AI 中文总结
本文证明抛物缺陷的修正可逆性，引入含余维一、二缺陷的编织理论，为曲面量子簇中标准坐标卡提供缺陷编织理论构造。
AI 中文摘要
我们证明了arXiv:2102.12283和arXiv:2505.14836中引入的抛物缺陷具有修正的可逆性。具体而言，我们表明Borel缺陷与其对偶相消，但需插入一个可逆的Weyl缺陷并限制到某个开子范畴。该主要结果对任意约化群$G$成立——为作说明，我们对$G=\mathrm{SL}_3$给出了详细计算。在此过程中，我们引入了一种缺陷编织理论，其中缺陷同时存在于余维数一和余维数二，该理论与缺陷3流形和缺陷曲面的粘合相容。我们的结果为量子簇代数中与曲面相关的某些标准坐标卡提供了一种潜在的缺陷编织理论构造。
英文摘要
We prove a modified invertibility property for the parabolic defects introduced in arXiv:2102.12283, arXiv:2505.14836. Namely, we show that the Borel defect cancels its dual, up to insertion of an invertible Weyl defect and up to restriction to an open-subcategory. The main result holds for an arbitrary reductive group $G$ -- for illustration we give extended computations for $G=\mathrm{SL}_3$. Along the way, we introduce a defect skein theory with defects in both codimension one and two, which is compatible with gluing of defect 3-manifolds and defect surfaces. Our results provide a potential defect skein theoretic construction of certain standard charts which appears frequently in quantum cluster varieties associated to surfaces.
Comments40 pages, many figures. Comments welcome!
