Gravity and Strings

586 String black holes in four and five dimensions
Now, introducing a parametrization similar to the five-dimensional Qi ≡ Ni − ¯Ni and
ℓ 2 s
ω 2
ω 2 R 4 1 R2 2 ···R2 6
ℓ 16
s ˆg4
ˆg2 ≡ 4ND6 ¯ND6,
≡ 4NW ¯NW,
we obtain for the mass and entropy
M = (ND2 + ¯ND2) R1 R6
ˆgℓ 3 s
+ (NS5 + ¯NS5) R1 ···R5
ˆg 2ℓ6 s
S = 2π √Ni
−
¯Ni .
ω 2 R 2 2 ···R2 5
ℓ 10
s ˆg2
≡ 4ND2 ¯ND2,
ω2 R2 6
ℓ2 ≡ 4ND6
¯ND6,
s ˆg2
+ (ND6 + ¯ND6) R1 ···R6
ˆgℓ 7 s
We can also express the moduli in terms of them, ℓs, and ˆg.
i
+ (NW + ¯NW) 1
,
20.2.3 Duality and black-hole solutions
R1
(20.43)
(20.44)
The solutions we have obtained are particular solutions that have only a few vectors and
scalars excited of maximal N = 4, d = 5 and N = 8, d = 4SUEGRA. These theories have,
respectively, 27 and 56 U(1) vector fields, which are rotated among themselves by the
U-duality groups E6(+6) and E7(+7) (see Table 16.1) and scalars that parametrize the coset
spaces E6(+6)/USp(8) and E7(+7)/SU(8) [263] and are also rotated the same U-duality
groups, but the Einstein metrics are U-duality-invariant and all unbroken supersymmetries
are preserved.
Several questions arise immediately. How does U duality act on these solutions and on
their d = 10 description? How does U duality act on physical parameters such as the masses
and entropies? What is the more general BH-type solution of these theories?
Most U-duality rotations correspond to T and S dualities in higher dimensions, whose
effects on the components are well known to us. We can use them to find configurations
that are more convenient for our purposes. For instance, we can dualize the d = 4BHwe
have obtained into one composed entirely of D-branes [76, 77], whose stringy description is
much better known than that of the S5s we have used. If we denote by T n aT-duality transformation
in the nth coordinate, ignoring time and the three overall transverse coordinates,
we find
1 2 3 4 5 6
D6 + +++++
S5 + ++++∼
D2 + ∼∼∼∼+
W →∼∼∼∼∼
−→
ST 2 T 1 T 5
1 2 3 4 5 6
D3 ∼∼++∼+
D5 +++++∼
D3 ∼+∼∼++
D1 +∼∼∼∼∼
(20.45)