Curvature of an exotic 7-sphere
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arXiv
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| Format: | Preprint |
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2024
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| author | Berman, David S. Cederwall, Martin Gherardini, Tancredi Schettini |
| author_facet | Berman, David S. Cederwall, Martin Gherardini, Tancredi Schettini |
| contents | We study the geometry of the Gromoll-Meyer sphere, one of Milnor's exotic $7$-spheres. We focus on a Kaluza-Klein Ansatz, with a round $S^4$ as base space, unit $S^3$ as fibre, and $k=1,2$ $SU(2)$ instantons as gauge fields, where all quantities admit an elegant description in quaternionic language. The metric's moduli space coincides with the $k=1,2$ instantons' moduli space quotiented by the isometry of the base, plus an additional $\mathbb{R}^+$ factor corresponding to the radius of the base, $r$. We identify a "center" of the $k=2$ instanton moduli space with enhanced symmetry. This $k=2$ solution is used together with the maximally symmetric $k=1$ solution to obtain a metric of maximal isometry, $SO(3)\times O(2)$, and to explicitly compute its Ricci tensor. This allows us to put a bound on $r$ to ensure positive Ricci curvature, which implies various energy conditions for an $8$-dimensional static space-time. This construction then enables a concrete examination of the properties of the sectional curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature of an exotic 7-sphere Berman, David S. Cederwall, Martin Gherardini, Tancredi Schettini High Energy Physics - Theory Mathematical Physics Differential Geometry We study the geometry of the Gromoll-Meyer sphere, one of Milnor's exotic $7$-spheres. We focus on a Kaluza-Klein Ansatz, with a round $S^4$ as base space, unit $S^3$ as fibre, and $k=1,2$ $SU(2)$ instantons as gauge fields, where all quantities admit an elegant description in quaternionic language. The metric's moduli space coincides with the $k=1,2$ instantons' moduli space quotiented by the isometry of the base, plus an additional $\mathbb{R}^+$ factor corresponding to the radius of the base, $r$. We identify a "center" of the $k=2$ instanton moduli space with enhanced symmetry. This $k=2$ solution is used together with the maximally symmetric $k=1$ solution to obtain a metric of maximal isometry, $SO(3)\times O(2)$, and to explicitly compute its Ricci tensor. This allows us to put a bound on $r$ to ensure positive Ricci curvature, which implies various energy conditions for an $8$-dimensional static space-time. This construction then enables a concrete examination of the properties of the sectional curvature. |
| title | Curvature of an exotic 7-sphere |
| topic | High Energy Physics - Theory Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2410.01909 |