Gravity and Strings

400 Unbroken supersymmetry
Observe that, like any other physical quantity that depends only on the metric, which
is invariant under the electric–magnetic duality of the theory, both T and S are also
duality-invariant: they can be written in terms of the single mass matrix skew eigenvalue
Z =|Q + iP|. Onthe other hand, in the extreme limit, which is also here (in the absence
of angular momentum) the supersymmetric limit, T → 0and there is no Hawking radiation,
in agreement with our previous arguments about stability, while S → π M 2 = π|Z| 2
is completely determined by the central charges. We discussed at length the arguments in
favor ofand against there being a finite value of the entropy for ERN BHs in Section 8.6.
In any case, we can always say that the area of the ERN BH is finite and given by a dualityinvariant
combination of the central charges. This observation is relevant because we will
find that, in general, charged supersymmetric spherically symmetric solutions have a regular
horizon with finite area only when they preserve the same amount of supersymmetry as
the ERN BH (i.e. four supersymmetries).
If we consider asymptotically Taub–NUT IWP solutions, we find that all of them satisfy
the saturated bound M +|N|=|Q + iP|=|Z| (with or without angular momentum).
The supersymmetry of some solutions of N = 2, d = 4 AdS supergravity have also been
studied in [203, 634, 811]. The supersymmetry of the largest class of solutions, given by
Plebański and Demiański in [772, 773], was studied in [27].
N = 4, d = 4 supergravity. The bosonic part of this theory is described by the action
Eq. (12.58). All the field configurations admitting Killing spinors were found by Tod
in [894] and they include pp-waves and the SWIP solutions of [130, 613] described in
Section 12.2.1. These have been studied more thoroughly. There are SWIP solutions that
preserve half of the supersymmetries (i.e. eight) and SWIP solutions that preserve a quarter
(i.e. four) (these include the IWP solutions).
Actually, to study them, it is convenient to focus on the asymptotically flat ones and use
the supersymmetry bounds of the theory. There are two BPS bounds: M 2 −|Z1,2| 2 ≥ 0,
neither of which is invariant under the dualities of the theory. However, we can construct a
duality-invariant generalized BPS bound by taking their product and dividing by M 2 :
|Z1Z2| 2
M 2 +
M2 −|Z1| 2 −|Z2| 2 ≥ 0. (13.127)
This bound is satisfied by all the regular static non-extreme SWIP solutions of [665] and
it is saturated by all the supersymmetric SWIP solutions of [130]: the skew eigenvalues of
the central-charge matrix correspond precisely to the combinations of electric and magnetic
charges of Eq. (12.83) and the product of the two supersymmetry bounds gives precisely
the supersymmetry parameter r 2 0 of the SWIP solutions, Eq. (12.84).
The bound can be saturated in two different ways: when M =|Z1,2| =|Z2,1|,aquarter of
the supersymmetries are unbroken, and when M =|Z1|=|Z2|,half of the supersymmetries
are unbroken. The only supersymmetric solutions with regular horizons are the static ones
with only a quarter of the supersymmetries preserved. All the supersymmetric solutions
have zero temperature.
The entropy (see the second of Eqs. (12.84)) and the temperature of the BHs of
N = 4, d = 4 can be expressed in a form that is manifestly invariant under the two duality
groups of the theory: SL(2, R) (S duality) and SO(6) (T duality). The entropy of the