Teichmüller spaces as degenerated symplectic leaves in Dubrovin--Ugaglia Poisson manifolds

Abstract

In this paper we study the Goldman bracket between geodesic length functions both on a Riemann surface ${\Sigma }_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere ${\Sigma }_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichmüller spaces ${\mathcal{T}}_{g,s,0}$ and ${\mathcal{T}}_{0,1,n}$ are realised as real slices of degenerated symplectic leaves in the Dubrovin--Ugaglia Poisson algebra of upper--triangular matrices $S$ with 1 on the diagonal.

27 pages, 7 figures, contribution to special issue of Physica D on Boris Dubrovin 60th birthday

References in corpus (3)

• On Quantizing Teichmüller and Thurston theories
• Teichmüller Theory of Bordered Surfaces
• Solving the Sixth Painleve' Equation: Towards the Classification of all the Critical Behaviours and the Connection Formulae (October 2010)

Cited by in corpus (1)

• Surfaces, braids, Stokes matrices, and points on spheres