nLab Connes-Lott-Chamseddine-Barrett model

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.

Related entries

• Gepner model

• higher category theory and physics: Spectral standard model and gravity

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)

Relation to the “neutrino-paradigm”:

• José Gracia-Bondia, On Marshak’s and Connes’ views of chirality, in: E. C. G. Sudarshan (ed.), A Gift of Prophecy – Essays in Celebration of the Life of Robert Eugene Marshak, pp. 208-217, World Scientific (1995) (arXiv:hep-th/9706200, doi:10.1142/9789812831408_0017)

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

• Urs Schreiber, Spectral Standard Model and String Compactifications, 2016

Other commentary in view of D-brane-physics:

• John Iliopoulos, Gauge Theories and non-Commutative Geometry: A review, EPJ Web Conf. Volume 182, 2018 6th International Conference on New Frontiers in Physics (ICNFP 2017) (doi:10.1051/epjconf/201818202055)

See also

• Wikipedia, Noncommutative standard model

Original articles

The basic mechanism was originally laid out in

• Alain Connes, Gravity coupled with matter and foundation of non-commutative geometry, Commun.Math.Phys. 182 (1996) 155-176 (arXiv:hep-th/9603053)

(see also the references at spectral action).

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

• John Barrett, A Lorentzian version of the non-commutative geometry of the standard model of particle physics, J.Math.Phys.48:012303,2007 (arXiv:hep-th/0608221, doi:10.1063/1.2408400)

• Alain Connes, Noncommutative Geometry and the standard model with neutrino mixing, JHEP0611:081,2006 (arXiv:hep-th/0608226)

• Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007) 991-1089 [arXiv:hep-th/0610241, doi:10.4310/ATMP.2007.v11.n6.a3]

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

• Latham Boyle, Shane Farnsworth, Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics New J. Phys. 16, 123027 (2014) (arXiv:1401.5083)

• Shane Farnsworth, Latham Boyle, Rethinking Connes’ approach to the standard model of particle physics via non-commutative geometry, New J. Phys. 17, 023021 (2015) (arXiv:1408.5367)

• Christian Brouder, Nadir Bizi, Fabien Besnard, The Standard Model as an extension of the noncommutative algebra of forms (arXiv:1504.03890)

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

• Ali Chamseddine, Alain Connes, Resilience of the Spectral Standard Model, JHEP 1209 (2012) 104 [arXiv:1208.1030, doi:10.1007/JHEP09(2012)104]

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

• Ali Chamseddine, Alain Connes, Walter D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification, JHEP 1311 (2013) 132 [arXiv:1304.8050]

• Hosein Karimi Khozani, Symmetry Breaking and Proton Decay in Spectral Pati-Salam Model (arXiv:1905.04533)

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)

See also:

• 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

• Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Quanta of Geometry: Noncommutative Aspects, Phys. Rev. Lett. 114 (2015) 9, 091302 (arXiv:1409.2471)

• Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum: Basics, JHEP 12 (2014) 098 (arXiv:1411.0977)

• Alain Connes, section 4 of Geometry and the Quantum, in Foundations of Mathematics and Physics One Century After Hilbert, Springer 2018. 159-196 (arXiv:1703.02470, doi:10.1007/978-3-319-64813-2)

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:

• Latham Boyle, Shane Farnsworth,

  The standard model, the Pati-Salam model, and “Jordan geometry” (arxiv:1910.11888)

• Shane Farnsworth, The geometry of physical observables (arXiv:2003.01708)

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]