The hexagon equations for dilogarithms and the Riemann-Hilbert problem

TL;DR

This paper derives hexagon equations for dilogarithms from their analytic continuation, framing them as a Riemann-Hilbert problem that uniquely characterizes the dilogarithm function.

Contribution

It introduces the hexagon equations for dilogarithms and connects them to the Riemann-Hilbert problem, providing a new characterization of the dilogarithm.

Findings

Hexagon equations are equivalent to coboundary relations for a 1-cocycle.

The equations are solved by the Riemann-Hilbert problem of additive type.

The dilogarithm is uniquely characterized by these equations under normalization.

Abstract

In this article we present the hexagon equations for dilogarithms which come from the analytic continuation of the dilogarithm $\mathrm{Li}_2(z)$ to ${\mathbf P}^1 \setminus {0,1,\infty}$. The hexagon equations are equivalent to the coboundary relations for a certain 1-cocycle of holomorphic functions on ${\mathbf P}^1$, and are solved by the Riemann-Hilbert problem of additive type. They uniquely characterize the dilogarithm under the normalization condition.