**Mathematical Surveys and Monographs**

Volume: 88;
2004;
400 pp;
Softcover

MSC: Primary 17;
Secondary 81; 14

Print ISBN: 978-0-8218-3674-3

Product Code: SURV/88.R

List Price: $80.00

Individual Member Price: $64.00

**Electronic ISBN: 978-1-4704-1315-6
Product Code: SURV/88.R.E**

List Price: $80.00

Individual Member Price: $64.00

#### Supplemental Materials

# Vertex Algebras and Algebraic Curves: Second Edition

Share this page
*Edward Frenkel; David Ben-Zvi*

Vertex algebras are algebraic objects that encapsulate the concept of
operator product expansion from two-dimensional conformal field theory. Vertex
algebras are fast becoming ubiquitous in many areas of modern mathematics, with
applications to representation theory, algebraic geometry, the theory of finite
groups, modular functions, topology, integrable systems, and combinatorics.

This book is an introduction to the theory of vertex algebras with a
particular emphasis on the relationship with the geometry of algebraic
curves. The notion of a vertex algebra is introduced in a
coordinate-independent way, so that vertex operators become well defined on
arbitrary smooth algebraic curves, possibly equipped with additional data, such
as a vector bundle. Vertex algebras then appear as the algebraic objects
encoding the geometric structure of various moduli spaces associated with
algebraic curves. Therefore they may be used to give a geometric interpretation
of various questions of representation theory.

The book contains many original results, introduces important new concepts,
and brings new insights into the theory of vertex algebras. The authors have
made a great effort to make the book self-contained and accessible to readers
of all backgrounds. Reviewers of the first edition anticipated that it would
have a long-lasting influence on this exciting field of mathematics and would
be very useful for graduate students and researchers interested in the
subject.

This second edition, substantially improved and expanded, includes several
new topics, in particular an introduction to the Beilinson-Drinfeld theory of
factorization algebras and the geometric Langlands correspondence.

The book is suitable for graduate students and research mathematicians
interested in representation theory, algebraic geometry, and mathematical
physics. For more information, visit www.math.berkeley.edu/ frenkel/book.

#### Readership

Graduate students and research mathematicians interested in representation theory, algebraic geometry, and mathematical physics.

#### Reviews & Endorsements

The authors give a deep new insight into the theory of vertex algebras … many original results, important new concepts and very nice interpretations of structural results in the theory of vertex algebras … provides a natural link with earlier approaches to vertex algebras … The authors also present an excellent introduction to the theory of Wakimoto modules and \(\mathcal W\)-algebras … contains many new concepts and results that are important for the modern theory of vertex algebras.

-- Mathematical Reviews, Featured Review

#### Table of Contents

# Table of Contents

## Vertex Algebras and Algebraic Curves: Second Edition

- Contents v6 free
- Preface to the Second Edition ix10 free
- Preface to the First Edition xi12 free
- Introduction 116 free
- Chapter 1. Definition of Vertex Algebras 1126 free
- Chapter 2. Vertex Algebras Associated to Lie Algebras 2540
- Chapter 3. Associativity and Operator Product Expansion 4762
- Chapter 4. Applications of the Operator Product Expansion 6378
- Chapter 5. Modules over Vertex Algebras and More Examples 7590
- 5.1. Modules over vertex algebras 7691
- 5.2. Vertex algebras associated to one-dimensional integral lattices 8095
- 5.3. Boson–fermion correspondence 85100
- 5.4. Lattice vertex algebras 88103
- 5.5. Rational vertex algebras 90105
- 5.6. Twisted modules 91106
- 5.7. Constructing new vertex algebras 94109
- 5.8. Bibliographical notes 97112

- Chapter 6. Vertex Algebra Bundles 99114
- Chapter 7. Action of Internal Symmetries 121136
- Chapter 8. Vertex Algebra Bundles: Examples 133148
- Chapter 9. Conformal Blocks I 151166
- 9.1. Defining conformal blocks for the Heisenberg algebra 151166
- 9.2. Definition of conformal blocks for general vertex algebras 154169
- 9.3. Comparison of the two definitions of conformal blocks 158173
- 9.4. Coinvariants for commutative vertex algebras 161176
- 9.5. Twisted version of conformal blocks 163178
- 9.6. Appendix. Proof of Proposition 9.3.2 164179
- 9.7. Bibliographical notes 166181

- Chapter 10. Conformal Blocks II 167182
- Chapter 11. Free Field Realization I 187202
- Chapter 12. Free Field Realization II 203218
- Chapter 13. The Knizhnik–Zamolodchikov Equations 221236
- Chapter 14. Solving the KZ Equations 233248
- Chapter 15. Quantum Drinfeld–Sokolov Reduction and W–algebras 245260
- Chapter 16. Vertex Lie Algebras and Classical Limits 267282
- 16.1. Vertex Lie algebras 267282
- 16.2. Vertex Poisson algebras 271286
- 16.3. Kac–Moody and Virasoro limits 273288
- 16.4. Poisson structure on connections 275290
- 16.5. The Virasoro Poisson structure 279294
- 16.6. Opers 281296
- 16.7. Classical Drinfeld–Sokolov reduction 285300
- 16.8. Comparison of the classical and quantum Drinfeld–Sokolov reductions 288303
- 16.9. Bibliographical notes 290305

- Chapter 17. Vertex Algebras and Moduli Spaces I 291306
- Chapter 18. Vertex Algebras and Moduli Spaces II 311326
- Chapter 19. Chiral Algebras 329344
- Chapter 20. Factorization 351366
- Appendix 375390
- Bibliography 383398
- Index 393408
- List of Frequently Used Notation 397412