With E. Frenkel. Vertex Algebras and Algebraic
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.