Double shuffle torsor of cyclotomic multiple zeta values and stabilizers of de Rham and Betti coproducts

Racinet described the double shuffle and regularization relations between multiple polylogarithm values at \(N\)th roots of unity via a \(\mathbb{Q}\)-scheme \(\mathsf{DMR}^{\iota}\) where \(\iota : G\hookrightarrow \mathbb{C}^{\times}\) is a group embedding from a finite cyclic group \(G\) of order \(N\) to \(\mathbb{C}^{\times}\). Then, Enriquez and Furusho proved, when \(N=1\), that a subscheme \(\mathsf{DMR}^{\iota}_{\times}\) is a torsor of isomorphisms between Betti and de Rham objects. In this talk, we establish a cyclotomic generalization of this result (\(N\geq 1\)). First, we explicit the torsor structure of \(\mathsf{DMR}^{\iota}_{\times}\). Then, we introduce in this context the adequate de Rham and Betti objects: the former arise from a crossed product algebra and enables a reformulation of Racinet's harmonic coproduct closer to the formalism introduced by Enriquez and Furusho; the latter, on the other hand, arise from a group algebra of the orbifold fundamental group \(\left(\mathbb{C}^{\times} \smallsetminus \mu_{N}\right) / \mu_{N}\), where \(\mu_{N}\) is the group of \(N\)th roots of unity. Finally, we show the existence of a coalgebra and Hopf algebra coproduct such that \(\mathsf{DMR}^{\iota}_{\times}\) is a torsor of isomorphisms relying these Betti coproducts to their de Rham counterparts.