Orbifolds and Stringy Topology

163 Pages·2007·1.1 MB·english

by  ALEJANDRO ADEM

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CAMBRIDGETRACTSINMATHEMATICS
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171 Orbifolds and Stringy
Topology
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145 IsoperimetricInequalities. ByI.CHAVEL
146 RestrictedOrbitEquivalenceforActionsofDiscreteAmenableGroups. ByJ.KAMMEYERandD.RUDOLPH
147 FloerHomologyGroupsinYang–MillsTheory. ByS.K.DONALDSON
148 GraphDirectedMarkovSystems. ByD.MAULDINandM.URBANSKI
149 CohomologyofVectorBundlesandSyzygies. ByJ.WEYMAN
150 HarmonicMaps,ConservationLawsandMovingFrames. ByF.HÉLEIN
151 FrobeniusManifoldsandModuliSpacesforSingularities. ByC.HERTLING
152 PermutationGroupAlgorithms. ByA.SERESS
153 AbelianVarieties,ThetaFunctionsandtheFourierTransform. ByA.POLISHCHUK
154 FinitePackingandCovering, K.BÖRÖCZKY,JR
155 TheDirectMethodinSolitonTheory. ByR.HIROTA.EditedandtranslatedbyA.NAGAI,J.NIMMO,andC.
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156 HarmonicMappingsinthePlane. ByP.DUREN
157 AffineHeckeAlgebrasandOrthogonalPolynomials. ByI.G.MACDONALD
158 Quasi-FrobeniusRings. ByW.K.NICHOLSONandM.F.YOUSIF
159 TheGeometryofTotalCurvature. ByK.SHIOHAMA,T.SHIOYA,andM.TANAKA
160 ApproximationbyAlgebraicNumbers. ByY.BUGEAD
161 EquivalenceandDualityforModuleCategories. ByR.R.COLBY,andK.R.FULLER
162 LévyProcessesinLieGroups. ByMINGLIAO
163 LinearandProjectiveRepresentationsofSymmetricGroups. ByA.KLESHCHEV
164 TheCoveringPropertyAxiom, CPA.K.CIESIELSKIandJ.PAWLIKOWSKI
165 ProjectiveDifferentialGeometryOldandNew. ByV.OVSIENKOandS.TABACHNIKOV
166 TheLévyLaplacian. ByM.N.FELLER
167 PoincaréDualityAlgebras,Macaulay’sDualSystems,andSteenrodOperations. ByD.M.MEYERandL.SMITH
168 TheCube:AWindowtoConvexandDiscreteGeometry. ByC.ZONG
169 QuantumStochasticProcessesandNoncommutativeGeometry. ByK.B.SINHAandD.GOSWAMI
Orbifolds and Stringy Topology
ALEJANDRO ADEM
UniversityofBritishColumbia
JOHANN LEIDA
UniversityofWisconsin
YONGBIN RUAN
UniversityofMichigan
CAMBRIDGEUNIVERSITYPRESS
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© A. Adem, J. Leida and Y. Ruan 2007
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without the written permission of Cambridge University Press.
First published in print format 2007
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Contents
Introduction pagevii
1 Foundations 1
1.1 Classicaleffectiveorbifolds 1
1.2 Examples 5
1.3 Comparingorbifoldstomanifolds 10
1.4 Groupoids 15
1.5 Orbifoldsassingularspaces 28
2 Cohomology,bundlesandmorphisms 32
2.1 DeRhamandsingularcohomologyoforbifolds 32
2.2 Theorbifoldfundamentalgroupandcoveringspaces 39
2.3 Orbifoldvectorbundlesandprincipalbundles 44
2.4 Orbifoldmorphisms 47
2.5 Classificationoforbifoldmorphisms 50
3 OrbifoldK-theory 56
3.1 Introduction 56
3.2 Orbifolds,groupactions,andBredoncohomology 57
3.3 OrbifoldbundlesandequivariantK-theory 60
3.4 AdecompositionfororbifoldK-theory 63
3.5 Projectiverepresentations,twistedgroupalgebras,
andextensions 69
3.6 TwistedequivariantK-theory 72
3.7 TwistedorbifoldK-theoryandtwistedBredon
cohomology 76
v
vi Contents
4 Chen–Ruancohomology 78
4.1 Twistedsectors 80
4.2 DegreeshiftingandPoincare´ pairing 84
4.3 Cupproduct 88
4.4 Someelementaryexamples 95
4.5 Chen–Ruancohomologytwistedbyadiscretetorsion 98
5 CalculatingChen–Ruancohomology 105
5.1 Abelianorbifolds 105
5.2 Symmetricproducts 115
References 138
Index 146
Introduction
Orbifoldslieattheintersectionofmanydifferentareasofmathematics,includ-
ing algebraic and differential geometry, topology, algebra, and string theory,
among others. What is more, although the word “orbifold” was coined rel-
atively recently,1 orbifolds actually have a much longer history. In algebraic
geometry,forinstance,theirstudygoesbackatleasttotheItalianschoolun-
dertheguiseofvarietieswithquotientsingularities.Indeed,surfacequotient
singularitieshavebeenstudiedinalgebraicgeometryformorethanahundred
years,andremainaninterestingtopictoday.Aswithanyothersingularvariety,
an algebraic geometer aims to remove the singularities from an orbifold by
eitherdeformationorresolution.Adeformationchangesthedefiningequation
ofthesingularities,whereasaresolutionremovesasingularitybyblowingitup.
Usingcombinationsofthesetwotechniques,onecanassociatemanysmooth
varietiestoagivensingularone.Incomplexdimensiontwo,thereisanatural
notionofaminimalresolution,butingeneralitismoredifficulttounderstand
therelationshipsbetweenallthedifferentdesingularizations.
Orbifolds made an appearance in more recent advances towards Mori’s
birational geometric program in the 1980s. For Gorenstein singularities, the
higher-dimensional analog of the minimal condition is the famous crepant
resolution, which is minimal with respect to the canonical classes. A whole
zoo of problems surrounds the relationship between crepant resolutions and
Gorensteinorbifolds:thisisoftenreferredtoasMcKaycorrespondence.The
McKaycorrespondenceisanimportantmotivationforthisbook;incomplexdi-
mensiontwoitwassolvedbyMcKayhimself.Thehigher-dimensionalversion
hasattractedincreasingattentionamongalgebraicgeometers,andtheexistence
ofcrepantresolutionsinthedimensionthreecasewaseventuallysolvedbyan
1 AccordingtoThurston[148],itwastheresultofademocraticprocessinhisseminar.
vii
viii Introduction
array of authors. Unfortunately, though, a Gorenstein orbifold of dimension
four or more does not possess a crepant resolution in general. Perhaps the
best-knownexampleofahigher-dimensionalcrepantresolutionistheHilbert
schemeofpointsofanalgebraicsurface,whichformsacrepantresolutionof
itssymmetricproduct.UnderstandingthecohomologyoftheHilbertschemeof
pointshasbeenaninterestingprobleminalgebraicgeometryforaconsiderable
lengthoftime.
Besidesresolution,deformationalsoplaysanimportantroleintheclassifi-
cation of algebraic varieties. For instance, a famous conjecture of Reid [129]
knownasReid’sfantasyassertsthatanytwoCalabi–Yau3-foldsareconnected
toeachotherbyasequenceofresolutionsordeformations.However,deforma-
tionsarehardertostudythanresolutions.Infact,therelationshipbetweenthe
topologyofadeformationofanorbifoldandthatoftheorbifolditselfisone
ofthemajorunresolvedquestionsinorbifoldtheory.
The roots of orbifolds in algebraic geometry must also include the theory
ofstacks,whichaimstodealwithsingularspacesbyenlargingtheconceptof
“space”ratherthanfindingsmoothdesingularizations.Theideaofanalgebraic
stackgoesbacktoDeligneandMumford[40]andArtin[7].Theseearlypapers
alreadyshowtheneedforthestacktechnologyinfullyunderstandingmoduli
problems,particularlythemodulistackofcurves.Orbifoldsarespecialcases
of topological stacks, corresponding to “differentiable Deligne and Mumford
stacks”intheterminologyof[109].
Manyoftheorbifoldcohomologytheorieswewillstudyinthisbookhave
rootsinandconnectionstocohomologytheoriesforstacks.Thebook[90]of
Laumon and Moret-Bailly is a good general reference for the latter. Orbifold
Chen–Ruan cohomology, ontheotherhand,iscloselyconnected toquantum
cohomology – it is the classical limit of an orbifold quantum cohomology
also due to Chen–Ruan. Of course, stacks also play an important role in the
quantum cohomology of smooth spaces, since moduli stacks of maps from
curves are of central importance in defining these invariants. For more on
quantumcohomology,wereferthereadertoMcDuffandSalamon[107];the
original works of Kontsevich and Manin [87, 88], further developed in an
algebraic context by Behrend [19] with Manin [21] and Fantechi [20], have
alsobeenveryinfluential.
Stackshavebeguntobestudiedinearnestbytopologistsandothersoutside
of algebraic geometry, both in relation to orbifolds and in other areas. For
instance, topological modular forms (tmf), a hot topic in homotopy theory,
haveagreatdealtodowiththemodulistackofellipticcurves[58].
Outside of algebraic geometry, orbifolds were first introduced into topol-
ogy and differential geometry in the 1950s by Satake [138, 139], who called