The Betti side of the double shuffle theory: a survey

TL;DR

This survey reviews recent work on the double shuffle theory, demonstrating how associator relations imply double shuffle relations and elucidating the bitorsor structure using topological interpretations of moduli spaces.

Contribution

It provides an alternative proof linking associator and double shuffle relations and explicitly describes the bitorsor structure on Racinet's torsor.

Findings

Associator relations imply double shuffle relations.

Explicit description of the bitorsor structure.

Topological interpretation of harmonic coproduct.

Abstract

This is a survey of arXiv:1803.10151v4, arXiv:1807.07786v2 and arXiv:1908.00444v2 by H. Furusho and the author. The purpose of this series of papers is: (1) to give a proof that associator relations imply double shuffle relations, alternative to Furusho's paper arXiv:0808.0319v3; (2) to make explicit the bitorsor structure on Racinet's torsor of double shuffle relations. The main tool is the interpretation of the harmonic coproduct in terms of the topology of the moduli spaces $\mathfrak M_{0,4}$ and $\mathfrak M_{0,5}$, introduced in Deligne and Terasoma's 2005 preprint, and its extension to the Betti setup.