Gravity and Strings

538 The extended objects of string theory
and is, like the M5-brane, regular everywhere. Using the first of Eqs. (18.78), the S5 tension,
Eqs. (19.12), and G (10)
N ,Eq. (19.26), we obtain
hS5 = ℓ2 s , (19.61)
which is independent of g and remains constant in the weak-coupling limit.
In the near-horizon limit ρ =|x4|→0inthe string frame (which is the dual S5-brane
frame) we obtain a metric that is the product of Minkowski 6 + 1 and that of a round S3 ,
d ˆs 2 = dt 2 − d y 2
5 − dz2 − hS5d 2 (3) , z = h 1 2
S5 ln
⎛
⎝ ρ
h 1 ⎞
⎠. (19.62)
2
S5
The S5-brane metric interpolates, then, between Minkowski spacetime at infinity and
the above regular metric at the horizon, which is at an infinite proper distance. There is no
need to continue the metric analytically beyond the horizon and, actually, it would be more
correct to say that, in the limit ρ → 0, one finds another asymptotic region with the above
metric.
19.2.6 The Dp-branes
The generic solution for black Dp-branes N = 2A, B, d = 10 SUEGRA with p < 7is
d ˜ ˆs 2
E
7−p
−
8 2 2 = HDp Wdt − d y p
d ˆs 2 s = H − 1 2
Dp
p+1
8 −1 2 2 2 − HDp W dρ + ρ d(8−p) ,
1
2 2 2 −1 2 2 2 Wdt − d y p − HDp W dρ + ρ d(8−p) ,
e−2 ˆφ −2 ˆφ0 = e H p−3
2
Dp , Ĉ (p+1) ty1 ···y p = αe− ˆφ0
H −1
Dp − 1 ,
HDp = 1 + hDp
ω
, W = 1 +
ρ7−p ρ7−p , ω= hDp[1 − α2 ].
(19.63)
The above solution is not entirely correct for p = 3 since it does not take into account
the self-duality of the 5-form field strength, but the only change that has to be made is in
the 4-form potential: the metric and dilaton fields are correct, as we will see.
In the extreme limit ω = 0,α=±1 the solutions are valid for all p = 0,...,9 (with the
same caveats in the case p = 3) for harmonic functions with poles of the right order. These
are the solutions usually known as Dp-brane solutions in the literature:
d ˜ ˆs 2
E
p−7
8 2 2 = HDp dt − d y p
d ˆs 2 s = H − 1 2
Dp
− H p+1
8
Dp
d x 2
9−p ,
1
2 2 2 2
dt − d y p − HDpd x 9−p ,
e−2 ˆφ −2 ˆφ0 = e H p−3
2
Dp , Ĉ (p+1) ty1 ···y p =±e− ˆφ0
H −1
Dp − 1 ,
HDp = 1 + hDp
|x9−p| 7−p , p < 7, HD7 = 1 + hD7 ln |x2|, HD8 = 1 + hD8|x|.
(19.64)