Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

Abstract

We present new functional equations in weights 5, 6 and 7 and use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity Q 4 from the recent work of Goncharov and Rudenko that was used in their proof of the weight 4 case of Zagier's Polylogarithm Conjecture.

Cluster Superalgebras and Stringy Integrals

“…For example, by expressing its 2-loop 7-particle MHV amplitude in two different ways, guaranteed to be equal to each other as a simple physical consequence of parity symmetry, the authors of [7] discovered a mathematically nontrivial 40-term functional equation for the trilogarithm function Li 3 (z) = z 0 dt t Li 2 (t) whose arguments are cluster Poisson coordinates on the moduli space of 6 cyclically ordered points in P 2 (the D 4 cluster algebra). More generally, numerous identities at various weights have emerged from the study of Feynman integrals in quantum field theory; for recent developments see for example [20,21] and references therein. In a recent mathematical breakthrough, Goncharov and Rudenko have used the link between cluster varieties and polylogarithms to prove Zagier's polylogarithm conjecture in weight 4 [22].…”