Representations of superconformal algebras and mock theta functions

Abstract

It is well known that the normalized characters of integrable highest weight modules of given level over an affine Lie algebra [care over g] span an SL[subscript 2](Z)-invariant space. This result extends to admissible [caret over g]-modules, where g is a simple Lie algebra or osp[subscript 1|n]. Applying the quantum Hamiltonian reduction (QHR) to admissible [caret over g]-modules when g = sl[subscript 2] (resp. = osp[subscript 1|2]) one obtains minimal series modules over the Virasoro (resp. N = 1 superconformal algebras), which form modular invariant families. Another instance of modular invariance occurs for boundary level admissible modules, including when g is a basic Lie superalgebra. For example, if g = sl[subscript 2|1] (resp. = osp[subscript 3|2]), we thus obtain modular invariant families of g-modules, whose QHR produces the minimal series modules for the N = 2 superconformal algebras (resp. a modular invariant family of N = 3 superconformal algebra modules). However, in the case when g is a basic Lie superalgebra different from a simple Lie algebra or osp[subscript 1|n], modular invariance of normalized supercharacters of admissible [caret over g]-modules holds outside of boundary levels only after their modification in the spirit of Zwegers’ modification of mock theta functions. Applying the QHR, we obtain families of representations of N = 2, 3, 4 and big N = 4 superconformal algebras, whose modified (super)characters span an SL[subscript 2](Z)-invariant space.