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Composite Fermi liquid metals arise at certain special filling fractions inthe quantum Hall regime and play an important role as parent states of gappedstates with quantized Hall response. They have been successfully described bythe Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermionscoupled to a $U(1)$ gauge field with a Chern-Simons term. However, the validityof the HLR description when the microscopic system is restricted to a singleLandau has not been clear. Here for the specific case of bosons at filling $\nu= 1$, we build on earlier work from the 1990s to formulate a low energydescription that takes the form of a {\em non-commutative} field theory. Thistheory has a Fermi surface of composite fermions coupled to a $U(1)$ gaugefield with no Chern-Simons term but with the feature that all fields aredefined in a non-commutative spacetime. An approximate mapping of the longwavelength, small amplitude gauge fluctuations yields a commutative effectivefield theory which, remarkably, takes the HLR form but with microscopicparameters correctly determined by the interaction strength. Extensions to someother composite fermi liquids, and to other related states of matter arediscussed.

Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems