Preface to the First Edition

The present 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. It is based on the lecture courses given by Edward Frenkel at

Harvard University in the Fall of 1996 and at UC Berkeley in the Spring of 1999.

The notes of these lectures were taken by David Ben-Zvi. They were subsequently

improved and expanded by both authors. The main goal of this book is to introduce

the concept of vertex algebra in a coordinate-independent way, and to define the

spaces of conformal blocks attached to an arbitrary vertex algebra and a smooth

algebraic curve, possibly equipped with some extra geometric data. From this

point of view vertex algebras appear as the algebraic objects that encode the local

geometric structure of various moduli spaces associated with algebraic curves.

In the fifteen years that have passed since they were introduced by R. Borcherds,

vertex algebras have turned out to be extremely useful in many areas of mathemat-

ics. They are by now ubiquitous in the representation theory of infinite-dimensional

Lie algebras. They have also found applications in such fields as algebraic geom-

etry, the theory of finite groups, modular functions, topology, integrable systems,

and combinatorics. The theory of vertex algebras also serves as the rigorous math-

ematical foundation for two-dimensional conformal field theory and string theory,

extensively studied by physicists.

In the literature there exist two essentially different approaches to vertex alge-

bras. The first is algebraic, following the original definition of Borcherds [Bl]. It

has been developed by I. Frenkel, J. Lepowsky, and A. Meurman [FLM] and more

recently by V. Kac [Kac3]. Vertex operators appear here as formal power series

acting on graded vector spaces. The second approach is geometric and more ab-

stract: this is the theory of chiral algebras and factorization algebras developed by

A. Beilinson and V. Drinfeld [BD4]. In this approach the main objects of study are

D-modules on powers of algebraic curves equipped with certain operations. Chiral

algebras have non-linear versions called factorization spaces which encode various

intricate structures of algebraic curves and bundles on them.

The present book aims to bridge the gap between the two approaches. It

starts with the algebraic definition of vertex algebras, which is close to Borcherds',

and essentially coincides with that of [FKRW, Kac3]. The key point is to make

vertex operators coordinate-independent, thus effectively getting rid of the formal

variable. This is achieved by attaching to each vertex algebra a vector bundle with

a flat connection on the (formal) disc, equipped with an intrinsic operation. The

formal variable is restored when we choose a coordinate on the disc; the fact that

the operation is independent of this choice follows from the vertex algebra axioms.

Once this is done and we obtain a coordinate-independent object, we can study

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