Symplectic coordinates on $\mathrm{PSL}_3(\mathbb{R})$-Hitchin components
Abstract
Goldman parametrizes the -Hitchin component of a closed oriented hyperbolic surface of genus by parameters. Among them, coordinates are canonical. We prove that the -Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we show a version of the action-angle principle and the Zocca-type decomposition formula for the symplectic form of H. Kim and Guruprasad-Huebschmann-Jeffrey-Weinstein given to symplectic leaves of the Hitchin component.
Cite
@article{arxiv.1901.04651,
title = {Symplectic coordinates on $\mathrm{PSL}_3(\mathbb{R})$-Hitchin components},
author = {Suhyoung Choi and Hongtaek Jung and Hong Chan Kim},
journal= {arXiv preprint arXiv:1901.04651},
year = {2019}
}
Comments
40 pages, 2 figures; correct typos and revise introduction; section 4.5 is largely revised; correct sign mistake in Lemma 5.2.3 and Corollary 5.2.1; correct Goldman's (s,t) coordinates