The Mock Modular Data of a Family of Superalgebras
  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. 
Claudia Alfes and Thomas Creutzig
math.NT hep-th math.QA