The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914943235784704 |
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| author | Bonahon, Francis Sözen, Yaşar Zeybek, Hat\.ıce |
| author_facet | Bonahon, Francis Sözen, Yaşar Zeybek, Hat\.ıce |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component Bonahon, Francis Sözen, Yaşar Zeybek, Hat\.ıce Geometric Topology 57S20, 20F34 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. |
| title | The symplectic structure of the $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component |
| topic | Geometric Topology 57S20, 20F34 |
| url | https://arxiv.org/abs/2409.04905 |