Integrable highest weight modules over affine superalgebras and Appell's function

TL;DR

This paper classifies integrable highest weight modules over affine superalgebras, provides a free field construction at level 1, and links characters of certain modules to Appell's elliptic function, highlighting differences from affine Lie algebra theory.

Contribution

It offers a classification of integrable modules over affine superalgebras and connects their characters to Appell's function, introducing new insights into their representation theory.

Findings

Characters of level 1 modules expressed via Appell's elliptic function

Distinct representation theory features from affine Lie algebras

Explicit free field construction at level 1

Abstract

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level~1 case. The analysis of this construction shows, in particular, that in the simplest case of the $s\ell (2|1)$ level~1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras.