2010/07/22 by Leonid Chekhov, Marta Mazzocco · 8 citations
Mathematics · Physics and Astronomy · #Action (physics) #Affine transformation #Bracket #Braid group #Geodesic #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Poisson bracket #Singularity #math-ph #math.AG #math.DG #math.MP #msc:17B37 #msc:17B63 #msc:32G15 #msc:32G34 #msc:32G81
paper · pdf · doi:10.1088/1751-8113/43/44/442002
published in Journal of Physics A Mathematical and Theoretical 43(44), 442002 (Institute of Physics) · 14 pages, 2 pictures
arxiv created 2010/07/22 · openalex publication_date 2010/10/13 · arxiv updated 2015/05/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl2. We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlev'e equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.