Negative index Jacobi forms and quantum modular forms. (English) Zbl 1349.11085
Summary: We consider the Fourier coefficients of a special class of meromorphic Jacobi forms of negative index considered by V. G. Kac and M. Wakimoto [Prog. Math. 123, 415–456 (1994; Zbl 0854.17028)]. 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 A. O. L. Atkin and F. G. Garvan [Ramanujan J. 7, No. 1–3, 343–366 (2003; Zbl 1039.11069)]. 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.
MSC:
| 11F03 | Modular and automorphic functions |
| 11F22 | Relationship to Lie algebras and finite simple groups |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11F50 | Jacobi forms |
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