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The mock modular data of a family of superalgebras


Authors: Claudia Alfes and Thomas Creutzig
Journal: Proc. Amer. Math. Soc. 142 (2014), 2265-2280
MSC (2010): Primary 11F22, 11F37, 81T40
DOI: https://doi.org/10.1090/S0002-9939-2014-11959-9
Published electronically: April 3, 2014
MathSciNet review: 3195752
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Abstract: The modular properties of characters of representations of a family of W-superalgebras extending $ \widehat {\mathfrak{gl}}(1\vert 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\vert 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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Additional Information

Claudia Alfes
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
Email: alfes@mathematik.tu-darmstadt.de

Thomas Creutzig
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
Address at time of publication: Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: creutzig@ualberta.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-11959-9
Received by editor(s): July 19, 2012
Published electronically: April 3, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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