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

Francis C. S. Brown

TL;DR

The paper proves Goncharov–Manin's conjecture that periods of $\mathfrak{M}_{0,n}$ are multiple zeta values by building a universal differential algebra of polylogarithms on $\mathfrak{M}_{0,n}$ and employing an iterated-Stokes strategy along the associahedron boundary to reduce period integrals to MZVs. It develops dihedral coordinates and a dihedral-cover of the compactified moduli space, together with a reduced bar construction whose de Rham cohomology vanishes, guaranteeing primitives exist and enabling a geometric interpretation of product relations. The work also shows that shuffle and stuffle relations arise as two extremal cases of geometric product maps between moduli spaces, linking periods to well-known combinatorial structures via the mosaic operad. The results deliver a self-contained framework for expressing periods of $\mathfrak{M}_{0,S}$ in terms of multiple zeta values and offer a scalable approach potentially extendable to other hyperplane arrangements and perturbative Feynman-type integrals.

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.

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

TL;DR

The paper proves Goncharov–Manin's conjecture that periods of are multiple zeta values by building a universal differential algebra of polylogarithms on and employing an iterated-Stokes strategy along the associahedron boundary to reduce period integrals to MZVs. It develops dihedral coordinates and a dihedral-cover of the compactified moduli space, together with a reduced bar construction whose de Rham cohomology vanishes, guaranteeing primitives exist and enabling a geometric interpretation of product relations. The work also shows that shuffle and stuffle relations arise as two extremal cases of geometric product maps between moduli spaces, linking periods to well-known combinatorial structures via the mosaic operad. The results deliver a self-contained framework for expressing periods of in terms of multiple zeta values and offer a scalable approach potentially extendable to other hyperplane arrangements and perturbative Feynman-type integrals.

Abstract

In this paper we prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces of Riemann spheres with marked points are multiple zeta values. In order to do this, we introduce a differential algebra of multiple polylogarithms on , 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.

Paper Structure

This paper contains 55 sections, 99 theorems, 596 equations, 15 figures.

Table of Contents

  1. Introduction
  2. General overview
  3. Detailed summary of results
  4. Dihedral coordinates on $\overline{\mathfrak{M}}_{0,n}(\mathbb R)$.
  5. Dihedral coordinates on $\mathfrak{M}_{0,S}$
  6. Forgetful maps between moduli spaces and projections
  7. Normal vertex coordinates on $\mathfrak{M}_{0,S}(\mathbb C)$.
  8. The real moduli space $\mathfrak{M}^{\,\delta}_{0,S}(\mathbb R)$
  9. The compactification $\overline{\mathfrak{M}}_{0,S}$ and its divisors at infinity
  10. Product maps
  11. The reduced bar construction and Picard-Vessiot theory
  12. Shuffle algebras and non-commutative formal power series
  13. Arrangements of hyperplanes and the bar construction
  14. Unipotent extensions of differentially simple algebras
  15. Base points at infinity
  16. ...and 40 more sections

Key Result

Theorem 1.1

The integral $I$ is a $\mathbb Q[2\pi i]$-linear combination of multiple zeta values of weight at most $\ell$.

Figures (15)

  • Figure 1: Dihedral coordinates on $\mathfrak{M}_{0,5}$. The scheme $\mathfrak{M}^{\,\delta}_{0,5}$ (right) is defined to be the Zariski closure of the image of the embedding $\{u_{ij}\}: \mathfrak{M}_{0,5} \hookrightarrow \mathbb {A}^5$ defined by the set of dihedral coordinates, which are indexed by chords in a pentagon (middle). This map has the effect of blowing up the points $(0,0)$ and $(1,1)$. A cell $X_{S,\delta}$ is given by the region $0< t_1< t_2<1$ (left). After blowing-up it becomes a pentagon with sides $F_{ij}=\{u_{ij}=0\}$.
  • Figure 2: Part of an oriented regular $n$-gon inscribed in a circle. Its edges are labelled with the elements of $S$, and its vertices are labelled with elements of $S$ in parentheses. Left - a chord $\{i,j\}\in \chi_{S,\delta}$ meets four edges $i,i+1,j,j+1$ which define the dihedral coordinate $u_{ij}=[i\,\,i\!+\!1\,|\,j\!+\!1\,\, j]$. Changing the orientation of the $n$-gon does not alter $u_{ij}$ by the last equation in $(\ref{['S4']})$. Right - a set of four edges $i,j,k,l$ breaks the $n$-gon into four regions as in lemma \ref{['lemmacrossratio']}, and defines a pair $A, B \subset \chi_{S,\delta}$ of completely crossing chords, depicted by the shaded rectangles (corollary \ref{['corcrossingchords']}).
  • Figure 3: Decomposition of the hexagon on setting $u_{25}=0$. The variables corresponding to chords which cross $\{2,5\}$, namely $u_{13}$, $u_{46}$, $u_{14}$, $u_{36}$, are all equal to $1$ (left). The system $(\ref{['id']})$ splits into the pair of equations, $u_{15}=1-u_{26}$ and $u_{35}=1-u_{24}$, which identifies $D_{25}$ with $\mathfrak{M}^{\,\delta}_{0,4}\times \mathfrak{M}^{\,\delta}_{0,4}$.
  • Figure 4: A partial decomposition $\alpha \in \chi^2_{8,\delta}$ of an octagon gives an isomorphism of $D_\alpha$ with $\mathfrak{M}^{\delta_1}_{0,5} \times \mathfrak{M}^{\delta_2}_{0,4} \times \mathfrak{M}^{\delta_3}_{0,3} = \mathfrak{M}^{\delta_1}_{0,5} \times \mathbb {A}^1\times\{\mathrm{pt}\}.$
  • Figure 5: The forgetful map $f_T$ contracts edges labelled $3,4,6,8$. The dihedral coordinates corresponding to the two chords in the square are pulled back by $f_T^*$ to $u_{15}u_{16}$ and $u_{27}u_{37}u_{47}u_{28}u_{38}u_{48}$.
  • ...and 10 more figures

Theorems & Definitions (209)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 199 more