Subsection 7.4.12. (0PHU) · ScalingStacks

7.4.12. Fukaya categories from strand algebras

Consider an oriented singular curve $Z$ with $n_{z}\in\{2,4\}$ for all $z$ and its corresponding chord diagram $({\mathcal{Z}},{\mathbf{a}})$ (cf §7.2.4). Let $(F,\Lambda,S^{+},S^{-})$ be the associated sutured surface. We assume that every component of $\partial F$ intersects $S^{+}$ non-trivially (cf §7.2.5). Choose for each component $E$ of $S^{-}$ a point $e_{E}\in E$ and let $S=\{e_{E}\}_{E}$ . We have obtained a pair $(F,S)$ where $S$ is a finite subset of $\partial F$ .

Consider the arcs $\omega_{z}$ for $z\in Z_{exc}$ (cf §7.2.4). Note that $F\setminus\bigl(\bigcup_{z\in Z_{exc}}\omega_{z}\bigr)$ is a union of discs, each of which contains one point of $S$ .

Auroux [Au2, Definition 8] considers a partially wrapped Fukaya category ${\mathcal{F}}(\mathrm{Sym}^{n}F,S)$ of the symmetric power $\mathrm{Sym}^{n}(F)$ of $F$ with set of stops $S\times\mathrm{Sym}^{n-1}(F)$ . This is an (ungraded) $A_{\infty}$ -category over $k$ .

Let $s,t\in Z_{exc}$ . A path in $Z$ gives rise to a path in $F$ and this defines a bijection $f$ from the set of admissible homotopy classes of paths $s\to t$ in $Z$ to the set $\bar{\chi}_{t}^{s}$ of [Au2, Proposition 11] (recall the orientation reversal, cf Convention 7.2.12). When $s=t$ , the trivial path is sent to the element $\bf{1}_{i}$ of Auroux.

Auroux [Au2, Proposition 11] relates the $A_{\infty}$ -category ${\mathcal{F}}(\mathrm{Sym}^{n}F,S)$ to the strand algebra associated with $Z$ .

0PC4

Theorem 7.4.37 (Auroux). There is a fully faithful $A_{\infty}$ -functor

\Phi:{\mathcal{A}}(Z,n)\to{\mathcal{F}}(\mathrm{Sym}^{n}F,S),\ I\mapsto\prod_{i\in I}\omega_{i},\ \theta\mapsto(\chi(\theta),(f(\theta_{s}))_{s})

inducing an equivalence of derived categories.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2