Lattice Formulation of Two Dimensional Topological

arXiv:hep-lat/0611011v321Nov2006hep-lat/0611011RIKEN-TH-90TU-779YITP-06-56LatticeFormulationofTwoDimensionalTopologicalFieldTheoryKazutoshiOhta1,∗)andTomohisaTakimi2,∗∗)1DepartmentofPhysics,GraduateSchoolofScience,TohokuUniversity,Sendai980-8578,Japan2TheoreticalPhysicsLaboratory,TheInstituteofPhysicalandChemicalResearch(RIKEN),Wako,Saitama351-0198,JapanandYukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,JapanAbstractWeinvestigateanintegrablepropertyandobservablesof2dimensionalN=(4,4)topologicalfieldtheorydefinedonadiscretelatticebyusingthe“orbifolding”and“deconstruction”methods.Weshowthatourlatticemodelpossessestheintegrabil-ityandthepartitionfunctionreducestomatrixintegralsofscalarfieldsonsitesinconsequence.Wemakeclearmeaningfuldifferencesbetweenthediscretelatticeanddifferentiablemanifold,whichwouldbeimportanttoastudyoftopologicalquantitiesonthelattice.WealsoproposeanewconstructionofN=(2,2)supersymmetriclatticetheory,whichisrealizedbyasuitabletruncationofscalarfieldsfromtheN=(4,4)theory.∗)E-mail:[e-mail]∗∗)E-mail:[e-mail]1§1.IntroductionNon-perturbativedynamicsingaugeandstringtheoryareveryimportantissues.Alatticeformulationofgaugetheoryisoneofcandidatetoolstoanalyzenon-perturbativelyinstrongcouplingregion,butthewholepictureincludingnon-trivialtopologicalconfigurationslikeinstantonsstillhasnotbeenknown.Ontheotherhand,somenon-perturbativedynamicscanbefoundexactlyincontinuumgaugefieldtheoryifthetheorypossessessupersymmetry.Especially,ifthereexistthesufficientnumbersofpreservingsupercharges,holomorphicobservablesincludingcorrectionsfrommanyinstantonshasbeenalreadyknownexactly.SeibergandWittenshowthataprepotentialin4dimensionalN=2supersymmetricgaugetheoryisexactlygivenintermsofahyper-ellipticcurve(Seiberg-Wittencurve)whichcontainsmultiinstantoncorrectionsinprinciple.1),2)Morerecently,Nekrasovalsoshowsthattheprepotentialarisesasafreeenergyofsomestatisticalpartitionfunction,wheretheauthorusesatechniqueoftheso-called“localization”.3),4)MoreoverDijkgraafandVafashowthataneffectivesuperpotentialofN=1supersymmetricgaugetheorycanbeobtainedfromaholomorphicmatrixmodel.5)–7)Theseexactresultsarethankstointegrabilityofthesupersymmetricgaugetheory,andthemoresuperchargesthesystemhas,themoreconspicuoustheintegrablenaturebecomes.Thenwemayaskanaturalquestion;ifwerealizethesupersymmetricgaugetheoriesonthediscretelattice,istheintegrablestructurestillhold?Ifso,arethevariousnon-perturbativeresults,whichcanbeobtainedbythealgebraiccurvesandmatrixmodels,alsorecoveredfromlatticetheory?Toinvestigatethisstructureisourmotivationandpurposeinthepresentpaper.Constructionsofsupersymmetricgaugetheoriesonthelatticeareconsideredinvariousways.8)−28)Mostofconstructionsconcentrateone’sattentiononlyonthoseactionsthem-selves,whichofcoursecoincidewiththecontinuumtheoryinthelimitthatthelatticespacinggoestozero,butthereareotherseveralstudies12),15)−18)beyondatreelevel.Aswesaid,wewouldliketoexploretheintegrabilityoflatticetheoryaswellascontinuumtheory.Weexpectthatsomeclassesoftheobservablesmaybeobtainedexactlyevenonthelattice.Topologicalfieldtheoryisatoymodelbutgoodexampletoseetheintegrablestructureoffield(lattice)theory.Basicalgebraicstructureoftopologicalfieldtheoryonthelatticeisconsideredin29)physicallyanddevelopedin30)fromthemathematicalpointofview.Moreexplicitconstructionoftopologicalfieldtheoryonthelatticeisfirstproposedin13),14).Toproceedanalysisoftopologicalfieldtheoryonthelattice,weutilizetheformulationvia“orbifolding”and“deconstruction”,whicharediscoveredby31)and34),andapplied2tosupersymmetriclatticetheoryby8)-10).ThisisbecausetheseconstructionsseemtobenaturalfromaD-braneconstructionpointofview,namelythedeconstructiontechniqueisessentiallyequivalenttoquivergaugetheoriesorD-branesatanorbifoldsingularity,whichareverycompatiblewithstringtheoreticalinterpretations.Sowethinkthatthedeconstructionmaypreservetheintegrablestructure(andpartialsupersymmetries)eveninthediscretelatticepicture.Inthispaper,weformulate2dimensionaltopologicalfieldtheoryandinvestigatechar-actersofthepartitionfunctionandtopologicalobservablesbyusingthedeconstructionandthelocalization.Wefirstconstruct2dimensionalN=(4,4)fromareducedmatrixmodelwith8supercharges.Andalsowediscussonthepropertiesoftheobservablesofthereducedmatrixmodel,theorbifoldedmodelandthedeconstructedmodel.Insection4.1,wederiveanexactpartitionfunction,whichcoincideswithbythelocalizationtechnique.Organizationofthepaperisasfollows:Insection2weexplainaboutthereducedmatrixmodelfromwhichweconstructthesupersymmetriclatticeformulation.Thereducedmodelisobtainedfromadimensionalreductionof6dimensionalN=1supersymmetricgaugetheory.Therewefixsomenotationsandanoriginalactiontouseinthelatticeconstruction.Andinsection3,weconstructa2dimensionalN=(4,4)supersymmetriclatticemodelfromthereducedmatrixmodelusingthe“orbifolding”and“deconstruction”methods.Wealsoexplainaboutthesemethodsthere.Andinthelastpartofthesection,weshowanewconstructionofanN=(2,2)supersymmetriclatticeformulationbythe“truncation”ofscalarfieldsfromthe