From Modular Graph Forms to Iterated Integrals

TL;DR

This paper develops a systematic method to convert modular graph forms into iterated integrals, facilitating calculations in string theory amplitudes, and provides a Mathematica package for practical implementation.

Contribution

It introduces a new procedure to transform lattice-sum representations of modular graph forms into iterated integrals and offers a computational tool for topologies up to four vertices.

Findings

Converted integrand of four-graviton one-loop amplitude at eighth order in '

Calculated '835 contribution to the amplitude

Provided a Mathematica package for modular graph form topologies

Abstract

Modular graph forms are a class of non-holomorphic modular forms that arise in the low-energy expansion of genus-one closed string amplitudes. In this work, we introduce a systematic procedure to convert lattice-sum representations of modular graph forms into iterated integrals of holomorphic Eisenstein series and provide a \textsc{Mathematica} package that implements all modular graph form topologies up to four vertices. To achieve this, we introduce specific tree-representations of modular graph forms. The presented method enables the conversion of the integrand of the four-graviton one-loop superstring amplitude at eighth order in the inverse string tension $\alpha^{\prime 8}$, which we use to calculate the $\alpha^{\prime 8}\zeta_3\zeta_5$ contribution to the analytic part of the amplitude.