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

Abstract

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V )-Hitchin component. We show that any ideal triangulation on a closed connected surface of genus at least 2, and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the Hitchin component. Using this, we prove that a large class of flows defined in the companion paper [SWZ17] are Hamiltonian. We also construct an explicit collection of Hamiltonian vector fields on the Hitchin component that give a symplectic basis at every point. These are used to show that the global coordinate system on the Hitchin component defined iin the companion paper is a global Darboux coordinate system.