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This page intentionally left blank CAMBRIDGETRACTSINMATHEMATICS GeneralEditors B.BOLLOBA´S,W.FULTON,A.KATOK,F.KIRWAN, P.SARNAK,B.SIMON,B.TOTARO 171 Orbifolds and Stringy Topology CAMBRIDGETRACTSINMATHEMATICS AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Foracompleteseries listingvisit http://www.cambridge.org/series/sSeries.asp?code=CTM 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. GILSON 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 Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB28RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521870047 © A. Adem, J. Leida and Y. Ruan 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 978-0-511-28528-8 eBook (Adobe Reader) ISBN-10 0-511-28288-5 eBook (Adobe Reader) ISBN-13 978-0-521-87004-7 hardback ISBN-10 0-521-87004-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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 topology and differential geometry in the 1950s by Satake [138, 139], who called