Elliptic modular graph forms II: Iterated integrals

TL;DR

This paper develops methods to represent elliptic modular graph forms as iterated integrals, revealing algebraic relations and providing new realizations of elliptic polylogarithms with applications in string theory.

Contribution

It introduces a novel approach to express eMGFs as iterated integrals, enabling algebraic and differential analysis and connecting to elliptic polylogarithms at arbitrary depth.

Findings

Derived basis dimensions of eMGFs from iterated integrals

Established algebraic and differential relations among eMGFs

Provided realizations of elliptic polylogarithms in terms of modular forms

Abstract

Elliptic modular graph forms (eMGFs) are non-holomorphic modular forms depending on a modular parameter $\tau$ of a torus and marked points $z$ thereon. Traditionally, eMGFs are constructed from nested lattice sums over the discrete momenta on the worldsheet torus in closed-string genus-one amplitudes. In this work, we develop methods to translate the lattice-sum realization of eMGFs into iterated integrals over modular parameters $\tau$ of the torus with particular focus on cases with one marked point. Such iterated-integral representations manifest algebraic and differential relations among eMGFs and their degeneration limit $\tau \rightarrow i\infty$. From a mathematical point of view, our results yield concrete realizations of single-valued elliptic polylogarithms at arbitrary depth in terms of meromorphic iterated integrals over modular forms and their complex conjugates. The basis dimensions of eMGFs at fixed modular and transcendental weights are derived from a simple counting of iterated integrals and a generalization of Tsunogai's derivation algebra.