The hexagon equations for dilogarithms and the Riemann-Hilbert problem

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.