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Mirror Symmetry and Algebraic Geometry, 1999 67 A http://dx.doi.org/10.1090/surv/068 Selected Titles in This Series 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel 'Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 (Continued in the back of this publication) Mirror Symmetry and Algebraic Geometry Mathematical Surveys and Monographs Volume 68 Mirror Symmetry and Algebraic Geometry David A. Cox Sheldon Katz American Mathematical Society Providence, Rhode Island ^VDED" Editorial Board Georgia M. Benkart Michael Renardy Peter Landweber Tudor Stefan Ratiu, Chair 2000 Mathematics Subject Classification. Primary 14-02; Secondary 81-02. ABSTRACT. This monograph is an introduction to the mathematics of mirror symmetry, with a special emphasis on its algebro-geometric aspects. Topics covered include the quintic threefold, toric geometry, Hodge theory, complex and Kahler moduli, Gromov-Witten invariants, quantum cohomology, localization in equivariant cohomology, and the recent work of Lian-Liu-Yau and Givental on the Mirror Theorem. The book is written for algebraic geometers and graduate students who want to learn about mirror symmetry. It is also a reference for specialists in the field and background reading for physicists who want to see the mathematical underpinnings of the subject. Library of Congress Cataloging-in-Publication Data Cox, David A. Mirror symmetry and algebraic geometry / David A. Cox, Sheldon Katz. p. cm. — (Mathematical surveys and monographs, 0076-5376; v. 68) Includes bibliographical references and index. ISBN 0-8218-1059-6 (alk. paper) 1. Mirror symmetry. 2. Geometry, Algebraic. 3. Mathematical physics. I. Katz, Sheldon, 1956- II. Title. III. Series: Mathematical surveys and monographs; no. 68. QC174.17.S9C69 1999 516.3'62—dc21 98-55564 CIP AMS softcover ISBN: 978-0-8218-2127-5 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. © 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 15 14 13 12 11 10 To our families, who have endured with good grace our preoccupation with this book. D. A. C. S. K. Contents Preface xiii Goal of the Book xiii Relation to Physics xiv How to Read the Book xv Acknowledgements xvii Our Hope xviii Notation xix Chapter 1. Introduction 1 1.1. The Physics of Mirror Symmetry 1 1.2. Three-Point Functions 6 1.3. Why Calabi-Yau Manifolds? 10 1.4. The Mathematics of Mirror Symmetry 10 1.5. What's Next? 12 Chapter 2. The Quintic Threefold 15 2.1. The A-Model Correlation Function of the Quintic Threefold 15 2.2. The Quintic Mirror 17 2.3. The Mirror Map 19 2.4. The B-Model Correlation Function 21 2.5. Putting It All Together 22 2.6. The Mirror Theorems 24 Chapter 3. Toric Geometry 31 3.1. Cones and Fans 31 3.2. Polytopes and Homogeneous Coordinates 33 3.3. Kahler Cones and Symplectic Geometry 38 3.4. The GKZ Decomposition 42 3.5. Fano Varieties and Reflexive Polytopes 46 3.6. Automorphisms of Toric Varieties 47 3.7. Examples 49 Chapter 4. Mirror Symmetry Constructions 53 4.1. The Batyrev Mirror Construction 53 4.2. The Quintic Threefold, Revisited 61 4.3. Toric Complete Intersections 62 4.4. The Voisin-Borcea Construction 65 Chapter 5. Hodge Theory and Yukawa Couplings 73 5.1. Hodge Theory 73 x CONTENTS 5.2. Maximally Unipotent Monodromy 78 5.3. The Griffiths-Dwork Method 83 5.4. Examples 87 5.5. Hypergeometric Equations 90 5.6. Yukawa Couplings 102 Chapter 6. Moduli Spaces 113 6.1. Complex Moduli 113 6.2. Kahler Moduli 127 6.3. The Mirror Map 148 Chapter 7. Gromov-Witten Invariants 167 7.1. Definition via Algebraic Geometry 168 7.2. Definition via Symplectic Geometry 184 7.3. Properties of Gromov-Witten Classes 191 7.4. Computing Gromov-Witten Invariants, I 196 Chapter 8. Quantum Cohomology 217 8.1. Small Quantum Cohomology 217 8.2. Big Quantum Cohomology 229 8.3. Computing Gromov-Witten Invariants, II 234 8.4. Dubrovin Formalism 239 8.5. The A-Variation of Hodge Structure 242 8.6. The Mirror Conjecture 257 Chapter 9. Localization 275 9.1. The Localization Theorem 275 r 9.2. Localization in M0,n(P , d) 282 9.3. Equivariant Gromov-Witten Invariants 298 Chapter 10. Quantum Differential Equations 301 10.1. Gravitational Correlators 301 10.2. The Givental Connection 311 10.3. Relations in Quantum Cohomology 320 Chapter 11. The Mirror Theorem 331 11.1. The Mirror Theorem for the Quintic Threefold 332 11.2. Givental's Approach 356 Chapter 12. Conclusion 397 12.1. Reflections and Open Problems 397 12.2. Other Aspects of Mirror Symmetry 400 Appendix A. Singular Varieties 407 A.l. Canonical and Terminal Singularities 407 A.2. Orbifolds 408 A.3. Differential Forms on Orbifolds 409 A.4. The Tangent Sheaf of an Orbifold 410 A.5. Symplectic Orbifolds 410 Appendix B. Physical Theories 411 B.l. General Field Theories 411 CONTENTS B.2. Nonlinear Sigma Models 419 B.3. Conformal Field Theories 423 B.4. Landau-Ginzburg Models 426 B.5. Gauged Linear Sigma Models 428 B.6. Topological Quantum Field Theories 430 bliography 437 Index 453 Preface The field of mirror symmetry has exploded onto the mathematical scene in recent years. This is a part of an increasing connection between quantum field theory and many branches of mathematics. It has sometimes been said that quantum field theory combines 20th century physics with 21st century mathematics. Physicists have gained much experience with mathematical manipulations in situations which have not yet been mathe­ matically justified. They are able to do this in part because experiment can help them differentiate between which manipulations are feasible, and which are clearly wrong. Those manipulations that survive all known tests are presumed to be valid until evidence emerges to the contrary. Based on this evidence, physicists are confident about the validity of mirror symmetry. One of the tools they use with great virtuosity is the Feynman path in­ tegral, which performs integration with complex measures over infinite dimensional spaces, such as the space of C°° maps from a Riemann surface to a Calabi-Yau threefold.
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