Gravity and Strings

13.3 N = 1, 2, d = 4 vacuum supersymmetry algebras 389
The KG4 superalgebra The solution is given in Eq. (10.27), but we have to take into ac-
count the factor of two due to the normalizations being different, which changes Fu1 = 1
2 λ.
The metric is that of a Cahen–Wallach symmetric space, whose coset construction is re-
viewed in Section 10.1.1, with Aij = 1
8 λ2 δij. There is one subtlety: in order to use the coset
method, we are forced to use mostly plus signature, in which the Killing-spinor equation
takes the form
d − 1
4ωa bγa b + i
4 σ 2 Fγae a
κ = 0. (13.93)
As in the RB case, not just the metric, but also the Maxwell field, can be constructed
using the vertical Maurer–Cartan 1-forms. In fact
A = 1
2 λϑ 1 . (13.94)
An electric–magnetic-duality rotation is equivalent, in this solution, to a rotation in the
wavefront plane. The new Maxwell field would be
A = 1
2 λϑ 2 . (13.95)
On substituting the background fields into the above Killing-spinor equation, we find that
it takes the form (d − V )κ = 0, with the spinorial representation of the Heisenberg algebra:
Ɣs(Pa) =− i
4 σ 2 γ u γ 1 γa, Ɣs(Mi) =− 1
8 λ2 γ u γ i , Ɣs(V ) = 0. (13.96)
The Killing spinors can be constructed immediately and the action of the Heisenbergalgebra
generators on them is trivial to find, using Eq. (13.40). However, note the following.
1. H(6) is not the whole isometry algebra of the KG4 metric: SO(2) rotations in the
wavefront plane leave the metric invariant and the full isometry group is their semidirect
product SO(2) ⋉ H(6) (because SO(2) acts on the H(6) generators). We would
have to include SO(2) in the coset construction in order to obtain the commutator
between the generator of SO(2) and the supercharges. However, SO(2) is not a symmetry
of the full solution because it does not leave the field strength invariant.
2. Owing to the non-semisimplicity of the Heisenberg algebra, it is not possible to find
a relation between γ a and the dual representation Ɣs(P a ). Thus, the Killing-spinor
bilinears have to be calculated by brute force, but we will not do it here.
13.3.5 The vacua of N = 2, d = 4 AdS supergravity
The integrability condition now imposes a third constraint in order for the terms with zero,
two, and four gammas to vanish: the Maxwell field-strength tensor has to vanish. Then,
the only maximally supersymmetric solutions are those of N = 1, d = 4 AdS supergravity,
i.e. AdS4 spacetime, and a basis of Killing spinors will be provided by
κ(αi) β j = Ɣs(u −1 ) β αδ j j. (13.97)