Modular graph forms from equivariant iterated Eisenstein integrals

TL;DR

This paper validates that equivariant iterated Eisenstein integrals encompass modular graph forms, advancing the mathematical understanding of their structure beyond the simplest cases, with implications for string theory amplitudes.

Contribution

It provides the first validation beyond depth one that equivariant iterated Eisenstein integrals include modular graph forms, with explicit systematic descriptions.

Findings

Validated Brown's conjecture beyond depth one

Explicitly described elements of Brown's construction at higher depths

Established systematic correspondence between integrals and modular graph forms

Abstract

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown's alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown's conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown's construction fully explicit to all orders.