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

TL;DR

This paper explores the role of ext{W}_{ ext{infinity}} algebra in characterizing incompressible quantum Hall states, providing explicit second quantized expressions for various multilayer and hierarchical states.

Contribution

It introduces a novel algebraic framework using ext{W}_{ ext{infinity}} algebra to describe quantum Hall states and derives explicit second quantized forms for key state classes.

Findings

Quantum Hall states can be characterized as highest weight states of ext{W}_{ ext{infinity}} algebra.

Explicit second quantized expressions for multilayer and hierarchical states are derived.

The framework unifies different quantum Hall states under a common algebraic structure.

Abstract

We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the \Winf algebra. Second quantized expressions of the \Winf generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation.