Gravity and Strings

154 N = 1, 2, d = 4 supergravities
In terms of these variables, the SO(2, 3) and supersymmetry transformations take the
forms
δσ ea µ = Dµσ a + σ a beb µ,
δɛea µ =−i¯ɛγ aψµ, δσ ωµ ab = Dµσ ab + 2g 2 e [a µσ b] ,
δɛωµ ab =−2g ¯ɛγ ab ψµ,
δσ ¯ψµ
1
=−¯ψµ 4σ ab ig
γab −
2 ¯ψµσ aγa, δɛ ¯ψµ = Dµɛ − ig
2 γµɛ,
and the components of the supercurvature are given by
ˆRµν ab = Rµν ab (ω) + 2g2e [a µeb] ν + g ¯ψ[µγ abψν], ˆR a,−1
µν =−gTµν a − i ¯ψ[µγ a
ψν] ,
ˆRµν = 2g 1
2
.
D[µψν] − ig
2 γ[µψν]
(5.16)
(5.17)
On substituting these components into the action, we find that the right normalization
of the Einstein–Hilbert term in the action requires α =−1/(16g 2 χ 2 ).Furthermore, the
explicit 3 terms quartic in fermions drop out from the action (after Fierzing and massaging
of some terms) if β =−8i. This is the value that will also make the action supersymmetryinvariant.
The result is the action for N = 1, d = 4 AdS4 SUGRA,
S[ea µ,ωµ ab ,ψµ] = 1
χ 2
d4
xe R(e,ω)+ 6g2 + 2e−1ɛ µνρσ ¯ψµγ5γν ˆDρψσ
, (5.18)
which, in the g→0 limit, gives the action for N =1, d =4 Poincaré SUGRA [315, 403]:
S[ea µ,ωµ ab ,ψµ] = 1
χ 2
d4xe R(e,ω)+ 2e−1ɛ µνρσ
¯ψµγ5γνDρψσ . (5.19)
These are first-order actions in which, as indicated, the fundamental variables are the
Vielbein, spin connection, and gravitino field. Thanks to our experience with the CSK
theory, 4 we know that, when we solve the spin connection equation of motion, which is
3 Later we will see that the on-shell spin connection contains terms quadratic in the fermions, so the action
contains implicitly terms quartic in fermions, just as in the CSK theory.
4 We can interpret these actions as the CSK theory coupled to gravitino fields. However, there is more to
it, because the consistency of the gravitino field theory requires its action to be invariant under the gauge
transformations (in flat spacetime) δψµ = ∂µɛ(x) in order to decouple unwanted spins. When we couple the
gravitino to gravity, consistency requires that the Vierbeins also transform under these fermionic transformations
(otherwise, that gauge symmetry is broken), which become the local supersymmetry transformations.
In this way local supersymmetry does not reduce any further the number of degrees of freedom (graviton
plus gravitino). The non-trivial part is the transformation of the Vierbeins under supersymmetry. We could
have tried to arrive at the N = 1, d = 4 supergravity action from the linearized action which is just the sum
of the Fierz–Pauli action and the Rarita–Schwinger action, decoupled, by asking for consistent interaction
and following the Noether method as we did in Chapter 3. Then, the full supersymmetry transformations
should arise as the consistency requirement.