m

the left-moving or right-moving ground state, which is always 16 for a type II

string. However, multiplicative numerical factors turn out to be completely

negligible.

The degeneracy is given by the coe¬cient of w nW in the generating func-

tion

∞

1 + wm 4

G(w) = N0 . (11.96)

1 ’ wm

m=1

The numerator takes account of the four fermions and the denominator

takes account of the four bosons. To be precise, in this formula the fermions

are taken here to be in the R sector. The NS sector would give an equal

contribution (after GSO projection).

The degeneracy is evaluated for large nW by representing it as a contour

integral and using a saddle-point evaluation, as was described in Chapter 2.

11.4 Statistical derivation of the entropy 585

It is already clear that the answer is a function of N = nW = nQ1 Q5 only.

However, while this is true for the single-string sector under consideration,

it is not true for the subdominant multiple-string con¬gurations that are

not being considered. Evaluation of the degeneracy for large N requires

knowing the behavior of G near w = 1. This is obtained using the Jacobi

theta function identity

1

θ4 (0|„ ) = √ θ2 (0| ’ 1/„ ), (11.97)

’i„

where w = eiπ„ , for the representations

∞

1 ’ wm

θ4 (0|„ ) = (11.98)

1 + wm

m=1

∞

2

w(n’1/2) .

θ2 (0|„ ) = (11.99)

n=’∞

This implies that as w ’ 1

2

π2

log w

G(w) ’ ’ exp ’ . (11.100)

π log w

Then writing the degeneracy in the form

1 G(w)dw

d(Q1 , Q5 , n) = , (11.101)

wN +1

2πi

and using a saddle-point approximation, one ¬nds for large N that

d(Q1 , Q5 , n) ∼ (Q1 Q5 n)’7/4 exp 2π Q1 Q5 n , (11.102)

and as a result the microcanonical black-hole entropy is given by

7

S = log d ∼ 2π Q1 Q5 n ’ log(Q1 Q5 n) + . . . (11.103)

4

Remarkably, the leading term in Eq. (11.103) reproduces the result obtained

earlier in Eq. (11.55) by computing the area of the horizon in the super-

gravity approximation. The exponential factor in the degeneracy factor is

multiplied by a power of Q1 Q5 n, and the ¬rst correction to the entropy for-

mula is proportional to the logarithm of this factor. This term is a stringy

correction to the entropy computed in the supergravity approximation. For

the particular black hole considered here, the leading correction to the BH

entropy formula is proportional to log(A/GD ). That seems to be the rule

quite generally. However, in contrast to the famous factor of 1/4 in the

leading term, A/4GD , the coe¬cient of the logarithm is not universal.

586 Black holes in string theory

Nonextremal black holes

It is natural to try to extend this analysis to nonextremal black holes. The

goal would be to reproduce the BH entropy formula by counting microstates.

However, in this case the black holes are no longer supersymmetric, and the

entropy formula is not guaranteed to extrapolate from weak coupling to

strong coupling without corrections. Because of this lack of control, the

result has not been derived in the general case using controlled approxi-

mations. What has been done successfully, in a mathematically controlled

way, is to compare the results for nearly extremal black holes for which the

nonextremality can be treated as a perturbation.

Let us consider the nonextremal D = 5 black holes described in Sec-

tion 11.3 in the special case that the only antibranes are n Kaluza“Klein

¯

excitations. In this case, the macroscopic entropy formula Eq. (11.69) be-

comes

S = 2π( Q1 Q5 n + Q1 Q5 n).

¯ (11.104)

The interpretation in terms of the world-volume theory of a string of winding

number W = Q1 Q5 is that the equations NL = nQ1 Q5 and NR = 0, which

were appropriate in the extremal case are now replaced by

NL = nQ1 Q5 and NR = nQ1 Q5 .

¯ (11.105)

In this case the degeneracy of states contains both a left-moving and a right-

moving factor

d ∼ exp(2π NL + 2π NR ). (11.106)

Taking the logarithm of both sides gives the microscopic entropy

S = 2π( NL + NR ), (11.107)

in exact agreement with the macroscopic formula! Surely this is better

agreement than one had any right to expect. At the very least, the approx-

imations that have been made require n ¯ n. We will not describe the

precise requirements for the approximations to be justi¬ed. Su¬ce it to say,

there is some region for which they are justi¬ed, but the result that one

obtains turns out to give agreement in an even larger region. It would be

nice if one could understand why this happened.

Hawking radiation

The nonextremal black holes have a ¬nite temperature and decay by the

emission of Hawking radiation. The brane picture makes the instability

11.5 The attractor mechanism 587

clear: it can be interpreted as brane-antibrane annihilation. Speci¬cally, for

the set-up in the preceding subsection, where there are both left-moving and

right-moving Kaluza“Klein excitations, they can collide to give a massless

closed-string state, which is then emitted from the black hole. The calcula-

tion has been carried out for n¯ n with the conclusion that the decay rate

as a function of frequency is

d4 k

A

d“(ω) = ω/T , (11.108)

’ 1 (2π)4

e

where the temperature is

√

2n¯

T= . (11.109)

πR

If one considers D1-brane anti-D1-brane or D5-brane/anti-D5-brane an-

nihilations instead, then a di¬erent viewpoint is convenient. When a brane

and an antibrane coincide, their common world volume contains a tachyonic

mode that arises as the lowest mode of the open string that connects the

brane to the antibrane. This tachyon signals an instability of the world-

volume theory, which results in the emission of closed-string radiation as

in the previous discussion. In fact, one can test this reasoning by using

Witten™s string ¬eld theory to describe the open string. Sen has argued per-

suasively that this theory gives a potential for the tachyon ¬eld, and that the

decay corresponds to sliding down this potential from a local maximum to a

local minimum, that is, tachyon condensation. Furthermore, the value of the

potential at the minimum should be lower than its value at the maximum

by exactly twice the brane tension. Thus the world-volume tachyon rolling

to the minimum of its potential precisely corresponds to brane“antibrane

annihilation. This results in the emission of closed-string quanta. In the

black-hole setting considered here, these quanta comprise the Hawking ra-

diation. This prediction for the gap between the maximum and minimum

of the potential has been tested numerically in Witten™s bosonic string ¬eld

theory, and it has been veri¬ed to high precision. Moreover, it has recently