even integer. In other words, since the charges are encoded in the internal

momenta (pR , pL ), where pL has 22 components and pR has six components,

p2 ’ p2 = 2N. (11.156)

R L

The mass formula for these states is

1 1 1

± M 2 = p2 + NR = p2 + NL ’ 1, (11.157)

2R 2L

4

where NL and NR are the usual oscillator excitation numbers.

Dabholkar“Harvey states

Most states with masses given by Eq. (11.157) are unstable, but the BPS

states are stable. The BPS states, that is, the state belonging to short

supermultiplets for which the mass saturates the BPS bound, have NR = 0.

In this case

± M 2 = 2p2 , (11.158)

R

while NL is arbitrary. This results in a whole tower of stable states, which

are sometimes called Dabholkar“Harvey states. For these states the level-

matching condition reduces to

NL ’ 1 = N. (11.159)

For example, if there is winding number w and Kaluza“Klein excitation num-

ber n on one cycle of the torus, then N = |nw|. In general, the degeneracy

of states for large N is given by

√

≈ exp 4π N ,

dN (11.160)

resulting in a leading contribution to the black-hole entropy given by

√

S = log dN ≈ 4π N . (11.161)

22 We assume generic positions in the moduli space so that there is no enhanced gauge symmetry.

11.6 Small BPS black holes in four dimensions 601

Counting states

Let us now compute the corrections to Eq. (11.161). The degeneracy dN

denotes the number of ways that the 24 left-moving bosonic oscillators can

give NL = N + 1 units of excitation. This can be encoded in a partition

function

16

dN e’βN =

Z(β) = , (11.162)

∆(q)

where

q = e’β = e2πi„ , (11.163)

and the factor of 16 is the degeneracy of right-moving ground states. The

factor ∆(q) is related to the Dedekind · function by

∞

24

(1 ’ q n )24 .

∆(q) = ·(„ ) =q (11.164)

n=1

The large-N degeneracy depends on the value of this function as q ’ 1 or

β ’ 0. Under a modular transformation the Dedekind · function transforms

as

√

·(’1/„ ) = ’i„ ·(„ ). (11.165)

As a result,

’12

β 2 /β

’β

∆(e’4π

∆(e )= ), (11.166)

2π

which, by using ∆(q) ≈ q for small q, gives the estimate

’12

β 2 /β

’β

e’4π

)≈

∆(e . (11.167)

2π

This result is extremely accurate, since all corrections are exponentially

suppressed.

Now one can compute dN , as in earlier chapters.

1 dq 1 16 dq

dN = Z(β) = . (11.168)

q N +1 ∆(q) q N +1

2πi 2πi

Using Eq. (11.166), this can be approximated for large N by

√

ˆ

dN ≈ 16 I13 (4π N ), (11.169)

where

µ+i∞

1 2 /4t

ˆ (t/2π)’ν’1 et+z

Iν (z) = dt (11.170)

2πi µ’i∞

602 Black holes in string theory

is a modi¬ed Bessel function. This formula includes all inverse powers of N ,

but it does not include terms that are exponentially suppressed for large N .

A saddle-point estimate for large N gives

√ √

27 15

≈ 4π N ’

S = log dN log N + log 2 + . . . (11.171)

2 2

This shows that the leading-order entropy of the black hole obtained by

counting microstates is proportional to the mass, S ∼ M . We could try to

compare this to the corresponding macroscopic black-hole solution, but the

black hole constructed of perturbative Dabholkar“Harvey states only excites

two of the four charges that are needed to get a nonvanishing area of the

event horizon. So the result is zero in the supergravity approximation. This

is the best that one could hope for in this approximation, because if the area

were nonzero, the entropy would be proportional to M 2 .

So how can we construct a macroscopic black hole that reproduces the

entropy (11.169)? The resolution lies in realizing that elementary string

states become heavy enough to form black holes at large coupling. As a

result, one should expect that string-theoretic corrections to supergravity,

such as terms in the action that are higher order in the curvature, modify the

macroscopic geometry and the associated entropy, yielding a nonvanishing

result.

Macroscopic entropy

The preceding analysis gave a very accurate result for the degeneracy of

states dN of a certain class of supersymmetric black holes. Remarkably, this

formula has been reproduced precisely from a dual macroscopic analysis.

The crucial point is that the supergravity approximation is inadequate for

this problem, and one must include higher-order terms in the string e¬ective

action. In general, this is a hopelessly di¬cult problem. However, in the case

at hand, it turns out that the relevant higher-order terms can be computed.

In order to compute these corrections, it is more convenient to work with

the type IIA string theory compacti¬ed on K3 — T 2 instead of the heterotic

string on T 6 . According to a duality discussed in Chapter 9, this is an

equivalent theory. In this description the machinery of special geometry is

applicable. The quantum gravity corrections are then encoded in corrections

to the prepotential. No closed expression for these corrections is known in

general, but luckily in this case there is N = 4 supersymmetry. When

there is this much supersymmetry, a nonrenormalization theorem implies

that only the ¬rst correction to the prepotential is nonvanishing, and this

11.6 Small BPS black holes in four dimensions 603

is enough to reproduce the microscopic entropy discussed in the previous

section. A key ingredient in the analysis is the attractor mechanism.

Type IIA superstring theory on K3 — T 2 has N = 4 supersymmetry in

four dimensions, but the attractor mechanism analysis is carried out most