Shear coordinate description of the quantised versal unfolding of D_4 singularity

TL;DR

This paper links Teichmuller theory, Poisson brackets, and quantisation to describe the versal unfolding of D_4 singularity using shear coordinates and geodesic functions.

Contribution

It introduces a shear coordinate framework for the D_4 singularity's versal unfolding and connects it with Poisson brackets and quantisation techniques.

Findings

Goldman bracket matches Etingof-Ginzburg Poisson bracket

Braid group action corresponds to Painleve VI analytic continuation

Explicit quantisation of the Goldman bracket achieved

Abstract

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 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 the direct sum of three copies of sl_2. 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.