Subsection 8.1.9. (0PI5) · ScalingStacks

8.1.9. Action on Fukaya categories

Assume now $Z$ is as in §7.4.12, so that we have an associated pair $(F,S)$ . We sketch a construction of the $2$ -representation on Fukaya categories of symmetric powers of $F$ via Auroux’s equivalences (§7.4.12). A rigorous construction would require a general theory of partially wrapped Fukaya categories and Lagrangian correspondences.

The surface associated with the singular curve $\hat{Z}_{r}$ of §8.1.8 can be identified with $F$ , with set of stops $\hat{S}_{r}$ obtained from $S$ by adding points $z_{1},\ldots,z_{r}$ . We have $(\hat{Z}_{r})_{exc}=Z_{exc}\sqcup\{z_{1},\ldots,z_{r}\}$ and we put $\omega_{i}=\omega_{z_{i}}$ . We denote by $z_{0}$ the point of $Z_{exc}\cap\partial F$ such that the interval $(z,z_{1})$ of $\partial F$ contains no point of $Z_{exc}$ .

Consider a positive integer $r$ . We have a fully faithful functor $\Xi^{\prime}_{r}:{\mathcal{F}}(\mathrm{Sym}^{n}F,\hat{S}_{r-1})\to{\mathcal{F}}(\mathrm{Sym}^{n}F,\hat{S}_{r})$ obtained by moving endpoints of Lagrangians so that they are not on the interval $[z_{r-1},z_{r}]$ of $\partial F$ . There is a commutative diagram where the vertical functors are Auroux’s functors:

The Lagrangian correspondence

\bigl\{(\{x_{1},\ldots,x_{n}\},\{x_{1},\ldots,x_{n},y\})\ |\ x_{1},\ldots,x_{n}\in F,\ y\in\omega_{r}\bigr\}\subset-\mathrm{Sym}^{n}F\times\mathrm{Sym}^{n+1}F

induces a functor

\Upsilon^{\prime}_{r}:\bar{{\mathcal{F}}}^{i}(\mathrm{Sym}^{n}F,\hat{S}_{r-1})\to\bar{{\mathcal{F}}}^{i}(\mathrm{Sym}^{n+1}F,\hat{S}_{r}),\ L\mapsto L\sqcup\omega_{r}

and there is a commutative diagram

We define a bimodule

	$\displaystyle L^{\prime}_{r}=L^{\prime}_{r,n}:{\mathcal{F}}(\mathrm{Sym}^{n}F,S)\otimes{\mathcal{F}}(\mathrm{Sym}^{n+r}F,S)^{\operatorname{opp}\nolimits}$	$\displaystyle\to k\operatorname{\!-diff}\nolimits$
	$\displaystyle\lambda_{1}\otimes\lambda_{2}$	$\displaystyle\mapsto\operatorname{Hom}\nolimits(\Xi^{\prime}_{r}\cdots\Xi^{\prime}_{1}(\lambda_{2}),\Upsilon^{\prime}_{r}\cdots\Upsilon^{\prime}_{1}(\lambda_{1})).$

We put $L^{\prime}_{r}=\bigoplus_{n\geq 0}L^{\prime}_{r,n}$ , a $({\mathcal{F}}(\mathrm{Sym}^{*}F,S),{\mathcal{F}}(\mathrm{Sym}^{*}F,S))$ -bimodule. We have an isomorphism of bimodules $L(-,-,r)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\prime}_{r}\circ(\Phi\otimes\Phi)$ .

Consider $t\in\operatorname{Hom}\nolimits_{{\mathcal{F}}(F,\hat{S}_{2})}(\omega_{1},\omega_{2})$ corresponding, via Auroux’s equivalence, to $[\xi(1)\to\xi(2)]$ . Similarly, we consider the two maps $u,v\in\operatorname{Hom}\nolimits_{{\mathcal{F}}(\mathrm{Sym}^{2}(F),\hat{S}_{3})}(\omega_{1}\sqcup\omega_{2},\omega_{2}\sqcup\omega_{3})$ corresponding, via Auroux’s equivalences, to $\{[\xi(1)\to\xi(2)],[\xi(2)\to\xi(3)]\}$ and to $\{\operatorname{id}\nolimits_{\xi(2)},[\xi(1)\to\xi(3)]\}$ respectively.