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

Topological Quantum Field Theor…

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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

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,forexclusiveusebythepurchaseroftheworkPrintedintheNetherlands©SpringerTableofContentsPrefaceviiTopologicalAspectsofFourManifoldsHomologyandcohomologyTheintersectionformSelfdualandantiselfdualformsCharacteristicclassesExamplesoffourmanifoldsComplexsurfacesSpinandSpincstructuresonfourmanifoldsTheTheoryofDonaldsonInvariantsYang–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,LA´lvarezGaume´,JDEdelstein,CLozano,JMas,GMoore,´MPernici,AVRamallo,andEWittenixSeveralcolleaguesagreedtoreadpartofthemanuscriptbeforepublication,providingimportantremarksWewouldliketogivespecialthanksinthisrespecttoCarlosLozano,GregoryMoore,andVicenteMun˜ozxChapterTopologicalAspectsofFourManifoldsThepurposeofthischapteristocollectaseriesofbasicresultsaboutthetopologyoffourmanifoldsthatwillbeusedintherestofthebookNoattempttobeselfcontainedismadeandthereadershouldconsultsomeoftheexcellentbooksonthesubjectreviewedattheendofthechapterThediscussionwillberestrictedtofourmanifoldswhichareclosed,compactandorientable,whichisthecaseconsideredintherestofthebookWewillalsoassumethatallthefourmanifoldsunderconsiderationareendowedwithaRiemannianmetricHomologyandcohomologyThemostbasicclassicaltopologicalinvariantsofafourmanifoldarethehomologyandcohomologygroupsHiHH(X,Z),Hi(X,Z)Thesehomologygroupsareabeliangroups,andtherankofHiHH(X,Z)iscalledtheithBettinumberofX,denotedbybiRememberthat,foranndimensionalmanifold,byPoincare´duality,onehas´Hi(X,Z)�HnHH−i(X,Z),()andalsobi=bn−iWewillalsoneedthe(co)homologygroupswithcoefficientsinothergroupssuchasZToobtainthesegroupsoneusestheuniversalcoefficienttheorem,whichstatesthatHiHH(X,G)�HiHH(X,Z)⊗ZG⊕Tor(HiHH−(X,Z),G)()LetusfocusonthecaseG=ZpGivenanelementxinHiHH(X,Z),onecanalwaysfindanelementinHiHH(X,Zp)bysendingx→x⊗Thisinfactgivesamap:HiHH(X,Z)→HiHH(X,Zp)()TopologicalQuantumFieldTheoryandFourManifoldswhichiscalledthereductionmodpoftheclassxNoticethat,byconstruction,theimageof()isinHiHH(X,Z)⊗ZpTherefore,ifthetorsionpartin()isnotzero,themap()isclearlynotsurjectiveWhenthetorsionpartiszero,anyelementinHiHH(X,Zp)comesfromthereductionmodpofanelementinHiHH(X,Z)ForthecohomologygroupswehaveasimilarresultPhysicistsaremorefamiliarwiththedeRhamcohomologygroups,H∗DR(X)whicharedefinedintermsofdifferentialformsThesegroupsaredefinedoverR,andthereforetheyareinsensitivetothetorsionpartofthesingularcohomologyFormally,onehasHiDR(X)�(Hi(X,Z)Tor(Hi(X,Z)))⊗ZRRememberalsothatthereisanondegeneratepairingincohomology,whichinthedeRhamcaseistheusualwedgeproductfollowedbyintegrationWewilldenotethepairingofthecohomologyclasses(ordifferentialformrepresentatives)α,βby(α,β)LetusnowfocusondimensionfourPoincare´dualitythengivesan´isomorphismbetweenHHH(X,Z)andH(X,Z)Italsofollowsthatb(X)=b(X)RecallthattheEulercharacteristicχ(X)ofanndimensionalmanifoldisdefinedasχ(X)=n∑i=(−)ibi(X)()ForaconnectedfourmanifoldX,wethenhave,usingPoincare´duality,that´χ(X)=−b(X)b(X)()TheintersectionformAnimportantobjectinthegeometryandtopologyoffourmanifoldsistheintersectionform,Q:H(X,Z)×H(X,Z)→Z,()whichisjustthepairingrestrictedtothetwoclassesByPoincare´duality´itcanbedefinedonHHH(X,Z)×HHH(X,Z)aswellNoticethatQiszeroifanyoftheargumentsisatorsionelement,thereforeonecandefineQonthetorsionfreepartsofhomologyandcohomologyAnotherusefulwayoflookingattheintersectionformispreciselyintermsoftheintersectionofsubmanifoldsinXOnefundamentalresultinTopologicalAspectsofFourManifoldsthisrespectisthatwecanrepresentanytwohomologyclassinafourmanifoldbyaclosedorientedsurfaceS:givenanembeddingi:S↪→↪↪X()wehaveatwohomologyclassi∗(S)∈HHH(X,Z),whereSisthefundamentalclassofSConversely,anya∈HHH(X,Z)canberepresentedinthisway,anda=SaOnecanalsoprovethatQ(a,b)=Sa∪Sb,()wheretherighthandsideisthenumberofpointsintheintersectionofthetwosurfaces,countedwithsignswhichdependontherelativeorientationofthesurfacesIf,moreover,ηSa,ηSbdenotethePoincare´dualsofthesubmanifolds´Sa,SbonehasQ(a,b)=∫X∫∫ηSa∧ηSb=Q(ηSa,ηSb)()Ifwechooseabasis{ai}i=,,b(X)forthetorsionfreepartofHHH(X,Z)wecanrepresentQbyamatrixwithintegerentriesthatwewillalsodenotebyQUnderachangeofbasis,weobtainanothermatrixQ→CTQC,whereCisthetransformationmatrixThematrixQisobviouslysymmetric,anditfollowsbyPoincare´dualitythatitisunimodular,´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()foranya∈HHH(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(−),whilstevenindefinitelatticesareequivalenttopH⊕qEDefinitelatticesaremorecomplicated,sincetheyinvolve‘exotic’casesTheintersectionform

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