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 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.

37 pages

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• Spectral dimensions and dimension spectra of quantum spacetimes
• Indefinite Kasparov modules and pseudo-Riemannian manifolds
• Asymptotic and exact expansions of heat traces
• Spectral Action in Noncommutative Geometry
• The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
• A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)
• Matrix Geometries Emergent from a Point
• The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples