Low energy dynamics of a CP^1 lump on the sphere

arXiv:hep-th/9712089v110Dec1997Low-energydynamicsofaCP1lumponthesphereJMSpeight∗DepartmentofMathematicalSciences,UniversityofDurham,DurhamDH13LE,UKAbstractLow-energydynamicsintheunit-chargesectoroftheCP1modelonsphericalspace(space-timeS2×R)istreatedintheapproximationofgeodesicmotiononthemodulispaceofstaticsolutions,asix-dimensionalmanifoldwithnon-trivialtopologyandmetric.Thestructureoftheinducedmetricisrestrictedbyconsiderationoftheisometrygroupinheritedfromglobalsymmetriesofthefullfieldtheory.Evaluationofthemetricisthenreducedtofindingfivefunctionsofonecoordinate,whichmaybedoneexplicitly.Sometotallygeodesicsubmanifoldsarefoundandthequalitativefeaturesofmotiononthesedescribed.1IntroductionTheCP1modelinflatspaceisascalarfieldtheorywhoseconfigurationspaceQconsistsoffiniteenergymapsfromEuclideanR2tothecomplexprojectivespaceCP1,theenergyfunctionalbeingconstructednaturallyfromtheRiemannianstructuresofthebaseandtargetspaces(thatis,themodelisapuresigmamodelinthebroadsense).Therequirementoffiniteenergyimposesaboundaryconditionatspatialinfinity,thatthefieldapproachesthesameconstantvalue,independentofdirectioninR2,sothatthefieldmayberegardedasamapfromtheonepointcompactificationR2∪{∞}∼=S2toCP1.SinceCP1∼=S2also,finiteenergyconfigurationsareeffectivelymapsS2→S2,thehomotopytheoryofwhichiswellunderstood,andtheconfigurationspaceisseentoconsistofdisconnnectedsectorsQnlabelledbyanintegern,the“topologicalcharge”(degree),Q=[n∈ZQn.(1)Eachconfigurationistrappedwithinitsownsectorbecausetimeevolutioniscontinuous.TheLorentzinvariant,time-dependentmodelisnotintegrablebutcompletesolutionofthestaticproblemhasbeenachievedbymeansofaBogomol’nyiargumentandthegeneralchargenmodulispace,thespaceofcharge-nstaticsolutionsMn⊂Qn,isknown(thatallstatic,finiteenergysolutionsoftheCP1modelsaturatetheBogomol’nyiboundisanon-trivialresult[1]).Eachstaticsolutionwithinthecharge-nsectorhasthesameenergy(minimumwithinthatsectorandproportionalton),andMnisparametrizedby4n+2parameters(themoduli),sosuchamodulispacemaybethoughtofasthe(4n+2)-dimensionallevelbottomofapotentialvalleydefinedontheinfinitedimensionalcharge-nsector,Qn.Lowenergydynamicsmaybeapproximatedbymotionrestrictedtothisvalleybottom,amanifoldembeddedinthefullconfigurationspace,andthusinheritingfromitanon-trivialmetricinducedbythekineticenergyfunctional.Theapproximatedynamicproblemisreducedtothegeodesicproblemwiththismetric,andhasbeeninvestigatedbyseveralauthors[2,3].Intheunit-chargesectoronehereencountersadifficulty:certaincomponentsofthemetricaresingularandtheapproximationisilldefined.Forexample,unit-chargestaticsolutionsarelocalizedlumpsofenergywitharbitraryspatialscale,sooneofthesixmoduliofM1isascaleparameter.Motionwhichchangesthisparameterisimpededbyinfiniteinertiainthegeodesicapproximation,aresultinconflictwithnumericalevidencewhichsuggeststhatlumpscollapseunderscalingperturbation[4].Thisproblemshouldnotbepresentinthemodeldefinedonacompacttwodimensionalphysicalspace.Theobviouschoiceisthe2-spherebecausethehomotopicpartitionoftheconfigurationspacecarriesthrough∗Presentaddress:DepartmentofMathematics,UniversityofTexasatAustin,AustinTX78712,USA1unchanged.Also,S2withthestandardmetricisconformallyequivalenttoEuclideanR2∪{∞},andthestaticCP1modelenergyfunctionalisconformallyinvariant,sothewholeflatspacestaticanalysisisstillvalidandallthemodulispacesareknown.However,thekineticenergyfunctionaldoeschangeandinducesanew,welldefinedmetricontheunit-chargemodulispace.Bymeansoftheisometrygroupderivedfromthespatialandinternalsymmetriesofthefullfieldtheorywecanplacerestrictionsonthepossiblestructureofthismetric,greatlysimplifyingitsevaluation.Thegeodesicproblemisstilltoocomplicatedtobesolvedanalyticallyingeneral,butbyidentifyingtotallygeodesicsubmanifolds,itispossibletoobtainthequalitativefeaturesofanumberofinterestingsolutions.Inparticular,thepossibilitiesforlumpstravellingaroundthespherearefoundtobeunexpectedlyvaried.2TheCP1modelonS2TheCP1modelonthe2-sphereisdefinedbytheLagrangianL[W]=ZS2dS∂μW∂ν¯W(1+|W|2)2ημν(2)whereWisacomplexvaluedfield,dSistheinvariantS2measureandημνarethecomponentsoftheinverseoftheLorentzianmetricη=dt2−d2Ω(3)onR(time)×S2(space),d2ΩbeingthenaturalmetriconS2.AlthoughthelanguageoftheCP1modelisanalyticallyconvenient,thehomotopicclassificationandphysicalmeaningofthefieldconfigurationsaremoreeasilyvisualizedifweexploitthewellknownequivalencetotheO(3)sigmamodel[5,6].Inthelatter,thescalarfieldisathreedimensionalisovectorφconstrainedtohaveunitlengthwithrespecttotheEuclideanR3norm(φ·φ≡1),thatis,thetargetspaceisthe2-sphereofunitradiuswithitsnaturalmetric,whichwewilldenoteS2isoforclarity.(Thesuffixrefersto“isospace”inanalogywiththeinternalspaceofnuclearphysicsmodels.)TheCP1fieldWisthenthoughtofasthestereographicimageofφintheequatorialplane,projectedfromtheNorthpole,(0,0,1).Explicitly,φ=W+¯W1+|W|2,W−¯Wi(1+|W|2),|W|2−11+|W|2(4)andW=φ1+iφ21−φ3.(5)ThenL[W]≡Lσ[φ]=14ZS2dS∂μφ·∂νφημν(6)thefamiliarO(3)sigmamodelLagrangian.AWconfiguration,then,maybevisualizedasadistributionofunitlengtharrowsoverthesurfaceofthephysical2-sphereS2sp.EachsmoothmapS2sp→S