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 sℓˆ2|1 can be modified, using Zwegers' real analytic corrections, to form an SL2(Z)SL2(Z)-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for ospˆ3|2. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the N=2N=2 (resp. N=3N=3) superconformal algebras with central charge c=3(1−2m+2M), where m∈Z≥0m∈Z≥0, M∈Z≥2M∈Z≥2, gcd⁡(2m+2,M)=1gcd⁡(2m+2,M)=1 if m>0m>0 (resp. c=−32m+1M, where m∈Z≥0m∈Z≥0, M∈Z≥2M∈Z≥2gcd⁡(4m+2,M)=1gcd⁡(4m+2,M)=1), whose modified supercharacters form an SL2(Z)SL2(Z)-invariant family of functions.