Feynman Path Integral Formulation

6.16 Lattice Gravitino 215
and involves both the standard affine connection Γ σ
νρ, as well as the vierbein connection
ω ν ab = 1 2 [e μ
a (∂ ν e bμ − ∂ μ e bν )+e ρ
a e σ
b (∂ σ e cρ )e c ν] ,
− (a ↔ b) (6.181)
with Dirac spin matrices σ ab = 1 2i [γ a,γ b ], and ε μνρσ the usual Levi-Civita tensor,
such that ε μνρσ = −gε μνρσ .
Next one considers just two neighboring simplices s 1 and s 2 , covered by a common
coordinate system x μ . When the two vierbeins in s 1 and s 2 are made to coincide,
one can then use a common set of gamma matrices γ μ within both simplices. Then
(in the absence of torsion) the covariant derivative D μ in Eq. (6.179) reduces to just
an ordinary derivative. The fermion field ψ μ (x) within the two simplices can then be
suitably interpolated, and one obtains a lattice action expression very similar to the
spinor case. One can then relax the condition that the vierbeins e μ a (s 1 ) and e μ a (s 2 )
in the two simplices coincide. If they do not, then they will be related by a matrix
R(s 2 ,s 1 ) such that
e μ a (s 2 )=R μ ν(s 2 ,s 1 ) e ν a (s 1 ) , (6.182)
and whose spinorial representation S was given previously in Eq. (6.24). But the
new ingredient in the spin-3/2 case is that, besides requiring a spin rotation matrix
S(s 2 ,s 1 ), now one also needs the matrix R ν μ(s,s ′ ) describing the corresponding
parallel transport of the Lorentz vector ψ μ (s) from a simplex s 1 to the neighboring
simplex s 2 .
An invariant lattice action for a massless spin-3/2 particle takes therefore the
form
I = − 1 2
with
∑
faces f(ss ′ )
V [ f (s,s ′ )]ε μνλσ ¯ψ μ (s)S[R(s,s ′ )]γ ν (s ′ )n λ (s,s ′ )R ρ σ (s,s ′ )ψ ρ (s ′ )
(6.183)
¯ψ μ (s)S[R(s,s ′ )]γ ν (s ′ )ψ ρ (s ′ ) ≡ ¯ψ μα (s)S α β [R(s,s′ )]γ β
ν γ(s ′ )ψ γ ρ(s ′ ) , (6.184)
and the sum ∑ faces f(ss ′ ) extends over all interfaces f (s,s ′ ) connecting one simplex
s to a neighboring simplex s ′ . When compared to the spin-1/2 case, the most important
modification is the second rotation matrix R ν μ(s,s ′ ), which describes the
parallel transport of the fermionic vector ψ μ from the site s to the site s ′ , which is
required in order to obtain locally a Lorentz scalar contribution to the action.
In supergravity one more term is needed in the action. In order to achieve local
supersymmetry invariance, one needs an additional quartic fermion self-interaction
term, given in Eq. (1.176), and which is of the form (Ferrara, Freedman and van
Nieuwenhuizen, 1976)
1
L 4 = −
32κ 2 √ g (εταβν ετ
γδμ + ε ταμν ετ
γδβ − ε τβμν ετ γδα )
×( ¯ψ α γ μ ψ β )( ¯ψ γ γ ν ψ δ ) , (6.185)