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 Trudy Mat. Inst. Steklova, 2020, Volume 309, Pages 99–109 (Mi tm4082)

Symplectic Structures on Teichmüller Spaces $\mathfrak T_{g,s,n}$ and Cluster Algebras

Leonid O. Chekhovab

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Michigan State University, 426 Auditorium Rd., East Lansing, MI 48824, USA

Abstract: We recall the fat-graph description of Riemann surfaces $\Sigma _{g,s,n}$ 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 $\lambda$-lengths. Then we can define, on the one hand, a Poisson bracket on $\lambda$‑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 $\Omega_\mathrm{WP}$ on the set of extended shear coordinates that is induced by Penner's symplectic structure on $\lambda$-lengths. We derive the symplectic structure $\Omega_\mathrm{WP}$, which turns out to be similar to Kontsevich's symplectic structure for $\psi$-classes in complex analytic geometry, and demonstrate that it is indeed inverse to Fock's Poisson structure.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00460 The work was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00460.

DOI: https://doi.org/10.4213/tm4082

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English version:
Proceedings of the Steklov Institute of Mathematics, 2020, 309, 87–96

Bibliographic databases:

UDC: 514.7+512.548
Revised: December 9, 2019
Accepted: February 11, 2020

Citation: Leonid O. Chekhov, “Symplectic Structures on Teichmüller Spaces $\mathfrak T_{g,s,n}$ and Cluster Algebras”, Modern problems of mathematical and theoretical physics, Collected papers. On the occasion of the 80th birthday of Academician Andrei Alekseevich Slavnov, Trudy Mat. Inst. Steklova, 309, Steklov Math. Inst. RAS, Moscow, 2020, 99–109; Proc. Steklov Inst. Math., 309 (2020), 87–96

Citation in format AMSBIB
\Bibitem{Che20} \by Leonid~O.~Chekhov \paper Symplectic Structures on Teichm\"uller Spaces $\mathfrak T_{g,s,n}$ and Cluster Algebras \inbook Modern problems of mathematical and theoretical physics \bookinfo Collected papers. On the occasion of the 80th birthday of Academician Andrei Alekseevich Slavnov \serial Trudy Mat. Inst. Steklova \yr 2020 \vol 309 \pages 99--109 \publ Steklov Math. Inst. RAS \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm4082} \crossref{https://doi.org/10.4213/tm4082} \elib{https://elibrary.ru/item.asp?id=45368001} \transl \jour Proc. Steklov Inst. Math. \yr 2020 \vol 309 \pages 87--96 \crossref{https://doi.org/10.1134/S0081543820030074} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000557522500007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85089224548} 

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• https://doi.org/10.4213/tm4082
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This publication is cited in the following articles:
1. L. O. Chekhov, “Fenchel–Nielsen coordinates and Goldman brackets”, Russian Math. Surveys, 75:5 (2020), 929–964
2. M. Bertola, D. A. Korotkin, “WKB expansion for a Yang–Yang generating function and the Bergman tau function”, Theoret. and Math. Phys., 206:3 (2021), 258–295
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