Geometric classification of totally stable stability spaces

We construct a geometric model for the root category of any Dynkin diagram Q, which is an h-gon V with cores, where h is the Coxeter number. As an application, we classify all spaces ToSt(D) of totally stable stability conditions on triangulated categories D, where D must be of the form D^b(Q). More precisely, we prove that ToStD^b(Q)/C is isomorphic to the moduli spaces of stable h-gons of type Q. In particular, an h-gon V of type Dn is a centrally symmetric doubly punctured 2(n−1)-gon. V is stable if it is convex and the punctures are inside the level-(n−2) diagonal-gon. Another interesting case is E6, where the (stable) 12-gon can be realized as a pair of planar tiling pattern. This is a joint work with Xiaoting Zhang.