Cluster Polylogarithms I: Quadrangular Polylogarithms

TL;DR

This paper introduces cluster polylogarithms on cluster varieties, classifies them in type A, and extends functional equations for multiple polylogarithms, contributing to the understanding of Goncharov's depth conjecture.

Contribution

It defines cluster polylogarithms on arbitrary cluster varieties and classifies them in type A, extending functional equations to higher weights.

Findings

Functional equations generalizing Abel, Kummer, Goncharov equations

Classification of cluster polylogarithms in type A

Partial proof of Goncharov depth conjecture in weight six

Abstract

We suggest a definition of cluster polylogarithms on an arbitrary cluster variety and classify them in type $A$. We find functional equations for multiple polylogarithms which generalize equations discovered by Abel, Kummer, and Goncharov to an arbitrary weight. As an application, we prove a part of the Goncharov depth conjecture in weight six.