Deriving motivic coactions and single-valued maps at genus zero from zeta generators
Hadleigh Frost, Martijn Hidding, Deepak Kamlesh, Carlos Rodriguez, Oliver Schlotterer, Bram Verbeek
TL;DR
The paper proves genus-zero reformulations of the motivic coaction and the single-valued map for multiple polylogarithms using zeta generators. It builds a generating-series framework for MPLs in any number of complex variables, and proves a new coaction formula ΔG1m = (Hn^dr)−1 G1m Hn^dr G1dr, with Hn = M Gn…G2, by adapting Brown’s multivariate Ihara machinery and braid-group actions, leveraging Drinfeld associators to connect de Rham and motivic data. A parallel result gives the sv map as sv G1m = (sv Hn^m)−1 ¯G1t (sv Hn^m) G1m, with two proofs: one via antipode relations in the motivic/coaction formalism and another via direct comparison with established sv constructions and a change of alphabet. The work preserves fibration bases, clarifies how different classes of MZVs enter through zeta generators, and sets the stage for genus-one and higher-genus extensions by making the coaction and sv maps operable in a genus-zero, multivariate setting. Overall, it provides a genus-zero, zeta-generator–based unification of motivic coactions and single-valued MPLs across multiple variables, with implications for simplifying higher-genus generalizations in related physical and mathematical contexts.
Abstract
Multiple polylogarithms are equipped with rich algebraic structures including the motivic coaction and the single-valued map which both found fruitful applications in high-energy physics. In recent work arXiv:2312.00697, the current authors presented a conjectural reformulation of the motivic coaction and the single-valued map via zeta generators, certain operations on non-commuting variables in suitable generating series of multiple polylogarithms. In this work, the conjectures of the reference will be proven for multiple polylogarithms that depend on any number of variables on the Riemann sphere.