From Modular Graph Forms to Iterated Integrals

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 $\text{textsc}Mathematica$ 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 ${\alpha }^{\mathrm{\prime }8}$, which we use to calculate the ${\alpha }^{\mathrm{\prime }8}{\zeta }_3{\zeta }_5$ contribution to the analytic part of the amplitude.