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

Abstract

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.