A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
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arXiv
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| Formato: | Preprint |
| Publicado: |
2014
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| _version_ | 1866908762531430400 |
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| author | Ćaćić, Branimir |
| author_facet | Ćaćić, Branimir |
| contents | We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group $G$, also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontrjagin dual of $G$, and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to $G$-equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_4429 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | A reconstruction theorem for Connes-Landi deformations of commutative spectral triples Ćaćić, Branimir Mathematical Physics High Energy Physics - Theory Operator Algebras Quantum Algebra 58B34 (Primary) 46L55, 46L65, 46L8, 81R60 (Secondary) We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group $G$, also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontrjagin dual of $G$, and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to $G$-equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense. |
| title | A reconstruction theorem for Connes-Landi deformations of commutative spectral triples |
| topic | Mathematical Physics High Energy Physics - Theory Operator Algebras Quantum Algebra 58B34 (Primary) 46L55, 46L65, 46L8, 81R60 (Secondary) |
| url | https://arxiv.org/abs/1408.4429 |