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43.8 86.3 47.4z"/> </svg> nLab orbifold </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3428/#Item_51" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_geometry">Higher Geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> / <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> Ingredients <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> </li> </ul> Concepts <ul> <li> geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es <ul> <li> <a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a> </li> </ul> </li> <li> geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> Constructions <ul> <li> <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a> </li> </ul> Examples <ul> <li> <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s </li> <li> <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> derived smooth geometry <ul> <li> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a> </li> </ul> </div></div> <h4 id="higher_lie_theory">Higher Lie theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>) Background Smooth structure <ul> <li> <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a> </li> </ul> Higher groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> Lie theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a> </li> </ul> </li> </ul> ∞-Lie groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a> </li> </ul> </li> </ul> ∞-Lie algebroids <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> Formal Lie groupoids <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> Cohomology <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a> </li> </ul> Homotopy <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> Related topics <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> Examples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a> </li> </ul> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#GlobalQuotientOrbifolds'>Global quotient orbifolds</a></li> <li><a href='#cohomology'>(Co)homology</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#ReferencesAsLieGroupoids'>As Lie groupoids</a></li> <li><a href='#ReferencesAsDiffeologicalSpaces'>As diffeological spaces</a></li> <li><a href='#orbifold_cobordism'>Orbifold cobordism</a></li> <li><a href='#ReferencesInStringTheory'>In string theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> An <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> is much like a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> but possibly with <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> of the form of <a class="existingWikiWord" href="/nlab/show/fixed+points">fixed points</a> of <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> <a class="existingWikiWord" href="/nlab/show/group">group</a>-<a class="existingWikiWord" href="/nlab/show/actions">actions</a>. <div class="float_right_image" style="margin: -30px 0px 10px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/OrbifoldTypes.jpg" width="240px" /> <figcaption style="text-align: center">(from <a href="#HydeRamsdenRobins14">Hyde-Ramsden-Robins 14</a>)</figcaption> </figure> </div> Where a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> is a <a class="existingWikiWord" href="/nlab/show/space">space</a> locally modeled on <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>/<a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, an orbifold is, more generally, a <a class="existingWikiWord" href="/nlab/show/space">space</a> that is locally modeled on <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/action+groupoids">action groupoids</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⫽</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n\sslash G</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/action">acting</a> on a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. This turns out to be broadly captured (<a href="#MoerdijkPronk97">Moerdijk-Pronk 97</a>, <a href="#Moerdijk02">Moerdijk 02</a>) by saying that an orbifold is a <a class="existingWikiWord" href="/nlab/show/proper+groupoid">proper</a> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>. (<a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalent</a> Lie groupoids correspond to the same orbifolds.) The word orbifold was introduced in (<a href="#Thurston92">Thurston 1992</a>), while the original name was <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-manifold (<a href="#Satake">Satake</a>), and was taken in a more restrictive sense, assuming that the <a class="existingWikiWord" href="/nlab/show/actions">actions</a> of <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a> on the charts are always <a class="existingWikiWord" href="/nlab/show/effective+group+action">effective</a>. Nowadays these are called effective orbifolds and those which are global quotients by a finite group are <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifolds">global quotient orbifolds</a>. There is also a notion of finite stabilizers in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities. <h2 id="definition">Definition</h2> An orbifold is a stack presented by an <a class="existingWikiWord" href="/nlab/show/orbifold+groupoid">orbifold groupoid</a>. <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <ul> <li> One can consider a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> of proper étale Lie groupoids and the orbifolds will be the objects of certain bicategorical <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of this bicategory (a result of <a href="#MoerdijkPronk97">Moerdijk-Pronk 97</a>). </li> <li> Equivalently, every orbifold is globally a quotient of a smooth manifold by an <a class="existingWikiWord" href="/nlab/show/action">action</a> of finite-dimensional <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> with finite <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizers</a> in each point. (eg (<a href="#ALR07">Adem-Leida-Ruan 2007</a>), Corollary 1.24) </li> </ul> <h3 id="GlobalQuotientOrbifolds">Global quotient orbifolds</h3> In (<a href="#ALR07">ALR 07, theorem 1.23</a>) it is asserted that every effective orbifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (paracompact, Hausdorff) is isomorphic to a <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifold">global quotient orbifold</a>, specifically to a global quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the dimension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>) acting on the <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (which is a manifold). <h3 id="cohomology">(Co)homology</h3> 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 <a class="existingWikiWord" href="/nlab/show/orbifold+Euler+characteristics">orbifold Euler characteristics</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a> etc. One of the constructions which is useful in this respect is the <a class="existingWikiWord" href="/nlab/show/inertia+orbifold">inertia orbifold</a> (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. <h2 id="examples">Examples</h2> <ul> <li> Some basic building blocks of orbifolds: The quotient of a <a class="existingWikiWord" href="/nlab/show/ball">ball</a> by a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> of rotations is an orbifold, and orbifolds may be obtained by cutting out balls from ordinary <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and gluing in these orbifold quotients. </li> <li> The <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> is an orbifold, being the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> of the <a class="existingWikiWord" href="/nlab/show/upper+half+plane">upper half plane</a> by the <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a> acting by <a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+transformations">Möbius transformations</a>. </li> <li> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> any orbifold, then the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒢</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>𝒢</mi> <mrow><mi>B</mi><mi>ℤ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{G}^{\Pi(S^1)} = \mathcal{G}^{B\mathbb{Z}}</annotation></semantics></math> is again an orbifold, called the <a class="existingWikiWord" href="/nlab/show/inertia+orbifold">inertia orbifold</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/pillowcase+orbifold">pillowcase orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spindle+orbifold">spindle orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-orbifolds">G₂-orbifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/lens+spaces">lens spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ADE+singularities">ADE singularities</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/metric+cones">metric cones</a> </li> <li> See also <a href="#Lange24">Lange 2024</a> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/presentable+orbifold">presentable orbifold</a>, <a class="existingWikiWord" href="/nlab/show/good+orbifold">good orbifold</a>, <a class="existingWikiWord" href="/nlab/show/very+good+orbifold">very good orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbispace">orbispace</a>, <a class="existingWikiWord" href="/nlab/show/topological+stack">topological stack</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Riemannian+orbifold">Riemannian orbifold</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/flat+orbifold">flat orbifold</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lorentzian+orbifold">Lorentzian orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+orbifold">complex orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+orbifold">symplectic orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super-orbifold">super-orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/discrete+torsion">discrete torsion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/motivation+for+higher+differential+geometry">motivation for higher differential geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbifold+Euler+characteristic">orbifold Euler characteristic</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+differential+K-theory">orbifold differential K-theory</a> </li> </ul> Orbifolds are in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> what <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stacks">Deligne-Mumford stacks</a> are in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>. See also at <a class="existingWikiWord" href="/nlab/show/geometric+invariant+theory">geometric invariant theory</a> and <a class="existingWikiWord" href="/nlab/show/GIT-stable+point">GIT-stable point</a>. Orbifolds may be regarded as a kind of <a class="existingWikiWord" href="/nlab/show/stratified+spaces">stratified spaces</a>. See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+map">stable map</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a> </li> </ul> Orbifolds in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/fractional+D-brane">fractional D-brane</a> <a class="existingWikiWord" href="/nlab/show/permutation+D-brane">permutation D-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ADE+singularity">ADE singularity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> The original articles: <ul> <li id="Satake"> <a class="existingWikiWord" href="/nlab/show/Ichiro+Satake">Ichiro Satake</a>, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 (<a href="https://doi.org/10.1073/pnas.42.6.359">doi:10.1073/pnas.42.6.359</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ichiro+Satake">Ichiro Satake</a>, The Gauss–Bonnet theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-manifolds, J. Math. Soc. Japan 9 (1957), 464–492 (<a href="https://projecteuclid.org/euclid.jmsj/1261153826">euclid:1261153826</a>) </li> <li id="Thurston92"> <a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>: Three-dimensional geometry and topology, preliminary draft, University of Minnesota (1992) [1979: <a href="https://archive.org/details/ThurstonTheGeometryAndTopologyOfThreeManifolds/mode/2up">ark:/13960/t3714t34v</a>, 1991: <a class="existingWikiWord" href="/nlab/files/Thurston-3dGeometry-1991.pdf" title="pdf">pdf</a>, 2002: <a href="https://www.math.unl.edu/~jkettinger2/thurston.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Thurston-3dGeometry-2002.pdf" title="pdf">pdf</a>] the first three chapters of which are published in expanded form as: </li> <li> <a class="existingWikiWord" href="/nlab/show/William+Thurston">William Thurston</a>: The Geometry and Topology of Three-Manifolds, Princeton University Press (1997) [<a href="https://press.princeton.edu/books/hardcover/9780691083049/three-dimensional-geometry-and-topology-volume-1">ISBN:9780691083049</a>, <a href="https://en.wikipedia.org/wiki/The_geometry_and_topology_of_three-manifolds">Wikipedia page</a>] in particular orbifolds are discussed in <a href="http://library.msri.org/books/gt3m/PDF/13.pdf">chapter 13</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Haefliger">André Haefliger</a>, Groupoides d’holonomie et classifiants, Astérisque no. 116 (1984), p. 70-97 (<a href="http://www.numdam.org/item/AST_1984__116__70_0/">numdam:AST_1984__116__70_0/</a>) </li> </ul> and specifically for orbifolds in <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Walter+Lewis+Baily">Walter Lewis Baily</a>, On the quotient of an analytic manifold by a group of analytic homeomorphisms, PNAS 40 (9) 804-808 (1954) (<a href="https://doi.org/10.1073/pnas.40.9.804">doi:10.1073/pnas.40.9.804</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Walter+Lewis+Baily">Walter Lewis Baily</a>, The Decomposition Theorem for V-Manifolds, American Journal of Mathematics Vol. 78, No. 4 (Oct., 1956), pp. 862-888 (<a href="https://www.jstor.org/stable/2372472">jstor:2372472</a>) </li> </ul> For careful comparative review of the definitions in these original articles see <a href="#IKZ10">IKZ 10</a>. Survey of basic orbifold theory: <ul> <li> Daryl Cooper, Craig Hodgson, Steve Kerckhoff, Three-dimensional Orbifolds and Cone-Manifolds, MSJ Memoirs Volume 5, 2000 (<a href="https://web.math.ucsb.edu/~cooper/37.pdf">pdf</a>, <a href="https://projecteuclid.org/euclid.msjm/1389985812">euclid:1389985812</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Janez+Mr%C4%8Dun">Janez Mrčun</a>, Section 2.4 of: <a class="existingWikiWord" href="/nlab/show/Introduction+to+foliations+and+Lie+groupoids">Introduction to foliations and Lie groupoids</a>, Cambridge Studies in Advanced Mathematics 91, 2003. x+173 pp. ISBN: 0-521-83197-0 (<a href="https://doi.org/10.1017/CBO9780511615450">doi:10.1017/CBO9780511615450</a>) </li> <li id="BoyerGalicki07"> <a class="existingWikiWord" href="/nlab/show/Charles+Boyer">Charles Boyer</a>, <a class="existingWikiWord" href="/nlab/show/Krzysztof+Galicki">Krzysztof Galicki</a>, Chapter 4 of: Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, 2007 (<a href="https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198564959.001.0001/acprof-9780198564959">doi:10.1093/acprof:oso/9780198564959.001.0001</a>) </li> <li id="Kaye07"> Adam Kaye, Two-Dimensional Orbifolds, 2007 (<a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Kaye.pdf">pdf</a>) </li> <li> Michael Davis, Lectures on orbifolds and reflection groups, 2008 (<a href="https://math.osu.edu/sites/math.osu.edu/files/08-05-MRI-preprint.pdf">pdf</a>) </li> <li id="Porti09"> Joan Porti, An introduction to orbifolds, 2009 (<a href="http://mat.uab.es/~porti/orbifoldLeiden.pdf">pdf</a>) </li> <li id="Snowden11"> Andrew Snowden, Introduction to orbifolds, 2011 (<a href="https://ocw.mit.edu/courses/mathematics/18-904-seminar-in-topology-spring-2011/final-paper/MIT18_904S11_finlOrbifolds.pdf">pdf</a>) </li> <li id="AdemKlaus"> <a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, Michele Klaus, Lectures on orbifolds and group cohomology (<a href="http://www.math.ubc.ca/~adem/hangzhou.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/AdemKlausOrbifolds.pdf" title="pdf">pdf</a>) </li> <li> Francisco C. Caramello Jr, Introduction to orbifolds (<a href="https://arxiv.org/abs/1909.08699">arXiv:1909.08699</a>) </li> </ul> See also <ul> <li> Wikipedia, <a href="http://en.wikipedia.org/wiki/Orbifold">Orbifolds</a> (which is mainly tailored toward <a href="#Thurston92">Thurston’s approach</a>). </li> </ul> On <a class="existingWikiWord" href="/nlab/show/good+orbifolds">good orbifolds</a>: <ul> <li id="Lange24">Christian Lange: Good, but not very good orbifolds [<a href="https://arxiv.org/abs/2404.14234">arXiv:2404.14234</a>]</li> </ul> Textbook account: <ul> <li id="Ratcliffe06"> <a class="existingWikiWord" href="/nlab/show/John+Ratcliffe">John Ratcliffe</a>, Geometric Orbifolds, chapter 13 in Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (<a href="https://doi.org/10.1007/978-0-387-47322-2">doi:10.1007/978-0-387-47322-2</a>, <a href="http://entsphere.com/pub/pdf/Ratcliffe%20-%20Foundations%20of%20hyperbolic%20manifolds%20(2e)%20-%20GTM%20149.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Michael+Kapovich">Michael Kapovich</a>, Chapter 6 of: Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics, Birkhäuser 2008 (<a href="https://link.springer.com/book/10.1007/978-0-8176-4913-5">doi:10.1007/978-0-8176-4913-5</a>) </li> </ul> Application to <a class="existingWikiWord" href="/nlab/show/moduli+spaces+of+curves">moduli spaces of curves</a> and <a class="existingWikiWord" href="/nlab/show/moduli+spaces+of+Riemann+surfaces">moduli spaces of Riemann surfaces</a>: <ul> <li>Lizhen Ji, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, Transformation Groups and Moduli Spaces of Curves, International Press of Boston 2011 (<a href="https://www.intlpress.com/site/pub/pages/books/items/00000353/index.php">ISBN:9781571462237</a>)</li> </ul> On <a class="existingWikiWord" href="/nlab/show/Riemannian+orbifolds">Riemannian orbifolds</a>: <ul> <li> Christian Lange, Orbifolds from a metric viewpoint (<a href="https://arxiv.org/abs/1801.03472">arXiv:1801.03472</a>) </li> <li> Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, Teichmüller theory and collapse of flat manifolds, Annali di Matematica (2018) 197: 1247 (<a href="https://arxiv.org/abs/1705.08431">arXiv:1705.08431</a>, <a href="https://doi.org/10.1007/s10231-017-0723-7">doi:10.1007/s10231-017-0723-7</a>) </li> <li id="HydeRamsdenRobins14"> S. T. Hyde, S. J. Ramsden and V. Robins, Unification and classification of two-dimensional crystalline patterns using orbifolds, Acta Cryst. (2014). A70, 319-337 (<a href="https://doi.org/10.1107/S205327331400549X">doi:10.1107/S205327331400549X</a>) </li> </ul> Survey of applications in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a> and notably in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>: <ul> <li id="AdemMoravaRuan02"><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, <a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a>, Contemporary Mathematics 310 American Mathematical Society, 2002</li> </ul> Orbifolds often appear as <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> in differential geometric setting: <ul> <li><a class="existingWikiWord" href="/nlab/show/Joel+W.+Robbin">Joel W. Robbin</a>, <a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar A. Salamon</a>, A construction of the Deligne–Mumford <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>, J. Eur. Math. Society 8, Nº 4 (2006) 611–699, arXiv:<a href="https://arxiv.org/abs/math/0407090">math/0407090</a> <a href="http://www.ams.org/mathscinet-getitem?mr=2009d:32012">MR2009d:32012</a>, Corrigendum, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 901–905, <a href="https://doi.org/10.4171/JEMS/101">doi</a></li> </ul> The generalization of orbifolds to weighted <a class="existingWikiWord" href="/nlab/show/branched+manifolds">branched manifolds</a> is discussed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, Groupoids, branched manifolds and multisections, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (<a href="http://projecteuclid.org/euclid.jsg/1180135649">project euclid</a>).</li> </ul> On <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a> and specifically on <a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> and <a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a>: <ul> <li id="ALR07"><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Leida">Johann Leida</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (<a href="https://doi.org/10.1017/CBO9780511543081">doi:10.1017/CBO9780511543081</a>, <a href="http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf">pdf</a>)</li> </ul> <h3 id="ReferencesAsLieGroupoids">As Lie groupoids</h3> Discussion of orbifolds as <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>: <ul> <li id="MoerdijkPronk97"> <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, Orbifolds, sheaves and groupoids, K-theory 12 3-21 (1997) (<a href="http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf">pdf</a>, <a href="http://dx.doi.org/10.4171/LEM/56-3-4">doi:10.4171/LEM/56-3-4</a>) </li> <li id="Moerdijk02"> <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Orbifolds as Groupoids: an Introduction, in: <a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a> (eds.) <a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a>, Contemporary Math 310, AMS (2002), 205–222 (<a href="http://arxiv.org/abs/math.DG/0203100">arXiv:math.DG/0203100</a>) </li> <li id="Lerman08"> <a class="existingWikiWord" href="/nlab/show/Eugene+Lerman">Eugene Lerman</a>, Orbifolds as stacks?, Enseign. Math. 56 3-4 (2010) 315-363 [<a href="http://arxiv.org/abs/0806.4160">arXiv:0806.4160</a>, <a href="http://dx.doi.org/10.4171/LEM/56-3-4">doi:10.4171/LEM/56-3-4</a>] </li> </ul> Review: <ul> <li>Alexander Amenta, The Geometry of Orbifolds via Lie Groupoids, ANU 2012 (<a href="https://arxiv.org/abs/1309.6367">arXiv:1309.6367</a>)</li> </ul> Analogous discussion for topological orbifolds as <a class="existingWikiWord" href="/nlab/show/topological+stacks">topological stacks</a>: <ul> <li>Vesta Coufal, <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, Carmen Rovi, <a class="existingWikiWord" href="/nlab/show/Laura+Scull">Laura Scull</a>, Courtney Thatcher, Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach, Contemporary Mathematics 641 (2015): 135-166 (<a href="https://arxiv.org/abs/1401.4772">arXiv:1401.4772</a>)</li> </ul> Discussion of the corresponding perspective in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, via <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stacks">Deligne-Mumford stacks</a>: <ul> <li id="Kresch09"><a class="existingWikiWord" href="/nlab/show/Andrew+Kresch">Andrew Kresch</a>, On the geometry of Deligne-Mumford stacks (<a href="https://doi.org/10.5167/uzh-21342">doi:10.5167/uzh-21342</a>, <a href="https://www.zora.uzh.ch/id/eprint/21342/1/geodm.pdf">pdf</a>), in: D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) Algebraic Geometry: Seattle 2005, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 (<a href="https://bookstore.ams.org/pspum-80-1">pspum-80-1</a>)</li> </ul> The <a class="existingWikiWord" href="/nlab/show/mapping+stacks">mapping stacks</a> of orbifolds are discussed in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Weimin+Chen">Weimin Chen</a>, On a notion of maps between orbifolds, I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 (<a href="https://arxiv.org/abs/math/0603671">arXiv:math/0603671</a>, <a href="https://doi.org/10.1142/S0219199706002246">doi:10.1142/S0219199706002246</a>). </li> <li id="RobertsVozzo18"> <a class="existingWikiWord" href="/nlab/show/David+Roberts">David Roberts</a>, <a class="existingWikiWord" href="/nlab/show/Raymond+Vozzo">Raymond Vozzo</a>, 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) (<a href="https://arxiv.org/abs/1610.05904">arXiv:1610.05904</a>, <a href="https://doi.org/10.1007/978-3-319-72299-3_3">doi:10.1007/978-3-319-72299-3_3</a>) </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a> over orbifolds: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Camille+Laurent-Gengoux">Camille Laurent-Gengoux</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Tu">Jean-Louis Tu</a>, <a class="existingWikiWord" href="/nlab/show/Ping+Xu">Ping Xu</a>, Chern-Weil map for principal bundles over groupoids, Math. Z. 255, 451–491 (2007) (<a href="https://arxiv.org/abs/math/0401420">arXiv:math/0401420</a>, <a href="https://doi.org/10.1007/s00209-006-0004-4">doi:10.1007/s00209-006-0004-4</a>) </li> <li> Christopher Seaton, Characteristic Classes of Bad Orbifold Vector Bundles, Journal of Geometry and Physics 57 (2007), no. 11, 2365–2371 (<a href="https://arxiv.org/abs/math/0606665">arXiv:math/0606665</a>) </li> </ul> An expected relation of orbifolds (<a class="existingWikiWord" href="/nlab/show/orbispaces">orbispaces</a>) to <a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, Orbispaces, orthogonal spaces, and the universal compact Lie group (<a href="https://arxiv.org/abs/1711.06019">arXiv:1711.06019</a>) (on the relation to <a class="existingWikiWord" href="/nlab/show/orbispaces">orbispaces</a>/<a class="existingWikiWord" href="/nlab/show/topological+stacks">topological stacks</a>)</li> </ul> <h3 id="ReferencesAsDiffeologicalSpaces">As diffeological spaces</h3> On <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> regarded as naive local <a class="existingWikiWord" href="/nlab/show/quotient+spaces">quotient spaces</a> (instead of <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a>/<a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>) but as such formed in <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>: <ul> <li id="IKZ10"> <a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <a class="existingWikiWord" href="/nlab/show/Yael+Karshon">Yael Karshon</a>, Moshe Zadka, Orbifolds as diffeologies, Transactions of the American Mathematical Society 362 (2010), 2811-2831 (<a href="https://arxiv.org/abs/math/0501093">arXiv:math/0501093</a>) </li> <li id="Watts15"> <a class="existingWikiWord" href="/nlab/show/Jordan+Watts">Jordan Watts</a>, The Differential Structure of an Orbifold, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 (<a href="https://arxiv.org/abs/1503.01740">arXiv:1503.01740</a>) </li> </ul> and as <a class="existingWikiWord" href="/nlab/show/stratified+space">stratified</a> <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Serap+G%C3%BCrer">Serap Gürer</a>, <a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, Orbifolds as stratified diffeologies, Differential Geometry and its Applications 86 (2023) 101969 [<a href="https://doi.org/10.1016/j.difgeo.2022.101969">doi:10.1016/j.difgeo.2022.101969</a>]</li> </ul> On this approach seen in the broader context of <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomology">Proper Orbifold Cohomology</a> (<a href="https://arxiv.org/abs/2008.01101">arXiv:2008.01101</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory [<a href="https://arxiv.org/abs/2205.15887">arXiv:2205.15887</a>] </li> </ul> <h3 id="orbifold_cobordism">Orbifold cobordism</h3> Orbifold <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> are discussed in <ul> <li> K. S. Druschel, Oriented Orbifold Cobordism, Pacific J. Math., 164(2) (1994), 299-319 (<a href="http://dx.doi.org/10.2140/pjm.1994.164.299">doi:10.2140/pjm.1994.164.299</a>, <a href="https://msp.org/pjm/1994/164-2/pjm-v164-n2-p04-p.pdf">pdf</a>) </li> <li> K. S. Druschel, The Cobordism of Oriented Three Dimensional Orbifolds, Pacific J. Math., bf 193(1) (2000), 45-55. </li> <li> Andres Angel, Orbifold cobordism (<a href="http://www.math.uni-bonn.de/people/aangel79/Orbifold%20cobordism.pdf">pdf</a>) </li> </ul> See also at <a class="existingWikiWord" href="/nlab/show/orbifold+cobordism">orbifold cobordism</a>. <a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a> on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> (in the context of <a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Tim+Weelinck">Tim Weelinck</a>, Equivariant factorization homology of global quotient orbifolds (<a href="https://arxiv.org/abs/1810.12021">arXiv:1810.12021</a>, <a href="https://www.maths.ed.ac.uk/~tweelinck/efhqsp.pdf">talk pdf</a>) </li> <li> John Pardon, Orbifold bordism and duality for finite orbispectra (<a href="https://arxiv.org/abs/2006.12702">arXiv:2006.12702</a>) </li> </ul> <h3 id="ReferencesInStringTheory">In string theory</h3> <div style="margin: -30px 0px 10px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/GreenStringOrbifold.jpg" width="700px" /> <figcaption style="text-align: center">(from <a href="string+theory#Green86">Green 86</a>)</figcaption> </figure> </div> In <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>, orbifolds as <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> for a <a class="existingWikiWord" href="/nlab/show/string">string</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> were first considered in <ul> <li id="DixonHarveyVafaWitten85"> <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Harvey">Jeff Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Strings on orbifolds, Nuclear Physics B Volume 261, 1985, Pages 678-686 (<a href="https://doi.org/10.1016/0550-3213(85)90593-0">doi:10.1016/0550-3213(85)90593-0</a>) </li> <li id="DixonHarveyVafaWitten86"> <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Harvey">Jeff Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Strings on orbifolds (II), Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (<a href="https://doi.org/10.1016/0550-3213(86)90287-7">doi:10.1016/0550-3213(86)90287-7</a>) </li> </ul> and then further developed notably in <ul> <li id="DVVV89"> <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Erik+Verlinde">Erik Verlinde</a>, <a class="existingWikiWord" href="/nlab/show/Herman+Verlinde">Herman Verlinde</a>, The operator algebra of orbifold models, Comm. Math. Phys. Volume 123, Number 3 (1989), 485-526 (<a href="https://projecteuclid.org/euclid.cmp/1104178892">euclid:1104178892</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Eric+Zaslow">Eric Zaslow</a>, Topological orbifold models and quantum cohomology rings, Comm. Math. Phys. 156 (1993), no. 2, 301–331. </li> </ul> See also: <ul> <li>Stefano Giaccari, <a class="existingWikiWord" href="/nlab/show/Roberto+Volpato">Roberto Volpato</a>, A fresh view on string orbifolds [<a href="https://arxiv.org/abs/2210.10034">arXiv:2210.10034</a>]</li> </ul> For more references on orbifolds in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> see also at <ul> <li> <a class="existingWikiWord" href="/nlab/show/Riemannian+orbifold">Riemannian orbifold</a>, <a class="existingWikiWord" href="/nlab/show/toroidal+orbifold">toroidal orbifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fractional+D-brane">fractional D-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gepner+model">Gepner model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/RR-field+tadpole+cancellation">RR-field tadpole cancellation</a> </li> </ul> Review of <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic</a> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> on orbifolds: <ul> <li><a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, Heterotic Orbifold Models, in: <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a>, Springer (2024) [<a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>, <a href="https://arxiv.org/abs/2401.03125">arXiv:2401.03125</a>] </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/blow-up">blow-up</a> of orbifold <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> in string theory: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, Resolution of Orbifold Singularities in String Theory (<a href="https://arxiv.org/abs/hep-th/9403123">arXiv:hep-th/9403123</a>)</li> </ul> In terms of <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Yi-Zhi+Huang">Yi-Zhi Huang</a>, Representation theory of vertex operator algebras and orbifold conformal field theory (<a href="https://arxiv.org/abs/2004.01172">arXiv:2004.01172</a>)</li> </ul> Review of orbifolds in the context of string <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a> and <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a>: <ul> <li id="BailibLove99"> D. Bailin, A. Love, Orbifold compactifications of string theory, Phys. Rept. 315 (1999) 285-408 (<a href="https://doi.org/10.1016/S0370-1573(98)00126-4">doi:10.1016/S0370-1573(98)00126-4</a>, <a href="https://inspirehep.net/literature/504382">spire:504382</a>) </li> <li id="Wendland01"> <a class="existingWikiWord" href="/nlab/show/Katrin+Wendland">Katrin Wendland</a>, Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry, in <a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a> (<a href="https://arxiv.org/abs/hep-th/0112006">arXiv:hep-th/0112006</a>) </li> <li id="Giedt02"> Joel Giedt, Heterotic Orbifolds (<a href="https://arxiv.org/abs/hep-ph/0204315">arXiv:hep-ph/0204315</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Dieter+L%C3%BCst">Dieter Lüst</a>, S. Reffert, E. Scheidegger, S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv.Theor.Math.Phys.12:67-183, 2008 (<a href="https://arxiv.org/abs/hep-th/0609014">arXiv:hep-th/0609014</a>) </li> <li id="Reffert07"> Susanne Reffert, The Geometer’s Toolkit to String Compactifications, lectures at <a href="https://www.ggi.infn.it/showevent.pl?id=11">String and M theory approaches to particle physics and cosmology</a>, 2007 (<a href="https://arxiv.org/abs/0706.1310">arXiv:0706.1310</a>) </li> <li id="IbanezUranga12"> <a class="existingWikiWord" href="/nlab/show/Luis+Ib%C3%A1%C3%B1ez">Luis Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, Chapter 8 of <a class="existingWikiWord" href="/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology">String Theory and Particle Physics – An Introduction to String Phenomenology</a>, Cambridge University Press 2012 (<a href="https://doi.org/10.1017/CBO9781139018951">doi:10.1017/CBO9781139018951</a>) </li> <li id="BlumenhagenLustTheisen13"> <a class="existingWikiWord" href="/nlab/show/Ralph+Blumenhagen">Ralph Blumenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+L%C3%BCst">Dieter Lüst</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Theisen">Stefan Theisen</a>, Chapter 10.5 Toroidal orbifolds, of Basic Concepts of String Theory Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 </li> </ul> and for orbifolds of <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifolds">G₂-manifolds</a> for <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>: <ul> <li id="Acharya98"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, M theory, Joyce Orbifolds and Super Yang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227-248 [<a href="http://arxiv.org/abs/hep-th/9812205">arXiv:hep-th/9812205</a>] </li> <li id="Reidegeld15"> <a class="existingWikiWord" href="/nlab/show/Frank+Reidegeld">Frank Reidegeld</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-orbifolds from K3 surfaces with ADE-singularities [<a href="http://arxiv.org/abs/1512.05114">arXiv:1512.05114</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Frank+Reidegeld">Frank Reidegeld</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-orbifolds with ADE-singularities [<a href="https://core.ac.uk/download/pdf/159317626.pdf">pdf</a>] </li> </ul> and for <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic</a> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, Heterotic Orbifold Models, in <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a>, Springer (2024) [<a href="https://arxiv.org/abs/2401.03125">arXiv:2401.03125</a>, <a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>]</li> </ul> For <a class="existingWikiWord" href="/nlab/show/topological+strings">topological strings</a> the <a class="existingWikiWord" href="/nlab/show/path+integral+as+a+pull-push+transform">path integral as a pull-push transform</a> for target orbifolds – in analogy to what <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a> is for <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stacks">Deligne-Mumford stacks</a> – has first been considered in <ul> <li id="ChenRuan01"><a class="existingWikiWord" href="/nlab/show/Weimin+Chen">Weimin Chen</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, Orbifold Gromov-Witten Theory, in <a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a> (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (<a href="http://arxiv.org/abs/math/0103156">arXiv:math/0103156</a>)</li> </ul> Review with further pointers: <ul> <li id="Abramovich05"><a class="existingWikiWord" href="/nlab/show/Dan+Abramovich">Dan Abramovich</a>, Lectures on Gromov-Witten invariants of orbifolds (<a href="http://arxiv.org/abs/math/0512372">arXiv:math/0512372</a>)</li> </ul> On non-supersymmetric <a class="existingWikiWord" href="/nlab/show/flat+orbifolds">flat orbifolds</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories: <ul> <li>Anamaria Font, Alexis Hernandez, Non-Supersymmetric Orbifolds, Nucl. Phys. B 634 (2002) 51-70 [<a href="https://arxiv.org/abs/hep-th/0202057">arXiv:hep-th/0202057</a>]</li> </ul> and specifically <a class="existingWikiWord" href="/nlab/show/flux+compactification">fluxed</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/D%3D6+supergravity">D=6 supergravity</a> on the <a class="existingWikiWord" href="/nlab/show/pillowcase+orbifold">pillowcase orbifold</a>: <ul> <li> Christoph Ludeling, 6D supergravity: Warped solution and gravity mediated supersymmetry breaking, PhD thesis, Hamburg (2006) [<a href="https://doi.org/10.3204/DESY-THESIS-2006-020">doi:10.3204/DESY-THESIS-2006-020</a>] </li> <li> Gero von Gersdorff, Anomalies on Six Dimensional Orbifolds, JHEP 0703:083 (2007) [<a href="https://arxiv.org/abs/hep-th/0612212">arXiv:hep-th/0612212</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Markus+Dierigl">Markus Dierigl</a>, Aspects of Six-Dimensional Flux Compactifications, PhD thesis, Hamburg (2017) [<a href="https://doi.org/10.3204/PUBDB-2017-09253">doi:10.3204/PUBDB-2017-09253</a>] </li> </ul> On <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactified</a> (and <a class="existingWikiWord" href="/nlab/show/branes">branes</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> on) <a class="existingWikiWord" href="/nlab/show/spindle+orbifolds">spindle orbifolds</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Pietro+Ferrero">Pietro Ferrero</a>, <a class="existingWikiWord" href="/nlab/show/Jerome+P.+Gauntlett">Jerome P. Gauntlett</a>, Juan Manuel Pérez Ipiña, <a class="existingWikiWord" href="/nlab/show/Dario+Martelli">Dario Martelli</a>, <a class="existingWikiWord" href="/nlab/show/James+Sparks">James Sparks</a>, D3-branes wrapped on a spindle, Phys. Rev. Lett. 126 111601 (2021) [<a href="https://arxiv.org/abs/2204.02990">arXiv:2204.02990</a>, <a href="https://doi.org/10.1103/PhysRevLett.126.111601">doi:10.1103/PhysRevLett.126.111601</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Pietro+Ferrero">Pietro Ferrero</a>, <a class="existingWikiWord" href="/nlab/show/Jerome+P.+Gauntlett">Jerome P. Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/Dario+Martelli">Dario Martelli</a>, <a class="existingWikiWord" href="/nlab/show/James+Sparks">James Sparks</a>, M5-branes wrapped on a spindle, J. High Energ. Phys. 2021 2 (2021) [<a href="https://arxiv.org/abs/2105.13344">arXiv:2105.13344</a>, <a href="https://doi.org/10.1007/JHEP11(2021)002">doi:10.1007/JHEP11(2021)002</a>] ] </li> <li> Federico Faedo, <a class="existingWikiWord" href="/nlab/show/Dario+Martelli">Dario Martelli</a>, D4-branes wrapped on a spindle, J. High Energ. Phys. 2022 101 (2022) [<a href="https://arxiv.org/abs/2111.13660">arXiv:2111.13660</a>, <a href="https://doi.org/10.1007/JHEP02(2022)101">doi:10.1007/JHEP02(2022)10</a>] </li> <li> Christopher Couzens, A tale of (M)2 twists, J. High Energ. Phys. 2022 78 (2022) [<a href="https://arxiv.org/abs/2112.04462">arXiv:2112.04462</a>, <a href="https://doi.org/10.1007/JHEP03(2022)078">doi:10.1007/JHEP03(2022)078</a>] </li> <li> K. C. Matthew Cheung, Jacob H. T. Fry, <a class="existingWikiWord" href="/nlab/show/Jerome+P.+Gauntlett">Jerome P. Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/James+Sparks">James Sparks</a>, M5-branes wrapped on four-dimensional orbifolds, J. High Energ. Phys. 2022 82 (2022) [<a href="https://arxiv.org/abs/2204.02990">arXiv:2204.02990</a>, <a href="https://doi.org/10.1007/JHEP08(2022)082">doi:10.1007/JHEP08(2022)082</a>] </li> </ul> On orbifolds by <a class="existingWikiWord" href="/nlab/show/2-groups">2-groups</a> in view of <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> inspired from <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Alonso+Perez-Lona">Alonso Perez-Lona</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, Three-dimensional orbifolds by 2-groups [<a href="https://arxiv.org/abs/2303.16220">arXiv:2303.16220</a>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/Lie+theory">Lie theory</a></div></body></html> </div> <div class="revisedby"> Last revised on July 18, 2024 at 13:12:04. 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