Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

TL;DR

This paper develops a comprehensive theory of harmonic Maass-Jacobi forms with singularities, including a theta-like decomposition, filling a significant gap in the understanding of real-analytic Jacobi forms in mathematics and physics.

Contribution

It introduces a new space of harmonic Maass-Jacobi forms with singularities and establishes a theta-like decomposition using Zwegers's $\widehat exteta$-functions, advancing the theoretical framework.

Findings

Provides structure results for harmonic Maass-Jacobi forms

Includes real-analytic Jacobi forms from Zwegers's thesis

Establishes a theta-like decomposition

Abstract

Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers's PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers's $\widehat\mu$-functions to establish a theta-like decomposition.