The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component

Francis Bonahon; Yaşar Sözen; Hat\.ıce Zeybek

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arXiv · 2409.04905

The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component

Francis Bonahon·Yaşar Sözen·Hat\.ıce Zeybek

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Abstract

The $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component of a closed oriented surface is a preferred component of the character variety consisting of homomorphisms from the fundamental group of the surface to the projective linear group $\mathrm{PGL}_n(\mathbb{R})$. It admits a symplectic structure, defined by the Atiyah-Bott-Goldman symplectic form. The main result of the article is an explicit computation of this symplectic form in terms of certain global coordinates for the Hitchin component. A remarkable feature of this expression is that its coefficients are constant.

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BibTeX RIS

Francis Bonahon, Yaşar Sözen, Hat\.ıce Zeybek. 2024-09-07. The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component. https://arxiv.org/abs/2409.04905

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