$W_{\infty}$ algebra in the integer quantum Hall effects

TL;DR

This paper explores the mathematical structure of the $W_{00}$ algebra in the context of the integer quantum Hall effect, revealing its connection to edge states and system incompressibility.

Contribution

It introduces the $W_{1+0}$ algebra with a central extension and links it to the Kac-Moody algebra governing edge state behavior.

Findings

Derived the central extension of the $W_{00}$ algebra.

Showed the $W_{1+0}$ algebra underpins edge state dynamics.

Connected the algebraic structure to the incompressibility of the quantum Hall system.

Abstract

We investigate the $W_{\infty}$ algebra in the integer quantum Hall effects. Defining the simplest vacuum, the Dirac sea, we evaluate the central extension for this algebra. A new algebra which contains the central extension is called the $W_{1+\infty}$ algebra. We show that this $W_{1+\infty}$ algebra is an origin of the Kac-Moody algebra which determines the behavior of edge states of the system. We discuss the relation between the $W_{1+\infty}$ algebra and the incompressibility of the integer quantum Hall system.