GEOMETRIC CLASSIFICATION OF TOTALLY STABLE STABILITY SPACES

Abstract. We construct a geometric model for the root category D∞(Q)/[2] of any Dynkin diagram Q, which is an hQ-gon VQ with cores, where hQ is the Coxeter number and D∞(Q) = D(Q) is the bounded derived category associated to Q. As an application, we classify all spaces ToStD of totally stable stability conditions on triangulated categories D, where D must be of the form D∞(Q). More precisely, we prove that ToStD∞(Q)/C is isomorphic to the moduli spaces of stable hQ-gons of type Q. In particular, an hQ-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) hQ-gon (dodecagon) can be realized as a pair of planar tiling pattern.