Applications of the Liouville symplectic form on the cotangent bundle of a loop group
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913988891115520 |
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| author | Biswas, Indranil Inaba, Michi-aki Komyo, Arata Mukhopadhyay, Swarnava Saito, Masa-Hiko |
| author_facet | Biswas, Indranil Inaba, Michi-aki Komyo, Arata Mukhopadhyay, Swarnava Saito, Masa-Hiko |
| contents | Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^*G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures -- on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing -- were constructed earlier by various authors. Our results show that they all have a common origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Applications of the Liouville symplectic form on the cotangent bundle of a loop group Biswas, Indranil Inaba, Michi-aki Komyo, Arata Mukhopadhyay, Swarnava Saito, Masa-Hiko Algebraic Geometry Differential Geometry Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^*G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures -- on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing -- were constructed earlier by various authors. Our results show that they all have a common origin. |
| title | Applications of the Liouville symplectic form on the cotangent bundle of a loop group |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2508.09714 |