General branching functions of affine Lie algebras

TL;DR

This paper derives explicit algebraic formulas for branching functions of affine Lie algebra cosets, extending previous results and providing a simple proof that applies to more complex algebraic structures.

Contribution

It provides a new algebraic derivation of branching functions for affine Lie algebra cosets, generalizing to more complex cases with sums of simple and $u(1)$ terms.

Findings

Explicit formulas for branching functions are obtained.

The derivation is purely algebraical and independent.

Method extends to more complex algebraic structures.

Abstract

Explicit expressions are presented for general branching functions for cosets of affine Lie algebras $\hat{g}$ with respect to subalgebras $\hat{g}^\prime$ for the cases where the corresponding finite dimensional algebras $g$ and $g^\prime$ are such that $g$ is simple and $g^\prime$ is either simple or sums of $u(1)$ terms. A special case of the latter yields the string functions. Our derivation is purely algebraical and has its origin in the results on the BRST cohomology presented by us earlier. We will here give an independent and simple proof of the validity our results. The method presented here generalizes in a straightforward way to more complicated $g$ and $g^\prime$ such as {\it e g } sums of simple and $u(1)$ terms.