still applicable when there are additional supersymmetries. When one goes

beyond the supergravity approximation and includes higher-genus contribu-

tions to the e¬ective action, the holomorphic prepotential F (X I ) generalizes

to a function

∞

I 2

Fh (X I )W 2h ,

F (X , W ) = (11.172)

h=0

where h denotes the genus and W is a chiral super¬eld that appears in the

description of the N = 2 supergravity multiplet.23 The ¬rst component of W

is the anti-self-dual part of the graviphoton ¬eld strength. The graviphoton

is the U (1) gauge ¬eld contained in the N = 2 supergravity multiplet. The

prepotential satis¬es the homogeneity equation

X I ‚I F (X I , W 2 ) + W ‚W F (X I , W 2 ) = 2F (X I , W 2 ), (11.173)

which generalizes the formula presented in Chapter 9. Topological string

theory techniques, which are not described in this book, enable one to com-

pute the coe¬cients of terms in the e¬ective action of the form

d4 xd4 θW 2h Fh (X I ), (11.174)

which is exactly what is required.

When terms of higher-order than the Einstein“Hilbert term contribute to

the action in a signi¬cant way, the BH entropy formula is no longer correct.

The appropriate generalization has been worked out by Wald. Wald™s for-

mula (see Problem 11.15) is applied to the R2 corrected action in the present

case.

The attractor equations that determine the moduli in terms of the charges,

and make the central charge extremal, are24

pI = Re (CX I )

(11.175)

qI = Re (CFI ),

23 Since we do not wish to describe this formalism, as well as other issues, the argument presented

here is sketchy. The reader is referred to hep-th/0507014 for further details.

24 The coordinates X I , FI in this section and those in section 11.5 di¬er by a rescaling of the

holomorphic three-form „¦ by a factor 2iZ/C, where C is an arbitrary ¬eld introduced here for

bookkeeping purposes.

604 Black holes in string theory

where pI denote magnetic charges and qI denote electric charges as before.

Moreover, in the conventions that are usually used, the graviphoton ¬eld

strength at the horizon takes the value

C 2 W 2 = 256. (11.176)

After taking the corrections into account, it can be shown that the black-hole

entropy is

πi π

I

qI CX ’ pI CF I + Im C 3 ‚C F .

S= (11.177)

2 2

The ¬rst term in this equation agrees with the attractor value S = π|Z |2

(for G4 = 1) derived in the previous section when one takes account the

rescaling mentioned in the footnote. The second term is a string theory

correction.

The ¬rst equation in (11.175) is solved by writing

iI

CX I = pI + φ. (11.178)

π

In order to solve the second equation, we de¬ne

iI

F(φ, p) = ’π ImF (pI + φ , 256). (11.179)

π

Using this de¬nition,

1 ‚

CFI + CF I = ’ I F(φ, p),

qI = (11.180)

2 ‚φ

where we have used

‚ i‚ i‚

’

= . (11.181)

πC ‚X I

‚φI πC ‚X I

The homogeneity relation for the prepotential then implies

256 ‚

XI , = XI F ’ 2F.

C‚C F (11.182)

C2 ‚X I

As a result, the corrected entropy can be written in the form

‚

S(p, q) = F(φ, p) ’ φI F(φ, p). (11.183)

‚φI

In other words, the entropy of the black hole is the Legendre transform of

F with respect to φI . So it is more convenient to specify the φI , which play

the role of chemical potentials, rather than the electric charges qI .

11.6 Small BPS black holes in four dimensions 605

For the reasons just described, it is natural to consider a mixed ensemble

with the partition function

I ,pI ) Iq

Z(φI , pI ) = eF(φ „¦(qI , pI )e’φ

= , (11.184)

I

qI

which is microcanonical with respect to the magnetic charges pI and canoni-

cal with respect to the electric charges qI . Moreover, „¦(qI , pI ) are the black-

hole degeneracies, and log „¦ is the microcanonical entropy. The black-hole

entropy is then obtained according to

S(q, p) = log „¦(q, p). (11.185)

The inverse transform is (formally)

I ,pI )+φI q

eF(φ

„¦(qI , pI ) = dφI , (11.186)

I

which, in principle, allows one to obtain the microscopic black-hole degen-

eracies by using amplitudes computed by topological string theory.

Heterotic compacti¬cation on T 6

In the special case of the heterotic string on T 6 , one can use these results by

going to the S-dual description in terms of the type IIA theory on K3 — T 2 .

In this description the Kaluza“Klein modes and winding modes map to D4-

branes wrapped on the K3 and D0-branes. The D0-branes are electrically

charged with respect to one gauge ¬eld and the D4-branes are magnetically

charged with respect to another one. Thus, only two charges, q0 and p1 say,

are nonzero.

The prepotential is particularly simple in this case. Since this theory

has an N = 4 supersymmetry, the only nonvanishing contributions to the

prepotential are F0 and F1 . For F0 one takes the tree-level amplitude given

by

1

1 abX

F0 = ’ Cab X X , a, b = 2, . . . , 23 (11.187)

X0

2

where Cab is the intersection matrix of two-cycles on K3, and

„ = „1 + i„2 = X 1 /X 0 (11.188)

is the K¨hler modulus of the torus.

a

The only additional contribution is F1 . Schematically, this term can be

obtained by taking the ten-dimensional interaction B § Y8 and compacti-

fying on K3 — T 2 . In the type IIA description there is an SL(2, ) T-duality

symmetry associated with the T 2 factor, which corresponds to an SL(2, )

606 Black holes in string theory

S-duality symmetry of the dual heterotic string theory in four dimensions.