REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra sℓˆ[subscript 2|1] (resp. psℓˆ[subscript 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 ∈ ℤ[subscript ≥0], M ∈ ℤ[subscript ≥2], gcd(2 m + 2, M) = 1 if m > 0 (resp. 6 (m/M − 1), where m ∈ ℤ[subscript ≥1], M ∈ ℤ[subscript ≥2], gcd(2 m, M) = 1 if m > 1), whose modified characters and supercharacters form a modular invariant family