Noncommutative geometry: lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, September 3-9, 2000 PDF
Preview Noncommutative geometry: lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, September 3-9, 2000
Lecture Notes in Mathematics 1831 Editors: J.--M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris Subseries: FondazioneC.I.M.E.,Firenze Adviser:PietroZecca 3 Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo A. Connes J. Cuntz E. Guentner N. Higson J. Kaminker J. E. Roberts Noncommutative Geometry Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 3-9, 2000 Editors: S. Doplicher R. Longo 1 3 EditorsandAuthors AlainConnes NigelHigson Colle`gedeFrance DepartmentofMathematics 11,placeMarcelinBerthelot PennsylvaniaStateUniversity 75231ParisCedex05,France UniversityPark,PA16802 e-mail:[email protected] USA e-mail:[email protected] JoachimCuntz MathematischesInstitut JeromeKaminker Universita¨tMu¨nster DepartmentofMathematicalSciences Einsteinstr.62 IUPUI,Indianapolis 48149Mu¨nster,Germany IN46202-3216,USA e-mail:[email protected] e-mail:[email protected] SergioDoplicher RobertoLongo DipartimentodiMatematica JohnE.Roberts Universita`diRoma"LaSapienza" DipartimentodiMatematica PiazzaleA.Moro5 Universita`diRoma"TorVergata" 00185Roma,Italy ViadellaRicercaScientifica1 e-mail:[email protected] 00133Roma,Italy e-mail:[email protected] ErikGuentner [email protected] DepartmentofMathematicalSciences UniversityofHawaii,Manoa 2565McCarthyMall Keller401A,Honolulu HI96822,USA e-mail:[email protected] Cataloging-in-PublicationDataappliedfor BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetathttp://dnb.ddb.de MathematicsSubjectClassification(2000):58B34,46L87,81R60,83C65 ISSN0075-8434 ISBN3-540-20357-5Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2004 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readyTEXoutputbytheauthors SPIN:10967928 41/3142/du-543210-Printedonacid-freepaper Preface IfonehadtosynthesizethenoveltyofPhysicsoftheXXcenturywithasinglemagic word,onepossibilitywouldbe“Noncommutativity”. Indeed the core assertion of Quantum Mechanics is the fact that observables oughttobedescribed“bynoncommutingoperators”;ifyouwishedmoreprecision andsaid“bytheselfadjointelementsinaC*-algebraA,whilestatesareexpectation functionalsonthatalgebra,i.e.positivelinearformsofnormoneonA”,youwould haveputdownthefullaxiomsforatheorywhichincludesClassicalMechanicsifA iscommutative,QuantumMechanicsotherwise. More precisely, Quantum Mechanics of systems with finitely many degrees of freedom would fit in the picture when the algebra is the collection of all compact operatorson the separable,infinite-dimensionalHilbert space (so that all, possibly unbounded, selfadjoint operators on that Hilbert space appear as “generalized ob- servables”affiliatedwiththeenvelopingvonNeumannalgebra);thedistinctionbe- tweendifferentvaluesofthenumberofdegreesoffreedomrequiresmoredetails,as theassignmentofadenseBanach*-algebra(thequotient,obtainedbyspecifyingthe valueofthePlanckconstant,oftheL1-algebraoftheHeisenberggroup). QuantumField Theory,as explainedin Roberts’ lecturesin this volume,fits in thatpicturetoo:thekeyadditionalstructureneededisthelocalstructureofA.This meansthatAhastobetheinductivelimitofsubalgebrasoflocalobservablesA(O), whereO(cid:1)→A(O)mapscoherentlyregionsinthespacetimemanifoldtosubalgebras ofA. Asaconsequenceoftheaxioms,asmorecarefullyexpoundedinthisbook,Ais muchmoredramaticallynoncommutativethaninQuantumMechanicswithfinitely manydegreesoffreedom:Acannotbeanylongeressentiallycommutative(inother words,itcannotbeanextensionofthecompactsbyacommutativeC*-algebra),and actuallyturnsouttobeasimplenontypeIC*-algebra. Inordertodealconvenientlywiththenaturalrestrictiontolocallynormalstates, it is also most often natural to let each A(O) be a von Neumann algebra, so that VI Preface A is notnorm separable:for the sake of both Quantum Statistical Mechanicswith infinitely many degrees of freedom and of physically relevant classes of Quantum Field Theories - fulfilling the split property, cf. Roberts’ lectures - A can actually be identified with a universal C*-algebra:the inductive limit of the algebras of all bounded operators on the tensor powers of a fixed infinite dimensional separable Hilbert space; different theories are distinguished by the time evolution and/or by the local structure, of which the inductive sequence of type I factors gives only a fuzzypicture.TheactuallocalalgebrasofQuantumFieldTheory,ontheotherside, canbeprovedingreatgeneralitytobeisomorphictotheunique,approximatelyfinite dimensionalIII factor(exceptforthepossiblenontrivialityofthecentre). 1 Despite this highly noncommutativeambient, the key axiom of QuantumField Theoryofforcesotherthangravity,isademandofcommutativity:localsubalgebras associated to causally separated regions should commute elementwise. This is the basicLocalityPrinciple,expressingEinsteinCausality. This principle alone is “unreasonablyeffective” to determine a substantialpart oftheconceptualstructureofQuantumFieldTheory.ThisappliestoQuantumField TheoryonMinkowskispace butalso on largeclasses of curvedspacetimes, where the pseudo-Riemann structure describes a classical external gravitational field on whichtheinfluenceofthequantumfieldsisneglected(cfRoberts’lectures).Butthe LocalityPrincipleisboundtofailinaquantumtheoryofgravity. Mentioninggravitybringsintheothermagicwordonecouldhavementionedat thebeginning:“Relativity”. Classical General Relativity is a miracle of human thought and a masterpiece of Nature;the accuracyof its predictionsgrowsmoreand more spectacularlywith years(binarypulsarsareafamousexample).Buttheformulationofacoherentand satisfactoryQuantumTheoryofallforcesincludingGravitystillappearstomanyas oneofthefewmostformidableproblemsforscienceoftheXXIcentury. InsuchatheoryEinsteinCausality islost, andwe donotyetknowwhatreally replacesit: fortherelation“causallydisjoint”is boundto lose meaning;moredra- matically,spacetimeitselfhastolookradicallydifferentatsmallscales.Here“small” meansatscalesgovernedbythe Plancklength,whichistremendouslysmallbutis there. Indeed Classical General Relativity and Quantum Mechanics imply Spacetime Uncertainty Relations which are most naturally taken into accountif spacetime it- selfispicturedasaQuantumManifold:thecommutativeC*-algebraofcontinuous functions vanishing at infinity on Minkowski space has to be replaced by a non- commutativeC*-algebra,insuchawaythatthespacetimeuncertaintyrelationsare implemented[DFR].ItmightwellturnouttobeimpossibletodisentangleQuantum FieldsandSpacetimefromacommonnoncommutativetexture. Quantum Field Theory on Quantum Spacetime ought to be formulated as a Gaugetheoryonanoncommutativemanifold;onemighthopethattheGaugeprinci- ple,atthebasisofthepointnatureofinteractionsbetweenfieldsonMinkowskispace andhenceofthe PrincipleofLocality,mightberigidenoughto replacelocalityin theworldofquantumspaces. Preface VII Gauge Theorieson noncommutativemanifoldsoughtto appear as a chapter of NoncommutativeGeometry[CR,C]. Thus NoncommutativeGeometry may be seen as a main avenue from Physics of the XX century to Physics of the XXI century;but since it has been created by Alain Connes in the late 70s, as expoundedin his lectures in this Volume, it grew to a central theme in Mathematics with a tremendouspower of unifying disparate problemsandofprogressingindepth. OnecouldwithgoodreasonsarguethatNoncommutativeTopologystartedwith thefamousGel’fand-NaimarkTheorems:everycommutativeC*-algebraisthealge- bra ofcontinuousfunctionsvanishingat infinityon a locally compactspace, every C*-algebracanberepresentedasanalgebraofboundedoperatorsonaHilbertspace; thusanoncommutativeC*-algebracanbeviewedas“thealgebraofcontinuousfunc- tionsvanishingatinfinity”ona“quantumspace”. ButitwaswiththeTheoryofBrown,DouglasandFillmoreofExt,withthede- velopmentoftheK-theoryofC*-algebras,andtheirmergingintoKasparovbivariant functorKKthatNoncommutativeTopologybecamearichsubject.Nowthissubject couldhardlybeseparatedfromNoncommutativeGeometry. It suffices to mention a few fundamental landmarks: the discovery by Alain Connesof Cyclic Cohomology,crucialfor the lift of De Rham Theoryto the non- commutative domain, the Connes-Chern Character; the concept of spectral triple provedtobecentralandthenaturalroadtothetheoryofnoncommutativeRieman- nianmanifolds. Sincehestartedtobreakthisnewground,Connesdiscoveredaparadigmwhich couldnothavebeenanticipatedjustonthebasisofGel’fand-Naimarktheory:Non- commutativeGeometrynotonlyextendsgeometricalconceptsbeyondpointspaces to“noncommutativemanifolds”,butalsopermitstheirapplicationtosingularspaces: suchspacesarebestviewedasnoncommutativespaces,describedbyanoncommu- tativealgebra,ratherthanasmerepointspaces. A famousclass of examplesof singular spaces are the spaces of leafs of folia- tions;suchaspaceisbestdescribedbyanoncommutativeC*-algebra,which,when thefoliationisdefinedasorbitsinthe manifoldM bytheactionofa LiegroupG andhasgraphM×G, coincideswhichthe (reduced)crossproductof thealgebra of continuousfunctionson the manifold by that action. The Atiyah - Singer Index Theoremhaspowerfulgeneralizations,whichculminatedintheextensionofitslocal form to transversallyelliptic pseudodifferentialoperatorson the foliation,in terms ofthecycliccohomologyofaHopfalgebrawhichdescribesthetransversegeometry [CM]. Thereisamazeofexamplesofsingularspaceswhichacquirethiswayniceand tractable structures [C]. But also discrete spaces often do: Bost and Connes asso- ciatedtothedistributionofprimenumbersanintrinsicnoncommutativedynamical system with phase transitions [BC]. Connes formulateda trace formula whose ex- tensiontosingularspaceswouldproveRiemannhypothesis[Co].Thegeometryof thetwopointset,viewedas“extradimensions”ofMinkowskispace,isthebasisfor theConnesandLotttheoryofthestandardmodel,providinganelegantmotivation fortheformoftheactionincludingtheHiggspotential[C];thislinehasbeenfurther VIII Preface developedbyConnesintoadeepspectralactionprinciple,formulatedonEuclidean, compactifiedspacetime,whichunifiesthe StandardmodelandtheEinsteinHilbert action[C1]. ThusNoncommutativeGeometryissurprisinglyeffectiveinprovidingtheform oftheexpressionfortheaction.ButifoneturnstotheQuantumTheoryithasalot to say also on Renormalization.Connes and Moscovicidiscovereda Hopf algebra associatedwiththedifferentiablestructureofamanifold,whichprovidesapowerful organizingprinciplewhichwascrucialtotheTransverseIndexTheorem;inthecase ofMinkowskispace,itprovedtobeintimatelyrelatedwithKreimer’sHopfalgebra associated to Feynman graphs. Developing this connection, Connes and Kreimer could cast Renormalization Theory in a mathematically sound and elegant frame, asaRiemann-Hilbertproblem[CK]. TherelationsofNoncommutativeGeometrytotheAlgebraicApproachtoQuan- tumFieldTheoryarestilltobeexploredindepth.Thefirstlinksappearedinsuper- symmetricQuantumFieldTheory:thenonpolynomialcharacteroftheindexmapon someKgroupsassociatedtothelocalalgebrasinafreesupersymmetricmassivethe- ory[C],andtherelationtotheChernCharacteroftheJaffeLesniewskiOsterwalder cycliccocycleassociatedtoasuperGibbsfunctional[JLO,C]. More generallyin the theory of superselectionsectors it has long been conjec- tured that localized endomorphisms with finite statistics ought to be viewed as a highlynoncommutativeanalogofFredholmoperators;thediscoveryoftherelation betweenstatisticsandJonesindexgavesolidgroundstothisview.WhileJonesindex defines the analytical indexof the endomorphism,a geometricdimension can also be introduced,where, in the case of a curvedbackground,the spacetime geometry enterstoo,andananalogoftheIndexTheoremholds[Lo].Onecanexpectthisisa fertilegroundtobefurtherexplored. Noncommutative spaces appeared also as the underlying manifold of a quan- tumgroupinthesenseofWoronowicz;noncommutativegeometrycanbeappliedto thosemanifoldstoo.Mostrecentdevelopmentsanddiscoveriescanbefoundinthe LecturesbyConnes. NoncommutativeGeometry and NoncommutativeTopology merge in the cele- bratedBaum-ConnesconjectureontheK-TheoryofthereducedC*algebraofany discretegroup.Whileithasbeenrealizedinrecentyearsthatonecannotextendthis conjecturetocrossedproducts(“Baum-Conneswithcoefficients”),theoriginalcon- jectureisstillstanding,apowerfulpropulsionofresearchinIndexTheory,Discrete Groups,NoncommutativeTopology.ThelecturesofHigsonandGuentnerexpound thatsubject,withageneralintroductiontoK-TheoryofC*-algebras,E-theory,and Bott periodicity.Aspectsof the Baum - Connesconjecturerelatedto exactnessare dealtwithbyGuentnerandKaminker. K-Theory,KK-Theory and Connes - Higson E-Theory are unified in a general approachdueto Cuntz andCuntz - Quillen;a comprehensiveintroductionto these theoriesandtocycliccohomologycanbefoundinCuntz’slectures. Besidesthefundamentalreference[C]wepointouttothereaderotherreferences relatedtothissubject[GVF,L,M].SincethetheoryofOperatorAlgebrasissointi- matelyrelatedtothesubjectofthese LectureNotes,wefeelitappropriatetobring Preface IX to the reader’sattentionthe newly completedspectaculartreatise onvon Neumann Algebras(“noncommutativemeasuretheory”)byTakesaki[T]. Ofcoursethisvolumecouldnotbyitselfcoverthewholesubject,butwebelieve it is a catching invitation to Noncommutative Geometry, in all of its aspects from Prime Numbers to Quantum Gravity, that we hope many readers, mathematicians andphysicists,willfindstimulating. SergioDoplicherandRobertoLongo References [BC] J.B.Bost&A.Connes,HeckeAlgebras,typeIIIfactorsandphasetransitionswith spontaneoussymmetrybreakinginnumbertheory,SelectaMath.3,411-457(1995). [C] A.Connes,“NoncommutativeGeometry”,Acad.Press(1994). [C1] A.Connes,Gravitycoupledwithmatterandthefoundationofnoncommutative ge- ometry,Commun.Math.Phys.182,155-176(1996),andrefs. [Co] A.Connes,TraceformulainnoncommutativegeometryandthezerosoftheRiemann zetafunction,SelectaMath.5(1999),no.1,29-106. [CK] A.Connes,SymetriesGaloisiennesetrenormalisation, SeminairePoincare´ Octobre 2002,math.QA/0211199andrefs. [CM] A. Connes & H. Moscovici, Hopf algebras, cyclic cohomology and the transverse indextheorem,Commun.Math.Phys.198,199-246(1998),andrefs. [CR] A.Connes&M.Rieffel,YangMillsfornoncommutativetwotori,in:“OperatorAl- gebrasandMathematicalPhysics”,Contemp.Math.62,237-266(1987). [DFR] S.Doplicher,K.Fredenhagen&J.E.Roberts,Thequantumstructureofspacetimeat thePlanckscaleandquantumfields,Commun.Math.Phys.172,187-220(1995). [GVF] J.M.Gracia-Bondia,J.C.Varilly&H.Figueroa:“ElementsofNoncommutativeGe- ometry”,Birkhaeuser(2000). [JLO] A.Jaffe,A.Lesniewski&K.Osterwalder,QuantumK-theoryI.TheCherncharacter, Commun.Math.Phys.118,1-14(1988). [L] G. Landi, “An Introduction to Noncommutative Spaces and their Geometries”, Springer,LNPmonographs51(1997). [Lo] R. Longo, Notes for a quantum index theorem, Commun. Math. Phys. 222, 45-96 (2001). [M] J.Madore:“AnIntroductiontoNoncommutativeDifferentialGeometryanditsPhys- icalApplications”,LMSLectureNotes(1996). [T] M.Takesaki,“TheoryofOperatorAlgebras”vol.I,II,III,SpringerEncyclopaediaof MathematicalSciences124(2002),125,127(2003). CIME’sactivityissupportedby: Ministerodell’Universita`RicercaScientificaeTecnologica,COFIN’99; Ministero degliAffariEsteri - Direzione Generale per la Promozionee la Cooper- azione-UfficioV; ConsiglioNazionaledelleRicerche; E.U.undertheTrainingandMobilityofResearchersProgramme; UNESCO-ROSTE,VeniceOffice.
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