Representations of affine superalgebras and mock theta functions

Abstract

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