Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 107, 34 pages      arXiv:2104.04344      https://doi.org/10.3842/SIGMA.2021.107
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Clean Single-Valued Polylogarithms

Steven Charlton a, Claude Duhr b and Herbert Gangl c
a) Fachbereich Mathematik (AZ), Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
b) Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany
c) Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK

Received April 13, 2021, in final form November 28, 2021; Published online December 12, 2021

Abstract
We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths one and two, we present some explicit formulas and results. We also give explicit formulas for the single-valued and clean single-valued version attached to the Nielsen polylogarithms $S_{n,2}(x)$, and we show how the clean single-valued functions give new evaluations of multiple polylogarithms at certain algebraic points.

Key words: multiple polylogarithms; Nielsen polylogarithms; Hopf algebras; Dynkin operator; functional equations; single-valued projection; special values.

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