Noncommutative gauge symmetry in fractional quantum Hall states

We show how to couple the lowest Landau level to a probe field in a way that the whole theory is invariant under a noncommutative U(1) gauge symmetry. While the time component of the probe field couples to the projected density operator, the spatial components are best interpreted as quantum displacements. In order to resolve the conflict between particle-hole symmetry at half filling and the presence of a mixed Chern-Simons term, and the ability to write down a fully noncommutative field theory on the lowest Landau level, we develop a map from the noncommutative U(1) gauge symmetry to a simpler, "baby noncommutative" gauge symmetry which is isomorphic to the group of volume preserving diffeomorphisms of flat space.