Representations of affine superalgebras and mock theta functions. III (Q2827309)

The paper under review is the third in the series of papers on modular invariance of modified normalized characters of irreducible highest weight representations of affine Lie superalgebras [Zbl 1372.17019 and Zbl 1372.17020]. In this paper, the authors 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 the authors use a multi-step modification process of multivariate mock theta functions. The main result of the paper asserts that the span of the resulting modified normalized supercharacters is \(\mathrm{SL}_2(\mathbb{Z})\)-invariant. Moreover, if the Killing form on \(\mathfrak{g}\) is non-degenerate, then it is shown that the transformation matrix is equal to that for the basic defect 0 subalgebra \(\mathfrak{g}^!\) of \(\mathfrak{g}\) orthogonal to a maximal isotropic set of roots of \(\mathfrak{g}\).