Gravity and Strings

512 Extended objects
Their mass is proportional to k0, that is, to the radius of the compact dimension, and
they couple to the winding vector.
Kaluza–Klein (KK) branes They are described by gauged σ -models and couple to a positive
power of the volume of the compact space associated with the gauging. In
the decompactification limit the tension becomes infinite. Thus, these objects exist
only when there are compact dimensions. The archetype of these objects is the KK
monopole, which is described by a U(1)-gauged σ -model and coupled both to the
dilaton and to the KK scalar k as follows:
S (KK)
NG [X µ (ξ)] =−T(p)
d p+1 ξ e −2φ k 2 |ij|. (18.59)
18.2 General p-brane solutions
In this section we are going to construct the simplest classical solutions that describe uncharged
(Schwarzschild) and charged p-branes in a d-dimensional spacetime. They are
solutions of a generalization proposed in [557] of the a-model discussed in Section 12.1
in which we will replace the 1-form potential adequate for BHs (which, in a sense, are
point-like objects, 0-branes) by a (p + 1)-form potential that is adequate for p-branes. This
p-brane a-model is, for specific asandps, a simplified version of most of the supergravity
actions we are dealing with and the solutions we obtain will be supergravity (superstring)
solutions.
We are also going to see how, according to [337], it is possible to find a p-brane worldvolume
source for the “extreme” ones, generalizing the results we obtained for ERN and
KK BHs (Sections 8.4 and 11.2.3, respectively), and for the fundamental string solution
(Section 15.3), which will be particular cases of our general solution.
18.2.1 Schwarzschild black p-branes
The Schwarzschild solution describes the gravitational field of a massive, point-like object
in vacuum. Is there an analogous solution of the d-dimensional Einstein–Hilbert action
describing the gravitational field of a massive p-brane in vacuum?
Let us consider the simplest p-brane configuration (a “flat” p-brane with trivial topology).
This configuration should give rise to an asymptotically flat spacetime characterized
by p + 1 translational isometries associated with the p-brane worldvolume. The requirement
that the solution be asymptotically flat is essential, if we want to describe an isolated
p-brane. However, we cannot impose asymptotic flatness in the direction along which the
isometries act, but only in the d − (p + 1) spacelike transverse dimensions. In what follows,
we will use the concept of asymptotic flatness in this restricted sense.
Solutions with these properties, Schwarzschild p-branes, were constructed using KK
techniques in Section 11.3.3, and the metric is given by Eqs. (11.148), although here we are
going to ignore all the hats. The first thing we can do is compute the p-brane tension9 using
9 The restricted asymptotic flatness of the metric does not allow us to define a finite d-dimensional mass.