8 INTRODUCTION
In Chapter 1 we lay down the basics of the vertex algebra formalism, including
several approaches to the crucial property of locality of fields. We then formulate
the axioms of vertex algebras and discuss the example of commutative vertex alge-
bras. In Chapter 2 we consider in great detail the vertex algebras associated to the
Heisenberg, affine Kac-Moody, and Virasoro Lie algebras. Chapter 3 is devoted to
the Goddard uniqueness theorem and the associativity property of vertex algebras.
Using this property, we introduce the concept of the operator product expansion
(OPE). Many examples of OPE are worked out in detail in this chapter. In Chapter
4 we attach to any vertex algebra a Lie algebra U(V) and an associative algebra
U(V). The functors sending V to U(V) and U(V) allow us to relate vertex alge-
bras to more familiar objects, Lie algebras and associative algebras (though they
are much larger and more difficult to study than the vertex algebras themselves).
At the end of the chapter we present two other applications of the associativity
property: we prove a strong form of the Reconstruction Theorem and explain a
construction (called functional realization) which assigns to any linear functional
on a vertex algebra a collection of meromorphic functions on the symmetric powers
of the disc ("n-point functions"). In Chapter 5 we introduce the notions of a mod-
ule and a twisted module over a vertex algebra and study the general properties
of modules. In particular, we explain that a module over a vertex algebra V is
the same as a (smooth) module over the associative algebra U(V) introduced in
Chapter 4. We then define the lattice vertex algebras and the free fermionic vertex
superalgebra, and establish the boson-fermion correspondence. We introduce the
concept of rational vertex algebra and discuss various examples. We also outline
several constructions of vertex algebras, such as the coset construction and the
theory of orbifolds.
The second, and the central, part of the book consists of Chapters 6-10. In
these chapters we develop the geometric theory of vertex algebras and conformal
blocks.
In Chapter 6 we give a coordinate-independent realization of the vertex oper-
ation. We introduce the group Aut 0 and show how to make it act on a conformal
vertex algebra by internal symmetries. Using this action, we attach to any confor-
mal vertex algebra V a vector bundle V on any smooth algebraic curve. We prove
that the vertex operation Y gives rise to a canonical section yx of V*\Dx with values
in EndV^, for any x G X. Finally, we equip V with a flat connection and show that
the section ^x is horizontal.
A generalization of this construction, involving more general internal symme-
tries of vertex algebras (such as those generated by affine Kac-Moody algebras),
is given in Chapter 7. Various examples of the geometric structures defined in
Chapters 6 and 7 are discussed in Chapter 8. In particular, we explain how such
familiar geometric objects as projective and affine connections appear naturally in
the geometric theory of vertex algebras.
We give the definitions of the spaces of conformal blocks and coinvariants in
Chapter 9. We start with the simplest case of the Heisenberg vertex algebra, for
which conformal blocks may be defined in a rather elementary way. We then grad-
ually proceed to define the spaces of one-point conformal blocks and coinvariants
for general vertex algebras. The generalization to the case of several points, with
arbitrary module insertions, is given in Chapter 10. Functorial properties and
functional realizations of the spaces of conformal blocks are also discussed in this
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