The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields

TL;DR

This paper analyzes the Dirac operator spectrum on compact coset spaces with homogeneous gauge fields, linking degeneracies to the Atiyah-Singer index theorem and implications for higher-dimensional quantum Hall effects.

Contribution

It provides a detailed analysis of the Dirac spectrum with homogeneous gauge fields on coset spaces, connecting geometric index theorems to physical degeneracies.

Findings

Degeneracy of the lowest Landau level relates to the Atiyah-Singer index theorem.

Spectrum analysis applies to higher-dimensional quantum Hall systems.

Homogeneous gauge fields compatible with symmetries influence Dirac operator properties.

Abstract

The spectrum and degeneracies of the Dirac operator are analysed on compact coset spaces when there is a non-zero homogeneous background gauge field which is compatible with the symmetries of the space, in particular when the gauge field is derived from the spin-connection. It is shown how the degeneracy of the lowest Landau level in the recently proposed higher dimensional quantum Hall effect is related to the Atiyah-Singer index theorem for the Dirac operator on a compact coset space.