The Standard Model in noncommutative geometry and Morita equivalence

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 $$\begin{gathered} A_F=\mathbb{C}\oplus \\ mathbbH\oplus M_3(\mathbb{C}) \end{gathered}$$. 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 $B_F=\mathbb{C}\oplus M_2(\mathbb{C})\oplus M_3(\mathbb{C})$.