With E. Frenkel. **Vertex Algebras and Algebraic
Curves**.

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.