Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces

摘要

We show that all strongly non-degenerate trigonometric solutions of theassociative Yang-Baxter equation (AYBE) can be obtained from triple Masseyproducts in the Fukaya category of square-tiled surfaces. Along the way, wegive a classification result for cyclic $A_infty$-algebra structures on acertain Frobenius algebra associated with a pair of spherical objects in termsof the equivalence classes of the corresponding solutions of the AYBE. As anapplication, combining our results with homological mirror symmetry forpunctured tori (cf. arXiv:1601.06141), we prove that any two simple vectorbundles on a cycle of projective lines are related by a sequence of sphericaltwists.