Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection

2017/08/10 by Alekseev, Anton, Naef, Florian · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1708.03119

Abstract

For an oriented 2-dimensional manifold Σ of genus g with n boundary components the space ℂπ₁(Σ)/[ℂπ₁(Σ), ℂπ₁(Σ)] 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 H₁(Σ) 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 ℂ. It uses the Knizhnik-Zamolodchikov connection on ℂ\backslash\ z₁, … 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.

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