Gravity and Strings

514 Extended objects
p-brane solutions of the p-brane a-model we are about to study can be built by “dressing”
it with appropriate factors.
18.2.2 The p-brane a-model
As we have already said, this is the simplest model that embodies the main characteristics
of the string effective action (and supergravity actions): gravity coupled to one scalar and
a (p + 1)-form to which the scalar couples non-minimally. It generalizes the a-model for
BHs that we used before, which appears here as the p = 0case, but here we canonically
normalize the (p + 1)-form field strengths. The action is 10
S =
1
16πG (d)
N
d d x
|g| R + 2(∂ϕ) 2 + (−1)p+1
2 · (p + 2)! e−2aϕ F 2
(p+2) , (18.61)
where F(p+2) = dA(p+1) is the field strength of the (p + 1)-form potential A(p+1).
The equations of motion are
Gµν + 2T ϕ µν
1
− 2e−2aϕT A(p+1)
µν = 0,
∇2ϕ + (−1)p+1
4 · (p + 2)! ae−2aϕ F 2
(p+2)
−2aϕ
∇µ e F(p+2) µν1···νp+1
= 0,
= 0,
(18.62)
where T A(p+1) is the (p + 1)-form energy–momentum tensor, given in Eq. (1.122). As
usual, not all of them are independent in general (they can be derived from the Bianchi
identities). On the other hand, the solutions we will find will be defined up to gauge transformations,
including large gauge transformations that change the asymptotic behavior of
the fields so the physics could be inequivalent. The classical equations of motion are insensitive
to these subtleties.
We want to find single-charged-black-p-brane solutions of an analogous nature to the
black Schwarzschild p-brane of the previous section. The method we used to construct
them in Section 11.3.3 cannot be used in the presence of p-forms: we cannot simply add
extra dimensions to a lower-dimensional charged BH metric. For instance, if we took a
four-dimensional RN BH and added several extra dimensions, we would obtain a higherdimensional
metric with vanishing scalar curvature (as in four dimensions) but now the
trace of the Maxwell energy-momentum tensor would not be zero in more than four dimensions.
Another way of expressing the same fact is that, if we dimensionally reduce
the above action to d − p dimensions, one does not simply obtain the above action written
directly in d − p dimensions, but one finds extra fields. In particular, one finds extra
10 This action is equivalent to the original one written in [557] (whose main results we reobtain in a different
fashion here) which was given in the string frame. Here we do not want to assume that the scalar is the
dilaton and thus we prefer to use an Einstein-frame action. The constant a is, then, not the same as in [557]
butweobtain simpler, more-symmetric expressions.