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. $c=-3frac{2m+1}{M}$, 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.

Representations of affine superalgebras and mock theta functions

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $hat{sell}_{2|1}$ (resp. $hat{psell}_{2|2}$) can be modified, using Zwegers real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-frac{2m+2}{M})$, where $m in mathbb{Z}_{geq 0}$, $Min mathbb{Z}_{geq 2}$, $gcd(2m+2,M)=1$ if $m>0$ (resp. $6(frac{m}{M}-1)$, where $m in mathbb{Z}_{geq 1}, Min mathbb{Z}_{geq 2}$, $gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.