Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

Anton Alekseev; Florian Naef

doi:10.1016/j.crma.2017.10.013

Lie algebras/Topology

Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection
[Connexion de Knizhnik–Zamolodchikov et formalité pour la bigèbre de Lie de Goldman–Turaev]

Présenté par : Michel Vergne

Anton Alekseev ¹ ; Florian Naef ¹

¹ Department of Mathematics, University of Geneva, 2-4, rue du Lièvre, C.P. 64, CH-1211 Geneva, Switzerland

Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1138-1147.

Résumé
Abstract

Soit Σ une variété orientable de dimension 2, de genre g et avec n composants de bord. L'espace $C π_{1} (Σ) / [C π_{1} (Σ), C π_{1} (Σ)]$ a une structure de bigèbre de Lie de Goldman–Turaev définie par les intersections et les autointersections des courbes sur Σ. La bigèbre de Lie graduée associée (par rapport à la filtration naturelle) est décrite par l'espace des mots cycliques en $H_{1} (Σ)$ . En genre zéro, l'isomorphisme entre ces deux bigèbres de Lie a été établi dans [13] en utilisant l'intégrale de Kontsevich, et dans [2] en utilisant les solutions du problème de Kashiwara–Vergne.

Dans cette note, nous donnons une démonstration élémentaire de cet isomorphisme sur $C$ . Notre démonstration utilise la connexion de Knizhnik–Zamolodchikov sur $C \ {z_{1}, \dots z_{n}}$ . Nous montrons que cet isomorphisme dépend naturellement de la structure complexe sur Σ. La preuve de l'isomorphisme pour le crochet de Lie est une version d'un résultat classique de Hitchin [9]. D'une manière surprenante, un argument similaire s'applique également au cocrochet.

De plus, nous montrons que l'isomorphisme de formalité construit dans cette note coïncide avec l'isomorphisme défini dans [2] si on choisit la solution du problème de Kashiwara–Vergne, qui correspond à l'associateur de Knizhnik–Zamolodchikov.

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space $C π_{1} (Σ) / [C π_{1} (Σ), C π_{1} (Σ)]$ carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in $H_{1} (Σ)$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.

In this note, we give an elementary proof of this isomorphism over $C$ . It uses the Knizhnik–Zamolodchikov connection on $C \ {z_{1}, \dots z_{n}}$ . We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.

Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.

Reçu le : 2017-10-03
Accepté le : 2017-10-26
Publié le : 2017-11-01

DOI : 10.1016/j.crma.2017.10.013

Affiliations des auteurs :

Anton Alekseev ¹ ; Florian Naef ¹

¹ Department of Mathematics, University of Geneva, 2-4, rue du Lièvre, C.P. 64, CH-1211 Geneva, Switzerland

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Anton Alekseev; Florian Naef. Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection. Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1138-1147. doi : 10.1016/j.crma.2017.10.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.013/

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