Monodromifold planes: Refining [p,q]-branes and orientifold planes in string theory

F-theory is the non-perturbative compactification of type IIB superstring theory. The compact space of F-theory has the structure of elliptic fibrations, which describe the compactification of type IIB string with the 7-branes background. The elliptic fibration is written by the Weierstrass form. In particular when we choose the base space as P^1, then the whole of the compact space become the rational elliptic surface (RES) or the K3 surface. Furthermore, the discriminant loci of the Weierstrass form correspond to the D7-branes. On the other hand, the singularities of such a RES or K3 surface is classified by Kodaira, and B-branes and C-branes (or [p,q]-brane in general) with a monodromy emerge at singular points. The origin of such a monodromy can be thought of as f- and g-locus planes (named monodromifold planes) in the Weierstrass form. In this talk, I will start with the review of type IIB superstring theory and F-theory compactification and discuss the properties of monodromifold planes.