the classical entropy, which is meaningful for strong coupling. This type of

reasoning could break down if short supermultiplets join up to give a long

supermultiplet. Strictly speaking, the quantity that can be continued safely

from weak coupling to strong coupling is an index, which typically counts

the number of bosonic states minus the number of fermionic states, whereas

the entropy is the logarithm of the sum of these numbers. Usually, this

distinction can be ignored.

The entropy

Using Eqs (11.48), (11.50) and (11.51), one ¬nds that the entropy is

2πgs 4

A

=√ s

S= M1 M2 M3 . (11.54)

4G5 RV

Using the relations in (11.52) to re-express this in terms of the charges, one

obtains the elegant result

S = 2π Q1 Q5 n. (11.55)

As was mentioned earlier, there are 27 possible electric charges, and this

is the result when only a speci¬c three of them are nonzero. The charges

transform as a 27 representation of the noncompact E6,6 symmetry group

of N = 8 supergravity in D = 5. The entropy should be invariant under

this symmetry group.10 In other words, there should be an E6,6 invariant ∆

that is cubic in the 27 electric charges such that the entropy takes the form

√

S = 2π ∆. (11.56)

The invariant ∆ generalizes the factor Q1 Q5 n appearing in Eq. (11.55).

The construction of the cubic invariant is relatively simple in this case.

The 27 representation is also an irreducible representation of the maximal

compact subgroup U Sp(8). That group has a unique cubic invariant, which

therefore must also be the E6,6 invariant. In the case of ¬ve dimensions the

central charge matrix ZAB is a real antisymmetric 8 — 8 matrix that is also

symplectic traceless. This means that, given a symplectic matrix „¦AB , one

has tr(„¦Z) = 0.11 This is one real condition, so Z contains 27 independent

real charges, as expected. The unique cubic invariant with manifest U Sp(8)

10 Stringy corrections to the formula need only be invariant under the discrete E6 (¢ ) U-duality

subgroup.

11 We can choose „¦AB to be the antisymmetric matrix whose nonzero matrix elements with

A < B are „¦12 = „¦34 = „¦56 = „¦78 = 1. A symplectic matrix A satis¬es AT „¦A = „¦.

11.3 Black holes in string theory 571

symmetry is

1

∆=’ tr(„¦Z„¦Z„¦Z), (11.57)

48

where the normalization is chosen for later convenience.

By a transformation of the form, Z ’ AT ZA, where A is a symplectic

matrix,12 the matrix ZAB can be brought to a canonical form in which its

only nonzero entries for A < B are Z12 = x1 , Z34 = x2 , Z56 = x3 , Z78 = x4 ,

where xi = 0 and the xi s are real. A symmetric way of writing this is

x1 = Q 1 ’ Q 2 ’ Q 3 , x2 = ’Q1 + Q2 ’ Q3 ,

x3 = ’Q1 ’ Q2 + Q3 , x4 = Q1 + Q2 + Q3 . (11.58)

If one evaluates ∆ for these choices, one ¬nds the desired result:

1

x3 = Q1 Q2 Q3 .

∆= (11.59)

i

24

Thus, up to a change of basis, the three-charge solution is completely general.

Duality and other black-hole con¬gurations

Three-charge supersymmetric black holes in ¬ve dimensions have been de-

scribed above as D1-D5-P bound states in the toroidally compacti¬ed type

IIB theory. Here D1 refers to the Q1 D1-branes wrapped on a y 1 circle,

D5 refers to the Q5 D5-branes wrapped on the y 1 · · · y 5 torus, and P refers

to the n units of Kaluza“Klein momentum on the y 1 circle. Using the var-

ious possible S and T dualities that exist for type II theories, this brane

con¬guration can be related to various dual con¬gurations describing black

holes that have an entropy given by Eq. (11.55), with the corresponding

charges of the dual brane con¬guration. For example, an S-duality transfor-

mation replaces the D1-branes by F1-branes (fundamental strings) and the

D5-branes by NS5-branes. The Kaluza“Klein momenta P are una¬ected.

Alternatively, a T-duality transformation along the y 1 direction maps the

type IIB con¬guration to a type IIA con¬guration with the D1-branes map-

ping to D0-branes and the D5-branes mapping to D4-branes. Moreover, the

Kaluza“Klein momentum maps to an F1-brane wrapped n times on the dual

y 1 circle. Further T dualities give a host of other equivalent type IIA and

type IIB con¬gurations. Exercise 11.6 works out an example of such a dual

description.

12 This is appropriate, because U Sp(8) is the automorphism group of the N = 8, D = 5 super-

symmetry algebra.

572 Black holes in string theory

M-theory interpretation

Starting from the Type IIA con¬guration described above, one can carry out

two more T-duality transformations along the y 2 and y 3 directions to obtain

a type IIA con¬guration consisting of Q1 D2-branes wrapped on y 2 and y 3 ,

Q5 D2-branes wrapped on y 4 and y 5 and n fundamental strings wrapped on

y 1 . This con¬guration can be interpreted at strong coupling as M-theory

compacti¬ed on a 6-torus. Calling the M-theory circle coordinate y 6 , the n

fundamental type IIA strings are then identi¬ed as n M2-branes wrapped

on the y 1 and y 6 circles. The two sets of D2-branes are then identi¬ed as

sets of M2-branes. Altogether, there are three sets of M2-branes wrapped

on three orthogonal tori. This is a satisfying picture in that it puts the

three sources of charges on a symmetrical footing, which nicely accounts for

their symmetrical appearance in the entropy formula. The veri¬cation that

this brane con¬guration gives the same entropy as before is a homework

problem.

Nonextremal three-charge black holes for D = 5

The extremal three-charge black-hole solutions in ¬ve dimensions given

above have nonextremal generalizations, which describe nonsupersymmetric

black holes with ¬nite temperature. These black holes are described by the

metric

2

1/3 dr

’2/3 2

2

+ r2 d„¦2 ,

ds = ’h » dt + » (11.60)

3

h

where

2

r0

h=1’ 2 (11.61)

r

and

3

ri 2

with ri = r0 sinh2 ±i ,

2 2

»= 1+ i = 1, 2, 3. (11.62)

r

i=1

This reduces to the extremal metric in Eq. (11.46) in the limit r0 ’ 0

with ri held ¬xed. Moreover, the limit ±i ’ 0 with r0 held ¬xed gives the

Schwarzschild metric in ¬ve dimensions given in Eq. (11.9).

The mass of this black hole can be read o¬ using the same rules as before

resulting in

2

πr0

M= (cosh 2±1 + cosh 2±2 + cosh 2±3 ) . (11.63)

8G5

11.3 Black holes in string theory 573

The inclusion of the factor h in the metric shifts the position of the event

horizon from r = 0 to r = r0 . At the horizon the factor » takes the value

3

cosh2 ±i .