Multiple zeta values and periods of moduli spaces ${\mathfrak{M}}_{0,n}$

Abstract

In this paper we prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces ${\mathfrak{M}}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. In order to do this, we introduce a differential algebra of multiple polylogarithms on ${\mathfrak{M}}_{0,n}$, and prove that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively, and to exploit the geometry of the moduli spaces to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extremal cases of general product formulae for periods which arise by considering natural maps between moduli spaces.

arXiv comment110 pages, 15 figures

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