Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913371718156288 |
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| author | Diez, Tobias Futaki, Akito Ratiu, Tudor |
| author_facet | Diez, Tobias Futaki, Akito Ratiu, Tudor |
| contents | We pioneer the development of a rigorous infinite-dimensional framework for the Kempf-Ness theorem, addressing the significant challenge posed by the absence of a complexification for the symmetry group in infinite dimensions, e.g, the diffeomorphism group. We propose a novel approach, based on Cartan bundles, to generalize Kempf-Ness theory to infinite dimensions, invoking the fundamental role played by the Maurer-Cartan form. This approach allows us to define and study objects essential for the Kempf-Ness theorem, such as the complex model for orbits and the Kempf-Ness function, as well as establishing its convexity properties and defining a generalized Futaki character. We show how our framework can be applied to the study of various problems in Kähler geometry, deformation quantization, and gauge theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory Diez, Tobias Futaki, Akito Ratiu, Tudor Differential Geometry Symplectic Geometry 53D20, 58D27, (58D19, 22E65, 53C55, 53D55) We pioneer the development of a rigorous infinite-dimensional framework for the Kempf-Ness theorem, addressing the significant challenge posed by the absence of a complexification for the symmetry group in infinite dimensions, e.g, the diffeomorphism group. We propose a novel approach, based on Cartan bundles, to generalize Kempf-Ness theory to infinite dimensions, invoking the fundamental role played by the Maurer-Cartan form. This approach allows us to define and study objects essential for the Kempf-Ness theorem, such as the complex model for orbits and the Kempf-Ness function, as well as establishing its convexity properties and defining a generalized Futaki character. We show how our framework can be applied to the study of various problems in Kähler geometry, deformation quantization, and gauge theory. |
| title | Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory |
| topic | Differential Geometry Symplectic Geometry 53D20, 58D27, (58D19, 22E65, 53C55, 53D55) |
| url | https://arxiv.org/abs/2405.20864 |