The fractional quantum Hall effect: Chern-Simons mapping, duality, Luttinger liquids and the instanton vacuum

We derive, from first principles, the complete Luttinger liquid theory of abelian quantum Hall edge states. This theory includes the effects of disorder and Coulomb interactions as well as the coupling to external electromagnetic fields. We introduce a theory of spatially separated (individually conserved) edge modes, find an enlarged dual symmetry and obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge and Laughlin's gauge argument are used to obtain unambiguously the Hall conductance. In resolving the problem of counter flowing edge modes, we find that the long range Coulomb interactions play a fundamental role. In order to set up a theory for arbitrary filling fractions $\nu$ we use the idea of a two dimensional network of percolating edge modes. We derive an effective, single mode Luttinger liquid theory for tunneling processes into the quantum Hall edge which yields a continuous tunneling exponent $1/\nu$. The network approach is also used to re-derive the instanton vacuum or $Q$-theory for the plateau transitions.