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. arXiv:1608.08992v2 [math.SG] 29 Aug 2018 1 Our convention is that in an A ∞ -category, we read the compositions from right-to-left (as in [24]). This affects certain signs in computations. In particular, the A ∞ -relations are given by: m,n (−1) |a 1 |+...+|an|−n m d−m+1 (a d , . . . , a n+m+1 , m m (a n+m , . . . , a n+1 ), a n , . . . a 1 ) = 0

Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of the classical Yang–Baxter equation

“…It follows from the analysis made by Schedler [27] that not every quasi-constant (quasi-)trigonometric solutions of CYBE for the Lie algebra sl n (C) can be lifted to a solution of the associative Yang-Baxter equation for Mat n×n (C). Therefore, the combinatorial patterns of the trigonometric solutions of CYBE for sl n (C) (see [4]), of the quasi-trigonometric solutions of the CYBE for sl n (C) (see [21,26]) and of the trigonometric solutions of the AYBE for Mat n×n (C) (see [25,22]) share similar features but are different.…”

“…In the light of the work [11], it is natural to expect that any trigonometric solution of the CYBE arises as the geometric r-matrix of a pair (E, A), where E is a nodal Weierstraß cubic (such realizability is known for all elliptic and rational solutions; see [11] and references therein). On the other hand, the appearance of the Fukaya categories of higher genus Riemann surfaces in the classification of trigonometric solutions of AYBE [22] indicates, that a geometrization of the trigonometric solutions of CYBE could lead to further surprises.…”

Geometrization of trigonometric solutions of the associative and classical Yang-Baxter equations

“…In the above form the AYBE was introduced in [11]; the constant version was introduced in [1]. In [12] we proved an analog of Belavin-Drinfeld classifications for nondegenerate skew-symmetric solutions of the AYBE for the matrix algebra Mat n (C) in terms of some combinatorial data, called associative Belavin-Drinfeld data (BD data) (for the definition of the nondegeneracy condition, which is stronger than the one for the CYBE, see [10,Def. 1.4.3]).…”

“…In [12] we showed that some of trigonometric solutions of the AYBE are realized geometrically using families of 1-spherical objects (V s ), (O x ) on some nodal Calabi-Yau curves, where V s are simple vector bundles. In [10] we realized all nondegenerate trigonometric solutions of the AYBE using objects in the Fukaya categories of square-tiled surfaces.…”

Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation

“…Based on the description of simple vector bundles on Kodaira cycles of projective lines given in [15], Polishchuk computed the corresponding trigonometric solutions of AYBE [40]. Recently, Lekili and Polishchuk realized all trigonometric solutions of AYBE in terms of certain triple Massey products in the Fukaya category of an appropriate punctured Riemann surface [34].…”