Towards Noncommutative Linking Numbers via the Seiberg-Witten Map. (18th October 2015)

• Record Type: Journal Article
• Authors: García-Compeán, H.; Obregón, O.; Santos-Silva, R.
• Other Names: Amelino-Camelia Giovanni Academic Editor.
• Is Part Of: Advances in mathematical physics. Volume 2015(2015)
• Publication Date: 2015-10-18
• Subjects: Mathematical physics -- Periodicals; Mathematical physics; Periodicals; 530.15
• DOI: 10.1155/2015/845328
• Languages: English
• ISSNs: 1687-9120

Abstract

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 of6 n new knots at then 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.