Gravity and Strings

5.7 Proofs of some identities 169
To prove the invariance of the action under the local supersymmetry transformations, one
has to check the N = 2, d = 4AdS gauge identity
˜ δS δS
ˆDµ =−i
δ ¯ψµ δea γ
µ
a + δS
σ
δ Aµ
2
ψµ, (5.105)
where, here,
˜ δS
ˆDµ
δ ¯ψµ
= ˆDµ + 1
4γµ ˜Fσ 2 δS
. (5.106)
δ ¯ψµ
To prove this identity we need only check the g-dependent terms (the g-independent
ones work, as we checked in the previous section). To check the g-dependent terms, we
need only the additional identities (see Section 5.7)
and
( ¯ψ[ν|γaψ|µ|)γ5γ a σ 2 ψ|ρ] = ( ¯ψ[ν|γaγ5ψ|µ|)γ a σ 2 ψ|ρ]
( ¯ψ[ν|γaψ|µ)γ5γ a γρψσ ] + ( ¯ψ[νσ 2 ψµ)γ5γρσ 2 ψσ ] − ( ¯ψ[ν|γ5σ 2 ψ|µ)γρσ 2 ψσ ]
(5.107)
=−2( ¯ψ[ν|γ5γa|ρψµ)γ a ψσ ] − 2( ¯ψ[ν|γ5γ|ρσ 2 ψµ)σ 2 ψσ ]. (5.108)
5.6.1 The local supersymmetry algebra
The commutator of two supersymmetry variations closes on-shell with the same parameters
as in the ungauged case except for
σ ab = ξ µ ωµ ab − g ¯ɛ2γ ab ɛ1 − i ¯ɛ2
˜F ab − iγ5 ⋆ ˜F ab
σ 2 ɛ1. (5.109)
From the point of view of the supersymmetry algebra, we are going from Poincaré supersymmetry
to AdS supersymmetry in which the generator of SO(2) rotations has to appear
in the anticommutator of two supersymmetry charges, for consistency. Although it appears
in the same position as a central charge, it should be stressed that it is not a central charge
because it does not commute with the supercharges.
5.7 Proofs of some identities
Using the N = 2 Fierz identities Eq. (B.57) we immediately find, for any spinor λ, the
following two identities:
( ¯ψ[ν|γ5γaλ)γ a ψ|µ] =− 1
2 ( ¯ψ[νγ5σ 2 ψµ])σ 2 λ − 1
4 ( ¯ψ[ν|γaγ5σ 2 ψ|µ])γ a σ 2 λ
+ 1
2 ( ¯ψ[νσ 2 ψµ])γ5σ 2 λ
⎛ ⎞ ⎛ ⎞
− 1
4 ( ¯ψ[ν|γa
⎝
0 σ
σ 1
σ 3
⎠
T
ψ|µ])γ a γ5
⎝
0 σ
σ 1
σ 3
⎠λ (5.110)