Quantum Hall Effect, Bosonization and Chiral Actions in Higher Dimensions

TL;DR

This paper reviews higher-dimensional quantum Hall effects, connecting them to fuzzy spaces and deriving a bosonized noncommutative field theory that includes bulk Chern-Simons and boundary Wess-Zumino-Witten actions.

Contribution

It introduces a novel bosonized noncommutative field theory framework for higher-dimensional quantum Hall systems, explicitly applying it to ${f CP}^k$ and analyzing the bulk-boundary anomaly cancellation.

Findings

Effective action contains a bulk Chern-Simons term.

Boundary term is a generalized chiral Wess-Zumino-Witten action.

Anomaly cancellation between bulk and boundary actions.

Abstract

We give a brief review of the Quantum Hall effect in higher dimensions and its relation to fuzzy spaces. For a quantum Hall system, the lowest Landau level dynamics is given by a one-dimensional matrix action. This can be used to write down a bosonized noncommutative field theory describing the interactions of higher dimensional nonrelativistic fermions with abelian or nonabelian gauge fields in the lowest Landau level. This general approach is applied explicitly to the case of QHE on ${\bf CP}^k$. It is shown that in the semiclassical limit the effective action contains a bulk Chern-Simons type term whose anomaly is exactly canceled by a boundary term given in terms of a chiral, gauged Wess-Zumino-Witten action suitably generalized to higher dimensions.