Cluster Polylogarithms I: Quadrangular Polylogarithms

Abstract

We suggest a definition of cluster polylogarithms on an arbitrary cluster variety and classify them on M0,m. We find functional equations for multiple polylogarithms which generalize equations found by Abel, Kummer, and Goncharov to an arbitrary weight. As an application, we prove a part of the Goncharov depth conjecture in weight six. Also, we define cluster versions of Grassmannian polylogarithms, whose existence was conjectured by Goncharov.