Geometric classification of total stability spaces

Abstract

We construct a geometric model for the root category $\mathcal{D}^b(Q)/[2]$ of any Dynkin diagram $Q$, which is an $h_Q$-gon $\mathbf{V}_Q$ with cores, where $h_Q$ is the Coxeter number and $\mathcal{D}^b(Q)$ is the bounded derived category associated to $Q$. As an application, we classify all spaces $\mathrm{ToSt}\mathcal{D}$ of total stability conditions on triangulated categories $\mathcal{D}$, where $\mathcal{D}$ must be of the form $\mathcal{D}^b(Q)$. More precisely, we prove that $\mathrm{ToSt}\mathcal{D}^b(Q)/[2]$ is isomorphic to a suitable moduli space of stable $h_Q$-gons of type $Q$. In particular, an $h_Q$-gon $\mathbf{V}$ of type $D_n$ is a (centrally) symmetric doubly punctured $2(n-1)$-gon. $\mathbf{V}$ is stable if it is convex and the punctures are inside the level-$(n-2)$ diagonal-gon. Another interesting case is $E_6$, where the (stable) $h_Q$-gon (dodecagon) can be realized as a pair of planar tiling pattern.