11.5 The attractor mechanism

Moduli ¬elds

As has been discussed in previous sections, black holes can appear when

a superstring theory or M-theory is compacti¬ed to lower dimensions and

when branes are wrapped on nontrivial cycles of the compact manifold. The

588 Black holes in string theory

compacti¬ed theories typically have many moduli of the type considered in

Chapters 9 and 10. These moduli appear as part of the black-hole solutions,

which turn out to exist for generic values of these moduli at in¬nity, that is,

far from the black hole where the geometry is essentially ¬‚at. As a result,

there is the dangerous possibility that the entropy of the black hole may

depend on parameters that are continuous, namely the moduli ¬elds at in-

¬nity, and not only on discrete black-hole charges. This would be a problem,

since the number of microstates with given charges is an integer, that should

not depend on parameters that can be varied continuously. It should only

depend on quantities that take discrete values, such as electric/magnetic

charges and angular momenta.16

The attractor mechanism

In order to resolve this puzzle, one has to realize that the entropy of a black

hole is determined by the behavior of the solution at the horizon of the

black hole and not at in¬nity. The obvious way to reconcile this with the

observations in the preceding paragraph is for the moduli ¬elds to vary with

the radius in such a way that their values at the horizon of the black hole

are completely determined by the discrete quantities, such as the charges,

regardless of their values at in¬nity. In other words, the radial dependence

of these moduli is determined by di¬erential equations whose solutions ¬‚ow

to de¬nite values at the horizon, regardless of their boundary values at

in¬nity. This solution is called an attractor and its existence is the essence

of the attractor mechanism. The existence of an attractor is necessary for a

microscopic description of the black-hole entropy to be possible.

The attractor equations arise from combining laws of black-hole physics

with properties of the internal compacti¬cation manifolds. To be speci¬c,

this section gives the derivation of the attractor equations for type IIB super-

string theory compacti¬ed on Calabi“Yau three-folds. A crucial ingredient

in these cases is special geometry, a tool used to describe the relevant moduli

spaces that was introduced in Chapter 9.

Black holes in type IIB Calabi“Yau compacti¬cations

As discussed in Chapter 9, when type IIB superstring theory is compacti¬ed

on a Calabi“Yau three-fold M , the resulting theory in four dimensions has

N = 2 supersymmetry. The four-dimensional theory is N = 2 supergrav-

ity coupled to h2,1 abelian vector multiplets and h1,1 + 1 hypermultiplets.

16 The entropy formulas given earlier depend on integers only, though they are not logarithms of

integers. The reason, of course, is that the formulas are not exact.

11.5 The attractor mechanism 589

The vector multiplets contain the complex-structure moduli, while the hy-

permultiplets contain the K¨hler moduli and the dilaton. The following

a

discussion focuses on the ¬elds contained in vector multiplets, since the en-

tropy does not depend on the hypermultiplets, at least in the supergravity

approximation, as will become clear.

Brief review of special geometry

An N = 2 vector multiplet contains a complex scalar, a gauge ¬eld and

a pair of Majorana (or Weyl) fermions. The moduli space describing the

scalars is h2,1 -dimensional and is a special-K¨hler manifold. The K¨hler

a a

potential for the complex-structure moduli space is

K = ’ log i „¦§„¦ , (11.110)

M

where „¦ is the holomorphic three-form of the Calabi“Yau manifold, as usual.

In this set up a black hole can be realized by wrapping a set of D3-branes on

a special Lagrangian three-cycle C. In order to describe this, let us introduce

the Poincar´ dual three-form to C, which we denote by “.

e

This black hole carries electric and magnetic charges with respect to the

h2,1 U (1) gauge ¬elds originating from the ten-dimensional type IIB self-dual

¬ve-form F5 as well as the graviphoton belonging to the N = 2 supergravity

multiplet. In order to describe the charges, let us introduce a basis of three-

cycles AI , BJ (with I, J = 1, . . . , h2,1 + 1), which can be chosen such that

the intersection numbers are

AI © BJ = ’BJ © AI = δJ

I

AI © AJ = BI © BJ = 0. (11.111)

and

The Poincar´ dual three-forms are denoted ±I and βI . The group of trans-

e

formations that preserves these properties is the symplectic modular group

Sp(2h2,1 + 2; ). The symplectic coordinates introduced in Chapter 9 are

X I = eK/2 FI = eK/2

„¦ and „¦. (11.112)

AI BI

Recall that the de¬nition of „¦ can be rescaled by a factor that is independent

of the manifold coordinates and that this corresponds to a rescaling of the

homogeneous coordinates X I . This freedom has been used to include the

factors of eK/2 , which will be convenient later.

The electric and magnetic charges, qI and pI , that result in four dimen-

sions are encoded in the homology class C = pI BI ’ qI AI or the equivalent

cohomology class “ = pI ±I ’ qI β I . Thus, in terms of a canonical homology

590 Black holes in string theory

basis AI , BI , one can write

“ § β I = pI “ § ±I = q I .

“= and “= (11.113)

AI M BI M

The central charge, which is determined by the charges, is given by

Z(“) = ei± |Z| = eK/2 “ § „¦ = eK/2 „¦. (11.114)

C

M

This expression for the central charge can be derived from the N = 2 su-

persymmetry algebra, as was shown in Chapter 9. it can be re-expressed as

follows:

Z(“) = eK/2 = p I FI ’ q I X I .

„¦’

“ “ „¦ (11.115)

AI AI

BI BI

I

The attractor equations and dyonic black holes

Let us now show that the complex-structure moduli ¬elds at the horizon

are determined by the charges of the black hole, independent of the values

of these ¬elds at in¬nity. In order to illustrate this, we will derive the

di¬erential equations satis¬ed by the complex-structure moduli ¬elds for

the case of four-dimensional spherically symmetric supersymmetric black

holes. These conditions restrict the space-time metric to be of the form

ds2 = ’e2U (r) dt2 + e’2U (r) dx · dx, (11.116)

where x = (x1 , x2 , x3 ) and r = |x| is the radial distance and r = 0 is the

event horizon. Note that this requires using a coordinate system that is

singular at the horizon like the one in Eq. (11.77), for example. Let us

also assume that the holomorphic complex-structure moduli ¬elds t± only

depend on the radial coordinate, so that t± = t± (r), with ± = 1, . . . , h2,1 .

Recall that these coordinates are related to the homogeneous coordinates

X I introduced above by t± = X ± /X 0 . It is convenient to introduce the

variable „ = 1/r. Then „ = 0 corresponds to spatial in¬nity, while „ = ∞