# "Vertex Algebras and Algebraic Curves"

## by Edward Frenkel and David Ben-Zvi

The first edition of this book has been published in August of 2001 by
the American Mathematical Society in the series "Mathematical Surveys
and Monographs" (vol. 88). The second edition, substantially improved
and expanded, has appeared in August of 2004. You can order it from the
AMS Bookstore (at a discount if you are a member of AMS; if you
reside outside of the US, you may use Oxford University
Press , which is the Worldwide distributor for the AMS),
Barnes & Noble.com, or
Amazon.com .
Read a review which appeared in the Bulletin of the American Mathematical
Society .

### Description

Vertex algebras are algebraic objects that encapsulate the concepts of
vertex operators and operator product expansion from two-dimensional
conformal field theory. In the fifteen years since they were
introduced by R. Borcherds, vertex algebras have turned out to be
extremely useful in many areas of mathematics. They have by now become
ubiquitous in the representation theory of infinite-dimensional Lie
algebras. They have also found applications in such fields as
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 between vertex algebras and the geometry
of algebraic curves. The authors make the first steps toward
reformulating the theory of vertex algebras in a way that is suitable
for algebraic-geometric applications.
The notion of a vertex algebra is introduced in the book in a
coordinate-independent way, allowing the authors to give global
geometric meaning to vertex operators on arbitrary smooth algebraic
curves, possibly equipped with some additional data. To each vertex
algebra and a smooth curve, they attach an invariant called the space
of conformal blocks. When the complex structure of the curve and other
geometric data are varied, these spaces combine into a sheaf on the
relevant moduli space. From this perspective, vertex algebras appear
as the algebraic objects that encode the geometric structure of
various moduli spaces associated with algebraic curves.

Numerous examples and applications of vertex algebras are included,
such as the Wakimoto realization of affine Kac-Moody algebras,
integral solutions of the Knizhnik-Zamolodchikov equations, classical
and quantum Drinfeld-Sokolov reductions, and the W-algebras. Among
other topics discussed in the book are vertex Poisson algebras,
Virasoro uniformization of the moduli spaces of pointed curves, the
geometric Langlands correspondence, and the chiral de Rham complex.
The authors also establish a connection between vertex algebras and
chiral algebras, recently introduced by A. Beilinson and V. Drinfeld.

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.

This book may be used by the beginners as an entry point to the modern
theory of vertex algebras, and by more experienced readers as a guide
to advanced studies in this beautiful and exciting field.

For a brief summary of this book, see the text of the talk given by
E. Frenkel at Seminaire Bourbaki in June of 2000.

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If you have comments or suggestions, please write to frenkel@math.berkeley.edu

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