论文标题
总稳定空间的几何分类
Geometric classification of total stability spaces
论文作者
论文摘要
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 $Q$.作为一个应用程序，我们将所有空格分类为$ \ MATHRM {tost} \ Mathcal {d} $在三角形类别上的总稳定性条件的$ \ m nathcal {d} $，其中$ \ mathcal {d} $必须是形式的$ \ mathcal {d} d}^b（q）$。更确切地说，我们证明$ \ mathrm {tost} \ Mathcal {d}^b（q）/[2] $是稳定的稳定$ h_q $ - 类型$ q $的合适模量的同构。 特别是，类型$ d_n $的$ h_q $ -gon $ \ mathbf {v} $是（中央）对称双重刺穿$ 2（n-1）$ - gon。 $ \ mathbf {v} $如果它是凸面，并且穿刺在级别内 - $（n-2）$对角线。另一个有趣的情况是$ e_6 $，其中（稳定）$ h_q $ -gon（dodecagon）可以实现为一对平面平铺图案。
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.
