Topological Fukaya category of tagged arcs

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.

57 pages, 18 figures