The branching rules for affine Lie algebras
The irreducible highest weight representations of affine Kac-Moody algebras are one of the main tools in the construction of conformal field theories. One of the good clues to the understanding of the structure of irreducible highest weight representations of affine Kac-Moody algebras is to calculate the string functions, which count the weight multiplicities. The problem of determining the string functions of all level one modules for \(C^{(1)}_ l\) was stated as an open problem in \textit{V. G. Kac's} book [Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press (1990; Zbl 0716.17022)]. In 1988, a formula for this was discovered by \textit{V. G. Kac} and \textit{M. Wakimoto} in terms of Virasoro characters [Adv. Math. 70, 156-236 (1988; Zbl 0661.17016)]. In this paper, the author gives a new formula for the string functions of all level one modules for \(C^{(1)}_ l\), which is expressed in terms of quadratic theta functions of degree \(l - 1\). Comparing his formula with the one obtained by Kac and Wakimoto, some curious theta function identities are discovered. He also finds a general formula expressing general branching functions in terms of string functions. As an application, the branching functions for the complementary decompositions are explicitly computed.
irreducible highest weight representations
scientific article; zbMATH DE number 611199