The Betti side of the double shuffle theory. I. The harmonic coproducts

TL;DR

This paper explores the algebraic and geometric structures underlying multiple zeta values, introducing Betti and de Rham formalisms, and demonstrating how associators connect these frameworks through harmonic coproducts.

Contribution

It introduces a Betti counterpart to Racinet's harmonic coproduct formalism and shows how associators relate Betti and de Rham structures in the context of multiple zeta values.

Findings

Introduces a Betti formalism as a counterpart to de Rham formalism.

Shows that associators relate Betti and de Rham algebra coproducts.

Provides geometric interpretations of the algebraic structures involved.

Abstract

This paper is the first in a series which aims at: (a) giving a proof that the associator relations between multizeta values imply the double shuffle and regularization (DSR) ones, alternative to that of the second-named author's 2010 paper; (b) enhancing Racinet's construction of a torsor structure over the Q-scheme of DSR relations to an explicit bitorsor structure. In this paper, we revisit Racinet's original DSR formalism, whose main character is an algebra coproduct, called the harmonic coproduct, and we introduce a variant which is a module coproduct; we explain the `de Rham' nature of this formalism and construct a `Betti' counterpart of it; we show how both formalisms can be interpreted in terms of geometry, following the ideas of Deligne and Terasoma's unfinished 2005 preprint; we use Bar-Natan's interpretation of associators as functors from the category of parenthesized braids to that of chord diagrams to show that any associator relates the Betti and de Rham geometric objects, both in the `algebraic' and in the `module' setups; we derive that any associator relates the Betti and de Rham algebra coproducts, as well as their module counterparts. These results will be used in the next parts of the series.