Gravity and Strings

general families of solutions.
18.1 Generalities 501
18.1 Generalities
The basic extended objects are known as p-branes, objects with p spatial dimensions that
sweep out (p + 1)-dimensional worldvolumes as they evolve in time in a d-dimensional
ambient (or target) spacetime. Strings, which we have already studied, are the simplest examples
of p-branes (p = 1), but there are many other examples that differ by their worldvolume
dimensions, their worldvolume fields, their couplings to background fields, and
other characteristics such as being associated with a compact dimension (such as the KK
monopole and more general KKp-branes [666]). We will discuss these variants of the basic
p-brane in order of increasing complexity and in the next chapter we will see which of them
occur in string/M theory.
The dynamics of all these objects is governed by their (p + 1)-dimensional worldvolume
actions and in what follows we are going to study them and use them as tools
to classify the objects. In this chapter we consider only bosonic actions, but in the next
chapter we will briefly discuss the κ-symmetric addition of fermions which is required by
the coupling to supergravity.
18.1.1 Worldvolume actions
The basic dynamical variables of a p-brane are the spacetime coordinates of the object
X µ (ξ) , µ = 0,...,d − 1(ξi , i = 0,...,p are the worldvolume coordinates), which give
the embedding of the worldvolume in the d-dimensional target spacetime and are worldvolume
scalar fields. Some p-branes may have additional worldvolume fields (scalars, vectors
(such as the BI vector of Dp-branes), and tensors), whose physical meanings will be discussed
in Section 19.6.
The simplest worldvolume and spacetime reparametrization-invariant action for a pbrane
is the generalization of the Nambu–Goto action Eq. (14.1) (in the same notation),
S (p)
NG [X µ
(ξ)] =−T(p)
d p+1 ξ |gij| , (18.1)
which is proportional to the volume swept out by the p-brane. The proportionality constant
T(p) is the p-brane tension and has natural dimensions of L −(p+1) or, equivalently, of
mass per unit of spatial p-dimensional volume. In fact, let us consider a spacetime that is
the direct product of a non-compact (d − p)-dimensional spacetime and a p-dimensional
compact space so the metric of the original spacetime ˆg ˆµˆν = diag(gµν, gmn) and a configuration
in which the p-brane 1 wraps the p-dimensional space so ˆX m = ξ m m = 1,...,p and
the remaining embedding coordinates are independent of the ˆX m = ξ m . The Nambu–Goto
(NG) action becomes the action of a massive particle moving in the (d − p)-dimensional
1 This is the straightforward generalization of a string winding-mode configuration.