Topological order in Quantum hall effect and two-dimensional conformal field theory

Abstract

After a brief summary of the general features of the Q.H.E. we review an approach to Fratcional Quantum Hall Effect (FQHE) based on two dimensional Conformal Field Theory (2D-CFT), more specifically on the use of Coulomb Gas Vertex Operators. We show that a consistent description of the Hall Fluid on the torus at filling v = 1/ m requires a set of m-Vertex Operators, leading to an m-fold degeneracy of the ground state wave functions. This result, as well as the value of the Hall conductivity, is shown to be related to the algebraic properties of a finite subgroup of the magnetic translation group for a N electron system; their explicit expression as the Verlinde operators which generate the modular transformations in CFT, are taken as a realization of topological long range order of the system. Then we discuss the U(1) Kac-Moody algebra which describes the massless edge states for an Hall fluid on a cylinder as states with the electric and magnetic charge of the Laughlin anyon (vortex); and finally we present a possible extension of this approach to the general case of v = p/ m.