Curvature and Weitzenbock formula for the Podleś quantum sphere
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909231903408128 |
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| author | Mesland, Bram Rennie, Adam |
| author_facet | Mesland, Bram Rennie, Adam |
| contents | We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere $S^{2}_{q}$. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of \cite{MRLC}. The scalar curvature is a constant, and as the parameter $q\to 1$, the scalar curvature converges to the classical value $2$. We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for $q\neq 1$, yet recovers it for $q\to 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature and Weitzenbock formula for the Podleś quantum sphere Mesland, Bram Rennie, Adam Operator Algebras Mathematical Physics Differential Geometry Quantum Algebra 46L87 We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere $S^{2}_{q}$. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of \cite{MRLC}. The scalar curvature is a constant, and as the parameter $q\to 1$, the scalar curvature converges to the classical value $2$. We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for $q\neq 1$, yet recovers it for $q\to 1$. |
| title | Curvature and Weitzenbock formula for the Podleś quantum sphere |
| topic | Operator Algebras Mathematical Physics Differential Geometry Quantum Algebra 46L87 |
| url | https://arxiv.org/abs/2406.18483 |