On the modularity of certain functions from the Gromov-Witten theory of elliptic orbifolds

TL;DR

This paper investigates the modular properties of functions from elliptic orbifold Gromov-Witten theory, providing modular completions and new formulas involving mock modular forms for indefinite theta functions.

Contribution

It introduces new modular completions for complex indefinite theta functions and derives novel closed formulas involving mock modular forms, advancing understanding of elliptic orbifold Gromov-Witten functions.

Findings

Provided modular completions for indefinite theta functions

Derived new formulas involving mock modular forms

Enhanced understanding of modularity in elliptic orbifold Gromov-Witten theory

Abstract

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulas for special indefinite theta functions of type $(1,2)$ in terms of products of mock modular forms. This formula is also of independent interest.