The Poisson Bracket of Length functions in the Hitchin Component

TL;DR

This paper provides a concise proof that generalizes Wolpert's cosine formula for the Poisson bracket of length functions from Teichmüller space to Hitchin components, using Goldman's formula for surface group representations.

Contribution

It offers a new, streamlined proof of the Poisson bracket formula in Hitchin components, extending Wolpert's classical result to a broader geometric setting.

Findings

Generalization of Wolpert's cosine formula to Hitchin representations

Application of Goldman's formula to prove the generalization

Simplified proof technique for Poisson brackets in higher Teichmüller theory

Abstract

Wolpert's cosine formula on Teichm\"uller space gives the Weil-Petersson Poisson bracket $\{l_\alpha, l_\beta\}$ for geodesic length functions $l_\alpha,l_\beta$ of closed curves $\alpha,\beta$ as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie. In this paper, we give a short proof of this generalization using Goldman's formula for the Poisson bracket on representation varieties of surface groups into reductive Lie groups.