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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
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
Article copyright:
© Copyright 2010
American Mathematical Society