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\section*{boundary conformal field theory}

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

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum Field Theory}}\label{quantum_field_theory}

[[!include AQFT and operator algebra 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{results}{Results}\dotfill \pageref*{results} \linebreak
\noindent\hyperlink{computer_scan_of_gepnermodel_compactifications}{Computer scan of Gepner-model compactifications}\dotfill \pageref*{computer_scan_of_gepnermodel_compactifications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_ktheory}{Relation to K-theory}\dotfill \pageref*{relation_to_ktheory} \linebreak
\noindent\hyperlink{for_rational_cft_wzw_models_and_gepner_models}{For rational CFT, WZW models and Gepner models}\dotfill \pageref*{for_rational_cft_wzw_models_and_gepner_models} \linebreak
\noindent\hyperlink{on_orbifolds}{On orbifolds}\dotfill \pageref*{on_orbifolds} \linebreak


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

The study of [[conformal field theory]] on [[manifolds with boundary]], hence conformal [[boundary field theory]], is known as \emph{boundary conformal field theory} (often abbreviated as BCFT).

For the special case of [[2d CFT]] understood as the [[quantum field theory]] on the [[worldsheet]] of [[strings]], boundary conformal field theory is the key ingredient of [[perturbative string theory]] with prescribed [[boundary conditions]] for the [[open strings]]. From the [[target spacetime]]-perspective these [[boundary conditions]] are interpreted as the [[D-branes]] that the [[open strings]] may ``end on''.

In this way 2d BCFT can serve to give an [[algebra|algebraic]] definition of [[D-branes]], independent of or complementary to their incarnation in the [[target space|target]]-[[spacetime]] [[geometry]]. In particular [[fractional D-branes]] may be described this way as boundary conditions for the [[2d CFTs]] describing [[open strings]] propagating in [[orbifolds]].

A key tool of BCFT is the \emph{boundary state} formalism, where one interprets [[boundary conditions]] for the [[open string]] as [[coherent states]] of the corresponding [[closed string]] [[2d CFT]]. The physical interpretation of this algebraic fact is that the [[D-brane]], being [[mass|massive]] (see also at \emph{[[black brane]]}) emits and absorbs [[gravitons]], which, in [[string theory]], are excitations of the [[closed string]].

It has been argued that, therefore, for [[sigma-model]] [[2d CFT]]s the classification of [[equivalence classes]] of BCFT boundary states under [[renormalization group flow]] should reproduce the expected classification of [[D-branes]] by the ([[equivariant K-theory|equivariant]], [[twisted K-theory]]) [[topological K-theory]] of the [[target spacetime]] (\hyperlink{Moore03}{Moore 03, section 3}).

While this gives, in principle, a precise algebraic definition of [[D-brane charge]] from the [[worldsheet]]-perspective, in practice it is hard to follow the [[renormalization group flow]] of the boundary states. The proposal has been checked in some special cases (e.g. \hyperlink{BraunSchaeferNameki05}{Braun \& Schaefer-Nameki 05}). Other authors reported some discrepancies (\hyperlink{QuirozStefanski01}{Quiroz-Stefanski 01})

Specifically for [[fractional D-branes]] at $G$-[[orbifold]] [[singularities]] the relevant boundary states are [[sums]] of boundary states for each ``$g$-twisted sector'' of the [[closed string]] [[2d CFT]], consisting of those string configurations that are closed in the [[covering space]] up to the [[action]] of the element $g \in G$:

\begin{quote}%
graphics grabbed from \hyperlink{BCR00}{BCR 00}


\end{quote}
Here the [[coefficient]] of the $g$-twisted sector in the total boundary state is proprotional to the value $\chi_V(g)$ of the [[character]] $\chi_V$ of some $G$-[[representation]] $V \in R_{\mathbb{C}}(G) = KU_G(\ast)$ (e.g. \hyperlink{RecknagelSchomerus13}{Recknagel-Schomerus 13 (4.102)}). This makes it plausible that the [[RG-flow]]-[[equivalence classes]] of the boundary states for [[fractional D-branes]] do coincide with the [[equivariant K-theory]] $KU_G(\ast) = R_{\mathbb{C}}(G)$ of the [[orbifold]] singularity (the [[representation ring]] of $G$).

\begin{quote}%
But maybe this has not yet been actually proven?


\end{quote}
\hypertarget{results}{}\subsection*{{Results}}\label{results}

There are some mistakes in the literature. A clean account is in \hyperlink{FuchsSchweigertWalcher00}{Fuchs-Schweigert-Walcher 00}, \hyperlink{FuchsKasteLercheLutkenSchweigert00}{Fuchs-Kaste-Lerche-Lutken-Schweigert 00}

\hypertarget{computer_scan_of_gepnermodel_compactifications}{}\subsection*{{Computer scan of Gepner-model compactifications}}\label{computer_scan_of_gepnermodel_compactifications}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with 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{DijkstraHuiszoonSchellekens04b}{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 [[boundary field theory]]


\item [[fractional brane]], [[permutation brane]]



\end{itemize}
[[!include field theory with boundaries and defects - table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The boundary state formalism in BCFT is due to

\begin{itemize}%
\item C. G. Callan, C. Lovelace, C. R. Nappi and S. A. Yost, \emph{Adding holes and crosscaps to the superstring}, Nucl. Phys. B 293 (1987) 83 ()


\item C. G. Callan, C. Lovelace, C. R. Nappi and S. A. Yost, \emph{Loop corrections to superstring equations of motion}, Nucl. Phys. B308 (1988) 221 (\href{https://www.researchgate.net/profile/Scott_Yost/publication/222468211_Loop_Corrections_to_Superstring_Equations_of_Motion/links/5a54d5eda6fdccd72f5b4583/Loop-Corrections-to-Superstring-Equations-of-Motion.pdf}{pdf})


\item N. Ishibashi, \emph{Boundary and Crosscap States in Conformal Field Theories}, Mod. Phys. Lett. A4(1989) 251 (\href{https://inis.iaea.org/collection/NCLCollectionStore/_Public/20/072/20072844.pdf#page=125}{pdf})


\item N. Ishibashi and T. Onogi, \emph{Open string model building}, Nucl. Phys. B318(1989) 239 ()


\item \hyperlink{DixonFriedanMartinecShenker87}{Dixon-Friedan-Martinec-Shenker 87}



\end{itemize}
See also

\begin{itemize}%
\item [[Andreas Recknagel]], [[Volker Schomerus]], \emph{Boundary Deformation Theory and Moduli Spaces of D-Branes}, Nucl.Phys. B545 (1999) 233-282 (\href{https://arxiv.org/abs/hep-th/9811237}{arXiv:hep-th/9811237})

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Andreas Recknagel]], [[Volker Schomerus]], \emph{Boundary Conformal Field Theory and the Worldsheet Approach to D-branes}, Cambridge 2013 (\href{http://inspirehep.net/record/1308990}{spire:1308990})

\end{itemize}
\hypertarget{relation_to_ktheory}{}\subsubsection*{{Relation to K-theory}}\label{relation_to_ktheory}

The suggestion that [[renormalization group flow]]-[[equivalence classes]] of boundary states in 2d BCFT should realize the expected classification of [[D-brane charge]] in ([[equivariant K-theory|equivariant]]) [[topological K-theory]] apparntly goes back to

\begin{itemize}%
\item [[Gregory Moore]], section 3 of \emph{K-Theory from a physical perspective} (\href{https://arxiv.org/abs/hep-th/0304018}{arXiv:hep-th/0304018})

\end{itemize}
This has been checked in examples in

\begin{itemize}%
\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})


\item [[Igor Kriz]], Leopoldo A. Pando Zayas, Norma Quiroz, \emph{Comments on D-branes on Orbifolds and K-theory}, Int.J.Mod.Phys.A23:933-974, 2008 (\href{https://arxiv.org/abs/hep-th/0703122}{arXiv:hep-th/0703122})



\end{itemize}
A mismatch was claimed in

\begin{itemize}%
\item \hyperlink{QuirozStefanski01}{Quiroz-Stefanski 01}

\end{itemize}
\hypertarget{for_rational_cft_wzw_models_and_gepner_models}{}\subsubsection*{{For rational CFT, WZW models and Gepner models}}\label{for_rational_cft_wzw_models_and_gepner_models}

For [[rational CFT]], specifically [[Gepner models]] and [[WZW models]]

\begin{itemize}%
\item [[Ilka Brunner]], [[Michael Douglas]], Albion Lawrence, Christian Romelsberger, \emph{D-branes on the Quintic}, JHEP 0008 (2000) 015 (\href{https://arxiv.org/abs/hep-th/9906200}{arXiv:hep-th/9906200})


\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})

(with emphasis on [[gauge enhancement]] on coincident D-branes)


\item Sergei E. Parkhomenko, \emph{Free Field Construction of D-Branes in Rational Models of CFT and Gepner Models} (\href{https://arxiv.org/abs/0802.3445}{arXiv:0802.3445})


\item [[Giovanni Felder]], [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Christoph Schweigert]], \emph{The geometry of WZW branes} (\href{https://arxiv.org/abs/hep-th/9909030}{arXiv:hep-th/9909030})


\item \hyperlink{BraunSchaeferNameki05}{Braun \& Schaefer-Nameki 05}



\end{itemize}
\hypertarget{on_orbifolds}{}\subsubsection*{{On orbifolds}}\label{on_orbifolds}

Discussion specifically concerning [[fractional D-branes]] on [[orbifolds]] includes

\begin{itemize}%
\item [[Lance Dixon]], [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], \emph{The conformal field theory of orbifolds}, Nucl. Phys. B 282 (1987) 13. ()


\item Diaconescu, [[Jaume Gomis]], \emph{Fractional Branes and Boundary States in Orbifold Theories}, JHEP 0010 (2000) 001 (\href{https://arxiv.org/abs/hep-th/9906242}{arXiv:hep-th/9906242})


\item M. Billo', B. Craps, F. Roose, \emph{Orbifold boundary states from Cardy's condition}, JHEP 0101:038, 2001 (\href{https://arxiv.org/abs/hep-th/0011060}{arXiv:hep-th/0011060})


\item N. Quiroz, [[Bogdan Stefanski]] Jr, \emph{Dirichlet Branes on Orientifolds}, Phys.Rev. D66 (2002) 026002 (\href{https://arxiv.org/abs/hep-th/0110041}{arXiv:hep-th/0110041})


\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})


\item Jaydeep Majumder, Subir Mukhopadhyay, Koushik Ray, \emph{Fractional Branes in Non-compact Type IIA Orientifolds}, JHEP0611:008, 2006 (\href{https://arxiv.org/abs/hep-th/0602135}{arXiv:hep-th/0602135})



\end{itemize}
[[!redirects boundary conformal field theories]]

[[!redirects boundary CFT]] [[!redirects boundary CFTs]]

[[!redirects BCFT]] [[!redirects BCFTs]]

[[!redirects boundary state]] [[!redirects boundary states]]

[[!redirects twisted sector]] [[!redirects twisted sectors]]



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