Indefinite theta functions arising in Gromov-Witten Theory of elliptic orbifolds

TL;DR

This paper completes the analysis of modularity properties of Gromov-Witten potentials for elliptic orbifolds, linking geometric, mirror symmetry, and mock modular form theories to deepen understanding of their mirror-symmetric features.

Contribution

It provides a comprehensive understanding of the modularity transformation properties of Gromov-Witten potentials for elliptic orbifolds, building on recent developments in mock modular forms and mirror symmetry.

Findings

Complete analysis of modularity properties of Gromov-Witten potentials

Identification of identities linking functions with mock modular properties

Enhanced understanding of mirror symmetry in elliptic orbifolds

Abstract

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and Lau's functions, inspired by recent work of Alexandrov, Banerjee, Manschot, and Pioline on similar functions. Combined with the work of Lau and Zhou, as well as the aforementioned work of Zwegers and two of the authors, this affords a complete understanding of the modularity transformation properties of the open Gromov-Witten potentials of elliptic orbifolds of the form $\mathbb P^1_{a,b,c}$ computed by Cho, Hong, Kim, and Lau. It is hoped that this will provide a fuller picture of the mirror-symmetric properties of these orbifolds in subsequent works.