Black holes in string theory
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to one brane and parallel to the other is a multiple of four and that a (p - 2)-brane
can lie within a p-brane [20]. In the present context, therefore, we may combine
the p = 5 solitons described earlier with the p = I soli tons, which we take
to have longitudinal coordinates xO,xl-the remaining four xQ coordinates are
transverse to the I-brane but parallel to the 5-brane. Then again just half of the
supersyrnmetry charges are preserved leaving just eight, corresponding to N = 2
supersyrnmetry in four dimensions.
In the first instance, the harmonic function HI is a function of l' where
1'2 == (x 2 )2 + (x 3 )2 + {x4)2 + (xS)2 + r2
(10.139)
and varies as 1'-6. However, when the four directions x 2 , x 3 , x4, x S are
compactified on T4 C TS of volume V4, this has the effect of 'smearing' the
D I-branes. For example, compactifying the coordinate x 2 on a circle of radius
R, the system of DI-branes is replaced by an infinite array of parallel branes
a distance 21f R apart. Because they are extreme states, the effective harmonic
function HI for the array is easily written down and, for r » R, this function
varies as [1'2 - (x2)2rS/2, as if the D I-branes also wrapped x 2 . The effective
number of D2-branes is ql# / R. This result obviously generalizes to the case
in which all four coordinates are compactified on T4. Then HI varies as r- 2 , just
like Hs(r), and is given by (10.130) with p = 5 but with the effective number qS
of D5-branes given by
qS = (21f #)4
(10.140)
V4
ql·
The composite system has warp factors that are given by the 'harmonic
function sum rule' [21]. With the proviso that the harmonic function HI for
the Dl-branes is replaced by the smeared version HI just described, so that both
of the harmonic functions are functions of the overall transverse distance r. the
warp factors of the composite system are just the product of the separate warp
factors in each of the three sectors, namely parallel to both, transverse to both and
transverse to the I-brane but parallel to the 5-brane. The dilaton is also given by
the product. Thus, for the combined system,
ds2 = Hs-I/4H~3/4[(dxO)2 _ (dx l )2] _ HS-I/4HII/4[(dx 2 )2 + ... + (dx S )2]
- Hi/4iill/4[(dyfi)2 + ... + (dl)2]
(10.141)
et; = H;I/2 HII/2
(10.142)
I
C(S) = [H s - I - l]dx° "dx l ... " dx s CO) = [H I - - l]dx°" dx I.
(10.143)
Then the dilaton is, as required, finite at the event horizon. (We note, incidentally,
that this is only possible when at least two 'charges' are activated.) In addition.
the warp factor associated with the yi directions now approximates the H warping
in (10.1 01). Thus. ignoring the directions x 2 , x 3 • x4. x S for the moment, all
301
to one brane and parallel to the other is a multiple of four and that a (p - 2)-brane
can lie within a p-brane [20]. In the present context, therefore, we may combine
the p = 5 solitons described earlier with the p = I soli tons, which we take
to have longitudinal coordinates xO,xl-the remaining four xQ coordinates are
transverse to the I-brane but parallel to the 5-brane. Then again just half of the
supersyrnmetry charges are preserved leaving just eight, corresponding to N = 2
supersyrnmetry in four dimensions.
In the first instance, the harmonic function HI is a function of l' where
1'2 == (x 2 )2 + (x 3 )2 + {x4)2 + (xS)2 + r2
(10.139)
and varies as 1'-6. However, when the four directions x 2 , x 3 , x4, x S are
compactified on T4 C TS of volume V4, this has the effect of 'smearing' the
D I-branes. For example, compactifying the coordinate x 2 on a circle of radius
R, the system of DI-branes is replaced by an infinite array of parallel branes
a distance 21f R apart. Because they are extreme states, the effective harmonic
function HI for the array is easily written down and, for r » R, this function
varies as [1'2 - (x2)2rS/2, as if the D I-branes also wrapped x 2 . The effective
number of D2-branes is ql# / R. This result obviously generalizes to the case
in which all four coordinates are compactified on T4. Then HI varies as r- 2 , just
like Hs(r), and is given by (10.130) with p = 5 but with the effective number qS
of D5-branes given by
qS = (21f #)4
(10.140)
V4
ql·
The composite system has warp factors that are given by the 'harmonic
function sum rule' [21]. With the proviso that the harmonic function HI for
the Dl-branes is replaced by the smeared version HI just described, so that both
of the harmonic functions are functions of the overall transverse distance r. the
warp factors of the composite system are just the product of the separate warp
factors in each of the three sectors, namely parallel to both, transverse to both and
transverse to the I-brane but parallel to the 5-brane. The dilaton is also given by
the product. Thus, for the combined system,
ds2 = Hs-I/4H~3/4[(dxO)2 _ (dx l )2] _ HS-I/4HII/4[(dx 2 )2 + ... + (dx S )2]
- Hi/4iill/4[(dyfi)2 + ... + (dl)2]
(10.141)
et; = H;I/2 HII/2
(10.142)
I
C(S) = [H s - I - l]dx° "dx l ... " dx s CO) = [H I - - l]dx°" dx I.
(10.143)
Then the dilaton is, as required, finite at the event horizon. (We note, incidentally,
that this is only possible when at least two 'charges' are activated.) In addition.
the warp factor associated with the yi directions now approximates the H warping
in (10.1 01). Thus. ignoring the directions x 2 , x 3 • x4. x S for the moment, all