Gravity and Strings

374 Unbroken supersymmetry
infinite-dimensional group of all GCTs. In this case, the structure of the finite-dimensional
supersymmetry group of a solution is inherited from that of the infinite-dimensional supergroup
of all local supersymmetry transformations, GCTs, etc. of the supergravity theory.
The commutator of two local supersymmetry transformations is a combination of all the
symmetries of the theory: for instance, in N = 2, d = 4 Poincaré supergravity, given by
Eq. (5.96), a GCT, a local Lorentz rotation, a U(1) gauge transformation, and a local supersymmetry
transformation with parameters that depend on ɛ1,2 and the fields of the theory.
Now, if κ1,2 are Killing spinors of a bosonic solution, the commutator will give bosonic
symmetries of the same solution. In particular, we find that the solution will be invariant 5
under GCTs generated by bilinears of the form 6
k µ =−i ¯κ1γ µ κ2. (13.8)
Other Killing spinor bilinears will be associated with generators of other (non-geometrical)
symmetries of the solution. This is how the bosonic generators of the supersymmetry group
of a bosonic solution arise.
Following our previous discussion of isometries in general-covariant theories, we can
associate solutions admitting a maximal number of Killing spinors (maximally supersymmetric
solutions) with vacua of the supergravity theory. Now, a given supergravity can have
more than one maximally supersymmetric solution (vacuum). Usually, one of the vacua
is also a maximally symmetric solution (Minkowski or AdS), but the other vacua are not
and have non-vanishing matter fields. Each of these vacua defines a class of solutions with
the same asymptotic behavior, which can be associated with states of the QFT that one
would construct on the corresponding vacuum. The vacuum supersymmetry algebras can
be used to define conserved quantities for those spacetimes/states. Thus, we can study the
supersymmetries of these spacetimes using knowledge of their conserved charges and the
superalgebra of the asymptotic vacuum spacetime or by solving the Killing spinor equation
directly. We will do this in Section 13.5.
Our immediate task is to develop a method by which to find the supersymmetry algebras
of the vacuum (or any other) solutions. Let us proceed by analogy with the nonsupersymmetric-gravity
case discussed in the previous section. There will be a bosonic
generator P(I ) of the abstract supersymmetry algebra for each Killing vector k(I ) µ that generates
a GCT that leaves invariant all the fields of the solution, there will be other “internal”
bosonic generators B(M) associated with each invariance of the matter fields, and there
will be a fermionic generator Q(A) of the abstract supersymmetry algebra for each Killing
spinor κ α (A) .
Now, we have to identify all the generators of the abstract supersymmetry algebra with
operators acting on the supergravity fields. The (anti)commutators of these operators will
give the corresponding (anti)commutators of the superalgebra generators.
Let us start with the bosonic generators P(I ). Onworld tensors, each P(I ) is represented
by (minus) the standard Lie derivative with respect to the corresponding Killing vector k(I ),
which transforms world tensors into world tensors of the same rank. However, most of the
5 This statement will be made more precise shortly.
6 If κ1 and κ2 are identical commuting Killing spinors, the bilinear does not vanish. Furthermore, it can be
shown that k µ =−i ¯κγ µ κ is always timelike or null in d = 4, null in N = 1, d = 10 supergravity, etc.