Noncommutative mirror symmetry for punctured surfaces
Noncommutative mirror symmetry for punctured surfaces
复制标题
穿孔表面的非交换镜面对称
DOI：
10.1090/tran/6375
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发表时间：
2011
影响因子：
1.3
通讯作者：
Raf Bocklandt
中科院分区：
文献类型：
--
作者：
Raf Bocklandt
2013年，Abouzaid，Auroux，Efimov，Katzarkov和Orlov证明了被穿透球面的包络Fukaya范畴和被穿孔球面的有限无分支覆盖等价于超势的奇点范畴。我们将这一结果推广到其他被穿孔的黎曼曲面上，并用来自二聚体模型的某些非对易代数来重新表述它。特别地，给定任何相容的二聚体模型，我们可以研究非交换矩阵分解的一个子范畴，并且证明这个范畴是$\mathtt{A}_\inty$-同构于穿孔黎曼曲面的包裹Fukaya范畴的一个子范畴。然后，二聚体模型和被穿透的黎曼面之间的联系在二聚体模型的对偶方面得到了很好的解释。
In 2013, 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.