An explicit study of a family of cellular integrals

TL;DR

This paper explicitly expresses a family of cellular integrals over moduli spaces in terms of multiple zeta values, explores relations between odd and even weights, and discusses implications related to Grothendieck's Period Conjecture.

Contribution

It provides explicit formulas for cellular integrals in terms of multiple zeta values and investigates their weight relations, addressing a question posed by Brown.

Findings

Explicit expressions for cellular integrals in terms of multiple zeta values

A relation connecting odd and even-dimensional integrals

A conceptual explanation aligned with Grothendieck's Period Conjecture

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.