Noncommutative mirror symmetry for punctured surfaces
Raf Bocklandt With an appendix by Mohammed Abouzaid Raf Bocklandt Korteweg de Vries institute University of Amsterdam (UvA) Science Park 904 1098 XH Amsterdam The Netherlands
Abstract
In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is ${\mathtt{A}}_{\infty}$ -isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.
中文速览
带标点黎曼面（punctured Riemann surfaces）与代数几何之间存在一种深刻的镜像对称关系，但此前只在球面挖孔这一特殊情形下得到严格证明。本文将这一结果推广到任意带至少三个穿孔的黎曼面：以"二聚体模型"（dimer model，嵌入黎曼面的带方向图）为出发点，构造对应的雅可比代数（Jacobi algebra）及其非交换矩阵分解范畴，并证明该范畴的一个子范畴与相应带标点曲面的缠绕福卡亚范畴（wrapped Fukaya category）的一个子范畴之间存在A∞-同构。两侧之间的对应关系可以用一种称为"二聚体对偶"（dimer duality）的显式操作来解释：将原始二聚体翻转、反向并重新粘合，就得到描述辛几何一侧的对偶二聚体。这项工作不仅大幅拓展了镜像对称的适用范围，还将非交换代数几何与辛拓扑之间的联系建立在一个统一的组合框架之上，为进一步理解朗道-金兹堡模型（Landau-Ginzburg model）的非交换推广提供了新工具。
原文 arXiv:1111.3392;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1111.3392v2