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
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