Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces
Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces
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DOI：
10.1016/j.aim.2018.11.018
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发表时间：
2016-08
影响因子：
1.7
通讯作者：
Yankı Lekili;A. Polishchuk
中科院分区：
文献类型：
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作者：
Yankı Lekili;A. Polishchuk
证明了结合型Yang-Baxter方程(AYBE)的所有强非退化三角解都可以由方块曲面Fukaya范畴中的三重Massey积得到。在此过程中，我们给出了与一对1-球面对象相关的某个∞代数上的循环A Frobenius-代数结构的分类结果。作为应用，结合我们的结果和穿孔环面的同调镜像对称性(参看[17])，我们证明了射影直线圈上的任意两个简单向量丛由一列1-球面扭曲及其逆联系在一起。
We show that all strongly non-degenerate trigonometric solutions of the associative Yang–Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya categories of square-tiled surfaces. Along the way, we give a classification result for cyclic A∞-algebra structures on a certain Frobenius algebra associated with a pair of 1-spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf.[17]), we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of 1-spherical twists and their inverses.