Gravity and Strings

578 String black holes in four and five dimensions
given mass M should agree. 2 The mass of a highly excited closed-string state is, according
to Eqs. (14.46) and (14.48), M ∼ N 1 2 /ℓs, while, at the correspondence point, the Schwarzschild
BH’s mass is M ∼ RS/(gℓs) 2 ∼ 1/(g 2 ℓs). Ifboth are the mass of the same object
then g ∼ N − 1 4 .
Let us now compute the entropy of this object in the string description and in the BH
description at the correspondence point RS ∼ ℓs, g ∼ N − 1 4 . The BH entropy is
S ∼ R 2 S /G(4)
N ∼ g−2 ∼ √ N. (20.11)
The string entropy is the logarithm of the degeneracy of states at the mass level M.
String theories are a particular case of two-dimensional CFTs [455], which in general have
an infinite spectrum of states. The degeneracy of states of a CFT characterized by a central
charge c, for large values of the two-dimensional energy E, isgivenbyCardy’s formula,
ρ(E) ∼ e √ π(c−24E0)EL/3 , (20.12)
where E0 is the lowest energy and L the size of the spatial coordinate of the twodimensional
theory. For string theories L ∼ ℓs and E = M2 ∼ N/ℓ2 s . Therefore, ρ ∼ eM/M0
and S = ln ρ ∼ √ N, which is in good qualitative agreement with the result in the BH picture.
The BH–string correspondence principle can be extended to higher-dimensional
Schwarzschild BHs and also to charged BHs, generalizing at the same time the string picture
to a string/brane picture characterized by the same conserved quantities.
Clearly, this principle underlies the logic of the calculation of entropies of stringy BHs
depicted in Figure 20.1. It works best when there is unbroken supersymmetry (extreme
BHs); it can be argued that the counting of states remains unmodified when we vary g.Inthe
next few sections we are going to construct these d = 4, 5 stringy extreme supersymmetric
BH solutions.
20.2.2 Black holes from wrapped intersecting branes
We have seen that, in order to construct d = 4extreme BH solutions with regular horizons,
we need at least four extended objects that break seven eighths of the supersymmetries.
In d = 5, three are necessary and, since this case is a bit simpler and the counting of microstates
for it clearer, we are going to start with it.
There are many possible configurations of three extended objects that give rise to a regular
BH in d = 5 upon compactification on T5 . They are related by string (U) dualities in
the five compact dimensions, which appear in d = 5ashidden symmetries of the maximal
N = 4, d = 5 SUEGRA. These symmetries do not act on the Einstein metric (although they
do act on the moduli), thus any of these configurations is equally good for obtaining a BH
metric (the issue of U duality will be studied in Section 20.2.3). Not all the corresponding
d = 10 configurations are equally simple to treat, basically because we do not have good
string descriptions of KK monopoles of S5-branes. D-brane configurations are clearly preferred.
The simplest configurations of this kind are D5 D1 W, as proposed in [208] as a
2 The BH–fundamental-string transition has been studied further in [284, 554, 618, 619].