2026/05/01 by Yanki Lekili · 1 voice
Mathematics · #math.SG #math.AG #math.RT
arxiv published 2026/05/01 · arxiv updated 2026/05/11
Given an immersion of a circle in a punctured surface Σ, we give an explicit (and finite) computation of the A_∞-algebra associated with this curve when viewed as an object in a (relative) Fukaya category of Σ in terms of the signed Gauss word recording the double points in a traversal of the curve and the visible polygons that it bounds in Σ. We illustrate our computational technique by fully determining the A_∞-products for immersions with up to three self-intersections. In particular, it is proved that, over an algebraically closed field, all associative algebras of dimension ≤ 4, with one exception, can be realized as the (degree 0) endomorphism algebra of some Lagrangian immersion of a circle equipped with a bounding cochain computed in some relative Fukaya category F(Σ,D). We also note that any finite-dimensional algebra with radical square zero arises as the (degree 0) endomorphism algebra of an object in the Fukaya category F(Σ) of some punctured surface Σ.