On fuzzy analogues of physical spaces in noncommutative geometryTools Laird, Thomas (2025) On fuzzy analogues of physical spaces in noncommutative geometry. PhD thesis, University of Nottingham.
AbstractSince the development of noncommutative geometry, it has become common to construct fuzzy versions of well-known geometric spaces, such as the fuzzy sphere. This work follows efforts to construct approximations to physical spaces that are as consistent as possible with the definitions of a spectral triple. The first part looks at a way of constructing lattice-like spectral triples that are able to approximate simple geometries such as the line and the circle. The latter model is successfully interpreted in terms of a fermionic state-sum model with a U(1) connection. The second and main part of the work develops a model for a fuzzy complex projective plane within the formalism of matrix geometries, itself a formulation of real spectral triples in terms of matrices. An argument is presented that justifies the expectation that such fuzzy spaces with symmetry groups may by necessity approximate coadjoint orbits. A Hilbert space and Dirac operator that approximate the complex spinor geometry of the complex projective plane are then constructed, which are much simpler and more concrete than those obtained by earlier efforts in the literature.
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