2

INTRODUCTION

such as the Wess-Zumino-Witten models. The physical theories that have led to

new predictions in enumerative geometry (such as mirror symmetry) are also based

on conformal field theories (such as the sigma models of Calabi-Yau manifolds). One

may hope that further development of the theory of vertex algebras will allow us to

build a suitable framework for understanding these theories as well. The first steps

in this direction are already being made. For example, F. Malikov, V. Schechtman

and A. Vaintrob [MSV] have recently constructed a sheaf of vertex superalgebras

on an arbitrary smooth algebraic variety, called the chiral de Rham complex (see

§ 18.5), which may hopefully be used in understanding the new "stringy" invariants,

such as the elliptic genera.

In another important development, A. Beilinson and V. Drinfeld have recently

introduced a geometric version of vertex algebras which they call chiral algebras

[BD4]. Chiral algebras give rise to some novel concepts and techniques which are

likely to have a profound impact on algebraic geometry. The formalism of vertex

and chiral algebras appears to be particularly suitable for the construction of the

conjectural geometric Langlands correspondence between D-modules on the moduli

space of G-bundles on a smooth projective curve X over C, and flat

LG-bundles

on

X, where G is a reductive algebraic group and LG is the Langlands dual group (see

[BD3]). We will see two examples of such constructions: one involves the affine

Kac-Moody vertex algebra of critical level (see § 18.4) and the other involves the

chiral Hecke algebra (see § 20.5).

These applications present ample evidence for the relevance of vertex alge-

bras not only in representation theory, where they originated, but also in other

fields, such as algebraic geometry. In this book, we make the first steps towards

reformulating the theory of vertex algebras in a way suitable for algebro-geometric

applications.

What is a vertex algebra?

In a nutshell, a vertex algebra is a vector space V equipped with a vector |0)

and an operation

Y :

V-+EndVUz*1]],

assigning to each A G V a formal power series, called a vertex operator,

(0.0.1) Y(A,z) =

J^A{n)z-n-1,

where each A^ is a linear operator on V, so that for any v £ V, we have A^v = 0

for n 0.

These data must satisfy a short list of axioms (see Chapter 1), the most im-

portant of which is the locality axiom. It states that for any A,B EV, the formal

power series in two variables, obtained by composing Y(A, z) and Y(B, w) in two

possible ways, are equal to each other, possibly after multiplying them with a large

enough power of (z — w). In other words, the commutator \Y(A,z),Y(B,w)] is a

formal distribution supported on the diagonal z = w.

If we ask instead that the equality Y(A, z)Y(J5, w) = Y(B,w)Y(A,z) holds

even before multiplying by a power of (z — w), then we obtain the structure equiv-

alent to that of a commutative associative algebra with a unit and a derivation.

Thus, vertex algebras may be thought of as "meromorphic" generalizations of com-

mutative algebras (for more on this point of view, see [B3]).