Gravity and Strings

570 The extended objects of string theory
find the solutions that would have BIons and their generalizations as sources (this was the
approach of [461]). Partially localized solutions and some special fully localized solutions
have been found in [48, 379, 503, 561, 598, 658, 874, 967], but in [688, 760] it was argued,
using AdS/CFT-correspondence arguments, that fully localized solutions might not exist in
general, and these arguments seem to be confirmed by the results in [461]. If the string in
the F1 ⊥ Dp(0) intersection can be seen as “hair” on the Dp-brane, then the absence of a
fully localized solution for that configuration can be seen as a sort of “no-hair theorem” for
Dp-brane solutions.
Nevertheless, an Ansatz for fully localized intersections of this and other kinds has been
given in [794]. The solutions depend on an unknown function that satisfies a highly nonlinear
differential equation, but it is not known whether this equation has solutions with the
appropriate boundary conditions.
19.6.4 The (a1–a2) model for p1- and p2-branes and black intersecting branes
This is a straightforward generalization of the p-brane a-model that includes (p1 + 1)- and
(p2 + 1)-form potentials, coupled to a scalar with parameters a1 and a2:
S =
1
16πG (d)
N
d d x √ |g|
R + 2(∂ϕ) 2 +
i=1,2
(−1) pi +1
2 · (pi + 2)! e−2ai
ϕ 2 F(pi +2) , (19.146)
and it is just a convenient simplification of the higher-dimensional supergravity actions we
are dealing with. Notice, in particular, the absence of Chern–Simons terms: the solutions
we will obtain will be solutions of the full supergravity action only when those terms do
not contribute to the equations of motion. This condition will be fulfilled in most cases.
The equations of motion corresponding to this action are
Gµν + 2T ϕ µν
+
∇ 2 ϕ +
i=1,2
i=1,2
(−1) pi +1
2 · (pi + 1)! e−2ai ϕ A(pi +1)
Tµν = 0,
(−1) pi +1
4 · (pi + 2)! aie −2ai ϕ 2
F(pi +2) = 0,
∇µ(e −2ai ϕ F(pi +2) µν1...νp i +1 ) = 0, i = 1, 2.
(19.147)
The harmonic-superposition rule and our experience indicate that an adequate Ansatz for
a p1- and a p2-brane intersecting over r spatial directions may depend on three functions:
Hi, i = 1, 2, which will be independent harmonic functions in the extreme limit and are
associated with the potentials, and the “Schwarzschild factor” W , which becomes 1 in the
extreme limit. The Ansatz must be such that, when a given Hi is set to 1 to recover a solution
for a single p j-brane, i = j. All these conditions are fulfilled by the metric Ansatz
ds 2 = H 2z1
1
H 2z2
2
− H −2y1
1
Wdt 2 − d y 2
r
− H 2z1
1
−2y2 H2 d y 2
(p1−r)
H −2y2
2
2 d y q + W −1 dρ 2 + ρ 2 d 2 (δ−2)
− H −2y1
1
H 2z2
2
d y 2
(p2−r)
. (19.148)