Subsection 1.3. (0PE7) · ScalingStacks

1.3. Fukaya categories

The main examples of higher representations of $\mathfrak{gl}(1|1)^{+}$ that we introduce here are on partially wrapped Fukaya categories of symmetric powers of surfaces. These are $A_{\infty}$ -categories and the results of §1.3 will be made more precise in §1.4, where we work with differential categories.

Let $\Sigma$ be a compact oriented surface with a finite collection $M$ of marked points in its boundary, and assume that each component of $F$ contains at least one point of $M$ .

In the pictures below, we expand each point of $M$ to an interval in the boundary of $\Sigma$ . We draw the complement of these intervals in dotted light orange. Here are two views of a genus-one surface with one boundary component and $M$ consisting of one point.

For any $k\geq 0$ , Auroux [Au2, Section 3.1] considers a partially wrapped Fukaya category $\mathcal{F}(\mathrm{Sym}^{k}(\Sigma),M)$ of $\mathrm{Sym}^{k}(\Sigma)$ with set of stops $M\times\mathrm{Sym}^{k-1}(\Sigma)$ . We write $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ for the direct sum of these categories over all $k\geq 0$ (they vanish for $k$ large enough).

Given a component $I$ of $\partial\Sigma\setminus M$ , we define a higher action of $\mathfrak{gl}(1|1)^{+}$ on $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ .

Consider $(\Sigma_{1},M_{1},I_{1})$ and $(\Sigma_{2},M_{2},I_{2})$ two surfaces with chosen intervals as above. We form a new surface $(\Sigma,M)$ with a chosen interval $I$ by gluing $I_{1}$ and $I_{2}$ to the two legs of an open pair of pants.

0P4I

Theorem 1.3.1. There is an equivalence of triangulated categories

\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)\simeq\mathcal{F}(\mathrm{Sym}^{*}(\Sigma_{1}),M_{1}){\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}\mathcal{F}(\mathrm{Sym}^{*}(\Sigma_{2}),M_{2})

compatible with the structure of higher representations of $\mathfrak{gl}(1|1)^{+}$ .

Theorem 1.3.1 extends a result of Douglas–Manolescu [DouMa]; their theorem corresponds to the special case of Theorem 1.3.1 in which $\Sigma_{1}$ and $\Sigma_{2}$ have only one boundary circle and one marked point each. They prove an equivalence of categories without the statement on compatibility of higher actions.

More generally, given $(\Sigma,M)$ with two disjoint chosen intervals $I_{1},I_{2}$ in $\partial\Sigma\setminus M$ , we form a new surface $(\overline{\Sigma},\overline{M})$ by gluing the two legs of an open pair of pants to $I_{1}$ and $I_{2}$ . Theorem 1.3.1 generalizes to say that $\mathcal{F}(\mathrm{Sym}^{*}\overline{\Sigma},\overline{M})$ is equivalent to $\Delta\mathcal{F}(\mathrm{Sym}^{*}\Sigma,M)$ with its diagonal action.

This makes it possible to recover $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ for any $(\Sigma,M)$ from the case of a disk with two points in the boundary. As an illustration, consider the genus-one surface $(\Sigma,M)$ shown above; the partially wrapped Fukaya category $\mathcal{F}(\mathrm{Sym}^{*}(\Sigma),M)$ is described by the “torus algebra”, a standard example in bordered Heegaard Floer homology. We can cut $(\Sigma,M)$ along three arcs as shown; we are left with two rectangles. Gluing the cuts back together, the torus algebra can then be recovered from one application of the tensor product followed by two applications of the more general $\Delta$ construction.

In the case of the first symmetric power $\operatorname{Sym}\nolimits^{1}(\Sigma)=\Sigma$ , a general construction of Fukaya categories by a gluing procedure is given by Haiden, Katzarkov and Kontsevich in [HaiKaKon].

A general theory of partially wrapped Fukaya categories and how they glue is provided by Ganatra, Pardon and Shende in [GaPaShe], but this doesn’t apply directly to our case.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2