Gravity and Strings

152 N = 1, 2, d = 4 supergravities
This is the superalgebra that one has to gauge in order to construct N = 1, d = 4Poincaré
supergravity. However, to follow Section 4.5, we prefer to start from the supersymmetrized
version of the AdS4 algebra and then perform a Wigner–Inönü contraction. To supersymmetrize
it, we need to add consistently a set of fermionic supersymmetry generators to those
of the bosonic algebra ˆM â ˆb .Tohaveconsistency, the fermionic generators have to transform
as AdS4 Majorana spinors, which, as discussed in Appendix B, have four real (or purely
imaginary) components. Denoting them by ˆQ α ,wefind the following (anti)commutation
relations for the AdS4 superalgebra:
ˆM â ˆb ,
ˆM ĉ ˆd =− ˆM ê ˆb Ɣv
ˆM êâ
ĉ ˆd − ˆMâêƔv ˆM ê
ĉ ˆd ˆb ,
ˆQ α
, ˆM â ˆb = Ɣs ˆM αβ ˆQ â ˆb
β ,
{Q α , Qβ
}= ˆM â
ˆb Cˆ −1
αβ ˆM â ˆb .
An infinitesimal transformation generated by this superalgebra is
Ɣs
(5.4)
ˆ ≡ 1
2 ˆσ â ˆb ˆM â ˆb + ¯ ˆɛα ˆQ α , (5.5)
where ˆσ â ˆb =−ˆσ ˆbâ is the infinitesimal parameter of an SO(2, 3) transformation and ˆɛ α ,an
anticommuting Majorana spinor, is the infinitesimal parameter of a supersymmetry transformation.
The bar indicates Dirac conjugation.
To construct theories that are invariant under local infinitesimal transformations ( ˆσ â ˆb =
ˆσ â ˆb (x), ¯ ˆɛα = ¯ ˆɛα(x)), we need to introduce a gauge field µ,
µ ≡ 1
2 ˆωµ â ˆb ˆM â ˆb + ¯ ˆψ µα ˆQ α , (5.6)
whose components are the standard bosonic SO(2, 3) connection ˆωµ â ˆb from which we will
obtain the Lorentz connection ωµ ab and the Vierbein e a µ that will describe the graviton. It
also contains a new fermionic field: the Rarita–Schwinger field ¯ˆψ µα, which has a vector
index and a spinor index. This field describes a particle of spin 3
; the gravitino, which is
2
the supersymmetric partner of the graviton, related to it by supersymmetry transformations,
and other excitations, which should be eliminated if there is enough gauge symmetry in its
action (as is the case).
By construction, the action of an infinitesimal transformation of the gauge field is the
supercovariant derivative of ˆ(x),
δ µ = ∂µ ˆ + [ ˆ, µ]. (5.7)
On expanding the commutator (which should be understood as the anticommutator between
the fermionic generators), we find the following transformation laws for the component