H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series

It was shown in previous work that the one-variable $\widehat{\mu }$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1,1)$ are both Heisenberg harmonic Maa\ss-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maa\ss-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of “product type” for non-degenerate lattices of signature $(r\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}s,s)$. We thus obtain a clean generalization of $\widehat{\mu }$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic Maa\ss-Jacobi forms, we obtain harmonic weak Maa\ss\ forms of higher depth in the sense of Zagier and Zwegers. In particular, we explain the modular completion of some, so-called degenerate indefinite theta series in the context of higher depth mixed mock modular forms. The structure theory for Heisenberg harmonic Maa\ss-Jacobi forms developed in this paper also explains a curious splitting of Zwegers’s two-variable $\widehat{\mu }$-function into the sum of a meromorphic Jacobi form and a one-variable Maa\ss-Jacobi form.