Affine Lie algebras and vertex operator algebras

Abstract

In this chapter we shall discuss standard (or integrable highest weight) representations of affine Kac-Moody algebras from the point of view of vertex operator algebra theory. This will also provide the setting for the next chapter. We restrict our attention to a simple Lie algebra G of type A, D or E. Let G be the corresponding affine Lie algebra and let l be a positive integer. We show that a certain distinguished one of the level l standard G-modules L(l,0) has the structure of a vertex operator algebra and that every level l standard G-module, at least every such module which can be obtained from the tensor product of l basic modules (i.e., level 1 standard modules for affine Lie algebras of type Â, D or Ê), is an irreducible module for this vertex operator algebra. Conversely, every irreducible module for the vertex operator algebra L(l, 0) is also a standard (G-module of level l. The Virasoro algebra for the vertex operator algebra L(l,0) comes from a canonical vertex operator associated with the (suitably normalized) Casimir operator for G. Using an abelian intertwining algebra (see Chapter 12) associated with the direct sum of l copies of the weight lattice, we also construct intertwining operators for irreducible L(l, 0)-modules. There is some overlap between the results in this chapter and those in [FZ], which uses a different approach, involving Verma modules.