and

X

H

where x5 has period 4π. The solution is then formulated in terms of a set

of two-center harmonic functions H de¬ned by

4 1 q0 q0

H0 = H0 = ’

+ + , (11.149)

2

RTN | x | | x ’ x0 |

L

and

p1 q1

1

H= , H1 = 1 + , (11.150)

| x ’ x0 | | x ’ x0 |

where x0 = (0, 0, L) and (p1 , q1 , q0 ) are constants. When compacti¬ed to

four dimensions, this background is a bound state of one D6-brane located

at | x |= 0 and a black hole with D4-D2-D0 brane charges (p1 , q1 , q0 ).

Using these harmonic functions, Eq. (11.145) is solved by

H1

1

dx5 + ω 0 1

˜ =d + 3 dH , (11.151)

H0

while the solution to Eq. (11.147) is provided by

(H 1 )2

f = H1 + 3 . (11.152)

H0

Moreover,

(H 1 )3 H1 H 1

(dx5 + ω 0 ) + ω (4) ,

ω = ’ H0 + 2 0 2 + (11.153)

0

(H ) H

where ω (4) is the solution of

dω (4) = HI dH I ’ H I dH I . (11.154)

3 3

The black-ring solution is then obtained by taking the limit RTN ’ ∞. This

is the same sort of limit considered earlier when we discussed the 4d/5d

connection relating a rotating black hole in ¬ve dimensions to one with

suitable charges in four dimensions.

The ¬ve-dimensional metric can then be written in the form

ds2 = G(4) dxµ dxν + »(dx5 ’ Aµ dxµ )2 , (11.155)

5 µν

598 Black holes in string theory

(4)

where Gµν is the four-dimensional metric, » is a scalar and Aµ is a U (1)

gauge ¬eld. The four-dimensional metric satis¬es the two-center attractor

equations.

EXERCISES

EXERCISE 11.10

Deduce Eq. (11.118) by projecting both sides of Eq. (11.120) on e’i±+K/2 „¦.

SOLUTION

Equation (11.120) is equivalent to

d ’U ’i±+K/2

„¦ ’ ei±+K/2 „¦ ∼ ’i“.

e e

d„

Taking the wedge product with e’i±+K/2 „¦, only the second term on the left

d

contributes since „¦ § „¦ = „¦ § d„ „¦ = 0. Thus

d ’U K d

e „¦ § „¦ ’ e’U e’i±+K/2 „¦ § ei±+K/2 „¦

’ e

d„ d„

∼ ’ie’i±+K/2 „¦ § “.

The imaginary part of this equation is now integrated over the manifold. A

useful identity that implies that the integral of the second term is real is

d d

e’i±+K/2 „¦ = e’i±+K/2 „¦ §

ei±+K/2 „¦ § ei±+K/2 „¦ ,

d„ d„

which is derived by di¬erentiating Eq. (11.110) written in the form

e’i±+K/2 „¦ § ei±+K/2 „¦ = ’i.

In this way, one obtains

d ’U K 1

„¦ § „¦ = ’ eK/2 e’i± „¦ + ei± „¦ § “.

i e e

d„ 2

Using Eq. (11.110) to simplify the left-hand side and Eq. (11.114) to simplify

the right-hand side, one obtains

d ’U

= |Z|,

e

d„

11.6 Small BPS black holes in four dimensions 599

which is equivalent to Eq. (11.118). 2

11.6 Small BPS black holes in four dimensions

Section 11.4 showed how the counting of microscopic degrees of freedom

reproduces the BH entropy of certain supersymmetric black holes. A crucial

requirement for this agreement is that Dp-branes wrapped on cycles of the

internal manifold excite enough di¬erent charges to give a solution with a

nonvanishing classical black-hole horizon.

Black holes can also be created using fundamental strings and their ex-

citations without invoking solitonic branes. The string spectrum consists

of an in¬nite tower of states with arbitrarily large masses. For su¬ciently

high excitations, or su¬ciently large coupling constant, gravitational col-

lapse becomes unavoidable. This implies that the Hilbert space of string

excitations should contain black holes. This opens up the interesting possi-

bility that certain string excitations admit an alternative interpretation as

black holes. In this section we discuss evidence that black holes are an al-

ternative description of certain elementary string excitations. The evidence

again follows from comparing the black-hole entropy obtained by counting

microscopic quantum states to the macroscopic black-hole entropy described

by geometry.

The di¬culty in making a black hole out of perturbative string excitations

is that an elementary string states do not excite all four types of charges in

the » factor of the metric in Eq. (11.77). Therefore, the area of the horizon

would vanish in the supergravity approximation, leaving a null singularity

at the origin. For large string excitation number N , the entropy is pro-

√ √2

portional to N , and the area of the horizon is A ∼ N p where p is

the four-dimensional Planck length. Even though the area of the horizon

is large in Planck units, it is of order one in string units, which explains

why the supergravity approximation gives zero. Therefore, ± corrections to

the supergravity approximation are important for obtaining a horizon of ¬-

nite radius that shields the singularity, as required by the cosmic censorship

conjecture.

Microstate counting

As a speci¬c example, let us consider the heterotic string compacti¬ed on a

six-torus to four dimensions, which was discussed in Chapter 7.21 This gives

21 The techniques discussed in this section are of more general applicability than this speci¬c

example.

600 Black holes in string theory

28 U (1) gauge ¬elds.22 These transform as a vector of the O(22, 6; ) duality

group. 22 of the gauge ¬elds belong to 22 vector multiplets, while the other

6 belong to the supergravity multiplet. The allowed charges of these gauge

¬elds are given by sites of the Narain lattice, as described in Chapter 7.