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

TL;DR

This paper proves that the periods of moduli spaces of Riemann spheres with marked points are multiple zeta values, using a differential algebra of polylogarithms and geometric techniques.

Contribution

It introduces a differential algebra of multiple polylogarithms on moduli spaces and proves their closure under primitives, confirming a conjecture relating periods to multiple zeta values.

Findings

Periods of $rak{M}_{0,n}$ are multiple zeta values.

Double shuffle relations are extremal cases of product formulas.

Geometric interpretation of period relations.

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.