Table Of ContentMathematical 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
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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.
