Non-compact spectral triples with finite volume

Abstract

In order to extend the spectral action principle to non-compact spaces, we propose a framework for spectral triples where the algebra may be non-unital but the resolvent of the Dirac operator remains compact. We show that an example is given by the supersymmetric harmonic oscillator which, interestingly, provides two different Dirac operators. This leads to two different representations of the volume form in the Hilbert space, and only their product is the grading operator. The index of the even-to-odd part of each of these Dirac operators is 1. We also compute the spectral action for the corresponding Connes-Lott two-point model. There is an additional harmonic oscillator potential for the Higgs field, whereas the Yang-Mills part is unchanged. The total Higgs potential shows a two-phase structure with smooth transition between them: In the spontaneously broken phase below a critical radius, all fields are massive, with the Higgs field mass slightly smaller than the NCG prediction. In the unbroken phase above the critical radius, gauge fields and fermions are massless, whereas the Higgs field remains massive.

36 pages

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Cited by in corpus (8)

• Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
• Index theory for locally compact noncommutative geometries
• 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory
• Spectral geometry of the Moyal plane with harmonic propagation
• On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
• Space-Time from the spectral point of view
• The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
• Infinite Noncommutative Covering Projections