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

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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 \({\mathbb {Q}}\)-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.

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Notes

  1. We set \(\mathrm {i}:=\sqrt{-1}\).

  2. An affine \({\mathbb {Q}}\)-scheme (resp., \({\mathbb {Q}}\)-group scheme) \({\mathsf {X}}\) can be identified with a representable functor from the category of \({\mathbb {Q}}\)-algebras to that of sets (resp. groups), denoted \({\mathbf {k}}\mapsto {\mathsf {X}}({\mathbf {k}})\), for \({\mathbf {k}}\) running over \({\mathbb {Q}}\)-algebras.

  3. \(\mathsf {DMR}\) stands for the French ‘double mélange et régularisation’.

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Acknowledgements

The collaboration of both authors has been supported by Grants JSPS KAKENHI JP15KK0159 and JP18H01110 as well as HighAGT ANR-20-CE40-0016.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Benjamin Enriquez

  2. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Hidekazu Furusho

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  1. Benjamin Enriquez
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  2. Hidekazu Furusho
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Correspondence to Benjamin Enriquez.

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Enriquez, B., Furusho, H. The Betti side of the double shuffle theory. I. The harmonic coproducts. Sel. Math. New Ser. 27, 79 (2021). https://doi.org/10.1007/s00029-021-00659-w

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