Feynman Path Integral Formulation

1.9 Supergravity 37
δA μ = −2g ¯λ a T a γ μ ε
δλ a = − i g σ μν Fμν a ε
δ ¯λ a = i g ¯ε σμν Fμν a , (1.174)
with σ μν = 1 2i [γ μ ,γ ν ], and ε(x) an arbitrary Grassmann parameter with Majorana
properties. The supersymmetry of Eq. (1.174) leaves the action locally invariant,
and at the same time relates fermions to bosons. The corresponding Noether current
is
S μ = −F a ρσ σ ρσ γ μ λ a , (1.175)
and satisfies ∂ μ S μ = 0 after using the field equations, as well as γ μ S μ = 0. Furthermore
it is known that in this theory gluino condensation occurs non-perturbatively,
giving rise to a non-vanishing condensate 〈¯λλ〉 ≠ 0.
1.9 Supergravity
When supersymmetry is promoted to a local invariance of the theory one obtains supergravity:
supergravity can therefore rightfully be considered as the gauge theory
of supersymmetry. In the simplest model one adds to the Einstein gravity Lagrangian
a spin-3/2 gravitino field, whose purpose is to exactly cancel loop divergences from
the Einstein contributions. The enhanced symmetry is built into the action so as to
ensure that such a cancellation does not just occur at one loop order, but propagates
to every order of the loop expansion. In these theories the gravitino therefore
emerges as the natural fermionic partner of the graviton, with zero mass for unbroken
supersymmetry. The intent of this section is more to provide the general flavor
of such an approach, and illustrate supergravity theories by a few specific examples
of suitable actions, without getting into elegant technical aspect such as superfields
and superpropagators. The reader is then referred to the vast literature on the subject
for further examples, as well as contemporary leading candidate theories.
As stated, in the simplest scenario, one adds to gravity a spin- 3 2
fermion field with
suitable symmetry properties. A generally covariant action describing the interaction
of vierbein fields e a μ(x) (with the metric field given by g μν = e a μe aν ) and Rarita-
Schwinger spin- 3 2 fields ψ μ(x), subject to the Majorana constraints ψ ρ = C ¯ψ T ρ ,was
originally given in (Ferrara, Freedman and van Nieuwenhuizen, 1976). In the second
order formulation it contains three contributions
∫
I = d 4 x (L 2 + L 3/2 + L 4 ) , (1.176)
with the usual Einstein term
L 2 = 1
4κ 2 √ gR , (1.177)