Gravity and Strings

19.3 The masses and charges of the p-brane solutions 547
the simplest generalizations involve either a modification of the worldvolume geometry,
which we have taken so far to be flat (p + 1)-dimensional spacetime, or a modification of
the geometry of the transverse space.
Since the supergravity equations of motion are local, it is clear that global modifications
of the worldvolume geometry such as imposing periodicity conditions on the n coordinates
will give new solutions describing the p-branes wrapped on a rectangular n-torus. Another
interesting possibility is to replace the flat worldvolume metric ηij by gij(t, yp) and the
transverse metric δmn by hmn(x):
ds 2 = H α (x)gij(x)dy i dy j − H β (x)hmn(x)dx m dx n . (19.86)
The equations of motion (with a minor modification of the (p + 1)-form potential Ansatz)
are still solved if gij and hmn are Ricci-flat and H is harmonic in the new transverse space
(see [179, 391, 418, 602]). This kind of solution can also be used to describe the wrapping
of branes on cycles of more complicated spaces and also intersections.
A possible choice of transverse metric hmn consists in the replacement of the round
S ( ˜p+2) metric d2 ( ˜p+2) by the metric of an Einstein space with the same curvature as the
round S ( ˜p+2) [341] (G/Hcoset spaces in the cases studied in [220]) in the flat metric written
in spherical coordinates dρ2 + ρ2d 2 ( ˜p+2) .Inthe M2, M5, and D3 cases, the near-horizon
limits are now the product of an AdS space and the Einstein space. This kind of solution
induces spontaneous compactification in the Einstein space and the theory is described by
agauged supergravity with a gauge group related to the isometry group of the Einstein
space (for a review, see e.g. [401]). Furthermore, since they do not preserve all the supersymmetries
(unlike the AdSn × Sm solutions), the supergravities will also have fewer
supercharges.
It is also possible to look directly for metrics that can be understood as near-horizon
limits (see, for instance, [389]).
19.3 The masses and charges of the p-brane solutions
In Section 19.1.1 we found the masses and charges of the extended objects of string/M theory
using duality arguments. We matched these with the coefficients of the harmonic functions
of the extreme solutions by studying the coupling of supergravity to the sources. This
procedure is, however, difficult or impossible to follow for generic solutions that represent
complex systems of extended objects or are “black” (non-extreme). In those cases we need
a procedure by which to calculate the masses and charges of the objects described by the
solutions using only the solution and the normalizations of the fields that appear in the
action. This is the subject of this section. We will follow [40, 677].
19.3.1 Masses
We need to collect here several pieces of data that are scattered over several chapters.
1. The closed-superstring worldvolume action is given in Eq. (14.1.1). T = 1/(2πα ′ )
is the string tension, α ′ = ℓ 2 s is the Regge slope, and ℓs is the string length. With
that normalization of the worldvolume fields, the low-energy effective action of the