Towards Noncommutative Linking Numbers via the Seiberg-Witten Map
Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of 6n new knots at the nth order of the Seiberg-Wi...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2015/845328 |
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| author | H. García-Compeán O. Obregón R. Santos-Silva |
| author_facet | H. García-Compeán O. Obregón R. Santos-Silva |
| author_sort | H. García-Compeán |
| collection | DOAJ |
| description | Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of 6n new knots at the nth order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the higher-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative “Jones-Witten” invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter; we also show the relation to the noncommutative Landau levels. |
| format | Article |
| id | doaj-art-6e70b471a08641fa91319164df670895 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-6e70b471a08641fa91319164df6708952025-02-03T05:44:36ZengWileyAdvances in Mathematical Physics1687-91201687-91392015-01-01201510.1155/2015/845328845328Towards Noncommutative Linking Numbers via the Seiberg-Witten MapH. García-Compeán0O. Obregón1R. Santos-Silva2Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, P.O. Box 14-740, 07000 Mexico City, DF, MexicoDepartamento de Física, DCI, Universidad de Guanajuato, 37150 León, GTO, MexicoDepartamento de Física, DCI, Universidad de Guanajuato, 37150 León, GTO, MexicoSome geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of 6n new knots at the nth order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the higher-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative “Jones-Witten” invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter; we also show the relation to the noncommutative Landau levels.http://dx.doi.org/10.1155/2015/845328 |
| spellingShingle | H. García-Compeán O. Obregón R. Santos-Silva Towards Noncommutative Linking Numbers via the Seiberg-Witten Map Advances in Mathematical Physics |
| title | Towards Noncommutative Linking Numbers via the Seiberg-Witten Map |
| title_full | Towards Noncommutative Linking Numbers via the Seiberg-Witten Map |
| title_fullStr | Towards Noncommutative Linking Numbers via the Seiberg-Witten Map |
| title_full_unstemmed | Towards Noncommutative Linking Numbers via the Seiberg-Witten Map |
| title_short | Towards Noncommutative Linking Numbers via the Seiberg-Witten Map |
| title_sort | towards noncommutative linking numbers via the seiberg witten map |
| url | http://dx.doi.org/10.1155/2015/845328 |
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