Gravity and Strings

19.6 Intersections 563
Maximally supersymmetric vacua of string and M theories. We have mentioned that the
AdS4 × S7 and AdS7 × S4 solutions of d = 11 supergravity Eqs. (19.54) and Eqs. (19.47)
are maximally supersymmetric solutions and, therefore, vacua of the theory. The metrics
of these spaces are products of those of symmetric spaces and the Killing spinors and
symmetry superalgebras can be constructed and studied using the methods of Chapter 13
(see [25]). The superalgebras are extended AdS superalgebras of the kind we studied in
Section 5.4, written in 11-dimensional notation. These are also the superalgebras of the
gauged SUEGRAs one obtains by compactification on S7 and S4 .
These are not the only maximally supersymmetric solutions of d = 11 supergravity since
we can always take the Penrose limit of any solution while preserving (or increasing) the
number of supersymmetries [158, 160, 495, 764]. The Penrose limits of the above two
vacua give the same KG11 solution (first found in [636]) which has an Hpp-wave metric of
the form Eq. (10.18) with
⎧
⎪⎨
ˆG ux 1 ···x 3 = λ, Aij =
⎪⎩
− 1
18 λ2 δij
− 1
72 λ2 δij
i, j = 1, 2, 3,
i, j = 4,...,9.
(19.137)
The symmetry superalgebra of this solution was studied in [392]. It does not seem to be
associated with any known supergravity. The same happens for the other KG solutions.
There are no more maximally supersymmetric vacua in d = 11 [393, 636]. Let us turn
now to the ten-dimensional theories. In the N = 2A, 1 cases the only maximally supersymmetric
vacuum is Minkowski spacetime [393]. In the N = 2B theory there is, as we have
seen, a maximally supersymmetric solution with the metric of AdS5 × S 5 Eq. (19.67). The
superalgebra is that of gauged N = 4, d = 5 SUEGRA with gauge group SO(6), butit is
naturally written in ten-dimensional notation. The Penrose limit gives the maximally supersymmetric
KG10 solution [159] which also has an Hpp-wave metric of the form Eq. (10.18)
with
ˆG (5)
ux 1 ···x 4 = ˆG (5)
ux 5 ···x 8 = λ, Aij =− 1
2 δijλ 2 , i, j, = 1,...,8, (19.138)
in our conventions.
There are no more maximally supersymmetric vacua in d = 10 [393], but there are other
vacua with fewer supersymmetries, which are perhaps more interesting from a phenomenological
point of view. We have mentioned some of them (those that can be obtained by replacing
the spheres by other Einstein spaces). A complete classification is still lacking, but
there is currently intense work in this direction (see, for instance, [423, 582]).
19.6 Intersections
Although now it may seem natural to look for solutions that represent simultaneously
branes of different kinds in equilibrium, the first solutions of that kind were only identified
in [753] among general solutions found years before in [494]. After that, very many
solutions were quickly constructed and the basic rules that govern their existence studied.
Trying to review all these solutions and the various approaches in depth in the space