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2 Linear Sigma Models 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 <f) and a Dirac fermion ij)±, ij)± — in 1 + 1 dimensions, with the following action + + (1-1) Here do and d\ are derivatives with respect to the time and space coordinates x1.
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