Integral quadratic forms, Kac-Moody algebras, and fractional quantum Hall effect. AnADE-O classification

Abstract

The purpose of this paper is to present a rather comprehensive classification of incompressible quantum Hall states in the limit of large distance scales and low frequencies. In this limit, the description of low-energy excitations above the groundstate of an incompressible quantum Hall fluid is intimately connected to the theory of integral quadratic forms on certain lattices which we call quantum Hall lattices. This connection is understood with the help of the representation theory of algebras of gapless, chiral edge currents or, alternatively, from the point of view of the bulk effective Chern-Simons theory. Our main results concern the classification of quantum Hall lattices in terms of certain invariants and their enumeration in low dimensions and for a limited range of values of those invariants. Among physical consequences of our analysis we find explicit, discrete sets of plateau values of the Hall conductivity, as well as the quantum numbers of quasiparticles in fluids corresponding to any one among those quantum Hall lattices. Furthermore, we are able to predict transitions between structurally different quantum Hall fluids corresponding to the same filling factor. Our general results are illustrated by explicitly considering the following plateau values: σ H =N/(2N±1),N=1, 2, 3,..., σ N =1/2.