AI 中文总结
该研究针对辛流形的浸入拉格朗日膜，利用Koszul对偶建立局部同调镜像对称的拟等价，构造出到形变预投射代数等非交换空间的镜像函子，重现相关HMS结果与紧对象分裂生成结论。
AI 中文摘要
我们研究辛流形X中浸入拉格朗日膜𝕃（可能配备高阶平坦丛）相关的局部同调镜像对称。在𝕃的Floer理论满足特定有限性假设下，利用Koszul对偶推导出拟等价关系：𝔻Fuk_𝕃(X) ≅ 𝔻_{fd}(Ã_𝕃)，其中Ã_𝕃是称为扩展局部镜像的对偶微分阶箭图代数。我们将此应用于球面余切丛 plumbing 结构，得到若干HMS结果，并重现紧对象分裂生成的已知结论。论文第二部分考虑X中与𝕃有非平凡相交的体形变周期，这会产生镜像的非交换形变。当应用于（带框的）plumbing结构时，我们得到到形变预投射代数（或一般复矩映射水平下的Nakajima箭图簇）的镜像函子。对于ADHM和仿射ADE型浸入，我们的构造生成到Kapustin-Kuznetsov-Orlov、Baranovsky-Ginzburg-Kuznetsov及Kawamata所研究非交换空间的镜像函子。
英文摘要
We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.
Comments52 pages. Comments are welcome!
