Contents

Contents

Idea

The Connes-Lott-Chamseddine-Barrett model (Connes-Lott 91, Barrett 07, Chamseddine-Connes-Marcolli 07) is a spectral triple that spectrally encodes a spacetime which is the product of 4d Minkowski spacetime $X$ with a an exotic non-commutative geometry space $F$ whose classical dimension is that of a point, but which has KO-dimension 6 (mod 8). Hence this is a non-commutative version of a Kaluza-Klein compactification of a spacetime of KO-dimension $4 + 6$. (See also the discussion at 2-spectral triple).

As in all (super-)Kaluza-Klein theory, what is pure pseudo-Riemannian geometry (albeit “spectral”) in 4+6 (KO-)dimension, hence pure gravity, effectively looks like a configuration of Einstein-Yang-Mills-Dirac-Higgs theory down in 4 dimensions, hence of gravity coupled to gauge fields and fermions and a Higgs boson.

The interest in the model lies in the fact that a comparatively simple algebraic choice in the spectral triple of the Connes-Lott-Chamseddine model this way reproduces the standard model of particle physics, and does so in quite some fine detail (even if some issues remain open).

For instance the fiber space $F$ is modeled as the formal dual to the algebra

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

which is the direct sum of the $\mathbb{R}$-algebras of complex numbers, quaternions and $3x3$ complex matrix algebra.

Moreover, the entire field content in one generation of fermions (i.e. electrons, quarks, neutrinos) is claimed to be encoded precisely in the bimodule over this algebra which is the direct sum of all inequivalent irreducible odd bimodules (Connes 06, prop. 2.2 - prop 2.5)

References

Review

Exposition highlighting the relation to KK-compactification and string theory-vacua includes

Other comment in view of D-brane-physics:

Original articles

The basic mechanism was originally laid out in