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)