Non-holomorphic modular forms from zeta generators
arxiv(2024)
摘要
We study non-holomorphic modular forms built from iterated integrals of
holomorphic modular forms for SL(2,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.
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