Trieste Lectures On Mirror Symmetry
Kentaro Hori*
Department of Physics and Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
Lectures given at the Spring School on Superstrings and Related Matters Trieste, 18-26 March 2002
LNS0313003
* [email protected] Abstract These are pedagogical lectures on mirror symmetry given at the Spring School in ICTP, Trieste, March 2002. The focus is placed on worldsheet descriptions of the physics related to mirror symmetry. We start with the introduction to general aspects of (2,2) supersymmetric field theories in 1 +1 dimensions. We next move on to the study and applications of linear sigma model. Finally, we provide a proof of mirror symmetry in a class of models. Contents
1 Aspects of A/* = 2 Supersymmetry 115 1.1 (2,2) Supersymmetry in 1 + 1 Dimensions 116 1.1.1 (2,2) Superspace and Superfields 120 1.1.2 Decoupling of Parameters 121 1.2 Non-Linear Sigma Models and Landau-Ginzburg Models . . . 122 1.2.1 The Models 122 1.2.2 Witten Index 124 1.2.3 R-Symmetry 125 1.2.4 Twisting 127 1.2.5 Supersymmetric Ground States 129 1.2.6 Renormalization Group Flow 130 1.2.7 Complexified Kahler Class and Complex Structure . . 131 1.3 D-Branes and Orientifolds 132 1.3.1 A-branes and B-branes 132 1.3.2 A-orientifolds and B-orientifolds 133 1.3.3 Overlaps with RR Ground States and BPS Basses . . 134 1.4 Summary 136
2 Linear Sigma Models 138 2.1 The Models 140 2.2 Non-Linear Sigma Models from Gauge Theories 143 2.2.1 QP^-1 143 2.2.2 0(-1) 0 0(-1) over OP1 (Resolved Conifold) 145 1 N 2.2.3 0{-d) over QP^" or C /Zd Orbifold 147 2.2.4 Toric Manifolds 148 2.2.5 Hypersurfaces and Complete Intersections 149 2.3 Low Energy Dynamics 152 2.3.1 The QP^-1 Model 156 2.3.2 The "Phases" 160 2.3.3 (LG) Orbifold as an ffi. Fixed Point 162 2.3.4 A Flow from (LG) Orbifold 163
3 Mirror Symmetry 164 3.1 T-duality 166 3.1.1 T-duality as Mirror Symmetry 167 3.1.2 T-duality for Charged Fields 168 3.2 Vortex Instanton Effects 170 3.3 The Proof 173 3.4 Some Applications 173 3.4.1 Projective Spaces 173 3.4.2 Resolved Conifolds 175 1 N 3.4.3 0{-d) over QP^'" or Perturbed Orbifold C /Zd . . 176 3.5 The Case of Hypersurfaces in Toric Manifolds 179 3.5.1 Low Energy Dynamics 182 3.5.2 Geometric Description of the Mirror 183 3.6 Dilatonic Backgrounds 186 3.6.1 2d Black hole and Liouville Theory 186 3.6.2 Squashed Toric Manifolds 189
3.6.3 Orbifolds 191
4 Bibliography 192
References 197 Mirror Symmetry 113
Introduction
For nearly a decade, duality has been playing the central role in most of the development in superstring theory and supersymmetric quantum field theories. Duality is the idea that the strong coupling limit of one theory is equivalent to the weak coupling limit of another apparently different theory. As such, duality can be used to learn about the non-perturbative aspects of a given theory. A typical example is the electric-magnetic duality in M — 4 supersymmetric Yang-Mills theory (Montonen-Olive duality). There are various generalizations, to other gauge theories in four-dimensions with lower supersymmetry (such as Seiberg duality of Af — 1 gauge theories), to supersymmetric field theories in other dimensions, and to superstring backgrounds such as S-duality of Type IIB superstring (string theory version of Montonen-Olive duality). Mirror symmetry is one of them — it is a duality in (2,2) supersymmetric field theories in 1 + 1 dimensions. Most of the dualities found so far remain conjectural. They have been tested by computing and comparing quantities that are protected from quan- tum corrections, but of course such tests, no matter how many, are insuf- ficient to prove the statement of duality which usually takes the form of exact equivalence of two theories (either at all scales or in the deep infra-red limit). Abelain dualities in free field theories, however, are different.1 The typical example is the electric-magnetic duality in Maxwell theory (U(l) gauge theory in 3 + 1 dimensions) without charged matter field. Duality in such theories can be proved by functional Fourier transform. Most of the non-trivial dualities mentioned above may be regarded as generalizations of such abelian duality. For example, mirror symmetry is a generalization of T-duality, or radius inversion duality, of compact scalar fields. It is then a natural idea to try to extend the functional Fourier transform (or "dualization") of free field theories to interacting theories. In fact, mirror symmetry for a certain class of (2,2) supersymmetric field theories can be proved in such a way. One can quantitatively analyze the effect of the inter- action to the dualization process and find out the mirror of a given theory. One purpose of this lecture is to describe this proof of mirror symmetry. As stated, mirror symmetry is just one example of the whole set of du- alities in (supersymmetric) quantum field theories and string theories. It is extremely interesting to see whether and how the proof presented here may
1By definition free fields have no interaction, but the "coupling constant" can be defined as the constant g that appears in front of the action as 1 /g2. 114 K. Hori be applied to other dualities. Actually, there is another aspect of mirror symmetry. (2,2) supersym- metric field theories, especially those with superconformal symmetry, can be used to construct string backgrounds. For example, supersymmetric sigma models on six dimensional Calabi-Yau manifolds have (2,2) superconfor- mal symmetry and can be used to compactify Type II superstring to four- dimensional Minkowski space with M — 2 supersymmetry. If D-branes and orientifolds are included one can also realize M — 1 supersymmetry in four- dimensions, which is extremely important for application to particle physics. Duality in such systems therefore provide examples of duality in important superstring backgrounds. In particular, this will be useful in understanding worldsheet non-perturbative effects. The lectures consists of three parts. In the first part (Section 1), I de- scribe general aspects of theories on 1 + 1 dimensions with (2,2) supersym- metry. We start with formal aspects derived from the algebraic structure of the symmetry generators. One important fact is that there are two special representations of the algebra — "chiral" and "twisted chiral"— and fields and parameters in different representations in a given system are in a cer- tain sense decoupled from each other. We then move on to more specific models including non-linear sigma models on Kahler manifolds and Landau- Ginzburg models. We discuss important ingredients such as Witten index, R- symmetry, twisting, supersymmetric ground states, renormalization groups ow (and non-renormalization theorem). Finally we study theories formu- lated on 1 +1 dimensional space with boundaries and crosscaps, and explore the boundary conditions and parity symmetries that preserve a half of the (2,2) supersymmetry. The second part (Section 2) is devoted to a class of models called (gauged) Linear Sigma Models which have been introduce in late 1970's but were revived by the seminal work by Witten in 1993. Such models provide a way to describe the non-linear sigma models globally, without relying on the choice of coordinate patches. This makes possible to study the non-perturbative aspects, which would have been difficult in patch-wise description of the models. Another virtue of Linear Sigma Models is that the chiral and twisted chiral parameters are concretely realized in F-terms and twisted F-terms of the classical Lagrangian, and one can make full use of the power of (2,2) supersymmetry — the decoupling of the two sectors in particular. The last and the main reason to introduce this subject here is that the set-up is used in an essential way in the proof of mirror symmetry. Mirror Symmetry 115
In the third part (Section 3), the proof of mirror symmetry given by Hori (the lecturer) and Vafa is described. We first introduce T-duality, the abelian duality in compact scalar fields in 1 + 1 dimensions, for bosonic as well as supersymmetric cases. We next discuss how the effect of interaction can be taken into account in the dualization process. We consider the sys- tem that can be viewed as the compact scalar field fibred over some base space. "Interaction" corresponds to the variation of the radius of the scalar (as a function of the base), including vanishing radius at some locus. We see that the most important role is played by the vortex instantons of the Linear Sigma Model that realizes the system. Once it is done, the proof of mirror symmetry is a simple manipulation of integrating out heavy fields. We provide some application of mirror symmetry to study dynamical aspects of the non-linear sigma models. We also describe the mirror symmetry for string backgrounds with non-trivial dilaton profiles. Finally, a remark on the bibliography of the subject is given. This include a brief description of the history of mirror symmetry.
1 Aspects of J\f = 2 Supersymmetry
Let us consider a field theory of a complex scalar field + + (1-1) Here do and d\ are derivatives with respect to the time and space coordinates x1. The system has an unusual symmetry which mixes the bosonic and fermionic fields: The action is invariant under 5 Sip± — ±iezf(do ± di), Sip± — =F«eT(ao ± di), where the transformation parameters e±, e± — are fermionic. The corre- sponding conserved charges are therefore fermionic: 1 Q± = -^F ± D^&X = Q*±. 116 K. Hori The "square" of such a transformation are proportional to the time and space translations [SiMO = ~ del) (5b + dJO + i(e\e\ - (5b - dy)0, (up to equation of motion). Such a symmetry is called a supersymmetry and the conserved charges are called supercharges. In the present case, there are two real left-handed supercharges (Re(Q+) and Im(Q+)) and two real right-handed supercharges (Re(Q_) and Im(Q_)). Hence the supersymme- tiy is called (2,2) supersymmetry. The model has also two kinds of U(l) symmetries: iQ U(l)v : i>± e- V±, i>± U(1)A : i>± i>± with the conserved charges Fy — ^ J(fp, FA — ^J(— 1 ip^ip^dx . The supercharges Q±1 transform in the same way as ip±1 ij)± under these U(l) symmetries. Such symmetries are called (vector and axial) R-symmetries. In this section, we study some general aspects of quantum field theories in 1 +1 dimensions with (2,2) supersymmetry. We start with formal aspects derived from the algebraic structure of the supercharges. One important fact is that there are two special representations of the algebra — "chiral" and "twisted chiral". We next move on to more specific models including non-linear sigma models on Kahler manifolds and Landau-Ginzburg mod- els. We discuss important ingredients such as Witten index, R-symmetry, twisting, supersymmetric ground states, renormalization groups flow (and non-renormalization theorem). Finally we study theories formulated on 1 + 1 dimensional space with boundaries and explore the boundary conditions that preserve a half of the (2,2) supersymmetry. 1.1 (2,2) Supersymmetry in 1 + 1 Dimensions In general, a 1 + 1 dimensional quantum field theory with (2,2) super- symmetry has four supercharges, Q+: Q+: Q-: Q_ which are complex but related to each other by hermitian conjugation — Q±. Together with Hamiltonian H, momemtum P and Lorentz generator M, they obey the following (anti-)commutation relations: 2 Ql = Q2 = Ql = Q _ = 0, (1.2) Mirror Symmetry 117 {Q±,Q±} = H±P, (1.3) {Q+,Q-} = Z, {Q+,Q_} = Z*, (1.4) {'Q-,Q+} = Z, {Q+,Q_} = Z*, (1.5) [iM, Q±] = =FQ±, [»M, g±] = =fQ±, (1.6) Z and Z are central charges. By Q± — Q± and (1.3), the spectrum is positive semi-definite, H > 0, and zero energy states are those annihilated by all the supercharges Q±: Q±, i.e. supersymmetric ground states. There are also vector and axial R-charges [iFv, Q±] = ~iQ±, [iFv, Q±] = %Q±, (1-7) [iFA, Q±] = TiQ±: [iFA,Q±] = ±iQ±• (1.8) Whether there are conserved R-charges depends on the system, although the fermion numbers emFv and emFA should always be conserved. This algebra may be considered as the dimensional reduction of d — 4 M — 1 supersymmetry: FA and Z are the rotation and momenta in the reduced directions while Fy descends from the H — 1 R-charge. The model with the action (1.1) has indeed (2,2) supersymmetry. It is a simple exercise to check, using the canonical commutation relations Mx^dofty1)] = 2tti5(xl-yl), {^(z1),^1)} = 2tvS(X1 -y1) etc., that the supercharges and the R-charges obey all of the above (anti-)commutation relations, in which the central charges are set equal to zero, Z — Z — 0. The (2,2) algebra has an interesting automorphism called the mirror automorphism: Fv • FA, (1.9) Z^Z. Mirror symmetry is an equivalence of two theories where the (2,2) generators are exchanged in this way. The algebra is invariant under two kinds of parity actions — A-parity and B-parity. In addition to P —» —P and M —> — M, they act on the generators as 118 K. Hori Like R-symmetries, whether there is a parity symmetry depends on the system. Generators that are invariant under an A-parity are H, FA, Z1 QA = Q+ + Q- and QA = Q+ + Q~- Generators invariant under a B-parity are H, FY1 Z1 QB-Q+ + Q- and Qjg = Q+ + Q_. A-parity and B-parity are exchanged under the mirror automorphism, and so are the two sets of operators. We now describe several general consequences of this algebra. These are strikingly powerful method in analyzing (2,2) supersymmetric theories. Supersymmetric Ground States Let us quantize the system on a circle with a periodic boundaiy condition which sets Z — Z — 0. The operators (Q,F) — (QA,FA) or {QB,FV) obey the following commutation relations {Q,Qt} = 2if, (1-10) Q2 = 0, (1.11) [F,Q] = Q. (1.12) By the second and the third equation, the Hilbert space of states TL can be regarded as the Q-complex; % 'Hq~1 Uq+1 A • • •, (1.13) where TLQ is the subspace of R-charge F — q. As noted above, F is not necessarily a conserved charge and the grading q may not be a Z-grading. However, the fermion number emF is always conserved and thus there is at least a Z2 grading. By the equation (1.10), Q-cohomology classes are in one to one correspondence with the supersymmetric ground states; ^ ^ HH0) ._ Ker Q-.W^W* "susy - ^ KQ) - Im Q . ^ w • Witten index is the Q-index (or the Euler characteristic of the complex (1.13)) TM-lfe-^ = ^(-l)«dimiT Chiral Ring An operator O is called a chiral operator if [QB, O} — 0 and twisted chiral operator if [QA,0} — 0. It represents a Q — QB/QA cohomology class of operators. One can show from the supersymmetiy algebra (with Z — Z = 0) that if O is a chiral operator, [QB,0} = 0, then [(H ±P),0] = {QB, [Q±, O}]. Thus, the worldsheet translations do not change the QB- cohomology classes. If 0\ and O2 are two chiral operators, the product O1O2 is also a chiral operator. Same can be said on twisted chiral operators. Thus, Q-cohomology classes of operators form a ring, called the chiral ring for Q — QB and twisted chiral ring for Q — QA- Twisting By Wick rotation, we obtain the Euclidean theory where the group of rota- tion is Spin(2) generated by Me — iM. It makes a perfect sense to formu- late the Euclidean theory on an arbitrary two dimensional surface S with a Riemannian metric h as long as the spin structure is chosen. However, it generically loses supersymmetiy. To see this, we note that a supervariation of the action would be given by where e is the fermionic variation parameter, which is a section of the spinor bundle, and G^ is the supercurrent, a vector with values in the spinor bun- dle. If (S, h) is curved, there is no covariantly constant spinor and thus the action is not invariant under any supervariation. The only way to preserve a supersymmetry is to change the system so that some of the supercharges Q±, Q± are scalars and the corresponding variation parameters (being scalars) can be covariantly constant, V^e = d^e — 0. This is the idea of twisting. If a vector R-charge Fy is conserved and integral, one can twist the theory by declaring that ME + Fy to be the new rotation generator. This is called the A-twist. The same procedure for axial R-symmetry is called the B-twist. Af- ter B(A)-twist, QB (QA) becomes scalar and thus there is a supersymmetry even when the worldsheet is curved. Correlation functions with insertions of only (twisted) chiral operators are independent of the choice of worldsheet metric. For this reason the twisted model is sometimes called topological B(A)-model. Sphere 3-point functions determine the structure constants of the (twisted) chiral ring. 120 K. Hori Field/State Correspondence When a B-twist is possible, there is a one to one correspondence with chiral ring elements and supersymmetric ground states. Consider a worlsheet of semi-infinite cigar geometry as in Fig. 1, and perform the B-twisted path- I • Figure 1: The semi-infinite cigar integral in the interior of the cigar, with a chiral ring element fa inserted at the tip. This leads to a wavefunction at the circle boundary. The flat cylinder region is not affected by twisting, and thus the wavefunction can be regarded as a state of the untwisted theory. Because of the twisting in the curved region, the fermions are periodic along the circle — namely the state belongs to RR sector. In the limit of infinite length, all the excited states are projected out and we are left with the zero energy state |'i). This is the supersymmetric ground state corresponding to fa. The same applies to A-twistable theories where the ground states corresponds to twisted chiral ring elements. 1.1.1 (2,2) Superspace and Superfields We now introduce superfield formalism which is a useful tool in constructing supersymmetric models but also important in making manifest the decou- pling of parameters. Let us extend the Minkowski space with time and space coordinates x°,xl by including four fermionic coordinates 6>+, 6>~, 6>+, 9 (re- lated by complex conjugations Superfields are functions on this superspace. We introduce differential operators ± Q± = dd± Q± = fflt —~M d±- where d± — — \ (-£p ± gfr). They obey the relations {Q±, Q±} = —2id± with all other anti-commutators vanishing. Another set of differential operators d9± Qfft Mirror Symmetry 121 anti-commute with Q± and Q±. These obey the similar relations {D±1 D±} — 2 id±. _ A chiral superfield $ is a superfield which satisfies D± — 0. It has the ± ± a + J ^-expansion of the form §{xi\e ,e ) = c/)(y+ 9 i)a(y») + 0 0~F(y> ) where y± — x± — id±6 . A twisted chiral superfield $ is a superfield which satisfy D+<& — — 0. It has a similar expansion as $ with 6~ and 9 exchanged. Holomorphic combinations of (twisted) chiral superfields are also (twisted) chiral superfields. We now construct action functionais of supefields which are invariant under the transformation 5 — e+ — e_ Q+ — e+ Q_+e_ Q+. There are three kinds of them: D-term, F-term and twisted F-terms, which are expressed as J d2xd40K^i) — j d2xd6+d6~dd~dt K{Ti), (1.16) 2 2 2 + J d xd 9W($i) = J d xd9-de W($i) _± Q, (1.17) J d2xd2ew($i) = J d2xde~de+W($i) ,. (1.18) Here K(—) is an arbitrary differentiable function of arbitrary superfields T^ W( ± iaqv J 1 ia ± <7/(2^, 9 , 0*) ^ e ^ G1(x' , e- 0 , e'^) (1.19) ± ± 1 gz(x^, e ,e ) ^ j^g^x* , e^^). (1.20) 1.1.2 Decoupling of Parameters An extremely strong property is the decoupling of the chiral and twisted chiral parameters. Suppose we have a (2,2) supersymmetric theory and 122 K. Hori integrate out heavy fields or high frequency modes to obtain a low energy effective theory. Then, parameters of superpotential at high energy theory cannot enter into twisted chiral superpotential of low energy theory. This can be shown by using the idea of promoting parameters to fields. As we have seen, for (2,2) supersymmetiy (twisted) superpotential has to be a holomorphic function of (twisted) chiral superfields. Thus a parameter of superpotential must be promoted to a chiral superfields, but that cannot enter into the twisted superpotential at lower energy. Likewise, parameters of twisted superpotential cannot enter into the superpotential of low energy theory. Using this fact, one can derive various kinds of supersymmetric non-renormalization theorems. An alternative way to see this "decoupling" is to look at the correlation functions of topological models: A(B)-modd correlation functions depend only on twisted chiral (chiral) parameters and depend on them holomor- phically. This is because all the supersymmetric deformations except by twisted chiral (chiral) parameters are by Q_A(Q.g)-exact operators. For ex- ample, consider deformation by a chiral parameter. It is given by an F-term f d0-d0+ SW. Using the lowest component Sw of SW, this can be written as where we have used [Q+, Sw] — 0, {Q+^ Q+} — H + P and stokes theorem. Since it is Q^-exact, it annihilates the correlation function of Q^-closed operators. 1.2 Non-Linear Sigma Models and Landau-Ginzburg Models We introduce supersymmetric non-linear sigma models on Kahler man- ifolds and Landau-Ginzburg models. We write down the classical action and supercharges, and compute the Witten index. We also study anomaly of some fermion number symmetry, the space of supersymmetric ground states, and renormalization group flow. 1.2.1 The Models Non-linear Sigma Model Let K( = J d*9K(&,W). (1.21) In terms of component fields, it is expressed up to total derivatives in xIJl as 1 Ckin - -g^fd^ + ig^_{DQ + + - D1)r+ (1.22) where :— c^f/4 + d^Y1^^ and R^i is the Riemannian curvature of the metric g^. The last term of (1.22) can be eliminated by the equation of motion, and the rest is invariant under coordinate change of C71. Thus, applying this construction for each coordinate patch of a Kahler manifold M, we can define an action for a map $ : S —» M, where ip±1 are spinors with values in Landau-Ginzburg Model l 1 For a holomorphic function W{cj) 1..., fa ), one can add the the F-term 2 i f d 9W{&) = F diW - didji>\iP_ to the Lagrangian: 2 C = J d(J d 9W(&) + c.c. ^ -gVfyWdjW - DidjW^L - DTdjWi>_i/+ (1.24) where we have eliminated F, F in the second step. We now have the potential term g'^d^WdjW plus the fermion mass (or Yukawa) term —W"( The model is called the M — 2 supersymmetric Landau-Ginzburg model on (M, g) with superpotential W. Supercharges The supersymmetiy variation of the component fields are 8ft = e+V- - ^ = -e+V^- + ^ = + e+JP% 0± = -2ie-d+~ft + e+F, (1-25) = + e-i^, ^ = 2ie+3_? + , where can be replaced by the solution F% — r^-^H-V7— — gll9[W. By Noether procedure, we find the following expression of the supercurrents G± and G+ G°± = gi3(do ± d^ftfi =F G± = ± " W (1.26) = ± d1)ft ± i,%diW, ci = ± W ± ^ W x x The supercharges are given by Q± — J dx G±i Q± — f dx (f±. 1.2.2 Witten Index F H Let us compute the Witten index I — Tr%RR(—l) e~P , regarded as the torus path-integral with periodic boundary condition in both directions. Us- ing the deformation invariance we can take the zero size limit of the torus. Then, it reduces to the integral of only constant modes. Let us first consider NLSM on a compact Kahler manifold (M,g). The space of bosonic constant modes is nothing but M itself. Fermionic constant modes span the sum of two copies of tangent bundles (one from (•?/;_, another from (•?/>+, V'-))- F°r a constant mode the Lagrangian (1.22) reduces to __ _ C = (1.27) By the substitution —» dz%, -> dz'1, and V>!_ —» gl3i>j, it 3 l, k 1 3 1 can be regarded as R ikfl> 4>jdz Adz — R ^ ^ where R}j is the curvature 2-form of the bundle T^M. Then the index is given by I — f = f det f=x(M) (1.28) jMfji JM v27r J Mirror Symmetry 125 The index is the Euler number of M. Let us next consider the LG model. We assume that the potential U — | dW\2 grows at infinity in the non-compact space M so that the index is well-defined. We also assume that all the critical points are non-degenerate — they are isolated and that the Hessian d^djW is of maximal rank. The classical action reduces on the constant modes to £ = - gVfyWdjW - DidjW^XirL - D-.d-jWf^. (1.29) It may appear difficult to perform this integration. However, we note that this O-dimensional system also has a supersymmetry: simply eliminate d^cj)1 in (1.25). Then, the integral localizes on the fixed point of the supersym- metry variation and the exact answer can be obtained by the quadratic approximation around each fixed point. The fixed points are at the critical point of W. Thus, the index is given by = #(critical points of W). (1.30) In the first expression, the numerator and denominator are from the fermionic and bosonic integrals respectively, both in the quadratic approximation. 1.2.3 R-Symmetry Recall that (2,2) algebra has vector and axial R-charges. In NLSM and LG model, we examine the condition for conservation of R-charges. Classical Level We start with examining the invariance of the classical Lagrangian under vector/axial R-rotations (1.19)/(1.20). The D-term is always invariant (for Qv — QA — 0) since d46> is invariant under phase rotations of 9±. Thus, the NLSM Lagrangian is invariant under both vector and axial R-rotations. LG model has also F-term f d?9W(<&). The measure d26 is neutral under axial R-rotation, and thus there is an axial R-symmetry at the classical level. On the other hand, d20 has charge —2 under the vector R-rotation. Thus, the F-term is invariant only if it is possible to transform the fields in such a way that W(<&'1) is rotated by charge 2. This transformation should be holomorphic and preserve the Kahler metric. Thus, the LG model has classical vector R-symmetry if there is a one-parameter family of holomorphic 126 K. Hori automorphisms fp:X—>X such that fjjg — g and f*W = (1.31) Such a superpotential is called quasi-homogeneous. Anomaly Exact symmetry of the quantum theory must preserve the path-integral measure VXe~s^x\ not just the classical action l -2 igtflrLDtp- + 2igrjip+Dzip' +. (1.32) The index of the Dirac operator is given by IndexD^ = — IndexD^ = Jsci( [B] [B] - 2j3a(M). (1.33) Thus, the axial R-symmetiy is anomalous if c\{M) ^ 0. It is anomaly free only if ci(M) = 0, i.e. only if M is Calabi-Yau. However, even for non-Calabi-Yau, not all the axial rotations are anomalous. For example, if f Summary In NLSM on M, there is always U(L)v R-symmetry but U(1)A may be broken to its discrete subgroup. For LG model, U(l)v is unbroken iff W is quasi-homogeneous, while U(1)A is not broken by superpotential. Typical Mirror Symmetry 127 cases are presented in the table. U(l)v U(1)A CY sigma model O O sigma model on M o X with c1(M)^0 LG model on CY X with generic W O LG model on CY with quasi-homogeneous W o O Notice the symmetry of the table: it is invariant under left-right/top-bottom exchange. This is actually not a coincidence — some NLSMs and some LG models are exchanged under mirror symmetry that exchanges U(l)y and U(\)AI mapping quantum effect of one to the classical property of the other, or vice versa. In particular, we note that the mirror of NLSM on M with ci(M) ^ 0 cannot have U(l)y symmetry and therefore cannot be a NLSM again; if it exists it should have some U(l)v breaking F-term. 1.2.4 Twisting We can consider A-twist of non-linear sigma models (where Fy is conserved) and B-twist of Landau-Ginzburg models on Calabi-Yau target spaces (where FA is conserved). We recall that ME+FV (ME+FA) is the rotation generator after A-twist (B-twist). The quantum numbers of the fermions change under twisting, as shown in the table. ME FV FA ME + FV ME + FA 1 -1 1 0 2 IFT+ -1 -1 -1 -2 -2 I>- 1 1 -1 2 0 -1 1 1 0 0 Let us first consider A-twist of NLSM on a Kahler manifold (M, g) with B-field B. The change of the spin motivates to rename the fields as xl — ^-i X1 — pi — i?- and = 'tp\. Then, under the scalar supersymmetry 128 K. Hori 8 — cQa, the fields transform as i k 6fa = e \ W = 0, 64 = 2%•. iedF.farhch + +eT)^ fT^., fi ,v . X (1.34) 7 k = ef , <55? = 0, 5ffz = -2iedz^ + eVl^zx - M tl tv 1 iq For a differential form a G f2 (M), Oa :— ai1...ivj1...jqX •••'x x' •••X obeys 50a — eOda Thus, the Q^-cohomology classes of fields are in one to one correspondence with the de Rham cohomology classes of M. Configurations invariant under QA are those with DZFA — 0, namely holomorphic maps of the worldsheet to M. Thus, A-model correlation functions receive contribu- tions only from holomorphic maps. We note that the classical action for a holomorphic map $ : S —» M is expressed as S= I gijh^dnftdvfiVhtfx-i f fa*B = f f iB) (OaiOa2Oa3)g=o = J2 [ai], [a2], [03] respectively. Let us next consider B-twist of LG model with superpotential W on a CY space M. This time, an appropriate renaming of the fermions is 0j — gij(ipJ- — i>\)-> rp — ~iftl_ + p\ — p| = Under the scalar supersymmetry S — tQs the fields transform as 5 fa = 0, 56j = -edjW, 0 = of, 8if = 0, (1-37) = 2eJ^dvfa. The Qs-cohomology group is isomorphic to the space of holomorphic func- tions on M modulo holomorphic derivatives of W. Q^-invariant configurar tions are constant maps to the critical points of W. In particular, genus zero three-point functions are f } = E t%wlt ' (1-38) psECrit(VK) rj Mirror Symmetry 129 where the sum is over the critical points of W and d2W the Hessian matrix of second derivatives of W. Note that the Hessian and the determinant both depend on the choice of the coordinates. The ambiguity is fixed by fixing a nowhere vanishing holomorphic top form f2 and choosing a coordinate system such that f2 is written as dz1 A • • • A dzn. One could also consider the case where M is a compact CY manifold and W — 0. In such a case, the QB- cohomology group is identified as if°'*(M, A*Tm), under the identification rf -H- dzl, 6,i -H- Geneus zero three-point functions for a CY 3-fold M with holomorphic 3-form f2 are given by / N\ A 4 A/i^yfc A(], (1.39) JM 0 for/ii,/i2,/i3 eH ^(M,TM). 1.2.5 Supersymmetric Ground States Since the supersymmetric ground states are in one to one correspondence with the QA(Qif)-cohomology classes of fields when the A(B)-twist is pos- sible, one can determine the space supersymmetric ground states using the above analysis of the twisted models. The space of supersymmetric ground states of NLSM on M is isomorphic to the cohomology group of M which is in turn the same as the space of harmonic forms on M; n %USY ~ 0 ir™(M). (1.40) p,q=1 Here Hp,q(M) is the space of harmonic (p, q) forms, or (p, q)-th Dolbeault cohomology group. If M is Calabi-Yau, the vector and axial R-charges of the ground states are qv = ~P + on Hv,q(M). (1.41) qA—P + q — n If M is not Calabi-Yau, the axial R-symmetry is anomalous, and only the expression for qy makes sense. In any case, Witten index is given by _p+<*(M) = X^-l^if^M) = x(M), reproduc- ing the formula (1.28). If two Calabi-Yau manifolds M and M are mirror to each other, the ground states in Hp,q(M) are mapped to the ground states 130 K. Hori in Hn p,q(M) so that the vector and axial R-charges are exchanged. In particular, there is a relation between M and M in the Hodge numbers hP,Q = dimHp,q: _ hp,q(M) = hn~™{M). (1.42) The supersymmetric ground states of LG model are in one to one corre- spondence with the critical points of the superpotential W, if all the critical points are non-degenerate. If M is a non-compact Calabi-Yau, the axial R-charges are conserved and they are all zero qA — 0 on the ground states. (1-43) The reason is that the ground state wavefunctions in the dimensionally reduced model (supersymmetric quantum mechanics) is given by middle- dimensional forms. This also reproduces the index formula I — #(crit. pts. afW). If a NLSM on M is mirror to a LG model, then the vector R-charge of the NLSM ground states has to be zero. Namely, Hp>q(M) — 0 if p ^ q (the Hodge diamond of M is diagonal). 1.2.6 Renormalization Group Flow The non-linear sigma model is scale invariant at the classical level. However, the metric is renormalized with the beta function given by where Ricci tensor term is the one-loop effect and H are from higher loops and are convention dependent. If Ricci tensor is positive definite, at higher energies the metric is larger and the sigma model coupling is weaker. Thus NLSM is asymptotically free for Ricci positive manifolds. For a Calabi-Yau manifold (RJJ — 0), the sigma model is scale invariant at the one-loop level. This applies also to supersymmetric NLSM on a Kahler manifold M. The H terms are modified but still convention dependent. However, there is a nicer story: The complex structure of M is not renormalized, and the Kahler class is renormalized only at the one-loop level. The latter means that the cohomology class of the Kahler form to — | g-tjdz'1 A dz^ flows exactly as = ci(M). (1.45) Mirror Symmetry 131 If M is Calabi-Yau, the Kahler class is invariant under RG flow. In fact, it is believed that a CY sigma model flows in the infrarred limit to a non-trivial superconformal field theory which is determined uniquely by the complex structure, Kahler class and the class of B-field. The Landau-Ginzburg superpotential is invariant under renormalization group flow, except for the overall scaling. This is the non-renormalization theorem of superpotentials, which is one of the strongest properties of (2,2) theories or any dimensional reduction of Ad H — 1 theoories. This was first shown by Grisaru-Seigel-Rocek using superfield perturbation theory and a simpler and sometimes stronger argument using holomorphy was found by Seiberg. 1.2.7 Complexified Kahler Class and Complex Structure Complex structure of M is parametrized by chiral parameters. This is ob- vious since the complex coordinates themselves are represented by chiral superfields, and the information of complex structure resides in the transi- tion function at the overlap of patches. In the LG model, the parameters of the superpotential are of course chiral. To see what are the twisted chiral parameters of the system, let us look at the A-model correlation functions which depend only on twisted chiral par rameters, holomorphically. They receive contributions of holomorphic maps (1.46) of the cohomology classes, and the depencence is holomorphic. This shows that the complexified Kahler class yields twisted chiral parameters. Thus, complex structure and the parameters in the LG superpotential are chiral, while the complexified Kahler class (1.46) is twisted chiral. We will see that the linear sigma model makes this result more transparent for a certain class of target spaces. Since Mirror Symmetry exchanges chiral and twisted chiral, the com- plexified Kahler class of the one manifold is mapped to the complex struc- ture of the mirror, and vice versa. This is consistent with the relation h1>1(M)=hn~1>1(M). 132 K. Hori 1.3 D-Branes and Orientifolds Recenetly, D-branes and orientifolds have become increasingly impor- tant in string theory. D-branes are boundary conditions on the open string worldsheet, and orientifolds are to gauge parity symmetries of the world- sheet. We will consider those preserving a half of the (2,2) supercharges. The relevant halves are QA — Q+ + Q- and Q\ or QB — Q+ + Q- and QG. A D-brane/orientifold preserving them will be called A-brane/A-orientifold or B-brane/B-orientifold respectively. If the worldsheet theory gives rise to space-time physics, we are also interested in D-branes/orientifolds that preserve some space-time supercharges. 1.3.1 A-branes and B-branes Let us consider a D-brane wrapped on a submanifold 7 of M. The open string boundaiy condition associated with this D-brane is d^cj)1 and (V>- + 'tp+Y are tangent to 7, die/)1 and (ip- — f/'+)/ are normal to 7. Here we are using the real coordinates of M. The condition on the fermions is required from Af — 1 supersymmetry. We examine the condition for an M — 2 supersymmetry — A or B. QA and generate the variation 5 with e+ = e_, which does 6A4>1 — e+V- — = ei(— + ^iW1- + i>+)i where ei and ti are the real and imaginary parts of e+. In the real coordinates this reads as SA QB and QG generate the variation 5 with e_ = — e+. The bosonic fields ft transforms as 5b ft — e+(V;- + i>+)i or in the real coordinates t K &B+) + te2(^- + (1-49) Combining with the condition from Af — 1 supersymmetry we find that J applied to tangent vectors to 7 are tangent to 7. Namely, 7 must be a complex submanifold of (M, J). In fact this is sufficient for B-type syper- symmetry. One can also have gauge field A on the brane: QB invariance requires the curvature FA to be a (1, l)-form. A superpotential W has to be locally constant on B-branes. If the model has space-time interpretation (e.g. Calabi-Yau sigma model), it is more interesting to consider D-branes preserving a part of space-time supersymmetry. This imposes an additional constraint. For A-branes, space- time supersymmetiy requires the Lagrangian submanifold 7 to be special Lagrangian, = e'^vol(7) for a constant phase elS where f2 is the top holomorphic form of M. (There is a farther condition of criticality of the space-time superpotential which is generated by disc instantons.) For B- branes, it requires the gauge field to obey Hermitian-Yang-Mills equation, which depends on the choice of Kahler structure. 1.3.2 A-orientifolds and B-orientifolds An orientifold is to gauge a parity symmetiy of the worldsheet theory. As we have seen there are two kinds of parities in (2,2) theories. A-parity exchanges Q_|_ and Q- whereas B-parity exchanges Q+ and Q_. The supercurrent are given in (1.26) and the supercharges takes the form 1 (do ± di)(/nl>± =F i^W' Q± = J dx (1.50) 1 Q± = f dx i,±(do±d1)(t>±i^W' We consider the parity symmetry of the form T~f2 where fI is the map 1 1 x —x , ^ and r is some isometiy of M. Exchange of Q+ and Q- requires that r maps holomorphic coordinates to anti-holomorphic coordinates. Since r is an isometry this is equivalent to the condition that T flips the sign of the Kahler form, T*CJ — —w. If there is a superpotential W, r : W —>• W is also required. Exchange of Q+ and Q_ occurs if r pre- serves the holomorphic structure of M. Also it should flip the sign of the superpotential, r : W —>• — W. 134 K. Hori 1.3.3 Overlaps with RR Ground States and BPS Basses The simplest two-dimensional surface with a boundary has the topology of a disc. In particular, we consider the ones with the geometry of semi-infinite cigar that appear in Fig. 1, which yield the overlap of the boundary state with the RR ground states. We denote by II" the diagram with boundary condition "a". Likewise, one can consider MP2 diagram Ilf associated with a n: =a() ~ >•> parity symmetry P,which yield the overlap of the crosscap state with the RR ground states. (Boundary and crosscap states are "states" that characterize the boundary condition and the parity symmetry from the closed string channel.) By the factorization of cylinder, Klein bottle and Mobius strip diagrams, we obtain the following formulae for the Witten indices (-1)^2 = nf^np, (i.5i) Here T-La,b is the space of states of the open string stretched from the brane a to the brane 6, and T-LPlP-i is the space of closed string states on the circle with the PiP^-twisted boundary condition; gl5 is the inverse of the ground state matrix gjL — (j]i); and n~ is the disc diagram in which the boundary is placed on the right of the semi-infinite cigar (or the orientation of the boundary circle is reversed). For boundary conditions and parity symmetries that preserve A- or B- type M — 2 supersymmetries, these diagrams depend on the chiral and twisted chiral parameters in a specific way. Let us consider an A-brane or an A-parity in a B-twistable model. We consider the diagrams in which the curved region of the hemisphere is B-twisted and the operators (pi inserted at the tip are chiral. Then, H; and II do not depend on the twisted chiral Mirror Symmetry 135 parameters, but depends on the chiral parameters tl in the following way Anj + tfcg-n^o, (1.52) DiUj-ipC%Uk = Q. (1.53) Here ft is the circumference of the circle of the flat-cylinder region, C^ is the structure "constant" of the chiral ring, which is actually a function of ti- Finally, Di is the natural connection (Berry's connection) of the bundle of RR-ground states. It has been known that the connection V,-^ DiSj + zC-fj is flat for any z. Thus, the overlaps II; and n, are parallel sections of the vacuum bundle with respect to the connection V^) for the values z = i{3 and —i/3. One can also consider A-branes/A-parities in an A-twistable model, where the curved region of the hemisphere is A-twisted and the inserted operators fy are twisted chiral. Then, the diagram can be identified as the disc/MP2 amplitudes of the topological field theory, and the path-integral can be performed using localization technique. The same thing can be said on B-branes and B-parities provided B- twist/twisted-chiral/chiral are exchanged with A-twist/chiral/twisted-chiral. Let us provide some examples. For A-branes in Calabi-Yau sigma models or Landau-Ginzburg models, which are B-twistable, the diagrams do not depend on the size of the target manifold which is a twisted chiral parameter. Thus, the computation can be made in the infinite size limit where the system reduces to quantum mechanics. Then, the diagrams are identified as the integration of the ground state wavefunctions Wj (which are differential forms) over the A-branes: n i=J<*. (1.54) This applies also to MP2 diagrams where L is understood to be the fixed point of the action of parity on the target space. For LG models, it is appropriate to put a constant weight factor Q-IP(W-W) jn integrand (this is associated with a modification of the boundary term so that the vacuum energy of the open string is zero). One can also show that the ground state wavefunctions can be expressed as Wj = li + (Q+ + Q-)cni, where are holomorphic top forms. Then the diagrams can be written as nf = J e-WQi, (1.55) where L~ is the infinitesimal deformation of L so that its VF-irnage, which extends in positive real direction, is rotated by small phase e~'lt. In the 136 K. Hori Calabi-Yau sigma model, in which we choose L to be special Lagrangian, the axial U( 1) R-symmetry is unbroken in both bulk and boundary theory. Then, only the ground states with zero axial R-charge can have non-zero overlaps with the boundary states for L. These are the states corresponding to middle dimensional forms. Of particular interest is the state "0" corre- sponding to the holomorphic volume form f2, with which the overlap is given by the period integral n% = J n. (1.56) This is identified as the mass of the D-brane wrapped on L, which is BPS, namely, preserved a half of the space-time supersymmetry. The bilinear relations (1.51) can be checked using (1.54) together with similar expressions for Ilf — they are nothing but Riemann's bilinear identities. Other examples are B-branes in non-linear sigma models, which are A- twistable. The diagrams depend on the size but one can obtain approximate answers which are good in the large volume limit. The idea is again to use reduction to quantum mechanics, where only the constant maps are taken into account. For simplicity, let us consider the brane wrapped totally on the target space X and supporting a holomorphic vector bundle E. The approximate result is nf = [ + (1-57) Jx The factor ch(i?v) simply emerges from the path-integral of the quantum mechanics living on the boundary circle. The corrections H come from the non-constant maps. The exact answer could be obtained by identifying the mirror of the brane and using the formulae (1-54) in the mirror theory. 1.4 Summary We have studied some aspects of (2,2) supersymmetric field theories. One thing we have noticed is that a (2,2) theory contains two sectors which are decoupled from each other — let us call them A-sector and B-sector. See the table below. Each of them has its own "holomorphy" and this enables us to control many things. The two "holomorphies" never mix. Introduction of A-brane/A-orientifold or B-brane/B-orientifold keeps one holomorphy and looses the other. Mirror Symmetry is an equivalence of two theories under which the two sectors are exchanged. It is usually the case that B-sector is easier to handle Mirror Symmetry 137 compared to A-sector. This is because the "size of a manifold" resides in the A-sector, and B-sector quantities can be computed by going to the large size limit where the sigma model is weakly coupled. Thus, if we know a mirror pair, by combining the knowledge on B-sector of the two we can learn about both sectors of both theories. This practical aspect is of vital importance in studying Calabi-Yau com- pactification of superstring theories, with and without D-branes/orientifolds. Note that Type II string theory on Calabi-Yau 3-folds yields Af — 2 theories in 3+1 dimensions, while addition of branes and taking orientifold projection break Af — 2 supersymmetry, in some cases keeping Af — 1 supersymmetry. Note that A-sector is of interest from the point of view of symplectic geometry while B-sector is of interest from complex analysis or algebraic ge- ometry. Thus one may say that Mirror Symmetry is a "symmetry" between symplectic geometiy and algebraic geometry. Regarding the long and differ- ent histories of these two fields in mathematics, this is quite an interesting and fascinating suggestion. If we pick up one aspect which is well-known in one side, it hits the other side with a complete surprise! 138 K. Hori A B QA = Q+ + Q- QB = Q+ + Q- twisted chiral chiral [QA,0] = 0 [QB,0] = 0 twisted superpotential term superpotential term Jd29W FD2ew A-twist B-twist Me Me + Fv Me ME + FA complexified Kahler class complex structure M -i[B] J A-model: B-model: GW-invariants Variations of Hodge Structure (counting holomorphic curves) (period integrals) A-brane : B-brane : Lagrangian submanifold of (M, w) complex submanifold of (M, J) with flat connection with holomorphic connection Im(l^) = const W — const A-orientifold: B-orientifold: involution to —> —w holomorphic involution W^W W^-W Exercise: Continue the table. 2 Linear Sigma Models In this section, we describe a class of models called (gauged) Linear Sigma Models with (2,2) supersymmetry in 1 + 1 dimensions. The basic idea is to realize a complicated non-linear sigma model as the low energy limit of a simple potential theory. Let us consider a field theory of N real scalar fields 2.1 The Models For a chiral superfield L = f d4# Then, it is invariant under the gauge transformation $ V ^V + i(A-A) (2.2) where A is an arbitrary chiral superfield. A real scalar superfield V that transforms as (2.2) is called a vector superfield. Gauge transformations can eliminate the low components of V and make it into the form + + + + v = e-e~(vo-v1) + e e (vo + v1)-e-e a-e e~a +ie~e+(e~\- + 0+A+) + ie+e~(e-a_ + 0+A+) + e~e+(te~D. Vp is a one-form, a is a complex scalar, (A±, A±) is a Dirac fermion, and D is a real scalar. This fix the gauge symmetry (2.2) up to the ordinary one A — that only transforms v^ as v^ —> v^ — d^a. This is called the Wess-Zumino gauge. The superfield S := D+D-V = a + i9+A+ - iTA_ + 9+9~[D - ivQi] + • • • (2.3) is gauge invariant and twisted chiral. Here VQI :— DQVI — 3\VQ is the gauge fieldstrength. Let us consider the Lagrangian of the form (2.4) where e is a real parameter of mass dimension 1, and t is a dimensionless complex parameter. This is manifestly gauge invariant and supersymmetric. In terms of the component fields in WZ gauge, it has kinetic terms for 2 2 L = -D^D^ + i^_(D0 + D1)^+i^+(D0-D1)^+-^(\ 2 2 — — |cr| |^| — V'-cn/M- it+oi*- — i + 2i2 + iJ-(do + + - + voi) + Ovoi (2.6) We see that there is a potential U — |cr|2|^|2 + y (|<^»|2 — r)2. It is straightforward to generalize this to the case of many charged chiral multiplets <&i,..., with various charges Qi,..., QN- The Lagrangian L = / d*9 ^E eQiV®i " SS) + ^ (-* / d2^S ) ' (2"7) is manifestly supersymmetric and invariant under gauge transformations —V V - i(A — A). One can also add an F-term 2 Lw = J d 9W($i) + c.c., (2.8) provided there is a gauge invariant superpotential — W($i). The component expression for the Lagrangian L + Lw is similar to (2.6), with the scalar potential N N N 2 - dW 2 Ufa = £ IQi^W + y (E QM\2 ~ r) + E (2.9) Oft i=1 i=1 i=1 One can also consider generalization to higher rank or non-abelian gauge groups. The system has two kinds of parameters; One is t which is a twisted chiral parameter, and the others are the parameters of the superpotential W that are chiral parameters. As stressed before, t and the superpotential parameters do not mix. The Lagrangian is invariant under axial R-rotations if we assign the axial R-charge 2 to S. It is also invariant under vector R- rotations as long as W(<&j) is quasi-homogeneous. Thus, in such a case the classical system has both U(1)A and U(L)V R-symmetries. 142 K. Hori Renormalization and Axial Anomaly The system is super-renormalizable with respect to the gauge coupling that has mass dimension 1. However, Fl-parameter will be renormalized in many cases as we will see. A related phenomenon is the axial anomaly. We first consider the U( 1) gauge theory with a single charge 1 chiral superfield Look at the terms in the Lagrangian involving the D field. At the cut-off scale Auv it is 2 2 ±;D + DM -r(Auv)). (2.10) Let us integrate out the modes of the fields with the frequencies in the range /i < \k\ < Auv- Then, the term D\(j)\2 in (2.10) is corrected by D{\4>{2) where ^<|fc| In other words, FI parameter is a function of the scale of the form r(/i) = log (/i/A). The dimensionless parameter r of the classical theory is replaced by the scale parameter A in the quantum theoiy. Dimensional transmutation is at work. A related quantum effect is the anomaly of the axial R-symmetry of the classical system. Counting the number of fermionic zero modes, we find that the axial R-rotation changes the measure as TtyThjj —e~'2kiaVi>Vi>, (2.13) in the background with quantized curvature (—l/2iv) J vudx^^dx'2 — k. Since the Theta term in the Euclidean action is i{6/2n) f vi^dx^dx'2 — —ikd (the path-integral weight is the rotation (2.13) amounts to the shift in Theta angle 6—-2a. (2.14) Thus, the U(1)A R-symmetry of the classical system is broken to in the quantum theory. Physics does not depend on the Theta angle 9 since a shift of 9 can be absorbed by the axial rotation, or a field redefinition. In the Mirror Symmetry 143 present model, the dimensionless parameters r and 9 of the classical theory are not any more a parameter of the quantum theory. They are replaced by the single scale parameter A. One can repeat this argument in the case where there are N chiral su- perfields of charge Qi (i — 1,... ,N). The term D\cf>\'2 in (2.10) is now replaced by D Qil&l'21 and thus the FI parameter rims as r(p) — r(jj!) + Qi logW/^')- The axial rotation shifts the Theta angle as 9 —> 9 — 2 Y^iLi Qia- Thus, if b\ •.— l Qi ^ 0; dimensional transmu- tation is at work and the U{1)A symmetry is anomalously broken to TL^. The FI and Theta parameters are replaced by the single scalar parameter A. If bi — 0, the FI parameter does not rim as a function of the scale and the fall U(1)A symmetiy is unbroken. The FI and Theta parameters r and 9 remain as the parameters of the quantum theory. The above argument applies independently of whether or not the super- potential term f cf9W(^i) is present. The interaction induced from this does not yield divergences that renormalize the FI parameters, which is the content of the decoupling theorem presented before. Furthermore, the su- perpotential W{<&ri) itself is not renormalized as long as we keep all the fields. 2.2 Non-Linear Sigma Models from Gauge Theories We show that the linear sigma models reduce at low enough energies to non-linear sigma models on a class of target spaces. We first consider models without superpotential for the chiral fields, in which case the target space is a toric manifold. We then turn on superpotentials. This will give us the sigma model on a submanifold of a toric manifold. 2.2.1 QP^-1 Let us consider the U(l) gauge theory with N chiral superfields <&i,..., of charge 1, with no superpotential. We first look at classical supersymmetric vacua given by configurations where the potential energy (2.15) vanishes. If r is positive U — 0 is attained by a configuration which obeys a — 0 and N (2.16) i=1 144 K. Hori The set of all supersymmetric vacua, modulo U( 1) gauge group action makes the vacuum manifold. It is nothing but the complex projective space of dimension N—l, QPA_1. The modes of (fo's tangent to this vacuum manifold are massless. The transverse modes and the field a have mass e\/2r, as can be seen from the potential in (2.15). The gauge field v^ acquires mass ey/2r by eating the Goldstone mode (Higgs mechanism). For fermions, the modes of and ifii± obeying N N i=1 i=1 are massless. The equations (2.17) mean that the vectors are tan- gent to — r aJid are orthogonal to the gauge orbit. Namely, they are tangent vectors of the vacuum manifold CPA_1 at fa. These together with the tangent modes of constitute massless supermultiplets. The massive bosonic and fermionic modes constitute a supermultiplet of mass This is the supersymmetric Higgs mechanism. At low energies com- pared to e\/2r, the classical theory reduces to that of the massless modes only. Namely, non-linear sigma model on CPA _1. Another way to see this is to note that the gauge kinetic term vanishes in the limit e —» oo and the vector multiplets fields become non-dynamical. The equations of motion for v^ and a give constraints on themselves; i YA=I {^idfi^i ~ d^i^ij (2.18) = e£i ^i+^i- (2.19) The equations of motion for D and X± yield the constraints (2.16) and (2.17). Thus, we are left with the supersymmetric non-linear sigma model on CPA_1. Eqn (2.18) means that is orthogonal to the gauge orbit. Thus, the metric YliLi \D(/>i\2 measures the length of a tangent vector of CPA_1 by lifting it to a tangent vector of \ ds2 = r gFS. (2.20) Mirror Symmetry 145 Using (2.18), one can also show that the Theta term {6/2n) f dv yields the B-field B = 0 wFS, (2.21) where wFS is the Kahler form for the Fubini-Study metric. Finally, the background value (2.19) for a yields the four-fermi term of the non-linear sigma model. Thus, the classical theory reduces in the limit e —» oo to the supersymmetric non-linear sigma model on CPA_1 with the metric (2.20) and the B-field (2.21). Let us examine the story at the quantum level. The main quantum effect is the one-loop renormalization of the Fl-parameter: r(/i) = r(/i') + iVlog (^j = Nlogfc/A). (2.22) At high enough energies /i > A, r is large positive and the theory can be regarded as the QPA_1 sigma model plus transverse modes, and the effect of the latter is suppressed by /i/ei/log(/i/A). Since the metric of QPA_1 is given by (2.20), the flow of FI parameter (2.22) means the flow of the metric gtj(n) — gij(p') + N log(jj)g~s. Since the Fubini-Study metric obey the Einstein equation R^ — Ngthe flow is written as 9ij(/*) = gijtt) + log (fy Rij. (2.23) This is nothing but the flow of the metric in NLSM (1.44). Finally, let us make an important observation. The Kahler form to for the metric (2.20) is proportional to the Fubini-Study form; to — rwFS. Noting also (2.21), we find that the complexified Kahler class is given by M - i[B] = t[toFS]. (2.24) In other words, t is nothing but the complexified Kahler class parameter. Thus, it is now manifest that the complexified Kahler class is parametrized by a twisted chiral parameter. We do not have to compute the topological correlation function to see this. This is a great advantage of the linear sigma model. 2.2.2 0(-1) © 0(-1) over OP1 (Resolved Conifold) We next consider the U(l) gauge theory with four chiral superfields $2; $3, $4 with charge 1,1,-1,-1. Since the sum of charges vanish, 1 + 1 — 1 —1 = 0, r 146 K. Hori is not renormalized and axial R-symmetry is anomaly free. Thus, t — r — id is a genuine twisted chiral parameter of the system. The scalar potential of the model is 2 2 + U = J2 M I*I + T - ^ - ^ - r)2 (2-25) i=l z The interpretation of the system is different depending on whether r 0, r — 0 or r 1 N 2.2.3 0{-d) over QP^" or C /Zd Orbifold We next consider [7(1) gauge theoiy with N+l fields <&i,..., of charge 1,..., 1, —d, without superpotential term. The scalar potential of the model is U = |a|2 g \fa? + \o?#\p? + y l^l2 " d\P\2 ~ " (2-26) The structure of the vacuum manifold depends on whether r > 0, r — 0 or r < 0: r > 0 The D-term constraint requires some fa / 0, which breaks the U(l) gauge symmetry and giving mass to the gauge multiplet. In particular a — 0. The vacuum manifold is the total space of the line bundle O(-d) over QPA_1 where (fa, ...,<^v) span the base CPA_1 and p is the fibre coordinate. The size of QP^-1 is r. r — 0 fa — p — 0 is allowed, at which the gauge symmetry is unbroken and there is a flat direction parameterized by a; let us call it a "Coulomb branch". There is also the standard "Higgs branch" where the gauge symmetry is broken by fa or p. It is 0(—d) with the zero size base. The total vacuum manifold is a union of the two branches attached at one-point. r < 0 The D-term constraint requires p ^ 0, breaking U( 1) gauge symmetry into Zrf. The gauge multiplet is again massive and a — 0 is enforced. The vacuum manifold has the orbifold singularity at fa — 0. The metric is exactly the flat orbifold CN in the limit r —> —oo. We recall that the FI parameter is renormalized as r(Ai) = r(Ai') + (JV-d)log(Ai/Ai')- Thus, the quantum theory depends crucially on whether d < N, d — N or d> N. • d • d>N. r < 0 at high energies and the model is interpreted as the sigma model on a space with orbifold singularity. In fact since r —> —oo in the continuum limit, the model is understood as some relevant perturbation of the CN/1>d orbifold CFT. In all these careful analysis is required to understand what really happens in the infra-red limit. This will be done shortly. 2.2.4 Toric Manifolds The vacuum manifolds of gauge theories considered so far, CPA_1, the to- tal space of 0(-1) © 0(-1) over CP1, 0{-D) over QP^'"1, are examples of a class of manifolds called toric manifolds (or toric varieties when the algebro-geometric aspects are the focus of attention). Let us consider the U(l)k gauge theory with N charged chiral superfields where the i-th field has charge under the a-th [7(1) gauge group (i — l,...,iV; a — 1, ...,&). The vacuum manifold depends on the k Fl-parameters r°: it is the U(l)k quotient of the subset of CN defined by the equation Qi\^2 — r° (a = 1,..., k). Depending on the choice of the values r°, the vacuum man- ifold X may have complex dimension (N — k). It inherits the structure of a Kahler manifold from 2.2.5 Hypersurfaces and Complete Intersections So far, we have been considering gauge theories without F-terms. We can actually obtain non-linear sigma models on a certain class of submanifolds of toric manifolds by turning on a certain type of superpotential. We focus on the basic example of hypersurfaces of CPA _1, which captures the essential point. Hypersurfaces in OP^"1 Let us consider a degree d polynomial of (pi,..., = Uii-idfai "'fad' (2"27) il,—4d We assume that G{ ci(M) = (N - d)[H]\M. So, M is Ricci positive for d < N, Calabi-Yau for d — N, and Ricci negative for d > N. The non-linear sigma model on M is asymptotically free, scale invariant, and infra-red free, respectively. Linear Sigma Model for the Hypersurface Now, let us again consider a U(l) gauge theory with N + 1 chiral multi- plets • • •, P of charge 1, • • •, 1, —d. This time, we include the gauge invariant superpotential W = PG{® i,-..,$iv)- (2.29) The scalar potential of the model is I 2 N +\G( Let us analyze the spectrum of the classical theory. The structure of the vacuum manifold is different for r > 0 and r < 0, and r — 0. 2 r > 0 17 = 0 if and only if a = p = 0, \fa\ = r and G( W = (p)G($ (2.31) where (p) is the vacuum value of p (say (p) — yf\r\fd). We should keep in mind that there is a residual gauge symmetry, and it acts non-trivially on (as charge 1 fields). Thus, the low energy theory is not the ordinary Landau-Ginzburg model but its Zd-orbifold, or "Landau-Ginzburg orbifold". r — 0 U — 0 if and only Mp — fai — 0. The vacuum manifold is the complex cr-plane. S multiplet fields are always massless. At a ^ 0 other modes are massive, but they become massless at a — 0. In the quantum theory, we must take into account the renormalization of the FI parameter r. It depends on whether b\ — N — d is positive, zero, or negative. We separate the discussion into these three cases. • d r(/i) = (N — d) log(/i/A). (2.32) Mirror Symmetry 151 At the scale much larger than A, the FI parameter is positive and very large: r> 1. Thus, the first case of the above argument applies. In particular, by taking the limit where e/A —> oo and ct^^/A —> oo, the theory reduces to the non-linear sigma model on the hypersurface M. Since ci(M) = (N - d)[H]| M (2.33) is positive, the sigma model is asymptotically free. The logarithmic running of the Kahler parameter of the non-linear sigma model is proportional to (2.33) and matches precisely with the logarithmic running (2.32) of the FI parameter. • d = N. In this case, the FI parameter does not rim and the theory is parametrized by t — r — id. In particular, we can choose the value of r as we wish. For r >> 0, the theory reduces in the limit ey/r —> oo and ailm..id —> oo to the non-linear sigma model on the hypersurface M. Since M is a Calabi-Yau manifold c\ (M) = 0, the Kahler class of the sigma model does not run, which agrees with the fact that r does not rim, either. The complexified Kahler class is identified as t at large r. For r 0, the theory reduces in the limit e-^/jrj —» oo to the LG orbifold. For r — 0, the a branch develops. It is a non-compact flat direction and the theory must exhibit some kind of singularity when approached from r 0 or r < 0. The behavior of the theory near r — 0 is modified by several quantum effects and the Theta angle 9 plays an important role. This will be discussed later in this section. • d> N. In this case, the FI parameter at the cut-off scale is large and negative. Thus, the theory at high energies does not describe the non-linear sigma model on the hypersurface M but looks closer to the LG orbifold. The LG orbifold itself is a superconformal field theory and must preserve the axial R- symmetry. On the other hand, the gauge theoiy preserves only the discrete subgroup 1j2{d—N) and contains a running coupling (the FI parameter). Thus, it would be appropriate to identify the model as the LG orbifold perturbed by a relevant operator that breaks the U( 1) axial R-symmetry to 152 K. Hori 2.3 Low Energy Dynamics In the previous discussion, we have identified the gauge theories as NLSMs (or LG models) by looking at energies which are smaller than the coupling ey/r but are considered as high energies from the point of view of the NLSMs. We now attempt to describe the physics at much lower energies. In the case where the theory undergoes the dimensional transmutation we will look at energies /i smaller than the dynamical scale A. It turns out that it is useful in many ways to look at the behavior of the theory where the lowest component a of the super-field-strength S is taken to be large and slowly varying. The a dependent terms in the kinetic term of the charged matter field $ are 2 2 \a\ \(/)\ — — il>+cril>-. (2.34) We see that a plays the role of the mass for the field <&. Taking a large means malting $ heavy. We are thus considering gauge theory with heavy charged matter fields. 1 + 1 Dimensional Gauge Thoery with Heavy Charged Particles To be specific, let us consider a U(l) gauge theory with several charged chiral superfields At large a the charged matter fields are heavy and the massless degrees of freedom are only the S multiplet itself. The theory is that of a U( 1) gauge theory in 1 + 1 dimensions with heavy charged fields. Let us compute the vacuum energy of the system. Since are heavy, they are frozen at the zero expectation value and one can set <&j = 0 classi- cally. Then, the potential energy is given by (2.35) In addition, there is a contribution from the gauge field v^. The v^-dependent terms in the action are Let us quantize the system formulated on on S1, x1 = x1 + 2iv. Using gauge transformations v^ —» v^ — £^7, one can set vq — 0 and v\ — a(t) where Mirror Symmetry 153 a(t) depends only on t — x°. Residual gauge symmetries are of the form ei-y _ gimi with mgZ, which yield the gauge equivalence relation a(t) = a(t) + m, rngZ. (2.36) In terms of this variable, the action is given by S — fdt( ^a2 + 9a) . The conjugate momentum for a is pa — ^ = Jj + 6 and Hamiltonian is H — paa — L — Pa ~ Since a is periodic with period 1, pa eigenvalues 2 are 2-JV times integers. Thus, the spectrum is En — y (27m — 6)' 1 n € 7L. The ground state energy is therefore given by = f 02 (2-37) where 92 is defined by 2 2 d := miiinez{(0 - 2-jm) }. (2.38) 2 Note that the fieldstrength is given by VQI — A — + E PA- In particular, the vacuum value of VQ\ is ^oiU = -e20. (2.39) Thus, Theta angle is essentially identified as the vacuum value of the field- strength, or the electric field. However, this value is discontinuous as a function of 6. There is an intuitive understanding of this discontinuity, due to Coleman, which applies when the theory is formulated on M2. We assume that the charged particles are much heavier than the gauge coupling e, so that they can be treated semi-classically. In the presence of a particle of charge Q at x1 — 0, the equation of motion is divoi = 2irQe28(x1). (2.40) Namely, VQI has a gap by 2nQe2. Now suppose 9 is positive but smaller than 7r. Then there is a unique ground state with the field strength VQ\ — —E26 and the energy density U — \02. One cannot have a single charged particle since that would make 'foi (+oo) to be different from 'foi(—oo) but VQ\ is required to take the (unique) vacuum value at both spatial infinity. However, one can have particles of total charge zero. For instance, let us consider the situation where we have one with charge 1 at x1 — —Lj2 and one with charge —1 at x1 — L/2. Outside the interval —L/2 < x1 < L/2 the field strength takes the 154 K. Hori vacuum value —e26 while it takes the value —e26 + 27re2 inside that interval. The energy of that configuration compared to the one for the vacuum state with vqi = —e29 is (2.41) As long as 9 < 7r, this is positive and is proportional to the separation L. To decrease the energy, the separation L is reduced to zero. Namely, there is an attractive force between the particles of opposite charge. Charged particles cannot exist in isolation; they are confined. Now let us increase 9 so that 9 > 7r. Then AE is negative. It is now energetically favorable for the separation L to be larger. There is a repulsive force now. Eventually, the two particles are infinitely separated and disappear from the finite x1 region. What is left is the field strength with the value voi — —e20 + 27re2. The absolute value is nothing but e2\6\ for 9 in the range 7r < 9 < 3ir. Even if we started without particles, a pair of particles of opposite charges can be created. Creating a pair costs an energy, but the negative energy AE for large L is enough to cancel it. Effectively, the field strength is reduced by 27re2. This is the intuitive explanation of the discontinuity. 1 The total energy density is thus the sum of (2.35) and (2.37) + (2-42) We notice that this expression is almost the same as the potential energy of the Landau-Ginzburg model obtained by setting to zero and considering S as the ordinary twisted chiral superfield having the twisted superpotential W(E) = -ffi. (2.43) That S is not really an ordinary twisted chiral superfield but the super-field- strength (the imaginary part of the auxiliary field is the curvature vol) has only a minor effect; the shift in 9 by 2iv times an integer. 1Here we are assuming that there is a matter field of charge 1, or the greatest common divisor of the charges Qi is 1. If the g.c.d. of Q^s is q > 1, the critical value of 9 is qir (times an odd integer) and the definition of 62 is replaced by 2 0 :- min„6Z{(6l + 2-Kqnf}. Thus, in such a case the physics is periodic in 9 with period 2irq. Mirror Symmetry 155 This stoiy, however, can be further modified by quantum effects. In the above discussion we have considered to be totally frozen. But of course we must take into account the quantum fluctuation of What it does is to modify the FI-Theta parameter as a function of a. Let us now analyze this. Effective Action for S We first consider the basic example of the U(l) gauge theory with a single chiral superfield $ of charge 1. Let us take a to be slowly varying and large compared to the energy scale /i where we look at the effective theory. The $ multiplet has a mass of order CT /i and therefore it is appropriate to describe the effective theory in terms of the low frequency modes of S only. By supersymmetry, the terms with at most two derivatives and not more than four fermions are constrained to be of the form Since the action 5(2, is quadratic in the ^-integral can be carried out exactly by the one-loop computation. The result is (2.44) (2.45) The superpotential captures the axial anomaly of the system; The axial rotation S e2i^S shifts the Theta angle as 9 9 - 2/3. We have yet to integrate out the high frequency modes of the S multiplet fields. The Kahler potential (2.45) will be further corrected by this, but the superpotential remains the same as (2.44). The reason is that any correction should depend on the gauge coupling constant e, but the superpotential cannot depend on e, which is not a twisted chiral parameter. We thus see that the effective superpotential is exactly given by daWeg — log(/i/cr) — i(/i) = \og(A/a) where A — A e10, or (2.46) In this theory teff(v) := -daWeff(a) = log(aM) (2.47) 156 K. Hori can be regarded as the effective FI-Theta parameter that varies as a function of a. Also the effective gauge coupling constant is given by (1/2e2^) = dadaKeff. Applying (2.42) we find that the energy density is given by (2.48) Here the hat in teff stands for the possible shift by 2im as explained above. This shift resolves the apparent problem of the superpotential (2.46) not being single valued. It is straightforward to generalize the above result to more interesting cases. If there are N chiral superfields of charge 1, the case for the QPA_1 model, the effective superpotential is U(l)k = Ua=i U(l)a with the chiral matter fields <$>i of charge (2.50) This is derived exactly using one-loop computation in the case where there is no superpotential term for However, even if there is such an F-term, by the decoupling theorem of F-terms and twisted F-terms, the result (2.50) will not be affected. 2.3.1 The QP^-1 Model As an application, we study in some detail the low energy dynamics of the CPA_1 model, which is realized by the U(l) gauge theory with N chiral superfields of charge 1. The axial R-symmetry U(1)A is anomalously broken to I*2N and the theoiy dynamically generates the scale parameter A. We look at the effective theoiy at energy fi«A. The region in the field space with large and slowly varying a is described by the theory of the S multiplet, with the effective twisted superpotential (2.49), or the effective FI-Theta parameter teff(a) := -daWeff(a) = N\og(a/A). (2.51) Mirror Symmetry 157 The supersymmetric ground states are found by looking for the value of a 2 which satisfy U — (^EG/2)\IEFF(cr)| = 0. Namely, we look for solutions to eteff(a) _ This is solve(J by a = A • e2""/^, n = 0,...,iV-l. (2.52) Thus, we find N supersymmetric vacua in this region. The "L^N axial R- symmetry cyclically permutes these N vacua. Namely, a choice of a vacuum spontaneously breaks the axial R-symmetry to 7L% %2N Z2. (2.53) From this analysis alone, however, we cannot exclude the possibility of other vacua in the region with small a. To describe the physics in such a region, we need to use a completely different set of variables. If we use the full variables and S, we need to find a minimum where the potential U in (2.15) vanishes. However, if /i The Dynamics at Large N We have seen that a has a non-zero expectation values at these N vacua. This shows that the matter fields which include massless modes (the Goldstone modes for SU(N)/ZN U(N — 1)/Zjv) classically, acquire a mass ~ A at the quantum level. This is consistent with the general fact that there cannot be Goldstone boson in 1 + 1 dimensions. Let us try to analyze the gauge dynamics of these massive charged fields. For this we need to know also the gauge kinetic terms, not only the superpo- tential terms. From (2.45) we see that the effective gauge coupling constant at the one-loop level is given by = 2 2 (2.54) P(1)2 e ' 2\a\ ' eeff 1 1 As we noted above, this is further corrected by S-integrals and we do not know the actual form of the effective gauge coupling constant. However, 158 K. Hori there is a limit in which one can actually use (2.54) to analyze the dynamics. It is the large N limit. Since there are N matter fields of the same charge, the matter integral simply yields N times AL^E). Thus, any correction to (2.54) is suppressed by powers of 1/N. Also, the gauge coupling near the vacua is of order A/y/N and can be made as small as one wishes, no matter how large is the bare gauge coupling e. In particular, in this limit, the mass of the charged matter fields is very large compared to the gauge coupling constant, m d ( VQI 1 1 ,2 + 9eff = 2tt ^(x - zj), (2.56) dx e eff J i= 1 where 9eff is the effective Theta angle 9eff — Im (teff(a)) — N axg(a/A): and 1 — ±1 is the charge of the particle at x — x\. Thus, 'f oi/e^ + 9eff has a gap of ±27r at the location of the particles. At any of the N vacua we have 2 2 vqi — e eff\0eff\' — 0, which means 9eff — 2im for some n E Z. Thus, in order to have a finite energy configuration, we need voi 0 | at x1 ±oo, (2.57) 9e ff 2im± where n± are some integers. For an arbitrary distribution of particles, we can find a solution to (2.56) obeying this condition. In particular, a particle (or a particle) can exist by itself. In the presence of a particle, the vacuum at left infinity x1 —> —oo is not the same as the vacuum at the right infinity x1 —>• +oo. This is because =/:; W (f+ If one particle and one particle are located at x1 = —Lj2 and x1 — Lj2 respectively, eqn.(2.56) can be solved by a configuration as shown in Fig.2. The configuration is at the vacuum in the region —Lj2 < x1 < L/2 2tt (nf 2rcn -L/2 L/2 Figure 2: The configuration of 9Eg — N /A) for a pair of particles, charge 1 at x1 = —L/2 and charge —1 at x1 = L/2. and the total energy does not grow linearly as a function of the separation L. Thus, there is no long range force between them. Namely, charged particles are not confined in this theory. This is essentially the effect of the coupling Naxg(a/A)VQ\. This coupling screens the long range interaction between the charged particles. Thus, the particle exists as a particle state in the quantum Hilbert space. From the classical story, we expect that these states constitute the fundamental representation of the group SU(N). Note that SU(N) is not quite the same as the classical global symmetry group, which is SU(N)/ZJV- The symmetry group of the quantum theory is not SU(N)/ZN but its uni- versal covering group. Such a phenomenon is common in quantum field theories (known as charge jractionalization). In the present case this hap- pens because there appeared a state transforming nontrivially under the "overlap" Zjv of SU(N) and the gauge group, U(L). Whether such a thing happens or not depends on the gauge dynamics. If the particles were confined, there would not be a state charged under U(L) gauge group, and therefore all the states would be neutral under ZN — SU(N) fl U(L). In that case, the global symmetry group would remain as SU(N)/ZN- 160 K. Hori 2.3.2 The "Phases" Let us consider a U(l) gauge theory with several chiral superfields <&i,..., <&M with charges Qi,..., QM that sum to zero: M (2.59) i= 1 In this case, the axial R-symmetry U{1)A is an exact symmetry of the quan- tum theory, and the FI parameter does not run along the RG flow. We have in mind two classes of theories: one is the linear sigma model for compact Calabi-Yau hypersurfaces in CPA_1 or weighted projective spaces; the other is the theoiy without F-terms, which yields the non-linear sigma model on non-compact Calabi-Yau manifolds. Since the FI parameter does not run, one can choose r to be whatever value one wants. As we have seen in the previous discussion, the theory at r 0 and the theory at r < 0 have completely different interpretations, and also at r — 0 the theory becomes singular due to a development of a new branch of vacuum manifold where a is unconstrained. Thus, it appears that the parameter space is completely separated by a singular point r — 0 into two regions with different physics. This picture is considerably modified when the Theta angle 9 is taken into account. The actual parameter of the theory (in addition to the real and chiral parameters that enter into D-terms and F-terms) is t — r — i9 and the parameter space is a complex torus or a cylinder. It may appear that the parameter space is still separated into two regions by the circle at r — 0. However, it turns out not to be the case when we think about the origin of the singularity at r = 0. The singularity is expected when there is a new branch of vacua where new massless degrees of freedom appears. In the classical analysis at r — 0, that is identified as the S multiplet since there is a non-compact flat direction where a is free. However, at large cr, as we have analysed the actual energy density receives also a contribution from the electric field or Theta angle as in (2.42). Taking into account the more refined quantum correction, the energy density at large a is (2.60) where M (2.61) i=1 Mirror Symmetry 161 Here we have used the formula (2.50) for Weff, where the E//i depen- dence disappears because of (2.59). Thus, the energy at large a vanishes at r — — Qi log Qi and at a single value of 0 which is 0 or 7r (mod 2-ir) depending on Q^s. Thus, except at a single point in the cylinder, there is no flat direction of a. This means that the singularity is expected only at the single point. This yields a significant change to our picture; The two regions, r 0 and r<0, are not any more separated by a singularity, but are smoothly connected along a path avoiding the singular point. These two regions can be considered as a sort of analytic continuation of each other. This change of picture has several applications, including correspondence between Calabi-Yau sigma models and Landau-Ginzburg orbifolds as well as analytic continuation to different topology. We now describe them here. Topology Change Let us revisit the U(l) gauge theory with chiral superfields of charge 1,1,-1,-1 without superpotential. We recall that the theory at r 0 and theoiy at r < 0 both yields sigma model on resolved conifold but the two are related by "Flop". In the semi-classical analysis, we found that the two regions are separated by a singularity at r = 0. Now we know that the genuine singularity is at t — 0 of complex codimension 1, and the two regions r >> 0 and r<0 are no longer separated. The present case is a special case where the two resolutions are isomorphic. However, if this is embedded as a part of some larger geometry, "flop" usualy changes the global topology. In such a case, what we have seen shows that the sigma model on topologically distinct manifolds can be smoothly connected. Calabi- Yau/Orbifold Correspondence Let us reconsider the U(l) gauge theory with chiral superfields <&i,..., P of charge 1,..., 1, —N, without superpotential. We have learned that the theory at r 0 describes the sigma model on the total space of O(-N) over CPA_1, which is a non-compact Calabi-Yau manifold. On the other hand, the theoiy at r —» —oo is the free C^/Zjv orbifold theory. Thus, the sigma-model on the total space of O(-N) over CPA_1 and the one on the orbifold CN/"LN are in the same moduli space of theories. In this one- parameter family of theories, there is one point t — N\og(—N) at which the theory is singular. 162 K. Hori Calabi- Yau/Landau- Ginzburg Correspondence Let us turn on the superpotential W — PG(<&i) to the model considered right above. As we have seen, the theory at r 0 is identified as the non-linear sigma model on the Calabi-Yau hypersurface G — 0 of CPA _1, whereas the theory at r —» —oo is identified as the LG orbifold with group Zjv and the superpotential W — (p)G(&i,..., Thus, the Calabi-Yau sigma model and the LG orbifold are smoothly connected to each other. In other words, the LG orbifold and the Calabi-Yau sigma model are in the same moduli space of theories. The two are interpretations of different regions of the moduli space. The theory is singular exactly at one point t — N\og(—N). 2.3.3 (LG) Orbifold as an IR Fixed Point As another example, let us consider the U(l) gauge theory with chiral super- fields ..., P of charge 1,..., 1, — d where we take d < N. For a degree d polynomial ..., the theory with superpotential W — PG(&i) describes the non-linear sigma model on the degree d hypersurface — 0 0£QpiV-i_ q^e axial R-symmetry U(1)A is anomalously broken to ^(N-d) and the theory dynamically generates the scale parameter A. The effective theory for large and slowly varying S is the theory of a U( 1) gauge multiplet with the effective FI-Theta parameter given by aN~d = {-d)dA (2.63) and we find (N — d) of them in the admissible region. These are massive, and a choice of vacuum spontaneously breaks the axial R-symmetry as Z2(jv-d) Z2. Now let us ask whether these (N — d) are the whole set of vacua. There is an obvious reason to doubt it; the direct analysis of non-linear sigma model shows that the number of vacua is equal to the dimension of the cohomology group H*(M), which is larger than (N — d). How can we find the rest? They must be in the region where the large a analysis does not apply. Let us examine the potential (2.30) in terms of the full set of variables once again, now at low energies. At /i < A the FI parameter is negative, and the analysis of supersymmetric vacua U — 0 is completely different Mirror Symmetry 163 from that at high energies. It is more like in the d — N case with r < 0 and we find a single supersymmetric vacuum at a — 0, ft — 0 and [p\ — •\f\r\Jd where the axial R-symmetry group %2{N-d) is not spontaneously broken. Thus, we find at least one extra supersymmetric vacuum besides those found at a ~ A. The theory around this vacuum is described by the LG orbifold of the fields <&i,..., with the group Z^ and the superpotential For d > 2 this LG orbifold is expected to flow to a non-trivial superconformal field theory where the axial !*2(N-d) discrete R- symmetry enhances to the fall [7(1) symmetry. One can actually analyze the spectrum of the supersymmetric vacua of this LG orbifold, which in fact yields extra vacua as many as dim H* (M) — (N — d). Thus, we expect that this extra vacua really exists in the quantum theory and is the only one that was missed by the large a analysis. We will see that the mirror theory shows this as well. The Case without Superpotential One could also consider the theory without the superpotential W — PG. As we have seen, such a model is interpreted as the non-linear sigma model whose target space is the total space of the line bundle O(-d) over CPA_1. Due to the decoupling of F-term and twisted F-term, the analysis of the effective superpotential for S goes through without modification, and in particular (2.62) applies. Thus, we find (N—d) massive vacua at the solutions of (2.63). As in the discussion above, one may also worry if there are extra vacuum from the region cr< A. Since r(/i) is large negative at /i A, we expect that there is another (degenerate) vacuum at p — i/|r(/i)|/d where the light modes are only and the gauge group is broken to Z^. The theory there is the Z^ orbifold of N chiral superfields without superpotential. In conclusion, the sigma model on the total space of O(-d) over QPA'-1 has (N — d) massive vacua plus another vacuum that flows to the free orbifold N C /Zd. 2.3.4 A Flow from (LG) Orbifold As a final example, let us consider the case d > N of the [7(1) gauge theory considered right above, with and without the superpotential W — PG(<&i). In either case the FI parameter at the cut-off scale is negative. With W — PG, the high energy theory describes the Zd LG orbifold with superpotential W ~ (p)G(§i,..., which is perturbed by an operator that breaks the 164 K. Hori U( 1) axial R-symmetry to 7L 3 Mirror Symmetry Let us consider a closed string moving on the circle S1 of radius R. The space of states is decomposed into sectors labeled by two conserved charges — the momentum Z e Z associated with the translation symmetry, and the winding number me Z which counts how many times the string winds around the circle. The energy E of the string without oscillation is given by E2 — p2 + M2 — (l/R)2 + (Rm)2 in the (/, m)-sector. This is invariant under 1 I m. In fact, the sigma model on the circle of radius R is equivalent to the sigma model on the circle of radius 1 /R1 where the momentum of one theory is exchanged with the winding number of the other. This is called T-ducdity. The same story applies to (2,2) supersymmetric theories on the worldsheet. As the simplest example, let us consider the supersymmetric sigma model on the cylinder Cx =Ix51 with radius R for S"1. T-duality applied to the S1 yields another cylinder Cx =Mx51 with radius l/R for S1. This is Mirror Symmetry 165 actually an example of mirror symmetry: Q- and Q_ are exchanged under the equivalence. Let us next consider a more interesting target space — the two-sphere S'2. It can be viewed as the circle fibration over a segment, and one may ask what happens if T-duality is applied fiber-wise. Since T-duality inverts the radius of the circle, a larger circle is mapped to a smaller circle and a smaller circle is mapped to a larger circle, and one may naively expect that the dual geometry is as in Fig. 3. Since the size of the dual circle Figure 3: Is this what T-duality does? blows up towards the two ends, two holes effectively open up and the dual geometry has the topology of a cylinder. This is consistent in one aspect: the conserved momentum associated with the [/(l)-isometry of S2 fiber- rotation) is mapped to the winding number of the dual system, which is conserved due to the cylindrical topology. However, another aspect is not clear. The winding number is not conserved in the original system because 7Ti S'2) — {1}, and this should mean in the dual theory that the momentum is not conserved or the translation symmetry is broken. But how can it be broken? Is it because the metric is secretly not invariant under rotation of the cylinder? What really happens under T-duality is as follows. It is true that the dual geometry has the topology of a cylinder. However, the dual theory is not just a sigma model but a Landau-Ginzburg model. Let us parametrize the dual cylinder by a complex coordinate Y which is periodic in the imaginary direction, Y = Y + 2wi. Then the superpotential of the dual LG model is given by W= e-Y+e~t+Y 3.1) Here t — r—id is the complexified Kahler class of S2. It is this superpotential 166 K. Hori that breaks the translation symmetry Im(Y) —> Im(Y)+ constant. This T- duality is a mirror symmetry, as in the example of the cylinder. In this section, we give a proof of this mirror symmetry. The proof uses linear sigma models and is applicable to all target spaces that can be realized as the vacuum manifolds of LSM. Namely, we will find the mirror of NLSM whose target space is a toric manifold or a hypersurface/complete intersection in a toric manifold. 3.1 T-duality Let us start with reviewing the path-integral derivation of T-duality. The Lagrangian for the sigma model on the circle of radius R is given by R2 (3.2) where y is a periodic scalar field of period 27r, ip = ip + 2TT. One can actually obtain this from a different system with a larger number of fields: we consider the system of the periodic scalar field (p = (p + 2-K and a one-form field B^ with the Lagrangian Integrating out BIJri we obtain the system of with Lagrangian (3.2). In other words, solving the equation of motion as B^ — -j^e^dyip and plugging this back to the Lagrangian (3.3), we recover (3.2). Now let us change the order of integration — integrate out first. Then, we obtain the constraint e^uduBn — 0, which can be locally solved as B^ — d^lp for some function (p. In fact, if we carefully perform the integration over the periodic field (p, we find that (3.4) This is the action for the sigma model on the circle of radius 1 /R. This shows the equivalence of the two models. Comparison of the two expressions for B^ yields the relation of variables ReSdvV = ^dplp. (3.5) Mirror Symmetry 167 The left hand side is the winding number current of the circle of radius R while the right hand side is the momentum current of the circle of radius 1/R. Thus, this relation exhibits the momentum-winding exchange. The momentum-winding exchange has the following consequence. Con- sider a defect at a point x of the worldsheet around which (p has a non-trivial winding, or a monodromy: ip(r: 9) —» ip(r: 9) + 2iv under 9 —» 9 + 2iv where (r, 9) are the polar coordinates around x. In the T-dual side, there has be an operator that creates a unit momentum at x. That is, (See Fig. 4.) This can be shown directly in the path-integral method: Consider inserting X \ (p • ' T » ^ P |Cp(X) Figure 4: Winding-momentum exchange the operator exp(—i fxB) in the system with the Lagrangian (3.3), where the integral is along a semi-infinite line emanating from the point x. Using Bn — dptpi we find that the inserted operator is One the other hand, the line integral can be rewritten as JXB — JE B A dB where 6 is a multi- valued function with a jump by 1 along the line. This will effectively replace ip in (3.3) by if/ — tp + 2ir@. After integrating out BIJri we find the action (3.2) for ip' which has a jump by 2iv along the path. 3.1.1 T-duality as Mirror Symmetry T-duality holds also in supersymmetric sigma model on the circle, since the fermion is decoupled from the boson in such a free theory and in particular does not see the periodicity. The same can be said for (2,2) supersymmetric sigma model where the target space is a complex torus, or more simply a cylinder M x S1. However, in this case, there is something interesting — if we start with the theory where the complex coordinate of the target cylinder is represented by a chiral superfield, as usual, then, the complex coordinate of the T-dual cylinder is represented by a twisted-chiral superfield. This can be most easily shown by performing the dualization in (2,2) superspace. We start with the following Lagrangian for a real superfield B and the chiral superfield where the latter is periodic in the imaginary 168 K. Hori direction $ = $ + 2m: We first integrate out the real superfield B. Solving for B as B — + <&) and inserting it back into L', we obtain L — J d40 the Lagrangian for the supersymmetric sigma model on cylinder of radius R. Now, reversing the order of integration, we consider integrating out $ and $ first. This yields the constraint D+D-B — D+D-B — 0 which is solved by B — $ + where $ is a twisted chiral superfield of periodicity $ = <& + 2m. Inserting this into the original Lagrangian we obtain z = f the Lagrangian for supersymmetric sigma model on the cylinder of radius 1 /R. This time, however, the complex coordinate is described by the twisted chiral superfield <&. We may flip the convention of the T-dual system into the standard one — complex coordinate is represented by a chiral superfield. This will exchange Q- and Q_. Then, the two systems, the sigma model on the cylinder of radius R in the original convention and the sigma model on the cylinder of radius 1 /R in the flipped convention, are equivalent but the equivalence involves the exchange of Q- and Q_. Namely, they are mirror to each other. 3.1.2 T-duality for Charged Fields The basic idea in the derivation of mirror symmetry is to apply T-duality to the phase of the chiral superfields in the linear sigma model. The new feature is that is charged. Let us first see what happens when we T- dualize the phase of a charged field in a simple theory of bosonic U(l) gauge theory with one complex charged field (p- In terms of the polar variables lip 2 2 defined by (j> — pe : the kinetic term — \Dn £ We now dualize (p. As before, we start with the system with extra one-form field Bp 2 £ -4^ W + ^B^dvtp + vv). Integration over B^ yields the Lagrangian Cv. If instead we first integrate over = + fTdplpvu = - ipv01, (3.6) where a partial integration is performed in the last step. The dual variable 1\p is not charged, but is coupled to the gauge field as a dynamical Theta angle. One can repeat the same thing in the supersymmetric gauge theory. Consider the (2,2) U(l) gauge theory with a single chiral superfield and dualize the phase of Then, the dual field couples to the gauge field as a dynamical Theta angle. Such a field must be in a twisted chiral superfield Y that couples to the superfieldstrength S in the twisted superpotential as YE. Together with the original FI-Theta term, we now have the following terms in the twisted superpotential (Y - (3.7) Let us look at the change of variables in more detail. The dual action includes the terms (p2 —r)D — {jp — 6)v01 where p2D is from \ y = p2 - Hp. In fact, this is encoded in the relation between the superfields = Re(Y) (3.8) that can be shown by a direct superspace dualization (which also shows (3.7)). As can be read from (3.6), the dual metric is * 2 (y + yY This is the expression for the metric in terms of the bare field y — yo- As we have seen, in this model the FI parameter rims as ro = r(/i) + log(Auv/V)- 170 K. Hori This induces the renormalization of the field yo — y(/i)+log(Auy//i), so that te coupling (3.7) is finite. This forces the renormalized field to look at the region yo ~ log(Auv//f) at which the metric is that of the flat cylinder ds2 ~ |dy|2/41og(Auv//^)- We recall that the system also have axial anomaly. This is reflected in the dual theory as the effect of the axial rotation y —» y — 2i/3: which can be shown using (3.8). 3.2 Vortex Instanton Effects By dualization of the phase of we obtained a theory written in terms of a twisted chiral superfield Y = Y+2wi with the linear coupling to S (3.7). The superpotential term {Y — t)Yi respects the standard requirements, such as holomorphy, periodicity in t and Y, and R-symmetry (where the axial 2z rotation Y —» Y — 2i0: S —» e ^S is assumed to act on Theta angle as 9—^9 + 2/5 as well). However, this is not the whole story: there is one important effect that modifies the story in an essential way. It is the vortex instantons. Finiteness of the Euclidean action requires the field to satisfy \(f>{2 — r, V12 — 0, cr = 0 as well as D^cf) — 0 at infinity. Let us consider a configuration with a non-trivial flux Since (d^ + iv^tj) — 0 is required at infinity, we find 00 00 In particular, k has to be an integer for X Figure 5: Profile of 121 and \ e-y (3.9) y 2 whose lowest component is e~ — e-p +'^_ We now claim that summing up vortex/anti-vortex configurations has an effect to generate the twisted super- potential term e-y. Note that it has the correct R-charge (qviQA) — (0,2) to be added to the twisted superpotential (qA — 2 because axial rotation transforms Y — 2i/3). In fact, we claim that the sum of (3.7) and (3.9) with some coefficient c, W = (Y-£)E + ce-y, (3.10) is the exact twisted superpotential of the dual theory. It is actually easy to see that the W in the dual theory has to have this form — it is simply impossible to find other expressions that respect holomorphy, periodicity in Y and t1 R-symmetry, and the condition that any correction to (Y — must be small at large Y and large t and analytic at S = 0. The question is whether c is zero or not. That c is non-zero can be shown by an explicit instanton computation. For instance, we may consider the two-point function (x+(xi)X-(x2)) of the fermionic components of Y. This would vanish if c were zero. This correlation function can be computed in the original gauge theory, using the relations x+ — and x~ — More General Case: Use of Extended Gauge Symmetry So far, we have been considering a very special system — U( 1) gauge the- oiy with a single charge 1 chiral matter field. What about the more gen- eral case? Let us consider the U(l)k gauge theory with N chiral matter fields ..., of charge Qf,..., Q% where a — 1,..., k labels the gauge group. The model has gauge coupling constants and FI-Theta parameters ta. Dualization of the phase maps each to a twisted chiral superfield Yi = Yi + 2m, and we are interested in the description of the theory in terms of (S, Yi,..., YN). We claim that the exact twisted superpotential is given by k / N \ N W Yi ta = £ £ Qi ~ M Sa + 5>e-«. (3.11) a=l \i=l / i=1 (KKKCtiKCKKKi/ S-linear terms arises just from the dualization proce- dure. Non-trivial is the fact that the non-perturbative effect is simply the sum over /i e~Yi. The idea is to gauge the flavor symmetry group U(l)N~k. Then, the total gauge group becomes U(1)N and the system can be regarded as consisting of N copies of the U(l) gauge theoiy with a single matter. For a suitable choice of gauge coupling, the N copies are decoupled from each other, and one can apply the basic story completed above. In particular, the exact twisted superpotential is given by N N w = - + i=1 i=1 This expression remains true even if the gauge coupling is chosen so that the iV-systems interact with each other. This is because the gauge coupling is not a twisted chiral parameter and cannot enter into the twisted F-term. Now, we can take the limit where the gauge coupling of the flavor group Mirror Symmetry 173 U(l)N~k vanishes. This will constrain some of the gauge fields but will not change the above twisted superpotential. The constraint takes the form Ej = X)o=l Qt^a + constant. The constant, however, is not small at large Yi and is excluded. This is how (3.11) is proved. 3.3 The Proof Now, we are in a position to prove the mirror symmetry between QP1 sigma model and the LG model with superpotential (3.1), and its generaliza- tions. Let us be general — consider U(l)k gauge theory with N matter fields of charge (i — 1,..., N, a — 1,..., k): which is parametrized by FI-Theta a parameters t . In the limit where the gauge coupling constants ea are taken to be large, the system reduces to the sigma model on the toric manifold, realized as the symplectic quotient Qil&i^2 — ra}/U(l)k, with the B-field determined by 9a. If we dualize the phase of the N matter fields, we obtain N twisted chiral superfields Y\ with periodicity Y\ = Yi+2wi. We have seen above that the theory described in terms of (E, Yi,..., Yv) has twisted superpotential given exactly by (3.11). From the Y-E mixing term E fields have masses and they are proportional to the gauge coupling constants. At this stage we take the limit ea —» oo that corresponds to the sigma model limit. Then, the E fields are heavy and it is appropriate to integrate them out. This yields the constraint J2QiYi = ta- (3.12) i=1 We are now left with the following twisted superpotential W= e~Yl + •••+ e~YN. (3.13) Namely, we have LG model whose target space is the subspace of (Cx )N de- termined by (3.12), which is isomorphic to (Cx )N~k, with the superpotential (3.13). 3.4 Some Applications 3.4.1 Projective Spaces Let us apply the above result to the sigma model on the complex projective space. Let us first consider OP1. Its linear sigma model is the U(l) gauge theory with two charge 1 matter fields, Qi — Qi — 1. Then the constraint 174 K. Hori equation is Y\ +12 = t and the superpotential is W — e_Yl + e~Y2. Solving the constraint as Y\ — Y and Y-2 — t — Y, we find that the superpotential is given by W= e~Y + e~t+Y, which is (3.1). For QP^-1, realized as the U(l) gauge theory with N charge 1 matter fields, the constraint is Y\ +Y2 H H YN — t which is solved by Yi — @I (i — 1,..., N - 1) and YN — t-@ 1 ®N-I- Thus, the mirror theory is the LG model on the (N — l)-dimensional cylinder parametrized by &i = &i + 2m (i — 1,..., TV — 1) with the superpotential @1 _0JV_1 t 01 0 1 W = e~ + h e + e- + +-+ ^- . (3.14) This function is known as the 1 affine Toda potential. Thus, we have shown that the CFA _1 sigma model is mirror to the M — 2 A^-i affine Toda theory. Let us examine this mirror theory of the QPA'-1 sigma model. The axial R-symmetry is classically broken by the superpotential to its N subgroup ± ± N ± ±7ri iV ± generated by ©i(0 ,0 ) &i(e^ 9 : e / 0 ) - 2tri/N. This corre- sponds to the axial anomaly of the sigma model. The supersymmetric ground states are in one-to-one correspondence with the critical points of the super- potential W, which are N points e-01 = ••• = e"0^-1 = (n— 1,..., N). Since S and e~Yi — e~@i are related by S = e~Yi — e~@i under the equations of motion, we find S = e-t/N+2mn/N ri_tli vacua. Thus, they correspond to the N vacua found in the linear sigma model anal- ysis, Eqn (2.52). A choice of vacuum breaks the v R-symmetry into Z2, again as in the linear sigma model analysis. In the linear sigma model analysis at large N, we have seen that there are solitons that connect different vacua, and the ones connecting neighboring vacua are interpreted as the fundamental electron fields In the LG model, solitons are kink configurations that interpolate different critical points of W. In particular, if a kink is projected by W to the straight segment connecting the respective critical values Wa: it is BPS, i.e. it preserves a half of the supersymmetry and has minimum energy. The energy (or the mass) is the absolute value of the difference of the critical values m — — Wa|- One can find exactly N such kinks for each pair of neighboring vacua. Thus, the elementary fields $1,..., are interpreted as such N BPS kinks. Since the -0i critical value of W at the n-th vacuum e = e-t/N+2mn/N _ ^ e2vrm/N -1S Mirror Symmetry 175 2nm N W^ln-th = NAe / : its mass is given by ^neighboring = - A\ ~ A if N > 1. Its approximate value at large N is nothing but the mass of predicted by LSM analysis at large N. In general, for a pair of vacua separated by ^-steps, there are BPS solitons whose masses are m^_step = ^rAsia(im/N). Finally, one can determine the (twisted) chiral rings using the superpo- tential. It is given by the ring of functions of (Cx modulo the relations dts^W — 0. It is easy to show that it is generated by E = e-01 = • • • = e-©iv-i has one relation E^ = e"*. One can also compute the topological correlation functions using (1.38): fc (E'E-^o = 5i+j,N-1, and (E''E^E )g=0 = 6i+j+k,N-i + ^^i+j+k^N-u which can also determine the chiral ring. 3.4.2 Resolved Conifolds Let us next consider the sigma model on the resolved conifold M, the total space of the vector bundle 0(—1) © O(-l) over QP1. It is realized as the IR limit of the U( 1) gauge theory with four matters of charge Qi — Q2 — 1 and Qi — QA — —1- The constraint is Yi + Y2 — Y3 — Yj = £ and is solved by Yi = Y0 + ©1, Y2 = Y0 + ©2, Y3 = Y0 + ©i + ©2, Y4 = Y0 - t, and the superpotential is W = e~Y° (e-01 + e-02 + e-01"02 + e*) . (3.15) The superpotential is homogeneous, with the weight of (e_y°, e-01, e-02) being (2,0,0), and therefore there is an axial U(l) R-symmetry. This cor- responds to the fact that the axial U(l) R-symmetry of the original sigma model is anomaly free because M is Calabi-Yau. We note that the superpo- tential has a run-away behaviour at large Rje(Yo), corresponding to the fact that M is non-compact. We recall from LSM analysis that the system is sin- gular at exactly one point t — 0. This can be seen in the mirror model as fol- lows. If t ^ 0, the critical point set is exactly at Re(Yo) = +oo (©i, ©2 free). At t — 0, the superpotential factorizes as W — e_y°(e_01 + l)(e-02 + 1) and the critical point set includes a new component ©i = ©2 = wi (Yq free) 176 K. Hori in addition to the usual one Re(Yo) = +00 (©1, ©2 free). This jump in the flat direction of W signifies the singularity of the theory. Actually, there is an alternative description of the mirror theory. This is found by computing the mass of supersymmetric D-branes. As explained in Section 1.3.3, the mass of a BPS A-brane is determined by the period integral, where the measure is the holomorphic volume form f2 in the case of Calabi-Yau sigma model and the weighted volume form e~'lWil in the case of Landau-Ginzburg model. Before applying this, we modify the system by adding two twisted chiral superfields U and V with the superpotential AW — UV. The added theory is massive and therefore the modification does not alter the IR fixed point. Now, let us apply the mass formula. The relevant period integral is given by II = f E~'LW &YQ&®I&Q-2&U&V. Under the change of variables U — e~Y°u and V — v, it is expressed as n = J e-ie-^(e-®i + e-®2+e-®i-®2+et+™) e-^dyod@ld©2ducb. We see that the total superpotential factories and also that the measure for Yo becomes —de_y° due to the Jacobian factor. Now, the integration over e_y° yields a delta function supported at the locus e-01 + e-02 + e-01-02 + e* + uv = 0. (3.16) x x This equation defines a submanifold W of C xC xCxC = {(©1, ©2,'")}• The manifold W is a non-compact Calabi-Yau manifold whose holomorphic volume form is given by f2yy = d©id©2dw/w. The period integral II is the same as a period integral of f2>V over a three-cycle in W, which is the mass of BPS D-branes in W. This indicates that the LG mirror may be identified as the sigma model on W. This argument does not prove the equivalence, but the relation is exact as long as the holomorphic information such as BPS masses are concerned. Let us see what happens at t — 0. There the equation defining W becomes (e-01 +1) (e-02 + l)-\-uv — 0, and we notice that there is a conifold singularity at ©1 = ©2 = m, u — v — 0. W is smooth except at this value of t. This is another manifestation of the singularity of the theory at t = 0. 1 N 3.4.3 0(-d) over QP^" or Perturbed Orbifold C /Zd Finally, we consider U(l) gauge theory with N charge 1 matters and one charge —d matter. As discussed in Section 2, the system describes the sigma Mirror Symmetry 177 model on O(-d) over OPA' 1 if d < N, a deformation of the free orbifold CN jljd if d> N, and a one parameter family of theories which interpolates A _1 N 0{-d) CP ' and C /Zd. If we denote the dual of the charge 1 fields by YI (I — 1, ...,N) and that of the charge —d fields by Yp, the constraint equation becomes Y\ + • • • + Yjv — DYP — t and the superpotential of the mirror theoiy is W — e-Yl H + e~YN + e~Yp. • d — N Let us start with the case d — N where t is the genuine parameter of the theory. Solving the constraint equation on the dual fields by YI — Yq + &i (i = 1,..., N - 1), YN = Y0 +1 - ©i ©iv-i and YP = Y0, we find that the superpotential is given by -y -01 _0JV_1 t+01 0 W = e °(e + • • • + e + e- +-+ ^-i + 1). (3.17) The superpotential is homogeneous and has run-away behaviour, correspond- ing to the fact that the total space of 0{—N) over QPA_1 is Calabi-Yau and non-compact. One can also see in the mirror theory the singularity at t = iVlog(-iV), found in the LSM analysis: If t ^ iVlog(-iV) the critical set of W is Re(Yo) = +oo (6j free), while at t — N\og(—N) there develops a new branch of the critical locus e-01 = • • • = e-0JV_1 = —1/N (Yo free). As in the case of resolved conifold, one can find a sigma model description of the mirror theory. The mirror manifold is the submanifold W of (Cx x C? defined by the equation _0 e-©i + ... + e iv-i + e-t+©i+-+©iv-i + i + uv = 0j (3.18) where &i is a Cx coordinate and (u,v) are C2 coordinates. This is a non- compact Calabi-Yau manifold whose holomorphic volume form is given by f2>V — d©i • • • d©jv-idu/u. At the point t — iVlog(—iV), the defining equa- tion behaves near ©j = log(—N) as _ N-l — (Sij + l)(«i - log(~M®3 - log(-N)) + uv + • • • = 0, ij=1 where H are cubic and higher order terms in (©j — log(—N)). We see that W has a conical singularity. In this sense, t — N\og(—N) is the "conifold" point of this family of theories. For other values of t, W is smooth. There is an alternative description of the mirror theory which applies also to the cases d ^ N. The constraint can be solved as YI — d x ZI 178 K. Hori (i — 1,..., N), Yp — Z\ H 1-Zm — ^- Note that there are many (-Zj)'s that correspond to the same (Yi,Yp): Transformations Zi^Zi + ^j^-; mi = 0 (mod d), (3.19) do not change the value of (1^, Yp), and two (-Zi)'s giving the same value of (Yi, Yp) are related by such a transformation. These transformations form a group isomorphic to (Z^)^-1. Thus, the mirror theory is identified as the (Zd)A -orbifold of the LG model with superpotential W = e_dZl + • • • + e~dZN + eZ~Zl Zn. (3.20) If d — N, there is U(l) axial R-symmetry since the superpotential is homo- geneous: Zi^Zi- 2i/3/N does W e2'^W. If d ^ N, W is not homogeneous and the axial R-symmetry is smaller: it ± ± is Z2|jv_d| generated by ^(#±,0*) e W^0 ) - + where rij are any integers such that ni — 1- This corresponds to the R-symmetry of the sigma model reduced to 7L _dZi dz.W = -rfe - ei-Zi—-zN = o. In addition to the obvious ones at infinity, there are some massive vacua. To find them, we define S — e~dZi which is ^-independent from the vacuum equations. The equations imply (•-d)dSd = elSN, which has (N — d) solutions. Note the similarity with the equation (2.63) found in LSM analysis. In fact, it is not a coincidence — the dual system involving both S and Yi has vacuum equations S = e~Yi, which identifies S and S. Now, let us analyze the IR and UV limits of the system, separating the cases into d < N and d> N. • d < N Since t is large negative at low energies, we expect that the last term of Zl Zn (3.20), ed~ : is small in the IR limit. In fact, if we assign weight 2 to each e-dZi, then e~Zl Zn has weight 2N/d > 2. Thus, the last term yields an irrelevant operator and we find the superpotential of the IR limit is WJR = e-dZl + • • • + e~dZN. (3.21) Mirror Symmetry 179 Namely, we identify the IR limit as the {Ld)N 1-orbifold of this LG model. We now recall that the LSM analysis suggests that the IR limit of the sigma N 1 model is the free orbifold C /Zd. Thus, the (Z^'" LG orbifold with superpotential e-dZl H + e-dZjv may be identified as the mirror of the N C /Zd orbifold. • d> N In this case, at high energies, t is large negative and the last term of (3.20) is negligible. Thus, we expect that the UV limit of the model is the (Z^)^-1 orbifold of the LG model with superpotential e~dZl H + e-dZjv. In the LSM analysis, we have seen that the model is identified as the free orbifold CN jTLd perturbed by an operator that breaks the axial U(l) R-symmetry into !*2(d-N)- This implies that the (Z^)^-1 LG orbifold with superpotential N Q-dZi _| 1_ e-dZN js the mirror of the free orbifold C /Zd. Moreover, the N operator perturbing the free orbifold C /!*d is concretely realized in the mirror description: it is e~Zl Zn. In fact, the LG orbifold in the UV limit has U(l) R-symmetiy that transforms the fields as Z;L Z;L — 2ifi/d, but that is broken by e~Zl Zn into those with £iNP/d = i.e. /3 = ndmj(d — N), m G Z. This is in general a subgroups of Z2(d-N) generated by = 7v/(d — N), which is the actual axial R-symmetry of the perturbed system. In all the above cases, including d — N, we are led to consider that N 1 C /Zd is mirror to the LG model with superpotential W — e + • • • + 1 e-dzN mQ(jded out by the (Z^'" orbifold action (3.19). In fact this is true irrespective of the relation of d and N, as will be proved at the end of Section 3.6.3. 3.5 The Case of Hypersurfaces in Toric Manifolds We would now like to find the mirror of the sigma models on hypersur- faces or complete intersections in toric manifolds. One should note that there is a difficulty to start with: there is no exact isometry in hypersurfaces/complete- intersections — the defining equations explicitly breaks the [/(l)-isometries of the ambient toric manifold. In short, we do not know how to dualize the fields, and we cannot tiy to directly find the exact mirror of the theory. Even in such a situation, however, there is still something one can do. It is to use the decoupling of chiral and twisted chiral sectors of the (2,2) theories — parameters that enter into the defining equations of hypersurfaces/complete- intersections cannot enter into twisted chiral sector, such as the twisted 180 K. Hori superpotential and the twisted chiral ring. We can in fact find a "mirror theory" that reproduces correctly the twisted chiral sector of the original sigma model. For concreteness, let us consider the hypersurface of GPA'-1, defined by a degree d polynomial equation G((/>I,..., t/d W = xf + --- + xff+ e X !---XN, (3.23) modded out by the (Z^)^-1 orbifold action 27rim,- »f Xi Xi mi e z, Y^i=l = 0 (mod d). (3.24) The change of variables (3.22) is justified by looking at the overlaps of D- branes with RR ground states, to be more precise, overlaps of B-branes with RR ground states constructed by the A-twisted semi-infinite cigar diagram. These quantities do not depend on the superpotential parameters but carry a lot of information on the twisted chiral sector of the theory. What we show is that the claimed mirror reproduces the same results as the original hypersurface sigma model, as long as these overlaps are concerned. To this end, we first show that such overlaps for the sigma model on the hypersyrface G — 0 can be computed in the sigma model on the ambient non-compact Mirror Symmetry 181 toric manifold, the total space of 0{—D) over QPA_1. The essential point is that the latter model has a normalizable RR-ground state that remains as a ground state even after turning on the superpotential W — PG. Let us denote the compact hypersurface by M and the non-compact ambient space by V. The normalized ground state of the V model is the one corresponding to the twisted chiral operator S — S x d where S is the field strength of the linear sigma model. Let us denoted the ground state by After turning on W — PG, this continues to be the RR ground state of the M model corresponding to the identity operator 1: Let 7 be a B-brane in the M-model. By embedding of M into V, this can also be regarded as a B-brane of the V model. Since the superpotential parameter cannot enter into the overlaps of B-brane and RR ground states defined via A-twist, we find the relation between the overlaps <7| Thus, the overlap II7 = (7|1)m of the M model is written purely in terms of the F-model. Now, we know the mirror of the V model and therefore the overlap (7|J)y can be expressed in the mirror LG theory. The mirror of the B-brane 7 is an A-brane which we denote by 7. We know that the overlap of an A-brane with RR ground states defined by B-twist is given by the weighted period integral. Thus, IF = <7|£)v = dYi • • • dYjvdYpdE*W>W)£+Ef=1 Sm (3.26) Here we have used the form of the mirror in which the field S is not yet integrated out, and 7' is the lift of 7 in such a description. This can also be written as IF = -id^- [ dYi • • • dYjydYpdS at Jtj, Integration over S yields a delta function that is solved by Yi — dZj, Yp — 1 —t/d+ Z\ + VZN (modulo (Z^'" ), and thus 182 K. Hori The i-derivative drops the factor —i Zx "' Zn from the exponent an we finally have Zl ZN TT7 - _Pt/d f e~ d£i • • • e~ dZN e-dzi+et/d-Zl--zN) = (-1)^+1 e'/d [ dX.l'e-K&xf+^MN). (3.27) This is the expression for the overlaps for the LG model of variables Xi,..., Xrsj X t d and superpotential W = i + e / Xi • • • XN (modulo orbifold action). We note that we are led to perform the change of variables (3.22) 3.5.1 Low Energy Dynamics Let us examine the low energy dynamics of this LG orbifold. The vacuum equation dxiW — 0 has |N — d\ solutions at p t/d d N d d xf = • • • = X N = ---XN=:S, S ~ = (-d) e"*, and, for d > 2, one solution at Xi = • • • = XN = 0. The former solutions are present if d ^ N and correspond to massive vacua, that breaks spontaneously the Z2\N-d\ axial R-symmetry to Z2- They corre- sponds to the |N — d\ vacua (2.63) found in the original linear sigma model approach. The critical point at X;L = 0 corresponds to the "missing vacua" suggested in LSM approach. The low energy behaviour of the theory de- pends on the value of d in relation to N. • d= 1 This is the case where M — CPN~2. There are only N — 1 massive vacua. If we integrate out Xjv, we obtain the constraint Xjv-i = — e_t/(X\ • • • Xjv-2) and the effective superpotential for the remaining fields is W = X1 + --- + Xjv-2 - e~7(Xi • • • XN_2). (3.28) This is nothing but the Av-2 affine Toda superpotential. Thus, we have reproduced the mirror symmetry of QP^-2 model and affine Toda theory. Mirror Symmetry 183 • 2 In particular, we see the enhancement of the axial Z2(jv-d) R-symmetry to U(1)A symmetry. For d — 2 the critical point at X;L — 0 is non-degenerate and does not corresponds to a non-trivial fixed point. However, because of the orbifolding, it could correspond to multiple vacua. • d = N This is the case where M is a Calabi-Yau manifold. There are no massive vacua but one massless vacuum at Xi — 0. At Xi — 0, the last term of (3.23) is equally relevant compared to the first N terms. Thus t remains as the parameter describing a marginal deformation of the SCFT. • d>N In this case M has negative first Chern class. There are (d — N) massive vacua and one massless vacuum at X;L — 0. At X;L — 0, the first N terms of (3.23) are irrelevant compared to the last term. Thus, the IR fixed point is described by the theory with superpotential d WiR=& X1---XN. (3.30) The vacuum equation dWIR — 0 is solved if two of X;'s vanish. Namely, the theory is a free SCFT on CN~2. This is expected since if c\ (M) < 0 the sigma model is IR free. Thus, we have seen that the LG orbifold with the superpotential (3.23) and group (Z^)^-1 captures the physics of the sigma model for all values of d. Also, it describes both the massive vacua and the massless vacuum that flows to a non-trivial (or trivial) IR fixed point. 3.5.2 Geometric Description of the Mirror The above manipulation applies only to a special class of hypersurfaces/complete- intersections in toric manifolds. We now present a manipulation that applies to a general model. 184 K. Hori As before, we look at the overlaps of D-branes with RR-ground states: r N n = d / dS dYP JJ dYi S exp (-»w) (3.31) J i= 1 Here, we again use the description including the S field; W — Yi — dYp — t) + e~Yi + e~Yp. Since oE is given by d/dYp of the linear terms in W we have (we henceforth ignor the factors of i) e~Yi _ie-YP n = JdX f[AYiAYP-^- exp Yl -dYP-t exp I Yp Yi Yp = J dS AYt dYp e~ exp ^ Yi - dYP - t^ exp ^-i ^ e~ — i e~ = / ndri dYp e~Yp s feYi -dYp - exp n e~Yi -ie~Yp i= 1 \i=l / \ i=1 (3.32) We make the following change of variables e~Yp — P, e~Yi — PAi, for i — 1,..., d, e~Yi - Aj, for j - d + 1,..., N. Then, = /S KH+«> (g a+1) «p £>) • (3.33) Thus we have obtained a submanifold M° of (Cx )N defined by N II (3.34) i=1 d J2Ai + 1=0" (3.35) i=l Mirror Symmetry 185 This is a non-compact manifold of dimension N — 2. The expression (3.33) is identical to the period of an LG model on M° with superpotential N WM° = E Ai- (3"36) i=d+1 This model is the mirror of the sigma model on M, at least when twisted to topological field theory. In the case d — N where M is a compact Calabi-Yau manifold, the superpotential (3.36) is trivial and the mirror is simply the non-linear sigma model on M°. The mirror manifold M° is actually an open subset of the compact hypersurface in QP^-1 = + ••• + $#+ = (3.37) modded out by the (ZJ\I)n~'2 action given by 7f = l, 71-"7iv = l. (3-3) To see this, we note that At = ~ (3.39) is invariant under the Cx x (ZN)n~2 action and solves the first equation (3.34). The second equation (3.35) becomes = 0. If and yields the same A^ it is easy to see that = and • • • = • • • x x N 2 modulo C action. Then, this means = modulo the C x (ZN) ~ action. Thus, the map from [<&j] to A;L is one to one. Under this identification, we have 1 £ d$ JG($ ...,$ ) n = f 1 tt d^j r i u N J vol(Cx) .N-1 ) = /"«, (3.40) G=0/ J which is the period of the holomorphic differential of the Calabi-Yau manifold M. This leads us to propose that the sigma model on M is mirror to the sigma model on the compactified manifold M. 186 K. Hori Repeating this procedure, we are led to propose the following as the mirror of the degree d hypersurface in QPA_1 for d < N. It is the LG model defined on the hypersurface in QPd_1 x CN~d — {(($1 : ... : x {([/!,..., jv_I)> d + + e^ i • • • • • • $d = 0 (3.41) modded out by the (Zd)N~2 orbifold action Uj^-yj+dPj, if = 1, 7i-"7iv = l, (3-42) with the superpotential W = U?+-+ (3.43) Note that the hypersurface (3.41) can be regarded as the degree d Calabi- Yau hypersurfaces in QPd-i fibred over It itself is a non-compact CY manifold whose holomorphic volume form is f2d-2d i • • • d N-d- 3.6 Dilatonic Backgrounds So far, we have been considering non-linear sigma models that would give a constant dilaton in case the model yields a superstring background. However, with a slight modification of the linear sigma models, one can also study models that yield string backgrounds with non-trivial dilaton profiles. In this section, we consider a simple but important class of dilatonic back- grounds — Euclidean 2d black holes, and determine their mirrors. This consideration will also provide a way to refine the statement of mirror sym- metry even for non-Calabi-Yau sigma models. 3.6.1 2d Black hole and Liouville Theory A while ago, Fateev, Zamolodchikov and Zamolodchikov conjectured a du- ality between the two-dimensional black hole background and a potential theory called sine-Liouville theory. The former has a semi-infinite cigar ge- ometry which asymptotes to a flat cylinder with a linear dilaton. The latter is a theory of a cylinder variable with (Liouville x cosine) potential. This also has an asymptotic region with a linear dilaton where the potential is exponentially small. Comparison of the asymptotic regions suggests that the two theories are related by T-duality. The degeneration of the circle in Mirror Symmetry 187 the 2d black hole corresponds to the growing sine-Liouville potential. This is strongly reminiscent of our mirror symmetry. There is a supersymmetric version of the FZZ duality. It is between the 2d black hole background for fermionic strings and the H — 2 Liouville the- oiy. The former is also known as the SL(2, M)fc+2 mod U(l) Kazama-Suzuki super-coset model, which have a semi-infinite cigar geometry which asymp- Figure 6: The cigar totes to a flat cylinder of radius y/k with a linear dilaton. The latter is the LG model of a periodic chiral superfield Y = Y + 2m with the superpoten- tial W — e-y and the Kahler potential K — \Y\'2/2k. The two theories are weakly coupled in the opposite regimes k >> 1 and k 1. We show that this duality is indeed a mirror symmetry and can be shown by applying the method explained in this section. The essential part is to show that the SCFT of the 2d black hole arises as the infra-red fixed point of the following U( 1) gauge theory. It is the theory with two chiral superfields $ and P = P + 2m which transforms as $ —» eza$ and P + ia under the U(l) gauge transformation, with the Lagrangian given by (3.44) Note that the U( 1) gauge group is broken at large values of (f> and we are left with a neutral field P whose geometry is just the cylinder of radius y/k. There are also a region with small Af — 2 superconformal algebra of the IR fixed point in the ring of left- chiral operators which is invariant under renormalization group flow. If there is a non-trivial IR fixed point, the right moving part of the Af — 2 superconformal algebra is in that ring, and therefore must be observable even at high energies. This idea has been introduced by Witten and also used by Silverstein and Witten in their study of LG models and heterotic string backgrounds. Technically essential point is to identify the right-moving R- current. The axial U( 1) R-symmetry is anomalous in the present system, but one can use the field P to modify the current in a gauge invariant way so that it is conserved. By reality of the currents in the asymptotic region, one can also fix the ambiguity due to the presence of the global symmetry. Thus, one can completely identify the Af — 2 algebra. This shows that the central charge of the IR fixed point is (3.45) which is the correct value for the super-coset model. The modified parts of the Af — 2 currents are linear in P and this exhibits the linear dilaton in the asymptotic region (with the correct slope). Finally, one excludes the possibility that the theory flows not to the super-coset itself but to some other nearby fixed point with the same central charge, symmetries, and asymptotic behaviour. This is done by showing that the super-coset has no supersymmetric, parity invariant marginal operator that is small at the asymptotic region. The rest is a straightforward generalization of the argument in the pre- vious section. Dualization of the phase of $ and the imaginary part of P turns them into twisted chiral superfields Y and Yp (both period 2wi). The exact superpotential of the dual system is W = E(y + Yp) + e-y. (3.46) No non-perturbative superpotential is generated for Yp since there is no P-vortex. The Kahler potential is (3.47) where • • • are small in the asymptotic region. In the limit e —» oo, it is appropriate to integrate out S and we have the constraint Y + Yp — 0. In fact, one can use the rigidity of the super-coset to show that the terms • • • in Mirror Symmetry 189 (3.47) vanishes. Thus, we conclude that the dual theory flows to the M — 2 Liouville theory with the Kahler potential K — — |Y|2/2k. This completes the proof of the claimed equivalence. 3.6.2 Squashed Toric Manifolds This story can be generalized to other models including higher dimensional target spaces. The starting point is the linear sigma model for the toric target space X— U(l)k gauge theory with N chiral fields of charge Q\. We modify the theory by gauging the U(l)N~k flavor group and introducing new N k fields Pq: each of which shifts under each U(l) factor of the U(l) ~ . We note that the flavor group II^i^ U(l)q acts on with the charge Riq (q — 1,..., N — k) which are complementary to Qf, rank(Q°, Riq) — N. Denoting the vector superfield for the new gauge symmetry by V^', the Lagrangian of the system reads J it! J J r r N-k , N—k 1 J .0= 1 4 <7=1 (3.48) where Ri-V' — Y^q=i RiqVq- Since a new field is introduced to each of the new U(l) gauge groups, the modification is mild — it does not change the dimension of the target space but changes the detail of the metric. In fact, the new target space X' is the same complex manifold as before (again a toric manifold) with the same Kahler class. Deep in the interior of the base of the torus fibration, the sizes of the fibers are constants ~ y/kq- Thus X' is a "squashed" version of the toric manifold. In the limit kq —> oo, the P-Tl pairs decouple, and we recover the sigma-model on the "round toric manifold" X. When all bf — 0, the theory is expected to flow to a non-trivial superconformal field theory. The M — 2 currents can be uniquely identified in the left-chiral ring if there are asymptotic regions, and they include linear terms in Pq proportional to bq:— Y^iLi Riqi implying a linear dilaton if bq ^ 0. The central charge is found as t \ (3.49) q q=l 190 K. Hori In the "round limit" kq —» oo, we recover c/3 —» iV — k — dimX. The dual theory is found as before. In the limit ea, eq —> oo, we obtain a LG model on the (Cx )N~k defined by ^=1 QT^i = ta with the following Kahler and superpotentials 1 N N e Yi 3 5 K = --J2gij¥iYj + ---, W = Y, ~ ' ( " °) i,j=1 i=1 where gij Y^q=i RiqRjq/kq- This is the mirror of the sigma-model on the squashed toric manifold X'. Note that the Kahler potential depends on the squashing parameters kq and tends to vanish in the "round limit" kq —» oo. Let us consider the simple case X — CP1 which is not conformal. The original linear sigma model is a U(l) gauge theory with two charge 1 matters To this we add one U( 1) gauge group and one field P, where the new gauge group acts on the fields as —» elA<&i and P —» P + iA. The parameter k for the kinetic term of P is a free parameter but FI parameter r runs precisely as before r(/i) = 2log(/i/A). At high energies we have r k. Then, \(j>\\ and can have large values which breaks both [7(1) gauge groups, leaving us with a single uncharged periodic field P, whose geometiy is a cylinder of radius y/k. Thus, we expect that the vacuum manifold X' to be of "sausage" shaped. Indeed, the metric of the classical vacuum manifold is d^ = 1 + 1 + ^, » 1/2C C~o, U~ < l/k k<£C<.r-k, 1/2 (r-C) C ~ r- They correspond to a neighborhood of a tip of the sausage, the flat cylinder (with metric ds2 = d('2/k + kdip'2) and a neighborhood of the other tip, respectively. The sausage is longer at higher energy and approaches the flat cylinder of radius Vk in the UV limit. In other words, it flows to shorter sausages at lower energies. This model is called the "sausage model". We Mirror Symmetry 191 r/re ... Figure 7: The "sausage" have shown that this model is mirror to the M — 2 sine-Gordon model with a finite Kahler potential, K — \Y\'2/2k + • • •. Since the UV limit of the sausage is the flat cylinder of radius Vk: the UV limit of the mirror has to be the flat cylinder of radius l/Vk, the one described by the Kahler potential K — \Y\'2/2k. Since the potential term is negligible in the UV limit, this is consistent with our claimed mirror symmetry. The equivalence of the M — 2 sausage model and H — 2 sine-Gordon model with finite Kahler potential has been conjectured by Fendley and Intriligator based on the study of scattering matrices. 3.6.3 Orbifolds Another class of generalizations of the model in Section 3.6.1 are orbifolds. he model with Lagrangian (3.44) has U(l) symmetry which corresponds to the rotation of the cigar. We consider orbifolding the system by the 7Ln subgroup of that symmetry group. In the UV gauge theory, this is to gauge the symmetry that shifts Im(P) by 1/n unit. In other words, the orbifold theory is obtained by replacing P by P'/n where P' has the ordinary periodicity P' = P' + 2wi. After dualization with vortex-instanton effects taken into account, we obtain the theory with superpotential W — y 2 2 E(Y + nYP,) + e- , and Kahler potential K = |E| - ^|Yp/| + • • In the limit e2 —» oo, we have the constraint Y + nYpt — 0, solved by Y — nY'i Ypi — — Y', and we obtain the LG theory with W = e~nY', (3.52) m his is the mirror of the Zn-orbifold of the One can consider more generalizations by taking the products and var- ious orbifolds, out of which we consider one example. Let us name the original SL(2, M)fc+2 mod U(l) supercoset by Ck and take the product of them, Ckl fcjV = Ckl x • • • CkN. Let us take the orbifold of the product by the Zd subgroup of the diagonal U(l) symmetry, Cjj? kN/Zd- We are in- terested in the mirror of such a model. As a trick to find it, we consider iV_1 orbifolding the model Ckl^kN/Zd by the (Zd) -symmetry. This would be N equivalent to C^kN/(id) = CkjZd x •••CfcjV/Zd, the product of N Zd orbifolds. We know the mirror of the individual orbifold — the mirror of Cfc/Zd is the LG model (3.52) in which n is replaced by d. Thus, the mirror 1 of (CG^JZAHZD)"- - CKL/ZD X ••• CkN/Zd is the LG model W = e-dyi + • • • + e~dYN, 2 (3.53) is _— d? \\n\2r d IVr ' |2 2fci l ll 2kN l iVl " At this point, we use the general fact in the orbifold theoiy — "the orb- ifold of the orbifold by the quantum symmetry is the original model". The model we are considering, ^/Zd)/(Zd)^_1, has the quantum (Zd)N_1 symmetry. The latter acts on the mirror fields by Y,' ^ Y,' + rm e Z, = 0 (mod d). (3.54) Thus, the model Cj^ fcjV/Z^ is mirror to the above LG model modded out by this (Zd)A _ 1 -orbifold action. Let us now take the limits ki —> oo. Each Cki approaches the CFT of the flat complex plane in the limit ki —> oo. Thus Cj^ fcjV/Z^ approaches the free orbifold CFT CN /Z^. By the above conclusion, we find that it is mirror to a LG model with superpotential W — e~dYi H 1- e~dY'v modded out by the (Z^'"1 orbifold action (3.54), in the limit of vanishingly small Kahler potential. This gives the derivation of the fact mentioned at the end of Section 3.4.3. Indeed the orbifold action (3.54) is identical to (3.19). 4 Bibliography We make some remarks on bibliography. We start with the historical remarks, and then move on to section by section discussion. Mirror symmetry was first suggested by Dixon [1] and Lerche, Vafa and Warner [2] from consideration of N — (2,2) superconformal algebra. Ex- amples of mirror pairs of Calabi-Yau manifolds were found by Greene and Mirror Symmetry 193 Plesser [3] based on Gepner' construction of exactly solvable M — 2 SCFT [4] and their geometric interpretation [5-7]. Search for candidate mirror pairs was also done in [8]. The examples were extended systematically using toric geometry by Batyrev [9]. Candelas et al [10] have shown how mirror symmetry can be used effectively to analyze non-perturbative effects involv- ing worldsheet instantons. This has led to a conjecture on the "number" of rational curves in a Calabi-Yau manifold. Mathematical proof of the con- jecture was initiated by Kontsevich [11] and completed by Givental [12] and [13]. Mirror symmetry between non-conformal theories first appeared in the work of Fendley and Intriligator [14], where it was shown that the supersym- metric QP1 model and M — 2 sine-Gordon model have the same scattering matrices. This mirror pair and generalizations have also been observed in many other ways: from the ground state metric [15], in topological sigma models [16,17] and in topological strings [18]. Mirror symmetry involving D-branes has a rich structure physically and mathematically. This was first discussed from mathematical point of view by Kontsevish [19]. A notable physical work is by Strominger, Yau and Zaslow [20] who proposed, using the transformation of D-branes under T- duality, that mirror symmetry of Calabi-Yau manifolds in nothing but du- alization of special Lagrangian torus fibrations. There are many review works on mirror symmetry, for example see [21, 22]. The recent book [23] includes introductory material as well as advanced topics from both physical and mathematical points of view. Section 1 (2,2) supersymmetry in 1 +1 dimensions can be regarded as the dimen- sional reduction of M — 1 supersymmetry in 3 + 1 dimensions. The basic references of the latter are [24,25]. Twisted chiral multiplet appears only after dimensional reduction and was found in [26]. For general aspects of supersymmetric ground states in supersymmetric quantum mechanics/field theories, see the works by Witten [27-29]. Twisting M — 2 supersymmetric field theories to topological field theories was introduced in [30,31]. The cor- respondence between supersymmetric ground states and chiral ring elements is formulated in the form of Section 1 in [32]. That supersymmetric non-liner sigma models on Kahler manifolds in 1+1 dimensions have (an equivalent of) M — (2,2) supersymmetry was found by Zumino [33]. Af — (2,2) Landau-Ginzburg model is the dimensional re- 194 K. Hori duction of the very first supersymmetric field theory discovered by Wess and Zumino [34] in 3+1 dimensions, and therefore sometimes called Wess-Zumino models. Non-renormalization theorem for superpotential is first proved per- turbatively by Grisaru, Siegel and Rocek [35] by super-graph formalism. A simpler but sometimes stronger argument is due to Seiberg [36]. Renormal- ization group flows of supersymmetric sigma models on Kahler manifolds (in 1 + 1 dimensions) was discussed in [37]. That the Kahler class is renormal- ized only at one loop is shown in [38] (for Calabi-Yau spaces) but a simpler proof is provided in [23]. M — 2 supersymmetric boundary conditions in Calabi-Yau sigma models are first discussed by Ooguri, Oz and Yin [39], where the terminology of A- branes and B-branes was introduced. Boundary conditions in more general target spaces as well as in LG models were studied in [40,41]. A-parity and B-parity were introduced and studied in [42]. The properties of the overlaps with RR ground states, discussed in Section 1.3.3, were found in [40,42]. In application to superstring theory, we ordinarily need to study systems with (2,2) superconformal symmetry, such as Calabi-Yau sigma models and LG models with homogeneous superpotentials. Recently, however, it was shown massive Af = (2,2) theories can also arise to describe special super- string backgrounds [43]. Section 2 To a large extent, this section is a review of the seminal work by Wit- ten [44] where linear sigma models are used in the modern context to study dynamical aspects of supersymmetric sigma models and LG models as well as the relation between them. Vector multiplets in (2,2) models are the dimensional reduction of 3 + 1 dimensional M — 1 vector multiplet. See [24,25] for the latter. (2,2) supersymmetric linear sigma models in 1 + 1 dimensions were originally formulated in [45,46]. Comprehensible reviews of toric geometry in the context of linear sigma model can be found in [47,23]. For those who are interested in more mathe- matical aspects, the books by Oda [48] and by Fulton [49] are recommended. For the equivalence of symplectic quotients and the algebro-geometric quo- tients (the latter are called Geometric Invariant Theory quotients [50]), see [51]. The SlogS type effective superpotential in non-scale-invariant linear sigma model was first computed in [52]. The relevant aspect of the quantum Mirror Symmetry 195 gauge theory in 1 +1 dimension, especially the identification of Theta angle as the background electric flux, is due to Coleman [53]. The discussion of solitons in CPA_1 model is from [45]. Calabi-Yau/(LG) orbifold correspon- dence is due to the paper [44]. Analog of this in the non-conformal case was discussed also by Witten in [54]. RG flow from orbifolds attracted recent interest from the space-time point of view [55]. Section 3 The proof of mirror symmetry discussed in this section is due to my work with Vafa [56]. T-duality was found by Kikkawa and Yamasaki [57] and Sakai and Senda [58]. A path-integral formulation of T-duality is due to Buscher [59]. A review of T-duality can be found in [60]. The treatment of T-duality in A/* = (2,2) superspace is due to Rocek and Verlinde [61]. Vortex solutions were found in abelian Higgs model by Nielsen and Ole- sen [62]. (Vortices are instantons in 1 + 1 dimensions but are strings in 3 + 1 dimensions [62].) Vortex-anti-vortex gas was used by Callan-Dashen- Gross [63] to study quantum gauge theories in 1 + 1 dimensions. In finding the dual superpotential, we were inspired by the work of Polyakov [64] where it was shown how the monopole-anti-monopole gas generates the potential for the dual photon. The proof of the duality between SL(2, M) mod U( 1) Kazama-Suzuki supercoset model and H — 2 Liouville theory is due to myself and Ka- pustin [65]. See also [67,68] for some earlier works. This is motivated by a work by Fateev, Zamolodchikov and amolodchikov [66] which pro- poses equivalence between the non-supersymmetric coset model and "sine- Liouville" theory (the latter is a bosonic LG model with the potential U — e~p cos(<^>)). The supersymmetric duality can be used to study the NS5-brane backgrounds. See for example [69-71]. After the proof of mirror symmetry [56], two independent "proofs" ap- peared. 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