
TL;DR
This paper develops a categorified noncommutative geometry framework using Fell bundle $C^*$-categories, enabling new insights into spectral triples, quantisation, and fermion mass matrices relevant to physics.
Contribution
It introduces a categorification approach to noncommutative geometry by replacing points with objects in Fell bundle categories, extending spectral triples and their applications.
Findings
Constructed a noncommutative geometry with a generalized tangent bundle.
Provided a categorification of real spectral triples with Dirac operators from morphisms.
Applied framework to quantisation and constraints on fermion mass matrices.
Abstract
We construct a noncommutative geometry with generalised `tangent bundle' from Fell bundle -categories () beginning by replacing pair groupoid objects (points) with objects in . This provides a categorification of a certain class of real spectral triples where the Dirac operator is constructed from morphisms in a category. Applications for physics include quantisation via the tangent groupoid and new constraints on (the fermion mass matrix).
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