An explicit study of a family of cellular integrals

TL;DR

We give an explicit description of the simplest family of cellular integrals on moduli spaces $\mathfrak{M}_{0,l+3}$ in terms of multiple zeta values, answering a question of Brown. The work proves a weight dichotomy: even $l$ yields weight $l$ while odd $l$ yields weight $l-1$, with the odd case explained via an algebraic primitive and motivated by Grothendieck's period conjecture. The main results express $\xi_l$ through precise combinatorics: for even $l=2m$, $\xi_{2m}=\sum_{k_1+\dots+k_s=m} \gamma_{k_1,\dots,k_s}\psi_{2k_1}\cdots\psi_{2k_s}$, and for odd $l=2m+1$, $\xi_{2m+1}=\sum_{h=0}^m \xi_{2h}\xi_{2m-2h}$. The proof combines Panzer's symbolic integration algorithm for hyperlogarithms, a recurrences sequence $\beta^{(l+1)}_m$, and a detailed polar/primitive analysis of the associated forms, yielding computable expressions with potential applications to irrationality proofs of zeta values.

Abstract

We express a family of basic cellular integrals over moduli spaces of curves explicitly in terms of multiple zeta values, answering a question of Brown. Moreover, we study a priori the weights appearing in these integrals and find a relation that expresses the odd-dimensional integrals in terms of the even-dimensional ones. We also sketch an explanation of this relation in the spirit of Grothendieck's Period Conjecture.

Figures (1)

• Figure 1: Example of the computation of $\beta^{(7)}_5$.

Theorems & Definitions (46)

• Theorem 1
• Remark 2
• Theorem 3: De_Concini-Procesi-Wonderful_models_of_subspace_arrangements
• Remark 4
• Proposition 5
• Remark 6
• Lemma 7
• proof
• Lemma 8
• proof