Abstract
We recall the fat-graph description of Riemann surfaces Σg,s,n and the corresponding Teichmüller spaces \({\mathfrak{T}_{g,s,n}}\) with s > 0 holes and n > 0 bordered cusps in the hyperbolic geometry setting. If n > 0, we have a bijection between the set of Thurston shear coordinates and Penner’s λ-lengths. Then we can define, on the one hand, a Poisson bracket on λ-lengths that is induced by the Poisson bracket on shear coordinates introduced by V. V. Fock in 1997 and, on the other hand, a symplectic structure ΩWP on the set of extended shear coordinates that is induced by Penner’s symplectic structure on λ-lengths. We derive the symplectic structure ΩWP, which turns out to be similar to Kontsevich’s symplectic structure for ψ-classes in complex analytic geometry, and demonstrate that it is indeed inverse to Fock’s Poisson structure.
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Funding
The work was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00460.
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Dedicated to my teacher Andrei Alekseevich Slavnov on the occasion of his 80th birthday
This article was submitted by the author simultaneously in Russian and English
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Vol. 309, pp. 99–109.
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Chekhov, L.O. Symplectic Structures on Teichmüller Spaces \({\mathfrak{T}_{g,s,n}}\) and Cluster Algebras. Proc. Steklov Inst. Math. 309, 87–96 (2020). https://doi.org/10.1134/S0081543820030074
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DOI: https://doi.org/10.1134/S0081543820030074