Representations of affine superalgebras and mock theta functions III

TL;DR

This paper investigates the modular invariance of supercharacters of affine Lie superalgebras, developing a multi-step modification of mock theta functions to establish SL_2(Z)-invariance, especially when the Killing form is non-degenerate.

Contribution

It introduces a novel multi-step modification process of mock theta functions to prove modular invariance of supercharacters for affine Lie superalgebras.

Findings

Modified supercharacters form an SL_2(Z)-invariant span.

Transformation matrices relate to subalgebras orthogonal to isotropic root sets.

The process applies to arbitrary basic Lie superalgebras.

Abstract

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type "modifier" is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak{g}$ is non-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $ \mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $ \mathfrak{g}. $