Towards Noncommutative Linking Numbers via the Seiberg‐Witten Map

H. García-Compeán, O. Obregón and R. Santos-Silva

Advances in Mathematical Physics, 2015, vol. 2015, issue 1

Abstract: 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.

Date: 2015
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https://doi.org/10.1155/2015/845328

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2015:y:2015:i:1:n:845328

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