Title: Representations of superconformal algebras and mock theta functions,
Abstract: It is well known that the normaized characters of integrable highest weight
modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an
$SL_2(\mathbf{Z})$-invariant space. This result extends to admissible
$\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or
$osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible
$\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. $=osp_{1|2}$) 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 $\frak{g}$ is a basic Lie superalgebra. For example, if
$\frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant
families of $\hat{\frak{g}}$-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 $\frak{g}$ is a basic Lie superalgebra different
from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized
supercharacters of admissible $\hat{\frak{g}}$-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 $SL_2(\mathbf{Z})$-invariant space.