Representations of superconformal algebras and mock theta functions

V. G. Kac ; M. Wakimoto

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 17-88

Résumé

It is well known that the normalized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\mathfrak{g}}$ span an $\mathrm{SL}_2(\mathbb{Z})$–invariant space. This result extends to admissible $\hat{\mathfrak{g}}$–modules, where $\mathfrak{g}$ is a simple Lie algebra or $\mathrm{osp}_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\mathfrak{g}}$–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 $\hat{\mathfrak{g}}$–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 $\hat{\mathfrak{g}}$–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.