DUALITY AND THE FRACTIONAL QUANTUM HALL-EFFECT

The edge states of a sample displaying the quantum Hall effect (QHE) c an be described by a (1+1)-dimensional (conformal) field theory of d m assless scalar fields taking values on a d-dimensional torus. It is kn own from the work of Naculich, Frohlich et al. and others that the req uirement of chirality of currents in this scalar field theory implies the Schwinger anomaly in the presence of an electric field, the anomal y coefficient being related in a specific way to Hall conductivity. Th e latter can take only certain restricted values with odd denominators if the theory admits fermionic states. We show that the duality symme try under the Old, d; Z) group of the free theory transforms the Hall. conductivity in a well-defined way and relates integer and fractional QHE's. This means, in particular, that the edge spectra for dually re lated Hall conductivities are identical, a prediction which may be exp erimentally testable. We also show that Haldane's hierarchy as well as certain of Jain's fractions can be reproduced from the Laughlin fract ions using the duality transformations. We thus find a framework for a unified description of the QHE's occurring at different fractions. We also give a simple derivation of the wave functions for fractions in Haldane's hierarchy.