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The Poisson Bracket of Length functions in the Hitchin Component (1502.05975v1)
Published 20 Feb 2015 in math.GT
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.