The Geometry of Quantum Hall Effect: An Effective Action for all Dimensions

TL;DR

This paper derives a comprehensive formula for the topological effective action of quantum Hall systems across multiple dimensions, accounting for gauge and metric fluctuations, and explores implications of gravitational anomalies.

Contribution

It introduces a unified approach to compute the topological effective action for quantum Hall effects in all dimensions, incorporating gauge and gravitational anomaly effects.

Findings

Derived a general formula for the topological effective action in higher dimensions

Analyzed the role of gravitational anomalies in quantum Hall systems

Discussed features of the action in dimensions 2+1, 4+1, 6+1

Abstract

We present a general formula for the topological part of the effective action for quantum Hall systems in higher dimensions, including fluctuations of the gauge field and metric around background fields of a specified topological class. The result is based on a procedure of integrating up from the Dolbeault index density which applies for the degeneracies of Landau levels, combined with some input from the standard descent procedure for anomalies. Features of the topological action in (2+1), (4+1), (6+1) dimensions, including the contribution due to gravitational anomalies, are discussed in some detail.