Gravity and Strings

386 Unbroken supersymmetry
The dual generators Ɣs(P a ) can be defined by
Tr [Ɣs(P a )Ɣs(Pb)] = δ a b, ⇒ Ɣs(P a ) =− i
2g γ a ,
Tr [Ɣs(M ab )Ɣs(Pcd)] = δ ab cd, ⇒ Ɣs(M ab ) =− 1
2 γ ab .
We see that the matrix S is just the identity in this case. The bilinears give
−i ¯κγ a κea = 2gƔs(u −1 ) T CƔs(P a )Ɣs(u −1 )ea
= gCƔs( ˆM ˆbĉ )ƔAdj(u −1 ) a ˆbĉ ea
= gCƔs( ˆM ˆbĉ )k( ˆbĉ) ,
and we recover the standard anticommutator of the supercharges,
(13.74)
(13.75)
{Q(α), Q(β)}=g[CƔs( ˆM â ˆb )]αβ ˆM â ˆb =−i(Cγ a )αβ Pa − g
2 (Cγ ab )αβ Mab. (13.76)
The commutators [Q(α), ˆM â ˆb ] are found using Eq. (13.40):
[Q(α), ˆM â ˆb ] =−Q(β)Ɣs( ˆM â ˆb )β α. (13.77)
13.3.4 The vacua of N = 2, d = 4 Poincaré supergravity
The integrability condition Eq. (13.59) now gives two independent conditions for maximal
supersymmetry: a vanishing Weyl tensor and a covariantly constant Maxwell field strength.
Only three solutions satisfy them: Minkowski spacetime, the Robinson–Bertotti (RB) solution
[146, 812] given in Eq. (8.90), whose metric is that of the AdS2 × S 2 symmetric
space, and the d = 4Kowalski–Glikman solution (KG4) [637] given in Eq. (10.27) with
a Hpp-wave metric (again, a symmetric space, that we have constructed as a coset space
in Section 10.1.1). The symmetry superalgebra of the Minkowski spacetime is identical to
that of the N = 1case, with additional indices i, j = 1, 2and, thus, we will focus on the RB
and KG4 solutions since they are the simplest of a series of maximally supersymmetric solutions
with metrics of the same form: AdSm × S n and Hpp whose symmetry superalgebras
can be calculated in a very similar fashion [25]. The five- and six-dimensional cases will be
discussed in Section 13.4 and the ten- and eleven-dimensional cases in Section 19.5.1, but
these four-dimensional examples already exhibit all the interesting features.
The Robinson–Bertotti superalgebra. The solution is given in Eq. (8.90), but we rewrite it
in a more convenient form, adapted to the normalization we used for the Maxwell field of
N = 2, d = 4supergravity (a factor of two difference):
ds 2 = R 2 2 d2 (2) − R2 2 d2 (2) , F =−R2 ωAdS2 . (13.78)
Here d 2 (2) is the metric of the AdS2 spacetime of unit radius, d 2 (2)
the metric of the