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arXiv · 1505.03943

Towards Noncommutative Linking Numbers Via the Seiberg-Witten Map

H. García-Compeán·O. Obregón·R. Santos-Silva
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Abstract

In the present work 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 $6^n$ new knots at the $n$-th order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the high-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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H. García-Compeán, O. Obregón, R. Santos-Silva. 2015-09-22. Towards Noncommutative Linking Numbers Via the Seiberg-Witten Map. https://arxiv.org/abs/1505.03943

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