Representations of affine superalgebras and mock theta functions III

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 $\mathfrak{g}.$