The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component

TL;DR

This paper explicitly computes the symplectic form on the $ ext{PGL}_n( ext{R})$-Hitchin component using global coordinates, revealing constant coefficients and deepening understanding of its geometric structure.

Contribution

It provides an explicit formula for the symplectic form on the Hitchin component in terms of global coordinates, with constant coefficients, enhancing geometric and algebraic understanding.

Findings

Explicit computation of the symplectic form

Coefficients of the form are constant

Deepens understanding of Hitchin component geometry

Abstract

The $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component of a closed oriented surface is a preferred component of the character variety consisting of homomorphisms from the fundamental group of the surface to the projective linear group $\mathrm{PGL}_n(\mathbb{R})$. It admits a symplectic structure, defined by the Atiyah-Bott-Goldman symplectic form. The main result of the article is an explicit computation of this symplectic form in terms of certain global coordinates for the Hitchin component. A remarkable feature of this expression is that its coefficients are constant.