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

Leonid O. Chekhov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematical and theoretical physics, Tome 309 (2020), pp. 99-109

Résumé

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.