Spectral geometry of the Moyal plane with harmonic propagation

TL;DR

This paper constructs a spectral triple for the Moyal plane incorporating harmonic propagation, revealing a spectral dimension d and KO-dimension 2d, and unifies gauge and Higgs fields in a novel geometric framework.

Contribution

It introduces a non-unital spectral triple for the Moyal plane with harmonic propagation and computes the spectral action, unifying gauge and Higgs fields within this geometric setting.

Findings

Spectral dimension is d, KO-dimension is 2d.

Unified potential for gauge and Higgs fields.

Deep unification of discrete and continuous geometry parts.

Abstract

We construct a `non-unital spectral triple of finite volume' out of the Moyal product and a differential square root of the harmonic oscillator Hamiltonian. We find that the spectral dimension of this triple is d but the KO-dimension is 2d. We add another Connes-Lott copy and compute the spectral action of the corresponding U(1)-Yang-Mills-Higgs model. We find that the `covariant coordinate' involving the gauge field combines with the Higgs field to a unified potential, yielding a deep unification of discrete and continuous parts of the geometry.