Gravity and Strings

5.2 N = 1, d = 4(Poincaré) supergravity 155
purely algebraic, we are going to find that there is torsion proportional to some expression
quadratic in fermions, making the ˆRµν a,−1 (Â) components of the supercurvature vanish.
Substituting the torsion into the action will give rise to terms that are quartic in fermions.
In what follows we are going to study these actions, their equations of motion, and their
symmetries separately. The most efficient way to do it is to treat them in the so-called
1.5-order formalism: we consider that we have solved the equation of motion of the spin
connection and we have substituted its solution back into the action, but we do not do it
explicitly, keeping the action in its first-order form. Then, in varying over the two remaining
fundamental fields (the Vierbein and gravitino), we use the chain rule, varying over the spin
connection first. That variation is its equation of motion, which has been solved, and simply
vanishes. In this way, many calculations are greatly simplified.
We are going to make this study as self-contained as possible and, thus, we will repeat
some of the general points explained in this introductory section.
5.2 N = 1, d = 4 (Poincaré) supergravity
The fields of N = 1, d = 4supergravity are the Vierbein and the gravitino {ea µ,ψµ}. The
gravitino is a vector of Majorana (real) spinors. The action is written in a first-order form,
in which the spin connection ωµ ab is also considered as an independent field and the action
contains only first derivatives. We rewrite the action here for convenience, setting χ = 1:
S[ea µ,ωµ ab
,ψµ] =
d4xe R(e,ω)+ 2e−1ɛ µνρσ
¯ψµγ5γνDρψσ . (5.20)
Here Dµ is the Lorentz-covariant derivative (rather than the completely covariant derivative,
which we denote as usual by ∇µ),
and
Dµψν = ∂µψν − 1
4 ωµ ab γabψν, ∇µψν = Dµψν − Ɣµν ρ ψρ, (5.21)
R(e,ω)= ea µ eb ν Rµν ab (ω), (5.22)
where Rµν ab (ω) is the Lorentz curvature of the Lorentz connection ωµ ab ,Eq. (1.81).
As usual, to obtain the second-order action we solve the spin-connection equation of
motion and substitute the solution for ωµ ab in terms of ea µ and ψµ back into the first-order
action. The spin-connection equation of motion is
δS
µνρ
= 3!eabc
δωµ
ab
Dνe c ρ + i
2 ¯ψνγ c ψρ
= 0. (5.23)
This equation implies that the expression in brackets, antisymmetrized in ν and ρ, iszero.
Looking at Eq. (1.86), we see that there is torsion in this theory and it is given by 5
5 The bilinear ¯ψµγ a ψν is automatically antisymmetric in µν.
Tµν a = i ¯ψµγ a ψν. (5.24)