under SL(2, ) transformations in the usual way. Its real part, „1 , which is

an axion-like ¬eld, arises from a duality transformation of the two-form B

in four dimensions. Accordingly, the ten-dimensional interaction gives rise

to a four-dimensional term of the form

1

„1 (trR § R ’ trF § F ) . (11.189)

8π

The normalization is ¬xed by the requirement that this should be well de-

¬ned up to a multiple of 2π when „1 is shifted by an integer, since such

shifts are part of the SL(2, ) group. To get the rest of the group working,

speci¬cally the transformation „ ’ ’1/„ , it is necessary to add higher-order

terms by the replacement

1 24 1

log q ’

„= log ·(„ ) = log ∆(q). (11.190)

2πi 2πi 2πi

In the heterotic viewpoint, the corrections given by this substitution have the

interpretation as instanton contributions due to Euclideanized NS5-branes

wrapping the six-torus.

It follows that the S-duality invariant and supersymmetric completion of

the trR § R term is25

1

Im log ∆(q)tr [(R ’ iR ) § (R ’ iR )] . (11.191)

16π 2

The factor involving the curvatures is part of d4 θW 2 , and its coe¬cient

determines F1 to be

i

F1 = log ∆(q). (11.192)

128π

This shows that F1 is independent of the K3 moduli. Moreover, Fh = 0

for h > 1. As a result, one ¬nds that the prepotential for this case takes a

particularly simple form, namely

X1 W2

1

F (X, W ) = ’ Cab X a X b

2

’ log ∆(q). (11.193)

X0

2 128πi

Using these formulas one can solve the attractor equations and the Legendre

transformation obtaining

p1

φ0 = ’2π . (11.194)

q0

25 In terms of two-forms, R— is de¬ned by a duality transformation of the Lorentz indices

(R— )mn = 1 µmn pq Rpq .

2

Homework Problems 607

One then reproduces the desired entropy formula

ˆ p1 q0 ) .

S ∼ log 16I13 (4π (11.195)

The analysis described above was restricted to supersymmetric black holes.

However, the analysis can be extended to the entropy of black holes that

are extremal, but not necessarily supersymmetric. Speci¬cally, the entropy

given by Wald™s formula is given by extremizing an entropy function with

respect to moduli ¬elds as well as electric ¬elds at the horizon. This im-

plies that the attractor mechanism is very general: if the entropy function

depends on a speci¬c modulus, that modulus is ¬xed at the horizon. If it

does not depend on a modulus, the entropy does not depend on it either.

HOMEWORK PROBLEMS

PROBLEM 11.1

Consider motion of a massive particle in an arbitrary D = 4 space-time.

The Newtonian limit can be obtained when the curvature of the space-time

is small and the velocity is small v 1. Expand the space-time metric

about ¬‚at Minkowski space, gµν = ·µν + gµν with |˜µν |

˜ g 1, to show that

the Newtonian potential ¦ is related to the metric by ¦ = ’˜tt /2.

g

PROBLEM 11.2

Verify that the metric in Eq. (11.9) has a vanishing Ricci tensor, so that D-

dimensional Schwarzschild black hole is a solution to Einstein™s equations.

PROBLEM 11.3

Derive Eq. (11.11).

PROBLEM 11.4

Re-express the metric in Eq. (11.9) in a higher-dimensional generalization of

Kruskal“Szekeres coordinates and verify that there is no singularity at the

horizon.

PROBLEM 11.5

Calculate the temperature of the nonextremal black hole (11.60). What

happens in the limit r0 ’ 0?

608 Black holes in string theory

PROBLEM 11.6

By similar reasoning to Exercise 11.6, show that the entropy of the three-

charge extremal D = 5 black hole is given correctly by M-theory on T 6 =

T 2 — T 2 — T 2 with Q1 M2-branes wrapping the ¬rst T 2 , Q2 M2-branes

wrapping the second T 2 and Q3 M2-branes wrapping the third T 2 .

PROBLEM 11.7

Verify that Eq. (11.90) follows from Eq. (11.88).

PROBLEM 11.8

Deduce Eq. (11.119) by projecting both sides of Eq. (11.120) on e’i±+K/2 Da „¦

and using reasoning similar to that in Exercise 11.12. Warning: this is a

di¬cult problem.

PROBLEM 11.9

Show that the K¨hler potential in Eq. (11.110) can be recast in the form

a

I

K = ’ log[2 Im(X FI )].

What form does this equation take when re-expressed in terms of t± =

X ± /X 0 and F (t± ) = (X 0 )’2 F (X I )?

PROBLEM 11.10

Show that the ¬ve-dimensional three-charge black hole with rotation dis-

cussed in Section 11.3 solves Eqs (11.144) to (11.154).

PROBLEM 11.11

Show that the horizon of the black-ring solution described by Eqs (11.148)

to (11.154) has the topology S 1 — S 2 . What is the area of the horizon and

what is the entropy of the corresponding black hole?

PROBLEM 11.12

The Dedekind · function can be represented in the form

∞ ∞

3 2

1/24 n

(’1)n q 2 (n’1/6) .

(1 ’ q ) =

·(„ ) = q

n=’∞

n=1

Use the Poisson resummation formula and this representation of the · func-