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

TL;DR

This paper demonstrates that many quantum Hall states can be described using \\Winf algebra representations, linking algebraic structures to physical states including multilayer and hierarchical filling fractions.

Contribution

It provides an explicit construction of \\Winf algebra generators in second quantization for various quantum Hall states, including multilayer and Jain states.

Findings

Quantum Hall states correspond to different \\Winf algebra representations.

Explicit second quantized generators are constructed for these states.

The approach includes multilayer and hierarchical filling fraction states.

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.