Geometric classification of total stability spaces

TL;DR

This paper introduces a geometric model for the root category of Dynkin diagrams, classifies total stability spaces using moduli of stable polygons, and provides explicit descriptions for types D and E.

Contribution

It constructs a geometric model for root categories of Dynkin diagrams and classifies total stability spaces via moduli of stable polygons, linking algebraic and geometric structures.

Findings

The geometric model is an $h_Q$-gon with cores for each Dynkin diagram.

Total stability spaces are classified as moduli spaces of stable polygons.

Explicit descriptions are provided for types D and E polygons.

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.