String Theory and M-Theory

of Eq. (11.46) is
where
11.3 Black holes in string theory 575
ds 2 = −λ −1/2 dt 2 + λ 1/2 dr 2 + r 2 dΩ 2 2 , (11.77)
λ =
4
1 + ri
. (11.78)
r
i=1
This reduces to Eq. (11.30) when all four ri are equal. We can read off the
mass of the black hole from the large distance behavior of gtt using Eqs (11.7)
and (11.8). The result is
M =
4
i=1
Mi with Mi = ri
The area of the horizon, which is located at r = 0, is
. (11.79)
4G4
A = 4π √ r1r2r3r4. (11.80)
Putting these facts together, the resulting entropy is
S = A
4G4
= 16πG4 M1M2M3M4. (11.81)
Type IIA brane construction
It still remains to relate the four masses to four electric (or magnetic)
charges. This requires some sort of brane construction involving four types
of branes or excitations. To be specific, let us consider the type IIA theory
compactified on a six torus that is a product of six circles with coordinates
y 1 , . . . , y 6 and radii R1, . . . , R6. A brane configuration that preserves 1/8 of
the N = 8 supersymmetry, and therefore is suitable, is the following: Q1
D2-branes wrapped on the y 1 and y 6 circles, Q2 D6-branes wrapped on all
six circles, Q3 NS5-branes wrapped on the first five circles, and Q4 units of
Kaluza–Klein momentum on the first circle. The masses that correspond to
these types of excitations are
M1 = (2πR1)(2πR6)TD2Q1 = 1
gsℓ3 (R1R6)Q1,
s
M2 = (2πR1) · · · (2πR6)TD6Q2 = 1
gsℓ7 (R1 · · · R6)Q2,
s
M3 = (2πR1) · · · (2πR5)TNS5Q3 = 1
g2 s ℓ6 (R1 · · · R5)Q3,
s
M4 = 1
R1 Q4.
(11.82)