The Mock Modular Data of a Family of Superalgebras

TL;DR

This paper investigates the modular properties of characters of representations of W-superalgebras extending gl(1|1), revealing their transformation behaviors, fusion rules, and algebraic structures through advanced mathematical analysis.

Contribution

It introduces a detailed analysis of the modular transformations of characters for a family of superalgebras, including non-generic modules, and connects these to fusion rules and algebraic products.

Findings

Characters of non-generic modules are expressed via Appell-Lerch sums.

Modular transformations form a representation of SL(2;Z).

The product from modular transformations matches the Grothendieck ring when fusion multiplicities are at most one.

Abstract

The modular properties of characters of representations of a family of W-superalgebras extending the affine Lie superalgebra of gl(1|1) are considered. Modules fall into two classes, the generic type and the non-generic one. Characters of non-generic modules are expressed in terms of higher-level Appell-Lerch sums. We compute the modular transformations of characters and interpret the Mordell integral as an integral over characters of generic representations. The \C-span of a finite number of non-generic characters together with an uncountable set of characters of the generic type combine into a representation of SL(2;\Z). The modular transformations are then used to define a product on the space of characters. The fusion rules of the extended algebras are partially inherited from the known fusion rules for modules of the affine Lie superalgebra of gl(1|1). Moreover, the product obtained from the modular transformations coincides with the product of the Grothendieck ring of characters if and only if the fusion multiplicities are at most one.