The Influence of Solomon Lefschetz in Geometry and Topology

621

The Inﬂuence of Solomon Lefschetz in Geometry and Topology 50 Years of Mathematics at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

American Mathematical Society

The Inﬂuence of Solomon Lefschetz in Geometry and Topology 50 Years of Mathematics at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

621

The Inﬂuence of Solomon Lefschetz in Geometry and Topology 50 Years of Mathematics at CINVESTAV

Ludmil Katzarkov Ernesto Lupercio Francisco J. Turrubiates Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classiﬁcation. Primary 14-06, 53-06, 55-06.

Library of Congress Cataloging-in-Publication Data The inﬂuence of Solomon Lefschetz in geometry and topology : 50 years of mathematics at CINVESTAV / Ludmil Katzarkov, Ernesto Lupercio, Francisco J. Turrubiates, editors. pages cm. – (Contemporary mathematics ; volume 621) Includes bibliographical references. ISBN 978-0-8218-9494-1 (alk. paper) 1. Lefschetz, Solomon, 1884–1972. 2. Centro de Investigaci´on y de Estudios Avanzados del I.P.N. 3. Institute for Geometry and Physics Miami-Cinvestav-Campinas. 4. Geometry, Al- gebraic. 5. Algebraic topology. 6. Topology. I. Katzarkov, Ludmil, 1961– . II. Lupercio, Ernesto, 1970– . III. Turrubiates, Francisco J., 1970– . IV. American Mathematical Society.

QA565.I54 2014 516.3’5–dc23 2013049429

DOI: http://dx.doi.org/10.1090/conm/621

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Contents

Preface vii Solomon Lefschetz and Mexico Michael Atiyah 1 Recent Progress in Symplectic Flexibility Yakov Eliashberg 3 Equivariant Extensions of Diﬀerential Forms for Non-compact Lie Groups Hugo Garc´ıa-Compean,´ Pablo Paniagua, and Bernardo Uribe 19 From Classical Theta Functions to Topological Quantum Field Theory Razvan˘ Gelca and Alejandro Uribe 35 Toric Topology Samuel Gitler 69 Beilinson Conjecture for Finite-dimensional Associative Algebras Dmitry Kaledin 77 Partial Monoids and Dold-Thom Functors Jacob Mostovoy 89 The Weak b-principle Rustam Sadykov 101 Orbit Conﬁguration Spaces Miguel A. Xicotencatl´ 113 Dynamical Systems and Categories G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich 133 The Nahm Pole Boundary Condition Rafe Mazzeo and Edward Witten 171

Preface

The purpose of this volume is to both celebrate half a century of Mathematics at the Center for Research and Advanced Studies in Mexico City (Cinvestav) and the opening of the Institute for Geometry and Physics Miami-Cinvestav-Campinas. Cinvestav is one of the most prestigious research centers in Latin America. It focuses most of its activities in exact sciences. It was founded in 1961, and one of the founding departments was the Mathematics Department under the direction of José Adem. At the time, there were two members: Adem, as a full professor, and Samuel Gitler, as an assistant professor (both working in the field of Algebraic Topology). This particular choice was influenced by the fact that both Adem and Gitler were recruited to study their Ph.D.s at Princeton by Solomon Lefschetz, who used to spend half a year in Mexico for many years lecturing at UNAM; he also began a Mexican school of Geometry and Topology. Solomon Lefschetz received the order of the Aguila´ Azteca from President López Mateos in 1964 for his contributions to Mexican Mathematics. Nowadays, the Mathematics Department at Cinvestav organizes the Lefschetz Memorial Lecture Series biyearly: the conference is one of the foremost mathematical events in Latin America drawing some of the most distinguished mathematicians in the world to deliver three talks at Cinvestav during the fall. The following is the list of all previous Lefschetz lecturers:

• Jack Hale • Michael Atiyah • Enrico Bombieri • Shing S. Chern • John Milnor • David Mumford • William Thurston • Ya G. Sinai • P. Maslov • Sergio Albeverio • Maxim Kontsevich • Dennis Sullivan • Graeme Segal • Michael Hopkins • Edward Witten • Yasha Eliashberg

This volume contains the paper by Eliashberg corresponding to his 2012 Lef- schetz Lectures.

vii

viii PREFACE

The celebration for the 50th anniversary of the Mathematics Department at Cinvestav took place in 2012 with three main events: TQFT, Langlands and Mirror Symmetry: the Opening Conference of the Institute for Geometry and Physics Miami-Cinvestav-Campinas (IGP-MCC), the 50th year anniversary of Cinvestav, (all together celebrating 50 years of Topology, Geometry and Physics at Cinvestav), and finishing with the International Conference on Symplectic Geometry 2012. This volume contains contributions of the participants from all these events. The Institute for Geometry and Physics Miami-Cinvestav-Campinas (IGP- MCC) is a joint virtual venture to stimulate a North-South collaboration in Geome- try and Physics. The directors are Elisabeth Gasparim, Maxim Kontsevich, Ernesto Lupercio, and Dennis Sullivan, and its founding coincided with the celebrations for the 50th anniversary of Cinvestav. The events were centered in the fields of Geometry (Algebraic and Symplectic) and Algebraic Topology, which are some of the strongest fields at Cinvestav and, as it turned out to be, they reflect quite nicely that they are still of current interest in the Mathematics Department 50 years after Lefschetz. This volume is thus a witness to his enormous influence in Mexican Mathematics. Let us describe very briefly the contents of this volume. In the opening paper, Eliashberg surveys the relation between flexibility and rigidity in Symplectic Ge- ometry and Topology, concentrating primarily on the former. This is an important survey of very recent developments in the area by the leading expert. The second paper by Garc´ıa-Compeán, Paniagua, and Uribe generalizes a result of Witten on the equivalence of absence of anomalies in gauged WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term in the case in which the gauge group is the special linear group. The third contribution by Gelca and Uribe provides a rigorous construction of Abelian Chern-Simons theory using only the classical theory of theta functions and recovering the Murakami-Ohtsuki-Okada formula for invariants of 3-dimensional manifolds. The volume also contains a survey paper by Samuel Gitler (a founding member of the Mathematics Department at Cinvestav 50 years ago) dealing with recent developments on the field of Toric Topology by him and his collaborators. In his paper Kaledin proposes a non-commutative version of Beilinson’s famous conjecture predicting that the regulator map induces an isomorphism between the higher Algebraic K-groups of an Arithmetic scheme and its so-called Beilinson’s cohomology. This non-commutative version of the conjecture predicts the existence of a certain exact triangle of complexes associated to smooth and proper dg-Algebras defined over Z, for which each term is related to either (negative or periodic) cyclic homology or Algebraic K-theory. The main result of this work is the statement that the non-commutative version of Beilinson’s conjecture holds for finite dimensional Algebras. This is a work in the field of non-commutative Algebraic Geometry. The paper by Mostovoy generalizes the classical Dold-Thom result in Algebraic Topology from ordinary homology to any generalized homology theory. Sadykov’s contribution deals with his b-principle. The h-principle is a general observation that differential Geometry problems can often be reduced to problems in (unstable) homotopy theory. Similarly, the b-principle is a general observation that differential Geometry problems can often be reduced to problems in stable

PREFACE ix homotopy theory; in his paper, Sadykov surveys some recent developments in the b-principle. Xicoténcatl contributes to the volume with a survey on orbit configuration spaces. Some of the results deal with the existence of certain natural fibrations extending those of Fadell and Neuwirth, Lie algebras associated to the lower central series of fundamental groups of these spaces, cohomology rings, the Longhoni- Salvatore result of the failure of configuration spaces to preserve homotopy equivalences of closed manifolds, and structures of loop-spaces. The volume continues with a paper written by Dimitrov, Haiden, Katzarkov, and Kontsevich. This is a paper densely packed with deep results. One of the main objects of study in this contribution is the entropy of automorphisms of non- commutative Algebraic varieties as exemplified by saturated A∞ categories. The paper is organized around two major results. The first is an explicit identification between the categorical entropy of an endofunctor of the wrapped Fukaya category of a punctured Riemann surface and the classical entropy of a pseudo-Anosov map in the corresponding mapping class. With their second result, the authors give a complete answer to the problem of detecting the density of phases of stable objects in categories of representations of quivers. Finally, Rafe Mazzeo and Edward Witten conclude this volume with a study of the Nahm pole boundary condition in an ambitious program to generalize the classical Chern-Simons theory of knot invariants. As the reader can notice, many active fields within Geometry and Topology have been represented as we have celebrated our first 50 years of Mathematics at Cinvestav.

Ernesto Lupercio Ludmil Katzarkov Francisco Turrubiates

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12412

Solomon Lefschetz and Mexico

Michael Atiyah

I first met Lefschetz when I went to Princeton as a postdoc in 1955, but I had already heard much about him from my supervisor Professor Hodge. When Hodge started his research, he followed the path blazed by Lefschetz on the topology of algebraic varieties. Hodge’s first significant result was that holomorphic 2-forms on algebraic surfaces had non-zero periods. Lefschetz at first refused to believe Hodge’s argument, even though it was closely modelled on arguments for 1-forms of Lefschetz him- self. After much exchange with Hodge, Lefschetz finally conceded and, like most converts, became one of Hodge’s strongest supporters. When Hodge later visited Princeton, he used to go for walks with Lefschetz who was always brimming with ideas, at least some of which would turn out to be fruitful. So, as I was Hodge’s student, it was natural that Lefschetz would take an interest in me, but he was always somewhat aggressive (in a friendly way). I had written a joint paper with Hodge, and I remember Lefschetz holding the paper in front of me, in the old Fine Hall common room, and challenging me to point to any significant result! At this time Lefschetz’s interests had moved to differential equations—a delib- erate shift on his part to aid the war effort—so my mathematical contacts were few. But he did invite me to the famous international conference in Mexico which he organized in the summer of 1956. It was a memorable event for my wife and I since we drove all round the U.S., ending up for a month in Mexico City (and then returning to Princeton). I also had the chance to meet many famous mathematicians, including Henri Cartan and Henry Whitehead. Lefschetz had a powerful personality combined with considerable charm. He used to boast that the many years he had spent in isolation in Kansas had been the most productive of his life, since there was no one there to disturb him! As we all know he was very active in Mexico, building up mathematics there, and it is very appropriate that his legacy there should be celebrated. School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JZ, United Kingdom E-mail address: [email protected]

c 2014 American Mathematical Society 1

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12413

Recent Progress in Symplectic Flexibility

Yakov Eliashberg

Abstract. This is a survey of recent advances on the ﬂexible side of symplectic topology.

1. Flexible vs. Rigid Symplectic Topology Many problems in Mathematics and its applications deal with partial differential equations, partial differential inequalities, of more generally with partial differential relations, i.e any conditions imposed on partial derivatives of an unknown function. A solution of such a partial differential relation R is any function which satisfies this relation. Any differential relation has an underlying algebraic relation which one gets by substituting all the derivatives entering the relation with new independent functions. A solution of the corresponding algebraic relation, called a formal solution of the original differential relation R, is a necessary condition for the solvability of R. Though it seems that this necessary condition should be very far from being sufficient, it was a surprising discovery in the 1950s of geometrically interesting problems where existence of a formal solution is the only obstruction for the genuine solvability. One of the first such non-trivial examples were the C1-isometric embedding theorem of J. Nash and N. Kuiper, [37, 31], and the immersion theory of S. Smale and M. Hirsch, [45, 29]. After Gromov’s remarkable series of papers beginning with his paper [26] and culminating in his book [27] the area crystal- lized as an independent subject. Rigid and flexible results coexist in many areas of geometry, but nowhere else they come so close to each other, as in symplectic geometry. 1.1. Symplectic and contact basics. To set the stage I begin with some basic definitions from symplectic and contact geometries. Symplectic geometry was born as a geometric language of classical mechanics, and similarly contact geometry emerged as a natural set-up for geometric optics and mechanics with non-holonomic constraints. The cotangent bundle T ∗M of any smooth n-dimensional manifold M carries a canonical Liouville 1-form λ, usually denoted pdq, which in any local coordinates (q1,...,qn)onM and dual coordinates (p1,...,pn) on cotangent fibers can be n n ∧ written as λ = 1 pidqi. The differential ω := dλ = 1 dpi dqi is called the

2010 Mathematics Subject Classiﬁcation. Primary 57R17. The author was partially supported by the NSF grant DMS-1205349.

c 2014 American Mathematical Society 3

4YAKOVELIASHBERG canonical symplectic structure on the cotangent bundle of M. In the Hamiltonian formalism of classical mechanics the cotangent bundle T ∗M is viewed as the phase space of a mechanical system with the conﬁguration space M.Thep-coordinates have a mechanical meaning of momenta. The full energy of the system expressed through coordinates and momenta, i.e. viewed as a function H : T ∗M → R on the ∗ cotangent bundle (or a time-dependent family of functions Ht : T M → R if the system is not conservative) is called the Hamiltonian of the system. The dynamics ∈ ∗ is then deﬁned by the Hamiltonian equationsz ˙ = XHt (z),z T M,wherethe

Hamiltonian vector ﬁeld XHt is determined by the equation i(XHt )ω = dHt,which in the canonical (p, q)-coordinates has the form

n ∂H ∂ ∂H ∂ X = − t + t . Ht ∂q ∂p ∂p ∂q 1 i i i i TheflowofthevectorfieldX preserves ω, i.e. X∗ ω = ω. The isotopy generated Ht Ht by the vector field XHt is called Hamiltonian. More generally, the Hamiltonian dynamics can be defined on any 2n-dimensional manifold endowed with a symplectic, i.e. a closed non-degenerate differential 2-form ω. According to a theorem of Darboux any such form admits local canonical n ∧ coordinates p1,...,pn,q1,...,qn in which it can be written as ω = 1 dpi dqi. Dif- feomorphisms preserving ω are called symplectomorphisms. Symplectomorphisms which can be included in a time dependent Hamiltonian flow are called Hamilton- ian.Whenn = 1 a symplectic form is just an area form, and symplectomorphisms are area preserving transformations. Though in higher dimensions symplectomorphisms are volume preserving but the subgroup of symplectomorphisms represents a small part of the group of volume preserving diffeomorphisms. The projectivized cotangent bundle PT∗M serves as the phase space in the geometric optics. It can be interpreted as the space of contact elements of the manifold M, i.e. the space of all tangent hyperplanes to M.Theformpdq does not descend to PT∗M but its kernel does, and hence the space of contact elements carries a canonical field of tangent to it hyperplanes. This field turns out to be completely non-integrable. It is called a contact structure. More generally, a contact structure on a (2n + 1)-dimensional manifold is a completely non-integrable field of tangent hyperplanes ξ, where the complete non-integrability can be expressed by the Frobenius condition α ∧ (dα)∧n = 0 for a 1-form α (locally) defining ξ by the Pfaffian equation α = 0. Though at first glance symplectic and contact geometries are quite different, they are in fact tightly interlinked and it is useful to study them in parallel. An important property of symplectic and contact structures is the following stability theorem, which is due to Moser [35] in the symplectic case and to Gray [22] in the contact one: Given a 1-parametric family ωt of symplectic or ξt of contact structures on a manifold X which coincide outside of a compact set, which in the symplectic case belong to the same cohomology class with compact support, there exists an isotopy ht : X → X with compact support which starts at the identity ∗ ∗ h0 =Idand such that ht ωt = ω0 or ht ξt = ξt. Maximalintegral(i.e.tangenttoξ) submanifolds of a (2n + 1)-dimensional contact manifold (V,ξ)havedimensionn and called Legendrian. Their symplectic counterparts are n-dimensional submanifolds L of a 2n-dimensional symplectic manifold (W, ω) which are isotropic for ω, i.e. ω|L = 0. They are called Lagrangian

RECENT PROGRESS IN SYMPLECTIC FLEXIBILITY 5 submanifolds. Here are two important examples of Lagrangian submanifolds. A diﬀeomorphism f : W → W of a symplectic manifold (W, ω) is symplectic if and only if its graph Γf = {(x, f(x)); x ∈ W }⊂(W × W, ω × (−ω)) is Lagrangian. A1-formθ on a manifold M viewed as a section of the cotangent bundle T ∗M is Lagrangian if and only if it is closed. For instance, if H1(M) = 0 then Lagrangian sections are graphs of diﬀerentials of functions, and hence the intersection points of a Lagrangian with the the 0-section are critical points of the function. A general Lagrangian submanifold corresponds to a multivalued function, called the front of the Lagrangian manifold. Given a submanifold N ⊂ M (of any codimension), the setofalltangenttoithyperplanesinTM is a Legendrian submanifold of the space of contact elements PT∗M.

1.2. Gromov’s alternative and discovery of symplectic rigidity. From the first steps of symplectic topology flexible and rigid methods were coexisting and competing. It was an original idea of H. Poincaré that Hamiltonian systems should satisfies special qualitative properties. In particular, his study of periodic orbits in the so- called restricted 3-body problem led him to the following statement, now known as the “last geometric theorem of H. Poincaré”: any area preserving transformation of an annulus S1 × [0, 1] which rotates the boundary circles in opposite directions should have at least two fixed points. Poincaré provided many convincing arguments why the statement should be true [42], but the actual proof was found by G.D. Birkhoff [4] in 1913, a few months after Poincaré’s death. Birkhoff’s proof was purely 2-dimensional and further development of Poincaré’s dream of what is now called symplectic topology had to wait till 1960s when V. I. Arnold [2] formulated a number of conjectures formalizing this vision of Poincaré. In particular, one of Arnold’s conjectures stated that the number of fixed points of a Hamiltonian diffeomorphism is bounded below by the minimal number of critical points of a function on the symplectic manifold. At about the same time Gromov was proving his h-principle type results. He realized that symplectic problems exhibited some remarkable flexibility. This called into question whether Arnold’s conjectures could be true in dimension > 2. Among remarkable results pointing towards symplectic flexibility which were proven by Gromov at the end of 60s and the beginning of 70s were: • h-principle for symplectic an contact structures on open manifolds: in any homotopy class of non-degenerate (not necessarily closed) 2-forms on an open manifold there is a symplectic form in any prescribed cohomology class. Moreover, any 2 such forms are homotopic as symplectic forms. Similarly, any almost contact structure, i.e. a pair (λ, η)of1-and2-forms on a (2k + 1)-dimensional open manifold which satisfies λ ∧ ηk =0,is homotopic through almost contact structures to a pair (α, dα). • h-principle for Lagrangian immersions which asserts that the Lagrangian regular homotopy classes of Lagrangian immersions L → X are in 1-1 correspondence with homotopy classes of injective Lagrangian homomorphisms TL → TX; • h-principle for -Lagrangian embeddings (i.e. embeddings whose tangent planes deviate from Lagrangian directions by an angle < ).

6YAKOVELIASHBERG

• h-principle for the iso-symplectic and iso-contact embeddings. For instance, in the symplectic case, if (M,ω) and (N,η) are two symplectic manifolds such that dim N ≥ dim M +4 then any smooth embedding f : M → N which pulls back the cohomology class of the form η to the cohomology class of ω, and whose diﬀerential df is homotopic to a symplectic bundle isomorphism, can be C0-approximated by an iso-symplectic embedding f: M → N, i.e. f∗η = ω. For iso-symplectic and iso-contact immersions the h-principle holds in codimension 2.

Gromov formulated (and proved) the following alternative: either the group of symplectomorphisms (resp. contactomorphisms) is C0-closed in the group of all diffeomorphisms, or its C0-closure coincides with the group of volume preserving (resp. all) diffeomorphisms. One of the corollaries of Gromov’s convex integration method was that there are no additional lower bounds for the number of fixed points of a volume preserving diffeomorphism of a manifold of dimension ≥ 3. Clearly the bound on the number of fixed points is a C0-property, and hence, if the second part of the alternative were true this would imply that Hamiltonian diffeomorphisms of symplectic manifolds of dimension > 2 have no special fixed point properties, and hence Poincaré’s theorem and Arnold’s conjectures reflected a pure 2-dimensional phenomenon. In fact, it was clear from this alternative, that all basic problems of symplectic topology are tightly interconnected. Here are some of such problems, besides Gromov’s alternative: – Extension of symplectic and contact structures to the ball from a neighborhood of the boundary sphere. – 1-parametric version of the previous question: is it true that two structures on the ball which coincide near the boundary and which are formally homotopic relative the boundary, are isotopic? – Fixed point problems for symplectomorphisms. More generally, Lagrangian intersection problem: Do Lagrangian manifolds under certain conditions have more intersection points than it is required by topology? – Are there any non-formal obstructions to Legendrian isotopy? Proving an h-principle type statement in one of these problems would imply that all symplectic problems have soft solutions, and hence a resolution of Gromov’s alternative became a question about existence of symplectic topology as a separate subject. At the beginning of 80s Gromov’s alternative was resolved in favor of rigidity in the series of works [3], [7], [14, 15], culminating in Gromov’s paper [25]where he introduced his method of (pseudo-)holomorphic curves in symplectic manifolds, which brought a genuine revolution into this subject. After Gromov’s paper the rigid side of symplectic topology began enraveling with an exponentially increasing speed. The discovery of Floer homology, Hofer’s metric, Gromov-Witten invariants followed. Symplectic Field Theory brought new powerful applications in contact geometry. There were discovered remarkable connections of Gromov-Witten theory with the theory of integrable systems. This theory is also an essential part of the mathematical theory of Mirror Symmetry, which predicts a wealth of information about symplectic manifolds and their Lagrangian submanifolds by looking at the mirror problems in complex geometry. Many of these predictions are now rigorously proven, and the picture continues to unravel.

RECENT PROGRESS IN SYMPLECTIC FLEXIBILITY 7

The Heegaard homology theory created by P. Ozsvath and Z. Szabo [41], and more recently embedded contact homology of M. Hutchings [28] and C.H. Taubes [48], which are defined using J-holomorphic curves, became one of the most powerful tools in low-dimensional topology and led to solutions of several long-standing classical problems in this area. Applications of holomorphic curves in Hamiltonian Dynamics brought us closer to the realization of Poincaré’s dream of establishing qualitative properties of mechanical systems (e.g. existence and the number of periodic trajectories) without actual solving the equations of motion. In particular, the Weinstein conjecture asserting existence of periodic trajectories of Reeb vector fields was proven in many cases, see [51, 30], and in dimension 3 in full generality (see [47]).

2. Flexible milestones after the resolution of Gromov’s alternative Though in a shadow of successes on the rigid side, over the years the flexible symplectic topology had also a number of success stories. Here are some of the interesting developments. Overtwisted contact structures. It was understood in 1989, see [16], that in the world of 3-dimensional contact manifolds there is an important dichotomy: if a contact manifold contains the so-called overtwisted disc, i.e. an embedded disc which along its boundary is tangent to contact structure, then the contact structure becomes very flexible and abides a certain h-principle: two overtwisted contact structures which are homotopic as plane fields are homotopic as contact structures, and hence in view of Gray’s theorem are isotopic. Non-overtwisted contact manifolds are called tight, and that is where the rigid methods of symplectic topology are applicable. The classification of overtwisted contact structures yields similar flexibility results for Legendrian knots in overtwisted contact 3-manifolds. Namely, Legenfdrian knots in the complement of an overtwisted disc, called loose in [18], also satisfy an h-principle. The high-dimensional analog of loose knots is discussed in Section 3.1 below. Despite a significant progress (see e.g. [38,39]), the high-dimensional analogues of the overtwisting phenomenon is far from being understood. Donaldson’s almost holomorphic sections. We already mentioned above Gro- mov’s h-principle for iso-symplectic embeddings in codimension > 2. Applying holomorphic curve technique it is not difficult to construct counter-examples to a similar h-principle in codimension 2. However, Simon Donaldson used his theory of almost holomorphic sections of complex line bundles over almost complex symplectic manifolds to prove, among other remarkable results, the following h-principle type theorem:

Theorem 2.1 ([8]). For any closed 2n-dimensional symplectic manifold (M,ω) with an integral cohomology class [ω] ∈ H2(M) and a suﬃciently large integer k there exists a codimension 2 symplectic submanifold Σ ⊂ M which represents the homology class Poincar´e dual to kω. Moreover, the complement M \ Σ has a homotopy type of an n-dimensional cell complex (asitisthecaseforcomplements of hyperplane sections in complex projective manifolds).

8YAKOVELIASHBERG

Furthermore, he proved the following symplectic Lefschetz pencil theorem: Theorem 2.2 ([9]). If (V,ω) is a symplectic manifold with integral cohomology class [ω] ∈ H2(M), then for sufficiently large integer k there exists a topological Lefshetz pencil in which the fibers are symplectic manifolds representing homology class dual to k[ω]. By definition the topological Lefshetz pencil is equivalent to the complex algebraic one near all the singularities. E. Giroux’s theory [23] of contact book decompositions of contact manifold can be viewed as an adaption of Donaldson’s technique for the contact case. Symplectic embeddings of polydiscs. LetusdenotebyP (r1,...,rn) the poly- n disc {|z1|≤r1,...,|zn|≤rn}⊂C , where we assume r1 ≤ r2 ≤ ··· ≤ rn. If P (r1,...,rn) symplectically embeds into P (R1,...,Rn) then famous Gromov’s non-squeezing theorem implies that r1 ≤ R1. We also have the volume constraint r1 ...rn ≤ R1 ...Rn. Many people tried to prove that when n>2 there should be more constraints on the radii besides the Gromov width and volume constraints. However, Larry Guth proved the following remarkable result on the flexible side, which showed that the room for additional constraints is very limited. Theorem 2.3 ([24]). There exists a constant C(n) depending on the dimension n such that if C(n)r1 ≤ R1 and C(n)r1 ...rn ≤ R1 ...Rn then a polydisc P (r1,...,rn) symplectically embeds into P (R1,...,Rn). Existence of Stein complex structure. Stein manifolds are complex manifolds which admit proper holomorphic embeddings into CN . According to a theorem of H. Grauert a Stein manifold can also be characterized as a manifold which admits an exhausting strictly plurisubharmonic function. Here the word exhausting means proper and bounded below, while a real-valued function φ : V → R on a complex manifold V is called strictly plurisubharmonic or i-convex if the Hermitian C form −dd φ =2i∂dφ¯ which in local holomorphic coordinates is given by a matrix 2 ∂ φ is positive definite. For an arbitrary complex manifold with a complex ∂zi∂zj structure J we will use the term J-convex instead of strictly plurisubharmonic, to C stress the dependence on the complex structure J. Here we denoted by√d φ(X):= dφ(iX) the differential twisted by the operator of multiplication by −1. It can be easily seen that critical points of a Morse strictly plurisubharmonic function on acomplexn-dimensional manifold have index ≤ n, and hence the Morse theory implies that a Stein manifold of complex dimension n has a homotopy type of a cell complex of real dimension n. The following theorem is proved in my paper [17]: Theorem 2.4 (Existence of Stein structures). Let (V,J) be an almost complex manifold of dimension 2n>4 and φ : V → R an exhausting Morse function without critical points of index >n. Then there exists an integrable complex structure J on V homotopic to J for which the function φ is target equivalent to a J-convex function. In particular, (V,J) is Stein. What is transpired from the proof of Theorem 2.4 is that it is useful to define a symplectic analog of Stein manifold. The corresponding notion of Weinstein manifold, crucial for understanding of Morse theoretic properties of Stein structures,

RECENT PROGRESS IN SYMPLECTIC FLEXIBILITY 9 was introduced in [19], formalizing the Stein handlebody construction from [17]and symplectic handlebody construction from Alan Weinstein’s paper [53]. We discuss this notion and related results in Section 3.4 below.

3. Renaissance of the h-principle in symplectic topology The results proven in the last 2 years signiﬁcantly advanced the ﬂexible side of the symplectic story.

3.1. Loose Legendrian knots. It turns out that in contact manifolds of dimension > 3 there is a remarkable class of Legendrian knots, discovered by E. Murphy in [36], which satisfies a certain form of an h-principle. These knots are called loose in analogy with loose knots in overtwisted contact manifolds. A remarkable fact about Murphy’s loose knots is that in contrast with the 3-dimensional case they exist in all contact manifolds of dimension > 3. Stabilization. The stabilization construction for Legendrian submanifolds, see [17, 5, 36], can be defined as follows. Consider standard contact R2n−1: n−1 R2n−1 R2n−1 − st = ,ξst =ker dz yidxi , 1 2n−1 where (x1,y1,...,xn−1,yn−1,z) are coordinates in R , and consider a diffeomorphic to Rn−1 Legendrian submanifold 1 2 1 3 Λ = (x ,y ,...,x − ,y − ,z): x = y ,y = ···= y − =0,z= y cu 1 1 n 1 n 1 1 2 1 2 n 1 3 1 One can check that given any Legendrian (n − 1)-submanifold Λ ⊂ Y in a contact (2n − 1)-manifold Y , any point p ∈ Λ has a neighborhood Ω ⊂ Y that admits a map ∩ → R2n−1 Φ: (Ω, Λ Ω) ( st , Λcu), Φ(p)=0, which is a contactomorphism onto a neighborhood of the origin. The stabilization construction is a local modification of a Legendrian knot in a neighborhood of a point. It replaces the image of Lcu by an image of another U Legendrian Lcu, which coincides with Lcu at infinity. We describe this modification below. The two branches of the front Γcu of the Legendrian Λcu, i.e. the projection to the (x1,...,xn−1,z)-coordinate subspace, are graphs of the functions ±h,where

√ 3 2 2 2 h(x)=h(x1,...,xn−1)= 3 x1 , Rn−1 { ≥ } deﬁned on the half-space + := x =(x1,...,xn−1): x1 0 . Rn−1 Let U be a domain with smooth boundary contained in the interior of + , ⊂ Rn−1 Rn+1 → R U Int ( + ). Pick a non-negative function φ: + with the following Rn−1 − properties: φ has compact support in Int ( + ), the function φ(x):=φ(x) 2h(x) −1 ∞ U is Morse, U = φ ([0, )), and 0 is a regular value of φ. Consider the front Γcu in n−1 R × R obtained from Γcu by replacing the lower branch of Γcu, i.e. the graph z = −h(x), by the graph z = φ(x) − h(x). Since φ has compact support, the U front Γcu coincides with Γcu outside a compact set. Consequently, the Legendrian U Rn−1 → R2n−1 U embedding Λcu : deﬁned by the front Γcu coincides with Λcu outside acompactset.

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It turns out that if (and only if) the Euler characteristics of the domain U is U equal to 0 then the Legendrian submanifolds Λcu and Λcu are formally Legendrian isotopic via a compactly supported Legendrian isotopy (however, they are never Legendrian isotopic if U = ∅). We recall that a formal Legendrian isotopy connecting Legendrian embeddings f0,f1 :Λ→ (Y,ξ)isapairft, Φt, t ∈ [0, 1], where ft is a smooth isotopy and Φt : T Λ → ξ is a family of Lagrangian homomorphisms connecting φ0 = df 0 and Φ1 = df 1, and such that the paths of homomorphisms df t, Φt are homotopic with fixed end points as paths of injective homomorphisms T Λ → TY. Now, given a Legendrian (n − 1)-submanifold Λ of a contact (2n − 1)-manifold Y and contactomorphism ∩ → R2n−1 Φ: (Ω, Λ Ω) ( st , Λ0), ⊂ ∈ ∩ −1 U Ω Y is a neighborhood of a point of p Λ, we replace Ω ΛwithΦ (Λ0 ). The resulted Legendrian embedding ΛU which coincides with Λ outside of Ω is called the U-stabilization of Λ in Ω. Murphy’s theorem. A Legendrian embedding Λ → Y of a connected manifold Λ (which we sometimes simply call a Legendrian knot) is called loose if it is isotopic to the stabilization of another Legendrian knot. We point out that looseness depends on the ambient manifold. A loose Legendrian embedding Λ into a contact manifold Y need not be loose in a smaller neighborhood Y ,Λ⊂ Y ⊂ Y . The above construction shows that a Legendrian submanifold Λ ⊂ Y can be made loose by stabilizing it in an arbitrarily small neighborhood of a point, and even without changing its formal Legendrian isotopy class. E. Murphy proved the following h-principle for loose Legendrian knots in contact manifolds of dimension 2n − 1 > 3: Theorem 3.1 ([36]). Any two loose Legendrian embeddings which coincide outside a compact set and which can be connected by a formal compactly supported Legendrian isotopy can be connected by a genuine compactly supported Legendrian isotopy. 3.2. Lagrangian caps. It turned out that Lagrangian embeddings with loose Legendrian boundary also satisfies an h-principle. The story begins with the following question: Let B be the round ball in the standard symplectic R2n. Is there an embedded Lagrangian disc Δ ⊂ R2n\Int B with ∂Δ ⊂ ∂B such that ∂Δ is a Legendrian submanifold and Δ transversely intersects ∂B along its boundary? If n = 2 then such a Lagrangian disc does not exist: its existence contradicts the so-called slice Bennequin inequality,see[43]. Until recently no such examples were known in higher dimensions either. Surprisinhgly, it turns out that when n>2 then Lagrangian discs with loose Legendrian boundary satisfy an h-principle, which, in particular, implies that such disc exist in abundance: In particular, we have the following result. Theorem 3.2 ([20]). Let L be a smooth manifold of dimension n>2 with non- empty boundary such that its complexified tangent bundle TL⊗ C is trivial. Then there exists an exact Lagrangian embedding f :(L, ∂L) → (R2n \ Int B,∂B) with f(∂Δ) ⊂ ∂B such that f(∂Δ) ⊂ ∂B is a Legendrian submanifold and f transverse to ∂B along its boundary ∂L.

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Note that the triviality of the bundle TL ⊗ C is a necessary (and according to Gromov’s h-principle for Lagrangian immersions [27] suﬃcient) condition for existence of any Lagrangian immersion L → Cn. Lagrangian embedding of a n-disc to the complement of a ball with Legendrian boundary in its boundary sphere can be completed to a Lagrangian embedding of a n-sphere with a conical singular point. More precisely, given a symplectic manifold (X, ω)wesaythatL ⊂ M is a Lagrangian submanifold with an isolated conical point if it is a Lagrangian submanifold away from a point p ∈ L,andthereexists −1 a symplectic embedding f : Bε → X such that f(0) = p and f (L) ⊂ Bε is a 2n Lagrangian cone. Here Bε is the ball of radius ε in the standard symplectic R . Note that this cone is automatically a cone over a Legendrian sphere in the sphere ∂Bε endowed with the standard contact structure given by the restriction to ∂Bε 1 n − of the Liouville form λst = 2 1 (pidqi qidpi). As a special case of Theorem 3.2 (when ∂L is a sphere) we get

Corollary 3.3. Let L be an n-dimensional, n>2, closed manifold such that the complexiﬁed tangent bundle T ∗(L \ p) ⊗ C is trivial. Then L admits an exact Lagrangian embedding into R2n with exactly one conical point. In particularly an n-sphere admits a Lagrangian embedding to R2n with one conical point for each n>2.

3.3. Lagrangian non-intersections. The conical singularity with an appropriate loose Legendrian asymptotics in Corollary 3.3 can be resolved into an immersion with 1 self-intersection point. This leads to surprising, and at ﬁrst glance going against popular Arnold type Lagrangian intersection conjectures, constructions of Lagrangian immersions with minimal number of self-intersection points. In particular, we get

Theorem 3.4 ([13]). Let L be an n-dimensional closed manifold with trivial bundle TL ⊗ C.Wedenotebys(L) the minimal number of double points of a Lagrangian immersion of L into the standard symplectic R2n. Then the following hold: (i) If n is odd or if L is non-orientable, then s(L) ∈{1, 2}. (ii) If n =3then s(L)=1. 1 | | (iii) If n is even and L is orientable, then for χ(L) < 0, s(L, σ)= 2 χ(L) , ≥ 1 1 and for χ(L) 0,eithers(L)= 2 χ(L) or s(L)= 2 χ(L)+2. The case n = 2 is due to D. Sauvaget, [46]. It is interesting to compare Theo- rem 3.4 with the results of [11,12] which show that for even value of n the standard n-sphere is the only homotopy n-sphere that admits a self-transverse Lagrangian immersion into Euclidean space with only one double point. This means in, particular, that in the case when dim(L)isevenandχ(L) > 0, s(L) is generally not determined by the homotopy type of L. The following result constrains the homotopy type of a manifold which admits a Lagrangian embedding with the minimal number of self-intersection points.

Theorem 3.5 ([13]). Let L be an even dimensional spin manifold with χ(L) > 1 0.Ifs(L)= 2 χ(L) then π1(L)=1and H2k+1(L)=0for all k.Inparticular if dim L>4 then L has the homotopy type of a CW-complex with χ(L) even- dimensional cells and no odd-dimensional cells.

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It is interesting to note that even for the standard odd-dimensional sphere S2k+1 the construction in Theorem 3.4 provides an immersion with a single double point of Maslov index, which is diﬀerent from the standard Lagrangian immersion S2k+1 → R4k+2. Using Polterovich’s surgery [49]wethenget

Corollary . 1 × 2k → R4k+2 3.6 There exists a Lagrangian embedding S S st for which the generator of the ﬁrst homology of positive action has non-positive Maslov − 1 × 2 → R6 index 2 2k. In particular, there exists a Lagrangian embedding S S st with zero Maslov class. 1 × 2 → R6 Existence of a Lagrangian embedding S S st with zero Maslov class was a well-known problem in symplectic topology.

3.4. Flexible Stein and Weinstein manifolds. 3.4.1. Weinstein manifolds.

Definition. A Weinstein structure on an open manifold V is a triple (ω, X, φ), where • ω is a symplectic form on V , • φ : V → R is an exhausting Morse (or generalized Morse, i.e. having either non-degenerate or birth-death critical points) function, • X is a complete vector ﬁeld which is Liouville for ω (i.e. LX ω = ω)and gradient-like for φ. The quadruple (V,ω,X,φ) is then called a Weinstein manifold.

Though any Weinstein structure (ω, X, φ) can be perturbed to make the function φ Morse, in 1-parameter families of Weinstein structures birth-death zeroes are generically unavoidable. We will also consider Weinstein structures on a cobordism, i.e., a compact manifold W with boundary ∂W = ∂+W ∂−W . A Morse (or generalized Morse) function φ : W → R is called defining if ∂±W are regular level sets of φ with | | φ ∂−W =minφ and φ ∂+W =maxφ. The definition of a Weinstein cobordism (W, ω, X, φ) differs from that of a Weinstein manifold only in replacing the condition that φ is exhausting by the requirement that φ is defining function, and replacing the completeness condition of X by the requirement that X points inward along ∂−W and outward along ∂+W . A Weinstein cobordism with ∂−W = ∅ is called a Weinstein domain. A J-convex function φ : V → R on a complex manifold (V,J)servesasa C potential of a Kähler form ωJ,φ = −dd φ. The gradient Xφ := ∇J,φφ of the function φ with respect to the metric which it defines is a Liouville field for ωJ,φ, i.e. LXJ,φ ωJ,φ = ωJ,φ.If(V,J) is Stein then for any exhausting J-convex function φ : V → R the vector field XJ,φ can be made complete by composing φ with any function h : R → R with positive first and second derivatives. Assuming that this is already done we associate with a Stein complex manifold (V,J) together with an exhausting J-convex (generalized) Morse function φ : V → R a Weinstein structure − C ∇ W(V,J,φ)=(V,ωφ,J = dd φ, XJ,φ = gJ,φ φ, φ).

Here the Riemannian metric gJ,φ is the real part of the Hermitian form HJ,φ = gJ,φ − iωJ,φ.

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By a Stein cobordism structure on a cobordism W , we understand a pair (J, φ) where J is an integrable complex structure on W and φ : W → R a defining J- convex function. A Stein cobordism with empty ∂−W is called a Stein domain. As in the manifold case, any Stein cobordism structure (J, φ)onW determine − C ∇ a Weinstein cobordism structure W(J, φ)=(ωφ,J = dd φ, XJ,φ = gφ φ, φ). Theorem 2.4 can be upgraded to the following more precise result: Theorem 3.7 ([5]). (i) Let W =(V,ω,X,φ) be a Weinstein structure. Then there exists - an integrable complex structure J on V for which φ is J-convex and - a homotopy of Weinstein structures Wt =(ωt,Xt,φ) connecting W0 = W and W1 = W(V,J,φ). (ii) Let V be a manifold of dimension 2n =4 , φ : V → R an exhausting Morse function without critical points of index >n,andη a non-degenerate 2- form. Then there exists a Weinstein structure (ω, X, φ) on V such that ω and η are homotopic as non-degenerate forms. A similar result holds in the cobordism case. 3.4.2. Flexibility. Each Weinstein manifold or cobordism can be cut along regular level sets of the function into Weinstein cobordisms that are elementary in the sense that there are no trajectories of the Liouville vector field connecting different critical points. An elementary 2n-dimensional Weinstein cobordism (W, ω, X, φ), n>2, is called flexible if the attaching spheres of all index n handles form in ∂−W a loose Legendrian link. i.e. each its component is loose in the complements of the others. A Weinstein cobordism or manifold structure (ω, X, φ) is called flexible if it can be decomposed into elementary flexible cobordisms. A2n-dimensional Weinstein structure (ω, X, φ), n ≥ 2, is called subcritical if all critical points of the function φ have index 4 is by definition flexible. Remark 3.8. The property of a Weinstein structure being subcritical is not preserved under Weinstein homotopies because one can always create a pair of critical points of index n and n − 1. It is an open problem whether or not flexibility is preserved under Weinstein homotopies. Theorem 3.9 ([5]). (i) Let W =(V,ω,X,φ) be a flexible Weinstein manifold of dimension 2n>4. Then there exists a flexible Weinstein homotopy W =(V,ωt,Xt,φt), t ∈ [0, 1],withW0 = W,whichisfixed outside a compact set and such that the Morse function φ1 has minimal number of critical points allowed by the Morse theory. If φ has finitely many critical points then the homotopy can be made fixed at infinity. (ii) Let W0 =(ω0,X0,φ0) and W1 =(ω1,X1,φ1) be two flexible Weinstein structures on a manifold V of dimension 2n. Suppose that η0 and η1 are homotopic as nondegenerate (not necessarily closed) 2-forms. Then W0 and W1 can be connected by a homotopy Wt =(ωt,Xt,φt), t ∈ [0, 1],of flexible Weinstein structures. An analog of Theorem 3.9 also holds for Weinstein cobordisms. Combining with Theorem 3.7 we also get an analog of Theorem 3.9 in the Stein case. We will formulate it here only for Stein cobordisms.

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Theorem 3.10 ([5]). (i) Let W =(W, J, φ) be a flexible Stein cobordism of dimension 2n>4. Then there exists a homotopy of defining J-convex functions φt : W → R, t ∈ [0, 1],withφ0 = φ, such that the Morse function φ1 has minimal number of critical points allowed by the Morse theory. (ii) Any two flexible Stein cobordism structures (W, J0) and (W, J1) which are homotopic as almost complex structures are homotopic as flexible Stein structures.

In particular, we have the following Weinstein/Stein version of the h-cobordism theorem.

Corollary 3.11 (Weinstein and Stein h-cobordism theorem). Any flexible Weinstein structure on a product cobordism W = Y × [0, 1] of dimension 2n>4 is homotopic to a Weinstein structure (W, ω, X, φ),whereφ : W → [0, 1] is a function without critical points. Similarly, any flexible Stein cobordism (W, J) which is diffeomorphic to Y × [0, 1] admits a defining J-convex function.

We note that without the flexibility assumption the above claim is wrong, see [50, 33]. Symplectomorphisms of flexible Weinstein manifolds. Theorem 3.9 has the following consequence for symplectomorphisms of flexible Weinstein manifolds.

Corollary 3.12. Let W =(V,ω,X,φ) be a flexible Weinstein manifold of dimension 2n>4,andf : V → V a diffeomorphism such that f ∗ω is homotopic to ω through nondegenerate 2-forms. Then there exists a diffeotopy ft : V → V , t ∈ [0, 1], such that f0 = f,andf1 is an exact symplectomorphism of (V,ω). Remark 3.13. Even if W is of finite type, i.e. φ has finitely many critical points, and f = id outside a compact set, the diffeotopy ft provided by Theo- rem 3.12 will be in general not equal the identity outside a compact set.

Equidimensional symplectic embeddings of ﬂexible Weinstein manifolds. The following result about equidimesional symplectic embeddings of ﬂexible Weinstein domainsisprovenin[20]:

Theorem 3.14 ([20]). Let (W, ω, X, φ) be a flexible Weinstein domain with Liouville form λ.LetΛ be any other Liouville form on W such that the symplectic forms ω and Ω:=dΛ are homotopic as non-degenerate (not necessarily closed) → ∗ 2-forms. Then there exists an isotopy ht : W W such that h0 =Idand h1Λ= ελ+dH for some small ε>0 and some smooth function H : W → R.Inparticular, h1 defines a symplectic embedding (W, εω) → (W, Ω). Corollary 3.15 ([20]). Let (W, ω, X, φ) be a flexible Weinstein domain and (X, Ω) any symplectic manifold of the same dimension. Then any smooth embedding → ∗ → f0 : W X such that the form f0 Ω is exact and the differential df : TW TX is homotopic to a symplectic homomorphism is isotopic to a symplectic embedding f1 :(W, εω) → (X, Ω) for some small ε>0. Moreover, if Ω=dΛ, then the ∗ − embedding f1 can be chosen in such a way that the 1-form f1 Λ iX ω is exact. If, moreover, the Liouville vector field dual to Λ is complete, then the embedding f1 exists for an arbitrarily large constant ε.

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3.5. Topology of polynomially and rationally convex domains. We finish this section by implications of the discussed above flexibility results for a problem of high-dimensional complex analysis concerning topology of polynomially and rationally convex domains. Polynomial, rational and holomorphic convexity. Recall the following complex analytic notions of convexity for domains in Cn. For a compact set K ⊂ Cn one defines its polynomial hull as n n KP := {z ∈ C |P (z)|≤max |P (u)| for all complex polynomials P on C }, u∈K and its rational hull as n P KR := {z ∈ C |R(z)|≤max |R(u)| for all rational functions R = ,Q|K =0 }. u∈K Q GivenanopensetU ⊃ K,theholomorphic hull of K in U is defined as U KH := {z ∈ U |f(z)|≤max |f(u)| for all holomorphic functions f on U}. u∈K n AcompactsetK ⊂ C is called rationally (resp. polynomially) convex if KR = n K (resp. KP = K). An open set U ⊂ C is called holomorphically convex if U n KH is compact for all compact sets K ⊂ U. A compact set K ⊂ C is called holomorphically convex if it is the intersection of its holomorphically convex open neighborhoods. We have polynomially convex =⇒ rationally convex =⇒ holomorphically convex. The first implication is obvious, while the second one follows from the fact that a rationally convex compact set K is an intersection of bounded rational polyhedra {|Ri| n(see e.g. [5]). It follows that any holomorphically, rationally or polynomially convex domain has the same property. We already stated above, see Theorem 2.4, that for n ≥ 3, any domain in Cn with such a Morse function is smoothly isotopic to an i-convex one. It turns out, in the spirit of Theorem 2.4, that for n ≥ 3 there are no additional constraints on the topology of rationally convex domains. Theorem 3.16 ([6]). A compact domain W ⊂ Cn, n ≥ 3, is smoothly isotopic to a rationally convex domain if and only if it admits a defining Morse function without critical points of index >n. The next result gives necessary and sufficient constraints on the topology of polynomially convex domains.

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Theorem 3.17 ([6]). A compact domain W ⊂ Cn, n ≥ 3, is smoothly isotopic to a polynomially convex domain if and only if it satisfies the following topological condition: (T) W admits a defining Morse function without critical points of index >n, and Hn(W ; G)=0for every abelian group G. The “only if” part is well known, see [1](seealso[21]). Note that, in view of the universal coefficient theorem, condition (T) is equivalent to the condition (T’) W admits a defining Morse function without critical points of index >n, Hn(W ) = 0, and Hn−1(W ) has no torsion. Further analysis of condition (T) yields Proposition 3.18. (a) If W is simply connected, then condition (T) is equivalent to the existence of a defining Morse function without critical points of index ≥ n. (b) For any n ≥ 3 there exists a (non-simply connected) domain W satisfying condition (T) with πn(W, ∂W ) =0 .Inparticular,W does not admit a defining function without critical points of index ≥ n. Theorems 3.16 and 3.17 are consequences of the following more precise result for flexible Stein domains: Theorem 3.19 ([6]). Let (W, J) be a flexible Stein domain of complex dimension n ≥ 3,andf : W→ Cn a smooth embedding such that f ∗i is homotopic to J through almost complex structures. Then (W, J) is deformation equivalent to a rationally convex domain in Cn.Moreprecisely,f is smoothly isotopic to an embedding g : W→ Cn such that g(W ) ⊂ Cn is rationally convex, and g∗i is Stein homotopic to J. If in addition Hn(W ; G)=0for every abelian group G,theng(W ) can be made polynomially convex.

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Department of Mathematics, Stanford University

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12414

Equivariant Extensions of Diﬀerential Forms for Non-compact Lie Groups

Hugo Garc´ıa-Compe´an, Pablo Paniagua, and Bernardo Uribe In celebration of the ﬁftieth anniversary of the Mathematics Department at the CINVESTAV.

Abstract. Consider a manifold endowed with the action of a Lie group. We study the relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and we show that the cohomology of the Cartan complex lies on the 0 − th row of the second page of a spectral sequence converging to the equivariant cohomology. We use this result to generalize a result of Witten on the equivalence of absence of anomalies in gauged WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term, to the case on which the gauge group is the special linear group with real coeﬃcients.

1. Introduction In certain situations, geometrical information of manifolds might be encoded in differential forms. In the presence of symmetries of the manifold via the action of a Lie group, the behavior of these differential forms under the group action may lead to a better understanding of the manifold itself. In some well known instances of actions of compact Lie groups, the action is of a particular type whenever the differential form may be extended to an equivariant one in the Cartan model of equivariant cohomology [5]; this is the case for example in Hamiltonian actions on symplectic manifolds [1], in Hamiltonian actions on exact Courant algebroids [4,6,12,14] or in gauged WZW actions which are anomaly free [16] and its generalizations [9, 13]. When the Lie group is not compact, the Cartan complex associated to the action in general does not compute the equivariant cohomology of the manifold, and therefore many of the results that hold for compact Lie groups may not hold for the non-compact case. But since the Cartan model is very well suited for studying the infinitesimal behavior of the differential forms with respect to the action of the Lie algebra, it would be worthwhile knowing more about the relation between the cohomology of the Cartan complex and the equivariant cohomology. In this paper we study this relation and we obtain some interesting results which in particular

2010 Mathematics Subject Classiﬁcation. Primary 57R91, 57T10, 81T40, 81T70. Key words and phrases. Equivariant cohomology, gauged WZW action, equivariant extension. The third author acknowledges and thanks the ﬁnancial support of the Alexander Von Hum- boldt Foundation.

c 2014 American Mathematical Society 19

20 H. GARCÍA-COMPEAN,´ P. PANIAGUA, AND B. URIBE permit us generalize the conditions for cancellation of anomalies on gauge WZW actions developed by Witten [16], to the case on which the gauge group is the non-compact group SL(n, R). The main ingredient of this work is the equivariant De Rham complex developed by Getzler [10] whose cohomology calculates the equivariant cohomology independent whether the group is compact or not. We show that there is an inclusion of complexes of the Cartan complex into the equivariant De Rham one, and therefore we obtain a homomorphism between the cohomology of the Cartan complex to the equivariant De Rham cohomology. This in particular implies that any closed form in the Cartan complex defines an equivariant cohomology class, but note that the converse may not be true. We furthermore show that there is a spectral sequence converging to the equivariant De Rham cohomology such that ∗,0 its E2 -term is isomorphic to the cohomology of the Cartan complex, and in this way an explicit relation between the cohomology of the Cartan complex and the equivariant De Rham cohomology is obtained. Now, since the first page of this spectral sequence could be understood in terms of the differential cohomology of a Lie group, we recall its definition and some of its properties and we reconstruct some calculations for the group SL(n, R). We conclude this work by applying our results to the gauged WZW actions whenever the gauge group is SL(n, R). We recall the result of Witten which claims that the gauged WZW action on compact Lie groups is anomaly free, if and only if 1 −1 3 the WZW term ω = 12π Tr(g dg) can be extended to a closed form in the Cartan complex, and we generalize it to the case on which the gauge group is SL(n, R); this is Theorem 4.1. We use this theorem to construct explicit examples where the equations representing the condition of anomalies cancelation hold and where they do not. The physical implications of our work will appear elsewhere.

Acknowledgments. The first author was partly supported by the CONACyT- México through grant number 128761. The third author would like to acknowledge and to thank the financial support provided by the Alexander Von Humboldt Foun- dation and also would like to thank the hospitality of Prof. Wolfgang Lück at the Mathematics Institute of the University of Bonn.

2. Equivariant cohomology Let G be a Lie group and M a manifold on which G acts on the left by diffeomorphisms. The G-equivariant cohomology of M could be defined as the singular cohomology of the homotopy quotient EG ×G M ∗ Z ∗ × Z HG(M; ):=H (EG G M; ), where EG is the universal G-principal bundle G → EG → BG. The previous definition works for any topological group and any continuous action, but sometimes it is convenient to have a De Rham version with differentiable forms for the equivariant cohomology whenever the group is of Lie type and the action is differentiable.When the group G is compact, the Weil and Cartan models provide a framework on which the equivariant cohomology could be obtained via a complex whose ingredients are the local action of the Lie algebra g, and the differentiable forms Ω•M. When the group G is not compact there is a more elaborate model for equivariant cohomology that we will describe in the next section.

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 21

2.1. De Rham model of Equivariant cohomology. One way to obtain a De Rham model for equivariant cohomology is through the total complex of the double complex ∗ Ω (N•(G M)) that is obtained after applying the differentiable forms functor to the simplicial space N•(G M) which is the nerve of the differentiable groupoid G M. By the works of Bott-Shulman-Stasheff [3]andGetzler[10] we know that one way to calculate the cohomology of the total complex of the double complex of differentiable forms ∗ Ω (N•(G M)) is through the differentiable cohomology groups H∗(G, Sg∗ ⊗ Ω•M) of the group G with values on the differentiable forms of M tensor the symmetric algebra of the dual of the Lie algebra g. In [10] Getzler has shown that there is a De Rham theorem for equivariant cohomology showing that there is an isomorphism of rings ∗ ∗ • ∼ ∗ H (G, Sg ⊗ Ω M) = H (EG ×G M; R) between the De Rham model for equivariant cohomology and the cohomology of the homotopy quotient. The De Rham model for equivariant cohomology defined by Getzler is described as follows. Consider the complex Ck(G, Sg∗ ⊗ Ω•M) with elements smooth maps k • f(g1,...,gk|X):G × g → Ω M, which vanish if any of the arguments gi equals the identity of G. The operators d and ι are defined by the formulas k (df )(g1,...,gk|X)=(−1) df (g1,...,gk|X)and k (ιf)(g1,...,gk|X)=(−1) ι(X)f(g1,...,gk|X), as in the case of the differential in Cartan’s model for equivariant cohomology [5, 11]. Recall that the elements in g∗ are defined to have degree 2, and therefore the operator ι has degree 1. Denote the generators of the symmetric algebra Sg∗ by Ωa where a runs over a base of g. The coboundary d¯: Ck → Ck+1 is defined by the formula k ¯ i (df)(g0,...,gk|X)=f(g1,...,gk|X)+ (−1) f(g0,...,gi−1gi,...,gk|X) i=1 − k+1 | −1 +( 1) gkf(g0,...,gk−1 Ad(gk )X), and the contraction ¯ι : Ck → Ck−1 is defined by the formula k−1 i ∂ tX (¯ιf)(g ,...,g − |X)= (−1) f(g ,...,g ,e i ,g ...,g − |X), 1 k 1 ∂t 1 i i+1 k 1 i=0 where Xi =Ad(gi+1 ...gk−1)X. If the image of the map f : Gk → Sg∗ ⊗ Ω•M

22 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE is a homogeneous polynomial of degree l, then the total degree of the map f equals deg(f)=k + l. It follows that the structural maps d, ι, d¯ and ¯ι are degree 1 maps, and the operator ¯ dG = d + ι + d +¯ι becomes a degree 1 map that squares to zero. Definition 2.1. The elements of the complex ∗ ∗ • (C (G, Sg ⊗ Ω M),dG) will be called equivariant De Rham forms and its cohomology H∗(G, Sg∗ ⊗ Ω•M) will be called the equivariant De Rham cohomology. ∗ ∗ • In [10]itwasshownthatthecomplex(C (G, Sg ⊗ Ω M),dG) together with the cup product

l(|a|−k) −1 (a ∪ b)(g1, ..., gk+l|X)=(−1) γa(g1, ..., gk|Ad(γ )X)b(gk+1, ..., gk+l|X) for γ = gk+1...gk+l, becomes a differential graded algebra, and moreover that there is a canonical isomorphism of rings ∗ ∗ • ∼ ∗ H (G, Sg ⊗ Ω M) = H (M ×G EG; R) with the cohomology of the homotopy quotient. 2.1.1. Cartan model for equivariant cohomology. The Cartan model for equivariant cohomology is the differential graded algebra ∗ ∗ ⊗ • G CG(M):=(Sg Ω M) endowed with the differential d + ι. Therefore there is a natural homomorphism of differential graded algebras ∗ → ∗ ∗ ⊗ • (2.1) i :(CG(M),d+ ι) (C (G, Sg Ω M),dG) given by the inclusion (Sg∗ ⊗ Ω•M)G ⊂ C0(G, Sg∗ ⊗ Ω•M)

∗ • G since the restriction of dG to (Sg ⊗ Ω M) is precisely d + ι because the operators d¯ and ¯ι act trivially on the invariant elements of C0(G, Sg∗ ⊗ Ω•M). The induced map on cohomologies ∗ ∗ → ∗ ∗ ⊗ • i : H (CG(M),d+ ι) H (G, Sg Ω M) R R∗ is far from being an isomorphism as the case of M = pt and G = GL(1, )+ = + shows. ∗ In this case CG(M)=S(gl(1, R)) = R[x]andd + ι = 0, hence H (CG(M)) = R[x] where |x| = 2. On the other hand H∗(G, Sg∗)=H∗(BG,R)=R since BG is contractible. Nevertheless, when the Lie group G is compact, the map i induces an isomorphism in cohomology [10]. Now, in order to understand in more detail the relation between the cohomology of the Cartan model and the equivariant cohomology we will introduce a spectral sequence suited for this purpose.

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 23

2.1.2. A spectral sequence for the equivariant De Rham complex. Let us filter the complex C∗(G, Sg∗ ⊗Ω•M) by the degree in Sg∗ ⊗Ω•M; namely, if we consider maps f : Gk → Sg∗ ⊗ Ω•M with image homogeneous elements of degree l, we will denote deg1(f)=k and deg2(f)=l. Then we can define the filtration

p ∗ ∗ • F := {f ∈ C (G, Sg ⊗ Ω M)|deg2(f) ≥ p} where F p+1 ⊂ F p. We have that the diﬀerentials have the following degrees:

deg1(d)=0 deg2(d)=1

deg1(ι)=0 deg2(ι)=1

deg1(d)=1 deg2(d)=0

deg1(ι)=−1 deg2(ι)=2 and therefore the filtration is compatible with the differentials. The spectral sequence associated to the filtration F ∗ has for page 0: p p+1 ∼ ∗ ∗ • E0 = F /F = C (G, Sg ⊗ Ω M) p and the 0-th differential is d0 = d because the other three differentials raise deg2. Therefore the page 1 is:

∗ ∗ ∗ • E1 = H (C (G, Sg ⊗ Ω M), d) the differentiable cohomology of G with coefficients in the representation Sg∗ ⊗ Ω•M. The 0-th row of the first page is precisely the Cartan complex ∗,0 0 ∗ ∗ ⊗ • ∗ ⊗ • G E1 = H (C (G, Sg Ω M), d)=(Sg Ω M) = CG(M) ∗,0 → ∗+1,0 and the first differential on this row d1 : E1 E1 becomes precisely the Cartan differential d + ι. Therefore we see that on the second page we get that

∗,0 ∼ ∗ E2 = H (CG(M),d+ ι), namely that the 0-th row of the second page is isomorphic to the cohomology of the Cartan complex. Therefore we conclude that the composition ∗,0 → ∗ ∗ ⊗ • E2 H (G, Sg Ω M) ∗,0 → ∗,0 of the surjective homomorphism E2 E∞ with the inclusion ∗,0 ∗ ∗ • E∞ ⊂ H (G, Sg ⊗ Ω M) is equivalent to the induced map on cohomologies ∗ ∗ → ∗ ∗ ⊗ • i : H (CG(M),d+ ι) H (G, Sg Ω M) deﬁned previously.

24 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE

2.2. Differentiable cohomology of Lie groups. Notice that in De Rham model for equivariant cohomology the operator d is defined in a similar fashion as the differential for group cohomology in the case of discrete groups. The cohomology groups defined by the differential d are called the differentiable cohomology groups and are defined for any G-module V ; i.e. if G is a Lie group and V is a G-module then the differentiable cohomology of G with values in V is the cohomology of the ∗ k k → complex Cd (G, V )whereCd (G, V ) consists of differentiable maps f : G V such that f vanishes if any of the arguments gi equals the identity of G,andthe differential is d is defined by k ¯ i (df)(g0,...,gk)=f(g1,...,gk)+ (−1) f(g0,...,gi−1gi,...,gk) i=1 k+1 +(−1) gkf(g0,...,gk−1). We denote this cohomology by ∗ ∗ ∗ Hd (G, V ):=H (C (G, V ), d). For V in the category of topological G-modules, the cohomology groups ∗ Hd (G, V ) can be seen as the relative derived functor associated to the G invariant submodule V G. In particular we have that 0 G Hd (G, V )=V , and whenever G is compact the functor of taking the G-invariant submodule is ∗>0 exact and therefore in that case Hd (G, V )=0. Remark 2.2. The first page of the spectral sequence that was defined in section §2.1.2 ∗ ∗ ∗ • E1 = H (C (G, Sg ⊗ Ω M), d) is precisely the differential cohomology defined above ∗ ∗ ⊗ • E1 = Hd (G, Sg Ω M) for the G-module Sg∗ ⊗ Ω•M. For V a vector space over R, by differentiating the functions from Gk to V , Van Est [15]provedthatforG connected ∗ ∼ ∗ Hd (G; V ) = H (g, k; V ) whenever K is the maximal compact subgroup of G, k and g are their corresponding Lie algebras, and H∗(g, k; V ) denotes the Lie algebra cohomology defined by Chevalley and Eilenberg in [7]. Whenever the group G hasacompactformGu, i.e. a compact Lie group whose complexification is isomorphic to the complexification of G ∼ GC = (Gu)C, then we have that ∗ ∼ ∗ ∼ ∗ H (g, k; V ) = H (gu, k; V ) = H (Gu/K; V ), where the second isomorphism was proved by Chevalley and Eilenberg in [7]for compact Lie groups, and the first isomorphism follows from the isomorphisms ∗ ∼ ∗ ∼ ∗ H (g, k; V ) ⊗R C = H (gC, kC; VC) = H (gu, k; V ) ⊗R C,

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 25 ∼ which follow from the isomorphism of complex Lie algebras gC = (gu)C.Inthiscase we have that the diﬀerentiable cohomology of G can be calculated by topological methods, i.e. ∗ ∼ ∗ Hd (G; V ) = H (Gu/K; V ). 2.2.1. Example G = SL(n, R). Let us consider the non-compact group G = SL(n, R). In this case we have

g = sl(n, R) gC = sl(n, C)

gu = su(n) k = so(n) kC = so(n, C) K = SO(n)

Gu = SU(n). Therefore for n>2wehavethat ∗ R R ∗ R Hd (SL(n, ), )=H (SU(n)/SO(n); )=Λ[h3,h5, ..., hn ], where the degree of hi is 2i − 1andn is the largest odd integer which is less or equal than n. The equivariant De Rham cohomology of this group is H∗(G, Sg∗). The spectral sequence defined in section §2.1.2 has for first page ∗ ∗ ∼ ∗ ∗ E1 = Hd (G, Sg ) = H (g, k; Sg ) and since the algebra g = sl(n, R) is reductive and Sg∗ is a finite dimensional semi-simple g-module in each degree [2], then we have that

∗ ∗ ∼ ∗ ∗ G H (g, k; Sg ) = H (g, k; R) ⊗ (Sg ) and therefore ∼ ∗ G E1 = Hd(G; R) ⊗ (Sg ) . The ideal of G-invariant polynomials is known to be

∗ G ∼ (Sg ) = R[c2,c3, ..., cn], where the degree of ci is 2i, and so we get that the ﬁrst page of the spectral sequence converging to H∗(G, Sg∗)is ∼ E1 = Λ[h3,h5, ..., hn ] ⊗ R[c2,c3, ..., cn]. Since we know that for n>2 ∗ ∗ ∗ ∗ ∼ H (G, Sg )=H (BG; R)=H (BSO(n); R) = R[c2,c4, ...c2[n/2]] is the free algebra on the Pontrjagin classes, we obtain that the (2i)-th diﬀerential of the spectral sequence maps the class hi to the class ci

d2i hi → ci. In particular we obtain that R 4 ∗ ∼ 4,0 ∼ 4,0 2 ∗ G (2.2) = H (G, Sg ) = E∞ = E1 =(S g ) , and therefore we see that the fourth cohomology group of BG is generated by the G-invariant quadratic forms in (S2g∗)G.

26 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE

3. Equivariant extension of differential forms In many instances in geometry, the action of a compact Lie group on a manifold being of a certain kind is equivalent to the existence of an equivariant lift of a specific invariant closed differential form on the Cartan model. Some examples of this phenomenon are the following: • The action of G on a symplectic manifold (M,ω) being Hamiltonian is a equivalent to the existence of a closed equivariant liftω ˜ = ω + μaΩ ∈ 2 → R CG(M) of the symplectic form; in this case the maps μa : M may ∞ be assembled into a map μ : g → C M, μ(a):=μa, which becomes the moment map (see [1]). ∗ • The action of G on an exact Courant algebroid (TM ⊕ T M,[, ]H ), with Courant-Dorfman bracket twisted by the closed three form H, is Hamil- a tonian provided there exists a closed equivariant lift H˜ = ω + ξaΩ ∈ 3 CG(M)(see[4, 6, 12, 14]). • Let Γ be a connected, simple, simply connected and compact matrix group, i.e. Γ ⊂ GL(N,R), and denote by ΓL × ΓR the product of two copies of Γ acting on Γ on the left by the action

(ΓL × ΓR) × Γ → Γ ((g, h),k) → gkh−1.

A subgroup G ⊂ ΓL × ΓR is called an anomaly-free subgroup if there the WZW action could be gauged with respect to the group action given by G.In[16] Witten showed that the anomaly-free subgroups are precisely the subgroups G on which the WZW term 1 ω = Tr(g−1dg)3 12π could be lifted to a closed equivariant 3-form in the Cartan complexω ˜ = − a ∈ 3 ω λaΩ CG(Γ). (We will elaborate this construction in the next chapter). The previous examples are not exhaustive, but they give the idea of the general principle. In these cases we have a G-invariant closed form ω and we need to find a closed equivariant liftω ˜. Note that since the group G is compact, the existence of a closed equivariant lift in the Cartan model is equivalent to the existence of a lift of the cohomology class [ω] ∈ H∗(M; R) on the equivariant cohomology ∗ H (EG ×G M; R). Therefore the obstructions of the existence of the equivariant lift could be studied via several methods, for instance, with the use of the Serre spectral sequence associated to the fibration M → EG ×G M → BG,orwiththe use of the spectral sequence associated to a filtration of the Cartan complex CG(M) given by the degree of Sg∗ as it is done in the papers [8, 17]. But what happens in the case that the Lie group G is not compact? We speculate that the situation should be similar, in the sense that the action being of certain kind is equivalent to the existence of a lift on the equivariant De Rham complex defined in section §2.1. This situation has not been explored so far but we believe that the equivariant De Rham complex provides a framework in which actions on non-compact Lie groups could be better understood. Since in many instances the geometric information is captured by closed forms in the Cartan complex, we would like to study the relation between the cohomology

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 27 of the Cartan complex and the equivariant De Rham cohomology whenever the group is not compact.

3.1. Cartan complex vs. equivariant De Rham complex. Let us consider the diagram of complexes

C∗ (M) i /C∗(G, Sg∗ ⊗ Ω•M) G m mmm mmm mmm vmmm Ω•M where the horizontal map is the injective map of complexes defined in (2.1) and the vertical maps are the natural forgetful maps. Let us take a closed G-invariant form H on M and let us suppose that we can lift this closed form to a closed form in the Cartan model H ∈ CG(M), then equivariant form iH becomes a closed lift for H in the complex of equivariant De Rham forms, i.e. iH ∈ C∗(G, Sg∗ ⊗ Ω•M). We have then Lemma 3.1. Take a closed G-invariant form H ∈ (Ω•M)G.IfH can be lifted toaclosedformH in the Cartan model of the equivariant cohomology, then the form i(H) is a closed lift for H in C∗(G, Sg∗ ⊗ Ω•M), the closed forms of the equivariant De Rahm complex. Note in particular that Lemma 3.1 implies that if one can extend an invariant closed form to a closed form in the Cartan model for equivariant cohomology, then the cohomology class [H] lies in the image of the canonical forgetful homomorphism H∗(G, Sg∗ ⊗ Ω•M) → H∗(M), and therefore the cohomology class [H] could be extended to an equivariant cohomology class in any model for the equivariant cohomology of M. The converse of Lemma 3.1 would say that if one knows that an invariant differentiable form H could be lifted to a closed form in C∗(G, Sg∗ ⊗ Ω•M)then a lift could be written as a closed element in the Cartan model. The converse of Lemma 3.1 is indeed true whenever the Lie group G is compact, but for general group actions it does not hold. From the spectral sequence defined in section §2.1.2 we have seen that in the first page we have ∗,0 ∼ ∗ ⊗ • G E1 = (Sg Ω M) = CG(M) ∗,0 → ∗+1,0 with differential d1 : E1 E1 equivalent to d + ι. Therefore on the second page we get that ∗,0 ∼ ∗ E2 = H (CG(M),d+ ι), namely that the 0-th row of the second page is isomorphic to the cohomology of the Cartan complex. ∗,0 → ∗,0 Since we have the surjective homomorphism E2 E∞ we can conclude that Proposition 3.2. For a closed G-invariant form H ∈ (Ω•M)G, it can be lifted to a closed form in the Cartan complex if, firstly the cohomology class [H] could be lifted to an equivariant cohomology class in H∗(G, Sg∗ ⊗ Ω•M), and secondly, if ∗,0 ∗ ∗ • the lift lies on the subgroup E∞ ⊂ H (G, Sg ⊗ Ω M).

28 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE

The second condition of Proposition 3.2 is more difficult to check than the first one, since it depends explicitly on the equivariant De Rham model for equivariant cohomology; for the first condition any model for the equivariant cohomology works. In certain specific situations, extensions in the Cartan model of closed forms may be obtained, and this is the subject of the next and final chapter. We note here that a sequence of obstructions for lifting a G-invariant form to a closed form in the Cartan complex can be determined, when studying the spectral sequence associated to appropriate filtrations of the Cartan complex, as it is carried out in [8, 17]. Our approach is different since we are interested in using the fact that an extension in the Cartan model can only exist if there is an equivariant extension, i.e. an extension in the homotopy quotient.

4. Equivariant extensions of the WZW term for SL(n, R) actions In the physics literature (see [16] and the references therein) it has been argued that the condition of anomaly cancelation for the gauged WZW action is given by the equation

Tr(Ta,LTb,L − Ta,RTb,R)=0 and this equation is moreover equivalent to the existence of an equivariant extension of the WZW term 1 ω = Tr(g−1dg)3 12π on the Cartan model for equivariant cohomology. In the case that the group that we are gauging is compact, the anomaly cancellation is equivalent to the existence of an equivariant lift of the cohomology class [ω], and therefore the anomaly cancellation becomes topological and could be checked with topological methods. In this section we study in detail the case on which the gauge group is G = SL(n, R) (or any subgroup of it) and we show that the anomaly cancelation condition for the WZW action is also topological and only depends on the existence of an equivariant lift of the cohomology class [ω]; in this way we ﬁnd a large family of SL(n, R) actions with anomaly cancellation. Let us start by recalling the explanation given by Witten [16]thatassertsthat the condition for anomaly cancelation is equivalent to the existence of a lift in the Cartan complex of the form ω.

4.1. Gauged WZW actions. Let Γ be a connected and simple matrix group, i.e. Γ ⊂ GL(N,R), such that its fundamental group is ﬁnite. Denote by ΓL × ΓR the product of two copies of Γ acting on Γ on the left by the action

(ΓL × ΓR) × Γ → Γ ((g, h),k) → gkh−1 and consider a subgroup G ⊂ ΓL ×ΓR acting on the left on Γ by the induced action of ΓL × ΓR. The embedding G ⊂ ΓL × ΓR determines a map at the level of Lie algebras which can be written as

a → (Ta,L,Ta,R),a∈ g

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 29 and the canonical vector fields Xa on Γ generated by the G action could be written as (Xa)g = Ta,Lg − gTa,R for all g ∈ Γ. The matrix 1-forms g−1dg and dgg−1 satisfy the equations −1 −1 − ιXa (g dg)=g Ta,Lg Ta,R, −1 − −1 ιXa (dgg )=Ta,L g Ta,Rg, d(dgg−1)2p+1 = −(dgg−1)2p+2, d(g−1dg)2p+1 =(g−1dg)2p+2 and we can take the differential form 1 ω = Tr(g−1dg)3, 12π 3 which defines the WZW action. The form ω ∈ Ω ΓisΓL × ΓR invariant, therefore ω ∈ (Ω3Γ)G, it is closed and is a generator of the cohomology group H3(Γ) = R. To find a closed extension of ω in the Cartan complex we need to find 1-forms λa such that the following equations are satisfied: − ιXa ω dλa =0,

ιXa λb + ιXb λa =0, L Xb λa = λ[b,a], a where the first two imply that the formω ˜ = ω − λaΩ is (d + ι)-closed, and the third one implies thatω ˜ is G-invariant. Calculating ι ω we obtain Xa ι Tr(g−1dg)3 =3Tr (g−1T g − T )(g−1dg)2 Xa a,L a,R =3Tr T (dgg−1)2 − T (g−1dg)2 a,L a,R −1 −1 = d 3Tr Ta,L(dgg )+Ta,R(g dg) and therefore we see that we can define 1 λ = Tr T (dgg−1)+T (g−1dg) a 4π a,L a,R − L satisfying the equation ιXa ω dλa =0.Thefactthat Xb λa = λ[b,a] is satisfied, is a tedious but straightforward computation. Now we compute 1 ι λ + ι λ = Tr (T T − T T ) Xa b Xb a 2π a,L b,L a,R b,R noting that for a, b ∈ g the function thus defined become constant, and therefore we have that a ∗ • G ω˜ = ω − λaΩ ∈ (Sg ⊗ Ω Γ) is a 3-form in the Cartan complex CG(Γ) and its failure to be closed is the quadratic form (d + ι)˜ω ∈ (S2g∗)G, where the coefficient of ΩaΩb of this quadratic form is precisely 1 − Tr (T T − T T ) . 2π a,L b,L a,R b,R

30 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE

Since it was known in the literature that the condition for the absence of anomalies was

Tr (Ta,LTb,L − Ta,RTb,R)=0, Witten concluded that the absence of anomalies was equivalent to the existence of a closed extension of ω in the Cartan complex. Whenever the Lie group G is compact, the existence of such an extension is equivalent to the existence of an equivariant extension of the cohomology class [ω], and this can be checked with the use of the Serre spectral sequence associated to the ﬁbration

Γ → EG ×G Γ → BG. The second page of the spectral sequence becomes p,q p q R ∼ p R ⊗ q R E2 = H (BG; H (Γ; )) = H (BG; ) H (Γ; ), and since H1(Γ; R)=H2(Γ; R) = 0, the only non-trivial differential that affects ∈ 0,3 [ω] E2 is d4 thus defining an element ∈ 4,0 ∼ 4 R d4([ω]) E2 = H (BG; ); this implies that the only obstruction to lift [ω] to an equivariant class is precisely d4([ω]). Since we assumed that G is compact, we know that 4 ∼ 2 ∗ G H (BG; R) = (S g ) and therefore we must have that

d4([ω]) = (d + ι)˜ω, namely that the two obstructions are the same. The previous argument permits to find several cases on which there is anomaly cancellation. The simplest of all is the adjoint action of G on itself Γ = G since in this case the spectral sequence associated to the fibration EG ×G Gad → BG always collapses at the second page, and therefore d4 =0. 4.2. WZW actions with gauge group G = SL(n, R). In this section we will argue that if the gauge group is G = SL(n, R) then the cancellation of anomalies is topological, and therefore it is equivalent to the existence of an equivariant lift of the cohomology class [ω]. Theorem 4.1. Let Γ be a connected, simple matrix group with finite fundamental group. Let G = SL(n, R) and consider an action on Γ defined by an injection 3 G ⊂ ΓL × ΓR. The existence of an equivariant extension on H (EG ×G Γ; R) of the cohomology class [ω] is equivalent to the existence of a closed lift to the Cartan model of ω. Proof. We know that if there is an extensionω ˜ in the Cartan model, then the cohomology class [˜ω] represents the lift in the cohomology group H3(G, Sg∗ ⊗Ω•Γ). To prove the converse we will make use of the constructions and results of sections §2.2.1, §3and§4.1. ∗ ∼ ∗ From the equivariant De Rham theorem we know that H (BG; R) = H(G, Sg ) and therefore we could take the class d4([ω]), which is the obstruction of extending [ω] to an equivariant class, to be an element in 4 ∗ d4([ω]) ∈ H (G, Sg ).

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 31

We already know that (d + ι)˜ω ∈ (S2g∗)G and its cohomology class in H4(G, Sg∗) represents the same obstruction for an equivariant lift, i.e.

d4([ω]) = [(d + ι)˜ω]. In general it may happen that the cohomology class [(d + ι)˜ω] is zero even though the form (d + ι)˜ω is diﬀerent from zero. But in the particular case of G = SL(n, R) we have already seen in (2.2) that the inclusion map of the Cartan complex into the equivariant De Rham complex (Sg∗)G → C∗(G, Sg∗) induces an isomorphism in degree 4 ∼ (S2g∗)G →= H4(G, Sg∗), (d + ι)˜ω → [(d + ι)˜ω] and therefore we have that the vanishing of the class d4([ω]) is equivalent to the vanishing of the quadratic form (d + ι)˜ω, i.e.

d4([ω]) = 0 if and only if (d + ι)˜ω =0.

We see that for the case on which the gauge group is SL(n, R), the equations of cancellation of anomalies, namely that for all a, b ∈ sl(n, R)

Tr (Ta,LTb,L − Ta,RTb,R)=0, 3 are equivalent to the existence of an equivariant extension on H (EG ×G Γ; R)of the cohomology class [ω]. Now we are ready to give examples on both the existence and the non existence of equivariant extensions of ω. 4.3. Examples. 4.3.1. Adjoint action. Let G = Γ and consider the diagonal injection

G ⊂ ΓL × ΓR,g→ (g, g) which induces the adjoint action of G on Γ = Gad. In this case the cohomology of ad the homotopy quotient EG ×G G is isomorphic to the cohomology of G tensor the cohomology of BG: ∗ ad ∼ ∗ ∗ H (EG ×G G ; R) = H (BG; R) ⊗ H (G; R). This isomorphism can be obtained from the Serre spectral sequence associated to the fibration ad G → EG ×G G → BG, which collapses at level 2 because the classes in H∗(G; R) can be lifted to classes ∗ ad ∗ to H (EG ×G G ; R): take a primitive class in H (BG; R) (namely a class in H∗(BG; R) which is in the image of a primitive element in H∗(G)ofoneofthe differentials of the Serre Spectral Sequence associated to the fibration G → EG → BG) pull it back to S1 ×LBG via the evaluation map where LBG is the space of free loops of BG, then integrate over S1 and get a class in H∗(LBG; R)ofdegree 1 less; this class in H∗(LBG; R) once restricted to the based loops ΩBG G is precisely the class in H∗(G) that defined the primitive class in H∗(BG)thatwe ad started with; finally recall that LBG EG ×G G . If we take G =Γ=SL(n, R), we know by Theorem 4.1 that the existence of an equivariant extension of [ω] is equivalent to the cancelation of anomalies for this

32 H. GARC´IA-COMPEAN,´ P. PANIAGUA, AND B. URIBE gauged action, and since the class [ω] can be extended to an equivariant one, we conclude that in this case there is an anomaly cancelation. In particular, for any subgroup F ⊂ SL(n, R) acting by the adjoint action on SL(n, R) there is also cancelation of anomalies. 4.3.2. G = SL(n, R) ⊂ ΓL for n>2. Whenever the action of G = SL(n, R)on Γ is obtained by a left action induced by an inclusion G = SL(n, R) ⊂ ΓL,wehave that the G action on Γ is free and therefore the homotopy quotient EG ×G Γand the quotient G\Γ are homotopy equivalent. For n>2 the Serre spectral sequence tells us that 4 d4([ω]) = c2 ∈ H (BSL(n, R); R) and therefore the class [ω] does not extend to an equivariant one. By Theorem 4.1 we know that this implies that there is no cancellation of anomalies. We conclude that for left free actions of the group SL(n, R)forn>2 there must exist a, b ∈ sl(n, R) such that

Tr (Ta,LTb,L − Ta,RTb,R) =0 .

4.3.3. G = SL(2, R) ⊂ ΓL. Following the same argument as before, we have 4 that d4([ω]) = 0 since H (BSL(2, R), R) = 0. Therefore for left free actions of the group SL(2, R) there is anomaly cancelation.

References [1] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28, DOI 10.1016/0040-9383(84)90021-1. MR721448 (85e:58041) [2] M. Berger et al. Séminaire “Sophus Lie” de l’Ecole Normale Supérieure, 1954/1955. Théorie des algèbres de Lie. Topologie des groupes de Lie. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1955. [3] R. Bott, H. Shulman, and J. Stasheff, On the de Rham theory of certain classifying spaces, Advances in Math. 20 (1976), no. 1, 43–56. MR0402769 (53 #6583) [4] Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765, DOI 10.1016/j.aim.2006.09.008. MR2323543 (2009d:53124) [5] Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibréprincipal (French), Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 57–71. MR0042427 (13,107f) [6] Alexander Caviedes, Shengda Hu, and Bernardo Uribe, Chern-Weil homomorphism in twisted equivariant cohomology, Differential Geom. Appl. 28 (2010), no. 1, 65–80, DOI 10.1016/j.difgeo.2009.09.002. MR2579383 (2011d:55011) [7] Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR0024908 (9,567a) [8] José M. Figueroa-O’Farrill and Sonia Stanciu, Gauged Wess-Zumino terms and equivariant cohomology, Phys. Lett. B 341 (1994), no. 2, 153–159, DOI 10.1016/0370-2693(94)90304-2. MR1311654 (96h:81069) [9] Hugo Garc´ıa-Compeán and Pablo Paniagua, Gauged WZW models via equivariant cohomology, Modern Phys. Lett. A 26 (2011), no. 18, 1343–1352, DOI 10.1142/S0217732311035754. MR2812789 (2012g:81188) [10] Ezra Getzler, The equivariant Chern character for non-compact Lie groups, Adv. Math. 109 (1994), no. 1, 88–107, DOI 10.1006/aima.1994.1081. MR1302758 (95j:57028) [11] Victor W. Guillemin and Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]. MR1689252 (2001i:53140) [12] Shengda Hu, Reduction and duality in generalized geometry, J. Symplectic Geom. 5 (2007), no. 4, 439–473. MR2413310 (2009m:53217)

EQUIVARIANT EXTENSIONS OF DIFFERENTIAL FORMS 33

[13] P. Paniagua. Anomal´ıas, Teor´ıa de Wess-Zumino-Witten y cohomolog´ıa equivariante.PhD thesis, CINVESTAV, M´exico, 2010. [14] Bernardo Uribe, Group actions on DG-manifolds and exact Courant algebroids, Comm. Math. Phys. 318 (2013), no. 1, 35–67, DOI 10.1007/s00220-013-1669-2. MR3017063 [15] W. T. van Est, Group cohomology and Lie algebra cohomology in Lie groups. I, II,Ned- erl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 (1953), 484–492, 493–504. MR0059285 (15,505b) [16] Edward Witten, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. 144 (1992), no. 1, 189–212. MR1151251 (93i:81259) [17] Siye Wu, Cohomological obstructions to the equivariant extension of closed invariant forms, J. Geom. Phys. 10 (1993), no. 4, 381–392, DOI 10.1016/0393-0440(93)90005-Y. MR1218557 (94g:55010)

Departamento de F´ısica, Centro de Investigacion´ y de Estudios Avanzados, Av. I.P.N 2508, Zacatenco, Delegacion´ Gustavo A. Madero, C.P. 07360, Mexico´ D.F., Mexico´ E-mail address: [email protected] Departamento de Matematicas,´ Escuela Superior de F´ısica y Matematicas´ del In- stituto Politecnico´ Nacional, Unidad Adolfo Lopez´ Mateos, Edificio 9, 07738, Mexico´ D.F., Mexico´ E-mail address: [email protected] Departamento de Matematicas,´ Universidad de los Andes, Carrera 1 N. 18A - 10, Bogota,´ Colombia E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12415

From Classical Theta Functions to Topological Quantum Field Theory

R˘azvan Gelca and Alejandro Uribe

Abstract. Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrödinger representation is obtained as an action on curves in a handlebody. A careful analysis of the discrete Fourier transform yields the Murakami-Ohtsuki-Okada formula for invariants of 3-dimensional manifolds. In this context, we give an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.

1. Introduction In this paper we construct the abelian Chern-Simons topological quantum field theory directly from the theory of classical theta functions, without the insights of quantum field theory. It has been known for years, within abelian Chern-Simons theory, that classical theta functions relate to low dimensional topology [2], [35]. Abelian Chern-Simons theory is considerably simpler than its non-abelian counterparts, and has been studied thoroughly (see e.g. [18], [19]). Here we do not start with abelian Chern- Simons theory, but instead give a direct construction of the associated topological quantum field theory based on the theory of theta functions, and arrive at skein modules from representation theoretical considerations. We consider theta functions in the representation theoretic point of view of AndréWeil[33]. As such, the space of theta functions is endowed with an action of a finite Heisenberg group (the Schrödinger representation), which induces, via a Stone-von Neumann theorem, the Hermite-Jacobi action of the modular group. All this structure is what we shall mean by the theory of theta functions. We show how the group algebra of the finite Heisenberg group and its Schrö- dinger representation on the space of theta functions, lead to algebras of curves on

2010 Mathematics Subject Classiﬁcation. 14K25, 57R56, 57M25, 81S10, 81T45. Research of the ﬁrst author was supported by the NSF, award No. DMS 0604694. Research of the second author was supported by the NSF, award No. DMS 0805878.

c 2014 American Mathematical Society 35

36 RAZVAN˘ GELCA AND ALEJANDRO URIBE surfaces and their actions on spaces of curves in handlebodies. These notions are formalized using skein modules. The Hermite-Jacobi representation of the modular group on theta functions is a discrete analogue of the metaplectic representation. The modular group acts by automorphisms that can be interpreted as discrete Fourier transforms. We show that these discrete Fourier transforms can be expressed as linear combina- tions of curves. A careful analysis of their structure and of their relationship to the Schrödinger representation yields the Murakami-Ohtsuki-Okada formula [21]of invariants of 3-manifolds. As a corollary of our discussion we obtain an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds obey the same formula. The paper uses results and terminology from the theory of theta functions, quantum mechanics, and low dimensional topology. To make it accessible to different audiences we include a fair amount of detail. A more detailed discussion of these ideas can be found in [8]. Section 2 reviews the theory of theta functions on the Jacobian variety of a surface. The action of the finite Heisenberg group on theta functions is defined via Weyl quantization of the Jacobian variety in a Kähler polarization. In fact it has been found recently that Chern-Simons theory is related to Weyl quantization [10], [1], and this was the starting point of our paper. The next section exhibits the representation theoretical model for theta functions. In Section 4 we show that this model for theta functions is topological in nature, and reformulate it using algebras of curves on surfaces, together with their action on skeins of curves in handlebodies which are associated to the linking number. In Section 5 we derive a formula for the discrete Fourier transform as a skein. This formula is interpreted in terms of surgery in the cylinder over the surface. Section 6 analizes the exact Egorov identity which relates the Hermite-Jacobi action to the Schrödinger representation. This analysis shows that the topological operation of handle slides is allowed over the skeins that represent discrete Fourier transforms, and this yields in the next section the abelian Chern-Simons invariants of 3-manifolds defined by Murakami, Ohtsuki, and Okada. We point out that the above-mentioned formula was introduced in an ad-hoc manner by its authors [21], our paper derives it naturally. Section 8 shows how to associate to the discrete Fourier transform a 4-dimensional manifold, and explains why the cocycle of the Hermite-Jacobi action is related to that governing the non-additivity of the signature of 4-manifolds [32]. Section 9 should be taken as a conclusion; it puts everything in the context of Chern-Simons theory.

2. Theta functions

We start with a closed genus g Riemann surface Σg, and consider a canonical basis a1,a2,...,ag,b1,b2,...,bg of H1(Σg, R), like the one in Figure 1. To it we associate a basis in the space of holomorphic differential 1-forms ζ1,ζ2,...,ζg,defined by the conditions ζj = δjk, j, k =1, 2,...,g. The matrix Π with entries ak πjk = ζj , j, k =1,...,g, is symmetric with positive definite imaginary part. bk ThismeansthatifΠ=X + iY ,thenX = XT , Y = Y T and Y>0. The g × 2g matrix (Ig, Π) is called the period matrix of Σg, its columns λ1,λ2,...,λ2g, called

FROM THETA FUNCTIONS TO TQFT 37

a ag a1 2

b b g 1 b2

Figure 1

g 2g periods, generate a lattice L(Σg)inC = R . The complex torus

g J (Σg)=C /L(Σg) is the Jacobian variety of Σg.Themap αj aj + βj bj → (α1,...,αg,β1,...,βg) j j induces a homeomorphism H1(Σg, R)/H1(Σg, Z) →J(Σg). g The complex coordinates z =(z1,z2,...,zg)onJ (Σg) are inherited from C . We introduce real coordinates (x, y)=(x1,x2,...,xg,y1,y2,...,yg) by imposing z = x +Πy. A fundamental domain for the period lattice in terms of the (x, y) 2g coordinates is {(x, y) ∈ [0, 1] }. J (Σg) has the canonical symplectic form,

g ω =2π dxj ∧ dyj . j=1

J (Σg) with the complex structure and symplectic form is a Kähler manifold. ∞ The symplectic form induces a Poisson bracket on C (J (Σg)), given by {f,g} = ω(Xf ,Xg), where Xf is the Hamiltonian vector field defined by df (·)=ω(Xf , ·).

Classical theta functions arise when quantizing J (Σg)inaKähler polarization in the direction of this Poisson bracket. In this paper we perform the quantization in the case where Planck’s constant is the reciprocal of an even positive integer: 1 ∈ N h = N where N =2r, r . The Hilbert space of the quantization consists of the holomorphic sections of a line bundle obtained as the tensor product of a line bundle with curvature Nω and the square root of the canonical line bundle. The latter is trivial for the complex torus and we ignore it. The line bundle with curvature Nω is the tensor product of a flat line bundle and the line bundle defined g ∗ by the cocycle Λ : C × L(Σg) → C ,

−2πiNzj −πiNπjj Λ(z,λj )=1, Λ(z,λg+j )=e , j =1, 2,...,g. (See e.g. §4.1.2 of [5] for a discussion of how this cocycle gives rise to a line bundle with curvature Nω.) We choose the trivial ﬂat bundle to tensor with. Then the Hilbert space can be identiﬁed with the space of entire functions on Cg satisfying the periodicity conditions

−2πiNzj −πiNπjj f(z + λj )=f(z),f(z + λg+j )=e f(z).

38 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Π 1 We denote this space by ΘN (Σg); its elements are called classical theta functions. Π A basis of ΘN (Σg) consists of the theta series 1 μ T μ μ T Π 2πiN[ 2 ( N +n) Π( N +n)+( N +n) z] ∈{ − }g θμ (z)= e ,μ 0, 1 ...,N 1 . n∈Zg The definition of theta series will be extended for convenience to all μ ∈ Zg,by g ∈ Zg Z θμ+Nμ = θμ for any μ . Hence the index μ is taken in N . The inner product that makes the theta series into an orthonormal basis is T (2.1) f,g =(2N)g/2 det(Y )1/2 f(x, y)g(x, y)e−2πNy Yydxdy. [0,1]2g That the theta series form an orthonormal basis is a corollary of the proof of Proposition 2.1 below. To define the operators of the quantization, we use the Weyl quantization method. This quantization method can be defined only on complex vector spaces, the Jacobian variety is the quotient of such a space by a discrete group, and the quantization method goes through. As such, the operator Op(f) associated to a − hΔΠ function f on J (Σg) is the Toeplitz operator with symbol e 4 f ([7] Proposition 2 2.97), where ΔΠ is the Laplacian on functions, ∗ ∞ 1 ΔΠ = −d ◦ d, d : C (J (Σg)) → Ω (J (Σg)). On a general Riemannian manifold this operator is given in local coordinates by the formula 1 ∂ jk ∂f ΔΠf = g det(g) , det(g) ∂xj ∂xk −1 jk where g =(gjk)isthemetricandg =(g ). In the Kähler case, if the Kähler form is given in holomorphic coordinates by i ω = h dz ∧ dz¯ , 2 jk j k j,k then 2 jk ∂ ΔΠ =4 h , ∂zj ∂z¯k j,k jk −1 where (h )=(hjk) . In our situation, in the coordinates zj , z¯j , j =1, 2,...,g, −1 −1 jk one computes that (hjk) = Y and therefore (h )=Y (recall that Y is the imaginary part of the matrix Π). For Weyl quantization one introduces a factor 1 of 2π in front of the operator. As such, the Laplace (or rather Laplace-Beltrami) operator ΔΠ is equal to g −1 ∇ − −1∇ − −1 ∇ −1∇ Yjk (Ig + iY X) x iY y j (Ig iY X) x + iY y k . j, k=1 (A word about the notation being used: ∇ represents the usual (column) vector of partial derivatives in the indicated variables, so that each object in the square

1The precise terminology is canonical theta functions, classical theta functions being deﬁned by a slight alteration of the periodicity condition.Weusethenameclassicalthetafunctionsto emphasize the distinction with the non-abelian theta functions. 2Thevariableoff is not conjugated because we work in the momentum representation.

FROM THETA FUNCTIONS TO TQFT 39 brackets is a column vector of partial derivatives. The subindices j, k are the corresponding components of those vectors.) A tedious calculation that we omit results in the following formula for the Laplacian in the (x, y) coordinates: −1 ∂2 −1 ∂2 jk ∂2 ΔΠ = (Y + XY X)jk − 2(XY )jk + Y . ∂xj ∂xk ∂xj ∂yk ∂yj ∂yk

We will only need to apply ΔΠ explicitly to exponentials, as part of the proof of the following basic proposition. Note that the exponential function

T T e2πi(p x+q y) deﬁnes a function on the Jacobian provided p, q ∈ Zg. Proposition 2.1. The Weyl quantization of the exponentials is given by T T − πi T − 2πi T 2πi(p x+q y) Π N p q N μ q Π Op e θμ (z)=e θμ+p(z). Proof. Let us introduce some useful notation local to the proof. Note that N and Π are ﬁxed throughout. (1) e(t):=exp(2πiNt), ∈ Zg ∈{ − }g μ (2) For n and μ 0, 1,...N 1 , nμ := n + N . 1 T (3) Q(nμ):= 2 (nμ Πnμ) 2πi(pT x+qT y) 1 T T (4) Ep,q(x, y)=e = e( N (p x + q y)). With these notations, in the (x, y)coordinates T θμ(x, y)= e(Q(nμ)) e(nμ (x +Πy)). n∈Zg

We ﬁrst compute the matrix coeﬃcients of the Toeplitz operator with symbol Ep,q, namely Ep,qθμ ,θν ,whichis g/2 1/2 −2πNyT Yy (2N) det(Y ) Ep,q(x, y)θμ(x, y)θν (x, y) e dxdy. [0,1]2g Then a calculation shows that

Ep,q(x, y)θμ(x, y)θν (x, y) qT = e Q(n ) − Q(m )+(n − m )T x + + nT Π − mT Π y . μ ν μ+p ν N μ ν m,n∈Zg The integral over x ∈ [0, 1]g of the (m, n) term will be non-zero iﬀ

N nμ+p − mν = μ + p − ν + N(n − m)=0, in which case the integral will be equal to one. Therefore Ep,qθμ ,θν = 0 unless Zg [ν]=[μ + p], where the brackets represent equivalence classes in N . This shows that the Toeplitz operator with multiplier Ep,q maps θμ to a scalar times θμ+p.We now compute the scalar. { ··· − }g Zg Taking μ in the fundamental domain 0, 1, ,N 1 for N , there is a unique representative, ν,of[μ + p] in the same domain. This ν is of the form ν = μ + p + Nκ for a unique κ ∈ Zg. With respect to the previous notation, κ = n − m.

40 RAZVAN˘ GELCA AND ALEJANDRO URIBE

It follows that g/2 1/2 Ep,qθμ ,θν =(2N) det(Y ) e Q(nμ) − Q(mν ) g ∈Zg [0,1] n qT + + nT Π − mT Π y + iyT Yy dy, N μ ν

− 1 where m = n κ in the nth term. Using that mν = nμ + N p, one obtains 1 1 Q(n ) − Q(m )=inT Yn − pT Πn − Q(p) μ ν μ μ N μ N 2 and 1 nT Π − mT Π=2inT Y − pT Π, μ ν μ N and so we can write g/2 1/2 − 1 Ep,qθμ ,θν =(2N) det(Y ) e 2 Q(p) dy N g ∈Zg [0,1] n 1 1 1 e inT Yn − pT Πn + qT +2inT Y − pT Π y + iyT Yy . μ μ N μ N μ N

Making the change of variables w := y + nμ in the summand n, the argument of the function e can be seen to be equal to 1 1 iwT Yw+ qT − pT Π w − qT n . N N μ Since q and n are integer vectors, 1 T e qT n = e−2πiq μ/N . N μ The dependence on n of the integrand is a common factor that comes out of the summation sign. The series now is of integral over the translates of [0, 1]n that tile the whole space. Therefore Ep,qθμ ,θν is equal to g/2 1/2 − 1 −2πiqT μ/N −2πNwT Yw+2πi qT −pT Π w (2N) det(Y ) e 2 Q(p) e e dw. N Rg

A calculation of the integral3 yields that it is equal to g/2 1 −1/2 − π (qT −pT Π)Y −1(q−Πp) det(Y ) e 2N . 2N and so

− πi pT Πp −2πiqT μ/N − π (qT −pT Π)Y −1(q−Πp) Ep,qθμ ,θν = e N e e 2N .

− T T g 1/2 1 T −1 3 x Ax+b x π 4 b A b Rg e dx = det A e

FROM THETA FUNCTIONS TO TQFT 41

The exponent on the right-hand side is (−π/N)times 1 2iqT μ + ipT (X − iY )p + [qT − pT (X − iY )]Y −1[q − (X − iY )p] 2 1 =2iqT μ + ipT (X − iY )p + [qT Y −1 − pT XY −1 + ipT ][q − Xp + iY p] 2 1 =2iqT μ + ipT (X − iY )p + qT Y −1q − 2qT Y −1Xp +2iqT p 2 1 +pT XY −1Xp − 2ipT Xp − pT Yp =2iqT μ + iqT p + R 2 where R := qT Y −1q − 2qT Y −1Xp + pT (XY −1X + Y )p. That is, − 2πi qtμ− πi qT p− π R (2.2) Ep,qθμ ,θν = e N N 2N . 2 On the other hand, it is easy to check that ΔΠ(Ep,q)=−(2π) REp,q, and therefore − ΔΠ π R e 4N (Ep,q)=e 2N Ep,q, so that, by (2.2) − ΔΠ − 2πi qtμ− πi qT p e 4N (Ep,q)θμ ,θν = e N N , as desired. Let us focus on the group of quantized exponentials. First note that the symplectic form ω induces a nondegenerate bilinear form on R2g, which we denote also by ω,givenby g pj qj (2.3) ω((p, q), (p ,q )) = . p q j=1 j j As a corollary of Proposition 2.1 we obtain the following result. Proposition 2.2. Quantized exponentials satisfy the multiplication rule T T T T Op e2πi(p x+q y) Op e2πi(p x+q y)

πi ω((p,q),(p,q)) 2πi((p+p)T x+(q+q)T y) = e N Op(e ). This prompts us to deﬁne the Heisenberg group H(Zg)={(p, q, k),p,q∈ Zg,k ∈ Z} with multiplication (p, q, k)(p,q,k)=(p + p,q+ q,k+ k + ω((p, q), (p,q))).

This group is a Z-extension of H1(Σg, Z), with the standard inclusion of H1(Σg, Z) into it given by pj aj + qkbk → (p1,...,pg,q1,...,qg, 0). The map πi k 2πi(pT x+qT y) (p, q, k) → Op e N e deﬁnes a representation of H(Zg) on theta functions. To make this representation faithful, we factor it by its kernel.

42 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Proposition 2.3. The set of elements in H(Zg) that act on theta functions as identity operators is the normal subgroup consisting of the Nth powers of elements of the form (p, q, k) with k even. The quotient group is isomorphic to a ﬁnite Heisenberg group. Recall (cf. [22]) that a ﬁnite Heisenberg group H is a central extension

0 → Zm → H → K → 0 where K is a ﬁnite abelian group such that the commutator pairing K × K → Zm, (k, k) → [k,˜ k˜](k˜,andk˜ being arbitrary lifts of k and k to H)identiﬁesK with the group of homomorphisms from K to Zm. Proof. By Proposition 2.1, − πi T − 2πi T π Π N p q N μ q+ N k Π (p, q, k)θμ (z)=e θμ+p(z). For (p, q, k) to act as the identity operator, we should have

− πi T − 2πi T N p q N μ q Π Π e θμ+p(z)=θμ (z) for all μ ∈{0, 1,...,N − 1}g.Consequently,p should be in NZg.ThenpT q is a − πi pT q− 2πi μT q+ πi k − 2πi μT q+ πi k multiple of N, so the coefficient e N N N equals ±e N N .This coefficient should be equal to 1. For μ =(0, 0,...,0) this implies that −pT q + k should be an even multiple of N. But then by varying μ we conclude that q is a multiple of N. Because N is even, it follows that pT q is an even multiple of N, and consequently k is an even multiple of N. Thus any element in the kernel of the representation must belong to NZ2g × (2N)Z. It is easy to see that any element of this form is in the kernel. These are precisely the elements of the form (p, q, k)N with k even. g The quotient of H(Z ) by the kernel of the representation is a Z2N -extension Z2g of the finite abelian group N , thus is a finite Heisenberg group. This group is isomorphic to { | ∈ Zg ∈ Z } (p, q, k) p, q N ,k 2N with the multiplication rule (p, q, k)(p,q,k)=(p + p,q+ q,k+ k +2pq). Zg → Z2g × Z The isomorphism is induced by the map F : H( ) N 2N , F (p, q, k)=(p mod N,q mod N,k + pq mod 2N). Zg We denote by H( N ) this finite Heisenberg group and by exp(pT P + qT Q + kE) Zg the image of (p, q, k) in it. The representation of H( N ) on the space of theta functions is called the Schrödinger representation. It is an analogue, for the case of the 2g-dimensional torus, of the standard Schrödinger representation of the Heisenberg group with real entries on L2(R). In particular we have T Π Π exp(p P )θμ (z)=θμ+p(z) T Π − 2πi qT μ Π (2.4) exp(q Q)θμ (z)=e N θμ (z) Π πi k Π exp(kE)θμ (z)=e N θμ (z).

FROM THETA FUNCTIONS TO TQFT 43

Theorem 2.4 (Stone-von Neumann). The Schr¨odinger representation of Zg H( N ) is the unique (up to an isomorphism) irreducible unitary representation πi k of this group with the property that exp(kE) acts as e N Id for all k ∈ Z.

Proof. Let Xj =exp(Pj), Yj =exp(Qj ), j =1, 2,...,g, Z =exp(E). 2 Then Xj Yj = Z Yj Xj , Xj Yk = YkXj if j = k, XjXk = XkXj , Yj Yk = YkYj , N N 2N ZXj = XjZ, ZYj = Yj Z, for all i, j,andXj = Yj = Z = Id for all j.Be- cause Y1,Y2,...,Yg commute pairwise, they have a common eigenvector v.And N because Yj = Id for all j, the eigenvalues λ1,λ2,...,λg of v with respect to the Y1,Y2,...,Yg are roots of unity. The equalities

− 2πi − 2πi Yj Xjv = e N XjYj = e N λj Xj v,

Yj Xkv = XkYj v = λj Xkv, if j = k show that by applying Xj ’s repeatedly we can produce an eigenvector v0 of the com- muting system Y1,Y2,...,Yg whose eigenvalues are all equal to 1. The irreducible n n1 n2 ··· g ∈{ − } representation is spanned by the vectors X1 X2 Xg v0, ni 0, 1,...,N 1 . Any such vector is an eigenvector of the system Y1,Y2,...,Yg, with eigenvalues 2πi n 2πi n 2πi n respectively e N 1 ,eN 2 ,..., e N g . So these vectors are linearly independent and form a basis of the irreducible representation. It is not hard to see that the Zg action of H( N ) on the vector space spanned by these vectors is the Schrödinger representation. T T Proposition 2.5. The operators Op e2πi(p x+q y) , p, q ∈{0, 1,...,N− 1}g Π form a basis of the space of linear operators on ΘN (Σg). Proof. For simplicity, we show that the operators πi pT q 2πi(pT x+qT y) g e N Op e ,p,q∈{0, 1,...,N − 1} , form a basis. Denote by Mp,q the respective matrices of these operators in the basis Π ∈{ − }g (θμ )μ.Forafixedp, the nonzero entries of the matrices Mp,q, q 0, 1,...,N 1 are precisely those in the slots (m, m + p), with m ∈{0, 1,...,N− 1}g (here m + p is taken modulo N). If we vary m and q and arrange these nonzero entries in a matrix, we obtain the gth power of a Vandermonde matrix, which is nonsingular. g We conclude that for fixed p, the matrices Mp,q, q ∈{0, 1,...,N − 1} form a basis for the vector space of matrices with nonzero entries in the slots of the form (m, m + p). Varying p, we obtain the desired conclusion.

Corollary . Π 2.6 The algebra L(ΘN (Σg)) of linear operators on the space of C Zg theta functions is isomorphic to the algebra obtained by factoring [H( N )] by the iπ relation (0, 0, 1) = e N . Let us now recall the action of the modular group on theta functions. The modular group, known also as the mapping class group, of a simple closed surface Σg is the quotient of the group of homemorphisms of Σg by the subgroup of homeomorphisms that are isotopic to the identity map. It is at this point where it is essential that N is even. The mapping class group acts on the Jacobian in the following way. An element h of this group induces a linear automorphism h∗ of H1(Σg, R). The matrix of h∗

44 RAZVAN˘ GELCA AND ALEJANDRO URIBE

T has integer entries, determinant 1, and satisﬁes h∗J h∗ = J ,where 0 0 0 Ig J0 = Ig 0 is the intersection form in H1(Σg, R). As such, h∗ is a symplectic linear automorphism of H1(Σg, R), where the symplectic form is the intersection form. Identifying ˜ J (Σg) with H1(Σg, R)/H1(Σg, Z), we see that h∗ induces a symplectomorphism h of J (Σg). The map h → h˜ induces an action of the mapping class group of Σg on the Jacobian variety. This action can be described explicitly as follows. Decompose h∗ into g × g blocks as AB h∗ = . CD

Then h˜ maps the complex torus defined by the lattice (Ig, Π) and complex variable z to the complex torus defined by the lattice (Ig, Π ) and complex variable z ,where Π =(ΠC + D)−1(ΠA + B)andz =(ΠC + D)−1z. This action of the mapping class group of the surface on the Jacobian induces an action of the mapping class group on the finite Heisenberg group by h · exp(pT P + qT Q + kE)=exp[(Ap + Bq)T P +(Cp + Dq)T Q + kE]. The nature of this action is as follows: Since h induces a diffeomorphism on the Jacobian, we can compose h with an exponential and then quantize; the resulting operator is as above. We point out that if N were not even, this action would be defined only for h∗ in the subgroup Spθ(2n, Z) of the symplectic group (this is because only for N even is the kernel of the map F defined in Proposition 2.3 preserved under the action of h∗). As a corollary of Theorem 2.4, the representation of the finite Heisenberg group · Π · Π on theta functions given by u θμ =(h u)θμ is equivalent to the Schrödinger Π representation, hence there is an automorphism ρ(h)ofΘN (Σg) that satisfies the exact Egorov identity: (2.5) h · exp(pT P + qT Q + kE)=ρ(h)exp(pT P + qT Q + kE)ρ(h)−1. (Compare with [7], Theorem 2.15, which is the analogous statement in quantum mechanics in Euclidean space.) Moreover, by Schur’s lemma, ρ(h) is unique up to multiplication by a constant. We thus have a projective representation of the mapping class group of the surface on the space of classical theta functions that statisfies with the action of the finite Heisenberg group the exact Egorov identity from (2.5). This is the finite dimensional counterpart of the metaplectic representation, called the Hermite-Jacobi action.

Remark 2.7. We emphasize that the action of the mapping class group of Σg on theta functions factors through an action of the symplectic group Sp(2n, Z). Up to multiplication by a constant, Π − T −1 Π (2.6) ρ(h)θμ (z)=exp[ πiz C(ΠC + D) z]θμ (z ) (cf. (5.6.3) in [22]). When the Riemann surface is the complex torus obtained as the quotient of the complex plane by the integer lattice, and h = S is the map induced by a 90◦ rotation around the origin, then ρ(S) is the discrete Fourier transform. In general, like for the metaplectic representation (see [17]), ρ(h) can be written as a

FROM THETA FUNCTIONS TO TQFT 45 composition of partial discrete Fourier transforms. For this reason, we will refer to ρ(h)asadiscrete Fourier transform.

3. Theta functions in the abstract setting In this section we apply to the finite Heisenberg group the standard construction which identifies the Schrödinger representation as a representation induced by an irreducible representation (i.e. character) of a maximal abelian subgroup (see for example [17]). Start with a Lagrangian subspace of H1(Σg, R) with respect to the intersection form, which for our purpose is spanned by the elements b1,b2,...,bg of the canonical basis. Let L be the intersection of this space with H1(Σg, Z). Under the standard g inclusion H1(Σg, Z) ⊂ H(Z ), L becomes an abelian subgroup of the Heisenberg Zg group with integer entries. This factors to an abelian subgroup exp(L)ofH( N ). Z Zg Let exp(L+ E) be the subgroup of H( N ) containing both exp(L)andthescalars exp(ZE). Then exp(L + ZE) is a maximal abelian subgroup. Being abelian, it has only 1-dimensional irreducible representations, which are its characters. In view of the Stone-von Neumann Theorem, we consider the induced repre- πi k sentation defined by the character χL :exp(L + ZE) → C, χL(l + kE)=e N .This representation is H(Zg ) N Zg C Indexp(L+ZE) = C[H( N )] C[exp(L+ZE)] Zg with H( N ) acting on the left in the first factor of the tensor product. Explicitly, C Zg the vector space of the representation is the quotient of the group algebra [H( N )] by the vector subspace spanned by all elements of the form −1 u − χL(u ) uu ∈ Zg ∈ Z H with u H( N )andu exp(L + E). We denote this quotient by N,g(L), and C Zg →H let πL : [H( N )] N,g(L) be the quotient map. Let also the inner product be defined such that πL(u) has norm 1, where u is an element of the finite Heisenberg group seen as an element of its group algebra. Zg The left regular action of the Heisenberg group H( N ) on its group algebra descends to an action on HN,g(L). Proposition . Π → T ∈ Zg 3.1 The map θμ (z) πL(exp(μ P )), μ N defines a unitary Π H map between the space of theta functions ΘN (Σg)and N,g(L), which intertwines the Schrödinger representation and the left action of the finite Heisenberg group. Proof. Π H It is not hard to see that ΘN (Σg)and N,g(L) have the same di- ∈ Zg T T mension. Also, for μ = μ N , πL(exp(μ P )) = πL(exp(μ P )), hence the map from the statement is an isomorphism of finite dimensional spaces. The norm of T πL(exp(μ P )) is one, hence this map is unitary. We have exp(pT P )exp(μT P )=exp((p + μ)T )P ) and T T − πi qT μ T T exp(q Q)exp(μ P )=e N exp(μ P )exp(q Q). It follows that T T T exp(p P )πL(exp(μ P )) = πL((p + μ) P ) T T − πi qT μ T exp(q Q)πL(exp(μ P )) = e N πL(exp(μ P )) in agreement with the Schrödinger representation (2.4).

46 RAZVAN˘ GELCA AND ALEJANDRO URIBE

We rephrase the Hermite-Jacobi action in this setting. To this end, fix an element h of the mapping class group of the Riemann surface Σg.LetL be the subgroup of H1(Σg, Z) associated to a canonical basis as explained in the beginning of this section, which determines the maximal abelian subgroup exp(L + ZE). The automorphism of H1(Σg, Z) defined by aj → h∗(aj), bj → h∗(bj ), j = 1, 2,...,g, maps isomorphically L to h∗(L), and thus allows us to identify in a canonical fashion HN,g(L)andHN,g(h∗(L)). Given this identification, we can view the discrete Fourier tranform as a map ρ(h):HN,g(L) →HN,g(h∗(L)). The discrete Fourier transform should map an element u mod ker(πL)inthe C Zg C Zg space [H( N )]/ker(πL)tou mod ker(πh∗(L))in [H( N )]/ker(πh∗(L)). In this form the map is not well defined, since different representatives for the class of u might yield different images. The idea is to consider all possible liftings of u and average them. For lifting the element u mod ker(πL) we use the section of πL defined as 1 (3.1) s (u mod ker(π )) = χ (u )−1uu . L L 2N g+1 L 1 1 u1∈exp(L+ZE) Then, up to multiplication by a constant 1 (3.2) ρ(h)(u mod ker(π )) = χ (u )−1uu mod ker(π ). L 2N g+1 L 1 1 h∗(L) u1∈exp(L+ZE) This formula identifies ρ(h) as a Fourier transform. That this map agrees with the one defined by (2.6) up to multiplication by a constant follows from Schur’s lemma, since both maps satisfy the exact Egorov identity (2.5).

4. A topological model for theta functions The finite Heisenberg group, the equivalence relation defined by the kernel of πL,andtheSchrödinger representation can be given topological interpretations, which we explicate below. First, a heuristical discussion. The Heisenberg group. The group H(Zg)isaZ-extension of the abelian group H1(Σg, Z). The bilinear form ω from (2.3), which defines the cocycle of this extension, is the intersection form in H1(Σg, Z). Cycles in H1(Σg, Z) can be represented by families of non-intersecting simple closed curves on the surface. As vector spaces, g −1 we can identify C[H(Z )] with C[t, t ]H1(Σg, Z), where t is an abstract variable whose exponent equals the last coordinate in the Heisenberg group. We start with an example on the torus. Here and throughout the paper we agree that (p, q) denotes the curve of slope q/p on the torus, oriented from the origin to the point (p, q) when viewing the torus as a quotient of the plane by integer translations. Consider the multiplication (1, 0)(0, 1) = t(1, 1), shown in Figure 2. The product curve (1, 1) can be obtained by cutting open the curves (1, 0) and (0, 1) at the crossing and joining the ends so that the orientations agree. This operation is called smoothing of the crossing. Itiseasytocheckthatthis works for arbitrary surfaces: whenever multiplying two families of curves introduce acoefficientoft raised to the algebraic intersection number of the two families then smoothen all crossings. Such algebras of curves, with multiplication related to polynomial invariants of knots, were first considered in [29].

FROM THETA FUNCTIONS TO TQFT 47

Figure 2

Figure 3

Zg Zg The group H( N )isaquotientofH( ), but can also be viewed as an extension Z C Zg of H1(Σg, N ). As such, the elements of [H( N )] can be represented by families of non-intersecting simple closed curves on the surface with the convention that any N parallel curves can be deleted. The above observation applies to this case iπ as well, provided that we set t = e N . Π It follows that the space of linear operators L(ΘN (Σg)) can be represented as an algebra of simple closed curves on the surface with the convention that any N parallel curves can be deleted. The multiplication of two families of simple iπ closed curves is defined by introducing a coefficient of e N raised to the algebraic intersection number of the two families and smoothing the crossings. Theta functions. Next, we examine the space of theta functions, in its abstract framework from Section 3. To better understand the factorization modulo the kernel of πL, we look again at the torus. If the canonical basis is (1, 0) and (0, 1) with L = Z(0, 1) , then an equivalence modulo ker(πL)isshowninFigure3.If we map the torus to the boundary of a solid torus in such a way that L becomes null-homologous, then the first and last curves from Figure 3 are homologous in the solid torus. To keep track of t we apply a standard method in topology which consists of framing the curves. A framed curve in a manifold is an embedding of an annulus. One can think of the curve as being one of the boundary components of the annulus, and then the annulus itself keeps track of the number of ways that the curve twists around itself. Changing the framing by a full twist amounts to multiplying by t or t−1 depending whether the twist is positive or negative. Then the equality from Figure 3 holds in the solid torus. It is not hard to check for a general surface Σg the equivalence relation modulo ker(πL)isofthisforminthe handlebody bounded by Σg in such a way that L is null-homologous.

The Schrödinger representation. One can frame the curves on Σg by using the blackboard framing, namely by embedding the annulus in the surface. As such, the Schrödinger representation is the left action of an algebra of framed curves on a surface on the vector space of framed curves in the handlebody induced by the inclusion of the surface in the handlebody. We will make this precise using the language of skein modules [23]. Let M be a compact oriented 3-dimensional manifold. A framed link in M is a smooth embedding of a disjoint union of finitely many annuli. The annuli are called link components. We consider oriented framed links. The orientation of a

48 RAZVAN˘ GELCA AND ALEJANDRO URIBE

t ; t −1

−1 t ; t

Figure 4 link component is an orientation of one of the circles that bound the annulus. When M is the cylinder over a surface, we represent framed links as oriented curves with the blackboard framing, meaning that the annulus giving the framing is always parallel to the surface. Let t be a free variable. Consider the free C[t, t−1]-module with basis the isotopy classes of framed oriented links in M including the empty link ∅.LetS be the the submodule spanned by all elements of the form depicted in Figure 4, where the two terms in each skein relation depict framed links that are identical except in an embedded ball, in which they look as shown. The ball containing the crossing can be embedded in any possible way. To normalize, we add to S the element consisting of the diﬀerence between the unknot in M and the empty link ∅. Recall that the unknot is an embedded circle that bounds an embedded disk in M and whose framing annulus lies inside the disk.

Definition 4.1. The result of the factorization of the free C[t, t−1]-module with basis the isotopy classes of framed oriented links by the submodule S is called the linking number skein module of M, and is denoted by L(M). The elements of L(M) are called skeins.

In other words, we are allowed to smoothen each crossing, to change the framing provided that we multiply by the appropriate power of t, and to identify the unknot with the empty link. The “linking number” in the name is motivated by the fact that the skein relations from Figure 4 are used for computing the linking number. These skein modules were ﬁrst introduced by Przytycki in [24] as one-parameter deformations of the group algebra of H1(M,Z). Przytycki computed them for all 3-dimensional manifolds.

Lemma 4.2. Any trivial link component, namely any link component that bounds a disk disjoint from the rest of the link in such a way that the framing is an annulus inside the disk, can be deleted.

Proof. The proof of the lemma is given in Figure 5.

t −1

Figure 5

FROM THETA FUNCTIONS TO TQFT 49

If M =Σg × [0, 1], the cylinder over a surface, then the identiﬁcation

Σg × [0, 1] ∪ Σg × [0, 1] ≈ Σ × [0, 1] obtained by gluing the boundary component Σg ×{0} in the first cylinder to the boundary component Σg ×{1} in the second cylinder by the identity map induces a multiplication on L(Σg × [0, 1]). This turns L(Σg × [0, 1]) into an algebra, called the linking number skein algebra. As such, the product of two skeins is obtained by placing the first skein on top of the second. The nth power of an oriented, framed, simple closed curve consists then of n parallel copies of that curve. We adopt the same terminology even if the manifold is not a cylinder, so γn stands for n parallel copies of γ. Additionally, γ−1 is obtained from γ by reversing orientation, and γ−n =(γ−1)n. Definition 4.3. For a fixed positive integer N, we define the reduced linking number skein module of the manifold M, denoted by LN (M), to be the quotient of L(M) obtained by imposing that γN = ∅ for every oriented, framed, simple closed πi curve γ, and by setting t = e N . As such, L = L whenever L is obtained from L by removing N parallel link components. Remark 4.4. As a rule followed throughout the paper, whenever we talk about skein modules, t is a free variable, and when we talk about reduced skein modules, t 3 ∼ −1 3 ∼ is a root of unity. The isomorphisms L(S ) = C[t, t ]andLN (S ) = C allow us to identify the linking number skein module of S3 with the set of Laurent polynomials in t and the reduced skein module with C.

For a closed, oriented, genus g surface Σg, consider a canonical basis of its first homology a1,a2,...,ag,b1,b2,...,bg (see Section 1). The basis elements are oriented simple closed curves on the surface, which we endow with the blackboard framing. Let Hg be a genus g handlebody and h0 :Σg → ∂Hg be a homeomorphism that maps b1,b2,...,bg to null homologous curves. Then a1,a2,...ag is a basis of the first homology of the handlebody. Endow these curves in the handlebody with the framing they had on the surface. The linking number skein module of a 3-manifold M with boundary is a module over the skein algebra of a boundary component Σg. The module structure is induced by the identification

Σg × [0, 1] ∪ M ≈ M where Σg × [0, 1] is glued to M along Σg ×{0} by the identity map. This means that the module structure is induced by identifying Σg × [0, 1] with a regular neighborhood of the boundary of M. The product of a skein in a regular neighborhood of the boundary and a skein in the interior is the union of the two skeins. This module structure descends to relative skein modules. In particular L(Σg ×[0, 1]) acts on the left on L(Hg) with action induced by the homeomorpism h0 :Σg → ∂Hg, and the action descends to relative skein modules.

Theorem 4.5. (a) The linking number skein module L(Σg × [0, 1]) is a free C[t, t−1]-module with basis

m1 m2 ··· mg n1 n2 ··· ng ∈ Z a1 a2 ag b1 b2 bg ,m1,m2,...,mg,n1,n2,...,ng . −1 (b) The linking number skein module L(Hg) is a free C[t, t ]-module with basis m1 m2 ··· mg ∈ Z a1 a2 ag ,m1,m2,...,mg .

50 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Figure 6

g (c) The algebras L(Σg × [0, 1]) and C[H(Z )] are isomorphic, with the isomorphism deﬁned by the map tkγ → ([γ],k). where γ ranges over all skeins represented by oriented simple closed curves on Σg 2g (with the blackboard framing) and [γ] is its homology class in H1(Σg, Z)=Z . Proof. Parts (a) and (b) are consequences of a general result in [24]; we include their proof for sake of completeness. (a) Bring all skeins in the blackboard framing of the surface. A skein tkL,where L is an oriented framed link in Σg × [0, 1] is equivalent modulo skein relations to a skein tk+mL where L is an oriented framed link such that the projection of L onto the surface has no crossings, and m is the diﬀerence between the number of positive and negative crossings of the projection of L. Moreover, since any embedded ball can be isotoped to a cylinder over a disk, any skein tnL that is equivalent to tkL, with L a framed link with no crossings, has the property that n = k + m. If L is an oriented link with blackboard framing whose projection onto the surface has no crossings, and if it is null-homologous in H1(Σg × [0, 1], Z), then L is equivalent modulo skein relations to the empty skein. This follows from the computations in Figure 6 since Σg can be cut into pairs of pants and annuli. View Σg as a sphere with g punctured tori attached. Then L is equivalent to a link L consisting of simple closed curves on the tori, which therefore is of the form

k1 k2 kg (p1,q1) (p2,q2) ···(pg,qg) , where (pj,qj ) denotes the curve of slope pj /qj on the jth torus. This last link is equivalent, modulo skein relations, to k p k p k q k q j kj pj qj 1 1 2 2 ··· kgpg 1 1 2 2 ··· kg qg (4.1) t a1 a2 ag b1 b2 bg . It is easy to check that if we change the link by a Reidemeister move, then resolve all crossings, we obtain the same expression (4.1). So the result only depends on the link and not on how it projects to Σg.Thisproves(a). Part (b) is analogous to (a), given that a genus g handlebody is the cylinder over a disk with g punctures. For (c) recall Corollary 2.6. That the speciﬁed map is a linear isomorphism follows from (a). It is straightforward to check that the multiplication rule is the same.

Remark 4.6. Explicitly, the map k m1 m2 ··· mg n1 n2 ··· ng → ∈ Z t a1 a2 ag b1 b2 bg (m1,m2,...,mg,n1,n2,...,ng,k),mj ,nj,k g deﬁnes an algebra isomorphism between L(Σg × [0, 1]) and C[H(Z )].

FROM THETA FUNCTIONS TO TQFT 51

Theorem 4.7. (a) The reduced linking number skein module LN (Σg × [0, 1]) is a ﬁnite dimensional vector space with basis

m1 m2 ··· mg n1 n2 ··· ng ∈ Z a1 a2 ag b1 b2 bg ,m1,m2,...,mg,n1,n2,...,ng N .

(b) The reduced linking number skein module LN (Hg) is a ﬁnite dimensional vector space with basis

m1 m2 ··· mg ∈ Z a1 a2 ag ,m1,m2,...,mg N .

L Π Moreover, there is a linear isomorphism of N (Hg) and ΘN (Σg) given by

n1 n2 ··· ng → Π ∈ Z a1 a2 ag θn1,n2,...,ng , for all n1,n2,...,ng N .

(c) The algebra isomorphism defined in Theorem 4.5 factors to an algebra iso- L × Π morphism of N (Σg [0, 1]) and L(ΘN (Σg)), the algebra of linear operators on the space of theta functions. The isomorphism defined in (b) intertwines the left action of LN (Σg × [0, 1]) on LN (Hg) and the Schrödinger representation.

g Proof. (a) By Theorem 4.5 we can identify L(Σg × [0, 1]) with C[H(Z )]. iπ Setting t = e N and deleting any N parallel copies of a link component are precisely the relations by which we factor the Heisenberg group in Proposition 2.3. The only question is whether factoring by this additional relation before applying the other skein relations factors any further the skein module. However, we see that when a curve is crossed by N parallel copies of another curve, there is no distinction between overcrossings and undercrossings. Hence if a link contains N parallel copies of a curve, we can move this curve so that it is inside a cylinder Σg × [0, ]thatdoes not contain other link components and we can resolve all self-crossings of this curve without introducing factors of t. Then we can delete the curve without introducing new factoring relations. This proves (a). For (b), notice that we factor LN (Σg × [0, 1]) to obtain LN (Hg)bythesame C Zg H relations by which we factor [H( N )] to obtain N,g(L) in Section 3. (c) An easy check shows that that the left action of the skein algebra of the cylinder over the surface on the skein module of the handlebody is the same as the one from Propositions 2.1 and 3.1.

In view of Theorem 4.7 we endow LN (Hg) with the Hilbert space structure of the space of theta functions. Now we turn our attention to the discrete Fourier transform, and translate in topological language formula (3.2). Let h be an element of the mapping class group of Σg. The action of the mapping class group on the ﬁnite Heisenberg group from Section 2 becomes the action on skeins in Σg × [0, 1] given by

σ → h(σ), where h(σ) is obtained by replacing each framed curve of the skein σ by its image through the homeomorphism h.

52 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Consider h1 and h2 two homeomorphisms of Σg onto the boundary of the handlebody Hg such that h2 = h ◦ h1. These homeomorphisms extend to embeddings of Σg × [0, 1] into Hg whichwedenotebyh1 and h2 as well. The homeomorphisms h1 and h2 define the action of LN (Σg × [0, 1]) on LN (Hg) in two different ways, i.e. they give two different constructions of the Schrödinger representations. By the Stone-von Neumann theorem, these are unitary equivalent; they are related by the isomorphism ρ(h). We now give ρ(h) a topological definition. For this, let us take a closer look at the lifting map sL defined in (3.1). First, it is standard to remark that one should only average over exp(L + ZE)/ exp(ZE)=exp(L), hence 1 s (u mod ker(π )) = uu . L L N g 1 u1∈exp(L) ∈ Zg If u = u H( N ), then, as a skein, u is defined by a framed oriented multicurve on Σg = ∂Hg and k is an integer. The equivalence classu ˆ = u mod ker(πL(u)) is just this skein viewed as lying inside the handlebody. n1 n2 ng On the other hand, as a skein, u1 is of the form b1 b2 ...bg ,andassuch, the product uu1 becomes, after smoothing all crossings, another lift of the skeinu ˆ to the boundary obtained by lifting γ to the boundary. Such a lift is obtained by pushingu ˆ inside a regular neighborhood of the boundary and then viewing it as an element in LN (Σg × [0, 1]). When u1 ranges over all exp(L) we obtain all possible lifts ofu ˆ to the boundary obtained by pushing γ to the boundary.

Theorem 4.8. For a skein in LN (Hg), defined by a multicurve in Hg,consider all possible liftings to LN (Σg × [0, 1]) using h1, obtained by pushing γ to the boundary. Take the average of these liftings and map the average by h2 to LN (Hg). This defines a linear endomorphism of LN (Hg) which is, up to multiplication by a constant, the discrete Fourier transform ρ(h). Proof. The map defined this way intertwines the Schrödinger representations defined by h1 and h2, so the theorem is a consequence of the Stone-von Neumann theorem.

Example: We will exemplify this by showing how the S-map on the torus acts on the theta series Π 1 2 1 Π 2πiN 2 ( N +n) +z( N +n) θ1 (z)= e n∈Z (in this case Π is a just a complex number with positive imaginary part). This theta series is represented in the solid torus by the curve shown in Figure 7. The N linearly independent liftings of this curve to the boundary are shown in Figure 8. The S-map sends these to those in Figure 9, which, after being pushed inside the solid torus, become the skeins from Figure 10.

Figure 7

FROM THETA FUNCTIONS TO TQFT 53

t , ... , N−1 , t ...

Figure 8

, t ,... ,tN−1 ...

Figure 9

, t 2 ,... , t2(N−1) ...

Figure 10

Note that in each skein the arrow points the opposite way as for θ1(z). Using the identity γN = ∅, we can replace j parallel strands by N − j parallel strands 2j with opposite orientation. Hence these skeins are t θN−j , j =1,...,N (note also that θ0(z)=θN (z)). Taking the average we obtain N−1 N−1 1 2πij 1 − 2πij ρ(S)θ (z)= e N θ − (z)= e N θ (z), 1 N N j N j j=0 j=0 which is, up to a multiplication by a constant, the standard discrete Fourier transform of θ1(z).

5. The discrete Fourier transform as a skein As a consequence of Proposition 2.5, ρ(h) can be represented as an element in C Zg [H( N )]. Furthermore, Theorem 4.7 implies that ρ(h) can be represented as left multiplication by a skein F(h)inLN (Σg × [0, 1]). The skein F(h)isuniqueupto a multiplication by a constant. We wish to ﬁnd an explicit formula for it. Theorem 4.7 implies that the action of the group algebra of the ﬁnite Heisenberg group can be represented as left multiplication by skeins. Using this fact, the exact Egorov identity (2.5) translates to

(5.1) h(σ)F(h)=F(h)σ for all σ ∈LN (Σg × [0, 1]). By the Lickorish twist theorem (Chapter 9 in [25]), every homeomorphism of Σg is isotopic to a product of Dehn twists along the 3g − 1 curves depicted in Figure 11. Recall that a Dehn twist is the homemorphism obtained by cutting the surface along the curve, applying a full rotation on one side, then gluing back. The curves from Figure 11 are nonseparating, and any two can be mapped into one another by a homeomorphism of the surface. Thus, to understand F(h)in general it suﬃces to consider the case h = T , the positive Dehn twist along the curve b1 from Figure 1. The word positive means that after we cut the surface along b1 we perform a full rotation of the part on the left in the direction of the

54 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Figure 11 arrow. Because T (σ)=σ for all skeins that do not contain curves that intersect b1, it follows that ρ(T ) commutes with all such skeins. It also commutes with the multiples of b1 (viewed as a skein with the blackboard framing). Hence ρ(T ) commutes with all operators of the form exp(pP + qQ+ kE) with p1, the ﬁrst entry of p, equal to 0. This implies that N−1 ρ(T )= cj exp(jQ1). j=0

To determine the coeﬃcients cj , we write the exact Egorov identity (2.5) for exp(P1). Since T · exp(P1)=exp(P1 + Q1) this identity reads N−1 N−1 exp(P1 + Q1) cj exp(jQ1)= cj exp(jQ1)exp(P1). j=0 j=0 We transform this further into N−1 N−1 πi j − πi j cj e N exp[P1 +(j +1)Q1]= cj e N exp(P1 + jQ1), j=0 j=0 or, taking into account that exp(P1)=exp(P1 + NQ1), N−1 N−1 πi (j−1) − πi j cj−1e N exp(P1 + jQ1)= cj e N exp(P1 + jQ1), j=0 j=0

πi (2j−1) where c−1 = cN−1. It follows that cj = e N cj−1 for all j. Normalizing so that −1/2 πi j2 ρ(T ) is a unitary map and c0 > 0weobtaincj = N e N , and hence N−1 −1/2 πi j2 F(T )=N e N exp(jQ1). j=0 Turning to the language of skein modules, and taking into account that any Dehn twist is conjugate to the above twist by an element of the mapping class group, we conclude that if T is a positive Dehn twist along the simple closed curve γ on Σg, then N−1 2 F(T )=N −1/2 tj γj . j=0 This is the same as the skein N−1 F(T )=N −1/2 (γ+)j j=0

FROM THETA FUNCTIONS TO TQFT 55

+ + ... + ...

Figure 12 where γ+ is obtained by adding one full positive twist to the framing of γ (the twist is positive in the sense that, as skeins, γ+ = tγ). This skein has an interpretation in terms of surgery. Consider the curve γ+ × + {1/2}⊂Σg × [0, 1] with framing deﬁned by the blackboard framing of γ on Σg. Take a solid torus which is a regular neighborhood of the curve on whose boundary the framing determines two simple closed curves. Remove it from Σg × [0, 1], then glue it back in by a homeomorphism that identiﬁes its meridian (the curve that is null-homologous) to one of the curves determined by the framing. This operation, called surgery, yields a manifold that is homeomorphic to Σg × [0, 1], such that the restriction of the homeomorphism to Σg ×{0} is the identity map, and the restriction to Σg ×{1} is the Dehn twist T . The reduced linking number skein module of the solid torus H1 is, by Theorem ∅ N−1 4.7, an N-dimensional vector space with basis ,a1,...,a1 . Alternately, it is the Π Π Π vector space of 1-dimensional theta functions with basis θ0 (z),θ1 (z),...,θN−1(z), where Π in this case is a complex number with positive imaginary part. We introduce the element N−1 N−1 −1/2 j −1/2 Π (5.2) Ω=N a1 = N θj (z) j=0 j=0 L Π in N (H1)=ΘN (Σ1). As a diagram, Ω is the skein depicted in Figure 12 multiplied by N −1/2.IfS is the homemorphism on the torus induced by the 90◦ rotation of the plane when viewing the torus as the quotient of the plane by the integer lattice, ∅ Π then Ω = ρ(S) . So Ω is the (standard) discrete Fourier transform of θ0 (z). For an arbitrary framed link L we denote by Ω(L) the skein obtained by replacing each link component by Ω. In other words, Ω(L)isthesumofframedlinks obtained from L by replacing its components, in all possible ways, by 0, 1,...,N−1 parallel copies. The skein Ω is called the coloring of L by Ω. Proposition 5.1. a) The skein Ω(L) is independent of the orientations of the components of L. b) The skein relation from Figure 13 holds, where the n parallel strands point in the same direction.

Proof. a) The computation in Figure 6 implies that if we switch the orien- Π Π tation on the j parallel curves that represent θj (z)weobtainθN−j (z). Hence by

n if n=0 Ω Ω 0 if 0

Figure 13

56 RAZVAN˘ GELCA AND ALEJANDRO URIBE changing the orientation on all curves that make up Ω we obtain the skein −1/2 Π Π Π ··· Π N [θ0 (z)+θN−1(z)+θN−2(z)+ + θ1 (z)], which is, again, Ω. b) When n = 0 there is nothing to prove. If n = 0, then by resolving all crossings in the diagram we obtain n vertical parallel strands with the coeﬃcient − N1 t2Nn − 1 N −1/2 t±2nj = N −1/2 · . t2n − 1 j=0 where the signs in the exponents are either all positive, or all negative. Since t2 is aprimitiveNth root of unity, this is equal to zero. Hence the conclusion. Up to this point we have proved the following result: Lemma 5.2. For a Dehn twist T , F(T ) is the skein obtained by coloring the surgery framed curve γ+ of T by Ω. Since by the Lickorish twist theorem every element h of the mapping class group is a product of twists, we obtain the following skein theoretic description of the discrete Fourier transform induced by the map h.

Proposition 5.3. Let h be an element of the mapping class group of Σg obtained as a composition of Dehn twists h = T1T2 ···Tn. Express each Dehn twist Tj by surgery on a curve γj as above, and consider the link Lh = γ1 ∪ γ2 ∪···∪γn which expresses h as surgery on the framed link Lh in Σg ×[0, 1]. Then the discrete Fourier transform ρ(h):LN (Hg) →LN (Hg)isgivenby

ρ(h)β =Ω(Lh)β. 6. The Egorov identity and handle slides Next, we give the Egorov identity a topological interpretation in terms of handle slides. For this we look at its skein theoretical version (5.1). We start again with an example on the torus. Example: For the positive twist T and the operator represented by the curve (1, 0) the exact Egorov identity reads ρ(T )(1, 0) = (1, 1)ρ(T ), which is shown in Figure 14. The diagram on the right is the same as the one in Figure 15. As such, the curve (1, 1) is obtained by sliding the curve (1, 0) along the surgery curve of the positive twist. Here is the detailed description of the operation of sliding a framed knot along another using a Kirby band-sum move. The slide of a framed knot K0 along the framed knot K, denoted by K0#K, is obtained as follows. Let K1 be a copy of K obtained by pushing K in the

Ω Ω

Figure 14

FROM THETA FUNCTIONS TO TQFT 57

Figure 15

Figure 16

3 direction of its framing. Take an embedded [0, 1] that is disjoint from K, K0, 2 and K1 except for the opposite faces Fi =[0, 1] ×{i}, i =0, 1andwhichare embedded in ∂K0 respectively ∂K1. Fi is embedded in the annulus Ki such that [0, 1]×{j}×{i} is embedded in ∂Ki. Delete from K0 ∪K1 the faces Fi and add the faces {j}×[0, 1] × [0, 1]. The framed knot obtained this way is K0#K. Saying it less rigorously but more intuitively, we cut the knots K0 and K1 and join together the two open strands by pulling them along the sides of an embedded rectangle (band) which does not intersect the knots. Figure 16 shows the slide of a trefoil knot over a ﬁgure-eight knot, both with the blackboard framing. When the knots are oriented, we perform the slide so that the orientations match. One should point out that there are many ways in which one can slide one knot along the other, since the band that connects the two knots is not unique. For a closed curve α in Σg =Σg ×{0},thecurveh(α) is obtained from α by slides over the components of the surgery link of h. Indeed, if h is the twist along + the curve γ, with surgery curve γ ,andifα and γ intersect on Σg at only one point, then h(α)=α#γ+. If the algebraic intersecton number of α and γ is ±k, then h(α) is obtained from α by performing k consecutive slides along γ+.The general case follows from the fact that h is a product of twists. It follows that the exact Egorov identity is a particular case of slides of framed knots along components of the surgery link. In fact, the exact Egorov identity covers all cases of slides of one knot along another knot colored by Ω, and we have

Theorem 6.1. Let M be a 3-manifold, σ a skein in LN (M) and K0 and K two oriented framed knots in M disjoint from σ.Then,inLN (M), one has

σ ∪ K0 ∪ Ω(K)=σ ∪ (K0#K) ∪ Ω(K), however one does the band-sum K0#K. Remark 6.2. The knots from the statement of the theorem should be understood as representing elements in LN (M). 3 Proof. Isotope K0 along the embedded [0, 1] that deﬁnes K0#K to a knot K0 that intersects K. There is an embedded punctured torus Σ1,1 in M, disjoint ∪ from σ, which contains K0 K on its boundary, as shown in Figure 17 a). In fact, by looking at a neighborhood of this torus, we can ﬁnd an embedded Σ1,1 × [0, 1]

58 RAZVAN˘ GELCA AND ALEJANDRO URIBE

a) b)

Figure 17

∪ ⊂ ×{ } such that K0 K Σ1,1 0 . The boundary of this cylinder is a genus 2 surface Σ2,andK0 and K1 lie in a punctured torus of this surface and intersect at exactly one point. By pushing oﬀ K0 to a knot isotopic to K0 (which we identify with K0),weseethatwecanplaceK0 and K in an embedded Σ2 × [0, 1] such that K0 ∈ Σ2 ×{0} and K ∈ Σ2 ×{1/2}. By performing a twist in Σ1,1 × [0, 1] we can change the framing of K in such a way that K0 and K look inside Σ2 × [0, 1] like in Figure 17 b). Then K0 is mapped to K0#K in Σ2 ×{1} by the Dehn twist of Σ2 with surgery diagram K. Hence the equality

K0 ∪ Ω(K)=(K0#K) ∪ Ω(K) in Σ2×[0, 1] is just the exact Egorov identity, which we know is true . By embedding Σ2 × [0, 1] in Σ1,1 × [0, 1] we conclude that this equality holds in Σ1,1 × [0, 1]. By applying the inverse of the twist, embedding Σ1,1 × [0, 1] in M, and adding σ,we conclude that the identity from the statement holds as well. The operation of sliding one knot along another is related to the surgery description of 3-manifolds (see [25]). We recall the basic facts. We use the standard notation Bn for an n-dimensional (unit) ball and Sn for the n-dimensional sphere. Every oriented, closed, 3-dimensional manifold is the boundary of a 4-dimensional manifold obtained by adding 2-handles B2 × B2 to B4 along the solid tori B2 × S1 [16]. On the boundary S3 of B4, when adding a handleweremoveasolidtorusfromS3 (the one identified with B2 × S1) and glue back the solid torus S1 × B2.Thecurve{1}×S1 in the solid torus B2 × S1 that was removed becomes the null-homologous curve on the boundary of S1 × B2. This procedure of constructing 3-manifolds is called Dehn surgery with integer coefficients. The curve {1}×S1 together with the core of B2 × S1 bound an embedded annulus which defines a framed link component in S3. So the information for Dehn surgery with integer coefficients is encoded in a framed link in S3. 3 If K0 is a knot inside a 3-dimensional manifold M obtained by surgery on S and if the framed knot K is a component of the surgery link, then K0#K is the slide of K0 over the 2-handle corresponding to K. Indeed, when we slide K0 along the handle we push one arc close to K, then move it to the other side of the handle by pushing it through the meridinal disk of the surgery solid torus. The meridian of this solid torus is parallel to the knot K (when viewed in S3), so by sliding K0 acrossthehandleweobtainK0#K. In particular, the operation of sliding one 2-handle over another corresponds to sliding one link component of the surgery link along another. In conclusion, we can say that the Egorov identity allows handle-slides along surgery link components colored by Ω. We will make use of this fact in Section 7.

FROM THETA FUNCTIONS TO TQFT 59

7. The topological quantum ﬁeld theory associated to theta functions Theorem 6.1 has two direct consequences: • the deﬁnition of a topological invariant for closed 3-dimensional manifolds, • the existence of an isomorphism between the reduced linking number skein modules of 3-dimensional manifolds with homeomorphic boundaries. We have seen in the previous section that handle-slides correspond to changing the presentation of a 3-dimensional manifold as surgery on a framed link. Kirby’s theorem [14] states that two framed link diagrams represent the same 3-dimensional manifold if they can be transformed into one another by a sequence of isotopies, handle slides, and additions/deletions of the trivial link components U+ and U− described in Figure 18. A trivial link component corresponds to adding a 2-handle to B4 in a trivial way, and on the boundary, to taking the connected sum of the original 3-dimensional manifold and S3.

Figure 18

Theorem 6.1 implies that, given a framed link L in S3, the element Ω(L) ∈ 3 LN (S )=C is an invariant of the 3-dimensional manifold obtained by performing surgery on L, modulo addition and subtraction of trivial 2-handles. This ambiguity can be removed by using the linking matrix of L as follows. Recall that the linking matrix of an oriented framed link L has the (i, j)entry equal to the linking number of the ith and jth components for i = j and the (i, i) entry equal to the writhe of the ith component, namely to the linking number of the ith component with a push-out of this component in the direction of the framing. The signature sign(L) of the linking matrix does not depend on the orientations of the components of L, and is equal to the signature sign(W )ofthe 4-dimensional manifold W obtained by adding 2-handles to B4 as speciﬁed by L. Here the signature of sign(W ) is the signature of the intersection form in H2(W, R). When adding a trivial handle via U+ respectively U−, the signature of the linking matrix, and hence of the 4-dimensional manifold, changes by +1 respectively −1. Proposition 7.1. In any 3-dimensional manifold, the following equalities hold πi − πi Ω(U+)=e 4 ∅, Ω(U−)=e 4 ∅.

Consequently Ω(U+) ∪ Ω(U−)=∅. Proof. We have N−1 −1/2 j2 Ω(U+)=N t ∅. j=0 Because N is even, N−1 N−1 N−1 2N−1 j2 πi j2 πi (N+j)2 j2 t = e N = e N = t . j=0 j=0 j=0 j=N

60 RAZVAN˘ GELCA AND ALEJANDRO URIBE

Hence N−1 2N−1 j2 1 2πi j2 t = e 2N . 2 j=0 j=0

πi 1/2 The last expression is a Gauss sum, which is equal to e 4 N (see [15, page 87]. This proves the ﬁrst formula. On the other hand, N−1 −1/2 − πi j2 Ω(U−)=N e N ∅ j=0 which is the complex conjugate of Ω(U+). Hence, the second formula.

Theorem 7.2. Given a closed, oriented, 3-dimensional manifold M obtained as surgery on the framed link L in S3,thenumber

− πi sign(L) Z(M)=e 4 Ω(L) is a topological invariant of the manifold M. Proof. Using Proposition 7.1 we can rewrite

−b+ −b− Z(M)=Ω(U+) Ω(U−) Ω(L) where b+ and b− are the number of positive, respectively negative eigenvalues of the linking matrix. This quantity is invariant under addition of trivial handles, and also under handleslides, by Theorem 6.1, so it is a topological invariant of M.

Remark 7.3. This is the Murakami-Ohtsuki-Okada invariant [21]! Here we derived its existence directly from the theory of theta functions. The second application of the exact Egorov identity is the construction of a Sikora isomorphism, which identiﬁes the reduced linking number skein modules of two manifolds with homeomorphic boundaries. Let us point out that such an isomorphism was constructed for reduced Kauﬀman bracket skein modules in [27].

Theorem 7.4. Let M1 and M2 be two 3-dimensional manifold with homeomorphic boundaries. Then ∼ LN (M1) = LN (M2).

Proof. Because the manifolds M1 and M2 have homeomorphic boundaries, there is a framed link L1 ⊂ M1 such that M2 is obtained by performing surgery on L1 in M1.LetN1 be a regular neighborhood of L1 in M, which is the union of several solid tori, and let N2 be the union of the surgery tori in M2.Thecoresof these tori form a framed link L2 ⊂ M2,andM1 is obtained by performing surgery on L in M .EveryskeininM , respectively M can be isotoped to one that misses 2 2 1 2 ∼ N1, respectively N2. The homeomorphism M1\N1 = M2\N2 yields an isomorphism

φ : LN (M1\N1) →LN (M1\N1).

However, this does not induce a well deﬁned map between LN (M1)andLN (M2) because a skein can be pushed through the Ni’s. To make it well deﬁned, the

FROM THETA FUNCTIONS TO TQFT 61 skein should not change when pushed through these regular neighborhoods. We use Theorem 6.1 and deﬁne

F1 : LN (M1) →LN (M1),F1(σ)=φ(σ) ∪ Ω(L1) −1 F2 : LN (M2) →LN (M1),F2(σ)=φ (σ) ∪ Ω(L2). By Proposition 5.1 b) we have −1 Ω(L1) ∪ Ω(φ (L2)) = ∅∈Lt(M1), −1 since each of the components of φ (L2) is a meridian in the surgery torus, hence it surrounds exactly once the corresponding component in L1. This implies that F2 ◦ F1 = Id. A similar argument shows that F1 ◦ F2 = Id. One should note that the Sikora isomorphism depends on the surgery diagram. Now it is easy to describe the reduced linking number skein module of any manifold. Proposition 7.5. For every oriented 3-dimensional manifold M having the boundary components Σgi , i =1, 2,...,n, one has n ∼ Ngi LN (M) = C . i=1 3 Proof. If M has no boundary then LN (M)=LN (S )=C,andifM is 3 3 bounded by a sphere, then LN (M)=LN (B )=C,whereB is the 3-dimensional ball. If M has one genus g boundary component with g ≥ 1, then LN (M)= Ng LN (Hg)=C by Theorem 4.7. To tackle the case of more boundary components we need the following result:

Lemma 7.6. Given two oriented 3-dimensional manifolds M1 and M2,let M1#M2 be their connected sum. The map

LN (M1) ⊗LN (M2) →LN (M1#M2) deﬁned by (σ, σ) → σ ∪ σ is an isomorphism. Proof. In M1#M2, the manifolds M1 and M2 are separated by a 2-dimen- 2 N−1 sional sphere Ssep.EveryskeininM1#M2 can be written as j=0 σj,whereeach σ intersects S2 in j strands pointing in the same direction. A trivial skein colored j sep 2 N−1 by Ω is equal to the empty link. But when we slide it over Ssep it turns j=0 σj into σ0. This shows that the map from the statement is onto. On the other hand, the reduced linking number skein module of a regular 2 C neighborhood of Ssep is since every skein can be resolved to the empty link. This means that, in M1#M2, if a skein that lies entirely in M1 canbeisotopedtoaskein that lies entirely in M2, then this skein is a scalar multiple of the empty skein. So ∪ ∪ L L if σ1 σ1 = σ2 σ2,thenσ1 = σ2 in N (M1)andσ1 = σ2 in N (M2). Hence the map is one-to-one, and we are done. Returning to the theorem, an oriented 3-manifold with n boundary components can be obtained as surgery on a connected sum of n handlebodies. The conclusion follows by applying the lemma. If M is a 3-manifold without boundary, then Theorem 7.4 shows that ∼ 3 LN (M) = LN (S )=C.

62 RAZVAN˘ GELCA AND ALEJANDRO URIBE

If we describe M as surgery on a framed link L with signature zero, which is always possible by adding trivial link components with framing ±1, then the Sikora isomorphism maps the empty link in M to the vector

3 Z(M)=Ω(L) ∈LN (S )=C.

More generally, M can be endowed with a framing deﬁned by the signature of the 4-dimensional manifold W that it bounds constructed as explained before. If L is the surgery link that gives rise to M and W , then to the framed manifold (M,sign(W )) = (M,m) we can associate the invariant

3 Z(M,m)=Ω(L) ∈LN (S ).

The Sikora isomorphism associated to L identifies this invariant with the empty link in M. All these can be generalized to manifolds with boundary. A 3-dimensional manifold M with boundary can be obtained by performing surgery on a framed link 3 L in the complement M0 of n handlebodies embedded in S , n ≥ 1. Endow M with a framing by filling in the missing handlebodies in S3 and constructing the 4-dimensional manifold W with the surgery instructions from L. To the manifold (M,sign(W )) = (M,m) we can associate the skein ∅∈LN (M). A Sikora isomorphism allows us to identify this vector with Ω(L) ∈LN (M0). Another Sikora isomorphism allows us to identify Lt(M0) with the reduced linking number skein ⊗n CNgi module of the connected sum of handlebodies, namely with i=1 ,wheregi, i =1, 2,...,n are the genera of the boundary components of M. Via Proposi- tion 7.5, Ω(L) can be identified with a vector

n Z(M,m) ∈ CNgi . i=1

There is another way to see this identiﬁcation of the linking number skein ⊗n CNgi module with i=1 , done in the spirit of [26]. In this alternative formalism, Ω(L) acts as a linear functional on

n → C Ω(L): LN (Hgi ) , i=1

3 by gluing handlebodies to M0 as to obtain S . In this setting, the formalism developed in [30] Chapter IV applies to show that the vector

∗ N n Z(M,m) ∈ CNgi = CNgi i=1 i=1 is well deﬁned once the framing m is ﬁxed. Let us point out that this can be modeled with the quantum group of abelian Chern-Simons theory in the framework of [9].

FROM THETA FUNCTIONS TO TQFT 63

The construction ﬁts Atiyah’s formalism of a topological quantum ﬁeld theory (TQFT) [3] with anomaly [30]. In this formalism • ∪ ∪···∪ to each surface Σ = Σg1 Σg2 Σgn we associate the vector space n V (Σ) = CNgi i=1 which is isomorphic to the reduced linking number skein module of any 3-dimensional manifold that Σ bounds. • to each framed 3-dimensional manifold (M,m) we associate the empty link in LN (M). As a vector in V (∂M), this is Z(M,m). Atiyah’s axioms are easy to check. Functoriality is obvious. The fact that Z is involutory namely that Z(Σ∗)=Z(Σ)∗ where Σ∗ denotes Σ with opposite orientation follows by gluing a manifold M bounded by Σ to a manifold M ∗ bounded by Σ∗ and using the standard pairing ∗ ∗ LN (M) ×LN (M ) →LN (M ∪ M )=C. Let us check the multiplicativity of Z for disjoint union. If Σ and Σ are two surfaces, we can consider disjoint 3-dimensional manifolds M and M such that ∂M =Σ and ∂M =Σ.Then Z(Σ ∪ Σ )=LN (M ∪ M )=LN (M) ⊗LN (M )=Z(Σ) ⊗ Z(Σ ). Also ∅∈LN (M ∪ M )equals∅⊗∅∈LN (M) ⊗LN (M ). If we now endow M and M with framings m, respectively m,then Z(M ∪ M ,m+ m)=Z(M,m) ⊗ Z(M ,m), since the Sikora isomorphism acts separately on the skein modules of M and M . What about multiplicativity for manifolds glued along a surface? Let M1 and ∗ M2 be 3-dimensional manifolds with ∂M1 =Σ∪ Σ and M2 =Σ ∪ Σ , and assume that M1 is glued to M2 along Σ. Then the empty link in M ∪ M is obtained as the union of the empty link in M with the empty link in M . It follows that − iπ τ Z(M1 ∪ M2,m1 + m2)=e 4 Z(M1,m1),Z(M2,m2) , where , is the contraction V (Σ) ⊗ V (Σ) ⊗ V (Σ)∗ ⊗ V (Σ) → V (Σ) ⊗ V (Σ), and τ expresses the anomaly of the TQFT and depends on how the signature of the surgery link changes under gluing (or equivalently, on how the signatures of the 4-dimensional manifolds bounded by the given 3-dimensional manifolds change under the gluing, see Section 8). Finally, Z(∅)=C, because the only link in the void manifold is the empty link. Also, if M =Σ× [0, 1], then the empty link in M is the surgery diagram of the identity homeomorphism of Σ and hence Z(M,0)canbeviewedastheidentity map in End(V (Σ)). This TQFT is hermitian because V (∂M), being a space of theta functions, has the inner product introduced in Section 2. And if M is a 3-dimensional manifold and M ∗ is the same manifold but with reversed orientation, then the surgery link L∗ of M ∗ is the mirror image of the surgery link L of M. The invariant of M is computed by smoothing the crossings in L while the invariant of M ∗ is computed by smoothing the crossings in L∗, whatever was a positive crossing in L becomes a

64 RAZVAN˘ GELCA AND ALEJANDRO URIBE negative crossing in L∗ and vice-versa. Hence sign(L∗)=−sign(L). Also, because iπ −1 ∗ for t = e N one has t = t¯, and hence Ω(L )=Ω(L). It follows that Z(M ∗, −m)=Z(M,m) as desired.

8. The Hermite-Jacobi action and the non-additivity of the signature of 4-dimensional manifolds An interesting coincidence in mathematics is the fact that the Segal-Shale-Weil cocycle of the metaplectic representation [17] and the non-additivity of the signature of 4-dimensional manifolds under gluings [32] are both described in terms of the Maslov index. We explain this coincidence by showing how to resolve the pro- jectivity of the Hermite-Jacobi action using 4-dimensional manifolds. Note that the theory of theta functions explains the coincidence of the cocycle of the metaplectic representation and the cocycle of the Hermite-Jacobi action. Each element h of the mapping class group of Σg can be represented by surgery on a link Lh ∈ Σg × [0, 1], meaning that surgery along Lh yields a manifold that is homeomorphic to Σg × [0, 1] is such a way that the homeomorphism is the identity maponΣg ×{0} and h on Σg ×{1}. The link Lh is not necessarily obtained from writing h as a composition of Dehn twists, as in Proposition 5.3.

Theorem 8.1. Let h be an element of the mapping class group of Σg obtained by surgery on the framed link Lh in Σg ×[0, 1]. Then the discrete Fourier transform ρ(h):LN (Hg) →LN (Hg) is given by

ρ(h)β =Ω(Lh)β.

Proof. Write h = T1T2 ···Tn,whereTj are Dehn twists obtained as surgeries along the curves γj , j =1, 2,...,n. If we glue two handlebodies so as to obtain 3 S , the gluing deﬁnes a nondegenerate pairing [·, ·]:LN (Hg) ×LN (Hg) → C.Ifwe consider basis elements ej ,ek in LN (Hg), then [Ω(Lh)ej ,ek] completely determines the operator deﬁned by Ω(Lh). Because of invariance under handle slides, we can transform Ω(Lh)intoΩ(γ1)Ω(γ2) ···Ω(γn) such that

[Ω(Lh)ej ,ek] = [Ω(γ1)Ω(γ2) ···Ω(γn)ej ,ek].

Hence the operators deﬁned by Ω(Lh)andΩ(γ1)Ω(γ2) ···Ω(γn) are equal. The conclusion follows from Proposition 5.3.

Fix a Lagrangian subspace L of H1(Σg, R) and consider the closed 3-manifold × 0 M obtained by gluing to the surgery of Σg [0, 1] along L the handlebodies Hg 1 0 ×{ } 1 ×{ } and Hg such that ∂Hg =Σg 0 , ∂Hg =Σg 1 ,andL respectively h∗(L)are 0 1 the kernels of the inclusion of Σg into Hg respectively Hg . M is the boundary of a 4-manifold W obtained by adding 2-handles to the ball B4 as prescribed by L. The discrete Fourier transform ρ(h)isaskeininLN (Σg × [0, 1]) which is uniquely determined once we ﬁx the signature of W . Hence to the pair (h, n) where h is an element of the mapping class group and n ∈ Z, we associate uniquely askeinF(h, n), its discrete Fourier transform. We identify the pair (h, n) with (h, sign(W )) where W is a 4-dimensional manifold deﬁned as above. Note that by adding trivial 2-handles we can enforce sign(W )tobeanyinteger.

FROM THETA FUNCTIONS TO TQFT 65

Consider the Z-extension of the mapping class group defined by the multiplication rule (h, sign(W )) ◦ (h, sign(W )) = (h ◦ h, sign(W ∪ W )) 0 ∈ 1 where W and W are glued in such a way that Hg W is identified with Hg in W .IfL is the Lagrangian subspace of H1(Σg, R) used for defining W ,then necessarily L = h∗(L). Recall Wall’s formula for the non-additivity of the signature of 4-dimensional manifolds sign(W ∪ W )=sign(W )+sign(W ) − τ(L,h∗(L),h∗ ◦ h∗(L)), where τ is the Maslov index. By using this formula we obtain F(h ◦ h, sign(W ∪ W )) = F(h ◦ h, sign(W )+sign(W ) − τ(L,h∗(L),h∗ ◦ h∗(L))

iπ − τ(L,h∗(L),h ◦h∗(L)) = e 4 ∗ F(h ◦ h, sign(W )+sign(W ))

iπ − τ(L,h∗(L),h ◦h∗(L)) = e 4 ∗ F(h , sign(W ))F(h, sign(W )), where for the second step we changed the signature of the 4-dimensional manifold associated to h ◦ h by adding trivial handles and used Proposition 7.1. Or equivalently, if we let ρ(h, sign(W )) be discrete the Fourier transform associated to h, normalized by the (signature of) the manifold W ,then

iπ − τ(L,h∗(L),h ◦h∗(L)) ρ(h ◦ h, sign(W ∪ W )) = e 4 ∗ ρ(h , sign(W ))ρ(h, sign(W )). In this formula we recognize the Segal-Shale-Weil cocycle

iπ − τ(L,h∗(L),h ◦h∗(L)) c(h, h )=e 4 ∗ used for resolving the projective ambiguity of the Hermite-Jacobi action, as well as for resolving the projective ambiguity of the metaplectic representation.

9. Theta functions and abelian Chern-Simons theory By Jacobi’s inversion theorem and Abel’s theorem [6], the Jacobian of a surface Σg parametrizes the set of divisors of degree zero modulo principal divisors. This is the moduli space of stable line bundles, which is the same as the moduli space Mg(U(1)) of flat u(1)-connections on the surface (in the trivial U(1)-bundle). The moduli space Mg(U(1)) has a complex structure defined as follows (see for 1 example [13]). The tangent space to Mg(U(1)) at an arbitrary point is H (Σg, R), which, by Hodge theory, can be identified with the space of real-valued harmonic 1-forms on Σg. The complex structure is given by Jα = −∗α, where α is a harmonic form. In local coordinates, if α = udx + vdy,thenJ(udx + vdy)=vdx − udy. If we identify the space of real-valued harmonic 1-forms with the space of holo- (1,0) morphic 1-forms H (Σg) by the map Φ given in local coordinates by Φ(udx + vdy)=(u − iv)dz, then the complex structure becomes multiplication by i in (1,0) H (Σg). The moduli space is a torus obtained by exponentiation 1 2g MU(1) = H (Σg, R)/Z .

66 RAZVAN˘ GELCA AND ALEJANDRO URIBE

If we choose a basis of the space of real-valued harmonic forms α1,α2,...,αg, β ,...,β such that 1 g

(9.1) αk = δjk, αk =0, βk =0, βk = δjk, aj bj aj bj then the above Z2g is the period matrix of this basis. On the other hand, if ζ1,ζ2,...,ζg are the holomorphic forms introduced in −1 −1 − Section 2, and if αj =Φ (ζj)andβj =Φ ( iζj ), j =1, 2,...,g then one can compute that αk = δjk, αk =Reπjk, βk =0, βk =Imπjk. aj bj aj bj The basis α1,...,αg,β1,...,βg determines coordinates (X ,Y ) in the tangent space to MU(1). If we consider the change of coordinates X + iY = X +ΠY , then the moduli space is the quotient of Cg by the integer lattice Z2g.Thisis exactly what has been done in Section 2 to obtain the jacobian variety. This shows that the complex structure on the Jacobian variety coincides with the standard complex structure on the moduli space of flat u(1)-connections on the surface. The moduli space Mg(Σg) has a symplectic structure defined by the Atiyah- Bott form [4]. This form is given by ω(α, β)=− α ∧ β, Σg where α, β are real valued harmonic 1-forms, i.e. vectors in the tangent space to Mg(Σg). If αj ,βj , j =1, 2,...,g are as in (9.1), then ω(αj ,αk)=ω(βj ,βj)=0 and ω(αj ,βk)=δjk (which can be seen by identifying the space of real-valued 1 harmonic 1-forms with H (Σg, R) and using the topological definition of the cup product). This shows that the Atiyah-Bott form coincides with the symplectic form introduced in Section 2. For a u(1)-connection A and curve γ on the surface, we denote by holγ (A)the holonomy of A along γ.ThemapA → trace(holγ (A)) induces a function on the 4 Jacobian variety called Wilson line. If [γ]=(p, q) ∈ H1(Σg, Z), then the Wilson line associated to γ is the function (x, y) → exp 2πi(pT x + qT y). These are the functions on the Jacobian variety of interest to us. The goal is to quantize the moduli space of flat u(1)-connections on the closed Riemann surface Σg endowed with the Atiyah-Bott symplectic form. One procedure has been outlined in Section 2; it is Weyl quantization on the 2g-dimensional torus in the holomorphic polarization. Another quantization procedure has been introduced by Witten in [35]using Feynman path integrals. In his approach, states and observables are defined by path integrals of the form i L(A) e h trace(holγ (A))DA, A where L(A) is the Chern-Simons lagrangian 1 2 L(A)= tr A ∧ dA + A ∧ A ∧ A . 4π Σg ×[0,1] 3

4Since we work with the group U(1), the holonomy is just a complex number of absolute value 1, and the trace is the number itself.

FROM THETA FUNCTIONS TO TQFT 67

According to Witten, states and observables should be representable as skeins in the skein modules of the linking number discussed in Section 4. Witten’s quantization model is symmetric with respect to the action of the mapping class group of the surface, a property shared by Weyl quantization in the guise of the exact Egorov identity (2.5). As we have seen, the two quantization models coincide. It was Andersen [1] who pointed out that the quantization of the Jacobian that arises in Chern-Simons theory coincides with Weyl quantization. For non-abelian Chern-Simons theory, this phenomenon was ﬁrst observed by the authors in [10]. In the sequel of this paper, [11], we will conclude that, for non-abelian Chern- Simons, the algebra of quantum group quantizations of Wilson lines on a surface and the Reshetikhin-Turaev representation of the mapping class group of the surface are analogues of the group algebra of the ﬁnite Heisenberg group and of the Hermite- Jacobi action. We will show how the element Ω corresponding to the group SU(2) can be derived by studying the discrete sine transform.

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Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 E-mail address: [email protected] Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12416

Toric Topology

Samuel Gitler

1. Introduction Toric Topology is the study of the following spaces: (1) Polyhedral product spaces. (2) Geometric realization of monomial ideal rings. (3) Toric manifolds. In algebraic geometry, toric varieties had been studied since the 1970’s, see [9]. In 1991 Davis and Januszkiewicz wrote a seminal paper [10], where they introduced a topological version of toric varieties, which they were always possible to obtain as a quotient of a free action of a torus on a space, which is a special kind polyhedral product space. In 2008, I attended a symposium in Japan in toric topology. I started a collaboration with Anthony Bahri, Martin Bendersky and Federick Cohen on the subject. This paper is a survey in our progress in the study of this ﬁeld.

2. Polyhedral product spaces Strickland [14] defined a family of spaces which are the polyhedral product spaces, but did not study them. The polyhedral product functor is a functor of two variables (1) A finite abstract simplicial complex K with n vertices { }n (2) A family (X,A)= (Xi,Ai,xi) i=1 of pointed pairs of CW-complexes. The morphisms in (1) are inclusions and the morphisms in (2) are based continuous maps of pairs. The concept of full subcomplexes of K were introduced by Hochster in [13]. Let K be a family of ordered subsets of [n]=(1, 2, ..., n) satisfying: (1) The empty subset ∅∈K and every i ∈ [n] belongs to K (2) If I is an ordered subset of [n]andI ⊂ σ,thenI ∈ K. K is thus an abstract simplicial complex with n vertices. Let I ⊂ [n], then define KI by

KI = {σ ∩ I | σ ∈ K}.

KI is called a full subcomplex of K.

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Höchster studied a particular family of polyhedral products, the moment angle complexes, denoted by Z(K;(D2,S1)) where (X,A)={(D2,S1)} and with D2 the 2diskandS1 in boundary circle. Höchster’s Theorem: The cohomology group of Z(K;(D2,S1)),Hq(Z(K;(D2,S1))), q−|I|−1 | | is isomorphic to I H ( KI ; Z) for all q. As topologists, we thought there should be a geometric result which implied Höchster’s theorem. We introduced an associated functor, which we called the smashed polyhedral product functor. { }n Given K and (X,A )= (Xi,Ai,xi) i=1 as above, n n Let i=1 Yi and i=1 Yi be the n-cartesian product and the n-smash product of spaces respectively. Now for every simplex σ of K,let

D(σ)=ΠYi Xi if i ∈ σ where Yi = Ai if i ∈ [n] − σ D(σ)= Yi. ∅ n ∅ n D( )= i=1 Ai and D( )= i=1 Ai. Now deﬁne the polyhedral product spaces Z(K;(X,A)) = D(σ) ⊂ Xn, σ∈K Z(K;(X,A)) = (D(σ)) ⊂ Xn. σ∈K

If I ⊂ [n], let (X,A)I = {(Xi,Ai,xi)}i∈I . 1 Let ΣX be the reduced suspension of X, i.e., ΣX = S∧ X,wherewetake 1 1 ∈ S as the base point and xi ∈ X is its base point, and Xi the wedge of the Xi, i.e. the one point union of the based Xi. Now we can state our main theorem. Theorem 2.1. Given K and (X,A) as above we have a natural homotopy equivalence: ΣZ(K;(X,A)) −→ Σ ∨I⊂[n] Z(KI ;(X,A)I ) ρI induced by the projection Z(K;(X,A)) −→ Z(KI ); (X,A)I ). Now we needed to identify the spaces Z(KI ;(XI ,AI )). We proved the next two theorems: Theorem 2.2. If A is contractible for all i,then i pt if X is not a simplex Z(K ;(X,A) )= I I I ∧ ∧ Xi1 ... Xik where I =(i1, ..., ik),otherwise.

Theorem 2.3. If Xi is contractible for all i,then ∼ I Z(KI ;(X,A)) = Σ | Ki |∧A .

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When we let (X,A)=(D2,S1), we obtain

Corollary 2.4. 2 1 |I|+1 Z(KI ;(D ,S )) ∼ Σ |KI |, where |I| is the length of I,and|KI | is the geometric realization of KI . ∼ − Using the fact that Hq(ΣX) = Hq 1(X) for any space X, we recover H¨ochster’s Theorem. A communication of the paper appeared in [1] and the full paper in [2], where many applications were obtained.

3. Cup products in polyhedral products H∗ ∗ Let (K;(X,A)= I H (Z(KI ;(X,A)I ))) then we have a degree preserving additive isomorphism

ϕ H∗(K;(X,A)) −→ H∗(Z(K;(X,A))).

We deﬁne partial diagonals Z(KJ∪L;(X,A)J∪L) −→ Z(KJ ;(X,A)J ) ⊗ Z(KL;(X,A)L) which induce a ∗-product satisfying

ϕ(u ∗ v)=ϕ(u) ϕ(v) making ϕ a ring isomorphism. This allowed us to compute cups products in some polyhedral products spaces. These results appeared in [3]

4. Rational homotopy of moment angle complexes Felix and Halperin proved a dichotomy on the rational homotopy of ﬁnite simply connected CW-complexes. If X is such a space, then:

(1) either the rational homotopy groups π∗(X) ⊗ Q is a ﬁnite vector space, in which case X is called rationally elliptic or (2) πq(X) ⊗ Q grows exponentially in which case X is rationally hyperbolic. In [6]weprove:

Theorem 4.1. A moment angle complex is rationally elliptic if and only if X is of the homotopy type of a product of odd spheres.

As a corollary we obtain:

Theorem 4.2. If L(X) denotes the free loop space of X. Then the rational homology Poincare series of L(Z(K)) is a rational functional if and only if Z(K) is of the homotopy type of S2d1+1 × ... × S2dr +1.

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5. Geometric realization of monomial ideal rings

Let us survey the results of [4]. Let Pn = Z[x1, ..., xn] be the integral polynomial algebra on n generators. Each generator will have geometric dimension 2. Let t t { }k i1 in ≥ n M = mi i=1 be k monomials mi = x1 ...xn with ti 0 and degree j=1 tij , such that

(0) xi appearsisatleastonemj for every 1 ≤ i ≤ n (1) degree mi ≥ 2 (2) no mi divides mj for all i = j.

Let I(M) be the ideal generated by M in Pn.ThenA(M)=Pn/I(M)isa monomial ideal ring. When all the mi are square free monomials, the algebra is called a Stanley Riesner algebra and will be denoted by SR(M). The algebra SR(M) has a geometric realization. Let K(M) be the abstract simplicial complex on n vertices with monomial ∞ non-faces given by the mi.ThenZ(K(M); (CP , ∗)) is a geometric realization of SR(M), such that ∗ ∞ ∼ H (Z(M), (CP , ∗)) = SR(M). We construct a geometric realization of A(M), but it will not be in general a polyhedral product space, denoted by W (A(M)), which is a CW-complex and satisﬁes ∗ ∼ H (W (A(M))) = A(M) as a graded algebra. { }k Given M a family of monomials with arbitrary powers M = mi i=1, with t t m = x i1 ...x in . i i1 in Let m(M)=LCM(mi ∈ M) where LCM is the least common multiple, then

t1 ··· tn m(M)=x1 xn . We introduce a set of new variables, its number being: n d(t)= ti, i=1

x11,...,x1t1 ,x21,x22,...,x2t2 ,...,xn1,...,xntn .

ti1 tin If mi = x1 ...xn is a monomial in M,let m = x ···x · x ...x ...x ...x , i i1,1 i1,ti1 i2,1 i2,ti2 in,1 in,tin { }k which is a monomial with ti1 + ... + tin variables. Let M = m¯ i i=1.ThenM is a set of square free monomials. It produces an abstract simplicial complex K(M). Let Z(K(M); (D2,S1)) = Z(K(M)). Then the torus T 1 = S1 acts on (D2,S1) since D2 and S1 are subspaces of C closed under multiplication by S1. Hence T d(t) acts on Z(K(M)). The Borel ﬁbration for the action is: −→ d(t) × −→ d(t) Z(K(M)) ET T d(T ) Z(K(M)) BT . We deﬁne a “diagonal map”

BTn −Δ→ BTd(t),

TORIC TOPOLOGY 73 where Δ(a1, ..., an)=(a1, ..., a1,a2, ..., a2, ..., an, ..., an).

t1 t2 tn Using Δ we obtain a diagram of ﬁbrations: Z(K(M)) /Z(K(M)) /∗ .

W (A(M)) /ETd(t) × Z(K(M)) /BTd(t)−n

∗ BTn Δ /BTd(t) /BTd(t)−n where W (A(M)) is the pullback. We prove that ∗ ∼ H (W (A(M))) = A(M). by looking at the middle ﬁbration, and showing that the Eilenberg-Moore spectral sequence collapses, giving that H∗(W (M)) is even dimensional and so is H∗(BTm), hence the Serre spectral sequence collapses and the result follows.

6. Infinite families of toric manifolds associated to a given one A toric manifold is a triple (M 2n,Tn,Pn)whereM 2n is a closed differentiable manifold of dimension 2n. T n is the n-torus acting on M 2n and P n is a simple convex polytope which is the orbit space of the action of T n on M 2n.LetK be the abstract simplicial complex dual to the boundary of the polytope, i.e, its vertices correspond to the facets of P n and the k-simplices correspond to the n−k −1 faces n of P .IfK has m vertices, let J =(j1, ..., jm) be a sequence of m positive integers. Let − − 2J 2J 1 { 2ji 2ji 1 }m (D ,S )= (D ,S ) i=1 then form the polyhedral product Z(K;(D2J ,S2J−1)). Let T 1 = S1 act on D2ji ⊂ C by · t (z1, ..., zji )=(tz1, ..., tzji ). This action extends to a T m action on Z(K;(D2J ,S2J−1)). Now the space Z(K)= Z(K;(D2,S1)) is also acted on by T m, and Davis-Januszkiewicz showed that there exists a subgroup G(M)ofT m isomorphic to T m−n, which acts differentiably on Z(K) with orbit space M 2n.NowG(M) acts freely and differentiably also on ∼ − Z(K(J)) = Z(K;(D2J ,S2J 1)), so taking the quotient of Z(K(J)) by G(M)we abtain a manifold of dimension 2d(J) − 2m − 2n, which we called M (J). We could not prove directly that it was a toric manifold so we constructed a polytope P n(J) and a toric manifold M(J), which was equivariantly diffeomorphic to M (J). The cohomology of a toric manifold M has a very nice compact description: ∗ 2n ∼ H (M ,Z) = Z[x1, ..., xm]/I(K)+L(M), where dim xi =2,K is the dual of the boundary of K, m number of vertices of K, I(K) is the ideal generated by the minimal non-faces of K and L(M)isanideal generated by n forms of dim 2, which are linearly independent, and are determined uniquely by M.

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The cohomology of M(J)hasareduced presentation and is ∗ H (M(J)) = Z[x1, ..., xm]IJ (K)+L(M), where m is as above, are the J-weighted minimal non-faces of K. The generators of IJ (K) correspond to the generators of I(K)asfollows:If j j x ···x is a generator of I(K), then x i1 ...x ik is the corresponding generator i1 ik i1 ik of IJ (K). In other words, IJ (K) is an ideal of a monomial ideal ring. In this case this monomial ideal algebra has geometric realization Z(K;(CP ∞, CP J−1)), where CP ∞ is the inﬁnite complex projective space and CP J−1 is a family of ﬁnite dimensional projective space. The manifold M 2n has stable tangent bundle isomorphic to a sum of complex line boundles α1 + ... + αm where c1(αi)=xi,wheretheαi are not independent as 2 2n rank H (M )=m − n. 2n The manifolds M (J) has stable tangent bundle jiαi and we have nests ⊂ ⊂ ⊂ ⊂ m−1C ∞ M M(J1) ... M(Jk) ...Πi=1 P of codimension 2 embedding with normal boundle. The complex projective spaces CP n are the canonical toric manifolds and they are obtained as CP 1(J)byvarying J. These sequences are generalization of those for CP 1.

7. Intersection of quadrics, moment angle manifolds, and connected sum The study of intersection of quadrics began by S. Lopez de Medrano [12]and continuedbyBocioandMeerssemanin[7], which showed that they correspond to moment angle manifolds, is used in the paper by Samuel Gitler and Santiago Lopez de Medrano [11]. In this paper, the main theorem is Theorem 7.1. If Z = Z(P ) is a connected sum of sphere products. Then the J following operation on Z: Z , Zv and Ze are also connected sum of sphere products. Here sphere products is a product of two spheres ZJ is the J-construction applied to Z, Zv is obtained by cutting a vertex v to the polytope P and Ze is cutting an edge e to the polytope P . We also describe a cup products in Z(K;(D1,S0)) showing that cup products in Z(K;(D2,S0)) are much harder to determine that those of Z(K;(D2,S1))which are determined by a Tor algebra [8]. In this paper we also showed that the cup products in Z(K;(D1,S0)) and in Z(K;(D2,S0)) are very different. We have that Z(K;(D2,S1)) and Z(K;(D1,S0)) have ungraded additively isomorphic cohomology rings, but in general not as rings. Studying moment angle manifolds by intersection of quadrics has allowed Lopez de Medrano, L. Merseman and A. Verjosky to study the differential topology of these spaces. This method of studying moment angle manifolds has produced very interesting new examples of non-Kählerian complex manifolds, (the so-called LV - M manifolds) and of contact manifolds (personal communication by Alberto Verjovsky).

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References [1] A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces,Proc.Natl. Acad.Sci.USA106 (2009), no. 30, 12241–12244, DOI 10.1073/pnas.0905159106. MR2539227 (2010j:57036) [2] A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), no. 3, 1634–1668, DOI 10.1016/j.aim.2010.03.026. MR2673742 (2012b:13053) [3]A.Bahri,M.Bendersky,F.R.CohenandS.Gitler.Cup-products in generalized moment- angle complexes. Math. Proc. Camb. Phil. Soc. 2012. [4]A.Bahri,M.Bendersky,F.R.CohenandS.Gitler.On monomial ideal rings and a theorem of Trevisan. arXiv:1205.6258 [5]A.Bahri,M.Bendersky,F.R.CohenandS.Gitler.Operations on Polyhedral products and a new topological construction of infinite families of toric manifolds associated to a given one. arXiv:1011.0094 [6]A.Bahri,M.Bendersky,F.R.CohenandS.Gitler.On the Rational Type of Moment Angle Complexes. arXiv:1206.2173 [7] Frédéric Bosio and Laurent Meersseman, Real quadrics in Cn, complex manifolds and convex polytopes,ActaMath.197 (2006), no. 1, 53–127, DOI 10.1007/s11511-006-0008-2. MR2285318 (2007j:32037) [8] V. M. Bukhshtaber and T. E. Panov, Actions of tori, combinatorial topology and homological algebra (Russian, with Russian summary), Uspekhi Mat. Nauk 55 (2000), no. 5(335), 3– 106, DOI 10.1070/rm2000v055n05ABEH000320; English transl., Russian Math. Surveys 55 (2000), no. 5, 825–921. MR1799011 (2002a:57051) [9] V. I. Danilov, The geometry of toric varieties (Russian), Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247. MR495499 (80g:14001) [10] Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451, DOI 10.1215/S0012-7094-91-06217-4. MR1104531 (92i:52012) [11] Samuel Gitler and Santiago López de Medrano, Intersections of quadrics, moment- angle manifolds and connected sums,Geom.Topol.17 (2013), no. 3, 1497–1534, DOI 10.2140/gt.2013.17.1497. MR3073929 [12] Santiago López de Medrano, Topology of the intersection of quadrics in Rn, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 280– 292, DOI 10.1007/BFb0085235. MR1000384 (90f:58015) [13] Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes,Ringthe- ory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), Dekker, New York, 1977, pp. 171–223. Lecture Notes in Pure and Appl. Math., Vol. 26. MR0441987 (56 #376) [14] N. Strickland. Private communication.

Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000, Mexico,´ D.F., Mexico´ E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12417

Beilinson Conjecture for Finite-dimensional Associative Algebras

Dmitry Kaledin

Introduction The phrase “Beilinson conjectures”, or, more precisely, “Beilinson’s conjectures on special values of L-functions” refers to a set of conjectures formulated by A. Beilinson in [Be]. The subject is, roughly speaking, relations between algebraic K- groups of a smooth proper algebraic variety X over Q and its de Rham cohomology groups equipped with the Hodge structure. The conjectures explain and tie together nicely several disparate known facts and present a unified picture of great beauty and great simplicity. One wants to believe in them to such an extent that there are some papers where the author states right away that everything takes place in a “Beilinson World” – the world where the conjectures are true. However, the conjectures are notoriously difficult: in the 30 years that passed since [Be], we are not much closer to knowing whether we live in a Beilinson World or not. Formally, the eventual goal of the conjectures is to predict, up to a rational multiple, the values of L-functions of the algebraic variety X at integer points. This follows earlier conjectures by Deligne, where the values was predicted in some cases, and expressed in terms of the “period matrix” relating natural Q-structures in the de Rham cohomology and the Betti cohomology of X. To get the individual values, one has to also take account of the Hodge filtration on de Rham cohomology; in the end, for every integer, one produces an R-vector space with two independent Q-structures, and the determinant of the corresponding transition matrix should give the L-values. However, this only works in some cases; in the general case, one ends up with a R-vector space with only one obvious Q-structure, and it is not clear how to produce a number out of that. Beilinson’s insight was to add algebraic K-groups to the picture. The vector space in question is the target of a certain natural “regulator map” from K-groups; Beilinson then conjectures that, roughly speaking, the regulator map becomes a isomorphism after tensoring with R, so that we obtain a second Q-structure, and that the determinant of the resulting period matrix gives the L-values. In the present note, we are only concerned with the first part of the conjectures – namely, with the prediction that the regulator map becomes an isomorphism after

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78 DMITRY KALEDIN tensoring with R (this is why we use the singular “conjecture” in the title). We only do two things: we observe that the conjecture can actually be stated so that it makes sense for a non-commutative algebraic variety, and then we observe that the conjecture thus stated is almost trivially true for finite dimensional associative algebras (we do not really give a complete proof, but we do give a detailed sketch). This is boring, but not as boring as the corresponding commutative statement – finite-dimensional associative algebras are not as trivial as the commutative ones. In fact, it is the basic principle of non-commutative algebraic geometry that every variety is derived-affine, so that were we able to extend our observation from algebras to DG algebras, we would be close to proving the conjecture in full generality. Needless to say, at this point we have no idea how to do that. The note consists of two parts. In Section 1, we review the relevant parts of Beilinson’s conjectures, with an elegant reformulation due to K. Kato presented in Section 2. In Section 3, we state the non-commutative version and show why it is true in the finite-dimensional algebra case.

Acknowledgments. This paper was presented as a talk at the annual Miami- Cinvestav-Campinas Homological Mirror Symmetry meeting, and written up during my stay at MSRI during the Non-commutative Geometry program in 2013. I am very grateful to the organizers of both events. I am also grateful to the referee for useful suggestions and corrections. This work was partially supported by RFBR grant 12-01-33024, Russian Federation government grant, ag. 11.G34.31.0023, and the Dynasty Foundation award.

1. Beilinson conjecture 1.1. R-Hodge structures. Recall that an R-mixed Hodge structure V is a triple R VR,VC,ϕ of an -vector space VR equipped with an increasing “weight” filtrationq W q,aC-vector space VC equipped with a decreasing “Hodge” filtration F ,andan ∼ q isomorphism ϕ : VR ⊗R C = VC such that W q and F satisfy certain non-degenracy conditions. The category R -MHS of R-mixed Hodge structures is abelian and has homological dimension 1. q q q R q An -mixed Hodge complex is a triple VR ,Vq C ,ϕ ofq a filtered complex VR ,W of R-vector spaces, a bifiltered complex VC ,Wq,F of C-vector spaces, and a filtered quasiisomorphism q ∼ q ϕ : VR ,Wq⊗R C = VC ,Wq q q i i such that in every degree i, the homology triple H (VR ),H (VC ),ϕ with the induced filtrations is an R-mixed Hodge structure. A map between mixed Hodge complexes is a quasiisomorphism if it induces a quasiisomorphism on the associated graded quotients with respect to filtration. One shows that inverting quasiisomorphisms in the category of R-mixed Hodge complexes, one obtains a triangulated category equivalent to the derived cate- D R R gory ( -MHS) (this is closely related to the fact thatq q -MHS hasq homologicalq dimension 1). For any R-mixed Hodge complex VR ,VC ,ϕ,Hom(R(0),V )in D(R -MHS) can be computed by the total complex of a bicomplex q q q q 0 (1.1) W0VR ⊕ (F VC ∩ W0VC ) −−−−→ W0VC , q where the horizontal differential is ϕ on W0VR and the embedding map on the second summand.

BEILINSON CONJECTURE FOR FINITE-DIMENSIONAL ASSOCIATIVE ALGEBRAS 79

The category R -MHS is a symmetric tensor category. The Hodge-Tate R- R Hodgeq structure (1) of weight 1 is an invertible objectq in thisq category, and for any V ∈D(R -MHS) and any integer i, one denotes V (i)=V ⊗ R(1)⊗i.

1.2. Absolute Hodge cohomology. Assume given a smooth projective algebraic variety X over Q.LetXan be the complex manifold underlying the algebraic variety X ⊗Q C. We then have comparison isomorphisms q q q ∼ ⊗ C ∼ ⊗ C HDR(Xan) = H (X Q ) = H (X) Q between de Rham cohomology groups. Moreover, by definition, all these groups are equipped with a Hodge filtration, and the comparison isomorphism q q q R ⊗ C ∼ C ∼ ϕ : H (Xan, ) R = H (Xan, ) = HDR(Xan) q equips the Betti cohomology groups H (Xan, R) with a mixed R-Hodge structure (pure of weight i in degree i). One can also refine the comparison isomorphism to a quasiisomorphism q q R ⊗ C ∼ ϕ : C (Xan, ) R = CDR(Xan) between the Betti and the de Rham cohomology complexes, and define an R-mixed Hodge complex q q q R (1.2) C (X)= C (Xan, ),CDR(Xan),ϕ , whereq the weight filtration W q is the canonical filtration, and the Hodge filtration F on the de Rham cohomology complex q q q CDR(Xan)=C (Xan, ΩX ) q is induced by the stupid filtration on the de Rham complex ΩX . The details of how one defines ϕ and the cohomology complexes are completely irrelevant, since the resulting mixed Hodge complex is anyway quasiisomorphic to the sum of its homology objects. Definition . q 1.1 For any integer j,theabsolute Hodge cohomology groups R HAH (X, (j)) are given by q q q R R HAH (X, (j)) = HomD(R -MHS)( (0),C (Xan)(j)), q where C (Xan) is the R-mixed Hodge complex of (1.2).

Note that by deﬁnition, absolute Hodge cohomology groups can be computedq by the bicomplex (1.1). We also note that since the Hodge ﬁltration on CDR(Xan) starts with F 0,wehave i R (1.3) HAH (X, (j)) = 0 for i>2j.

1.3. The conjecture. Beilinson formulates his conjectures in terms of the associated graded quotients of K-groups with respect to the γ-ﬁltration (“motivic cohomology” in current parlance). For our purposes, it will be convenient to stick to K-groups themselves.

80 DMITRY KALEDIN

To formulate the conjecture, one needs one additional ingredient. Consider ι the category R -MHS of mixed R-Hodge structures VR,VC,ϕ that are equipped → → with a additionalq involution ι : VR VR and an anticomplex involution ι : VC VC preserving F and W q and compatible with ϕ. This is again a symmetric tensor category of homological dimension one. Equip the Tate object R(1) with an involution ι acting by −1, and let R(i)=R(1)⊗i, i ∈ Z. The derived category D R ι ( -MHS ) can then again be obtained from an obvious ι-versionq q of the category of mixed Hodge complexes, and for any two such complexes V1 , V2 ,wehave q q q q q q ∼ ι RHomD(R -MHSι)(V1 ,V2 ) = RHomD(R -MHS)(V1 ,V2 ) , where in the right-hand side, weq take invariantsq with respect to the involution ι induced by the involutions on V1 and V2 . Assume given a smooth projective algebraic variety X over Q.Notethatsince X is defined over Q ⊂ R, the complexification X ⊗Q C carries a natural anticomplex involution ι, and it induces an anticomplex involutionq of the underlying analytic variety Xan. Therefore the mixed Hodge complex C (Xan) of Definition 1.1 becomes an object in D(R -MHSι), and the absolute Hodge cohomology groups acquire a ∼ natural involution ι. We note that under our convention, H2i(Pn) = R(−i)as objects in R -MHSι for 0 ≤ i ≤ n. Then Beilinson shows that there exists a canonical functorial regulator map ⊗ R → 2j−i R ι (1.4) r : Ki(X) Z HAH (X, (j)) j for any integer i ≥ 0, and he conjectures that it is an isomorphism for i ≥ 2. By virtue of (1.3), the conjecture can be extended to negative i, and in this case it is trivially true. The situation is different for i =0, 1, “critical values” in Beilinson’s terminology. This can be easily seen in the following two examples. • Let X =SpecF be the spectrum of a number field. Then by Borel Theorem, the ranks of higher K-groups of X are given by ⎧ ⎨⎪0,i=2j, j ≥ 1,

dim Ki(X) ⊗Z R = r ,i=4j − 1,j ≥ 1, ⎩⎪ 2 r1 + r2,i=4j +1,j ≥ 1,

where r1 and r2 are the number of real and complex embeddings of the ﬁeld K. This matches exactly the right-hand side of (1.4). However, ∗ K0(X)=Z,andK1(F )=F ,sothatK1(X)⊗ZR is inﬁnite-dimensional. • Let X be an elliptic curve. Then dim K0(E) ⊗Z R is an extremely deli- cate invariant predicted by the Birch-Swinnerton-Dyer Conjecture; it is expected to be related to the pole order of the L-function of X at 1, and it depends very essentially on the arithmetic properties of X.Onthe other hand, the right-hand side of (1.4) is the same for all elliptic curves.

To solve the obvious problem with K1(F ), Beilinson introduces a further conjecture – he conjectures that every smooth projective variety X over Q has a regular model XZ projective and ﬂat over Z. He then replaces K q(X) in (1.4) with the image of the natural map K q(XZ) → K q(X). In the case X =SpecF , XZ can be taken to be the spectrum of the ring of integers OF ⊂ F , and after tensoring with R,the procedure does not aﬀect higher K-groups. However, it changes K1 drastically: the

BEILINSON CONJECTURE FOR FINITE-DIMENSIONAL ASSOCIATIVE ALGEBRAS 81

DirichletUnitTheoremsaysthat ⊗ R O∗ ⊗ R K1(XZ) Z = F Z has dimension r1 + r2 − 1. This is still diﬀerent from the right-hand side of (1.4), but the diﬀerence of dimensions is now only 1; incidentally, it coincides with dimR K0(XZ) ⊗Z R. To take account of the remaining discrepancies, and to make things plausible for elliptic curves, too, Beilinson includes the so-called “height pairing” into the picture. We will now recall this. However, we prefer to repackage the whole story in a way that the author learned from K. Kato’s brilliant lectures at UChicago in 2010.

2. Kato’s reformulation We again start with linear algebra. Let us denote by R -HS the category of triples VR,VC,ϕ of an R-vector space VR,aC-vector space VC equipped with a q ∼ decreasing filtration F , and an isomorphism ϕ : VR ⊗R C = VC.Inotherwords, R R objects in -HS are “ -Hodge structures withoutq the weight filtration” – and without any conditions at all on the filtration F . The category R -HS is not an abelian category; however, it has a natual structure of an exact tensor category, so D R that one can define the tensor derived category ( -HS). As in the mixedq q Hodge R structure case,q we can also define an “ -Hodgeq complex” as a triple VR ,VC ,ϕ of acomplexVR of R-vector spaces, a complex VC of filtered C-vector spaces, and a q ∼ q quasiisomorphism ϕ : VR ⊗R C = VC ; then inverting filtered quasiisomorphisms in the category of R-Hodge complexes, we obtain the derived category D(R -HS). We also have the obvious ι-versions R -HSι, D(R -HSι) of these categories, and we have the forgetful functors R -MHS → R -HS, D(R -MHS) →D(R -HS), (2.1) R -MHSι → R -HSι, D(R -MHSι) →D(R -HSι). q q q For any R-Hodge complex V ,Hom(R(0),V )inD(R -HS) can be computed by the total complex of a bicomplex q q q 0 (2.2) VR ⊕ F VC −−−−→ VC , q q a version of (1.1) without W0, and to obtain Hom in D(R -HS ), one has to take ι-invariants in this bicomplex. q For any smooth proper variety X over Q, C (Xan) becomes an object of D(R -HSι) by applying the forgetful functor (2.1). The corresponding version q q q R R HD(X, (j)) = HomD(R -HS)( (0),C (Xan)(j)) of absolute Hodge cohomology groups of Definition 1.1 is known as Deligne cohomology of X, and it actually predates the absolute Hodge cohomology. As in the absolute Hodge cohomology case, Deligne cohomology can be computed by the bicomplex (2.2), but in this case, one can actually take the cone of the hori- zontalq differential locally on Xan, that is, before taking the cohomology complex C (Xan, −). Then one gets actual complexes R(j)ofsheavesonXan, and one has q q HD(X, R(j)) = H (Xan, R(j)). It is in this form that the Deligne cohomology groups were originally introduced by Deligne.

82 DMITRY KALEDIN

The forgetful functors (2.1) provide natural maps q q − → − HAH (X, ) HD(X, ), so that the regulator map (1.4) induces a regulator map 2j−i ι (2.3) r : Ki(XZ) ⊗Z R → HD (X, R(j)) , j where XZ is a good Z-model of X which we tacitly assume given. The drawback of Deligne cohomology is that this map clearly has no chance of being an isomorphism for all i – one checks easily that the right-hand side does not vanish for i<0. This is one of the reasons Beilinson had to refine Deligne cohomology and introduce absolute Hodge cohomology. However, Kato handles this discrepancy in a different way. q Namely, note that the complex C (Xan) is equipped with a Poincare pairing q q C (Xan) ⊗ C (Xan) → R(d)[2d], where d =dimX, and both sides are considered as objects of the tensor category D(R -HSι). If one lets q q (2.4) C (Xan)= C (Xan)(j)[2j], j then this extends to a pairing q q ι ι (2.5) C (Xan) ⊗ C (Xan) → R(i)[2i] for any particular choice of an integer i in the right-hand side. Let us take i =1. Then since D(R -HSι) is a tensor category, (2.6) induces a pairing q q q (2.6) CD(Xan) ⊗ CD(Xan) → RHom (R(0), R(1)[2]) → R[1], wherewedenote q q q ι R C CD(Xan) =RHomD(R -HSι)( (0), (Xan)), considered as a complex of R-vector spaces, and we use the identification 1 ∼ Hom (R(0), R(1)) = R easily deduced from (2.2). q We note note that CD(Xan) coincides, up to a quasiisomorphism, with the target of the regulator map (2.3). Then (2.6) induces a natural map q ∗ ι ∗ (2.7) r : CD(Xan) → (K q(XZ) ⊗Z R) [1] to the complex of R-vector spaces dual to K(XZ) ⊗Z R shifted by 1. Using this map, we can now formulate the conjecture. Conjecture 2.1. For any smooth projective variety X over Q with a regular model XZ flat and projective over Z, there exists a natural distinguished triangle q ∗ r ι r ∗ (2.8) K q(XZ) ⊗Z R −−−−→ CD(Xan) −−−−→ (K q(XZ) ⊗Z R) [1] −−−−→ of complexes of R-vector spaces, where r is the regulator map (2.3),andr∗ is the dual regulator map (2.7).

BEILINSON CONJECTURE FOR FINITE-DIMENSIONAL ASSOCIATIVE ALGEBRAS 83

We note that for dimension reasons, the conjectural connecting diﬀerential ∗ δ :(K q(XZ) ⊗Z R) → K q(XZ) ⊗Z R in the triangle (2.8) can only be non-trivial in degree 0. It is further expected to coincide with the inverse to the non-degenerate height pairing on the kernel of the regulator map on K0(XZ) ⊗Z R. This explain the story for elliptic curves: K0(XZ) ⊗Z R can be large, but the height pairing on the codimension-one subspace of numerically trivial cycles is non-degenerate, so that δ kills oﬀ this subspace. On the other hand, in the case X =SpecF , XZ =SpecOF ,themapδ is actually trivial. The right-hand side of (2.8) vanishes in homological degrees ≥ 2, while in degrees 0 and 1 we get two exact sequences − 0 −−−−→ R −−−−→ Rr1+r2 −−−−→ Rr1+r2 1 −−−−→ 0,

− 0 −−−−→ Rr1+r2 1 −−−−→ Rr1+r2 −−−−→ R −−−−→ 0 of R-vector spaces, in perfect agreement with Dirichlet Unit Theorem.

3. Non-commutative version 3.1. DG algebras. Let us now present a version of Conjecture 2.1 for non- commutative algebraic varieties. To begin with, we should explain what does a “non-commutative algebraic variety” mean; following the current practice, we take it to mean a small DG category considered up to a Morita equivalence, as in [Ke]. In fact, in the main conjecture we will even restrict our attention to DG categories with one object, that is, toq DG algebras. However, for now, let us recall that for any small DG categoryq A over a commutative ring k, one defines itsq periodic cyclic homology HPq(A ). This is the homology of a complex CPq(A )of k-modules, well-defined up to a quasiisomorphism. We have an additional periodicity quasiisomorphism q ∼ q (3.1) u : CPq(A ) = CPq(A )[2]. q Moreover,q CPq(A ) can be refined to a filtered complex; the descreasing filtration F is such that q q F 0CPq(A )=CP−q (A ), the negative cyclic homology comlpex, and q q F iCPq(A )=uiF 0(A ) for any integer i. In the filtered sense, (3.1) becomes a filtered quasiisomorphism q ∼ q (3.2) u : CPq(A ) = CPq(A )(1)[2], where (1) indicates renumbering of the filtration. q Up to a filtered quasiisomorphism,q theq complex CPq(A ) is Morita-invariant D ∼ D – if we have an equivalence (A ) = q(B )q between the derivedq q category of left modules over two small DG categories A , B given by an A -B -bimodule M,then we have a natural identification q ∼ q CPq(A ) = CPq(A ), compatible with the filtration and the periodicity isomorphism (3.2). Moreover,q if the base ring k is a field of characteristic 0, then if the derived category D(A ) is equivalent to the derived category D(X) of quasicoherent sheaves on a smooth proper variety X over k, and the equivalence is again given by a bimodule, – loosely

84 DMITRY KALEDIN q speaking, when A is Morita-equivalent to X, – we have a natural identification of filtered periodic complexes q ∼ q CPq(A ) = CDR(X), whereweset q q CDR(X)= CDR(X)(j)[2j], j as in (2.4). Thus periodic cyclic homology classes can be thought of as a non- commutative generalization of de Rham cohomology classes.q To set up an R-Hodge complex structure on CPq(A ) in the sense of Section 2, we also need a version of Betti cohomology for non-commutative varieties, and this is much more problematic. At this point, the best candidate seems to be the semitopological K-theory of B. Toën. We will not reproduce its definition here (for a very brief overview with further references,q see e.g. [Ka, Section 8]). Let us just C say that to any small DGq category A over , one associates a certain group-likeq infinite loop space M(A ), and one defines the semitopological K-groups Kstq (A ) as q q Kstq (A )=π q(M(A )). q Since M(A ) is a grouplike infinite loop space, it defines a spectrum, and semitopological K-groups are the homotopy groups of this spectrum. Rationally, every spectrum is an Eilenberg-Maclane spectrum, that is,q a complex; thereforeq we can st st effectively assume that we are given a complex K q (A )Q = K q (A )⊗Q of Q-vector spaces. Semitopological K-theory is compatible with productsq of DG categories, so that st st in particular, K q (C)Q is an algebra, and K q (A )Q is a module over this algebra. It is easy to prove that in fact, st ∼ K q (C)Q = Q[β], the algebra of polynomials in one variable β of homological degree 2 (see [Ka, Corollary 8.4]); the element β is aq version of the Bott periodicity generator. For any small DG category A , one can invert the Bott periodicity and consider the complex q q st −1 st −1 K q (A )Q(β )=K q (A )Q ⊗Q[β] Q[β,β ]. Then a natural functorial Chern character map q q st −1 (3.3) K q (A )Q(β ) ⊗Q C → CPq(A ) has been constructed by A. Blanc in [Bl1]. This maps sends β to the periodicity endomorphism u on the right-hand side, and it is expected to an isomorphism in good cases. More precisely, there is the following expectation. q Conjecture 3.1 (Blanc-Toën). For any smooth and proper DG category A , the map ch of (3.3) is a quasiisomorphism. The precise meaning of smoothness and properness for DG categories can be foundq e.g. in [Ke]; let us just say that both properties hold automatically when A is Morita-equivalent to a smooth projective algebraic variety X over C.Inthis highly non-trivial special case, the conjecture has been recently proved by Blanc in [Bl2, §4.1]. The general case seems to be wide open.

BEILINSON CONJECTURE FOR FINITE-DIMENSIONAL ASSOCIATIVE ALGEBRAS 85

3.2. The conjecture. To define a non-commutative versionq of the middle term Z of the triangle (2.8), weq observe that for any DG category A over ,thesemitopo- st −1 logical K-theory K q (A ⊗ C)Q ⊗Q C[β,β ] by its very definition carries a natural R-structure (and even a Q-structure). One then has the following two choices: (i) one can assume Blanc-Toën conjecture, so that the triple q q st −1 K q (A q ⊗ C)Q ⊗Q R[β,β ],CPq(A ⊗ C), ch is an R-Hodge complex in the sense of Section 2, or R (ii) one can define a weakq -Hodgeq complex by dropping the requirement that the map ϕ : VR ⊗R C → VC is a quasiisomorphism, and note that the bicomplex (2.2) still makes perfect sense even in this more general situation. Following the path ofq least resistance, we will make theq second choice. For any small DG category A over C, we will denote by CPDq (A ) the total complex of the bicomplex q q q − st −1 id − ch (3.4) CPq (A ) ⊕ (K q (A ⊗ C)Q ⊗Q R[β,β ]) −−−−→ CPq(A ), where ch is the Chern character map (3.3).q Now, assume given a DG algebra A over Z, and recall that it defines two small DG categories: q q pf (i) the category C q (A )q of perfect modules over A ,and q (ii) the category Cpspfq (A ) of pseudoperfect modules over A – that is, modules that are perfect when considered as modules over Z. q q q Here Cpfq (A ) is Morita-equivalent to A . The category Cpspfq (A )canbevery different, but at least up to a Morita equivalence, it is well-defined. The Hom- pairing then gives a functor q q Cpfq (A ) × Cpspfq (A ) → Cpfq (Z) into the DG category of perfect complexes of abelian groups, and taking the periodic cyclic homology and semitopological K-theory, we obtain a pairing q q CPq(Cpfq (A ) ⊗ C) ⊗ CPq(Cpspfq (A ) ⊗ C) → R(i)[2i] compatible with the weak R-Hodge complex structure on both sides. Here i is an arbitrary integer, as in (2.5). Taking i = 1, we obtain a pairing q q (3.5) CPDq (Cpfq (A ) ⊗ C) ⊗ CPDq (Cpspfq (A ) ⊗ C)[1] → R, a version of (2.6). q At the same time, on theq K-theory side, we have algebraic K-groups K q(A ) for any small DG category A over Z – for their definition, see e.g. [Ke, Subsection 5.2] – and we have a Chern character map q q K q(A ) ⊗ Q → CPq(A ⊗ C). q This map factors both through CP−q (A ⊗ Q) (this is classic) and through q st −1 K q (A ⊗ C)Q ⊗Q Q[β,β ] (this has been proved in [Bl1]). Therefore it defines a natural regulator map q q r : K q(A ) ⊗ R → CPDq (A ⊗ C).

86 DMITRY KALEDIN q Z ⊂ R ⊂ C R Moreover,q since A is deﬁned over ,theweak -Hodge complex CPq(A ⊗ C) has a natural ι-version, and the regulator map actually gives a map q q (3.6) r : K q(A ) ⊗ R → CPDq (A ⊗ C)ι. q q q Since A is Morita-equivalent to Cpfq (A ), we may replace it with Cpfq (A )inthe q right-hand side of (3.6). Now deﬁne K q(A )R by

q q ∗ pspf K q(A )R = K q(C q (A )) ⊗ R , q R pspf ⊗ R the complex of -vector spaces dual to K q(C q (A )) . Then by virtueq of the pairing (3.5), the regulator map r of (3.6) for the DG category Cpspf (A ) induces a natural map q q ∗ D ι (3.7) r : CPq (A ⊗ C) → K q(A )R[1], a version of (2.7). q Conjecture 3.2. For any DG algebra A over Z, we have a natural distinguished triangle

q q q ∗ q r D ι r (3.8) K (A ) ⊗ R −−−−→ CPq (A ⊗ C) −−−−→ K q(A )R[1] −−−−→ , where r and r∗ are the maps (3.6) and (3.7). q Remark 3.3. For a general DG algebra A , the meaning of the invariant q q K q(A )R in Conjectureq 3.2 seems very unclear. If A is actuallyq smoothq and proper Z pf pspf over ,thenC q (A ) is Morita-equivalentq to C q (A ), and K−i(A )R are the dual R-vector spaces to Ki(A ) ⊗ R. However, in practice, what one cares about are DG algebras smooth and proper over Q, and it is certainly too much to expect that every such DG algebra has a smooth and proper model over Z.Notethatin the commutative case, Beilinson only asks for a model XZ which is regular as an abstract scheme, not smooth over Z. It might very well be that as stated, Con- jecture 3.2 admits easy counterexamples, and one needs to impose some version of regularity; unfortunately, at this moment we have no idea what it could possibly be. q q Remark 3.4. Every map f : A → B between DG algebras induces a pair of adjoint functors between the DG categories of DG modules over them. How- ever, only one functor of the pair always sends perfect modules to perfect modules, and the other one always sends pseudoperfect modules to pseudoperfect modules. Therefore we have natural DG functors q q q q f ∗ : Cpfq (A ) → Cpfq (B ),f∗ : Cpspfq (B ) → Cpspfq (A ).

Since K-groups are functorial with respect to DG functors, this shows that both K q(−)andK q(−)R are covariantly functorial with respect to DG algebra maps. So are the regulator maps r and r∗. By adjunction, the Hom-pairing is compatible with functoriality, soq that the triangles (3.8) of Conjecture 2.1 should also be covariantly functorial in A .

BEILINSON CONJECTURE FOR FINITE-DIMENSIONAL ASSOCIATIVE ALGEBRAS 87

3.3. Finite-dimensional algebras. Let us now illustrate Conjecture 3.2 in the simple particular case of finite-dimensional algebras. We thus assume given a ring A of finite rank as a Z-module, and consider the additive invariants of A considered Z as a DG algebra over . q We first observe that Conjecture 3.1 is almost trivially true for A = A ⊗ C. Here is a sketch of a possible proof (see also [Bl2, Section 4.2]). q Sketch of a possible proof of Conjecture 3.1 for A = A ⊗ C. First of all, by the same argument as in [Ka, Lemma 8.3], the infinite loop space M(A⊗C) is the group completion of the space * B Aut(P ), where the sum is over the isomorphism classes of finitely generated projective A⊗C- modules P ,Aut(P ) stands for the group of automorphisms of such a module P considered as a complex Lie group, and B Aut(P ) is its classifying space. Next, note that since A⊗C is a finite-dimensional algebra over C,itisanilpotentextension of a semisimple algebra Ass.ThenAss is the sum of several matrix algebras over C, and since both Kstq and HPq are Morita-invariant and compatible with sums, the statement for Ass follows from the corresponding statement for C itself (that is, [Ka, Corollary 8.4]). It remains to notice that on one hand, HPq does not change under nilpotent extensions of algebras over a field of characteristic 0 by a ss ss theorem of Goodwillie [G], and on the other hand, sending P to P = P ⊗A⊗C A gives a one-to-one correspondence between isomorphism classes of finitely generated projective modules over A ⊗ C and Ass, and for any such projective P ,Aut(P )isa unipotent extension of Aut(P ss), so that the natural map B Aut(P ) → B Aut(P ss) is a homotopy equivalence.

We now consider Conjecture 3.2, and we note that it can be handled by the same argument. Namely, changing the Z-lattice A in A ⊗ Q does not affect any of the terms in (3.8), and doing such a change if necessary, we may assume that A is a nilpotent extension of an algebra Ass of the form + ss A = Ai, i where Ai is a maximal order in an algebra Ai ⊗ Q over Q, a matrix algebra over a skew-field of finite rank over Q.ForAss, K q(Ass) ⊗ R and Kq(Ass) ⊗ R are dual vector spaces, and they can be computed by the Borel Theorem. Thus verifying the conjecture for Ass is straightforward, and we only need to show that Conjecture 3.2 is stable under nilpotent extensions. For this, one combines the following two observations. (i) After tensoring with Q, the cone of the natural map K q(A) → K q(Ass)is identified by the Chern character map with the cone of the natural map CP−q (A ⊗ Q) → CP−q (Ass) – this is a version of the famous Goodwillie Theorem of [G]. (ii) The other terms in the bicomplex (3.4) do not change inder nilpotent extensions, and by Quillen Devissage Theorem, neither does the right- hand side K q(A ⊗ C)R of the triangle (3.8).

88 DMITRY KALEDIN

References [Be] A. A. Be˘ılinson, Higher regulators and values of L-functions (Russian), Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238. MR760999 (86h:11103) [Bl1] A. Blanc, Topological K-theory and its Chern character for non-commutative spaces, arXiv:1211.7360. [Bl2] A. Blanc, Invariants topologiques des Espaces non commutatifs, arXiv:1307.6430. [G] Thomas G. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402, DOI 10.2307/1971283. MR855300 (88b:18008) [Ka] D. Kaledin, Motivic structures in non-commutative geometry, Proceedings of the Interna- tional Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 461–496. MR2827805 (2012h:14055) [Ke] Bernhard Keller, On diﬀerential graded categories, International Congress of Mathemati- cians. Vol. II, Eur. Math. Soc., Z¨urich, 2006, pp. 151–190. MR2275593 (2008g:18015)

Steklov Math Institute, Algebraic Geometry section, and Laboratory of Alge- braic Geometry, NRU HSE E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12418

Partial Monoids and Dold-Thom Functors

Jacob Mostovoy

Abstract. Dold-Thom functors are generalizations of inﬁnite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of Γ-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting an abelian partial monoid corresponding to a connective spectrum.

1. Introduction Let SP∞X be the infinite symmetric product of a pointed connected cell complex X. Then, according to the Dold-Thom Theorem [4], the homotopy groups of SP∞X coincide, as a functor, with the reduced singular homology of X. Although there is no computational advantage in this definition of singular homology, it is important since it can be extended to the algebro-geometric context; in particular, the motivic cohomology of [9] is defined in this way. The construction of the infinite symmetric product can be generalized so as to produce an arbitrary connective homology theory. Such generalized symmetric products were defined by G. Segal in [11]: essentially, these are labelled configuration spaces, with labels in a Γ-space (see also [2, 3, 6]).IftheΓ-spaceoflabels is injective (see [13]) it gives rise to a partial abelian monoid; it has been proven in [13] that for each connective homology theory there exists an injective Γ-space. In this case Segal’s generalized symmetric product can be thought of as a space of configurations of points labelled by composable elements of a partial monoid. An explicit example discussed in [12] is the space of configurations of points labelled by orthogonal vector spaces: it produces connective K-theory. We shall call the generalized symmetric product functor with points having labels, or “multiplicities”, in a partial monoid M, the Dold-Thom functor with coefficients in M. Certainly, the construction of such functors via Γ-spaces is most appealing. However, if we start with a spectrum, constructing the corresponding Dold-Thom functor using Γ-spaces is not an entirely straightforward procedure since the Γ-space naturally associated to a spectrum is not injective. The purpose of the present note is to show how a connective spectrum M gives rise to an explicit partial monoid M such that the homotopy of the Dold-Thom functor with coefficients in M coincides, as a functor, with the homology with coefficients in M.

2010 Mathematics Subject Classiﬁcation. Primary 55P47.

c 2014 American Mathematical Society 89

90 JACOB MOSTOVOY

2. Partial monoids and infinite loop spaces 2.1. Partial monoids. Most of the following definitions appear in [10]. A partial monoid M is a topological space equipped with a subspace M(2) ⊆ M × M andanadditionmapM(2) → M, written as (m1,m2) → m1 + m2, and satisfying the following two conditions: • there exists 0 ∈ M such that 0 + m and m + 0 are defined for all m ∈ M andsuchthat0+m = m +0=m; • for all m1,m2 and m3 the sum m1 +(m2 + m3) is defined whenever (m1 + m2)+m3 is defined, and both are equal.

We shall say that a partial monoid is abelian if for all m1 and m2 such that m1 +m2 is defined, m2 + m1 is also defined, and both expressions are equal. The classifying space BM of a partial monoid M is defined as follows. Let k M(k) be the subspace of M consisting of composable k-tuples. The M(k) form a simplicial space, with the face operators ∂i : M(k) → M(k−1) and the degeneracy operators si : M(k) → M(k+1) defined as

∂i(m1,...,mk)=(m2,...,mk)ifi =0 =(m1,...,mi + mi+1,...,mk)if01 can be deﬁned inductively for all the pairs of maps from R to the partial monoid Ωn−1X whose values are composable

PARTIAL MONOIDS AND DOLD-THOM FUNCTORS 91 at each point. This is, of course, the same as treating the points of ΩnX as maps Rn → X and deﬁning the composable pairs of maps as those with disjoint supports in Rn. Explicitly, for two maps f,g : Rn → X with disjoint supports their sum f +g is equal to f on the support of f,tog on the support of g and to the basepoint everywhere else. Let now M be a connective spectrum. We construct a partial abelian multiplication on its inﬁnite loop space as follows. First, let us replace inductively M0 by a point and the spaces Mi for i>0 by the mapping cylinders of the structure maps

ΣMi−1 →Mi, obtaining a spectrum M. This, in particular, allows us to assume that all the structure maps are inclusions. As a consequence, we have inclusion maps i−1M → iM Ω i−1 Ω i, whichsendcomposablek-tuples of elements to composable k-tuples for all k.The union of all the spaces [i] iM M =Ω i is the inﬁnite loop space for the spectrum M and it naturally has the structure of a partial abelian monoid.

2.3. Dold-Thom functors. Let M be an abelian partial monoid and X a topological space with the base point ∗. We define the configuration space Mn[X] of at most n points in X with labels in M as follows. For n>0letWn be the n ∧ n subspace of the symmetric product SP (X M) consisting of points i=1(xi,mi) such that the mi are composable; W0 is defined to be a point. The space Mn[X]is the quotient of Wn by the relations

(x, m1)+(x, m2)+...=(x, m1 + m2)+..., where the omitted terms on both sides are understood to coincide, and (x, 0) = (∗,m) for all x and m. This quotient map commutes with the inclusions of Wn into Wn+1 coming, in turn, from the inclusions SPn(X ∧ M) → SPn+1(X ∧ M)whichadd n the basepoint (∗, 0) to a point of SP (X ∧ M). Therefore, Mn[X] is a subspace of Mn+1[X]. The Dold-Thom functor of X with coefficients in M is the space M[X]= Mn[X]. n≥0 The Dold-Thom functor with coefficients in the monoid of non-negative integers is the infinite symmetric product. If M has trivial multiplication, we have

M[X]=M1[X]=X ∧ M. The composability of labels in a conﬁguration is essential for the functoriality of M[X]. A based map f : X → Y induces a map M[f]:M[X] → M[Y ] as follows: apoint (xi,mi)issenttothepoint (yj ,nj )wherethelabelnj is equal to the sum of all the mi such that f(xi)=yj . Note that the space M[X] is itself a partial monoid, with the multiplication induced by that of M. The induced map M[f]is a homomorphism of partial monoids.

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Apart from the infinite symmetric products, Dold-Thom functors generalize classifying spaces: for any partial monoid M its classifying space BM is homeomorphic to M[S1]. To construct the homeomorphism, take the lengths of the intervals between the particles to be the barycentric coordinates in the simplex in BM defined by the labels of the particles. Similarly, the classifying space of an arbitrary Γ-space can be constructed in this way (modulo some technical details), see Section 3 of [11]. When X = S1 the space M[X] can be defined even for non- abelian M; the identification of BM with M[S1] also makes sense for non-abelian monoids. The construction of a classifying space for a monoid as a space of particles on S1 was first described in [8]. In the case when M is a partial abelian monoid coming from a spectrum M it is convenient to consider a different topology on Mn[X]. Namely, we define Mn[X] [i] as the the union of the spaces Mn [X] with the weak topology. Then, as above, the Dold-Thom functor with coefficients in M associates to a space X the space M[X]=∪ Mn[X]. The technical advantage provided by this modification is that [i] for any compact space Y the map Y → M[X] factorizes through Mn [X]forsome i and n. We have the following generalization of the Dold-Thom theorem: Theorem 1. Let M be partial abelian monoid corresponding to a connective spectrum M. Then the spaces M[Sn] form a connective spectrum weakly equivalent to M.Thefunctor X → π∗M[X] coincides with the reduced homology with coefficients in M. 2.4. Other partial monoids. The subject of this note are the partial monoids coming from spectra. Nevertheless, it is worth pointing out that there are many examples of partial abelian monoids, apart from infinite loop spaces, such that the corresponding Dold-Thom functors are linear. It is not hard to prove using the methods similar to those of Dold and Thom that if a partial monoid M has a good base point and if the complement of the subspace of composable pairs has, in a certain sense, infinite codimension in M × M, then the corresponding Dold-Thom functor defines a homology theory. We have already mentioned Segal’s partial monoid of vector subspaces of R∞ defined in [12]. Among other examples we have: 1. The configuration space of several (between 0 and ∞)distinctpointsin R∞, with the sum of two disjoint configurations being defined as their union. For configurations with points in common the sum is not defined. The unit is the point ∅ thought of as the configuration space of 0 points. More generally, one can consider the configuration spaces of distinct particles in R∞, labelled by points of afixedspaceM. This was done in the paper [14] by K. Shimakawa; the homology theory produced by this construction assigns to X the stable homotopy of X ∧ M. (In fact, Shimakawa considers a more general situation of configurations in R∞ with partially summable labels belonging to some partial abelian monoid. Such configuration space is then itself a partial abelian monoid which, as Shimakawa proves, always gives rise to a homology theory.) 2. The space of all n-dimensional closed compact submanifolds of R∞ with the sum being the union of the submanifolds, whenever they do not intersect. This

PARTIAL MONOIDS AND DOLD-THOM FUNCTORS 93 construction, however, adds nothing substantially new to the previous example. Indeed, since the dimension of submanifolds is finite and their codimension is infinite, their connected components can be shrunk in size simultaneously (at least in a compact family of submanifolds) and we see that the partial monoid of n- submanifolds of R∞ is weakly homotopy equivalent to the labelled configuration ∞ space of particles in R , with labels in ∪M BDiff(M), the space of all connected n-submanifolds of R∞. 3. The space of all (piecewise smooth) spheres in R∞. The operation is the join of two spheres inside R∞, and it is defined whenever any two intervals connecting points of the two summands are disjoint.

3. Properties of Dold-Thom functors coming from spectra [i] In what follows M = ∪i M is a partial abelian monoid coming from a connective spectrum. [i] Ri+1 →M If a point of M is regarded as a map i+1, its support lies in i+1 the region −1

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The deformation ui pushes the points of the conical neighbourhood of zero towards the apex; outside of this neighbourhood it is constant. The precise formulae are easy to write down, but we do not need them.

Lemma 4. The set π0(M) is an abelian group with the addition induced by the partial addition in M. Proof. It is suﬃcient to notice that if points of M are thought of as maps of R i−1M − to Ω i for some i,aninversetoamapγ in π0M will be given by γ(C t) for suﬃciently big C.

For a space X let Ma [X] ⊂ M[X] be the subset of configurations whose coefficients are composable with a ∈ M. Lemma . 5 The inclusion of Ma [X] into M[X] is a weak homotopy equivalence. Moreover, if (xi,mi) ∈ Ma [X],themap Ma+ m [X] → Ma [X] i given by summing with (xi,mi), is also a weak homotopy equivalence. Proof. We first need to show that the image of any map f of a finite cell complex Y into M[X]canbedeformedintoMa [X] by means of a homotopy that does not take points out of Ma [X]. Assume that both a and the image of f is [i] i contained in M [X]. Then, applying the deformation dt given by (1) to all the labels of the configurations f(y), where y ∈ Y , we get the required homotopy of f to a map of Y into Ma [X]. In order to establish the second claim, for each mi choosem ˜ i in such a way that the class ofm ˜ i is inverse to that of mi in π0M and so that the sum a+ mi + m˜ i is defined. It is then easy to see that adding (xi,mi)andthen (xi, m˜ i) to a configuration in Ma+ mi+ m˜ i [X] is homotopic to the natural inclusion → Ma+ mi+ m˜ i [X] Ma [X].

Since all the natural inclusions between the spaces Ma [X]fordiﬀerenta are homotopy equivalences, it follows that the map in the statement of the lemma is surjective on the homotopy groups. Replacing in this argument a by a + mi we see that the map M [X] → M [X], a+ mi+ m˜ i a+ mi given by adding (xi, m˜ i), is also surjective on homotopy and this proves the lemma.

4. Quasifibrations and Dold-Thom functors The proof of Theorem 1 is based on the original argument of Dold and Thom [4], see also [1, 5]. We have the following fact: Proposition 6. Let M be a partial monoid coming from a connective spectrum. If X is a cell complex and A ⊂ X is a subcomplex, the map M[X] → M[X/A] is a quasifibration with the fibre M[A]. The rest of this section is dedicated to the proof of this statement. Note that we do not require A to be connected.

PARTIAL MONOIDS AND DOLD-THOM FUNCTORS 95

We shall need the following criterion for quasiﬁbrations. Let p : E → B be a map which is quasiﬁbration over B ⊂ B. Assume that for any compact C ⊂ B there is a homotopy ft of the inclusion map i : C→ B toamapC → B ,which maps C ∩ B to B for all t. Further, suppose that for any compact C ⊆ p−1(C) ˜ −1 there is a homotopy ft of the inclusion map C → E to a map C → p (B )which maps C ∩ p−1(B)top−1(B) for all t,andsuchthat ˜ p ◦ ft = ft ◦ p.

Moreover, assume that ft and f˜t are well-defined up to homotopy. Take a point b ∈ B and ˜b ∈ p−1(b). Then we have two paths: from b to b ∈ B and from ˜b ∈ p−1(b)to˜b ∈ p−1(b), the latter covering the former. This gives a map of the homotopy groups −1 −1 πi(p (b), ˜b) → πi(p (b ), ˜b ). Lemma 7. Assume that all the above maps of homotopy groups of the fibres are isomorphisms for all i ≥ 0. Then the map p is a quasifibration. This lemma is a version of Hilfssatz 2.10 of [4] and the proof is, essentially, the same. We denote by p the projection map X → X/A and by π the induced map M[X] → M[X/A].

We shall prove by induction on n that π is a quasifibration over Mn[X/A]. This, by Satz 2.15 of [4] (or Theorem A.1.17 of [1]) will imply that π is a quasifibration over the whole M[X/A]. Assume that π is a quasifibration over Mn−1[X/A]. According to Satz 2.2 of [4] (or Theorem A.1.2 of [1]) it is sufficient to prove that π is a quasifibration over Mn[X/A]−Mn−1[X/A], over a neighbourhood of Mn−1[X/A]inMn[X/A]andover the intersection of this neighbourhood with Mn[X/A] − Mn−1[X/A]. It will be convenient to speak of delayed homotopies. A delayed homotopy is amapft : A × [0, 1] → B such that for some ε>0wehaveft = f0 when t ≤ ε. Amapp : E → B is said to have the delayed homotopy lifting property if it has the homotopy lifting property with respect to all delayed homotopies of finite cell complexes into B. It is clear that a map that has the delayed homotopy lifting property is a quasifibration.

Lemma 8. Let B be an arbitrary subspace of Mn[X/A]−Mn−1[X/A].Themap π, when restricted to π−1(B), has the delayed homotopy lifting property. Proof. Let ft : Z × [0, 1] → B be a delayed homotopy of a ﬁnite cell complex Z into B, such that ft = f0 for t ≤ ε, and let ˜ −1 f0 : Z → π (B) be its lifting at t =0. Notice that Mn[X/A]−Mn−1[X/A] can be thought of as the subspace of Mn[X] consisting of conﬁgurations of n distinct points which lie outside A, all with non- trivial labels. In particular, we can think of B as a subspace of Mn[X].

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We have that −1 (2) π Mn[X/A]=Mn[X]+M[A], where the sum is taken whenever it is defined. In particular, we can write ˜ f0(z)=f0(z)+g0(z) with g0(z) ∈ M[A]. The map g0 : Z → M[A] ⊆ M[X] is well-defined and continuous. If ft(z)werecomposablewithg0(z) for all t and z, the homotopy ft +g0 would be the required lifting of ft. In general, one can deform the map g0 to a homotopy gt so that ft is composable with gt for all t, in the following manner. [i] Since Z is compact, the image of f˜0 belongs to M [X]forsomei; the coeffi- [i] cients of g0(z) are composable with the coefficients of f0(z) inside M for all z ∈ Z. i [i] [i+1] Recall that (1) defines a deformation dt of M inside M such that each point i [i] [i+1] × in d1(M )iscomposableinM with each point in the image of Z [0, 1] under [i] i ft. There is an induced deformation of M [A] which we also denote by dt. Define the homotopy g : Z → M[A]as t i d − ◦ g for 0 ≤ t<ε g = tε 1 0 t i ◦ ≤ ≤ d1 g0 for ε t 1.

Lemma 2 implies that gt is well-defined and is composable with ft for all t.Consider ˜ the map ft = ft + gt : Z → M[X]. By construction, it lifts ft. It remains to see that the map π is a quasifibration over some neighbourhood of Mn−1[X/A]. For this we shall use the criterion given in Lemma 7. Let us show how to construct a neighbourhood B of Mn−1[X/A]sothatLemma7canbeapplied with B = Mn−1[X/A]. Recall from Lemma 3 that there is a homomorphism of partial monoids [i] hi : M → I. It gives rise to a map [i] → vi : Mn [X] In[X]. If X and A are cell complexes, let Δ be the subspace of In[X] consisting of configurations which either contain a point of A,orhavelessthann points. It is not hard to show that In[X] is a cell complex and Δ is a subcomplex. In particular, Δ is a strong deformation retract of its neighbourhood U ⊂ In[X]. Write Vi for the −1 ⊂ [i] preimage vi (U)in Mn [X]. Let dt : In[X] × [0, 1] → In[X] be the deformation that retracts U to Δ. We can assume that dt is a homeomorphism for all t<1 and that it does not increase the values of the labels of the points. Then dt can be lifted to a deformation Dt : Vi × [0, 1] → Vi, −1 ⊂ [i] that retracts the open subset Vi to the subspace vi (Δ) Mn [X], which consists of configurations which either contain a point of A,orhavelessthann points. −1 [i] Now, denote by Bi the open neighbourhood π(vi (U)) of Mn−1[X/A] inside [i] Mn [X/A]. By construction, there exists a deformation

Dt : Bi × [0, 1] → Bi,

PARTIAL MONOIDS AND DOLD-THOM FUNCTORS 97 such that Dt ◦ π = π ◦ Dt [i] for all t.Inparticular,Dt retracts Bi onto Mn−1[X/A]. Let B be the union of all the neighbourhoods Bi for all i in M[X/A]. Then B is open in M[X/A] and it follows from Lemma 5 that the projection π−1(B) → B satisﬁes the conditions of Lemma 7.

5. On the spectrum M[S] 5.1. The spectrum M[S] and the proof of Theorem 1. Proposition 6 with X = Dn and A = ∂Dn gives the quasifibration M[Sn−1] → M[Dn] → M[Sn]. The space M[Dn] is contractible, and therefore, we have weak homotopy equivalences M[Sn−1] ΩM[Sn] and the spaces M[Si]fori ≥ 0 form an Ω-spectrum whichwedenotebyM[S]. More generally, given X, the cofibration X → CX → ΣX givesrisetoaweak homotopy equivalence M[X] ΩM[ΣX], and given an inclusion map i : A→ X, the cofibration A → Cyl(i) → X ∪i CA gives rise to an exact sequence

...π∗M[A] → π∗M[X] → π∗M[X ∪i CA] → π∗−1M[A] → .... Here CX is the cone on X and Cyl(i) is the mapping cylinder of the map i.Since π∗M[X] is, clearly, a homotopy functor (see Section 2.3), this means that the groups π∗M[X] form a reduced homology theory. There is a natural transformation of the homology with coefficients in M[Si] i i toπ∗M[X], induced by the obvious map M[S ] ∧ X → M[S ∧ X] that sends mizi,x to mi(zi,x): i i lim πk+i M[S ] ∧ X → lim πk+iM[S ∧ X]=πkM[X]. i→∞ i→∞ If X is a sphere, the Freudenthal Theorem implies that this is an isomorphism. Hence, π∗M[X] coincides as a functor, on connected cell complexes, with the homology with coefficients in M[S]. 5.2. The weak equivalence of M[S] and M. We shall first construct a weak homotopy equivalence between the infinite loop spaces of the spectra M and M[S] and then show that there exists an inverse to this equivalence, which is induced by a map of spectra. This will establish Theorem 1. Let I be the interval [−1/2, 1/2]. By Proposition 6 the map I → S1 which identifies the endpoints of I induces a quasifibration M[I] → M[S1] with the fibre ∞ M Ω M∞. Since M[I] is contractible, it follows that M is weakly homotopy equivalent to ΩM[S1]; the weak equivalence is realized by the map ψ that sends m ∈ M to the loop (parameterized by I) whose value at the time t ∈ I is the configuration consisting of one point with coordinate −t and label m. Let us now define the map of spectra Φ, inverse to the above weak equivalence ψ. For n>0 identify Sn with the n-dimensional cube In =[−1/2, 1/2]n ⊂ Rn, modulo its boundary. Fix a homeomorphism of the interior of In with Rn,say, n by sending each coordinate uk to tan πuk. Think of a point in M[S ]asasum ∈ n n M (xα,mα)wherexα I and mα are maps from I to n. Assume that the

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− Ri M maps mα( xα) are composable as maps from some to n+i. Then their sum M is a well-defined point of n which does not depend on i.WesetΦn( (xα,mα)) to be equal to this sum. n →M The map Φn : M[S ] n is only partially defined, but this problem can be ∗ n n circumvented as follows. Let M [S ] ⊂ M[S ] be the subspace consisting of the ∈ iM configurations (qα,mα) such that for some i the points mα(qα) Ω n+i are i n composable (that is, have disjoint supports in R ) for any choice of the qα ∈ I . ∗ n ∗ The spaces M [S] form a sub-spectrum M [S]ofM[S]. Indeed, the structure n map of M[S] sends (xα,mα) ∈ M[S ]totheloop t → ((xα,t),mα),

∗ n where t ∈ I.ThemapΦn is well-defined on M [S ]. If we define the map Φ0 simply as ΩΦ1, it is clear from the construction that Φ0 ◦ ψ is the identity map on M0. The proof will be finished as soon as we prove the following Proposition 9. The inclusion map M ∗[Sn] → M[Sn] is a weak homotopy equivalence for all n>0. Proof. Set ∗ n ∗ n ∩ n Mk [S ]=M [S ] Mk[S ]. It is sufficient to prove that the inclusion ∗ n → n Mk [S ] Mk[S ] is a weak homotopy equivalence for all k. For k = 1 this inclusion is the identity map. Also, for k>1themap ∗ n ∗ n → n n (3) Mk [S ]/Mk−1[S ] Mk[S ]/Mk−1[S ] is a weak homotopy equivalence. Indeed, take a map j n n f : S → Mk[S ]/Mk−1[S ] and let us show that the labels of the points in each configuration in the image of f can be pushed away from each other, thus deforming the image of f into ∗ n ∗ n Mk [S ]/Mk−1[S ]. By forgetting the labels in the configurations we get a map j −1 n (S − f (∗)) → Bk(S ) to the configuration space of k distinct particles in Sn. Without loss of generality we can assume that the boundary of Sj −f −1(∗) is a collared (for instance, smooth) hypersurface in Sj , so that this map gives rise to a bundle ξ of k-element sets over the closure C of Sj − f −1(∗)inSj . In turn, this bundle of sets gives rise to a k-vector bundle η whose fibre is spanned by the elements of the corresponding fibre of ξ.SinceC is compact, η can be considered as a subbundle of some trivial bundle. What this means is that there exists an N such that given a configuration in the image of f we can assign a unit vector in some RN toeachofitspointssothatthe vectors assigned to all the points are mutually orthogonal. [i] n [i] n Now, the image of f is contained in Mk [S ]/Mk−1[S ]forsomei.Wecan treat the labels of the configurations in the image of f as elements of [i+N] i N M M =ΩΩ i+N ,

PARTIAL MONOIDS AND DOLD-THOM FUNCTORS 99

i N and write them as functions m(x, y) with x ∈ R and y ∈ R .Forz ∈ C write n [i+N] f(z)= (qα,mα)whereqα ∈ S and mα ∈ M , and define ft(z)as ft(z)= (qα,mα(x, y + tvα)), where vα are the orthogonal vectors associated to the points qα of the configuration ∗ n ∗ n f(z). Then for t sufficiently large, the image of ft will lie in Mk [S ]/Mk−1[S ]. This argument shows that (3) induces surjective maps on homotopy groups. In the same way one shows that these maps are injective: instead of a map of Sj to n n j Mk[S ]/Mk−1[S ] one considers a map of S × [0, 1] to the same space with the j × { }∪{ } ∗ n ∗ n property that S ( 0 1 )issenttoMk [S ]/Mk−1[S ]. n ⊂ n ∗ n ⊂ ∗ n The subspaces Mk−1[S ] Mk[S ]andMk−1[S ] Mk [S ] are not, strictly speaking, neighbourhood deformation retracts, but each of these subspaces has a neighbourhood such that the map of any compact into this neighbourhood can be retracted onto the subspace. This is sufficient to claim that the fact that the maps (3) are weak homotopy equivalences implies, via induction, that the inclusion ∗ n → n Mk [S ] Mk[S ] induces isomorphisms in homology. It remains to show that both spaces are simple. In order to do this, notice that any two elements of the ∗ n homotopy groups of Mk [S ] can be represented by a pair of maps such that each value of one map is composable with any value of the other; the same is true for n Mk[S ]. (Such composability can be achieved by using a deformation such as (1).) Then the usual argument for the simplicity of H-spaces applies.

Acknowledgments. I would like to thank Ernesto Lupercio for conversations that provided the motivation for this paper, Norio Iwase, Dai Tamaki and Peter Teichner for pointing out useful references, and the referee for numerous helpful comments. I am also grateful to Max-Planck-Institut f¨ur Mathematik, Bonn for hospitality during the stay at which this paper was written. This work was partially supported by the CONACyT grant no. 44100.

References [1] Marcelo Aguilar, Samuel Gitler, and Carlos Prieto, Algebraic topology from a homotopical viewpoint, Universitext, Springer-Verlag, New York, 2002. Translated from the Spanish by Stephen Bruce Sontz. MR1908260 (2003c:55001) [2]D.W.Anderson,Chain functors and homology theories, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971), Springer, Berlin, 1971, pp. 1–12. Lecture Notes in Math., Vol. 249. MR0339132 (49 #3895) [3] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimpli- cial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR513569 (80e:55021) [4] Albrecht Dold and RenéThom,Quasifaserungen und unendliche symmetrische Produkte (German), Ann. of Math. (2) 67 (1958), 239–281. MR0097062 (20 #3542) [5] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR1867354 (2002k:55001) [6] Nicholas J. Kuhn, The McCord model for the tensor product of a space and a commutative ring spectrum, Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math., vol. 215, Birkhäuser, Basel, 2004, pp. 213–236. MR2039768 (2004m:55013) [7] J. Peter May, Categories of spectra and infinite loop spaces, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 448–479. MR0248823 (40 #2073) [8] M. C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273–298. MR0251719 (40 #4946)

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[9] Vladimir Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604 (electronic). MR1648048 (99j:14018) [10] Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213– 221. MR0331377 (48 #9710) [11] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR0353298 (50 #5782) [12] Graeme Segal, K-homology theory and algebraic K-theory, K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), Springer, Berlin, 1977, pp. 113–127. Lecture Notes in Math., Vol. 575. MR0515311 (58 #24242) [13] R. Schwänzl and R. M. Vogt, E∞-spaces and injective Γ-spaces, Manuscripta Math. 61 (1988), no. 2, 203–214, DOI 10.1007/BF01259329. MR943537 (89j:55012) [14] Kazuhisa Shimakawa, Configuration spaces with partially summable labels and homology theories, Math. J. Okayama Univ. 43 (2001), 43–72. MR1913872 (2003g:55026)

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Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12419

The Weak b-principle

Rustam Sadykov

Abstract. We review the bordism version of the h-principle established in an earlier work by the author and discuss its generalization.

1. The h-principle The h-principle is a general observation that differential geometry problems can often be reduced to problems in (unstable) homotopy theory. For example, let us consider three fairly different problems in differential geometry: does a given smooth manifold M admit a symplectic/contact structure? a foliation of a given codimension? a Riemannian metric of positive/negative scalar curvature? Each of the mentioned differential geometry problems reduces to a purely homotopy theory problem provided that the given manifold M is open. The h-principle reduction is important since in contrast to the original differential geometry problems, the obtained homotopy theory problems can be approached by standard homotopy theory methods. The general h-principle is formulated in terms of differential relations. Definition 1.1. A differential relation R on smooth functions f : Rm → Rn is defined to be a system of differential equations and inequalities. A solution of R is a function satisfying all equations and inequalities of R. Given a differential relation R, we may convert it into a functional relation F(R) by replacing each partial derivative ∂I f in equations and inequalities of R by a new unknown continuous function fI . Definition 1.2. Solutions of F(R) are called formal solutions of R. In contrast to solutions of R, formal solutions are easy to find. Example 1.3 (A relation on f : R → R). A formal solution of f (5) −f −f =0 is a set (f0,f1, ..., f5) of arbitrary continuous functions with f5 = f0 + f1. In the case of maps of manifolds differential relations are given in terms of local charts [6].

2010 Mathematics Subject Classiﬁcation. 55N22, 55P35, 57R45. The author was supported by CONACyT grant 179823.

c 2014 American Mathematical Society 101

102 RUSTAM SADYKOV

H-principle. Let R be a diﬀerential relation imposed on smooth maps f : M → N of manifolds. The strongest version of the h-principle asserts that the inclusion Sol(R) −→ hSol(R), ∂f ∂f f → (f, , , ...) ∂x1 ∂x2 of the space Sol(R) of genuine solutions into the space hSol(R) of formal solutions is a weak homotopy equivalence. In particular, each family of formal solutions admits a deformation into a family of genuine solutions.

Note, the statement of the h-principle is a conjecture; for a given R,itmayor may not be true. Early examples of the h-principle type results include the Nash-Kuiper isometric immersion theorem, Smale-Hirsch immersion theorem and Phillips submersion theorem. However the potential of the h-principle remained hidden until the ap- pearance of the general Gromov theorem on open coordinate invariant differential relations. Definition 1.4. A differential relation R imposed on functions Rm → Rn is open if it is given by strict inequalities. We say that R is coordinate invariant if for all coordinate changes g : Rm → Rm and h: Rn → Rn a function f is a solution of R whenever h ◦ f ◦ g is. A coordinate invariant differential relation imposed on functions Rm → Rn also imposes a differential relation on maps of any manifold M of dimension m into any manifold N of dimension n (by using arbitrary coordinate charts). Example 1.5. A submersion f : M → N is a smooth map such that the differential of f is a fiberwise epimorphism TM → TN of tangent vector bundles. Submersions are solutions of a certain open coordinate invariant differential relation R. A formal submersion is a continuous map f : M → N together with a fiberwise epimorphism TM → TN covering f. Similarly, f is an immersion if its differential is a fiberwise monomorphism. A formal immersion is a continuous map f : M → N together with a fiberwise monomorphism TM → TN covering f. Theorem 1.6 (Gromov, 1969). Let R be an open coordinate invariant differential relation imposed on maps M → N of smooth manifolds. Then the h-principle for R holds true provided that M is an open manifold. The statement of Theorem 1.6 is counter-intuitive. Indeed, a priori the space of solutions of a differential relation is an extremely complicated space. However, by Theorem 1.6, for a large class of differential relations their spaces of solutions are weakly homotopy equivalent to relatively simple topological spaces of formal solutions. Example 1.7. One of the historically first surprising manifestations of the h- principle is the Smale paradox: the standard embedding of the sphere S2 into R3 can be deformed to the inside-out embedding through immersions. To deduce the Smale paradox from Theorem 1.6, note that the space of immersions S2 R3 is homotopy equivalent to the space of immersions of the open manifold S2 × R into R3.SinceS2 × R is parallelizable, the latter space is homotopy equivalent to the 2 path connected space of maps S → SO3 by Theorem 1.6. Consequently, the space of immersions S2 R3 is path connected.

THE WEAK B-PRINCIPLE 103

A diﬀerential relation is not bound to be a relation on maps; it can be imposed on tensors and other related structures. Because of that among striking consequences of the h-principle for maps of open manifolds are the existence theorems for contact and symplectic structures on open manifolds, foliations of given codimension, and Riemannian metrics of positive/negative scalar curvature on open manifolds. For an extensive account of the theory see the original book of Gromov [6], and a more recent book by Eliashberg and Mishachev [3].

2. The b-principle Similarly, the b-principle is a general observation that differential geometry problems can often be reduced to problems in stable homotopy theory. The known instances of the b-principle type phenomena include the Barratt-Priddy-Quillen theorem, the Audin-Eliashberg theorem on Legendrian immersions, and the standard Mumford Conjecture on moduli spaces of Riemann surfaces. In [10] I proposed a general setting for the b-principle and showed that the b-principle phenomenon occurs for a fairly arbitrary differential relation imposed on smooth maps, i.e., in many cases a differential geometry problem does reduce to a stable homotopy theory problem. Remark 2.1. In most of the cases invariants of solutions to differential relations computed by means of the h-principle reduction are stronger than their b-principle counterparts. On the other hand, the h-principle invariants are often considerably harder to compute, than the invariants obtained by the b-principle reduction. The general b-principle is formulated in terms of open stable differential relations R of dimension d.In[10] these are defined to be certain open coordinate invariant differential relations imposed on maps R∞ → R∞−d. Herewegivea different definition emphasizing their pullback property. Definition 2.2. An open stable differential relation R of dimension d is a family {Rm}m≥0 of open coordinate invariant differential relations imposed on m m−d functions R → R . We require that the differential relations Rm in the family m m−d R are compatible in the sense that if a map f : R → R is a solution of Rm and h: Rk−d → Rm−d is a map transversal to f, then the pullback h∗f is a solution of Rk. In particular, if R is an open stable differential relation, then the map m m−d f × idR : R × R → R × R satisfies R for every solution f of R. An open stable differential relation R of dimension d imposes a differential relation on maps of manifolds M of arbitrary dimension m to manifolds N of dimension m − d. Example 2.3. The simplest open stable differential relations are those of submersions and immersions of a fixed dimension d. In the b-principle setting, formal solutions are replaced with stable formal solutions. Definition 2.4. A stable formal solution of an open stable differential relation R is a continuous map f : M → N of manifolds of dimension m and n respectively together with an equivalence class of formal solutions F of Rm+k imposedonmaps k k M × R → N × R for k ≥ 0 such that F covers f × idRk . The equivalence relation is generated by identifying F with F × idR,andF with G if F = G over M ×{0}.

104 RUSTAM SADYKOV

Example 2.5. A stable formal solution of the submersion (respectively, immersion) differential relation of dimension d essentially is a continuous map f : M → N of dimension dim M − dim N = d together with an equivalence class of fiberwise epimorphisms (respectively, monomorphisms) F : TM ⊕ εk → TN ⊕ εk covering f, where εk is a trivial vector bundle of dimension k ≥ 0. The equivalence relation is generated by F F ⊕ idε. The spaces of solutions and formal solutions of the h-principle are replaced in the b-principle setting with moduli spaces of solutions and stable formal solutions respectively. The moduli space MR of solutions of an open stable differential relation R is an appropriate quotient space of all proper solutions of R;in§3we will give a simplicial model for MR. The b-principle approach is based on the observation that the disjoint union of two solutions f1 : M1 → N and f2 : M2 → N of R is again a solution M1 M2 → N, and therefore, the moduli space MR of solutions of R is an H-space. In fact the H-space operation on MR satisfies certain homotopy coherence conditions (homotopy associativity, homotopy commutativity and other Segal Γ-space conditions), and therefore by the Segal theorem [11]the moduli space MR admits a group completion. It also turns out that a similarly constructed moduli space hMR of all proper stable formal solutions of R is an infinite loop space [10]; we will describe its homotopy type in §5. Definition 2.6. Let R be an open stable differential relation. The b-principle is a conjecture asserting that the canonical map

MR −→ hMR of the moduli space MR of solutions into the moduli space hMR of stable formal solutions is a group completion.

Remark 2.7. In many cases the monoid π0(MR) is actually a group, and therefore the b-principle statement is equivalent to the assertion that the map MR → hMR is a homotopy equivalence. Early b-principle type results include the Barratt-Priddy-Quillen theorem, the Wells theorem for immersions, Eliashberg theorems for k-mersions, Audin-Eliash- berg theorems for Legendrian and Lagrangian immersions, and general theorems of Ando, Szucs and the author. In §3 we will see that the b-principle reduces to the weak b-principle which holds true under mild assumptions.

3. The weak b-principle A collection C of smooth maps f : M → N with fixed dim M − dim N = d is said to be a class of maps of dimension d if the pullback h∗g is in C for every map g ∈Cand every map h transversal to g. An appropriate quotient space of all proper maps in C is called the moduli space for C. We will use its simplicial model M = |X•|. Namely, recall that the opening of a subset X of a manifold V is an arbitrarily small but non-specified open neighborhood Op(X)ofX in V .Consider m+1 the affine subspace {x1 +···+xm+1 =1} in R . It contains the standard simplex m ≤ ≤ n Δ bounded by all additional conditions 0 xi 1. Let Δe denote the opening of Δm in the considered affine subspace. Then every morphism δ in the simplicial ˜ m → n C category extends linearly to a map δ :Δe Δe .LetXm denote the subset of of m proper maps to Δe transversal to all extended face maps. Then X• is a simplicial set with structure maps X(δ) given by the pullbacks f → δ˜∗f.

THE WEAK B-PRINCIPLE 105

n f

k g

Figure 1. Identiﬁcation of simplicies in the case of immersions.

The moduli space M is homotopy equivalent to the semi-simplicial geometric n n C n realization of X•:takeacopyΔf of Δ ,oneforeachpropermapf in to Δe k ⊂ n transversal to each face Δe Δe . Such a solution f restricts to a solution over k n k each face Δe .Ifasolutionf over Δe restricts to a solution g over a face Δe , k n n ∼ identify Δg with the corresponding face of Δf . The obtained space ( Δf )/ is the semi-simplicial realization of X•; see Figure 1. The moduli space M possesses an important classifying property. Let f : M → N be a proper map in the class C. Choose a smooth triangulation of N such that each face of each simplex of the triangulation is transversal to f. Choose an ordering on the set of verticies of the triangulation. Then each simplex Δ of the triangulation comes with a proper map f|f −1(Op(Δ)) transversal to all extended face maps, and therefore has a unique counterpart in M. This defines a map τf : N →M.The map τf depends on the choices of a smooth triangulation of N and an ordering of the set of verticies. However, the homotopy class of τf depends only on f. Example 3.1. Let f be a proper map in the class C.Ifκ is a cohomology class ∗ M ∗ in H ( ), then τf κ is an invariant of f. Example 3.2. For the class of oriented submersions of dimension d, the moduli space is the disjoint union of classifying spaces of orientation preserving diffeomorphism groups of all oriented closed manifolds of dimension d,see[8]. The similarly constructed moduli space for the class of formal oriented submersions of dimension d is homotopy equivalent to the infinite loop space of the Madsen-Tillman spectrum MTSO(d), see [4]. In the case of oriented submersions of dimension d =2,the ∗ rational cohomology invariants τf κ of Example 3.1 are precisely the Miller-Morita- Mumford classes, see [9]. To study C we consider derived collections C1 and hC1 of maps. The collection hC1 consists of proper maps (f,α): M → N × R with f ∈Csuch that every regularfiberof(f,α) is null-cobordant, while C1 ⊂ hC1 is a subset of pairs with α ◦ f −1(x) = R for all x ∈ N. The spaces M1 and hM1 are the geometric m × R realizations of simplicial sets of maps (f,α)toΔe such that f is transversal to all extended face maps and (f,α)isinC1 and hC1 respectively. Definition 3.3. The weak b-principle for C is said to hold true if the inclusion M1 → hM1 is a homotopy equivalence. Example 3.4. For the class C of embeddings, a regular fiber of (f,α)isempty. Hence M1 = hM1. Thus, the weak b-principle for embeddings holds true. It can be shown that M is homotopy equivalent to the Thom space MOd of the

106 RUSTAM SADYKOV universal vector bundle of dimension d over BOd. In fact, the points of M are of the two types: the ones with empty preimage and ones with one point preimage. The points of the former type form a contractible space, while the points of the latter type form a space homotopy equivalent to BOd. Also note that there is no h-principle or b-principle for embeddings since embeddings are not solutions to a differential relation. Now let us explain the relation between the b-principle and the weak b-principle. The collections C of solutions and hC of stable formal solutions to an open stable differential relation R are classes of maps with moduli spaces M and hM respectively. The homotopy type of hM is known [10]; it is a certain infinite loop space. We also proved [10]thathM is the loop space of hM1, while ΩM1 is a group completion of M. In fact, there is a commutative diagram b-principle group completion M −−−−−−−−−−−−−−−−−−→ hM ⏐ ⏐ ⏐ ⏐ group completion- -

Ω(weak b-principle map) ΩM1 −−−−−−−−−−−−−−−−→ ΩhM1. Thus, the b-principle reduces to the weak b-principle. Example 3.5. If C is the class of submersions of dimension d =0,thenM is the disjoint union BΣk of the classifying spaces of permutation groups Σk of k elements, one space for each k ≥ 0, cf. Example 3.2, while hM is the infinite loop space Ω∞S∞ of the sphere spectrum, see Example 5.1. In this case the b-principle is ∞ ∞ the Barratt-Priddy-Quillen theorem [1]: the b-principle map BΣk → Ω S is a group completion. We emphasize that the h-principle for submersions of dimension d = 0 imposed on smooth maps does not hold in general. For example, there exists no submersion S1 → R, while the projection TS1 S1 × R → R is a formal solution. Example 3.6. If C is the class of all smooth maps of dimension d, then the formal moduli space is the Thom space Ω∞+dMO, see Example 5.2, and the b- principle essentially reduces to the Pontrjagin-Thom theorem. Example 3.7. Let R be the open stable differential relation of immersions of dimension d. Then the b-principle reduces to the weak b-principle. The latter holds true by a dimensional argument as above, see Example 3.4. Hence the moduli space for immersions is homotopy equivalent to the moduli space of stable formal immer- ∞ ∞ sions, which in its turn is homotopy equivalent to Ω Σ MOd by the description in §5. This allows us to perform computations of invariants of immersions. Generalizing the previous example, we get the following consequence of a reduction of the b-principle to the weak b-principle. Theorem 3.8. Let R be a open stable differential relation imposed on maps of non-positive dimension. Then the b-principle for R holds true. Example 3.9. Let F be a connected oriented closed surface of genus g ≥ 2. It is known that the set Sg of complex structures on F coincides with the set of almost complex structures on F . Recall that an almost complex structure on F is a smooth fiberwise isomorphism J : TF → TF of the tangent bundle such that J 2 = −id and for every tangent vector v at x ∈ F the pair (v, Jv)isanoriented

THE WEAK B-PRINCIPLE 107 basis of the tangent space at x. In other words an almost complex structure on F is a smooth section of an appropriate bundle over F . The set of such sections is endowed with the Whitney C∞ topology, and therefore the space of complex structures Sg is a topological space.

Definition 3.10. The moduli space of Riemann surfaces Mg is the orbit + space Sg/ Diff F of the action of the orientation preserving diffeomorphism group + Diff F on the space Sg. According to the standard Mumford conjecture, in dimensions g the rational cohomology ring of Mg is isomorphic to the polynomial algebra Q[κ1,κ2, ...] in classes κi of degrees 2i. The Mumford conjecture has been established by Mad- senandWeissin[8](seeproofsalsoin[2, 4, 5, 7]). Namely, it can be shown that the rational cohomology ring of Mg is isomorphic to that of the classifying space + + BDiff Fg of Diff Fg. Madsen and Weiss showed that in a stable range of dimen- ∗ + sions H (BDiff Fg; Q)isisomorphictoQ[κ1,κ2, ...]. As has been mentioned in Example 3.5, the b-principle map for oriented submersions of dimension 2 is a map BDiffF → hM where the disjoint union ranges over diffeomorphism types of all (possibly non-connected) oriented closed surfaces F . This map is not a group completion, and therefore the b-principle does not hold ∗ true. However, H (hM; Q) Q[κ1,κ2, ...] and the Madsen-Weiss theorem shows that the b-principle holds true on some stable homology level. In the conclusion of this section let us compare the Smale-Hirsch h-principle for immersions and the Wells b-principle for immersions. Example 3.11. According to the Smale-Hirsch h-principle, a continuous map f : M → N of negative dimension covered by a fiberwise monomorphism TM → TN is homotopic to an immersion. According to the Wells b-principle, a continuous map f : M → N of negative dimension covered by a fiberwise monomorphism TM ⊕ ε → TN ⊕ ε is concordant to an immersion; we will recall the notion of concordance in §4.1.

4. Broken solutions Given a smooth map f : M → N, a point x ∈ M is said to be regular if in a neighborhood of x the map f is a submersion. If in a neighborhood of x the map f is the product of a Morse function and the identity map on an open disc of dimension dim N − 1, then x is a fold point. It immediately follows that the set of fold points of f is a submanifold of M.Wesaythatf is a fold map if each point x ∈ M is either regular or fold. Suppose that a path component σ of fold points of f is closed in M and the restriction f|σ is an embedding. We say that σ is a breaking component if there are a map τM of a neighborhood d+1 of σ into R trivializing the normal bundle of σ,andamapτN of a neighborhood of f(σ)toR trivializing the normal bundle of f(σ) such that τN ◦ f = gσ ◦ τM on d+1 the common domain, where gσ is a Morse function on R with one critical point of index =0 ,d+ 1; the maps τM and τN are parts of the structure. The minimum of the indices of the critical point of gσ and −gσ is called the index of σ.Note that in general the normal bundle in M of a component σ of fold points of f is not trivial, and therefore not every component of fold points admits a structure of a breaking component.

108 RUSTAM SADYKOV

× d+1 Rd+1 6 R M M

f g6

f(6(×R N N R

GivenanopenstablediﬀerentialrelationR imposed on maps of dimension d, suppose a map f away of the set of its breaking folds is a solution. Then we say that f is a broken solution of R. Example 4.1. Let W be a compact manifold of dimension d,andf a positive- valued proper Morse function on the interior of W such that f −1[1, ∞) is diﬀeo- morphic to ∂W × [1, ∞) and the restriction

R Int W

Int W × Sn

Rn of f to the latter is the projection onto [1, ∞). Then, by Proposition 4.2 below, f×id ⊂ (g, α): IntW × Sn −−−→ (0, ∞) × Sn −→ Rn+1 Rn × R is a map in hC1 where C is any class containing the underlying maps of broken submersions. Here the inclusion (0, ∞) × Sn ⊂ Rn+1 takes a scalar r and a vector v ∈ Rn+1 of length 1 to rv. Note that all singular points of f × id are fold points; they constitute submanifolds p × Sn ⊂ W × Sn,wherep ranges over all critical points of f.By Proposition 4.2 below, the map g is also a fold map, and the set of fold points of g consists of components p × Sn−1 ⊂ W × Sn,whereSn−1 stands for the equator of Sn. Proposition 4.2. Let f : M → Rn+1 be a fold map and π : Rn+1 → Rn the projection. Let Σ denote the set of fold points of f. Suppose that the composition π ◦ f|Σ is a fold map on Σ.Thenπ ◦ f is a fold map on M, and the fold points of π ◦ f coincide with the fold points of π ◦ f|Σ. Proof. Given a map g : M → N, a point x ∈ M is a fold point if it is not regular and the quadratic form Kx ⊗ Kx → Cx canonically deﬁned by means of the second derivatives of g is non-degenerate. Here Kx and Cx are the kernel and cokernel of the diﬀerential dxg of g at x.

THE WEAK B-PRINCIPLE 109

Since π is a submersion, every regular point of f is a regular point of π ◦ f. Furthermore, every regular point of π ◦ f|Σ is a regular point of π ◦ f as well. Let x be a fold point of π ◦ f|Σ. Then −1 Kx(π ◦ f)=dxf (Kf(x)(π)) = Kx(f) ⊕ Kx(π ◦ f|Σ), and π restricted to Cx(f) is an isomorphism onto Cx(π ◦ f)=Cx(π ◦ f|Σ). By the hypotheses of Proposition 4.2, the quadratic forms on the two summands on the right hand side are non-degenerate. Thus, the quadratic form on Kx(π ◦ f)isalso non-degenerate. 4.1. Concordance of maps. We say that a class C satisﬁes the sheaf property if for every smooth map f : M → N and every covering of N by open sets −1 Ui the map f is in C whenever the restrictions f|f (Ui)aremapsinC.Ifthe class C satisﬁes the sheaf property, then for every manifold N the set [N,M(C)] is isomorphic [8] to the set of concordance classes of proper maps in C to N. Recall that two proper maps f0,f1 in C to N are concordant if there is a proper map f to N × I, I =[0, 1], such that over N × (0, 1) the map f is in C, while over ε-collar neighborhoods of N ×{0} and N ×{1},themapf is given by f0 × id[0,ε) and f1 × id(1−ε,1] respectively for some ε>0.

Example 4.3. Let iA denote the inclusion of a subset A into R,andlet(gA,αA) denote the pullback of the map (g, α): IntW × Sn → Rn × R in Example 4.1 with respect to the inclusion n−1 n (iA × idRn−1 ) × idR :(A × R ) × R → R × R. 1 Then (g[0,1],α[0,1]) is a concordance between (g0,α0)and(g1,α1). Recall that hC consists of pairs (f,α) with f ∈Csuch that every regular ﬁber of (f,α) is null- cobordant, while C1 is its subset with α◦f −1(x) = R for all x. Thus the constructed broken submersion is a concordance between an element (g0,α0) in the derived class 1 1 of submersions C and an element (g1,α1)inhC . In fact, note that every map (f,β)inanyhC1 in a neighborhood F × Rn × R ⊂ M of a regular ﬁber F coincides n n with the projection F × R × R → R × R ⊂ N × R which is precisely (g1,α1)for F = ∂W where W is a null-cobordism W of F . Thus, locally (f,β) is concordant ◦ −1 to (g0,α0) satisfying α0 g0 (0) = 0. The concordance from (f,β)to(g0,α0)is called a breaking concordance. We say that a class C is monoidal if the empty map ∅→∗toapointisamap in C and the class C is closed with respect to taking disjoint unions of maps, i.e., if f1 : M1 → N and f2 : M2 → N are maps in C,thenf1 f2 : M1 M2 → N is also amapinC. It immediately follows that for a monoidal class C satisfying the sheaf property, the spaces M, M1 and hM1 are H-spaces. In particular, their homotopy groups of degree n are isomorphic to the groups of homotopy classes of free maps n n of S , e.g., πn(M) [S , M]. Now we are in position to formulate a very general b-principle type theorem. Theorem 4.4. Let C be a class of maps satisfying the sheaf property. Suppose that every breaking concordance of every map in hC1 is itself in hC1. Then the weak b-principle for C holds true. Proof of Theorem 4.4. Givenamap(f,α)inhC1 to a disc Dk of dimension k ≥ 1 that restricts over ∂Dk to a map in C1, we need to show that there is a concordance of (f,α)trivialover∂Dk to a map in C1. In other words, we need a

110 RUSTAM SADYKOV concordance that breaks the fibers of f. In a neighborhood of a regular fiber, (f,α) is isomorphic to (g1,α1) in Example 4.3. It remains to observe that the concordance of Example 4.3 is local in nature and breaks the fiber. In terms of the b-principle we deduce the following Theorem from [10]. Theorem 4.5. Let R be an open stable differential relation imposed on maps of positive dimension such that every Morse function on manifolds of dimension d +1 is a solution of R. Then the b-principle for R holds true.

5. The homotopy type of hMR Let R be an open stable differential relation. In this section we will show that invariants of hMR can be computed by standard stable homotopy methods. In fact, hMR is homotopy equivalent to a certain infinite loop space; we will describe this infinite loop space. Let B = BR be the topological space of equivalence classes of pairs (i, P )ofa map germ i:(Rn+d, 0) → (R∞, 0) and a subspace P ⊂ R∞ of dimension n ≥ 0such that the map germ α = π ◦ i is a solution germ of R,whereπ :(R∞, 0) → (P, 0) is the orthogonal projection germ onto P . The equivalence relation is generated by equivalences (i, P ) ∼ (i,P) with × ≡ i :(R × Rn+d, 0) id−→R i (R × R∞, 0) −→ (R∞, 0), ≡ ⊂ ≡ (P , 0) −→ (R × P, 0) −→ (R × R∞, 0) −→ (R∞, 0). The set of pairs (i, P ) is a subset of the product of the space of map germs i endowed with the Whitney C∞ topology and BO(n). In particular, the set of pairs (i, P ) inherits a topology. The topology on the space B is defined to be the quotient of the topology on the space of pairs (i, P ). ⊂ Let Bn+d B denote the space of pairs (i, P ) with P of dimension n.Thereis → → ∗ a projection fn+d : Bn+d BO(n) defined by (i, P ) P .Letfn+dξn denote the pullback of the universal vector bundle over BO(n). Then the space Tn is defined ∗ to be the Thom space of fn+dξn. Then the sequence of spaces Tn forms a Thom spectrum, whose infinite loop space is equivalent to hMR,see[10]. Let us recall an example from [10]. Example 5.1. For the submersion differential relation R imposed on maps of dimension d,thespaceB can be identified with the space of equivalence classes of linear maps α: Rn+d → R∞ of rank n for some n ≥ 0; the equivalence relation is generated by equivalences α ∼ α,whereα: Rn+d −→ R∞ and ≡ × ≡ α : Rn+d+1 −→ R × Rn+d id−→R α R × R∞ −→ R∞. There is a vector bundle ker α of dimension d over B whose fiber over the class of α is given by the kernel of α. Thus, there is a well-defined map ker α: B → BOd.The fiber of this map is contractible and therefore B BOd.Themapf : B → BO ×{d} of the tangential structure is the standard inclusion BOd → BO since it classifies n+d ∗ the stable vector bundle T0R ! α P represented by the kernel of the map α. In particular, if d =0,thenBOd is a point and hMR is homotopy equivalent to the infinite loop space Ω∞S∞ of the sphere spectrum. For oriented submersion of dimension d the spaces BOd and BO are replaced with BSOd and BSO respectively. For example, for oriented submersions of dimension d = 2, the space BSOd coincides

THE WEAK B-PRINCIPLE 111 with CP ∞ and hM is homotopy equivalent to the inﬁnite loop space of the Madsen- Tillman spectrum MTSO(2). Example 5.2. If all smooth maps of dimension d are solutions of a diﬀerential relation R,thenBR is homotopy equivalent to the space BO and hM is homotopy equivalent to Ω∞+d MO.

Since hMR is equivalent to the inﬁnite loop space of a spectrum, its rational cohomology can be computed by standard methods of stable homotopy theory.

Theorem 5.3. The rational cohomology ring of hMR is a tensor product of polynomial and exterior algebras

Q[x1,x2, ...] ⊗ Λ[y1,y2, ...], where there is one class xi for each even-dimensional generator of the vector space Q ⊗ H∗(BR) and one class yi for each odd-dimensional generator of Q ⊗ H∗(BR). Corollary 5.4. For oriented submersions of dimension 2, there is an isomor- ∗ phism H (hMR; Q) Q[τ1,τ2, ...],whereeachτi is a class of dimension 2i.

Proof. Recall that in the case of oriented submersions, the space BR is homotopy equivalent to the space BSO2 and Hj (BSO2; Q)istrivialifj is odd and is a one-dimensional vector space if j is even. The b-principle also allows us to perform computations of certain bordism groups. Example 5.5. The Wells theorem asserts that modulo finite group the bordism n+d group of immersions of codimension d into R is isomorphic to H∗ MOd.Toprove the Wells theorem, observe that the bordism group of immersions of codimension d n+d into R is isomorphic to πn+dMR,whereR is the open stable differential relation of immersions of codimension d. By the b-principle, we have M ⊗ Q M ⊗ Q st ⊗ Q Q πn+d( R) πn+d(h R) πn+d(MOd) Hn+d(MOd; ). 6. Final remarks I would like to thank the referee for a nice observation which I include here in a slightly modified/extended form. The referee pointed out that there are higher analogues of the weak b-principle; these are always hold true and are very closely related to [5, Proposition 3.21] and [7, Proposition 3.1]. In fact, the geometric construction in the current note (and in the proof of the earlier b-principle in [10]) is closely related to the constructions in the mentioned papers by Galatius and Randal-Williams and Hatcher; we break surfaces using allowed singularities, while the mentioned authors break the surfaces by pushing them to infinity and, thus, avoiding singularities. In detail, for each class C of maps, we defined derived collections C1 and hC1. Similarly, we may define higher derived analogues Ck and hCk. Indeed, let hCk k be the collection that consists of proper maps (f,α1, ..., αk): M → N × R such that f is in C and every regular fiber of (f,α1, ..., αk) is null-cobordant, while k k −1 C ⊂ hC is its subspace that consists of tuples with αk ◦ f (x) = R for all x ∈ N. −1 In other words, informally, we consider families of manifolds Mx = f (x) with singularities—which are prescribed by a choice of C—together with proper maps k k (α1, ..., αk): Mx → R . In the case of the bigger family hC we require that each

112 RUSTAM SADYKOV

fiber of the proper map is null-cobordant, while in the case of the smaller family k C we require that αk misses at least one value. We say that the k-th weak b-principle for a class C holds true if the inclusion Mk → hMk of the moduli spaces of Ck into that of hCk is a homotopy equivalence. The main theorem of this note, Theorem 4.4, establishes the k-th weak b-principle for k = 1 under certain conditions on C. Galatius and Randal-Williams, and Hatcher show in the mentioned papers that the k-th weak b-principle for k ≥ 2 holds true for submersions; while, as we have seen, the b-principle for submersions of dimension d = 2 fails to be true. In fact, their argument shows that the k-th weak b-principle holds true for k ≥ 2, for example, for the class C of solutions to any given open stable differential relation. Here is a sketch of a slightly modified proof. m × Rk Let (f,α)=(f,α1, ..., αk)beamaptoΔe representing a relative homotopy k k class in π∗(hM , M ). We need to construct a concordance that breaks the fibers of (f,αk). As in the proof of Theorem 4.4 we have a breaking fold concordance such that for the resulting map (f,¯ α¯)wehave¯α◦f −1(x) = Rk for all x ∈ Δm.The breaking fold components of the resulting map (f,¯ α¯) are spheres S of dimension m + k − 1, and the corresponding new singularities of f¯ are the singularities of f¯|S, see Proposition 4.2. Since k ≥ 2, we can push the singularities of f¯|S ⊂ S,and the corresponding singularities of the concordance, in the (k − 1)-st direction to infinity. Finally, we can stretch the holes Rk \ α¯ ◦ f −1(x)inthe(k − 1)-st direction, so that the fibers of the map (f,αk) are broken. This completes the proof. References [1] Michael Barratt and Stewart Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972), 1–14. MR0314940 (47 #3489) [2] Yakov Eliashberg, Søren Galatius, and Nikolai Mishachev, Madsen-Weiss for geometrically minded topologists,Geom.Topol.15 (2011), no. 1, 411–472, DOI 10.2140/gt.2011.15.411. MR2776850 (2012d:55008) [3] Y. Eliashberg and N. Mishachev, Introduction to the h-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR1909245 (2003g:53164) [4] Søren Galatius, Ulrike Tillmann, Ib Madsen, and Michael Weiss, The homotopy type of the cobordism category,ActaMath.202 (2009), no. 2, 195–239, DOI 10.1007/s11511-009-0036-9. MR2506750 (2011c:55022) [5] Søren Galatius and Oscar Randal-Williams, Monoids of moduli spaces of manifolds,Geom. Topol. 14 (2010), no. 3, 1243–1302, DOI 10.2140/gt.2010.14.1243. MR2653727 (2011j:57047) [6] Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Gren- zgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR864505 (90a:58201) [7] A. Hatcher, A short exposition of the Madsen-Weiss theorem, preprint. [8] Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941, DOI 10.4007/annals.2007.165.843. MR2335797 (2009b:14051) [9] Shigeyuki Morita, Geometry of characteristic classes, Translations of Mathematical Mono- graphs, vol. 199, American Mathematical Society, Providence, RI, 2001. Translated from the 1999 Japanese original; Iwanami Series in Modern Mathematics. MR1826571 (2002d:57019) [10] R. Sadykov, The bordism version of the h-principle, arxiv:1002.1650v2. [11] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR0353298 (50 #5782)

Department of Mathematics, CINVESTAV, Mexico E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12420

Orbit Conﬁguration Spaces

Miguel A. Xicot´encatl Dedicated to the 50th anniversary of the Mathematics Department of CINVESTAV

Abstract. The purpose of this article is to analyze equivariant analogues Fk(M; G) of conﬁguration spaces, for a manifold M with a free G-action. We describe their basic properties and several examples and applications that include: complements of hyperplane arrangements, loop space homology and its relation to the Lie algebra of the descending central series, cohomology of conﬁguration spaces of projective spaces and braid groups on surfaces.

1. Introduction Given a topological space M, the configuration space of k-tuples of distinct points is defined as k Fk(M)={(m1,...,mk) ∈ M | mi = mj if i = j}. Since their introduction by Fadell and Neuwirth [FN], configuration spaces have proven to be very useful in algebraic topology. They admit applications in many diverse ways from fixed point theory to function spaces, braid groups and cohomology of groups. See [Co4]and[FH] for a survey. The purpose of this paper is to explore properties of a related construction. Namely, let M have a free action of a topological group G such that M → M/G is a principal G-bundle. Associated with this action, there is the so called “orbit configuration space” of k-tuples of points in M, lying in different orbits, defined as k Fk(M; G)={(m1,...,mk) ∈ M | Gmi = Gmj for i = j}. These orbit configuration spaces have special features in their loop space homology and features of certain Lie algebras. To the best knowledge of the author, these spaces were introduced in [Xi1] and have since then attracted interest because of their relation to equivariant mapping spaces, to fiber-type arrangements [DC], to ordinary configuration spaces [FZ], [Xi3] and to surface braid groups [GP], [Xi5]. In [LS] R. Longoni and P. Salvatore studied Massey products on certain orbit configuration spaces associated to lens spaces Lp,q, to show that configuration spaces are not homotopy invariants. More recently, J. Chen, Z. LüandJ.Wu

2010 Mathematics Subject Classiﬁcation. 16S30, 20F36, 55P35, 55R80, 57S30. Key words and phrases. Orbit conﬁguration spaces, descending central series, Lie algebras, surface braid groups. The author would like to thank CONACyT grant 168349 and ABACUS, CONACyT grant EDOMEX 2011-C01-165873.

c 2014 American Mathematical Society 113

114 MIGUEL A. XICOTENCATL´ considered the case of non-free actions in [CLW], namely they study the homotopy type and the homology of orbit configuration spaces of small covers and quasi-toric manifolds, naturally endowed with a non-free action. Also, since ordinary configuration spaces are closely related to mapping class groups (see for instance [Co3], [Co4], [BCP], [Co5]), it seems reasonable to think there are some further connections between orbit configuration spaces and mapping class groups, but they will be explored elsewhere. The purpose of this article is to present a survey of some natural examples and applications of orbit configuration spaces already in the literature, with the exception of section 4 and part of section 5. The article is organised as follows. In section 2 we recall the basic properties of orbit configuration spaces, in particular the analogues of the Fadell-Neuwirth fibrations, and give several examples. In section 3 we study loop spaces of orbit configuration spaces, and explore the connection between their homology and the Lie algebra associated to the descending central series of related groups. Section 4 presents some unpublished results from the author’s Ph.D. thesis [Xi1], namely n the homology and the loop space homology of the space Bk(R ) defined as the n n orbit configuration space Fk(R \{0}; Z2), where R \{0} is equipped with the Z2-antipodal action. In section 5 we review the computation of the cohomology n algebra of Fk(S ; Z2) given in [Xi3], and its relation to the configuration space n k ∗ n Fk(RP ). We also describe completely the (Z2) -action on H Fk(S ; Z2), recently found A. Guzmán in his Ph.D. thesis [Gu], as well as the ring of invariants in thecasewhenn is odd. This describes completely the rational cohomology of n Fk(RP )forn odd. Section 6 describes the pure braid group Pk(M)andthebraid 2 2 k k group Bk(M)ofasurfaceM = S , RP ,asextensionsofπ1(M) and π1(M) Σk, respectively, by the fundamental group of the orbit configuration space of the . universal cover of M, Fk(M; G), where G = π1(M). Section 7 recalls a recent result of R. Longoni and P. Salvatore [LS] on the non-homotopy invariance of configuration spaces of lens spaces. Namely, they show the configuration spaces Fk(L7,1)andFk(L7,2) are not homotopy equivalent, even though L7,1 L7,2.This is done by studying Massey products on the corresponding universal covers, which 3 turn out to be orbit configuration spaces of the form Fk(S ; Z7). Finally, in section 8 we prove Theorem 2.2.

2. Definition and main properties Let M be an n-dimensional manifold, G a topological group and let us assume that G acts freely on M.LetGm denote the orbit of an element m ∈ M under the action of G. Inspired by [FN], define the orbit configuration space k Fk(M; G)={(m1,...,mk) ∈ M | Gmi = Gmj for i = j} There is an obvious relation to the ordinary configuration spaces of M/G. Namely, notice that the group Gk = G × ...× G acts coordinate-wise on the space Fk(M; G) and the quotient is homeomorphic to the configuration space Fk(M/G). Thus, an equivalent definition for Fk(M; G) can be given in terms of ordinary configuration spaces. Let f : M/G → BG a map that classifies the principal bundle G → M → M/G.Then

Theorem 2.1. The space Fk(M; G) is homeomorphic to the total space of the pull-back of the principal ﬁbration Gk → (EG)k → (BG)k along the composition

ORBIT CONFIGURATION SPACES 115

k k f k k Fk(M/G) → (M/G) −→ (BG) . Thus there are maps of G -bundles: Gk Gk Gk

/ k / k Fk(M; G) M (EG)

k / k f / k Fk(M/G) (M/G) (BG)

k Therefore, there is a principal G -bundle Fk(M; G) → Fk(M/G). Proof. Notice the Gk-action on the product M k induces a free action of Gk on Fk(M; G), and the orbit space can be identified with Fk(M/G). Now, it follows from the definition of f that the principal bundle Gk → M k → (M/G)k is classified by the map f k :(M/G)k → (BG)k and thus we have ∗ k ∗ k ∗ k Fk(M; G)=i (M )=i (f ) (EG) k where i : Fk(M/G) → (M/G) is the natural inclusion. Obviously, the orbit configuration space can also be defined in the case when the action is not free, but then the natural projection Fk(M; G) → Fk(M/G)and the analogues of the Fadell-Neuwirth fibrations (see Theorem 2.2) become singular fibrations at the points where the action has nontrivial isotropy. Examples of non- free orbit configuration spaces were recently studied in [CLW] in the context of small covers and quasi-toric manifolds. Some examples of manifolds with free group-actions are: n (1) R \{0} with the Z2-action given by the antipodal map. The homology n and loop-space homology of the orbit configuration space Fk(R \{0}; Z2) is computed in section 4 below. n (2) C \{0} with a Zp-action given by multiplication by a primitive p-th root of unity, ζp.Ifn =1andp = 2, the orbit configuration space Fk(C\{0}; Z2) is the complement of a hyperplane arrangement of type Bk. See sections 3, 4 and [OT]. (3) C with the natural action of the Gaussian integers Z + iZ by translation. In this case, the space Fk(C; Z + iZ) was studied in [CX] n (4) S with the Z2 antipodal action. In this case, the corresponding orbit con- n k figuration space Fk(S ; Z2)isa(Z2) -cover of the ordinary configuration n space Fk(RP ). (5) If M is a compact orientable surface, then Γ = π1(M) can be identified with a discrete subgroup of PSL(2, R), acting properly discontinuously on the hyperbolic plane H2 via Möbius transformations. In this case, the 2 fundamental group of the space Fk(H ; Γ) is the normal closure of Pk,the pure braid group of the plane, inside Pk(M), the pure braid group of M. See section 6 below. In the case when the G-action on M is free and properly discontinuous, the natural projection maps p : Fk(M; G) → F(M; G) give analogues of the Fadell and G ⊂ Neuwirth fibrations [FN]. For any natural number ,letQ M be the union of disjoint G-orbits in M.

116 MIGUEL A. XICOTENCATL´

Theorem 2.2. For k>, the projection p : Fk(M; G) → F(M; G) onto the \ G first coordinates, is a locally trivial bundle with fibre Fk−(M Q ; G). Just as in the case of ordinary configuration spaces, Theorem 2.2 is a very useful tool in the description of orbit configuration spaces, specially when computing their homology, as it will be shown the next sections. The proof of this statement will be postponed until section 8. One can also consider the case of unordered configurations. This is specially useful when studying surface braid groups, see section 6 below. Let Σk be the symmetric group in k letters and notice that Σk acts freely on Fk(M; G) by permutation of coordinates. Moreover, the semidirect k product G Σk acts freely on Fk(M; G)by

(g1,...,gk; σ)(m1,...,mk)=(g1 · mσ−1(1),...,g1 · mσ−1(k)) Then, the following result is immediate. Theorem 2.3. If G acts freely on M such that G → M → M/G is a principal k G-bundle, then G Σk acts freely on Fk(M; G) and there is a principal bundle k (G Σk) → Fk(M; G) → Fk(M/G)/Σk. k Thus the quotient Fk(M; G)/(G Σk) is homeomorphic to Fk(M/G)/Σk, the unordered conﬁguration space of M/G. In addition, if G acts properly discontinuously k on M,thenG Σk acts properly discontinuously on Fk(M; G),andtheﬁbration above is a covering space.

3. Loop spaces of orbit conﬁguration spaces n The integral homology of the based loop space ΩFq(R ) has been computed by E. Fadell, S. Husseini (unpublished), and subsequently by F. Cohen and S. Gitler [CG], who showed that the Lie algebra of primitives is given by a non-trivial extension of free Lie algebras. Namely there is an isomorphism from the module of n ∼ primitives PH∗(ΩFq(R )) = L2 ⊕ ...⊕ Lq,whereLi = L[Ai,1,...,Ai,i−1]isafree Lie algebra on i − 1 generators, and the relations among the Ai,j are of the form:

[Ai,j ,Ak,i]=[Ak,i,Ak,j] for 1 ≤ j

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Now for every p and q the commutator [a, b]=aba−1b−1 in π induces a bilinear map 0 × 0 −→ 0 Ep Eq Ep+q 0 0 which makes E∗ (π) into a Lie algebra. Moreover, E∗ is a functor from the category 0 of groups to the category of Lie algebras. Two elementary properties of E∗ are: 0 Z Z (1) E1 (π)=H1(π; )(with -trivial coeﬃcients). 0 (2) If F [X] is the free group generated by a set X,thenE∗(F [X]) is isomorphic to L[X], the free Lie algebra generated by X (see [Se]). In [FR], M. Falk and R. Randell have proven the following result:

j Theorem 3.1. Let 1 → A −→i B −→ C → 1 be a split extension of groups such that C acts trivially on H1(A). Then, the induced sequence 0 0 0 0 → E∗(A) → E∗ (B) → E∗(C) → 0 is split exact (as graded abelian groups, but not necessarily as Lie algebras). Proof. Let σ : C → B be a cross-section, i.e. a right-inverse of j. Then there is a well deﬁned map τ : B → A given by τ(b)=i−1[σj(b−1)·b] which need not be a homomorphism, but still is a left-inverse of i. Now, the triviality of the action is equivalent to [A, C] ⊂ [A, A]=Γ2A,whereweidentifyA and C with their images under i and σ. One can prove inductively that [ΓnA, ΓkC] ⊂ Γn+kA and use this to show that τ(ΓnB) ⊂ ΓnA. Now one can show that the induced sequence 1 → ΓnA → ΓnB → ΓnC → 1 is (split) exact, since b ∈ ΓnB ∩ ker j implies b = i(τ(b)). The theorem follows easily from this last statement.

0 ∼ 0 0 Thus, there is an isomorphism E∗ (B) = E∗(A) ⊕ E∗(C) of abelian groups. In general, this is not a trivial extension of Lie algebras but under the hypothesis of 0 0 0 Theorem 3.1, one can prove that if a ∈ E∗ (A)andc ∈ E∗ (C)then[a, c] ∈ E∗ (A). 2 Now, recall that the configuration space Fk(R )isaK(π, 1) and its fundamental group can be identified with the pure braid group on k strands, Pk.From[FN] and [Co1] we know that there is a fibration: 2 2 2 R \ Qk−1 → Fk(R ) → Fk−1(R ) which has a cross-section and trivial local coefficients in homology. Then looking at the exact sequence of homotopy groups, we obtain a split extension of fundamental groups

1 → Fk−1 → Pk → Pk−1 → 1 which satisﬁes the hypothesis of the theorem. Here Fk−1 is a free group on (k − 1) generators, usually denoted Ak,1,Ak,2,...,Ak,k−1. Applying inductively Theorem 3.1 we get 0 ∼ 0 0 0 E∗ (Pk) = E∗(F1) ⊕ E∗(F2) ⊕ ...⊕ E∗ (Fk−1) ∼ = L2 ⊕ L3 ⊕ ...⊕ Lk with Li = L[Ai,1,...,Ai,i−1]. We can work out the Lie algebra extension by using an explicit presentation of Pk (see [Xi2] for example) and obtain the inﬁnitesimal braid relations among the Ai,j ’s.

118 MIGUEL A. XICOTENCATL´

In section 4, we compute the homology of the loop space of some natural examples of orbit configuration spaces. Using a very well known result about fibrations with cross-sections (see section 4.2), one can prove the following as a starting point: Theorem . \ G → → 3.2 If the fibration M Q(i−1) Fi(M; G) Fi−1(M; G) has a cross-section for 2 ≤ i ≤ k, then there is a homotopy equivalence: k+−1 \ G ΩFk(M; G) Ω(M Qi ) i=0

Remark 3.3. This is always the case when M = M − {∗} is a manifold with a “puncture” or M = N × R. This product decomposition however is not multiplicative, i.e. it is not a homotopy equivalence of H-spaces. The precise algebraic n extension in homology is given next for the space Bk(R ) which is defined below. n n n Set Bk(R )=Fk(R \{0}; Z2), where R \{0} is given the Z2 antipodal action. k The notation comes from the fact Bk(C) ⊂ C is the complement of an arrangement of hyperplanes of type Bk (see [OT]). The arrangement in question is the union of the complexifications of the reflecting hyperplanes in a Coxeter group of type Bk. n In the next section we study the homology of Bk(R ) as well as the homology ring of its loop space. A sample of the homological calculation is given by the following theorem. For any non-zero integer j, define sgn(j)=j/|j|. Theorem 3.4. For n even, n ≥ 4, the Lie algebra of primitives in the Hopf n algebra H∗(ΩBq(R )) is given as a graded module by: n ∼ PH∗(ΩBq(R )) = L[V1] ⊕ L[V2] ⊕ ...⊕ L[Vq] where L[Vi] is the free Lie algebra generated by the set Vi = { Bi,j | i>|j|} (to be defined in section 4). The Lie products [a, b] with a ∈ Vi and b ∈ Vk for 1 ≤ i

[Bi,0,Bk,l]=[Bk,l, (Bk,−i + Bk,0 + Bk,i)]. The case for general n is given in Theorem 4.2. By [MM] the integral homol- n ogy ring H∗(ΩBk(R )) can be recovered as the universal algebra of the graded Lie n algebra PH∗(ΩBq(R )). Summarizing, the Lie brackets among primitive elements are given in terms of the “inﬁnitesimal braid relations” with signs (a), and a “new n relation” (b) and they provide the set of commutation rules in H∗(ΩBq(R )). We know that the presence of the signs is due to the Z2-action considered, since analogous calculations have been carried out for the Zp case, with the generators of g ∈ Z the form Bi,j , g p (see [Xi1]). The new relations appeared naturally since the manifold M considered here was not contractible. For completeness we include the case for general p: Theorem 3.5. For n ≥ 2, the Lie algebra of primitives elements in the Hopf n algebra H∗(ΩFq(C \{0}; Zp)) is given as a graded module by: n ∼ PH∗(ΩFq(C \{0}; Zp)) = L[V1] ⊕ L[V2] ⊕ ...⊕ L[Vq]

ORBIT CONFIGURATION SPACES 119

{ }∪{ g | − ∈ Z } where for each i, Vi = Bi,0 Bi,j j =1,...,i 1 and g p is certain set of homology classes of cardinality (i − 1)p +1,indimension2n − 2.TheLie products [a, b] with a ∈ Vi and b ∈ Vk for 1 ≤ i

0 Theorem 3.6. For n ≥ 2, with appropriate regrading, the Lie algebra E∗ P (p, q) associated to the descending central series of P (p, q) is isomorphic to the Lie algebra n n of primitives, PH∗(ΩFq(C \{0}; Zp)), in the Hopf algebra H∗(ΩFq(C \{0}; Zp)). Some further connections between the descending central series and the the homology of loop spaces of orbit configuration spaces were considered in [CX] and [CKX]. In [CX] F. Cohen and the author gave an explicit presentation for the Lie algebra associated with the descending central series of the fundamental group of Fk(M; G), in the case when M = C acted by the Gaussian integers G = Z+iZ. They also showed this Lie algebra is isomorphic, after regrading, and modulo torsion, to the homotopy Lie algebra q E = πr(ΩFk(M × C ; G)), r≥1 where G acts trivially on Cq,andthebracketonE is the Samelson product. In [CKX] F. Cohen, T. Kohno and the author looked at the case when G is a discrete subgroup of PSL(2, R) acting on the hyperbolic plane. Finally, we mention that some other results of the author give product decompositions for the loop-space of ordinary configuration spaces of projective spaces in terms of known spaces and loop-spaces of orbit configuration spaces. Recall n n n 2n+1 1 n 4n+3 3 RP = S /Z2, CP = S /S and HP = S /S . The following result was proven in [Xi4]. Theorem 3.7. For every k ≥ 1 there are homotopy equivalences n k n ΩFk(RP ) (Z2) × ΩFk(S ; Z2) if n ≥ 3 n 1 k 2n+1 1 ΩFk(CP ) (S ) × ΩFk(S ; S ) if n ≥ 2 n 3 k 4n+3 3 ΩFk(HP ) (S ) × ΩFk(S ; S ) if n ≥ 2.

n 4. The homology ring H∗(ΩBq(R )) n n 4.1. The homology of Bq(R ). In this section we compute H∗Bq(R )and Z Rn 2 ∪{ } construct an explicit basis for Hn−1(Bq( )). Let Q2q−1 denote the set Qq−1 0 , n the union of q − 1 disjoint Z2-orbits in R \{0} plus 0. We already know that the n n n n ﬁbration (R \Q2q−1) → Bq(R ) → Bq−1(R ) has a section; therefore H∗Bq−1(R )

120 MIGUEL A. XICOTENCATL´ injects in the homology of the total space and the corresponding Serre spectral sequence collapses for dimensional reasons, so n ∼ n n H∗(B (R )) = H∗(B − (R )) ⊗ H∗(R \ Q − ) q q 1 / 2q 1 ∼ n n−1 = H∗(Bq−1(R )) ⊗ H∗( S ) 2q−1 q / ∼ n−1 = H∗( S ) (by induction) i=1 2i−1 2 n and in particular, Hn−1 hasrank1+3+...+(2q − 1) = q .Lete1 ∈ R be the ﬁrst canonical unit vector. Now, for i =0,...,q put xi = i · e1.For0≤ j0. i,j 1 2 i 1 j 2 i+1 q n−1 Let ι be the fundamental class in H∗(S ). Then, the set of homology classes

{ Ci,j∗(ι) | 0 ≤ j

xi − xj pi,j (x)= if j ≥ 0 |xi − xj | + xi + xj pi,j (x)= if j>0 |xi + xj | Then, the induced maps + Rn −→ Z pi,j∗,pi,j∗ : Hn−1(Bq( )) n−1 n represent elements in H (Bq(R )) dual to Ci,j∗(ι)andC¯i,j∗(ι). This can be seen + C C by evaluating directly pi,j∗; ,where runs through the set constructed above. u Finally, if we choose (x1,...,xq) to be the base point of Bq(R ), we can identify the set {Cq,0∗(ι)}∪{Cq,j∗(ι) | 1 ≤ j

n 4.2. The Lie algebra of primitives in H∗(ΩBq(R )). We recall the following well known fact (see for example [Co2]). Let F → E → B be a ﬁbration with a cross-section σ : B → E. Then there is a homotopy equivalence: ΩB × ΩF ΩE. A choice of equivalence is

Ω(σ)×Ω(inc) ΩB × ΩF −−−−−−−−−−−−→ ΩE × ΩE −−−−mult→ ΩE Therefore (1) The inclusions ΩB → ΩE and ΩF → ΩE are multiplicative and thus, the induced maps in homology are morphims of algebras.

ORBIT CONFIGURATION SPACES 121

(2) The induced map H∗(ΩB) ⊗ H∗(ΩF ) → H∗(ΩE) is an isomorphism if coefficients are taken such that the strong form of the Künneth theorem holds (where Tor = 0). n This implies Theorem 3.2 and gives the module structure of H∗(ΩBq(R )). Indeed, (2) can be applied to the tower of fibrations n n Bq(R ) ←− (R \ Q2q−1) ↓ n n Bq−1(R ) ←− (R \ Q2q−3) ↓ . . . . ↓ n n B2(R ) ←− (R \ Q3) ↓ n n B1(R ) ←− (R \{0}) togetupanisomorphism q / ∼ n−1 = n H∗ Ω( S ) −−−→ H∗(ΩBq(R )). i=1 2i−1 n−1 ∼ ¯ n−2 By the Bott-Samelson Theorem, we have H∗Ω( 2i−1 S ) = T [H∗( 2i−1 S )] and then as graded modules, there is an isomorphism n ∼ H∗(ΩBq(R )) = T [V1] ⊗ T [V2] ⊗ ...⊗ T [Vq] where T [V ] denotes the tensor algebra generated by the set V and Vi is a basis n−2 ∼ n−1 for the group Hn−2(∨2i−1S ) = Hn−1(∨2i−1S ) (the isomorphism is given by n the homology suspension). Thus, to calculate the ring structure of H∗(ΩBq(R )) it suffices to compute the commutators [a, b] with a ∈ Vi and b ∈ Vk for 1 ≤ i

Lemma 4.1. Let X be 1-connected and assume that H∗(ΩX; Z) is primitively generated (as a Hopf algebra) and torsion-free. Then ∼ H∗(ΩX; Z) = U(P (H∗(ΩX; Z))) is the universal enveloping algebra of the Lie algebra of primitives.

Proof. By hypothesis, the natural morphism U(PH∗(ΩX; Z)) → H∗(ΩX; Z) is a surjection. Notice that after tensoring with Q it also becomes an injection (see for example [MM]). Since there is no torsion this implies the original map was an injection.

n−1 Now, the module of primitives in H∗Ω(∨2i−1S ; Z)isgivenbyL[Vi], the free Lie algebra generated by Vi. Finally, the module of primitives in H∗X ⊗ H∗Y (for path connected X and Y )isPH∗X ⊕ PH∗Y . Thus there is an isomorphism n ∼ PH∗(ΩBq(R )) = L[V1] ⊕ L[V2] ⊕ ...⊕ L[Vq] n as graded modules and for every i, L[Vi] is a Lie subalgebra of PH∗(ΩBq(R )). The n problem is now reduced to determining the Lie algebra structure on PH∗(ΩBq(R )). Consider now the maps induced in homology by the loops of the Ci,j ’s : n−1 n (ΩCi,j )∗, (ΩC¯i,j)∗ : H∗(ΩS ) −→ H∗(ΩBq(R ))

122 MIGUEL A. XICOTENCATL´ and use them to deﬁne classes

Bi,j =(ΩCi,j)∗(ιn−2)

B¯i,j =(ΩC¯i,j)∗(ιn−2).

We take now Vk to be the set { Bk,0}∪{Bk,j | 1 ≤ j

n Theorem 4.2. The Lie algebra of primitives in the Hopf Algebra H∗(ΩBq(R )) is given as a graded module by:

n PH∗(ΩBq(R )) = L[V1] ⊕ L[V2] ⊕ ...⊕ L[Vq]

The Lie products [a, b] with a ∈ Vi and b ∈ Vk for 1 ≤ i

[Bi,j ,Bk,l]=[Bi,j , B¯k,l]=[B¯i,j ,Bk,l]=[B¯i,j , B¯k,l]=0

Otherwise, if [B ,B ]=[B ,B ](j =0) (a) j = l then i,j k,j k,j k,i ¯ [Bi,0,Bk,0]=[Bk,0,Bk,i, ]+α[Bk,0, Bk,i] [B ,B ]=α[B ,B ](j =0) (b) i = l then i,j k,i k,i k,j [Bi,0,Bk,i]=α[Bk,i,Bk,0]+[Bk,i, B¯k,i] ¯ ¯ ¯ (c) j = l then [Bi,j , Bk,j]=α[Bk,j, Bk,i](j =0) [B , B¯ ]=[B¯ , B¯ ](j =0) (d) i = l then i,j k,i k,i k,j [Bi,0, B¯k,i]=[B¯k,i,Bk,0]+[B¯k,i,Bk,i]

(e) j = l then [B¯i,j ,Bk,j]=α[Bk,j, B¯k,i] (f) i = l then [B¯i,j ,Bk,i]=α[Bk,i, B¯k,j](j =0) (g) j = l then [B¯i,j , B¯k,j]=[B¯k,j,Bk,i] (h) i = l then [B¯i,j , B¯k,i]=[B¯k,i,Bk,j](j =0) . In the case of n even, most of the relations are redundant and the statement simpliﬁes considerably if we introduce a change of notation. For |j|

The Lie products [a, b] with a ∈ Vi and b ∈ Vk for 1 ≤ i

[Bi,0,Bk,l]=[Bk,l, (Bk,−i + Bk,0 + Bk,i)]

ORBIT CONFIGURATION SPACES 123

4.3. Lie relations. To compute the Lie brackets [Bi,j ,Bk,l], with |j|

n−1 n−1 n ϕ : S × S −→ Bq(R ) given by ϕ(z,w)=(y1,y2,...,yq)where ⎧ ⎨⎪xs if s = i, q

ys = sgn(j) · x + z/2ifs = i ⎩⎪ j sgn(l) · yi + w/4ifs = q

n−1 n−1 and compute the images of the fundamental cycles in H∗(S × S ), with the + aid of the dual classes pi,j and pi,j ϕ∗(ιn−1 ⊗ 1) = Cr,s∗(ι) ϕ∗(1 ⊗ ιn−1)= Ct,u∗(ι).

We know that after looping, the cycles (ιn−2 ⊗ 1) and (1 ⊗ ιn−2) commute (in the n−1 n−1 graded sense) in H∗(Ω(S × S )) and it follows by naturality that Br,s, Bt,u =0.

This method provides us with all the relations. See [Xi1] for the details.

n 5. The cohomology of Fk(RP ) n Let Z2 act on S via the antipodal map. In [Xi3] the orbit configuration space n Fk(S ; Z2) was studied. In particular, a complete description of its cohomology algebra was given with coefficients in Z or Q. Similar results in this direction were obtained independently by E.M. Feichtner and G. Ziegler [FZ]. The space n k Fk(S ; Z2) admits a natural (Z2) -action and the quotient is homeomorphic to the n ordinary configuration space Fk(RP ). Therefore, there is a spectral sequence p,q p Z k q n Z E2 = H (( 2) ; H Fk(S ; 2)) ∗ n with non-trivial local coefficients that converges to H Fk(RP ). In the rational n case the spectral sequence above collapses and the cohomology of Fk(RP )isgiven by the algebra of invariants:

k ∗ n ∼ ∗ n (Z2) H (Fk(RP ); Q) = H (Fk(S ; Z2); Q)

n The cohomology of Fk(S ; Z2) is computed in several steps. First, notice that n n the fibre of the projection onto the first coordinate π : Fk+1(S ; Z2) → S is home- n omorphic to the Z2-orbit configuration space Fk(R \{0}; Z2), where the induced n 2 Z2-action on R \{0} is given through the involution τ(x)=−x/|x| . In turn, the n cohomology algebra of Fk(R \{0}; Z2) was computed in [Xi3]. The relations obtained there are analogues of the 3-term relation As,uAs,t − At,uAs,t + At,uAs,u =0 n in the cohomology of Fk(R ) as described by F. Cohen [Co1], [Co4]. In addition to these, an extra relation was obtained.

124 MIGUEL A. XICOTENCATL´

n Theorem 5.1. The integral cohomology of Fk(R \{0}; Z2) is generated, as a graded algebra, by the set of (n − 1)-dimensional classes: Ai,0, i =1,...,k and Ai,j, A¯i,j for 1 ≤ j

As,0As,t = At,0(As,t − As,0) n As,0A¯s,t =(−1) At,0(A¯s,t − As,0) n As,tA¯s,t =(−1) At,0(A¯s,t − As,t) (c) For 1 ≤ u

As,uAs,t = At,u(As,t − As,u) n As,uA¯s,t =(−1) (Au,0 + At,0 − A¯t,u)(A¯s,t − As,u) n As,tA¯s,u =(−1) A¯t,u(A¯s,u − As,t) n A¯s,uA¯s,t = Au,0(A¯s,t − A¯s,u)+(−1) (At,0 − At,u)(A¯s,t − A¯s,u) The next step is to work out the differentials in the Serre spectral sequence for the fibration n n π n Fk(R \{0}; Z2) −→ Fk+1(S ; Z2) −−→ S whose E2-term is given by: p,q p n ⊗ q Rn \{ } Z (5.1) E2 = H (S ) H Fk( 0 ; 2) Let R be the ring of coefficients. Since the cohomology of the fibre is generated multiplicatively by the classes in degree n−1, it enough to determine the differential 0,n−1 → n,0 n n dn : En En = H (S ). Let us denote by x the n-dimensional class in Hn(Sn) which corresponds to the natural orientation of Sn. Then the following result was proven in [Xi3]: Lemma 5.2. For 0 ≤ j

Bi,0 = Ai,0 − A1,0 for 2 ≤ i ≤ k

Bs,t = As,t − A1,0 for 1 ≤ t

B¯s,t = A¯s,t − A1,0 and deﬁne K to be the free R-module generated by the all the classes Bi,0,Bs,t, B¯s,t above. Notice that K is the kernel of the homomorphism 0,n−1 → n,0 dn : En En

ORBIT CONFIGURATION SPACES 125

For j = −1, 0,...,letKj be the R-module spanned by the products of length j of elements in K. Here we set K0 = R as usual and K−1 is understood to be 0. Also, − M → M let ( )2 : FreeR- od R- od be the “mod 2 reduction” functor, which takes i R to i(R/2R). Then we have: n Theorem 5.3. The cohomology of Fk+1(S ; Z2) is given by Z Q Z 1 (a) For R = , ,or [ 2 ] and n odd, there is an isomorphism of algebras: ∗ n ∼ ∗ n ∗ n H Fk+1(S ; Z2) = H (S ) ⊗ H Fk(R \{0}; Z2) (b) For R = Z and n even: ⎧ ⎪ j ⎪K if q = j(n − 1),j≥ 0 ⎪ ⎨⎪ ∗ n Z Z · Kj−2 ⊕ · Kj−1 − ≥ H (Fk+1(S ; 2); )=⎪xA1,0 (x )2 if q = j(n 1) + 1,j 1 ⎪ ⎪ ⎩⎪ 0 otherwise. (c) For R = Q or Z[ 1 ] and n even: 2 ⎧ ⎪ j ⎪K if q = j(n − 1),j≥ 0 ⎪ ⎨⎪ ∗ n Z · Kj−2 − ≥ H (Fk+1(S ; 2); R)=⎪xA1,0 if q = j(n 1) + 1,j 2 ⎪ ⎪ ⎩⎪ 0 otherwise. k n Now recall that (Z2) = 1,..., k acts coordinate-wise on Fk(S ; Z2), namely for x =(x1,...,xk), i(x)=(x1,...,−xi,...,xk). Thus, the maps i induce a k (Z2) -action in cohomology, which was completely worked out recently by A. Guz- m´an in his Ph.D. thesis [Gu]. Notice that for i =2,...,kthe maps i commute with n n the projection π : Fk(S ; Z2) → S onto the ﬁrst coordinate and 1 commutes up k to homotopy. Therefore, there is an induced (Z2) -action on the spectral sequence 3 (5.1). For k =3,the(Z2) -actionontheE2 term is given in Table 1. In the general case, the action can be determined by looking at the various projections of n n the space Fk(S ; Z2)ontoF3(S ; Z2) and using naturality.

3 ∗ n Table 1. The (Z2) - action on H F3(S ; Z2)

1 2 3 x (−1)n+1x x x

n A1,0 −A1,0 (−1) A1,0 A1,0

n A2,0 −A2,0 A2,0 (−1) A2,0

n−1 n n−1 n A2,1 (−1) A1,0 −A2,0 +A2,1 A¯2,1 (−1) A1,0 +(−1) A¯2,1 +(−1) A2,0

n n−1 A¯2,1 −A1,0 + A¯2,1 − A2,0 A2,1 A1,0 +(−1) A2,0 +(−1) A2,1

∗ n We now specialize to the case when n is odd. Recall in this case H Fk+1(S ; Z2) ∗ n ∗ n k+1 is the tensor product H (S ) ⊗ H Fk(R \{0}; Z2) and notice the (Z2) -action ﬁxes the fundamental class x of Sn. In this case, the algebra of invariants was completely determined in [Gu]. Namely,

126 MIGUEL A. XICOTENCATL´

m(n−1) n Theorem 5.4 ([Gu]). For n odd, the invariants in H Fk(R \{0}; Z2) k+1 under the (Z2) -action is the submodule generated by the products of the form ¯ − ¯ − ¯ − (Ai1,j1 + Ai1,j1 Ai1,0)(Ai2,j2 + Ai2,j2 Ai2,0) ...(Aim,jm + Aim,jm Aim,0) with i1 >i2 > ···>im. ∗ n Moreover, the algebra of invariants in H Fk(R \{0}; Z2) is related to the n cohomology of the ordinary conﬁguration space Fk(R ) in the following way. For 1 ≤ j

Corollary 5.5 ([Gu]). For n odd, there is an isomorphism of algebras ∗ n ∼ ∗ n ∗ n H (Fk(RP ); Q) = H (S ; Q) ⊗ H (Fk−1(R ); Q).

6. Braids groups on surfaces 2 Let Σk be the symmetric group acting on Fk(R ) by permuting coordinates and 2 2 set Pk = π1Fk(R )andBk = π1(Fk(R )/Σk). Then the group Bk is isomorphic to Artin’s braid group on k strands and Pk is the pure braid group, that is, the kernel of the natural epimorphism Bk → Σk. By analogy with the classical case, given a smooth manifold M one defines the pure braid group of M as Pk(M)=π1Fk(M) and the full braid group Bk(M)=π1(Fk(M)/Σk). Notice that the natural inclusion k ϕ : Fk(M) ⊂ M induces a homomorphism k ϕ∗ : π1Fk(M) → (π1M) which is an isomorphism if dim M>2 and an epimorphism if dim M =2,bya result of J. Birman [Bi1]. In the case when M is a surface, one obtains a short exact sequence ϕ∗ k 1 −→ ker ϕ∗ −→ Pk(M) −−→ (π1M) −→ 1 2 and a description of the group ker ϕ∗ is given next. Let V ≈ R be an euclidean neighborhood in M. Then the inclusion V ⊂ M induces an inclusion at the level of 2 configuration spaces i : Fk(R ) → Fk(M) and thus a homomorphism of pure braid groups 2 i∗ : Pk(R ) −→ Pk(M) which is a monomorphism for any compact surface different from S2 or RP 2,by a theorem of Birman [Bi1] and Goldberg [Go]. Thus Pk canberegardedasa subgroup of Pk(M). In [Go] C. Goldberg has given a group theoretic description for ker ϕ∗ in terms of Pk.Namely, Theorem 6.1. For any closed surface different from S2 or RP 2,thesubgroup ker ϕ∗ is equal to Pk, the normal closure of Pk in Pk(M).

This result provides a combinatorial description of ker ϕ∗, and therefore of k Pk(M) as an extension of (π1M) by Pk. A more geometrical description of ker ϕ∗ was given in [Xi5], since this group appears as the fundamental group of a natural orbit conﬁguration space. Namely,

ORBIT CONFIGURATION SPACES 127

Theorem 6.2. For a closed surface M = S2, RP 2,letM. be the universal cover . of M and G = π1(M), such that M ≈ M/G. Then there is a natural isomorphism ∼ . Pk = π1Fk(M; G). Proof. Notice that G acts properly discontinuously on M. and M/G. ≈ M. k . From Theorem 2.1, there is a covering G → Fk(M; G) → Fk(M) whose long homotopy exact sequence reduces to: . k (6.1) 1 −→ π1Fk(M; G) −→ Pk(M) −→ G −→ 1 since M is a K(G, 1). Now there is a map of extensions

/ / ϕ∗ / k / 1 P k Pk(M) (π1M) 1 ∼ = loop lifting / . / / k / 1 π1Fk(M; G) Pk(M) G 1 where the right hand side homomomorphism is given by the usual lifting of loops . ∼ . in the covering G → M → M. This induces an isomorphism Pk = π1Fk(M; G). . We can use this description of Pk as the fundamental group of Fk(M; G)to show

Theorem 6.3. The group Pk ⊂ Pk(M) can be expressed as an iterated semi-direct product of (inﬁnitely generated) free groups ∼ Pk = Fk Fk−1 (··· (F3 F2) ...)). Proof. By Theorem 2.2 there are ﬁbrations of the form . o . \ G F(M; G) M Q−1

. F−1(M; G) G − . where Q−1 denotes the union of 1 different orbits in M. Notice all the spaces . \ G involved are K(π, 1)’s and, in particular, M Q−1 is homotopy equivalent to an . \ G infinite wedge of circles. Thus, F = π1(M Q−1) is an infinitely generated free group. On the other hand, it is easy to show that in this case the fibration above has a section. Therefore ∼ P = F P−1 ∼ Finally, for =2wehaveP2 = F2 P1 = F2 and the theorem follows.

Next we consider the case of the full braid group Bk(M). Recall from Theorem k . 2.3 that the semidirect product G Σk acts freely on Fk(M; G)andthereisa principal bundle k . G Σk −→ Fk(M; G) −→ Fk(M)/Σk whose homotopy exact sequence is given by . k 1 −→ π1Fk(M; G) −→ Bk(M) −→ G Σk −→ 1 2 2 . in the case when when M = S , RP ,sinceFk(M; G) is easily seen to be a K(π, 1).

128 MIGUEL A. XICOTENCATL´

Sumarizing, we have the following result: Theorem 6.4. For a closed surface M = S2, RP 2, there is a short exact sequence k 1 →Pk → Bk(M) → (π1M) Σk → 1 which ﬁts into a commutative diagram whose rows and columns are short exact sequences / / k Pk Pk(M) (π1M)

/ / k Pk Bk(M) (π1M) Σk

/ 1 Σk Σk

Here Pk stands for the normal closure of the pure braid group Pk inside Pk(M). ∼ . Moreover, Pk = π1Fk(M; G) with G = π1(M) acting properly discontinuously on M., the universal cover of M. Theorems 6.3 and 6.4 were also obtained by J. Gonz´alez-Meneses and L. Paris in [GP] using a more algebraic approach. A detailed proof of Theorem 6.4 using orbit conﬁguration spaces appeared in the master’s thesis of A. Tochimani [To], in thecasewhenM is an orientable surface.

7. Non-formality of configuration spaces In this section we recall a recent result of R. Longoni and P. Salvatore on the non-formality of configuration spaces [LS]. A long standing conjecture asserted that, when M is a closed manifold, the homotopy type of the configuration space Fk(M) depends only on the homotopy type of M.In[LS] R. Longoni and P. Salvatore disproved this conjecture by showing that for the lens spaces L7,1 and L7,2,which are known to be homotopy equivalent, the associated configuration spaces Fk(L7,1) and Fk(L7,2) are not homotopy equivalent, for k ≥ 2. Recall the lens spaces are three-dimensional oriented manifolds defined as 3 2 2 Lm,n = S /Zm = {(z,w) ∈ C × C ||z| + |w| =1}/Zm m 3 2πi/m 2πin/m where Zm = ζ | ζ =1 acts freely on S by ζ(z,w)=(e z,e w). Thus 3 the orbit configuration space Fk(S ; Zm) is defined and we have 3 k Fk(S ; Zm)/(Zm) = Fk(Lm,n) 3 Moreover,itiseasytoseethatFk(S ; Zm) is simply connected (using for instance . Theorem 2.2), and thus it is the universal cover Fk(Lm,n) of the configuration space Fk(Lm,n). In this paper, the authors study and compare the universal covers of F2(L7,1)andF2(L7,2). Namely, in the first case they show the fibration given by Theorem 2.2 Z 3 \ 7 −→ 3 Z −→ 3 S Q1 F2(S ; 7) S 3 2 3 splits as a product and thus F2(S ; Z7) is homotopy equivalent to (∨6S ) × S . . Threrfore, all Massey products in the cohomology of F2(L7,1) are trivial. On the . other hand, they show F2(L7,2) supports non-trivial Massey products a1,a2,a3 2 with a1,a2,a3 ∈ H . This is done geometrically by using intersection theory on the

ORBIT CONFIGURATION SPACES 129

Poincar´e dual cycles. It follows from here that the conﬁguration spaces F2(L7,1) and F2(L7,2) are not homotopy equivalent. This result can be extended to show

Theorem 7.1 ([LS]). The conﬁguration spaces Fk(L7,1) and Fk(L7,2) are not homotopy equivalent for any k ≥ 2.

Proof. For j =1, 2, the forgetful map (x1,...,xk) → (x1,x2) defines a bun- . . dle Fk(L7,j ) → F2(L7,j ) which admits a section. By naturality one deduces that . 2 . Fk(L7,2) has a non-trivial Massey product on H . On the other hand, Fk(L7,1) 3 splits as a product S × Yk−1,whereYk−1 is the orbit configuration space of k − 1 Z Z 3 \ 7 points of the 7-space S Q1 . Using Theorem 2.2 one gets a tower of fibrations expressing Yk−1 as a twisted product, up to homotopy, of wedges of spheres 2 2 ∨6S , ∨13S , and so on. The additive homology of Yk−1 splits as a tensor product of the homology of the factors, by the Serre spectral sequence. In particular, 2 there is a map ∨(k−1)(7k−2)/2S → Yk−1 inducing isomorphism on H2. The prod- 3 2 . uct map S ×∨(k−1)(7k−2)/2S → Fk(L7,1) induces isomorphism on cohomology 2 3 5 2 . groups H ,H ,H . Thus all Massey products on elements of H (Fk(L7,1)) must vanish.

8. Proof of Theorem 2.2 Lemma 8.1. Let V =(Dn)◦ ⊂ Rn be the interior of the n-dimensional unit ∈ ¯ → ¯ disc. Then for every x0 V there is a homeomorphism φx0 : V V such that

(a) φ|∂V = id∂V (ﬁxes the boundary).

(b) φx0 (x0)=0. (c) The map ψ : V × V¯ → V¯ deﬁned by ψ(x, y)=φx(y), is continuous. n n n Proof. For v ∈ R ,letTv : R → R be the translation by −v, Tv(x)=x−v. → Rn x Let g : V be the homeomorphism: g(x)= 1−|x| . Now, the composition

_ _ _φ˜_ _ _/ V VO

− g g 1

T Rn g(xo) /Rn gives a homeomorphism of V and can be extended continuously to φ : V¯ → V¯ by deﬁning φ(x)=x, ∀x ∈ ∂V .Thusφ is the desired map. It is now easy to express the restriction ψ|V ×V as the composite of continuous maps, and we can extend it to V × V¯ by ψ(x, y)=y ∀x ∈ ∂V .

Proof of Theorem 2.21. We adapt the proof in [FN]toprovethecase = k − 1. The general case is similar. Fix a base point (ξ1,...,ξk−1) ∈ Fk−1(M; G). For i =1,...,k− 1 consider euclidean neighborhoods ξi ∈ Vi ⊂ M such that

(1) Vi ∩ Vj = ∅ i = j (2) Vi ∩ gVj = ∅ ∀ i, j and g = e where gVj = { gx | x ∈ Vj }. Conditions (1) and (2) are easily satisﬁed since M is a regular space, locally euclidean and the action of G is properly discontinuous.

130 MIGUEL A. XICOTENCATL´

Notice now that V = V1 × ...× Vk−1 is a neighborhood about (ξ1,...,ξk−1). Using the previous lemma, construct maps θi : Vi × V¯i → V¯i satisfying:

(i) θi(x, −):V¯i → V¯i is a homeomorphism ﬁxing ∂Vi (ii) θi(x, x)=ξi

(in some sense, θi|x : V¯i → V¯i are homeomorphisms parametrized by x), and use them to deﬁne the map θ : V × M → M, 0 0 ∈ y if y/ i g gVi − θ(x1,...,xk 1,y)= −1 gθi(xi,g y)ify ∈ gVi

By construction, θ has the following property: y/∈{gxi | i =1,...,k− 1; g ∈ G } if and only if θ(x, y) ∈{/ gξi | i =1,...,k−1; g ∈ G }. Now, the local trivialization

−1 φ / × \ G p (V ) ≈ V (M Qk−1)

p proj

V = /V is given by: φ(x, y)=(x, θ(x, y)) with inverse φ−1(x, z)=(x, θ−1(x, z)).

References [Bi1] Joan S. Birman, On braid groups, Comm. Pure Appl. Math. 22 (1969), 41–72. MR0234447 (38 #2764) [Bi2] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974. Annals of Mathematics Studies, No. 82. MR0375281 (51 #11477) [BCP] C.-F. Bödigheimer,F.R.Cohen,andM.D.Peim,Mapping class groups and function spaces, Homotopy methods in algebraic topology (Boulder, CO, 1999), Con- temp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 17–39, DOI 10.1090/conm/271/04348. MR1831345 (2002f:55036) [CLW] J. Chen, Z. Lü, J. Wu, Orbit configuration spaces of small covers and quasi-toric manifolds, arXiv:1111.6699v2. [DC] Daniel C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001), no. 4, 505–530, DOI 10.1515/form.2001.020. MR1830245 (2002i:52016) [Co1] F. R. Cohen, The homology of Cn+1spaces, n ≥ 0, Lecture Notes in Math., Vol. 533, Springer-Verlag (1976), 207–351. [Co2] F. R. Cohen, A course in some aspects of classical homotopy theory, Algebraic topology (Seattle, Wash., 1985), Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 1–92, DOI 10.1007/BFb0078738. MR922923 (89e:55027) [Co3] F. R. Cohen, On the mapping class groups for punctured spheres, the hyperelliptic mapping class groups, SO(3),andSpinc(3), Amer. J. Math. 115 (1993), no. 2, 389–434, DOI 10.2307/2374863. MR1216436 (94k:57021) [Co4] F. R. Cohen, On configuration spaces, their homology, and Lie algebras, J. Pure Appl. Algebra 100 (1995), no. 1-3, 19–42, DOI 10.1016/0022-4049(95)00054-Z. MR1344842 (96d:55005)

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132 MIGUEL A. XICOTENCATL´

[Xi4] Miguel A. Xicoténcatl, Product decomposition of loop spaces of configuration spaces,Pro- ceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 33–38, DOI 10.1016/S0166-8641(01)00107-9. MR1903680 (2003c:55009) [Xi5] Miguel A. Xicoténcatl, Onthepurebraidgroupofasurface, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. Special Issue, 525–528. MR2199368 (2006i:55019)

Departmento de Matematicas,´ Centro de Investigacion´ y de Estudios Avanzados del IPN. Mexico City 07360, Mexico E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12421

Dynamical Systems and Categories

G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich

Abstract. We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A∞-categories. First, entropy is deﬁned for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.

Contents 1. Introduction 2. Entropy of endofunctors 2.1. Complexity in triangulated categories 2.2. Entropy of exact endofunctors 2.3. The case of saturated A∞-categories 2.4. Example: semisimple categories 2.5. Regular maps 2.6. Entropy of the Serre functor 2.7. Pseudo-Anosov maps 3. Density of phases 3.1. Preliminaries 3.2. Dynkin, Euclidean quivers and Kronecker quiver 3.3. Kronecker pairs 3.4. Application to quivers 3.5. Further examples of Kronecker pairs 4. Open questions 4.1. Algebraicity of entropy 4.2. Pseudo-Anosov autoequivalences 4.3. Birational maps 4.4. Dynamical spectrum 4.5. Complexity and mass 4.6. Questions related to Kronecker pairs and density of phases References

2010 Mathematics Subject Classiﬁcation. Primary 18E30; Secondary 37B40, 16G20, 14F05.

c 2014 American Mathematical Society 133

134 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

1. Introduction Recent work of Cantat-Lamy [16] on the Cremona group and Blanc-Cantat [8] on dynamical spectra suggests that there exists a deep parallel between the study of groups of birational automorphisms on one hand, and mapping class groups on the other. Under this parallel, the dynamical degree of a birational map plays the role of the entropy of pseudo-Anosov maps. In the present paper we consider these developments from the perspective of derived categories and their groups of autoequivalences, making direct connections with the classical theory. In another striking series of papers Gaiotto-Moore-Neitzke [28] and Bridgeland- Smith [15] have established a connection between Teichmüller theory and the theory of stability conditions on triangulated categories. An analogy between Teichmüller geodesic flow and the space of stability conditions had been noticed previously in the paper [36] by Soibelman and the fourth author. One of the results is a correspondence between geodesics and stable objects, with slopes of the former giving the phases of the latter. Motivated by this, we study the set of phases of stable objects for general stability conditions, leading to some surprising results. Let us describe the contents of the paper in more detail. First, we define and study entropy in the context of triangulated and A∞-categories, more specifically dynamical entropy, as a measure of the complexity of a dynamical system. This notion comes in a variety of flavors: Let us mention Kolmogorov-Sinai (measure- theoretic) entropy [50], topological entropy [2], and algebraic entropy [6]. In analogy with these notions, we define in Section 2 the entropy of an exact endofunctor of a triangulated category with generator. By taking into account the Z-graded nature of the category one gets not just a single real number, but a function on the real line. In the case of saturated (smooth and proper) A∞-categories, the following foundational results are proven (see Theorems 2.7 and 2.9 for the precise statements)

Theorem. In the saturated case, the entropy of an endofunctor may be computed as a limit of Poincar´e polynomials of Ext-groups.

Theorem. In the saturated case (under a certain generic technical condition), there is a lower bound on the entropy given by the logarithm of the spectral radius of the induced action on Hochschild homology.

We compute the entropy of endofunctors in various examples. In the most basic case of semisimple categories the entropy of any endofunctor is described, in log-coordinates, as a branch of a real algebraic curve deﬁned over Z.Next, we show — under a certain generic technical condition — that the entropy of a regular endomorphism of a smooth projective variety is given by the logarithm of the spectral radius of the induced map on cohomology. As another direction, we determine the entropy of the Serre functor in a number of cases. The results suggest some relation to the “dimension” of the category. Finally, we show that the entropy of an autoequivalence induced by a pseudo-Anosov map on the Fukaya category of a surface coincides with its topological entropy, the logarithm of the stretch factor. This concludes Section 2 of the paper. Next, in Section 3, the topological properties of the set of phases of a stability condition on a triangulated category are investigated. This notion of stability was introduced by T. Bridgeland [13] based on a proposal by M. Douglas [21,22]. It is

DYNAMICAL SYSTEMS AND CATEGORIES 135 an essential ingredient in the theory of motivic Donaldson-Thomas invariants developed by Y. Soibelman and the fourth author [36]. As we have mentioned, guided by work of Gaiotto, Moore, and Neitzke [28], Bridgeland and Smith [15] identify spaces of meromorphic quadratic differentials with spaces of stability conditions on categories associated with quivers with potential. Moreover, under this correspondence, stable objects are identified with geodesics of finite length. Motivated by the density of the set of slopes of closed geodesics on a Riemann surface we investigate in Section 3 the question whether a given triangulated category admits a stability condition such that the set of phases of stable objects is dense somewhere in the circle. In case of stability conditions non-dense behavior is possible (Lemma 3.13 and Corollary 3.15):

Theorem. The phases are never dense in an arc for Dynkin and Euclidean quivers.1

Similarly as in the case of geodesics density property is expected to hold in general. We introduce the notion of Kronecker pair (Deﬁnition 3.23) and show that (Theorem 3.27)

Theorem. If a C-linear triangulated category T contains a Kronecker pair, s.t. a certain family of stability conditions on it is extendable to the whole category, then the extended stability conditions have phases dense in some arc.

Using this theorem we obtain (Proposition 3.32):

Theorem. Any connected quiver Q, which is neither Euclidean nor Dynkin has a family of stability conditions with phases which are dense in an arc.

We record these ﬁndings on density property in the following table:

(1)

Dynkin quivers Pσ is always finite Euclidean quivers Pσ is either finite or has exactly two limit points All other quivers Pσ is dense in an arc for a family of stability conditions where Pσ denotes the set of stable phases (see Definition 3.4). By the non-dense behavior of stability conditions we find that on Dynkin and Euclidean quivers the dimensions of Hom spaces of exceptional pairs are strictly smaller than 3 (Corollary 3.31). Further examples of density of phases (blow ups of projective spaces) are given in subsection 3.5. We summarize the guiding analogies between the classical theory of dynamical systems and the theory of triangulated and A∞-categories in the following table.

1i.e. acyclic quivers with underlying graph a Dynkin or an extended Dynkin diagram

136 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Geodesics Thurston dense set compactiﬁcation Entropy Classical Geodesics

Density of phases Behaviour Stable Stability Categorical of the mass objects conditions of a generator

In this paper we only scratch the surface of a rather promising, in our opinion, area of future research. In the final section we suggest some open problems and possible new directions related to stability conditions, birational geometry, and Fukaya categories. First, inspired by the classical theory of pseudo-Anosov maps, we suggest a definition of a pseudo-Anosov functor in the context of triangulated categories with stability conditions. This is conjecturally a direct generalization of the usual notion which has a geometric interpretation in higher dimensions. Furthermore, we propose a connection with the works of Cantat-Lamy [16] and Blanc-Cantat [8]. In [16], based on Gromov’s ideas in geometric group theory, the authors prove the nonsimplicity of Cremona group in dimension two. Later, in the paper [8], Blanc and Cantat consider the dynamical spectra of the group of birational transformations as a tool for studying groups of birational automorphisms of surfaces. We propose that this dynamical spectrum is part of a categorical dynamical spectrum. We also briefly explore Gromov’s idea that entropy measures growth of volume of submanifolds under the action of a smooth map. From this perspective we conjecture that categorical complexity is related to the mass of objects with respect to a stability condition. Finally, we pose several questions pertaining to the behavior of phases of stable objects.

Acknowledgments. We are grateful to Denis Auroux for many useful discus- sions. The authors were funded by NSF DMS 0854977 FRG, NSF DMS 0600800, NSF DMS 0652633 FRG, NSF DMS 0854977, NSF DMS 0901330, FWF P 24572 N25, by FWF P20778 and by an ERC Grant.

2. Entropy of endofunctors In this section we define and study the entropy of an exact endofunctor of a triangulated category with generator. Our definition, see Subsection 2.2, is based on a notion of complexity of one object relative to another in such a category discussed in Subsection 2.1. In Subsection 2.3 the case of saturated A∞-categories is treated, for which entropy is more easily computed from dimensions of Ext-groups. The rest of the section is devoted to various examples. 2.1. Complexity in triangulated categories. Suppose T is a triangulated category with (split-) generator G. By definition, this means that for every E ∈ T there is an E and a tower of triangles - - - ∼ 0=F0 F1 F2 Fk−1 Fk = E ⊕ E ··· (2) G[n1] G[n2] G[nk]

DYNAMICAL SYSTEMS AND CATEGORIES 137 with k ≥ 0, ni ∈ Z. One can ask, for each E ∈ T, what the least number of shifted copies of G needed is, in order for such a relation to hold. Taking into account the ni’s as well, one arrives at the following deﬁnition.

Definition 2.1. Let E1,E2 be objects in a triangulated category T.Thecom- plexity of E relative to E is the function δ (E ,E ):R → [0, ∞] given by 2 1 t 1 2 1 k - - - 0 A A A − E ⊕ E nit 1 2 k 1 2 2 δt(E1,E2)=inf e ··· . i=1 E1[n1] E1[n2] E1[nk] Note that δ (E ,E )=+∞ iﬀ E does not lie in the thick triangulated sub- t 1 2 2 ∼ category generated by E1.Also,whenT is Z/2-graded in the sense that [2] = idT, only the value at zero, δ0(E1,E2), will be of any use. Remark 2.2. After a ﬁrst version of this article appeared, P. Seidel brought to our attention his recent lecture notes [49], in which he also condsiders the quantity δ0(G, E). We collect some basic inequalities in the following proposition.

Proposition 2.3. Let T be a triangulated category, E1,E2,E3 ∈ T.Then (a) (Triangle inequality) δt(E1,E3) ≤ δt(E1,E2)δt(E2,E3), (b) (Subadditivity) δt(E1,E2 ⊕ E3) ≤ δt(E1,E2)+δt(E1,E3), (c) (Retraction) δt(F (E1),F(E2)) ≤ δt(E1,E2) for any exact functor F : T → T. Proof. We will prove only (a). Take >0 and let - - - ⊕ 0 A1 A2 Ak−1 E2 E2 ··· E1[n1] E1[n2] E1[nk]

- - - ⊕ 0 B1 B2 Bp−1 E3 E3 ··· E2[m1] E2[m2] E2[mp] be such that k p nit mj t (3) e <δt(E1,E2)+ , e <δt(E2,E3)+ . i=1 j=1 Using the given sequences of triangles one can construct a sequence of the type - - - ⊕ 0 C1 C2 Ckp−1 E3 E3 ··· E1[n1 + m1] E1[n2 + m1] E1[nk + mp] , where E = E ⊕ E [m1] ⊕ E [m2] ···⊕E [mp]. Hence 3 3 2 2 2 (ni+mj )t (4) δt(E1,E3) ≤ e ≤ (δt(E1,E2)+ )(δt(E2,E3)+ ). i,j Since this holds for any >0, the inequality follows. For complexes of vector spaces the complexity with respect to the ground ﬁeld is just the Poincar´e (Laurent-) polynomial in e−t.

138 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Lemma 2.4. Let T = Db(k) be the bounded derived category of finite-dimensional vector spaces over a field k.Then n −nt δt(k, E)= dim(H E)e n for any E ∈ T. Proof. Any object in T is isomorphic to its cohomology, which is a direct sum with dim(HnE) copies of k[−n]foreachn. This shows the inequality “≤”. For the reverse inequality, use the fact that for any exact triangle E1 → E2 → E3 → E1[1] n n n we have dim H E2 ≤ dim H E1 +dimH E3. 2.2. Entropy of exact endofunctors. We now come to the main definition of this section. It is based on the observation that if F is an endofunctor of a trian- n gulated category T with generator G,thenδt(G, F G) grows at most exponentially, and the growth constant is independent of the choice of generator. Definition 2.5. Let F : T → T be an exact endofunctor of a triangulated category T with generator G.Theentropy of F is the function ht(F ):R → [−∞, +∞) of t given by

1 n ht(F ) = lim log δt(G, F G). n→∞ n The next lemma shows that ht(F ) is well-deﬁned. Lemma . T 1 n 2.6 With ,F as in the deﬁnition, the limit limn→∞ n log(δt(G, F G)) exists in [−∞, +∞) for every t and is independent of the choice of generator G. Proof. Using Proposition 2.3, m+n m m m+n δt(G, F (G)) ≤ δt(G, F (G))δt(F (G),F (G)) (5) m n ≤ δt(G, F (G))δt(G, F (G)) hence existence of the limit follows from Fekete’s Lemma: for any subadditive { } an { an | ≥ } sequence an n≥1 we have limn→∞ n =inf n n 1 .Furthermore,ifG, G are generators then n n n n (6) δt(G ,F (G )) ≤ δt(G ,G)δt(G, F (G))δt(F (G),F (G )) n (7) ≤ δt(G ,G)δt(G, G )δt(G, F (G)) which implies the second claim.

We conclude this subsection with some remarks on the behaviour of ht with respect to composition of functors. Clearly, ht is constant on conjugacy classes, n and ht(F )=nht(F )forn ≥ 1. If T =0,then h0(idT) = 0, but this may fail for other values of t. In general, nothing can be said about ht(F ◦ H)fromht(F )and ht(H) alone, but if F and H commute, then ht(F ◦ H) ≤ ht(F )+ht(H).

2.3. The case of saturated A∞-categories. For background on A∞-categories see for example [38]and[35]. We will work over a fixed field k.AnA∞- category C is triangulated if every compact object in Mod(C)isquasi-representable. 0 The homotopy category, H C, of a triangulated A∞-category is triangulated in the sense of Verdier. An A∞-category is saturated if it is triangulated and Morita- equivalent to a smooth and compact A∞-algebra A. In the case of saturated A∞-categories one can, for the purpose of computing ∗ −t the entropy, replace δt(A, B) with the Poincaré polynomial of Ext (A, B)ine .

DYNAMICAL SYSTEMS AND CATEGORIES 139

Theorem 2.7. Let C be a saturated A∞-category and F an endofunctor of C. Then 1 n N −nt ht(F ) = lim log dim Ext (G, F G)e N→∞ N n∈Z for any generator G of C. Proof. Note ﬁrst that N n N −nt (8) δt(k, RHom(G, F G)) = dim Ext (G, F G)e n by Lemma 2.4. We will show that there exist C1,C2 : R → R>0, depending on G, such that

(9) C1(t)δt(G, E) ≤ δt(k, RHom(G, E)) ≤ C2(t)δt(G, E) for every E ∈ C. The theorem follows from (9) by setting E = F N G,takingthe logarithm, dividing by N, and passing to the limit N →∞. For the second inequality, apply Proposition 2.3 to the functor RHom(G, ): C → Db(k), which is well deﬁned by local properness of C.Weget

(10) δt(k, RHom(G, E)) ≤ δt(k, RHom(G, G))δt(RHom(G, G),RHom(G, E))

(11) ≤ δt(k, RHom(G, G))δt(G, E) where the ﬁrst factor is a function C2(t):R → R>0 independent of E.Notethat we did not use the assumption that C is smooth. On the other hand

(12) δt(G, E) ≤ δt(G, G ⊗ RHom(G, E))δt(G ⊗ RHom(G, E),E)

(13) ≤ δt(k, RHom(G, E))δt(G ⊗ RHom(G, ), idC) by the retraction map property of the functors G ⊗ : Db(k) → C and Φ → Φ(E): Fun(C, C) → C. It remains to show that the second factor, δt(G ⊗ RHom(G, ), idC) is in fact ﬁnite, i.e. that idC lies in the subcategory of Fun(C, C) generated by G ⊗ RHom(G, ). But this is just equivalent to the smoothness of C. Indeed, let A =End(G), then the functor G ⊗ RHom(G, ) corresponds to the bimodule op A ⊗k A and IdC corresponds to the diagonal bimodule A. This completes the proof of (9) and the theorem.

Suppose that C is a saturated A∞-category over C.SinceC is smooth and proper, its Hochschild homology, HH∗(C), is ﬁnite-dimensional. Any endofunctor F of C induces a linear map, HH∗(F ), on HH∗(C) and we may consider the logarithm of its spectral radius, ρ(HH∗(F )), as a kind of “homological entropy”. We will show that, under a certain generic condition discussed in the lemma below, this number is bounded above by h0(F ). Lemma 2.8. Let V be a ﬁnite-dimensional Z/(2)-graded C-vector space, A = A+ ⊕ A− a homogeneous endomorphism of V . Suppose that subsets of those eigenvalues of A+ and A− which lie on the circle {|λ| = ρ(A)} do not coincide, with multiplicity. Then lim sup |sTrAn|1/n = ρ(A) n→∞

140 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Proof. We may assume that ρ(A) =0.Let α1,...,αm be the eigenvalues of A+ and β1,...,βn those of A−. Consider the identity ∞ m 1 n 1 (14) sTrAnzn = − 1 − αkz 1 − βlz n=0 k=1 l=1 which holds for |z| suﬃciently small. If R denotes the radius of convergence of the series on the left hand side of (14), then (15) R−1 = lim sup |sTrAn|1/n . n→∞ On the other hand, it follows from the conditions on A that (14), extended mero- morphically to C, has a pole on {|z| =1/ρ(A)}, and no poles closer to the origin. Hence R = ρ(A), and the proof of the lemma is complete.

Theorem 2.9. Let C be a saturated A∞-category over C and F : C → C a functor. Assume that the induced map on Hochschild homology, HH∗(F ),satisﬁes the condition of Lemma 2.8,then

log ρ(HH∗(F )) ≤ h0(F ). Proof. Let S denote the Serre functor of C. By Lemma 2.4, N n N (16) δ0(k, RHom(F ,S)) = dim Ext (F ,S) n thus N N (17) χ(RHom(F ,S)) ≤ δ0(k, RHom(F ,S)). Applying the triangle inequality, N δ0(k, RHom(F ,S)) ≤ δ0(k, RHom(G ⊗ RHom(G, ),S)) (18) N × δ0(RHom(G ⊗ RHom(G, ),S),RHom(F ,S)). The first factor is independent of N, while the second is bounded above by N δ0(G⊗RHom(G, ),F ) as follows from the retraction map property of the functor RHom( ,S):Fun(C, C) → Db(k-Vect)op. Now, (19) N N δ0(G ⊗ RHom(G, ),F ) ≤ δ0(G ⊗ RHom(G, ),F N N ◦ (G ⊗ RHom(G, )))δ0(F ◦ (G ⊗ RHom(G, )),F ). By the retraction property of F N ◦ , the second factor is bounded above by δ0(G ⊗ RHom(G, ), idC) which is independent of N and finite by smoothness of N C. The first factor is bounded above by δ0(G, F G), hence 1/N (20) lim sup χ(RHom(F N ,S)) ≤ eh0(F ). N→∞ By the Lefschetz fixed point theorem for Hochschild homology [39]wehave N N (21) χ(RHom(F ,S)) = sTr(HH∗(F )) and by Lemma 2.8, N 1/N (22) lim sup sTr(HH∗(F ) ) = ρ(HH∗(F )) N→∞ which completes the proof.

DYNAMICAL SYSTEMS AND CATEGORIES 141

Remark 2.10. We expect that the conclusion of Theorem 2.9 holds without the condition on F . 2.4. Example: semisimple categories. Fix a finite set X and a field k. Let Db(X) denote the triangulated dg-category of X-indexed families of bounded complexes over k. Any (additive, graded) endofunctor of Db(X) is isomorphic to one of the form ⊗ ∗ (23) ΦK (V )=(p1)∗(K p2V ) for some K ∈ Db(X ×X) with trivial differential. Such a K corresponds to a matrix of Poincaré (Laurent–) polynomials n n (24) PK (z)= dim(Kx,y)z n∈Z x,y∈X with non-negative integer coefficients. Note that composition of functors corresponds to multiplication of matrices. b Let G ∈ D (X) be the generator with Gx = k for x ∈ X.Then n N −nt % N −t % (25) dim Ext (G, ΦK (G))e = PK (e ) 1 n∈Z % % | | N −t where A 1 = aij is the matrix 1-norm and we use non-negativity of PK (e ). We obtain −t (26) ht(ΦK ) = log ρ(PK (e )) by Gelfand’s formula for the spectral radius n 1 (27) ρ(A) = lim %A % n n→∞ which holds for any matrix norm. We can say a bit more about the function ρ(PK (x)). By Perron–Frobenius theory, the spectral radius of a non-negative matrix is an eigenvalue. Hence, if we consider the spectral curve

(28) C = {(x, λ) ∈ R>0 × R | det(PK (x) − λ)=0} then the graph of ρ(PK (x)) : R>0 → R is the maximal branch of C, the points in C which maximize λ for fixed x. 2.5. Regular maps. Let X be a smooth projective variety over C, f : X → X a regular map, and f ∗ : Db(X) → Db(X) the pullback functor. We will show that, under the generic condition of Lemma 2.8, the entropy of f ∗ is constant and equal to the logarithm of the spectral radius of H∗(f; Q), the induced map on cohomology. By a result of S. Friedland [27], log ρ(H∗(f; Q)) is also equal to the topological entropy of f, more generally for compact Kähler manifolds. We begin with a lemma which gives a sufficient condition for the entropy ht(F ) to be constant in t.

Lemma 2.11. Let C be a saturated A∞-category and F an endofunctor of C. Suppose there exists a generator G of C and M ≥ 0 such that Extn(G, F N G)=0 for |n| >M,N≥ 0 then ht(F ) is a constant function.

142 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Proof. Use (29) dim Extn(G, F N G)e−M|t| ≤ dim Extn(G, F N G)e−nt ≤ dim Extn(G, F N G)eM|t| to conclude h0(F ) ≤ ht(F ) ≤ h0(F ). Note that Lemma 2.11 applies in particular to functors preserving a bounded t-structure of finite homological dimension, as the generator can be chosen to lie in the heart of the t-structure. Theorem 2.12. Let X be a smooth projective variety over C, f : X → X a regular map, and f ∗ : Db(X) → Db(X) the pullback functor. Suppose that the induced map on cohomology, H∗(f; Q), satisfies the condition of Lemma 2.8.Then ∗ ∗ ht(f ) is constant and equal to log ρ(H (f; Q)). Proof. Clearly, f ∗ preserves coh(X) ⊂ Db(X) which has finite homological ∗ dimension, since X is smooth and proper. Hence, by Lemma 2.11, ht(f ) is constant in t. We will show that ∗ ∗ (30) h0(f ) ≤ log ρ(H (f,Q)) without the assumption of Lemma 2.8. The claim then follows from Theorem 2.9 by the Hochschild-Kostant-Rosenberg isomorphism and Hodge theory which give b ∼ ∗ (31) HH∗(D (X)) = H (X; C) as Z/2-graded vector spaces. It remains to show (30). Choose L ∈ Pic(X) very ample and consider d+1 (32) G = Lk k=1 where d =dimX.BothG and G∗ are generators of Db(X)by[42]. Since X is 1 ∗ N ∗ projective, f preserves the nef cone in N (X)R. In particular, (f ) G is a sum of anti-nef line bundles for every N ≥ 0. Note that G∗ itself is a sum of anti-ample line bundles, hence, using nef + ample = ample, we conclude that (33) G∗ ⊗ (f ∗)N G∗ = L−k ⊗ (f ∗)N L−l 1≤k,l≤d+1 is a sum of anti-ample line bundles. Thus, by the Kodaira vanishing theorem, Ext∗(G, (f ∗)N G∗) is concentrated in degree d.Weget (34) dim Ext∗(G, (f ∗)N G∗)=χExt∗(G, (f ∗)N G∗) ∗ ∗ N ∗ (35) = ch(G )(f ) ch(G )td(X) where f ∗ is the induced map on H∗(X; Q). It follows that 2 (36) lim N dim Ext∗(G, (f ∗)N G∗) ≤ ρ(H∗(X; Q)) N→∞ ∗ which shows the claim, since the left hand side is exp(h0(f )) by Theorem 2.7. More precisely, we use here a variant of this theorem, with the same proof, where one is allowed to take two different generators.

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2.6. Entropy of the Serre functor. The notion of a Serre functor was introduced by Bondal and Kapranov in [7]. If it exists, it is unique up to natural isomorphism, and thus its entropy an invariant of the triangulated category. In the examples below, the entropy of the Serre functor will always turn out to be of the linear form at + b, with a having some interpretation as a dimension. We note however, that this can fail in general, for example for categories which are orthogonal sums. 2.6.1. Fractional Calabi-Yau categories. A triangulated category T with Serre m ∈ Q functor S is called fractional Calabi-Yau of dimension n if n ∼ (37) S = [m] c.f. [32]. Suppose that T is the homotopy category of a saturated A∞-category C. Using

n (38) ht(F )=nht(F )forn ≥ 1,ht([n]) = nt for n ∈ Z we see that m (39) h (S)= t t n in this case. 2.6.2. Smooth projective varieties. Let X be a smooth projective variety over aﬁeldk. The bounded derived category of coherent sheaves on X, Db(X)= Db(coh(X)) is saturated with Serre functor

(40) S = ⊗ ωX [dim X] where ωX is the canonical sheaf.

Proposition 2.13. ht(S)=dim(X)t

Note that in particular, if C is a saturated category and ht(S) is not of the form nt for some n ∈ Z≥0,thenC cannot come from an algebraic variety.

Proof. This is equivalent to ht( ⊗ ωX ) = 0, which is a special case of Lemma 2.14 below.

Lemma 2.14. Let X be a smooth projective variety over a ﬁeld k, L ∈ Pic(X). Then ht( ⊗ L)=0.

Proof. By lemma 2.11, ht( ⊗ L) is constant. Let G ∈ coh(X)beagenerator of Db(X). It suﬃces to show that (41) dim Extn(G, G ⊗ LN )=O(N d),N→∞ n∈Z for some d. This holds by the standard fact that

n N dim X (42) dim H (OX , F ⊗ L )=O(N ),N→∞ for any coherent sheaf F, n ≥ 0, see [37].

144 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

2.6.3. Quivers. Let Q be a quiver. We assume throughout that Q is connected and does not have oriented cycles. Fix a field k and form the path algebra A = kQ of Q.LetA be the category of finite dimensional left modules over A. Its bounded derived category, DbA, is saturated with Serre functor isomorphic to the derived functor of Hom( ,A)∗. Lemma 2.15. If X ∈ A is projective, then S(X) ∈ A and S(X) is injective. If X has no projective summands, then S(X) ∈ A[1]. Proof. Note first that A is hereditary and thus any indecomposable object in DbA is contained in A[n]forsomen ∈ Z.Furthermore,ifM ∈ A is indecomposable, then S(M) is contained either in A or in A[1]. Suppose P is projective, then (43) Extn(P, M)=Ext−n(M,S(P )) = 0 for n =0 ,M ∈ A hence S(P ) ∈ A and S(P ) is injective. If N ∈ A is indecomposable and not projective, then there exists an M ∈ A with (44) Ext1(N,M)=Ext−1(M,S(N)) =0 thus S(N) ∈/ A.

Let ei be the idempotent corresponding to the vertex i ∈ Q0. Recall that Aei, ∗ i ∈ Q0 is a complete list of indecomposable projectives and (eiA) , i ∈ Q0 is a complete list of indecomposable injectives. Lemma 2.16. If Q is not Dynkin, then Φ−N (P ) ∈ A for any projective module P and N ≥ 0. Proof. By the previous lemma, Φ−1(M) ∈ A for an indecomposable M ∈ A as long as M is not injective. Suppose that P is projective and M =Φ−N (P ) is injective. Then M is both preprojective and preinjective, thus A of ﬁnite representation type by a result of [3] , contradicting the assumption that Q is not Dynkin.

If Q is Dynkin, then DbA is fractional Calabi-Yau, see [32]. We assume from now on that Q is not Dynkin. Then, by the lemma (45) Extn(A, Φ−N (A)) = 0 for N ≥ 0,n=0 and using dim Hom(A, M)=dimM we get

−1 1 −N (46) ht(Φ ) = lim log dim Φ A. N→∞ N Furthermore, by duality, (47) dim Extn(A, ΦN A)=dimExt1−n(A, (Φ−1)N−1A) hence −1 (48) ht(S)=ht(Φ) + t = ht(Φ )+t. −1 At this point, the problem of computing ht(Φ ) is one of linear algebra, since dim M depends only on

∼ Q0 (49) M ∈ K0(A) = Z

DYNAMICAL SYSTEMS AND CATEGORIES 145 where a natural basis of K0(A) is given by the simple modules. Recall that the Euler form (50) M,N = (−1)n dim Extn(M,N) n∈Z is a bilinear form on K0(A) with matrix E given by

(51) Eij = δij − nij where nij is the number of arrows from the j-th to the i-th vertex. Let [S]denotethe induced map on K0(A). By the deﬁning property of the Serre functor, E = E ·[S], hence (52) [S]=E−1E, [Φ] = −E−1E, [Φ−1]=−E−E. The linear map [Φ] is classically the Coxeter transformations for the Cartan matrix E +E. Its spectral properties were investigated by Dlab and Ringel in [23]. They ﬁnd that ρ = ρ([Φ]) = ρ([Φ−1]) is an eigenvalue of [Φ], ρ ≥ 1, and ρ = 1 if and only if Q is of extended Dynkin type. Furthermore, if P is projective, then

1 (53) lim dim Φ−N P N = ρ. N→∞ We summarize our results. Theorem 2.17. Let Q be a connected quiver without oriented cycles, not of Dynkin type, and S the Serre functor on Db(kQ).Then

(54) ht(S)=t +logρ([S]) and ρ([S]) ≥ 1 with equality if and only if Q is of extended Dynkin type. 2.7. Pseudo-Anosov maps. A great insight of Thurston was that a “typical” element of the mapping class group of a surface M can be represented by a pseudo- Anosov map [26, 51], which is by deﬁnition a homeomorphisms φ : M → M such that there exist transverse measured foliations (Fs,μs), (Fu,μu)andλ>1, such that 1 (55) φ(Fs,μs)=(Fs, μs),φ(Fu,μu)=(Fu,λμu). λ The stretch factor λ can be computed as follows. Let S be the set of isotopy classes of simple closed curves on M which do not contract to a point. For α, β ∈ S the geometric intersection number i(α, β) is the minimum number of intersection points of simple closed curves representing α and β.Thenforφ, λ as above and α, β ∈ S one has (56) lim n i(α, φn(β)) = λ. n→∞ We restrict to the case when M is closed, orientable, and of genus g>1. The Fukaya category Fuk(M)isaZ/2-graded A∞-category over the Novikov ﬁeld ΛR. For our purposes, objects are embedded curves with orientation and the bound- ing spin structure. Let DπFuk(M) denote the triangulated hull, which is locally proper (in the Z/2-graded sense), and has a generator given by a collection of loops [24, 48]. In fact, an explicit resolution of the diagonal is constructed in [24, 48], proving homological smoothness. Thus DπFuk(M) is saturated. An element φ of π the mapping class group of M induces an autoequivalence φ∗ of D Fuk(M).

146 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Theorem 2.18. Let φ be a pseudo-Anosov map with stretch factor λ,then h0(φ∗) = log λ. It is a classical result, see [26], that the topological entropy of a pseudo-Anosov map φ is log λ,andthatφ minimizes topological entropy in its isotopy class.

Proof. Note ﬁrst that Theorem 2.7 is also valid, with essentially the same proof, for saturated Z/2-graded A∞-categories in the sense that 1 0 N 1 N (57) h0(F ) = lim log dim Ext (G, F G)+dimExt (G, F G) N→∞ N for any endofunctor F . On the other hand, if α, β are simple closed loops in M then (58) dim Ext0(α, φN (β)) + dim Ext1(α, φN (β)) = i(α, φN (β)) for N & 0 by Lemma 2.19 below. Now, DπFuk(M) has a generator G which is a direct sum of 2g + 1 simple loops. In view of (56) the theorem follows.

The following is considered well-known to experts in Floer theory. Lemma 2.19. Let M be a closed surface of positive genus with symplectic structure, α and β simple closed loops on M which are not isotopic. Then (59) dim HF0(α, β)+dimHF1(α, β)=i(α, β) where HFk(α, β) denotes Floer cohomology over the Novikov field. Proof. If α and β are already in minimal position, i.e. i(α, β)=|α ∩ β|,then (59) is clear, since there are no immersed bi-gons with boundary on α ∪ β,and thus the Floer differential vanishes on CF∗(α, β) which has a basis given by the intersection points. In general, there is an embedded loop α which is isotopic to α and intersects β minimally. Moreover, we can construct an isotopy from α to α which is a composition of Hamiltonian isotopies and isotopies which do not create or remove any intersection points between α and β. The invariance of HF∗(α, β) under the former is true by the general theory, we will show that invariance also holds for the latter. Let us observe first that near each intersection point p of α and β there is a chart taking α to the x-axis and β to the y-axis. In such a chart, any bi-gon contributing to d[p] must map locally to either the second or fourth quadrant, see [1]. Furthermore, the assumption on α and β ensures that bi-gons contributing to d0 (resp. d1) cannot form a closed chain. Let M be the matrix representing the Floer differential d0 or d1, with respect to the basis given by intersection points. Since α and β are not isotopic, there can be at most one bi-gon contributing to any entry of M, hence the position of the non-zero entries does not change under the isotopy. The observations made above translate into the following properties of M. (1) Any column or row of M has at most two non-zero entries. (2) Let Q be the quiver with vertices the columns and rows of M and an arrow from i to j if the entry in the i-thcolumnandj-th row of M is non-zero. Then the underlying undirected graph of Q has no loops.

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One shows that the rank of a matrix satisfying the second property is determined by the positions of the non-zero entries alone, and not the speciﬁc values. We conclude that dim HF∗(α, β) is invariant under isotopy.

3. Density of phases 3.1. Preliminaries. 3.1.1. On Bridgeland stability conditions. We start by recalling some definitions and results by Bridgeland. Here we introduce the notations HA(see Definition 3.2 and Remark 3.3) and Pσ (the set of semistable phases of a stability condition σ), which are useful later. In [13] a stability condition on a triangulated category T is defined as a pair σ =(P,Z) satisfying certain axioms, where the first component is a family of full additive subcategories {P(t)}t∈R (by the axioms it follows that they are abelian) Z- and the second component is a group homomorphism K0(T) C . The non- zero objects in P(t)aresaidtobeσ-semistable of phase t and the phase φσ(E)of a semistable E ∈ P(t)iswelldefinedbyφσ(E)=t. To give a description of the Bridgeland stability conditions on T, we recall first the following definition: Z- Definition 3.1. Let (A,K0(A) C ) be an abelian category and a stability function on it.2 A non-zero object X ∈ A is said to be Z-semistable of phase t if every A-monic X → X satisfies3 arg Z(X) ≤ arg Z(X)=πt (if equality is ∼ attained only for X = X then X is said to be stable).

If A ⊂ T is a bounded t-structure and Z : K0(A) → C is a stability function, which satisfies the HN property ([13, Definition 2.3]) then the pair σ =(P,Ze) defined by(see the proof of [13, Proposition 5.3]): a) If t ∈ (0, 1], define P(t) ⊂ A as the set of Z-semistable objects of phase t in A(as defined in definition 3.1). If t ∈ (n, n +1], n ∈ Z, define P(t)=T n(P(t − n)). Ze- b) Define K0(T) C , such that: A⊂T K0( -) Z-e Z- K0(A) K0(T) C = K0(A) C is a stability condition on T (furthermore all stability conditions on T are obtained by this procedure). Let us denote: Definition 3.2. Let A ⊂ T be a bounded t-structure in a triangulated category T.WedenotebyHA the family of stability conditions in T obtained by a), b) above varying Z in the set of all stability functions on A with HN property. In particular HA ' P → HA ( ,Z) Z|K0(A) is a bijection between and this set. Bridgeland proved that the set of all stability conditions, satisfying a property called locally finiteness, on a triangulated category T is a complex manifold, denoted by Stab(T).

2I.e. Z is homomorphism, s.t. Z(X) ∈ H = {r exp(iπt)|r>0and0

148 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Remark 3.3. Let A ⊂ T be as in the previous definition. If A is an abelian category of finite length, then any stability function Z : K0(A) → C satisfies the HN property ([13, Proposition 2.4]). If in addition A has finitely many, say A s1,s2,...,sn, simple objects then all stability conditions in H are locally finite. Hence, in this setting we have HA ⊂ Stab(T) and bijection HA ' (P,Z) → n (Z(s1),...,Z(sn)) ∈ H . Finally, we introduce the notation: Definition 3.4. Let T be a triangulated category and σ =(P,Z) ∈ Stab(T) a stability condition on it. We denote:4 T { ∈ R|P { }} ⊂ 1 (60) Pσ =exp(iπ t (t) = 0 ) S . P P 5 − T T By (t +1)= (t)[1] it follows Pσ = Pσ . 3.1.2. On θ-stability and a theorem by King. In the next subsection we use a result by King. We recall first

Definition 3.5 (θ-stability). Let θ : K0(A) → R be a non-trivial group homomorphism, where A is an abelian category. Then X ∈ A is called θ-semistable if θ(X)=0and for each monic arrow X → X in A we have θ(X) ≥ 0 (if θ(X)=0 only for the sub-objects 0 and X then it is called θ-stable). Remark 3.6. Z-semistable of phase t (as defined in Definition 3.1)isthesame as θ-semistable with θ = −Im(e−iπtZ). From Proposition 4.4 in [34] it follows Proposition 3.7 (A. King). Let A be a finite dimensional, hereditary C- algebra. Let α ∈ K0(A-Mod). Then the following conditions are equivalent:

(1) There exist X ∈ A-Mod and a non-trivial θ : K0(A-Mod) → R,s.t. [X]=α and X is θ-stable. (2) α is a Schur root, which by deﬁnition means that some Y ∈ A-Mod with [Y ]=α satisﬁes EndA-Mod(Y )=C. This Proposition will be used in the proof of Corollary 3.19. 3.2. Dynkin, Euclidean quivers and Kronecker quiver. In this subsection we comment on the set Pσ as σ varies in the set of stability conditions on Dynkin, Euclidean quivers and on the Kronecker quiver. The main results here are Lemma 3.13, Corollary 3.15 and Corollary 3.20. 3.2.1. Quivers and Kac’s theorem. For any quiver Q we denote its set of vertices by V (Q), its set of arrows by Arr(Q) and the underlying non-oriented graph by Γ(Q). Let (61) Arr(Q) → V (Q) × V (Q) a → (s(a),t(a)) ∈ V (Q) × V (Q) be the function assigning to an arrow a ∈ Arr(Q) its origin s(a) ∈ V (Q) and its end t(a) ∈ V (Q). A vertex v ∈ V (Q) is called source/sink if all arrows touching it start/end at it (more precisely v = t(a)/v = s(a)foreacha ∈ Arr(Q)). Throughout this section 3 the term Dynkin quiver means a quiver Q, s.t. Γ(Q) is one of the simply laced Dynkin diagrams Am,m ≥ 1, Dm, m ≥ 4, E6, E7, E8

4 When the triangulated category T is ﬁxed in advance we write just Pσ. 5which is one of Bridgeland’s axioms

DYNAMICAL SYSTEMS AND CATEGORIES 149

(see for example [5, p. 32]) and the term Euclidean quiver means an acyclic quiver Q, s.t. Γ(Q) is one of the extended Dynkin diagrams Am,m ≥ 1, Dm, m ≥ 4, E6, E7, E8 (see for example [5, fig. (4.13)]). By K(l), l ≥ 1 we denote the quiver, which consists of two vertices with l parallel arrows between them. Note that K(1) is Dynkin, K(2) is Euclidean. We call K(l), l ≥ 3anl-Kronecker quiver. Recall the Kac’s Theorem. Remark 3.8 (On Kac’s Theorem). Let Q be a connected quiver without edges- loops. In [31] is defined the positive root system of Q. We denote this root system by V (Q) V (Q) Δ+(Q) ⊂ N .ForX ∈ RepC(Q) we denote by dim(X) ∈ N its dimension vector. The main result of [31] (we consider only the field C)is:

(62) {dim(X)|X ∈ RepC(Q),X is indecomposable} =Δ+(Q). The Euler form of any quiver Q is deﬁned by − ∈ NV (Q) (63) α, β Q = αj βj αs(j)βt(j),α,β . j∈V (Q) j∈Q1

The set Δ+(Q) has a simple description for Dynkin, extended Dynkin or hyperbolic quivers (K(l), l ≥ 3 are hyperbolic quivers) as shown by Kac in [31]. It is determined by the Euler form as follows { ∈ NV (Q) \{ }| ≤ } ∩ NV (Q) (64) Δ(Q)= r 0 r, r Q 1 , Δ+(Q)=Δ(Q) . If Q is acyclic,6 then the path algebra CQ is ﬁnite dimensional and we have ∼ V (Q) an isomorphism K0(RepC(Q)) = Z determined by K0(RepC(Q)) ' [X] → V (Q) dim(X) ∈ Z for X ∈ RepC(Q). In particular, for any homomorphism Z : K0(RepC(Q)) → C and any X ∈ RepC(Q)wehave { } (65) Z(X)= dimi(X) Z(si)=(v, dim(X)), vi = Z(si) i∈V (Q), i∈V (Q) where si is the simple representation with C in the vertex i ∈ V (Q) and 0 in the V (Q) other vertices. Throughout this section 3(, ) denotes the bilinear form on C × CV (Q) ∈ CV (Q) deﬁned by (α, β)= i∈V (Q) αiβi, α, β , NOT the symmetrization α, β Q + β,α Q of , Q. We mention once this symmetrization and denote it by (, )Q. ⊆ 3.2.2. The inclusion Pσ Rv,Δ+ . Lemma 3.9. Let T be any triangulated category. Then for each σ =(P,Z) ∈ Stab(T) we have:

{t ∈ R|P(t) =0 } = {φσ(I)|I is T-indecomposable and σ-semistable}.

Proof. Let X ∈ P(t) be non-zero. Since P(t) is of ﬁnite length , we have a P ∼ n decomposition in (t)oftheformX = i=1 Xi,whereXi are indecomposable in P(t), therefore (here we use that P(t) is abelian) there are no non-trivial idempotents in EndP(t)(Xi)=EndT(Xi), hence Xi is indecomposable in T. Thus, we see that t = φσ(Xi), where Xi is an indecomposable in T and σ-semistable. The lemma follows.

6i.e. it has not oriented cycles.

150 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Corollary 3.10. Let A be a hereditary abelian category. For each σ = (P,Z) ∈ Stab(Db(A)) holds the inclusion: Z(X) (66) P ⊆ ± |X is indecomposable in A,Z(X) =0 . σ |Z(X)| Proof. Take any t ∈ R with P(t) = {0}. From the previous lemma there is a b semi-stable, indecomposable X ∈ D (A), s.t. φσ(X)=t.SinceA is hereditary, it follows that X = X[i] for some indecomposable X ∈ A, i ∈ Z.Nowwecanwrite i (−1) Z(X )=Z(X)=m(X)exp(iπφσ(X)) = m(X)exp(iπt) m(X) > 0, where we use that X is σ-semistable and one of the Bridgeland’s axioms ([13, Deﬁnition 1.1 a)]). The corollary is proved.

When A = RepC(Q) with Q-acyclic we can rewrite this corollary in a useful form. Putting3 (62) and (65) in the righthand4 side of (66) with A = RepC(Q)we ± (v,r) | ∈ ∈ CV (Q) get a set |(v,r)| r Δ+(Q), (v, r) =0 ,wherev is a non-zero vector. It is useful to deﬁne Definition 3.11. For any ﬁnite set F , any subset A ⊂ NF \{0} and any ∈ CF 7 non-zero vector v we denote (v, r) (67) RF = ± |r ∈ A, (v, r) =0 ⊂ S1, where (v, r)= v r . v,A |(v, r)| i i i∈F Then we can rewrite (66) as follows (we assume that Q is an acyclic, because we used (65), which holds only for acyclic quivers) : Corollary 3.12. Let Q be an acyclic quiver. For any b σ =(P,Z) ∈ Stab(D (RepC(Q))) holds the inclusion ⊆ { } (68) Pσ Rv,Δ+(Q) v = vi = Z(si) i∈V (Q).

3.2.3. On the set Rv,Δ+(Q). Lemma 3.13. Let Q be a Dynkin quiver. For any stability condition σ ∈ b Stab(D (RepC(Q))) the set of semi-stable phases Pσ is finite. Proof. It is well known that for a Dynkin quiver Q the positive root system ∈ CV (Q) Δ+(Q) is finite. Hence for any non-zero v the set Rv,Δ+(Q) is finite. Now the lemma follows from Corollary 3.12. Lemma 3.14. Let Q be an Euclidean quiver (see subsubsection 3.2.1 for defini- ∈ CV (Q) tion). For any non-zero v the set Rv,Δ+(Q) is either finite or there exist ∈ N ∈ 1 { i ⊂ 1} { i }m m , p S and sequences pj S i=1,...,m;j∈N,s.t. limj→∞ pj = p i=1 and ∪m {± i } Rv,Δ+ = i=1 pj j∈N. Proof. The root system Δ of an Euclidean quiver Q (as described in the first ∈ NV (Q) ∪{ } Z ⊂ equality of (64)) has an element δ ≥1 with the properties Δ 0 + δ Δ ∪{0} and Δ ∪{0}/Zδ is finite (see0 [18, p. 18]). Hence there is a finite set { }⊂ ∪{ } m Z ∈{ } α1,α2 ...,αm Δ, s.t. Δ 0 = i=1(αi + δ). If for any i 1, 2,...,m

7 When the set F is clear we write just Rv,A.

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∈ Z ∈ we choose0 the minimal ni , s.t. αi + niδ Δ+ and denote βi = αi + niδ,then m N Δ+ = i=1(βi + δ). From the definition (67) of Rv,Δ+ we see that (69) m (v, β )+n(v, δ) R = ± i |n ∈ N,i=1, 2,...,m,(v, β )+n(v, δ) =0 . v,Δ+ |(v, β )+n(v, δ)| i i=1 i If (v, δ) = 0, then the set is finite. Otherwise for i = {1, 2,...,m} we have (v,βi)+n(v,δ) (v,δ) limn→∞ = . |(v,βi)+n(v,δ)| |(v,δ)| From this lemma and Corollary 3.12 it follows: b Corollary 3.15. Let Q be an Euclidean quiver. Then for any σ ∈D (Repk(Q)) 8 the set Pσ is either finite or has exactly two limit points of the type {p, −p}. Proof. ⊂ { If Pσ is infinite, then by the previous lemma and Pσ Rv,Δ+ , vi = } ∪m {± i } i Z(si) i∈V (Q) it follows that Rv,Δ+ = i=1 pj j∈N with limj→∞ pj = p for i = 1, 2,...,m. In particular Pσ can not have more than two limit points. Since Pσ is ∩{ i } ∩{− i } infinite, the sets Pσ pj j∈N, Pσ pj j∈N are infinite for some i (recall that −Pσ = Pσ). Hence {p, −p} are limit points of Pσ. The corollary follows.

Next we discuss the set Rv,Δ+ for the l-Kronecker quiver K(l) (two vertices with l parallel arrows between them), l ≥ 3. In this case the vertices are two, so v nz1+mz2 has two complex coordinates. I.e. Rv,Δ consists of fractions like ,where + |nz1+mz2| z1,z2 ∈ C, n, m ∈ N.Itisusefultonote Remark 3.16. Let z = r exp(iφ ), r > 0, i =1, 2, 0 <φ <φ ≤ π.Then i i i i 2 1 αz + βz α ≥ 1 2 exp if β α 0,β >0, (70) | | = αz1 + βz2 exp(iφ1) α>0,β =0, where f :[0, ∞) → [φ ,φ ) ⊂ (0,π) is the strictly increasing smooth function: 2 1 xr1 cos(φ1)+r2 cos(φ2) f(x) = arccos ,f(0) = φ2, lim f(x)=φ1. 2 2 2 − x→∞ x r1 + r2 +2xr1r2 cos(φ1 φ2) From (63) we see that the Euler form for the quiver K(l)is − (α1,α2), (β1,β2) K(l) = α1β1 + α2β2 lα1β2. Hence the positive roots are 2 2 2 (71) Δl+ =Δ+(K(l)) = {(n, m) ∈ N |n + m − lmn ≤ 1}\{(0, 0)}. Remark 3.17. Since the root systems Δ(K(l)) with l ≥ 3 will play an important role, we reserve for them the notation Δl =Δ(K(l)), respectively Δl+ =Δ+(K(l)). The roots with n2 +m2−lmn = 1 are called real roots and with n2 +m2−lmn ≤ 0 - imaginary roots. We can represent the real and the imaginary roots as follows: (72) 5 1 re n 1 4 Δ = {(1, 0)}∪{(0, 1)}∪ = l ± l2 − 4+ |n, m ∈ N≥ , (n, m)=1 l+ m 2 m2 1 im 1 n 1 (73) Δ = l − l2 − 4 ≤ ≤ l − l2 − 4 |n ∈ N≥ ,m∈ N≥ . l+ 2 m 2 0 1

8 In [20] are shown examples of both the cases: Pσ is ﬁnite and Pσ is with two limit points.

152 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Lemma . ≤ 3.18 Let v √=(z1,z2), zi = riexp(i φi√), ri > 0, 0 <φ2 <φ1 π.Let 1 − 2 − 1 2 − us denote u = f 2 l l 4 , v = f 2 l + l 4 ,wheref is deﬁned in Remark 3.16.Then {± } ∪± ∪{± } Rv,Δl+ = cj j∈N D aj j∈N, 1 where D is a dense subset in the arc exp(i[u, v]) ⊂ S , {aj }j∈N is a sequence with a0 =exp(iφ2) and anti-clockwise monotonically converges to exp(iu), {cj }j∈N is a sequence with c0 =exp(iφ1) and clockwise monotonically converging to exp(iv). 3 4 nz1+mz2 Proof. Now Rv,Δ = ± |(n, m) ∈ Δl+ . We have a disjoint union l+ |nz1+mz2| re ∪ im re im Δl+ =Δl+ Δl+ ,whereΔl+,Δl+ are taken from (72), (73). Recall also that if m ≥ 1then nz1+mz2 =exp(if (n/m)) (see Remark 3.16). Therefore we can write |nz1+mz2| for R : v,Δl+ ± nz1 + mz2 | ∈ re | | (n, m) Δl+ nz1 + mz2 ∪ ± nz1 + mz2 | ∈ im (n, m) Δl+ |nz1 + mz2|

= {± exp(iφ1)}∪{±exp(iφ2)} 5 1 1 4 ∪ ± exp if l ± l2 − 4+ |m ∈ N≥ 2 m2 1 ∪{±exp (if (n/m)) |n/m ∈ [u, v]} . Now the lemma follows from the properties of f given in Remark 3.16 and the fact that Q ∩ [u, v] is dense in [u, v].

3.2.4. Stability conditions σ on K(l) with Pσ = Rv,Δl+ . In this subsection l ≥ 3isﬁxedandQ = K(l), Δl+ is the positive root system of K(l), A = RepC(Q), b T = D (RepC(Q)). For a representation j . X = Cn . Cm ∈ A . *

we write dim(X)=(n, m), dim1(X)=n,dim2(X)=m. The simple objects of the b standard t-structure A = RepC(Q) ⊂ D (RepC(Q)) are:

j j . . (74) s1 = C . 0,s2 = 0 . C. . * . *

To A ⊂ T wecanapplyRemark3.3andthenwehaveHA ⊂ Stab(T) and bijection A 2 H ' (P,Z) → (Z(s1),Z(s2)) ∈ H . For any (P,Z) ∈ HA , t ∈ (0, 1] P(t) consists of the objects in A satisfying 2 the condition in Deﬁnition 3.1. If we denote v =(Z(s1),Z(s2)) ∈ H ,thenby Z(X)=(v, dim(X)): X ∈ P(t),t∈ (0, 1] ⇐⇒ (75) for any A-monic X → X arg(v, dim(X)) ≤ arg(v, dim(X)) = πt.

DYNAMICAL SYSTEMS AND CATEGORIES 153

A Lemma 3.19. Let σ =(P,Z) ∈ H and arg(Z(s1)) > arg(Z(s2)).Let(n, m) ∈ nz1+mz2 Δl+ be a Schur root. Then ∈ Pσ,wherezi = Z(si), i =1, 2. |nz1+mz2|

Proof. So, let (n, m) ∈ Δl+ be a Schur root. We show that there exists a σ- semistable X with dim(X)=(n, m). Then the lemma follows because X ∈ P(t) = {0} for some t ∈ (0, 1] and by the formula m(X)exp(iπt)=Z(X)=nz1 + mz2. If m =0,thenn = 1 (recall (71)) and then X = s1 is the semistable, which we need (it is even stable in σ, since it is a simple object in A). Hence we can assume that m ≥ 1. Denote arg(zi)=φi, i =1, 2, v =(z1,z2). Then 0 <φ2 <φ1 ≤ π. By (70) for any X with dim(X)=(n, m)wehavearg(v, dim(X)) = arg(nz1 + mz2)=f(n/m). Then by (75) such a X is semi-stable in σ iff any A-monic X → X satisfies n arg(dim (X)z +dim (X)z ) ≤ f . 1 1 2 2 m Recall that f(n/m) <φ1 (see Remark (3.16)). From the last inequality we get dim2(X ) = 0 and then by (70) this inequality can be rewritten as ≤ f(dim1(X )/dim2(X )) f (n/m) . So, we see that X ∈ A with dim(X)=(n, m)isσ-semistable iff any A-monic arrow X → X satisfies: dim1(X ) ≤ n (76) dim2(X ) =0, . dim2(X ) m Now since (n, m) is a Schur root, by Proposition 3.7 there exists X ∈ A with dim(X)=(n, m) and a non-zero θ : K0(A) → R, s.t. X is θ-semistable (see definition 3.5). We will show that this X is the σ-semistable, which we need. By θ-semistability of X we have θ(1, 0)n + θ(0, 1)m =0.Bym =0wehavea 9 monic map s2 → X and then again by θ-semistability θ(0, 1) ≥ 0, which together with θ(1, 0)n + θ(0, 1)m =0,θ = 0 implies n θ(1, 0) < 0,θ(0, 1) > 0,θ(1, 0) + θ(0, 1) = 0. m Let us take now any monic arrow X → X in A with X =0.By θ-semistability ≤ 0 θ(X )=θ(1, 0)dim1(X )+θ(0, 1)dim2(X ). Hence by θ(1, 0) < 0weobtain dim2(X ) = 0. Therefore we can write dim1(X ) ≥ n θ(1, 0) + θ(0, 1) 0=θ(1, 0) + θ(0, 1). dim2(X ) m dim1(X ) n By θ(1, 0) < 0 it follows ≤ . Hence, we verified (76) and the lemma dim2(X ) m follows.

RepC(K(l)) b Corollary 3.20. Let σ =(P,Z) ∈ H ⊂ StabD (RepC(K(l))) and arg(Z(s1)) > arg(Z(s2)).ThenPσ = Rv,Δl+ ,wherev =(Z(s1),Z(s2)).

Proof. The good luck is that all elements of Δl+ are Schur roots. Indeed im [31, Theorem 4 a)] says that Δl+ are Schur roots. In [46] one can read that the indecomposable representations of Q with dimension vectors real roots are also Schur. Therefore, we see that any (n, m) ∈ Δl+ is a Schur root and we can apply the previous lemma to it. The corollary follows.

9 Since the vertex corresponding to s2 is a sink, s2 is a subobject of any X ∈ RepC(K(l)) with dim2(X) =0.

154 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Remark 3.21. Recently it was noted in [29] that there is a connection between [52], [53], [44] and the density in an arc for the Kronecker quiver. 3.3. Kronecker pairs. In this subsection we generalize Corollary 3.20. The most general statement is Theorem 3.27, but we use further only its Corollary 3.29 (corollary 3.28 is intermediate). The ﬁrst step is: 10 Lemma 3.22. Let T be a k-linear triangulated category, where k is any ﬁeld. Let (E1,E2) be a full exceptional pair, s.t. ≤0 1 Hom (E1,E2)=0, 0 < dimk(Hom (E1,E2)) = l<∞.

Let A be the extension closure of (E1,E2) in T. Then A is a heart of a bounded t-structure in T and there exists an equivalence of abelian categories: F : A → Repk(K(l)),s.t. F (E1)=s1, F (E2)=s2 (s1, s2 are as in (74)). ≤0 Proof. In [17,p.6]or[45, section 3] it is shown that by hom (E1,E2)=0 11 and T = E1,E2 it follows that A is a heart of a bounded t-structure of T (see also [33, section 8]). In particular A is an abelian category. Let DT(T) be the category of distinguished triangles in T (objects are the distinguished triangles and morphisms are triple of arrows between triangles making commutative the corresponding diagram). Using the semi orthogonal decomposition T = E1 , E2 one can construct three functors:

(77) G : T → DT(T),λ1 : T →E1 ,λ2 : T →E2 s.t. the triangle G(X) ∈ DT(T) for any X ∈ T is:

uX- v-X w-X (78) G(X)=λ2(X) X λ1(X) λ2(X)[1],

λ1(X) ∈E1 ,λ2(X) ∈E2 .

It is well known that λ2 is right adjoint to the embedding functor E2→T and λ1 is left adjoint to E1→T (see for example [30, p. 279]). The adjoint functor (left or right) to an exact functor is also an exact functor ([7, Proposition 1.4]). Therefore λ1 and λ2 are exact functors If we restrict λ1, λ2 to A then we obtain exact functors between abelian categories A A → A λi : i i =1, 2, ∼ where Ai = k-Vectis the additive closure of Ei. We deﬁne the functor F : A → Repk(K(l)) as follows. First choose a basis of 1 Hom (E1,E2) and a decomposition of any Y ∈ Ai into dim(Hom(Ei,Y)) number of copies of Ei, i =1, 2. Take any X ∈ A, then we get a distinguished triangle G(X)as A wX A A - λ1 (X) λ2 (X)[1] in (78) with λi(X)=λi (X), in particular we get an arrow . 1 This arrow, using the chosen decompositions and the basis of Hom (E1,E2), can be expressed by la2 × a1 matrices over k,whereai = dim(Ei,λi(X)), i =1, 2.

10It is motivated by Bondal’s result in [9] for equivalence between triangulated category generated by a strong exceptional collection and the derived category of modules over an algebra of homomorphisms of this collection and by a note on this equivalence in [40]. Observe however that we do not have restriction on (E1,E2) to be a strong pair and we construct equivlanece between t-structures. 11If S is a subset of objects in a triangulated category T we denote by S the triangulated subcategory generated by S.

DYNAMICAL SYSTEMS AND CATEGORIES 155

In particular these l matrices are a representation of K(l) with dimension vector (a1,a2) and we deﬁne F (X) to be this representation. Let f : X → Y be an arrow in A then, as far as G : T → DT(T) is a functor, G(f) is a morphism of triangles, hence the diagram:

A w-X A λ1 (X) λ2 (X)[1] A A λ1 (f) ? λ2 (f)[1] ? A w-Y A λ1 (Y ) λ2 (Y )[1]

A A is commutative. Let M1, M2 be the matrices of λ1 (f), λ2 (f). The commutativity of the diagram above implies that (M1,M2):F (X) → F (Y )isanarrowin Repk(K(l)) and our deﬁnition of F (f)isF (f)=(M1,M2). A By the exactness of λi , i =1, 2 it follows that F is an exact functor between abelian categories. Now, by straightforward computations one can show that F is an equivalence.

This lemma prompts the following deﬁnition

Definition 3.23. Apairofobjects(E1,E2) in a k-linear (k is any ﬁeld) triangulated category T is called Kronecker pair if:

• (E1,E2) is an exceptional pair ≤0 • Hom (E1,E2)=0 1 • 3 ≤ dimk(Hom (E1,E2)) < ∞.

Corollary 3.24. Let (E1,E2) be a Kronecker pair in a C-linear triangulated 1 category D.Denotel =dim(Hom(E1,E2)), T = E1,E2⊂D and A -the extension closure of (E1,E2). A b Then any σ =(P,Z) ∈ H ⊂ StabD (T) with arg(Z(E1)) > arg(Z(E2)) satisﬁes Pσ = Rv,Δl+ ,wherev =(Z(E1),Z(E2)).InparticularPσ is dense in an arc of non-zero length.

Proof. We take the equivalence F : A → RepC(K(l)) constructed in Lemma b 3.22, A ⊂ T, RepC(K(l)) ⊂ D (RepC(K(l))). This equivalence induces a natural bijection F ∗ : HRepC(K(l)) → HA.Forσ =(P,Z) ∈ HA, σ =(P,Z) ∈ HRepC(K(l)), ∗ from F (σ )=σ it follows Z(Ei)=Z (si) (because F (Ei)=si)andPσ = Pσ . Then the corollary follows from Corollary 3.20.

Thus, in this Corollary we obtained σ ∈ Stab(E1,E2) with Pσ dense in a nontrivial arc. To obtain σ ∈ Stab(D) with such a property, we want to extend the given σ ∈ Stab(E1,E2) to a stability condition on D ⊃ T in the following sense: Definition 3.25. Let T ⊂ D be a triangulated subcategory in a triangulated category D. We say that σ =(P,Z) ∈ Stab(T) can be extended to D (or extendable to D)ifthereexistsσe =(Pe,Ze) ∈ Stab(D),s.t. Ze ◦ K0(T ⊂ D)=Z and {P(t) ⊂ Pe(t)}t∈R. In this case σe is called extension of σ.

Remark 3.26. From Deﬁnition 3.4 it follows that if σe is an extension of σ, ⊃ then Pσe Pσ. By Corollary 3.24 it follows:

156 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Theorem 3.27. Let (E1,E2) be a Kronecker pair in a C-linear triangulated 1 category D.Denotel =dim(Hom(E1,E2)), T = E1,E2⊂D and A -the extension closure of (E1,E2). Then any σ ∈ StabDb(D), which is an extension of a stability condition (P,Z) ∈ HA ⊂ b T ⊃ StabD ( ) with arg(Z(E1)) > arg(Z(E2)) satisﬁes Pσ Rv,Δl+ ,where v =(Z(E1),Z(E2)).InparticularPσ is dense in an arc of non-zero length. One setting, where we can extend these stability conditions, is as follows. 12 Assume that (E0,E1,...,En) is a full Ext-exceptional collection in D.Then for any 0 ≤ i

Corollary 3.28. Let (E0,E1,...,En) be a full Ext-exceptional collection in a C-linear triangulated category D.Let(Ei,Ei+1) be a Kronecker pair for some 1 0 ≤ i ≤ n − 1.Denotel =dim(Hom(Ei,Ei+1)) and the extension closure of (E0,E1,...,En) by A. A Then any σ =(P,Z) ∈ H with arg(Z(Ei)) > arg(Z(Ei+1)) satisﬁes Pσ ⊃

Rv,Δl+ ,wherev =(Z(Ei),Z(Ei+1)).InparticularPσ is dense in an arc of non- zero length.

Corollary 3.29. Let (E0,E1,...,En) be any full exceptional collection in a 14 C-linear triangulated category D of ﬁnite type. Let (Ei,Ej ) be a Kronecker pair for some 0 ≤ i

Proof. First by mutations of the exceptional collection (E0,E1,...,En)we can obtain a full exceptional collection (Ei,Ej,C2,...,Cn). Then, because T is of finite type, after shifts of C2,C3,...,Cn we can obtain a full exceptional collection B = {B0,B2,...,Bn}, which is Ext and B0 = Ei, B1 = Ej . SowegetafullExt- exceptional collection B for which (B0,B1) is a Kronecker pair. Now if we denote by A the extension closure of B by Corollary 3.28 it follows that any σ =(P,Z) ∈ HA ⊃ with arg(Z(B0)) > arg(Z(B1)) satisfies Pσ Rv,Δl+ ,wherev =(Z(B0),Z(B1)) 1 and l =dimC(Hom (B0,B1)). Remark 3.30. A more general setting, where the stability conditions HA in Theorem 3.27 can be extended, is that there exists a semi-orthogonal decomposition (D , E1,E2) of D with additional assumptions, specified in [17, Theorem 3.6] and [17, Proposition 3.5]. 3.4. Application to quivers. In this subsection we apply the results of the previous subsection 3.3 to quivers and obtain Corollary 3.31, Proposition 3.32. Table (1) contains Proposition 3.32 and the results of subsection 3.2. Let Q be an acyclic quiver. The notations V (Q), Arr(Q), Γ(Q)areex- plained in subsubsection 3.2.1. It is shown in [19] that any exceptional collection

12 ≤0 The “Ext-” means Hom (Ei,Ej )=0for0≤ i

DYNAMICAL SYSTEMS AND CATEGORIES 157

(E1,E2,...,En)inRepC(Q)oflengthn =#(V (Q)) is a full exceptional collec- b tion of D (RepC(Q)). Furthermore, any exceptional collection (E1,E2,...,Ei)in RepC(Q) with imax imal nonzero degree Hom (E1,E2) =0,Hom (E1,E2) = 0 has dimension max dimC(Hom (E1,E2)) ≤ 2.

For any exceptional pair (E1,E2)itiswellknownthat(LE1 (E2),E1) is also an exceptional pair, where LE1 (E2) is determined by the distinguished triangle: ev L (E ) - Hom∗(E ,E ) ⊗ E E1,E-2 E - L (E )[1]. (79) E1 2 1 2 1 2 E1 2 i ∼ −i Take any i ∈ Z.WeshowbelowthatHom(LE (E2),E1) = Hom (E1,E2), which 1 ∗ means that the maximal non-zero degree of Hom (E1,E2) is the minimal non-zero ∗ degree of Hom (LE1 (E2),E1) and they are isomorphic. Then the corollary follows by the proved inequality for the minimal non-vanishing degrees. i ∗ We apply Hom ( ,E1) to the triangle above and by Hom (E2,E1) = 0 it follows i ∗ ⊗ ∼ i (80) Hom (Hom (E1,E2) E1,E1) = Hom (LE1 (E2),E1).

On the other hand (recall that E1 is an exceptional object) j ∗ ∼ dim(Hom (E1,E2)) Hom (E1,E2) ⊗ E1 = E1[−j] j i ∗ ⇒ Hom (Hom (E1,E2) ⊗ E1,E1) j ∼ dim(Hom (E1,E2)) = Hom(⊕jE1[−j] ,E1[i]) ∼ dim(Hom−i(E ,E )) = C 1 2 , i ∼ −i which together with (80) give Hom (LE1 (E2),E1) = Hom (E1,E2) and the corollary is proved. Next we want to prove: Proposition 3.32. Any acyclic connected quiver Q, which is neither Euclidean b nor Dynkin has a family of stability conditions σ on D (RepC(Q)),s.t.Pσ is dense in an arc of non-zero length.

158 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Remark 3.33. If there are oriented cycles in Q, then one can show that there 15 is a family {sλ}λ∈C of non isomorphic simple objects in A = RepC(Q).Thenif we define for simple object s ∈ A λ if s = s ,λ∈ H (81) Z(s)= |λ| λ i otherwise we obtain a stability function Z : K0(A) → C, which has HN property, since A is of finite length. One can show that the corresponding stability condition σ =(P,Z) ∈ A H is locally finite. Since all {sλ}λ∈C are simple in A,theyareσ-semistable. Hence 1 Pσ = S . So let us fix a quiver Q, satisfying the conditions of Proposition 3.32. By the arguments given in the beginning of this subsection (before Corollary 3.31) we b reduce the proof to finding a Kronecker pair in D (RepC(Q)). From here till the end of this subsubsection we present the proof of the following: Proposition 3.34. Any Q, satisfying the conditions in Proposition 3.32, has a Kronecker pair in RepC(Q)(i.e. a pair or representations (ρ, ρ ) in RepC(Q) with ∗ ≤0 1 b HomT(ρ ,ρ)=HomT (ρ, ρ )=0, dimC(HomT(ρ, ρ ))≥3,whereT =1D (RepC(Q))). 16 Recall (see page 8 in [18]) that for ρ, ρ ∈ RepC(Q), we have the formula − 1 (82) dimC(Hom(ρ, ρ )) dimC(Hom (ρ, ρ )) = dim(ρ), dim(ρ ) Q , ∈ where , Q is defined in (61), (63). Let us denote for ρ RepC(Q): { ∈ | } (83) supp(ρ)=supp(dim(ρ)) = i V (Q) dimi(ρ) =0 .

For i ∈ V (Q) the simple representation si is characterized by supp(si)={i}, { | ∈ } b dimi(si) = 1. Obviously si i V (Q) are exceptional objects in D (RepC(Q)). One of the representations in the Kronecker pair (ρ, ρ), which we shall obtain, is among the exceptional objects {si|i ∈ V (Q)}. It is useful to denote A, B ⊂ V (Q) Arr(A, B)={a ∈ Arr(Q)|s(a) ∈ A, t(a) ∈ B}, (84) Ed(A, B)=Ed(B,A)=Arr(A, B) ∪ Arr(B,A).

b To ﬁnd Kronecker pairs in D (RepC(Q)) we observe ﬁrst, that for ρ, ρ ∈ ≤−1 b RepC(Q)wehaveHom (ρ, ρ )=0inD (RepC(Q)) and supp(ρ) ∩ supp(ρ)=∅⇒Hom(ρ, ρ)=Hom(ρ,ρ)=0, (85) 1 dimC(Hom (ρ, ρ )) = dims(a)(ρ)dimt(a)(ρ ) a∈Arr(supp(ρ),supp(ρ)) which follows by (82). Another useful statement is Lemma 3.35. Let Q be an Euclidean quiver. Then for each n ∈ N there exists ∈ ≥ ∈ an exceptional representation ρ RepC(Q),s.t.dimv(ρ) n for each v V (Q).

15 -λ For example the representations { C C }λ∈C of the quiver with one vertex and one loop are all simple and mutually non isomorphic 16 i i b throughout the proof Hom (, )meansHomT(, ), T = D (RepC(Q))

DYNAMICAL SYSTEMS AND CATEGORIES 159

Proof. ∈ NV (Q) Let δ ≥1 be the minimal imaginary root of Δ+(Q), used in 17 V (Q) the proof of Lemma 3.14. One property of δ is that ( ,δ)Q =0onN , where (α, β)Q = α, β Q + β,α Q is the symmetrization of , Q. We find below ∈ 1 avertexv V (Q), s.t. 1v, 1v Q = 2 (1v, 1v)Q =1, 1v,δ Q =0.Thenfor ∈ N 1 any m we have 1 = 2 (1v + mδ, 1v + mδ)Q = 1v + mδ, 1v + mδ Q =1, 1v + mδ, δ Q = 0, hence, for big enough m, r =1v + mδ is a real positive root ∈ { ≥ } r Δ+(Q) with r, δ Q =0, rv n v∈V (Q). Hence by [18, p. 27] there is an exceptional representation ρ with dim(ρ)=r and the lemma follows. 18 If Γ(Q)=Am, m ≥ 1: As far as Q is not an oriented cycle then there is a sink s ∈ V (Q) (i.e. both the arrows touching s end at it). Hence by (63) − 1s, 1s Q = 1s,δ Q =1. If Γ(Q)=Dm, m ≥ 4, E6, E7, E8.In[5, fig. (4.13)] are given the coordinates of δ for all these options for Γ(Q). We take v ∈ V (Q) to be the extending vertex, in [5, fig. (4.13)] this vertex is denoted by , i.e. v = . Then by (63) and the ± given in [5, fig. (4.13)] coordinates of δ one computes 1v, 1v Q =1, 1v,δ Q = 1, depending on whether is a sink/source in Q.

Definition 3.36. Let A ⊂ V (Q), A = ∅.ByQA we denote the quiver with V (QA)=A and Arr(QA)=Arr(A, A)={a ∈ Arr(Q)|s(a) ∈ A, t(a) ∈ A}.For any ρ ∈ RepC(QA) we denote by the same letter ρ the representation in RepC(Q), which on A, Arr(A, A) coincides with ρ andiszeroelsewhere. We say that a vertex v ∈ V (Q) is adjacent to QA if v ∈ A and Ed(A, {v}) = ∅.

Remark 3.37. If ρ ∈ RepC(QA) is an exceptional representation then the corresponding extended representation in RepC(Q) is also exceptional.

Remark 3.38. If v ∈ V (Q) is adjacent to QA then Arr(QA∪{v})=Arr(QA) ∪ Ed(A, {v}). In the following two corollaries we consider a conﬁguration of a subset A ⊂ V (Q)andanadjacenttoitvertexv ∈ V (Q), s.t. the arrows connecting v and A are all directed either from v to A or from A to v, which means that v is either a source or a sink in QA∪{v}. In Corollary 3.39 we show that if QA is an Euclidean quiver then we get a 1 Kronecker pair (E1,E2)inRepC(Q) with dimC(Hom (E1,E2)) as big as we want (without additional assumption on the quiver Q). In Corollary 3.40 we show that if Γ(QA) is either An(n ≥ 1) or Dn(n ≥ 4) then, under the additional assumption that there are at least three edges between 1 v and A, we get a Kronecker pair (E1,E2)inRepC(Q) with dimC(Hom (E1,E2)) equal to this number of edges.

Corollary 3.39. Let A ⊂ V (Q) be such that QA is Euclidean. Let v ∈ V (Q) be a vertex, which is adjacent to QA and either a sink or a source in QA∪{v}. Then for any n ≥ 3 there exists a Kronecker pair (E1,E2) in RepC(Q) with 1 dim(Hom (E1,E2)) ≥ n.

17In [5, ﬁg. (4.13)] are given the coordinates of δ for all euclidean graphs 18 Recall that by Γ(Q) we denote the underlying non-oriented graph and that A1 is graph with two vertices and two parallel edges connecting them, Am with m ≥ 2isaloopwithm +1 vertices and m edges connecting them forming a simple loop

160 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Proof. From Lemma 3.35 and Remark 3.37 we get an exceptional represen- ∈ { ≥ } tation ρ RepC(Q), s.t. supp(ρ)=A and dimi(ρ) n i∈A. If v is a sink in QA∪{v} then Arr({v},A)=∅, and, since v is adjacent to ∗ QA, Arr(A, {v}) = ∅.From{v}∩A = ∅ and (85) we get Hom (sv,ρ)=0, ≤0 1 Hom (ρ, sv)=0,dimC(Hom (ρ, sv)) ≥ n. Therefore (ρ, sv) is the Kronecker pair we need. If v is a source in QA∪{v} then the same arguments show that (sv,ρ)issucha Kronecker pair.

Corollary 3.40. Let A ⊂ V (Q) be such that Γ(QA) is either An(n ≥ 1)or Dn(n ≥ 4). Let v ∈ V (Q) be adjacent to QA and either a sink or a source in QA∪{v}.Let#(Ed({v},A)) = n ≥ 3. Then there exists a Kronecker pair (E1,E2) 1 in RepC(Q) with dim(Hom (E1,E2)) = n.

Proof. Using that Γ(QA) is either An(n ≥ 1) or Dn(n ≥ 4) we see that the representation ρ, with A ' i → C, Arr(A, A) ' a → IdC and zero elsewhere is an { } exceptional representation in RepC(Q) with supp(ρ)=A and dimi(ρ)=1 i∈A. If v is a sink in QA∪{v} then Arr({v},A)=∅ and

#(Arr(A, {v})) = #(Ed({v},A)) = n.

From {v}∩A = ∅ and (85) we get

∗ ≤0 1 Hom (sv,ρ)=0, Hom (ρ, sv)=0, dimC(Hom (ρ, sv)) = n.

So that (ρ, sv) is the Kronecker pair we need. If v is a source in QA∪{v} then the same arguments show that (sv,ρ)issucha Kronecker pair.

An immediate consequence of this corollary is

Corollary 3.41. If there are n ≥ 3 parallel arrows in Arr(Q). Then there 1 exists a Kronecker pair (E1,E2) in RepC(Q) with dim(Hom (E1,E2)) = n.

Proof. Let these arrows start at a vertex i and end at a vertex j,then #Arr({i}, {j})=n ≥ 3, Arr({j}, {i})=∅ and we apply the previous Corollary to A = {j}, v = {i}.

Hence, we can assume that there are no more than two parallel arrows in Arr(Q). We consider next the case that two parallel arrows do occur.

Remark 3.42. In the considerations, that follow, we refer most often to Corol- 1 lary 3.39 (i.e. then we get Kronecker pairs with arbitrary big dimC(Hom (E1,E2))), but there are three situations, where we need Corollary 3.4019 with the minimal ad- missible number of edges connecting v and A,namely3, and then the produced 1 Kronecker pair is with minimal possible dimC(Hom (E1,E2)) = 3.

19we already used it once in Corollary 3.41

DYNAMICAL SYSTEMS AND CATEGORIES 161

20 The quiver QA∪{v} observed in these three special situations , in which we use Corollary 3.40, is as follows (we denote the set A by A = {a1,a2,...,an}):

a4 - 6 a - 2 a1 a2 a3 a1 a2 a3. - - 6 - 6 - - v - a S1 = 1 S2 = v S3 = v .

In S1 Γ(QA)=A2,inS2 Γ(QA)=A3,inS3 Γ(QA)=D4. 3.4.1. If there are 2 parallel arrows in Arr(Q). Let these two arrows start at avertexi and end at a vertex j. Thus, throughout this subsubsection we have #(Arr({i}, {j})) = 2, Arr({j}, {i})=∅. Then (recall Deﬁnition 3.36): - i - j (86) Q{i,j} = , Γ(Q{i,j})=A1. By our assumption that Q is connected and not Euclidean there is a vertex k ∈ V (Q) whichisadjacenttoQ{i,j}. If either Arr({k}, {i, j})=∅ or Arr({i, j}, {k})=∅ then by Corollary 3.39 we get a Kronecker pair. Hence we can assume Arr({k}, {i, j}) = ∅ and Arr({i, j}, {k}) = ∅. From the condition that there are no oriented cycles we reduce to (87) Arr({k}, {j}) = ∅ Arr({i}, {k}) = ∅⇒Arr({j}, {k})=Arr({k}, {i})=∅. - k - j If Arr({k}, {j}) has two elements then Q{k,j} = ,Γ(Q{k,j})=A1 , i is adjacent to Q{k,j} and Arr({k, j}, {i})=Arr({k}, {i}) ∪ Arr({j}, {i})=∅, hence Corollary 3.39 produces a Kronecker pair. Therefore we can assume that Arr({k}, {j}) has only one element. - If Arr({i}, {k})hastwoelements,thenQ{i,k} = i k ,Γ(Q{i,k})=A1, j is adjacent to Q{i,k} and Arr({j}, {i, k})=Arr({j}, {i}) ∪ Arr({j}, {k})=∅, hence Corollary 3.39 produces a Kronecker pair. Hence we can assume that Arr({i}, {k}), - i - j Arr({k}, {j}) are single element sets. Using that Q{i,j} = we obtain that Q{i,j,k} isthesameasthequiverS1 in Remark 3.42 with v = i, a1 = j, a2 = k and then by Corollary 3.40 we obtain a Kronecker pair. We reduce to the case 3.4.2. If there are no parallel arrows in Arr(Q). In this case for any pair i, j ∈ V (Q), i = j we have #(Arr(i, j)) = #(Ed(i, j)) ≤ 1. In this subsubsection the term loop with m vertices, m ≥ 1 in Γ(Q) means a sequence a1,a2,...,am in V (Q), s.t. { } { ∅}m−1 ∅ # a1,a2,...,am = m and Ed(ai,ai+1) = i=1 , Ed(am,a1) = . Since there are no edges-loops and there are no parallel arrows in Q, any loop in Γ(Q) (if there is such) must be with m ≥ 3 vertices. First we show quickly how to get a Kronecker pair if there are no loops.

20 in Corollary 3.41 the quiver QA∪{v} is the Kronecker quiver, i.e. QA∪{v} = K(n), n ≥ 3

162 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

If there are no loops in Γ(Q) (recall also that by assumption Q is neither Dynkin nor Euclidean) then one can show that for some proper subset A ⊂ V (Q) QA is an Euclidean quiver of the type Dm, m ≥ 4, E6, E7, E8. Take an adjacent to QA vertex v ∈ V (Q). The assumption that there are no loops in Γ(Q) imply that there is a unique edge between v and QA, i.e. either Arr({v},A)=∅ or Arr(A, {v})=∅, and then we can apply Corollary 3.39 to obtain a Kronecker pair. Till the end of this subsubsection we assume that there is a loop in Γ(Q). Let us ﬁx a loop with minimal number of vertices a1,a2,...,am, i.e. m is the minimal possible number of vertices in a loop. Denote A = {a1,a2,...,am},#(A)=m. From the minimality of m it follows that Ed(ai,aj )=∅,if1≤ i

a 2 - 6 a3. - a QA = 1 By #(Ed(A, {v})) = 3 it follows that v ∈ V (Q) must be connected to all the three vertices {a1,a2,a3} and by (88) one of the arrows must start at v and another must end at it. We have either Arr({v}, {a2}) = ∅ or Arr({a2}, {v}) = ∅. If Arr({v}, {a2}) = φ, then by the assumption that there are no oriented cycles we have Arr({a3}, {v})=φ, Arr({v}, {a3}) = φ, so we can only choose the direction of Ed(v, a1), i.e. we have two options for Q{a1,a2,a3,v}:

a a 2 - 2 - 6 6 a3 - v a3 v - - a1 a1 .

In both the cases a3 is a sink in Q{a1,a2,a3,v}, hence we can apply Corollary 3.39 to ∈ Q{a1,a2,v} and a3 V (Q) and get Kronecker pairs. If Arr({a2}, {v}) = φ, now the direction of Ed(v, a1) is ﬁxed and both the options for Q{a1,a2,a3,v} are:

a2 a2 - - - - 6 6 - a3 - v a3 - v - - a1 a1 .

DYNAMICAL SYSTEMS AND CATEGORIES 163

∈ In the ﬁrst case we apply Corollary 3.39 to Q{a1,a2,v} and a3 V (Q) and in the ∈ second we apply it to Q{a1,a2,a3} and v V (Q). The case #(Ed(A, {v})) = 2, m =3. Letusconsider

a3 a Γ(QA)= 1 without ﬁxing the orientation. Now there are only two edges between A and v. Hence by (88) there are exactly two arrows between A and v and one of them must start at v and the other must end at it. As long as we have not ﬁxed the orientations of the arrows in QA, we can assume that Arr({a1}, {v}) = ∅, Arr({v}, {a2}) = ∅.

So that Q{a1,a2,a3,v}, up to a choice of orientation in QA,is

a3- v.

We consider now the possible choices of directions of the arrows in QA.Ifa3 is a source/sink in QA, then it is a source/sink in QA∪{v} and we can apply Corollary ∈ 3.39 to Q{a1,a2,v} and a3 V (Q). Hence, by the condition that Q is acyclic, we reduce to a 2 - 6 a3 a2 v a3- v 6 - - a1 = a1 and this is a permutation of the special case S2 of Remark 3.42. In this case we ∈ obtain a Kronecker pair by Corollary 3.40 applied to Q{v,a2,a3} and a1 V (Q). The case m =4. In this case

a4 a3

a a Γ(QA)= 1 2 and by the minimality of m = 4 it follows #(Ed(A, {v})) = 2 (recall that we have reduced to (88)). In other words the adjacent vertex v must be connected to two of the vertices of the quadrilateral Γ(QA). Again by the minimality of m =4these two vertices must be diagonal and, as long as we have not ﬁxed the orientations of the arrows, we can assume that Arr({a1}, {v}) = ∅, Arr({v}, {a3}) = ∅.Sothat

Q{a1,a2,a3,a4,v}, up to a choice of orientation in QA,is

a4 a2 v. - a1

It follows to assign directions of the arrows in QA.Ifa4 or a2 is a source/sink in QA ∈ then we can apply Corollary 3.39 to Q{a1,a2,a3,v} and a4 V (Q)ortoQ{a1,a4,a3,v}

164 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH and a2 ∈ V (Q), respectively. Therefore, we reduce to

a3 - 6 a4 a2 v, 6- a1 Q{a1,a2,a3,a4,v} = which is a permutation of the quiver S3 in Remark 3.42. The case m ≥ 5. If m =2k + 1 is odd, k ≥ 2, then we can depict Γ(QA),v ∈ V (Q)andone edge in Ed({v},QA)asfollows: ak+1 ak+2 . . . . a3 a2k

a2 a2k+1

a1 v

We have reduced to the case #(Ed({v},QA)) ≥ 2 (see (88)). If we add another edge between v and A then we obtain another loop with number of vertices less or equal to k +2.Byk ≥ 2wehavek +2< 2k + 1, which contradicts the minimality of m =2k +1. If m =2k is even, k ≥ 3, then we can depict Γ(QA),v ∈ V (Q) and one edge in Ed({v},QA)asfollows: ak+1

ak ak+2 . . . . a3 a2k−1

a2 a2k

a1 v Again, another edge between v and A produces another loop with number of vertices less or equal to k+2. By k ≥ 3wehavek+2 < 2k, which contradicts the minimality of m =2k. Proposition 3.34 is completely proved and it implies Proposition 3.32. Having Proposition 3.32, Remark 3.33, Corollary 3.15 and Lemma 3.13 we obtain table 1. 3.5. Further examples of Kronecker pairs. Here we give some more examples of Kronecker pairs. In all the cases Corollary 3.29 can be applied. 3.5.1. Markov triples. It is shown in [11, Example 3.2] that if X is a smooth projective variety (we assume over C), such that Db(Coh(X)) is generated by a strong exceptional collection of three elements (for example X = P2) then for any such collection (E0,E1,E2) the dimensions

a = dim(Hom(E0,E1)),b= dim(Hom(E0,E2)),c= dim(Hom(E1,E2))

DYNAMICAL SYSTEMS AND CATEGORIES 165 satisfy Markov’s equation a2 + b2 + c2 = abc.If(a, b, c) =(0 , 0, 0) and a, b, c ≤ 3 then (a, b, c) satisfy Markov’s equation iﬀ a = b = c = 3, i.e. the “minimal” such b triple is (3, 3, 3). Hence for any strong collection (E0,E1,E2)onD (Coh(X)) for some i

4. Open questions In the previous sections we have described the foundations of some new directions in the theory of derived categories and their connection with “classical” geometry and dynamical systems. We believe this is a promising ﬁeld and conclude by formulating possible future directions of research.

4.1. Algebraicity of entropy. In all examples we have considered so far the graph of the entropy function t → ht(F ) is an algebraic curve deﬁned over Q in coordinates (exp(t), exp(ht(F ))). Also, in the Z/2-graded case exp(h0(F )) is an algebraic number. (The stretch factor of a pseudo-Anosov map is an algebraic integer of degree bounded above by a quantity depending only on the genus, see [26].)

Question 4.1. Is algebraicity of entropy a general phenomenon, and if so, what are natural suﬃcient conditions for it?

4.2. Pseudo-Anosov autoequivalences. Recall that pseudo-Anosov maps are characterized by the fact that they preserve a quadratic differential up to rescaling in the horizontal and vertical direction by factors 1/λ and λ respectively. Conjecturally, such a quadratic differential defines a stability condition on the (wrapped) Fukaya category of the complement of its zeros. This stability condition would then be preserved by the autoequivalence up to rescaling of the real and imaginary parts of the central charge. In the case of the torus this follows from homological mirror symmetry and the description of the space of stability conditions of the bounded derived category of an elliptic curve. With this in mind, we propose the following.

166 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

Definition 4.1. An autoequivalence φ of a triangulated category C is pseudo- Anosov if there exists a Bridgeland stability condition σ on C and λ>1 such that λ−1 0 (89) φσ = σ 0 λ where we use the action of the universal cover of GL+(2, R) on Stab(C). An example is given by the shifted Serre functor S◦[−1] on the bounded derived category of the Kronecker quiver with m ≥ 3 arrows. A σ satisfying (89) lies in the chamber where the set of phases is dense in an arc. The stretch factor is given by √ m2 + m m2 − 4 (90) λ = − 1 2 and log λ is the entropy of S ◦ [−1]. Suppose M is a symplectic manifold and f : M → M a symplectomorphism inducing a pseudo-Anosov autoequivalence, in the above sense, on the Fukaya category of M. Question 4.2. Are there, in analogy with the case of surfaces, two transverse singular lagrangian foliations of M which are preserved by some symplectomorphism Hamiltonian isotopic to f? 4.3. Birational maps. For birational automorphisms the maximal dynamical degree provides a useful notion of entropy, c.f. the work of Cantat–Lamy [16] and Blanc–Cantat [8] for the case of surfaces. In order to incorporate this into the categorical framework discussed here, one would first need to find a suitable triangulated category. Question 4.3. Suppose X is a smooth projective variety. Is there a suitable triangulated category C associated with X so that birational automorphisms of X give rise to autoequivalences of C, and moreover the maximal dynamical degree conincides with the entropy h0 under this correspondence? Recall that cluster transformations are (often) related to automorphisms of 3- Calabi-Yau categories. Consider a cluster collection of spherical objects as in [36], generating a CY3 category C. A sequence of mutations which produces a collection isomorphic to the original one defines an element in Aut(C). Let Γ ⊂ Aut(C)be the subgroup of all such elements, and N ⊂ Γ the normal subgroup generated by spherical twists by objects in the collection. Associating a (birational) cluster transformation to a mutation we get a sequence of groups

(91) Γ −→ Γ/N −→ Crn where n is the number objects in the cluster collection. Question 4.4. Is there natural splitting s of the map Γ → Γ/N so that for γ ∈ Γ/N the entropy h0(s(γ)) coincides with the maximal dynamical degree of the corresponding element in Crn? 4.4. Dynamical spectrum. In recent work of Blanc and Cantat [8]theset of all dynamical degrees of birational automorphisms of an algebraic surface X is studied. Analogously, the set of entropies of Pseudo-Anosov maps of a closed

DYNAMICAL SYSTEMS AND CATEGORIES 167 surface has been studied, see [25] for an overview. In the context of the present work one could thus investigate the set

{h0(F )|F ∈ Aut(C)} of all entropies of autoequivalences of a given triangulated category C.Afirst question is Question 4.5. Under what conditions is the above set of entropies discrete in R? 4.5. Complexity and mass. Suppose T is a triangulated category with sta- ∈ T bility condition σ.LetE be an object with semistable factors Gi, then the | | mass m(E)ofE is by definition i Z(Gi) . Intuitively, m(E) measures the “size” of E, analogous to δ (G, E) for a fixed generator G. Let us consider 0 1 n (92) hσ(F )= sup lim sup log (mσ(F E)) 0= E∈T n→∞ n T for an endofunctor F of . Recall first that | − − − | | + − + | mσ(E) (93) d(σ, τ )= sup φσ (E) φτ (E) , φσ (E) φτ (E) , log 0= E∈T mτ (E) defines a generalized metric on Stab(T), see [13]. In particular, if σ and τ have finite distance, then there are C1,C2 > 0 such that

(94) mσ(E)

(95) m(E) ≤ m(G)δ0(G, E) which follows from the “triangle inequality” for mass, i.e. (96) m(E) ≤ m(A)+m(B) if there is a distinguished triangle A −→ E −→ B −→ . Consequently, the maximum is attained in (92) for a generator and hσ(F ) ≤ h0(F ). If the reverse inequality δ0(G, E) ≤ Cm(E) where to hold as well for some C>0, then we could conclude hσ(F )=h0(F ). Question 4.6. Under what conditions does there exist a lower bound for mass in terms of complexity, i.e. δ0(G, E) ≤ Cmσ(E) where C>0 is independent of E? The question can easily be answered positively in the following special case. Assume A is a smooth and proper dg-algebra which is non-positive, i.e. HiA =0 for i>0. There is a natural bounded t-structure on T =Perf(A)withheartA the category of ﬁnite-dimensional modules over H0A. This t-structure deﬁnes a stability condition σ with central charge Z(E)=i dim(E)forE ∈ A.Thuswe ∗ have mσ(E)=dimH (E) ≥ Cδ0(A, E), since A is smooth. Finally let us point out that as with complexity one can also introduce a parameter t and consider φ(Gi) t (97) mt(E)= |Z(Gi)|e π i where Gi are the semistable components of E and φ(Gi) ∈ R is the phase of Gi.

168 G. DIMITROV, F. HAIDEN, L. KATZARKOV, AND M. KONTSEVICH

4.6. Questions related to Kronecker pairs and density of phases. The results of Section 3 are a motivation for the following questions. Recall that in any of the quivers listed in Remark (3.42) we found Kronecker k pairs (E1,E2) with dim(Hom (E1,E2)) = 3 for k =1and0fork = 1. We expect that the ﬁrst part of the following question has a positive answer: k Question 4.7. Do the inequalities {dim(Hom (E1,E2)) ≤ 3}k∈Z hold for any exceptional pair (E1,E2) in any of the quivers listed in Remark (3.42)? k Determine all the quivers Q, s.t. the dimensions {dim(Hom (E1,E2))}k∈Z, where (E1,E2) vary through all exceptional pairs, are bounded above. Recall that by Corollary 3.31 these dimensions are strictly smaller than 3 if Q is either Euclidean or a Dynkin quiver, and they are not bounded above if Corollary 3.39 can be applied to Q. We expect that the categories with non-dense behaviour of phases form a “thin” b set of categories. Among these, the categories D (RepC(Q)) with Q an Euclidean quiver have somehow remarkable behaviour of Pσ (see the second row of Table 1). So the next question is: Question 4.8. Which are the triangulated categories T, s.t. for any σ ∈ Stab(T) the set of phases Pσ is either ﬁnite or has two limit points?

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Universitat¨ Wien, Garnisongasse 3, 1090, Wien, O¨ sterreich E-mail address: [email protected]

Universitat¨ Wien, Garnisongasse 3, 1090, Wien, O¨ sterreich E-mail address: [email protected] IHES, G35 route de Chartres, F-91440, France E-mail address: [email protected]

Contemporary Mathematics Volume 621, 2014 http://dx.doi.org/10.1090/conm/621/12422

The Nahm Pole Boundary Condition

Rafe Mazzeo and Edward Witten

Abstract. The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard elliptic boundary conditions. We also establish a uniqueness theorem for the solution of the relevant equations on a half-space with Nahm pole boundary conditions. These results are expected to have a generalization involving knots, with applications to the Jones polynomial and Khovanov homology.

Contents 1. Introduction 2. Uniqueness Theorem For The Nahm Pole Solution 2.1. Solutions On A Four-Manifold Without Boundary 2.2. A Weitzenbock Formula Adapted To The Nahm Pole 2.3. The Indicial Equation 2.4. The Nahm Pole Boundary Condition 2.5. The Boundary Terms And The Vanishing Theorem 2.6. Behavior At Inﬁnity 2.7. Extension To Five Dimensions 3. The Linearized Operator On A Half-Space 3.1. Overview 3.2. Vanishing Theorem For The Kernel 3.3. Index 3.4. Pseudo Skew-Adjointness 3.5. Extension To Five Dimensions 4. The Nahm Pole Boundary Condition On A Four-Manifold 4.1. Boundary Conditions on the Connection 4.2. The Index 5. Analytic theory 5.1. Uniformly Degenerate Operators 5.2. Elliptic Weights 5.3. Structure of the Generalized Inverse 5.4. Algebraic Boundary Conditions and Ellipticity 5.5. Regularity of Solutions of the KW Equations

2010 Mathematics Subject Classiﬁcation. Primary 81T13, 57R56, 57M25, 53C07.

c 2014 American Mathematical Society 171

172 RAFE MAZZEO AND EDWARD WITTEN

Appendix A. Some Group Theory Acknowledgments References

1. Introduction Nahm’s equations are a system of ordinary diﬀerential equations for three functions φ =(φ1,φ2,φ3) of a real variable y that take values in the Lie algebra g of a compact Lie group G. These functions satisfy dφ (1.1) 1 +[φ ,φ ]=0, dy 2 3 along with cyclic permutations of these equations. More succinctly, we write dφ (1.2) + φ × φ =0 dy or dφ 1 (1.3) i + ε [φ ,φ ]=0, dy 2 ijk j k j,k where εijk is the antisymmetric tensor with ε123 = 1. These ways of writing the equation show that if we view φ as an element of g ⊗ R3, then Nahm’s equation is invariant under the action of SO(3) on R3. In Nahm’s work on magnetic monopole solutions of gauge theory [1], a key role was played by a special singular solution of Nahm’s equations on the open half-line y>0. The solution reads t (1.4) φ(y)= , y where t =(t1, t2, t3)isatripletofelementsofg,obeying

(1.5) [t1, t2]=t3, and cyclic permutations thereof. In other words, the ti obey the commutation relations of the Lie algebra su(2); we can think of them as the images of a standard basis of su(2) under a homomorphism1 : su(2) → g. We will call this solution the Nahm pole solution. The Nahm pole solution has been important in many applications of Nahm’s equations; for example, see [2], which is also relevant as background for the present paper. Nahm’s work on monopoles was embedded in D-brane physics in [3]. The Nahm pole therefore plays a role in D-brane physics, and this was explained conceptually in [4]. Results about D-branes often have implications for gauge theory, and in the case at hand, by translating the D-brane results to gauge theory language, one learns [5] that the Nahm pole should be used to deﬁne a natural boundary condition not just for Nahm’s 1-dimensional equation but for certain gauge theory equations in higher dimensions. The equations in question include second order equations of supersymmetric Yang-Mills theory, and associated ﬁrst-order equations that are

1We are primarily interested in the case that is non-zero and hence is an embedding of Lie algebras, but our considerations also apply for = 0. See Appendix A for some background and examples concerning homomorphisms from su(2) to a simple Lie algebra g.

THE NAHM POLE BOUNDARY CONDITION 173 relevant to the geometric Langlands correspondence [6] and the Jones polynomial and Khovanov homology of knots [7, 8]. Our aim in this paper is to elucidate the Nahm pole boundary condition. Though we will also discuss generalizations (including a ﬁve-dimensional equation [7,9] that is important in the application to Khovanov homology), we will primarily study a certain system of ﬁrst-order equations in four dimensions for a pair A, φ. Here A is a connection on a G-bundle E → M, with M an oriented Riemannian four-manifold with metric g,andφ is a 1-form on M valued in ad(E) (the adjoint bundle associated to E). The equations read

F − φ ∧ φ + dAφ =0

(1.6) dA φ=0, where is the Hodge star and dA =d+[A, ·] is the gauge-covariant extension of the exterior derivative. Alternatively, in local coordinates x1,...,x4, kl Fij − [φi,φj ]+εij Dkφl =0 i (1.7) Diφ =0,

i where Di = D/Dx is the covariant derivative (defined using the connection A and the Riemannian connection on the tangent bundle of M), εijkl is the Levi- Civita antisymmetric tensor, and indices are raised and lowered using the metric g. (Summation over repeated indices is understood.) These equations (or their generalization to t = 1; see eqn. (2.11) below) have sometimes been called the KW equations and we will use this name for lack of another one. For recent work on these equations, see [10–12]. To explain the relation to the Nahm pole, take M to be the half-space x4 ≥ 0 4 1 4 in a copy of R with Euclidean coordinates x ,...,x (oriented with ε1234 =1). R4 1 2 3 4 We denote this half-space as + and write x =(x ,x ,x )andy = x .TheKW equations have a simple exact solution 3 t dxa (1.8) A =0,φ= a=1 a , y where the ta obey the su(2) commutation relations (1.5). This gives an embedding of the basic Nahm pole solution (1.6) in four-dimensional gauge theory, for any choice of the homomorphism : su(2) → g. However, in many applications, the basic case is that defines a principal embedding of su(2) in g, in the sense of Kostant. For G = SU(N), this means that the N-dimensional representation of G is an irreducible representation of (su(2)); in general, the principal embedding is the closest analog of this for any G. If is a principal embedding, we also say that is regular or that φ has a regular Nahm pole. The motivation for this terminology is that if is a principal embedding, then any nonzero complex linear combination of the ta is a regular element of the complex Lie algebra gC = g ⊗R C; for instance, t1 + it2 is a regular nilpotent element. For every , one defines [5] a natural boundary condition on the KW equations that we call the Nahm pole boundary condition, but in this introduction, we consider only the case of a principal embedding. (For more detail and the generalization R4 to any , see section 2.4.) For M = + and a principal embedding, the Nahm pole boundary condition is defined by saying that one only allows solutions that

174 RAFE MAZZEO AND EDWARD WITTEN coincide with the Nahm pole solution (1.8) modulo terms that are less singular for y → 0; the equation then implies that in a suitable gauge these less singular terms actually vanish for y → 0. The Nahm pole boundary condition can be generalized, with some care, to a more general four-manifold with boundary. See section 3.4 of [7] and also section 4 of the present paper. There are two main results of the present paper. The first is that the Nahm pole boundary condition is elliptic. Since the equation and solutions contain singular terms, this is not the standard notion of ellipticity of boundary problems, formulated for example using the Lopatinski-Schapiro conditions, but is the analog of this in the framework of uniformly degenerate operators [13]. In fact, we verify the ellipticity of the linearization of this problem. The data prescribing the Nahm pole boundary condition are inherently discrete, so the linearization measures the fluctuations of the solution relative to this principal term. The boundary conditions for this linear operator simply require solutions to blow up less quickly than the Nahm pole; see section 2.4 for a precise statement. The steps needed to verify that this linearization with such boundary conditions is elliptic involve first computing the indicial roots of the problem, and then showing that the linear operator in the R4 model setting of the upper half-space + is invertible on a certain space of pairs (a, ϕ) satisfying these boundary conditions. The indicial roots measure the formal rates of growth or decay of solutions as y → 0. One of the key consequences of ellipticity is that the actual solutions of this linearized problem, and eventually also the nonlinear equations, possess asymptotic expansions with exponents determined by these initial roots. This is a strong regularity statement which allows us to manipulate solutions to these equations rather freely. The second main result here is a uniqueness theorem for the KW equations with Nahm pole boundary condition. This states that a solution of these equa- R4 tions on M = + which satisfies the Nahm pole condition at y =0andwhichis also asymptotic at a suitable rate to the Nahm pole solution for (x, y) large must actually be the Nahm pole solution. This uniqueness theorem is important in the application to the Jones polynomial [7] and corresponds to the expected result that the Jones polynomial of the empty link is trivial. The proof of the uniqueness theorem involves finding a suitable Weitzenbock formula adapted to the Nahm pole solution, and showing that the fluctuations around the Nahm pole solution decay at a rate sufficient to justify that boundary terms in the Weitzenbock formula vanish. Essentially the same reasoning leads to an analogous uniqueness theorem for the related five-dimensional equation that is expected to give a description of Khovanov homology. In this case, the uniqueness theorem corresponds to the statement that the Khovanov homology of the empty link is of rank 1. This Weitzenbock formula can be linearized, and this version of it is used to establish the second part of the proof that the linearized boundary problem is elliptic. The uniqueness theorem and the ellipticity both hold for arbitrary . The uniqueness theorem means roughly that solutions of the KW equations with Nahm pole boundary condition do not exhibit “bubbling” along the boundary. R4 The basis for this statement is that on +, the KW equations and also their Nahm pole solution are scale-invariant, that is invariant under (x, y) → (λx, λy), λ>0. R4 If there were a non-trivial solution on + with the appropriate behavior at infinity, it could be “scaled down” by taking λ very small and glued into any given solution that obeys the Nahm pole boundary conditions. Ths would give a new approximate

THE NAHM POLE BOUNDARY CONDITION 175 solution that obeys the same boundary conditions and coincides with the given solution except in a very small region near the boundary; the behavior for λ → 0 would be somewhat similar to bubbling of a small Yang-Mills instanton. The Nahm pole boundary condition can be naturally generalized to include knots. In the framework of [7], this is done by modifying the boundary conditions in the equations (1.6) along a knot or link K ⊂ ∂M. The appropriate general procedure for this is only known if is a principal embedding. The model case R4 R ⊂ R3 is that M = + and K is a straight line = ∂M. To every irreducible representation R∨ of the Langlands or GNO dual group G∨ of G, one associates a model solution of eqns. (1.6) that coincides with the Nahm pole solution away from K and has a more complicated singular behavior along K. This more complicated behavior depends on R∨. Solutions for the model case were found in section 3.6 of [7]forG of rank 1 and in [14] for any G. A boundary condition on eqns. (1.6) is then defined by saying that a solution should be asymptotic to this model solution along K, and should have a Nahm pole singularity elsewhere along ∂M. This boundary condition can again be extended naturally, with some care, to the case that M has a product structure W × R+ near its boundary, with an arbitrary embedded knot or link in W = ∂M. (In the case of a link with several connected components, each component can be labeled by a different representation of G∨, corresponding to a different singular model solution.) The Nahm pole boundary condition in the presence of a knot is again subject to a uniqueness the- R4 R ⊂ orem, which says that for M = +, with K = ∂M, and for any representation R∨, a solution that agrees with the model solution near ∂M and has appropriate behavior at infinity must actually coincide with the model solution. This more general type of uniqueness theorem and the closely related ellipticity of the boundary condition in the presence of a knot will be described elsewhere.

2. Uniqueness Theorem For The Nahm Pole Solution In this section we lay out the strategy for proving uniqueness of the Nahm pole solution. The centerpiece of this is the introduction of the Nahm pole boundary condition, and the analysis which shows that this is an elliptic boundary condition, so that solutions have well controlled asymptotics near the boundary. A proper statement of this boundary condition requires a somewhat elaborate calculation of the indicial roots of the problem. These are the formal growth rates of solutions, but without further analysis, there is no guarantee that solutions grow at these precise rates. This further analysis rests on the verification of the ellipticity of the linearized KW operator acting on fields with a certain imposed growth rate at the boundary. The second main result here is a Weitzenbock formula for these equations. There are a number of such formulas, in fact, and the subtlety is to choose one which is well adapted to solutions with Nahm pole singularities. A linearization of this formula plays an important role in understanding ellipticity of the Nahm pole boundary condition. These results and ideas are somewhat intertwined, and we present them in a way that is perhaps not the most logical from a strictly mathematical point of view, but which emphasizes the essential points as quickly as possible. Thus we first explain the Weitzenbock formula, and then proceed to the calculation of indicial roots. We are then in a position to give a precise definition of the Nahm pole boundary conditions. At this point, we use the Weitzenbock formula to prove the

176 RAFE MAZZEO AND EDWARD WITTEN uniqueness theorem. This is only a formal calculation unless we prove that solutions do have these asymptotic rates. This is established in the remainder of the paper.

2.1. Solutions On A Four-Manifold Without Boundary. We first review how to characterize the solutions of the KW equations when formulated on an oriented four-manifold M without boundary. (See section 3.3 of [7].) The details are not needed in the rest of the paper. This material is included only to motivate the way we will search for a uniqueness theorem in the presence of the Nahm pole. As a preliminary, we give a brief proof of ellipticity of the KW equations. By definition, a nonlinear partial differential equation is called elliptic if its linearization is elliptic. For a gauge-invariant equation, this means that the linearization is elliptic if supplemented with a suitable gauge-fixing condition. In the case of the KW equations linearized around a solution A(0),φ(0), a suitable gauge-fixing − condition is dA(0) (A A(0)) = 0, or equivalently D (2.1) (A − A )i =0, Dxi (0) i i where D/Dx = ∂i+[A(0) i, ·] is the covariant derivative defined using the connection A(0). Any other gauge condition that differs from this one by lower order terms also gives an elliptic gauge-fixing condition; a convenient choice turns out to be D (2.2) (A − Ai )+ [φ ,φi − φi ]=0. Dxi (0) (0) i (0) i i It is convenient to regard the linearized KW equations as equations for a pair Φ=(A − A(0),(φ − φ(0))) consisting of a 1-form and 3-form on M both valued in ad(E). With this interpretation, the symbol of the linearized and gauge-fixed KW equations is the same as the symbol of the operator d + d∗ mapping odd-degree differential forms on M valued in ad(E) to even-degree forms valued in ad(E). This is a standard example of an elliptic operator, so the KW equations are elliptic. For future reference, we observe that the d + d∗ operator admits two standard and very simple elliptic boundary conditions, which are much more straightforward than the Nahm pole boundary condition which is our main interest in the present paper.2 These conditions are respectively (2.3) i∗(Φ) = 0 and (2.4) i∗(Φ) = 0, where i : ∂M → M is the inclusion and, for a differential form ω on M, i∗(ω)is the pullback of ω to ∂M. Now set kl 0 i (2.5) Vij = Fij − [φi,φj ]+εij Dkφl, V = Diφ , so that the KW equations are 0 (2.6) Vij = V =0.

2As we explain in section 2.4, these boundary conditions can be viewed as special cases of the Nahm pole boundary condition with =0.

THE NAHM POLE BOUNDARY CONDITION 177

These equations arise in a twisted supersymmetric gauge theory in which the bosonic part of the action is √ 4 1 ij i j i j 1 i j (2.7) I = − d x gTr FijF + Diφj D φ + Rijφ φ + [φi,φj][φ ,φ ] , M 2 2 where sums over repeated indices are understood and Rij is the Ricci tensor. Also Tr is an invariant, nondegenerate, negative-definite quadratic form on the Lie algebra g of G. For example, for G = SU(N), Tr can be the trace in the N-dimensional representation; the precise normalization of the quadratic form will not be important in this paper. The simplest way to find a vanishing theorem for the KW equations is to form a Weitzenbock formula. We take the sum of the squares of the equations and integrate over M. After some integration by parts, and without assuming the boundary of M to vanish, we find (2.8) √ 4 1 ij 0 2 3 abc 1 − d x gTr VijV +(V ) = I + d xε Tr φa[φb,φc] − φaFbc . M 2 ∂M 3 (We write i, j, k =1,...,4 for indices tangent to M and a, b, c =1,...,3 for indices tangent to ∂M.) In evaluating the boundary term, we assume that near its boundary, M is a product ∂M × [0, 1). We also assume that, if n is the normal vector to ∂M,thenn φ = 0 along ∂M, or equivalently, that the pullback of the 3-form φ to ∂M vanishes: (2.9) i∗(φ)=0. This condition is needed to get a useful form for the boundary contribution in the Weitzenbock formula (or alternatively because of supersymmetric considerations explained in [5]), so it will be part of the Nahm pole boundary condition. However, for the rest of this introductory discussion, we assume that ∂M is empty. 0 If the KW equations Vij = V = 0 hold and ∂M vanishes, it follows from the formula above that I = 0. This immediately leads to a vanishing theorem: if the Ricci tensor of M is non-negative, then each term in (2.7) must separately vanish. Thus, the curvature F must vanish; φ must be covariantly constant and its components must commute, [φi,φj ] = 0; and finally φ must be annihilated by the j Ricci tensor, Rijφ =0. Still on an oriented four-manifold without boundary, the KW equations are actually subject to a stronger vanishing theorem than we have just explained, because of a fact that is related to the underlying supersymmetry: modulo a topological invariant, the functional I can be written as a sum of squares in multiple ways. To explain this, we first generalize the KW equations to depend on a real parameter t. Given a two-form Λ on M,wewriteΛ=Λ+ +Λ−,whereΛ+ and Λ− are the selfdual and anti-selfdual projections of Λ. Then we define V+ − − + ij (t)=(Fij [φi,φj ]+t(Diφj Dj φi)) V− − − −1 − − ij (t)=(Fij [φi,φj ] t (Diφj Dj φi)) 0 i (2.10) V = Diφ . The equations V+ V− V0 (2.11) ij (t)= ij (t)= =0

178 RAFE MAZZEO AND EDWARD WITTEN are a one-parameter family3 of elliptic differential equations that reduce to (1.7) for t = 1. All considerations of this paper can be extended to generic4 t, but to keep the formulas simple and because this case has the closest relation to Khovanov homology, we will generally focus on the case t =1. The generalization of eqn. (2.8) to generic t reads √ t−1 t − d4x gTr V+(t)V+ ij(t)+ V−(t)V− ij(t)+(V0)2 t + t−1 ij t + t−1 ij M − −1 t t 4 ijkl (2.12) = I + −1 d xε TrFijFkl. 4(t + t ) M In writing this formula, we have assumed that the boundary of M vanishes. (For a more general formula for ∂M non-empty, see eqn. (2.60) of [7].) Notably, the expression I that appears on the right hand side on (2.12) is the functional defined in eqn. (2.7), independent of t. This immediately leads to very strong results about possible solutions. Suppose, for example, that we find A, φ obeying the original KW equations (1.7) at t = 1. Then setting t = 1 in (2.12), the left hand side vanishes, and − −1 P t t 4 ijkl (2.13) = −1 d xε TrFijFkl 4(t + t ) M certainly also vanishes at t = 1, so therefore I = 0. Now suppose that the integral 4 ijkl M d xε Tr FijFkl – a multiple of which is the first Pontryagin class p1(E)–is nonzero. Then we can choose t =1tomake P < 0, and we get a contradiction: the left hand side of (2.12) is non-negative, and the right hand side is negative. Hence any solution of the original equations at t = 1 is on a bundle E with p1(E)=0. The same is actually true for a solution of the more general eqn. (2.11) at any value of t other than 0 or ∞. To show this, starting with a solution of (2.11) at, say, t = t0, one observes that unless p1(E) = 0, one would reach the same contradiction as before by considering eqn. (2.12) at a value t = t1 at which P is more negative than it is at t = t0. Such a t1 always exists for t0 =0 , ∞ unless p1(E)=P =0. Once we know that P = 0, it follows that the right hand side of (2.12) is independent of t, and hence vanishes for all t if it vanishes for any t.Buttheleft hand side of (2.12) vanishes if and only if the eqns. (2.11) are satisfied. So if A, φ obey the eqns. (2.11) at any t =0 , ∞, they satisfy those equations for all t.This leads to a simple description of all the solutions (away from t =0, ∞). Combine A, φ to a complex connection A = A + iφ. We view A asaconnectiononaGC-bundle EC → M;hereGC is a complex simple Lie group that is the complexification of G,andEC → M is the GC bundle that is obtained by complexifying the G-bundle E → M. We also define the curvature of A as F =dA + A∧A. The condition that eqns. (2.11) are satisfied for all t is that F =0anddA φ =0.Bya well-known result [16], solutions of these equations are in 1-1 correspondence with homomorphisms ψ : π1(M) → GC that satisfy a certain condition of semistability.

3One can naturally think of t as taking values in R ∪∞ = RP1.Fort → 0, one should multiply V−(t)byt,andfort →∞, one should multiply V+(t)byt−1. The proof of ellipticity given above at t =1canbeextendedtoallt. One approach to this uses the formula (2.12) below, supplemented by some special arguments at t =0, ∞. 4The Nahm pole boundary condition is deﬁned for generic t starting with a model solution in which the Nahm pole appears in A as well as φ,witht-dependent coeﬃcients.

THE NAHM POLE BOUNDARY CONDITION 179

(This condition says roughly that if the holonomies of ψ are triangular, then they are actually block-diagonal.) The key to getting these simple results was the fact that (modulo a multiple of p1(E)) the same functional I can be written as a sum of squares in more than one way. This fact is related to the underlying supersymmetry. We will look for something similar to ﬁnd a uniqueness theorem associated to the Nahm pole.

2.2. A Weitzenbock Formula Adapted To The Nahm Pole. Now suppose that M has a non-empty boundary, and consider a solution with a Nahm pole along ∂M. The formulas above do not lead to a useful conclusion directly because the Nahm pole causes the boundary term in (2.8) to diverge. R4 To make this more precise, let us specialize to the case M = +.Asinthe introduction, introduce coordinates x =(x1,x2,x3)andy = x4 on R4, with M the half-space y ≥ 0. The familiar Nahm pole solution is given by

3 t · dxa (2.14) A =0,φ= a . y a=1 (Indices i, j, k =1,...,4 will refer to all four coordinates x1,...,x4, and indices a, b, c =1,...,3 will refer to x1,x2,x3 only.) For this solution, the commutators 2 [φa,φb] and covariant derivatives Dyφ are all of order 1/y , hence not square- integrable near y =0(oras|(x, y)|→∞), and thus the functional I in (2.7) diverges. Accordingly, the boundary terms in the Weitzenbock√ formula, which we repeat here for convenience (omitting the factor of g on the left because M is Euclidean), (2.15) 4 1 ij 0 2 3 abc 1 − d xTr VijV +(V ) = I + d xε Tr φa[φb,φc] − φaFbc , M 2 ∂M 3 are also divergent. A standard way to regularize such divergences is to replace M by Mε = {y>ε, |(x, y)| < 1/ε}, carry out the integrations by parts, and discard the terms which diverge as ε → 0. For the purposes of the present exposition, let us focus only on the portion of the boundary where y = ε; arguments are given in section 2.6 to show that the contributions from the other part of the boundary are negligible. The bulk and boundary terms on the right hand side of (2.15) are both of order 1/ε3 near this lower boundary, and since the left side vanishes (for the Nahm pole solution), these various diverging contributions on the right must cancel. However, when such a cancellation comes into play, it is very diﬃcult to deduce any positivity of the remaining terms on the right, so this formula is not well-suited to deduce a vanishing theorem. It is inevitable that the boundary contribution in (2.15) is at least nonzero for the Nahm pole solution, since otherwise, we could prove that I = 0 for this solution, contradicting the fact that I is a sum of squares of quantities (such as [φa,φb]) not all of which vanish for the Nahm pole solution. Observe that once we know that both I and the boundary term are nonvanishing, scale-invariance implies that they must diverge as ε → 0. To learn something in the presence of the Nahm pole, we need a diﬀerent way to write the left hand side of (2.15) as a sum of squares plus a boundary term, where the boundary term will vanish for any solution that obeys the Nahm pole boundary condition. This will imply a vanishing theorem for such solutions. Of

180 RAFE MAZZEO AND EDWARD WITTEN course, for this to be possible, the objects whose squares appear on the right hand side of the new formula must vanish in the Nahm pole solution. So let us write down a set of quantities that vanish in the Nahm pole solution. 3 a It is convenient to expand φ = a=1 φadx + φydy. The Nahm pole solution is characterized by A = φy = 0 and hence trivially

(2.16) F = Diφy =[φi,φy]=0. Somewhat less trivially, the Nahm pole solution also satisﬁes

(2.17) Wa =0=Daφb, wherewedefine 1 (2.18) W = D φ + ε [φ ,φ ]. a y a 2 abc b c Conversely, these equations characterize the Nahm pole solution, in the following sense. The equations (2.16) and (2.17) imply immediately that in a suitable gauge A =0andφa and φy are functions of y only. Moreover, ∂yφy = 0 (in the gauge with A =0),soifφy is required to vanish at y = 0 (which will be part of the Nahm pole boundary condition) then it vanishes identically. Finally, the condition Wa = 0 means that the functions φa(y) obey the original 1-dimensional Nahm equation dφa/dy +(1/2)εabc[φb,φc]=0. This discussion motivates us to replace the functional I of eqn. (2.7) by a new functional I which is the sum of squares of objects which vanish for the Nahm pole solution: ⎛ − 4 ⎝1 2 2 I = d x Tr Fij + (Daφb) R3×R 2 + i,j a,b 2 2 2 (2.19) + (Diφy) + [φy,φa] + Wa . i a a 2 The only difference between I and I is that we have replaced a(Dyφa) + 1 2 2 [φa,φb] by W .Since 2 a,b ⎛ a a ⎞ 1 1 (2.20) Tr ⎝ (D φ )2 + [φ ,φ ]2⎠ = Tr W 2 − ∂ εabcTr φ [φ ,φ ], y a 2 a b a 3 y a b c a a,b a the sole effect of this is to change the boundary term in (2.8), in fact to cancel the cubic terms in φ that cause the divergence as y → 0 in the Nahm pole solution. Eqn. (2.8) is now replaced by the new identity: (2.21) 4 1 ij 0 2 3 abc − d x Tr VijV +(V ) = I − − d xε Tr φaFbc +Δ, 3 R ×R+ 2 y=0 y=∞ where 4 ∂ j i − i j (2.22) Δ = d x i Tr φj D φ φ Dj φ . 3 R ×R+ ∂x For the time being, we do not replace Δ by a boundary integral. If M is a compact manifold with boundary, then Δ vanishes for a solution that is regular along ∂M and satisfies (2.9), which explains why Δ does not appear in eqn. (2.8). However, R4 for M = +, the use of (2.9) in eliminating the boundary contribution is less simple

THE NAHM POLE BOUNDARY CONDITION 181 in the presence of the Nahm pole, so the term Δ cannot be dropped trivially and will be analyzed later. Now there is a clear strategy for proving a uniqueness theorem for the Nahm pole solution. We must show that any solution that is asymptotic to the Nahm pole solution for y → 0andfor|(x, y)|→∞approaches the Nahm pole solution quickly enough that the boundary terms in eqn. (2.21) (including Δ) vanish. It will then follow that I = 0 for any such solution. Since I is a sum of squares of quantities that vanish only for a solution derived from the 1-dimensional Nahm solution, the given solution will coincide with the Nahm pole solution everywhere. 2.3. The Indicial Equation. 2.3.1. Overview. Our next task is to examine in detail the possible behavior of a solution of the KW equation that is asymptotic to the Nahm pole solution (with some )asy → 0. This analysis is necessary before we can properly define the Nahm pole boundary condition, and will also be essential for showing that the boundary terms in eqn. (2.21) vanish. In making this analysis, we need to supplement the KW equation with a gauge condition. In the Nahm pole boundary condition, we only allow gauge transformations that are trivial5 at y = 0, and we are interested in a gauge condition that fixes this gauge invariance. A gauge transformation that vanishes at y = 0 can be chosen in a unique fashion to make Ay = 0, and for understanding the asymptotic behavior of perturbations of the Nahm pole solution near y = 0, this is a natural boundary condition. However, for other purposes (including proving that the Nahm pole boundary condition is well-posed, but also studying the boundary terms at infinity in the Weitzenbock formula), it is necessary to choose an elliptic gauge condition, i.e. one which aug- ments the KW equations to an elliptic system. Two examples of elliptic gauge conditions were given in equations (2.1) and (2.2). The Nahm pole solution is A(0) =0,φ(0) = t · dx/y, and we consider nearby solutions, which we write as A = a, φ = t · dx/y + ϕ,soa and ϕ are the fluctuations about the Nahm pole. The i gauge conditions (2.1) and (2.2) are ∂ia =0and 1 (2.23) ∂ ai + [t ,ϕ ]=0, i y a a respectively. Both of these gauge conditions are elliptic, but we use (2.23) as it simplifies the later analysis considerably. Technically, we assume that a and ϕ admit asymptotic expansions as y → 0, and consider solutions of the KW equations (with a gauge condition) such that a and ϕ are less singular than 1/y there. Writing the putative expansion around the Nahm pole solution as 3 t dxa (2.24) A = yλa (x)+..., φ= a=1 a + yλϕ (x)+..., 0 y 0 where a0(x), ϕ0(x) depend only on x, and the ellipses refer to terms that are less singular than yλ for y → 0, we ask which exponents λ>−1 are allowed if this expression satisfies the equations formally.

5At the end of section 2.4, we explain that in the case of a nonregular Nahm pole, one can deﬁne a more general boundary condition in which gauge transformations are not required to be trivial at y =0.

182 RAFE MAZZEO AND EDWARD WITTEN

In greater detail, write the KW equations along with a ﬁxed gauge condition as (2.25) KW(A, φ)=0. Expanding this about the Nahm pole solution yields

(2.26) KW(a, φ(0) + ϕ)=L(a, ϕ)+Q(a, ϕ), where L is the linearization of KW at (0, t · dx/y) and the remainder term Q vanishes quadratically in a suitable sense. Assuming that a and ϕ have expansions as above, and that these expansions may be differentiated, multiplied, etc., we see that the most singular terms in L(a, ϕ) are of order yλ−1, while Q(a, ϕ)isnomore singular than y2λ.Sinceλ>−1, this is less singular than yλ−1. Furthermore, only λ λ λ−1 certain terms in L(y a0,y ϕ0) are as singular as y . Specifically, the terms which λ−1 include a ∂y yield a singular factor y , as do the terms containing a commutator with the unperturbed Nahm pole solution. On the other hand, terms containing λ ∂xa are O(y ) and hence may be dropped for these considerations. What remains is a linear algebraic equation involving a0,ϕ0 and the exponent λ. Thisisknown as the indicial equation for the problem. We have been somewhat pedantic about separating the steps of first passing to the linearization and then the indicial operator of this linearization. The same sets of equations can be obtained by directly inserting the putative expansions for a and ϕ into the nonlinear equations and retaining only the leading terms. The reason for our emphasis will become clear later. At this level, the dependence of these coefficients on x is irrelevant, and because of this, the indicial equation respects the symmetry Aa →−Aa, φy →−φy, with φa and Ay left unchanged. This means that the indicial equation uncouples into a system of equations for ϕa,ay and another for aa,ϕy.Theseread

λaa +[ta,ϕy] − εabc[tb,ac]=0

(2.27) λϕy − [ta,aa]=0, and

λϕa − [ta,ay]+εabc[tb,ϕc]=0

(2.28) λay +[ta,ϕa]=0,

i respectively. (Had we used the gauge condition ∂ia = 0 instead of eqn. (2.23), the only difference would be that the [ta,ϕa] term would be missing in the second line of (2.28).) The determination of the indicial roots of the problem, which are defined as the values of λ for which these equations have nontrivial solutions, requires a foray into group theory. 2.3.2. Some Useful Group Theory. One obvious ingredient in these equations is the su(2) subalgebra of g that is generated by the ta. This depends on the choice of homomorphism : su(2) → g; we call its image su(2)t ⊂ g. From the representation theory of su(2), we know that up to isomorphism, su(2) has one irreducible complex module of dimension n for every positive integer n. Itisconvenienttowriten = 2j+1, where j (which is a non-negative half-integer) is called the spin. In particular under the action of su(2)t, the complexification gC = g ⊗R C of g decomposes as a

THE NAHM POLE BOUNDARY CONDITION 183

6 direct sum of irreducible modules rσ,ofdimensionnσ =2jσ +1. For a principal embedding, the jσ are positive integers of which precisely one is equal to 1 (the submodule of g of spin jσ = 1 is precisely su(2)t ⊂ g). For example, G = SU(N) 7 has rank N − 1 and the values of the jσ are 1, 2, 3,...,N − 1. At the opposite extreme, if = 0, so that the ta all vanish, then g is the direct sum of trivial 1-dimensional su(2)t modules, all of spin 0. The indicial equation does not intertwine the su(2)t submodules rσ ⊂ gC since none of the terms in the equation do; hence the equation can be restricted to any one of the rσ. For example, in (2.27), it suﬃces to consider ϕy and all the aa taking values in the same submodule rσ. A general solution of the indicial equation is asumofrσ-valued solutions over all the diﬀerent σ. This is useful because for solutions taking values in a given rσ, the endomorphisms appearing in (2.27) and (2.28) reduce to diagonal operators, so that the equations then completely decouple. The calculation making this explicit occupies the remainder of this subsection. An important property of the algebra su(2) is the existence of a quadratic Casimir operator that commutes with the algebra. In general, given any su(2) algebra with a basis ba, a =1,...,3, obeying the su(2) relations

(2.29) [ba, bb]=εabcbc, we deﬁne the Casimir as 3 − 2 (2.30) Δ = ba. a=1 On a module of spin j, one has (2.31) Δ = j(j +1).

In the case of su(2)t ⊂ g, we usually write the action of the generators on g as w → [ta,w] (rather than w → ta(w)). So we can write the Casimir ΔT as 3 (2.32) ΔT = − [ta, [ta, ·]], a=1 or more abstractly, 3 − 2 (2.33) ΔT = ta, a=1 as in (2.30). It is often best to think of the triple a =(a1,a2,a3) or similarly the triple ∼ 3 ϕ =(ϕ1,ϕ2,ϕ3) as a single element of g ⊗ N where N = R . Another useful su(2) algebra acts on the three-dimensional vector space N. (This is simply inherited from invariance of the original KW equations under rotations of x =(x1,x2,x3).) Explicitly, we deﬁne 3 × 3 matrices sa, a =1,...,3by

(2.34) (sa)bc = −εabc.

6 When the jσ are all integers, for example in the case of a principal su(2) embedding, this statement is true without having to replace g by its complexification gC. The complexification is needed in case some jσ are half-integers. It can happen in general that several of the rσ’s areisomorphicandinthatcasethedecompositionofgC as a direct sum of irreducible su(2)t submodules rσ is not unique. This does not affect the following analysis. 7For this and additional group-theoretic background, see Appendix A.

184 RAFE MAZZEO AND EDWARD WITTEN

These matrices obey the su(2) commutation relations

(2.35) [sa, sb]=εabcsc, and generate an su(2) algebra that we call su(2)s. We define the quadratic Casimir − 3 2 ΔS = a=1 sa and find that ΔS = 2. The value 2 is j(j + 1) with j =1,and reflects the fact that N is an irreducible su(2)s module of spin 1. Finally, we can define a third su(2) algebra that we call su(2)f, generated by fa = ta + sa. The importance of su(2)f is that, since the Nahm pole solution is invariant under su(2)f but not under su(2)t or su(2)s,itisonlysu(2)f that is a symmetry of the indicial equation. To be more exact, to make su(2)f a symmetry of the indicial equation, we let su(2)f act on g⊗N as just described, while in acting on g itself, we declare that sa =0andfa = ta + sa = ta. Then interpreting aa and ϕa as elements of g ⊗ N and ay and ϕy as elements of g,withthesu(2)f action just described, the indicial equation is invariant under su(2)f. − 3 2 To exploit this, it is useful to again define a quadratic Casimir ΔF = a=1 fa. · · Now we have a very useful formula for the su(2)f-invariant operator s t = a sa ta: 1 (2.36) s · t = − (ΔF − ΔT − ΔS) . 2

To make this formula explicit for the module rσ ⊗ N, we need to know how to decompose this module under su(2)f. The answer is given by the representation theory of su(2). Provided that jσ ≥ 1, the tensor product rσ ⊗N decomposes under su(2)f as a direct sum of modules rσ,η of spin fσ,η = jσ + η where η ∈{1, 0, −1}. For jσ < 1, the decomposition is the same except that the range of values of η is smaller; for jσ =1/2, one has only η ∈{1, 0},andforjσ = 0 one has only η =1. In any event, it follows from (2.36) that in acting on rσ,η, the value of s · t is ⎧ ⎨⎪−jσ if η =1 (2.37) s · t = 1ifη =0 ⎩⎪ jσ +1 if η = −1. This result is useful because the object s · t appears in the indicial equation. For example, understanding a =(a1,a2,a3)asanelementofg ⊗ N,sothats · t (a)is also a triple of elements (s · t (a))a, a =1, 2, 3ofg, we have from the deﬁnitions

(2.38) (s · t (a))a = εabc[tb,ac]. The right hand side appears in (2.27), and now we have a convenient way to evaluate it. Similarly, the analogous object εabc[tb,ϕc] appears in (2.28). When we decompose the rσ-valued part of the indicial equation (2.27) under the action of the symmetry group su(2)f, modules with spin jσ ± 1 (in other words η = ±1) appear only in the su(2)f decomposition of aa, while spin jσ (or η =0) appears both in aa and in ϕy. It follows that the terms in (2.27) involving ϕy,and likewise, the terms in (2.28) involving ay, only appear when η =0. 2.3.3. The Indicial Roots. It is now straightforward to determine the indicial roots. First we consider the pair aa,ϕy and we restrict to the rσ-valued part of the equation. For η =0,wecanset ϕy = 0, as explained at the end of section 2.3.2, and so the equation (2.27) reduces to λaa = εabc[tb,ac]. The right hand side was analyzed in eqn. (2.37) and (2.38), and so λ = −jσ for η =1andλ = jσ +1for η = −1. For η = 0, we have to work a little harder. We solve the second equation

THE NAHM POLE BOUNDARY CONDITION 185 in (2.27) with8 λ (2.39) aa = − [ta,ϕy] jσ(jσ +1) and then after also using the Jacobi identity and the su(2) commutation relations, the ﬁrst equation in (2.27) becomes λ2 λ (2.40) − +1+ =0. jσ(jσ +1) jσ(jσ +1)

So for η = 0, the possible values of λ are jσ +1and−jσ.Insumforaa,ϕy,the indicial roots are ⎧ ⎨⎪−jσ if η =1 (2.41) λ = j +1, −j if η =0 ⎩⎪ σ σ jσ +1 if η = −1.

These results need correction for jσ < 1, since some modes are missing. For jσ = 1/2, the λ = jσ +1=3/2 mode with η = −1 should be dropped, and for jσ =0, both modes with λ = jσ + 1 = 1 should be dropped. Inspection of eqns. (2.27) and (2.28) shows that the indicial roots for the pair ϕa,ay are obtained from those for aa,ϕy by just changing the sign of λ. So with no need for additional calculations, the indicial roots for the pair ϕ ,a are as follows: ⎧ a y ⎨⎪jσ if η =1 (2.42) λ = j , −j − 1ifη =0 ⎩⎪ σ σ −jσ − 1ifη = −1.

Again, some modes should be omitted for jσ < 1. It is notable that all of these modes have ay = 0, and therefore make sense in the gauge Ay = 0, except the η = 0 modes in (2.42). Those particular modes are spurious in the sense that they are pure gauge: they are of the form ay = λ+1 ∂yu, ϕa =[ta/y, u] with u(x, y)=y v(x). After finding a solution of the KW equations, one can always make a gauge transformation that sets Ay =0and eliminates these modes. However, this can only be usefully done after finding a global solution: to develop a general theory of solutions of the KW equations, which can predict the existence of solutions, one needs an elliptic gauge condition, and such a gauge condition will always allow pure gauge modes, such as the ones we have identified. Unlike the pure gauge modes with ay = 0, the perturbations we have found with ay = 0 have gauge-invariant content; they cannot be removed by a gauge transformation that is trivial at y = 0, since there are no gauge transformations that are trivial at y = 0 and preserve the condition ay = 0. So the gauge-invariant content of the possible perturbations of the Nahm pole solution near y =0is precisely contained in the modes in (2.41) and those in (2.42) with η =0. One more mode in (2.42) has a simple interpretation. Nahm’s equations have the familiar Nahm pole solution φa = ta/y, but since Nahm’s equations are invariant under shifting y by a constant, they equally well have a solution φa = ta/(y − y0) for any constant y0. Differentiating with respect to y0 and setting y0 = 0, we find

8 This solution is not valid if jσ = 0, because of the factor of jσ in the denominator. For jσ = 0, eqns. (2.27) and (2.28) become trivial, since all commutator terms vanish, and tell us that all modes have λ = 0. This agrees with the result we ﬁnd in eqn. (2.41) below, except that some modes – the ones with λ = jσ + 1 – do not exist for jσ =0.

186 RAFE MAZZEO AND EDWARD WITTEN that the linearization of Nahm’s equations around the Nahm pole solution can be 2 satisfied by ϕa = ta/y . This accounts for the mode in (2.42) with jσ =1,λ = −2, and η = −1. Each value of λ that is indicated in (2.41) or (2.42) represents a space of fluctuations of dimension 2(jσ + η) + 1, transforming as an irreducible su(2)t module. Allowing for these multiplicities, the sum of all indicial roots is 0 for aa,ϕy and likewise for ϕa,ay. This is a check on the calculations: the indicial roots are eigenvalues of matrices that appear in (2.27) and (2.28) and are readily seen to be traceless. 2.4. The Nahm Pole Boundary Condition. We are now in a position to give a precise formulation of the Nahm pole boundary condition. This boundary condition depends on the choice of homomorphism : su(2) → g, which determines the most singular behavior of the solution at ∂M. As explained below, in favorable cases (for example, when is a regular embedding), the Nahm pole boundary condition simply requires that a solution coincides with the Nahm pole solution modulo less singular terms, but in general (when jσ = 0 appears in the decomposition of g) the formulation of the boundary condition involves some further details. Without exploring any of the questions concerning existence of solutions that obey the Nahm pole boundary condition, the expectation is that any such solution has an asymptotic expansion at ∂M, where the leading term is precisely the Nahm pole singularity, and that all fluctuations are well-behaved lower order terms in the expansion with strictly less singular rates of blowup. As already suggested by the discussion in section 2.3.1, the growth rates of these fluctuation terms are governed by the indicial roots, which are themselves determined by the linearization of the KW equations at the Nahm pole solution, as in (2.26). In fact, one might try to construct solutions of KW(A, φ) = 0 by fixing the Nahm pole singularity, then setting the right side of (2.26) to zero to solve for the fluctuation terms. This would involve inverting the linearized operator L, and it is thus important to understand the possible invertibility properties of this operator. In summary, the actual boundary condition we want to discuss is one for the linear operator L which requires solutions to blow up at some rate strictly less than y−1. The indicial root calculation above is what leads us to specify a growth (or decay) rate such that L is as close to invertible as possible when acting on fields with this growth rate. 2.4.1. The Regular Case. First assume that is a principal embedding of su(2) in g, or more generally, that in the decomposition of gC under su(2)t, the minimum value of jσ is 1. (The two conditions are equivalent for G = SU(N) but not in general, as explained in Appendix A.) In this case there is a simplification stemming from the fact that there are no indicial roots with −1 <λ<1. We certainly want to exclude fluctuations around the Nahm pole solution with λ ≤−1, and allow fluctuations with λ>0. Accordingly, for a principal embedding and more generally whenever jσ =1is the smallest value in the decomposition of gC, we can state the Nahm pole boundary condition in either of the following two equivalent ways: (1) A solution satisfies the Nahm pole boundary condition if in a suitable gauge it has an asymptotic expansion as y → 0 with leading term the Nahm pole solution, and all remaining terms less singular than 1/y. (2) A solution satisfies the Nahm pole boundary condition if in a suitable gauge it has an asymptotic expansion as y → 0 with leading term the Nahm pole solution, and with all remaining terms vanishing as y → 0.

THE NAHM POLE BOUNDARY CONDITION 187

Condition (1) is aprioriweaker than condition (2), but they are equivalent because under our assumption on the values of jσ, there are no indicial roots in the range (−1, 0]. The full explanation of ellipticity of the Nahm pole boundary condition is in section 5. However, a preliminary observation that plays an important role is that this boundary condition allows half of the perturbations near y =0:thosewith η =1inϕa,ay,thosewithη = −1inaa,ϕy, and half of the η = 0 perturbations in both sets of fields. (One of the two pure gauge modes that appear at η =0ineqn. (2.42) vanishes at y = 0 and one diverges, so the pure gauge modes did not affect this counting.) 2.4.2. The General Case. As explained in Appendix A, if any value jσ < 1 occurs in the decomposition of gC under su(2), then in fact jσ = 0 occurs in this decomposition. (If jσ = 0 occurs in the decomposition, then jσ =1/2mayormay not occur.) So let us consider the case that jσ = 0 does occur in the decomposition. This simply means that there is a nonzero subspace c of g that commutes with the ta; c is automatically a Lie subalgebra of g, the Lie algebra of a subgroup C ⊂ G. We write ϕc, ac for the c-valued parts of ϕ and a. The formulas (2.41), (2.42) for the indicial roots show that all modes of jσ =0haveλ = 0. Actually, this is clear without a detailed calculation; for jσ = 0, the commutator terms can be dropped in (2.27) and (2.28), which just say that λ = 0. This also shows that for jσ =0, the equations for aa and ϕy decouple and the modes with η = 1 or 0 describe fluctuations of only aa or ϕy, respectively; a similar remark applies, of course, for ϕa and ay. Exactly what one means by the Nahm pole boundary conditions depends on how one treats these λ = 0 modes; this will be discussed momentarily. When both jσ =1/2andjσ = 0 occur in the decomposition of gC,thereis a further subtlety. In this case, to preserve the counting mentioned at the end of section 2.4.1, we have to define the Nahm pole boundary condition to not allow the perturbation with an indicial root λ = −1/2 that appears in (2.41) for jσ =1/2 and η = 0. In other words, we have to require that a solution departs from the Nahm pole solution by a correction that is less singular than 1/y1/2. As for the modes with jσ = 0, there are different physically motivated choices of how to treat them [5], and these correspond to different boundary conditions. The general possibility is explained at the end of this subsection. However, for every , there is a natural boundary condition that we call the strict Nahm pole boundary condition; for G = SU(N), this is the half-BPS boundary condition that can be naturally realized via D-branes. For this boundary condition, we want to c c c leave φa unconstrained at y = 0 but to constrain φy, Aa to vanish at y =0. Thus, we can state the strict Nahm pole boundary condition for general in either of the following two equivalent ways:

(1) A solution satisﬁes the strict Nahm pole boundary condition if in a suitable gauge it has an asymptotic expansion as y → 0 with leading term the Nahm pole solution and with remaining terms less singular than 1/y1/2 (or 1/y if jσ =1/2 does not occur in the decomposition of gC), with the c c further restriction that φy and Aa vanish at y =0. (2) A solution satisﬁes the strict Nahm pole boundary condition if in a suitable gauge it has an asymptotic expansion with leading term the Nahm pole → c solution and with remaining terms vanishing as y 0, except that φa is regular at y = 0 but does not necessarily vanish there.

188 RAFE MAZZEO AND EDWARD WITTEN

To illustrate the strict Nahm pole boundary condition when c =0,letus consider the extreme case = 0, so that there is no Nahm pole and c = g.Inthis case, the strict Nahm pole boundary condition just means that φa is regular at y =0 while φy and Aa are constrained to vanish. This is an elementary elliptic boundary condition on the KW equation, already formulated in eqn. (2.3) above. For general , the strict Nahm pole boundary condition is a sort of hybrid of this case with the opposite case of a principal embedding. Of course, the well-posedness of the strict Nahm pole boundary condition is elementary at = 0, while understanding it for = 0 is the main goal of the present paper. Finally, we describe a generalized Nahm pole boundary condition associated to a more general treatment of the jσ = λ = 0 modes. For this, we pick an arbitrary subalgebra h ⊂ c, corresponding to a subgroup H ⊂ C ⊂ G, and we denote as h⊥ the orthocomplement of h in c (h⊥ is a linear subspace of c but generically not a subalgebra). Then, in addition to allowing only perturbations that are less 1/2 ⊥ singular than 1/y , we declare that the h -valued parts of Aa and φy vanish at y = 0, while the h-valued part of these ﬁelds is unconstrained; and reciprocally, we ⊥ place no constraint on the h -valued part of φa, but require the h-valued part of φa to vanish at y = 0. The strict Nahm pole boundary condition is the case that h = 0. To get the generalized Nahm pole boundary condition for h =0,werelax the requirement that a gauge transformation should be trivial at y =0;instead, we allow gauge transformations that are H-valued at y =0.(ForG = SU(N), any , and some speciﬁc choices of h, this boundary condition can be realized via a combination of D-branes with an NS5-brane [5].) For a simple illustration of this more general boundary condition, take =0andh = c = g. Then the boundary condition is simply that i∗φ = 0. This is actually a second elementary elliptic boundary condition on the KW equations, which was formulated in eqn. (2.4). We will show in section 5.4 that the boundary condition described in this paragraph is well-posed for all and h.

2.5. The Boundary Terms And The Vanishing Theorem. We can now easily show the vanishing of the boundary terms at y = 0 in the Weitzenbock- like formula (2.21). (In doing this, we can ignore the spurious modes that can be removed by a gauge transformation – the η = 0 modes in (2.42). Since the boundary terms are gauge-invariant, they do not receive a contribution from the spurious modes.) 3 For example, let us first look at the boundary contribution d xεabcTr Fabφc. The dominant part of φc is the Nahm pole term tc/y. To avoid a contribution involving this term, we need the part of Fab that is valued in su(2)t (and thus not orthogonal to the coefficient of this Nahm pole) to vanish faster than y for y → 0. 2 The relevant part of Fab is of order y for y → 0, since for example the part of F that is su(2)t-valued and linear in the connection A is a jσ = 1 mode with an indicial root jσ + 1 = 2. The part of F that is quadratic in A vanishes equally fast. (Note that since all of this appears in a boundary integral, we discard the component Fay, which only vanishes like y.) So the contribution to εabcTr Fabφc involving the 2 −1 Nahm pole in φc is of order y · y = y. We can also consider contributions to εabcTr Fabφc that come from fluctuations in φc around the Nahm pole solution (as well as fluctuations in the connection Aa). As long as we do not consider modes with jσ = 0, all fluctuations in either φc or Aa are controlled by strictly positive indicial roots, so the fluctuations vanish at y = 0 and their contribution to εabcTr Fabφc

THE NAHM POLE BOUNDARY CONDITION 189 vanishes. The last case to consider is the case of jσ = 0 fluctuations in both φc and Aa. The general boundary condition formulated in the last paragraph of section 2.4 ensures that the contributions of these fluctuations to εabcTr Fabφc vanishes, because Fab when restricted to y = 0 is valued in a subalgebra h ⊂ c, while φc is valued in an orthogonal subspace h⊥ ⊂ c. Thus the general construction with arbitrary , h ensures the vanishing of εabcTr Fabφc at y =0. The other possible boundary term in the Weitzenbock formula at y =0comes from the expression Δ, defined in eqn. (2.22). Here the integral we have to consider 3 at y =0is d xTr (φaDaφy − φyDaφa). Again, we first consider the Nahm pole contribution with φa = ta/y. A contribution from this term is avoided for reasons similar to what we found in the last paragraph. Indeed, the only contribution to Tr (φaDaφy −φyDaφa) that is linear in fluctuations around the Nahm pole solution comes from the jσ =1partofφy. The corresponding indicial root is 2, so the 2 relevant piece of φy vanishes as y , too quickly to contribute for y → 0evenwhen multiplied by the Nahm pole ta/y. Alternatively, we can consider contributions to Tr (φaDaφy − φyDaφa) that are bilinear in fluctuations around the Nahm pole. Here, for group-theoretic reasons, only modes with jσ > 0 are relevant. Modes with jσ > 0 have indicial roots of at least 3/2 for aa,ϕy or 1/2forϕa,ay,soan expression bilinear in such modes and linear in the Nahm pole part of φa vanishes at least as fast as y3/2y1/2 · y−1 ∼ y. Finally, we can consider contributions to Tr (φaDaφy − φyDaφa) that do not involve the Nahm pole part of φa at all. Since all relevant indicial roots are nonnegative, the only possible contribution come from the jσ = 0 modes for which the indicial root vanishes. These contributions vanish for much the same reason as in the last paragraph: the restriction of Aa and φy to y = 0 is valued in a subalgebra h, while the jσ =0partofφa is valued in an orthogonal subspace h⊥. To complete the proof of the uniqueness theorem for the Nahm pole solution, we need to know that the surface terms in (2.21) also vanish for x and/or y going to ∞.Letr = |x|2 + y2. To ensure vanishing of the surface terms for x, y →∞,we 3 need εabcTr Fabφc and Tr (φiDiφj − φj Diφi) to vanish at infinity faster than 1/r . For example, this is so if the deviation of A and φ from the Nahm pole solution A =0,φ = t · dx/y vanishes at infinity faster than 1/r and the curvature F and the covariant derivatives of φ vanish faster than 1/r2. For a solution of the KW equations with this property, the boundary terms for x, y →∞vanish, so if such a solution obeys the Nahm pole boundary condition, it actually is the Nahm pole solution.

2.6. Behavior At Inﬁnity. To decide if the uniqueness result stated in section 2.5 is strong enough to be useful, we need to know if the rate of approach to the Nahm pole solution that we had to assume for x, y →∞is natural. The goal of the following analysis is to show that it is. We start with the same reasoning with which we began the analysis for y → 0. If a solution of the KW equations does approach the Nahm pole solution for r →∞, then its leading deviation from that solution satisﬁes, at large r, the linear equation obtained by linearizing around the Nahm pole solution. We will show that any solution of that linear equation that vanishes for r →∞vanishes at least as fast as 1/r2 (and its derivative vanishes at least as fast as 1/r3). This holds irrespective of what singularities the solution might have if continued in to small r (where we do not

190 RAFE MAZZEO AND EDWARD WITTEN assume the linearized KW equations to be valid). Vanishing of the perturbations as 1/r2 for r →∞is more than was needed in section 2.5. As before, we will write ai and ϕi for the perturbations of Ai and φi around the Nahm pole solution. To explain the idea of the analysis, we first describe the simplest case, which is the behavior of ϕy. The linearized KW equations imply a linear equation for ϕy independent of all other modes. This perhaps surprising fact can be proved by taking a certain linear combination of derivatives of the KW equations. However, a quicker route is to go back to eqn. (2.8). At a solution of the 0 KW equations Vij = V = 0, the left hand side of (2.8) is certainly stationary under variations of A and φ, so the right hand side is also. If we consider variations of A and φ whose support is in the interior of M, the boundary term can be ignored and therefore the functional I is stationary at a solution of the KW equations. In other words, the KW equations imply the Euler-Lagrange equations δI/δA = δI/δφ =0. (This statement is part of the relation of the KW equations to a four-dimensional supersymmetric gauge theory.) It is straightforward to work out the Euler-Lagrange equation for ϕy from the explicit formula for the action I in eqn. (2.7). A convenient way to do this is to expand the action in powers of the fluctuations a and ϕ, around the Nahm pole R4 solution on +. There are no linear terms – since the Nahm pole solution is a solution – but there are quadratic terms. The part of I that is second order in the fluctuations and has a nontrivial dependence on ϕy is 4 2 2 (2.43) − d xTr (∂iϕy) + [φa,ϕy] . i a

Importantly, there are no terms that are bilinear in ϕy and the other fluctuations a and ϕa; that is why at the linearized level one finds an equation that involves only ϕy and not the other fields. This equation is just the Euler-Lagrange equation that arises in varying the functional (2.43) with respect to ϕy.Wewrite 4 − 2 (2.44) Δ = ∂xi i=1 R4 for the Laplacian on + (with the gauge connection A taken to vanish, as in the Nahm pole solution), and of course we set φa = ta/y. The equation for ϕy is 1 (2.45) Δ − [t , [t , ·]] ϕ =0. y2 a a y a In four dimensions, 2 ∂ 3 ∂ Δ 3 (2.46) Δ = − − + S , ∂r2 r ∂r r2 where ΔS3 is the Laplacian on the three-sphere r = 1. Since we are working on the R4 R4 half-space + rather than all of ,weconsiderΔS3 as an operator defined on a hemisphere in S3. It is convenient to introduce the polar angle ψ where y/r =cosψ, so that ψ = 0 along the positive y-axis where x = 0, and the hemisphere is defined by ψ ≤ π/2. One has

1 2 1 (2.47) Δ 3 = − ∂ψ sin ψ∂ψ + Δ 2 , S sin2 ψ sin2 ψ S

THE NAHM POLE BOUNDARY CONDITION 191 where ΔS2 is the Laplacian on a unit two-sphere. The boundary condition at ψ = π/2 is determined by the four-dimensional boundary condition at y =0and hence can be read oﬀ from section 2.4. In our context, this generally means that ϕy h vanishes at ψ = π/2, except possibly if jσ =0,inwhichcaseϕy is not required to vanish at y =0orψ = π/2. (Rather, the KW equation Dyφy + Daφa =0,where h h φa =0aty =0,andA is h-valued at y = 0, implies that Dyφy =0aty =0,so h for ϕy, eqn. (2.45) should be supplemented with Neumann boundary conditions at y = 0.) Actually, as we will see in a moment, for jσ > 0, there is a potential that enforces the vanishing of ϕy at ψ = π/2. − · To make (2.45) more explicit, we replace a[ta, [ta, ]] with jσ(jσ +1),where jσ is deﬁned as in section 2.3.2. The equation for ϕy becomes ∂2 3 ∂ W (2.48) − − + ϕ =0, ∂r2 r ∂r r2 y where

jσ(jσ +1) 1 2 1 jσ(jσ +1) (2.49) W =Δ 3 + = − ∂ψ sin ψ∂ψ + Δ 2 + . S cos2 ψ sin2 ψ sin2 ψ S cos2 ψ W is a self-adjoint operator on the hemisphere with a discrete and non-negative spectrum (strictly positive except for jσ =0andϕy ∈ h). Any solution of (2.48) is s a linear combination of solutions of the form ϕy = r f,wheref is an eigenfunction of W ,obeyingWf = γf for some γ ≥ 0, and s(s +2)− γ =0or (2.50) s = −1 ± 1+γ. Actually, the spectrum of the operator W can be found in closed form. In particular, for jσ > 0, the ground state (which is the unique everywhere positive eigenfunction) jσ +1 is f =cos ψ, with eigenvalue γ =(jσ +1)(jσ +3). So from (2.50), if s is negative –asitmustbeifϕy is to vanish for r →∞–thens = −3 − jσ. The perturbations −3−jσ thus decay for large r as r . Thus for example if is principal so that jσ ≥ 1 for all modes, then in a solution that is asymptotic to the Nahm pole solution, ϕy vanishes for r →∞as 1/r4. This analysis of the fluctuations is valid at large r even for y → 0, so we can jσ +1 jσ +3 compare to our study of the indicial roots. The wavefunction ϕy =cos ψ/r vanishes for y → 0asyjσ +1, in agreement with (2.41), where the positive indicial roots are λ = jσ +1. As usual, for jσ = 0, there is more to say as there are actually two types of ⊥ mode. For ϕy ∈ h , ϕy obeys Dirichlet boundary conditions at y =0,andallthe previous formulas are valid, including the asymptotic behavior ϕ ∼ 1/r3+jσ =1/r3. But for ϕy ∈ h, we want Neumann boundary conditions at ψ = π/2. The lowest 2 eigenvalue of W is γ = 0, with eigenfunction 1, leading to s = −2andϕy ∼ 1/r . We have analyzed here a second order equation, not all of whose solutions are necessarily associated to solutions of the linearized KW equation, which is first order. In practice, we need not explore this issue here in detail since all modes we have found decay more rapidly at infinity than was needed for the vanishing argument of action 2.5. The same remarks apply in what follows. Fluctuations in the other fields can be analyzed along the same lines. For this, it is convenient to write a general formula for the expansion of the action I around the Nahm pole solution. We write I2 for the part of I that is quadratic in the

192 RAFE MAZZEO AND EDWARD WITTEN

ﬂuctuations a, ϕ, and compute that

(2.51) I2 = I2,0 + I2,1 + I2,2 with ⎛ ⎞ 2 1 I = − d4x Tr ⎝ (∂ a )2 +(∂ ϕ ) + [t ,a ]2 +[t ,ϕ ]2 ⎠ 2,0 i j i j y2 a i a i i,j a,i 2 4 (2.52) I = − d4x Tr ε [t ,ϕ ]ϕ ]+ a [t ,ϕ ] 2,1 y2 abc a b c y2 y a a 2 1 I = d4x Tr ∂ ai + [t ,ϕ ] = d4x Tr S2, 2,2 i y a a i where the gauge condition (2.23) was S = 0. This illuminates one of the advantages of that gauge condition: because I2,2 is homogeneous and quadratic in S,itis automatically stationary when S = 0 and hence does not contribute to the Euler- Lagrange equations. This significantly simplifies the analysis. Since the spatial part aa of the connection does not appear in I2,1,itobeys an Euler-Lagrange equation that comes entirely from I2,0. This equation coincides with the equation for fluctuations of ϕy, which we have already analyzed. This is in keeping with the fact that ϕy and aa have the same indicial roots, so their fluctuations must have the same behavior for ψ → π/2. The equation for fluctuations of ϕa,ay does receive a contribution from I2,1. This contribution, which only arises for jσ > 0(sinceI2,1 =0forjσ = 0), slightly modifies the behavior of the perturbations for r →∞. It can be analyzed using methods similar to those that we used in computing the indicial roots. For the same reason as in that analysis, the term in I2,1 that involves ay contributes only for η =0.Forη = ±1, we only have to consider the term involving εabcTr [ta,ϕb]ϕc which we evaluate using (2.38) and (2.37), to find that − − 4 1 · 2jσ η =1 (2.53) I2,1 = d x 2 2 Tr φaφa r cos ψ 2(jσ +1) η = −1.

The equation (2.48) is modiﬁed only by a shift in W , − → 1 2jσ η =1 (2.54) W W + 2 cos ψ 2(jσ +1) η = −1.

This is equivalent to replacing jσ by jσ − η in the definition (2.49) of W ,sothat 3+jσ −η the perturbations in φa with η = ±1 vanish at infinity as 1/r . The modes that decay most slowly are those with η = 1; they decay for r →∞as 1/r2+jσ and for ψ → π/2ascosjσ ψ. The last statement is in accord with the value found in (2.42) for the indicial root at η =1. For η = 0, we can express ϕa in terms of a new field u by ϕa =[ta,u]/ jσ(jσ +1) (recall that we can assume that jσ > 0, since otherwise the perturbation vanishes). In terms of these variables, the Euler-Lagrange equation turns out to be ∂2 3 ∂ W 2 u (2.55) − − + + M =0, ∂r2 r ∂r r2 r2 cos2 ψ ay

THE NAHM POLE BOUNDARY CONDITION 193 where the contribution of I2,1 is the term proportional to 1 j (j +1) (2.56) M = σ σ . jσ(jσ +1) 0

The eigenvalues of M are jσ +1 and −jσ. Upon substituting one of these eigenvalues for M in (2.55), one gets precisely the same shifts of W as described in eqn. (2.54). So the two modes with η = 0 obey precisely the same equations as the two modes with η = ±1. In particular, the mode that decays most slowly for r →∞again decays as 1/r2+jσ , while vanishing as cosjσ ψ for ψ → π/2. The last statement corresponds to the indicial root λ = jσ found at η = 0 in (2.42). Though we motivated this analysis by asking if the conditions needed to get a uniqueness theorem for the Nahm pole solution are reasonable, the results are applicable more widely. For example, we may be interested in a solution of the KW equations in which the Nahm pole boundary condition is modified by inclusion of knots at y = 0. As long as the knots are compact, one can look for solutions of the KW equation that coincide with the Nahm pole solution for r →∞.Theirrateof approach to that solution will be as we have just described. We observed in section 2.1 that the symbol σ of the linearized KW equations is the symbol of the operator D =d+d∗ mapping odd degree forms to even degree forms. Let D† be the adjoint of D and let σ† be the adjoint of σ.SinceD†D is the Laplacian on odd degree differential forms, it follows that σ†σ is the symbol of the Laplacian. This is reflected in the above formulas: the fluctuations are annihilated by a second order differential operator that is equal to the Laplacian plus corrections of lower order.

2.7. Extension To Five Dimensions. The four-dimensional KW equations are closely related to a certain system of elliptic differential equations in five dimensions [9], [7, 8] and this relationship is crucial in the application to Khovanov homology. To explain the relationship, we first specialize the KW equations to a four- manifold of the form M = W ×I, with W an oriented three-manifold, and I an oriented one-manifold, possibly with boundary. We endow M with a product a b 2 a metric gab(x)dx dx +dy ,wherethex , a =1,...,3, parametrize W and y I a parametrizes , and as usual we expand φ = a φadx +φydy. The KW equations have the property that φy enters only in commutators – either covariant derivatives Daφy =[Da,φy], or commutators [φa,φy] with other components of φ. This enables us to do the following. We replace the four-manifold M by the five-manifold Y = R × M = R × W ×I,whereR is parametrized by a new coordinate x0.Then wherever a commutator with φy appears in the KW equations, we simply replace 0 it by a commutator with D0 = D/Dx . So we replace Daφy with [Da,D0]=Fa0, and [φa,φy]with[φa,D0]=−D0φa. In this way, we get some partial differential equations in five dimensions. Most of what we have said in this paper about the KW equations carries over to them. For example, the five-dimensional equations have a Weitzenbock formula quite analogous to (2.12). One simply has to replace [φy, ·]with[D0, ·] in the formula for the action functional I of equation (2.7). This Weitzenbock formula can be used to prove that the five-dimensional equations are elliptic for t =0 , ∞. (See eqn. (5.44) of [7] for the Weitzenbock formula at t = 1.) The proof of ellipticity in the interior amounts to showing that – similarly to what was explained for the KW equations

194 RAFE MAZZEO AND EDWARD WITTEN at the end of section 2.6 – if σ is the symbol of the five-dimensional equations, then σ†σ is the symbol of the Laplacian (times the identity matrix) and in particular is invertible; hence σ is invertible and the equations are elliptic. Though the equations are elliptic for generic t, something nice happens precisely for t =1(ort = −1, which is equivalent to t = 1 modulo φ →−φ). Just in this case, the equations acquire four-dimensional symmetry. From the way we described these equations, they are formulated on a five-manifold of the particular form Y = R × W ×I. However, at t = 1, there is more symmetry, a fact that is essential in the application to Khovanov homology. One can replace R × W by a general oriented Riemannian four-manifold X with no additional structure, 9 and formulate the equations on Y = X ×I. WetakeonY a product metric 3 μ ν 2 μ μ,ν=0 gμν dx dx +dy ,wherex , μ =0,...,3 are local coordinates on X and I is parametrized by y. The equations on Y can be described as follows. Let Ω2,+ → X be the bundle of self-dual two-forms, and using the natural projection X ×I →X, pull this bundle back to a bundle over Y = X ×I that we also denote as Ω2,+. The fields appearing in the five-dimensional equations are a connection → 2,+ 2,+ ⊗ A on a G-bundle E Y , and a section B of Ω (ad(E)) = Ω ad(E). (For R × a X = W , the relation between B and the object φ = a φadx that appears in the KW equations is B0a = φa, Bab = εabcφc.) The five-dimensional equations can be written 1 1 F + − B × B − D B =0 (2.57) 4 2 y ν Fyμ + D Bνμ =0. Here F + is the orthogonal projection of the curvature F onto the part valued in Ω2,+(ad(E)), and B × B is defined as follows. Since Ω2,+ is a rank 3 real ∼ bundle with structure group SO(3), there is a natural isomorphism10 ∧2Ω2,+ = Ω2,+. By composing this with the Lie bracket ∧2g → g, we get a natural map Sym2Ω2,+(ad(E)) → Ω2,+(ad(E)). The image of B ⊗ B under this map is what we call B × B. An explicit formula, viewing B as a self-dual two-form valued in ad(E), is 3 στ (2.58) (B × B)μν = g [Bμσ,Bντ]. σ,τ =0

In view of all this, any solution of the KW equations on W × R+ can be viewed as a “time”-independent solution of the ﬁve-dimensional equations on R × W × 0 R+ (where we identify x as time, as is natural in the application to Khovanov homology), with φy reinterpreted as A0. In particular, the Nahm pole solution on 3 4 R ×R+ can be regarded as a solution of the ﬁve-dimensional equations on R ×R+. The solution is simply

(2.59) A =0,B0a = ta/y, Bab = εabctc/y. Given the classical Nahm pole solution, we then ask if we can define a boundary condition on the five-dimensional equations by allowing only solutions that are asymptotic for y → 0 to the Nahm pole. The first step is to compute the indicial

9Still more generally, one can formulate these equations – and they remain elliptic – on an arbitrary ﬁve-manifold Y with an everywhere non-zero vector ﬁeld [9]. 10For a vector space V , we denote the symmetric and antisymmetric parts of V ⊗V as Sym2V and ∧2V , respectively.

THE NAHM POLE BOUNDARY CONDITION 195 roots. These are precisely the same for the five-dimensional equation as for the four-dimensional KW equations, since the indicial roots are defined in terms of solutions that depend only on y and so in particular are time-independent. The only real difference between the computation of indicial roots in five dimensions and the four-dimensional computation described in section 2.3.3 is that the five- dimensional interpretation, with φy reinterpreted as A0, gives a better explanation of the symmetry of the equations between aa and ϕy. (This symmetry is visible in the formula (2.52) for I2, and accounts for the fact that in eqn. (2.41), the indicial roots for different values of η are pairwise equal.) Given the indicial roots, one can imitate the discussion in section 2.4 to define precisely the Nahm pole boundary condition, and as in section 2.5 it follows that the boundary terms at y =0in R4 × R the Weitzenbock formula vanish. The Nahm pole solution on + is therefore unique if one requires sufficiently fast convergence as r = x2 + y2 becomes large. The analysis in section 2.6 can be repeated to show that the expected rate of convergence at infinity of a solution that does converge to the Nahm pole solution is fast enough to make the uniqueness result concerning the Nahm pole solution relevant. Just a few modifications are needed. In equation (2.45), Δ should now be the five-dimensional Laplacian on a half-space, 2 ∂ 4 ∂ Δ 4 (2.60) Δ = − − + S . ∂r2 r ∂r r2 We expand the fluctuation in the connection around the Nahm pole as 3 s (2.61) a = asdx + aydy. s=0 Theequationobeyedbya is now s ∂2 4 ∂ W (2.62) − − + a =0,s=0,...,3 ∂r2 r ∂r r2 s where

jσ(jσ +1) 1 3 1 jσ(jσ +1) (2.63) W =Δ 4 + = − ∂ψ sin ψ∂ψ + Δ 3 + . S cos2 ψ sin3 ψ sin2 ψ S cos2 ψ

The lowest eigenvalue of W is now (jσ +1)(jσ + 4), again with eigenfunction − − cosjσ +1 ψ, and now the fluctuations decay as r 4 jσ for r →∞, with precisely one extra power of 1/r compared to the four-dimensional case. The corresponding formulas for B and ay are similar to the analysis of ϕa and ay in the four-dimensional case, and again, the fluctuations in five dimensions decay with one extra power of r compared to what we found in four dimensions.

3. The Linearized Operator On A Half-Space 3.1. Overview. A nonlinear partial differential equation is said to be elliptic if its linearization is elliptic. An important property of a linear elliptic differential operator on a closed manifold is that its kernel and cokernel are finite-dimensional. If a linear elliptic differential equation is considered on a manifold M with a nonempty boundary, then it is necessary to impose some sort of boundary condition to make the problem similarly well posed. A boundary condition such that the accompanying problem has a finite dimensional kernel and cokernel, and so that in addition solutions enjoy optimal regularity properties, is called an elliptic boundary

196 RAFE MAZZEO AND EDWARD WITTEN condition. As before, for a nonlinear elliptic differential equation on a manifold M with boundary, a choice of boundary condition is called elliptic if it (or its linearization if the boundary condition is also nonlinear) is an elliptic boundary condition for the linearized operator. For standard nondegenerate elliptic operators, the theory of elliptic boundary conditions is now classical, and the criterion for ellipticity of a boundary condition (which involves both the interior and boundary operators) is called the Lopatinski-Schapiro condition. The linearized operator in our setting cannot be treated with this classical theory since its coefficients of order 0 are singular at y = 0. It is, however, a uniformly degenerate elliptic operator, as introduced in [13]. There is a suitable notion of ellipticity for boundary conditions in this setting as well, to which we shall be appealing here. As a way to motivate the definition of ellipticity of a given boundary condition, Rn { n ≥ } consider the model case where M is the half-space + = x 0 . If a linear differential operator on this half-space and a boundary condition on the boundary Rn−1 are both invariant under rotations and translations in the boundary variables, then the kernel and cokernel of this boundary problem (amongst tempered solutions Rn on +) are finite-dimensional if and only if they are actually trivial. Conversely, a boundary condition with this property, and so that the accompanying linear operator as a map between appropriate Sobolev spaces has closed range, is elliptic. We have stated this formulation for operators and boundary conditions with substantial symmetry; more generally, if L is a general linear elliptic operator with variable coefficients and B a possibly non-constant operator giving the boundary conditions, then we may apply this condition to the constant coefficient problem Rn L ∈ Rn−1 on + obtained by freezing the coefficients of and B at q . These remarks are relevant to the Nahm pole boundary condition on the KW equations. Our task now is to show that the linearization L of the KW operator around a solution with Nahm pole boundary data on a four-manifold M with boundary satisfies this ellipticity condition. By the remarks above, this is actually equivalent to proving the corresponding property for the linearization of the KW R4 operator around the actual model Nahm pole solution on the half-space +.We shall explain in section 5 how this fits into the analytic theory which justifies the main consequences of this paper, namely the regularity at y = 0 of more general solutions satisfying Nahm pole boundary conditions and the uniqueness theorem. Thus the aim of this section is to study the linearized KW operator on the half-space, and to show that it is an isomorphism on the space of L2 fields which satisfy the Nahm pole boundary conditions. As suggested above, this consists of two rather separate parts: one involves showing that the kernel and cokernel of L vanish, while the other requires showing that L has closed range as a map between the appropriate function spaces. We undertake the first of these in the present section. Section 3.2 contains a proof that the kernel of L vanishes. This turns out to be a rather direct consequence of the formulas that were used in section 2 to establish the uniqueness theorem. As for vanishing of the cokernel, a standard strategy, once the kernel is known to vanish, is to show, after reducing to an ODE via a Fourier transform, that the index of L (defined as the difference in dimension between the kernel and cokernel) vanishes. We do this in two essentially separate ways. The first involves some fairly elementary algebraic considerations; see section 3.3. The second is more direct (and much more useful in generalizations). It turns out, see section 3.4, that the adjoint of L is conjugate to −L, a property that we

THE NAHM POLE BOUNDARY CONDITION 197 call pseudo skew-adjointness. This immediately gives an isomorphism between the kernel and cokernel of L, so the cokernel vanishes if the kernel does. Finally, in section 3.5, we show that these arguments carry over more or less immediately to the ﬁve-dimensional extension of the KW equations that is relevant to Khovanov homology. The remaining task, to show that the range of L is closed, turns out to follow using general machinery that will be explained in section 5.

3.2. Vanishing Theorem For The Kernel. The uniqueness theorem for the Nahm pole solution on a half-space was deduced from an identity (2.21) which reads schematically − 4 V2 − 4 W2 (3.1) d x Tr λ = d x Tr σ, λ σ where we omit surface terms since we have shown them to vanish in a solution of the KW equations that is asymptotic to the Nahm pole solution for y → 0andat 0 infinity. The Vλ are the g-valued quantities Vij and V that appear on the left hand side of (2.21). The Wσ are the objects Fij, Daφb, Diφy,[φy,φa], and Wa whose squares appear on the right hand side of the definition (2.19). If the KW equations Vλ = 0 are obeyed, then the identity (3.1) shows that the Wσ vanish, which implies that the solution is constructed from a solution of Nahm’s equations. Now let us see what this formula tells us about the linearization of the KW equations about the Nahm pole solution. Schematically, let us combine the fields A, φ to an object Φ (one can think of Φ = A + φ as an odd-degree differential R4 form on + valued in ad(E)). We write Φ0 for the Nahm pole solution, and we consider a family of fields Φs =Φ0 + sΦ1,wheres is a parameter and Φ1 is a perturbation. Expanding the identity (3.1) in powers of s, the linear term vanishes because Vλ = Wσ =0ats = 0. Taking the second derivative with respect to s, V 2V V terms such as λ∂s λ vanish at s =0since λ =0ats =0.Soweget ∂V 2 ∂W 2 (3.2) − d4x Tr λ = − d4x Tr σ . ∂s ∂s λ σ

Hence the equations ∂Vλ/∂s = 0 are satisfied if and only if the equations ∂Wσ/∂s = 0 are satisfied. The equations ∂Vλ/∂s = 0 are the linearization of the KW equations around the Nahm pole solution. In other words, these equations are LΦ1 =0,whereL is the linearization of the KW equations and Φ1 is the perturbation around the Nahm pole solution. The equations ∂Wσ/∂s = 0 imply that Φ1 actually vanishes if it vanishes at infinity. For example, the equation ∂sFij = 0 implies that the fluctuation in the connection A can be gauged away, and upon doing so, the equations ∂s(Daφb)= ∂s(Daφy) = 0 imply that the perturbation in φ is independent of x and so vanishes if it vanishes at infinity. The other conditions ∂sWσ = 0 imply that Φ1 actually comes from a solution of the linearization of Nahm’s equation. Of course, such a perturbation (or any perturbation that is independent of x) is not square-integrable in four dimensions. Hence if the linearization L of Nahm’s equations is understood as an operator acting on a Hilbert space of square-integrable wavefunctions, its kernel vanishes.

198 RAFE MAZZEO AND EDWARD WITTEN

3.3. Index. Here we will sketch a standard strategy to prove that the cokernel of L vanishes once one knows that the kernel vanishes. We only provide a sketch since in the particular case of the KW equations, there is a more powerful and direct method that we explain in section 3.4. R3 R4 First of all, using the translation symmetries of = ∂( +), we can look for ik·x a momentum eigenstate, that is a perturbation of the form Φ1(x, y)=e F (y) where F depends on y only and k is a real “momentum” vector. F (y)isafunction on R+ with values in a finite-dimensional complex vector space Y .Letd =dimY ; for the KW equations, d =8dimg. To show that the cokernel of L is trivial for square-integrable wavefunctions, it suffices to show that it vanishes for momentum eigenstates with k =0. On momentum eigenstates, the linearized KW equation LΦ1 = 0 reduces to an equation L1(k)F (y)=0,with d (3.3) L (k)= + B(y,k), 1 dy where B(y,k) is a self-adjoint matrix-valued function of y and k.Infact, B (3.4) B(y,k)= 0 + B (k), y 1 where B0 is a constant matrix (independent of y and k)andB1 is independent of y and homogeneous and linear in k. Actually, B0 is the matrix whose eigenvalues are the indicial roots, which we computed in section 2.3.3. B1(k) is the symbol of the operator d + d∗ on ad(E)-valued dfferential forms on R3 of all possible degrees; ∗ in other words, B1(k) is the momentum space version of the d + d operator. Let Y be the space of all solutions of the linear equation L1F (y)=0onR+, with no condition on the behavior near y =0or∞.WecanidentifyY with Y by, for example, mapping a solution F (y) ∈ Y to its value F (1) ∈ Y ,soY has dimension d.LetY0 be the subspace of Y consisting of solutions that obey the Nahm pole boundary condition at y = 0 (and in particular are square-integrable near y =0),andletY∞ be the subspace consisting of solutions that are square- ∗ integrable at infinity. Also, let d0 =dimY0, d1 =dimY1. Finally, let Y be the space of solutions of (3.3) that obey the Nahm pole boundary condition at y =0, and in addition are square-integrable, and set d∗ =dimY∗. ∗ Y is simply the intersection Y0 ∩ Y∞ of the spaces of solutions that are well- behavedat0andat∞.Soifd0 + d∞ − d ≥ 0 and the subspaces Y0, Y∞ ⊂ Y are ∗ ∗ generic, then the dimension of Y is d = d0 + d∞ − d. Even if these conditions do not hold, the index of the operator L1 is d0 + d∞ − d. In the case of the KW equations with Nahm pole boundary conditions, d0 = d∞ = d/2 and therefore the index of L1 is 0. Hence if the kernel of L1 vanishes – as shown in section 3.2 – then the cokernel also vanishes. The fact that d0 = d/2 was explained in section 2.4, basically as a consequence of the fact that the indicial roots of ϕa,ay are the negatives of those of aa,ϕy (with some care when some roots vanish). Since B(y, k) can be approximated by B1(k)fory →∞, solutions of L1F (y) = 0 that are square-integrable for y →∞correspond to positive eigenvalues of B1(k). So to show that d∞ = d/2, we must show that B1(k) has equal numbers of positive and negative eigenvalues. This follows from the fact that B1(k)isthe symbol of the d + d∗ operator; its square is |k|2, so its eigenvalues are ±|k|,and

THE NAHM POLE BOUNDARY CONDITION 199 as it is traceless, precisely half of the eigenvalues are positive. Alternatively, by rotation symmetry, the number of positive eigenvalues of B1(k) is invariant under k →−k; but since B1(−k)=−B1(k), it has equally many positive and negative eigenvalues. We have omitted various details here, since we turn next to a more direct (and much more widely applicable) proof of the vanishing of the cokernel of the linearized KW operator L. There are two specific reasons to have included the above material. First, this explains why the counting of positive indicial roots in section 2.4 is important. Second, even without a direct calculation of the index of the operator L1, the fact that the problem can be formulated in terms of the vanishing of the cokernel of this 1-dimensional operator will make it easy to go from 4 to 5 dimensions in section 3.5. 3.4. Pseudo Skew-Adjointness. Inspection of the equations (2.27) and (2.28) that determine the indicial roots shows that these roots (in the gauge (2.23), which we assume in what follows) are odd under exchange of aa,ϕy with ϕa,ay. We make this exchange via the linear transformation a ϕ N a = a ϕ a y y ϕ a (3.5) N a = − a . ay ϕy The minus sign in the second line does not affect what we have said so far, but will be important shortly. Taking this minus sign into account, we have (3.6) N 2 = −1,N† = −N, † where N is the transpose of N in the usual basis given by ai and ϕi,ormore invariantly the adjoint of N with respect to the quadratic form 4 − 2 2 (3.7) Tr ai + ϕi . i=1 Perturbations of the Nahm pole solution that depend only on y are governed by an equation that we schematically write L1(0)Φ = 0, where Φ combines all the fields and d B (3.8) L (0) = + 0 1 dy y is obtained from L1(k) (eqn. (3.3)) by setting k =0.ThematrixB0 can be read off from (2.27) and (2.28) (which are derived from the equation L1(0)Φ = 0 by replacing d/dy with λ/y), and by inspection we see that B0 is a real symmetric matrix in the usual basis, or in other words is real and self-adjoint for the quadratic form (3.7). On the other hand, d/dy is skew-adjoint. The adjoint of L1(0)isthus d B (3.9) L (0)† = − + 0 . 1 dy y

Since the indicial roots are the eigenvalues of −B0, the statement that the matrix N reverses the sign of the indicial roots is equivalent to NB0 = −B0N.Wecan combine this with (3.9) as the statement that † −1 (3.10) L1(0) = −NL1(0)N .

200 RAFE MAZZEO AND EDWARD WITTEN

So far we have just reformulated the symmetry that changes the sign of the indicial roots. It turns out, however, that (3.10) holds without change for k =0: † −1 (3.11) L1(k) = −NL1(k)N .

Once one knows that the kernel of L1(k) is trivial, it immediately follows from (3.11) that the cokernel of this operator is also trivial. Indeed, the cokernel of L1(k)isthe † † kernel of L1(k) , but (3.11) implies that the kernel of L1(k) is obtained by acting with N on the kernel of L1(k). Since eqn. (3.11) holds for any k, this statement can be formulated without introducing momentum eigenstates. If L is the linearization of the KW equations around the Nahm pole solution, then (3.12) L† = −NLN −1, a property that we describe by saying that L is pseudo skew-adjoint. We can write 11 L = ∂y +B,whereB is a self-adjoint first-order differential operator that contains † derivatives only along W .ThenL = −∂y +B and (3.12) amounts to the statement that (3.13) NBN−1 = −B. Another equivalent statement is that L = NL is actually self-adjoint. These assertions hold in much greater generality than perturbing around the 3 Nahm pole solution on R × R+. They hold, as we will see, in perturbing around any solution of the KW equations on W ×I for any oriented three-manifold W and a b 2 one-manifold I (endowed with a product metric gabdx dx +dy ), provided only that φy (which as usual is the component of φ in the I direction) vanishes. This claim can be verified by inspection of the linearized KW equations. In doing this, to minimize clutter, we write simply A, φ (rather than A(0),φ(0) as in section 2.1) for a solution of the KW equations about which we wish to perturb. As usual, we denote the perturbations about this solution as a, ϕ, so we consider the condition that A + sa, φ + sϕ obeys the KW equations to first order in the small parameter s.ThesymbolDi will denote a covariant derivative defined using the unperturbed connection A (and the Levi-Civita connection of W if W is not flat). As already explained, we assume that the solution about which we expand obeys φy = 0, and we describe the background in the gauge Ay = 0. However, the gauge condition that we impose on the fluctuations is that of eqn. (2.23): i a (3.14) Dia +[φa,ϕ ]=0.

Given that Ay =0,thisisequivalentto ∂a (3.15) y + D aa +[φ ,ϕa]=0. ∂y a a

11Self-adjointness of B is not hard to verify by inspection, and is clear in the relation [8]ofthe KW equations on W ×I, for a one-manifold I, to gradient flow equations for the complex Chern- Simons functional on W . Linearization of the gradient flow equation for any Morse function h on a Riemannian manifold X always produces a differential operator d/dy + B,whereB (which is derived from the matrix of second derivatives of the function h and the metric of X) is self-adjoint. In the present example, X is essentially the space of complex-valued connections on the bundle EC → W ,whereEC is the complexification of E,andh is the imaginary part of the Chern-Simons functional of such a connection.

THE NAHM POLE BOUNDARY CONDITION 201

Now let us compare this gauge condition to the linearization of one of the KW i equations, namely the condition Diφ = 0. With Ay = φy = 0, the linearization of this equation gives ∂ϕ (3.16) y + D ϕa − [φ ,aa]=0. ∂y a a When we transform a and ϕ via (3.5), these two equations are exchanged except that the signs are reversed for all terms not proportional to ∂y, as predicted in eqn. (3.13). k l The other KW equations Fij − [φi,φj ]+εijklD φ = 0 behave similarly. We write the linearization of these equations in detail12 in a way adapted to the split M = W ×I: b c b c ∂yaa − Daay − [ϕy,φa] − εabcD ϕ − εabc[a ,φ ]=0 b c b c (3.17) ∂yϕa +[ay,φa] − Daϕy − εabcD a + εabc[φ ,ϕ ]=0. Again when we transform a and ϕ via (3.5), these two equations are exchanged except that the signs are reversed for all terms not proportional to ∂y. 3.5. Extension To Five Dimensions. As explained in section 2.7, the Nahm pole solution can also be used to define a boundary condition on certain elliptic differential equations in five dimensions that are expected to be relevant to Khovanov 3 3 homology. We simply replace R × R+ by R × R × R+, where the first factor is 0 parametrized by a new “time” coordinate x . We reinterpret φy as the component 0 0 A0 oftheconnectioninthex direction, and replace [φy, · ] with D/Dx .The equations acquire a rotation symmetry in R4 = R × R3. For the Nahm pole boundary condition in five dimensions to be elliptic, the linearization L of the five-dimensional equations around the Nahm pole solution R4 must have trivial kernel and cokernel. Using the translation symmetries of ,we 3 j can consider momentum eigenstates, proportional to exp i j=0 kj x with a real four-vector k =(k0,...,k3). Acting on wavefunctions of this kind, L becomes a 1-dimensional operator L1(k), acting on functions that depend only on y,andit suffices to show that for k =0, L1(k) has trivial kernel and cokernel. Using the rotation symmetries of R4, we can assume that the “time” component k0 of k vanishes. In that case, we are dealing with a time-independent perturbation. By definition, the five-dimensional equations reduce to the KW equations in the time-independent case (with A0 interpreted as φy), so for k0 =0,L1(k)coincides precisely with the corresponding operator L1(k) of the four-dimensional poblem. But we already know that the kernel and cokernel of L1(k) vanish; the kernel vanishes by the vanishing result of section 3.2, and the cokernel vanishes because of pseudo skew-adjointness. So in expanding around the Nahm pole solution in five dimensions, the kernel and cokernel of L vanish, as we aimed to show.

4. The Nahm Pole Boundary Condition On A Four-Manifold So far, we have described the Nahm pole boundary condition for certain four- R4 R5 or ﬁve-dimensional equations on a half-space + or +. The purpose of the present

12 Our orientation convention is such that the antisymmetric tensors εijkl and εabc obey εabcy = εabc = −εyabc.

202 RAFE MAZZEO AND EDWARD WITTEN section is to explore the Nahm pole boundary condition on a general manifold with boundary. In section 4.1, we formulate the Nahm pole boundary condition for theKWequationonanorientedfour-manifoldM with boundary W . The five- dimensional case is similar but will not be described here. Once the KW equation with Nahm pole boundary condition is defined on a four-manifold with boundary, one can inquire about the index of the linearization L of this equation. Assuming certain foundational results that we postpone to section 5 (such as the fact that L is Fredholm), a simple formal computation that we present in section 4.2 determines this index. The analogous index problem on a five-manifold with boundary will not be treated in the present paper. The index of an elliptic operator on a five-manifold without boundary is always 0, but this is not necessarily the case on a five-manifold with boundary.

4.1. Boundary Conditions on the Connection. For a homomorphism : su(2) → g, let us say that is quasiregular if jσ = 0 does not occur in the decomposition of g or equivalently if the commutant C is a finite group. (This is so if is principal; for additional examples see Appendix A.) On a half-space, for quasiregular , the Nahm pole boundary condition on the KW equations implies that the connection A vanishes along the boundary, since the relevant indicial roots are strictly positive. More generally, on any four-manifold M with boundary W = ∂M, the leading order behavior of the connection A along W is coupled by the KW equations to the leading order behavior of φ. In particular, if KW(A, φ)=0andφ and A have −1 a expansions φ = y ( tadx )+..., A = A(0) + ... near the boundary, then for quasiregular , the restriction A(0) of the connection to W is uniquely determined, as we will explain. In expanding the solution near y = 0 and analyzing A(0), we implicitly use the regularity theorem of section 5.5, that solutions (A, φ)have asymptotic expansions as y → 0. In the calculations below we only use the first few terms of this expansion. The KW equations determine an entire sequence of relationships between the higher coefficients in the expansion of φ and A,witha certain number of these left undetermined because of the gauge freedom. However, we focus here on the leading order relationships, which signify the rigidity of the Nahm pole boundary condition. The main subtlety is to understand the generalization of the formula φ = a a tadx /y + ... to the case that the boundary manifold W is not flat. For this, we view φ as a section of the bundle Hom(TW,ad(E)). For a point x ∈ W ,let{ea} be any orthonormal basis of TxW . Suppose that the leading term in the expansion −1 ∗ ∈ of φ is of order y . This leading term must be a taea/y for some ta ad(E)x, −2 and the y term in the first of eqns. (4.3) below implies that the ta satisfy the commutation relations of su(2). The classification of su(2) subalgebras of g up to conjugacy is discrete and hence the conjugacy class of ta is independent of x;thisis the conjugacy class of some homomorphism : su(2) → g. It also follows from the ∗ theory of su(2) that φ = a taea, viewed as a homomorphism from TxW to the subspace of ad(E)x spanned by the ta,isanisometry.ForG = SO(3) or SU(2), assuming that =0,the ta span all of ad(E) and we have learned that the polar part of φ determines an isomorphism between ad(E)andTW. (We assume that = 0 to avoid many exceptions in the following remarks, but the discussion below of the non-quasiregular case applies in particular to =0.)

THE NAHM POLE BOUNDARY CONDITION 203

For G = SO(3), a knowledge of ad(E) is equivalent to a knowledge of the principal bundle E.ForG = SU(2), this is not quite true; the possible choices of E – once the identification of ad(E) with TW isknown–correspondtospin structures on W .ForG of higher rank, the full story is more complicated, and includes the possibility of twisting by a C-bundle, where C is the commutant of (su(2)) in G. The possibility of this twisting will be reflected in eqn. (4.5) below. We henceforth fix φ and consider any solution pair (A, φ) with Nahm pole ∗ given by φ = a taea. We assume that both φ and A are polyhomogeneous (this −1 will be proved in section 5.5), where φ has leading term φ(0) = y φ and A has leading term A(0). Insert the expansions for A and φ into the two equations in (1.6) and collect the terms with like powers. We discuss the coefficients of the powers y−2 and y−1 in turn. We first discuss the quasiregular case, which means that −1 (4.1) φ ∼ y φ + yϕ1 + ..., A∼ A(0) + ya1 + .... We compute

FA = O(1), −2 φ ∧ φ = y φ ∧ φ + O(1), (4.2) − −2 ∧ −1 O dAφ = y (dy φ)+y dA(0) φ + (1), − −2 ∧ −1 O dA φ= y dy φ + y dA(0) (φ)+ (1), so the KW equations become −2 −1 −y (dy ∧ φ + φ ∧ φ)+y (dA φ)+...=0 (4.3) (0) − −2 ∧ −1 y dy φ + y dA(0) (φ)+...=0. The gauge condition (2.2) does not include any terms with y−2 or y−1. The coef- −2 ficient of y inthefirstofeqns.(4.3)isjustNahm’sequation,forcingtheta to generate an su(2) subalgebra of ad(E), while in the second, φ already contains a dy,sobothtermsvanish. The coefficients of y−1 reduce to

(4.4) i)dA(0) φ =0, and ii)dA(0) φ =0

Since φ along W is proportional to dy,eqn.ii) involves only tangential derivatives and does not involve the y component A(0). Equation i), on the other hand, may have a dy term, but let us first examine its pullback to W . This restricted equation implies that φ intertwines the Levi-Civita connection on TW and the connection A(0) on ad(E). Indeed, this equation shows that the part of this connection induced on the image of φ is torsion-free, and since it is a g connection, it is also compatible with the Killing metric on ad(E). Hence its pullback to TW must be the Levi-Civita connection: i.e. in terms of the orthonormal frame ea, ∇ ∇ φ ( ea eb)= ea (φ(eb)) for all a, b. Equivalently, with respect to the product connection on T ∗W × ad(E), ∇φ = 0. The second equation in (4.4) is then automatically satisfied. Eqn. i) further implies that A(0) is valued in (TW) ⊂ ad(E); its projection onto the orthocomplement of this space vanishes. Thus, if is quasiregular or equivalently if the commutant C is a finite group, the restriction of ad(E)toW and its connection A(0) are uniquely determined locally in terms of TW with its Levi-Civita

204 RAFE MAZZEO AND EDWARD WITTEN connection. (Globally, depending on C, there may be some discrete choices that generalize the choice of a spin structure for G = SU(2).) The discussion up to this point is independent of how φ or A(0) is extended A into M.Thedy component of eqn. i) in (4.4) states that ∇ (0) φ =0aty =0.This ∂y is not only a gauge-invariant condition, but it is also independent of the extension of A(0) into the interior. In particular, if we choose the gauge so that (A(0))y =0, 0 then ∂yφ = 0 (which vindicates the fact that we have omitted the y term in the expansion for φ). In any case, the normal derivative of φ at W is pure gauge. The remaining coefficients in the expansions of A and φ are then determined by these leading coefficients and the successive equalities determined by the vanishing of the coefficients of each yλ. However, it is increasingly hard to extract information from the higher terms; even the coefficients of y0 are not so easy to interpret. Finally, consider the non-quasiregular case. Let C, with Lie algebra c,bethe commutant in G of (su(2)). The KW equations would allow the above description of φ and A to be modified by c-valued terms that would be O(1) for y → 0. However, as in section 2.4.2, we pick an arbitrary subgroup H ⊂ C, with Lie algebra h,and we write h⊥ for the orthocomplement of h in c. Then we restrict the expansion to take the form −1 (4.5) φ ∼ y φ + ϕ0 + ..., A∼ A(0) + a0 + ..., ⊥ where (ϕ0)a ∈ h and (ϕ0)y,aa ∈ h,andφ, A(0) are as above. (In general, in the non-quasiregular case, the next term in the expansion is of order y1/2.) In ⊥ this expansion, we have set to 0 the h -valued part of aa, (ϕ0)y and the h-valued part of (ϕ0)a, and then, in an appropriate global setting, the KW equations will ⊥ determine globally the h-valued part of aa, (ϕ0)a and the h -valued part of (ϕ0)a. The justification for this assertion is provided in section 5, where we show that the boundary problem just stated is well-posed. Calculating the first few terms in the expansions of FA, φ ∧ φ, dAφ and dA φ as before, we see that the coefficient of 1/y2 in KW(A, φ) vanishes just as before. −1 The y coefficient in the term φ∧φ now equals φ ∧ϕ0. This involves terms of the form [ta, (ϕ0)b]or[ta, (ϕ0)y], and these vanish since ϕ0 is valued in the commutant c of (su(2)). Similarly, A(0) must be replaced by A(0) + a0 in each of the two equations in (4.4), but again this does not modify the above considerations, since a0 is valued in c.

4.2. The Index. Here, assuming foundational results proved in section 5, we calculate the index of the linearization L of the KW equations. We do this ﬁrst on a closed four-manifold, and then on a compact four-manifold with boundary, with arbitrary Nahm pole boundary conditions. The ﬁrst case is included simply to isolate the contribution from the interior topology of M. The computation in the second case relies on a short argument to show that the index is actually independent of the choice of Nahm pole boundary condition, or in other words, of the representation . This reduces the computation to one for the special case = 0, where the computation reduces to a well-known one.

4 Proposition 4.1. Let (M ,g) be closed, and suppose that KW(A(0),φ(0))=0. Writing L for the linearization of KW at this solution, then ind(L)=−(dim g) χ(M).

THE NAHM POLE BOUNDARY CONDITION 205

We have already remarked in section 2.1 that the symbol of L is the same as ∗ odd that of the twisted Gauss-Bonnet operator (d + d ) ⊗ Idad(E) : ∧ M ⊗ ad(E) → ∧evenM ⊗ ad(E). The formula here follows directly from the fact that the index of d+d∗, acting from even forms to odd forms, equals χ(M), the Euler characteristic of M. (Twisting by E does not aﬀact the index of the twisted Gauss-Bonnet operator even if E is topologically non-trivial.) Proposition 4.2. Let (M 4,g) be a manifold with boundary, with g cylindrical near ∂M. Then ﬁxing the Nahm pole boundary conditions at ∂M with =0and h = {0}, the index of the linearization about any solution is given by ind(L)=−(dim g) χ(M).

This choice of Nahm pole condition is the same as the absolute boundary condition for d + d∗. This is again a classical formula. We could equally well have chosen h = g, corresponding to relative boundary conditions for the Gauss-Bonnet operator, in which case the index equals −(dim g) χ(M,∂M), but by Poincar´e duality, χ(M,∂M)=χ(M). This is a special case of the next result, which is the main one of this section. Proposition 4.3. Let (M 4,g) be an arbitrary compact manifold with boundary, with g cylindrical near ∂M, and ﬁx any choice of Nahm pole boundary condition at W = ∂M.Thenonceagain ind(L)=−(dim g) χ(M).

We prove this in two steps. First consider the special case where M = W × I with a product metric, and with Nahm pole boundary condition given by any at y = 0 and with trivial Nahm pole boundary condition ( =0,h =0)aty =1. We see immediately, using the pseudo skew-hermitian property of L in this product setting, that the index vanishes. Now let M and be arbitrary and denote by L the linearized KW operator about any solution satisfying the Nahm pole boundary condition associated to , and similarly let Lrel denote the linearized KW operator relative to =0andh = g, i.e. with relative boundary conditions. We apply a standard excision theorem for the index, for example [17, Prop 10.4], which shows that

ind(L) − ind(Lrel)=ind(L,rel), where the operator on the right is the linearized KW operator on the cylinder ∂M×I with Nahm pole boundary condition at one end and with relative boundary condtions on the right. We have already shown that the index on the right vanishes, whence the claim.

5. Analytic theory We now turn to a more careful description of the analytic theory which under- pins many of the preceding considerations. More speciﬁcally, we brieﬂy describe some aspects of the theory of linear elliptic uniformly degenerate equations, all taken from [13]and[18], explain how the results and calculations obtained above

206 RAFE MAZZEO AND EDWARD WITTEN

fit into this theory, and then prove a regularity theorem for solutions of the nonlinear KW equations which justifies the calculations in the uniqueness theorem. Let us make two comments before proceeding. The first is that in the special case where = 0, the Nahm pole boundary condition reduces to a classical elliptic boundary problem, and it is well-known that solutions are then smooth up to W . The theory described below is the natural extension of those ideas which allows us to handle the case where =0and L is no longer a uniformly elliptic operator. The second is that to keep the exposition simpler, we focus on the problem in four dimensions. All of the theory below generalizes immediately to the linearized problem in five dimensions, as does the application of these results to the regularity of solutions of the nonlinear equations as in section 5.5. The changes required are strictly notational.

5.1. Uniformly Degenerate Operators. Let M be a manifold with boundary, and choose coordinates (x, y) near a boundary point, where x ∈ U ⊂ Rn−1 and y ≥ 0. A diﬀerential operator L0 is called uniformly degenerate if in any such coordinate chart near the boundary it takes the form j α (5.1) L0 = Ajα(x, y)(y∂y) (y∂x) ; j+|α|≤m

α α1 αn−1 here (y∂x) =(y∂x1 ) ...(y∂xn−1 ) . Such operators are also called 0-differential operators. The key point in this definition is that every derivative is accompanied by a factor of y. In our setting, the order m equals 1 and the coefficients Ajα are matrices. Observe that a uniformly degenerate can never be uniformly elliptic in the standard sense at ∂M because all the coefficients of the highest order terms vanish when y = 0. However, there is an extended notion of ellipticity for such operators: L0 is said to be an elliptic uniformly degenerate operator if it is elliptic in the standard sense at interior points, where y>0, and if in addition, near points of the boundary, the matrix-valued polynomial obtained by replacing each y∂xa and y∂y with multiplication by the linear variables −ika and −ikn is invertible when (k1,...,kn) = 0. (This formal replacement actually has an invariant meaning; see [13].) The linearized KW operator L is not quite of the form (5.1); instead, yL = L0 is an elliptic uniformly degenerate operator. This is close enough so that the methods described here can be applied to its analysis equally well. To put this into perspective, note that if Δ is the standard Laplacian on a half-space, then y2Δ is elliptic uniformly degenerate, which indicates that the uniformly degenerate theory for the latter operator must therefore reflect the well-known properties of the former. In other words, the theory described below subsumes and generalizes the classical theory of boundary problems for nondegenerate elliptic operators. Unlike −2 2 −1 Δ=y (y Δ), however, the operator L = y L0 is still degenerate because of the presence of terms involving 1/y; hence (contrary to the study of Δ), it is necessary to draw on this uniformly degenerate theory. The mapping and regularity properties of solutions of an elliptic uniformly degenerate operator L0 hinge on the study of two simpler model operators. The first of these: j α (5.2) N(L0)= Ajα(x, 0)(s∂s) (s∂w ) , j+|α|≤m

THE NAHM POLE BOUNDARY CONDITION 207 is called the normal operator. This is invariantly defined (up to a linear change Rn of variables) as an operator on the half-space +, naturally identified with the inward-pointing tangent space at the boundary point (x, 0) ∈ ∂M. To emphasize that it acts on functions defined on this entire half-space, rather than just on a coordinate chart, we write it using the linear variables s ≥ 0, w ∈ Rn−1,whichare L Rn globally defined on this half-space. The global behavior of N( 0)on + plays a central role in the analysis below. As a matter of notation, we define the normal operator of the linearized KW operator as −1 (5.3) N(L)=s N(L0).

Notice that N(L0) only depends on x ∈ ∂M as a parameter; there is a diﬀerent normal operator at each point of the boundary, each of which is again uniformly degenerate and elliptic in this sense. This whole collection of operators is called the normal family of L0. In some cases, certain crucial features of each Nx(L0) vary with the parameter x. Fortunately this does not happen in our setting; the normal operators at diﬀerent boundary points all ‘look the same’, and so we shall usually omit the dependence on x. The normal operator N(L0) enjoys considerably n−1 more symmetries than L0 itself; namely, it is translation invariant in w ∈ R and invariant under dilations (s, w) → (λs, λw), λ>0. Because of these symmetries, it is relatively elementary to study directly, and the goal of this entire theory is to show that key properties of these normal operators carry over to L0 itself. The second model operator is a further reduction, called the indicial operator j (5.4) I(L0)= Aj0(x, 0)(s∂s) , j≤m

L αj obtained from N( 0) by dropping all of the terms (s∂wj ) with αj > 0. Following −1 the convention above, we also write I(L)=s I(L0), when L is the KW operator. There is a further, purely algebraic, reduction of the indicial operator obtained λ by letting I(L0) act on the elementary functions s . This yields the indicial family: j −λ λ (5.5) I(L0,λ)= Aj0(x, 0)λ = s I(L0)s , j≤m where each s∂s has been replaced by a factor of λ. The reader will notice that the normal operator N(L) was effectively already introduced in section 3 when we considered the linearized KW operator at the R4 special Nahm pole solution on +. Moreover, we also encountered the indicial family of the linearized KW operator in the matrices on the left hand side of (2.27) −1 and (2.28). The indicial roots of L0 (or equivalently, of L = s L0) are the finite setofvaluesofλ for which I(L0,λ) is not invertible. As we have seen, their computation in our setting involves nontrivial algebraic subtleties. For the rest of this discussion, let us consider only the case where L is the linearized KW operator. Everything we say here has analogues for operators of higher order. With this assumption, one obvious simplification is that the equation characterizing the indicial roots is a simple (generalized) eigenvalue problem, namely that the matrix

(5.6) A10λ + A00 has nontrivial kernel. Writing the ﬁelds on which L0 acts as Ψ, then corresponding to each indicial root λ there is an eigenvector Ψλ. Equivalently, there is a solution

208 RAFE MAZZEO AND EDWARD WITTEN

λ of the indicial operator of the form Ψλs . In general the indicial roots may depend on the basepoint x ∈ ∂M, and this introduces substantial analytic complications. Fortunately, in our case, this does not occur and we assume henceforth that the indicial roots are constant in x. The importance of these indicial roots can be explained at various levels. At the simplest level, they provide the expected growth or decay rates of solutions to the equation L0Ψ = 0. There is no a priori guarantee that actual solutions to this linear PDE actually do grow or decay at these precise rates, and the fact that they do in some cases is a regularity theorem. The discussion in the earlier part of this paper assumes that these growth rates are legitimate, and we are now in the process of showing that this is so for the linearized KW equation with the Nahm pole boundary condition. To proceed further, we pass from the normal operator N(L)tothesameoper- ator conjugated by the Fourier transform in w, just as in section 3.3. This leads to the matrix-valued ordinary differential operator 1 (5.7) N(L)=A ∂ − iA ka + A 10 s 0a s 00 ik·w −ik·w a The factor of i appears since e (s∂w )e = −ik . There are two key facts about solutions of this operator, each following from elementary considerations: i) Any solution Ψ( s,k)toN(L)Ψ = 0 either decays exponentially or else grows exponentially as s )∞; ii) Any solution of this equation near s = 0 has a complete (and in fact convergent) asymptotic expansion ∞ λ+j (5.8) Ψ(s, k)= Ψλjs , λ j=0 where the first sum is over indicial roots of L. (In exceptional cases, where the difference between different indicial roots is an integer, this sum may include extra logarithmic factors. This actually happens in the case of the KW equations.) The second of these assertions is an immediate consequence of the classical theory of Frobenius series of solutions of equations with analytic coefficients near regular singular points. The first assertion is slightly more subtle in that it depends on the ellipticity of the normal operator N(L). The dominant terms in (5.7) as s →∞are the first two on the right. Dropping the third term A00, we obtain the a λs constant coefficient operator A10∂s −iA0ak , which has solutions of the form Ψλe a where λ and Ψλ satisfy the algebraic eigenvalue equation (A10λ − iA0ak )Ψλ =0. The fact that there are no solutions of this equation with purely imaginary λ (or with λ =0,k = 0) is a restatement of the ellipticity, in the ordinary sense, of the operator A10∂s + A0a∂wa . Before proceeding with the formulation of boundary conditions, we recall the general notions of conormality and polyhomogeneity of a field Ψ near ∂M;these are simple and useful extensions of the notion of smoothness up to the boundary of λ0 −λ0 Ψ. We say that Ψ is conormal of order λ0,andwriteΨ∈A ,ify |Ψ|≤C with −λ0 j α a similar estimate for all its derivatives, i.e. y |(y∂y) (∂x) Ψ|≤Cjα for all j, α. AnysuchfieldissmoothintheinteriorofM, but these estimates√ give only a very limited sort of smoothness near the boundary: for example, both y −1 and 1/ log y

THE NAHM POLE BOUNDARY CONDITION 209 lie in A0. A more tractable subclass consists of the space of polyhomogeneous ﬁelds Ψ. Here Ψ is said to be polyhomogeneous at y = 0 if it is conormal and in addition has an asymptotic expansion γj p (5.9) Ψ ∼ y (log y) Ψjp(x).

The exponents γj on the right lie in some discrete set E ⊂ C, called the index set of Ψ, which has the following properties: Re γj →∞as j →∞,thepowersp of log y are all nonnegative integers, and there are only finitely many such log terms γj accompanying any y . Notice that the conormality of Ψ implies that each Ψjp(x) ∈ C∞(∂M). The meaning of ∼ is the classical one for an asymptotic expansion: namely, γj p ReγN+1 q |Ψ − y (log y) Ψjp|≤Cy (log y) , j≤N where the term on the right is the next most singular term in the expansion. The corresponding statement must hold for the series obtained by differentiating any finite number of times. If the γj are all nonnegative integers and the log terms are absent, this is just the standard notion of smoothness up to y = 0. Solutions of uniformly degenerate equations L0Ψ = 0 are typically polyhomogeneous (at least in favorable circumstances), but essentially never smooth in the classical sense. In our specific problem, the exponents γj are of the form γj = j/2, j =0, 1, 2,...;log terms, if they appear at all, do not occur in the leading terms. 5.2. Elliptic Weights. We now turn to the various sorts of boundary conditions that can be imposed on the operator L and a description of what makes a boundary condition elliptic (relative to L). General types of boundary conditions can be local and of ‘mixed’ Robin type, or nonlocal, such as an Atiyah-Patodi- Singer type boundary condition. We shall focus, however, on the particular local algebraic boundary conditions which arise in the Nahm pole setting. In this section we describe the simplest of these boundary conditions, where L acts on fields with a prescribed rate of vanishing or blowup at y = 0. This is analogous to a homogeneous Dirichlet condition (which is tantamount to considering solutions which vanish like yε at the boundary for any 0 <ε<1). This type of boundary condition is relevant in our setting only when none of the jσ = 0; in particular, this is the correct type of boundary condition when is a regular representation. This case is simpler to state, and considerably simpler to analyze, than the more general one when some of the jσ = 0, which we come to only in section 5.4. As before, we continue to focus exclusively on the linearized KW operator L, or its uniformly degenerate associate L0 = yL, although all the results below have analogs for more general elliptic uniformly degenerate operators. L λ0+1/2 k We shall study the action of on weighted 0-Sobolev spaces y H0 (M), and so we start by defining these. First consider yλ0+1/2L2(M), which consists of λ0+1/2 2 all fields Ψ = y Ψ1 where Ψ1 ∈ L (M). The measure is always assumed to equal dx dy up to a smooth nonvanishing multiple. Next, for k ∈ N,let k { ∈ 2 α j ∈ 2 ∀ | |≤ } H0 (M)= Ψ L (M):(y∂x) (y∂y) Ψ L (M), j + α k ,

λ0+1/2 k { λ0+1/2 ∈ k} and finally, define y H0 = Ψ=y Ψ1 :Ψ1 H0 . The spaces λ0+1/2 k Rn s H0 ( +) are defined similarly. The subscript 0 on these Sobolev spaces indicates that they are defined relative to the 0-vector fields y∂y and y∂xa ;itdoes not connote that the fields have compact support. A key feature of these spaces is

210 RAFE MAZZEO AND EDWARD WITTEN that their norms have a scale invariance coming from the invariance of y∂y and y∂xa under dilations (y,x) → (cy, cx), c>0. In fact, N(L0) does not act naturally on the more standard Sobolev spaces defined using the vector fields ∂y, ∂xa . The shift by 1/2 in these weight factors is for notational convenience only and corresponds to the fact that the function yλ0 lies in yλ0+1/2+εL2 locally near y =0whenε<0 but not when ε ≥ 0. In other words, yλ0 just marginally fails to lie in yλ0+1/2L2 (near y =0). It is evident that − L λ0+1/2 1 −→ λ0 1/2 2 (5.10) : y H0 (M) y L (M) is a bounded mapping for any λ0 ∈ R. Notice that the weight on the right has dropped by 1, reflecting that the operator L involves the terms ∂y and 1/y;ifwe were formulating this using L0, then it would be appropriate to use the same weight on the left and the right. Our main concern is whether this mapping is Fredholm, i.e. has closed range and a finite dimensional kernel and cokernel, and to describe the regularity of solutions of LΨ=0(orLΨ=f for fields f which have better − regularity and decay as compared to yλ0 1/2L2). It is not hard to show that (5.10) does not have closed range when λ0 is an indicial root of L. Indeed, in this case, an appropriate sequence of compactly supported cutoffs of the function yλ0 can be used to create a Weyl sequence Ψj , i.e. an orthonormal sequence of fields such that

|| || ||L || − → Ψj λ0+1/2 1 =1, Ψj λ0 1/2 2 0. y H0 y L

Hence (5.10) is certainly not Fredholm then. When λ0 0, then (5.10) has an infinite dimensional kernel, while if λ0 & 0, its cokernel is infinite dimensional. Thus the only chance for (5.10) to be Fredholm is when λ0 is nonindicial and lies in some intermediate range. In some cases (as described in section 5.4), it is not Fredholm for any weight λ0. Closely related is the fact that fields in the kernel of (5.10) when λ0 0 are, in general, not regular, i.e. polyhomogeneous; indeed, for such λ0, most solutions are quite rough at ∂M. All of this motivates the following definition.

Definition 5.1. The weight λ0 is called elliptic for the linearized KW operator L if its normal operator deﬁnes an invertible mapping: − L λ0+1/2 1 Rn −→ λ0 1/2 2 Rn (5.11) N( ):s H0 ( +;dw ds) s L ( +;dw ds). The two main consequences of the ellipticity of a weight are stated in the following propositions. Proposition 5.2. Let L be the linearized KW operator on a compact manifold with boundary M, and suppose that λ0 is an elliptic weight. Then the mapping (5.10) is Fredholm.

Recalling that we are in the case where no jσ =0,letλ and λ be the largest negative and smallest positive indicial roots of L, respectively. Thus (see Appendix A) λ = −1andλ = 1. We assert that any λ0 ∈ (λ, λ) is an elliptic weight. We shall prove this later; in fact, all of the necessary facts for the proof come from the considerations in sections 2 and 3. It will follow from this proof that if some λ0 is an elliptic weight, then so is any other λ0 which lies in a maximal interval around λ0 containing no indicial roots.

THE NAHM POLE BOUNDARY CONDITION 211

Proposition 5.3. With all notation as above, let λ0 be an elliptic weight for − L, and suppose that LΨ=f ∈ yλ0 1/2L2 where Ψ ∈ yλ0+1/2L2.Iff is smooth in a neighborhood of some point q ∈ ∂M, or slightly more generally, if it has a polyhomogeneous asymptotic expansion as y → 0, then in that neighborhood, Ψ admits a polyhomogeneous expansion γj (5.12) Ψ ∼ y Ψj , where the exponents γj are of the form λ + , ∈ N, where either λ is an indicial root of L or else is an exponent occurring in the expansion of f.

The expansion (5.12) may contain terms of the form yγj (log y)p, p>0. These can only appear when the differences between indicial roots are integers (as happens in our setting), or when there is a coincidence between the indicial roots of L and the terms in the expansion of the inhomogeneous term f. However, the key fact is simply that Ψ has an expansion at all; once we know this, then the precise terms in its expansion can be determined by matching like terms on both sides of the equation LΨ=f. The existence of such an asymptotic expansion for solutions should be regarded as a satisfactory replacement for smoothness up to the boundary. For an ordinary (nondegenerate) elliptic operator L0, if the standard Dirichlet condition (requiring solutions to vanish at y = 0) is an elliptic boundary condition in the classical sense, then solutions of L0Ψ=0withΨ(0,x) = 0 are necessarily smooth and vanish to order 1 at the boundary. Proposition 5.3 is the exact analogue of this. For the linearized KW operator, the nonnegative indicial roots lie in the set {j/2:j = 0, 1, 2,...},soifLΨ = 0 in some neighborhood of a boundary point, then j/2 Ψ ∼ y Ψj , (we are neglecting log terms which might appear); in general, there are half-integral exponents, so Ψ is genuinely not smooth at y =0. We now verify the invertibility of (5.11) for every λ0 ∈ (λ, λ) in our particular example, which proves the assertion that every such λ0 is an elliptic weight. First conjugate N(L) with the Fourier transform in w, thus passing to the simpler ordinary differential operator N(L) as in (5.7). We must show that − L λ0+1/2 1 R+ −→ λ0 1/2 2 R+ (5.13) N( ):s H0 ( ;ds) s L ( ;ds), is invertible for each k, and that the norm of its inverse is bounded independently of k. The first step is to use the scaling properties of N(L) to reduce to the case a |k| = 1. Indeed, set t = s|k| and write B(L)=A10∂t − itA0ak /|k| + A00.(The “B” refers to the fact that this operator which has many features in common with the Bessel equation, and so we call B(L) the model Bessel operator of L.) Applying this change of variables replaces B(L)by|k|−1N(L). Suppose that we have already shown that the version of (5.13) with B(L) replacing N(L) is invertible for every k with |k| = 1, and let B(G)(t, t,k)denote the Schwartz kernel of this inverse. We then recover the Schwartz kernel G(s, s,k) of the inverse of (5.13) for any k =0as N(G)(s, s,k)=B(G)(s|k|,s|k|,k/|k|).

212 RAFE MAZZEO AND EDWARD WITTEN

To see that this is the case, we ﬁrst compute that N(L) B(G)(s|k|,s|k|,k/|k|)f(s,k)ds = (B(L)B(G))(s|k|,s|k|,k/|k|)|k|f(s,k)ds = δ(s|k|−s|k|)|k|f(s,k)ds = f(s,k), since the δ function in one dimension is homogeneous of degree −1. This result may seem counterintuitive since one expects that ||N(L)−1|| ∼ 1/|k|, but that expectation is false because we are letting N(L) act between spaces with diﬀerent weight factors. In fact, the norm of N(G) is bounded uniformly in k, but does not decay as |k|→∞. To this end, observe that we must estimate the norm of −λ −1/2 λ −1/2 2 2 H(s, s ,k):=s 0 N(G)(s, s ,k)(s ) 0 : L → L . This is done by calculating 2 H(s, s ,k)f(s ,k)ds ds 2 −λ −1/2 λ −1/2 = B(G)(s|k|,s|k|,k/|k|)(s|k|) 0 (s |k|) 0 |k|f(s ,k)ds ds 2 −λ −1/2 λ −1/2 −1/2 = B(G)(t, t ,k/|k|)t 0 (t ) 0 |k| f(t /|k|,k)dt

≤ C|||k|−1/2f(t/|k|,k)|| = C||f(·,k)||. − The inequality in the fourth line reflects the boundedness of B(G):tλ0 1/2L2 → tλ0+1/2L2. Beyond all this, compactness of the unit sphere in k shows that the norm of B(G) can be bounded independently of k. As for showing that (5.13) is invertible for each k, we first show that it is Fredholm. This can be done by a standard ODE analysis of the operator. First construct approximate local inverses near s =0ands = ∞; the existence of these shows that (5.13) is Fredholm precisely when λ0 is not an indicial root of L0.(As noted earlier, when λ0 is an indicial root, this mapping does not have closed range.) Now fix λ0 to be any nonindicial value and recall the fact i) that any element of the kernel either grows or decays exponentially as s →∞. Then injectivity of this map means precisely that the solutions which decay exponentially as s →∞must blow up faster than sλ0 as s → 0. One can perform a similar analysis for the adjoint operator, or by showing by other methods that the index vanishes, to show that (5.13) is surjective too. For the linearized KW operator L, the discussion in section 2.4 implies directly λ0+1/2 2 that the kernel of N(L) has only trivial kernel on s L when λ0 > 0. Indeed, perform the integrations by parts (which are now only in the s variables), using the decay of solutions both as s → 0andass →∞to rule out contributions from the boundary terms. We can extend this to allow any λ0 >λsimply by observing using the fact ii) about solutions that if N(L)Ψ=0andψ ∈ sλ0+1/2L2, then Ψ vanishes like sλ, so we may integrate by parts as before. The fact that

THE NAHM POLE BOUNDARY CONDITION 213 the index vanishes when λ ∈ (λ, λ) follows by using the pseudo skew-adjointness established in section 3.4. Note that those arguments are for L on the model space R4 L +, which is canonically identified with the normal operator N( ) of the linearized KW equations on any manifold with boundary, and the pseudo skew-adjointness passes directly to N(L)aswell. We have now proved that in the quasiregular case, when no jσ =0,anyλ0 ∈ (λ, λ) is an elliptic weight for L. The results just stated are not well suited for our nonlinear problem simply because these weighted L2 spaces do not behave well under nonlinear operations. One hope might be to use L2-(orLp-) based scale-invariant Sobolev spaces with sufficiently high regularity. These do have good multiplicative properties locally in the interior, but not near the boundary. This leads us to introduce several related Hölder-type spaces, and then describe the mapping properties of L acting on them. Ck j α We start with the spaces 0 which consist of all fields Ψ such that (y∂y) (y∂x) Ψ is bounded on M and continuously differentiable in the interior of M for all j+|α|≤ k.TheHölder seminorm is defined by |Ψ(y,x) − Ψ(y,x )|(y + y)γ [Ψ]0;0,γ =sup γ γ (y,x)=( y,x ) |y − y | + |x − x | Ck,γ ∈Ck j α ∞ Then 0 consists of all Ψ 0 such that [(y∂y) (y∂x) Ψ]0;0,γ < . Finally, k,γ k,γ λ0 C λ0 ∈C y 0 consists of all Ψ = y Ψ1 with Ψ1 0 . These spaces capture no information about regularity in the x directions at y = 0, so we also introduce hybrid spaces Ck,,γ { ∈Ck,γ α ∈Ck−|α|,γ | |≤ } 0 = Ψ 0 :(∂x) Ψ 0 , for all α . λ λ p Note that all of these spaces contain elements like y or y (log y) ,providedλ>λ0 (or λ = λ0 if p =0). The mapping property of L on these spaces is much the same as in Proposition 5.2.

Proposition 5.4. Let L be the linearized KW operator. Suppose that no jσ =0 and let λ0 ∈ (λ, λ) be an elliptic weight. Then the mapping − − L λ0 Ck,,γ −→ λ0 1Ck 1,,γ (5.14) : y 0 y 0 is Fredholm for 0 ≤ ≤ k − 1 and k ≥ 1. We explain in the next section how this result is essentially a corollary of Proposition 5.2. More speciﬁcally, both results are proved by parametrix methods; this parametrix is constructed using L2 methods and it is initially proved to be bounded between weighted Sobolev spaces, but it is also bounded between certain of these weighted H¨older spaces, which leads directly to the proof of Proposition 5.4.

5.3. Structure of the Generalized Inverse. We now briefly describe the technique behind the proofs of these results. The main step in each is the construction and use of the generalized inverse G for (5.10). The ellipticity of the weight enters directly into this construction. By definition, a generalized inverse for (5.10) − λ0 1/2 2 → λ0+1/2 1 is a bounded operator G : y L y H0 which satisfies

(5.15) GL =Id− R1, LG =Id− R2,

214 RAFE MAZZEO AND EDWARD WITTEN where R1 and R2 are finite rank projections onto the kernel and cokernel of L, respectively. The nonuniqueness here is mild and results only from the different possible choices of projectors. Since we are working on a specific weighted L2 space, it is natural to demand that R1 and R2 be the orthogonal projectors onto the kernel and cokernel with respect to that inner product, and we make this choice henceforth. If we already know that (5.10) is Fredholm, then general functional analysis tells us that a generalized inverse exists. Conversely, the existence of an operator G with these properties (the boundedness of G is particularly important) implies that (5.10) is Fredholm. In fact, it is only necessary to find a bounded operator G such that the ‘error terms’ R1 and R2 defined as in (5.15) are compact operators, for then standard abstract arguments imply that (5.10) is Fredholm and show that G can be corrected to an operator such that (5.15) holds, with R1 and R2 the actual projectors. This observation is important because it is certainly easier to construct an intelligently designed approximation to the generalized inverse than to construct the precise generalized inverse directly. The criterion by which one judges the approximation to be good enough is simply that the remainder terms R1 and R2 are compact. An approximation of this type is called a parametrix for L. A parametrix can be constructed within the framework of geometric microlocal analysis, as carried out in full detail in [13]. The key point is to work within a class of pseudodifferential operators on M adapted to the particular type of singularity exhibited by L. This is the class of 0- (or uniformly degenerate) pseudodifferential ∗ ∗ operators, Ψ0(M). We wish that Ψ0(M) is sufficiently large to contain parametrices of elliptic uniformly degenerate differential operators, but not so large that the ∗ individual operators in Ψ0(M) are too unwieldy to analyze. We describe these operators in sufficient detail for the present purposes, but refer to [13]forfurther details. ∗ The elements of Ψ0(M) are characterized by the singularity structure of ∈ ∗ their Schwartz kernels. Thus, an operator A Ψ0(M)hasaSchwartzkernel 2 κA(y,x, y ,x ), which is a distribution on M . We expect it to have a a standard pseudodifferential singularity (generalizing that of the Newtonian potential, for example) along the diagonal {y = y,x = x }, but we also require a very precise regularity along the boundaries of M 2, {y =0, or y =0}, and at the intersection of 2 the diagonal with the boundary. To formalize this, introduce the space M0 obtained by taking the real blowup of the product M 2 at the boundary of the diagonal. In local coordinates this means that we replace each point (0,x, 0,x) in the boundary of the diagonal with its inward-pointing normal sphere-bundle. Alternatively, in polar coordinates

2 2 | − |2 1/2 − ∈ 4 R =(y +(y ) + x x ) ,ω=(ω0,ω0, ω)=(y, y ,x x )/R S+, 4 R5 ≥ where S+ consists of the unit vectors in with ω0,ω0 0, we replace each point (0,x, 0,x) by the quarter-sphere at R =0.Wecanthenuse(R, ω, x )asafull set of coordinates. This new space is a manifold with corners up to codimension three, and has a new hypersurface boundary at R = 0, which is called the front face. Its two other codimension one boundaries ω0 =0andω0 = 0 are called its 2 → 2 left and right faces. There is an obvious blowdown map M0 M .Wenowsay that A is a 0-pseudodiﬀerential operator if κA is the pushforward under blowdown

THE NAHM POLE BOUNDARY CONDITION 215

2 of a distribution on M0 (which we denote by the same symbol) which decomposes in the following fashion as a sum κA(R, ω, x )=κA + κA.HereκA is supported away from the left and right faces and has a pseudodifferential singularity of some { } −4 order m along the lifted diagonal ω0 = ω0, ω =0 ,andifwefactorκA = R κA, then κA (along with its conormal diagonal singularity) extends smoothly across the 2 − front face of M0 .Thisexponent 4 is dimensional; in general it should be replaced 2 by the dimension of M. On the other hand κA is smooth in the interior of M0 and has polyhomogeneous expansions at the left, right and front faces of this space, with product type expansions at the corners. Altogether, if the expansions at these a b faces commence with the terms ω0 (at the left face), (ω0) (at the right face) and R−4+s (atthefrontface),thenwewrite ∈ m,s,a,b A Ψ0 (M). Slightly more generally we could replace the superscripts a, b, denoting the leading exponents of the polyhomogeneous expansions at the left and right faces, by index sets, but we do not need this more refined notation here. This elaborate notation simply specifies the precise vanishing or blowup properties of κA in each of these regimes. We have introduced it out of some necessity since at least some features of this precise structure will be used in an important way below. Before proceeding, note one very special case: the identity operator Id 0,0,∅,∅ is an element in this class, and lies in Ψ0 . The fact that it has order 0 along the diagonal is expected, and since its Schwartz kernel δ(y − y)δ(x − x )issupported on the diagonal, its expansion is trivial at the left and right faces, which explains the third and fourth superscripts. Finally, the second superscript is explained by noting that in polar coordinates − − −4 − δ(y y )δ(x x )=R δ(ω0 ω0)δ(ω) Having introduced this general class of pseudodifferential operators, we now ∈ ∗ explain the parametrix construction. We aim to find an operator G Ψ0(M)such that LG is equal to the identity up to some compact remainder terms. Rewriting this as the distributional equation L −4 − κG = R δ(ω0 ω0)δ(ω) we see that the singularity of κG along the diagonal can be obtained by classical methods (the symbol calculus), and this construction is uniform as R → 0once we have removed the appropriate factors of R. In fact, writing L in these same polar coordinates and noting that it lowers homogeneity in R by 1, we expect κG to only blow up like R−3 at the front face. In addition, L must kill the terms in the expansion of κG at the left face, which means that the terms in the expansion in this face should involve the indicial roots; in particular, the leading exponent at this face must equal λ. It is not apparent here where the ellipticity of the weight λ0 enters. The answer is as follows. After first solving away the diagonal singularity using the symbol calculus, we must then improve the initial guess for the parametrix to another one L − L 2 for which κG δId vanishes at the front face as well. Because the lift of to M0 acts tangentially to the boundary faces of that space, this equation restricts to an elliptic equation on the front face. Using a natural identification of each quarter- sphere fiber of the front face with a half-Euclidean space, and a few other steps which we omit, we are led to having to find the exact solution to N(L)Ψ = f where

216 RAFE MAZZEO AND EDWARD WITTEN

R4 f is some smooth compactly supported function on +. If we are able to do this, we can then correct the parametrix to all orders so that the remainder terms are 4 clearly compact. The natural identification used here is that the quarter-sphere S+ fiber in the front face over each point (0,x, 0,x) of the boundary of the diagonal R4 can be identified with the half-space +, where this identification is unique up to L 2 a projective map, and the restriction to the front face of the lift of to M0 is transformed to N(L) in this identification. Section 2 of [13] (especially around eqn. (2.10)) explains more about these identifications. This explains why the exact invertibility of the normal operator plays a crucial role. The vindication that this all works is that by carrying out this parametrix construction, one proves that the generalized inverse G for L is an element of −1,1,λ,b Ψ0 (M), where the final index b is some positive number related to the indicial roots of the adjoint of L, and that the remainder terms in (5.15) satisfy −∞,λ,b −∞,b,λ R1 ∈ Ψ (M), R2 ∈ Ψ (M), where this notation (note the absence of the subscript 0) means that their Schwartz kernels are smooth in the interior and polyhomogeneous at the two boundary hypersurfaces of M 2, rather than being 2 polyhomogeneous on the blown up space M0 . We now explain how to use all of this for our purposes. Granting this structure of the Schwartz kernel of the generalized inverse G, the boundedness of the map − λ0 1/2 2 −→ λ0+1/2 1 (5.16) G : y L y H0 can be deduced from the standard local boundedness of pseudodifferential operators of order −1 between L2 and H1, and the following inequalities for the orders of −3 vanishing of κG at the various boundary faces. The fact that κG blows up like R at the front face, one order better than the Schwartz kernel of the identity, partly explains why G raises the exponent in the weight factor by 1. The other aspect which affects the weight on the right in (5.16) is the leading exponent λ at the left face, since it is clear that the decay profile of (5.17) κG(y,x, y ,x )f(y ,x )dy dx M as y → 0 must incorporate that exponent. Recalling the earlier discussion that κG decomposes into the near-diagonal and off-diagonal parts, κG and κG,aclose analysis of the integrals from each of these terms proves that − − λ0 1/2 2 −→ λ0+1/2 1 λ+1/2 ε k (5.18) G : y L y H0 + y H0 . ε>0 k≥0 In other words, the near-diagonal part raises regularity and the weight parameter by exactly 1, while the off-diagonal part improves the 0-regularity to an arbitrarily large amount, and has growth/decay rate of the outcome dictated by the leading term of κG on the left face. We must intersect over all ε>0 here simply because yλ ∈ yλ+1/2−εL2 when ε>0, but not otherwise. From all of this, and observing that when λ>λ0, the range of (5.18) is con- λ0+1/2 1 tained in y H0 , we see that (5.10) is Fredholm, which is Proposition 5.2. The proof of Proposition 5.3 is obtained by a more refined examination of the mapping properties of G, in particular the fact that if f is polyhomogeneous, then so is the outcome of the integral (5.17). Finally, the proof of Proposition 5.4 can be explained as follows. The preceding discussion has been based on L2 considerations, which is natural since, for example, Fourier analysis has been used at several points.

THE NAHM POLE BOUNDARY CONDITION 217

However, the precise pointwise behavior of the Schwartz kernel of G makes it possible to read oﬀ its mapping properties on other function spaces. In particular, we obtain the analog of (5.18) on weighted H¨older spaces: − λ0 1Ck,γ −→ λ0 Ck+1,γ λCm,γ (5.19) G : y 0 y 0 + y 0 . m

k+1,γ λ0 C As before, this range lies in y 0 when λ0 < λ. There is a slight refinement of this which we shall need later, namely that μ−1Ck,γ −→ μCk+1,γ λCm,γ (5.20) G : y 0 y 0 + y 0 m for any μ>λ0 (and for simplicity of the statement, μ not an indicial root). Since the equations (5.15) are satisfied as distributions, and every operator in them is bounded between the appropriate Hölder spaces, we see that (5.14) is in fact Fred- holm, at least for = 0. To prove that (5.14) is Fredholm for ≥ 0, we need one −1,1,λ,b extra fact, which is that each of the commutators [G, ∂xa ] lies in Ψ0 , i.e. is a 0-pseudodifferential operator with the same indices, hence has the same mapping properties as G itself. This simple transition from Sobolev to Hölder spaces is a good exemplar of the parametrix method; if we were working solely with a priori estimates, then it is no simple matter to deduce Fredholmness on one type of function space from the corresponding property on another type of function space.

5.4. Algebraic Boundary Conditions and Ellipticity. There are many natural operators, however, for which there are no ellliptic weights, i.e. so that for any nonindicial λ0, the map (5.13) has either nontrivial kernel or cokernel, or both. This is the case for the linearized KW operator L when some of the jσ =0.Wenow describe the somewhat more complicated formulation of the ellipticity criterion for boundary conditions in these cases. As before, we consider only the parts of this story relevant to the Nahm pole boundary conditions for L0. The argument in the last section (slightly after (5.13)) shows that even when − the lowest nonnegative indicial root λ is 0, then N(L):sλ0+1/2L2 → sλ0 1/2L2 has no kernel if λ0 > 0, although the cokernel of this mapping is nontrivial. On the other hand, when λ0 < 0, the nullspace has positive dimension, though the map is surjective. To be definite, suppose that −1/2 <λ0 < 0, which rules out solutions which blow up like sλ where λ is any one of the strictly negative indicial roots of L0. Using the fact ii) that solutions of N(L)Ψ = 0 have (convergent) expansions at 0 s = 0, the leading coefficient Ψ0 =(aa, ϕy, ay, ϕa), i.e. the coefficient of s , is well- defined. This coefficient lies in the eigenspace corresponding to the indicial root λ = 0, and hence, following the language at the end of section 2.4, is an element of c8. We call this the Cauchy data of Ψ and write it as C(Ψ). At this ODE level, the Nahm pole boundary condition intermediates between λ0+1/2 2 the spaces y L when −1/2 <λ0 < 0and0<λ0 < 1/2. Recall that for this boundary condition, we fix a subalgebra h ⊂ c and its orthogonal complement h⊥ in c, and then consider the linear map

⊥ ⊥ B h h h h ∈ ⊥ 3 ⊕ ⊥ ⊕ ⊕ 3 (5.21) h(aa, ϕy, ay, ϕa)=(aa , ϕy , ay , ϕa ) (h ) h h h .

218 RAFE MAZZEO AND EDWARD WITTEN

This determines a boundary condition for N(L) and we shall study the problem

L ∈ λ0+1/2 1 B C (5.22) N( )Ψ=f, Ψ s H0 , h( (Ψ)) = 0. As a first observation, following the same integrations by parts as above, there are no nontrivial fields Ψ which decay at infinity and satisfy (5.22) with f =0; h⊥ h⊥ h h indeed, the conditions aa =0,ϕy =0,ay =0,ϕa = 0 make all boundary terms at s = 0 in this integration by parts vanish. On the other hand, if we only assume λ0−1/2 2 that f ∈ s L for some −1/2 <λ0 < 0, then this problem is not well posed. Indeed, although there is a solution to the first equation in (5.22), there is no reason for the leading coefficient C(Ψ) to have any meaning, so the boundary condition may not have any sense. Thus it is necessary to suppose that f lies in a slightly better space, as we now describe. With λ0 ∈ (−1/2, 0) as before, define

H { ∈ λ0+1/2 2 R+ L ∈ λ0+1/2 2 R+ } (5.23) λ0 = Ψ s L ( ):N( )Ψ s L ( ) . The right hand side is one order less singular than might be expected, and us- 0 λ0+3/2 ing standard ODE techniques, one sees that Ψ=s Ψ0 + O(s ), and hence the leading coeﬃcient Ψ0 is well deﬁned. We may now legitimately consider the mapping

L H ∩{ B C }−→ λ0+1/2 2 (5.24) N( ): λ0 Ψ: h( (Ψ)) = 0 s L . Note that the weighted L2 restriction on Ψwhen s is large precludes the exponentially growing solutions of N(L)Ψ=0. We have so far suppressed the dependence of N(L), and hence also of the map Bh, on the parameters x ∈ ∂M and k = 0. As observed earlier, the only essential part of the dependence on k is on the direction k/|k|, and hence we assume for the remainder of this discussion that |k| = 1, i.e. k ∈ S2. For this particular operator, the dependence on x is not very serious in that the operator ‘looks the same’ in appropriate local coordinates at any point of ∂M. However, formally we should be considering (x,k) as a point in S∗∂M, the cosphere bundle of ∂M.If ∗ → B ∗ 8 π : S ∂M ∂M is the natural projection, then h is a bundle map between π c ∗ ⊥ 3 ⊥ 3 and π (h ) ⊕ h ⊕ h ⊕ (h) .ThefactthatBh is independent of k allows us to call this an algebraic boundary condition. The additional ingredient we need in this discussion is the space { ∈ H L } (5.25) V = Vx,k = Ψ λ0 : N( )(Ψ) = 0 of homogeneous solutions in sλ0+1/2L2 which do not necessarily satisfy the boundary condition. The dependence of V on x is again negligible, but its dependence on k is genuine since k appears in the coeﬃcients of N(L) and these kernel elements do vary nontrivially with k. However, the dimension of Vx,k does not depend on k, andinfactV varies smoothly with k (and x). This means that we can regard V ∗ λ0+1/2 2 as a vector bundle over S ∂M. The injectivity of N(L)ony L when λ0 > 0 shows that the restriction of the Cauchy data map C to V is injective. This means that C(V ) is a subbundle of c8; this is sometimes called the Calderon subbundle. We can now ﬁnally state the property which makes Bh an elliptic boundary condition.

THE NAHM POLE BOUNDARY CONDITION 219

Definition 5.5. Let B be any bundle map from π∗c8 to another vector bundle W over S∗∂M.ThenB is said to be an elliptic boundary condition for N(L) (and hence ultimately for L)iftherestrictionofB to the subbundle V is bijective onto W , i.e. so that B| −→ V : V W is a bundle isomorphism. This condition can be phrased in various obviously equivalent ways; the one we use below is to require that B is injective and that the ranks of V and W are the same. However, in the end, this condition is precisely what is needed to construct a good parametrix for the actual boundary problem on M.Wecansee − this rather easily at the level of this ODE. If f ∈ sλ0+1/2L2 ⊂ sλ0 1/2L2, then there ∈ λ0+1/2 1 L is a solution Ψ s H0 to N( )Ψ=f. This is not unique, since there is a nullspace. In any case, this solution has a leading coefficient Ψ0, which is however unlikely to satisfy Bh(Ψ0) = 0. Now modify Ψ by subtracting an element Φ ∈ V .By definition, N(L)(Ψ − Φ) = f, and provided we choose ΦsothatBh(Φ0)=Bh(Ψ0), then Ψ − Φ satisfies the boundary condition too. The fact that there is a unique such choice of Φ is precisely the content of Definition 5.5. Let us now check that this condition holds for the map Bh which appears in the general Nahm pole boundary condition. We have proved above that Bh is injective on V . Furthermore, it follows from the results of section 3 that the rank of V is half the rank of c8, i.e. rk(V )=4dimc. Since this is the same as dim((h⊥)3⊕h⊥⊕h⊕h3), B | we see that h V is also surjective, and hence an isomorphism. As noted earlier, B is called an algebraic boundary condition if W = π∗W where W is a bundle over ∂M. If this is the case, then the analytic theory of the boundary problem for the actual operator L is simpler because the boundary conditions are local (of mixed Dirichlet-Neumann type), rather than nonlocal (pseudodifferential). It is clear that Bh is an algebraic boundary condition. Return now to the linearized KW operator, and assume that some jσ =0, so that we can augment the operator L with the boundary condition Bh.Fix λ0 ∈ (−1/2, 0) and consider

H { ∈ λ0+1/2 1 L ∈ λ0+1/2 2} (5.26) λ0 = Ψ y H0 (dxdy): Ψ y L . − As before, the expected behavior for Ψ ∈ yλ0+1/2L2 is that LΨ ∈ yλ0 1/2L2. H This means that ﬁelds in λ0 must possess some special properties to ensure that − LΨ is one order less singular than yλ0 1/2L2. Although we no longer have ODE ∈H arguments to fall back upon, it is still possible to show that any Ψ λ0 has a weak partial expansion 0 Ψ ∼ Ψ0 y + Ψ, w where Ψ0 is a distribution of negative order which lies in the Sobolev space Hλ0 (∂M). The remainder term Ψ vanishes like yλ0+1 in a similar distributional sense. We do not pause to make this more precise (see section 7 of [13]), but note only that the actual meaning of the weak expansion above is that if we ‘test’ Ψ against some χ ∈C∞(∂M), then 0 Ψ(y,x)χ(x)dx = Ψ0,χy + Ψ,χ, where the second term on the right vanishes like yλ0+3/2.

220 RAFE MAZZEO AND EDWARD WITTEN

The point of belaboring all of this is that it is possible to make sense of the ⊕8 leading coeﬃcient Ψ0 of a general element Ψ ∈Has a c -valued distribution of negative order on ∂M. Because the boundary condition is algebraic, we can then make sense of the projection Bh(Ψ0), again as a distribution. In particular, when Ψ ∈H, there is now a good meaning of the condition Bh(C(Ψ)) = 0. We can now state analogs of all the main results. Proposition 5.6. Let L be the linearized KW operator on a compact manifold M with boundary and suppose that ρ is not quasiregular, so that some jσ =0.Using the elliptic boundary condition given by the bundle map Bh, then for −1/2 <λ0 < 0, the mapping

L { ∈H B }−→ λ0+1/2 2 (5.27) : Ψ λ0 : h(Ψ0)=0 s L (M) is Fredholm. Proposition 5.7. With all notation as above, suppose that LΨ=f where f is smooth in a neighborhood of some point q ∈ ∂M, or slightly more generally, where f has an asymptotic expansion as y → 0, and in addition Bh(Ψ0)=0near q.Then in that neighborhood, Ψ has a polyhomogeneous expansion γj (5.28) Ψ ∼ y Ψj , where the exponents γj are of the form λ + , ∈ N, where either λ is an indicial root of L or else is an exponent occurring in the expansion of f. As before, this expansion may include log terms.

Proposition 5.8. Let L be the linearized KW operator. If some jσ =0,then for −1/2 <λ0 < 0 and relative to any choice of subalgebra h, the mapping − − L λ0 Ck,,γ −→ λ0 1Ck 1,,γ (5.29) : y 0 y 0 is Fredholm when 0 ≤ ≤ k − 1 and k ≥ 1. These results are proved, as before, using parametrix methods. Unlike the earlier case, this is a slightly more involved process which requires the introduction of generalized Poisson and boundary trace operators; see [18]. We find along the way that the analog of the refined mapping property of the generalized inverse (5.20) still holds. 5.5. Regularity of Solutions of the KW Equations. We have now described enough of the linear theory that we can formulate and prove the main result needed earlier in the paper, that solutions of the full nonlinear gauge-fixed KW equations KW(A, φ) = 0 are also polyhomogeneous at ∂M. The implication of this regularity is that all the calculations in section 2 which led to the uniqueness R4 theorem when M = + are fully justified. (Of course, this polyhomogeneity is far more than is really needed to carry out those calculations, but it is very useful to have this sharp regularity for other purposes too.) Proposition 5.9. Let KW(A, φ)=0, and suppose that near y =0, A = A(0) + a, φ = φ(0) + ϕ,where(a, ϕ) satisfy the Nahm pole boundary conditions. Then (a, ϕ) is polyhomogeneous. The proof begins by using (2.26) to write KW(A, φ)=0asL(a, ϕ)=−Q(a, ϕ). ObservethatsincethetermsinKW are at most quadratic, Q is a bilinear form 1,γ λ0 C in (a, ϕ). We suppose from the beginning that a and ϕ lie in y 0 ,wherethe

THE NAHM POLE BOUNDARY CONDITION 221 rate of blowup (or decay) λ0 is as dictated by the Nahm pole boundary condition. The details of the proof are essentially the same in the simpler quasiregular case and in the more general case where some jσ =0.Intheformer,λ =1,λ = −1, and we take any elliptic weight λ0 ∈ (λ, λ), while in the latter, generically we take λ0 ∈ (−1/2, 0) (if jσ =1/2 does not occur in the decomposition of gC,we can take λ0 ∈ (−1, 0)), and the generalized inverse is constructed using the more elaborate considerations of section 5.4. The key facts that we use below, however, are the existence of a generalized inverse satisfying (5.15), in particular so that the remainder term R1 maps into a finite dimensional space of polyhomogeneous fields and the fact that G satisfies (5.20). Although the construction of G is more complicated in the second case, we still end up with the result that both these properties hold then too. There are two main steps. The first is to prove that (a, ϕ)isconormaloforder λ, i.e. (a, ϕ) ∈Aλ, which we recall means that j α ∈ λ0 Ck,γ (5.30) (y∂y) ∂x (a, ϕ) y 0 k for all j and all multi-indices α. In the second step we improve this to the existence of a polyhomogeneous expansion. Since λ0 is an elliptic weight, there exists a generalized inverse G for L which provides an inverse to (5.14) up to finite rank errors. Applying G to KW(A, φ)=0 gives

(5.31) (a, ϕ)=−GQ(a, ϕ)+R1(a, ϕ).

The finite rank operator R1 has range in the space of polyhomogeneous functions (with leading term yλ), maps into a finite dimensional space of polyhomogeneous functions, so the second term on the right is polyhomogeneous and hence negligible. We are thus free to concentrate on proving the regularity of the first term on the right in (5.31). ∈ λCk,γ ≥ We first assert that (a, ϕ) y 0 for all k 0. (Note that this is not conormality since we are not yet applying the tangential vector fields ∂xa without 1,γ ∈ 2λ0 C the extra factor of y.) Since Q is bilinear, we see first that Q(a, ϕ) y 0 . so that from (5.20) with k = 1, and since 2λ0 >λ0 − 1, we obtain (a, ϕ) ∈ 2,γ m,γ 2λ0+1C λC y 0 + y 0 for all m. After a finite number of iterations, the right hand λCm,γ side is contained in y 0 for some m, and bootstrapping further shows that it lies in this space for all m. Revisiting this iteration, we can improve the regularity with respect to the vector fields ∂xa as well. This relies on a structural fact about 0-pseudodifferential operators already quoted at the end of section 5.3, nmely that ∈ −1,1,λ,b [∂xa ,G] Ψ0 , and hence this commutator satisfies the same mapping properties as G itself. We apply this as follows. Write

∂xa (a, ϕ)=−G(∂xa Q(a, ϕ)) − [∂xα ,G]Q(a, ϕ).

We are discarding the term R1(a, ϕ) since it is already fully regular. It is convenient m,γ λ0 C − now to regard (a, ϕ)aslyinginy 0 for λ0 = λ ε, since we want to use the mapping properties of G at a nonindicial weight. By the mapping properties of m,γ ∩ λ0 C the commutator, the second term on the right lies in my 0 . On the other

222 RAFE MAZZEO AND EDWARD WITTEN

− − m,γ 1 ∩ 2λ0 1C hand, we write ∂xa Q(a, ϕ)=y∂xa (y Q(a, ϕ)). This lies in my 0 ,since − − m,γ 1 ∈∩ 2λ0 1C y Q(a, ϕ) my 0 and y∂xa preserves this property. Since G acts on this m,γ m,1,γ ∩ λ0 C ∈∩ C space, the entire ﬁrst term lies in my 0 .Thisprovesthat(a, ϕ) m 0 . The same argument improves the tangential regularity incrementally, so (a, ϕ) ∈ k,,γ λ0 C ≤ ≤ ∞ y 0 for all 0 k< . Recalling (5.20) again, we can now replace λ0 by λ. This proves that (a, ϕ) ∈Aλ. The second main step of the proof is easier. We wish to prove that (a, ϕ) has an expansion. This relies on the observation that we can treat the nonlinear equation KW(A(0) + a, φ(0) + ϕ) = 0 as a nonlinear ODE in y, regarding the dependence on x as parametric. This is reasonable since (a, ϕ) is completely smooth in this tangential variable. Thus we can decompose the linear term in (2.26) further using the indicial operator I(L) (introduced in eqn. (5.4)) at any given boundary point to get (5.32) I(L)(a, ϕ)=(I(L) −L)(a, ϕ) − Q(a, ϕ).

The two terms on the right lie in yλ0 A and y2λ0 A, respectively. (We recall that I(L)−L has no ∂y or 1/y terms; the coefficients of this difference are smooth to y = λj 0.) Integrating this ODE shows that (a, ϕ) is a finite sum of terms (aj,ϕj)y ,where the λj are the indicial roots of L which lie between λ0 and μ =min{2λ0 +1,λ0 +1}, and an error term vanishing at this faster rate yμ. At the next step, inserting this new information into (5.32) shows that this is now an ODE where the right side has a partial expansion up to order min{μ, 2μ} plus an error term vanishing at that rate, and so (a, ϕ) has a partial expansion up to order μ2 =min{μ+1, 2μ+1}.This completes the proof of the existence of the expansion of (a, ϕ) in the case where λ0 ∈ (0, λ) is an elliptic weight.

Appendix A. Some Group Theory The purpose of this appendix is to describe some basic facts and examples in group theory as background to the paper. First of all, up to isomorphism, the group SU(2) or equivalently the Lie algebra su(2) has precisely one irreducible representation of dimension n,foreachpositive integer n.Itisconvenienttowriten =2j +1 where j is a non-negative integer or half-integer called the spin. If vj denotes an irreducible su(2) representation of spin j,thenforj ≥ j,wehave j+j ⊗ ∼ ⊕ (A.1) vj vj = j=j−j vj . Now, for N ≥ 2, we will describe group homomorphisms : SU(2) → G = SU(N), or equivalently Lie algebra homomorphisms : su(2) → su(N). To describe such a homomorphism amounts to describing how the fundamental N-dimensional representation of SU(N), which we denote V , transforms under (SU(2)). As an SU(2)-module, V will have to be the direct sum of a number of irreducible

SU(2) modules vji of dimension ni =2ji +1,forsomeji. The possibilities simply correspond to partitions of N,thatistowaysofwritingN as an (unordered) sum of positive integers,

(A.2) N = n1 + n2 + ···+ ns. For example, the trivial homomorphism : su(2) → su(N), which maps su(2) to 0, corresponds to the partition N =1+1+···+1withN terms. At the other

THE NAHM POLE BOUNDARY CONDITION 223 extreme, a principal embedding of su(2) in su(N) (which is the most important example for the present paper) corresponds to the partition with only one term, the integer N. In general, we deﬁne the commutant C of as the subgroup of SU(N)that commutes with (SU(2)); its Lie algebra c is the subalgebra of su(n)thatcommutes with (su(2)). If corresponds as in eqn. (A.2) to a partition with s terms, then C is a Lie group of rank s − 1. (It is abelian if and only if the ni are all distinct.) In particular, for G = SU(N), the only case that C is a ﬁnite group (or equivalently a group of rank 0) is that s = 1, meaning that is a principal embedding. In this case, C is simply the center of G. Whenever s>1, C has a non-trivial Lie algebra, and this means, in the language of section 2, that jσ = 0 occurs in the decomposition of gC under su(2). Thus, for G = SU(N), the only case that jσ =0 does not occur in this decomposition is the case that is a principal embedding. To explicitly decompose su(N) under (su(2)), we use the fact that su(N) is the traceless part of Hom(V,V ); equivalently it can be obtained from V ⊗ V ∨ by omitting a 1-dimensional trivial representation. (Here V ∨ is the dual of V .) Any su(2)-module is isomorphic to its own dual, so as a su(2) module, su(N) is V ⊗ V with a copy of the trivial module v0 removed. For example, if is a principal embedding, so that V is an irreducible (su(2)) module vj with N =2j+1, then we use (A.1) to learn that su(N) is the direct sum of su(2)-modules of spins jσ =1, 2,...,N − 1. As a corollary, we note that if jσ = 0 does not occur in the decomposition of su(n) (which happens only if is principal), then the jσ’s are integers and in particular jσ =1/2 does not occur in the decomposition of su(n). As explained below, this statement has an analog for any simple Lie group G. A few additional facts that follow from the above discussion of SU(N) hold for all G.Thenumberof summands in the decomposition of g under a principal su(2) subalgebra is always the rank of G (this rank is N − 1forG = SU(N)). Also, the minimum value of jσ for a principal embedding is always jσ = 1, and this value occurs with multiplicity 1, corresponding to the su(2) submodule (su(2)) ⊂ g. With similar elementary methods, we can analyze the other classical groups SO(N)andSp(2k). Here the following is useful. An SU(2) module v is said to be real, or to admit a real structure, if there is a symmetric, non-degenerate, and SU(2)-invariant map v⊗v → C; it is said to be pseudoreal, or to admit a pseudoreal structure, if there is an antisymmetric, non-degenerate, and SU(2)-invariant map v ⊗ v → C. The representation vj is real (but not pseudoreal) if j is an integer, or equivalently the dimension n =2j + 1 is odd, and pseudoreal (but not real) if j is a half-integer, or equivalently the dimension n =2j + 1 is even. If w is a 2-dimensional complex vector space (with trivial SU(2) action), then w admits both a symmetric nondegenerate map w ⊗ w → C and an antisymmetric one. So if ∼ v is either real or pseudoreal, then v ⊕ v = v ⊗ w admits both a real structure and a pseudreal one. Suppose that

(A.3) v = ⊕j≥0aj vj ,aj ∈ Z is an SU(2) module that is the direct sum of aj copies of vj (with almost all aj vanishing). The criterion for v to be real or pseudoreal reduces to separate conditions on each aj : v is real precisely if aj is even for half-integer j (with no restriction for integer j), and v is pseudoreal precisely if aj is even for integer j (with no restriction for half-integer j).

224 RAFE MAZZEO AND EDWARD WITTEN

Now let us consider homomorphisms : SU(2) → G for G = SO(N). Such a homomorphism can be described by giving the decomposition of the fundamental N-dimensional representation V of SO(N)asanSU(2)-module. Thus, such a homomorphism again determines a partition of the integer N, as in (A.2). However, now we must impose the condition that the representation V of SO(N)isreal.In view of the statements in the last paragraph, the condition that this imposes on the partition is simply that even integers ni in (A.2) must occur with even multiplicity. The condition that the commutant C of SU(2) – or more precisely of (SU(2)) – is a finite group, and hence that jσ = 0 does not occur in the decomposition of 13 the Lie algebra so(N), is that the integers ni in the partition must be all distinct. But since even integers must occur with even multiplicity, this implies that the ni must be odd. For example, if N is odd, a principal embedding of su(2) in so(N) corresponds to a partition with only one term, the integer N.ButifN is even, a principal embedding corresponds to the two-term partition N = 1+(N − 1). For SO(N), in contrast to SU(N), an embedding that is not principal can still have a trivial commutant. For example, for N = 9, the partition 9 = 1 + 3 + 5 represents 9 as the sum of distinct odd integers; this embedding is not principal, but its commutant is a finite group. When the commutant is not a finite group, it has a Lie algebra of positive dimension and hence jσ = 0 occurs in the decomposition of so(n) under su(2). For G = SO(N), rather as we found for SU(N), there is also a useful elementary criterion that ensures that jσ =1/2 does not appear in the decomposition of so(n). In fact, there is a useful criterion that ensures that no half-integer value of j occurs ∼ σ in this decomposition. For this, we first recall that so(N) = ∧2V ⊂ V ⊗ V .For agivensu(2) embedding, the decomposition of V ⊗ V under su(2) can be worked out using (A.1). One finds that half-integer values of j occur in V ⊗ V (and also 2 in ∧ V ) if and only if both odd and even integers ni occur in the chosen partition of N. But we have already observed that if even integers appear in this partition, then jσ = 0 occurs in the decomposition of so(n) under su(2). So if jσ =0does not occur in the decomposition of so(n), then jσ =1/2 also does not occur. ThecasethatG = Sp(2k)forsomek can be analyzed similarly, with the words “even” and “odd” exchanged in some statements. A homomorphism from SU(2) to Sp(2k) can be described by giving the decomposition of the 2k-dimensional representation V of Sp(2k) as a direct sum of SU(2) modules. Thus, such a homomorphism determines a partition 2k = n1 + n2 + ···+ ns.Nowthefactthe representation V of Sp(2k) is pseudoreal implies that odd integers occur in this partition with even multiplicity. The condition that the commutant C is a finite group, so that jσ = 0 does not occur in the decomposition of sp(2k) under su(2), is again that the integers appearing in the partition should be distinct. But now this implies that these integers are all even. A principal embedding is the case that the partition consists of only of a single integer 2k.JustasforSO(N), there are non-principal embeddings with the property that jσ = 0 does not occur in the

13 If an integer ni appears with multiplicity di in the partition of N, then the commutant of su(2) in SO(N) contains a factor of SO(di)ifni is odd or Sp(di)ifni is even. (The last statement makes sense because di is always even when ni is even.) Hence the group C is ﬁnite if and only if the ni are all distinct, so that the di are 0 or 1. The last statement is true for G = Sp(2k) for similar reasons: if an integer ni appears with multiplicity di in the partition of 2k, then the commutant contains a factor of SO(di)ifni is even and of Sp(di)ifni is odd. (For G = Sp(2k), di is even when ni is odd.)

THE NAHM POLE BOUNDARY CONDITION 225 decomposition of sp(2n); these correspond to partitions of 2k as the sum of distinct even integers, for example 6 = 2 + 4. By the same argument as for SO(N), one can show that if jσ =0doesnot occur in the decomposition of sp(2k), then half-integer values of j do not occur σ ∼ and in particular jσ =1/2 does not occur. For this, one uses the fact that sp(2k) = Sym2V ⊂ V ⊗ V , along with the rule (A.1) for decomposition of tensor products. To understand homomorphisms from SU(2) to an exceptional Lie group G,it is probably best to use less elementary methods, and we will not explore this here. We remark, however, that the following feature of the above examples is actually true for any simple Lie group G:ifjσ = 0 does not occur in the decomposition of gC under su(2), then only integer values of jσ occur in this decomposition and in particular jσ =1/2 does not occur. (For a proof, see the next paragraph.) Given this, it follows from the formulas of section 2.3.3 that if jσ = 0 does not occur in the decomposition of gC, then there are no indicial roots in the gap between λ = −1 and λ =1.Both−1 and 1 always are indicial roots in this situation, since jσ =1 always occurs in the decomposition of gC, the corresponding subspace of gC being (su(2)). A proof of a claim in the last paragraph was sketched for us by B. Kostant. The complexiﬁcation of is a homomorphism of complex Lie algebras : sl2(C) → gC. We take a standard basis (h, e, f)ofsl2(C) and write simply (h, e, f)fortheir images in gC. The hypothesis that half-integer values of jσ occur in the decomposition of gC means, in the terminology of [19], p. 165, that e is not even. In this case, according to Proposition 5.7.6 of that reference, e is not distinguished, and therefore, according to Proposition 5.7.4 of the same reference, the homomorphism ad(e):g(0) → g(2) has a non-trivial kernel. This kernel is the commutant c of (sl2(C)).

Acknowledgments. Research of RM supported in part by NSF Grant DMS- 1105050. Research of EW supported in part by NSF Grant PHY-1314311.

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[10] C. H. Taubes, Compactness theorems for SL(2; C) generalizations of the anti-self dual equations, Part I, arXiv:1307.6447. [11] C. H. Taubes, Compactness theorems for SL(2; C) generalizations of the anti-self dual equations, Part II, arXiv:1307.6451. [12] Michael Gagliardo and Karen Uhlenbeck, Geometric aspects of the Kapustin-Witten equations, J. Fixed Point Theory Appl. 11 (2012), no. 2, 185–198, DOI 10.1007/s11784-012-0082-3. MR3000667 [13] Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664, DOI 10.1080/03605309108820815. MR1133743 (93d:58152) [14] Victor Mikhaylov, On the solutions of generalized Bogomolny equations,J.HighEnergyPhys. 5 (2012), 112, front matter+17. MR3042947 [15] Davide Gaiotto and Edward Witten, Knot invariants from four-dimensional gauge theory, Adv. Theor. Math. Phys. 16 (2012), no. 3, 935–1086. MR3024278 [16] Kevin Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361–382. MR965220 (89k:58066) [17] Bernhelm Booss-Bavnbek and Krzysztof P. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1993. MR1233386 (94h:58168) [18] R. Mazzeo and B. Vertman Elliptic theory of differential edge operators, II: boundary value problems. arXiv:1307.2266. [19] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR794307 (87d:20060)

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CONM 621 0Yaso ahmtc tCINVESTAV at Mathematics of Years 50

The influence of Solomon Lefschetz (1884–1972) in geometry and topology 40 years after his death has been very profound. Lefschetz’s influence in Mexican mathematics has been even greater. In this volume, celebrating 50 years of mathematics at Cinvestav-Mexico,´ many of the fields of geometry and topology are represented by some of the leaders of their respective fields. This volume opens with Michael Atiyah reminiscing about his encounters with Lef- schetz and Mexico.´ Topics covered in this volume include symplectic flexibility, Chern– Simons theory and the theory of classical theta functions, toric topology, the Beilinson conjecture for finite-dimensional associative algebras, partial monoids and Dold–Thom functors, the weak b-principle, orbit configuration spaces, equivariant extensions of differential forms for noncompact Lie groups, dynamical systems and categories, and the Nahm pole boundary condition. • azro ta. Editors al., et Katzarkov

ISBN 978-0-8218-9494-1

9 780821 894941

CONM/621 AMS