The ¬rst-order di¬erential equations satis¬ed by U („ ) and t± („ ) can be

derived by solving the conditions for unbroken supersymmetry

δψµ = δ»± = 0, (11.117)

where ψµ is the gravitino, and »a represents the gauginos. These equations

11.5 The attractor mechanism 591

Fig. 11.4. The pendulum with a dissipative force acting on it evolves towards θ = 0

independently of the initial conditions.

imply a set of ¬rst-order di¬erential equations17

dU („ )

= ’eU („ ) |Z|, (11.118)

d„

dt± („ ) ¯

= ’2eU („ ) G±β ‚β |Z|. (11.119)

¯

d„

¯

Recall that G±β is the inverse of G±β = ‚± ‚β K. In this form the conditions

¯ ¯

for unbroken supersymmetry can be interpreted as di¬erential equations

describing a dynamical system with „ playing the role of time.

The physical scenario described by these equations has a nice analogy with

dynamical systems. Consider, for example, a pendulum with a dissipative

force acting on it. In general, the ¬nal position of the pendulum is inde-

pendent of its initial position and velocity. The point at θ = 0 in Fig. 11.4

represents the attractor in this simple example. Solving the equations in the

near-horizon limit is then equivalent to solving the late-time behavior of the

dynamical system. It will turn out that the horizon represents an attractor,

that is, a point (or surface) in the phase space to which the system evolves

after a long period of time. This means that the moduli approach ¬xed

values at the horizon that are independent of the initial conditions.

Solution of the attractor equations

In order to solve Eqs (11.118) and (11.119) explicitly near the horizon, let us

¬rst note that these di¬erential equations can be written in the alternative

equivalent form

d ’U („ )+K/2

Im e’i± „¦ ∼ ’“.

2 e (11.120)

d„

17 The derivations are given in hep-th/9807087. Since Eq. (11.114) is homogeneous of degree one

in the X s, Z(t± ) means (X 0 )’1 Z(X I ).

592 Black holes in string theory

The ∼ symbol means that two sides are cohomologous. In other words, they

are allowed to di¬er by an exact three-form, though this freedom can be

eliminated by choosing harmonic representatives. Equation (11.118) is one

real equation and (11.119) is h2,1 complex equations, making 2h2,1 + 1 real

equations altogether. Equation (11.120) can be projected along each of the

classes of H 3 , and there are 2h2,1 + 2 of these. So, if it really is equivalent to

Eqs (11.118) and (11.119), it is necessary that there is a redundancy among

these equations.

Consider integrating Eq. (11.120) over the A cycles and the B cycles.

Using Eqs (11.112) and (11.113) this gives

d ’U („ )

Im e’i± X I = ’pI

2 e (11.121)

d„

and

d ’U („ )

Im e’i± FI = ’qI .

2

e (11.122)

d„

Contracting the ¬rst equation with qI and subtracting the second equation

contracted with pI gives

d ’U („ )

Im e’i± (qI X I ’ pI FI )

2

e = 0. (11.123)

d„

However, Eqs (11.114) and (11.115) imply that

e’i± (qI X I ’ pI FI ) = ’|Z|, (11.124)

so that Eq. (11.123) is automatically satis¬ed. This is the required redun-

dancy that leaves 2h2,1 + 1 nontrivial equations.

Equation (11.118) can be obtained from Eq. (11.120) by projecting both

sides on e’i± eK/2 „¦. This means taking the wedge product with this three-

form and then integrating over the manifold. The derivation of Eq. (11.118)

by this method is given in Exercise 11.10. To deduce the complex equations

in Eq. (11.119), one should project along e’i± eK/2 D± „¦. Together with the

previous result, this extracts the full information content of Eq. (11.120).

The derivation of Eq. (11.119) is left as a homework problem.

The di¬erential equation (11.120) can be integrated, since “ does not

depend on „ . Its expansion in a real cohomology basis only depends on the

electric and magnetic charges carried by the black hole.18 The result is

2e’U („ )+K/2 Im e’i± „¦ ∼ ’“„ + 2 e’U („ )+K/2 Im e’i± „¦ . (11.125)

„ =0

This equation yields implicitly the solution for the moduli ¬elds t± = t± („ ).

18 Of course, its Hodge decomposition depends on the complex structure and thus on „ , but this

is not relevant to the argument.

11.5 The attractor mechanism 593

Equation (11.119) implies that

¯¯

dt± („ ) dt± („ )

d|Z| ¯

‚± |Z| = ’4eU G±β ‚± |Z|‚β |Z| ¤ 0. (11.126)

‚± |Z| +

= ¯

¯

d„ d„ d„

As a result, |Z| is a monotonically decreasing function of „ converging to a

minimum. The ¬xed point is then determined by

d|Z|

’ 0 as „ ’ ∞. (11.127)

d„

In order to solve for the moduli ¬elds near the horizon, we assume that the

central charge has a nonvanishing value Z = Z = 0 at the ¬xed point.

Therefore, Eq. (11.118) can be integrated to give, for large „ ,

„ ’1 e’U („ ) ’ |Z |. (11.128)

Substituting into the metric, this implies that the near-horizon geometry is

AdS2 — S 2 , just as in Exercise 11.3,

r2 2

2 dr

2

dt + |Z | 2 + |Z |2 (dθ2 + sin2 θdφ2 ),

2

ds ’ ’ (11.129)

2

|Z | r

and it determines the area of the horizon to be

A = 4π|Z |2 . (11.130)

In the near-horizon limit Eq. (11.125) can be solved giving rise to the at-

tractor equation, which is a determining equation for the complex-structure

moduli in the near-horizon limit. In this limit Eq. (11.125) implies that

2eK/2 Im Z „¦ ∼ ’“ (11.131)

at the horizon. This implies that

“ = “(3,0) + “(0,3) (11.132)

at the horizon, that is, the only nonvanishing terms in the Hodge decom-

position of “ are (3, 0) and (0, 3), while the (1, 2) and (2, 1) parts vanish.

This is a property of the ¬xed-point, and it need not be true away from the

horizon. Therefore, the attractor mechanism can be viewed as a method to

determine „¦ at the horizon in terms of the charges of the black hole.

The attractor condition (11.131) and the charges de¬ned in Eq. (11.113)

give the alternative formulas19

pI = ’2Im ZX I qI = ’2Im ZFI .

and (11.133)

This form of the attractor equations is used in the following sections.

19 These equations often appear with plus signs. Clearly, the signs depend on conventions that

have been made along the way.

594 Black holes in string theory