Representations of affine superalgebras and mock theta functions III

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}. $