M4 = (2π)6 (R1 · · · R6 )TD6 P0 =

M3 = Q3 , P0 .

gs 3 gs 7

s s

Therefore, the entropy is

gs 8

2

A s

S= = 16π M1 M2 M3 M4 = 2π Q1 Q2 Q3 P0 ,

8R1 · · · R6

4G4

which reproduces Eq. (11.84). 2

EXERCISE 11.9

Construct the nonextremal generalization of the four-charge black hole by

analogy with the construction given for nonextremal black holes in ¬ve di-

mensions. Interpret the masses, charges, and entropy in terms of branes and

antibranes, as was done in the ¬ve-dimensional case.

582 Black holes in string theory

SOLUTION

The formulas analogous to Eqs (11.60) “ (11.62) for D = 4 black holes take

the form

’1

r0 r0

’1/2

2

dt2 + »1/2 dr2 + r2 d„¦2 ,

ds = ’» 1’ 1’ 2

r r

4

r0 sinh2 ±i

»= 1+ .

r

i=1

We can then extract the values of the masses in the usual way,

4 4

r0 r0 r0

2

M= sinh ±i + = cosh 2±i ,

4G4 2G4 8G4

i=1 i=1

which gives

r0 cosh 2±i

Mi = .

8G4

The outer horizon, located at r = r0 , has an area

4

2

A= 4πr0 cosh ±i .

i=1

This result can be interpreted as signaling the presence of Qi antibranes in

addition to the Qi branes. Identifying Qi ∼ exp(’2±i ) and Qi ∼ exp(2±i ),

we see that the result for the mass comes from the sum of the masses of the

branes and the antibranes, while the net charge comes from the di¬erence

Qi = Qi ’ Qi .

The result for the entropy can then be written in terms of these charges

4

A

S= = 2π ( Qi + Qi ).

4G4

i=1

2

11.4 Statistical derivation of the entropy

Extremal black holes

Now let us turn to the microscopical derivation of the entropy of the three-

charge supersymmetric black hole in ¬ve dimensions. The four-charge su-

persymmetric black hole in four dimensions can be analyzed in a similar

11.4 Statistical derivation of the entropy 583

manner, but that is left as a homework problem. The derivation was ¬rst

given by Strominger and Vafa in the context of type IIB compacti¬cations

on K3—S 1 . The discussion that follows analyzes the somewhat simpler case

of the toroidal compacti¬cation described in Section 11.3. The analysis can

be carried out either for the D1-D5-P system or for the S-dual F1-NS5-P

system. For de¬niteness, the discussion that follows refers to the former

set-up.

The fact that there are Q1 units of charge associated with D1-branes

means that there are Q1 windings of D1-branes around the circle. However,

the way this is achieved has not been speci¬ed. The two extreme possibil-

ities are (1) there are Q1 D1-branes each of which wraps around the circle

once and (2) there is a single D1-brane that wraps around the circle Q1

times. Altogether, the distinct possibilities correspond to the partitions of

Q1 . When there is more than one D1-brane, it is important that they form

a bound state in order to give a single black hole. The Q5 units of D5-brane

charge also can be realized in various ways. In all cases, one wants the D1-

D5-P system to form a bound state, so that one ends up with a localized

object in the noncompact dimensions.

The low-energy physics of these bound states is described by an orbifold

conformal ¬eld theory that is de¬ned on the circle of radius R. The ¬elds

in the conformal ¬eld theory correspond to the zero modes of open strings

that connect the D1-branes to the D5-branes. There are Q1 Q5 distinct such

strings, since each strand of D1-branes can connect to each strand of D5-

branes. That is the picture locally. However, imagine displacing this (small)

connecting string repeatedly around the circle. If there is a single multiply

wound D1-brane and a single multiply wound D5-brane (along the circle),

and if Q1 and Q5 have no common factors, then one must go around the

circle Q1 Q5 times to get back to where one started. Thus, the excitations

of this system are the same as what one gets from having a single string

wound around the circle Q1 Q5 times. Since this string is localized in the

noncompact dimensions, the only bosonic zero modes in its world-volume

theory correspond to its position in the four transverse compact dimensions.

Since the system is supersymmetric, there must therefore be four boson and

four fermion zero modes on the string world volume.

The system described above can be represented as an orbifold conformal

¬eld theory that is obtained by taking the tensor product of Q1 Q5 theo-

ries describing singly wound strings and then modding out by all of their

(Q1 Q5 )! permutations. This orbifold theory has many twisted sectors,15 and

15 They are given by the conjugacy classes of the permutation group SQ1 Q5 .

584 Black holes in string theory

just one of them corresponds to a single string wound Q1 Q5 times. How-

ever, this sector gives more low-energy degrees of freedom than any of the

other sectors, all of which involve multiple strings. Excitations of shorter

strings have higher energy, which suppresses them entropically. Therefore,

one obtains an excellent approximation to the entropy by only counting the

excitations of this long string, which is what we will do.

In view of the preceding, let us consider a single string wound Q1 Q5

times around a circle of radius R that is only allowed to oscillate in four

transverse directions. The question to be answered is how many di¬erent

ways are there of constructing a supersymmetric excitations of energy n/R.

The string can have left-moving and right-moving excitations, and the level-

matching condition is NL ’ NR = nW , where the winding number is

W = Q1 Q5 . (11.94)

Supersymmetry requires that either NL or NR vanishes, since then that

sector contributes a short (supersymmetric) representation in the tensor

product of left-movers and right-movers that gives the physical states of the

closed string. Whether NL or NR should vanish is determined by the sign

of nW .

If Nm and ni denote excitation numbers of the four transverse bosonic

i

m

and fermionic oscillators, respectively, then evaluation of NL or NR gives

∞

4

m(Nm + ni ).

i

|nW | = (11.95)

m

i=1 m=1

The degeneracy d(Q1 , Q5 , n) is then given by N0 times the number of choices

for Nm and ni that gives|nQ1 Q5 |. The factor N0 denotes the degeneracy of