Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy

TL;DR

This paper explores the higher-dimensional quantum Hall effects on odd-dimensional spheres, revealing a dimensional hierarchy, quantum geometry, and topological field theories that connect even and odd dimensions, with implications for topology and D-brane physics.

Contribution

It introduces a novel dimensional ladder linking odd and even-dimensional quantum Hall effects using quantum Nambu geometry and topological field theories.

Findings

Degeneracy of Landau levels equals the winding number.

Quantum Nambu geometry underpins odd-dimensional quantum Hall effects.

Extended hierarchy relates to quantum anomalies and D-branes.

Abstract

We demonstrate dimensional ladder of higher dimensional quantum Hall effects by exploiting quantum Hall effects on arbitrary odd dimensional spheres. Non-relativistic and relativistic Landau models are analyzed on $S^{2k-1}$ in the $SO(2k-1)$ monopole background. The total sub-band degeneracy of the odd dimensional lowest Landau level is shown to be equal to the winding number from the base-manifold $S^{2k-1}$ to the one-dimension higher $SO(2k)$ gauge group. Based on the chiral Hopf maps, we clarify the underlying quantum Nambu geometry for odd dimensional quantum Hall effect and the resulting quantum geometry is naturally embedded also in one-dimension higher quantum geometry. An origin of such dimensional ladder connecting even and odd dimensional quantum Hall effects is illuminated from a viewpoint of the spectral flow of Atiyah-Patodi-Singer index theorem in differential topology. We also present a BF topological field theory as an effective field theory in which membranes with different dimensions undergo non-trivial linking in odd dimensional space. Finally, an extended version of the dimensional hierarchy for higher dimensional quantum Hall liquids is proposed, and its relationship to quantum anomaly and D-brane physics is discussed.