The Standard Model in Noncommutative Geometry and Morita equivalence

TL;DR

This paper explores the properties of spectral triples in noncommutative geometry models of the Standard Model, focusing on Morita equivalence, modifications to the Dirac operator, and implications of degenerate representations.

Contribution

It introduces necessary modifications to the Dirac operator for Morita equivalence and analyzes the impact of degenerate representations in noncommutative geometric models.

Findings

Additional terms are required in the Dirac operator for Morita equivalence.

The relation between the Dirac operator and orientability is clarified.

Degenerate representations alter the structure of the spectral triple.

Abstract

We discuss some properties of the spectral triple $(A_F,H_F,D_F,J_F,\gamma_F)$ describing the internal space in the noncommutative geometry approach to the Standard Model, with $A_F=\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$. We show that, if we want $H_F$ to be a Morita equivalence bimodule between $A_F$ and the associated Clifford algebra, two terms must be added to the Dirac operator; we then study its relation with the orientability condition for a spectral triple. We also illustrate what changes if one considers a spectral triple with a degenerate representation, based on the complex algebra $B_F=\mathbb{C}\oplus M_2(\mathbb{C})\oplus M_3(\mathbb{C})$.