Novel Aspects in p-Brane Theories Weyl-Invariant Light-Like Branes

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

NovelAspectsinp-BraneTheories:

Weyl-InvariantLight-LikeBranes

EduardoGuendelmanandAlexanderKaganovich

DepartmentofPhysics,Ben-GurionUniversity,Beer-Sheva,Israel

email:guendel@bgumail.bgu.ac.il,alexk@bgumail.bgu.ac.il

arXiv:hep-th/0409208v4 28 Jul 2005EmilNissimovandSvetlanaPachevaInstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,So a,Bulgariaemail:nissimov@inrne.bas.bg,svetlana@inrne.bas.bgAbstractWeconsideranovelclassofWeyl-conformallyinvariantp-branetheorieswhichdescribein-trinsicallylight-likebranesforanyoddworld-volumedimension,hencetheacronymWILL-branes(Weyl-InvariantLight-Likebranes).WediscussinsomedetailthepropertiesofWILL-branedy-namicswhichsigni cantlydi ersfromordinaryNambu-Gotobranedynamics.WeprovideexplicitsolutionsofWILL-membrane(i.e.,p=2)equationsofmotioninarbitraryD=4sphericallysym-metricstaticgravitationalbackgrounds,aswellasinproductspacesofinterestinKaluza-Kleincontext.Inthe rstcasewe ndthattheWILL-membranematerializestheeventhorizonofthecorrespondingblackholesolutions,thusprovidinganexplicitdynamicalrealizationofthemembraneparadigminblackholephysics.Inthesecond“Kaluza-Klein”contextwe ndsolu-tionsdescribingWILL-braneswrappedaroundtheinternal(compact)dimensionsandmovingasawholewiththespeedoflightinthenon-compact(space-time)dimensions.Keywords:Weyl-conformalinvariantp-braneactions,light-likep-branes,non-Riemannianvolumeforms,variablestring/branetension,Kaluza-Klein,eventhorizons,membraneparadigm.

1Introduction

TheideaofreplacingthestandardRiemannianintegrationmeasure(Riemannianvolume-form)withanalternativenon-Riemannianvolume-formor,moregenerally,employingonequalfootingbothRieman-nianandnon-Riemannianvolume-formstoconstructnewclassesofmodelsinvolvinggravity,calledtwo-measuretheories,hasbeenproposedfewyearsago[1]andsincethenitisasubjectofactiveresearchanddevelopments[2](forrelatedideas,see[3]).

Two-measuretheoriesaddressvariousbasicproblemsincosmologyandparticlephysics,andprovideplausiblesolutionsforabroadarrayofissues,suchas:scaleinvarianceanditsdynamicalbreakdown;spontaneousgenerationofdimensionfullfundamentalscales;thecosmologicalconstantproblem;theproblemoffermionicfamilies;applicationsinmodernbrane-worldscenarios.Foradetaileddiscussionwerefertotheseriesofpapers[1,2].

Subsequently,theideaofemployinganalternativenon-Riemannianintegrationmeasurewasappliedsystematicallytostring,p-braneandDp-branemodels[4](forabackgroundonstringandbranetheories,seerefs.[5]).Themainfeatureofthesenewclassesofmodi edstring/branetheoriesistheappearanceofthepertinentstring/branetensionasanadditionaldynamicaldegreeoffreedombeyondtheusualstring/branephysicaldegreesoffreedom,insteadofbeingintroducedadhocasadimensionfullscale.Thedynamicalstring/branetensionacquiresthephysicalmeaningofaworld-sheetelectric eldstrength(inthestringcase)orworld-volumep+1-form eldstrength(inthep-branecase)andobeysMaxwell(Yang-Mills)equationsofmotionortheirhigher-rankantisymmetrictensorgauge

1

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

eldanalogues,respectively.Asaresultofthelatterpropertythemodi ed-measurestringmodelwithdynamicaltensionyieldsasimpleclassicalmechanismof“color”chargecon nement.

Inthenextsectionweproceedtoourmaintaskwhichisthestudyofanovelclass( rstproposedinourprecedingwork[6])ofp-branetheorieswhichareWeyl-conformalinvariantforanypandwhichdescribeintrinsicallylight-likebranesforanyodd(p+1).Thus,theirdynamicssigni cantlydi ersbothfromthestandardNambu-Goto(orDirac-Born-Infeld)branesaswellasfromtheirmodi edversionswithdynamicalstring/branetensions[4]mentionedabove.

2

2.1Weyl-Invariantp-BraneTheoriesStandardNambu-GotoBranes

HereγabistheordinaryRiemannianmetriconthep+1-dimensionalbraneworld-volumewithγ≡det||γab||.Theworld-volumeindicesa,b=0,1,...,p;GµνdenotestheRiemannianmetricintheembeddingspace-timewithspace-timeindicesµ,ν=0,1,...,D 1.Tisthegivenadhocbranetension;theconstantΛcanbeabsorbedbyrescalingT(seebelowEq.(7)).Theequationsofmotionw.r.t.γabandXµread:

1(p 1)=0,(2)Tab≡ aXµ bXν 2

√ γγab aXν bXλΓµ a(3)νλ=0,

where:

Γµ

νλ=1Letus rstbrie yrecallthestandardPolyakov-typeformulationofthebosonicp-braneaction: TabµνS= γγ aX bXGµν(X) Λ(p 1).(1)

p+1γabγ cX dXcdµν Gµν=0.(6)

Furthermore,using(5)thePolyakov-typebraneaction(1)becomeson-shellequivalenttotheNambu-Goto-typebraneaction:

S= TΛ p 1 det|| aXµ bXνGµν||.(7)

2.2Weyl-InvariantBranes:ActionandEquationsofMotion

withFab(A)= aAb bAa,and:

Φ( )≡1Inref.[6]weproposedthefollowingnovelp-braneactions: 1p+1S= dσΦ( )Fab(A)Fcd(A)γacγbd(8)

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

Newnon-Riemannianintegrationmeasuredensity(volume-form)Φ( )(9)insteadoftheusual√

γ)in(8).

Theaction(8)ismanifestlyWeyl-conformalinvariantforanyp;hereWeyl-conformalsymmetryisgivenbyWeylrescalingofγabsupplementedwithaspecialdi eomorphisminthetargetspaceofauxiliary - elds:

γab →′γab=ργab, ′i → = ( )withdet i′i′i

2(1 p)√χ.

Theaction(8)containsanadditionalworld-volumegauge eldAaina“square-root”Maxwell(Yang-Mills)Lagrangian1;thelattercanbestraightforwardlygeneralizedtothenon-Abelian case:

2γcd( cX dX)

FFγγ≡

(13)FacγcdFdb=0,( aX bX)+2FFγγ

whichupontakingthetraceimplyM=0inEq.(11).

Furtherweobtainthefollowingequationsofmotionw.r.t.world-volumegauge eldAaandw.r.t.braneembeddingcoordinatesXµ,respectively:

Fcdγacγbd

bΦ( )=0,(14)FFγγ

(15) aΦ( )γab bXµ+Φ( )γab aXν bXλΓµ

νλ=0,

whereΓµ

νλisthesameasin(4).

2.3Light-LikeBranes

Now,letusconsidertheγab-equationsofmotion(13).SinceFabisananti-symmetric(p+1)×(p+1)matrix,itisthereforenotinvertibleinanyodd(p+1),i.e.Fabhasatleastonezero-eigenvaluevectorVa(FabVb=0).Thus,foranyodd(p+1)theinducedmetric:

gab≡( aX bX)≡ aXµ bXνGµν(16)

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

ontheworld-volumeoftheWeyl-invariantbrane(8)issingularasopposedtotheordinaryNambu-GotobranewheretheinducedmetricisproportionaltotheintrinsicRiemannianworld-volumemetric(cf.Eq.(5)).Inotherwords:

( aX bX)Vb=0,i.e.( VX VX)=0,( ⊥X VX)=0,(17)

where V≡Va aand ⊥arederivatesalongthetangentvectorsinthecomplementofthetangentvector eldVa.

Theconstraints(17)implythefollowingimportantconclusion:everypointonthe( xed-time)world-surfaceoftheWeyl-invariantp-brane(8)(forodd(p+1))movesinorthogonaldirectionw.r.t.itselfwiththespeedoflightinatime-evolutionalongthezero-eigenvaluevector- eldVaoftheworld-volumeelectromagnetic eld-strengthFab.Therefore,wewillcall(8)(forodd(p+1))bytheacronymWILL-brane(Weyl-InvariantLight-Like-brane)model.

2.4DualFormulationofWILL-Branes

TheAa-equationsofmotion(14)canbesolvedintermsof(p 2)-formgaugepotentialsΛa1...ap 2dualw.r.t.Aa.Therespective eld-strengthsarerelatedasfollows:

Fab(A)= 1 γεabc1...cp 1

√

(p 1)2γa1b1...γap 1bp 1Fa1...ap 1(Λ)Fb1...bp 1(Λ).(20)

Now,theBiancchiidentitiesforAaturnintodynamicalequationsofmotionforthedual(p 2)-formgaugepotentialsΛa1...ap 2:

√ a(21)γabγa1b1...γap 2bp 2Fbb1...bp 2(Λ)γcd( cX dX)=0χ(γ,Λ)

Allequationsofmotion(13),(15)and(21)canbeequivalentlyderivedfromthefollowingdualWILL-braneaction:1Sdual= γγab aXµ bXνGµν(22)withχ(γ,Λ)givenin(20)above.

3TheWILL-Membrane

1 γγab( aX bX),

χ(γ,u)≡

√TheWILL-membranedualaction(particularcaseof(22)forp=2)reads:Sdual= (23)2χ(γ,u)

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

Theequationsofmotionw.r.t.γab,u(orAa),andXµreadaccordingly:

1 γab=0,( aX bX)+γef eu fu

a

√ aχ(γ,u) √χ(γ,u)γcd( cX dX) =0,(26)(27)(28)

The rstequationaboveshowsthattheinducedmetricgab≡( aX bX)haszero-modeeigenvectorVa=γab bu.

Theinvarianceunderworld-volumereparametrizationsallowstointroducethefollowingstandard(synchronous)gauge- xingconditions:

γ0i=0(i=1,2),γ00= 1.(29) γγab aXν bXλΓµνλ=0.

Inspiteofthehighnon-linearityofEq.(27)forthedual“gaugepotential”u,wecaneasily ndsolutionsbyusingthefollowingansatz:

u(τ,σ1,σ2)=T0τ,2(30)

whereT0isanarbitraryintegrationconstantwiththedimensionofmembranetension.Inparticular:

χ≡

(33)2

NotethatEqs.(33)lookexactlyliketheclassical(Virasoro)constraintsforanEuclideanstringtheorywithworld-sheetparameters(σ1,σ2).

Thegaugechoicefor(29)togetherwiththeansatz(30),aswellastakingintoaccount(32),bringthetheequationsofmotionw.r.t.utotheform:

√ 0γijγkl( kX lX)=0,

γ(2) 0 +1

γ(2) i

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

4

4.1WILL-MemraneSolutionsinNon-TrivialGravitationalBack-groundsExample:WILL-MembraneinSpherically-SymmetricStaticBackgroundsLetusconsiderageneralspherically-symmetricstaticgravitationalbackgroundinD=4embeddingspace-time:

(ds)2= A(r)(dt)2+B(r)(dr)2+r2[(dθ)2+sin2(θ)(dφ)2].(37)

Speci callywehave:

A(r)=B 1(r)=1 2GM

r+Q2

τ

τrr=±A(r), =0whichuponcombiningwith(43)gives:

r=r0≡const,whereA(r0)=0.(44)

TheX0-equationofmotion(Eq.(35)forµ=0)impliesfortheintrinsicWILL-membranemetric:

10,(45) γij =c0e τ/r00sin2(σ1)

wherec0isanarbitraryintegrationconstant.

From(44)weconcludethattheWILL-membranewithsphericaltopology(andwithexponentiallyblowing-up/de atingradiusw.r.t.internalmetric)“sits”on(materializes)theeventhorizonofthepertinentblackholeinD=4embeddingspace-time.

4.2Example:WILL-membraneinProduct-SpaceBackgrounds

HereweconsiderWILL-membranemovinginageneralproduct-spaceD=(d+2)-dimensionalgravi-tationalbackgroundMd×Σ2withcoordinates(xµ,ym)(µ=0,1,...,d 1,m=1,2)andRiemannianmetric(ds)2=f(y)gµν(x)dxµdxν+gmn(y)dymdyn.

WeassumethattheWILL-branewrapsaroundthe“internal”spaceΣ2andusethefollowingansatz(recallτ≡σ0):

Xµ=Xµ(τ),Ym=σm,γmn=a(τ)gmn(σ1,σ2)(46)

Thentheequationsofmotionandconstraints(32)–(36)reduceto:

τXµ τXνgµν(X)=0,1

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

wherea(τ)istheconformalfactorofthespace-likepartoftheinternalmembranemetric(lastEq.(46)).Eqs.(47)areofthesameformastheequationsofmotionforamasslesspoint-particlewithaworld-line“einbein”e=a 1movinginMd.Inotherwords,thesimplesolutionabovedescribesamembranelivingintheextra“internal”dimensionsandmovingasawholewiththespeedoflightin“ordinary”space-time.

NoticethatalthoughtheWILL-braneiswrappingtheextradimensionsinatopologicallynon-trivialway(cf.secondEq.(46)),itsmodesremainmasslessfromtheprojectedd-dimensionalspace-timepointofview.Thisisahighlynon-trivialresultsincewehavehereparticles(membranemodes),whichaquireinthiswaynon-zeroquantumnumbers,whileatthesametimeremaingmassless.Incontrast,oneshouldrecallthatinordinaryKaluza-Kleintheory(forareview,see[11]),non-trivialdependenceontheextradimensionsispossibleforpointparticlesorevenstandardstringsandbranesonlyataveryhighenergycost(eitherbymomentummodesorwindingmodes),whichimpliesaveryhighmassfromtheprojectedD=4space-timepointofview.

4.3Example:WILL-MembraneinaPP-WaveBackground

Asa nalnon-trivialexampleletusconsiderWILL-membranedynamicsinexternalplane-polarizedgravitationalwave(pp-wave)background:

(ds)2= dx+dx F(x+,xI)(dx+)2+dxIdxI,

andemployin(32)–(36)thefollowingnaturalansatzforXµ(hereσ0≡τ;I=1,...,D 2):

X =τ,X+=X+(τ,σ1,σ2),XI=XI(σ1,σ2).(49)

Thenon-zeroa neconnectionsymbolsforthepp-wavemetric(48)are:Γ ++= +F,Γ+I= IF,

1ΓI++=(48)

2γijγ kX lX=0,klII i

G R

4Fµν(A)Fκλ(A)GµκGνλ+SWILL brane,

(52)whereFµν(A)= µAν νAµ,and:

1SWILL brane= d3σΦ( )FabFcdγacγbd qd3σεabcAµ aXµFbc.(53)

NotetheappearanceofanaturalWeyl-conformalinvariantcouplingoftheWILL-branetotheexternalspace-timeelectromagnetic eldAµ–thelastChern-Simmons-liketermin(53).ThelatterisaspecialcaseofaclassofChern-Simmons-likecouplingsofextendedobjectstoexternalelectromagnetic eldsproposedinref.[10].

TheEinstein-Maxwellequationsofmotionareofthestandardform:

Rµν 1

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

ν √

(56)

(brane)Tµν

FortheWILL-membranesubsystemwecanuseinsteadoftheaction(53)itsdualone(similartothesimplercaseEq.(8)versusEq.(23)):

dualSWILL brane= FρκFσλGρσGκλ,4 (4) x X(σ)δ3Φ( )γab aXκ bXλ,≡ GµκGνλdσ G µ3(4)x X(σ)εabcFbc aXµ.j≡qdσδ(57)(58)1 γγab( aX bX),

isgivenby:(59)wherethevariablebranetensionχ≡

χ(γ,u,A)≡ Φ( ) γ

2χ(γ,u,A)√

2γcd( cX dX) ( au qAa)( bu qAb)

γγab( bu qAb)

√ γγab bXµ+χ(γ,u,A)

r,forr<r0≡rhorizon=2GM1.(66)

Reissner-Norstr¨omblackholeoutsidethehorizon:

A(r)≡A+(r)=1 2GM2

r2,forr>r0≡rhorizon,(67)

44whereQ2=8πq2rhorizon≡128πq2G4M1;

FortheMaxwellsubsystemwehaveA1=...=AD 1=0everywhereand:

8

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

Coulomb eldoutsidehorizon:

A0=

Noelectric eldinsidehorizon:√A0=√r,forr≥r0≡rhorizon.(68)

A+ rr=rhorizon

matchingcondition(71)correspondstothestaticallysolderingconditionsinthephenomenologicaltheoryoflight-likethinshelldynamicsingeneralrelativity[12].2The

9

We consider a novel class of Weyl-conformally invariant p-brane theories which describe intrinsically light-like branes for any odd world-volume dimension, hence the acronym WILL-branes (Weyl-Invariant Light-Like branes). We discuss in some detail the prop

Whenputinagravitationalblackholebackground,theWILL-membrane(p=2)sitson(“ma-terializes”)theeventhorizon.

Whenmovinginbackgroundproduct-spaces(“Kaluza-Klein”context)theWILL-membranede-scribesmasslessmodes,eventhoughthemembraneiswrappingtheextradimensionsandthere-foreaquiringnon-trivialKaluza-Kleincharges.

ThecoupledEinstein-Maxwell-WILL-membranesystem(52)possessesself-consistentsolutionwheretheWILL-membraneservesasamaterialandelectricallychargedsourceforgravityandelectromagnetism,andit“sits”on(materializes)thecommoneventhorizonforaSchwarzschild(intheinterior)andReissner-Nordstr¨om(intheexterior)blackholes.Thusourmodel(52)providesanexplicitdynamicalrealizationofthesocalled“membraneparadigm”inthephysicsofblackholes[13].

TheWILL-branescouldbegoodrepresentationsforthestring-likeobjectsintroducedby’tHooftinref.[14]todescribegravitationalinteractionsassociatedwithblackholeformationandevapo-ration,sinceasshownabovetheWILL-braneslocatethemselvesautomaticallyinthehorizonsand,therefore,theycouldrepresentdegreesoffreedomassociatedparticularlywithhorizons.ThenovelclassofWeyl-conformalinvariantp-branesdiscussedabovesuggestsvariousphysicallyinterestingdirectionsforfurtherstudy:quantization(Weyl-conformalanomalyandcriticaldimen-sions);supersymmetricgeneralization;possiblerelevancefortheopenstringdynamics(similartotheroleplayedbyDirichlet-(Dp-)branes);WILL-branedynamicsinmorecomplicatedgravitationalblackholebackgrounds(e.g.,Kerr-Newman).

Acknowledgements.Twoofus(E.G.andE.N.)aresincerelygratefultoPlamenFizievandtheorganizersoftheSecondWorkshoponGravity,AstrophysicsandStringsforthekindinvitationtopresenttheretheaboveresults.E.N.andS.P.arealsothankfulforhospitalityandsupporttotheorganizersofthe2ndAnnualMeetingoftheEuropeanRTNEUCLID,Sozopol(Bulgaria),2004.Oneofus(E.G.)thankstheInstituteforNuclearResearchandNuclearEnergy(So a)andTriesteUniversityforhospitality.HealsoacknowledgesusefulconversationswithGerard‘tHooft,EuroSpallucciandStefanoAnsoldi.

E.N.andS.P.arepartiallysupportedbyBulgarianNSFgrantsF-904/99andF-1412/04.Finally,allofusacknowledgesupportofourcollaborationthroughtheexchangeagreementbetweentheBen-GurionUnivesityoftheNegev(Beer-Sheva,Israel)andtheBulgarianAcademyofSciences.

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