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\begin{document}

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\section*{Gepner model}

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

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BoundaryStates}{Boundary states}\dotfill \pageref*{BoundaryStates} \linebreak
\noindent\hyperlink{Phenomenology}{Phenomenology}\dotfill \pageref*{Phenomenology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Gepner model} (\hyperlink{Gepner87}{Gepner 87}) is a [[rational CFT|rational]] [[2d SCFT]] which is a [[tensor product]] of $N = 2$ super-[[minimal model CFT]].

This means that Gepner models are ``[[non-geometric string vacua]]'' in that they do not arise as [[sigma-models]] with [[target space]] a [[smooth manifold]]. Indeed the Gepner models appear as the limiting cases of sigma-models with [[target space]] a 6d [[Calabi-Yau manifold]] at singular points in the [[moduli space]] of the CY target: the \emph{[[flop transition]]}.

As such the Gepner models are directly analogous to the purely algebraically defined non-classical fibers in the [[Connes-Lott-Chamseddine-Barrett model]] (it is a ``[[2-spectral triple]]''-analog of the [[spectral triples]] in the Connes-Lott model, see \href{Connes-Lott-Chamseddine-Barrett+model#Schreiber16}{there}) and, accordingly, plays a central role in [[string phenomenology]] (for review see e.g. \hyperlink{Reppel07}{Reppel 07}).

The Gepner models are a basic building block for [[rational conformal field theory]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{BoundaryStates}{}\subsubsection*{{Boundary states}}\label{BoundaryStates}

\begin{quote}%
All the known [[rational CFT|rational]] [[boundary states]] for [[Gepner models]] can be regarded as [[permutation branes]].


\end{quote}
(\hyperlink{EngerRecknagelRoggenkamp05}{Enger-Recknagel-Roggenkamp 05})

\hypertarget{Phenomenology}{}\subsubsection*{{Phenomenology}}\label{Phenomenology}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with [[non-geometric background|non-geometric]] [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of \emph{\href{https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/}{Spectral Standard Model and String Compactifications}}) is in (\hyperlink{DijkstraHuiszoonSchellekens04a}{Dijkstra-Huiszoon-Schellekens 04a}, \hyperlink{DijkstraHuiszoonSchellekens04a}{Dijkstra-Huiszoon-Schellekens 04b}):

\begin{quote}%
A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to \hyperlink{DijkstraHuiszoonSchellekens04b}{Dijkstra-Huiszoon-Schellekens 04b}.

The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

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


\item [[Kazama-Suzuki model]]


\item [[heterotic string]], [[type II superstring]]


\item [[KK-compactification]]



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

The original article is

\begin{itemize}%
\item [[Doron Gepner]], \emph{Space-time supersymmetry in compactified string theory and superconformal models}, Nucl. Phys. B296 (1987) 757.

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Doron Gepner]], \emph{Lectures On $N=2$ String Theory}, 1989 ([[Gepner89.pdf:file]], \href{http://inspirehep.net/record/277718/}{spire:277718/})

\end{itemize}
Further discussion in

\begin{itemize}%
\item [[Maximilian Kreuzer]], \emph{Heterotic $(0,2)$ Gepner Models and Related Geometries}, in \emph{Fundamental Interactions}, pp. 335-362 World Scientific (2009) (\href{https://arxiv.org/abs/0904.4467}{arXiv:0904.4467}, \href{https://doi.org/10.1142/9789814277839_0019}{doi:10.1142/9789814277839\_0019})

\end{itemize}
See also the references at \emph{[[flop transition]]} for more.

Review of application in [[string phenomenology]] includes

\begin{itemize}%
\item Christian Reppel, \emph{Phenomenological Aspects of Gepner Models}, 2007 (\href{https://www.ru.nl/publish/pages/760962/christiaan_reppel.pdf}{pdf})

\end{itemize}
[[D-branes]] in [[string theory]] [[vacua]] defined by Gepner model SCFTs are discussed, via [[boundary conformal field theory]] in

\begin{itemize}%
\item [[Jürgen Fuchs]], [[Christoph Schweigert]], J. Walcher, \emph{Projections in string theory and boundary states for Gepner models}, Nucl.Phys. B588 (2000) 110-148 (\href{https://arxiv.org/abs/hep-th/0003298}{arXiv:hep-th/0003298})

(with emphasis on [[GSO projections]])


\item [[Jürgen Fuchs]], P. Kaste, [[Wolfgang Lerche]], C. Lutken, [[Christoph Schweigert]], J. Walcher, \emph{Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3}, Nucl.Phys.B598:57-72, 2001 (\href{https://arxiv.org/abs/hep-th/0007145}{arXiv:hep-th/0007145})



\end{itemize}
See also

\begin{itemize}%
\item [[Andreas Recknagel]], [[Volker Schomerus]], \emph{D-branes in Gepner models}, Nucl.Phys. B531 (1998) 185-225 (\href{http://arxiv.org/abs/hep-th/9712186}{arXiv:hep-th/9712186})


\item [[Volker Braun]], Sakura Schafer-Nameki, \emph{D-Brane Charges in Gepner Models}, J.Math.Phys. 47 (2006) 092304 (\href{https://arxiv.org/abs/hep-th/0511100}{arXiv:hep-th/0511100})



\end{itemize}
Discussion of [[permutation D-branes]] for Gepner models, via [[boundary conformal field theory]], includes

\begin{itemize}%
\item Håkon Enger, [[Andreas Recknagel]], [[Daniel Roggenkamp]], \emph{Permutation branes and linear matrix factorisations}, JHEP0601:087, 2006 (\href{https://arxiv.org/abs/hep-th/0508053}{arXiv:hep-th/0508053})

\end{itemize}
Gepner model [[orientifolds]]:

\begin{itemize}%
\item Brandon Bates, [[Charles Doran]], Koenraad Schalm, \emph{Crosscaps in Gepner Models and the Moduli space of T2 Orientifolds}, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5, 839-912, 2007 (\href{https://arxiv.org/abs/hep-th/0612228}{arXiv:hep-th/0612228})

\end{itemize}
Specifically [[string phenomenology]] and the [[landscape of string theory vacua]] of Gepner model [[orientifold]] compactifications:

\begin{itemize}%
\item T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], \emph{Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models}, Phys.Lett. B609 (2005) 408-417 (\href{https://arxiv.org/abs/hep-th/0403196}{arXiv:hep-th/0403196})


\item T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], \emph{Supersymmetric Standard Model Spectra from RCFT orientifolds}, Nucl.Phys.B710:3-57,2005 (\href{https://arxiv.org/abs/hep-th/0411129}{arXiv:hep-th/0411129})



\end{itemize}
[[!redirects Gepner models]]

[[!redirects Gepner point]] [[!redirects Gepner points]]



\end{document}