Gravity and Strings

13.4 The vacua of d = 5, 6 supergravities with eight supercharges 391
where d2 (3) is the metric of an AdS3 spacetime of unit radius and ωAdS3 is its volume
3-form. Remarkably, the electric and magnetic components of the 2-form potential can be
written in terms of the vertical Maurer–Cartan 1-forms in AdS3 and S3 ,which are SO(2, 1)
and SO(3) Yang–Mills solutions (the ca are constants):
The KG6 solution can be written as follows:
ˆB − a = c(a)ɛ (a)bc ϑ i fibc. (13.101)
d ˆs 2
= 2du dv + λ2 6
8
2 x (4) du
− d x 2 (4) ,
ˆB − =−λ6du ∧ (x 1 dx 2 − x 3 dx 4 ).
The potential can be also written as a Yang–Mills field:
(13.102)
ˆB − i ∼ ϑ j f j iu . (13.103)
The calculation of the Killing spinors and superalgebras can be done following the general
method. It is, however, more interesting to see how the dimensional reduction of these
solutions gives all the maximally supersymmetric vacua of N = 1, d = 5 supergravity.
13.4.2 N = 1, d = 5 supergravity
The dimensional reduction of N = (1, 0), d = 6 supergravity gives N = 1, d = 5 supergravity
(Section 11.2.5) coupled to a vector multiplet. We are interested in reducing maximally
supersymmetric six-dimensional solutions preserving all their unbroken supersymmetries.
When is this possible?
Let us consider the component of the gravitino in the compact direction w (which gives
rise to a five-dimensional spin − 1
“gaugino”) and its supersymmetry variation,
2
δ ˆɛ
ˆψ + w ∼ (∂w + M)ˆɛ + , (13.104)
where M is a combination of gamma matrices (different from the unit matrix). This equation
has to vanish identically for a w-independent Killing spinor ˆκ + in order to have fivedimensional
unbroken supersymmetry, which implies that M has to vanish identically. If
we reduce a maximally supersymmetric six-dimensional solution and M does not vanish
identically, then, since the above equation vanishes for some Killing vectors, they must
depend on w and five-dimensional supersymmetry is broken. The amount of supersymmetry
broken depends on the rank of M, which tells us how many non-trivial solutions of
M ˆκ + = 0exist, and how many six-dimensional Killing spinors are independent of w.
Up to this point, this discussion carries over to any other case without modification. However,
there are two different possibilities concerning the vanishing of M:inthe present case,
we obtain a five-dimensional reducible theory of which we can always truncate consistently
the matter multiplet. Matter fields in a given multiplet are identified in the supersymmetry