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 $\widehat {\mathfrak {g}}$ span an $\mathrm {SL}_2(\mathbb {Z})$-invariant space. This result extends to admissible $\widehat {\mathfrak {g}}$-modules, where $\mathfrak {g}$ is a simple Lie algebra or $\mathrm {osp}_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\widehat {\mathfrak {g}}$-modules when $\mathfrak {g} =s\ell _2$ (resp. $=\mathrm {osp}_{1|2}$) 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 $\mathfrak {g}$ is a basic Lie superalgebra. For example, if $\mathfrak {g}=s\ell _{2|1}$ (resp. $=\mathrm {osp}_{3|2}$), we thus obtain modular invariant families of $\widehat {\mathfrak {g}}$-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 $\mathfrak {g}$ is a basic Lie superalgebra different from a simple Lie algebra or $\mathrm {osp}_{1|n}$, modular invariance of normalized supercharacters of admissible $\widehat {\mathfrak {g}}$-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 $\mathrm {SL}_2(\mathbb {Z})$-invariant space.