Fermions in the fuzzy sphere Kaluza-Klein model

TL;DR

This paper explores fermions in a Kaluza-Klein model with fuzzy sphere fibers, showing how massless spinors manifest as multiplets with specific mass ratios and gauge charges, demonstrating a proof of concept for noncommutative geometry applications.

Contribution

It introduces a novel approach to modeling fermions in fuzzy sphere Kaluza-Klein spaces, analyzing their mass spectra and gauge charges, and demonstrating potential for noncommutative fiber algebra applications.

Findings

Massless spinors appear as multiplets with specific mass ratios.

Fuzzy sphere models produce distinct gauge charge multiplets.

The approach can be extended to other noncommutative fiber algebras.

Abstract

We consider spinors on the total space of a Kaluza-Klein model with fuzzy sphere fibre and geometrically realised Dirac operator on the product. We show that a single massless spinor on the product appears on spacetime as multiplets of spinors with a particular signature of differing masses and $SU(2)$ Yang-Mills charges. For example, for the reduced fuzzy sphere isomorphic to $M_2(\mathbb{C})$, a massless spinor appears as two $SU(2)$ doublets and an $SU(2)$ quadruplet in mass ratios $1:5/3:7/3$. Although such signatures do not appear to exactly match the Standard Model, the paper provides proof of concept of the approach, which can be applied to other noncommutative fibre algebras.