8 INTRODUCTION

In Chapter 1 we lay down the basics of the vertex algebra formalism, including

several approaches to the crucial property of locality of fields. We then formulate

the axioms of vertex algebras and discuss the example of commutative vertex alge-

bras. In Chapter 2 we consider in great detail the vertex algebras associated to the

Heisenberg, affine Kac-Moody, and Virasoro Lie algebras. Chapter 3 is devoted to

the Goddard uniqueness theorem and the associativity property of vertex algebras.

Using this property, we introduce the concept of the operator product expansion

(OPE). Many examples of OPE are worked out in detail in this chapter. In Chapter

4 we attach to any vertex algebra a Lie algebra U(V) and an associative algebra

U(V). The functors sending V to U(V) and U(V) allow us to relate vertex alge-

bras to more familiar objects, Lie algebras and associative algebras (though they

are much larger and more difficult to study than the vertex algebras themselves).

At the end of the chapter we present two other applications of the associativity

property: we prove a strong form of the Reconstruction Theorem and explain a

construction (called functional realization) which assigns to any linear functional

on a vertex algebra a collection of meromorphic functions on the symmetric powers

of the disc ("n-point functions"). In Chapter 5 we introduce the notions of a mod-

ule and a twisted module over a vertex algebra and study the general properties

of modules. In particular, we explain that a module over a vertex algebra V is

the same as a (smooth) module over the associative algebra U(V) introduced in

Chapter 4. We then define the lattice vertex algebras and the free fermionic vertex

superalgebra, and establish the boson-fermion correspondence. We introduce the

concept of rational vertex algebra and discuss various examples. We also outline

several constructions of vertex algebras, such as the coset construction and the

theory of orbifolds.

The second, and the central, part of the book consists of Chapters 6-10. In

these chapters we develop the geometric theory of vertex algebras and conformal

blocks.

In Chapter 6 we give a coordinate-independent realization of the vertex oper-

ation. We introduce the group Aut 0 and show how to make it act on a conformal

vertex algebra by internal symmetries. Using this action, we attach to any confor-

mal vertex algebra V a vector bundle V on any smooth algebraic curve. We prove

that the vertex operation Y gives rise to a canonical section yx of V*\Dx with values

in EndV^, for any x G X. Finally, we equip V with a flat connection and show that

the section ^x is horizontal.

A generalization of this construction, involving more general internal symme-

tries of vertex algebras (such as those generated by affine Kac-Moody algebras),

is given in Chapter 7. Various examples of the geometric structures defined in

Chapters 6 and 7 are discussed in Chapter 8. In particular, we explain how such

familiar geometric objects as projective and affine connections appear naturally in

the geometric theory of vertex algebras.

We give the definitions of the spaces of conformal blocks and coinvariants in

Chapter 9. We start with the simplest case of the Heisenberg vertex algebra, for

which conformal blocks may be defined in a rather elementary way. We then grad-

ually proceed to define the spaces of one-point conformal blocks and coinvariants

for general vertex algebras. The generalization to the case of several points, with

arbitrary module insertions, is given in Chapter 10. Functorial properties and

functional realizations of the spaces of conformal blocks are also discussed in this