Geometric construction of the Quantum Hall Effect in all even dimensions

TL;DR

This paper provides a unified geometric framework for Quantum Hall Effects in all even dimensions, demonstrating that their existence does not rely on division algebras and explicitly describing their Hamiltonians and ground states.

Contribution

It introduces a geometric construction of QHE in all even dimensions, challenging previous assumptions about the role of division algebras and providing explicit Hamiltonians and wave-functions.

Findings

QHE exists in all even dimensions without division algebras

Explicit Hamiltonians and ground states are constructed for flat spaces

Higher-dimensional QHE shares key features like incompressibility

Abstract

The Quantum Hall Effects in all even dimensions are uniformly constructed. Contrary to some recent accounts in the literature, the existence of Quantum Hall Effects does not {\it crucially} depend on the existence of division algebras. For QHE on flat space of even dimensions, both the Hamiltonians and the ground state wave-functions for a single particle are explicitly described. This explicit description immediately tells us that QHE on a higher even dimensional flat space shares the common features such as incompressibility with QHE on plane.