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, 1-12

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 new knots at the th 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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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlamp:845328

DOI: 10.1155/2015/845328

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