Matrix geometries and fuzzy spaces as finite spectral triples

TL;DR

This paper develops a framework for finite spectral triples resembling Riemannian manifolds, explores fuzzy spaces like fuzzy 2-spheres, and analyzes their Dirac operators and eigenvalues.

Contribution

It introduces a class of finite spectral triples, including fuzzy spaces, and provides detailed analysis of fuzzy 2-spheres and their Dirac operators.

Findings

Fuzzy 2-spheres correspond to two spinor fields on the sphere.

Adding a mass mixing matrix aligns eigenvalues with the commutative case.

A general form for the Dirac operator in finite spectral triples is determined.

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.