Gravity and Strings

378 Unbroken supersymmetry
transformation with parameter χ = kν Aν + with =−i¯ɛ2σ 2ɛ1 and it can be checked
that, for Killing spinors, we have, precisely,
= k(I ) λ Fµλ k(I ) µ ≡−i¯κ(A)γ µ κ(B). (13.25)
∂µ −i ¯κ(B)σ 2 κ(A)
This exercise is useful because there can be other gauge-dependent fields in the supergravity
theory: in N = 2, d = 4 AdS (gauged) supergravity the gravitinos ψµ and the supersymmetry
parameters ɛ (and, therefore, the Killing spinors, if any) are electrically charged
and transform according to
A ′ µ = Aµ + ∂µχ, ψ ′ µ
= e−igχσ2ψµ,
ɛ ′ = e −igχσ2
ɛ, (13.26)
and we need to define a U(1) and Lorentz-covariant Lie derivative for them. For the supersymmetry
parameters ɛ and Killing vectors k,wedefine
Lkɛ ≡ Lkɛ + ig(k µ Aµ + )σ 2 ɛ, (13.27)
where has been defined above and exists if k is Killing and Eq. (13.21) is satisfied. This
derivative has the Lie algebra property and also preserves the N = 2, d = 4 AdS supercovariant
derivative
[Lk, ˆ ˜Dv] ɛ = ˆ ˜D[k,v] ɛ, (13.28)
under the condition Eq. (13.21), which is necessary anyway in order for the associated P(I )
to be a symmetry of the whole solution.
A non-Abelian generalization of all these formulae can be found in Appendix A.4.1.
For the gravitinos ψµ we expect problems similar to those we found for Aµ since they can
be considered (super) gauge fields and transform inhomogeneously under supersymmetry.
The role of the supersymmetry transformation that appears in the commutators Eqs. (5.45),
(5.58) and (5.96) will clearly be that of compensating the effect of the GCT.
13.2.2 Calculation of supersymmetry algebras
We have developed all the tools we need to calculate the symmetry superalgebra of any
supergravity solution. Now we just have to follow this six-step recipe [390].
1. First we have to solve the Killing and Killing-spinor equations. We keep only the
Killing vectors that leave invariant all the fields of the solution. Furthermore, we have
to find any other “internal” invariance of the fields.
2. With each Killing vector k(I ) µ we associate a bosonic generator of the superalgebra
P(I ), with any internal symmetry of the fields another bosonic generator B(M), and
with each Killing spinor κ(A) α we associate a fermionic generator (supercharge) Q(A).
The bosonic subalgebra is in general the sum of two subalgebras generated by the
P(I )s andthe B(M)s with structure constants f IJ K and fMN P . The fermionic generators
are in representations of these bosonic subalgebras. These representations are
determined by the structure constants f AI B and f AM B that appear in
[Q(A), P(I )] = f AI B Q(B), [Q(A), B(M)] = f AM B Q(B). (13.29)