  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.

High Energy Physics - Theory;Number Theory