Chiral Fermions and Spinc structures on Matrix approximations to manifolds

TL;DR

This paper explores the application of the Atiyah-Singer index theorem to finite matrix approximations of manifolds, demonstrating how twisted bundles on certain spaces can reproduce aspects of the standard model spectrum.

Contribution

It shows how to realize chiral fermions and standard model-like spectra on fuzzy spaces using $Spin^c$ structures and twisting, extending the geometric framework to non-spin manifolds.

Findings

One generation of the electroweak sector obtained from $ ext{CP}^2$.

Correct fermion charges achieved with the Grassmannian $U(5)/(U(3) imes U(2))$.

Spectrum with varying multiplicities matching standard model fermions.

Abstract

The Atiyah-Singer index theorem is investigated on various compact manifolds which admit finite matrix approximations (``fuzzy spaces'') with a view to applications in a modified Kaluza-Klein type approach in which the internal space consists of a finite number of points. Motivated by the chiral nature of the standard model spectrum we investigate manifolds that do not admit spinors but do admit $Spin^c$ structures. It is shown that, by twisting with appropriate bundles, one generation of the electroweak sector of the standard model, including a right-handed neutrino, can be obtained in this way from the complex projective space $\CP^2$. The unitary Grassmannian $U(5)/(U(3)\times U(2))$ yields a spectrum that contains the correct charges for the Fermions of the standard model, with varying multiplicities for the different particle states.