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.
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