The hexagon equations for dilogarithms and the Riemann-Hilbert problem

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.

6 pages, 1 figure