Compactification of string theory gives rise to plethora of vacua, at least as long as we adopt the understanding of string theory in the last decade or so. This is usually regarded as lack of precise low-energy prediction, and certainly it is, but there is also a positive side; the landscape of vacua may be regarded as theoretical foundation for eternal inflation and also for the notion of naturalness.

It has been known well for a decade that flux compactification of F-theory / Type IIB string theory stabilizes 7-brane configuration as well as complex structure moduli of the internal geometry. This means that the statistics of gauge group and matter multiplicity of low-energy effective theories can be determined, in principle. In practice, though, not much has been done so far to make this possibility come to reality.

We begin with a question when two F-theory vacua on elliptic fibred geoemtries describe physically distinct vacua, and then move on to study how to derive statistics of gauge groups by using K3 x K3 compactification of F-theory. With F-theory compactification on more general Calabi–Yau 4-folds in mind, analytic vacuum distribution formula by Ashok-Denef-Douglas is also generalized so that the formula can be used to address various questions of phenomenological interest. As an example, we will discuss the statistics of number of generations.

This talk is based on a joint work with Andreas Braun and Yusuke Kimura.