Mathematical Surveys and Monographs Volume 88 Vertex Algebras and Algebraic Curves Edward Frenkei David Ben-Zvi American Mathematical Society
Contents Preface xi Introduction 1 Chapter 1. Definition of Vertex Algebras 11 1.1. Formal distributions 11 1.2. Locality 16 1.3. Definition of a vertex algebra 20 1.4. First example: commutative vertex algebras 22 1.5. Bibliographical notes 23 Chapter 2. Vertex Algebras Associated to Lie Algebras 25 2.1. Heisenberg Lie algebra 25 2.2. The vertex algebra structure ontt 28 2.3. Checking vertex algebra axioms 32 2.4. Affine Kac-Moody algebras and their vertex algebras 38 2.5. The Virasoro vertex algebra 41 2.6. Bibliographical notes 46 Chapter 3. Associativity and Operator Product Expansion 47 3.1. Uniqueness theorem 47 3.2. Associativity. ' 48 3.3. Operator product expansion 51 3.4. Examples and applications of OPE 56 3.5. A Lie algebra attached to a vertex algebra 60 3.6. Strong Reconstruction Theorem 63 3.7. Correlation functions 65 3.8. Bibliographical notes 67 Chapter 4. Rational Vertex Algebras 69 4.1. Modules over vertex algebras 69 4.2. Vertex algebras associated to one-dimensional integral lattices 72 4.3. Boson-fermion correspondence 77 4.4. Lattice vertex algebras 80 4.5. Introducing rational vertex algebras. 82 4.6. Constructing new vertex algebras 83 4.7. Bibliographical notes 86 Chapter 5. Vertex Algebra Bundles 87 5.1. The group AutO 87 5.2. Exponentiating vector fields 89
viii CONTENTS 5.3. Primary fields 92 5.4. The main construction 96 5.5. A flat connection on the vertex algebra bundle 102 5.6. Bibliographical notes 103 Chapter 6. Action of Internal Symmetries 105 6.1. Affine algebras, revisited 105 6.2. The general twisting property.- 107 6.3. Description of the n-point functions and modules 112 6.4. Bibliographical notes 114 Chapter 7. Vertex Algebra Bundles: Examples 115 7.1. The Heisenberg algebra and affine connections 115 7.2. The Virasoro algebra and projective connections 119 7.3. Pseudodifferential operators and kernels 123 7.4. The gauge action on the Heisenberg bundle 128 7.5. The affine Kac-Moody vertex algebras and connections 130 7.6. Bibliographical notes 131 Chapter 8. Conformal Blocks I 133 8.1. Defining conformal blocks for the Heisenberg algebra 133 8.2. Definition of conformal blocks for general vertex algebras 136 8.3. Comparison of the two definitions of conformal blocks 140 8.4. Coinvariants for commutative vertex algebras 143 8.5. Twisted version of conformal blocks 145 8.6. Appendix. Proof of Proposition 8.3.2 146 8.7. Bibliographical notes 148 Chapter 9. Conformal Blocks II 149 9.1. Multiple points 149 9.2. Functoriality of conformal blocks 151 9.3. Chiral correlation functions 153 9.4. Conformal blocks in genus zero 158 9.5. Functional realization of Heisenberg conformal blocks ' ' 164 9.6. Bibliographical notes 168 Chapter 10. Free Field Realization I 169 10.1. The idea 169 10.2. Finite-dimensional setting 171 10.3. Infinite-dimensional setting 177 10.4. Bibliographical notes 184 Chapter 11. Free Field Realization II ' 185 11.1. Weyl algebras in the infinite-dimensional case 185 11.2. Local completion 190 11.3. Wakimoto realization 196 11.4. Bibliographical notes 201 Chapter 12. The Knizhnik-Zamolodchikov Equations 203 12.1. Conformal blocks in the Heisenberg case 203 12.2. Moving the points., 207
CONTENTS ix 12.3. Conformal blocks for affine Kac-Moody algebras 211 12.4. Bibliographical notes 214 Chapter 13. Solving the KZ Equations 215 13.1. Conformal blocks from the point of view of free field realization 215 13.2. Generalization: singular vectors 219 13.3. Finding solutions 222 13.4. Bibliographical notes 226 Chapter 14. Quantum Drinfeld-Sokolov Reduction and W-algebras 227 14.1. The BRST complex 227 14.2. Proof of the main theorem 231 14.3. Examples 234 14.4. The second computation 237 14.5. Bibliographic notes 246 Chapter 15. Vertex Lie Algebras and Classical Limits 247 15.1. Vertex Lie algebras 247 15.2. Vertex Poisson algebras 252 15.3. Kac-Moody and Virasoro limits 254 15.4. Poisson structure on connections 256 15.5. The Virasoro Poisson structure 260 15.6. Opers 261 15.7. Classical Drinfeld-Sokolov reduction 265 15.8. Comparison of the classical and quantum Drinfeld-Sokolov reductions 268 15.9. Bibliographical notes 270 Chapter 16. Vertex Algebras and Moduli Spaces I 271 16.1. The flat connection on the vertex algebra bundle, revisited 272 16.2. Harish-Chandra pairs. 275 16.3. Moduli of curves 280 16.4. Bibliographical notes 288 Chapter 17. Vertex Algebras and Moduli Spaces II 291 17.1. Moduli of bundles 291 17.2. Local structure of moduli spaces 297 17.3. Global structure of moduli spaces 298 17.4. Localization for affine algebras at the critical level 299 17.5. Chiral de Rham complex 304 17.6. Bibliographical notes 307 Chapter 18. Chiral Algebras 309 18.1. Some sheaf theory 309 18.2. Sheaf interpretation of OPE ' 313 18.3. Chiral algebras 316 18.4. Lie* algebras 320 18.5. Factorization 323 18.6. Global Kac-Moody and Virasoro algebras 324 18.7. Bibliographical notes 328
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