Integrable representations of affine A(m, n) and C(m) superalgebras

TL;DR

This paper completes the classification of irreducible integrable modules with finite dimensional weight spaces for affine superalgebras of types A(m, n) and C(m), revealing they are of highest weight type but not necessarily loop modules.

Contribution

It provides a complete classification of zero-level integrable modules for affine superalgebras of types A(m, n) and C(m), highlighting their highest weight structure.

Findings

Modules are of highest weight type.

Modules are not necessarily loop modules.

Classification is completed for these superalgebras.

Abstract

Rao and Zhao classified the irreducible integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras which are not $\hat{A}(m,n)$ ($m\ne n$) or $\hat{C}(m)$. Here we treat the latter affine superalgebras to complete the classification. The problem boils down to classifying the irreducible zero-level integrable modules with finite dimensional weight spaces for these affine superalgebras, which is solved in this paper. We note in particular that such modules for $\hat{A}(m,n)$ ($m\ne n$) and $\hat{C}(m)$ must be of highest weight type, but are not necessarily loop modules. This is in sharp contrast to the cases of ordinary affine algebras and the other types of affine superalgebras.