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Clay Mathematics Monographs 1 Volume 1 Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the M enumeration of holomorphic curves inside complex manifolds by solving differ- i ential equations obtained from a “mirror” geometry. The inclusion of D-brane r states in the equivalence has led to further conjectures involving calibrated r o submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds:the Gopakumar Vafa invariants. r This book aims to give a single, cohesive treatment of mirror symmetry from S both the mathematical and physical viewpoint. Parts 1 and 2 develop the neces- y sary mathematical and physical background “from scratch,”and are intended for m readers trying to learn across disciplines. The treatment is focussed,developing only the material most necessary for the task. In Parts 3 and 4 the physical and m mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic e physics. Each physical theory can be described geometrically, and thus mirror t r symmetry gives rise to a “pairing” of geometries. The proof involves applying y MIRROR SYMMETRY R↔1/R circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic MIRROR SYMMETRY setting,beginning with the moduli spaces of curves and maps,and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero,as is predicted by mirror symmetry. Part C 5 is devoted to advanced topics in mirror symmetry, including the role of u D-branes in the context of mirror symmetry, and some of their applications in m Kentaro Hori physics and mathematics:topological strings and large N Chern-Simons theory; r Sheldon Katz u geometric engineering; mirror symmetry at higher genus; Gopakumar-Vafa n Albrecht Klemm invariants; and Kontsevich’s formulation of the mirror phenomenon as an equiv- V alence of categories. Rahul Pandharipande a f This book grew out of an intense,month-long course on mirror symmetry at Pine a Richard Thomas Manor College,sponsored by the Clay Mathematics Institute. The lecturers have a Cumrun Vafa tried to summarize this course in a coherent,unified text. n d Ravi Vakil E Eric Zaslow r i For additional information c and updates on this book, visit Z a www.ams.org/bookpages/cmim-1 s l o w , E d i t o r s www.ams.org American Mathematical Society AMS CMIM/1 www.claymath.org CMI Clay Mathematics Institute 3 color cover:This plate PMS 123 Yellow;This plate PMS 187 Red;This plate Black Ink 952 pages pages on 50 lb stock • 2 5/8 spine cmim-1-title.qxp 6/11/03 11:03 AM Page 2 MIRROR SYMMETRY cmim-1-title.qxp 6/11/03 11:01 AM Page 1 Clay Mathematics Monographs Volume 1 MIRROR SYMMETRY Kentaro Hori Sheldon Katz Albrecht Klemm Rahul Pandharipande Richard Thomas Cumrun Vafa Ravi Vakil Eric Zaslow American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 14J32; Secondary 14-02, 14N10, 14N35, 32G81,32J81, 32Q25, 81T30. For additional informationand updates on this book, visit www.ams.org/bookpages/cmim-1 Library of Congress Cataloging-in-Publication Data Mirrorsymmetry/KentaroHori...[etal.]. p.cm. —(Claymathematicsmonographs,ISSN1539-6061;v. 1) Includesbibliographicalreferencesandindex. ISBN0-8218-2955-6(alk. paper) 1.Mirrorsymmetry. 2.Calabi-Yaumanifolds. 3.Geometry,Enumerative. I.Hori,Kentaro. II.Series. QC174.17.S9M5617 2003 530.14(cid:1)3–dc21 2003052414 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication ispermittedonlyunderlicensefromtheClayMathematicsInstitute. Requestsforsuchpermission should be addressed to the Clay Mathematics Institute, 1770 Massachusetts Ave., #331, Cam- bridge,MA02140,[email protected]. (cid:1)c 2003bytheauthors. Allrightsreserved. PublishedbytheAmericanMathematicalSociety,Providence,RI, fortheClayMathematicsInstitute,Cambridge,MA. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 080706050403 Contents Preface xi Introduction xiii A History of Mirror Symmetry xv The Organization of this Book xvii Part 1. Mathematical Preliminaries 1 Chapter 1. Differential Geometry 3 1.1. Introduction 3 1.2. Manifolds 4 1.3. Vector Bundles 5 1.4. Metrics, Connections, Curvature 11 1.5. Differential Forms 18 Chapter 2. Algebraic Geometry 25 2.1. Introduction 25 2.2. Projective Spaces 25 2.3. Sheaves 32 2.4. Divisors and Line Bundles 38 Chapter 3. Differential and Algebraic Topology 41 3.1. Introduction 41 3.2. Cohomology Theories 41 3.3. Poincaré Duality and Intersections 42 3.4. Morse Theory 43 3.5. Characteristic Classes 45 3.6. Some Practice Calculations 53 Chapter 4. Equivariant Cohomology and Fixed-Point Theorems 57 4.1. A Brief Discussion of Fixed-Point Formulas 57 v vi CONTENTS 4.2. Classifying Spaces, Group Cohomology, and Equivariant Cohomology 58 4.3. The Atiyah–Bott Localization Formula 62 4.4. Main Example 64 Chapter 5. Complex and Ka¨hler Geometry 67 5.1. Introduction 67 5.2. Complex Structure 67 5.3. Kähler Metrics 71 5.4. The Calabi–Yau Condition 74 Chapter 6. Calabi–Yau Manifolds and Their Moduli 77 6.1. Introduction 77 6.2. Deformations of Complex Structure 79 6.3. Calabi–Yau Moduli Space 82 6.4. A Note on Rings and Frobenius Manifolds 87 6.5. Main Example: Mirror Symmetry for the Quintic 88 6.6. Singularities 95 Chapter 7. Toric Geometry for String Theory 101 7.1. Introduction 101 7.2. Fans 102 7.3. GLSM 111 7.4. Intersection Numbers and Charges 114 7.5. Orbifolds 121 7.6. Blow-Up 123 7.7. Morphisms 126 7.8. Geometric Engineering 130 7.9. Polytopes 132 7.10. Mirror Symmetry 137 Part 2. Physics Preliminaries 143 Chapter 8. What Is a QFT? 145 8.1. Choice of a Manifold M 145 8.2. Choice of Objects on M and the Action S 146 8.3. Operator Formalism and Manifolds with Boundaries 146 8.4. Importance of Dimensionality 147 CONTENTS vii Chapter 9. QFT in d = 0 151 9.1. Multivariable Case 154 9.2. Fermions and Supersymmetry 155 9.3. Localization and Supersymmetry 157 9.4. Deformation Invariance 160 9.5. Explicit Evaluation of the Partition Function 162 9.6. Zero-Dimensional Landau–Ginzburg Theory 162 Chapter 10. QFT in Dimension 1: Quantum Mechanics 169 10.1. Quantum Mechanics 169 10.2. The Structure of Supersymmetric Quantum Mechanics 182 10.3. Perturbative Analysis: First Approach 197 10.4. Sigma Models 206 10.5. Instantons 220 Chapter 11. Free Quantum Field Theories in 1+1 Dimensions 237 11.1. Free Bosonic Scalar Field Theory 237 11.2. Sigma Model on Torus and T-duality 246 11.3. Free Dirac Fermion 254 11.4. Appendix 268 Chapter 12. N = (2,2) Supersymmetry 271 12.1. Superfield Formalism 271 12.2. Basic Examples 276 12.3. N = (2,2) Supersymmetric Quantum Field Theories 282 12.4. The Statement of Mirror Symmetry 284 12.5. Appendix 285 Chapter 13. Non-linear Sigma Models and Landau–Ginzburg Models 291 13.1. The Models 291 13.2. R-Symmetries 294 13.3. Supersymmetric Ground States 299 13.4. Supersymmetric Sigma Model on T2 and Mirror Symmetry 307 Chapter 14. Renormalization Group Flow 313 14.1. Scales 313 14.2. Renormalization of the Kähler Metric 315 viii CONTENTS 14.3. Superspace Decouplings and Non-Renormalization of Superpotential 331 14.4. Infrared Fixed Points and Conformal Field Theories 335 Chapter 15. Linear Sigma Models 339 15.1. The Basic Idea 339 15.2. Supersymmetric Gauge Theories 348 15.3. Renormalization and Axial Anomaly 353 15.4. Non-Linear Sigma Models from Gauge Theories 356 15.5. Low Energy Dynamics 378 Chapter 16. Chiral Rings and Topological Field Theory 397 16.1. Chiral Rings 397 16.2. Twisting 399 16.3. Topological Correlation Functions and Chiral Rings 404 16.4. Examples 408 Chapter 17. Chiral Rings and the Geometry of the Vacuum Bundle 423 ∗ 17.1. tt Equations 423 Chapter 18. BPS Solitons in N=2 Landau–Ginzburg Theories 435 18.1. Vanishing Cycles 437 18.2. Picard–Lefschetz Monodromy 439 18.3. Non-compact n-Cycles 441 18.4. Examples 443 ∗ 18.5. Relation Between tt Geometry and BPS Solitons 447 Chapter 19. D-branes 449 19.1. What are D-branes? 449 19.2. Connections Supported on D-branes 452 19.3. D-branes, States and Periods 454 Part 3. Mirror Symmetry: Physics Proof 461 Chapter 20. Proof of Mirror Symmetry 463 20.1. What is Meant by the Proof of Mirror Symmetry 463 20.2. Outline of the Proof 464 20.3. Step 1: T-Duality on a Charged Field 465 20.4. Step 2: The Mirror for Toric Varieties 472 CONTENTS ix 20.5. Step 3: The Hypersurface Case 474 Part 4. Mirror Symmetry: Mathematics Proof 481 Chapter 21. Introduction and Overview 483 21.1. Notation and Conventions 483 Chapter 22. Complex Curves (Non-singular and Nodal) 487 22.1. From Topological Surfaces to Riemann Surfaces 487 22.2. Nodal Curves 489 22.3. Differentials on Nodal Curves 491 Chapter 23. Moduli Spaces of Curves 493 23.1. Motivation: Projective Space as a Moduli Space 493 23.2. The Moduli Space M of Non-singular Riemann Surfaces 494 g 23.3. The Deligne–Mumford Compactification M of M 495 g g 23.4. The Moduli Spaces M of Stable Pointed Curves 497 g,n Chapter 24. Moduli Spaces M (X,β) of Stable Maps 501 g,n 24.1. Example: The Grassmannian 502 24.2. Example: The Complete (plane) Conics 502 24.3. Seven Properties of M (X,β) 503 g,n 24.4. Automorphisms, Deformations, Obstructions 504 Chapter 25. Cohomology Classes on M and M (X,β) 509 g,n g,n 25.1. Classes Pulled Back from X 509 25.2. The Tautological Line Bundles L , and the Classes ψ 512 i i 25.3. The Hodge Bundle E, and the Classes λ 516 k 25.4. Other Classes Pulled Back from M 517 g,n Chapter 26. The Virtual Fundamental Class, Gromov–Witten Invariants, and Descendant Invariants 519 26.1. The Virtual Fundamental Class 519 26.2. Gromov–Witten Invariants and Descendant Invariants 526 26.3. String, Dilaton, and Divisor Equations for M (X,β) 527 g,n 26.4. Descendant Invariants from Gromov–Witten Invariants in Genus 0 528 26.5. The Quantum Cohomology Ring 530 Chapter 27. Localization on the Moduli Space of Maps 535