Conformal topological Yang-Mills theory and de Sit

arXiv:hep-th/0111190v225Jan2002PreprinttypesetinJHEPstyle.-HYPERVERSIONEMPG-01-04,QMUL-PH-01-16,USYD-01-04ConformaltopologialYangMillstheoryanddeSitterholographyPauldeMedeiros,ChristopherHull,BillSpeneDepartmentofPhysis,QueenMary,UniversityofLondon,LondonE14NS,England{p.demedeiros,.m.hull,w.j.spene}qmul.a.ukJosØFigueroa-O’FarrillDepartmentofMathematisandStatistis,TheUniversityofEdinburgh,EdinburghEH93JZ,Sotlandj.m.figueroaed.a.ukAbstrat:AnewtopologialonformaleldtheoryinfourEulideandimensionsisonstrutedfromN=4superYangMillstheorybytwistingthewholeoftheon-formalgroupwiththewholeoftheR-symmetrygroup,resultinginatheorythatisonformallyinvariantandhastwoonformallyinvariantBRSToperators.AurvedspaegeneralisationisfoundonanyRiemannian4-fold.Thisformulationhaslo-alWeylinvarianeandtwoWeyl-invariantBRSTsymmetries,withanationandenergy-momentumtensorthatareBRST-exat.Thistheoryisexpetedtohaveaholographidualin5-dimensionaldeSitterspae.Keywords:TopologialFieldTheories,Holography.Contents1.Introdution12.Twistingandgrouptheory53.ThediagonallytwistedN=4theoryandonformalinvariane94.Redeningtheelds155.Conservedurrents186.Couplingtogravity207.ThetaAngleandS-Duality278.Topologialholography291.IntrodutionIn[1℄itwasproposedthattheoriesinD-dimensionaldeSitterspaeouldhaveaholographidualwhihisaonformaleldtheoryinD-1dimensionalEulideanspae.Inpartiular,in[1,2℄supersymmetritheorieswereidentiedforwhihanargumentanalogoustothatof[3℄foranti-deSitterholographyouldbemade.TheseinludedavedimensionaldeSittervauumarisingfromasolutionofthetypeIIB∗stringtheory.Thissolutionpreserves32supersymmetriesandalsoarisesasasolutionofavedimensionalgaugedsupergravity.TheproposeddualtheoryistheN=4superonformalYang-MillstheoryinfourEulideandimensions.Unfortunately,boththeD=5supergravityandtheD=4super-Yang-Millstheoriesarenon-standardinthattheyhavesomeeldswithkinetitermsofthewrongsign.However,itwaspointedoutin[1℄thatthesuper-Yang-Millstheorythatarisesinthiswayispreiselytheonethatanbetwistedtoobtainatopologialeldtheory.Thisshouldorrespondtoatwistingofthevedimensionalsupergravitytheory,sothatthe’tHooftlimitofthetopologialeldtheoryshouldhaveadualdesriptionasatopologialsupergravitytheoryinvedimensionaldeSitterspae.InfourEulideandimensions,theLorentzgroupisSpin(4)∼=SU(2)×SU(2).Intheusualtwistings,Spin(4)oranSU(2)subgroupthereofisidentiedwitha1subgroupoftheR-symmetrygroup.IntheN=4theories,thisneessarilybreakstheR-symmetrygroup.Itwaspointedoutin[1℄thattheR-symmetrygroupisinfatthesameastheonformalgroupinsuhtheories,sothereisthepossibilityoftwistingthewholeoftheR-symmetrygroupwiththeonformalgroup,andthisseemsthemostnaturaltwistingtouseintheholographiontext.Thepurposeofthispaperistofurtherdeveloptheseproposals,andinpartiulartoonstrutanewtopologialonformaleldtheoryinwhihtheonformalgroupistwistedwiththewholeoftheR-symmetrygroup.Itwillbeusefultobeginbyreviewingtwowaysofviewingtopologialeldtheories.ConsiderrstthetwistingofN=2supersymmetriYangMillstogiveatopologialtheorywhoseobservablesaretheDonaldsoninvariants[4℄.TheN=2supersymmetriYangMillstheoryin3+1dimensionsanbeobtainedbyreduingD=6N=2supersymmetriYangMillstheoryonaspatial2-torus.TheD=6theoryhasanSU(2)R-symmetryandtheredutiongivesatheorywithU(2)R-symmetry,theextraSO(2)beingrelatedtorotationsinthe5-6plane.(ThisSO(2)wouldbebrokenifoneweretokeepthemassiveKaluza-Kleinmodes.)ThebosonisetorisLie-algebravaluedandonsistsofavetoreldandtwosalars,whihtransformasa2-vetorundertheSO(2).TheWikrotatedversionofthistheoryisusuallytakentobeatheoryinfourEulideandimensionswithU(2)R-symmetryandthesamebosonisetor.Therearetheusualsubtletiesastohowonetreatsthefermions,buthoweveroneproeeds,theresultingtheoryinfourEulideandimensionshasnoonventionalsupersymmetry.However,thereisasimplewayofobtainingasupersymmetritheoryinfourEulideandimensions.OneansimplystartwiththeD=6theoryandredueononespaeandonetimedimension,andtheresultingtheoryinfourEulideandimensionswillautomatiallybeinvariantunderN=2supersymmetry[5,6℄.TheR-symmetryofthistheoryisSU(2)×SO(1,1)andoneofthesalarelds(theonearisingfromthetimeomponentoftheD=6vetoreld)hasakinetitermofthewrongsign;thissignisneessaryforthenon-ompatR-symmetryandforinvarianeunderEulideansupersymmetry.Following[1℄,itwillbeonvenienttorefertothisasaEulideaneldtheoryandtotheWik-rotatedoneasaEulideanisedeldtheory.Thetwistingof[4℄wasoriginallyformulatedintermsoftheEulideanisedthe-ory,withsymmetrySpin(4)×U(2)∼=SU(2)×SU(2)×SU(2)×SO(2).AnSU(2)subgroupoftheSpin(4)LorentzsymmetryistwistedwiththeSU(2)subgroupoftheU(2)R-symmetry,sothatoneofthesuperhargesbeomesasalarBRSTharge.Equivalently,dierenttwistingsorrespondtoregardingdierentembed-dingsofSpin(4)∼=SU(2)×SU(2)inSpin(4)×SU(2)∼=SU(2)×SU(2)×SU(2)astheLorentzsymmetrygroup.However,inthisapproah,therearesomesubtletiesinndingtheationinvariantunderthetwistedsupersymmetry,andinpartiular,thesignofthekinetitermofoneofthetwooriginalsalarsmustbehanged,sothatthetwistedtheoryhasanSO(1,1)symmetryinsteadoftheoriginalSO(2)[4℄;this2SO(1,1)symmetryistheghost-numbersymmetry,wit