The mock modular data of a family of superalgebras

Alfes, Claudia; Creutzig, Thomas

doi:10.1090/S0002-9939-2014-11959-9

The mock modular data of a family of superalgebras
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by Claudia Alfes and Thomas Creutzig PDF

Proc. Amer. Math. Soc. 142 (2014), 2265-2280 Request permission

Abstract:

The modular properties of characters of representations of a family of W-superalgebras extending $\widehat {\mathfrak {gl}}(1|1)$ 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 $\mathbb {C}$-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 $\mathrm {SL}(2;Z)$. 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 $\widehat {\mathfrak {gl}}(1|1)$. 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.

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