hide
Free keywords:
High Energy Physics - Theory, hep-th,Mathematics, Algebraic Geometry, math.AG,Mathematics, Number Theory, math.NT
Abstract:
We study non-holomorphic modular forms built from iterated integrals of
holomorphic modular forms for SL$(2,\mathbb Z)$ known as equivariant iterated
Eisenstein integrals. A special subclass of them furnishes an equivalent
description of the modular graph forms appearing in the low-energy expansion of
string amplitudes at genus one. Notably the Fourier expansion of modular graph
forms contains single-valued multiple zeta values. We deduce the appearance of
products and higher-depth instances of multiple zeta values in equivariant
iterated Eisenstein integrals, and ultimately modular graph forms, from the
appearance of simpler odd Riemann zeta values. This analysis relies on
so-called zeta generators which act on certain non-commutative variables in the
generating series of the iterated integrals. From an extension of these
non-commutative variables we incorporate iterated integrals involving
holomorphic cusp forms into our setup and use them to construct the modular
completion of triple Eisenstein integrals. Our work represents a fully explicit
realisation of the modular graph forms within Brown's framework of equivariant
iterated Eisenstein integrals and reveals structural analogies between
single-valued period functions appearing in genus zero and one string
amplitudes.
