Flows on the PGL(V)-Hitchin component

TL;DR

This paper introduces new geometric flows on Hitchin components for PGL(V), including generalized twist and eruption flows, and constructs a global Darboux coordinate system to better understand their symplectic structure.

Contribution

It defines novel flows on Hitchin components, including eruption flows for pairs of pants, and establishes a global Darboux coordinate system for these moduli spaces.

Findings

Defined new flows on Hitchin components, including twist and eruption flows.

Constructed a global coordinate system that is a Darboux coordinate system.

Connected the flows to the Goldman symplectic form in a companion paper.

Abstract

In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. Using these flows, we construct a global coordinate system on the Hitchin component. In a companion paper to this article two of the authors develop new tools to compute the Goldman symplectic form on the Hitchin component, and prove that this global coordinate system is a Darboux coordinate system.