INTRODUCTION

9

chapter. In particular, we explain that the vertex operation may itself be viewed

as a special conformal block.

The third part of the book, Chapters 11-16, contains some important construc-

tions and applications of vertex algebras. In Chapters 11 and 12 we motivate and

then define the free field realization of affme Kac-Moody algebras following the

works of M. Wakimoto [Wak] and B. Feigin-E. Prenkel [FF1, FF2, F6]. Then

in Chapters 13 and 14 we use this realization to obtain an explicit description of

the spaces of conformal blocks for affine Kac-Moody algebras in the genus zero

case. This gives us a systematic way to produce the Schechtman-Varchenko inte-

gral solutions of the Knizhnik-Zamolodchikov equations [FFR]. Next, we introduce

in Chapter 15 a family of vertex algebras, called W-algebras, using the quantum

Drinfeld-Sokolov reduction and the screening operators [FF7, FF8, dBT]. The

W-algebras became important recently because of their role in the geometric Lang-

lands correspondence (see below). The W-algebras generalize the Virasoro vertex

algebra, but in contrast to the Virasoro vertex algebra, a general W-algebra is not

generated by a Lie algebra. Thus, the formalism of vertex algebras developed in

this book is really essential in this case.

Poisson analogues of vertex algebras, in particular, of the W-algebras, are

discussed in detail in Chapter 16. At the beginning of the chapter we introduce the

notions of vertex Lie algebras and vertex Poisson algebras. Important features of the

formalism of vertex Poisson algebra can be traced back to the work by I.M. Gelfand,

L.A. Dickey and others on the Hamiltonian structure of integrable hierarchies of

soliton equations (see, e.g., [Di]). We explain how to obtain non-trivial examples

of vertex Poisson algebras by taking suitable" classical limits of vertex algebras, in

particular, of the Virasoro and Kac-Moody vertex algebras.

Classical limits of vertex algebras give rise to hamiltonian structures of famil-

iar integrable systems, such as the KdV hierarchy, which is closely related to the

Virasoro algebra. In the mid-80s, V. Drinfeld and V. Sokolov [DS] attached an in-

tegrable hierarchy of soliton equations generalizing the KdV system to an arbitrary

simple Lie algebra. The objects encoding the Hamiltonian structure of these hier-

archies are classical limits of the W-algebras, which may be viewed as the algebras

of functions of the spaces of opers on the disc. The concept of an oper, which is

implicit in [DS], has been generalized by A. Beilinson and V. Drinfeld to the case

of an arbitrary curve [BD2, BD3]. We recall their definition of opers and explain

the connection with the classical Drinfeld-Sokolov reduction and W-algebras.

Finally, in the more advanced last four chapters of the book, we consider "local-

ization" of vertex algebras on the moduli spaces and the relationship between vertex

algebras and the Beilinson-Drinfeld formalism of chiral algebras and factorization

algebras.

In Chapter 17 we review the formalism of Harish-Chandra localization following

A. Beilinson and J. Bernstein. We apply it to construct sheaves of coinvariants of

vertex algebras on the moduli spaces of pointed curves. The key to this construction

is the "Virasoro uniformization" of the moduli spaces of pointed curves. This

construction is generalized in Chapter 18 to the case of moduli spaces of bundles, for

which we have an analogous "Kac-Moody uniformization". We discuss the sheaves

of coinvariants corresponding to 0-modules of critical level and their relationship to

the geometric Langlands correspondence, following Beilinson and Drinfeld [BD3].

In establishing this correspondence, the classical W-algebras and the center of a