hide
Free keywords:
Mathematics, Quantum Algebra, math.QA,High Energy Physics - Phenomenology, hep-ph,High Energy Physics - Theory, hep-th,Mathematics, Algebraic Geometry, math.AG,Mathematics, Number Theory, math.NT
Abstract:
Zeta generators are derivations associated with odd Riemann zeta values that
act freely on the Lie algebra of the fundamental group of Riemann surfaces with
marked points. The genus-zero incarnation of zeta generators are Ihara
derivations of certain Lie polynomials in two generators that can be obtained
from the Drinfeld associator. We characterize a canonical choice of these
polynomials, together with their non-Lie counterparts at even degrees $w\geq
2$, through the action of the dual space of formal and motivic multizeta
values. Based on these canonical polynomials, we propose a canonical
isomorphism that maps motivic multizeta values into the $f$-alphabet. The
canonical Lie polynomials from the genus-zero setup determine canonical zeta
generators in genus one that act on the two generators of Enriquez' elliptic
associators. Up to a single contribution at fixed degree, the zeta generators
in genus one are systematically expanded in terms of Tsunogai's geometric
derivations dual to holomorphic Eisenstein series, leading to a wealth of
explicit high-order computations. Earlier ambiguities in defining the
non-geometric part of genus-one zeta generators are resolved by imposing a new
representation-theoretic condition. The tight interplay between zeta generators
in genus zero and genus one unravelled in this work connects the construction
of single-valued multiple polylogarithms on the sphere with
iterated-Eisenstein-integral representations of modular graph forms.
