Towards Motivic Coactions at Genus One from Zeta Generators

2025/08/04 by Axel Kleinschmidt, Kleinschmidt, Axel, Franziska Porkert +3 · 1 voice
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT) #hep-th #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.2508.02800

arxiv published 2025/08/04 · arxiv updated 2026/05/04

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.

Citations

• Higher-genus multiple zeta values
• Meromorphic higher-genus integration kernels via convolution over homology cycles
• Positive geometries and canonical forms via mixed Hodge theory
• Schottky-Kronecker forms and hyperelliptic polylogarithms
• Canonicalizing zeta generators: genus zero and genus one
• Motivic coaction and single-valued map of polylogarithms from zeta generators
• Motivic geometry of two-loop Feynman integrals
• Higher genus polylogarithms on families of Riemann surfaces
• Feynman Integrals
• Towards closed strings as single-valued open strings at genus one
• The diagrammatic coaction and the algebraic structure of cut Feynman integrals
• Elliptic multizetas and the elliptic double shuffle relations
• Identities between modular graph forms
• Mathematical aspects of scattering amplitudes
• Mixed Tate motives over \Z
• Elliptic associators
• Groupes fondamentaux motiviques de Tate mixte
• The multiple zeta value algebra and the stable derivation algebra

Discussions

• Man this looks like a badass paper arxiv.org/abs/2508.02800 [bsky, 3 points, 1 comments]

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• Deriving motivic coactions and single-valued maps at genus zero from zeta generators
• Motivic coaction and single-valued map of polylogarithms from zeta generators
• Non-holomorphic modular forms from zeta generators
• Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes
• Canonicalizing zeta generators: genus zero and genus one