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Fermionic Chern-Simons Field Theory for the Fractional Hall Effect

We review the fermionic Chern-Simons field theory for the Fractional Quantum Hall Effect (FQHE). We show that in this field theoretic approach to the problem of interacting electrons moving in a plane in the presence of an external magnetic field, the FQHE states appear naturally as the semiclassical states of the theory. In this framework, the FQHE states are the unique ground states of a system of electrons on a fixed geometry. The excitation spectrum is fully gapped and these states can be viewed as infrared stable fixed points of the system. It is shown that the long distance, low energy properties of the system are described exactly by this theory. It is further shown that, in this limit, the actual ground state wave function has the Laughlin form. We also discuss the application of this theory to the problem of the FQHE in bilayers and in unpolarized single layer systems.