Towards Motivic Coactions at Genus One from Zeta Generators

TL;DR

This paper proposes coaction formulae for iterated integrals over holomorphic Eisenstein series at genus one, revealing structural insights into multiple zeta values and modular forms relevant to quantum field theory and string theory.

Contribution

It introduces a new coaction framework for genus-one iterated integrals based on zeta generators, extending motivic coaction concepts to genus one.

Findings

The proposed coaction exhibits expected algebraic properties.

Derived $f$-alphabet decompositions of multiple modular values.

Shows formal similarities with genus zero multiple polylogarithms.

Abstract

The motivic coaction of multiple zeta values and multiple polylogarithms encodes both structural insights on and computational methods for scattering amplitudes in a variety of quantum field theories and in string theory. In this work, we propose coaction formulae for iterated integrals over holomorphic Eisenstein series that arise from configuration-space integrals at genus one. Our proposal is motivated by formal similarities between the motivic coaction and the single-valued map of multiple polylogarithms at genus zero that are exposed in their recent reformulations via zeta generators. The genus-one coaction of this work is then proposed by analogies with the construction of single-valued iterated Eisenstein integrals via zeta generators at genus one. We show that our proposal exhibits the expected properties of a coaction and deduce $f$-alphabet decompositions of the multiple modular values obtained from regularized limits.