On Hamiltonian structure of the spin Ruijsenaars-S

OnHamiltonianstructureofthespinRuijsenaars-SchneidermodelG.E.ArutyunovandS.A.FrolovyAbstractTheHamiltonianstructureofspingeneralizationoftherationalRuijsenaars-SchneidermodelisfoundbyusingtheHamiltonianreductiontechnique.Itisshownthatthemodelpossessesthecurrentalgebrasymmetry.ThepossibilityofgeneralizingthefoundPoissonstructuretothetrigonometriccaseisdiscussedanddegenerationtotheEuler-Calogero-Mosersystemisexamined.1IntroductionRecentlyaspingeneralizationoftheellipticRuijsenaars-Schneidermodel(spinRSmodel)wasintroducedasadynamicalsystemdescribingthepoleevolutionoftheellipticsolutionsofthenon-abelian2DTodachain[1].EquationsofmotionproposedforthemodelgeneralizetheonesfortheEuler-Calogero-Mosersystem(ECM)[2]-[6],whichisanintegrablesystemofNparticleswithinternaldegreesoffreedominteractingbyaspecialpairwisepotential.AnimportanttoolfordealingwithclassicalintegrablesystemsandespeciallyforquantizingthemistheHamiltonianformalism.AlthoughequationsofmotiondeningthespinRSmodelcanbeintegratedintermsofRiemanntheta-functions,thequestionabouttheirHamiltonianformremainsopen.Theaimofthepresentpaperistogiveapartialanswertothisquestion,whichliesinconstructingtheexplicitHamiltonianformulationfortherationalspinRSmodel.OurconstructionisbasedontheHamiltonianreductionprocedureacknowledgedastheuni-fyingapproachtodynamicalsystemsofCalogeroorRuijsenaarstype[7]-[16].InthisapproachonestartswithalargeinitialphasespaceandasimpleHamiltonianpossessingasymmetrygroup.Thenfactorizingthecorrespondingmotionbythissymmetryoneisleftwithanontrivialdynamicalsystemdenedonareducedphasespace.Inparticular,therationalRSmodelandthetrigonometricCalogero-MosersystemappearinthiswayifoneusesthecotangentbundleTGoveraLiegroupGastheinitialphasespace[9].AnaturalgeneralizationofthisapproachallowingustoincludespinvariablesconsistsinreplacingTGbyamoregeneralphasespacePthatwechoosetobeTGJ,whereJSteklovMathematicalInstitute,Gubkina8,GSP-1,117966,Moscow,Russia;[e-mail]stitute,Gubkina8,GSP-1,117966,Moscow,Russia;[e-mail]miltonianHRandperformingtheHamiltonianreductionbyG-action,weobtainthePoissonstructureoftherationalspinRSmodel.Letusbrieflydescribethecontentofthepaperandtheresultsobtained.Forsimplicity,werestrictourselvestothecaseofG=GL(N;C).InSection2wedeneonPtwodynamicalsystemsgovernedbyHamiltoniansHCandHRandshowthatthecorrespondingintegralsofmotioncombineintogeneratorsoftheYangianandthecurrentalgebrarespectively.Sincealltheseintegralsaregauge-invariant,thecorrespondingsymmetrieswillsurviveafterthereduction.Asisknown[11]thedynamicalsystemonthereducedphasespacecorrespondingtoHCisthetrigonometricECMmodel.Thisimmediatelyreproducestheresultfoundin[4,5]thatthemodelpossessestheYangiansymmetry.Section3isdevotedtotherationalspinRSmodel.FirstweintroduceG-invariantspinvariablesthataftersolvingthemomentmapequationcanbeidentiedwithcoordinatesonthereducedphasespacePr.EquationsofmotionfordynamicalvariablesofPrproducedbyHRcoincidewiththeonesintroducedin[1]fortherationalcase.ThatisthewayweobtainanexplicitHamiltonianformulationofthespinRSmodel.ThePoissonstructureofthemodelisfoundtoberathernontrivialandadmitsatleasttwoequivalentdescriptionsintermsofdierentphasevariables.Moreover,itdependsonaparameterγbeingacouplingconstantofthemodel.ItturnsoutthatthespinRSmodeladmitssuch(spectral-independent)L-operator(Laxmatrix)thatsatisesthesameL-operatoralgebraasforthecorrespondingspinlessmodel.WealsoshowthattheHamiltonianreductionprovidesanalternativewayofsolvingequationsofmotionwithoutusingspectralcurves.Finally,wepresentanexplicitexpressionforgeneratorsofthecurrentalgebraviaphasevariablesofthespinRSmodelanddenethegauge-invariantmomentumvariables.InSection4degenerationoftherationalspinRSmodeltotheECMsystemisexamined.AninterestingfeaturewecomehereistheappearanceofspinvariablesobeyingthedeningrelationsoftheFrobeniusLiealgebra.WeobservethatthegeneralellipticECMsystemcanbealsoformulatedintermsofFrobeniusspinvariables.2CurrentandYangiansymmetriesInthissectionweconstructrepresentationsoftheYangianandcurrentalgebrasrelatedtothecotangentbundleTGoverthematrixgroupG=GL(N;C)anddescribetheirconnectiontotheECMandthespinRSmodelsrespectively.ConsiderthefollowingmanifoldP=TGG,whereGisadualspacetotheLiealgebraG=Mat(N;C)ofG.WeparametrizeanelementfromGbyamatrixS2GduetotheisomorphismGG.ThespaceTGisnaturallyisomorphictoGGandweparametrizeitbypairs(A;g),whereA2Gandg2G.ThealgebraofregularfunctionsonPissuppliedwithaPoissonstructure,whichcanbewrittenintermsofvariables(A;g;S)asfollowsfA1;A2g=12[C;A1−A2](2.1)fA1;g2g=g2C;fg1;g2g=0;(2.2)2fS1;g2g=fS1;A2g=0(2.3)fS1;S2g=−12[C;S1−S2]:(2.4)HereweusethestandardtensornotationandC=Pi;jEij⊗Ejiisthepermutationoperator.ThePoissonstructureisinvariantunderthefollowingactionofthegroupG:A!hAh−1;g!hgh−1;S!hSh−1:(2.5)Wereferto(2.5)astogaugetransformations.Themomentmapofthisactionisoftheform=gAg−1−A+S:(2.6)Thesimplestgauge-invariantHamiltoniansareHC=trA2andHR=tr