Gravity and Strings

13.3 N = 1, 2, d = 4 vacuum supersymmetry algebras 385
AdS4 can be identified with the coset space SO(2, 3)/SO(1, 3).Weuse the index conventions
of Section 4.5. g = so(2, 3), the Lie algebra of G = SO(2, 3),iswritten in Eq. (4.152).
It is convenient to rescale and rename the generators as in Eq. (4.154), and the commutation
relations take the form Eqs. (4.155).
The Mabs generate the subalgebra h = so(1, 3) of the Lorentz subgroup. The orthogonal
complement k is generated by the Pas and, on looking at the commutation relations
Eqs. (4.155), we see that we have a symmetric pair.
Now we construct the coset representative u,
and the Maurer–Cartan 1-form V ,
u(x) = e x3 P3 e x 2 P2 e x 1 P1 e x 0 P0 , (13.66)
V =−u −1 du
=−P0dx 0 − e −x0 P0 P1e x0 P0 1 −x
dx − e 1 P1 −x
e 0 P0 P2e x0 P0 x
e 1 P1 2
dx
− e −x2 P2 −x
e 1 P1 −x
e 0 P0 P3e x0 P0 x
e 1 P1 x
e 2 P2 3
dx . (13.67)
Using the definition of the adjoint action of the group on the algebra, we see that
e −x0 P0 P1e x0 P0 = TI ƔAdj(e −x0 P0 ) I 1, (13.68)
etc. and, by projecting onto the horizontal subspace, we find the Vierbeins,
e 0 =−dx 0 , e 1 =−cos x 0 dx 1 , e 2 =−cos x 0 cosh x 1 dx 2 ,
e 3 =−cos x 0 cosh x 1 cosh x 2 dx 2 , (13.69)
which,with the Killing metric (+−−−),give the following form of the AdS4 metric:
ds 2 = (dx 0 ) 2 − cos 2 x 0 {(dx 1 ) 2 + cosh 2 x 1 [(dx 2 ) 2 + cosh 2 x 2 (dx 3 ) 2 ]}. (13.70)
The explicit form of the vertical 1-forms ϑ ab is not necessary, but we need to know how
they enter the spin connection. According to Eq. (A.117)
The Killing spinor equation is
dx µ ˆDµκ =
ω a b = 1
2 ϑ cd fcd −1b −1a = ϑ ac ηcb. (13.71)
d − 1
2
4 ωabγ ab − ig
and can be written in the form (d − V )κ = 0 with 12
a
γae κ = 0, (13.72)
Ɣs(Pa) = ig
2 γa, Ɣs(Mab) = 1
2γab. (13.73)
The Killing spinors are thus of the form Eqs. (13.38) and (13.39).
12 Compare this with Eq. (B.132).