We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{ s\ell}_{ 2|1}$ (resp. $\hat{ ps\ell}_{ 2|2}$) 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 $m \in \mathbb{ Z}_{ \geq 0}$, $M\in \mathbb{ Z}_{ \geq 2}$, $\gcd(2m+2,M)=1$ if $m>0$ (resp. $6(\frac{ m}{ M}-1)$, where $m \in \mathbb{ Z}_{ \geq 1}, M\in \mathbb{ Z}_{ \geq 2}$, $\gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.
Categories: math.RT