根栈与三维镜像对称的GIT
GIT for root stacks and 3d mirror symmetry
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中文总结 AI 辅助
本文将根栈构造推广到范畴层面，结合三维镜像对称的GIT框架，推导了根栈导出范畴的半正交分解性质，证明了不同阶互素根栈凝聚层范畴的嵌入关系与周期性。
中文摘要 AI 辅助
Bodzenta-Donovan利用窗口理论证明了根栈√[n]{X/D}的导出范畴具有2n-周期的2项半正交分解。我们定义了根栈构造的范畴推广，并将其解释为Gammage-Hilburn-Mazel-Gee三维镜像对称等价B侧的拉回。我们运用Ben-Zvi-Francis-Nadler及Ben-Zvi-Nadler-Preygel关于半正交分解（SOD）的范畴表示论结果分析该拉回，还证明此拉回等价于A侧偏施伯的推前，可从A侧拉格朗日骨架的简单分解推得周期性。此外，我们将Bodzenta-Donovan的构造适配到新的GIT问题，得到当互素的m<n时，凝聚层范畴Coh(√[m]{X/D})到Coh(√[n]{X/D})的嵌入，并证明所得2项SOD具有2n-周期性。
英文摘要
Using window theory, Bodzenta--Donovan showed that the derived category of a root stack $\sqrt[n]{X/D}$ has a $2n$-periodic $2$-term semiorthogonal decomposition. We define a categorical generalization of the root stack construction and interpret this as a pullback on the B-side of the 3d mirror symmetry equivalence of Gammage--Hilburn--Mazel-Gee. We analyze this pullback using categorical representation theoretic results of Ben-Zvi--Francis--Nadler and Ben-Zvi--Nadler--Preygel applied to SODs. We also show that the pullback is equivalent to a pushforward of perverse schobers on the A-side, allowing us to deduce periodicity from a simple decomposition of an A-side Lagrangian skeleton. Additionally, we adapt the construction of Bodzenta--Donovan to a new GIT problem, which yields an embedding of Coh($\sqrt[m]{X/D}$) into Coh($\sqrt[n]{X/D}$) for $m<n$ coprime, and prove $2n$-periodicity of the resulting $2$-term SOD.
