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