Quantum mock modular forms arising from eta-theta functions

TL;DR

This paper constructs new quantum mock modular forms from eta-theta functions with even characters, revealing their shadows as eta-theta functions with odd characters, and demonstrates their quantum modularity with applications to Eichler integrals and algebraic identities.

Contribution

It unifies eta-theta functions into mock modular forms with explicit shadows and proves their quantum modularity, providing new tools for evaluating Eichler integrals and discovering algebraic identities.

Findings

Mock modular forms with shadows given by eta-theta functions with odd characters.

Proof that these mock modular forms are quantum modular forms.

Finite hypergeometric expressions for Eichler integrals and algebraic identities.

Abstract

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities.