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Vertex Algebras and Algebraic Curves: Second Edition
Edward Frenkel, University of California, Berkeley, CA, and David Ben-Zvi, University of Chicago, IL
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Mathematical Surveys and Monographs
2004; 400 pp; softcover
Volume: 88
ISBN-10: 0-8218-3674-9
ISBN-13: 978-0-8218-3674-3
List Price: US$80 Member Price: US$64
Order Code: SURV/88.R

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.

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

Reviews

From a review of the First Edition:

"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

• Introduction
• Definition of vertex algebras
• Vertex algebras associated to Lie algebras
• Associativity and operator product expansion
• Applications of the operator product expansion
• Modules over vertex algebras and more examples
• Vertex algebra bundles
• Action of internal symmetries
• Vertex algebra bundles: Examples
• Conformal blocks I
• Conformal blocks II
• Free field realization I
• Free field realization II
• The Knizhnik-Zamolodchikov equations
• Solving the KZ equations
• Quantum Drinfeld-Sokolov reduction and $$\mathcal{W}$$-algebras
• Vertex Lie algebras and classical limits
• Vertex algebras and moduli spaces I
• Vertex algebras and moduli spaces II
• Chiral algebras
• Factorization
• Appendix
• Bibliography
• Index
• List of frequently used notation