$${\mathrm {H}}$$ H -Harmonic Maaß-Jacobi forms of degree 1

Abstract

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 \((r\!+\!1,1)\) are both Heisenberg harmonic Maas-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maas-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of “product type” for non-degenerate lattices of signature \((r\!+\!s,s)\). We thus obtain a clean generalization of \(\widehat{\mu }\) to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic Maas-Jacobi forms, we obtain harmonic weak Maas 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 Maas-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 Maas-Jacobi form.