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

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

2011-05-23

arxiv.org/abs/1105.4501v1

Physics

Mathematical Physics

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

Scientific paper

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 Teichm\"uller 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.

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