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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*{Riemannian orbifold} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry} [[!include Riemannian geometry - contents]] \hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry} [[!include higher geometry - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{noncompact_orbifolds}{Non-compact orbifolds}\dotfill \pageref*{noncompact_orbifolds} \linebreak \noindent\hyperlink{CompactFlatOrbifoldsFromCrystallographicGroups}{Compact flat orbifolds from crystallographic groups}\dotfill \pageref*{CompactFlatOrbifoldsFromCrystallographicGroups} \linebreak \noindent\hyperlink{FlatCompact2DimensionalOrbifolds}{Flat compact 2-dimensional orbifolds}\dotfill \pageref*{FlatCompact2DimensionalOrbifolds} \linebreak \noindent\hyperlink{FlatCompact4dOrbifolds}{Flat compact 4-dimensional orbifolds}\dotfill \pageref*{FlatCompact4dOrbifolds} \linebreak \noindent\hyperlink{flat_compact_6dimensional_orbifolds}{Flat compact 6-dimensional orbifolds}\dotfill \pageref*{flat_compact_6dimensional_orbifolds} \linebreak \noindent\hyperlink{7dimensional_orbifolds}{7-Dimensional $G_2$-orbifolds}\dotfill \pageref*{7dimensional_orbifolds} \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{flat_orbifolds}{Flat orbifolds}\dotfill \pageref*{flat_orbifolds} \linebreak \noindent\hyperlink{of_dimension_2}{Of dimension 2}\dotfill \pageref*{of_dimension_2} \linebreak \noindent\hyperlink{of_dimension_4}{Of dimension 4}\dotfill \pageref*{of_dimension_4} \linebreak \noindent\hyperlink{of_dimension_6}{Of dimension 6}\dotfill \pageref*{of_dimension_6} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The concept of \emph{Riemannian orbifolds} is the joint generalization of the concepts of \emph{[[Riemannian manifolds]]} and \emph{[[orbifolds]]}: A Riemannian orbifold is an [[orbifold]] equipped with an orbifold [[atlas]] where each [[chart]] $(\widehat{U}_i, G)$ is equipped with a [[Riemannian metric]] such that the [[action]] of $G$ is by [[isometries]], and such that the transition functions from one chart to the other are isometries. A key aspect is that the orbifold [[singularities]] behave like carrying singular [[curvature]], notably there are [[flat orbifolds]] (also ``Euclidean orbifolds'', i.e. Riemannian orbifolds with vanishing [[Riemann curvature]] away from the [[singularities]]) whose underlying [[topological spaces]] are [[n-spheres]] (see \hyperlink{FlatCompact2DimensionalOrbifolds}{below}). Key examples of flat orbifolds are global [[homotopy quotients]] $\mathbb{T}^n \sslash G$ of the [[n-torus]] $\mathbb{T}^n$ equipped with its canonical flat [[Riemannian metric]]. These flat orbifolds are called \emph{toroidal orbifolds}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \begin{quote}% under construction \end{quote} Every flat orbifold whose underlying [[metric space]] is [[connected topological space|connected]] and [[complete metric space|complete]]) is a global quotient of [[Euclidean space]]/[[Cartesian space]] $\mathbb{R}^n$ (\hyperlink{Ratcliffe06}{Ratcliffe 06, 13.3.10}) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{noncompact_orbifolds}{}\subsubsection*{{Non-compact orbifolds}}\label{noncompact_orbifolds} Basic examples of non-compact Riemannian orbifolds are [[conical singularities]]. In the flat case these are [[homotopy quotients]] of the form $V\sslash G$ for $G$ a [[finite group]] and $V \in RO(G)$ a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] [[linear representation]] of $G$. $\backslash$begin\{center\} $\backslash$end\{center\} \begin{quote}% graphics grabbed from \hyperlink{BlumenhagenLustTheisen13}{Blumenhagen-Lüst-Theisen 13} \end{quote} For $V = \mathbb{H}$ equipped with the canonical action of [[finite subgroups of SU(2)]] these are the [[ADE-singularities]]. \hypertarget{CompactFlatOrbifoldsFromCrystallographicGroups}{}\subsubsection*{{Compact flat orbifolds from crystallographic groups}}\label{CompactFlatOrbifoldsFromCrystallographicGroups} \begin{example} \label{CompactFlatOrbifoldFromCrystallographicGroup}\hypertarget{CompactFlatOrbifoldFromCrystallographicGroup}{} \textbf{([[compact topological space|compact]] [[flat orbifolds]] from [[crystallographic groups]])} Let $E$ be a [[Euclidean space]] and $S \subset Iso(E)$ a [[crystallographic group]] [[action|acting]] on it, with translational [[normal subgroup]] [[lattice (discrete subgroup)|lattice]] $N \subset S$ and corresponding [[point group]] $G = S/N$. \begin{displaymath} \itexarray{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 } \end{displaymath} Then the [[action]] of $G$ on $E$ descends to the [[quotient space]] [[torus]] $E/N$ (\href{crystallographic+group#InducedPointGroupActionOnTorus}{this Prop.}) \begin{displaymath} \itexarray{ E &\overset{g}{\longrightarrow}& E \\ \big\downarrow && \big\downarrow \\ E/N &\underset{g}{\longrightarrow}& E/N } \end{displaymath} The resulting [[homotopy quotient]] $(E/N)\sslash G$ is a compact flat orbifold. \end{example} The following is the class of special cases of Example \ref{CompactFlatOrbifoldFromCrystallographicGroup} for [[point group]] being the [[involution]]-[[action]] by [[reflection]] at a point: \begin{example} \label{CoordinateReflectionOnNTorus}\hypertarget{CoordinateReflectionOnNTorus}{} \textbf{([[coordinate function|coordinate]] [[reflection]] on [[n-torus]])} Let $\mathbb{T}^d \coloneqq \mathbb{R}^d / \mathbb{Z}^d$ be the [[n-torus|d-torus]] and consider the [[action]] of the [[cyclic group]] $\mathbb{Z}_2$ by canonical [[coordinate function|coordinate]] [[reflection]] \begin{displaymath} \itexarray{ \mathbb{Z}_2 \times \mathbb{T}^d &\longrightarrow& \mathbb{T}^d \\ (\sigma, \vec x) &\mapsto& - \vec x } \,. \end{displaymath} The resulting [[homotopy quotient]] [[orbifold]] $\mathbb{T}^d\sslash\mathbb{Z}_2$ has $2^d$ [[singularities]]/[[fixed points]], namely the points with all coordinates in $\{0\,,\, 1/2\, \mathrm{mod} \mathbb{Z}\}$. In applications to [[string theory]] orbifolds of the form $\mathbb{R}^{p,1} \times \mathbb{T}^d\sslash \mathbb{Z}_2$ play the role of [[orientifold]] [[spacetimes]] with $2^d$ [[Op-planes]]. \end{example} \hypertarget{FlatCompact2DimensionalOrbifolds}{}\subsubsection*{{Flat compact 2-dimensional orbifolds}}\label{FlatCompact2DimensionalOrbifolds} In 2 dimensions the [[crystallographic groups]] are the ``[[wallpaper groups]]''. Hence, as a special case of Example \ref{CompactFlatOrbifoldFromCrystallographicGroup}, the flat [[compact topological space|compact]] 2-dimensional orbifolds may be classified as [[homotopy quotients]] of the [[2-torus]] by [[wallpaper groups]] (for review see e.g. \hyperlink{Guerreiro09}{Guerreiro 09}): \begin{quote}% graphics grabbed from \hyperlink{BettiolDerdzinskiPiccione18}{Bettiol-Derdzinski-Piccione 18} \end{quote} \hypertarget{FlatCompact4dOrbifolds}{}\subsubsection*{{Flat compact 4-dimensional orbifolds}}\label{FlatCompact4dOrbifolds} The orbifold quotient of the [[4-torus]] by the sign [[involution]] on all four canonical [[coordinates]] is the flat compact 4-dimensional orbifold known as the \emph{[[Kummer surface]]} $T^4 \sslash \mathbb{Z}_2$ -- the special case of Example \ref{CoordinateReflectionOnNTorus} for $d = 4$. This is a singular [[K3-surface]] (e.g. \hyperlink{BettiolDerdzinskiPiccione18}{Bettiol-Derdzinski-Piccione 18, 5.5}) $\backslash$begin\{center\} $\backslash$end\{center\} \begin{quote}% graphics grabbed from \href{orbifold#Snowden11}{Snowden 11} \end{quote} Also $\mathbb{T}^4\sslash\mathbb{Z}_4$ gives a toroidal orbifold. As [[orientifolds]] with [[D5-branes]] in [[type IIB string theory]] these are discussed in \hyperlink{BuchelShiuTye99}{Buchel-Shiu-Tye 99, Sec. II}. $\backslash$linebreak \hypertarget{flat_compact_6dimensional_orbifolds}{}\subsubsection*{{Flat compact 6-dimensional orbifolds}}\label{flat_compact_6dimensional_orbifolds} see \hyperlink{FRTV12}{FRTV 12} \hypertarget{7dimensional_orbifolds}{}\subsubsection*{{7-Dimensional $G_2$-orbifolds}}\label{7dimensional_orbifolds} see \emph{[[G2-orbifold]]} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[crystallographic group]] \item [[branched cover]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item Joseph Ernest Borzellino, \emph{Riemannian Geometry of Orbifolds}, 1992 (\href{http://mathnet.kaist.ac.kr/mathnet/paper_file/California/Polytech/Borze/dis.pdf}{pdf 1}, \href{https://web.calpoly.edu/~jborzell/Publications/Publication%20PDFs/phd_thesis.pdf}{pdf 2}) \item [[John Ratcliffe]], \emph{Geometric Orbifolds}, chapter 13 in \emph{Foundations of Hyperbolic Manifolds}, Graduate Texts in Mathematics 149, Springer 2006 (\href{https://doi.org/10.1007/978-0-387-47322-2}{doi:10.1007/978-0-387-47322-2}, ) \item Christian Lange, \emph{Orbifolds from a metric viewpoint} (\href{https://arxiv.org/abs/1801.03472}{arXiv:1801.03472}) \item Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, \emph{Teichmüller theory and collapse of flat manifolds}, Annali di Matematica (2018) 197: 1247 (\href{https://arxiv.org/abs/1705.08431}{arXiv:1705.08431}, \href{https://doi.org/10.1007/s10231-017-0723-7}{doi:10.1007/s10231-017-0723-7}) \end{itemize} Discussion of [[gravity]] and maybe [[quantum gravity]] on orbifolds: \begin{itemize}% \item Helio V. Fagundes, Teofilo Vargas, \emph{Orbifolds, Quantum Cosmology, and Nontrivial Topology} (\href{https://arxiv.org/abs/gr-qc/0611048}{arXiv:gr-qc/0611048}) \end{itemize} Discussion of [[perturbative string theory]] on toroidal orbifolds \begin{itemize}% \item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Chapter 10.5 \emph{Toroidal orbifolds}, of \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 \end{itemize} For more see the references at \emph{[[orbifold]]}. \hypertarget{flat_orbifolds}{}\subsubsection*{{Flat orbifolds}}\label{flat_orbifolds} \hypertarget{of_dimension_2}{}\paragraph*{{Of dimension 2}}\label{of_dimension_2} In 2 dimensions \begin{itemize}% \item [[John Milnor]], \emph{On Lattès Maps} (\href{https://arxiv.org/abs/math/0402147}{arXiv:math/0402147}) \item João Guerreiro, \emph{Orbifolds and Wallpaper Patterns} (\href{https://faculty.math.illinois.edu/~ruiloja/Estudantes/TrabalhoJGuerreiro.pdf}{pdf}) \end{itemize} 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} \hypertarget{of_dimension_4}{}\paragraph*{{Of dimension 4}}\label{of_dimension_4} Flat (toroidal) orbifolds of dimension 4: In the the context of [[black holes in string theory]]: \begin{itemize}% \item Justin R. David, Gautam Mandal, Spenta R. Wadia, \emph{Microscopic Formulation of Black Holes in String Theory}, Phys.Rept.369:549-686,2002 (\href{https://arxiv.org/abs/hep-th/0203048}{arXiv:hep-th/0203048}) \end{itemize} In the context of [[RR-field tadpole cancellation]] for [[intersecting D-brane models]] on toroidal orientifolds: 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} In the context of [[Mathieu moonshine]] from [[string]] [[sigma models]] on [[K3]]s: \begin{itemize}% \item [[Matthias Gaberdiel]], [[Stefan Hohenegger]], [[Roberto Volpato]], \emph{Symmetries of K3 sigma models}, Commun.Num.Theor.Phys. 6 (2012) 1-50 (\href{https://arxiv.org/abs/1106.4315}{arXiv:1106.4315}) \item [[Matthias Gaberdiel]], Roberto Volpato, \emph{Mathieu Moonshine and Orbifold K3s} (\href{https://arxiv.org/abs/1206.5143}{arXiv:1206.5143}) \end{itemize} \hypertarget{of_dimension_6}{}\paragraph*{{Of dimension 6}}\label{of_dimension_6} In 6 dimensions (mostly motivated as singular [[supersymmetry and Calabi-Yau manifolds|Calabi-Yau compactifications]] of [[heterotic string theory]] to 4d) \begin{itemize}% \item Jens Erler, [[Albrecht Klemm]], \emph{Comment on the Generation Number in Orbifold Compactifications}, Commun. Math. Phys. 153:579-604, 1993 (\href{https://arxiv.org/abs/hep-th/9207111}{arXiv:hep-th/9207111}) \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}) \item S. Reffert, \emph{Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology} (\href{https://arxiv.org/abs/hep-th/0609040}{arXiv:hep-th/0609040}) \item [[Ron Donagi]], [[Katrin Wendland]], \emph{On orbifolds and free fermion constructions}, J. Geom. Phys. 59:942-968, 2009 (\href{https://arxiv.org/abs/0809.0330}{arXiv:0809.0330}) \item Maximilian Fischer, Michael Ratz, Jesus Torrado, Patrick K.S. Vaudrevange, \emph{Classification of symmetric toroidal orbifolds}, JHEP 1301 (2013) 084 (\href{https://arxiv.org/abs/1209.3906}{arXiv:1209.3906}) \end{itemize} [[!redirects Riemannian orbifolds]] [[!redirects flat orbifold]] [[!redirects flat orbifolds]] [[!redirects toroidal orbifold]] [[!redirects toroidal orbifolds]] \end{document}