Preface to the Second Edition

This is a new edition of the book, substantially rewritten and expanded. We

would like to mention the most important changes that we have made.

Throughout the book we have dropped the requirement that a vertex algebra

be Z-graded with finite-dimensional graded components. The exposition of asso-

ciativity and operator product expansion (Chapter 3 of the old edition) has been

completely redone. A new chapter has been added (Chapter 4) in which we discuss

in more detail the Lie algebra U(V) attached to a vertex algebra V. In particular,

we show that when V is the affine Kac-Moody vertex algebra Vfc(g), the natural

map from U(Vk(g)) to a completion Uk(&) of the universal enveloping algebra of " g

of level k is a Lie algebra homomorphism. We also define for an arbitrary vertex

algebra a topological associative algebra U(V) (when V = Vfc(g) this algebra is

isomorphic to Uk($)).

In Chapter 5 (Chapter 4 of the old edition) we show that there is an equivalence

between the category of ^-modules and the category of smooth U(V)-mod\iles.

We have added a new section in Chapter 5 in which we introduce twisted modules

associated to vertex algebras equipped with an automorphism of finite order. The

following chapter, Chapter 6 (Chapter 5 of the old edition), has also been rewritten.

We have added a new motivational section at the beginning of the chapter and have

supplied a direct algebraic proof of coordinate-independence of the connection on

the vertex algebra bundle. In Chapter 19 (old Chapter 18) on chiral algebras we

have added a new motivational section and examples of chiral algebras that do not

arise from vertex algebras. We have also explained how to attach to modules and

twisted modules over vertex algebras certain modules over the corresponding chiral

algebras.

Finally, we have added a new Chapter 20, on factorization algebras and factor-

ization spaces. Factorization algebras, introduced by A. Beilinson and V. Drinfeld,

provide a purely geometric reformulation of the definition of vertex algebras. Here

we present an informal introduction to factorization algebras and give various ex-

amples. The most interesting examples come from factorization spaces, such as the

Beilinson-Drinfeld Grassmannians which are moduli spaces of bundles on a curve

equipped with trivializations away from finitely many points. We explain how the

concept of factorization naturally leads us to Hecke correspondences on bundles and

to the geometric Langlands correspondence. At the end of the chapter we discuss

the chiral Hecke algebras introduced by Beilinson and Drinfeld which provide a tool

for establishing the geometric Langlands conjecture. This chapter brings together

and illuminates the material of several other chapters of the book.

We wish to thank those who kindly pointed out to us various typos in the

first edition and pointed out additional references, especially, Michel Gros, Kenji

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