Gravity and Strings

534 The extended objects of string theory
M2 M5, D3 Dp
Fig. 19.1. Penrose diagrams of different (extreme) string/M-brane solutions: the M2-brane
which has a timelike singularity covered by a horizon which does not allow it to be seen
from the asymptotic region covered by the isotropic coordinates (shaded), the M5- and
D3-brane that are regular everywhere and have two asymptotic regions separated by the
horizon, and the Dp-branes with p = 3 which have singular horizons. In all cases the angular
coordinates of the transverse spheres and the spacelike worldvolume coordinates have
been ignored.
In fact, on taking the near-horizon limit ρ =|x8|→0(which consists in the deletion
of the constant 1 in HM2) inspherical coordinates, we obtain a solution whose metric is
the direct product of those of AdS4 and S 7 with radii R4 and 2R4 (after a rescaling of the
worldvolume coordinates),
d ˆs 2 = R 2 4 d2 (4) − (2R4) 2 d 2 (7) ,
3 r
Ĉ ty1y2 = , R4 =
R4
h 1 2
M2
2 ,
where we are using the following form of the metric of the AdSn space with radius Rn:
R 2 n d2 (n) ≡
r
2dt 2 2
− d y n−2 −
Rn
(19.47)
2 Rn
dr
r
2 . (19.48)
The dual 7-form field strength is given by the S 7 volume form ˜ ˆG0 = 6(2R4) 6 ω(7).
All the extreme string/M-theory solutions that we are going to study preserve half of the
supersymmetries, but this near-horizon limit preserves all the supersymmetry and can be
considered a vacuum of M theory. The M2-brane can then be seen as a soliton interpolating
between two maximally supersymmetric vacua, Minkowski at infinity and AdS4 × S 7 at
the horizon [452]. Since the S 7 is a compact space, this vacuum can be seen to induce
spontaneous compactification to d = 4 [406]. The compactification of D = 11 supergravity
on S 7 gives rise to a gauged N = 8, d = 4 SUEGRA with gauge group SO(8) (the isometry
group of the compact space) and an AdS4 vacuum [343] (see also [342] and references
therein).