Non-commutative Complex Projective Spaces and the Standard Model

TL;DR

This paper constructs a non-commutative geometric model using fuzzy complex projective spaces to replicate the standard model fermion spectrum, including three generations, within a matrix approximation framework.

Contribution

It introduces a novel non-commutative geometric approach using fuzzy spaces to model the standard model's fermion spectrum and generations.

Findings

Reproduces the standard model fermion spectrum with a right-handed neutrino.

Models three generations as three copies of space-time within fuzzy space approximation.

Incorporates Higgs fields and Yukawa couplings in the non-commutative geometric framework.

Abstract

The standard model fermion spectrum, including a right handed neutrino, can be obtained as a zero-mode of the Dirac operator on a space which is the product of complex projective spaces of complex dimension two and three. The construction requires the introduction of topologically non-trivial background gauge fields. By borrowing from ideas in Connes' non-commutative geometry and making the complex spaces `fuzzy' a matrix approximation to the fuzzy space allows for three generations to emerge. The generations are associated with three copies of space-time. Higgs' fields and Yukawa couplings can be accommodated in the usual way.