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.
Categories: math.CA hep-th math.NT