Gravity and Strings

20.1 Composite dilaton black holes 575
and the potentials à (2) m correspond to the dual field strengths
˜F (2) 1 = e −2(φ+σ −ρ)⋆ F (2) 1, ˜F (2) 2 = e −2(φ−σ +ρ)⋆ F (2) 2. (20.3)
The harmonic functions appropriate to describe a single BH are
H (i) m = 1 + |q(i) m|
. (20.4)
|x3|
The q (1) ms are electric charges and the q (2) ms are magnetic charges. Their signs are given
by the α (i) m constants. The signs in the harmonic functions are chosen in order to have a
M =
= 1)
1
4
|q (i) m|,
A = 4π |q (i) m|. (20.5)
regular metric. The ADM mass and horizon are (G (4)
N
i,m=1,2
i,m=1,2
The area is non-zero (and the horizon is regular) only when the four charges are finite.
The metrics can be related to those of the extreme a-model dilaton BHs of Section 12.1.1:
when only one charge is different from zero, the metric is that of the extreme a = √ 3BH,
Eq. (12.22). If there are two non-vanishing charges that are equal, the metric is that of the
extreme a = 1BH, Eq. (12.23). Three identical non-vanishing charges give the metric of
the extreme a = 1/ √ 3BH, Eq. (12.24); and four identical non-vanishing charges give the
metric of the a = 0BH(the ERN BH), Eq. (12.25). This fact suggests the interpretation
of ERN BHs as objects composed of four extreme a = √ 3“BHs” [298], each of which
breaks/preserves separately half of the supersymmetries while the ERN preserves an eighth
as a type-II (i.e. N = 8, d = 4 SUEGRA) solution.
It is interesting to study in a bit more detail the preservation of supersymmetries in terms
of BPS bounds. As we discussed in Section 13.5.1, there are four central-charge skew
eigenvalues Zi in N = 8, d = 4 SUEGRA. Their absolute values are in this case [611]
|Z1| = 1
4 |q11 + q2 1 + q1 2 + q2 2|, |Z2| = 1
4 |q11 − q2 1 + q1 2 − q2 2|,
|Z3| = 1
4 |q11 + q2 1 − q1 2 − q2 2|, |Z4| = 1
4 |q11 − q2 1 − q1 2 + q2 (20.6)
2|.
If only one of the charges q is different from zero (the extreme a = √ 3 dilaton BH), M =
|Zi|, i = 1,...,4, and half of the supersymmetries are preserved. If two are different from
zero (the extreme a = √ 3 dilaton BH) (say q1 1 and q2 1, both positive), then M =|Z1,2| <
|Z3,4| and a quarter of the supersymmetries are preserved. For three (say q1 1, q2 1 and q1 2,
all positive), M =|Z1| < |Z2,3,4| and an eighth of the supersymmetries are preserved. If we
add a fourth charge q2 2, then, if it is positive, no additional supersymmetries are broken,
butall are broken if it is negative.
This discussion parallels the discussion of the addition of branes to a type-II configuration
on page 568 and we need only establish the link between the d = 4solutions and
d = 10 brane solutions wrapped on T6 to arrive at the conclusion that d = 4 BHs can be
understood as composed of wrapped branes and that, in order to obtain d = 4BHs with
regular horizons, we need to include enough branes to break seven eighths of the supersymmetries.
After reaching this conclusion, it is natural to try the construction of d = 4
and d = 5 BHs directly from d = 10 extended objects in order to identify precisely which
elementary string-theory objects these BHs are made of. This information will later be used
in the entropy calculation.