The flat DHS connection 𝒥_DHS constructed in arXiv:2602.01461 is smooth on the configuration space of n points on a fixed compact Riemann surface Σ of arbitrary genus h, takes values in an infinite-dimensional Lie algebra 𝔱̂_h,n and is invariant under the modular group Sp(2h,ℤ). This paper is the first in a series for a program whose goal is to extend the connection 𝒥_DHS to a global flat connection on the Teichmüller space 𝒯_h,n valued in the Lie algebra of derivations of 𝔱̂_h,n. Upon the choice of local coordinates adapted to the map 𝒯_h,n→𝒯_h, such a connection splits into three pieces: 𝒥_DHS, a piece ℒ corresponding to holomorphic directions of 𝒯_h and a third piece corresponding to anti-holomorphic directions in 𝒯_h. In this paper, we isolate the system of equations satisfied by ℒ and obtain its solution locally and explicitly. The construction of a global extension of 𝒥_DHS to 𝒯_h,n and the extension of the meromorphic connection of arXiv:1112.0864 to 𝒯_h,n, are relegated to future publications in this series.

If 𝔲 and 𝔳 are Lie algebras, then the product Out(𝔲)×Out(𝔳) of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from 𝔲 to 𝔳; the stabilizer of the outer class of a given such morphism is then a subgroup of Out(𝔲)×Out(𝔳). We show that this leads to two related interpretation of the Grothendieck-Teichmüller group 𝖦𝖱𝖳_1(𝐤), where 𝔲,𝔳 are the Lie algebra of infinitesimal braids on the plane (resp. framed infinitesimal braids on the sphere) with 3 and 4 (resp. 4 and 5) strands: namely, it can be expressed as the joint intersection of the stabilizer groups of the outer classes of certain strand doubling morphisms ϕ and ψ with Out^*(𝔲)×Out(𝔳), where Out^*(𝔲) is a subgroup of Out(𝔲) of outer classes of inertia-preserving automorphisms of 𝔲.

In this work, we relate two recent constructions that generalize classical (genus-zero) polylogarithms to higher-genus Riemann surfaces. A flat connection valued in a freely generated Lie algebra on a punctured Riemann surface of arbitrary genus produces an infinite family of homotopy-invariant iterated integrals associated to all possible words in the alphabet of the Lie algebra generators. Each iterated integral associated to a word is a higher-genus polylogarithm. Different flat connections taking values in the same Lie algebra on a given Riemann surface may be related to one another by the composition of a gauge transformation and an automorphism of the Lie algebra, thus producing closely related families of polylogarithms. In this paper we provide two methods, which are inverses of one another, to explicitly relate in this way the meromorphic multiple-valued connection introduced by Enriquez in e-Print 1112.0864 and the non-meromorphic single-valued and modular-invariant connection introduced by D’Hoker, Hidding and Schlotterer, in e-Print 2306.08644.

Racinet's double shuffle Lie algebra 𝔡𝔪𝔯_0 is a Lie subalgebra of the Lie algebra 𝔱𝔡𝔢𝔯 of tangential derivations of the free Lie algebra with generators x_0,x_1, i.e. of derivations such that x_1↦ 0 and x_0↦ [a,x_0] for some element a. We prove: (1) 𝔡𝔪𝔯_0 is contained in the Lie subalgebra 𝔰𝔡𝔢𝔯 of 𝔱𝔡𝔢𝔯 of special derivations, i.e. satisfying the additional condition that x_∞↦ [b,x_∞] for some element b, where x_∞:=x_1-x_0; (2) 𝔡𝔪𝔯_0 is stable under the involution of 𝔰𝔡𝔢𝔯 induced by the exchange of x_0 and x_∞. The first statement: (a) says that any element of 𝔡𝔪𝔯_0 satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion 𝔡𝔪𝔯_0⊂𝔨𝔯𝔳_2 (which was proved by Schneps in 2012 only conditionally to the truth of (1)). We also derive the analogues of statements (a) and (b) respective to Racinet's “double shuffle schemes” 𝖣𝖬𝖱_μ(𝐤) and to the Betti double shuffle group 𝖣𝖬𝖱^B(𝐤) introduced in our earlier work.

In this paper, we introduce the notion of a bimodule with a factorization structure (BFS) and show that such a structure gives rise to an algebra morphism. We then prove that this framework offers an interpretation of the geometric construction underlying both the Betti and de Rham harmonic coproducts of the double shuffle theory developed in .

For $C$ a smooth affine complex curve, there is a unique minimal subalgebra $A_C$ of the algebra $\mathcal O_{hol}(\tilde C)$ of holomorphic functions on its universal cover $\tilde C$, which is stable under all the operations $f\mapsto \int f\omega$, for $\omega$ in the space $\Omega(C)$ of regular differentials on $C$. We identify $A_C$ with the image of the iterated integration map $I_{x_0} : \mathrm{Sh}(\Omega(C))\to\mathcal O_{hol}(\tilde C)$ based at any point $x_0$ of $\tilde C$ (here $\mathrm{Sh}(-)$ denotes the shuffle algebra of a vector space), as well as with the unipotent part, with respect to the action of $\mathrm{Aut}(\tilde C/C)$, of a subalgebra of $\mathcal O_{hol}(\tilde C)$ of moderate growth functions. We show that any regular Maurer-Cartan (MC) element $J$ on $C$ with values in the topologically free Lie algebra over $\mathrm H^1_{\mathrm{dR}}(C)^*$ gives rise to an isomorphism of $A_C$ with $\mathcal O(C) \otimes\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$, where $\mathcal O(C)$ is the algebra of regular functions on $C$, leading to the assignment of a subalgebra $\mathcal H_C(J)$ of $A_C$ (isomorphic to $\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$) to any MC element. We also associate a MC element $J_\sigma$ to each section $\sigma$ of the projection $\Omega(C)\to \mathrm H^1_{\mathrm{dR}}(C)$; when $C$ has genus $0$, we exhibit a particular section $\sigma_0$ for which $\mathcal H_C(J_{\sigma_0})$ is the algebra of hyperlogarithm functions (Poincar\'e, Lappo-Danilevsky).

In earlier work, we constructed a pair of ”Betti” and ”de Rham” Hopf algebras and a pair of module-coalgebras over this pair, as well as the bitorsors related to both structures (which will be called the ”module” and ”algebra” stabilizer bitorsors). We showed that Racinet’s torsor constructed out of the double shuffle and regularization relations between multiple zeta values is essentially equal to the ”module” stabilizer bitorsor, and that the latter is contained in the ”algebra” stabilizer bitorsor. In this paper, we show the equality of the ”algebra” and ”module” stabilizer bitorsors. We reduce the proof to showing the equality of the associated ”algebra” and ”module” graded Lie algebras. The argument for showing this equality involves the relation of the ”algebra” Lie algebra with the kernel of a linear map, the expression of this linear map as a composition of three linear maps, the relation of one of them with the ”module” Lie algebra and the computation of the kernel of the other one by discrete topology arguments.

The category of topological spaces endowed with two marked points is equipped with two families $\mathbf F_n$ and $\mathbf H_n$ of functors to the category of abelian groups, indexed by a nonnegative integer $n$: namely, the functor $\mathbf F_n$ takes the object $(X,x,y)$ to the quotient of $\mathbb Z\pi_1(X,x,y)$ by an abelian subgroup associated with the $n+1$-st power of the augmentation ideal of the group algebra $\mathbb Z\pi_1(X,x)$, and the functor $\mathbf H_n$ takes the same object to the $n$-th singular homology group of $X^n$ relative to a subspace defined in terms of partial diagonals. We construct a family of natural transformations $\nu_n : \mathbf F_n\to \mathbf H_n$. We identify the natural transformation obtained by restricting $\nu_n$ to the subcategory of algebraic varieties with a natural equivalence due to Beilinson.

Let ℰ be a complex elliptic curve and S be a non-empty finite subset of ℰ. We show that the functions introduced in arXiv:1712.07089 out of string theory motivations give rise to a basis of the minimal algebra A_ℰ∖ S of holomorphic multivalued functions on ℰ∖ S which is stable under integration, introduced in arXiv:2212.03119; this basis is alternative to the basis of A_ℰ∖ S constructed in loc. cit. using elliptic analogues of the hyperlogarithm functions.

In the first two parts of the series, we constructed stabilizer subtorsors of a ‘twisted Magnus’ torsor, studied their relations with the associator and double shuffle torsors, and explained their ‘de Rham’ nature. In this paper, we make the associated bitorsor structures explicit and explain the ‘Betti’ nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro- p versions of the ‘Betti’ group of the double shuffle bitorsor.

For $C$ a complex curve and $n \geq 1$, a pair $(\mathcal{P},\nabla_\mathcal{P})$ of a principal bundle $\mathcal{P}$ with meromorphic flat connection over $C^n$, holomorphic over the configuration space $C_n(C)$ of $n$ points over $C$, was introduced in arXiv:1112.0864. For any point $\infty \in C$, we construct a trivialisation of the restriction of $\mathcal{P}$ to $(C\setminus\infty)^n$ and obtain a Maurer-Cartan element $J$ over $C_n(C\setminus\infty)$ out of $\nabla_\mathcal{P}$, thus generalising a construction of Levin and Racinet when the genus of $C$ is higher than one. We give explicit formulas for $J$ as well as for $\nabla_\mathcal{P}$. When $n=1$, this construction gives rise to elements of Hain's space of second kind iterated integrals over $C$.

We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.

We recall the cohomological interpretation of the unipotent quotients of the fundamental groupoid of an algebraic complex variety (Beilinson, Deligne-Goncharov). We then give a construction of the resutting transition morphisms in terms of singular homology.

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.

Apres un rappel de l'interpretation cohomologique des quotients unipotents du groupoide fondamental d'une variete algebrique sur un sous-corps de C (Beilinson, Deligne-Goncharov), nous proposons une construction homologique des morphismes de transition.

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 𝔐_0,4 and 𝔐_0,5, introduced in Deligne and Terasoma's 2005 preprint, and its extension to the Betti setup.

Let n ≥ 1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E # of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.

According to Racinet's work, the scheme of double shuffle and regularization relations between cyclotomic analogues of multiple zeta values has the structure of a torsor over a pro-unipotent $\mathbb Q$-algebraic group $\sf{DMR}_0$, which is an algebraic subgroup of a pro-unipotent $\mathbb Q$-algebraic group of outer automorphisms of a free Lie algebra. We show that the harmonic (stuffle) coproduct of double shuffle theory may be viewed as an element of a module over the above group, and that $\sf{DMR}_0$ identifies with the stabilizer of this element. We identify the tangent space at origin of $\sf{DMR}_0$ with the stabilizer Lie algebra of the harmonic coproduct, thereby obtaining an alternative proof of Racinet's result stating that this space is a Lie algebra (the double shuffle Lie algebra).

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.