THEBERRYCONNECTIONANDOTHERASPECTSOFTHEGINZBURG-LANDAUTHEORYINDIMENSION2ByÁkosNagyADISSERTATIONSubmittedtoMichiganStateUniversityinpartialful˝llmentoftherequirementsforthedegreeofMathematicsDoctorofPhilosophy2016ABSTRACTTHEBERRYCONNECTIONANDOTHERASPECTSOFTHEGINZBURG-LANDAUTHEORYINDIMENSION2ByÁkosNagyInthe˝rstchapter,weanalyzethe2-dimensionalGinzburg-Landauvorticesatcriticalcou-pling,andestablishasymptoticformulasforthetangentvectorsofthevortexmodulispaceusingtheoremsofTaubesandBradlow.WethencomputethecorrespondingBerrycurvatureandholonomyinthelargearealimit.Inthesecondchapter,wegeneralizeBradlow'stheoremaboutexistenceofirreducibleabsoluteminimizersoftheGinzburg-Landaufunctional.Tomyparents,EllaandLajos.iiiACKNOWLEDGMENTSFirstandforemost,IwishtothankmyadvisorTomParkerforguidingmethroughgraduateschool.IwasextremelyluckytostudygeometryfromtheexcellentteachersofMSU,likeMattHedden,RonFintushel,andJonWolfson.IlearntagreatdealofmathfrommyManosMaridakis.Finally,Ialsobene˝tedalotfromdiscussionswithDavidDuncan,TimNguyen,andLukeWilliams.Iwouldalsoliketothankmyfriendsallaroundtheworld,butmostimportantlytheonesin,andaroundEastLansing.Youguyshavebeenmyfamilyonthiscontinentforthepast5years.FinallyIthankmyparents,EllaandLajos,fortheirconstant,unquestionable,andsel˛esssupport,notjustthroughoutgraduateschool,butmywholelife.ivTABLEOFCONTENTSLISTOFFIGURES...................................viChapter1Ginzburg-Landauvortices.......................11.1Introduction....................................11.2Ginzburg-Landautheoryonclosedsurfaces...................41.2.1The˝-vortexequations..........................41.2.2The˝-vortexmodulispace........................61.2.3Thehorizontalsubspaces.........................71.3Theasymptoticformofhorizontalvectors...................121.4TheBerryconnection...............................201.5TheasymptoticBerrycurvature.........................231.6TheasymptoticBerryholonomy.........................261.7Thelargearealimit................................39Chapter2IrreducibleGinzburg-Landau˝elds.................422.1Non-existence...................................432.2Existence.....................................46REFERENCES......................................51vLISTOFFIGURESFigure1.1:Singlevortexloops:Onedivisorpointmovesalongoneofthei's.Allotherdivisorpointsare˝xed....................28Figure1.2:Vortexinterchange:Onedivisorpointmovesalong1andanotherdivisorpointmovesalong2.Allotherdivisorpointsare˝xed...28Figure2.1:PhaseDiagram:InRegionI,de˝nedby64ˇjdj,onlyreduciblesolutionsexist(InsulatorPhase).InthecomplementaryRegionII(shaded),thereexistirreduciblesolutions(SuperconductorPhase).42viChapter1Ginzburg-Landauvortices1.1IntroductionTheGinzburg-Landautheoryisaphenomenologicalmodelforsuperconductivity,introducedin[GL50];foramoremodernreviewsee[AK02].ThetheorygivesvariationalequationstheGinzburg-LandauequationsforanAbeliangauge˝eldandacomplexscalar˝eld.Thegauge˝eldistheEMvectorpotential,whilethenormofthescalar˝eldistheorderparameterofthesuperconductingphase.Theorderparametercanbeinterpretedasthewavefunctionoftheso-calledBCSgroundstate,asinglequantumstateoccupiedbyalargenumberofCooperpairs.Thispaperfocusesoncertainstaticsolutionsofthe2-dimensionalGinzburg-Landauequationscalled˝-vortices.Physicistsregardthenumber˝asacouplingconstant,sometimescalledthevortex-size.Mathematically,˝isascalingparameterforthemetric.Thegeometryof˝-vorticeshavebeenstudiedsince[JT80,B90],andthereisalargeliteratureonthesubject;cf.[MN99,MS03,CM05,B06,B11,DDM13,BR14,MM15].Familiesofoperatorsinquantumphysicscarrycanonicalconnections.ThisideawasintroducedbyBerryin[B84],andgeneralizedbyAharonovandAnandanin[AA87].Theseso-calledBerryconnectionswereused,forexample,tounderstandtheQuantumHallE˙ect[K85].1IngaugetheoriesincludingtheGinzburg-LandautheorytheBerryconnectioncanbeunderstoodgeometricallyasfollows:thespacePofsolutionsofgaugeinvariantequationsisanin˝nitedimensionalprincipalbundleoverthepartofmodulispaceMwheretheactionofthegaugegroupGisfree.Thusifallsolutionsareirreducible,thenPisanin˝nitedimensionalprincipalG-bundlePGMoverM.ThecanonicalL2-metricofPde˝nesahorizontaldistributiontheorthogonalcomplementofthegaugedirectionswhichde˝nestheBerryconnection.AconnectiononaprincipalG-bundleP!Xde˝nesparalleltransport:foreachsmoothmap:[0;1]!X,paralleltransportaroundisaG-equivariantisomorphismfromthe˝berattothe˝berat.Ifisaclosedloop,then=,andthecorrespondingparalleltransportiscalledholonomy.IfGisAbelian,holonomyisgivenbytheactionofanelementinG.HolonomiesoftheBerryconnectionaregaugetransformations,whichhaveaphysicalinterpretation:theydescribetheadiabaticevolutionofthestateofthesystem,thatisitsbehaviorunderslowchangesinthephysicalparameterssuchasexternal˝elds,orcouplingconstants.ThisdissertationinvestigatestheBerryconnectionofthe˝-vortexmodulispaceassoci-atedtoadegreedhermitianlinebundleoveraclosed,orientedRiemanniansurface.TheBerryholonomyassignsagaugetransformationg˝toeachclosedcurveinM˝.ThesegaugetransformationsareU(1)-valuedsmoothfunctionson.Whendispositiveand˝isgreaterthanthegeometry-dependentconstant˝0=4ˇd,thenthemodulispace,M˝,2isidenti˝edwiththed-foldsymmetricpowerofthesurfaceSymd.AclosedcurveinSymdde˝nesa1-cyclein,called,theshadowof,whichisde˝nedbychoosingaliftoftod!Symd,andtakingtheunionofthenon-constantcurvesappearinginthelift(seeequation(1.50)forprecisede˝nition).Themaintheoremofthispapergivesacompletetopologicalandanalyticaldescriptionofthesegaugetransformationsintermsoftheshadow:MainTheoremofChapter1.[TheBerryholonomyofthe˝-vortexprincipalbundle]Letg˝2GbetheBerryholonomyofasmoothcurveinthe˝-vortexmodulispaceM˝,andbetheclosed1-cycleinde˝nedinequation(1.50).Thenthefollowingpropertiesholdas˝!1:1.[Convergence]g˝!1intheC1-topologyoncompactsetsof.2.[Crossing]Letj:[0;1]!beasmoothpaththatintersectstransversallyandpositivelyonce,andwriteg˝j=exp(2ˇi'˝).Then'˝(1)'˝(0)!1.3.[Concentration]Asa1-current,12ˇig1˝dg˝convergestothe1-currentde˝nedby.Themap7!g˝inducesapairinghol?:H1Z)!H1Z);de˝nedin(1.53).(4)[Duality]Forall˝>˝0,thehomomorphismhol?isPoincaréduality.Whenisapositivelyoriented,boundingsinglevortexloop,orapositivelyorientedvortexinterchange(seeSection1.6forprecisede˝nitions)ourmaintheoremimpliesthat3thecorrespondingholonomycanbewrittenasg˝=exp(2ˇif˝),forarealfunctionf˝on.Moreover,f˝canbechosensothatitconvergesto1ontheinsideofthecurve,andto0ontheoutside.Thismakesphysicists'intuitionabouttheholonomyprecise;cf.[I01].Asmentionedabove,˝isascalingparameterforthemetric:onecanlookattheGinzburg-Landautheorywith˝=1˝xed,buttheKählerform!scaledas!t=t2!.Ourresults,includingtheMainTheoremabove,canbereinterpretedasstatementsaboutthelargearealimit(i.e.t!1),whichcanbemoredirectlyrelatedtophysics(seeSection1.7fordetails).Thischapterisorganizedasfollows.InSection1.2,wegiveabriefintroductiontothegeometryofthe˝-vortexequationsonaclosedsurface,derivethetangentspaceequationsofthe˝-vortexmodulispace,andthenrecasttheminacompactform.InSection1.3,weusetheoremsofTaubesandBradlowtoproveatechnicalresult,Theorem1.8,thatestablishesasymptoticformulasforthetangentvectorsoftheofthe˝-vortexmodulispace.InSection1.4,weintroducetheBerryconnectionassociatedtothisproblem.WethenproveasymptoticformulasfortheBerrycurvatureinSection1.5.InSection1.6,weproveourMainTheorem;theproofsareapplicationsofTheorem1.8.Section1.7discussesthelargearealimit.1.2Ginzburg-Landautheoryonclosedsurfaces1.2.1The˝-vortexequationsLetbeaclosedsurfacewithKählerform!,compatiblecomplexstructureJ,andRie-mannianmetric!(;J()).LetL!beasmoothcomplexlinebundleofpositivedegreedwithhermitianmetrich.Foreachunitaryconnectionr,andsmoothsection˚,consider4theGinzburg-Landaufreeenergy:E(r;˚)=ZjFrj2+jr˚j2+2!;(1.1)where˝2R+arecouplingconstants,Fristhecurvatureofr,andw=12˝j˚j2:(1.2)TheEuler-Lagrangeequationsoftheenergy(1.1)aretheGinzburg-Landauequations:dFr+iIm(h(˚;r˚))=0(1.3a)rr˚˚=0:(1.3b)When=1,theenergy(1.1)canbeintegratedbypartsandrewrittenasdi˙erentsumofnon-negativeterms,andgetthelowerbound2ˇ˝d.Theminimizerssatisfythe˝-vortexequations:iFrw=0(1.4a)@r˚=0;(1.4b)whereFristheinnerproductoftheKählerform!andthecurvatureofr,and@r=r0;1istheCauchy-Riemannoperatorcorrespondingtor.Solutions(r;˚)tothe˝rstorderequations(1.4a)and(1.4b)automaticallysatisfythesecondorderequations(1.3a)and(1.3b).51.2.2The˝-vortexmodulispaceAsisstandardingaugetheory,weworkwiththeSobolevWk;p-completionsofthespaceofconnectionsand˝elds.LetCLbetheW1;2-closureofthea˚nespaceofsmoothuni-taryconnectionsonLand0LbetheW1;2-closureofthevectorspaceofsmoothsectionsofL.Similarly,letk,andkLbetheW1;2-closureofk-forms,andL-valuedk-formsrespectively.ThecorrespondinggaugegroupGistheW2;2-closureofAut(L)intheW2;2-topology.Thegaugegroupiscanonicallyisomorphictothein˝nitedimensionalAbelianLiegroupW2;2;U(1)),whoseLiealgebraisW2;2iR).Elementsg2Aut(L)actonCL0Lasg(r;˚)=grg1;g˚,andthisde˝nesasmoothactionofGonCL0L.Finally,theenergy(1.1)extendstoasmoothfunctiononCL0L.ThespaceP˝ofallcriticalpointsoftheextendedenergyisanin˝nitedimensionalsubmanifoldofCL0L.Duetothegaugeinvarianceofenergy(1.1),GactsonP˝,andeverycriticalpointisgaugeequivalenttoasmoothonebyellipticregularity.The˝-vortexmodulispaceisthequotientspaceM˝=P˝=G.ElementsofP˝arecalled˝-vortex˝elds,whileelementsofM˝(gaugeequivalenceclassesof˝-vortex˝elds)arecalled˝-vortices.Forbrevity,wesometimeswrite˝-vortex˝eldsas˛=(r;˚)2P˝andthecorresponding˝-vorticesas[˛]=[r;˚]2M˝.Thereisageometry-dependentconstant˝0=4ˇd,calledtheBradlowlimit,withthepropertythatif˝<˝0,thenthemodulispaceisemptyandif˝>˝0,thenthereisacanonicalbijectionbetweenM˝andthespaceofe˙ective,degreeddivisors;cf.[B90,Theorem4.6].Thisspaceisalsocanonicallydi˙eomorphictothed-foldsymmetricproductofthesurfaceSymd,whichisthequotientofthed-foldproductd=:::bytheactionofthepermutationgroupSd.Althoughthisactionisnotfree,thequotientisasmoothKählermanifoldofrealdimension2d.Foreachvalueof˝>˝0,thereisa6canonicalL2-Kählerstructure(seeSection1.2.3).Intheborderline˝=˝0case,the˚-˝eldvanisheseverywhereandthemodulispaceisinone-to-onecorrespondencewiththemodulispaceholomorphiclinebundlesofdegreed[B90,Theorem4.7].Accordingly,wefocusonthe˝>˝0caseinthispaper.When˝>˝0,Bradlow'smapfromM˝toSymdiseasytounderstand:Byintegratingequation(1.4a),oneseesthattheL2-normof˚ispositive.Ontheotherhand,˚isaholomorphicsectionofLbyequation(1.4b).Since˚isanon-vanishingholomorphicsectionitde˝nesane˙ective,degreeddivisor,givingusthedesiredmap.Theinverseofthismapismuchhardertounderstandaninvolvesnon-linearelliptictheory.Muchofthispicturecarriesovertoopensurfaces,evenwithin˝nitearea(forexample=C),ifoneimposesproperintegrabilityconditions;cf[T84].Forsimplicitywewillalwaysassumethatiscompact.Furthermorethemodulispaceisemptyford<0,andasinglepointford=0.Thuswewillalwaysassumethatd>0inthispaper.1.2.3ThehorizontalsubspacesThetangentspaceatanypointofthea˚nespaceCL0Listheunderlyingvectorspacei10L.ThetangentspaceofP˝isdescribedinthenextlemma.Lemma1.1.ThetangentspaceofP˝atthe˝-vortex˝eld˛=(r;˚)isthevectorspaceofpairs(a; )2i10Lthatsatisfyida+Re(h( ;˚))=0(1.5a)@r +a0;1˚=0:(1.5b)7Proof.Thelinearizationofequations(1.4a)and(1.4b)inthedirectionof(a; )is:limt!01tiFr+ta12˝j˚+t j2=ida+Re(h( ;˚))limt!01t@r+ta(˚+t )=@r +a0;1˚;whereweusedthat˛isa˝-vortex˝eld.Thiscompletestheproof,sincethetangentspaceisthekernelofthelinearizationofequations(1.4a)and(1.4b).Thea˚nespaceCL0LhasacanonicalL2-metricgivenby(a; )a0; 0=Za^a0+Reh ; 0!(1.6)whereisthe(conjugate-linear)HodgeoperatoroftheRiemannianmetricof.OnecancheckthattherestrictionoftheL2-metric(1.6)tothesolutionsofequations(1.5a)and(1.5b)makesP˝asmooth,weakRiemannianmanifold,andthatgaugetransformationsactisometricallyonP˝.ThepushforwardofthetangentspaceT1Gbythegaugeactioniscalledtheverticalsub-spaceofT˛P˝.Wede˝nethehorizontalsubspaceofT˛P˝tobetheorthogonalcomplementoftheverticalsubspacebytheL2-metric(1.6).SinceM˝=P˝=G,thehorizontalsubspaceiscanonicallyisomorphictothetangentspaceT[˛]M˝ofthemodulispace.Thenextlemmashowsthatthehorizontalsubspaceisalsothekernelofa˝rstorderlinearellipticoperator.Lemma1.2.ThehorizontalsubspaceofT˛P˝,atthe˝-vortex˝eld˛=(r;˚)2P˝,isthe8vectorspaceofpairs(a; )2i10Lthatsatisfy(id+d)a+h( ;˚)=0(1.7a)@r +a0;1˚=0:(1.7b)Proof.Therealpartofequation(1.7a)isequation(1.5a)andequation(1.7b)isequa-tion(1.5b);thussolutionsofequations(1.7a)and(1.7b)areinT˛P˝.To˝nishtheproof,wemustcheckthatapair(a; )inT˛P˝isorthogonaltotheverticalsubspaceat˛ex-actlyifequations(1.7a)and(1.7b)hold.Thepushforwardofif2Lie(G)at˛isgivenbyXf(˛)=(idf;if˚),hencehorizontalvectorsarepairs(a; ),thatsatisfythefollowingequationforeveryf2C1R):0=h(a; )j(idf;if˚)i=Z(a^(idf)+Re(h( ;if˚))!):Integratingtheright-handsidebypartsyields0=Z(da+iIm(h( ;˚)))if!:Becausethisholdsforallf,weconcludethat(a; )isorthogonaltotheverticalsubspaceat˛exactlyifda+iIm(h( ;˚))=0holds.Addingthis(purelyimaginary)equationtothe(purelyreal)equation(1.5a)givesequation(1.7a).Equations(1.7a)and(1.7b)dependonthechoiceof˛,butif(a; )isasolutionofequations(1.5b)and(1.7a)for˛andg2G,then(a;g )isasolutionofequations(1.5b)and(1.7a)forg(˛)=r+gdg1;g˚.Sincegaugetransformationsactisometrically,the9L2-metriconthehorizontalsubspacesofTP˝descendstoaRiemannianmetriconM˝.LetKbetheanti-canonicalbundleof,and0;1=0K,theW1;2-completionofthespaceofsmoothsectionsofK.Werecastequations(1.7a)and(1.7b)inamoregeometricwayinthenextlemma.Lemma1.3.Equations(1.7a)and(1.7b)areequivalenttothefollowingpairofequationson(; )20;10L:p2@h(˚; )=0(1.8a)p2@r +˚=0:(1.8b)Moreover,theunitarybundleisomorphism(a; )7!1p2(a+ia); (1.9)interchangessolutionsofequations(1.7a)and(1.7b)withsolutionsofequations(1.8a)and(1.8b).Proof.Acomplex1-formisin0;1exactlyif=i.Fora2i1,de˝netheunitarymapubyu(a)=1p2(a+ia):(1.10)Using2a=a=a,weseethatu(a)=iu(a),andthusu(a)20;1.Set=u(a).Withthisnotationa0;1=p2,whichprovestheequivalenceofequations(1.7b)and(1.8b).TheKähleridentitiesyield(id+d)a=p2@,whichisequivalenttoequation(1.8a).Thevectorspaceofsolutionstoequations(1.8a)and(1.8b)hasacanonicalalmost10complexstructurecomingfromthecomplexstructuresofKandL,andthisde˝nesanalmostcomplexstructureforM˝.MundetiRiera[R00]showedthatthisstructureisintegrable,andtogetherwiththeL2-metricitmakesM˝aKählermanifold.Toputequations(1.8a)and(1.8b)inamorecompactform,notethattheyareequivalenttothesingleequationL˛(a; )=0;whereL˛=Dr+A˚isde˝nedasDr:0;10L!00;1L;(; )7!p2@;p2@r A˚:0;10L!00;1L;(; )7!(h(˚; );˚);TheoperatorDrisa˝rstorderellipticdi˙erentialoperator,andtheoperatorA˚isabundlemap.StraightforwardcomputationshowsthatforallZ20;10LA˚A˚(Z)=j˚j2Z:(1.11)ThusA˚isnon-degenerateonthecomplementofthedivisorof˚.NotethatDr,A˚,andhenceL(r;˚)makesenseforanypair(r;˚)2CL0L.Lemma1.4.Let˛=(r;˚)2CL0Lbeapairsuchthat@r˚=0.ThentheoperatorDrA˚+A˚Drisidenticallyzero.11Proof.TheadjointoperatorsareDr(f;˘)=p2@f;p2@r˘(1.12a)A˚(f;˘)=(h(˚;˘);f˚)(1.12b)forany(f;˘)200;1L.Thelemmafollowsfromequations(1.12a)and(1.12b),theholomorphicityof˚,andthede˝nitionsofDrandA˚.Corollary1.5.Let˛=(r;˚)2CL0Lbeapairsuchthat@r˚=0.Thenker(L˛)istrivial.Proof.FromLemma1.4andequation(1.11),weobtainL˛L˛=DrDr+A˚A˚.HenceifZisinthekernelofL˛,then0=kL˛(Z)k2L2=kDr(Z)k2L2+kA˚(Z)k2L2;whichimpliesthatbothtermsontherightvanish.Byequation(1.11),Zvanisheswhere˚doesnot,whichisthecomplementofa˝niteset.ButthenZvanisheseverywherebycontinuity.HencethekernelofL˛istrivial.1.3TheasymptoticformofhorizontalvectorsInthissectionwewilluseofthefollowingresultsofTaubesandBradlowaboutthelarge˝behaviorof˝-vortex˝elds.Recallfromequation(1.2)thatw=12˝j˚j2:12Theorem1.6.[BradlowandTaubes]Thereisapositivenumbercsuchthateach˝-vortex˝eld˛=(r;˚)2P˝satis˝esj˚j26˝(1.13a)w+jr˚j6c˝expp˝distDc;(1.13b)wheredistDisthedistancefromthedivisorD=˚1(0),andwisde˝nedinequation(1.2).Proof.In[B90,Proposition5.2]Bradlowshowedinequality(1.13a),usingthefactthat˝-vortex˝eldssatisfytheellipticequation+j˚j2w=j@r˚j2:(1.14)Theright-handsideispositiveawayfroma˝niteset,sothemaximumprincipleandequa-tion(1.2)impliesinequality(1.13a).Inequality(1.13b)wasprovedin[T99,Lemma3.3].Wecalladivisorsimpleifthemultiplicityofeverydivisorpointis1.Lemma1.7.FixasimpledivisorD2Symdandacorresponding˝-vortex˝eld˛=(r;˚)with˝>˝0.ThesmoothfunctionhD;˝=12ˇ˝j@r˚j2+2w2(1.15)dependsonlyonDand˝,butnotonthechoiceof˛.Moreover,lim˝!1hD;˝=Xp2Dp;(1.16)13inthesenseofmeasures,wherepistheDiracmeasureconcentratedatthepointp2.Proof.Everyterminequation(1.15)isgaugeinvariant,whichprovestheindependenceofthechoiceof˛forD.Usingequation(1.14)wegethD;˝=12ˇ˝w+˝w),henceforanysmoothfunctionf:ZhD;˝f!=12ˇ˝Zw+˝w)f!=12ˇ˝Zwf)!+12ˇZwf!:By[HJS96,Theorem1.1],wconvergesto2ˇDinthesenseofmeasuresas˝!1.Thusthe˝rsttermconvergesto0,andthesecondtermconvergestoPp2Df(p),whichcompletestheproof.Thespaceofsimpledivisors,Symds,isanopendensesetinSymd,anditscomple-mentiscalledthebigdiagonal.WhenDissimple,atangentvectorinTDSymdcanbegivenbyspecifyingatangentvectortoateachdivisorpoint.ThustherankdcomplexvectorbundleK!Symdsde˝nedbyKD=p2DKpisisomorphictoT0;1Symds.Wenextuseideasof[T99,Lemma3.3]toconstructanalmostunitaryisomorphismfromKtoT0;1Symds.FixasimpledivisorD,andlet˛beacorresponding˝-vortex˝eld.De˝neˆD=min(fdist(p;q)jp;q2D&p6=qg[f;!)g);(1.17)14where;!)istheinjectivityradiusofthemetric.Let˜beasmoothfunctionon[0;1)thatsatis˝es06˜61,˜j[0;1]=1,and˜j[2;1)=0,andset˜p=˜2distpˆD:(1.18)Foreach=pp2D2KDletbpbetheextensionofptotheopenballofradius;!)centeredatpusingtheexponentialmap.De˝neasmoothsection˙ofKsupportedinneighborhoodofDbysetting˙=Xp2D˜pbp(1.19)andextendingby0toallof.Notethat˙satis˝es˙(p)=p&jr˙j=O(distD)8p2D:(1.20)Finally,foreachsuch˛andde˝neY˛;asY˛;=1p2ˇ˝p2w˙;i˙^@r˚)20;10L;(1.21)whereagainw=12˝j˚j2.NotethatY˛;isgaugeequivariant,thatis,foreveryg2G:gY˛;=Yg(˛);:(1.22)ThefollowinganalyticresultisthekeyingredientneededtocomputetheasymptoticcurvatureinTheorem1.11andholonomiesintheMainTheorem.Theorem1.8.[Theasymptoticformofhorizontalvectors]Forevery˛2P˝and2KD15asabove,thereisauniqueZ˛;200;1LsuchthatX˛;=Y˛;L˛Z˛;(1.23)isahorizontaltangentvectorat˛.Moreover,thefollowingasymptoticestimateshold:1.[L2-estimate]kY˛;k2L2!Pp2Dp2as˝!1.2.[Pointwisebound]X˛;Y˛;=O˝1=2expp˝distDc,wheredistDisthedistancefromD,andcisthepositivenumberfromTheorem1.6.Equation(1.23)de˝nesabundlemapfromKtoT0;1Symdsby(D;7!D;X˛;;(1.24)where˛isany˝-vortex˝eldcorrespondingtothedivisorD,andistheprojectionfromP˝toM˝˘=Symd.Equation(1.22)impliesthat(1.24)doesnotdependonthechoiceof˛.Furthermore,thismapisalmostunitarybyStatements(1)and(2).Similarresultshaveonlybeenknownfor˛atmetrics[T99,Lemma3.3].ProofofTheorem1.8:FixY˛;asinequation(1.21).SinceL˛iselliptic,andker(L˛)=f0g,byCorollary1.5,theoperatorL˛L˛ishasaboundedinverse(L˛L˛)1.ThustheequationL˛Y˛;L˛Z˛;=0(1.25)hasauniquesolutionforZ˛;givenbyZ˛;=(L˛L˛)1L˛Y˛;(1.26)16ConsequentlyX˛;inequation(1.23)ishorizontal.ThepointwisenormofY˛;satis˝esY˛;2=hD;˝j˙j2;wherehD;˝isde˝nedinequation(1.15).Usingequation(1.16),onegetsStatement(1).InordertoproveStatement(2),weputZ˛;=f˛;;˘˛;200;1Linequation(1.25)toobtaintheequations12+12j˚j2f˛;=1p8ˇ˝w@˙;(1.27a)@r@r+12j˚j2˘˛;=ip4ˇ˝r0;1˙^@r˚:(1.27b)LetG[˚]betheGreen'soperatorofthenon-degenerateellipticoperatorH[˚]=12+12j˚j2on0.BothH[˚]andG[˚]dependonlyonthegaugeequivalenceclassof˚.ByanabuseofnotationG[˚]willalsodenotethecorrespondingGreen'sfunction,whichisapositive,symmetricfunctiononwithalogarithmicsingularityalongthediagonal.Withthisde˝nitions,wecanwritef˛;asf˛;=1p8ˇ˝ZG[˚]w@˙!:(1.28)StandardelliptictheorygivesthefollowingboundsontheGreen'sfunction:G[˚](x;y)6c1+lnp˝dist(x;y)expp˝dist(x;y)c(1.29a)dG[˚](x;y)6cp˝dist(x;y)expp˝dist(x;y)c(1.29b)17forsomec2R+independentof˝orD(see[T99,Equation(6.10)]).Usingequation(1.28)andinequalities(1.29a)and(1.29b),togetherwiththeboundonwininequality(1.13b)andonjr˙jinequation(1.20)weget(afterpossiblyincreasingc)f˛;6c˝expp˝distDc(1.30a)@f˛;6cp˝expp˝distDc:(1.30b)Beforeturningourattentiontoequation(1.27b),notethatwehavethefollowingtwoscalaridentities˘˛;2=2Re˘˛;rr˘˛;2r˘˛;2:(1.31a)˘˛;2=2˘˛;˘˛;2d˘˛;2;(1.31b)UsingtheKähleridentityon0;1Lrr=2@r@riFr=2@r@r12˝j˚j2;(1.32)inequation(1.27b),togetherwiththeCauchy-Schwarzinequalityinequation(1.31a),andKato'sinequality(cf.[FU91,Equation(6.20)])d˘˛;6r˘˛;(1.33)givesus+12˝˘˛;6cp˝jr˙j@r˚:(1.34)18Equation(1.34),togetherwiththeboundon@r˚ininequality(1.13b)andtheboundonjr˙jinequation(1.20)givesus(againafterpossiblyincreasingc)˘˛;6c˝expp˝distDc:(1.35)Applying@rtoequation(1.27b)givesanellipticequationon@r˘˛;.Similarlytothepreviouscomputationwegetthefollowinginequality:+12˝@r˘˛;6cp˝expp˝distDc:Thus(afterpossiblyincreasingconelasttime)@r˘˛;6cp˝expp˝distDc:(1.36)Finally,inequalities(1.30a),(1.30b),(1.35)and(1.36)giveusX˛;Y˛;=L˛Z˛;6p2@f˛;+j˚jjfj+p2@r˘+j˚jj˘j=O˝1=2expp˝distDc;whichcompletestheproofofStatement(2).191.4TheBerryconnectionThe˝-vortexprincipalbundleistheprincipalG-bundle:P˝!M˝describedinSec-tion1.2,with˛)=[˛].InLemma1.1weconstructedahorizontaldistributiononthe˝-vortexprincipalbundle,whichistheorthogonalcomplementofthekernelof.ThisdistributionisG-invariant,soisaconnectioninthedistributionalsense(cf.[KN63,Chap-terII]),whichwecalltheBerryconnection.Thecorrespondingconnection1-formistheuniqueLie(G)-valued1-formAthatsatis˝esthethreeconditions:1.ker(A˛)isthehorizontalsubspaceat˛2P˝,2.(gA)g(˛)=adg(A˛),forallg2G,3.AXf=if,forallif2W2;2iR)˘=Lie(G),whereXf(˛)=(idf;if˚),asde˝nedinLemma1.2.ThenextlemmagivesaformulaforA˛.Recallthatforeach˝-vortex˝eld˛=(r;˚),theGreen'soperatorG[˚]istheinverseofthenon-degenerateellipticoperatorH[˚]=12+12j˚j2.Lemma1.9.TheLie(G)-valued1-formonP˝de˝nedasA˛(a; )=12G[˚](da+iIm(h( ;˚)))(1.37)istheconnection1-formcorrespondingtotheBerryconnection.Proof.Theright-handsideofequation(1.37)isthecompositionofthenon-degenerateGreen'soperatorandaLie(G)-valued1-form.IntheproofofLemma1.1wesawthatthekernelofthis1-formisexactlythehorizontalsubspace.ThisprovesCondition(1)above.20BecauseGisAbelian,theadjointrepresentationofGistrivial,andhencetheCondition(2)reducesto(gA)g(˛)=A˛.Sinceg(a; )=(a;g ),wehavegAg(˛)(a; )=12G[˚](da+iIm(h(g ;g˚)))=12G[˚](da+iIm(h( ;˚)));thusgAg(˛)=A˛.ThisprovesCondition(2).Finally,weshowthatAisthecanonicalisomorphismbetweenthe˝bersoftheverticalbundleandtheLiealgebraofG,thatisAXf=ifforeveryf2C1R):A˛Xf=12G[˚](d(idf)+iIm(h(if˚;˚)))!=iG[˚]12f+12j˚j2f=if;thusCondition(3)holds.WecanuseLemma1.9tocomputethecurvature2-formoftheBerryconnection.SinceGisAbelian,thecurvature,calledtheBerrycurvature,isaLie(G)-valued2-formwhichdescendstothebasespaceM˝.Theorem1.10.Thecurvature2-formoftheBerryconnectionat[˛]2M˝is[˛](X;Y)=G[˚](iIm(h( X; Y)))(1.38)where(aX; X)and(aY; Y)arethehorizontalliftsofXandY,respectively,at˛.More-over,equation(1.38)doesnotdependonthechoiceofthe˝-vortex˝eld˛representing[˛].21Proof.Theclaimabouttheindependenceofthechoice˛isimmediatesinceeverythingontheright-handsideisgaugeinvariant.ThecurvatureistheuniqueLie(G)-valued2-formonM˝thatsatis˝es=dA,whereistheprojectionfromP˝toM˝.ThusitisenoughtocomputetheexpressiondA˛((aX; X);(aY; Y)),andcompareitwithequation(1.38).Recall,thattheformulafortheexteriorderivativedAeX;eX=eXAeYeYAeXAeX;eY;(1.39)whereeXandeYaresmoothlocalextensionsof(aX; X)and(aY; Y)respectively.ChoosetheextensionssothattheirLiebracketvanishesat˛.Lettbethelocal˛owgeneratedbyeX,sot(˛)=˛+t(aX; X)+Ot2.SinceAeY=0at0(˛)=˛,wehaveeX˛AeY=limt!01tAt(˛)eYt(˛)):NotethateYt(˛))=(t)(aY; Y))+Ot2,becauseheX;eYi˛=0.Finally,letuswriteGh˚+t X+Ot2=G[˚]+tGX[˚]+Ot2:22Keepingonlythelinearterms,weobtaineX˛AeY=limt!01tAt(˛)eYt(˛))=limt!012tGh˚+t X+Ot2daY+iIm(h( Y;˚+t X))+Ot2=limt!012titG[˚](Im(h( Y; X)))+tGX[˚](daY+iIm(h( Y;˚)))=i2G[˚](Im(h( Y; X)));whereweusedthefactthatdaY+iIm(h( Y;˚))=0fortangentvectors.InterchangingeXandeYchangessign,sinceIm(h( Y; X))isskew.Substitutingtheseintoequation(1.39),andnotingthatthecommutatorvanishes,givesequation(1.38).1.5TheasymptoticBerrycurvatureInthissectionweuseTheorems1.8and1.10toanalyzetheBerrycurvatureinthelarge˝limit.Asbefore,letDbeasimpledivisor,and˛=(r;˚)bethcorresponding˝-vortex˝eld.Foreachp2D,choosep=fp;qgq2D2KD,sothatjp;qj=p;q.Let˙p=˙pbethecorrespondingsectionde˝nedbyequation(1.19),andletX˛;p=ap; p,asde˝nedinTheorem1.8.Byequation(1.23),X˛;p=Y˛;pL˛Z˛;p:(1.40)whereZ˛;p=fp;˘p200;1L.ItiseasytoseethatinStatement(2)ofTheorem1.8wecannowreplacedistDwithdistp,thedistancefromthesinglepointp.23ThesetfXpgp2D,whereXp=X˛;p2T[˛]M˝,isanasymptoticallyorthonormalbasisforthehorizontalsubspaceat˛,inthesensethatas˝!1XpXq=p;q+Oexpp˝ˆDc!p;q:Finally,foreachtangentvectorX,letX[=hXbethemetric-dualcovector.Theorem1.11.[TheasymptoticBerrycurvature]Thereisapositivenumbercsuchthatif˝>˝0=4ˇdand[˛]isasimple˝-vortex,thentheBerrycurvaturesatis˝es[˛]=Xp;q2D˜pp;qiwˇ˝+iAp;q˝X[p^iXq[+iBp;q˝X[p^X[q+iCp;q˝iXp[^iXq[;(1.41)where˜pasde˝nedinEquation(1.18),andAp;q˝;Bp;q˝,andCp;q˝arerealfunctions,withAp;q˝+Bp;q˝+Cp;q˝=O˝1expp˝ˆDc:(1.42)Proof.Forp6=q,Theorems1.6and1.8implythath p; q=j pjj qj6cexpp˝(distp+distq)c6cexpp˝ˆDc:ThisinequalitytogetherwiththefactthatG[˚](1)=O˝1frominequality(1.29a),givesequation(1.42)inthiscase.Ingeneral,foreveryp2D,Theorem1.10showsthat1i[˛]Xp;iXp=G[˚] p2:(1.43)24Thisisnon-negative,becauseG[˚]isgivenbyconvolutionswiththepositiveGreen'sfunction.Byequation(1.40),wecanwrite p=1p2ˇ˝i˙p^@r˚p2@r˘p+fp˚:(1.44)ApplyingtheboundsinTheorems1.6and1.8toequations(1.43)and(1.44),weobtain1i[˛]Xp;iXp=12ˇ˝G[˚]˙p2j@r˚j2+G[˚]Oexpp˝distpc:(1.45)Byinequality(1.29a),andthepositivityoftheGreen'sfunctionthelasttermisG[˚]Oexpp˝distpc=O˝1expp˝distpc:(1.46)Theorem1.6andequation(1.14)givesusH[˚]˜pw=12˜pj@r˚j2+O˝expp˝distpc:(1.47)Thuswecanwritethemainterminequation(1.45)asG[˚]j˙pj2j@r˚j22ˇ˝=G[˚]˜pj@r˚j22ˇ˝+OG[˚]˝dist2pexpp˝distpc=˜pwˇ˝+O˝1expp˝distpc;(1.48)since˙p2˜p=Odist2pby(1.20).Combiningequations(1.45),(1.46)and(1.48)yields1i[˛]Xp;iXp=˜pwˇ˝+O˝1expp˝distpc:25Thiscompletestheproofofequations(1.41)and(1.42).1.6TheasymptoticBerryholonomyAconnectiononaprincipalG-bundleP!Xde˝nesthenotionofparalleltransport;cf.[KN63,ChapterII].Paralleltransportaroundaloopiscalledholonomy.HolonomycanbeviewedasamapfromtheloopspaceofXtothespaceofconjugacyclassesofG.ForAbelianG,thelaterspaceiscanonicallyisomorphictoG.Inourcase,the˝-vortexprincipalbundle,P˝!M˝,isaprincipalG-bundleequippedwiththeBerryconnection.ThephysicalinterpretationisthatifoneadiabaticallymovesthedivisorpointsalongacurveinSymd,thenthecorresponding˝-vortex˝eldevolvesbytheparalleltransportde˝nedtheBerryconnection(cf.[K50],and[B84]).Inparticular,whenisaloop,theholonomyoftheBerryconnection,calledtheBerryholonomy,isagaugetransformation.Inthissection,wegiveanalyticandtopologicaldescriptionsofthegaugetransformationsthatariseasBerryholonomies.SincetheBerryholonomyisamapfromtheloopspaceof˝-vortexmodulispace,werecallsomewell-knownpropertiesofloopsinM˝˘=Symd.WecallaloopinSymdasinglevortexloopifonlyoneofthedivisorpointsmoves,andallotherdivisorpointsare˝xed.Inotherwords,singlevortexloopsareinducedbyloopsinthatarebasedatoneofthedivisorpoints.EveryloopinSymdcanbedecomposeduptohomotopy(andthushomology)toaproductofsinglevortexloops.Moreover,H1(M˝;Z)˘=H1SymdZ˘=H1Z);(1.49)26wherethelastisomorphismisgivenbysendingsinglevortexloopstotheirhomologyclassesbytheHurewiczhomomorphism.RecallthatthecomplementofSymdsiscalledthebigdiagonal.AloopinSymdisregularifitisasmooth,embedded(immersed,ifd=1)loopthatdoesnotintersectthebigdiagonal.Thebigdiagonalisemptywhend=1.Whend>1thebigdiagonalisasubvarietyofcodimensionatleast2,thuseverysmoothloopinSymdcanbemaderegularafterasmallsmoothperturbation.Nowconsiderthecanonicalcoveringmapds!Symds,wheredsisthespaceoforderedd-tuplesinwithoutrepetition.GivenaregularloopthatstartsatthesimpledivisorD=2Symds,eachlifteD2dofDdeterminesauniquelifteof.Theliftecanberegardedasad-tuple(1;:::;d)ofcurves(notnecessarilyloops)in.Theshadowof,ˆis=[image(i);(1.50)wheretheunionisoverallnon-constanti.Thesethasanaturalorientationcomingfromtheorientationof.Sinceisregular,isaunionofimmersed,orientedloops,henceitisaninteger1-cyclein.ThehomologyclassinH1Z)representedbyisindependentofthechoiceofthelifteD.Wedenotethisclassby.ItiseasytocheckthehomotopyclassofinSymdissenttothehomologyclassbytheisomorphism(1.49).Ingeneral,forasinglevortexloop,onlyoneofthei'sisnotconstant,say,and=[]2H1Z).Example1.12.AnexampleofisseenonFigure1.1,where=[1]+[2]+[3]=[3];27sinceboth1and2arenull-homologous.Thusishomologoustoasinglevortexloop.Wecallaloopa(positivelyoriented)vortexinterchangeif,asinFigure1.2,onlytwoi's,say1and2,arenotconstant,andthecomposition=12isthe(oriented)boundaryofadisk.123Figure1.1:Singlevortexloops:Onedivi-sorpointmovesalongoneofthei's.Allotherdivisorpointsare˝xed.12Figure1.2:Vortexinterchange:Onedi-visorpointmovesalong1andanotherdivisorpointmovesalong2.Allotherdivisorpointsare˝xed.SincetheBerryholonomyhasvaluesinthegaugegroupG,wealsorecallacouplewell-knownpropertiesofgaugetransformations.Elementsg2GrepresentclassesinH1Z)asfollows:ForaclosedmanifoldXanda˝nitelygeneratedAbeliangroupG,HnG)iscanonicallyisomorphictothespace[X;K(G;n)]ofhomotopyclassesofcontinuousmapsfromXtotheEilenberg-MacLanespaceK(G;n)(cf.[H02,Theorem4.57]).SinceK(Z;1)˘=U(1)andGishomotopyequivalentto;U(1)],wegetthatH1Z)iscanonicallyisomor-phictoˇ0(G),whichisalsoagroup,becauseGis.Infact,ifG0istheidentitycomponentofG,thenˇ0(G)'G=G0,andtheshortexactsequencef0g!G0,!GH1Z)!f0g(1.51)isnon-canonicallysplit.Theisomorphismbetweenˇ0(G)andH1Z)canbeunderstoodonthe(co)cyclelevel;sinceisaclosed,orientedsurface,H1Z)iscanonicallyisomorphictoHom(H1Z);Z).28Anelementg2Gde˝nesanelement[g]2Hom(H1Z);Z)via[g]([])=g()=12ˇiZg1dg2Z:(1.52)TheBerryholonomycanbeviewedasamapfromtheloopspaceM˝ofM˝toG.Ittheninducesamapholontheconnectedcomponents:ˇ1(M˝)˘!ˇ0M˝)hol!ˇ0(G)˘!H1Z):SincecohomologygroupsareAbelian,theabovemapfactorsdowntothehomology,andthusde˝nesahomomorphism:hol?:H1Z)˘!H1(M˝;Z)hol!H1Z);(1.53)wherethe˝rstisomorphismisfrom(1.49).Usingequation(1.52),anexplicitformulaforhol?canbegivenasfollows:ifg=,thenhol?evaluatesonany1-cyclebyhol?])=12ˇiZg1dg:(1.54)Finally,recallthatak-currentisacontinuouslinearfunctionalonk.A1-forma21de˝nesa1-currentbyCa(b)=Za^b:(1.55)29Similarly,asmooth1-chainde˝nesa1-currentbyC(b)=Zb:(1.56)Wesaythatthe1-currentsinequations(1.55)and(1.56)arethe1-currentsde˝nedbyaandrespectively.NowwearereadytoproveourmaintheoremabouttheBerryholonomy,statedintheintroduction.TheproofoftheMainTheorem.Sinceeverysmoothpathcanbemaderegularbyanarbi-trarilysmallsmoothperturbation,itisenoughtocheckregularloops,.WeproveStatement(1)˝rst:Let˛bea˝-vortex˝eldcorrespondingtoD,andbbethehorizontalliftofstartingat˛2P˝.Sinceisregular,t)issimpleforallt,thuswecanapplyTheorem1.8tothevelocityvectorb0(t)=(at; t),whichishorizontal,andhenceobtainjatj+j tj6c˝dXi=10i(t)expp˝disti(t)c:(1.57)Byde˝nitionoftheholonomy,g˝(˛)=˛+1Z0b0(t)dt=0@r+1Z0atdt;˚+1Z0 tdt1A:Ontheotherhand,bythede˝nitionofthegaugeaction(forAbeliangroups),g˝(˛)=r+g˝dg1˝;g˝˚:30Thuswehaveg˝dg1˝=1Z0atdt;(1.58)and(g˝1)˚=1Z0 tdt:(1.59)LetVˆbeanycompactsetinthecomplementof.Since;V)>0,Theorem1.6showsthatj˚j>p˝2onVforalllarge˝.Henceequations(1.57)and(1.59)implythatforx2Vj˚jjg˝1jx6p˝2jg˝1jx61Z0j t(x)jdt6c˝dXi=11Z00i(t)expp˝dist(i(t);x)cdt:(1.60)Usingjg˝j=1,dg1˝=jdg˝j,andequations(1.57)and(1.58),wealsoobtainjdg˝jx6dXi=11Z0jat(x)jdt6c˝dXi=11Z00i(t)expp˝dist(i(t);x)cdt:(1.61)Combiningthelasttwoinequalitiesgivesp˝jg˝1jx+jdg˝jx6c0˝expp˝;V)c;(1.62)foralllarge˝,whichimpliesthatjg˝1jandjdg˝jconvergeto0,uniformlyonV,as˝!1.ThisprovesStatement(1).InordertoproveStatement(2),we˝rstassumethatisasinglevortexloop,forwhichisanembeddedloop,thatboundsanembeddeddiskBin,andD==hasnodivisorpointsintheinteriorB,andletpbethedivisorpointinDthatismovedby31.ThereisacanonicalembeddingofBintoM˝thatsendsapointx2Btothedivisorx+(Dp).Theimageofthismap,bB,isan(oriented)diskinM˝,whose(oriented)boundaryis.WewilldenotethisembeddingbyˇB:B!bB.Sinceisnull-homotopic,g˝isintheidentitycomponentofG,andsocanbewrittenasg˝=exp(2ˇif˝),wheref˝isasmooth,realfunctionon.ByStokes'Theorem,f˝=12ˇiZbB:(1.63)IfjisapathasinStatement(2)oftheMainTheorem,then'˝=f˝j.Usingequation(1.63)weseethat'˝(1)'˝(0)=f˝(j(1))f˝(j(0))=12ˇiZbBj(1))j(0)));wherej(1)isinB,andj(0)isnot.Toevaluatethisintegral,wereparametrizeusingˇB.ByTheorem1.8,if!˝isthepullbackoftheKählerclassofM˝frombBtoBusingˇB,then!˝=ˇ˝!+O(1):Foreachx2B,letwxbethefunctionw,de˝nedinequation(1.2),correspondingtothedivisorD=ˇB(x),andletdx=dist(x;fj(0);j(1)g).By[HJS96,Lemma1.1],wxjBconvergesto2ˇx,inmeasure,sinceD\B=fxg.NowusingTheorem1.11,thelastintegral32aboveequalsto'˝(1)'˝(0)=12ˇZBwx(j(1))wx(j(0))ˇ˝+O˝1exp2p˝dxc(ˇ˝+O(1))!x=ZB(x;j(1))+Oexpp˝dxc!x+O˝1=1+O˝1:ThisimpliesStatement(2)inthecasewhereisasimplevortexloopthatboundsadisk.Inthegeneralcase,letI(2")bethetubular2"-neighborhoodofI=image(j),where"issmallenoughsothatI(2")\isasingleembeddedarc.Letbeasingle,embeddedboundingloopin,asinthepreviouscase,forwhichand)coincideonI(2").Letoutandoutdenotethepartsofand,respectively,forwhichout)andshoutlieinthecomplementofI(2").Similarlyin=inistheircommonpart.Nowonecanjoinoutwiththereverseofouttogetapiecewisesmoothloopnew,suchthatnew)isdisjointfromI(2"),andfurthermore,theloopsumnewandydi˙eronlybyoutanditsreverse,thusg˝holnew=hol:OnI(")wehavethatholnewconvergesto1intheC1-topologyas˝!1.Sincethecrossingformulaholdsfor,itmustholdforaswell.ThisestablishesthegeneralcaseofStatement(2).InordertoproveStatement(3)wepicka˝nitecoverUofbycoordinatechartssuchthatforevery(U;2Uthepreimageoftheintersection1\U)ˆR2iseither(i)empty,(ii)thex-axis,or(iii)theunionofthetwoaxes.Byusingasubordinatepartitionof33unity,itisenoughtoproveStatement(3)for1-formsthataresupportedinsideoneofthesecharts.Fixonesuchchart(U;2U,andletb21a1-formwithsupp(b)ˆU.Letusalsowritea˝=df˝(1.64a)b=Adx+Bdy;(1.64b)whereAandBarecompactlysupportedfunctionsonR2.If1\U)isempty,thenthesupportofbandaredisjoint,henceC(b)=Zb=Z\supp(b)b=Z;b=0:Ontheotherhand,as˝!1,a˝!0onsupp(b)byStatement(1),andhencejCa˝(b)j6maxsupp(b)fja˝jgkbkL1!0;whichprovesStatement(3)inthecase(i).Next,if1\U)isthex-axis,thenbyequation(1.64b),C(b)=Z1\U)b=1ZA(x;0)dx:(1.65)WealsohaveCa˝(b)=Za˝^b=ZUa˝^b=ZR2df˝^b:(1.66)34Againusingequation(1.64b),thisbecomesCa˝(b)=ZR2df˝^(Adx+Bdy)=ZR2d(f˝Adx+f˝Bdy)ZR2f˝@B@x@A@ydx^dy:The˝rstintegralontheright-handsideiszerobyStokes'TheoremandthefactthatAandBarecompactlysupported.ByStatement(2),wechoosef˝sothatitconvergesto0whenyispositiveandto1whenyisnegative.As˝!1,wethenhaveCa˝(b)=ZR2f˝@A@y@B@xdx^dy!1Z0Z@A@y(x;y)@B@x(x;y)dydx=1ZA(x;0)dx:Togetherwithequation(1.65)thisprovesStatement(3)forcase(ii).Finally,if1\U)istheunionofthetwoaxes,thenagainbyequation(1.64b),C(b)=Z1\U)b=1ZA(x;0)dx+1ZB(0;y)dy:(1.67)Asbefore,wealsohaveCa˝(b)=ZR2f˝@A@y@B@xdx^dy:(1.68)ButnowStatement(2)showsthat,as˝!1,f˝convergesto0intheupperleftquadrant,to1intheupperrightandthelowerleftquadrants,andto2inthelowerrightquadrant.35Hencebyequation(1.68),Ca˝(b)!2ZR+ZR@A@y(x;y)@B@x(x;y)dydx+ZR+ZR+@A@y(x;y)@B@x(x;y)dydx+ZRZR@A@y(x;y)@B@x(x;y)dydx=1ZA(x;0)dx+1ZB(0;y)dy;wherethelaststepisanelementarycomputation.Togetherwith(1.67),thisprovesState-ment(3)forthecase(iii).ToproveStatement(4),accordingtoequation(1.54)andthede˝nitionofthePoincaréduality,weneedtoshowthatforany[]2H1Z),hol?])=12ˇiZg1˝dg˝=[]:(1.69)Sinceeverythingin(1.69)ishomotopyinvariant,wehavethefreedomtochangebyahomotopy.Recallfrom(1.49)thatcanbedecomposed,uptohomotopy,toaproductofsinglevortexloops;thusitsu˚cestocheckStatement(4)forabasisofˇ1Symd˘=H1Z).Weuseabasis:asetofsimpleclosedcurvesfi;ig16i6gsuchthatiintersectsitransversallyandpositivelyonce,andi\j=i\j=i\j=;fori6=j.Thereisalwayssuchaset,andf[i];[i]g16i6gisabasisofH1Z),with[i][j]=0;[i][j]=0;[i][j]=i;j;36whereisthehomologyintersection.DenotethecorrespondingsinglevortexloopsinSymdbyfbi;big16i6g.ToproveStatement(4),weneedonlytoverify(1.69)foreverypairinthebasis.When2fi;ig,and2fbj;bjgwithi6=j,wehavebyStatement(1)thatZig1˝dg˝=O˝expp˝dist(;c:(1.70)When=iand=bi,forsomei,wecanchoseanotherrepresentative0ifor[]=[i]thatisdisjointfromi.Thus,againbyStatement(1),wehaveZig1˝dg˝=O ˝exp p˝disti0ic!!:(1.71)Thusallintegralsin(1.70)and(1.71)convergeto0as˝!1.Ontheotherhand,theseintegralsareintegermultiplesof2ˇ,sotheyhadtobe0forall˝>˝0.Finally,assumethati=jand=iand=bi,or=iand=bi.Inordertoprovethe˝rstofthesecases,letus˝xasmallembeddedsegmentjon=ithatintersects=ioncepositively.Suchpathsexistbytheconstructionofthebasis.Writeg˝jI=exp(2ˇi'˝),andsog1˝dg˝jI=2ˇid'˝.Thusbyequation(1.54),hol?])=ZId'˝+12ˇiZIg1˝dg˝='˝(1)'˝(0)+Op˝expp˝;fj(0);j(1)g)c:ByStatement(2),thisconvergesto1as˝!1.Ontheotherhand,hol?])isaninteger,soithadtobe1forall˝>˝0.Thesameargumentcanbeusedinthecase37of=eiand=i,whichcompletestheproofofStatement(4)andtheMainTheorem.Corollary1.13.Forall˝>˝0,thespaceP˝isanin˝nite-dimensionalvectorbundleoveraconnected,orientedandsmoothmanifoldwithoutboundary,cM˝.Thismanifoldhasrealdimension2d+1andisaU(1)-principalbundleovertheuniversalcoverofM˝.InparticularP˝ishomotopyretractstocM˝.Proof.FirstwewillimposetheCoulombgauge:˝xa˝-vortex˝eld˛0=(r;˚)2P˝.Wesaythat˛=r0;˚0isinCoulombgaugewithrespectto˛0ifthe1-forma=r0rissatis˝es:da=0:Foreach˛2P˝thereisinfactagaugetransformationgthatisintheidentitycomponentofG,andisuniqueuptoconstantgaugetransformations(factorsinU(1)),suchthatg(˛)isinCoulombgaugewithrespectto˛0.Aproofofthis,whichappliestoourcasetoo,canbefound,forexample,in[EN11,Lemma2.1].ThesetcM˝ˆP˝,calledtheCoulombslice,consistingthe˝-vortex˝eldsthatareinCoulombgaugewithrespectto˛0intersectseach˝ber.Fixthenapointx2,andrequireg(x)=1.Suchag=g˛isthenunique,moreover,canbewrittenasg˛=exp(if˛),andf˛isalsouniqueifoneprescribesf˛(x)=0.Setgt;˛=exp(itf˛):Thenthemapde˝nedasr(t;˛)=gt;˛(˛)isahomotopyretractionofP˝totheCoulombslice.Theintersectionofeach˝berwiththeCoulombsliceisacollectionofcircles,duetotheU(1)ambiguitymentionedabove.Moreover,thesecirclesareinbijectionwithˇ0(G)˘=38ˇ1(M˝).ThusfM˝=cM˝=U(1)isaˇ1(M˝)-coverofM˝,whichistheuniversalcoverifconnected.OurMainTheoremimpliesthatP˝isconnected,bythefollowingargument:let˛and˛0betwoarbitrary˝-vortex˝elds.SincesimpledivisorsaredenseinSymd,wecanassume,thatthecorrespondingdivisorsaresimple.Jointhetwodivisorsbyaregularpath0.Thenhol0(˛)isequaltog˛0forsomeg2G.IfgisnotintheidentitycomponentofG,thenitrepresentsanon-zerocohomologyclass[g]2H2Z).Letbeasmoothloopbasedatadivisorpointof˛thatrepresentsthePoincarédualof[g];letbbetheinducedsinglevortexloop,andset=b10.Now˛andhol(˛)=˛00areconnectedbythepathinP˝givenbyparalleltransport.Ontheotherhand,˛00and˛aregaugeequivalent,andtheconnectinggaugetransformationisintheidentitycomponentofG,whichmeansthatthereisapathfrom˛00to˛.ThusP˝isconnected,butthensoiscM˝,whichcompletestheproof.1.7ThelargearealimitConsidertheenergy(1.1)forthecriticalcouplingconstant=1andwith˝=1.Bradlow'scriterionfortheexistenceofirreduciblevorticesinthiscasebecomes˝0=2ˇd<1;(1.72)usingtheareawithrespecttothegivenarea2-form!.Evenwheninequality(1.72)doesnotholdfor!,itstillholdsfor!t=t2!ift>t0=p˝0.LetPtbethespaceofallsolutionsofthe1-vortexequationswithKählerform!tfor39t>t0.Apair(r;˚)2CL0LisinPtifitFr=121j˚j2(1.73a)@r˚=0;(1.73b)wheret==t2.LetMtbethecorrespondingmodulispacePt=G.Bradlow'sTheoremstillholds,henceMt˘=Symd,wherethedi˙eomorphismisagaingivenbythedivisorofthe˚-˝eld.By[B90,Proposition5.1],thefollowingdiagramiscommutativewhent2=˝:Ptt//$$P˝zzSymd;wheretistheisomorphismofprincipalbundlesgivenbyt(r;˚)=(r;t˚).TheL2-metriconPtisde˝nedbyequation(1.6),butwithHodgeoperatorandareaformgivenby!t.ForallX2TPt,wenowhave:kt)XkP˝=tkXkPt:(1.74)ThustheL2-metricofPtisconformallyequivalenttothepullbackoftheL2-metricofP˝viathebundleisomorphismt.TheBerryconnectiononPt!Mtisagainde˝nedastheorthogonalcomplementoftheverticalsubspaces,henceitisthesameasthepullbackoftheBerryconnectiononP˝viat.ThustheresultsofTheorems1.8and1.11andourMainTheoremholdinthelargearea40limit(t!1):MainTheoremforthelargearealimit.TheconclusionsoftheMainTheoremintheintroductionholdfortheprincipalG-bundlePt!Mtwith˝replacedeverywherebyt2.41Chapter2IrreducibleGinzburg-Landau˝eldsInthischapterweshowthattheGiznburg-Landauequations(1.3a)and(1.3b)admitirre-duciblesolutionsundercertainconditions.ThisgeneralizesBradlow'sresultforthecritical=1case[B90,Theorem4.3].WeusethehypothesesandnotationofChapter1.AsmentionedintheintroductionofChapter1,theGinzburg-Landautheoryisamathe-maticalmodelforsuperconductivity.Thereduciblesolutionscorrespondtoinsulatingstates,whileirreduciblesolutionsmodelsuperconductingstates.Theresultsofthischapteraresummarizedinoursecondmaintheorem.MainTheoremofChapter2.TheGinzburg-Landauequations(1.3a)and(1.3b)admitirreduciblesolutionsifandonlyif>4ˇjdj.Thecontentofthemaintheoremisillustratedbythefollowingphase-diagram:1Area˝-vortexsolutionsexistIIIFigure2.1:PhaseDiagram:InRegionI,de˝nedby64ˇjdj,onlyreduciblesolutionsexist(InsulatorPhase).InthecomplementaryRegionII(shaded),thereexistirreduciblesolutions(SuperconductorPhase).Forsimplicity,weassumethatd,thedegreeofthelinebundle,isnon-negative.Theresultsfornegativedegreescanbeprovensimilarly.42Thischapterisorganizedasfollows.InSection2.1weusestandardgeometricanalyticmethodstoshowthatthereareonlyreduciblesolutionsforparametersinRegionIinFig-ure2.1.InSection2.2,weprovetheexistenceofirreduciblesolutionsforparametersinRegionII.ThisisdoneusingagaugedversionofthePalais-Smalecompactnessproperty.2.1Non-existenceFirstofall,anycriticalpointoftheenergy(1.1)isgaugeequivalenttoasmoothcriticalpoint,whichsatis˝esequations(1.3a)and(1.3b)(see[JT80]).Furthermore,ifasmoothconnectionr2CLsatis˝esthevacuumMaxwellequations:dFr=0:(2.1)WecallsuchaconnectionaMaxwellconnection.Notethatapair(r;0),consistingaMaxwellconnectionandtheidenticallyzerosectionofL,isalwaysasolutionofequations(1.3a)and(1.3b).WecallsuchasolutionaMaxwellsolution.Asolution(r;˚)iscalledreducible,ifthereisagaugetransformationg2Gsuchthatg(r;˚)=(r;˚),otherwiseirreducible.Maxwellsolutionsareallreducible,becauseconstantgaugetransformationsactine˙ectivelyonthem.Ontheotherhand,if(r;˚)isapairwithnon-vanishing˚-˝eld,thentheonlywayanon-trivialU(1)-valuedgaugetransformationactine˙ectivelyonthispairisif˚vanishesonanopensubsetof.Sinceequation(1.3b)iselliptic,byuniquecontinuation,anysuchsolutionwouldvanisheverywhere.ThusMaxwellsolutionsaretheonlyreducibleones.Toprovethattherearenoirreduciblesolutionsandhencetheonlysolutionsarethe43Maxwellsolutionswhen64ˇd,we˝rstprovethefollowingtechnicallemma.Lemma2.1.If(r;˚)2CL0LisacriticalpointofE,then4ˇdk˚k2L262k˚k4L4:(2.2)Proof.FixaMaxwellconnectionr0,andwrite(r;˚)=(r0+a;˚),wherea2i1.Withthisnotation,equation(1.3a)becomesdd+j˚j2a=iImhr0˚;˚:(2.3)TaketheL2-innerproductofbothsideswitha,usethefactthataisimaginary-valued,andintegratebypartstoget06kdak2L2+ka˚k2L2=Reha˚jr0˚iL2:Inparticular,thisshowsthattherighthandsideisnon-negative.Next,rewriteequation(1.3b)intermsofr0,a,and˚:r0r0˚+ar0˚+r0(a˚)+jaj2˚=2˝˚j˚j2˚:(2.4)TaketheL2-innerproductofbothsideswith˚,andintegratebypartstogetkr0˚k2L2+2Reha˚jr0˚iL2+ka˚k2L2=2k˚k2L22k˚k4L4:(2.5)Toobtainathirdequation,notethatthatallMaxwellconnectionshavethesamecurvature,44givenbyFr=2ˇdi!;(2.6)orequivalentlythatiFr0=2ˇd.CombiningthiswiththeKähleridentityyieldskr0˚k2L2=2k@r0˚k2L2+2ˇdk˚k2L2:(2.7)Usingequation(2.7),equation(2.5)becomes2k@r0˚k2L2+2ˇdk˚k2L2+2Reha˚jr0˚iL2+ka˚k2L2=2k˚k2L22k˚k4L4:(2.8)Everytermonthelefthandsideisnon-negative.Leavingthe˝rst,thethird,andthefourthtermgivestheinequality2ˇdk˚k2L262k˚k2L22k˚k4L4;whichisequivalenttoinequality(2.2).Thenon-existenceofirreduciblesolutionsisanimmediateconsequenceofLemma2.1.Corollary2.2.If64ˇd,and(r;˚)2CL0LisacriticalpointofE,then˚isidenticallyzero.Proof.If64ˇdand˚isnotidenticallyzero,thenthelefthandsideofinequality(2.2)isnon-negative,whereastherighthandsideisstrictlynegative,whichisacontradiction.452.2ExistenceWenowturntotheproofoftheexistenceofirreduciblesolutionsofequations(1.3a)and(1.3b)when>4ˇd.Wewillshowtwothings:(i)absoluteminimizersoftheenergy(1.1)exist,and(ii)thatwhen>4ˇd,theMaxwellsolutionsareunstable,inthesensethattherearecon˝gurationsarbitrarilyclosewithlowerenergy.ItfollowsthattheabsoluteminimizerscannotbeMaxwellsolutions,andthereforeareirreducible.FirstweprovethatthefunctionalEsatis˝esagaugedversionofthePalais-Smalecompactnesscondition.Lemma2.3.Let(rn;˚n)2CL0Lbeasequencewiththefollowingproperties:1.ThesequenceE(rn;˚n)isbounded,2.ThesequenceofderivativesE0(rn;˚n)convergestozerointheW1;2.Thenthereisasubsequence(rnk;˚nk)2CL0L,togetherwithasequenceofgaugetrans-formationsgnk2G,suchthatgnk(rnk;˚nk)convergesinCL0L.Proof.SincesmoothpairsinCL0Laredense,itisenoughtoprovethestatementforsmoothsequences.Fixareferenceconnectionr0andde˝netheWk;pnorms,ask kWk;p=kXn=0kr0n kLp(2.9)Wepickr0tobeasmoothMaxwellconnection,andwritean=rnr0.BytheHodgedecomposition,theimaginaryvalued1-formsancanbedecomposedasan=idfn+idgn+ihn;(2.10)46wherefnisasmoothfunction,gnisasmooth2-form,andhnisaharmonic1-form.Wecaneliminatethe˝rsttermbyapplyingthegaugetransformationsexp(ifn)tothepair(rn;˚n).Wecanalsoassume,aftersuitablegaugetransformations,thatthesequencehnisaboundedsequenceinL2.Byellipticregularity,thespaceofharmonic1-formsis˝nitedimensional,sowecanchoseasubsequence,sothathnisconvergent.Becauseallhnareharmonic,thisimpliesconvergenceintheW1;2aswell.TheGinzburg-Landaufreeenergy(1.1)cannowbewrittenasEr0+an;˚n=EM+kdank2L2+kr0+an˚nk2L2+4Z˝j˚j22!:(2.11)Sinceallofthetermsontherighthandsidearenon-negative,andtheenergyofthesequenceisbounded,wegetthefollowing:(1)Thesequencekdank2L2isbounded,.Becauseanhasnoexactpart,andhasconvergentharmonicpart,thismeansthatanisboundedinW1;2.BytheSobolevinequalitythesequenceanisalsoboundedinL4.(2)Thesequencekr0+an˚nk2L2isbounded,andthussoisjkr0˚nk2L2kan˚nk2L2j(3)ThesequenceR˝j˚j22!isbounded.Jensen'sinequalityimpliesthat˝k˚k2L226Z˝j˚j22!;so˚isboundedinL2.Thisinturnimmediatelyimpliesthat˚nisboundedinL4as47well,becauseoftheequalityk˚k4L4=Z˝j˚j22!˝2+2k˚nk2L2:(2.12)(4)Combining(1)and(3),togetherwiththeCauchy-Schwarzinequality,showsthatan˚nisboundedinL2.(5)Combining(2)and(4),wegetthatr0˚nisboundedinL2,andthus˚nisboundedinW1;2.UsingthatE0r0+an;˚nconvergestozero,andthefactthatdan=0,wegetan=˚nj2an+iImhr0˚;˚+bn;(2.13)wherebnconvergestozeroini1,andr0r02˚=ar0˚r0(a˚)+jaj2˚+2˝˚j˚j2˚+ n;(2.14)where=dd+ddistheLaplace-deRahmoperator,and nconvergestozeroin0L.Itiseasytosee,usingthepreviousobservations,thattherighthandsidesareboundedinL2.Ellipticregularitythenimplies,(an;˚n)isboundedinW2;2.SincetheembeddingW2;2,!W1;2iscompactindimension2,weconcludethatthesequence(an;˚n)hasaconvergentsubsequenceinW1;2.Corollary2.4.Thein˝mumofEisachievedbysome˝eld(r;˚)2CL0L.Proof.ThefunctionalEnon-negative,thussoisitsin˝mum.Pickaminimizersequence48(rn;˚n)2CL0L.ByLemma2.3,thereisasubsequence,andgaugetransformationsgn,suchthatthesequencegn(rn;˚n)convergestoalimit.SinceEisgaugeinvariant,andanalyticintheW1;2norm,thelimitisanabsoluteminimizer.Finallyweshowthatthein˝mumcannotbeaMaxwellsolutionwhen>4ˇd.Tothisend,notethatallMaxwellsolutionshavethesameGinzburg-Landaufreeenergy,whichwecalltheMaxwellenergy:EM=4ˇ2d2+24:(2.15)Thenextlemmacomputesthesecondvariationoftheenergyfunctional(1.1)aroundaMaxwellsolution.Lemma2.5.Letr0beaMaxwellconnection.Thenforany(a;˚)2i1L)thefollowingsecondvariationformulaholds:Er0+a;˚EM=kdak2L2+2k@r˚k2L2+2ˇd2k˚k2L2+Ok(a;˚)k3W1;2:(2.16)Proof.IntheexpansionofEr0+a;˚thelineartermsin(a;˚)vanish,because(r0;0)isacriticalpointofE.ThusuptoquadraticorderwegetEr0+a;˚EM=kdak2L2+kr0˚k2L22k˚k2L2+Ok(a;˚)k3W1;2:49BytheKähleridentitythemiddletermcanberewrittenaskr0˚k2L2=hr0˚jr0˚iL2=h˚jr0r0˚iL2=h˚j2@r0@r0˚+2ˇd˚iL2=2k@r0˚k2L2+2ˇdk˚k2L2;whichconcludestheproof.Corollary2.6.When>4ˇdtheminimumoftheGinzburg-Landaufreeenergy,EisnottheMaxwellenergy(2.15).Proof.Intheformula(2.16)leta=0and˚beanon-zeroholomorphicsectionwithrespecttotheholomorphicstructureinducedbyr0.If>4ˇd,thenfort>0smallenoughEr0;t˚=EM+t22ˇd2k˚k2L2+Ot3<EM;(2.17)becausethecoe˚cientofthequadratictermisnegative.Lemma2.3andCorollary2.6implytheexistencepartofthemaintheoremofChapter2:Corollary2.7.Thereexistirreduciblesolutionsoftheequations(1.3a)and(1.3b)when>4ˇd:50REFERENCES51REFERENCES[AA87]Y.AharonovandJ.Anandan.Phasechangeduringacyclicquantumevolution.Phys.Rev.Lett.,1987.[AK02]I.S.AransonandL.Kramer.TheworldofthecomplexGinzburg-Landauequa-tion.Rev.Mod.Phys.,2002.[B06]J.M.Baptista.Vortexequationsinabeliangaugedsigma-models.Commun.Math.Phys.,2006.[B11]J.M.Baptista.OntheL2-metricofvortexmodulispaces.Nucl.Phys.,B,333,2011.[B84]M.V.Berry.Quantalphasefactorsaccompanyingadiabaticchanges.Proceed-ingsoftheRoyalSocietyofLondonA:Mathematical,PhysicalandEngineeringSciences,1984.[B90]S.B.Bradlow.VorticesinholomorphiclinebundlesoverclosedKählermanifolds.Commun.Math.Phys.,1990.[BR14]M.BokstedtandN.M.Romao.Onthecurvatureofvortexmodulispaces.Math.Z.,2014.[CM05]H.Y.ChenandM.S.Manton.TheKählerpotentialofabelianHiggsvortices.J.Math.Phys.,46:052305,2005.[DDM13]D.Dorigoni,M.Dunajski,andN.S.Manton.Vortexmotiononsurfacesofsmallcurvature.AnnalsPhys.,2013.[EN11]G.EtesiandÁ.Nagy.S-dualityinabeliangaugetheoryrevisited.J.Geom.Phys.,2011.[FU91]DanielS.FreedandKarenK.Uhlenbeck.Instantonsandfour-manifolds,volume1ofMathematicalSciencesResearchInstitutePublications.Springer-Verlag,NewYork,secondedition,1991.52[GL50]V.L.GinzburgandL.D.Landau.Onthetheoryofsuperconductivity.Zh.Eksp.Teor.Fiz.,1950.[H02]A.Hatcher.Algebraictopology.CambridgeUniversityPress,Cambridge,2002.[HJS96]M.Hong,J.Jost,andM.Struwe.AsymptoticlimitsofaGinzburg-Landautypefunctional.InGeometricanalysisandthecalculusofvariations,pagesInt.Press,Cambridge,MA,1996.[I01]D.A.Ivanov.Non-abelianstatisticsofhalf-quantumvorticesinp-wavesupercon-ductors.Phys.Rev.Lett.,86(2):268,2001.[JT80]A.Ja˙eandC.H.Taubes.Vorticesandmonopoles.ProgressinPhysics.Birkhäuser,Boston,Mass.,1980.[K50]T.Kato.OntheAdiabaticTheoremofQuantumMechanics.Jour.Phys.Soc.Japan,1950.[K85]M.Kohmoto.TopologicalinvariantandthequantizationoftheHallconductance.AnnalsofPhysics,160(2):343354,1985.[KN63]S.KobayashiandK.Nomizu.Foundationsofdi˙erentialgeometry.VolI.Inter-sciencePublishers,NewYork-London,1963.[MM15]R.MaldonadoandN.S.Manton.Analyticvortexsolutionsoncompacthyperbolicsurfaces.J.Phys.,A,48(24):245403,2015.[MN99]N.S.MantonandS.M.Nasir.Volumeofvortexmodulispaces.Commun.Math.Phys.,1999.[MS03]N.S.MantonandJ.M.Speight.Asymptoticinteractionsofcriticallycoupledvortices.Commun.Math.Phys.,2003.[R00]I.MundetiRiera.AHitchin-KobayashicorrespondenceforKähler˝brations.J.ReineAngew.Math.,2000.[T84]C.H.Taubes.OntheYang-Mills-Higgsequations.Bull.Amer.Math.Soc.(N.S.),1984.[T99]C.H.Taubes.GR=SW:countingcurvesandconnections.J.Di˙.Geom.,1999.53