Curves on surfaces and moduli of associative algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916000657571840 |
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| author | Lekili, Yanki |
| author_facet | Lekili, Yanki |
| contents | Given an immersion of a circle in a punctured surface $Σ$, we give an explicit (and finite) computation of the $A_\infty$-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 $A_\infty$-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 $\leq 4$, 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 $\mathcal{F}(Σ,D)$. 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 $\mathcal{F}(Σ)$ of some punctured surface $Σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curves on surfaces and moduli of associative algebras Lekili, Yanki Symplectic Geometry Algebraic Geometry Representation Theory Given an immersion of a circle in a punctured surface $Σ$, we give an explicit (and finite) computation of the $A_\infty$-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 $A_\infty$-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 $\leq 4$, 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 $\mathcal{F}(Σ,D)$. 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 $\mathcal{F}(Σ)$ of some punctured surface $Σ$. |
| title | Curves on surfaces and moduli of associative algebras |
| topic | Symplectic Geometry Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2605.00715 |