Electromagnetic interactions of higher dimensional quantum Hall droplets

TL;DR

This paper develops an effective action framework for electromagnetic interactions in higher-dimensional quantum Hall systems using $W_N$-gauge theory, linking it to the Seiberg-Witten map and ensuring gauge invariance through boundary terms.

Contribution

It introduces a novel $W_N$-gauge theory approach to describe electromagnetic interactions in higher-dimensional quantum Hall droplets, including explicit calculations on ${f CP}^k$.

Findings

Derived the effective bulk action as a Kähler-Chern-Simons term.

Showed gauge invariance is maintained via boundary contributions.

Connected the formalism to the Abelian Seiberg-Witten map.

Abstract

Using a $W_{N}$-gauge theory to describe electromagnetic interactions of spinless fermions in the lowest Landau level, where the $W_{N}$ transformations are nonlinear realizations of U(1) gauge transformations, we construct the effective action describing electromagnetic interactions of a higher dimensional quantum Hall droplet. We also discuss how this is related to the Abelian Seiberg-Witten map. Explicit calculations are presented for the quantum Hall effect on ${\bf CP}^k$ with U(1) background magnetic field. The bulk action is a K\"ahler-Chern-Simons term whose anomaly is cancelled by a boundary contribution so that gauge invariance is explicitly satisfied.