Gravity and Strings

502 Extended objects
spacetime,
S (p)
NG [X µ (ξ)] =−T(p)V(p)
dξ 0
|gµν ˙X µ ˙X ν | , V(p) = dξ 1 ···dξ p√ |gmn|,
(18.2)
where V(p) is the volume of the internal manifold and T(p)V(p) is the mass of the particle if
the spacetime metric is asymptotically flat in the d − p directions orthogonal (“transverse”)
to the worldvolume. Transverse space, the space in which the wrapped p-brane moves as
a particle, plays a very important role in the definition of mass (as we have just seen) and
charge.
As we have discussed several times, on the one hand, the NG action is highly non-linear
and, on the other hand, it cannot describe massless (tensionless) objects (also known as null
branes). These problems are solved by introducing auxiliary fields. Several possibilities
have been proposed in the literature. For instance, we can introduce a scalar density field v
and write the action
S (p) [X µ
(ξ), v] =
d p+1 ξ 1 2 2
|g|+v T(p) , (18.3)
2v
which is equivalent to the NG action upon elimination of v using its equation of motion.
In this action we can take consistently the tensionless limit to obtain a null brane action
[657]. Although this action is still highly non-linear, it is useful for certain purposes: we
can replace the tension (a constant) by a worldvolume p-form potential 2 c(p) i1···i p whose
equation of motion tells us that the dual of its field strength G(p+1) = (p + 1)∂c(p) is just a
constant. The action is [896]
S (p) [X µ
(ξ), v, c(p)] =
d p+1 ξ 1 ⋆G(p+1) 2
|g|+ , (18.4)
2v
where here ⋆G(p+1) = [1/(p + 1)!]ɛi1···i p+1G(p+1) i1···i p+1 and the equation of motion of c(p)
has the solution
T(p)
= , (18.5)
G(p+1) i1···i p
v ɛi1···i p
where T(p) arises as just an integration constant. On substituting this solution into the action,
we recover exactly the action Eq. (18.3) and then T(p) is identified as the p-brane tension.
One can also consider solutions in which ⋆ G(p+1)/v is only piecewise constant, the discontinuities
being associated with brane intersections (see, for instance, [902] for an application
involving string and D-string junctions). On the other hand, these actions are also suitable
for supersymmetric objects and can be made κ-symmetric [135, 141].
2 This is similar to the replacement of the mass parameter by the RR 9-form potential in Romans’ massive
supergravity, Section 16.2.