Diffeomorphisms on the fuzzy sphere
Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of...
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| Published in | Progress of theoretical and experimental physics Vol. 2020; no. 1 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Oxford University Press
01.01.2020
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| Online Access | Get full text |
| ISSN | 2050-3911 2050-3911 |
| DOI | 10.1093/ptep/ptz151 |
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| Abstract | Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms. |
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| AbstractList | Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms. |
| Author | Ishiki, Goro Matsumoto, Takaki |
| Author_xml | – sequence: 1 givenname: Goro surname: Ishiki fullname: Ishiki, Goro organization: Tomonaga Center for the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan – sequence: 2 givenname: Takaki surname: Matsumoto fullname: Matsumoto, Takaki email: takaki@stp.dias.ie organization: Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan |
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| CitedBy_id | crossref_primary_10_1103_PhysRevD_101_106009 crossref_primary_10_1103_PhysRevD_103_126003 |
| Cites_doi | 10.1007/JHEP11(2015)146 10.1016/S0550-3213(98)00291-0 10.1007/978-3-642-61981-6 10.1007/978-1-4612-0005-5 10.1088/1126-6708/2002/05/056 10.1007/JHEP11(2015)089 10.1142/S0217751X16501177 10.1088/1126-6708/2006/11/089 10.1088/1126-6708/2005/08/078 10.1088/0264-9381/23/6/005 10.1103/PhysRevD.92.104009 10.1088/1126-6708/2005/10/043 10.1016/j.nuclphysb.2018.06.023 10.1007/s002200050011 10.1007/JHEP08(2016)042 10.1007/JHEP07(2018)008 10.1088/1751-8113/49/28/285301 10.1088/1126-6708/1999/09/032 10.1088/0264-9381/27/13/133001 10.1007/s12220-008-9022-2 10.1093/oso/9780198536734.001.0001 10.1103/PhysRevD.55.5112 10.1088/1126-6708/2002/03/034 10.1103/PhysRevD.98.026002 10.1126/science.294.5543.823 10.1016/S0550-3213(97)00290-3 10.1007/s002090100393 10.1007/BF02099210 10.1103/PhysRevD.92.046009 10.4310/jdg/1343133699 10.1016/0550-3213(88)90116-2 10.1103/PhysRevD.78.106001 10.1007/s002200050126 10.1088/0264-9381/9/1/008 10.1143/PTP.114.1295 10.1103/PhysRevD.86.086001 10.1007/BF02099460 10.1088/1126-6708/2005/03/015 10.1088/1126-6708/2004/12/009 10.1007/BF02099772 |
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| Copyright | The Author(s) 2020. Published by Oxford University Press on behalf of the Physical Society of Japan. 2020 The Author(s) 2020. Published by Oxford University Press on behalf of the Physical Society of Japan. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Title | Diffeomorphisms on the fuzzy sphere |
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