Characters of super-Kac-Moody algebras

Abstract

The partition functions for super-Wess-Zumino-Witten models can be expressed in terms of characters of super-Kac-Moody algebras. These characters are examined with the emphasis on maintaining supersymmetry explicitly. It is shown that an analogoue of Borel-Weil theory is at least formally relevant to the representation theory of super-Kac-Moody algebras, and that the characters have an interpretation in terms of fixed points of the action of the corresponding group on a homogeneous superspace. Characters with nontrivial dependence on the supermodular parameters of superconformal and supersymmetric Yang-Mills backgrounds on the torus with (++) spin structure are computed, and for the case of SU(2), they are used to extend the conventional GKO construction for the characters of the discrete series of unitary representations of the superconformal algebra with c< 3 2 to accomodate the odd superconformal supercharacters of Cohn and Friedan. This extension of the GKO construction requires the incorporation of a “spectator” space of free fermions in the standard GKO construction of superconformal characters relevant to the (+−), (−+) and (−−) spin structures.