Spectral geometry of the Moyal plane with harmonic propagation

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 Yang–Mills–Higgs model. As result, 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.