Gravity and Strings

5.2 N = 1, d = 4(Poincaré) supergravity 157
One may think that the gauging of the supersymmetry algebra should give us the firstorder
supersymmetry transformation rule for the spin connection, but it does not: it just
gives δɛωµ ab = 0. Nevertheless, to check the invariance of the action in the 1.5-order formalism
we do not need this variation, as we are going to see.
Let us check the invariance of the action Eq. (5.20) under these transformations in the
1.5-order formalism. This is not a complicated calculation if we construct the right setup,
which is the general setup explained in Chapter 2 for theories that are invariant under local
symmetries. There we showed that a given theory would be invariant up to total derivatives
under a local transformation if a certain gauge identity was satisfied by its equations of
motion. Thus, all we have to do is to identify the gauge identity that has to be satisfied in
this case by the Vierbein and gravitino equations of motion.
Under a general variation of the fields, the N = 1, d = 4SUGRA action Eq. (5.20) trans-
forms as follows:
δS = d 4
δS
x
δea δe
µ
a µ + δS
δωµ ab δωµ ab
δS
+ δ ¯ψµ . (5.32)
δ ¯ψµ
Here the variations are only with respect to explicit appearances of each field in the firstorder
action. The variation of the second-order action would be obtained by applying the
chain rule to the variation with respect to the spin connection, using Eq. (5.25). However,
these additional terms are proportional to the equation of motion of the spin connection
δS/δωµ ab , which we have assumed is satisfied (the 1.5-order formalism). Thus, the term
containing δωµ ab will always vanish (for any kind of variation) because it is proportional
to that equation of motion and we need only vary explicit appearances of the Vierbein and
gravitino in the first-order action Eq. (5.20),
δS = d 4
δS
x
δea δe
µ
a
δS
µ + δ ¯ψµ . (5.33)
δ ¯ψµ
Consider now the local supersymmetry transformations Eqs. (5.30). On substituting into
the above the explicit form of these transformations and integrating by parts the partial
derivative in
Dµ ¯ɛ = Dµɛ = ∂µ ¯ɛ + 1
4 ¯ɛωµ ab γab, (5.34)
we obtain, up to total derivatives,
δɛ S = d 4
x ¯ɛ −i δS
δea µ
The theory will be locally supersymmetric, then, if
γ a ψµ − Dµ
δS
. (5.35)
δ ¯ψµ
δS
Dµ
δ ¯ψµ
=−i δS
δea γ
µ
a ψµ, (5.36)
which will be, at the same time, the supersymmetry gauge identity. Let us prove it:
Dµ
δS
= 4ɛ
δ ¯ψµ
µνρσ γ5(Dµγν)Dρψσ + 4ɛ µνρσ γ5γνDµDρψσ
+ ɛ µνρσ γ5γaDµTνρ a ψσ + ɛ µνρσ γ5γaTνρ a Dµψσ , (5.37)