Symplectic Reduction in Infinite Dimensions
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912019170459648 |
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| author | Diez, Tobias Rudolph, Gerd |
| author_facet | Diez, Tobias Rudolph, Gerd |
| contents | This paper develops a theory of symplectic reduction in the infinite-dimensional setting, covering both the regular and singular case. Extending the classical work of Marsden, Weinstein, Sjamaar and Lerman, we address challenges unique to infinite dimensions, such as the failure of the Darboux theorem and the absence of the Marle-Guillemin-Sternberg normal form. Our novel approach centers on a normal form of only the momentum map, for which we utilize new local normal form theorems for smooth equivariant maps in the infinite-dimensional setting. This normal form is then used to formulate the theory of singular symplectic reduction in infinite dimensions. We apply our results to important examples like the Yang-Mills equation and the Teichmüller space over a Riemann surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symplectic Reduction in Infinite Dimensions Diez, Tobias Rudolph, Gerd Differential Geometry Mathematical Physics Symplectic Geometry 53D20, (58B25, 58D27, 53C55, 70H45, 58E50) This paper develops a theory of symplectic reduction in the infinite-dimensional setting, covering both the regular and singular case. Extending the classical work of Marsden, Weinstein, Sjamaar and Lerman, we address challenges unique to infinite dimensions, such as the failure of the Darboux theorem and the absence of the Marle-Guillemin-Sternberg normal form. Our novel approach centers on a normal form of only the momentum map, for which we utilize new local normal form theorems for smooth equivariant maps in the infinite-dimensional setting. This normal form is then used to formulate the theory of singular symplectic reduction in infinite dimensions. We apply our results to important examples like the Yang-Mills equation and the Teichmüller space over a Riemann surface. |
| title | Symplectic Reduction in Infinite Dimensions |
| topic | Differential Geometry Mathematical Physics Symplectic Geometry 53D20, (58B25, 58D27, 53C55, 70H45, 58E50) |
| url | https://arxiv.org/abs/2409.05829 |