Gravity and Strings

588 String black holes in four and five dimensions
• x 78 = 1 √ 2 Z (0) corresponds to D0 charge,
• y 78 =− 1 √ 2 Z (6)
123456 corresponds to a D6 wrapped on T6 ,
• xi7 = 1
5! √ 2 ɛijklmnZ (5)
jklmn correspond to S5As wrapped on jklmn,
• x i8 = 1
√ 2 p i correspond to KK momentum in the directions i,
• yi7 = 1
5! √ 2 ɛ(i) jklmn Z (6)
ric direction i, and
jklmn(i)
correspond to KK6As wrapped on jklmn with isomet-
• y i8 = 1
√ 2 Z (1) i correspond to F1s wrapped in the directions i.
The d = 4extremal BH that we have constructed corresponds to x 18 = (1/ √ 2)NW,
y 16 = (1/ √ 2)ND2, x 67 = (1/ √ 2)NS5, and y 78 =−(1/ √ 2)ND6, and the diamond formula
immediately gives the right value for the entropy. The dual configurations give exactly the
same result. In fact, the above identification between entries of the central-charge matrix
and charges of extended objects is clearly not unique, but is defined only up to U-duality
rotations. We can also look at it from a different point of view: the objects we have considered
are all wrapped on T 6 and are T dual to each other and, essentially, they cannot be
distinguished from the d = 4central-matrix point of view. Their masses may be different
if they are related by S dualities, though. On the other hand, since U duality acts on the
moduli, too, we have to use the description in which compactification radii are bigger than
the critical self-dual radius and the coupling constant is small.
It would be very interesting to have explicitly the most general U-duality-invariant BHtype
solutions of these theories consistent with the no-hair theorems, which would be similar
to the SWIP solutions of pure N = 4, d = 4 SUEGRA that we studied in Section 12.2.1
to check these formulae, but obtaining them turns out to be an extremely difficult problem.
A simpler problem consists in finding a generating solution that would give the general solution
when we act on it with a general U-duality transformation, which would preserve the
metric. Simple arguments [75, 271] tell us that such solutions must have, respectively, four
and five independent charge parameters in d = 4 and 5 dimensions. It is not hard to find
brane configurations with these parameters and generating solutions have been proposed in
[144, 145].
20.3 Entropy from microstate counting
In the previous section we constructed BH solutions of maximal d = 4, 5 SUEGRA and
identified the N = 2A/B, d = 10 SUEGRA solutions they originate from. In the extreme
cases, we identified unambiguously their “components,” which places us in the upper-lefthand
box of Figure 20.1, and allows us to move clockwise in the figure and calculate the