论文标题
拓扑类别类别的群集理论
Cluster theory of topological Fukaya categories
论文作者
论文摘要
我们在与标记表面相关的集群类别与表面拓扑类别相关的群集类别之间建立了新的关系。我们考虑对表面的三角群集类别的概括为$ 2 $ -CALABI-YAU的外侧/精确$ \ infty $ - 类别，该类别是通过Amiot的构造来从三角形表面的相对Ginzburg代数中构造而来的。该类别被证明等同于标记表面的拓扑类别的$ 1 $周期版本以及Wu的Higgs类别。我们将簇倾斜的对象分类为该外节群集类别，并将簇字符描述为标记表面的上群集代数，并在边界弧中具有系数。此外，我们还提供了$ 2 $ -CALABI-YAU FROBENIUS外部结构/精确的$ \ infty $ - 结构稳定的$ \ infty $ - 类别，配备了相对权利$ 2 $ -Calabi-yau结构，具有勇敢的dyckerhoff的意义，可能具有独立的利益。
We establish a novel relation between the cluster categories associated with marked surfaces and the topological Fukaya categories of the surfaces. We consider a generalization of the triangulated cluster category of the surface by a $2$-Calabi-Yau extriangulated/exact $\infty$-category, which arises via Amiot's construction from the relative Ginzburg algebra of the triangulated surface. This category is shown to be equivalent to the $1$-periodic version of the topological Fukaya category of the marked surface, as well as to Wu's Higgs category. We classify the cluster tilting objects in this extriangulated cluster category and describe a cluster character to the upper cluster algebra of the marked surface with coefficients in the boundary arcs. We furthermore give a general construction of $2$-Calabi-Yau Frobenius extriangulated structures/exact $\infty$-structures on stable $\infty$-categories equipped with a relative right $2$-Calabi-Yau structure in the sense of Brav-Dyckerhoff, that may be of independent interest.
