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It was shown in previous work that the one-variable $\widehat\mu$-functiondefined by Zwegers (and Zagier) and his indefinite theta series attached tolattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maa\ss-Jacobiforms. We extend the concept of Heisenberg harmonicity to Maa\ss-Jacobi formsof arbitrary many elliptic variables, and produce indefinite theta series of"product type" for non-degenerate lattices of signature $(r\!+\!s,s)$. We thusobtain a clean generalization of $\widehat\mu$ to these negative definitelattices. From restrictions to torsion points of Heisenberg harmonicMaa\ss-Jacobi forms, we obtain harmonic weak Maa\ss\ forms of higher depth inthe sense of Zagier and Zwegers. In particular, we explain the modularcompletion of some, so-called degenerate indefinite theta series in the contextof higher depth mixed mock modular forms. The structure theory for Heisenbergharmonic Maa\ss-Jacobi forms developed in this paper also explains a curioussplitting of Zwegers's two-variable $\widehat\mu$-function into the sum of ameromorphic Jacobi form and a one-variable Maa\ss-Jacobi form.

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