# Seminar by Marta Mazzocco (Loughborough University)

Title: Shear coordinate description of the quantised versal unfolding of $D_4$ singularity

2013.11.01 | Christine Dilling

 Date Tue 01 Mar Time 16:15 — 17:15 Location Aud. D3

Abstract:

In this paper by using Teichmüller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a $D_4$ 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 $D_4$ cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on
$\oplus_{1}3\mathfrak{sl}^\ast(2,{\mathbb C})$. 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.

Seminar