Representations of affine superalgebras and mock theta functions II

Abstract

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}_{2|1}$ can be modified, using Zwegers' real analytic corrections, to form an $SL_2(\mathbf{Z})$-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for $\widehat{osp}_{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-\frac{2m+2}{M})$, where $m \in \mathbf{Z}_{\geq 0}, M \in \mathbf{Z}_{\geq 2}$, gcd$(2m+2,M)=1$ if $m>0$ (resp. $c=-3\frac{2m+1}{M}$, where $m \in \mathbf{Z}_{\geq 0}, M \in \mathbf{Z}_{\geq 2}$ gcd$(4m +2, M) =1)$, whose modified supercharacters form an $SL_2(\mathbf{Z})$-invariant family of functions.