1

M4 = R1 Q4 .

576 Black holes in string theory

Inserting these masses into the entropy formula given above and using

gs 8

2

G10 s

G4 = = (11.83)

(2πR1 ) · · · (2πR6 ) 8R1 · · · R6

yields the ¬nal formula for the entropy

S = 2π Q1 Q2 Q3 Q4 . (11.84)

This result bears a striking resemblance to Eq. (11.55).

Dual brane con¬gurations

As in the ¬ve-dimensional three-charge case, there are many other equivalent

brane con¬gurations that are related by various S- and T-duality transforma-

tions, and Eq. (11.84) applies to all of them. For example, a T-duality along

directions 1,2,3 gives a type IIB con¬guration. Following this by an S-duality

gives a type IIB con¬guration that has Q1 D3-branes wrapping directions

2,3,6, Q2 D3-branes wrapping directions 4,5,6, Q3 D5-branes wrapping di-

rections 1“5, and Q4 D1-branes wrapping direction 1. A further T-duality

along direction 6 gives a type IIA con¬guration consisting of three sets of

D2-branes wrapping orthogonal two-tori and a set of D6-branes wrapping

the entire 6-torus.

N = 8 supergravity in D = 4

The e¬ective four-dimensional theory in this case is N = 8 supergravity,

which has a noncompact E7,7 duality group. This is a continuous symmetry

in the supergravity approximation, though it is broken to the in¬nite discrete

U-duality group E7 ( ) by string theory corrections. Since we are working in

the supergravity approximation, the entropy of extremal black holes should

be invariant under the continuous symmetry group. Writing the entropy

√

in the form S = 2π ∆, we found ∆ = Q1 Q2 Q3 Q4 for a certain speci¬c

four-charge black hole in Eq. (11.84). We can use group theory to ¬gure out

how this should generalize.

N = 8 supergravity has 28 U (1) gauge ¬elds. There are therefore 28

distinct electric and magnetic charges that a black hole can carry. These

charges form a 56 representation of the E7,7 duality group. There is a

unique E7,7 -symmetric quartic invariant that can be constructed out of these

charges, and the product Q1 Q2 Q3 Q4 corresponds to a special case of that

invariant. The way this works is as follows: The matrix of central charges

is an 8 — 8 complex antisymmetric matrix

ZAB = qAB + ipAB , (11.85)

11.3 Black holes in string theory 577

where the qAB denote the 28 electric charges and the pAB denote the 28

magnetic charges. The invariant ∆ is a quartic expression in these central

charges. The E7,7 duality group has an SU (8) subgroup, which can be made

manifest. Subscripts A, B label an 8 and superscripts label an ¯ of SU (8).

8

AB

Thus the complex conjugate of the central charge is denoted Z . Now

consider the formula

1 2

∆ = tr(ZZZZ) ’ trZZ + 4(PfZ + PfZ), (11.86)

4

where the Pfa¬an is

1

µABCDEF GH ZAB ZCD ZEF ZGH .

PfZ = (11.87)

4 · 4!

2

Each of the terms in Eq. (11.86) has manifest SU (8) symmetry. The claim

is that this particular combination is the unique one (up to normalization)

for which this extends to E7,7 symmetry.

By a transformation of the form Z ’ U T ZU , where U is a unitary ma-

trix,14 Z can be brought to a canonical form in which the only nonzero

entries (with A < B) are z1 = Z12 , z2 = Z34 , z3 = Z56 , z4 = Z78 . The zi s

are complex, in general. In this basis one has

2

4 2

|zi | ’ |zi |

∆=2 + 8 Re(z1 z2 z3 z4 ). (11.88)

As a matter of fact, by an SU (8) transformation, it is possible to remove

three phases. So, for example, all four zi could be chosen to have the same

phase or else three of the zi could be chosen to be real. Thus, the ¬ve-charge

case discussed below, is the generating solution for the arbitrary case in the

same sense that the three-charge solution was in ¬ve dimensions.

To make contact with the four-charge black hole considered previously,

all four zi can be chosen to be real in order to give four electric charges. For

the speci¬c choices

1 1

z2 = (Q1 + Q2 ’ Q3 ’ Q4 ),

z1 = (Q1 + Q2 + Q3 + Q4 ),

4 4

1 1

(Q1 ’ Q2 + Q3 ’ Q4 ), (Q1 ’ Q2 ’ Q3 + Q4 ),

z3 = z4 = (11.89)

4 4

one ¬nds after some algebra that ∆ = Q1 Q2 Q3 Q4 in agreement with what

we found earlier by other methods.

14 This is appropriate because U (8) is the automorphism group of the N = 8, D = 4 supersym-

metry algebra.

578 Black holes in string theory

Five-charge con¬guration

It is possible to add P1 D0-branes to the D2-D2-D2-D6 con¬guration de-

scribed above without breaking any additional supersymmetry. The result-

ing 5-charge con¬guration di¬ers from the con¬gurations considered so far

in an important respect. Namely, a D0-brane and a wrapped D6-brane are

mutually nonlocal. In other words, they are electric and magnetic with re-

spect to the same gauge ¬eld. Let us not attempt to write down the solution

that describes such a black hole. It is given by an E7,7 transformation of the

solution that we presented. Rather, let us simply note that the E7 quartic

invariant can be evaluated for all possible choices of electric and magnetic

charges, so it is simply a matter of reading o¬ what it gives. To do this we

should simply replace Q1 ’ Q1 + iP1 in each of the four zi s and re-evaluate

∆. After some algebra one ¬nds that Eq. (11.88) gives

12

∆ = Q1 Q2 Q3 Q4 ’ P1 Q2 . (11.90)

1

4

Thus

12

Q1 Q2 Q3 Q4 ’ P1 Q2 .

S = 2π (11.91)

1

4

If one chooses to make the more common convention of calling D0-branes

electrically charged and D6-branes magnetically charged, then we should

make an electric“magnetic duality transformation, which amounts to re-

naming the charges as follows: Q1 = P0 and P1 = ’Q0 . Written this way,

the entropy takes the form

12

P0 Q2 Q3 Q4 ’ P0 Q2 .

S = 2π (11.92)

0

4

The 4d/5d connection

The astute reader may have noticed a resemblance between the entropy

of a rotating black hole in ¬ve dimensions, given in Eq. (11.76), and the

four-dimensional entropy describing a four-charge black hole Eq. (11.92).

Speci¬cally, the two formulas agree if one sets P0 = 1 and makes the identi-