Kac-Moody vertex algebra play an important role. At the end of Chapter 18 we
briefly review the construction of the chiral de Rham complex [MSV].
In Chapter 19, we give a motivated introduction to the Beilinson-Drinfeld
theory of chiral algebras and Lie* algebras. In particular, we show that any quasi-
conformal vertex algebra gives rise to a chiral algebra on an arbitrary smooth
curve. Finally, in Chapter 20 we introduce factorization algebras and explain their
connection with the vertex and chiral algebras. We then discuss the motivating
example of a factorization space, the Beilinson-Drinfeld Grassmannian, describing
the modifications of a G-bundle on a curve. This and other naturally occurring
factorization spaces serve as a geometric source for vertex algebras, through the
process of linearization, which we illustrate with several examples. We conclude
with a brief sketch of the role of factorization in the study of Hecke correspondences
on the moduli spaces of bundles. We introduce the notion of a Hecke eigensheaf on
the moduli space of G-bundles and formulate the geometric Langlands conjecture.
We then sketch the Beilinson-Drinfeld construction of the chiral Hecke algebra.
The Appendix contains useful information about ind-schemes, CD-modules, and
Lie algebra cohomology.
All objects in this book (vector spaces, algebras, curves, schemes, etc.) are
considered over the field C of complex numbers, unless specified otherwise. It
would be highly desirable to develop the arithmetic theory of vertex algebras, as
indicated by the existing works on this subject [KSU1, KSU2, Mo].
Corrections and updates to the material of this book will be posted at
http : //www. math. berkeley. edu/~f renkel
We end each chapter of the book with bibliographical notes. They are meant to
indicate the sources of the results presented in the chapter as well as some references
for further reading (sometimes the sources are mentioned inside the chapter, and
sometimes this is deferred until the end). We apologize in advance for any omissions
or inaccurate attributions.
Many of the results presented in this book are original and previously unpub-
lished. These results are pointed out in the corresponding bibliographical notes.
For a brief summary of the material in this book, the reader may consult the
Seminaire Bourbaki talk [F5] given by the first author in June of 2000.
Several other books on the theory of vertex algebras are available at present.
The first book on the subject, by I. Frenkel, J. Lepowsky, and A. Meurman [FLM],
laid the foundations of the theory with the applications to the Monster group as
a motivation. Further results were presented in [FHL] and [DL]. A connection
between the algebraic theory and the geometric approach to conformal field theory
in genus zero through operads is discussed, among other things, in Y.-Z. Huang's
book [HI]. The recent book by V. Kac [Kac3] is an excellent introduction to the
subject, containing a number of interesting examples and new techniques, such as
the theory of conformal algebras and their cohomology theory. The foundations
of the theory of chiral algebras and factorization algebras are laid out in the book
[BD4] by A. Beilinson and V. Drinfeld. Finally, a good source for the study of
conformal field theory from the physics perspective is the book [dPMS].
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