Gravity and Strings

156 N = 1, 2, d = 4 supergravities
Furthermore, we see that the solution to the new equation is just that the Lorentz connection
consists of two pieces: the one that solves the standard equation D[µe a ν] = 0, which we
denote by ωµ ab (e) because it is completely determined by the Vierbein, and the contorsion
tensor Kµ ab , which depends on the gravitino through the torsion. It is convenient to write
the solution as follows:
ωabc =−abc + bca − cab, µν a = µν a (e) + 1
2 Tµν a , µν a (e) = ∂[µe a ν]. (5.25)
The other two equations of motion that the first-order action gives are
where we have used
δS
δe a µ
=−2e Ga µ − 2Tcan a µ (ψ) = 0,
Tcan a µ (ψ) = 1
2e ɛρµσν ¯ψργ5γaDσ ψν, (5.26)
δS
δ ¯ψµ
= 4ɛ µνρσ γ5γνDρψσ + 1
4 Tνρ a γ5γaψσ
= 0,
D[µγν] =− 1
2 Tµν a γa. (5.27)
The second-order equations of motion follow from the substitution of Eq. (5.25) into the
first-order ones.
The action Eq. (5.20) and equations of motion are manifestly invariant under
general coordinate transformations,
δξ x µ = ξ µ , δξe a µ =−ξ ν ∂νe a µ − ∂µξ ν e a ν, δξψµ =−ξ ν ∂νψµ − ∂µξ ν ψν,
(5.28)
and local Lorentz transformations,
δσ e a µ = σ a be b µ, δσ ψµ = 1
2 σ ab γabψµ, (5.29)
where σ ab =−σ ba .Ontop of this, if we eliminate the spin connection as an independent
field by substituting the solution of its equation of motion, there is invariance
under
local N = 1 supersymmetry transformations:
δɛe a µ =−i ¯ɛγ a ψµ, δɛψµ = Dµɛ. (5.30)
This requires some explanation. The first-order action is also invariant under the same
transformations supplemented by the supersymmetry transformation of the spin connection.
In the second-order formalism, the supersymmetry variation of the spin connection is
completely different and can be found by varying Eq. (5.25) with respect to the Vierbein
and gravitino:
δɛωµ ab =−i ¯ɛγµψ ab + i ¯ɛγ a ψ b µ − i ¯ɛγ b ψµ a , ψµν ≡ D[µψν]. (5.31)