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 non-geometric (formal dually: 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).

The Connes-Lott models realize a non-commutative version of gauge-Higgs unification.

## References

### Review

• Robert Brout, Notes on Connes’ Construction of the Standard Model, Nucl. Phys. Proc. Suppl. 65 (1998) 3-15 (arXiv:hep-th/9706200)

• Agostino Devastato, Maxim Kurkov, Fedele Lizzi, Spectral Noncommutative Geometry, Standard Model and all that (arXiv:1906.09583)

Exposition highlighting the relation to KK-compactification and string theory-vacua (2-spectral triples) includes

Other commentary in view of D-brane-physics:

### Original articles

The basic mechanism was originally laid out in

The early version of the model is due to

• Alain Connes, John Lott, Particle models and noncommutative geometry, Nuclear Physics B - Proceedings Supplements 18(2): 29-47 (1991) (web)

The modern version of the model that produces the correct fermionic content (and finds the KO-dimension of the compactification space to be 6 mod 8) is due to

A more succinct version of the axioms of the model is claimed in

Introduction of a scalar field to fix the prediction of the Higgs particle mass:

Implementation of Pati-Salam model ($SU(5)$-GUT):

and specifically discussion of leptoquarks and possible relation to flavour anomalies:

• Ufuk Aydemir, Djordje Minic, Chen Sun, Tatsu Takeuchi, $B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry, JHEP 09 (2018) 117 (arXiv:1804.05844)

• Fabien Besnard, Extensions of the noncommutative Standard Model and the weak order one condition (arXiv:2011.02708)

Relation to actual quantum gravity induced by quantized maps from spacetime to the 4-sphere (see also at Cohomotopy) is claimed in

With spin^c structure:

• Arkadiusz Bochniak, Andrzej Sitarz, A spectral geometry for the Standard Model without the fermion doubling (arXiv:2001.02902)

Discussion of spectral triples over Jordan algebras in the Connes-Lott model:

Discussion in Lorentzian signature:

• Fabien Besnard, Christian Brouder, Noncommutative geometry, the Lorentzian Standard Model and its B-L extension (arXiv:2010.04960)

• Manuele Filaci, Pierre Martinetti, A critical survey of twisted spectral triples beyond the Standard Model [arXiv:2301.08346]

Last revised on January 23, 2023 at 05:57:57. See the history of this page for a list of all contributions to it.