String Theory and M-Theory

306 M-theory and string duality
The spin connection can be expressed in terms of the elfbein by
where
ωMAB = 1
2 (−ΩMAB + ΩABM − ΩBMA), (8.19)
ΩMN A = 2∂ [NE A M] . (8.20)
In fact, these relations are valid in any dimension. Depending on conventions,
the spin connection may also contain terms that are quadratic in fermi
fields. Such terms are neglected here, since they are not relevant to the issues
that we discuss.
Supersymmetric solutions
One might wonder why the supersymmetry transformations have been presented
without also presenting the fermionic terms in the action. After all,
it is the complete action including the fermionic terms that is supersymmetric.
The justification is that one of the main uses of this action, and
others like it, is to construct classical solutions. For this purpose, only the
bosonic terms in the action are required, since a classical solution always
has vanishing fermionic fields.
One is also interested in knowing how many of the supersymmetries survive
as vacuum symmetries of the solution. Given a supersymmetric solution,
there exist spinors, called Killing spinors, that characterize the supersymmetries
of the solution. The concept is similar to that of Killing vectors,
which characterize bosonic symmetries. Killing vectors are vectors that appear
as parameters of infinitesimal general coordinate transformations under
which the fields are invariant for a specific solution. In analogous fashion,
Killing spinors are spinors that parametrize infinitesimal supersymmetry
transformations under which the fields are invariant for a specific field configuration.
Since the supersymmetry variations of the bosonic fields always
contain one or more fermionic fields, which vanish classically, these variations
are guaranteed to vanish. Thus, in exploring supersymmetry of solutions,
the terms of interest are the variations of the fermionic fields that do not
contain any fermionic fields. In the case at hand this means that Killing
spinors ε are given by solutions of the equation
δΨM = ∇Mε + 1
12
ΓMF (4) − 3F (4)
M
ε = 0, (8.21)
and the bosonic terms that have been included in Eq. (8.12) determine the
possible supersymmetric solutions.