Towards Noncommutative Linking Numbers Via the Seiberg-Witten Map

TL;DR

This paper investigates the geometric and topological effects of noncommutativity in gauge theories using the Seiberg-Witten map, revealing new knots, linking number expansions, and noncommutative invariants with applications to the Aharonov-Bohm effect.

Contribution

It introduces a novel analysis of noncommutative Wilson loops and linking numbers via the Seiberg-Witten map, including explicit computations of noncommutative invariants up to first order.

Findings

Noncommutativity induces 6^n new knots at the n-th order of expansion.

Linking numbers can be expanded in terms of noncommutative gauge fields.

Explicit first-order noncommutative Jones-Witten invariants are computed.

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 Poincar\'e dual to the high-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincar\'e dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincar\'e 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.