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