Mathematical Structures 1274.3.2 Dieter Lust: A short remark on flux and D-brane vacua and
their statistics
String compactifications provide a beautiful link between particle physics and the
geometrical and topological structures of the corresponding background geometries.
Already from the “old days” of heterotic string compactifications we know that they
exist a large number of consistent string compactifications with or without (being
tachyon free) space-time supersymmetry, non-Abelian gauge groups and chiral mat-
ter field representations, i.e. with more or less attractive phenomenological features.
In particular, a concrete number of the order of N,,, N of heterotic string
models within the covariant lattice constructions was derived [l]. More recently, a
detailed analysis of type I1 orientifold string compactifications with D-branes, their
spectra, their effective actions and also of their statistical properties was performed,
and also the study of heterotic string models and their landscape was pushed for-
ward during the last years. In this note we will comment on type I1 orientifold
compactifications with closed string background fluxes and with open strings end-
ing on D-branes. Two questions will be central in our discussion: first we will
briefly discuss the procedure of moduli stabilization due to background fluxes and
non-perturbative superpotentials. Second, we will be interested in the question what
is the fraction of all possible open string D-brane configurations within a given class
of orientifold models (like the 22 x 22 orientifold with background fluxes) that have
realistic Standard Model like properties, such as gauge group SU(3) x SU(2) x U(1),
three generations of quarks and leptons, etc. More concretely, the following steps
will be important:
0 We begin with choosing a toroidal ZN resp. ZN x ZM type I1 orbifold which
preserves N = 2 space-time supersymmtry in the closed string sector.
0 A consistent orientifold projection has to be performed. This yields 0-planes
and in general changes the geometry. The bulk space-time supersymmetry
is reduced to N = 1 by the orientifold projection. The tadpoles due to the
0-planes must be cancelled by adding D-branes and/or certain background
fluxes. Then the resulting Ramond-Ramond tadpole equations as well as the
NS-tadpoles, which ensure N = 1 space-time supersymmetry on the D-branes,
together with constraints from K-theory provide restrictions for the allowed
D-brane configurations. For each of the allowed D-brane model one has to
determine the corresponding open string spectrum, namely the gauge groups
and the massless matter fields, where the chiral N = 1 matter fields are located
at the various brane intersections.
0 In order to stabilize the moduli one is turning on certain background fluxes that
generate a potential for the moduli. According to the KKLT proposal [2], 3-form
fluxes in type IIB can fix all complex structure moduli and the dilaton. On the
other hand, the Kahler moduli can be fixed by non-perturbative effects. In case
the fluxes or the non-perturbative superpotential break N = 1 supersymmetry,