REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS

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???(2?m?+?2)/M), where m ∈ ?_≥0, M ∈ ?_≥2, gcd(2?m?+?2, M)?=?1 if m?>?0 (resp. 6 (m/M???1), where m ∈ ?_≥1, M ∈ ?_≥2, gcd(2?m, M)?=?1 if m?>?1), whose modified characters and supercharacters form a modular invariant family.