Gravity and Strings

390 Unbroken supersymmetry
The superalgebra will now include a bosonic generator associated with the constantgauge
transformations that rotate the spinor doublets and leave the vector field invariant.
This generator must appear in the anticommutator of two supercharges and the corresponding
structure constants are given by the bilinears −i ¯κ(αi)σ 2 κ(β j) = Cαβɛij.
13.4 The vacua of d = 5, 6 supergravities with eight supercharges
To complete our overview of supergravity vacua in lower dimensions, we are going to
study the vacua of the minimal supergravities in d = 5 and 6 dimensions that are related by
dimensional reduction with N = 2, d = 4 Poincaré supergravity and have the same number
of supercharges. Almost all the maximally supersymmetric vacua of these theories also
happen to be related by dimensional reduction [664].
13.4.1 N = (1, 0), d = 6 supergravity
The fields of this theory are the metric êâ ˆµ , 2-form field ˆB −
with anti-self-dual field
ˆµˆν
strength ˆH − = 3∂ ˆB − and positive-chirality symplectic Majorana–Weyl gravitino ˆψ +
ˆµ [727]
(we use the gamma matrices of Appendix B).
To write an action for an anti-self-dual 3-form, one has to introduce auxiliary fields.
Alternatively, one can write an action for a generic 3-form,
ˆS =
d6
ˆx | ˆg| ˆR + 1
12
ˆH 2
, (13.98)
and impose the anti-self-duality constraint ⋆ ˆH − =− ˆH − on the equations of motion. The
Killing-spinor equation is
ˆ∇
1
− â 48 ˆH −
ˆγ ˆκ â
+ = 0, (13.99)
where ˆκ + is a spinor of positive chirality.
Three maximally supersymmetric vacua of this theory are known: 13 Minkowski spacetime,
the near-horizon limit of the extreme anti-self-dual string solution [441] that has an
AdS3 × S 3 geometry, and the KG6 Hpp-wave solution [690]. The latter can be obtained by
taking the Penrose limit of the former [158, 495, 764].
The AdS3 × S 3 solution can be written as follows:
d ˆs 2 = R 2 3 d2 (3) − R2 3 d2 (3) ,
ˆH − = 4R3(ωAdS3 + ω S 3),
(13.100)
13 Actually, it has been shown in [229] that the maximally supersymmetric vacua of this theory and of N =
(2, 0), d = 6 supergravity are in one-to-one correspondence, and their metrics are locally isometric to biinvariant
Lorentzian metrics of six-dimensional Lie groups with anti-self-dual parallelizing torsion. The
three solutions that we present have this property and exhaust all the possibilities.