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

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.