Gravity and Strings

13.4 The vacua of d = 5, 6 supergravities with eight supercharges 393
The above solution admits a description as a homogeneous, reductive but non-symmetric
spacetime that can be used to compute its symmetry superalgebra [26].
Before the KG6 solution can be reduced, preserving all its supersymmetries, we need
to identify isometric directions satisfying the truncation condition. One of the directions is
found after performing a u-dependent rotation,
x 3
λ6
= cos
2 u
x 3 ′
λ6
+ sin
2 u
x 4 ′ , x 4
λ6
=−sin
2 u
x 3 ′
λ6
+ cos
2 u
x 4 ′ ,
v = v ′ + λ6
2 x 3 ′ x 4 ′ ,
that leaves the solution in the form
d ˆs 2
= 2du dv ′ + λ2 6
8 (x 2 + y2 )du + λ6x 3 ′
4 ′ dx
ˆB − =−λ6du ∧ (x 1 dx 2 − x 3 ′ dx 4 ′ ).
′ 2 − d x (4) ,
(13.110)
On reducing now in the isometric coordinate w ≡ x 4 ′ , we obtain the KG5 solution
Eq. (10.31) with λ5 =− √ 3λ6.Asimilar rotation in x 1 and x 2 produces the same result. The
isometric direction w = (1/ √ 2)(u + v) can also be used. If we perform the two rotations
and reduce in the direction w, weobtain a d = 5 maximally supersymmetric Gödel-like
solution [420]:
d ˆs 2 = (dt + ω) 2 − d x 2
(4) , ˆV = √ 3 ω, ω = λ6(x 1 dx 2 − x 3 dx 4 ). (13.111)
It is not known whether there are more maximally supersymmetric solutions in N = 1,
d = 5 supergravity, although in principle it is known how to construct all the solutions of
this theory that preserve some supersymmetry [420]. The relations among all the known
vacua are represented in Figure 13.1.
13.4.3 Relation to the N = 2, d = 4 vacua
The dimensional reduction of N = 1, d = 5 supergravity gives N = 2, d = 4 supergravity
coupled to a vector multiplet that can be consistently truncated as shown in Section 11.2.5.
The discussion at the beginning of the previous section applies to this situation: maximal
supersymmetry survives the dimensional reduction only if no matter fields are generated.
This condition cannot be satisfied for the Gödel-like solution Eq. (13.111) and only the
KG5 Eq. (10.31) and the near-horizon limit of the rotating BH Eq. (13.109) and string give
maximally supersymmetric four-dimensional solutions: the KG4 solution Eq. (10.27) with
λ4 = (2/ √ 3)λ5 and the dyonic RB solution in which the rotation parameter sin ξ now plays