论文标题
稳定条件空间的瑟斯顿紧凑
A Thurston compactification of the space of stability conditions
论文作者
论文摘要
我们提出了三角形类别的Bridgeland稳定性条件的模量空间的压缩。我们的构造是由稳定条件视为基础类别的度量标准的，灵感来自于表面上双曲线指标的Teichmüller空间的瑟斯顿紧凑型。构造中的关键成分是从稳定性歧管到无限投影空间的地图。我们证明，在合适的假设下，这些地图是注入性的，它们的图像具有紧凑的闭合。我们确定边界中的一个点家族，与Teichmüller理论中的相交功能类似。 我们详细研究了2-卡拉比 - YAU类别所得的紧凑型的几何形状，并充分解决\（a_2 \）和\（\ wideHat {a_1} \）砂的情况。为此，我们仔细研究了该类别自动等效性下的更艰难的纳拉西姆汉（Narasimhan）多重性的动态。我们介绍了一个有限的自动机来研究此动态，并在对\（a_ {2} \）和\（\ wideHat {a_1} \）类别的分析中使用它。
We propose compactifications of the moduli space of Bridgeland stability conditions of a triangulated category. Our construction arises from a viewing a stability condition as a metric on the underlying category and is inspired by the Thurston compactification of the Teichmüller space of hyperbolic metrics on a surface. The key ingredient in the construction are maps from the stability manifold to an infinite projective space. We prove that, under suitable hypotheses, these maps are injective and their image has a compact closure. We identify a family of points in the boundary that are categorical analogous to the intersection functionals in Teichmüller theory. We study in detail the geometry of the resulting compactification for the 2-Calabi--Yau categories of quivers, and fully work out the cases of the \(A_2\) and \(\widehat{A_1}\) quivers. To do so, we carefully examine the dynamics of Harder--Narasimhan multiplicities under auto-equivalences of the category. We introduce a finite automaton to study this dynamics and employ it in our analysis of the \(A_{2}\) and \(\widehat{A_1}\) categories.
