Multi-toric geometries with larger compact symmetry
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916312066818048 |
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| author | Madsen, Thomas Bruun Swann, Andrew |
| author_facet | Madsen, Thomas Bruun Swann, Andrew |
| contents | We study complete, simply-connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $\mathrm{Spin}(7)$-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $T^{3} \times \mathrm{SU}(2)$. We then specialise the analysis to holonomy $G_{2}$, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $\mathbb{R} \times \mathrm{SU}(2)$, and their extensions over singular orbits, to give local examples in the $\mathrm{Spin}(7)$-case that have singular orbits where the stabiliser is of rank one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-toric geometries with larger compact symmetry Madsen, Thomas Bruun Swann, Andrew Differential Geometry 53C25 (Primary) 32Q25, 53C26, 53C29, 53C55, 53D20 (Secondary) We study complete, simply-connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $\mathrm{Spin}(7)$-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $T^{3} \times \mathrm{SU}(2)$. We then specialise the analysis to holonomy $G_{2}$, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $\mathbb{R} \times \mathrm{SU}(2)$, and their extensions over singular orbits, to give local examples in the $\mathrm{Spin}(7)$-case that have singular orbits where the stabiliser is of rank one. |
| title | Multi-toric geometries with larger compact symmetry |
| topic | Differential Geometry 53C25 (Primary) 32Q25, 53C26, 53C29, 53C55, 53D20 (Secondary) |
| url | https://arxiv.org/abs/2407.03693 |