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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1687-9120 1687-9139 |