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Kac-Wakimoto characters and universal mock theta functions


Author: Amanda Folsom
Journal: Trans. Amer. Math. Soc. 363 (2011), 439-455
MSC (2000): Primary 11F22, 11F37, 17B67, 11F50
DOI: https://doi.org/10.1090/S0002-9947-2010-05181-5
Published electronically: August 31, 2010
MathSciNet review: 2719689
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Abstract: In recent work, Bringmann and Ono answer a question of Kac and show that character formulas for $ s\ell(r+1,1)^{\wedge}$ modules due to Kac and Wakimoto are ``holomorphic parts'' of nonholomorphic modular functions. Here, we confirm a speculation of Ono that these characters are, up to a simple $ q$-series, the universal mock theta functions $ g_2(\omega,q)$ and $ g_3(\omega,q)$ of Gordon and McIntosh. Using recent work of Bringmann-Ono, Kang, Zwegers, and Gordon-McIntosh, we show that $ g_2(\omega;q)$ and $ g_3(\omega;q)$ are, up to classical theta functions and $ \eta$-products, the characters of Kac and Wakimoto. As a consequence, we include a ``dictionary'' that gives a character formula for every classical mock theta function of Ramanujan, as well as subsequent natural generalizations.


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Additional Information

Amanda Folsom
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, Yale University, New Haven, Connecticut 06520
Email: folsom@math.wisc.edu, amanda.folsom@yale.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05181-5
Received by editor(s): April 21, 2009
Received by editor(s) in revised form: May 4, 2009
Published electronically: August 31, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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