Kac-Wakimoto characters and universal mock theta functions

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.