Categorified Noncommutative manifolds

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 $C^*$-categories ($E$) beginning by replacing pair groupoid objects (points) with objects in $E$. 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 $D_{\mathrm{finite}}$ (the fermion mass matrix).