Symplectic structures on Teichm\"uller spaces $\mathfrak T_{g,s,n}$ and cluster algebras

TL;DR

This paper explores the symplectic and Poisson structures on Teichmüller spaces of bordered Riemann surfaces, establishing connections with cluster algebras and Penner's lambda-lengths, and deriving a symplectic form analogous to Kontsevich's structure.

Contribution

It introduces a symplectic structure on Teichmüller spaces related to cluster algebras, linking shear coordinates, lambda-lengths, and Poisson brackets, and demonstrates its inverse relation to the Fock Poisson structure.

Findings

Derived the symplectic form 5WP on Teichmfcller spaces.

Showed 5WP is inverse to the Fock Poisson structure.

Connected 5WP to Kontsevich symplectic structure for 5-classes.

Abstract

We recall the fat-graph description of Riemann surfaces $\Sigma_{g,s,n}$ and the corresponding Teichm\"uller 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 and we can induce, on the one hand, the Poisson bracket on $\lambda$-lengths from the Poisson bracket on shear coordinates introduced by V.V.Fock in 1997 and, on the other hand, a symplectic structure $\Omega_{\text{WP}}$ on the set of extended shear coordinates from Penner's symplectic structure on $\lambda$-lengths. We derive $\Omega_{\text{WP}}$, which turns out to be similar to the Kontsevich symplectic structure for $\psi$-classes in complex-analytic geometry, and demonstrate that it is indeed inverse to the Fock Poisson structure.