Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection (Florian Naef, UniGe)

17.10.2017 15:30

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 talk we give an elementary proof of this isomorphism over C. It uses the Knizhnik-Zamolodchikov connection on C∖{z1,…zn}. 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.

Lieu

Bâtiment: Battelle

Séminaire "Groupes de Lie et espaces des modules"

entrée libre