String Theory and M-Theory

590 Black holes in string theory
basis AI , BI, one can write
Γ = Γ ∧ β I = p I
A I
M
and
BI
Γ =
M
Γ ∧ αI = qI. (11.113)
The central charge, which is determined by the charges, is given by
Z(Γ) = e iα |Z| = e K/2
Γ ∧ Ω = e K/2
Ω. (11.114)
M
This expression for the central charge can be derived from the N = 2 supersymmetry
algebra, as was shown in Chapter 9. it can be re-expressed as
follows:
Z(Γ) = e
K/2
I
A I
Γ
BI
Ω −
BI
C
Γ
AI
Ω = p I FI − qIX I . (11.115)
The attractor equations and dyonic black holes
Let us now show that the complex-structure moduli fields at the horizon
are determined by the charges of the black hole, independent of the values
of these fields at infinity. In order to illustrate this, we will derive the
differential equations satisfied by the complex-structure moduli fields for
the case of four-dimensional spherically symmetric supersymmetric black
holes. These conditions restrict the space-time metric to be of the form
ds 2 = −e 2U(r) dt 2 + 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 fields t α only
depend on the radial coordinate, so that t α = t α (r), with α = 1, . . . , h 2,1 .
Recall that these coordinates are related to the homogeneous coordinates
X I introduced above by t α = X α /X 0 . It is convenient to introduce the
variable τ = 1/r. Then τ = 0 corresponds to spatial infinity, while τ = ∞
corresponds to the horizon of the black hole.
The first-order differential equations satisfied by U(τ) and t α (τ) can be
derived by solving the conditions for unbroken supersymmetry
δψµ = δλ α = 0, (11.117)
where ψµ is the gravitino, and λ a represents the gauginos. These equations