Modular graph forms from equivariant iterated Eisenstein integrals

Authors: Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Nils Matthes, Oliver Schlotterer and Bram Verbeek Preprint number: UUITP-37/22 The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms dubbed ``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.