ferent kinds of charges. These can be studied in the context of compacti¬-

cations of the type IIB superstring theory on a ¬ve-torus T 5 . The analysis

is carried out in the approximation that ¬ve of the ten dimensions of the

IIB theory are su¬ciently small and the black holes are su¬ciently large so

that a ¬ve-dimensional supergravity analysis can be used.

N = 8 supergravity for D = 5

The supergravity theory in question is N = 8 supergravity in ¬ve dimen-

sions. This contains a number of one-form and two-form gauge ¬elds. How-

ever, by duality transformations, the two-forms can be replaced by one-

forms. Once this is done, the resulting theory contains 27 U (1) gauge ¬elds.

Furthermore, the theory has a noncompact E6,6 global U-duality symme-

try.8 The 27 U (1) s belong to the fundamental 27 representation of this

group. Therefore, a charged black hole in this theory can carry 27 di¬erent

types of electric charges. Some of these electric charges can be realized by

wrapping branes and exciting Kaluza“Klein excitations. A speci¬c example

is discussed below.

The black-hole solution

Three-charge black holes in ¬ve dimensions can be obtained by taking Q1 D1-

branes wrapped on an S 1 of radius R inside the T 5 , Q5 D5-branes wrapped

on the T 5 = T 4 —S 1 , and n units of Kaluza“Klein momentum along the same

circle. Each of these objects breaks half of the supersymmetry, so altogether

7/8 of the supersymmetry is broken, and one is left with solutions that have

four conserved supercharges. Other equivalent string-theoretic constructions

of these black-hole solutions are related to the one considered here by U-

duality transformations. Some examples are given later.

There are a variety of ways to analyze this system. One of them is in terms

of a ¬ve-dimensional gauge theory. Since the Q1 D1-branes are embedded

inside the Q5 D5-branes, this con¬guration can be described entirely in

terms of the U (Q5 ) world-volume gauge theory of the D5-branes. In this

description a D-string wound on a circle is described by a U (Q5 ) instanton

that is localized in the other four directions. So, altogether, there are Q1

such instantons. The Kaluza“Klein momentum can also be described as

excitations in this gauge theory.

8 In the supergravity approximation it is a continuous symmetry.

568 Black holes in string theory

The ¬ve-dimensional metric describing this black-hole can be obtained

from the ten-dimensional type IIB theory by wrapping the corresponding

branes as described above, or it can be constructed directly. In either case,

the resulting metric can be written in Einstein frame in the form

ds2 = ’»’2/3 dt2 + »1/3 dr2 + r2 d„¦2 , (11.46)

3

where

3

ri 2

»= 1+ . (11.47)

r

i=1

The relation between the parameters ri and the charges Qi is derived below.

This solution describes an extremal three-charge black hole with a vanish-

ing temperature T = 0. Note that this formula reduces to the extremal

Reissner“Nordstr¨m black-hole metric given in Eq. (11.32) in the special

o

case r1 = r2 = r3 , that is, when the three charges are equal. The dilaton

is a constant, so there is a globally well de¬ned string coupling constant gs .

Thus, the string-frame metric di¬ers from the one given above only by a

constant factor.

The horizon of the black hole in Eq. (11.46) is located at r = 0, and its

area is

A = 2π 2 r1 r2 r3 . (11.48)

This vanishes when any of the three charges vanishes, which is the reason

that three charges have been considered in the ¬rst place. Put di¬erently,

one needs to break 7/8 of the supersymmetry in order to form a horizon

that has ¬nite area in the supergravity approximation, and this requires

introducing three di¬erent kinds of excitations. When there is only one or

two nonzero charges, there still is a horizon of ¬nite area, but its depen-

dence on the string scale is such that its area vanishes in the supergravity

approximation. For the supergravity approximation to string theory to be

valid, it is necessary that the geometry is slowly varying at the string scale.

This requires ri s.

The black hole mass

Using Eq. (11.11) one can read o¬ the mass of the black hole M to be

2

πri

M = M1 + M2 + M3 where, Mi = . (11.49)

4G5

The fact that the masses are additive in this way is a consequence of the

form of the metric. However, this had to be the case, because the BPS

condition is satis¬ed, and the charges are additive.

11.3 Black holes in string theory 569

To express the result for ri in terms of ten-dimensional quantities, the

value of G5 needs to be determined. Letting (2π)4 V denote the volume of

the T 4 and R be the radius of the S 1 one obtains

G10

G5 = . (11.50)

(2π)5 RV

As explained in Chapter 8, G10 = 8π 6 gs 8 is the 10-dimensional Newton

2

s

constant in string units. Putting these facts together gives the relation

gs 8

2

s

2

ri

= Mi . (11.51)

RV

The masses Mi can be computed at weak string coupling using string-

theoretic considerations, namely the formulas for the mass of winding and

momentum modes derived in Chapter 7. In the string frame, the masses are

Q1 R

M1 = 2πRTD1 Q1 = ,

gs 2

s

Q5 RV

M2 = (2π)5 RV TD5 Q5 = (11.52)

,

gs 6

s

n

M3 = R.

Here Q1 and Q5 are the numbers of wrapped D1-branes and D5-branes,

respectively, and hence the values of the corresponding charges. Similarly,

n is the integer that speci¬es momentum on the circle.

The quantities TD1 and TD5 are the tensions of a single D1-brane and

2 2

D5-brane given in Chapter 6. Using these relations, the conditions ri s

become

R2 V

V 2

g s Q1 , g s Q5 1, gs n . (11.53)

4 6

s s

If R and V are of order string scale, and one wants gs 1, so as to be in

the perturbative string theory regime, then all three charges must be large.

Since the e¬ective expansion parameters in string perturbation theory are

actually gs Q1 and gs Q5 ,9 this takes one out of the perturbative regime. On

the other hand, when the couplings are small, the mass and the spectrum

of excitations can be computed by string-theoretic considerations.

The crucial fact is that supersymmetry allows us to extrapolate certain

properties from weak coupling to strong coupling reliably, so that results

that are obtained in the two limits can be compared meaningfully. The

property of this type that is of most interest is the number of quantum

9 These correspond to the ™t Hooft couplings in the corresponding large-N world-volume gauge

theories.

570 Black holes in string theory