INTRODUCTION

7

space or the space of global sections of an appropriate line bundle on the moduli

space. Conformal blocks may therefore be used to gain insights into the structure

of moduli spaces.

Chiral algebras and factorization algebras

As we already mentioned above, A. Beilinson and V. Drinfeld have recently

introduced the notion of chiral algebra. A chiral algebra on a smooth curve X is a

D-module on X equipped with a certain operation. This operation was designed

as a geometric analogue of the operation of operator product expansion. It was

expected that any vertex algebra V would give rise to a "universal chiral algebra",

i.e., an assignment of a chiral algebra to any curve in such a way that their fibers

are isomorphic to V. However, as far as we know, this has not been proved, except

for the case when the curve is of genus zero, equipped with a global coordinate,

when this was shown by Y.-Z. Huang and J. Lepowsky [HL].

For general curves, one first needs to develop the formalism of vertex algebras in

a coordinate-independent way. This is done in this book, and using this formalism,

we show in Chapter 19 that each (quasi-conformal) vertex algebra indeed gives rise

to a universal chiral algebra.

Perhaps, the most exciting and far-reaching part of the theory of chiral algebras

is the Beilinson-Drinfeld description of chiral algebras as factorization algebras, i.e.,

sheaves on the Ran space of finite subsets of a curve, satisfying certain compatibili-

ties. The formalism of factorization algebras is closely related to that of functional

realizations discussed in Chapter 10.

In Chapter 20 we give the definition of factorization algebras and explain their

connection to chiral algebras and vertex algebras following Beilinson and Drinfeld

[BD4]. The beauty of factorization algebras is that they have "non-linear" ana-

logues, factorization spaces which arise in "nature". For instance, the Beilinson-

Drinfeld Grassmannians which are the moduli spaces of G-bundles on a smooth

curve X together with trivializations away from finitely many points of X form a

typical factorization space. One can obtain factorization algebras (and hence chiral

algebras) on X by "linearizing" factorization spaces, i.e., by considering the coho-

mology of various sheaves on them. This formalism provides a novel method of

constructing chiral and vertex algebras that underscores their intimate relation to

moduli spaces. As one of the applications of this method, Beilinson and Drinfeld

have produced a fascinating object: the chiral Hecke algebra which we review at

the end of Chapter 20. It is expected that the chiral Hecke algebra may be used

for the construction of the Hecke eigensheaves whose existence is predicted by the

geometric Langlands conjecture.

Overview of the book

This book may be roughly broken down into four large parts. Each of them

may be read independently from the rest of the book (although there are many

connections between them).

The first part, comprising Chapters 1-5, contains a self-contained elementary

introduction to the algebraic theory of vertex algebras, intended mainly for begin-

ners. We give the definition of vertex algebras, study their properties, and consider

numerous examples.