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\section*{Connes-Lott-Chamseddine-Barrett model}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesReview}{Review}\dotfill \pageref*{ReferencesReview} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Connes-Lott-Chamseddine-Barrett model} (\hyperlink{ConnesLott91}{Connes-Lott 91}, \hyperlink{Barrett07}{Barrett 07}, \hyperlink{ChamseddineConnesMarcolli07}{Chamseddine-Connes-Marcolli 07}) is a [[spectral triple]] that spectrally encodes a [[spacetime]] which is the [[product]] of 4d [[Minkowski spacetime]] $X$ with a \emph{[[non-geometric background|non-geometric]]} ([[Isbell duality|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 \emph{[[2-spectral triple]]}).

As in all ([[supersymmetry|super]]-)[[KK-compactification|Kaluza-Klein theory]], what is pure [[pseudo-Riemannian geometry]] (albeit ``[[spectral geometry|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

\begin{displaymath}
\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})
\end{displaymath}
which is the [[direct sum]] of the $\mathbb{R}$-[[associative algebra|algebras]] of [[complex numbers]], [[quaternions]] and $3x3$ complex [[matrices|matrix algebra]].

Moreover, the entire field content in one [[generation of fundamental particles|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 (\hyperlink{Connes06}{Connes 06, prop. 2.2 - prop 2.5}).

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

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Gepner model]]


\item [[higher category theory and physics]]: \emph{\href{higher+category+theory+and+physics#SpecStandModAndGravity}{Spectral standard model and gravity}}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{ReferencesReview}{}\subsubsection*{{Review}}\label{ReferencesReview}

\begin{itemize}%
\item Robert Brout, \emph{Notes on Connes' Construction of the Standard Model}, Nucl. Phys. Proc. Suppl. 65 (1998) 3-15 (\href{https://arxiv.org/abs/hep-th/9706200}{arXiv:hep-th/9706200})


\item Agostino Devastato, Maxim Kurkov, [[Fedele Lizzi]], \emph{Spectral Noncommutative Geometry, Standard Model and all that} (\href{https://arxiv.org/abs/1906.09583}{arXiv:1906.09583})



\end{itemize}
Relation to the ``[[neutrino]]-paradigm'':

\begin{itemize}%
\item [[José Gracia-Bondia]], \emph{On Marshak's and Connes' views of chirality}, in: [[E. C. G. Sudarshan]] (ed.), \emph{A Gift of Prophecy -- Essays in Celebration of the Life of [[Robert Eugene Marshak]]}, pp. 208-217, World Scientific (1995) (\href{https://arxiv.org/abs/hep-th/9706200}{arXiv:hep-th/9706200}, \href{https://doi.org/10.1142/9789812831408_0017}{doi:10.1142/9789812831408\_0017})

\end{itemize}
Exposition highlighting the relation to [[KK-compactification]] and [[string theory]]-[[perturbative string theory vacuum|vacua]] ([[2-spectral triples]]) includes

\begin{itemize}%
\item [[Urs Schreiber]], \emph{\href{https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/}{Spectral Standard Model and String Compactifications}}, 2016

\end{itemize}
Other commentary in view of [[D-brane]]-physics:

\begin{itemize}%
\item [[John Iliopoulos]], \emph{Gauge Theories and non-Commutative Geometry: A review}, EPJ Web Conf. Volume 182, 2018 6th International Conference on New Frontiers in Physics (ICNFP 2017) (\href{https://doi.org/10.1051/epjconf/201818202055}{doi:10.1051/epjconf/201818202055})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Noncommutative_standard_model}{Noncommutative standard model}}

\end{itemize}
\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

The basic mechanism was originally laid out in

\begin{itemize}%
\item [[Alain Connes]], \emph{Gravity coupled with matter and foundation of non-commutative geometry}, Commun.Math.Phys. 182 (1996) 155-176 (\href{https://arxiv.org/abs/hep-th/9603053}{arXiv:hep-th/9603053})

\end{itemize}
(see also the references at \emph{[[spectral action]]}).

The early version of the model is due to

\begin{itemize}%
\item [[Alain Connes]], [[John Lott]], \emph{Particle models and noncommutative geometry}, Nuclear Physics B - Proceedings Supplements 18(2): 29-47 (1991) (\href{https://deepblue.lib.umich.edu/handle/2027.42/29524}{web})

\end{itemize}
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

\begin{itemize}%
\item [[John Barrett]], \emph{A Lorentzian version of the non-commutative geometry of the standard model of particle physics}, J.Math.Phys.48:012303,2007 (\href{http://arxiv.org/abs/hep-th/0608221}{arXiv:hep-th/0608221})


\item [[Alain Connes]], \emph{Noncommutative Geometry and the standard model with neutrino mixing}, JHEP0611:081,2006 (\href{https://arxiv.org/abs/hep-th/0608226}{arXiv:hep-th/0608226})


\item [[Ali Chamseddine]], [[Alain Connes]], [[Matilde Marcolli]], \emph{Gravity and the standard model with neutrino mixing}, Adv.Theor.Math.Phys.11:991-1089,2007 (\href{http://arxiv.org/abs/hep-th/0610241}{arXiv:hep-th/0610241})



\end{itemize}
A more succinct version of the axioms of the model is claimed in

\begin{itemize}%
\item [[Latham Boyle]], [[Shane Farnsworth]], \emph{Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics} New J. Phys. 16, 123027 (2014) (\href{http://arxiv.org/abs/1401.5083}{arXiv:1401.5083})


\item [[Shane Farnsworth]], [[Latham Boyle]], \emph{Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry}, New J. Phys. 17, 023021 (2015) (\href{https://arxiv.org/abs/1408.5367}{arXiv:1408.5367})


\item Christian Brouder, Nadir Bizi, Fabien Besnard, \emph{The Standard Model as an extension of the noncommutative algebra of forms} (\href{https://arxiv.org/abs/1504.03890}{arXiv:1504.03890})



\end{itemize}
Introduction of a [[scalar field]] to fix the prediction of the [[Higgs particle]] [[mass]]:

\begin{itemize}%
\item [[Ali Chamseddine]], [[Alain Connes]], \emph{Resilience of the Spectral Standard Model}, JHEP 1209 (2012) 104 (\href{https://arxiv.org/abs/1208.1030}{arXiv:1208.1030})

\end{itemize}
Implementation of [[Pati-Salam model]] ($SU(5)$-[[GUT]]):

\begin{itemize}%
\item [[Ali Chamseddine]], [[Alain Connes]], Walter D. van Suijlekom, \emph{Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification}, JHEP 1311 (2013) 132 (\href{https://arxiv.org/abs/1304.8050}{arXiv:1304.8050})


\item Hosein Karimi Khozani, \emph{Symmetry Breaking and Proton Decay in Spectral Pati-Salam Model} (\href{https://arxiv.org/abs/1905.04533}{arXiv:1905.04533})



\end{itemize}
and specifically discussion of [[leptoquarks]] and possible relation to [[flavour anomalies]]:

\begin{itemize}%
\item Ufuk Aydemir, Djordje Minic, Chen Sun, Tatsu Takeuchi, \emph{$B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry}, JHEP 09 (2018) 117 (\href{https://arxiv.org/abs/1804.05844}{arXiv:1804.05844})

\end{itemize}
Relation to actual [[quantum gravity]] induced by quantized maps from [[spacetime]] to the [[4-sphere]] (see also at \emph{[[Cohomotopy]]}) is claimed in

\begin{itemize}%
\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Quanta of Geometry: Noncommutative Aspects}, Phys. Rev. Lett. 114 (2015) 9, 091302 (\href{https://arxiv.org/abs/1409.2471}{arXiv:1409.2471})


\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Geometry and the Quantum: Basics}, JHEP 12 (2014) 098 (\href{https://arxiv.org/abs/1411.0977}{arXiv:1411.0977})


\item [[Alain Connes]], section 4 of \emph{Geometry and the Quantum}, in \emph{Foundations of Mathematics and Physics One Century After Hilbert}, Springer 2018. 159-196 (\href{https://arxiv.org/abs/1703.02470}{arXiv:1703.02470}, \href{https://www.springer.com/gp/book/9783319648125}{doi:10.1007/978-3-319-64813-2})



\end{itemize}
[[!redirects Connes-Lott-Chamseddine-Barrett models]]

[[!redirects Connes-Lott-Chamseddine model]] [[!redirects Connes-Lott-Chamseddine models]]

[[!redirects Connes-Lott model]] [[!redirects Connes-Lott models]]

[[!redirects non-commutative standard model]] [[!redirects non-commutative standard models]]



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