Functional equations for Rogers dilogarithm

Abstract

This paper proves a ``new'' family of functional equations (Eq(n)) for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus 0, M-0,M-n. For n = 4 we find back the duality relation while n = 5 gives back the 5 terms relation. It is then proved that the whole family reduces to the 5 terms relation. In the author's knownledge, it is the first time that an infinite family of functional equations for the dilogarithm with an increasing number of variables (n - 3 for (Eq(n))) is reduced to the 5 terms relation. This reduction explains the quotation marks around ``new'' at the beginning of this abstract.