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

TL;DR

This paper connects solutions of the associative Yang-Baxter equation to Fukaya categories of square-tiled surfaces, providing a classification of algebraic structures and applications to vector bundles via homological mirror symmetry.

Contribution

It demonstrates that all strongly non-degenerate trigonometric solutions of the AYBE arise from Fukaya categories and classifies cyclic A-infinity structures on related Frobenius algebras.

Findings

All such AYBE solutions derive from triple Massey products.

Classification of cyclic A-infinity structures on specific Frobenius algebras.

Any two simple vector bundles on a cycle of projective lines are related by spherical twists.

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 category of square-tiled surfaces. Along the way, we give a classification result for cyclic $A_\infty$-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.