String Theory and M-Theory

12.1 Black-brane solutions in string theory and M-theory 619
in order to incorporate self-duality. Using the formula G10 = 8π 6 g 2 s ℓ 8 s from
Chapter 8 and
TDp = (2π) −p ℓ −(p+1)
s g −1
s , (12.26)
from Chapter 6, one obtains
(rp/ℓs) 7−p = (2 √ π) 5−p
7 − p
Γ gsN. (12.27)
2
The extremal black D3-brane
In the special case of p = 3 the formulas above give a constant dilaton. In
this case, letting r3 = R, Eq. (12.27) takes the form
R 4 = 4πgsNα ′2 . (12.28)
Furthermore, the near-horizon limit of the metric takes the form
ds 2 ∼ (r/R) 2 dx · dx + (R/r) 2 dr 2 + R 2 dΩ 2 5. (12.29)
The change of variables z = R 2 /r brings this to the form
ds 2 2 dx · dx + dz2
∼ R
z2 + R 2 dΩ 2 5. (12.30)
This shows that the near-horizon geometry is AdS5 ×S 5 , where both factors
have radius R.
Nonextremal black D-branes
The extremal black D-brane solutions, which describe the geometry and
other fields generated by a set of coincident D-branes, are supersymmetric.
However, the equations of motion following from the action Eq. (12.18) also
have nonsupersymmetric charged solutions, which are called nonextremal
black p-branes (see Problem 12.6). We only consider p < 7 here, since the
other cases are somewhat special and not relevant to the discussion in the
remainder of this chapter. 6 For p < 7 the line element is given by
ds 2 = −∆+(r)∆−(r) −1/2 dt 2 + ∆−(r) 1/2 dx i dx i
+∆+(r) −1 ∆−(r) γ dr 2 + r 2 ∆−(r) γ+1 dΩ 2 8−p, (12.31)
6 7-branes have a conical deficit angle at their core, like point particles in D = 3. Their geometry
is discussed in Chapter 9 in connection with F-theory. 8-branes are domain walls in ten
dimensions that divide the space-time into disjoint regions and 9-branes are space-time-filling.