Vertical versus Diagonal Dimensional Reduction for

arXiv:hep-th/9605082v216Sep1996CTP-TAMU-18/96Imperial/TP/95–96/36hep-th/9605082VerticalVersusDiagonalDimensionalReductionforp-branesH.L¨u†,C.N.Pope†CenterforTheoreticalPhysics,TexasA&MUniversity,CollegeStation,Texas77843K.S.Stelle⋆TheBlackettLaboratory,ImperialCollege,PrinceConsortRoad,LondonSW72BZ,UKABSTRACTInadditiontothedouble-dimensionalreductionprocedurethatemploysworld-volumeKillingsymmetriesofp-branesupergravitysolutionsandactsdiagonallyonaplotofpver-susspacetimedimensionD,thereexistsasecondprocedureof“vertical”reduction.Thisreducesthetransverse-spacedimensionviaanintegralthatsuperposessolutionstotheunderlyingLaplaceequation.Weshowthatverticalreductionisalsocloselyrelatedtotherecently-introducednotionofintersectingp-branes.Weillustratethiswithexamples,andalsoconstructanewD=11solutiondescribingfourintersectingmembranes,whichpreserves1/16ofthesupersymmetry.Giventhetworeductionschemesplusdualitytrans-formationsatspecialpointsofthescalarmodulusspace,onemayrelatemostofthep-branesolutionsofrelevancetosuperstringtheory.WearguethatthemaximumclassifyingdualitysymmetryforthispurposeistheWeylgroupofthecorrespondingCremmer-Juliasuper-gravitysymmetryEr(+r).Wealsodiscussaseparateclassofduality-invariantp-braneswithp=D−3.†ResearchsupportedinpartbyDOEGrantDE-FG05-91-ER40633⋆ResearchsupportedinpartbytheCommissionoftheEuropeanCommunitiesundercontractSCI*-CT92-07891IntroductionInthispaper,weshallexplorethetwodifferenttypesofdimensionalreductionthatmaybeusedtorelatep-branesolutionsofsupergravitytheoriesindifferentdimensions.Thebetter-knownofthesetypesofreductionemploysKillingsymmetriesofthep-banesolutionsinDdimensionstoeffectasimultaneousreductionontheworld-volumeandinthetargetspace,yieldinga(p−1)-branein(D−1)dimensions.Thisisthusthefield-theoreticanalogueofthedoubledimensionalreductionprocedureforp-braneactions[1],whichisalsoapplicabletosupergravitysolutions,causing“diagonal”movementontheDversusp“brane-scan.”AsidefromaWeylrescalingofthemetricneededtomaintaintheconventionalEinstein-frameformofthegravitationalaction,thisdiagonalreductionproceduredoesnotchangetheasymptoticfalloffbehaviourofasolutioninthedirectionstransversetotheworld-volume.Thesecondtypeofreductionhasbeenreferredtointheliteratureas“constructingpe-riodicarrays”[2,3,4].Thisprocedureemploysthezero-forcepropertyofp-branesolutions,whichpermitsmulti-centersolutionsaswellassingle-centerones.Weshallpresent,insomemoredetailthanpreviouslygiven,justwhatthissecondprocedureentails.Inparticular,itrequiresthepossibilityofstackingupa“deck”ofaninfinitenumberofp-branesinDdimensions,sothatthesumofpotentialsinamulti-centersolutiongoesovertoanintegral,thuschangingtheasymptoticfalloffbehaviourinthetransversedirectionstothatcorre-spondingtoap-braneinD−1dimensions.Thus,weshallcallthis“vertical”dimensionalreduction,correspondingtoverticalmovementonthebrane-scan.Incontrasttosomeim-pressionsgatheredfromtheliterature,weshallshowthatwhilethezero-forcepropertiesofsupersymmetricp-branesindeedallowthistypeofreduction,thereareinfactmanymorenon-supersymmetricbutextremalsolutionsthatalsosatisfytheno-forcecondition.Verticaldimensionalreductioncanalsobeappliedtothesecases.Thisstacking-upprocedurebreaksdownforthecaseof(D−3)-branesinDdimensions,whichhaveconicalasymptoticspacetimesinwhicheachincrementtothenumberofcentersinamulti-centersolutioncausesaportionofthesolidangleattransverseinfinitytoberemoved,eventuallyleadingtoacompactificationofthetransversespacetime.Thus,theknownexamples[5,6,7]of(D−2)-branesseemtoresideinadifferentcategoryfromtheotherp-branesfromthisperspective.Thisisdueinparttothefactthattheyoccurindifferenttheories[8]:insteadofthestandardsupergravitytheoriesthatweconsiderhere,theyrequireadilatonpotentialterm,whichmaybeinterpretedastheprefactorforazero-formfieldstrength[7].Theverticalreductionprocedureinvolvestwostages,namelyanintegrationovera1continuumofchargesofthemulti-centersolutiondistributeduniformlyoverlines,planesorhyperplanes,followedbyanordinaryKaluza-KleinreductionovertheresultingKillingdirections.Weshowthatattheintermediatestage,themetricconfigurationinthehigherdimensioncansometimesbeinterpretedasaspecialcaseofintersectingp-branes.Thusweseethatsomep-branesolitonsthatarevertical-dimensionalreductionsofp-branesinhigherdimensionscanalsobeviewedasintersectionsofextendedobjectsinthehigherdimension.Inparticular,allthemulti-scalaranddyonicsolutionswhosechargesarecarriedbyfieldstrengthsderivedfromthefieldstrengthsofthehigher-dimensionaltheoryadmitsuchaninterpretation.Inpractice,thehigherdimensionofgreatestinterestisD=11,inwhichcasetheselower-dimensionalp-branescanbeinterpretedasintersectionsofM-branes.Thetwotypesofdimensionalreductioncanbecombinedwithdualitytransformationsindimensionswheretheverticalanddiagonalreductiontrajectoriescross.Forthispurpose,itisnecessarytofocusonthesubgroupofdualitytransformationsthatmapbetweendistinctp-branesolutions,i.e.theclassifyingsymmetryforp-branesolutions.WeshallarguethatthemaximalrelevantdualitygrouphereistheWeylgroupoftheCremmer-JuliasymmetryEr(+r),where