Topological Fukaya category of tagged arcs

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 Konts...

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Bibliographic Details
Published inarXiv.org
Main Authors Cheol-Hyun Cho, Kim, Kyoungmo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.04.2024
Subjects
Topology
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Summary: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.
ISSN:2331-8422