Gravity and Strings

380 Unbroken supersymmetry
The Killing spinor equation takes the generic form ˜Dµκ = 0, where ˜Dµ = ∂µ − µ (the
supercovariant derivative) can be understood as a standard covariant derivative with a connection
µ that is the combination of the spin connection and other supergravity fields
contracted with gamma matrices:
µ = µ I Ɣs(TI ), (13.34)
where Ɣs(TI ) stands for different antisymmetrized products of gamma matrices that constitute
a (spinorial) representation of some of the generators of some algebra. Thus, the Killing
spinor equation can be understood as an equation of parallelism. This is why Killing spinors
are sometimes called parallel spinors.
The integrability condition says that the commutator of the supercovariant derivative on
the Killing spinor has to be zero, that is
[ ˜Dµ, ˜Dν]κ = 0, ⇒ Rµν()κ = 0, (13.35)
where Rµν() is the curvature associated with the connection . This is a homogeneous
equation. The space of non-trivial solutions is determined by the rank of the matrix Rµν(),
which is a linear combination of Ɣs(TI )s with coefficients that depend on the values of the
supergravity fields in the solution. In particular, we can have maximal supersymmetry only
if Rµν() = 0 identically (the connection is flat), which means that all the coefficients in
the linear combination have to vanish.
All the maximally supersymmetric solutions known have homogeneous reductive spacetimes
with invariant metrics and the connection 1-form turns out to be the Maurer–Cartan
1-form V defined in Eq. (A.106) in a spinorial representation [25, 26]. In symmetric spaces,
the spin connection contributes with the vertical components of V :
− 1
4 ωabγ ab =−ϑ i Ɣs(Mi), Ɣs(Mi) ≡ 1
4 fia b γb a , (13.36)
due to Eq. (A.117) and the fact that the structure constants fia b are a representation of h on
k, which makes the above Ɣs(Mi) a spinorial representation of h.
All the horizontal components of V must come from the contribution of the supergravity
fields. In the non-symmetric case [26] a combination of the two contributions gives V .
The curvature of the 1-form V is identically zero: in the language of differential forms
˜D = d − V, ⇒ R(V ) = dV − V ∧ V = 0, (13.37)
which are precisely the Maurer–Cartan equations. The Killing spinor equations admit a
maximal number of solutions and, actually, since V =−Ɣs(u −1 )dƔs(u) where Ɣs(u) is
the coset representative defined in Eq. (A.104) using the spinorial representation Ɣs(P(a))
dictated by the supergravity theory, the Killing spinors take the form
κ = Ɣs(u −1 )κ0, (13.38)
where κ0 is any constant spinor. Choosing independent constant spinors we find the following
basis of Killing spinors:
κ(α) β = Ɣs(u −1 ) β α. (13.39)