Mathematical Structures 129
the requirement that all flat directions are lifted in the Minkowski vacuum leads to
similar constraints as the absence of tachyon condition in the Ads case.
In more concrete terms, the moduli stabilization procedure was studied in [15]
for the T6/22 x 22 orientifold, with the result that all moduli indeed can be fixed.
Moreover in [12, 16, 171 all other ZN and ZN x ZM orientfolds were studied in great
detail, where it turns out that in order to have divisors, which contribute to the non-
perturbative superpotential, one has to consider the blown-up orbifold geometries.
Then the divisors originating from the blowing-ups give rise to D3-instantons and/or
gaugino condensates, being rigid and hence satisfying the necessary topological
conditions. As a result of this investigation of all possible orbifold models, it turns
out that the 22 x 22, 22 x 24, 24, 26-11 orientifolds are good candidates where all
moduli can be completely stabilized.
The statistical approach to the flux vacua amounts to count all solutions of
the N = 1 supersymmetry condition eq.(3) refs. In fact, it was then shown
that the number of flux vacua on a given background space is very huge [18, 191:
Nu,, N lo5''. In addition there is another method to assign a probability mea-
sure to flux compactifications via a black hole entropy functional S. This method
however does not apply to 3-form flux compactifications but rather to Ramond
5-form compactifications on S2 x X, hence leading to Ad& vacua. Specifically a
connection between 4D black holes and flux compactifications is provided by type
N = 2 black hole solutions, for which the near horizon condition DZ = 0 can be
viewed as the the extremization condition of a corresponding 5-form superpotential
W N &zxx(F5 A 0) [20]. In view of this connection, it was suggested in [20, 211
to interpret II, = es as a probability distribution resp. wave function for flux com-
pactifications, where II, essentially counts the microscopic string degrees of freedom,
which are associated to each flux vacuum. Maximization of the entropy S then
shows that points in the moduli space, where a certain number of hypermultiplets
become massless like the conifold point, are maxima of the entropy functional [22].
Now let us also include D-branes and discuss the statistics of D-brane models with
open strings [23-251. To be specific we discuss the toroidal type IIA orientifold
T6/Zz x Z2 at the orbifold point. We have to add D6-branes wrapping special
Langrangian 3-cycles. They are characterized by integer-valued coefficients XI, YI
(I = 0,... ,3). The supersymmetry conditions, being equivalent to the vanishing of
the D-term scalar potential, have the form:
3
=o, CX'Ur>O
i-g I=O I=O
(4)
The Ramond tadpole cancellation conditions for k stacks of Na D6-branes are given
bY
k
a=l
