稳定条件空间的瑟斯顿紧化
A Thurston compactification of the space of stability conditions
AI总结：
受Thurston紧化启发，研究者通过将稳定条件视为范畴度量、构造到无限射影空间的单射，提出Bridgeland稳定条件模空间的紧化方法，还引入有限自动机研究自等价下的动力学，求解了两类箭图的2-Calabi–Yau范畴情形。
摘要：
我们提出了三角范畴的Bridgeland稳定条件模空间的紧化方法。我们将稳定条件视为底层范畴上的度量，受曲面上双曲度量的Teichmüller空间的Thurston紧化启发，构建了该紧化。构造的关键要素是从稳定流形到无限维射影空间的映射。我们证明，在适当假设下，这些映射是单射的，且其像具有紧的闭包。我们还确定了边界上的一族点，它们在范畴层面类似于Teichmüller理论中的相交泛函。我们详细研究了箭图的2-Calabi–Yau范畴所得紧化的几何性质，并完整求解了\boldsymbol{A_2}和\boldsymbol{\boldsymbol{\widehat{A_1}}}箭图的情形。为此，我们仔细考察了范畴自等价下Harder–Narasimhan重数的动力学，引入了一个有限自动机来研究该动力学，并将其用于\boldsymbol{A_2}和\boldsymbol{\widehat{A_1}}范畴的分析中。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
