The Poisson Bracket of Length functions in the Hitchin Component

Abstract

Wolpert's cosine formula on Teichmüller 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.