To the cusp and back: Resurgent analysis for modular graph functions

TL;DR

This paper applies resurgent analysis to modular graph functions in string theory, revealing non-perturbative corrections near the cusp of the moduli space and constraining their behavior through SL(2,Z) invariance.

Contribution

It introduces a resurgent analysis framework to study non-perturbative effects in modular graph functions relevant to string amplitudes.

Findings

Constructed non-perturbative corrections near the cusp.

Demonstrated SL(2,Z) invariance constrains non-perturbative behavior.

Provided insights into the low-energy expansion of string scattering amplitudes.

Abstract

Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are ${\rm SL}(2,\mathbb{Z})$-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising when the argument of the modular graph function approaches the cusp on this moduli space. ${\rm SL}(2,\mathbb{Z})$-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space.