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]).
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