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arXiv · 2404.10294

Topological Fukaya category of tagged arcs

Cheol-Hyun Cho·Kyoungmo Kim
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Abstract

A tagged arc on a surface is introduced by Fomin, Shapiro, and Thurston to study cluster theory on marked surfaces. Given a tagged arc system on a graded marked surface, we define its $\mathbb{Z}$-graded $\mathcal{A}_\infty$-category, generalizing the construction of Haiden, Katzarkov, and Kontsevich for arc systems. When a tagged arc system arises from a non-trivial involution on a marked surface, we show that this $\mathcal{A}_\infty$-category is quasi-isomorphic to the invariant part of the topological Fukaya category under the involution. In particular, this identifies tagged arcs with non-geometric idempotents of Fukaya category.

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Cheol-Hyun Cho, Kyoungmo Kim. 2024-04-16. Topological Fukaya category of tagged arcs. https://arxiv.org/abs/2404.10294

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