New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces
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| Format: | Preprint |
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2025
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| _version_ | 1866917430022897664 |
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| author | Bertola, Marco Korotkin, Dmitry Pillet, Jordi |
| author_facet | Bertola, Marco Korotkin, Dmitry Pillet, Jordi |
| contents | We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_06830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces Bertola, Marco Korotkin, Dmitry Pillet, Jordi Symplectic Geometry High Energy Physics - Theory We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones. |
| title | New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces |
| topic | Symplectic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2505.06830 |