Symplectic coordinates on \PSL₃(\ℝ)-Hitchin\n components

2019/01/14 by Suhyoung Choi, Choi, Suhyoung, Hongtaek Jung +3
Mathematics · #57M50 (Primary) 53D30 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1901.04651

openalex publication_date 2019/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Goldman parametrizes the \PSL₃(\ℝ)-Hitchin component of a\nclosed oriented hyperbolic surface of genus g by 16g-16 parameters. Among\nthem, 10g-10 coordinates are canonical. We prove that the\n\PSL₃(\ℝ)-Hitchin component equipped with the\nAtiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system\nsuch that the half of its coordinates are canonical Goldman coordinates. To\nthis end, we show a version of the action-angle principle and the Zocca-type\ndecomposition formula for the symplectic form of H. Kim and\nGuruprasad-Huebschmann-Jeffrey-Weinstein given to symplectic leaves of the\nHitchin component.\n

Citations

• Moduli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. Goncharov

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