Integrable Highest-Weight Modules over Affine Algebras. Application to η-Function Identities. Sugawara Operators and Branching Functions

Abstract

§12.0. In the last three chapters we developed a representation theory of arbitrary Kac–Moody algebras. From now on we turn to the special case of affine algebras. We show that the denominator identity (10.4.4) for affine algebras is nothing else but the celebrated Macdon aid identities. Historically this was the first application of the representation theory of Kac–Moody algebras. The basic idea is very simple: one gets an interesting identity by computing the character of an integrable representation in two different ways and equating the results. In particular, Macdonald identities are deduced via the trivial representation. Furthermore, we show that specializations (11.11.5) of the denominator identity turn into identities for q -series of modular forms, the simplest ones being Macdonald identities for the powers of the Dedekind η-function. We study the structure of the weight system of an integrable highest-weight module over an affine algebra in more detail. This allows us to write its character in a different form to obtain the important theta function identity. This identity involves classical theta functions and certain modular forms called string functions, which are, essentially, generating functions for multiplicities of weights in “strings.” Furthermore, we consider branching functions, which are a generalization of string functions when instead of the Cartan subalgebra a general reductive subalgebra is considered. Finally, we introduce one of the most powerful tools of conformal field theory, the Sugawara construction and the coset construction, in relation to the study of general branching rules.