Noncommutative gauge symmetry in the fractional quantum Hall effect

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. By using this map, we resolve the apparent contradiction between the noncommutative gauge symmetry, on the one hand, and the particle-hole symmetry of the half-filled Landau level and the presence of the mixed Chern-Simons terms in the effective Lagrangian of the fractional quantum Hall states, on the other hand. We outline the general procedure which can be used to write down effective field theories which respect the noncommutative U(1) symmetry.