Two-Dimensional-Gauge-Theoriesand-Quantum-Integrable-Sys：两维规范理论量子可积系统课件

TwoDimensionalGaugeTheories

and

QuantumIntegrableSystems

NikitaNekrasovIHESImperialCollegeApril10,2008

TwoDimensionalGaugeTheorie1BasedonNN,S.Shatashvili,toappearPriorwork:E.Witten,1992;A.Gorsky,NN;J.Minahan,A.Polychronakos;M.Douglas;~1993-1994;A.Gerasimov~1993;G.Moore,NN,S.Shatashvili~1997-1998;A.Losev,NN,S.Shatashvili~1997-1998;A.Gerasimov,S.Shatashvili~2006-2007BasedonNN,S.Shatashvili,to2

Wearegoingtorelate

2,3,and4dimensional

susygaugetheories

withfoursupersymmetries

N=1d=4

AndquantumintegrablesystemssolublebyBetheAnsatztechniques.

Wearegoingtorelate

2,3,3

Mathematicallyspeaking,thecohomology,K-theoryandellipticcohomologyofvariousgaugetheorymodulispaces,likemoduliofflatconnectionsandinstantonsAndquantumintegrablesystemssolublebyBetheAnsatztechniques.

Mathematicallyspeaking,the4Forexample,weshallrelatetheXXXHeisenbergmagnetand2dN=2SYMtheorywithsomematterForexample,weshallrelatet5(pre-)HistoryIn1992E.WittenstudiedtwodimensionalYang-Millstheorywiththegoaltounderstandtherelationbetweenthephysicalandtopologicalgravitiesin2d.(pre-)History6(pre-)HistoryTherearetwointerestingkindsofTwodimensionalYang-Millstheories(pre-)HistoryTherearetwo7Yang-Millstheoriesin2d(1)

CohomologicalYM=twistedN=2super-Yang-Millstheory,withgaugegroupG,whoseBPS(orTFT)sectorisrelatedtotheintersectiontheoryonthemodulispaceMGofflatG-connectionsonaRiemannsurfaceYang-Millstheoriesin2d(1)8Yang-Millstheoriesin2dN=2super-Yang-MillstheoryFieldcontent:

Yang-Millstheoriesin2dN=2s9Yang-Millstheoriesin2d(2)PhysicalYM=N=0Yang-Millstheory,withgaugegroupG;ThemodulispaceMGofflatG-connections=minimaoftheaction;Thetheoryisexactlysoluble(A.Migdal)withthehelpofthePolyakovlatticeYMactionYang-Millstheoriesin2d(2)10Yang-Millstheoriesin2dPhysicalYMFieldcontent:Yang-Millstheoriesin2dPhysi11Yang-Millstheoriesin2dWittenfoundawaytomaptheBPSsectoroftheN=2theorytotheN=0theory.Theresultis:Yang-Millstheoriesin2dWitte12Yang-Millstheoriesin2dTwodimensionalYang-MillspartitionfunctionisgivenbytheexplicitsumYang-Millstheoriesin2dTwod13Yang-Millstheoriesin2dInthelimitthepartitionfunctioncomputesthevolumeofMG

Yang-Millstheoriesin2dInth14Yang-Millstheoriesin2dWitten’sapproach:addtwistedsuperpotentialanditsconjugateYang-Millstheoriesin2dWitte15Yang-Millstheoriesin2dTakealimitInthelimitthefieldsareinfinitelymassiveandcanbeintegratedout:oneisleftwiththefieldcontentofthephysicalYMtheory

Yang-Millstheoriesin2dTake16Yang-Millstheoriesin2dBothphysicalandcohomologicalYang-Millstheoriesdefinetopologicalfieldtheories(TFT)Yang-Millstheoriesin2dBoth17Yang-Millstheoriesin2dBothphysicalandcohomologicalYang-Millstheoriesdefinetopologicalfieldtheories(TFT)Vacuumstates+deformations=quantummechanicsYang-Millstheoriesin2dBoth18YMin2dandparticlesonacirclePhysicalYMisexplicitlyequivalenttoaquantummechanicalmodel:freefermionsonacircleCanbecheckedbyapartitionfunctiononatwo-torusGrossDouglasYMin2dandparticlesonaci19YMin2dandparticlesonacirclePhysicalYMisexplicitlyequivalenttoaquantummechanicalmodel:freefermionsonacircleStatesarelabelledbythepartitions,forG=U(N)YMin2dandparticlesonaci20YMin2dandparticlesonacircleForN=2YMthesefreefermionsonacircleLabelthevacuaofthetheorydeformedbytwistedsuperpotentialWYMin2dandparticlesonaci21YMin2dandparticlesonacircleThefermionscanbemadeinteractingbyaddingalocalizedmatter:forexampleatime-likeWilsonloopinsomerepresentationVofthegaugegroup:YMin2dandparticlesonaci22YMin2dandparticlesonacircleOnegetsCalogero-Sutherland(spin)particlesonacircle(1993-94)A.Gorsky,NN;J.Minahan,A.Polychronakos;YMin2dandparticlesonaci23HistoryIn1997G.Moore,NNandS.Shatashvilistudiedintegralsovervarioushyperkahlerquotients,withtheaimtounderstandinstantonintegralsinfourdimensionalgaugetheoriesHistoryIn1997G.Moore,NNand24HistoryIn1997G.Moore,NNandS.Shatashvilistudiedintegralsovervarioushyperkahlerquotients,withtheaimtounderstandinstantonintegralsinfourdimensionalgaugetheoriesThiseventuallyledtothederivationin2002oftheSeiberg-WittensolutionofN=2d=4theoryInspiredbytheworkofH.NakajimaHistoryIn1997G.Moore,NNand25Yang-Mills-HiggstheoryAmongvariousexamples,MNSstudiedHitchin’smodulispaceMHYang-Mills-HiggstheoryAmongv26Yang-Mills-HiggstheoryUnlikethecaseoftwo-dimensionalYang-MillstheorywherethemodulispaceMGiscompact,Hitchin’smodulispaceisnon-compact(itisroughlyT*MGmodulosubtleties)andthevolumeisinfinite.Yang-Mills-HiggstheoryUnlike27Yang-Mills-HiggstheoryInordertocurethisinfnityinareasonablewayMNSusedtheU(1)symmetryofMHThevolumebecomesaDH-typeexpression:WhereHistheHamiltonianYang-Mills-HiggstheoryInorde28Yang-Mills-HiggstheoryUsingthesupersymmetryandlocalizationtheregularizedvolumeof

MHwascomputedwiththeresultYang-Mills-HiggstheoryUsingt29Yang-Mills-HiggstheoryWheretheeigenvaluessolvetheequations:Yang-Mills-HiggstheoryWhere30YMHandNLSTheexpertswouldimmediatelyrecognisetheBetheansatz(BA)equationsforthenon-linearSchroedingertheory(NLS)NLS=largespinlimitoftheSU(2)XXXspinchainYMHandNLSTheexpertswouldi31YMHandNLSMoreovertheNLSHamiltoniansarethe0-observablesofthetheory,likeTheVEVoftheobservable=TheeigenvalueoftheHamiltonianYMHandNLSMoreovertheNLSHa32YMHandNLSSince1997nothingcameoutofthisresult.Itcouldhavebeensimplyacoincidence.…….YMHandNLSSince1997nothing33In2006

A.GerasimovandS.ShatashvilihaverevivedthesubjectHistoryIn2006

A.GerasimovandS.Sha34YMHandinteractingparticlesGSnoticedthatYMHtheoryviewedasTFTisequivalenttothequantumYangsystem:Nparticlesonacirclewithdelta-interaction:YMHandinteractingparticlesG35YMHandinteractingparticlesThus:YMwiththematter--fermionswithpair-wiseinteractionYMHandinteractingparticlesT36HistoryMoreimportantly,GSsuggestedthatTFT/QISequivalenceismuchmoreuniversalHistoryMoreimportantly,37TodayWeshallrederivetheresultofMNSfromamodernperspectiveGeneralizetocovervirtuallyallBAsolublesystemsbothwithfiniteandinfinitespinSuggestnaturalextensionsoftheBAequationsTodayWeshallrederivetheres38HitchinequationsSolutionscanbeviewedasthesusyfieldconfigurationsfortheN=2gaugedlinearsigmamodelForadjoint-valuedlinearfieldsHitchinequationsSolutionscan39HitchinequationsThemodulispaceMHofsolutionsisahyperkahlermanifoldTheintegralsoverMHarecomputedbythecorrelationfunctionsofanN=2d=2susygaugetheoryHitchinequationsThemodulisp40HitchinequationsThekahlerformonMHcomesfromtwistedtreelevelsuperpotentialTheepsilon-termcomesfromatwistedmassofthemattermultipletHitchinequations41GeneralizationTakeanN=2d=2gaugetheorywithmatter,InsomerepresentationR

ofthegaugegroupGGeneralizationTakeanN=2d=242GeneralizationIntegrateoutthematterfields,computetheeffective(twisted)super-potentialontheCoulombbranchGeneralizationIntegrateoutth43MathematicallyspeakingConsiderthemodulispaceMRofR-HiggspairswithgaugegroupGUptotheactionofthecomplexifiedgaugegroupGCMathematicallyspeakingConside44MathematicallyspeakingStabilityconditions:UptotheactionofthecompactgaugegroupGMathematicallyspeakingStabili45MathematicallyspeakingPushforwardtheunitclassdowntothemodulispaceMGofGC-bundlesEquivariantlywithrespecttotheactionoftheglobalsymmetrygroupKonMR

MathematicallyspeakingPushfor46MathematicallyspeakingThepushforwardcanbeexpressedintermsoftheDonaldson-likeclassesofthemodulispaceMG2-observablesand0-observablesMathematicallyspeakingThepus47MathematicallyspeakingThepushforwardcanbeexpressedintermsoftheDonaldson-likeclassesofthemodulispaceMG2-observablesand0-observablesMathematicallyspeakingThepus48MathematicallyspeakingThemassesaretheequivariantparametersFortheglobalsymmetrygroupK

MathematicallyspeakingThemas49VacuaofthegaugetheoryDuetoquantizationofthegaugefluxForG=U(N)VacuaofthegaugetheoryDuet50VacuaofthegaugetheoryEquationsfamiliarfromyesterday’slectureForG=U(N)partitionsVacuaofthegaugetheoryEquat51VacuaofthegaugetheoryFamiliarexample:CPNmodel(N+1)chiralmultipletofcharge+1Qii=1,…,N+1U(1)gaugegroupN+1vacuumFieldcontent:Effectivesuperpotential:VacuaofthegaugetheoryFamil52VacuaofgaugetheoryGaugegroup:G=U(N)Matterchiralmultiplets:1

adjoint, massfundamentals, massanti-fundamentals, massFieldcontent:Anotherexample:VacuaofgaugetheoryGaugegro53VacuaofgaugetheoryEffectivesuperpotential:VacuaofgaugetheoryEffective54VacuaofgaugetheoryEquationsforvacua:VacuaofgaugetheoryEquations55VacuaofgaugetheoryNon-anomalouscase:Redefine:VacuaofgaugetheoryNon-anoma56VacuaofgaugetheoryVacua:VacuaofgaugetheoryVacua:57Gaugetheory--spinchainIdenticaltotheBetheansatzequationsforspinXXXmagnet:Gaugetheory--spinchainIden58Gaugetheory--spinchainVacua=eigenstatesoftheHamiltonian:Gaugetheory--spinchainVacu59TableofdualitiesXXXspinchainSU(2)LspinsNexcitationsU(N)d=2N=2Chiralmultiplets:1adjointLfundamentalsLanti-fund.Specialmasses!TableofdualitiesXXXspincha60Tableofdualities:mathematicallyspeakingXXXspinchainSU(2)LspinsNexcitations(Equivariant)IntersectiontheoryonMR

for

Tableofdualities:mathematic61TableofdualitiesXXZspinchainSU(2)LspinsNexcitationsU(N)d=3N=1Compactifiedonacircle

Chiralmultiplets:1adjointLfundamentalsLanti-fund.TableofdualitiesXXZspincha62Tableofdualities:

mathematicallyspeakingXXZspinchainSU(2)LspinsNexcitationsEquivariantK-theoryofthemodulispace

MRTableofdualities:

mathemati63TableofdualitiesXYZspinchainSU(2),L=2NspinsNexcitationsU(N)d=4N=1Compactifiedona2-torus=ellipticcurveE

Chiralmultiplets:1adjointL=2NfundamentalsL=2Nanti-fund.Masses=wilsonloopsoftheflavourgroup=pointsontheJacobianofETableofdualitiesXYZspincha64Tableofdualities:

mathematicallyspeakingXYZspinchainSU(2),L=2NspinsNexcitationsEllipticgenusofthemodulispaceMRMasses=KbundleoverE=pointsontheBunKofETableofdualities:

mathemati65TableofdualitiesItisremarkablethatthespinchainhaspreciselythosegeneralizations:rational(XXX),trigonometric(XXZ)andelliptic(XYZ)thatcanbematchedtothe2,3,and4dimcases.

TableofdualitiesItisremark66AlgebraicBetheAnsatzThespinchainissolvedalgebraicallyusingcertainoperators,WhichobeyexchangecommutationrelationsFaddeevetal.Faddeev-Zamolodchikovalgebra…AlgebraicBetheAnsatzThespin67AlgebraicBetheAnsatzTheeigenvectors,Bethevectors,areobtainedbyapplyingtheseoperatorstothe«

fake

»vacuum.AlgebraicBetheAnsatzTheeige68ABAvsGAUGETHEORYForthespinchainitisnaturaltofixL=totalnumberofspinsandconsidervariousN=excitationlevelsInthegaugetheorycontextNisfixed.ABAvsGAUGETHEORYForthespi69ABAvsGAUGETHEORYHowever,ifthetheoryisembeddedintostringtheoryviabranerealizationthenchangingNiseasy:bringinanextrabrane.Hanany-Hori’02ABAvsGAUGETHEORYHowever,if70ABAvsGAUGETHEORYMathematicallyspeakingWeclaimthattheAlgebraicBetheAnsatzismostnaturallyrelatedtothederivedcategoryofthecategoryofcoherentsheavesonsomelocalCYABAvsGAUGETHEORYMathematica71ABAvsSTRINGTHEORYTHUS:BisforBRANE!isforlocation!ABAvsSTRINGTHEORYTHUS:i72MoregeneralspinchainsTheSU(2)spinchainhasgeneralizationstoothergroupsandrepresentations.IquotethecorrespondingBetheansatzequationsfromN.ReshetikhinMoregeneralspinchainsTheSU73Generalgroups/repsForsimply-lacedgroupHofrankrGeneralgroups/repsForsimply-74Generalgroups/repsForsimply-lacedgroupHofrankrLabelrepresentationsoftheYangianofHA.N.Kirillov-N.ReshetikhinmodulesCartanmatrixofHGeneralgroups/repsForsimply-75Generalgroups/reps

fromGAUGETHEORYTaketheDynkindiagramcorrespondingtoHAsimply-lacedgroupofrankrGeneralgroups/reps

fromGAUGE76

QUIVERGAUGETHEORYSymmetries

QUIVERGAUGETHEORYSymmetries77

QUIVERGAUGETHEORYSymmetries

QUIVERGAUGETHEORYSymmetries78

QUIVERGAUGETHEORY

ChargedmatterAdjointchiralmultipletFundamentalchiralmultipletAnti-fundamentalchiralmultipletBi-fundamentalchiralmultiplet

QUIVERGAUGETHEORY

Chargedm79QUIVERGAUGETHEORYMatterfields:adjointsQUIVERGAUGETHEORYMatterfiel80QUIVERGAUGETHEORYMatterfields:fundamentals+anti-fundamentalsQUIVERGAUGETHEORYMatterfiel81QUIVERGAUGETHEORYMatterfields:bi-fundamentalsQUIVERGAUGETHEORYMatterfiel82QUIVERGAUGETHEORYQuivergaugetheory:fullcontentQUIVERGAUGETHEORYQuivergaug83QUIVERGAUGETHEORY:MASSESAdjointsiQUIVERGAUGETHEORY:MASSESAdj84QUIVERGAUGETHEORY:MASSESFundamentalsAnti-fundamentalsia=1,….,Li

QUIVERGAUGETHEORY:MASSESFun85QUIVERGAUGETHEORY:MASSESBi-fundamentalsijQUIVERGAUGETHEORY:MASSESBi-86QUIVERGAUGETHEORYWhatissospecialaboutthesemasses?QUIVERGAUGETHEORYWhatisso87QUIVERGAUGETHEORYFromthegaugetheorypointofviewnothingspecial…..QUIVERGAUGETHEORYFromthega88QUIVERGAUGETHEORYThemasspuzzle!QUIVERGAUGETHEORYThemasspu89ThemasspuzzleTheBetheansatz--likeequationsCanbewrittenforanarbitrarymatrixThemasspuzzleTheBetheansat90ThemasspuzzleHowevertheYangiansymmetryY(H)wouldgetreplacedbysomeuglyinfinite-dimensional«

free

»algrebawithoutnicerepresentations

ThemasspuzzleHowevertheYan91ThemasspuzzleThereforeweconcludethatourchoiceofmassesisdictatedbythehiddensymmetry--thatofthedualspinchain

ThemasspuzzleThereforeweco92TheStandardModelhasmanyfreeparametersAmongthemarethefermionmassesIstherea(hidden)symmetryprinciplebehindthem?TheStandardModelhasmanyfr93TheStandardModelhasmanyfreeparametersInthesupersymmetricmodelsweconsideredthemasstuningcanbe«

explained

»usingadualitytosomequantumintegrablesystemTheStandardModelhasmanyfr94Furthergeneralizations:

Superpotential

fromprepotentialTreelevelpartInducedbytwistFluxsuperpotential(Losev,NN,Shatashvili’97)TheN=2*theoryonR2XS2Furthergeneralizations:

Super95Superpotential

fromprepotentialMagneticfluxElectricfluxInthelimitofvanishingS2themagneticfluxshouldvanishSuperpotential

fromprepotent96InstantoncorrectedBAequationsEffectiveS-matrixcontains2-body,3-body,…interactionsInstantoncorrectedBAequatio97InstantoncorrectedBAe