论文标题
循环类别的稳定条件I：基本定义和示例
Stability conditions on cyclic categories I: basic definitions and examples
论文作者
论文摘要
带有规范的同构$ [2] \ xrightArrow {\ sim} id $的三角形类别$ \ MATHCAL {C} $在本文中称为环状类别。我们在$ k $ - 线krull-schmidt循环类别上给出了新的稳定条件概念。给定这样的稳定性条件$σ$，我们可以为此类类别中的每个基本循环分配Maslov索引。如果所有Maslov都消失了，我们将获得$ \ MATHCAL {C}'，σ'$作为$ \ Mathbb {z} $ - $ \ Mathcal {C} $的提升，分别为$ \ Mathcal {c} $ a $ \ mathbb is a $ \ mathbb is a $ rid andiane $ and $ and $ and $在$ \ Mathcal {C}'$上。此外，我们表明存在一个同构$$ stab^{0，e}（\ Mathcal {c}）\ xrightArrow {\ simeq} bStab（\ Mathcal {C}'）$ $σ$，和$ bstab（\ Mathcal {C}'）$表示Bridgeland稳定条件的空间，$ \ Mathcal {C}'$。 我们提供了简单的循环类别的稳定性条件的示例。我们还讨论了这些示例中的一些有趣现象，例如手性对称性破坏现象和非平凡的单曲。手性对称性破坏现象涉及稳定条件，无法提高布里奇兰稳定性条件。
A triangulated category $\mathcal{C}$ with a canonical Bott's isomorphism $[2]\xrightarrow{\sim}id$ is called a cyclic category in this paper. We give a new notion of stability conditions on a $k$-linear Krull-Schmidt cyclic category. Given such a stability condition $σ$, we can assign a Maslov index to each basic loop in such a category. If all Maslov indexes vanish, we get $\mathcal{C}',σ'$ as the $\mathbb{Z}$-lifts of $\mathcal{C},σ$ respectively such that $\mathcal{C}'$ is a $\mathbb{Z}$-graded triangulated category and $σ'$ is a Bridgeland stability condition on $\mathcal{C}'$. Moreover, we showed that there is an isomorphism $$Stab^{0,e}(\mathcal{C})\xrightarrow{\simeq} BStab(\mathcal{C}')$$ where $Stab^{0,e}(\mathcal{C})$ denotes the equivalence classes of stability conditions which are deformation equivalent to $σ$, and $BStab(\mathcal{C}')$ denotes the space of Bridgeland stability conditions on $\mathcal{C}'$. We provide examples of stability conditions on a simple cyclic category. We also discuss some interesting phenomena in these examples, such as the chirality symmetry breaking phenomenon and nontrivial monodromy. The chirality symmetry breaking phenomenon involves stability conditions which can not be lifted to Bridgeland stability conditions.
