The first examples of (non-commutative) vertex algebras are the induced rep-
resentations of infinite-dimensional Lie algebras related to the punctured disc. The
simplest of them is the Heisenberg Lie algebra CK, which is a central extension of
the commutative Lie algebra C((t)) of functions on the punctured disc. It has a
topological basis consisting of the elements bn =
£ Z, and the central element
1. The Fock representation TT of % is induced from the trivial one-dimensional
representation of its Lie subalgebra C[[i]] 0 1, and carries the structure of a vertex
algebra. This vertex algebra is generated, in an appropriate sense, by the vertex
6(S) = $ n * -
which is nothing but the generating function of the basis elements of
All other
vertex operators in TT may be expressed in terms of b(z) using the operations of
normally ordered product and differentiation with respect to z.
It is straightforward to generalize this construction to the affine Kac-Moody
algebras, which are central extensions of the Lie algebras g((i)) of functions on the
punctured disc with values in a simple Lie algebra g, and the Virasoro algebra,
which is a central extension of the Lie algebra DerC((£)) of vector fields on the
punctured disc. In each case, the generating functions of basis elements of the Lie
algebra play the role of the generating vertex operators in the corresponding vertex
algebra. The formalism of vertex algebras provides a compact and uniform way for
handling these generating functions. For example, the operator product expansion
gives us a convenient tool for recording the commutation relations between the
Fourier coefficients of various vertex operators.
But the formalism of vertex algebras goes far beyond computational conve-
nience. In fact, the most interesting examples of vertex algebras, such as the lattice
vertex algebras, the Monster Moonshine vertex algebra, and the W-algebras, are
not "finitely generated" by Lie algebras. There are still a finite number of vertex
operators, which generate each of these vertex algebras, but their Fourier coeffi-
cients are no longer closed under the commutator. Therefore they are not easily
accessible without the formalism of vertex algebras.
There is a special class of vertex algebras, which play a prominent role in
conformal field theory, called rational vertex algebras. These vertex algebras are
distinguished by the property that the category of their modules is semi-simple,
with finitely many simple objects, up to isomorphism. Examples of rational vertex
algebras are the lattice vertex algebras, the Moonshine Module vertex algebra, and
the integrable vacuum modules over affine Kac-Moody algebras. We briefly review
them in Chapter 5.
It is expected that the category of representations of a rational vertex algebra
carries the rich structure of a modular tensor category. In particular, one can
attach to it a collection of vector bundles with projectively flat connection on the
moduli spaces of stable pointed curves. Though this picture is still conjectural for
a general rational vertex algebra, in the genus one case it has already led to the
following beautiful result of Y. Zhu [Zl]: the characters of all simple modules over
a rational vertex algebra, considered as functions on the upper-half plane, span a
representation of SL2CZ).
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