Curvature and Weitzenbock formula for spectral triples
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917706791387136 |
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| author | Mesland, Bram Rennie, Adam |
| author_facet | Mesland, Bram Rennie, Adam |
| contents | Using the Levi-Civita connection on the noncommutative differential one-forms of a spectral triple $(\B,\H,\D)$, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenbock formula for them. We apply these tools to $θ$-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under $θ$-deformation, whereas the connection Laplacian, Clifford representation of the curvature and the scalar curvature are all invariant under deformation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07957 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature and Weitzenbock formula for spectral triples Mesland, Bram Rennie, Adam Operator Algebras Mathematical Physics Differential Geometry Quantum Algebra 58B34 Using the Levi-Civita connection on the noncommutative differential one-forms of a spectral triple $(\B,\H,\D)$, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenbock formula for them. We apply these tools to $θ$-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under $θ$-deformation, whereas the connection Laplacian, Clifford representation of the curvature and the scalar curvature are all invariant under deformation. |
| title | Curvature and Weitzenbock formula for spectral triples |
| topic | Operator Algebras Mathematical Physics Differential Geometry Quantum Algebra 58B34 |
| url | https://arxiv.org/abs/2404.07957 |