Multiplicative convolution and double shuffle relations

TL;DR

This paper presents a geometric, topological approach to understanding the regularized double shuffle relations for multiple zeta values, linking them to the homological pentagon equation.

Contribution

It introduces semi-holonomy isomorphisms and proves their compatibility with convolution is equivalent to the double shuffle relations, providing a new geometric proof.

Findings

Compatibility of semi-holonomy isomorphisms with convolution is equivalent to the pentagon equation.

The geometric approach avoids Hodge-theoretic and Tannakian methods.

The pentagon equation implies the regularized double shuffle relations.

Abstract

We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods.