Noncommutative gauge symmetry in the fractional quantum Hall effect

Understand

We show that a system of particles on the lowest Landau level can be coupled to a probe U(1) gauge field $\mathcal A_\mu$ in such a way that the theory is invariant under a noncommutative U(1) gauge symmetry.

• While the temporal component $\mathcal A_0$ of the probe field is coupled to the projected density operator, the spatial components $\mathcal A_i$ are best interpreted as quantum displacements, which distort the interaction potential between the particles.
• We develop a Seiberg-Witten-type map from the noncommutative U(1) gauge symmetry to a simpler version, which we call "baby noncommutative" gauge symmetry, where the Moyal brackets are replaced by the Poisson brackets.
• The latter symmetry group is isomorphic to the group of volume preserving diffeomorphisms.