Gravity and Strings

398 Unbroken supersymmetry
and prove that the mass is bounded from below by the skew eigenvalues of the central
charges matrix Z ij [384, 963],
M ≥|Zi|, ∀i = 1,...,[N/2], (13.121)
and the results can be generalized to other dimensions. These bounds are known as
Bogomol’nyi, BPS,orsupersymmetry bounds and play a crucial role in the stability of states
and theories. Broken supersymmetry is restored when one of these bounds is saturated, as
we have seen. Supersymmetric (“BPS”) states then have the minimal masses allowed for
given values of the central charges. The central charges cannot change, because there are
no (perturbative) states in the theory that carry them and, then, the masses of BPS states
cannot diminish and the states cannot decay and are stable.
As for the associated BPS solutions and SUEGRA theories, some stability properties
can also be proven, such as the positive-energy theorem of GR whose proof, inspired by
N = 1, d = 4 SUGRA, we gave in Section 6.3. The relation between positivity of the energy
and the supersymmetry algebra was studied in [575]. There are generalizations based
on the WNI construction (see e.g. [442] for N = 2, d = 4). As for the stability of solutions,
it manifests itself, most remarkably, in the absence of Hawking radiation (T = 0)
from ERN and other supersymmetric BHs. This establishes an interesting link between BH
thermodynamics and supersymmetry (see e.g. [745]), which we will use in Chapter 20.
In the next section we are going to review the (not maximally) supersymmetric solutions
of the simplest theories, N = 1, 2, 4, d = 4 SUGRA, and their relation to the supersymmetry
bounds one can derive from the superalgebras.
13.5.2 Examples
N = 1, d = 4 Poincaré supergravity. The only solutions with partially unbroken supersymmetry
in this theory are the purely gravitational pp-waves given by Eqs. (10.24) with
C = 0. The Killing-spinor equation ∇µκ = 0issolved for any constant spinor κ satisfying
the constraint γ uκ = 0, which is precisely what we expected from the superalgebra.
The absence of massive supersymmetric solutions can be understood as a consequence of
the N = 1supersymmetry bound M ≥ 0. This is in agreement with the finite temperature
and entropy of all Schwarzschild BHs. The bound is also in agreement with the cosmiccensorship
conjecture.
Things are different in Euclidean N = 1, d = 4supergravity: the integrability condition
of the Killing-spinor equation,
Rµν ab γabκ = 0, (13.122)
also admits solutions when the curvature is (anti-)self-dual because then it is proportional
to a projector 1
2 (1 ± γ 1γ 2γ 3γ 4 ),
Rµν ab γab = Rµν ab γab 1
2 (1 ± γ 1 γ 2 γ 3 γ 4 )κ, (13.123)
and half of the supersymmetries are preserved. Then all metrics (in any dimension!) with
(anti-)self-dual curvature (special SU(2) holonomy, see footnote 5) preserve half of the
supersymmetries. These are the metrics of (anti-)self-dual gravitational instantons, which
will be discussed in Section 9.2.1. In the frame in which the spin connection is also