Kac-Wakimoto characters and universal mock theta functions
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- by Amanda Folsom PDF
- Trans. Amer. Math. Soc. 363 (2011), 439-455 Request permission
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.References
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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
- MR Author ID: 690449
- Email: folsom@math.wisc.edu, amanda.folsom@yale.edu
- Received by editor(s): April 21, 2009
- Received by editor(s) in revised form: May 4, 2009
- Published electronically: August 31, 2010
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2719689