Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research PDF
Preview Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research
Mathematical Lectures from Peking University Yi-Zhi Huang et al. Editors Conformal Field Theories and Tensor Categories Proceedings of a Workshop Held at Beijing International Center for Mathematical Research Mathematical Lectures from Peking University Forfurthervolumes: www.springer.com/series/11574 Chengming Bai (cid:2) Jürgen Fuchs (cid:2) Yi-Zhi Huang (cid:2) Liang Kong (cid:2) Ingo Runkel (cid:2) Christoph Schweigert Editors Conformal Field Theories and Tensor Categories Proceedings of a Workshop Held at Beijing International Center for Mathematical Research Editors ChengmingBai LiangKong ChernInstituteofMathematics InstituteforAdvancedStudy NankaiUniversity TsinghuaUniversity Tianjin,People’sRepublicofChina Beijing,People’sRepublicofChina JürgenFuchs IngoRunkel TheoreticalPhysics DepartmentofMathematics KarlstadUniversity UniversityofHamburg Karlstad,Sweden Hamburg,Germany Yi-ZhiHuang ChristophSchweigert DepartmentofMathematics DepartmentofMathematics RutgersUniversity UniversityofHamburg Piscataway,NJ,USA Hamburg,Germany ISSN2197-4209 ISSN2197-4217(electronic) ISBN978-3-642-39382-2 ISBN978-3-642-39383-9(eBook) DOI10.1007/978-3-642-39383-9 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013952731 MathematicsSubjectClassification: 16T05,17B37,17B69,17B81,18D10,81T40,81T45 ©Springer-VerlagBerlinHeidelberg2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface In the past thirty years, (two-dimensional) conformal field theory has been devel- opedintoadeep,richandbeautifulmathematicaltheoryandthestudyofconformal fieldtheoriesandtheirapplicationsinmathematicsandphysicshasbecomeanex- citingareaofmathematics.Ithasledtonewideas,surprisingresultsandbeautiful solutionsofvariousproblemsindifferentbranchesinmathematicsandphysics,in- cluding,butnotlimitedto,algebra,numbertheory,combinatorics,topology,geom- etry, critical phenomena, quantum Hall systems, disorder systems, quantum com- putingandstringtheory,andisexpectedtoleadtomanymore. DuringJune13toJune17,2011,aworkshop“Conformalfieldtheoriesandten- sorcategories”washeldatBeijingInternationalCenterforMathematicalResearch, PekingUniversity,Beijing,China.Thisworkshopwasoneofthemainactivitiesof aone-semesterprogramonquantumalgebrafromFebruarytoJuly,2011atBeijing InternationalCenterforMathematicalResearch.Itwastheaimoftheworkshopto bringtogetherexpertsfromseveraldifferentareasofmathematicsandphysicswho areinvolvedinthenewdevelopmentsinconformalfieldtheories,tensorcategories andrelatedresearchdirections.Correspondinglytheareascoveredbytheworkshop were broad, including conformal field theories, tensor categories, quantum groups and Hopf algebras, representation theory of vertex operator algebras, nets of von Neumann algebras, topological order and lattice models and other related topics. Eachofthesefieldswasrepresentedbyleadingexperts. The simplest class of conformal field theories are rational conformal field the- ories. In 1988, Moore and Seiberg obtained polynomial equations for the fusing, braiding and modular transformations in rational conformal field theory. They ob- servedthatsomeoftheseequationsareanalogoustosomepropertiesoftensorcat- egories.Later,anotionofmodulartensorcategorywasformulatedmathematically and examples of modular tensor categories were constructed from representations ofquantumgroups.TheworkofMooreandSeibergcanbeinterpretedasderivinga modulartensorcategorystructurefromarationalconformalfieldtheory.Sincethen, thetheoryofvarioustensorcategorieshasbeengreatlydevelopedandhasbeenap- pliedtodifferentareasofmathematicsandphysics.Nowthetheoryoftensorcate- goriesnotonlyprovidesaunifyinglanguageforvariouspartsofmathematicsand v vi Preface applicationsofmathematicsinphysics,butalsogivesdeepresultsandfundamental structuresindifferentbranchesofmathematicsandphysics. On the other hand, though a number of examples of modular tensor categories were constructedat thetime thatthe notionof modulartensor categorywas intro- duced,ittookmanyyearsandalotofeffortsformathematicianstodirectlyconstruct themodulartensorcategoriesconjecturedtoappearinrationalconformalfieldthe- ories. Many mathematicians, including in particular Kazhdan-Lusztig, Beilinson- Feigin-Mazur, Finkelberg, Huang-Lepowsky, and Bakalov-Kirillov, contributed in 1990’sandearly2000’stotheconstructionoftheparticularclassofexamplesofthe modulartensorcategoriesassociatedwiththeWess-Zumino-Novikov-Wittenmod- els(therationalconformalfieldtheoriesassociatedwithsuitablerepresentationsof affine Lie algebras). However, the construction of even this particular class of ex- ampleswasnotcompleteuntil2005whenHuanggaveageneralconstructionofall themodulartensorcategoriesconjecturedtobeassociatedwithrationalconformal fieldtheories.Thecorrespondingchiralrationalconformalfieldtheoriesarethereby largely under control. Indeed, many problems in rational conformal field theories havemeanwhilebeensolved. Intheworkshop,newdevelopmentsbeyondrationalconformalfieldtheoriesand modular tensor categories and new applications in mathematics and physics were presentedbytopexperts.Herewewouldliketomentionespeciallythefollowing: 1. Construction of interesting tensor categories from representation categories of Hopfalgebras,asreviewedbyAndruskiewitschinhisoverviewtalkandalsoin the contribution by Andruskiewitsch, Angiono, García Iglesias, Torrecillas and Vayinthisvolume. 2. New categorical techniques and structures in tensor categories, as reviewed by Ostrikinhisoverviewtalk.OnealsoshouldincludeheretheWittgroupasdis- cussed by Nikshych and Davydov and Hopf-monadic techniques as explained byVirelizier.In a sense, Semikhatov’scontributionin this volumeusingHopf- algebraic structures in representation categories interpolates between this point andtheprecedingpoint. 3. Applications to topological phases and gapped systems as reviewed by Wen in hisoverviewtalkandalsointhecontributionbyWenandWanginthisvolume. ThestudyoftheLevin-WenmodelasdiscussedbyWuisanimportantexample ofsuchapplications. 4. Realizationofthetensor-categoricalstructuresinlatticemodelsasinFendley’s overviewandGainutdinov’stalk. 5. New developments in the representation theory of vertex operator algebras, es- peciallythenonsemisimpletheorycorrespondingtologarithmicconformalfield theory, as reviewed by Lepowsky’s overview talk and in the contribution by Huang, Lepowsky and Zhang in this volume. Recent results on representations and the structure of the representation category were reported by Adamovic, Arike,MilasandMiyamotoandalsointhecontributionsbyAdamovicandMilas and by Miyamoto in this volume. To some extent, Tsuchiya’s talk also went in thisdirection.Connectionstologarithmicconformalfieldtheorywerediscussed Preface vii by Runkel and Semikhatov and also in the contributions by Runkel, Gaberdiel andWoodandbySemikhatovinthisvolume. In the workshop, there were 21 invited talks by mathematicians and physicists fromArgentina,China,Croatia,France,Germany,Japan,RussiaandUSA.Someof the invited talks were given by young researchers. The participants benefited a lot from communicating results between the various disciplines and from the attempt to understand them in the framework of conformal field theories and tensor cate- gories.Theseattemptsgaverisetofurtherquestionsduringandafterthetalks,and, maybeevenmoreimportantly,alsoresultedinnumerousandlivelyprivatediscus- sionsamongtheparticipants. Theworkshopalsohadimportanttrainingimpactonstudents.Anumberofun- dergraduateandbeginninggraduatestudentsintheEnhancedProgramforGraduate StudyinBeijingInternationalCenterforMathematicalResearchparticipatedinthe workshop.Theybenefitedgreatlyfromthetalks,especiallythefiveoverviewtalks, andfromdiscussionswithactiveresearchersintheworkshop. The present volume is a collection of seven papers that are either based on the talks presented in the workshop or are extensions of the material presented in the talks in the workshop. We believe that the papers in this volume will be useful to everyonewhoisinterestedinconformalfieldtheories,tensorcategoriesandrelated topics.Wehopethatthesepaperswillalsoinspiremoreresearchactivitiesinthese directions. WeareverygratefultoBeijingInternationalCenterforMathematicalResearch and the National Science Foundation in USA for the funding and support of the workshop.WethankthestaffatBeijingInternationalCenterforMathematicalRe- searchfortheirhelpduringtheworkshop.Wethankalltheparticipants,thespeakers and,especially,theauthorswhosepapersareincludedinthisvolumeandtheanony- mousrefereesfortheircarefulreviewsofthepapersincludedinthisvolume. Tianjin,People’sRepublicofChina ChengmingBai Karlstad,Sweden JürgenFuchs Piscataway,NJ,USA Yi-ZhiHuang Beijing,People’sRepublicofChina LiangKong Hamburg,Germany IngoRunkel ChristophSchweigert Contents FromHopfAlgebrastoTensorCategories . . . . . . . . . . . . . . . . . 1 N.Andruskiewitsch,I.Angiono,A.GarcíaIglesias,B.Torrecillas,and C.Vay Pattern-of-ZerosApproachtoFractionalQuantumHallStatesand aClassificationofSymmetricPolynomialofInfiniteVariables . . . 33 Xiao-GangWenandZhenghanWang VirasoroCentralChargesforNicholsAlgebras . . . . . . . . . . . . . . 67 A.M.Semikhatov LogarithmicBulkandBoundaryConformalFieldTheoryandtheFull CentreConstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 93 IngoRunkel,MatthiasR.Gaberdiel,andSimonWood Logarithmic Tensor Category Theory for Generalized Modules foraConformalVertexAlgebra,I:IntroductionandStrongly GradedAlgebrasandTheirGeneralizedModules . . . . . . . . . . 169 Yi-ZhiHuang,JamesLepowsky,andLinZhang C -CofiniteW-AlgebrasandTheirLogarithmicRepresentations . . . 249 2 DraženAdamovic´ andAntunMilas C -CofinitenessandFusionProductsforVertexOperatorAlgebras . . 271 1 MasahikoMiyamoto ix From Hopf Algebras to Tensor Categories N.Andruskiewitsch,I.Angiono,A.GarcíaIglesias,B.Torrecillas,andC.Vay Abstract This is a survey on spherical Hopf algebras. We give criteria to decide whenaHopfalgebraissphericalandcollectexamples.Wediscusstiltingmodulesas ameantoobtainafusionsubcategoryofthenon-degeneratequotientofthecategory ofrepresentationsofasuitableHopfalgebra. MathematicsSubjectClassification(2000) 16W30 1 Introduction It follows from its very definition that the category RepH of finite-dimensional representations of a Hopf algebra H is a tensor category. There is a less obvious way to go from Hopf algebras with some extra structure (called spherical Hopf algebras)totensorcategories.SphericalHopfalgebrasandtheproceduretoobtain a tensor category from them were introduced by Barrett and Westbury [19, 20], inspiredbypreviousworkbyReshetikhinandTuraev[69,70],inturnmotivatedto giveamathematicalfoundationtotheworkofWitten[76]. AsphericalHopfalgebrahasbydefinitionagroup-likeelementthatimplements thesquareoftheantipode(calledapivot)andsatisfiestheleft-righttracesymmetry N.Andruskiewitsch(B)·I.Angiono·A.GarcíaIglesias·C.Vay FaMAF-CIEM(CONICET),UniversidadNacionaldeCórdoba,MedinaAllendes/n,Ciudad Universitaria,5000Córdoba,RepúblicaArgentina e-mail:[email protected] I.Angiono e-mail:[email protected] A.GarcíaIglesias e-mail:[email protected] C.Vay e-mail:[email protected] B.Torrecillas Dpto.ÁlgebrayAnálisisMatemático,UniversidaddeAlmería,04120Almería,Spain e-mail:[email protected] C.Baietal.(eds.),ConformalFieldTheoriesandTensorCategories, 1 MathematicalLecturesfromPekingUniversity,DOI10.1007/978-3-642-39383-9_1, ©Springer-VerlagBerlinHeidelberg2014 2 N.Andruskiewitschetal. condition (3.2). The classification (or even the characterization) of spherical Hopf algebrasisfarfrombeingunderstood,buttherearetwoclassestostartwith.Letus firstobservethatsemisimplesphericalHopfalgebrasareexcludedfromourconsid- erations,sincethetensorcategoriesarisingfromtheprocedureareidenticaltothe categoriesofrepresentations.Anotherremark:anyHopfalgebrais embeddedina pivotalone,sothatthetracecondition(3.2)isreallythecrucialpoint.Nowthetwo classeswemeanare • Hopf algebras with involutory pivot, or what is more or less the same, with S4=id. Here the trace condition follows for free, and the quantum dimensions willbe(positiveandnegative)integers. • RibbonHopfalgebras[69,70]. It is easy to characterize pointed or copointed Hopf algebras with S4 =id; so we have many examples of (pointed or copointed) spherical Hopf algebras with involutorypivot,mostofthemnotevenquasi-triangular,seeSect.3.6.Ontheother hand,anyquasitriangularHopfalgebraisembeddedinaribbonone[69];combined with the construction of the Drinfeld double, we see that any finite-dimensional Hopfalgebragivesrisetoaribbonone.So,wehaveplentyofexamplesofspherical Hopfalgebras,althoughofaratherspecialtype. TheproceduretogetatensorcategoryfromasphericalHopfalgebraH consists intakingasuitablequotientRepH ofthecategoryRepH.Thisappearsin[20]but similarideascanbefoundelsewhere,seee.g.[35,55].Theresultingsphericalcat- egoriesaresemisimplebutseldomhaveafinitenumberofirreducibles,thatis,they are seldom fusion categories in the sense of [32]. We are interested in describing fusion tensor subcategories of RepH for suitable H. This turns out to be a tricky problem.First,ifthepivotisinvolutive,thenthefusionsubcategoriesofRepH are integral, see Proposition 3.12. The only way we know is through tilting modules; but it seems to us that there is no general method, just a clever recipe that works. This procedure has a significant outcome in the case of quantum groups at roots ofone,wherethecelebratedVerlindecategoriesareobtained[3];seealso[72]for a self-contained exposition and [62] for similar results in the setting of algebraic groupsoverfieldsofpositivecharacteristic.OneshouldalsomentionthattheVer- linde categories can be also constructed from vertex operator algebras related to affineKac-moodyalgebras,see[18,45,46,50–53]andreferencestherein;thecom- parisonofthesetwoapproachesishighlynon-trivial.Anotherapproach,atleastfor SL(n),wasproposedin[40]viafacealgebras(anotionpredecessorofweakHopf algebras). The paper is organized as follows. Section 2 contains some information about thestructureofHopfalgebrasandnotationusedlaterinthepaper.Section3isde- votedtosphericalHopfalgebras.InSect.4wediscusstiltingmodulesandhowthis recipewouldworkforsomefinite-dimensionalpointedHopfalgebrasassociatedto Nicholsalgebrasofdiagonaltype,thatmightbethoughtofasgeneralizationsofthe smallquantumgroupsofLusztig.
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