Bosonization of the lowest Landau level in arbitrary dimensions: edge and bulk dynamics

TL;DR

This paper develops a bosonization framework for nonrelativistic fermions in the lowest Landau level within higher-dimensional quantum Hall systems, linking bulk and edge dynamics through noncommutative field theory and gauge invariance.

Contribution

It introduces a novel bosonic action as a noncommutative field theory invariant under $W_N$ transformations, and connects gauge invariance with a Seiberg-Witten map in higher dimensions.

Findings

Derivation of a bosonic matrix action for fermions in the lowest Landau level.

Establishment of a bulk-boundary correspondence via Chern-Simons and Wess-Zumino-Witten terms.

Demonstration of gauge invariance through a noncommutative field theory framework.

Abstract

We discuss the bosonization of nonrelativistic fermions interacting with non-Abelian gauge fields in the lowest Landau level in the framework of higher dimensional quantum Hall effect. The bosonic action is a one-dimensional matrix action, which can also be written as a noncommutative field theory, invariant under $W_N$ transformations. The requirement that the usual gauge transformation should be realized as a $W_N$ transformation provides an analog of a Seiberg-Witten map, which allows us to express the action purely in terms of bosonic fields. The semiclassical limit of this, describing the gauge interactions of a higher dimensional, non-Abelian quantum Hall droplet, produces a bulk Chern-Simons type term whose anomaly is exactly cancelled by a boundary term given in terms of a gauged Wess-Zumino-Witten action.