W_{\infty} Gauge Transformations and the Electromagnetic Interactions of Electrons in the Lowest Landau Level

TL;DR

This paper develops a $W_{ olinebreak\infty}$ gauge field theory for electrons in the lowest Landau level, linking electromagnetic interactions to a non-linear realization of an external gauge potential, with applications to quantum Hall systems.

Contribution

It introduces a novel $W_{ olinebreak\infty}$ gauge framework to describe electron interactions in the lowest Landau level, connecting it to electromagnetic phenomena in quantum Hall systems.

Findings

Derived effective Lagrangians for quantum Hall droplets.

Showed how $W_{ olinebreak\infty}$ gauge transformations cancel with electron gauge transformations.

Connected electromagnetic interactions to a non-linear realization of external gauge potentials.

Abstract

We construct a $W_{\infty}$ gauge field theory of electrons in the lowest Landau level. For this purpose we introduce an external gauge potential $\cal A $ such that its $W_{\infty}$ gauge transformations cancel against the gauge transformation of the electron field. We then show that the electromagnetic interactions of electrons in the lowest Landau level are obtained through a non-linear realization of $\cal A$ in terms of the $U(1)$ gauge potential $A^{\m}$. As applications we derive the effective Lagrangians for circular droplets and for the $\n =1$ quantum Hall system.