$W_{\infty}$ Algebras in the Quantum Hall Effect

Abstract

We show that a large class of incompressible quantum Hall states correspond to different representations of the \Winf algebra by explicit construction of the second quantized generators of the algebra in terms of fermion and vortex operators. These are parametrized by a set of integers which are related to the filling fraction. The class of states we consider includes multilayer Hall states and the states proposed by Jain to explain the hierarchical filling fractions. The corresponding second quantized order parameters are also given.