Representations of affine superalgebras and mock theta functions

TL;DR

This paper demonstrates that modified supercharacters of certain affine superalgebra modules can be made modular invariant, leading to new positive energy modules over superconformal algebras with specific central charges.

Contribution

It introduces a method to modify supercharacters of affine superalgebra modules for modular invariance, connecting them to superconformal algebra modules via quantum Hamiltonian reduction.

Findings

Supercharacters can be modified using Zwegers' corrections for modular invariance.

New families of positive energy superconformal modules are constructed.

Modified characters form a modular invariant family.

Abstract

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{s\ell}_{2|1}$ (resp. $\hat{ps\ell}_{2|2}$) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-\frac{2m+2}{M})$, where $m \in \mathbb{Z}_{\geq 0}$, $M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m+2,M)=1$ if $m>0$ (resp. $6(\frac{m}{M}-1)$, where $m \in \mathbb{Z}_{\geq 1}, M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.