Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

Symplectic coordinates on $\operatorname{PSL}_3(\mathbb{R})$-Hitchin components

Pages: 1321 – 1386

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a1

Authors

Suhyoung Choi (Department of Mathematical Sciences, KAIST, Daejeon, Korea)

Hongtaek Jung (Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea)

Hong Chan Kim (Department of Mathematics Education, Korea University, Seoul, South Korea)

Abstract

Goldman parametrizes the $\operatorname{PSL}_3(\mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g - 16$ parameters. Among them, $10g - 10$ coordinates are canonical. We prove that the $\operatorname{PSL}_3(\mathbb{R})$-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.

Keywords

Hitchin component, Goldman coordinates, Darboux coordinates

2010 Mathematics Subject Classification

Primary 57M50. Secondary 53D30.

Received 31 January 2020

Accepted 15 March 2020

Published 17 February 2021