h) be the algebra of polynomial functions of σ. The following natural operator acts on W =Ω∗(Y ) ⊗ Sym∗(h):

(5.16) D0 =d+ιV (σ), where d is the usual exterior derivative acting on Ω∗(Y ), V (σ) is the vector ﬁeld on M corresponding to σ ∈ h,andιV (σ) acts by contraction of a diﬀerential form with V (σ). Note that each term in D0 has degree 1 (the contraction operation has degree −1 but σ has degree 2). Evidently, 2 (5.17) D0 = LV (σ), where

(5.18) LV (σ) =dιV (σ) + ιV (σ)d is the Lie derivative with respect to the vector field V (σ), or in other words, the generator of the symmetry of W that corresponds to σ. The formula (5.17) corresponds to eqn. (5.3) in the field theory construction. It means that if we restrict to the H-invariant subspace of W, which we denote as H 2 W , then D0 = 0. So we can define the cohomology of D0 acting in that subspace. To explain just how D0 is related to the construction in section 5.1.1, let uI be local (real) coordinates on Y and set ψI =duI . Then we compute I I I I (5.19) [D0,u ]=ψ , {D0,ψ } = V (σ), [D0,σ]=0. This is in perfect parallel with eqns. (5.10) and (5.4), if we understand the uI to be the coordinates xi, xi of Z. And we can find another multiplet of 396 E. WITTEN

I the same kind if we consider the u to be A1 and interpret λ1 as dA1; then (5.19) matches (5.5). So if we set Y = Z ×h (where h is parametrized by A1), then D0 is very similar to the topological supercharge Q that was considered in section 5.1.1. There are two essential differences between the construction in section 5.1.1 and what we have said so far here. First of all, rather than being h- valued, σ in section 5.1.1 took values in the complexification hC of this Lie algebra. The formulas of section 5.1.1 also involve the complex conjugate σ of σ, and a fermionic field η such that [Q, σ]=η. To include these fields in the construction, we view hC as a complex manifold (which is isomorphic to Cn where n =dimH). We view the components of σ as linear holomorphic ∗ functions on hC, we interpret η as dσ, and we replace Sym (h) with the 0,∗ 0,∗ space Ω (hC)of(0,q) forms on hC,for0≤ q ≤ n. (We grade Ω (hC)by considering σ, σ,andη to have degrees 2, −2, and −1.) Finally we define

a ∂ a (5.20) D1 =dσ +[σ, σ] ιdσa ∂σa and replace D0 by

(5.21) D = D0 + D1,

∗ 0,∗ acting on Ω (Y ) ⊗ Ω (hC). This incorporates the commutation relations 2 for σ and η and obeys D = LV (σ) (where LV (σ) is now the symmetry generator in the bigger space). Furthermore, the cohomology of D acting on ∗ 0,∗ the H-invariant part of Ω (Y ) ⊗ Ω (hC) coincides with the cohomology of H D0 acting on the smaller space W . To prove this, a first observation is that if DΨ = 0, then the term in Ψ of highest degree in dσ is annihilated by a a dσ = dσ ∂/∂σ .Thisissobecausedσ is the part of D of highest degree in 2 dσ (namely degree 1). Note that dσ = 0, and that the cohomology of dσ vanishes for states of strictly positive degree in dσ. Now suppose that DΨ=0 and Ψ has degree k>0indσ. Then by a transformation Ψ → Ψ =Ψ+DΛ, for some Λ, we can replace Ψ by a state Ψ that is of degree at most k − 1 in dσ; this is possible because the cohomology of dσ vanishes for states of positive degree in dσ. Repeating this process, we reduce to the case k =0, in other words the case that Ψ is independent of dσ. Then the condition DΨ = 0 tells us that Ψ is holomorphic in σ. To have finite degree, Ψ must be polynomial in σ. But then Ψ can be regarded as an element of WH ,and the action of D on Ψ coincides with the action of D0. In constructing the operator D, we have incorporated all of the bosons σ, σ, A1,x, and x that should be present in describing physical states. (We have also properly taken into account the time component A0 of the gauge fields: in a Hamiltonian framework, this field is set to zero but is associated to a Gauss law constraint that physical states should be H-invariant, and we have imposed this constraint.) The fermions that we have incorporated i i j j explicitly so far are η =dσ, ψ =dx , ψ =dx ,andλ1 =dA1.Wehavenot THE PATH INTEGRAL OF QUANTUM MECHANICS 397

i j yet discussed explicitly the other fermions λ0, χ , χ ,andρ, but they are present implicitly as they are canonically conjugate to the fermion fields that have been discussed. However, we will need to make an important adjustment to incorporate those fields correctly. Fermions in this second group are contraction operators, so they have definite commutation relations with D, but those commutation relations are not the ones we want. For i j example, χ is canonically conjugate to ψ , which appears in D only in the j i ij i i term ψ ∂/∂xj.So{D,χ } = g ∂/∂xj = −dx /ds, while we want {Q, χ } = −(dxi/ds + gij∂h/∂xj). To achieve this result, and its analogs for the other fields, we conjugate D by exp(−h) and define

(5.22) Q = exp(h)D exp(−h). This is our final result for the description of Q in terms of differential geometry. Because Q and D are conjugate, computing the cohomology of Q is equivalent to computing that of D. However, we have to be careful to describe the class of wavefunctions in which we wish to take the cohomology. What we ultimately want to do with a wavefunction Ψ that represents a cohomology class of D is to pair it as in eqn. (5.28) below with another wavefunction Ψ that we will allow to have exponential growth for h →∞. To ensure that such integrals converge, we want Ψ to be supported on a region on which h is bounded above. (Without changing anything essential, we could instead require Ψ to decay faster than exp(−h)forh → +∞.) We refer to a region in which h is bounded above as a region with h<<∞,andin the above definitions, with Y = Z × h, we replace Ω∗(Y ) by what we will ∗ call Ωh<<∞(Y ), the space of differential forms on Y on whose support h is bounded above. First we will take a mathematical approach to the cohomology, and then a physics-based approach. For the mathematical approach, we begin with the fact that, by arguments already explained (these arguments are unaffected by the support condition), the desired cohomology of D is the H same as the cohomology of D0 acting on Wh<<∞, the H-invariant part of ∗ ∗ Ωh<<∞(Y ) × Sym (h). Ignoring the support condition for a moment, the ∗ ∗ cohomology of D0 acting on Ω (Y )⊗Sym (h) is known as the H-equivariant cohomology of Y . In general, if Y is any space with action of H, the complex ∗ ∗ Ω (Y ) ⊗ Sym (h) with action of the differential D0 is known as the Cartan model of the H-equivariant cohomology of Y . For example, if H acts freely on Y , the equivariant cohomology of Y is the same as the ordinary cohomology of the quotient Y/H. At the opposite extreme, if Y isapoint,thenD0 =0 H and its cohomology is just Sym∗(h) , the algebra of H-invariant polynomials on h. In the present case, the support condition means that we actually want not the usual equivariant cohomology of Y , but the equivariant cohomology with values in differential forms supported at h<<∞. This version of 398 E. WITTEN the equivariant cohomology is naturally computed by using the H-invariant Morse function h as an equivariant Morse function in the sense of [30]. The reason for this is that h is bounded above on any cycle generated by flows with respect to the Morse function h. First let us compute dh:

∂h ∂W j (5.23) = + ω V (A1) ∂xi ∂xi ji ∂h = μ. ∂A1 a Here V (A1)=A1Va is the vector field on Z corresponding to A1 ∈ h. Con- i i tracting the first equation with V (A1) and using the fact that V (A1) ∂iW = 0(sinceW is H-invariant and holomorphic), we deduce that if ∂h/∂xi =0, i j then V (A1) V (A1) ωij = 0, which implies (because of the relation of ω to the Kahler metric g and the positivity of g) that V (A1) = 0. So the conditions for a critical point are ∂W (5.24) = V (A1)=μ =0. ∂xi Since W is H-invariant and holomorphic, it is invariant under the complexification HC of the compact Lie group H. The critical points of W thus form HC orbits. We will assume that W is sufficiently generic that it has only finitely many critical HC orbits, which moreover are nondegenerate. (We call a critical orbit nondegenerate if W is nondegenerate in the directions transverse to the orbit.) As explained in [2], if W is sufficiently generic, there are no flows between distinct critical orbits. In this case, h is an equivariantly perfect Morse function in the sense of [30], and the desired equivariant cohomology is a direct sum of contributions from distinct critical orbits. Among the critical orbits of W , only those that admit points with μ = 0 contribute to the equivariant cohomology of Y ; this is clear from the fact that μ = 0 is one of the conditions for a critical point of h.Wecall a critical orbit unstable if it contains no point with μ =0.(In[2], it was shown that for understanding Stokes phenomena, such unstable orbits can consistently be excluded.) If an orbit OC does contain points with μ =0, then it is contractible to its subspace with μ = 0, and this subspace is an orbit O of the compact group H. In this case, OC is equivalent topologically to T ∗O.TheH orbit O is isomorphic to H/P for some subgroup P ⊂O.We call the orbit stable if P is a finite group, and (strictly) semi-stable otherwise. If O is stable, then the equivariant cohomology of O is the ordinary cohomology of the quotient O/H. That quotient is a point, so a stable orbit contributes a one-dimensional space to the equivariant cohomology of Y = Z × h. This contribution appears in a degree that is equal to the Morse index of the Morse function h at the orbit O;forh of the form we have considered, this index is always one-half of the real dimension of Z.This THE PATH INTEGRAL OF QUANTUM MECHANICS 399

∼ will be explained shortly. In general, if O = H/P, where P is a Lie group of Lie algebra p, then the equivariant cohomology of O is the same as the equivariant cohomology of P acting on a point. That equivariant cohomology ∗ is the cohomology of D0 = 0 acting on the P -invariant part of Sym (p); in other words, it is the ring of P -invariant polynomials on p. The contribution of such an orbit to the equivariant cohomology of Y is this polynomial ring, 1 shifted in degree by the Morse index of h, which again is 2 dim Z. This means that such an orbit contributes to the equivariant cohomology of Y asingle 1 class of degree 2 dim Z, and infinitely many classes of higher degree. To show that the Morse index of h at a nondegenerate critical orbit is 1 always 2 dim Z (where the real dimension is understood), independent of the dimension of the stabilizer P , we first observe that the function h0 =2ReW , understood as a Morse function on Z, has Morse index equal to one-half the 1 real codimension of the critical orbit OC,or 2 dim Z−dim O. Now the critical orbit OC ⊂ Z of the Morse function h0 pulls back, under the projection a Z × g → Z,toOC × g ⊂ Z × g. Let us regard h1 = A1μa as a function on OC × g; it has a critical set, defined by the equations μ = V (A1) = 0, that is a p bundle over O (where O⊂OC is defined by μ = 0). The Morse index of h1 is dim O, and the Morse index of h = h0 + h1 is the sum of the Morse 1 indices of h0 and of h1,or 2 dim Z. Now let us explain these facts from a physical point of view. The potential energy of the model is

2 2 2 2 (5.25) V = |dh| + |V (σ)| + |[A1,σ]| + |[σ, σ]| 2 2 2 2 2 2 =2|dW | + |μ| + |V (A1)| + |V (σ)| + |[A1,σ]| + |[σ, σ]| .

(In any expression such as |dh|2, the symbol ||refers to the norm with respect to the appropriate metric on the space in question.) In the first line, 2 we write V as the sum of |dh| (a contribution that is familiar in supersymmetric quantum mechanics without gauge fields) and additional terms whose origins in dimensional reduction of gauge fields from four dimensions are hopefully recognizable. (These terms are the squares of the σ-dependent quantities that appear in (5.12) and must vanish for unbroken supersymmetry.) To go to the second line, we used the explicit form of dh,andwe i also used again the identity V (A1) ∂iW = 0 to eliminate a cross term. Not coincidentally, V is invariant under SO(3) rotations acting on A1,A2,A3 (where σ = A2 − iA3). The underlying SO(3) symmetry of the reduction of the gauge theory from four dimensions to one has been broken by the choice of Q, but this does not affect the formula for V. In the classical approximation, a supersymmetric vacuum corresponds to a zero of V. Such zeroes evidently correspond to semistable critical orbits of W , that is to HC orbits OC on which dW = 0 and we can also solve the equation μ = 0. Consider first the case of a free or semi-free critical orbit. In this case, since the vector fields Va are linearly independent along 400 E. WITTEN

2 2 such an orbit, the terms |V (A1)| + |V (σ)| in the potential energy give 2 masses to all components of A1 and σ.The|μ| term in the energy, when restricted to OC, vanishes only on an H-orbit O⊂OC and gives a mass to those fluctuations away from O that lie in OC; nondegeneracy of W means 2 that the |dW | term gives mass to fluctuations normal to OC.Souptoa gauge transformation (that is, up to the action of H), a stable critical orbit contributes one classical vacuum, and in expanding around this vacuum, all fluctuations are massive. Because of the mass gap, there is no subtlety in quantization: a stable critical orbit contributes one supersymmetric state to the cohomology. By much the same computation as in supersymmetric quantum mechanics without gauge fields, the eigenvalue of F for this state is the Morse index of the function h at the given critical orbit OC.Fora Morse function of the type described in eqn. (5.15), this index, as explained above, is always one-half of the real dimension of Z. More generally, we consider a semistable critical orbit OC such that the locus O⊂OC with μ = 0 breaks the gauge symmetry from H to a subgroup P of positive dimension. Fluctuations normal to O are still massive. All classical vacua corresponding to points in O are gauge-equivalent, and by partially fixing the gauge symmetry, we can select a particular point p ∈O, leaving a residual unbroken gauge group P .ButA1 and σ now have flat directions. The conditions V (A1)=V (σ) = 0 now tell us that A1 and σ must lie in the Lie algebra p of P . The operator Q whose cohomology we wish to compute reduces at low energies to the operator D defined in eqn. (5.21), except that the symmetry group H is replaced by P and the space Y with action of P is just the Lie algebra p, parametrized by A1.As has already been discussed, the cohomology of this operator is the ring of P -invariant polynomials on the complex Lie algebra pC, graded in such a way that the linear functions σ on this Lie algebra are of degree 2. (A1 does not contribute to this cohomology, since it takes values in an equivariantly contractible space.) The contribution of a semistable orbit OC to the space of supersymmetric states is hence the indicated ring of P -invariant polynomials, shifted in degree by the Morse index of h, which again is one-half the real dimension of Z. In the last paragraph, we did not worry about whether the cohomology classes of Q that we described are square-integrable. In fact, polynomial functions of pC are not square-integrable in the natural metric on that Lie algebra. It seems quite likely, though not completely clear, that the Q-cohomology classes associated to strictly semistable critical orbits (with P of positive dimension) have no square-integrable representatives. In a somewhat similar compact situation, the cohomology of Q would coincide with the space of normalizable states annihilated by the Hamiltonian H = QQ† + Q†Q, but the case in which the classical potential admits noncompact flat directions at zero energy is quite different and it seems natural to conjecture that the kernel of H, in the space of square-integrable wave- functions, has a basis corresponding to the stable critical orbits only. THE PATH INTEGRAL OF QUANTUM MECHANICS 401

There is, however, a natural dual of the version of equivariant cohomology that we have considered so far; the dual consists of the equivariant cohomology with compact support along the A1 and σ directions, but with no growth condition for h → +∞, and decaying exponentially for h →−∞. Cohomology classes of this type can be described conveniently if one allows distributional wavefunctions. The most basic Q-invariant delta function is

(5.26) δ(A1)δ(λ1)δ(σ)δ(σ)δ(η), which certainly has compact support in the A1 and σ directions. (Here δ(σ)δ(σ) is an alternative notation for δ(Re σ)δ(Im σ).) This delta function is of degree zero, since δ(λ1)andδ(η) have opposite degrees. A more general Q-invariant distributional wavefunction is as follows. Take any H-invariant polynomial P on h∗ (the dual of h) and consider the wavefunction

(5.27) P(∂/∂σ) δ(A1)δ(λ1)δ(σ)δ(σ)δ(η) which again is Q-invariant. It has degree −2k if P is homogeneous of degree k. To satisfy the decay condition along Y , we should multiply one of these delta functions by a diﬀerential form on Y = Z × h that vanishes rapidly for h →−∞. For instance, if Z is a Calabi-Yau manifold with H-invariant holomorphic volume form Ω, and h =ReW , then an example of a class Ψ in the dual version of the equivariant cohomology is obtained by multiplying exp(W )Ω by the basic delta function given in (5.26). This product is 1 annihilated by D and has degree 2 dim Z. Now we can deﬁne a duality pairing. If Ψ is a class in the “ordinary” equivariant cohomology of Z × h (with polynomial growth allowed in the σ directions, but supported at h<<∞),andΨ is a class in the equivariant cohomology of Z ×h with compact support in the A1 and σ directions (with no growth condition for h →∞, and decaying at h →−∞), then there is a natural pairing (5.28) (Ψ, Ψ) = dA1 dλ1 dσ dσ dη dx dx dψ dψ ΨΨ. Z×h×hC

If Ψ and Ψ have deﬁnite degree, the sum of these degrees must equal the real dimension of Z in order for this integral to be nonzero. A typical such case is that Ψ is a class associated to a stable critical orbit, and Ψisas 1 described in the last paragraph; both are of degree 2 dim Z. Going back to the “ordinary” equivariant cohomology, one lesson from the above analysis is that in general, any cohomology class of D can be represented by a wavefunction Ψ(x, x, ψ, σ), independent of the other ﬁelds. In addition, cohomology classes associated to semi-free orbits have representatives that are independent of σ. 402 E. WITTEN

5.1.3. Gauge-invariant integration cycles. Next we will analyze the gauge-invariant version of something familiar from section 4.1: how to represent ordinary integrals over cycles obtained by solving flow equations by one-dimensional path integrals. We consider the path integral of the supersymmetric gauge theory that we have described above on a one-manifold L.WetakeL to be the half-line s ≤ 0. We define the boundary condition at s = −∞ by saying that, for s → −∞, x(s) approaches a specified H-orbit O⊂Z consisting of critical points of W on which μ = 0. The path integral on L with this behavior imposed at s →−∞, when viewed as a function of the boundary values of the fields at s = 0, gives a state Ψ in the equivariant cohomology of Y = Z × h.This class can be localized on the H-invariant space CO ⊂ Z ×h that parametrizes solutions of the flow equations that start on O. (In particular, therefore, it 1 is supported for h<<∞.) The codimension of CO is 2 dim Z, the index of the Morse function h. The description we have given of the behavior at s = −∞ is sufficiently precise if O is a stable orbit, for then the fields A1 and σ are massive in expanding around O and naturally vanish at s = −∞. More fundamentally, there is no essential ambiguity about the initial conditions because a critical orbit of this type is associated to an equivariant cohomology class that is unique, up to a constant multiple. The path integral actually fixes the multiple in a natural way, generating, just as in section 4.1, the Poincaré dual of CO rather than a multiple of this. If, however, O is strictly semi-stable, then the potential energy for A1 and σ has flat directions in expanding around O. The input conditions at s = −∞ require choosing a Q-invariant wavefunction on the space of flat directions. This wavefunction is not supposed to have distributional support at σ = 0 (this would land us in the wrong sort of equivariant cohomology), but rather will have polynomial dependence on σ. The initial conditions at s = −∞ depend on the choice of such a wavefunction. The state Ψ obtained from the path integral on L is much more complicated in the strictly semistable case, because its support has more than one branch. This is because of eqn. (5.12), according to which a Q-invariant field configuration on L must lie on the locus in Z × h that is invariant under the symmetry generated by σ. This condition is trivial if σ = 0, in which case all flows are allowed, but becomes non-trivial as soon as σ is non-zero. The branches of the space of Q-invariant field configurations are classified by the conjugacy class of the subgroup of P that is left unbroken by σ. Having picked a critical orbit O (and some additional data if O is only semi-stable) to fix the initial conditions at s = −∞, there are, as in section 4.1, several slightly different path integrals that we can do on L.Ifwetake the action of the sigma-model to be the physical action of the supersymmetric gauge theory – a gauge-extended version of (4.6) – then the state defined by the path integral at s = 0 is annihilated by Q = exp(h/)D exp(−h/). THE PATH INTEGRAL OF QUANTUM MECHANICS 403

Just as in eqn. (4.7), it is slightly more convenient to add to the action I an exact form 1 dh (5.29) I → I + ds . L ds The modified action can be written more naturally in terms of the supersymmetric flow equations, as in eqn. (4.7). The modification has the effect of multiplying the quantum state at s = 0 by exp(−h/), so that it is annihilated by D rather than Q. We will use this approach. The path integral on L, as a function of the values of the fields at s =0, will determine a state Ψ in the equivariant cohomology. This state will be localized on the cycle CO in Z × h obtained by solving the flow equations. However, using the path integral to compute a state associated to a critical orbit O is not really what we want to do. Our real aim is to represent an ordinary integral over a middle-dimensional cycle in Z as a one-dimensional path integral, just as we did in section 4.1 in the absence of gauge-invariance. As in eqn. (4.17) and the example given after eqn. (5.27), we assume that Z is a Calabi-Yau manifold with holomorphic volume-form Ω. We also pick on Z an H-invariant holomorphic function S. The path integral that we will describe is always convergent if S coincides with the superpotential W of the sigma-model. In general, we want to pick S to be close enough to W so that the path integral will be convergent. (For example, if Z = Cn and W is a polynomial whose terms of highest degree are sufficiently generic, then as in eqn. (4.20), it will suffice if S differs from W by subleading terms.) The ordinary integral that we want to represent by a one-dimensional path integral is (5.30) Ω exp(S). CO∩{A1=0}

The meaning of this is as follows. The cycle CO ⊂ Z × h is of codimension 1 1 2 dim Z, and hence of dimension 2 dim Z +dimH. By intersecting this cycle with the locus A1 = 0, whose codimension is dim H, we get the cycle CO ∩ 1 {A1 =0}⊂Z. The dimension of this cycle is 2 dim Z, the correct dimension for the integral (5.30) to make sense. The restriction to A1 = 0 will come from the boundary condition at s = 0, as discussed presently. In fact, ΓO = CO ∩{A1 =0} is an H-invariant, middle-dimensional cycle in Z which was introduced in [2] in a more naive fashion. There, it was characterized as the cycle that parametrizes points in Z that can be reached at s = 0 by solving the ﬂow equations for the Morse function h0 =2ReW on Z, starting at s = −∞ on the H-orbit O. In the present paper, the gauge theory machinery has led us to consider not the Morse function h0 on Z, but a the more sophisticated Morse function h = h0 + A1μa on Z × h. However, upon setting A1 = 0, we reduce to the cycle in Z that was analyzed in [2]. We now want to show that it is possible to pick a supersymmetric boundary condition at s = 0 so that the path integral on the half-line will indeed 404 E. WITTEN compute the integral (5.30). Just as in section 4.1, part of the construction will be an insertion at s = 0 of the operator

i1 i2 in (5.31) exp(S(x(0))) Ωi1i2...in ψ (0)ψ (0) ...ψ (0), and this implies a boundary condition (5.32) ψ(1,0) =0. s=0

The boundary conditions on other fermions in the chiral multiplets are the same as before, and for similar reasons: we set χ(0,1) =0ats = 0 and leave other components of ψ, χ unrestricted. What is new relative to the previous analysis is that we have to find the right boundary conditions for the vector multiplet. One essential point here is that since {Q, ψi} = V i(σ), where V i(σ) is generically non-zero, the boundary insertion (5.31) is Q-invariant only if we set σ =0ats =0.Soalso σ =0ats = 0, and since {Q, σ} = η, this is only consistent if η also vanishes at s =0. There is another way to motivate the boundary condition σ(0) = 0. We will describe a formalism that works for semistable as well as stable critical orbits O. In the semistable case, the conditions at s = −∞ allow σ to be unbounded, so the boundary condition σ(0) = 0 is needed for convergence of the path integral. The same reasoning motivates us to pick a boundary condition A1(0) = 0, which in any case is needed if we hope to arrive at the integral (5.30). For consistency, as [Q, A1]=λ1, we must also set λ1(0) = 0. At this point, we have set to zero the boundary values of half the fermions in the gauge multiplet, namely η and λ1, so we must leave unconstrained the other fermions in that multiplet, namely ρ and λ0. This boundary condition is Q-invariant, for reasons similar to what was explained at the end of section 4.1. As there, some contributions to Q vanish because of the fermionic boundary conditions. For example, one contribution to Q from a the vector multiplet is a ηa dσ /ds, which upon quantization becomes a − a ηa ∂/∂σ . This vanishes at s = 0, because η does. From the point of view of classical differential geometry, what the path integral with these boundary conditions computes is the pairing (5.28) between a state Ψ in the equivariant cohomology of Z supported at h<<∞, defined by the boundary condition at s = −∞, and a state Ψ in the equivariant cohomology with compact support in the A1 and σ directions, defined by the boundary condition at s = 0. (These two dual forms of equivariant cohomology were defined more precisely in section 5.1.2.) The state Ψis given by (5.31) multiplied by δ(χ(0,1)) and the delta function in (5.26):

 (1,0) (5.33) Ψ = exp(S(x(0)) δ(ψ )δ(χ(0,1))δ(A1)δ(λ1)δ(σ)δ(σ)δ(η). THE PATH INTEGRAL OF QUANTUM MECHANICS 405

i1 i2 in (1,0) (Here we have written Ωi1i2...in ψ (0)ψ (0) ...ψ (0) as δ(ψ ).) The bosonic boundary condition (4.24) is needed in verifying Q-invariance of δ(χ(0,1)). The pairing (Ψ, Ψ) deﬁned by the path integral (or in diﬀerential geometry by eqn. (5.28)) reduces to the desired integral (5.30). The localization on CO comes from Ψ, and the localization at A1 = 0 comes from Ψ. The nat- uralness of this pairing gives another motivation for the boundary condition that we have placed on the vector multiplet. The reason that we have given such a detailed explanation of this gauge- invariant generalization of the result of section 4.1 is that it has an interesting application to three-dimensional Chern-Simons gauge theory, to which we turn next. We will be able to express the path integral of Chern-Simons gauge theory in three dimensions in terms of a path integral of N =4super Yang-Mills theory in four dimensions. This in turn has an interesting application: it gives a new perspective on the relation [15] between Khovanov homology [16] and spaces of BPS states. We leave the details for [14]and remark here only that once one re-expresses the Chern-Simons path integral in terms of N = 4 super Yang-Mills theory, one can study it using standard techniques of electric-magnetic duality and related string theory dualities.

5.2. Application to Chern-Simons gauge theory. 5.2.1. The Chern-Simons form as a superpotential. We will apply what we have learned in section 5.1 to the following special situation. We compactify N = 4 super Yang-Mills theory from four dimensions to one on a three-manifold W . Thus, we formulate the theory on the four-manifold M = R × W , making an R-symmetry twist so that some supersymmetry is preserved. The twist is accomplished by embedding the holonomy group SO(3) of W in the SO(6) R-symmetry group in the obvious way (the vector representation of SO(6) transforms as the vector representation of SO(3) plus three singlets). Of the six adjoint-valued scalar fields Φ of the N =4 theory, the twist converts three, which we will call φ, into an adjoint-valued one-form on M, while the other three, which we will call φ, remain as scalar fields. The twist ensures that the compactified theory is independent of the choice of a metric g on W , but we do need to make a choice to define N =4 super Yang-Mills theory and to make the following constructions. For gauge group G = U(n), the twisted theory can be realized in string theory. One considers Type IIB superstring theory on R4 ×T ∗W (here T ∗W may be replaced by any Calabi-Yau three-fold X that admits W as a special Lagrangian submanifold), with n D3-branes wrapped on M = R×W ⊂ R4 × T ∗W . Then, by arguments similar to those in [31], upon scaling up the metric of T ∗W relative to the string scale, the low energy theory along M is the twisted theory described in the last paragraph. To get the twisted N =4 theory on a half-space (as we will want presently), one can let the n D3- branes end on an NS5-brane that is supported on R3 × W .ForG = SO(n) 406 E. WITTEN or Sp(n), a similar construction is possible using an orientifold three-plane supported on R × W . These brane constructions will be used in [14], but are not needed for our present purposes. In the reduction from four dimensions to one, the four-dimensional gauge field A splits up as a one-dimensional gauge field A0 ds (we parametrize the first factor of M = R × W by s) and a gauge field tangent to W that we will call B. The twisted compactification on W leaves four unbroken supersymmetries and the resulting theory can be interpreted as a one-dimensional supersymmetric gauge theory of precisely the sort studied in section 5.1.1, but with an infinite-dimensional gauge group. If we assume that the underlying four-dimensional theory is formulated on a trivial G-bundle, then the gauge group in the reduced one-dimensional theory is H = Maps(W, G), the space of maps from W to the finite-dimensional gauge group G. Somewhat more intrinsically, if the compactified theory on W isdefinedintermsofcon- nections on a G-bundle E → W , then H =Aut(E) is the group of bundle automorphisms of E. The bosonic fields of the vector multiplet of the H gauge symmetry are the untwisted fields A0 and φ. (Once we make a particular choice of a topological supercharge Q, φ splits up into the fields called A1, σ,andσ in section 5.1.1.) The other bosonic fields are usefully combined to a complex-valued field B = B + iφ on W , which we can think of as a connection on a bundle EC → W whose structure group is the complexification GC of G.(IfE was understood as a principal G-bundle, then EC is its complexification,which is a principal GC-bundle.) The fields B are the bosonic components of chiral supermultiplets that are acted on by the group H and in fact by its complexification HC, consisting of GC-valued gauge transformations. Let B be the space of all possible B fields. Once we endow Wwith a Riemannian 3 a b metric g, which we write in local coordinates as g = a,b=1 gabdy dy ,and pick a gauge coupling constant e, B acquires a Kahler metric √ 2 − 1 3 ab B ⊗ B (5.34) ds = 2 d y gg Tr δ a δ b 2e W and an associated symplectic form that is readily written in terms of the real fields B,φ: √ − 1 3 ab ∧ (5.35) ω = 2 d y gg Tr δBa δφb. e W Relative to this symplectic form, the moment map for the action of H on B is

1 1 a (5.36) μ = − dB φ= − Daφ , e2 e2 where is the Hodge star operator on W ,dB =d+[B,·] is the gauge- covariant exterior derivative defined with the real connection B, and similarly Di is the covariant derivative with respect to B. A gauge theory with THE PATH INTEGRAL OF QUANTUM MECHANICS 407 four supercharges in general has also a superpotential, which is a gauge- invariant holomorphic function W on the space that parametrizes the chiral superfields – in our case, this space is B. To be more precise, W is not quite uniquely determined as a complex-valued function. It is defined only up to an additive constant, because only derivatives of W appear in the action, and only up to an arbitrary phase factor, which is possible because the mapping from the underling N = 4 theory in four dimensions to a low energy description with vector and chiral multiplets (and four supercharges) is only uniquely determined up to an R-symmetry rotation, which can change the phase of W. In our case, the superpotential is the integral of the complex Chern-Simons form −exp(iα) 3 abc B B 2B B B (5.37) W = 2 d y Tr a∂b c + a b c . e W 3 where exp(iα) is the arbitrary phase. To verify that this is the right superpotential, one may for example start with the potential energy of the N = 4 theory in four dimensions and express it in the language of the reduced one-dimensional theory. In ten-dimensional notation, the bosonic part of the action of N = 4 super Yang-Mills theory 2 4 9 2 is −(1/2e ) d y I,J=0 Tr [DI ,DJ ] , where DI is a covariant derivative (if I =0,...,3) or the commutator with a scalar field (if I =4,...,9). The part of this action that involves only the chiral superfields is the time integral of a potential energy function V on B; this function can be written (after 2 2 some integration by parts) as V = |dW| + |μ| , using the above formulas for W, μ, and the metric on B. This is the expected form, independent of the phase α, and gives one way to verify the claimed formula for W. The fact that the Chern-Simons function arises in this way as a superpotential has been important in recent work [32] (this paper actually involves reduction from five to two dimensions rather than from four to one), and related observations have been made, for example, in [33]. In section 4.1 of [2], the gradient flow equations were studied for the flow on B with the Morse function

(5.38) h =2ReW.

Of course, these equations depend on the phase α. It was shown that these ﬂow equations have a natural interpretation in N = 4 super Yang-Mills theory. They express invariance under the topological supercharge Q of twisted N = 4 super Yang-Mills theory with the twist related to geometric Lang- lands that was studied in [6]. The supercharge Q depends on a twisting parameter, which was called t in [6]andu in [2]. According to eqn. (4.9) of [2], the twisting parameter is related to the phase α of W by 1 − cos α (5.39) t = . sin α 408 E. WITTEN

5.2.2. The Chern-Simons path integral from four dimensions. By simply adapting what we said in section 5.1.3, we can now represent a Chern-Simons path integral in three dimensions – over a suitable integration cycle – in terms of a four-dimensional path integral. We formulate the N = 4 theory on M = R+ × W , where R+ is the half- line s ≥ 0. As long as the gauge theory theta-angle vanishes (we defer the generalization to [14]), the theory can be regarded as an infinite-dimensional version of the one-dimensional gauge-invariant theory studied in section 5.1.3. In what follows, we simply imitate that analysis. So, as s →∞,we require B to approach a stable critical orbit of the superpotential. (As we discuss in detail presently, these orbits correspond to complex flat connections obeying a mild restriction.) At s = 0, we impose the boundary conditions that were used in section 5.1.3 to interpret an ordinary integral with gauge symmetry in terms of a path integral. In particular, to maintain Q-invariance at s = 0, we make the modification in the action that was described in eqn. (5.29), replacing the usual N = 4 action IN =4 by a Q- top invariant action IN =4 = IN =4 + h(0). Here h(0) is simply h evaluated at s = 0. (It does not matter whether we subtract from the action the value of h at s = −∞, because the boundary condition there fixes h(s = −∞)tobe a constant, and in particular Q-invariant. We omit here a factor 1/ multiplying h in (5.29); it has already been incorporated as the factor of 1/e2 in the definition of W.) Just as in section 5.1.3, we can now use the path integral of the N =4 theory on M to compute the integral of exp(W) over a real cycle ΓO ⊂ B: top (5.40) DA DΦ Dλ exp(−IN =4) exp(W)|s=0 = DB exp(W). ΓO

On the left of eqn. (5.40), A,Φ,andλ are the bose and fermi fields of N =4 super Yang-Mills theory, and W is to be evaluated at s = 0. The boundary conditions on fermions at s = 0 are those of section 5.1.3 and are discussed in detail, in the present infinite-dimensional context, in an appendix. On the right hand side, the integral is over a middle-dimensional cycle ΓO in the space of complex-valued connections B. This cycle, which corresponds to CO ∩{A1 =0} in (5.30), is found by solving the gradient flow equations with respect to the Morse function h =2ReW, starting at s = ∞ from the critical orbit O that is used to define the boundary conditions on the left hand side of (5.40). Because W is a multiple of the Chern-Simons action of the complex-valued connection B, the integral on the right hand side of (5.40) is simply the Chern-Simons path integral, carried out over an integration cycle that differs from the usual one. (The usual integration cycle is defined by setting φ = 0 or in other words by taking B to be real.) As explained in [2], the usual quantization of the Chern-Simons coupling constant, which is required to make sense of the integral over the usual integration cycle, is not necessary for defining the integral over a cycle obtained by gradient THE PATH INTEGRAL OF QUANTUM MECHANICS 409

flow, and indeed this quantization is not satisfied in the present context, as the superpotential contains an arbitrary factor exp(iα)/e2. An important generalization of (5.40) – discussed at several points in this paper – involves including on both sides an additional factor T, where T is a holomorphic function on B that grows too slowly at infinity to affect the convergence of the path integral. In the present context, there is a natural class of such functions. Let K be a knot (an embedded oriented circle) in W and let R be an irreducible representation of G, which we analytically continue to a holomorphic representation of GC. Then we define the holonomy function or Wilson loop operator

(5.41) WK,R(B)=TrR P exp B. K

WK,R(B) is, roughly speaking, the exponential of a linear function on B, while the superpotential W is cubic. So inclusion of a factor WK,R(B)does not aﬀect the convergence of the path integral. In the N = 4 theory, to preserve the topological symmetry, this factor must be inserted at s = 0 (where the boundary condition ψ(1,0) = 0 ensures Q-invariance of any holomorphic function). So we get a generalization of (5.40): − top B (5.42) DA DΦ Dλ exp( IN =4) exp(W)WK,R( ) s=0 = DB exp(W) WK,R(B). ΓO

This has an obvious generalization involving a link – a union of disjoint embedded oriented circles Ki ⊂ W – rather than a knot. Labeling the Ki by irreducible representations Ri,wehave D − top B (5.43) DA DΦ λ exp( IN =4) exp(W) WKi,Ri ( ) i s=0 B B = D exp(W) WKi,Ri ( ). ΓO i

There is a further extension in which links are replaced by a certain class of labeled graphs, as discussed for example in [34]. Another extension involving the theta-angle of the four-dimensional gauge theory is important in relation to Khovanov homology and will be described in [14]. Now let us discuss a few more details concerning these formulas. Since the superpotential is the integral of the complex Chern-Simons form, a critical point is a complex-valued ﬂat connection, characterized by the vanishing of the curvature

(5.44) F =dB + B∧B. 410 E. WITTEN

The monodromies of such a flat connection give a representation of the fundamental group of W into the finite-dimensional gauge group GC,orin other words a homomorphism ρ : π1(W ) → GC. The flat connections associated with a given monodromy form an orbit for the group HC of GC-valued gauge transformations. A critical orbit is nondegenerate if the corresponding representation of the fundamental group has no first-order deformations. In terminology introduced in section 5.1.2, a critical orbit is called semistable if it admits points at which the moment map μ (defined in eqn. (5.36)) vanishes. It is called stable if in addition the subgroup of GC that commutes with the monodromies of the flat connection is a finite group. A critical orbit containing no zero of μ is unstable. As shown in [35], the unstable critical orbits are exactly those that correspond to strictly triangular representations of the fundamental group. If W is compact, the space M = R+ × W is macroscopically one- dimensional. In this case, if there are flat directions in the bosonic potential of the theory, their quantum fluctuations are important and need to be treated carefully in defining the path integral. This tends to cause compli- cations in applications of (5.43). Such infrared questions are entirely absent if the boundary condition in the path integral at s = −∞ is set by a stable and nondegenerate flat connection. For a stable flat connection corresponds – just as in the context of section 5.1.3 with finitely many degrees of freedom – to a massive vacuum of the one-dimensional theory that arises by compactification on W . There is another way to avoid infrared subtleties: one can take W to be noncompact. The simplest choice is W = R3.InfieldtheoryonR3 (or 3 R ×R+) it is natural to consider only fields and gauge transformations that are trivial at infinity. So we define H to be the group of G-valued gauge transformations on R3 that are 1 at infinity,18 and we require the complex- valued connection B to vanish at infinity on R3. H acts on the space B of such connections, and we run the above theory with this choice of H and B. Since the fundamental group of R3 is trivial, the only critical orbit is the one that contains the trivial connection. This critical orbit is nondegenerate and stable. Nondegeneracy means that after gauge fixing, in expanding around the trivial flat connection, there are no zero modes that vanish at infinity. Stability expresses the fact that H – and its complexification HC – act freely on the orbit in B that contains the trivial flat connection. Indeed, H was defined as a group of gauge transformations that are 1 at infinity; no nontrivial gauge transformation obeying this condition leaves fixed the trivial flat connection. For W = R3, the topological field theory under discussion here has not much to say in the absence of knots, and the formula (5.40) is not terribly interesting – both sides equal 1. However, knots in R3 are interesting and

18 Since the fields φ –orA1 and σ – take values in the adjoint representation of H, this definition of H implies that those fields vanish at infinity on R3. THE PATH INTEGRAL OF QUANTUM MECHANICS 411

(after being extended to include the gauge theory theta-angle) the formulas (5.42), (5.43) will be the starting points for the study of Khovanov homology in [14]. A generalization, with infrared divergences avoided in a similar way, is to let W0 be a rational homology three-sphere with p apointinW0 and to take W = W0\p (that is, W is W0 with p omitted). We take on W a metric in which p is projected to infinity, so that the region near infinity in W looks like the region near infinity in R3. We again define H to consist of gauge transformations that are 1 at infinity, and B to parametrize complex- valued connections that vanish at infinity. The orbit containing the trivial flat connection is nondegenerate and stable. For G = SU(2), all critical orbits are stable (but not always nondegenerate); this is not necessarily so for G of higher rank. 5.2.3. Comparison to a sigma-model. A very illuminating example of the gauge theory construction that we have just considered arises if we take W = S1 × C, where C is a Riemann surface. We take the metric on C to be much larger than that on S1, and then instead of simply compactifying on W from four dimensions to one, we can think in terms of compactification on C from four to two dimensions, the two- 1 manifold being in this case R+ ×S . Compactification of N = 4 super Yang- Mills theory on a Riemann surface – with precisely the same topological twist as in our present discussion – was analyzed in [6], following [36, 37]. The low energy theory is a sigma-model with target space MH (G, C), the moduli space of Higgs bundles on C with structure group G. So the left 1 hand side of eqn. (5.40) reduces to a sigma-model path integral on R+ × S , with target MH (G, C). What about the right hand side of eqn. (5.40)? Chern-Simons theory in three dimensions with gauge group G, when compactified on a Riemann surface C, reduces to quantum mechanics with phase space the moduli space M(G, C) of flat (or holomorphic) G-bundles over C. To define an exotic integration cycle in this quantum mechanics, we must first complexify the target space and then pick the integration cycle. The complexification of M(G, C) is precisely MH (G, C), and so the right hand side of (5.40) is the quantum path integral of M(G, C) with an exotic integration cycle. Thus, the compactification of our present discussion on C gives a special case of the construction in section 2. 5.2.4. Elliptic boundary conditions. To arrive at (5.40) and its generalizations, we took a finite-dimensional construction, described in section 5.1.3, and adapted it to a case that the gauge group H and the space B that parametrizes the chiral superfields are infinite-dimensional. In eqn. (4.26), we have seen how such a generalization to infinite dimensions that may have sounded plausible can fail: it implied a boundary condition that is not elliptic. The formula (5.40) is not afflicted with this sort of problem, basically because the function W that appears in the exponent has derivatives in the W directions, which moreover are sufficiently generic. In this respect, (5.40) 412 E. WITTEN is a generalization not of (4.26), but of the sigma-model analysis of section 2 (to which it can be dimensionally reduced in some circumstances, as we noted in section 5.2.3). In the appendix, we show in detail that the boundary condition that is implicit on the left hand side of eqn. (5.40) is elliptic. This essentially means that the expressions that one would encounter in expanding the left hand side of (5.40) in perturbation theory have properties similar to what one would find in the finite dimensional construction of section 5.1.3, strongly suggesting that it would be possible to explicitly demonstrate the validity of (5.40) in perturbation theory. (As always, a rigorous nonperturbative proof would be much harder.)

5.3. Quantization with constraints. Here we will very brieﬂy consider the question of how one would describe integration cycles in the Feyn- man integral of a quantum mechanics problem with ﬁrst class constraints. We recall the setting with which we began in section 2.1. M is a classical phase space of dimension 2n with a real symplectic form that locally can be written n r (5.45) f = dpr ∧ dq . r=1

The first class constraints are functions μa that, via their Poisson brackets, c (5.46) {μa,μb} = fabμc generate the action on M of a Lie group G. Classically, imposing the constraints means restricting to the locus with μa =0, a =1,...,dim G,and dividing by G. Quantum mechanically, it means that one quantizes and restricts to the G-invariant subspace of the physical Hilbert space. We want to approach this process from the point of view of an analytically continued path integral. The constrained system can be described classically by the action r a (5.47) S = (prdq − φ μadt) , where the φa are Lagrange multiplier fields that impose the constraints μa =0. The first step in describing new integration cycles is analytic continuation. We complexify M to a complex symplectic manifold M , with complex structure J and holomorphic symplectic form Ω. (For more details, see sec- r tion 2.3.) pr and q analytically continue to holomorphic functions Pr and r Q on M, and similarly the μa analytically continue to holomorphic functions Ma. The action of G on M analytically continues to an action of the complex Lie group GC; the Ma generate the action of GC via Poisson brackets (computed using the symplectic form Ω). The Lagrange multiplier fields THE PATH INTEGRAL OF QUANTUM MECHANICS 413

φa become complex ﬁelds that we denote ϕa, and the complexiﬁed action is s a (5.48) S = (PsdQ − ϕ Ma dt) .

The integrand of the path integral is exp(iS). So to find a novel integration cycle, we introduce a metric on M and consider gradient flow with respect to the Morse function (5.49) h =Re(iS). Without repeating all the steps from sections 2 and 3.2, we can anticipate what will happen. The gradient flow equation will involve a new almost complex structure I on M that cannot coincide with J but that we can usefully take to anticommute with J. The nicest case will be that I and J and the holomorphic symplectic form Ω are all part of a G-invariant hyper-Kahler structure on M . In this case, the sigma-model with target M has N =4 supersymmetry in the two-dimensional sense, with eight supercharges. This is twice as much supersymmetry as was assumed in our study in section 5.1.3 of an ordinary integral (as opposed to a path integral) with gauge symmetry. All constructions in this paper based on two-dimensional path integrals, rather than one-dimensional ones, have involved this doubling of supersymmetry. We gauge the G symmetry preserving N = 4 supersymmetry. The flow equations of the resulting gauge-invariant sigma model have full two-dimensional symmetry, and the sigma-model gives a natural way to describe novel integration cycles in the path integral of the original constrained quantum system. Concretely, to gauge the G symmetry while preserving N = 4 supersymmetry, we introduce a vector multiplet, whose bosonic components are a two- dimensional gauge field A and four scalars in the adjoint representation of G. Upon topological twisting, A combines with two of the scalars to a complex- 1 valued gauge field A, leaving a complex scalar σ.OnS × R+, with the two factors parametrized respectively by t and s, we write A = Atdt+Asds.The complex field ϕ of eqn. (5.48) corresponds to At. As for the other bosonic fields As and σ, they were present in the case of an ordinary integral before doubling of supersymmetry, and correspond to the fields A0 + iA1 and σ in section 5.1.

Appendix A. Details on the four-dimensional boundary condition Here we will describe in detail the boundary conditions on the fermions of N = 4 super Yang-Mills theory that are implicit in our main results such as eqn. (5.43) relating this theory to Chern-Simons theory in three dimensions. We will also verify that these boundary conditions are elliptic. In the notation of section 3.1 of [6], the fermions of the twisted theory are as follows. The fermions of F = 1 are one-forms ψ and ψ,valuedin 414 E. WITTEN the adjoint representation. And the fermions of F = −1 are zero forms η and η and a two-form χ, all valued in the adjoint representation. We write χ = χ+ + χ−, where χ+ and χ− are respectively selfdual and anti-selfdual. The part of the fermion action that contains derivatives is (from eqns. (3.39) and (3.46) of [6]) 2i + − (A.1) 2 Tr η dA ψ+ η dA ψ + χ dAψ + χ dAψ , e M where is the Hodge star, dA is the gauge-covariant exterior derivative, and wedge products of differential forms are understood. The reason that we write only the derivative part of the fermion action is that nonderivative terms will not affect the boundary conditions or the condition of ellipticity. It is convenient to re-express ψ and ψ in terms of an adjoint-valued one- form ψ1 and three-form ψ3 by ψ = ψ1 + ψ3, ψ = −ψ1 + ψ3. Similarly, we re-express η and η in terms of an adjoint-valued zero-form η0 and four-form η4 by η =(η0 + η4)/2, η =(η0 − η4)/2. Then we combine the even degree forms to a field Ω = η0 + χ + η4 that is a sum of all differential forms of even degree. And similarly we combine ψ1 and ψ3 to a field Θ = ψ1 + ψ3 that is a sum of differential forms of all odd degree. Both Ω and Θ are adjoint-valued, of course. The derivative part of the fermion action becomes 2i 2i (A.2) 2 Tr (Ω(dA + dA )Θ) = 2 Tr Ω ( dA +dA )Θ. e M e M

The operator whose boundary conditions we need to consider is hence (modulo terms of order zero) the operator

(A.3) D = dA +dA mapping adjoint-valued forms of odd degree to those of even degree. Now we want to view N = 4 super Yang-Mills theory as a topological field theory with topological supercharge Q. As described in [6], the choice of Q depends on a parameter t = v/u. From eqns. (3.23), (3.24) of [6], and setting u = 1, the transformation law of the gauge field is δA = i(ψ + tψ)= i((1 − t)ψ1 +(1+t) ψ3) and that of the adjoint-valued one-form φ is δφ = i(tψ − ψ)=i((1 + t)ψ1 − (1 − t) ψ3). Introducing the complex-valued connection A = A + iφ, we find that δA =(−1+i)(1 + it)(ψ1 − i ψ3). To ensure Q-invariance of the left hand side of (5.43), the boundary condition must be such that δA| = 0. Here, for any differential form ϑ, we write ϑ| for the restriction of ϑ to the boundary. (This is the restriction in the sense of differential forms, so for example if ϑ is a one-form, then ϑ| is the part of ϑ that is tangent to the boundary.) Since δA is a multiple of ψ1 − i ψ3,to ensure that δA| = 0, the boundary condition must be, in part, that

(A.4) ψ1|−i ψ3| =0. THE PATH INTEGRAL OF QUANTUM MECHANICS 415

Another condition comes from the fact that in the constructions of section 5.1.3, we want σ| = 0 and hence also σ| = 0. To be compatible with this, the fermion boundary condition must set δσ| = 0. From eqn. (3.25) of [6], we have δσ = i(η + tη)=(i/2)((1 + t)η0 +(1− t) η4), so the boundary condition must tell us that

(A.5) (1 + t)η0| +(1− t) η4| =0.

When we vary the action (A.1) with respect to Θ, we get a surface term 2i ∧ ∧ (A.6) 2 Tr (Ω δΘ+ Ω δΘ) . e ∂M The fermion boundary condition must ensure vanishing of (A.6), and it must ensure the vanishing of half of the components of Ω and half of the components of Θ. Together with (A.4) and (A.5), these conditions uniquely determine the boundary conditions for fermions. For example, we need χ|∧ ψ1| + χ|∧ ψ3| = 0. (Here, for example, χ| is the tangential part of χ at the boundary; in other words, one acts with before restricting to the boundary.) In view of (A.4), this implies that the boundary condition on χ is

(A.7) χ|−i χ| =0.

And ﬁnally, in addition to (A.4), ψ1 and ψ3 must obey

(A.8) (1 + t) ψ1|−(1 − t)ψ3| =0.

A.1. Ellipticity. It remains to show that the boundary condition just described is elliptic for generic t, in fact for t = −i. (As explained in [6], the model has properties similar to those of a two-dimensional A-model for all values of t except t = ±i. A boundary condition related to the one we consider by φ →−φ is elliptic for t = i.) This is actually a straightforward exercise, the only difficulty being that the criterion for a boundary condition to be elliptic may not be familiar. In general, consider fields Φ on a manifold M with boundary that obey an elliptic differential equation DΦ = 0. (In our application, the operator D is dA +dA , acting from differential forms of odd degree to those of even degree; this is a standard example of an elliptic operator.) A general boundary condition is given by an equation UΦ = 0, where U is a linear map from the space of boundary data of Φ to some linear space V . (By boundary data, we mean the boundary values if D is a Dirac-like equation, the boundary values and normal derivatives at the boundary if D is a Laplace-like, etc.) We can assume that M looks near its boundary like ∂M ×R+, where R+ is the half-line y ≥ 0. The notion of ellipticity is local along ∂M,sowecan take ∂M = R3. We can then also work in momentum space, that is consider wavefunctions of definite momentum k along R3. A boundary condition is 416 E. WITTEN called elliptic if for every f ∈ V with nonzero momentum k, there is a unique Φ of momentum k that obeys DΦ = 0, decays exponentially with increasing y, and also obeys a shifted version of the boundary condition

(A.9) UΦ=f. The importance of this criterion is that an elliptic differential equation on a manifold with boundary, endowed with an elliptic boundary condition, has properties similar to those of an elliptic differential equation on a manifold without boundary. As a result, quantum perturbation theory has similar properties to those that are familiar in the absence of a boundary. (One expects the same to be true for the nonperturbative quantum theory, though this is certainly much harder to prove.) Before considering our problem, let us practice with the case of Dirichlet or Neumann boundary conditions for the scalar Laplace equation d dφ =0, where φ is a scalar field. Let u be Euclidean coordinates along ∂M = R3,and y a normal coordinate. A general solution of d dφ = 0 whose dependence on u is exp(ik ·u) (for some non-zero momentum vector k) and that vanishes · −| | exponentially with increasing y is a multiple of φk = exp(ik u) exp( k y). Dirichlet boundary conditions mean that the map U takes a solution of the scalar Laplace equation on the half-space R3 × R+ to its restriction to y = 0. So the criterion for ellipticity is as follows: for any constant c,and any nonzero k, there must exist a unique solution φ of the scalar wave equation, decaying exponentially with y, and with φ|y=0 = c exp(ik · u). Clearly, these conditions have the unique solution φ = cφk, so Dirichlet boundary conditions are elliptic. The case of Neumann boundary conditions is similar. Let us consider in the context of this definition the boundary conditions for Θ = ψ1 + ψ3 in our problem. These boundary conditions were given in eqns. (A.4) and (A.8):

ψ1|−i ψ3| =0 (A.10) (1 + t) ψ1|−(1 − t)ψ3| =0.

We generalize these equations to include adjoint-valued forms γ1,γ3 of odd degree:

ψ1|−i ψ3| = γ1 (A.11) (1 + t) ψ1|−(1 − t)ψ3| = γ3.

3 Concretely, for ∂M = R ,wetakeγ1 = θ1 exp(ik ·u), γ3 = θ3 exp(ik ·u), with constants θ1, θ3. We have to show that for any nonzero k,andanyθ1,θ3, there is a unique choice of ψ1,ψ3, with plane wave dependence on u and exponential decay in y, obeying DΘ = 0 along with (A.11). Since all fields have the same exp(ik · u) dependence on u, everything reduces to algebra. The algebra can be carried out as follows. First of all, we can replace the first equation in (A.11) by a pair of equations saying that the difference THE PATH INTEGRAL OF QUANTUM MECHANICS 417 between the left and right hand sides is both closed and coclosed (in other words, annihilated by both d and d 3, where 3 is the Hodge star operator of the boundary). The complete set of conditions then becomes:

d 3 (ψ1|−i ψ3|)=d 3 (θ1 exp(ik · u))

(A.12) d(ψ1|−i ψ3|)=d(θ1 exp(ik · u))

(1 + t) ψ1|−(1 − t)ψ3| = θ3 exp(ik · u).

Now, for wavefunctions of momentum k, we can solve the ﬁrst and third equations in (A.12), without aﬀecting the second one, as follows. One class of solutions of DΘ=0is

(A.13) ψ1 =d(aφk),ψ3 = d(bφk), with φk as above and constants a, b. These constants can be adjusted in a unique way to satisfy the first and third equations in (A.12), as long as (1 − t)/(1 + t) = i or t = −i. (The condition t = −i is needed, since otherwise the first and third equations in (A.12) depend on the same linear combination of a and b and we cannot solve both equations.) If we set ψ1 = ± ψ3, the equation DΘ = 0 becomes the selfdual or anti- selfdual Maxwell equations, together with a Lorentz gauge condition (for a connection defined by a one-form α, the Lorentz gauge condition is d α= 0). The selfdual and anti-selfdual equations are first order equations, so initial data are given by prescribing a gauge connection on a three-manifold. 3 In fact, any one-form γ1 on R can be written uniquely as γ1 = a+ +a− +dw where a+ and a− are coclosed and are the boundary values of solutions of 3 the selfdual and anti-selfdual Maxwell equations on R ×R+ that decay with increasing y,andw is a zero-form. Shifting ψ1 and ψ3 by suitable linear combinations of a+ and a−, we can solve the second equation in (A.12) with fields that decay for increasing y. After doing this, the procedure of the last paragraph can be used to solve the other two equations. This establishes what we wanted for Θ. The boundary conditions for Ω can be treated similarly.

A.2. A generalization. One last remark is as follows. In this appendix, we have constructed a boundary condition that is compatible with the topological supersymmetry of twisted N = 4 super Yang-Mills theory and ensures that A| is invariant, where A = A + iφ. Instead, we may pick any complex number κ with a nonzero imaginary part and ask for invariance of Aκ|, where Aκ = A + κφ. Aκ can be regarded as a complex-valued connection, just like A. In this appendix, we have set κ = i because this case seems natural in the context of the derivation in section 5.1.3. However, more general values of κ are useful in the application to Khovanov homology [14]. The above derivation works for general κ with minor modiﬁcations. 418 E. WITTEN

Acknowledgements I thank S. Cherkis, R. Dijkgraaf, E. Frenkel, D. Gaiotto, S. Gukov, L. Hollands, R. Mazzeo, G. Moore, N. Nekrasov, and L. Rozansky for discussions and the physics department of Stanford University for hospitality. Research supported in part by NSF Grant PHY-0969448.

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[25] E. Witten, “Supersymmetry And Morse Theory,” J. Diff. Geom. 17 (1982) 661–692. [26] R. Bott and L. Tu, Differential Forms In Algebraic Topology (Springer-Verlag, 1982). [27] E. Witten, “Chern-Simons Gauge Theory As A String Theory,” Prog. Math. 133 (1995) 637–678, hep-th/9207094. [28] L. Alvarez-Gauméand D. Z. Freedman, “Potentials For The Supersymmetric Non- linear Sigma Model,” Commun. Math. Pys. 91 (1983) 87. [29] E. Witten, “Phases Of N = 2 Theories In Two Dimensions,” Nucl. Phys. B403 (1993) 159–222, hep-th/9301042. [30] M. F. Atiyah and R. Bott, “The Yang-Mills Equations Over Riemann Surfaces,” Phil. Trans. R. Soc. Lond. A308 (1983) 523–615. [31] M. Bershadsky, C. Vafa, and V. Sadov, “D-Branes and Topological Field Theories,” Nucl. Phys. B463 (1996) 420-434, hep-th/9511222. [32] T. Dimofte, S. Gukov, and L. Hollands, “Vortex Counting And Lagrangian 3- Manifolds,” arXiv:1006.0977. [33] E. Witten, “New Issues In Manifolds Of SU(3) Holonomy,” Nucl. Phys. B268 (1986) 79. [34] E. Witten, “Gauge Theories, Vertex Models, And Quantum Groups,” Nucl. Phys. B330 (1990) 285–346. [35] K. Corlette, “Flat G-Bundles With Canonical Metrics,” J. Diff. Geom. 28 (1988) 361–382. [36] M. Bershadsky, A. Johansen, V. Sadov, and C. Vafa, “Topological Reduction Of 4-d SYM To 2-d Sigma-Models,” Nucl. Phys. B448 (1995) 166–186, hep-th/9501096. [37] J. A. Harvey, G. W. Moore, and A. Strominger, “Reducing S Duality To T Duality,” Phys. Rev. D52 (1995) 7161–7167.

School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 USA

Department of Physics, Stanford University, Palo Alto, CA 94305

Surveys in Diﬀerential Geometry XV

Quasi-Local Mass in General Relativity

Shing-Tung Yau

It is a great honor for me to give this talk in honor of my mentor and my old friend Is Singer. We have known each other since 1973. He had a deep influence in my mathematical thinking and my career. When I first met him at a big conference in geometry in Stanford, my advisor Chern told me that Is wanted to see me. I was somewhat nervous as Is is a big name in geometry. To my surprise, Is just smiled and said that he just wanted to get to know me. But he did something for me then that neither of us realized at the time. It was his proposal to invite Robert Geroch to explain the mathematics of general relativity to geometers during that conference. Geroch explained the positive mass conjecture for a special case in very simple terms. It was fascinating as nobody knew how to approach this problem, despite its elegance. I kept this problem in mind until 1978, when I visited Berkeley for one year, again at the invitation of Chern and of Is. In that year, I solved the first major part of the positive conjecture with Richard Schoen. Before this visit, Is invited me to visit MIT for a month during 1976, after I solved the Calabi conjecture. During that period, Is was very busy with a family issue. Nonetheless, we had dinner and he described to me what he heard in Stony Brook when he visited the physicists there. The main question was concerning self-dual instantons. Although I did not work on it immediately, it is such an interesting subject, that I finally worked on the higher dimensional generalization of it known today as the Hermitian Yang-Mills connections. In 1987, Is thought that it was important for mathematicians working on physics related issues to have the ability to obtain funding to hire postdoctoral fellows. He was able to work this out with the Department of Energy. And I have benefitted much by his efforts on this. I was able to hire postdoctoral fellows who earned physics degrees. Having them around have been really useful for my career.

Talk given at the Perspectives in Mathematics and Physics Conference, in celebration of Professor I.M. Singer’s 85th birthday. c 2011 International Press

421 422 S.-T. YAU

Is and I have worked together on numerous occasions, including the time that we initiated the mirror symmetry workshop in MSRI. There, we were able to attract the interest of a whole generation of mathematicians and got them excited about mirror symmetry. The mathematics inspired by physics is truly exciting and inspiring. It was Is Singer who helped me to direct my research in this beautiful direction. It is almost forty years since I met Is. I am truly grateful to his leadership in this field. Let me now begin on my recent work with Muo-Tao Wang on general relativity. The ideas went back to the work with Schoen on positive mass conjecture. In general relativity, Einstein’s equation is obtained by taking the variation of 1 R + L 16π where R is the scalar curvature of the spacetime and L is the Lagrangian of matter coupled to gravity. The gravitational interaction is described by means of a spacetime Lorentzian metric gij which has indefinite signature (−, +, +, +). For instance, the metric of the Minkowski spacetime R3,1 which is the vacuum with zero matter is 2 i j 2 2 2 2 ds = gij dx dx = −dt +(dx) +(dy) +(dz) . The variational equation has the form 1 Rij − Rgij = Tij 2 where Rij is the Ricci tensor, and Tij is called the matter energy-momentum tensor. In classical relativity, the matter tensor satisfies the weak energy condition i j Tij l l ≥ 0 for any four-vector li that is time-like i j gij l l < 0.

There are two very important solutions of the Einstein equation that are used extensively in general relativity. The Schwarzschild metric −1 2M 2M ds2 = − 1 − dt2 + 1 − dr2 + r2 dθ2 +sin2 θdφ2 r r and the Kerr metric 2 Δ 2 dr ds2 = − dt − a sin2 θdφ + U + dθ2 U Δ 2 sin θ 2 + adt− (r2 + a2) dφ2 U QUASI-LOCAL MASS IN GENERAL RELATIVITY 423 where U = r2 + a2 cos2 θ Δ=r2 − 2 Mr+ a2 −∞

Both solve the Einstein equation with Tij = 0 (vacuum). The first one is spherically symmetric and the second one is axially symmetric. They respectively describe static and stationary black holes. The constants M and a are the mass and angular momentum of the black hole. The famous black hole uniqueness theorem says that stationary black holes with no matter are exactly described by the Kerr metric. Both metrics have a null hypersurface at r = M + M 2 − a2 which is the event horizon of the black hole. This is the spacetime boundary of the black hole where any event occurring inside can not be detected ∂ by an outside observer. Note that the vector field ∂t is a Killing vector field as it preserves the metric. ∂ ∂ ∂ The Killing field ∂t is time-like (i.e. g( ∂t, ∂t) < 0) when r2 − 2 Mr+ a2 cos2 θ>0 ∂ ∂ but space-like (i.e. g( ∂t, ∂t) > 0) when r2 − 2 Mr+ a2 cos2 θ<0. This last region is called the ergosphere. It is a bounded region outside the event horizon except at θ =0andπ. We can consider the dynamics of a scalar field Φ(t, r, θ, φ) in the Kerr spacetime. Its propagation is described by a scalar wave equation. Since ∂ 2 the Kerr spacetime has a Killing vector field ∂t, the Lagrangian |∇Φ| associated to the wave equation defines a local energy density. It has the form 2 2 2 r + a 2 2 2 2 2 E = − a sin θ |∂tΦ| +Δ|∂rΦ| +sin θ |∂cos θΦ| Δ 2 1 a 2 + − |∂φΦ| sin2 θ Δ This density is positive except within the ergosphere, where it is negative. In the Kerr geometry, no first-order or higher-order positive conserved energy density exists for the scalar wave equation as was observed by Finster, Kamran, Smoller and Yau [7]. The major classical method to study wave equation is called the energy method. One exploits the conservation of total energy and Sobolev type inequality to control local behavior in terms of the 424 S.-T. YAU total energy. However, the energy method for the scalar wave equation breaks down due to the negativity of the energy density within the ergosphere unless the angular momentum is small relative to the mass. (In this case, Dafermos and Rodnianski [5] proved the solution is bounded in t if the initial data has compact support outside the event horizon.) An important question for the wave equation is whether the wave will decay in time if initially it does not spread out. The decay of the solution of the scalar wave equation is a special case of the decay of solutions to the Teukolsky equation which describes the linear stability of the Kerr black hole. In this connection, we quote Frolov and Novikov [9]: “Linear stability of the Kerr black hole is one of the few truly outstand- ing problems that remain in the field of black hole under gravitational wave perturbations.” The problem of linear stability of Kerr is still open and it is not even clear whether linear stability of the Kerr holds true if the angular momentum is not small relative to the mass. A remarkable property of the Kerr metric is that the Teukolsky equation is separable by the ansatz

e−iωt−ikφ R(r)Θ(θ)

Chandrasekhar called this property of Kerr geometry “having the aura of the miraculous” [3]. Making use of this, in 2001, Finster, Kamran, Smoller and Yau [6] proved that the propagation of waves described by a Dirac equation in Kerr space decays in time like t−5/6. We also proved that for the scalar wave equation, the wave with fixed angular momentum mode k also decays. In principle, we can sum up the modes to conclude the decay of the scalar wave equation. This can be done for the Schwarzschild geometry. However, the negativity of the energy density for the scalar wave in the ergosphere of the Kerr geometry causes problems. Indeed, the ergosphere has many strange properties including the energy extraction process proposed by Penrose [18]. The wave analogue was proposed by Zeldovich [29] and Starobinsky [23]. It is called superradiance. A complete rigorous treatment for the latter case was finally achieved recently by Finster, Kamran, Smoller and Yau [8]. We have seen now that in the ergosphere, the energy density of the scalar wave can be negative and has give peculiar properties. The problem comes from the fact that the gravitational field itself must have energy. After all, the potential energy of a pair of gravitating particles depends on their separation distance. Hence the total energy depends on the gravitational field configuration. In special relativity, the tensor Tij can be used to define the energy- momentum vector for a domain in space in the following manner: Take any Killing vector vj which preserves the metric. j Ji = Tijv QUASI-LOCAL MASS IN GENERAL RELATIVITY 425 defines a divergence free current. The Hodge dual of J is a closed three form. When we integrate this three form over a compact space-like hypersurface, we obtain the energy-momentum vector P associated to this hypersurface. However, since Tij accounts only for matter, the energy-momentum vector defined in this way does not account for the gravitational energy. A more serious problem is that a general spacetime does not admit any Killing field. If the spacetime admits a time-like Killing field, Noether’s theorem applied to the Lagrangian provides a current associated to this Killing field. This current can be used to define a mass called the Komar mass. Let us now recall Noether’s theorem: Let L be the Lagrangian which can be considered as a function defined on the tangent bundle of a manifold M. Suppose we have a one parameter family of diffeomorphisms ht : M → M so that

L ((ht)∗v)=L(v)

Then the Euler-Lagrange equation associated to L admits a ﬁrst integral I : TM → R given by

∂L dht(q) I(q, q˙)= . ∂q˙ dt t=0 For most mechanical systems, the Lagrangian is invariant under translation of time and space and the resulting conserved quantity is a four-vector defining energy and linear momentum. However, in general relativity, most spacetimes do not admit translational symmetry, so it is difficult to define energy-momemtum using Noether’s theorem. However, one can define a total energy-momentum vector for an isolated physical system if there is asymptotic translational symmetry. When the spacetime is asymptotically flat, there is a space-like hypersurface which outside a compact set is diffeomorphic to R3 minus a ball and the metric gij has the form 1 gij = δij + O , r and the second fundamental form pij of this hypersurface satisfies: 1 pij = O . r2

We expect that for such an asymptotically-flat spacetime, there is Lorentzian symmetry at infinity, and the approximate symmetry at infinity can be used to define the total mass (or energy) 1 j MADM = lim (∂igij − ∂j tr g) v 16π r→∞ S2(r) 426 S.-T. YAU and the total linear momentum 1 j Pi = lim (pij − pkkδij)v . r→∞ 8π S2(r) M The four-vector is called the ADM (Arnowitt-Deser-Misner) energy- Pi momentum four-vector. It was proved to be a time-like vector except for Minkowski spacetime which of course is trivially null. The total energy in general relativity cannot be obtained by integrating any local density along a hypersurface. The reason is that the density would depend on the first order differentiation of the metric gij. But there is a coordinate system where such quantities are zero at that point. The equivalence principle says that physical quantities should be independent of the choice of the coordinate system. Nonetheless, one can still ask whether it is still possible to have a quasi-local mass, where a total energy-momentum four-vector is assigned to any space-like sphere bounding a compact portion of a space-like hypersurface. In 1982, Penrose [20] listed the search for a definition of such quasi-local mass as his number one problem in classical general relativity besides his famous question of cosmic censorship. There are many reasons to search for such a concept. Many important statements in general relativity make sense only with the presence of a good definition of quasi-local mass. For example, it allows us to talk about the binding energy of two bodies rotating around each other. More importantly, a good definition of quasi-local mass should help us to control the dynamics of the gravitational field. Hopefully, this may be used to generalize the energy method in hyperbolic equations where difficulties were encountered even in the study of linearized stability of the Kerr metric. There were many attempts, including approaches given by Penrose [19], Hawking [11], and Brown-York [2] to give the definition of quasi-local mass. We shall use an approach which seems to be most promising. Recall that for a Lorenztian manifold M with boundary ∂M, the action in general relativity should be 1 1 I(g, Φ) = R + L(g, Φ) + K M 16π 8π ∂M where K is the trace of the second fundamental form of ∂M. The last term is needed to give rise to the right variational equation if we fix the metric and the matter field on the boundary. If we demand that a certain background (g0, Φ0) is a static solution to the field equation, we replace I by

I(g, Φ) − I(g0, Φ0). QUASI-LOCAL MASS IN GENERAL RELATIVITY 427

Figure 1

Hence, for ﬂat spacetime background with g0 being ﬂat and Φ0 =0,weuse 1 1 R + L(g, Φ) + (K − K0) . M 16π 8π ∂M

Suppose we take a family of space-like surface Σt generated by a time-like μ vector ﬁeld t such that t ∇μt = 1. We can write tμ = Nnμ + N μ μ μ where n is the normal to Σt, N is called the lapse function, N is called the shift vector. In this notation, we ﬁnd 1 μν 2 1 2 I(g, Φ) = Ndt R + pμνp − p +16πL + K , 2 16π Σt 8π St 2 where pμν is the second fundamental form of Σt and p is its trace, K is the mean curvature of ∂Σt = St. μν 3 If one introduces the canonical momenta k , k conjugate to gμν, Φ, we can rewrite the action to be μν μ 1 2 μ ν dt k g˙μν + kΦ˙ − N H−N Hμ + N K − N pμνr , Σt 8π St where H is the Hamiltonian constraint 1 μν 2 T00 − R − pμνp + p , 2 and Hμ is the momentum constraint

T0μ − pμν,ν + p,μ.

Note that H =0 and Hμ = 0 when the equation of motion is satisﬁed. We ν will let r to be the spacelike unit normal of St which is tangent to Σt. The Hamiltonian is then derived to be μ 1 2 μ ν H = (N H + N Hμ) − N K − N pμνr . Σt 8π St 428 S.-T. YAU

If we take the background so that pμν = 0, we see that the Hamiltonian relative to the background is given by μ 1 2 2 μ ν (N H + N Hμ) − N( K − K0) − N pμνr . Σt 8π St Hence associated to each time-like vector field t, we have the physical Hamil- tonian 1 2 2 μ ν − N( K − K0) − N pμνr 8π St This expression was derived by Brown-York [2] and Hawking-Horowitz [12]. They proposed to simply choose N =1, N μ = 0 for the definition of quasi-local mass. In general, the definition does not give positivity except in the time symmetric case (pμν = 0) which was proved by Shi-Tam [17]. The definition of Brown-York is gauge dependent. Liu-Yau [14, 15] defined a gauge independent mass to be 1 2 2 2 2 − ( K) − (trSp) − K0 8π S and proved that it is positive whenever the mean curvature vector of S is space-like and the Gauss curvature is positive. The proof combined arguments of Schoen-Yau [21, 22] and Witten [28]). We needed to handle metrics where the mean curvature may jump along the boundary. The discontinuity 2 2 2 of the Dirac spinor required nontrivial analysis. Note that ( K) − (trSp) is the Lorentian norm of the mean curvature vector 2 ν ν H = − Kr +(trSp)n .

Let me now describe the work that I did with Mu-Tao Wang [24, 25, 26, 27]. Given a surface S, we assume that its mean curvature is positive. We embed S isometrically into R3,1. Given any constant unit future time-like vector w (observer) in R3,1, we can deﬁne a future directed time-like vector ﬁeld w along S by requiring

H0,w = H, w

3,1 where H0 is the mean curvature vector of S in R and H is the mean curvature vector of S in spacetime. This means that the expansion of the body in R3,1 relative to the observed w is the same as the expansion of the body relative to w. Note that given any surface S in R3,1 and a constant future time-like unit vector wν, there exists a canonical gauge nμ (future timelike unit normal along S) such that 2 μ ν N K0 + N (p0)μνr S QUASI-LOCAL MASS IN GENERAL RELATIVITY 429

Figure 2

Figure 3 is equal to the total mean curvature of Sˆ, the projection of S onto the orthogonal complement of wμ, wμ = Nnμ + N μ, rμ is the spacelike unit μ normal orthogonal to n , p0 is the second fundamental form calculated by the three surface deﬁned by S and rμ. From the matching condition and the correspondence (wμ,nν) → (wν, nν), we can deﬁne a similar quantity from our data in spacetime 2 μ ν N K + N (p)μνr . S

Making use of the work of Liu-Yau [14, 15], Wang and I proved that 2 μ ν 2 μ ν 8πE(w)= N K + N (p)μνr − N K0 + N (p0)μνr S S is non-negative [25, 26]. 430 S.-T. YAU

We deﬁne a quasi-local mass to be

inf E(w) where the infimum is taken among all isometric embeddings into R3,1 and timelike unit constant vector w ∈ R3,1. In summary, given a closed spacelike 2-surface in spacetime whose mean curvature vector is space-like, we associate an energy-momentum four-vector to it that depends only on the first fundamental form, the mean curvature vector and the connection of the normal bundle such that 1. It is Lorentzian invariant. 2. It is trivial for surfaces sitting in Minkowski spacetime and future time-like for surfaces in spacetime which satisfies the local energy condition. We shall compare our definition with the classical definitions. First of all, let us look at some special cases. Spherical symmetric spacetime are foliated by the orbits of SU(2). We can define a function on the spacetime by associating to its orbit the area 4πr2. The mean curvature vector of the orbit is 2 − ∇r r where ∇ is with respect to the quotient Lorentzian (1, 1) metric. If this vector is space-like, the quasi-local mass of this orbit sphere is

M = r(1 −|∇r|).

Note that in 1964, Misner and Sharp [16] defined a mass r m = 1 −|∇r|2 2 which is the same as the Hawking mass [11] A 1 1 − |H|2 . 16π 16π S The relation with our mass is M 2 m = M − . 2r From the formula of quasi-local mass, which we proved to be positive, we derived a corollary that the mass m (Hawking mass) is also positive. (This was proved by Christodoulou and Yau [4] under extra assumptions.) Note that 1 M ≤ m ≤ M 2 QUASI-LOCAL MASS IN GENERAL RELATIVITY 431 where on the apparent horizon M =2m, and at space-like infinity M = m. Hence our quasi-local mass is equivalent to the standard definition in the case of spherically symmetric spacetime. In the spatial direction, the Hawking mass is monotonically increasing along the inverse mean curvature flow [10] and this is important in Huisken- Ilmanen’s work [13]. The quasi-local mass is not monotonically increasing in this sense. However, the spherical symmetric case indicates that such property may still hold, up to a constant depending on the initial surface. In the future time-like or null direction, the quasi-local mass is expected to decrease up to a constant depending on the initial surface if we choose the equation of motion for the 2-surfaces carefully. In the case when pij ≡ 0, there is also a definition of the quasi-local mass by Bartnik [1] which is obtained by minimizing the ADM mass among all asymptotically flat extension of the data which does not contain an apparent horizon and which extends the original data. Our definition of the quasi-local mass also satisfies the following important properties: 3. When we consider a sequence of spheres on an asymptotically flat space-like hypersurface, in the limit, the quasi-local mass (energy- momentum) is the same as the well-understood ADM mass (energy- momentum). 4. When we take the limit along a null cone, we obtain the Bondi mass. These properties of the quasi-local mass is likely to characterize the definition of quasi-local mass, i.e. any quasi-local mass that satisfies all these properties may be equivalent to the one that we have defined. Strictly speaking, we associate each closed surface not a four-vector, but a function defined on the light cone of the Minkowski spacetime. Note that if this function is linear, the function can be identified as a four-vector. It is a remarkable fact that for the sequence of spheres converging to spatial infinity, this function becomes linear, and the four-vector is defined and is the ADM four-vector that is commonly used in asymptotically flat spacetime. In any case, we are still in the process of deriving more properties of the quasi-local mass that we just introduced. We hope it can be used in applications for astronomy and physics.

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Department of Mathematics, Harvard University, MA 02138, USA