Extended Goldman symplectic structure in Fock-Goncharov coordinates

TL;DR

This paper expresses the extended Goldman symplectic structure on SL(n) character varieties of punctured Riemann surfaces using Fock-Goncharov coordinates, revealing its algebraic and geometric properties and connections to graph transformations.

Contribution

It provides a new formulation of the Goldman symplectic structure in Fock-Goncharov coordinates, linking it to the Cartan matrix and Rogers' dilogarithm, and explores its relation to graph-based Poisson structures.

Findings

Symplectic form has integer coefficients via inverse Cartan matrix.

Established connection between Goldman and Fock-Goncharov Poisson structures.

Identified Rogers' dilogarithm as a generating function for symplectomorphisms.

Abstract

The goal of this paper is to express the extended Goldman symplectic structure on the $SL(n)$ character variety of a punctured Riemann surface in terms of Fock-Goncharov coordinates. The associated symplectic form has integer coefficients expressed via the inverse of the Cartan matrix. The main technical tool is a canonical two-form associated to a flat graph connection. We discuss the relationship between the extension of the Goldman Poisson structure and the Poisson structure defined by Fock and Goncharov. We elucidate the role of the Rogers' dilogarithm as generating function of the symplectomorphism defined by a graph transformation.