Matrix geometries and fuzzy spaces as finite spectral triples

Abstract

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are investigated in detail, and it is shown that the fuzzy analogues correspond to two spinor fields on the commutative sphere. In some cases it is necessary to add a mass mixing matrix to the commutative Dirac operator to get a precise agreement for the eigenvalues.

39 pages, final version

References in corpus (3)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Noncommutative Geometry and the standard model with neutrino mixing
• Why the Standard Model

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