On Geometries and Monodromies for Branes of Codimension Two

We study geometries for the NS5-, the KK5- and the $5^2_2$-branes of codimension two in type II and heterotic string theories. The geometries are classified by monodromies that each brane has. They are the $B$-, the general coordinate and the $\beta$-transformations of the spacetime metric, the $B$-field and the dilaton (and the gauge fields). We show that the monodromy nature appears also in the geometric quantities such as the curvature and the complex structures of spacetime. They are linearly realized in the doubled (generalized) structures in the doubled space.