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

Abstract

Goldman parametrizes the ${\mathrm{PSL}}_3(\mathbb{R})$-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 ${\mathrm{PSL}}_3(\mathbb{R})$-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.

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

References in corpus (1)

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

Cited by in corpus (1)

• Volume of the moduli space of unmarked bounded positive convex ${\mathbb{RP}}^2$ structures