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Benjamin Enriquez

Publications and source records attributed to Benjamin Enriquez.

At least 19 recordsLinked to original sources
arXiv2026

Flat connections on moduli spaces I: Local (1,0)-extension of the DHS connection

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

hep-th↗
arXiv2026

A stabilizer interpretation of the Grothendieck-Teichm\"uller group $\mathsf{GRT}_1(\mathbf k)$

If $\mathfrak u$ and $\mathfrak v$ are Lie algebras, then the product $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$ of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from $\mathfrak u$ to $\mathfrak v$; the stabilizer of the outer class of a given such morphism is then a subgroup of $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$. We show that this leads to two related interpretation of the Grothendieck-Teichm\"uller group $\mathsf{GRT}_1(\mathbf k)$, where $\mathfrak u,\mathfrak v$ 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 $\phi$ and $\psi$ with $\mathrm{Out}^*(\mathfrak u)\times\mathrm{Out}(\mathfrak v)$, where $\mathrm{Out}^*(\mathfrak u)$ is a subgroup of $\mathrm{Out}(\mathfrak u)$ of outer classes of inertia-preserving automorphisms of $\mathfrak u$.

math.QA↗
arXiv2025

On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations

In a previous paper, the second author introduced a Betti counterpart of $N$-cyclotomic double shuffle theory for any $N \geq 1$. The construction is based on the group algebra of the free group $F_2$, endowed with a filtration relative to a morphism $F_2 \to \mu_N$ (where $\mu_N$ is the group of $N$-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ on the relative completion of a specific subalgebra $\mathcal{W}^\mathrm{B}$ of the group algebra of $F_2$. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct $\Delta^{\mathcal{W}, \mathrm{B}}$ defined in \cite{EF1} for the classical case ($N=1$) by the first author and Furusho remains compatible with the filtration structure on $\mathcal{W}^\mathrm{B}$ induced by the relative completion for arbitrary $N$. This compatibility suggests that the completion corresponding to $\Delta^{\mathcal{W}, \mathrm{B}}$ is a candidate for an explicit realization of $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$.

math.AT↗
arXiv2025

A categorical formulation of the Deligne-Terasoma approach to double shuffle theory

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 Enriquez-Furusho inspired by an unpublished preprint of Deligne-Terasoma.

math.NT↗
arXiv2025

Double shuffle Lie algebra and special derivations

Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ is a Lie subalgebra of the Lie algebra $\mathfrak{tder}$ of tangential derivations of the free Lie algebra with generators $x_0,x_1$, i.e. of derivations such that $x_1\mapsto 0$ and $x_0\mapsto [a,x_0]$ for some element $a$. We prove: (1) $\mathfrak{dmr}_0$ is contained in the Lie subalgebra $\mathfrak{sder}$ of $\mathfrak{tder}$ of special derivations, i.e. satisfying the additional condition that $x_\infty\mapsto [b,x_\infty]$ for some element $b$, where $x_\infty:=x_1-x_0$; (2) $\mathfrak{dmr}_0$ is stable under the involution of $\mathfrak{sder}$ induced by the exchange of $x_0$ and $x_\infty$. The first statement: (a) says that any element of $\mathfrak{dmr}_0$ satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion $\mathfrak{dmr}_0\subset \mathfrak{krv}_2$ (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'' $\mathsf{DMR}_\mu(\mathbf k)$ and to the Betti double shuffle group $\mathsf{DMR}^{\mathrm{B}}(\mathbf k)$ introduced in our earlier work.

math.AG↗
arXiv2025

Relating flat connections and polylogarithms on higher genus Riemann surfaces

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 to explicitly construct this correspondence between 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.

hep-th↗
arXiv2024

A survey on the Le-Murakami-Ohtsuki invariant for closed 3-manifolds

We review the original approach to the Le-Murakami-Ohtsuki (LMO) invariant of closed 3-manifolds (as opposed to the later approach based on the Aarhus integral). Following the ideas of surgery presentation, we introduce a class of combinatorial structures, called Kirby structures, which we prove to yield multiplicative 3-manifold invariants. We illustrate this with the Reshetikhin-Turaev invariants. We then introduce a class of combinatorial structures, called pre-LMO structures, and prove that they give rise to Kirby structures. We show how the Kontsevich integral can be used to construct a pre-LMO structure. This yields two families of multiplicative 3-manifolds invariants $\{\Omega_n^{\mathfrak{c}}\}_{n\geq 1}$ and $\{\Omega_n^{\mathfrak{d}}\}_{n\geq 1}$. We review the elimination of redundant information in the latter family, leading to the construction of the LMO invariant. We also provide uniqueness results of some aspects of the LMO construction. The family of invariants $\{\Omega_n^{\mathfrak{c}}\}_{n\geq 1}$ is not discussed explicitly in the literature; whereas $\Omega_n^{\mathfrak{c}}$ enables one to recover $\Omega_n^{\mathfrak{d}}$ for any $n \geq 1$, we show that these invariants coincide for $n = 1$.

math.GT↗
arXiv2023

The fundamental group of surfaces parametrizing cuboids

We prove that an irreducible projective complete intersection of dimension at least two with isolated singularities has trivial fundamental group. As an application, the surface $\Upsilon$ parametrizing cuboids and its minimal resolution of singularities are simply connected. By an independent argument we also show that the surface $V$ parametrizing face cuboids and its resolution are simply connected as well. We then introduce two smooth open subvarieties $S_{1}$ and $S_{2}$ of the surface parametrizing face cuboids, show that each has fundamental group isomorphic to $\mathbb{F}_{3}\ltimes \mathbb{Z}^{2}$, and prove that their Malcev completions reduce to the free pro-unipotent group on three generators. In an appendix we treat the corresponding real loci, whose fundamental groups, in contrast, are far from trivial.

math.AG↗
arXiv2023

Elliptic hyperlogarithms

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

math.AG↗
arXiv2023

Natural transformations relating homotopy and singular homology functors

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.

math.AT↗
arXiv2022

Analogues of hyperlogarithm functions on affine complex curves

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).

math.AG↗
arXiv2022

The stabilizer bitorsors of the module and algebra harmonic coproducts are equal

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.

math.AG↗
arXiv2021

Construction of Maurer-Cartan elements over configuration spaces of curves

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 $C$.

math.AG↗
arXiv2020

The Betti side of the double shuffle theory: a survey

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.

math.AG↗
arXiv2019

The Betti side of the double shuffle theory. III. Bitorsor structures

In the two first 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.

math.AG↗
arXiv2018

The Betti side of the double shuffle theory. II. Double shuffle relations for associators

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.

math.AG↗
arXiv2018

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

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.

math.AG↗