The Eight Dimensional Quantum Hall Effect and the Octonions

TL;DR

This paper generalizes the quantum Hall effect to eight dimensions using octonionic mathematics, revealing two distinct quantum liquids with unique geometric and gauge properties.

Contribution

It introduces a novel 8D quantum Hall effect model based on octonions, with two different liquids characterized by their holonomy groups and configuration spaces.

Findings

Two quantum liquids with different configuration spaces

One liquid on a 20D manifold with SO(7) holonomy

Another on a 14D manifold with G2 holonomy

Abstract

We construct a generalization of the quantum Hall effect where particles move in an eight dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configurations spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20 dimensional manifold of with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14 dimensional manifold with an internal component of $G_2$ holonomy.