Quantum Modular Forms and Resurgence

TL;DR

This paper unifies quantum modular forms with resurgence theory, strengthening known results and introducing median resummation techniques to recover quantum modular forms from their asymptotic expansions.

Contribution

It provides a comprehensive framework linking quantum modular forms to resurgence, extending Zagier's examples, and introduces median resummation results for holomorphic quantum modular forms.

Findings

Unified quantum modular forms under resurgence framework

Strengthened modularity results for Eichler integrals

Median resummation techniques recover quantum modular forms from asymptotics

Abstract

In 2010, Zagier described a new phenomenon which he called quantum modularity. This connected various examples coming from disparate fields which exhibit near-modular behavior. In the fifteen years since, Zagier's philosophy has informed new developments in areas such as knot theory, 3-dimensional topology, combinatorics, and physics. More recently, the concept of holomorphic quantum modularity has emerged, pointing to a clearer structure for Zagier's original examples. These new developments suggest connections to perturbative quantum field theory, like the theory of resurgence. In 2024, Fantini and Rella proposed a means of codifying some of these connections under their program of ``modular resurgence." Inspired by their work, we unify all of the examples of quantum modular forms in Zagier's original paper under the umbrella of resurgence. In doing so, we strengthen known quantum modularity results for holomorphic Eichler integrals of half-integer weight modular forms. Our main addition to the literature is a collection of median resummation type results which show that the examples of holomorphic quantum modular forms we consider can be recovered from their asymptotics, or the asymptotics of closely related functions.