String Theory and M-Theory

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 first-order differential equations 17
dU(τ)
dτ = −eU(τ) |Z|, (11.118)
dt α (τ)
dτ = −2eU(τ) G α ¯ β ∂ ¯β |Z|. (11.119)
Recall that G α ¯ β is the inverse of Gα ¯ β = ∂α∂ ¯ β K. In this form the conditions
for unbroken supersymmetry can be interpreted as differential equations
describing a dynamical system with τ 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 final position of the pendulum is independent
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 fixed
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
first note that these differential equations can be written in the alternative
equivalent form
2 d
e
dτ
−U(τ)+K/2 Im e −iα Ω
∼ −Γ. (11.120)
17 The derivations are given in hep-th/9807087. Since Eq. (11.114) is homogeneous of degree one
in the X s, Z(t α ) means (X 0 ) −1 Z(X I ).