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Vertex algebras and algebraic curves

Author: Edward Frenkel; David Ben-Zvi
Publisher: Providence, R.I. : American Mathematical Society, ©2001.
Series: Mathematical surveys and monographs, no. 88.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:

This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of moduli spaces of algebraic curves. The authors  Read more...

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Additional Physical Format: Online version:
Frenkel, Edward, 1968-
Vertex algebras and algebraic curves.
Providence, R.I. : American Mathematical Society, c2001
(OCoLC)623809179
Document Type: Book
All Authors / Contributors: Edward Frenkel; David Ben-Zvi
ISBN: 0821828940 9780821828946
OCLC Number: 46872350
Description: xii, 348 p. ; 27 cm.
Contents: 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 --
Chapter 2. Vertex Algebras Associated to Lie Algebras 25 --
2.1. Heisenberg Lie algebra 25 --
2.2. Vertex algebra structure on [pi] 28 --
2.3. Checking vertex algebra axioms 32 --
2.4. Affine Kac-Moody algebras and their vertex algebras 38 --
2.5. Virasoro vertex algebra 41 --
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 --
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 --
Chapter 5. Vertex Algebra Bundles 87 --
5.1. Group Aut O 87 --
5.2. Exponentiating vector fields 89 --
5.3. Primary fields 92 --
5.4. Main construction 96 --
5.5. A flat connection on the vertex algebra bundle 102 --
Chapter 6. Action of Internal Symmetries 105 --
6.1. Affine algebras, revisited 105 --
6.2. General twisting property 107 --
6.3. Description of the n-point functions and modules 112 --
Chapter 7. Vertex Algebra Bundles: Examples 115 --
7.1. Heisenberg algebra and affine connections 115 --
7.2. Virasoro algebra and projective connections 119 --
7.3. Pseudodifferential operators and kernels 123 --
7.4. Gauge action on the Heisenberg bundle 128 --
7.5. Affine Kac-Moody vertex algebras and connections 130 --
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 --
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 --
Chapter 10. Free Field Realization I 169 --
10.2. Finite-dimensional setting 171 --
10.3. Infinite-dimensional setting 177 --
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 --
Chapter 12. Knizhnik-Zamolodchikov Equations 203 --
12.1. Conformal blocks in the Heisenberg case 203 --
12.2. Moving the points 207 --
12.3. Conformal blocks for affine Kac-Moody algebras 211 --
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 --
Chapter 14. Quantum Drinfeld-Sokolov Reduction and W-algebras 227 --
14.1. BRST complex 227 --
14.2. Proof of the main theorem 231 --
14.4. Second computation 237 --
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. Virasoro Poisson structure 260 --
15.6. Opers 261 --
15.7. Classical Drinfeld-Sokolov reduction 265 --
15.8. Comparison of the classical and quantumC Drinfeld-Sokolov reductions 268 --
Chapter 16. Vertex Algebras and Moduli Spaces I 271 --
16.1. Flat connection on the vertex algebra bundle, revisited 272 --
16.2. Harish-Chandra pairs 275 --
16.3. Moduli of curves 280 --
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 --
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 --
A.1. 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.
Series Title: Mathematical surveys and monographs, no. 88.
Responsibility: Edward Frenkel, David Ben-Zvi.

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