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Topological Quantum Field Theory and Four Manifolds - Jose Labastida.pdf

Topological Quantum Field Theor…

上传者: 星星4268 2013-04-05 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《Topological Quantum Field Theory and Four Manifolds - Jose Labastidapdf》,可适用于高等教育领域,主题内容包含TOPOLOGICALQUANTUMFIELDTHEORYANDFOURMANIFOLDSMATHEMATICALPHYSICSSTUDIESEdi符等。

TOPOLOGICALQUANTUMFIELDTHEORYANDFOURMANIFOLDSMATHEMATICALPHYSICSSTUDIESEditorialBoard:MaximKontsevich,IHES,BuressurYvette,FranceMassimoPorrati,NewYorkUniversity,NewYork,USADanielSternheimer,UniversitéBourgogne,Dijon,FranceVOLUMEVladimirMatveev,UniversitéBourgogne,Dijon,FranceTopologicalQuantumFieldTheoryandFourManifoldsbyJOSELABASTIDAandMARCOSMARINOACIPCataloguerecordforthisbookisavailablefromtheLibraryofCongressISBN(HB)ISBN(ebook)PublishedbySpringer,POBox,AADordrecht,TheNetherlandsSoldanddistributedinNorth,CentralandSouthAmericabySpringer,PhilipDrive,Norwell,MA,USAInallothercountries,soldanddistributedbySpringer,POBox,AHDordrecht,TheNetherlandsPrintedonacidfreepaperAllRightsReservedNopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwrittenpermissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaseroftheworkPrintedintheNetherlandsSpringerTableofContentsPrefaceviiTopologicalAspectsofFourManifoldsHomologyandcohomologyTheintersectionformSelfdualandantiselfdualformsCharacteristicclassesExamplesoffourmanifoldsComplexsurfacesSpinandSpincstructuresonfourmanifoldsTheTheoryofDonaldsonInvariantsYang–MillstheoryonafourmanifoldSU()andSO()bundlesASDconnectionsReducibleconnectionsAlocalmodelforthemodulispaceDonaldsoninvariantsMetricdependenceTheTheoryofSeiberg–WittenInvariantsTheSeiberg–WittenequationsTheSeiberg–WitteninvariantsMetricdependenceSupersymmetryinFourDimensionsThesupersymmetryalgebraN=superspaceandsuperfieldsN=supersymmetricYang–MillstheoriesN=supersymmetricYang–MillstheoriesN=supersymmetrichypermultipletsN=supersymmetricYang–MillstheorieswithmatterTopologicalQuantumFieldTheoriesinFourDimensionsBasicpropertiesoftopologicalquantumfieldtheoriesTwistofN=supersymmetryDonaldson–WittentheoryTwistedN=supersymmetrichypermultipletExtensionsofDonaldson–WittentheoryMonopoleequationsTheMathai–QuillenFormalismvEquivariantcohomologyThefinitedimensionalcaseAdetailedexampleMathai–Quillenformalism:InfinitedimensionalcaseTheMathai–QuillenformalismfortheorieswithgaugesymmetryDonaldson–WittentheoryintheMathai–QuillenformalismAbelianmonopolesintheMathai–QuillenformalismTheSeiberg–WittenSolutionofN=SUSYYang–MillsTheoryLowenergyeffectiveaction:semiclassicalaspectsSl(,Z)dualityoftheeffectiveactionEllipticcurvesTheexactsolutionofSeibergandWittenTheSeiberg–WittensolutionintermsofmodularformsTheuplaneIntegralThebasicprinciple(or,‘CoulombHiggs=Donaldson’)EffectivetopologicalquantumfieldtheoryontheuplaneZeromodesFinalformfortheuplaneintegralBehaviorundermonodromyanddualitySomeApplicationsoftheuplaneIntegralWallcrossingTheSeiberg–WittencontributionTheblowupformulaFurtherDevelopmentsinDonaldson–WittenTheoryMoreformulaeforDonaldsoninvariantsApplicationstothegeographyoffourmanifoldsExtensionstohigherrankgaugegroupsAppendixASpinorsinFourDimensionsAppendixBEllipticFunctionsandModularFormsBibliographyviPrefaceTheemergenceoftopologicalquantumfieldtheoryhasbeenoneofthemostimportantbreakthroughswhichhaveoccurredinthecontextofmathematicalphysicsinthelastcentury,acenturycharacterizedbyindependentdevelopmentsofthemainideasinbothdisciplines,physicsandmathematics,whichhasconcludedwithtwodecadesofstronginteractionbetweenthem,wherephysics,asinpreviouscenturies,hasactedasasourceofnewmathematicsTopologicalquantumfieldtheoriesconstitutethecoreofthesephenomena,althoughthemaindrivingforcebehindithasbeentheenormouseffortmadeintheoreticalparticlephysicstounderstandstringtheoryasatheoryabletounifythefourfundamentalinteractionsobservedinnatureThesetheoriessetupanewrealmwherebothdisciplinesprofitfromeachotherAlthoughthemoststrikingresultshaveappearedonthemathematicalside,theoreticalphysicshasclearlyalsobenefitted,sincethecorrespondingdevelopmentshavehelpedbettertounderstandaspectsofthefundamentalsoffieldandstringtheoryTopologyhasplayedanimportantroleinthestudyofquantummechanicssincethelatefifties,afterdiscoveringthatglobaleffectsareimportantinphysicalphenomenaManyaspectsoftopologyhavebecomeordinaryelementsinstudiesinquantummechanicsaswellasinquantumfieldtheoryandinstringtheoryTheoriginoftopologicalquantumfieldtheorycanbetracedbackto,althoughthetermitselfappearedforthefirsttimeinInEWittenstudiedsupersymmetricquantummechanicsandsupersymmetricsigmamodelsprovidingaframeworkthatledtoageneralizationofMorsetheoryThisframeworkwaslaterconsideredbyAFloerwhoconstructeditsmathematicalsettingandenlargedittoamoregeneralcontextThis,inturn,wasreconsideredbyEWittenwho,influencedbyMAtiyah,proposedthefirsttopologicalquantumfieldtheoryitselfinHisconstructionconsistedofaquantumfieldtheoryrepresentationofthetheoryofDonaldsoninvariantsonfourmanifoldsproposedinAfterthefirstformulationofatopologicalquantumfieldtheorybyEWittenmanyothershavebeenconsideredAnewareaofactiveresearchhasdevelopedsincethenInthisbookwewillconcentrateourattentiononviiaspectsrelatedtothatfirsttheory,nowadaysknownasDonaldson–Wittentheory,whichisthemostrelevanttheoryinfourdimensionsOtherimportanttopologicaltheories,suchasChern–Simonsgaugetheoryinthreedimensionsortopologicalstringtheory,fallbeyondthescopeofthisbookWewilldealwithmanyaspectsofDonaldson–Wittentheory,emphasizinghowitsformulationhasallowedDonaldsoninvariantstobeexpressedintermsofasetofnewsimplerinvariantsknownasSeiberg–WitteninvariantsTopologicalquantumfieldtheoryisresponsibleforthediscoveryofSeiberg–WitteninvariantsandtheirrelationtoDonaldsoninvariantsIngeneral,quantumfieldtheoriescanbestudiedbydifferentmethodsprovidingseveralpicturesofthesametheoryTherelationbetweenDonaldson–WittentheoryandDonaldsoninvariantswasfoundusingperturbativemethodsinthecontextofquantumfieldtheoryTheapplicationofnonpertubativemethodstothesametheorywaitedseveralyearsbutledtothediscoveryoftherelevanceofSeiberg–WitteninvariantsasbuildingblocksofDonaldsoninvariantsThisconnectionwaspossibleowingtotheprogressachievedinbyNSeibergandEWitteninunderstandingnonperturbativepropertiesofsupersymmetrictheoriesFromthemathematicalsidetheemergenceofSeiberg–WitteninvariantsconstitutesoneofthemostimportantresultsobtainedintheninetiesinthestudyoffourmanifoldsTheseinvariantsturnouttobemuchsimplerthanDonaldsoninvariantsandcontainalltheinformationprovidedbythelatterTounderstandtheconnectionbetweentheseinvariantsoneneedstoregardDonaldson–WittentheoryasatheorywhichoriginatesafterthetwistingofcertainsupersymmetricquantumfieldtheoriesOtherpicturesoftopologicalquantumfieldtheory,suchastheoneintheframeworkoftheMathai–Quillenformalism,whichisalsodescribedinthisbook,donotprovideusefulinformationinthisrespectHowever,itisimportanttobecomeacquaintedwiththispicturesinceitprovidesaninterestinggeometricalsettingThemaingoalofthisbookistoprovideaunifyingtreatmentofalltheaspectsofDonaldson–WittentheoryasastemtheoryforDonaldsonandSeiberg–WitteninvariantsAnimportantefforthasbeenmadesothatitcanbereadbytheoreticalphysicistsandmathematiciansThefocusofthebookisontheinterplayofmathematicalandphysicalaspectsofthetheory,andalthoughwehaveincludedexpositionsofbackgroundmaterialsuchasthemoremathematicalaspectsofDonaldsontheoryorthephysicsoftheviiiSeiberg–Wittensolutionwehavenotprovidedallthedetails,andwereferthereadertomorespecificreferencesforanexhaustivetreatmentofsomeofthesubjectsThebookstartswithachapterthatcollectsbasicmathematicalresultsaboutthetopologyoffourmanifoldswhichwillbeneededintherestofthechaptersChaptersandcontainreviewsofthetheoriesofDonaldsonandSeiberg–Witteninvariants,respectivelyChapterpresentssupersymmetryinfourdimensionsanddescribesthesupersymmetrictheorieswhichwillberelevantforDonaldson–WittentheoryChapterdealswiththetwistingofsupersymmetrictheoriesandconstructsallthetopologicalquantumfieldtheorieswhichwillbeofinterestinotherchaptersThereisshown,inparticular,insectionsand,theconnectionbetweenthesetheoriesandtheDonaldsonandSeiberg–WitteninvariantsintroducedinChaptersandInChapteradifferentframeworkfordealingwithtopologicalquantumfieldtheories,theMathai–Quillenformalism,isintroducedThisformalismprovidesaninterestinggeometricalframeworkforthesetheorieswhichisworthbeingbeconsideredHowever,itscontentisnotneededfortherestofthebookThechaptercouldbeomittedinafirstreadingChapterdealswithnonperturbativeaspectsofsupersymmetrictheoriesAdetailedanalysisoftheresultingsolution,theSeiberg–Wittensolution,ispresentedThestructureofthissolutionisusedinDonaldson–WittentheoryinChapterItallowsonetowritetheDonaldson–WitteninvariantsasanintegralonthesocalleduplaneintroducedbyMooreandWittenTheuplaneintegralisthemostsystematicphysicalframeworkinwhichtounderstandDonaldson–Wittentheory,anditalsoleadstotheconnectionbetweenDonaldsoninvariantsandSeiberg–WitteninvariantsChapterdealswithseveralapplicationsoftheuplaneintegral,andChaptersummarizesfurtherdevelopmentsofDonaldson–WittentheoryFinally,twoappendicescontainusefulformulaeaboutspinorsinfourdimensions,ellipticfunctionsandmodularformsAcknowledgementsWewouldliketothankourcollaboratorsandcolleaguesoveralltheyearswehavedevotedtothestudyoftopologicalquantumfieldtheoryWehavebenefittedfromtheirknowledgeandtheirinsighthascertainlyinfluencedourworkItisnotpossibletolistallofthemherebutwewishtothankspeciallyMAlvarez,LAlvarezGaume,JDEdelstein,CLozano,JMas,GMoore,MPernici,AVRamallo,andEWittenixSeveralcolleaguesagreedtoreadpartofthemanuscriptbeforepublication,providingimportantremarksWewouldliketogivespecialthanksinthisrespecttoCarlosLozano,GregoryMoore,andVicenteMunozxChapterTopologicalAspectsofFourManifoldsThepurposeofthischapteristocollectaseriesofbasicresultsaboutthetopologyoffourmanifoldsthatwillbeusedintherestofthebookNoattempttobeselfcontainedismadeandthereadershouldconsultsomeoftheexcellentbooksonthesubjectreviewedattheendofthechapterThediscussionwillberestrictedtofourmanifoldswhichareclosed,compactandorientable,whichisthecaseconsideredintherestofthebookWewillalsoassumethatallthefourmanifoldsunderconsiderationareendowedwithaRiemannianmetricHomologyandcohomologyThemostbasicclassicaltopologicalinvariantsofafourmanifoldarethehomologyandcohomologygroupsHiHH(X,Z),Hi(X,Z)Thesehomologygroupsareabeliangroups,andtherankofHiHH(X,Z)iscalledtheithBettinumberofX,denotedbybiRememberthat,foranndimensionalmanifold,byPoincareduality,onehasHi(X,Z)HnHHi(X,Z),()andalsobi=bniWewillalsoneedthe(co)homologygroupswithcoefficientsinothergroupssuchasZToobtainthesegroupsoneusestheuniversalcoefficienttheorem,whichstatesthatHiHH(X,G)HiHH(X,Z)ZGTor(HiHH(X,Z),G)()LetusfocusonthecaseG=ZpGivenanelementxinHiHH(X,Z),onecanalwaysfindanelementinHiHH(X,Zp)bysendingxxThisinfactgivesamap:HiHH(X,Z)HiHH(X,Zp)()TopologicalQuantumFieldTheoryandFourManifoldswhichiscalledthereductionmodpoftheclassxNoticethat,byconstruction,theimageof()isinHiHH(X,Z)ZpTherefore,ifthetorsionpartin()isnotzero,themap()isclearlynotsurjectiveWhenthetorsionpartiszero,anyelementinHiHH(X,Zp)comesfromthereductionmodpofanelementinHiHH(X,Z)ForthecohomologygroupswehaveasimilarresultPhysicistsaremorefamiliarwiththedeRhamcohomologygroups,HDR(X)whicharedefinedintermsofdifferentialformsThesegroupsaredefinedoverR,andthereforetheyareinsensitivetothetorsionpartofthesingularcohomologyFormally,onehasHiDR(X)(Hi(X,Z)Tor(Hi(X,Z)))ZRRememberalsothatthereisanondegeneratepairingincohomology,whichinthedeRhamcaseistheusualwedgeproductfollowedbyintegrationWewilldenotethepairingofthecohomologyclasses(ordifferentialformrepresentatives)α,βby(α,β)LetusnowfocusondimensionfourPoincaredualitythengivesanisomorphismbetweenHHH(X,Z)andH(X,Z)Italsofollowsthatb(X)=b(X)RecallthattheEulercharacteristicχ(X)ofanndimensionalmanifoldisdefinedasχ(X)=ni=()ibi(X)()ForaconnectedfourmanifoldX,wethenhave,usingPoincareduality,thatχ(X)=b(X)b(X)()TheintersectionformAnimportantobjectinthegeometryandtopologyoffourmanifoldsistheintersectionform,Q:H(X,Z)H(X,Z)Z,()whichisjustthepairingrestrictedtothetwoclassesByPoincaredualityitcanbedefinedonHHH(X,Z)HHH(X,Z)aswellNoticethatQiszeroifanyoftheargumentsisatorsionelement,thereforeonecandefineQonthetorsionfreepartsofhomologyandcohomologyAnotherusefulwayoflookingattheintersectionformispreciselyintermsoftheintersectionofsubmanifoldsinXOnefundamentalresultinTopologicalAspectsofFourManifoldsthisrespectisthatwecanrepresentanytwohomologyclassinafourmanifoldbyaclosedorientedsurfaceS:givenanembeddingi:SX()wehaveatwohomologyclassi(S)HHH(X,Z),whereSisthefundamentalclassofSConversely,anyaHHH(X,Z)canberepresentedinthisway,anda=SaOnecanalsoprovethatQ(a,b)=SaSb,()wheretherighthandsideisthenumberofpointsintheintersectionofthetwosurfaces,countedwithsignswhichdependontherelativeorientationofthesurfacesIf,moreover,ηSa,ηSbdenotethePoincaredualsofthesubmanifoldsSa,SbonehasQ(a,b)=XηSaηSb=Q(ηSa,ηSb)()Ifwechooseabasis{ai}i=,,b(X)forthetorsionfreepartofHHH(X,Z)wecanrepresentQbyamatrixwithintegerentriesthatwewillalsodenotebyQUnderachangeofbasis,weobtainanothermatrixQCTQC,whereCisthetransformationmatrixThematrixQisobviouslysymmetric,anditfollowsbyPoincaredualitythatitisunimodular,ie,ithasdet(Q)=IfweconsidertheintersectionformontherealvectorspaceHHH(X,R)weseethatitisasymmetric,bilinear,nondegenerateform,andthereforeitisclassifiedbyitsrankanditssignatureTherankofQ,rk(Q),isclearlygivenbyb(X),thesecondBettinumberThenumberofpositiveandnegativeeigenvaluesofQwillbedenotedbyb(X),b(X),respectively,andthesignatureofthemanifoldXisthendefinedasσ(X)=b(X)b(X)()WewillsaythattheintersectionformisevenifQ(a,a)modOtherwiseitisoddAnelementxofHHH(X,Z)Tor(HHH(X,Z))iscalledcharacteristicifQ(x,a)Q(a,a)mod()foranyaHHH(X,Z)Tor(HHH(X,Z))AnimportantpropertyofcharacteristicelementsisthatQ(x,x)σ(X)mod()TopologicalQuantumFieldTheoryandFourManifoldsInparticular,ifQiseventhenthesignatureofthemanifoldisdivisiblebyExamples:()Thesimplestintersectionformis:n()m()=diag(,,,,,),()whichisoddandhasb=n,b=m()Anotherimportantformisthehyperboliclattice,H=(),()whichisevenandhasb=b=()Finally,onehastheevenpositivedefiniteformofrankE=,()whichistheDynkindiagramoftheexceptionalLiealgebraEFortunately,unimodularlatticeshavebeenclassifiedTheresultdependsonwhethertheintersectionformisevenoroddandwhetheritisdefinite(positiveornegative)ornotOdd,indefinitelatticesareequivalenttop()q(),whilstevenindefinitelatticesareequivalenttopHqEDefinitelatticesaremorecomplicated,sincetheyinvolve‘exotic’casesTheintersectionform

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