A note on the Supergravity Description of Dielectr

arXiv:hep-th/0106206v22Jul2001AnoteontheSupergravityDescriptionofDielectricBranesD.Brecher∗andP.M.Saffin†CentreforParticleTheory,DepartmentofMathematicalSciences,UniversityofDurham,SouthRoad,DurhamDH13LE,UnitedKingdom.PreprintDCPT-01/512July2001AbstractWecommentontherecentpapersbyCostaetalandEmparan,whichshowhowonemightgeneratesupergravitysolutionsdescribingcertaindielectricbranesintendimensions.The“basic”suchsolutionsdescribeeitherNfundamentalstringsorND4-branesexpandingintoaD6-brane,withtopologyM2⊗S5orM5⊗S2respectively.Treatingthesesolutionsinaunifiedway,wenotethattheyallowforpreciselytwovaluesoftheradiusoftherelevantsphere,andthatthesolutionwiththesmallervalueoftheradiushasthelowerenergy.Moreover,thepossibleradiiinbothcasesagreeuptonumericalfactorswiththecorrespondingsolutionsoftheD6-braneworldvolumetheory.WethusarguethatthesesupergravitysolutionsarethecorrectgravitationaldescriptionofthedielectricbranesofEmparanandMyers.∗email:[e-mail]†email:p.m.saffi[e-mail]11IntroductionThehigher-dimensionalcounterpartsoftheMelvinuniverse[1],whichdescribegravitatingmagneticfluxbranes,haveappearedinvariousstring-andM-theoreticguisesoverthelastfewyears(see,e.g.,[2]–[15]).WithintheKaluza-Kleincontext,themostnaturalwaytogeneratesuchfluxbranesisviaatwistedcompactificationofflatspace[16,17,18].Moreover,startinginsteadwiththeEuclideanSchwarzschildsolutioncrossatrivialtimedirection,onecangeneratesphericalbranesviajustsuchatwistedcompactification[18].Inslightlydifferentlanguage,onecanthinkofthisasfollows:takeaD-dimensionalunchargedblackstring;analyticallycontinuetheworldvolumecoordinates;andperformadoubledimensionalreductionalongatwisteddirection.Theresulting(D−1)-dimensionalsolutiondescribesaspherical(D−5)-braneinthecoreofamagneticfluxbrane.Ithaslocal,butzeronet,magneticcharge1.IfonenowaddsNunitsof2-formchargetotheD-dimensionalblackstring,the(D−1)-dimensionalsolutionwillexhibitNunitsof1-formcharge,soismorerightlyinterpretedasacollectionofNparticles“expanding”intoasphereundertheinfluenceofthebackground(D−3)-formmagneticfield.Ofcourse,thesetechniquesneednotonlybeappliedtoblackstringsand,inrecentwork,Costaetal[19]andEmparan[20]consideredtheirapplicationinanM-theoreticcontext.Startingwiththebasicbranesofeleven-dimensionalsupergravity,then,theseauthorsshowedhowtogeneratesolutionsoftypeIIAsupergravitywhichdescribevariousten-dimensionalbranesexpandingintoaD6-braneundertheinfluenceofabackground8-formmagneticfield.Inparticular,thereductionofcertainM2-orM5-branesolutionsgeneratesten-dimensionalsolutionsdescribingF-stringsorD4-branesexpandingintoaD6-branewithtopologyM2⊗S5orM5⊗S2respectively,whereMndenotesann-dimensionalMinkowskispace.ApplyingT-dualitygeneratesmoregeneralconfigurations[20]–Dp-branesexpandingintoaD(p+2)-brane,andF-stringsexpandingintoaDp-brane,forarbitraryvaluesofp–butthesesolutionswillnecessarilybesmearedalongcertaindirections2.OneofthemotivationsbehindthisworkwastotrytogetahandleonthepossiblesupergravitydescriptionofthedielectricbranesofEmparan[21]andMyers[22].ImposingT-dualityinvarianceonthenon-abelianactionrelevanttothedescriptionofmultipleDp–branes,givesrisetoawholehostofworldvolumecouplingswhicharenotpresentintheabeliantheory[22,23].Inparticular,theChern-SimonspieceoftheactionmustincludecouplingstoRamond-Ramond(R-R)potentialsofdegreegreaterthan(p+1),givingrisetopossiblecouplingstohigher-dimensionalbranes.Thepresenceofsuchtermsallowsforthe“dielectriceffect”,thesimplestexampleofwhichisthatofNDp-branesexpandingintoaD(p+2)-braneundertheinfluenceofthe(p+3)-formR-Rpotentialassociatedwiththelatter[22,24].Theresultingdielectricbrane,whichisaminimumoftheworldvolumeenergyfunctional,hasworldvolumeMp+1⊗S2NC,whereS2NCisthenon-commutativeorfuzzytwo-sphere.Althoughthisprocessismostproperlydescribedfromwithinthenon-abelianworldvolumetheoryoftheDp-branes,onecanalternativelyconsiderthedualdescriptionfromwithintheabeliantheoryoftheD(p+2)-brane.InthelargeNlimit,thenon-commutativenatureofthesphereislost,andthesetwodescriptionsagree[22].Sinceitisunlikelythatsuchnon-commutativestructureswillappearwithinasupergravityperspective,anysupergravitysolutionpurportingtodescribethisdielectriceffectshouldbecomparedtosolutionsoftheD(p+2)-braneworldvolumetheory.Fromthisperspective,thestandardcouplingofa(p+1)-formR-RpotentialtotheD(p+2)-brane1Moregenerally,spherical(D−3−2n)-branesinthecoreofnintersectingfluxbranescanbegeneratedbyapplyingntwists[18].2Furthersolutions,generatedfromcertainintersectingconfigurationsofM-branes,wereconsideredbyEmparanin[20]butwewillsaynomoreaboutthesehere.2givesriseto“dissolved”Dp-braneswithintheworldvolumeoftheD(p+2)-brane[25].AnditisthepresenceofthisdissolvedDp-branechargewhichallowsforsphericalsolutionsoftheD(p+2)-branetheory.TheexistenceofsuchsphericalD(p+2)-braneswithdissolvedDp-branechargewasanticipatedbyEmparanin[21].Themainpointofthiswork,however,wastoanalysesphericalD(p+2)-branesolutionswithdissolvedF-stringcharge.TheinterpretationofsuchsolutionswasthatofN“Born-Infeldstrings”[26,27]expandingintoasphereunderthei