The Quantum Structure of Space and Time (293 pages)

128 The Quantum Structure of Space and Time

one can compute the soft mass terms for the open string states on the D-branes.

In the statistical search for D-brane models with Standard Model like proper-

ties one first has to count all possible solutions of the tadpole and K-theory

constraints. Then one applies certain physical thresholds, i.e. counting those

models with Standard Model gauge group or those models with a certain IIUIII-

ber of chiral matter fields. Of particular interest is the question whether certain

physical observables are statistically correlated.

First let us give a few comments on the moduli stabilization process due to back-
ground fluxes and non-perturbative effects. To be specific consider type IIB Ramond

and NS 3-form fluxes through 3-cycles of a Calabi-Yau space X. They give rise to

the following effective flux superpotential in four dimensions [3-5] :

It depends on the dilaton T and also on the complex structure moduli U. How-

ever, since Wfl,, does not depend on the Kahler moduli, one needs additional non-

perturbative contributions to the superpotential in order to stabilize them. These
are provided by Euclidean D3-instantons [6], which are wrapped around 4-cycles
(divisors) D inside X, and/or gaugino condensations in hidden gauge group sectors
on the world volumes of D7-branes, which are also wrapped around certain divisors
D. Both give rise to terms in the superpotential of the form

wn.p. - IJie-aJi , (2)

where V, is the volume of the divisor Di, depending on the Kahler moduli T. Note
that the prefactor gi is in general riot a wristarit, but rather depends on the complex
structure moduli U. The generation of a non-perturbative superpotential crucially
depends on the D-brane zero modes of the wrapping divisors, i.e. on the topology
of the divisors together with their interplay with the 0-planes and also with the
background fluxes [7-101.
The moduli are stabilized to discrete values by solving the N = 1 supersymmetry
conditions

DAW = 0 (vanishing F - term). (3)

Then typically the superpotentials of the form Wfl,, + Wn.p. lead to stable super-

symmetric Ads4 minima. Additional restrictions on the form of the possible super-

potential arise [ll, 121 requiring that the mass matrix of all the fields (S, T, V) is

already positive definite in the AdS vacuum (absence of tachyons), as it is necessary,

if one wants to uplift the AdS vacua to a dS vacuum by a (constant) shift in the

potential. These conditions cannot be satisfied in orientifold models without any

complex structure moduli, i.e. for Calabi-Yau spaces with Hodge number h2?l = 0.

Alternatively one can also look for supersymmetric 4D Minkowski mimima which

solve eq.(3) [13, 141. They may exist if Wn,p, is of the racetrack form. In this case