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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
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)
Exposition highlighting the relation to KK-compactification and string theory-vacua includes
Other comment in view of D-brane-physics:
See also
The basic mechanism was originally laid out in
(see also the references at spectral action).
The early version of the model is due to
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)
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.11:991-1089,2007 (arXiv:hep-th/0610241)
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)
Last revised on January 13, 2019 at 03:39:16. See the history of this page for a list of all contributions to it.