Negative index Jacobi forms and quantum modular forms
Bringmann K, Creutzig T, Rolen L (2014)
Publication Type: Journal article
Publication year: 2014
Journal
Book Volume: 1
Article Number: 11
Journal Issue: 1
DOI: 10.1186/s40687-014-0011-8
Abstract
In this paper, we consider the Fourier coefficients of a special class of meromorphic Jacobi forms of negative index considered by Kac and Wakimoto. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family of rank-crank type partial differential equations generalizing the famous example of Atkin and Garvan. We then describe the modularity properties of these coefficients, showing that they are ‘mixed partial theta functions’, along the way determining a new class of quantum modular partial theta functions which is of independent interest. In particular, we settle the final cases of a question of Kac concerning modularity properties of Fourier coefficients of certain Jacobi forms.
Involved external institutions
Germany (DE)
Canada (CA)How to cite
APA:
Bringmann, K., Creutzig, T., & Rolen, L. (2014). Negative index Jacobi forms and quantum modular forms. Research in the Mathematical Sciences, 1(1). https://doi.org/10.1186/s40687-014-0011-8
MLA:
Bringmann, Kathrin, Thomas Creutzig, and Larry Rolen. "Negative index Jacobi forms and quantum modular forms." Research in the Mathematical Sciences 1.1 (2014).
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