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\section*{orbifold}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{GlobalQuotientOrbifolds}{Global quotient orbifolds}\dotfill \pageref*{GlobalQuotientOrbifolds} \linebreak
\noindent\hyperlink{cohomology}{(Co)homology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{ReferencesAsLieGroupoids}{As Lie groupoids}\dotfill \pageref*{ReferencesAsLieGroupoids} \linebreak
\noindent\hyperlink{ReferencesAsDiffeologicalSpaces}{As diffeological spaces}\dotfill \pageref*{ReferencesAsDiffeologicalSpaces} \linebreak
\noindent\hyperlink{orbifold_cobordism}{Orbifold cobordism}\dotfill \pageref*{orbifold_cobordism} \linebreak
\noindent\hyperlink{ReferencesInStringTheory}{In string theory}\dotfill \pageref*{ReferencesInStringTheory} \linebreak


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

An [[orbifold]] is much like a [[smooth manifold]] but possibly with [[singularities]] of the form of [[fixed points]] of [[group]]-[[actions]].

Where a [[smooth manifold]] is a [[space]] locally modeled on [[Cartesian spaces]]/[[Euclidean spaces]] $\mathbb{R}^n$, an \emph{orbifold} is, more generally, a [[space]] that is locally modeled on [[Lie groupoid|smooth]] [[action groupoids]] ([[homotopy quotients]]) $\mathbb{R}^n\sslash G$ of a [[finite group]] $G$ [[action|acting]] on a [[Cartesian space]].

\begin{quote}%
graphics grabbed from \hyperlink{HydeRamsdenRobins14}{Hyde-Ramsden-Robins 14}


\end{quote}
This turns out to be broadly captured(\hyperlink{MoerdijkPronk97}{Moerdijk-Pronk 97}, \hyperlink{Moerdijk02}{Moerdijk 02}) by saying that an orbifold is a [[proper groupoid|proper]] [[étale groupoid|étale]] [[Lie groupoid]]. ([[Morita equivalence|Morita equivalent]] Lie groupoids correspond to the same orbifolds.)

The word \emph{orbifold} was introduced in (\hyperlink{Thurston}{Thurston 1992}), while the original name was \emph{$V$-manifold} (\hyperlink{Satake}{Satake}), and was taken in a more restrictive sense, assuming that the [[actions]] of [[finite groups]] on the charts are always [[effective group action|effective]]. Nowadays these are called \emph{effective orbifolds} and those which are global quotients by a finite group are \emph{global quotient orbifolds}.

There is also a notion of finite stabilizers in [[algebraic geometry]]. A singular variety is called an (algebraic) \emph{orbifold} if it has only so-called \emph{orbifold singularities}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An orbifold is a stack presented by an [[orbifold groupoid]].

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

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

\begin{itemize}%
\item One can consider a [[bicategory]] of proper \'e{}tale Lie groupoids and the orbifolds will be the objects of certain bicategorical [[localization]] of this bicategory (a result of \hyperlink{MoerdijkPronk97}{Moerdijk-Pronk 97}).


\item Equivalently, every orbifold is globally a quotient of a smooth manifold by an [[action]] of finite-dimensional [[Lie group]] with finite [[stabilizer subgroup|stabilizers]] in each point. (eg (\hyperlink{ALR07}{Adem-Leida-Ruan 2007}), Corollary 1.24)



\end{itemize}
\hypertarget{GlobalQuotientOrbifolds}{}\subsubsection*{{Global quotient orbifolds}}\label{GlobalQuotientOrbifolds}

In (\hyperlink{ALR07}{ALR 07, theorem 1.23}) is asserted that every effective orbifold $X$ (paracompact, Hausdorff) is isomorphic to a global quotient orbifold, specifically to a global quotient of $O(n)$ (where $n$ is the dimension of $X$) acting on the [[frame bundle]] of $X$ (which is a manifold).

\hypertarget{cohomology}{}\subsubsection*{{(Co)homology}}\label{cohomology}

It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to [[orbifold Euler characteristics]], [[orbifold cohomology]] etc. One of the constructions which is useful in this respect is the [[inertia orbifold]] (the inertia stack of the original orbifold) which gives rise to ``twisted sectors'' in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Some basic building blocks of orbifolds:

The quotient of a [[ball]] by a [[discrete group|discrete]] [[subgroup]] of the [[special orthogonal group]] of rotations. Is an orbifold, and orbifolds may be obtained by cutting out balls from ordinary [[smooth manifolds]] and gluing in these orbifold quotients.


\item The [[moduli stack of elliptic curves]] over the [[complex numbers]] is an orbifold, being the [[homotopy quotient]] of the [[upper half plane]] by the [[special linear group]] acting by [[Möbius transformations]].


\item For $\mathcal{G}$ any orbifold, then the [[mapping space]] $\mathcal{G}^{\Pi(S^1)} = \mathcal{G}^{B\mathbb{Z}}$ is again an orbifold, called the [[inertia orbifold]].


\item [[G2-orbifolds]]


\item [[lens spaces]]


\item [[ADE singularities]]


\item [[metric cones]]



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

\begin{itemize}%
\item [[Riemannian orbifold]]

\begin{itemize}%
\item [[flat orbifold]]

\end{itemize}

\item [[discrete torsion]]


\item [[motivation for higher differential geometry]]


\item [[orbifold Euler characteristic]]



\end{itemize}
Orbifolds are in [[differential geometry]] what [[Deligne-Mumford stacks]] are in [[algebraic geometry]]. See also at \emph{[[geometric invariant theory]]} and \emph{[[GIT-stable point]]}.

If the finiteness condition is dropped one also speaks of \emph{[[orbispaces]] and generally of [[stacks]].}

Orbifolds may be regarded as a kind of \emph{[[stratified spaces]]}.

See also

\begin{itemize}%
\item [[étale groupoid]]


\item [[stable map]]


\item [[orbifold cohomology]]


\item [[Gromov-Witten theory]]



\end{itemize}
Orbifolds in [[string theory]]:

\begin{itemize}%
\item [[fractional D-brane]]

[[permutation D-brane]]


\item [[ADE singularity]]


\item [[orientifold]]



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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

The concept originates in

\begin{itemize}%
\item I. Satake, \emph{On a generalisation of the notion of manifold}, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359--363 (\href{https://doi.org/10.1073/pnas.42.6.359}{doi:10.1073/pnas.42.6.359})


\item I. Satake, \emph{The Gauss--Bonnet theorem for $V$-manifolds}, J. Math. Soc. Japan 9 (1957), 464--492.


\item [[William Thurston]], \emph{Three-dimensional geometry and topology,} preliminary draft, University of Minnesota, Minnesota, (1992)

which in completed and revised form is available as his book:

\emph{The Geometry and Topology of Three-Manifolds;} (\href{http://library.msri.org/books/gt3m/}{web})

in particular the orbifold discussion is in \href{http://library.msri.org/books/gt3m/PDF/13.pdf}{chapter 13}.



\end{itemize}
Survey of basic orbifold theory:

\begin{itemize}%
\item Daryl Cooper, Craig Hodgson, Steve Kerckhoff, \emph{Three-dimensional Orbifolds and Cone-Manifolds}, MSJ Memoirs Volume 5, 2000 (\href{https://web.math.ucsb.edu/~cooper/37.pdf}{pdf}, \href{https://projecteuclid.org/euclid.msjm/1389985812}{euclid:1389985812})


\item Adam Kaye, \emph{Two-Dimensional Orbifolds}, 2007 (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Kaye.pdf}{pdf})


\item Michael Davis, \emph{Lectures on orbifolds and reflection groups}, 2008 (\href{https://math.osu.edu/sites/math.osu.edu/files/08-05-MRI-preprint.pdf}{pdf})


\item Joan Porti, \emph{An introduction to orbifolds}, 2009 (\href{http://mat.uab.es/~porti/orbifoldLeiden.pdf}{pdf})


\item Andrew Snowden, \emph{Introduction to orbifolds}, 2011 (\href{https://ocw.mit.edu/courses/mathematics/18-904-seminar-in-topology-spring-2011/final-paper/MIT18_904S11_finlOrbifolds.pdf}{pdf})


\item [[Alejandro Adem]], Michele Klaus, \emph{Lectures on orbifolds and group cohomology} (\href{http://www.math.ubc.ca/~adem/hangzhou.pdf}{pdf}, [[AdemKlausOrbifolds.pdf:file]])


\item Francisco C. Caramello Jr, \emph{Introduction to orbifolds} (\href{https://arxiv.org/abs/1909.08699}{arXiv:1909.08699})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Orbifold}{Orbifolds}}

(which is mainly tailored toward \hyperlink{Thurston}{Thurston's approach}).



\end{itemize}
Textbook account:

\begin{itemize}%
\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}, )

\end{itemize}
On [[Riemannian orbifolds]]:

\begin{itemize}%
\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})


\item S. T. Hyde, S. J. Ramsden and V. Robins, \emph{Unification and classification of two-dimensional crystalline patterns using orbifolds}, Acta Cryst. (2014). A70, 319-337 (\href{https://doi.org/10.1107/S205327331400549X}{doi:10.1107/S205327331400549X})



\end{itemize}
Survey of applications in [[mathematical physics]] and notably in [[string theory]]:

\begin{itemize}%
\item [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], \emph{[[Orbifolds in Mathematics and Physics]]}, Contemporary Mathematics 310 American Mathematical Society, 2002

\end{itemize}
Orbifolds often appear as [[moduli spaces]] in differential geometric setting:

\begin{itemize}%
\item [[Dietmar Salamon]], [[Joel Robbin|Joel W. Robbin]], A construction of the Deligne--Mumford orbifold, J. Eur. Math. Society, ISSN 1435-9855, Vol. 8, N\textordmasculine{} 4, 2006, 611-699 (\href{http://arxiv.org/abs/math/0407090}{arXiv})

\end{itemize}
The generalization of orbifolds to \emph{weighted [[branched manifolds]]} is discussed in

\begin{itemize}%
\item [[Dusa McDuff]], \emph{Groupoids, branched manifolds and multisections}, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (\href{http://projecteuclid.org/euclid.jsg/1180135649}{project euclid}).

\end{itemize}
A reference dealing with the [[string topology]] of orbifolds is

\begin{itemize}%
\item A. Adem, J. Leida and Y. Ruan, \emph{Orbifolds and Stringy Topology}, Cambridge Tracts in Mathematics \textbf{171} (2007) (\href{http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf}{pdf})

\end{itemize}
\hypertarget{ReferencesAsLieGroupoids}{}\subsubsection*{{As Lie groupoids}}\label{ReferencesAsLieGroupoids}

Discussion of orbifold as [[Lie groupoids]]/[[differentiable stacks]] is in

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Dorette Pronk]], \emph{Orbifolds, sheaves and groupoids}, K-theory 12 3-21 (1997) (\href{http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf}{pdf}, \href{http://dx.doi.org/10.4171/LEM/56-3-4}{doi:10.4171/LEM/56-3-4})


\item [[Ieke Moerdijk]], \emph{Orbifolds as Groupoids: an Introduction}, [[Alejandro Adem]], [[Jack Morava]], Yongbin Ruan (eds.) \emph{[[Orbifolds in Mathematics and Physics]]}, Contemporary Math 310 , AMS (2002), 205–222 (\href{http://arxiv.org/abs/math.DG/0203100}{arXiv:math.DG/0203100})


\item [[Eugene Lerman]], \emph{Orbifolds as stacks?}, Enseign. Math. (2), 56 3-4 (2010) (\href{http://arxiv.org/abs/0806.4160}{arXiv:0806.4160}, \href{http://dx.doi.org/10.4171/LEM/56-3-4}{doi:10.4171/LEM/56-3-4})



\end{itemize}
Discussion of the corresponding perspective in [[algebraic geometry]], via [[Deligne-Mumford stacks]]:

\begin{itemize}%
\item [[Andrew Kresch]], \emph{On the geometry of Deligne-Mumford stacks} (\href{https://doi.org/10.5167/uzh-21342}{doi:10.5167/uzh-21342}, \href{https://www.zora.uzh.ch/id/eprint/21342/1/geodm.pdf}{pdf}), in: D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) \emph{Algebraic Geometry: Seattle 2005}, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 (\href{https://bookstore.ams.org/pspum-80-1}{pspum-80-1})

\end{itemize}
The [[mapping stacks]] of orbifolds are discussed in

\begin{itemize}%
\item W. Chen, \emph{On a notion of maps between orbifolds}, I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569--620.


\item [[David Roberts]], [[Raymond Vozzo]], \emph{The Smooth Hom-Stack of an Orbifold}, In : Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham (2018) (\href{https://arxiv.org/abs/1610.05904}{arXiv:1610.05904}, \href{https://doi.org/10.1007/978-3-319-72299-3_3}{doi:10.1007/978-3-319-72299-3\_3})



\end{itemize}
An expected relation of orbifolds to [[global equivariant homotopy theory]] is discussed in

\begin{itemize}%
\item [[Stefan Schwede]], \emph{Orbispaces, orthogonal spaces, and the universal compact Lie group} (\href{https://arxiv.org/abs/1711.06019}{arXiv:1711.06019}) (on the relation to [[orbispaces]]/[[topological stacks]])

\end{itemize}
\hypertarget{ReferencesAsDiffeologicalSpaces}{}\subsubsection*{{As diffeological spaces}}\label{ReferencesAsDiffeologicalSpaces}

Discussion of orbifolds regarded as naive local quotient [[diffeological spaces]]:

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, \emph{Orbifolds as diffeologies}, Transactions of the American Mathematical Society 362 (2010), 2811-2831 (\href{https://arxiv.org/abs/math/0501093}{arXiv:math/0501093})


\item [[Jordan Watts]], \emph{The Differential Structure of an Orbifold}, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 (\href{https://arxiv.org/abs/1503.01740}{arXiv:1503.01740})



\end{itemize}
\hypertarget{orbifold_cobordism}{}\subsubsection*{{Orbifold cobordism}}\label{orbifold_cobordism}

Orbifold [[cobordisms]] are discussed in

\begin{itemize}%
\item K. S. Druschel, \emph{Oriented Orbifold Cobordism}, Pacific J. Math., 164(2) (1994), 299-319.


\item K. S. Druschel, \emph{The Cobordism of Oriented Three Dimensional Orbifolds}, Pacific J. Math., bf 193(1) (2000), 45-55.


\item Andres Angel, \emph{Orbifold cobordism} (\href{http://www.math.uni-bonn.de/people/aangel79/Orbifold%20cobordism.pdf}{pdf})



\end{itemize}
See also at \emph{[[orbifold cobordism]]}.

\hypertarget{ReferencesInStringTheory}{}\subsubsection*{{In string theory}}\label{ReferencesInStringTheory}

In [[perturbative string theory]], orbifolds as [[target spaces]] for a [[string]] [[sigma-model]] were first considered in

\begin{itemize}%
\item [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], \emph{Strings on orbifolds}, Nuclear Physics B Volume 261, 1985, Pages 678-686 ()


\item [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], \emph{Strings on orbifolds (II)}, Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 ()



\end{itemize}
and then further developed notably in

\begin{itemize}%
\item [[Robbert Dijkgraaf]], [[Cumrun Vafa]], [[Erik Verlinde]], [[Herman Verlinde]], \emph{The operator algebra of orbifold models}, Comm. Math. Phys. Volume 123, Number 3 (1989), 485-526 (\href{https://projecteuclid.org/euclid.cmp/1104178892}{euclid:1104178892})


\item [[Eric Zaslow]], \emph{Topological orbifold models and quantum cohomology rings}, Comm. Math. Phys. 156 (1993), no. 2, 301--331.



\end{itemize}
And discussion of [[blow-up]] of orbifold [[singularities]] in string theory:

\begin{itemize}%
\item [[Paul Aspinwall]], \emph{Resolution of Orbifold Singularities in String Theory} (\href{https://arxiv.org/abs/hep-th/9403123}{arXiv:hep-th/9403123})

\end{itemize}
For [[orbifolds]] in [[string theory]] also the references at

\begin{itemize}%
\item \emph{[[Riemannian orbifold]]}, \emph{[[toroidal orbifold]]}


\item \emph{[[fractional D-brane]]}


\item \emph{[[Gepner model]]}


\item \emph{[[orientifold]]}


\item \emph{[[RR-field tadpole cancellation]]}



\end{itemize}
Review of orbifolds in the context of string [[KK-compactifications]] and [[intersecting D-brane models]] includes

\begin{itemize}%
\item D. Bailin, A. Love, \emph{Orbifold compactifications of string theory}, Phys.Rept. 315 (1999) 285-408 ()


\item [[Katrin Wendland]], \emph{Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry}, in \emph{[[Orbifolds in Mathematics and Physics]]} (\href{https://arxiv.org/abs/hep-th/0112006}{arXiv:hep-th/0112006})


\item Joel Giedt, \emph{Heterotic Orbifolds} (\href{https://arxiv.org/abs/hep-ph/0204315}{arXiv:hep-ph/0204315})


\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 Susanne Reffert, \emph{The Geometer's Toolkit to String Compactifications}, lectures at \emph{\href{https://www.ggi.infn.it/showevent.pl?id=11}{String and M theory approaches to particle physics and cosmology}}, 2007 (\href{https://arxiv.org/abs/0706.1310}{arXiv:0706.1310})


\item [[Luis Ibáñez]], [[Angel Uranga]], Chapter 8 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012 (\href{https://doi.org/10.1017/CBO9781139018951}{doi:10.1017/CBO9781139018951})


\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}
and for orbifolds of [[G2-manifolds]] for [[M-theory on G2-manifolds]]

\begin{itemize}%
\item [[Frank Reidegeld]], \emph{$G_2$-orbifolds from K3 surfaces with ADE-singularities} (\href{http://arxiv.org/abs/1512.05114}{arXiv:1512.05114})


\item [[Frank Reidegeld]], \emph{$G_2$-orbifolds with ADE-singularities} (\href{https://core.ac.uk/download/pdf/159317626.pdf}{pdf})



\end{itemize}
For [[topological strings]] the [[path integral as a pull-push transform]] for target orbifolds -- in analogy to what [[Gromov-Witten theory]] is for [[Deligne-Mumford stacks]] -- has first been considered in

\begin{itemize}%
\item Weimin Chen, [[Yongbin Ruan]], \emph{Orbifold Gromov-Witten Theory}, in \emph{[[Orbifolds in Mathematics and Physics]]} (Madison, WI, 2001), 25--85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (\href{http://arxiv.org/abs/math/0103156}{arXiv:math/0103156}) A review with further pointers is in


\item [[Dan Abramovich]], \emph{Lectures on Gromov-Witten invariants of orbifolds} (\href{http://arxiv.org/abs/math/0512372}{arXiv:math/0512372})



\end{itemize}
category: Lie theory

[[!redirects orbifolds]]

[[!redirects effective orbifold]] [[!redirects effective orbifolds]]



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