Vertex Algebras and Algebraic Curves

Transcription

1 Mathematical Surveys and Monographs Volume 88 Vertex Algebras and Algebraic Curves Edward Frenkei David Ben-Zvi American Mathematical Society

2 Contents Preface xi Introduction 1 Chapter 1. Definition of Vertex Algebras Formal distributions Locality Definition of a vertex algebra First example: commutative vertex algebras Bibliographical notes 23 Chapter 2. Vertex Algebras Associated to Lie Algebras Heisenberg Lie algebra The vertex algebra structure ontt Checking vertex algebra axioms Affine Kac-Moody algebras and their vertex algebras The Virasoro vertex algebra Bibliographical notes 46 Chapter 3. Associativity and Operator Product Expansion Uniqueness theorem Associativity. ' Operator product expansion Examples and applications of OPE A Lie algebra attached to a vertex algebra Strong Reconstruction Theorem Correlation functions Bibliographical notes 67 Chapter 4. Rational Vertex Algebras Modules over vertex algebras Vertex algebras associated to one-dimensional integral lattices Boson-fermion correspondence Lattice vertex algebras Introducing rational vertex algebras Constructing new vertex algebras Bibliographical notes 86 Chapter 5. Vertex Algebra Bundles The group AutO Exponentiating vector fields 89

3 viii CONTENTS 5.3. Primary fields The main construction A flat connection on the vertex algebra bundle Bibliographical notes 103 Chapter 6. Action of Internal Symmetries Affine algebras, revisited The general twisting property Description of the n-point functions and modules Bibliographical notes 114 Chapter 7. Vertex Algebra Bundles: Examples The Heisenberg algebra and affine connections The Virasoro algebra and projective connections Pseudodifferential operators and kernels The gauge action on the Heisenberg bundle The affine Kac-Moody vertex algebras and connections Bibliographical notes 131 Chapter 8. Conformal Blocks I Defining conformal blocks for the Heisenberg algebra Definition of conformal blocks for general vertex algebras Comparison of the two definitions of conformal blocks Coinvariants for commutative vertex algebras Twisted version of conformal blocks Appendix. Proof of Proposition Bibliographical notes 148 Chapter 9. Conformal Blocks II Multiple points Functoriality of conformal blocks Chiral correlation functions Conformal blocks in genus zero Functional realization of Heisenberg conformal blocks ' ' Bibliographical notes 168 Chapter 10. Free Field Realization I The idea Finite-dimensional setting Infinite-dimensional setting Bibliographical notes 184 Chapter 11. Free Field Realization II ' Weyl algebras in the infinite-dimensional case Local completion Wakimoto realization Bibliographical notes 201 Chapter 12. The Knizhnik-Zamolodchikov Equations Conformal blocks in the Heisenberg case Moving the points., 207

4 CONTENTS ix Conformal blocks for affine Kac-Moody algebras Bibliographical notes 214 Chapter 13. Solving the KZ Equations Conformal blocks from the point of view of free field realization Generalization: singular vectors Finding solutions Bibliographical notes 226 Chapter 14. Quantum Drinfeld-Sokolov Reduction and W-algebras The BRST complex Proof of the main theorem Examples The second computation Bibliographic notes 246 Chapter 15. Vertex Lie Algebras and Classical Limits Vertex Lie algebras Vertex Poisson algebras Kac-Moody and Virasoro limits Poisson structure on connections The Virasoro Poisson structure Opers Classical Drinfeld-Sokolov reduction Comparison of the classical and quantum Drinfeld-Sokolov reductions Bibliographical notes 270 Chapter 16. Vertex Algebras and Moduli Spaces I The flat connection on the vertex algebra bundle, revisited Harish-Chandra pairs Moduli of curves Bibliographical notes 288 Chapter 17. Vertex Algebras and Moduli Spaces II Moduli of bundles Local structure of moduli spaces Global structure of moduli spaces Localization for affine algebras at the critical level Chiral de Rham complex Bibliographical notes 307 Chapter 18. Chiral Algebras Some sheaf theory Sheaf interpretation of OPE ' Chiral algebras Lie* algebras Factorization Global Kac-Moody and Virasoro algebras Bibliographical notes 328

5 x CONTENTS Appendix 329 A.I. Discs, formal discs and ind-schemes 329 A.2. Connections 331 A.3. Lie algebroids and D-modules 332 A.4. Lie algebra cohomology 334 Bibliography 335 Index 343 List of Frequently Used Notation 345