Affine Lie algebras, theta functions and modular forms

Abstract

We begin this chapter with an exposition of a theory of theta functions. Using the classical transformation properties of theta functions and the theta function identity (12.7.13), we show that the string functions are modular forms and find a transformation law for these forms. Furthermore, using the theory of modular forms, we prove the “very strange” formula (12.3.7), which in turn, is used to show that the string functions, multiplied by a “standard” cusp-form, are cusp-forms. All this is applied to find explicit formulas for the weight multiplicities and characters of integrable highest weight modules.