Towards Noncommutative Linking Numbers via the Seiberg-Witten Map

Abstract EN

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.