The Goldman symplectic form on the PGL(V)-Hitchin component

TL;DR

This paper establishes a symplectic trivialization and proves that a specific coordinate system on the PGL(V)-Hitchin component is a global Darboux coordinate, advancing understanding of its geometric structure.

Contribution

It introduces a symplectic trivialization based on ideal triangulations and bridge systems, and proves the global Darboux coordinate property for the PGL(V)-Hitchin component.

Findings

Symplectic trivialization of the tangent bundle using ideal triangulations.

Identification of a large class of Hamiltonian vector fields.

Confirmation that the coordinate system is a global Darboux coordinate.

Abstract

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-Hitchin component of S. Using this, we prove that a large class of vector fields defined in the companion paper [SWZ20] are Hamiltonian. This is then used to prove that the explicit global coordinate system defined in the companion paper [SWZ20] is a global Darboux coordinate system for the PGL(V)-Hitchin component.