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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{orientifold} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - 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{lifts_to_mtheory_ftheory}{Lifts to M-theory F-theory}\dotfill \pageref*{lifts_to_mtheory_ftheory} \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{in_terms_of_kotheory}{In terms of KO-theory}\dotfill \pageref*{in_terms_of_kotheory} \linebreak \noindent\hyperlink{examples_and_models}{Examples and Models}\dotfill \pageref*{examples_and_models} \linebreak \noindent\hyperlink{orientifold_gepner_models}{Orientifold Gepner models}\dotfill \pageref*{orientifold_gepner_models} \linebreak \noindent\hyperlink{ReferencesInMTheory}{Lift to M-theory}\dotfill \pageref*{ReferencesInMTheory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} An \emph{orientifold} (\hyperlink{DaiLinPolchinski89}{Dai-Lin-Polchinski 89, p. 12}) is a [[target space|target]] [[spacetime]] for [[string theory|string]] [[sigma-models]] that combines aspects of $\mathbb{Z}_2$-[[orbifold]]s with \emph{[[orientation]] reversal} on the worldsheet, whence the name. In [[type II string theory]] orientifold backgrounds (inducing [[type I string theory]]) with $\mathbb{Z}_2$-[[fixed points]] -- called \emph{[[O-planes]]} (see there for more) -- are required for [[RR-field tadpole cancellation]]. This is a key consistency condition in particular for [[intersecting D-brane models]] used in [[string phenomenology]]. Where generally ([[higher gauge field|higher gauge]]) [[field (physics)|fields]] in [[physics]]/[[string theory]] are [[cocycles]] in ([[differential cohomology theory|differential]]) [[cohomology theory]] and typically in [[complex oriented cohomology theory]], fields on orientifolds are cocycles in genuinely $\mathbb{Z}_2$-[[equivariant cohomology]] and typically in [[real-oriented cohomology theory]]. For instance, the [[B-field]], which otherwise is a (twisted) cocycle in ([[ordinary differential cohomology|ordinary]]) [[differential cohomology]], over an orientifold is a cocycle in (twisted) [[HZR-theory]], and the [[RR-fields]], which usually are cocycles in ([[twisted cohomology|twisted]] [[differential K-theory|differential]]) [[K-theory]], over an orientifold are cocycles in [[KR-theory]] (\hyperlink{Witten98}{Witten 98}). An explicit model for [[B-fields]] for the [[bosonic string]] on orientifolds (differential [[HZR-theory]]) is given in (\hyperlink{Jandl}{Schreiber-Schweigert-Waldorf 05}) and examples are analyzed in this context in (\hyperlink{GawedzkiSuszekWaldorf08}{Gawedzki-Suszek-Waldorf 08}). See also (\hyperlink{HMSV16}{HMSV 16}, \hyperlink{HMSV19}{HMSV 19}). The claim that for the [[superstring]] the B-field is more generally a cocycle with coefficients in the [[Picard infinity-group]] of [[complex K-theory]] ([[super line 2-bundles]]) and a detailed discussion of the orientifold version of this can be found in (\hyperlink{Precis}{Distler-Freed-Moore 09}, \hyperlink{DistlerFreedMooreII}{Distler-Freed-Moore 10}) with details in (\hyperlink{Freed}{Freed 12}). The quadratic pairing entering the [[11d Chern-Simons theory]] that governs the [[RR-field]] here as a [[self-dual higher gauge field]] is given in (\hyperlink{Precis}{DFM 10, def. 6}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{lifts_to_mtheory_ftheory}{}\subsubsection*{{Lifts to M-theory F-theory}}\label{lifts_to_mtheory_ftheory} Lifts of orientifold background from [[type II string theory]] to [[F-theory]] go back to (\hyperlink{Sen96}{Sen 96}, \hyperlink{Sen97a}{Sen 97a}). Lifts of [[type IIA string theory]] orientifolds of [[D6-branes]] to [[ADE singularities|D-type ADE singularities]] in [[M-theory]] (through the [[duality between M-theory and type IIA string theory]]) goes back to (\hyperlink{Sen97b}{Sen 97b}). See at \emph{[[heterotic M-theory on ADE-orbifolds]]}. A more general scan of possible lifts of type IIA orientifolds to M-theory is indicated in (\hyperlink{HananyKol00}{Hanany-Kol 00, around (3.2)}), see (\hyperlink{HuertaSatiSchreiber18}{Huerta-Sati-Schreiber 18, Prop. 4.7}) for details. For instance the [[O4-plane]] lifts to the [[MO5-plane]]. (\ldots{}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[fractional D-brane]] [[permutation D-brane]] \item [[O-plane]], [[O-plane charge]], [[RR-field tadpole cancellation]] \item [[worldsheet parity operator]] \item [[type I string theory]] \item [[real-oriented cohomology theory]] \item [[higher orientifold]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The concept originates around \begin{itemize}% \item Jin Dai, R.G. Leigh, [[Joseph Polchinski]], p. 12 of \emph{New Connections Between String Theories}, Mod.Phys.Lett. A4 (1989) 2073-2083 (\href{http://inspirehep.net/record/25758}{spire:25758}, \href{https://lib-extopc.kek.jp/preprints/PDF/1989/8905/8905564.pdf}{pdf scan}) \end{itemize} Early accounts include \begin{itemize}% \item [[Sunil Mukhi]], \emph{Orientifolds: The Unique Personality Of Each Spacetime Dimension}, \href{http://cds.cern.ch/record/323857}{Workshop on Frontiers of Field Theory, Quantum Gravity and String Theory, Puri, India, 12 - 21 Dec 1996} (\href{https://arxiv.org/abs/hep-th/9710004}{arXiv:hep-th/9710004}, \href{http://cds.cern.ch/record/335233}{cern:335233}) \item [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of \emph{Triples, Fluxes, and Strings}, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (\href{https://arxiv.org/abs/hep-th/0103170}{arXiv:hep-th/0103170}) \end{itemize} Traditional lecture notes include \begin{itemize}% \item Atish Dabholkar, \emph{Lectures on Orientifolds and Duality} (\href{https://arxiv.org/abs/hep-th/9804208}{arXiv:hep-th/9804208}) \item Carlo Angelantonj, [[Augusto Sagnotti]], \emph{Open Strings}, Phys. Rept. 371:1-150,2002; Erratum ibid.376:339-405, 2003 (\href{https://arxiv.org/abs/hep-th/0204089}{arXiv:hep-th/0204089}) \end{itemize} Textbook discussion is in \begin{itemize}% \item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Section 15.3 of \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics, Springer 2013 \end{itemize} and specifically in the context of [[intersecting D-brane models]] with an eye towards [[string phenomenology]] in \begin{itemize}% \item [[Luis Ibáñez]], [[Angel Uranga]], sections 5.3.4 and 10.1.3 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012 \end{itemize} Exposition: \begin{itemize}% \item Marcus Berg, \emph{Introduction to Orientifolds} (\href{https://tp.hotell.kau.se/marcus/physics/talks/orienti_short.pdf}{pdf}, [[BergOrientifolds.pdf:file]]) \end{itemize} The original observation that [[D-brane charge]] for orientifolds should be in [[KR-theory]] is due to \begin{itemize}% \item [[Edward Witten]], section 5 of \emph{D-branes and K-theory}, J. High Energy Phys., 1998(12):019, 1998 (\href{http://arxiv.org/abs/hep-th/9810188}{arXiv:hep-th/9810188}) \end{itemize} and was re-amplified in \begin{itemize}% \item [[Sergei Gukov]], \emph{K-Theory, Reality, and Orientifolds}, Commun.Math.Phys. 210 (2000) 621-639 (\href{http://arxiv.org/abs/hep-th/9901042}{arXiv:hep-th/9901042}) \end{itemize} \hypertarget{in_terms_of_kotheory}{}\subsubsection*{{In terms of KO-theory}}\label{in_terms_of_kotheory} Discussion of orbi-orienti-folds in terms of [[equivariant K-theory|equivariant]] [[KO-theory]] is in \begin{itemize}% \item N. Quiroz, [[Bogdan Stefanski]], \emph{Dirichlet Branes on Orientifolds}, Phys.Rev. D66 (2002) 026002 (\href{https://arxiv.org/abs/hep-th/0110041}{arXiv:hep-th/0110041}) \item [[Volker Braun]], [[Bogdan Stefanski]], \emph{Orientifolds and K-theory}, Braun, Volker. ``Orientifolds and K-theory.'' Progress in String, Field and Particle Theory. Springer, Dordrecht, 2003. 369-372 (\href{https://arxiv.org/abs/hep-th/0206158}{arXiv:hep-th/0206158}) \item H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, \emph{D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory}, JHEP 0812:007, 2008 (\href{https://arxiv.org/abs/0809.4238}{arXiv:0809.4238}) \end{itemize} A definition and study of orientifold [[bundle gerbes]], modeling the [[B-field]] [[background gauge field|background]] for the [[bosonic string]] (differential [[HZR-theory]]), is in \begin{itemize}% \item [[Urs Schreiber]], [[Christoph Schweigert]], [[Konrad Waldorf]], \emph{Unoriented WZW models and Holonomy of Bundle Gerbes}, Communications in Mathematical Physics August 2007, Volume 274, Issue 1, pp 31-64 (\href{http://arxiv.org/abs/hep-th/0512283}{arXiv}) \item [[Krzysztof Gawedzki]], Rafal R. Suszek, [[Konrad Waldorf]], \emph{Bundle Gerbes for Orientifold Sigma Models} Adv. Theor. Math. Phys. 15(3), 621-688 (2011) (\href{http://arxiv.org/abs/0809.5125}{arXiv:0809.5125}) \end{itemize} see also \begin{itemize}% \item [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], \emph{Real bundle gerbes, orientifolds and twisted KR-homology} (\href{http://arxiv.org/abs/1608.06466}{arXiv:1608.06466}) \item [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], \emph{Sign choices for orientifolds} (\href{https://arxiv.org/abs/1905.06041}{arXiv:1905.06041}) \end{itemize} An elaborate formalization in terms of [[differential cohomology]] in general and [[twisted K-theory|twisted]] [[differential K-theory]] in particular that also takes the spinorial degrees of freedom into account is briefly sketched out in \begin{itemize}% \item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides}) \end{itemize} based on stuff like \begin{itemize}% \item [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], \emph{Orientifolds, RR Torsion, and K-theory}, JHEP 0105:047, 2001 (\href{https://arxiv.org/abs/hep-th/0103183}{arXiv:hep-th/0103183}) \end{itemize} Details on the computation of [[string scattering amplitudes]] in such a background: \begin{itemize}% \item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Spin structures and superstrings}, Surveys in Differential Geometry, Volume 15 (2010) (\href{http://arxiv.org/abs/1007.4581}{arXiv:1007.4581}, \href{http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4}{doi:10.4310/SDG.2010.v15.n1.a4}) \end{itemize} Related lecture notes / slides include \begin{itemize}% \item [[Daniel Freed]], \emph{Dirac charge quantiation, K-theory, and orientifolds}, talk at a workshop \emph{Mathematical methods in general relativity and quantum field theories}, Paris, November 2009 (\href{http://www.ma.utexas.edu/users/dafr/paris_nt.pdf}{pdf}, [[FreedK09.pdf:file]]) \item [[Greg Moore]], \emph{The RR-charge of an orientifold}, Oberwolfach talk 2010 (\href{https://www.physics.rutgers.edu/~gmoore/Oberwolfach_June2010_FINAL.pdf}{pdf}, [[MooreOrientifold2010.pdf:file]], \href{http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt}{ppt}) \item [[Daniel Freed]], \emph{Lectures on twisted K-theory and orientifolds}, lecures at \emph{\href{http://www.esi.ac.at/activities/events/2012/k-theory-and-quantum-fields}{K-Theory and Quantum Fields}}, ESI 2012 ([[FreedESI2012.pdf:file]]) \end{itemize} A detailed list of examples of [[KR-theory]] of orientifolds and their [[T-duality]] is in \begin{itemize}% \item [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], \emph{T-duality For Orientifolds and Twisted KR-theory} (\href{http://arxiv.org/abs/1306.1779}{arXiv:1306.1779}) \item [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], \emph{String theory on elliptic curve orientifolds and KR-theory} (\href{http://arxiv.org/abs/1402.4885}{arXiv:1402.4885}) \end{itemize} A formulation of some of the relevant aspects of (bosonic) orientifolds in terms of the [[schreiber:differential cohomology in an (∞,1)-topos -- survey|differential nonabelian cohomology]] with coefficients in the [[2-group]] $AUT(U(1))$ coming from the [[crossed module]] $[U(1) \to \mathbb{Z}_2]$ is indicated in \begin{itemize}% \item [[Urs Schreiber]], talk [[schreiber:Background fields in twisted differential nonabelian cohomology|Background fields in twisted differential nonabelian cohomology]] at [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] \end{itemize} More on this in section 3.3.10 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} \hypertarget{examples_and_models}{}\subsubsection*{{Examples and Models}}\label{examples_and_models} Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIB string theory]], hence for [[D9-branes]] and [[D5-branes]]: \begin{itemize}% \item Eric G. Gimon, [[Joseph Polchinski]], Section 3.2 of: \emph{Consistency Conditions for Orientifolds and D-Manifolds}, Phys. Rev. D54: 1667-1676, 1996 (\href{https://arxiv.org/abs/hep-th/9601038}{arXiv:hep-th/9601038}) \item [[Eric Gimon]], [[Clifford Johnson]], \emph{K3 Orientifolds}, Nucl. Phys. B477: 715-745, 1996 (\href{https://arxiv.org/abs/hep-th/9604129}{arXiv:hep-th/9604129}) \item Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, \emph{Anomaly Cancelations in Orientifolds with Quantized B Flux}, Nucl.Phys. B569 (2000) 329-361 (\href{https://arxiv.org/abs/hep-th/9907203}{arXiv:hep-th/9907203}) \item P. Anastasopoulos, A. B. Hammou, \emph{A Classification of Toroidal Orientifold Models}, Nucl. Phys. B729:49-78, 2005 (\href{https://arxiv.org/abs/hep-th/0503044}{arXiv:hep-th/0503044}) \end{itemize} Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]: \begin{itemize}% \item J. Park, [[Angel Uranga]], \emph{A Note on Superconformal N=2 theories and Orientifolds}, Nucl. Phys. B542:139-156, 1999 (\href{https://arxiv.org/abs/hep-th/9808161}{arXiv:hep-th/9808161}) \item G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], \emph{D=4 Chiral String Compactifications from Intersecting Branes}, J. Math. Phys. 42:3103-3126, 2001 (\href{https://arxiv.org/abs/hep-th/0011073}{arXiv:hep-th/0011073}) \item G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], \emph{Intersecting Brane Worlds}, JHEP 0102:047, 2001 (\href{https://arxiv.org/abs/hep-ph/0011132}{arXiv:hep-ph/0011132}) \item H. Kataoka, M. Shimojo, \emph{$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes}, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 (\href{https://arxiv.org/abs/hep-th/0112247}{arXiv:hep-th/0112247}, \href{https://doi.org/10.1143/PTP.107.1291}{doi:10.1143/PTP.107.1291}) \end{itemize} \begin{quote}% The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$ Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. (\href{https://arxiv.org/pdf/hep-th/0112247.pdf#page=4}{p. 4}) \end{quote} \begin{itemize}% \item [[Gabriele Honecker]], \emph{Non-supersymmetric Orientifolds with D-branes at Angles}, Fortsch.Phys. 50 (2002) 896-902 (\href{https://arxiv.org/abs/hep-th/0112174}{arXiv:hep-th/0112174}) \item [[Gabriele Honecker]], \emph{Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds}, JHEP 0201 (2002) 025 (\href{https://arxiv.org/abs/hep-th/0201037}{arXiv:hep-th/0201037}) \item [[Gabriele Honecker]], \emph{Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes}, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]]) \end{itemize} The [[Witten-Sakai-Sugimoto model]] on [[D4-D8-brane bound states]] for [[QCD]] with [[orthogonal group|orthogonal]] [[gauge groups]] on O-planes: \begin{itemize}% \item Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], \emph{$O(N)$ and $USp(N)$ QCD from String Theory}, Prog.Theor.Phys.122:1433-1453, 2010 (\href{https://arxiv.org/abs/0907.2968}{arXiv:0907.2968}) \item Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, \emph{5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry}, JHEP 1210 (2012) 142 (\href{https://arxiv.org/abs/1206.6781}{arXiv:1206.6781}) \end{itemize} Specifically D5 brane models [[T-duality|T-dual]] to D6/D8 models: \begin{itemize}% \item [[Angel Uranga]], \emph{A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points}, Nucl. Phys. B577:73-87, 2000 (\href{https://arxiv.org/abs/hep-th/9910155}{arXiv:hep-th/9910155}) \item Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], \emph{Orientifold dual for stuck NS5 branes}, JHEP 0106:065, 2001 (\href{https://arxiv.org/abs/hep-th/0103177}{arXiv:hep-th/0103177}) \end{itemize} Specifically for [[D6-branes]]: \begin{itemize}% \item S. Ishihara, H. Kataoka, Hikaru Sato, \emph{$D=4$, $N=1$, Type IIA Orientifolds}, Phys. Rev. D60 (1999) 126005 (\href{https://arxiv.org/abs/hep-th/9908017}{arXiv:hep-th/9908017}) \item [[Mirjam Cvetic]], Paul Langacker, Tianjun Li, Tao Liu, \emph{D6-brane Splitting on Type IIA Orientifolds}, Nucl. Phys. B709:241-266, 2005 (\href{https://arxiv.org/abs/hep-th/0407178}{arXiv:hep-th/0407178}) \end{itemize} Specifically for [[D3-branes]]/[[D7-branes]]: \begin{itemize}% \item \hyperlink{FengHeKarchUranga01}{Feng-He-Karch-Uranga 01} \end{itemize} Specifically 2d [[toroidal orbifold|toroidal]] orientifolds: 2d toroidal [[orientifolds]]: \begin{itemize}% \item Dongfeng Gao, [[Kentaro Hori]], Section 7.3 of: \emph{On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds} (\href{https://arxiv.org/abs/1004.3972}{arXiv:1004.3972}) \item [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], \emph{String theory on elliptic curve orientifolds and KR-theory} (\href{http://arxiv.org/abs/1402.4885}{arXiv:1402.4885}) \end{itemize} Various: \begin{itemize}% \item [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, \emph{Resolved Toroidal Orbifolds and their Orientifolds}, Adv.Theor.Math.Phys.12:67-183, 2008 (\href{https://arxiv.org/abs/hep-th/0609014}{arXiv:hep-th/0609014}) \end{itemize} \hypertarget{orientifold_gepner_models}{}\subsubsection*{{Orientifold Gepner models}}\label{orientifold_gepner_models} Orientifolds of [[Gepner models]] \begin{itemize}% \item Brandon Bates, [[Charles Doran]], Koenraad Schalm, \emph{Crosscaps in Gepner Models and the Moduli space of $T^2$ 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} \hypertarget{ReferencesInMTheory}{}\subsubsection*{{Lift to M-theory}}\label{ReferencesInMTheory} Lifts of orientifolds to [[M-theory]] ([[MO5]], [[MO9]]) and [[F-theory]] are discussed in \begin{itemize}% \item [[Ashoke Sen]], \emph{F-theory and Orientifolds} (\href{http://arxiv.org/abs/hep-th/9605150}{arXiv:hep-th/9605150}) \item [[Ashoke Sen]], \emph{Orientifold Limit of F-theory Vacua} (\href{http://arxiv.org/abs/hep-th/9702165}{arXiv:hep-th/9702165}) \item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123}) \item [[Kentaro Hori]], \emph{Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold}, Nucl. Phys. B539:35-78, 1999 (\href{https://arxiv.org/abs/hep-th/9805141}{arXiv:hep-th/9805141}) \item [[Eric Gimon]], \emph{On the M-theory Interpretation of Orientifold Planes} (\href{https://arxiv.org/abs/hep-th/9806226}{arXiv:hep-th/9806226}, \href{http://inspirehep.net/record/472499}{spire:472499}) \item Changhyun Ahn, Hoil Kim, Hyun Seok Yang, \emph{$SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$}, Phys.Rev. D59 (1999) 106002 (\href{https://arxiv.org/abs/hep-th/9808182}{arXiv:hep-th/9808182}) \item [[Amihay Hanany]], [[Barak Kol]], section 4 of \emph{On Orientifolds, Discrete Torsion, Branes and M Theory}, JHEP 0006 (2000) 013 (\href{https://arxiv.org/abs/hep-th/0003025}{arXiv:hep-th/0003025}) \item Philip C. Argyres, [[Ron Maimon]], Sophie Pelland, \emph{The M theory lift of two O6 planes and four D6 branes}, JHEP 0205 (2002) 008 (\href{https://arxiv.org/abs/hep-th/0204127}{arXiv:hep-th/0204127}) following \item [[Edward Witten]], \emph{Solutions Of Four-Dimensional Field Theories Via M Theory}, (\href{https://arxiv.org/abs/hep-th/9703166}{arXiv:hep-th/9703166}) \end{itemize} The [[MO5]] is originally discussed in \begin{itemize}% \item Keshav Dasgupta, [[Sunil Mukhi]], \emph{Orbifolds of M-theory}, Nucl. Phys. B465 (1996) 399-412 (\href{https://arxiv.org/abs/hep-th/9512196}{arXiv:hep-th/9512196}) \item [[Edward Witten]], \emph{Five-branes And M-Theory On An Orbifold}, Nucl. Phys. B463:383-397, 1996 (\href{https://arxiv.org/abs/hep-th/9512219}{arXiv:hep-th/9512219}) \end{itemize} The classification in \hyperlink{HananyKol00}{Hanany-Kol 00 (3.2)} also appears, with more details, in Prop. 4.7 of \begin{itemize}% \item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]} (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987}) \end{itemize} The ``higher orientifold'' appearing in [[Horava-Witten theory]] with circle 2-bundles replaced by the [[circle n-bundle with connection|circle 3-bundles]] of the [[supergravity C-field]] is discussed towards the end of \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The moduli 3-stack of the C-field|The E8 moduli 3-stack of the C-field in M-theory]]} (\href{http://arxiv.org/abs/1202.2455}{arXiv:1202.2455}) \end{itemize} [[!redirects orientifolds]] \end{document}