124 The Quantum StmctuTe of Space and Time
defined by the central charge Z.
of moduli is
In case of BPS black holes the attractor equation relating fluxes to fixed valuesF3 = 21m (Z fi3)~~=0. (3)
It has been studied extensively over the last 10 years and many interesting solu-
tions have been found. One of the most curious solutions of the black hole attractor
equation is the so-called STU black holes with three moduli [lo]. It was discovered
recently [ll] that the entropy of such black holes is given by the Caley's hyper-
determinant of the 2x2~2 matrix describing also the 3-qubit system in quantum
information theory.
The non-BPS black holes under certain conditions also exhibit the attractor
phenomenon: the moduli near the horizon tend to fixed values defined by fluxes
[9, 12, 131. The corresponding attractor equation is
This equation can be also used in the form
Stabilization of moduli is equivalent to minimization of effective N=l supergrav-
ity potential
VflzLz = (D2(2 - (^31212) (6)
defined by the effective central charge Z. All supersymmetric flux vacua in type IIB
string theory compactified on a Calabi-Yau manifold are subject to the attractor
equations defining the values of moduli in terms of fluxes.
F4 = 2Re [Z fi4 + D"Z r)~1fi4]~~=~ (7)
We may rewrite these equations in a form in which it is easy to recognize them as
generalized attractor equations. The dependence on the axion-dilaton T is explicit,
whereas the dependence on complex structure moduli is un-explicit in the section
(L, M).
Pah ZLa+ ZL" 201~~~" + zQIDILa
= ZMa + zaa ] + zQ'DIMa + ZO'DIM~ 1
(8)
TZLa + TZLa TZQID~L" + TZQID~L~TZM~ + TZMa Dz=o 7zQ1DIMa + TZQIDIMa Dz=o
The second term in this equation is absent in BH case. In the black hole case Z = 0
and DZ = 0 conditions lead to null singularity and runaway moduli. In flux vacua,