Gravity and Strings

5.3 N = 1, d = 4 AdS supergravity 159
Thus, using the value of the torsion field in this theory, we find
[δɛ1 ,δɛ2 ]ea µ = a
δξ + δσ + δɛ e µ, (5.48)
where
σ a b = ξ ν ων a b, ɛ = ξ µ ψµ. (5.49)
The same algebra is realized on all the fields of the theory.
5.3 N = 1, d = 4 AdS supergravity
The simplest N = 1, d = 4 Poincaré supergravity theory that we have just described can
be generalized in essentially two ways: adding N = 1 supersymmetric matter or generalizing
the Lorentz connection. Adding certain matter supermultiplets sometimes produces
enhancement of supersymmetry and in this way one obtains extended supergravities. We
will review N = 2, d = 4(gauged and ungauged)supergravity later.
The only generalizations of the four-dimensional Poincarégroup which are usually studied
are the four-dimensional (anti-)de Sitter groups dS4 = SO(1, 4) and AdS4 = SO(2, 3).
Of these, only AdS4 is compatible with consistent supergravity. We have obtained at the
beginning of this chapter the action for N = 1, d = 4 AdS supergravity in the first-order
form
where
S[ea µ,ωµ ab
,ψµ] =
d4
xe R(e,ω)+ 6g2 + 2e−1ɛ µνρσ ¯ψµγ5γν ˆDρψσ
, (5.50)
ˆDµ = Dµ − ig
2 γµ
(5.51)
is the AdS4-covariant derivative and Dµ is the Lorentz-covariant derivative in the spinor
representation.
This theory contains a negative cosmological constant proportional to the square of the
Wigner–Inönü parameter g, =−3g 2 .The vacuum will be anti-de Sitter spacetime.
The equation of motion for ωµ ab takes the same form as in the g = 0(Poincaré) case
and therefore has the same solution, Eq. (5.25). The other two equations of motion suffer
g-dependent modifications:
δS
δe a µ
=−2e Ga µ − 3g 2 ea µ − 2Tcan a µ = 0,
Tcan a µ = 1
2e ɛρµσν ¯ψργ5γa ˆDσ ψν − ig
2e ɛµνρσ ¯ψνγ5γρaψσ ,
δS
= 4ɛ
δ ¯ψµ
µνρσ
γ5γν ˆDρψσ + 1
4 Tνρ a
γ5γaψσ = 0.
The torsion term can be shown to vanish on-shell using Fierz identities. 7
7 This is also true in the Poincaré case.
(5.52)