2016/08/31 by Yankı Lekili, Yanki Lekili, Lekili, Yanki +2 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.AG #math.QA #math.RT #math.SG
paper · pdf · doi:10.48550/arxiv.1608.08992
37 pages, 9 figures. Minor revision after a referee's comments. To appear in Advances in Mathematics
arxiv created 2018/08/29 · arxiv updated 2018/08/30
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 category 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. arXiv:1601.06141), 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.