Representations of affine superalgebras and mock theta functions II

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.