Gravity and Strings

158 N = 1, 2, d = 4 supergravities
where we have used Dµγa = 0. Using Eq. (5.27) in the first term on the r.h.s. of the above
equation, we obtain minus two times the last term. In the second term we first use the Ricci
identity for the anticommutator of Lorentz-covariant derivatives, then expand the product
of gammas in antisymmetrized products γ (3) and γ (1) , reexpress the γ (3) in terms of γ (1) γ5
and the antisymmetric symbol, and, finally, use the identity
We keep the third term as it is and obtain the total result
Ga µ =− 3
2 gabc µνρ Rνρ bc . (5.38)
δS
Dµ = 2eiGa
δ ¯ψµ
µ γ a ψµ − ɛ µνρσ γ5γaTµν a Dρψσ + ɛ µνρσ Rµνρ a + DµTνρ a γ5γaψσ .
(5.39)
The first term is one of the two we want. The second term is equal to the other term we
want, due to the Fierz identity
¯ψνγ5γaDρψσ
a
(γ ψµ) =− 1
2 ( ¯ψνγ a ψµ)(γaγ5Dρψσ ). (5.40)
The expression in brackets vanishes due to the Bianchi identity 6
and this proves the supersymmetry gauge identity.
R[µνρ] a + D[µTνρ] a = 0, (5.44)
5.2.1 Local supersymmetry algebra
An important check to be performed is the confirmation that we have on-shell closure of
the N = 1 supersymmetry algebra on the fields. Let us first consider the Vierbein. Using
the supersymmetry rules (Dµɛ =∇µɛ), it is easy to obtain
where ξ a is the bilinear
[δɛ1 ,δɛ2 ]ea µ =−∇µξ a , (5.45)
ξ a =−i ¯ɛ1γ a ɛ2. (5.46)
The effect of the GCT generated by ξ µ = ξ a ea µ can be rewritten in this form:
δξe a µ =−∇µξ a − ξ ν Tµν a − ξ ν ων a be b µ. (5.47)
6 This identity can be related to the standard Bianchi identity as follows. First,
DµTνρ a =∇µTνρ a − Ɣµν λ Tρλ a + Ɣµρ λ Tνλ a . (5.41)
Antisymmetrizing and using the definition of torsion Ɣ[µν] ρ =− 1 2 Tµν ρ gives
DµTνρ a =∇[µTνρ] a + T[µν λ Tρ]λ a . (5.42)
Finally,
R[µνρ] a + D[µTνρ] a = R[µνρ] a +∇[µTνρ] a + T[µν λ Tρ]λ a , (5.43)
which vanishes on account of the usual Bianchi identity Eq. (1.30).