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29 March 2024
 
  » arxiv » 1708.3119

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Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection
Anton Alekseev ; Florian Naef ;
Date 10 Aug 2017
AbstractFor an oriented 2-dimensional manifold $Sigma$ of genus $g$ with $n$ boundary components the space $mathbb{C}pi_1(Sigma)/[mathbb{C}pi_1(Sigma), mathbb{C}pi_1(Sigma)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtration) is described by cyclic words in $H_1(Sigma)$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [G. Massuyeau, Formal descriptions of Turaev’s loop operations] using Kontsevich integrals and in [A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef, The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem] using solutions of the Kashiwara-Vergne problem.
In this note we give an elementary proof of this isomorphism over $mathbb{C}$. It uses the Knizhnik-Zamolodchikov connection on $mathbb{C}ackslash{ z_1, dots z_n}$. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin. Surprisingly, it turns out that a similar proof applies to cobrackets.
Source arXiv, 1708.3119
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