Locality in the Fukaya category of a hyperkähler manifold

2.1 Geometric framework

We start with a brief overview of Floer cohomology and Fukaya $A_{\infty }$ operations for Lagrangian submanifolds with vanishing Maslov class. We refer the reader to [Reference Fukaya, Oh, Ohta and OnoFOOO09, Reference SeidelSei08] for a detailed treatment. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}=\Big\{\sum a_{i}T^{E_{i}}\,\Big|\,a_{i}\in \mathbb{R},E_{i}\in \mathbb{R},\lim _{i\rightarrow \infty }E_{i}=\infty \Big\}\end{eqnarray}$$

denote the Novikov field. Let $\Vert \cdot \Vert :\unicode[STIX]{x1D6EC}\rightarrow \mathbb{R}_{{>}0}$ be the non-Archimedean norm given by $\Vert 0\Vert =0$ and

(2) $$\begin{eqnarray}\biggl\|\mathop{\sum }_{i}a_{i}T^{E_{i}}\biggr\|=\exp \Bigl(\!-\text{min}_{i}E_{i}\Bigr).\end{eqnarray}$$

Let $(M,\unicode[STIX]{x1D714})$ be a symplectic manifold of dimension $2n$ , either compact or with appropriately bounded geometry. Let $\unicode[STIX]{x1D6E9}\in {\mathcal{J}}_{\unicode[STIX]{x1D714}}$ . Assume there exists a non-vanishing complex $n$ -form $\unicode[STIX]{x1D6F7}$ that is of type $(n,0)$ with respect to $\unicode[STIX]{x1D6E9}$ . A grading for an oriented Lagrangian submanifold $L\subset M$ is the choice of a function $\unicode[STIX]{x1D703}_{L}:L\rightarrow \mathbb{R}$ such that $\unicode[STIX]{x1D6F7}|_{L}=e^{\sqrt{-1}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}_{L}}\unicode[STIX]{x1D70C}$ where $\unicode[STIX]{x1D70C}$ is a positive real $n$ -form on $L$ . Consider a collection ${\mathcal{L}}$ of graded spin compact Lagrangian submanifolds $L_{i}\subset M$ for $i$ belonging to an index set $A$ . Assume that the $L_{i}$ intersect each other cleanly. That is, for $B\subset A$ , the intersection $L_{B}=\bigcap _{i\in B}L_{i}$ is a smooth manifold, and

$$\begin{eqnarray}\mathit{TL}_{B}=\mathop{\bigcap }_{i\in B}\mathit{TL}_{i}.\end{eqnarray}$$

For each connected component $C\subset L_{i}\,\cap \,L_{j}$ , the gradings on $L_{i},L_{j}$ , give rise to an integer $m_{C}$ , called the Maslov index, in the following way. Let $p\in C$ . Choose a direct sum decomposition $T_{p}L_{i}=\bigoplus _{k}V_{k}$ and $\unicode[STIX]{x1D6FC}_{k}\in [0,1)$ such that $T_{p}L_{j}=\bigoplus _{k}e^{\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}_{k}\unicode[STIX]{x1D6E9}}V_{k}$ . Then,

$$\begin{eqnarray}m_{C}=\mathop{\sum }_{k}\unicode[STIX]{x1D6FC}_{k}\dim V_{k}-\unicode[STIX]{x1D703}_{L_{j}}(p)+\unicode[STIX]{x1D703}_{L_{i}}(p).\end{eqnarray}$$

One verifies that $m_{C}$ does not depend on $p$ or the direct sum decomposition.

We define the Floer complex $\mathit{CF}^{\ast }(L_{i},L_{j})$ as follows. For $N$ a smooth manifold, let $A^{k}(N)$ denote the differential $k$ -forms on  $N$ . Define

$$\begin{eqnarray}\mathit{CF}^{\ast }(L_{i},L_{j})=\bigoplus _{C\subset L_{i}\,\cap \,L_{j}}A^{\ast }(C)[-m_{C}]\otimes \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

The norm $\Vert \cdot \Vert$ on $\unicode[STIX]{x1D6EC}$ induces a norm, also denoted $\Vert \cdot \Vert$ , on $\mathit{CF}^{\ast }(L_{i},L_{j})$ , which makes $\mathit{CF}^{\ast }(L_{i},L_{j})$ a normed $\unicode[STIX]{x1D6EC}$ vector space. Here and below, all normed vector spaces are understood to be completed with respect to the norm. If $\unicode[STIX]{x1D6FC}\in \mathit{CF}^{m}(L_{i},L_{j})$ for some $m$ , write $|\unicode[STIX]{x1D6FC}|=m$ .

The vector spaces $\mathit{CF}^{\ast }(L_{i},L_{j})$ come with a family of multilinear operations constructed using $\unicode[STIX]{x1D6E9}$ -holomorphic curves. Let $D\subset \mathbb{C}$ denote the closed unit disk equipped with the complex orientation, and let $\unicode[STIX]{x1D6E5}\subset (\unicode[STIX]{x2202}D)^{k+1}$ denote the pairwise diagonal. For

$$\begin{eqnarray}\vec{z}=(z_{0},\ldots ,z_{k})\in (\unicode[STIX]{x2202}D)^{k+1}\setminus \unicode[STIX]{x1D6E5},\end{eqnarray}$$

let $\mathbf{z}$ denote the corresponding subset of $\unicode[STIX]{x2202}D$ . Say that $\vec{z}$ is cyclically ordered if its order agrees with the cyclic order given by the induced orientation of $\unicode[STIX]{x2202}D$ . Let $(z_{i},z_{i+1})\subset \unicode[STIX]{x2202}D$ denote the open interval starting at $z_{i}$ and ending at $z_{i+1}$ with respect to the induced orientation of $\unicode[STIX]{x2202}D$ . For ${\mathcal{I}}=(i_{0},\ldots ,i_{k})\in A^{k+1}$ and $E\in \mathbb{R}$ , define

$$\begin{eqnarray}\widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})=\left\{(u,\vec{z})\left|\begin{array}{@{}l@{}}\displaystyle \vec{z}\in (\unicode[STIX]{x2202}D)^{k+1}\setminus \unicode[STIX]{x1D6E5},\text{ cyclically ordered,}\\ \displaystyle u:D\setminus \mathbf{z}\rightarrow M\text{ smooth},\\ \displaystyle \overline{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D6E9}}u=0,\\ \displaystyle u((z_{j},z_{j+1}))\subset L_{i_{j}},\\ \displaystyle \int u^{\ast }\unicode[STIX]{x1D714}=E\end{array}\right.\right\}.\end{eqnarray}$$

For $(u,\vec{z})\in \widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ , we have [Reference McDuff and SalamonMS12, Lemma 2.2.1]

$$\begin{eqnarray}E=\int u^{\ast }\unicode[STIX]{x1D714}=\frac{1}{2}\int |du|_{g_{\unicode[STIX]{x1D6E9}}}^{2}\,\operatorname{dvol}_{D}\geqslant 0.\end{eqnarray}$$

So, if $E<0$ , then $\widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})=\emptyset$ . Furthermore, if $E=0$ , then $\widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ consists of constant maps.

An automorphism of the disk $\unicode[STIX]{x1D711}\in \operatorname{PSL}(2,\mathbb{R})$ acts on

$$\begin{eqnarray}(u,\vec{z})\in \widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})\end{eqnarray}$$

by $u\mapsto u\circ \unicode[STIX]{x1D711}$ and $z_{j}\mapsto \unicode[STIX]{x1D711}^{-1}(z_{j})$ . A pair $(I,E)$ is called stable if either $E>0$ or $k\geqslant 2$ and $E=0$ . If $(I,E)$ is stable, then the action of $\operatorname{PSL}(2,\mathbb{R})$ on ${\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})$ has finite order stabilizers. For stable $(I,E)$ , set ${\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})=\widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})/\operatorname{PSL}(2,\mathbb{R})$ . Otherwise, set ${\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})=\emptyset$ . For $[u,\vec{z}]\in {\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})$ , the condition

$$\begin{eqnarray}\int u^{\ast }\unicode[STIX]{x1D714}=E<\infty\end{eqnarray}$$

and the assumption that the Lagrangians $L_{i}$ intersect cleanly imply that $u$ extends to a continuous map $\overline{u}:D\rightarrow M$ . See [Reference WoodwardWoo11, Lemma 2.5]. So, we define evaluation maps

$$\begin{eqnarray}ev_{j}^{E}:{\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})\rightarrow L_{i_{j-1}}\,\cap \,L_{i_{j}},\quad j=0,\ldots ,k,\end{eqnarray}$$

by $ev_{j}^{E}((u,\vec{z}))=\overline{u}(z_{j})$ . The spaces ${\mathcal{M}}({\mathcal{I}},E,{\mathcal{L}})$ admit stable map compactifications $\overline{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ to which the evaluation maps extend naturally, and we denote these extensions by $ev_{j}^{E}$ as well.

Although moduli spaces of stable maps are usually singular, it can been shown in many cases that they admit a virtual fundamental chain, which behaves like the fundamental chain of a smooth manifold. For the purpose of this discussion, we assume that operations of virtual pull-back (respectively push-forward) of differential forms along the maps $ev_{i}^{E}$ (respectively the map $ev_{0}^{E}$ ) have been defined. These operations should satisfy the same properties as push-forward and pull-back along smooth maps of manifolds, with $ev_{0}^{E}$ behaving like a proper submersion. When the moduli spaces are smooth of expected dimension, virtual pull-back and push-forward reduce to the usual pull-back and push-forward. Such operations were introduced in a similar context by Fukaya [Reference FukayaFuk10]. It is expected that polyfolds [Reference Hofer, Wysocki and ZehnderHWZ10] will give an alternate approach. We implicitly extend all operations on differential forms $\unicode[STIX]{x1D6EC}$ -linearly to the Floer complex $\mathit{CF}^{\ast }(L_{i},L_{j})$ .

Define maps

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}:\mathit{CF}^{\ast }(L_{i_{0}},L_{i_{1}})\otimes \cdots \otimes \mathit{CF}^{\ast }(L_{i_{k-1}},L_{i_{k}})\rightarrow \mathit{CF}^{\ast }(L_{i_{0}},L_{i_{k}})[2-k]\end{eqnarray}$$

by

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\unicode[STIX]{x1D6FF}_{k,1}d\unicode[STIX]{x1D6FC}_{1}+(-1)^{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})}\mathop{\sum }_{E}T^{E}ev_{0\ast }^{E}(ev_{1}^{E\ast }\unicode[STIX]{x1D6FC}_{1}\wedge \cdots \wedge ev_{k}^{E\ast }\unicode[STIX]{x1D6FC}_{k}),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\mathop{\sum }_{j=1}^{k}j(|\unicode[STIX]{x1D6FC}_{j}|+1)+1.\end{eqnarray}$$

The sum is well defined by Gromov’s compactness theorem [Reference GromovGro85]. The maps $\unicode[STIX]{x1D707}_{k}$ preserve grading by a virtual dimension calculation along the lines of [Reference Fukaya, Oh, Ohta and OnoFOOO09, § 3.7.5]. It follows from the structure of the compactification $\overline{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ and the properties of pull-back and push-forward that the operations $\unicode[STIX]{x1D707}_{k}$ satisfy the $A_{\infty }$ relations,

(3) $$\begin{eqnarray}\mathop{\sum }_{\substack{ k_{1}+k_{2}=k+1 \\ 1\leqslant q\leqslant k_{1}}}(-1)^{\star }\unicode[STIX]{x1D707}_{k_{1}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{q-1},\unicode[STIX]{x1D707}_{k_{2}}(\unicode[STIX]{x1D6FC}_{q},\ldots ),\unicode[STIX]{x1D6FC}_{q+k_{2}},\ldots ,\unicode[STIX]{x1D6FC}_{k})=0,\end{eqnarray}$$

where $\star =q-1+\sum _{j=1}^{q-1}|\unicode[STIX]{x1D6FC}_{j}|$ . See [Reference Solomon and TukachinskyST16] for a detailed derivation including signs in the case of a single Lagrangian. The generalization to our setting is not hard.

Considering (3) for $k=1$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{1}\circ \unicode[STIX]{x1D707}_{1}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D707}_{0},\unicode[STIX]{x1D6FC})+(-1)^{|\unicode[STIX]{x1D6FC}|+1}\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707}_{0}).\end{eqnarray}$$

So, if $\unicode[STIX]{x1D707}_{0}=0\in \mathit{CF}(L_{i},L_{i})$ for $i\in A$ , then $\unicode[STIX]{x1D707}_{1}^{2}=0$ , and we can define the Floer cohomology of $L_{i},L_{j}$ , by

$$\begin{eqnarray}\mathit{HF}^{\ast }(L_{i},L_{j})=H^{\ast }(\mathit{CF}^{\bullet }(L_{i},L_{j}),\unicode[STIX]{x1D707}_{1}).\end{eqnarray}$$

In this case, we say the Lagrangians $L_{i},i\in A$ , are tautologically unobstructed with respect to $\unicode[STIX]{x1D6E9}$ . Floer [Reference FloerFlo88] introduced $\mathit{HF}^{\ast }(L_{i},L_{j})$ under the assumption $\unicode[STIX]{x1D70B}_{2}(M,L_{i})=0$ for $i\in A$ . Then, a holomorphic disk with boundary in $L_{i}$ must have energy zero, so the stability condition implies $\overline{{\mathcal{M}}}(i,E,{\mathcal{L}})=\emptyset$ for $i\in A$ . It follows that the Lagrangians $L_{i}$ are tautologically unobstructed for all tame $\unicode[STIX]{x1D6E9}$ . A similar case is when the ambient manifold  $M$ and the Lagrangians $L_{i}$ are exact. That is, there exist $\unicode[STIX]{x1D706}\in A^{1}(M)$ with $d\unicode[STIX]{x1D706}=\unicode[STIX]{x1D714}$ and $f_{i}:L_{i}\rightarrow \mathbb{R}$ with $\unicode[STIX]{x1D706}|_{L_{i}}=df_{i}$ . Then, Stokes’ theorem implies that a holomorphic disk with boundary in $L_{i}$ must have energy zero, so the stability condition implies $\overline{{\mathcal{M}}}(i,E,{\mathcal{L}})=\emptyset$ for $i\in A$ . Again, it follows that the Lagrangians $L_{i}$ are tautologically unobstructed for all tame $\unicode[STIX]{x1D6E9}$ . This is the setting of [Reference SeidelSei08].

To deal with general Lagrangian submanifolds, which may not be tautologically unobstructed, Fukaya et al. [Reference Fukaya, Oh, Ohta and OnoFOOO09] introduce the notion of bounding cochains. A bounding cochain for $L_{i}$ is a cochain $b\in \mathit{CF}^{1}(L_{i},L_{i})$ with $\Vert b\Vert <1$ that satisfies the Maurer–Cartan equation

(4) $$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\unicode[STIX]{x1D707}_{k}(b^{\otimes k})=0.\end{eqnarray}$$

Let $b_{i}$ be a bounding cochain for $L_{i}$ . We define

$$\begin{eqnarray}\mathit{CF}^{\ast }((L_{i},b_{i}),(L_{j},b_{j}))=\mathit{CF}^{\ast }(L_{i},L_{j}),\end{eqnarray}$$

and we define

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D707}}_{k}:\bigotimes _{j=1}^{k}\mathit{CF}^{\ast }((L_{i_{j-1}},b_{i_{j-1}}),(L_{i_{j}},b_{i_{j}}))\longrightarrow \mathit{CF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{k}},b_{i_{k}}))\end{eqnarray}$$

by

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D707}}_{k}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\mathop{\sum }_{\mathop{\sum }_{i}m_{i}+k=l}\unicode[STIX]{x1D707}_{l}(b_{i_{0}}^{\otimes m_{0}}\otimes \unicode[STIX]{x1D6FC}_{1}\otimes b_{i_{1}}^{\otimes m_{1}}\otimes \unicode[STIX]{x1D6FC}_{2}\otimes \cdots \otimes \unicode[STIX]{x1D6FC}_{k}\otimes b_{i_{k}}^{\otimes m_{k}}).\end{eqnarray}$$

One verifies algebraically that (3) holds with $\unicode[STIX]{x1D707}_{k}$ replaced by $\hat{\unicode[STIX]{x1D707}}_{k}$ . Moreover, (4) is the same as $\hat{\unicode[STIX]{x1D707}}_{0}=0$ . Consequently, $\hat{\unicode[STIX]{x1D707}}_{1}^{2}=0$ . Thus, we define

$$\begin{eqnarray}\mathit{HF}^{\ast }((L_{i_{j-1}},b_{i_{j-1}}),(L_{i_{j}},b_{i_{j}}))=H^{\ast }(\mathit{CF}^{\bullet }((L_{i_{j-1}},b_{i_{j-1}}),(L_{i_{j}},b_{i_{j}})),\hat{\unicode[STIX]{x1D707}}_{1}).\end{eqnarray}$$

Equation (3) for $k=2$ implies that the composition map

$$\begin{eqnarray}\circ :\mathit{HF}^{\ast }((L_{i_{1}},b_{i_{1}}),(L_{i_{2}},b_{i_{2}}))\otimes \mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{1}},b_{i_{1}}))\rightarrow \mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{2}},b_{i_{2}}))\end{eqnarray}$$

given by $[\unicode[STIX]{x1D6FC}_{2}]\circ [\unicode[STIX]{x1D6FC}_{1}]=(-1)^{|a_{1}|}[\hat{\unicode[STIX]{x1D707}}_{2}(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2})]$ is well defined. This composition is associative by (3) for $k=3$ . In particular, the Floer cohomology $\mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{0}},b_{i_{0}}))$ of a Lagrangian with itself is an associative algebra.

A Hamiltonian flow $\unicode[STIX]{x1D711}_{t}:M\rightarrow M$ gives rise to a series of maps

$$\begin{eqnarray}f_{k}^{\unicode[STIX]{x1D711}}:\mathit{CF}^{\ast }(L_{i_{0}},L_{i_{1}})\otimes \cdots \otimes \mathit{CF}^{\ast }(L_{i_{k-1}},L_{i_{k}})\rightarrow \mathit{CF}^{\ast }(\unicode[STIX]{x1D711}_{1}(L_{i_{0}}),\unicode[STIX]{x1D711}_{1}(L_{i_{k}}))\end{eqnarray}$$

defined using moduli spaces of holomorphic disks. The geometric construction is similar to [Reference Fukaya, Oh, Ohta and OnoFOOO09, § 4.6.1] and [Reference SeidelSei08, § 10e]. The maps $f_{k}^{\unicode[STIX]{x1D711}}$ satisfy $\Vert f_{k}\Vert \leqslant 1$ with strict inequality for $k=0$ , and

(5) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ l \\ m_{1}+\cdots +m_{l}=k}}\unicode[STIX]{x1D707}_{l}(f_{m_{1}}^{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{m_{1}}),\ldots ,f_{m_{l}}^{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D6FC}_{k-m_{l}+1},\ldots ,\unicode[STIX]{x1D6FC}_{k}))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ k_{1}+k_{2}=k+1 \\ 1\leqslant i\leqslant k_{1}}}(-1)^{\star }f_{k_{1}}^{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{i-1},\unicode[STIX]{x1D707}_{k_{2}}(\unicode[STIX]{x1D6FC}_{i},\ldots ,\unicode[STIX]{x1D6FC}_{i+k_{2}-1}),\ldots ,\unicode[STIX]{x1D6FC}_{k}),\end{eqnarray}$$

where $\star =q-1+\sum _{j=1}^{q-1}|\unicode[STIX]{x1D6FC}_{j}|$ . One verifies algebraically that if $b$ solves the Maurer–Cartan equation (4), then so does

$$\begin{eqnarray}f_{\ast }^{\unicode[STIX]{x1D711}}(b)=\mathop{\sum }_{k}f_{k}^{\unicode[STIX]{x1D711}}(b^{\otimes k}).\end{eqnarray}$$

A fundamental property of Floer cohomology is that any Hamiltonian flow $\unicode[STIX]{x1D711}_{t}:M\rightarrow M$ gives rise [Reference SeidelSei08, § 8c,8k] to a canonical element

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\in \mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(\unicode[STIX]{x1D711}_{1}(L_{i_{0}}),f_{\ast }^{\unicode[STIX]{x1D711}}(b_{i_{0}}))).\end{eqnarray}$$

If $\unicode[STIX]{x1D711}_{t}=\mathsf{id}_{M}$ , then $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}$ is the unit of the algebra $\mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{0}},b_{i_{0}}))$ . Moreover, if $\unicode[STIX]{x1D713}_{t}:M\rightarrow M$ is another Hamiltonian flow, then

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D713}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D713}\circ \unicode[STIX]{x1D711}}.\end{eqnarray}$$

It follows that

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}^{-1}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}=\mathsf{id}_{(L_{i_{0}},b_{i_{0}})},\quad \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}^{-1}}=\mathsf{id}_{(\unicode[STIX]{x1D711}(L_{i_{0}}),f_{\ast }^{\unicode[STIX]{x1D711}}(b_{i_{0}}))}.\end{eqnarray}$$

In particular, composition with $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}$ induces a canonical isomorphism

$$\begin{eqnarray}\mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(L_{i_{1}},b_{i_{1}}))\overset{{\sim}}{\longrightarrow }\mathit{HF}^{\ast }((L_{i_{0}},b_{i_{0}}),(\unicode[STIX]{x1D711}(L_{i_{1}}),f_{\ast }^{\unicode[STIX]{x1D711}}(b_{i_{1}}))).\end{eqnarray}$$

If $L,L^{\prime }$ , are tautologically unobstructed but do not intersect cleanly, we define

$$\begin{eqnarray}\mathit{HF}^{\ast }(L,L^{\prime }):=\mathit{HF}^{\ast }((L,0),(\unicode[STIX]{x1D711}_{1}(L^{\prime }),f_{\ast }^{\unicode[STIX]{x1D711}}(0)))\end{eqnarray}$$

where $\unicode[STIX]{x1D711}_{t}:M\rightarrow M$ is a Hamiltonian flow such that $\unicode[STIX]{x1D711}_{1}(L^{\prime })$ is transverse to $L$ . Since the choice of bounding chain is canonical, we omit it from the notation. By the preceding discussion, this definition does not depend on the choice of $\unicode[STIX]{x1D711}$ .

2.2 First applications

Let $(M,I,J,K,g)$ be a hyperkähler manifold, and let $L_{i}\subset M$ be $I$ -holomorphic Lagrangians for $i$ belonging to a countable index set $A$ . In particular, the submanifolds $L_{i}$ are Lagrangian with respect to the symplectic form $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{J}$ . In the following, all Floer-theoretic constructions will be carried out with respect to this symplectic form.

Corollary 2.1. For all $\unicode[STIX]{x1D6E9}\in R_{I}\,\cap \,{\mathcal{J}}_{\unicode[STIX]{x1D714}}$ except a countable set, the $A_{\infty }$ operations

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}:\mathit{CF}^{\ast }(L_{i},L_{i})^{\otimes k}\rightarrow \mathit{CF}^{\ast }(L_{i},L_{i})\end{eqnarray}$$

are given by

(6) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\left\{\begin{array}{@{}ll@{}}d\unicode[STIX]{x1D6FC}_{1}\quad & k=1,\\ (-1)^{|\unicode[STIX]{x1D6FC}_{1}|}\unicode[STIX]{x1D6FC}_{1}\wedge \unicode[STIX]{x1D6FC}_{2}\quad & k=2,\\ 0\quad & k\neq 1,2.\end{array}\right.\end{eqnarray}$$

In particular, the Lagrangians $L_{i}$ are tautologically unobstructed.

Corollary 2.2. For all $\unicode[STIX]{x1D6E9}\in R_{I}\,\cap \,{\mathcal{J}}_{\unicode[STIX]{x1D714}}$ except a countable set, the Floer coboundary operator $\unicode[STIX]{x1D707}_{1}:\mathit{CF}^{\ast }(L_{i},L_{j})\rightarrow \mathit{CF}^{\ast }(L_{i},L_{j})$ coincides with the exterior derivative $d$ . Thus,

$$\begin{eqnarray}\mathit{HF}^{\ast }(L_{i},L_{j})\simeq \bigoplus _{\substack{ C\subset L_{i}\,\cap \,L_{j} \\ \text{a}\;\text{component}}}H^{\ast }(C)[m_{c}]\otimes \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

Proof of Corollaries 2.1 and 2.2.

Constant maps have energy zero, so Theorem 1.6 implies that the moduli spaces $\overline{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ are empty unless $E=0$ . On the other hand, for ${\mathcal{I}}\in A^{k+1}$ with $k\leqslant 1$ , the stability condition implies that $\overline{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}})$ is empty unless $E>0$ . Consequently, we obtain (6) when $k=0,1$ as well as Corollary 2.2.

When $k\geqslant 2$ , constant maps can be stable. Let $\overline{{\mathcal{M}}}_{k+1}$ denote the moduli space of stable disks with $k+1$ cyclically ordered boundary marked points, up to biholomorphism. For ${\mathcal{I}}=(i,i,\ldots ,i)$ we have $\overline{{\mathcal{M}}}({\mathcal{I}},0,{\mathcal{L}})\simeq L_{i}\times \overline{{\mathcal{M}}}_{k+1}$ , and $ev_{j}^{0}:\overline{{\mathcal{M}}}({\mathcal{I}},0,{\mathcal{L}})\rightarrow L_{i}$ is the projection to the first factor for $j=0,\ldots ,k$ . Since $\overline{{\mathcal{M}}}({\mathcal{I}},0,{\mathcal{L}})$ is a smooth manifold of expected dimension, the virtual fundamental class coincides with the usual fundamental class. We have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{k}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k}) & = & \displaystyle (-1)^{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})}ev_{0\ast }^{0}\biggl(\bigwedge _{j=1}^{k}ev_{j}^{0\ast }\unicode[STIX]{x1D6FC}_{j}\biggr)\nonumber\\ \displaystyle & = & \displaystyle (-1)^{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})}ev_{0\ast }^{0}ev_{0}^{0\ast }\biggl(\bigwedge _{j=1}^{k}\unicode[STIX]{x1D6FC}_{j}\biggr)\nonumber\\ \displaystyle & = & \displaystyle (-1)^{\unicode[STIX]{x1D700}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})}\bigwedge _{j=1}^{k}\unicode[STIX]{x1D6FC}_{j}\wedge ev_{0\ast }^{0}(1).\nonumber\end{eqnarray}$$

The fiber of $ev_{0}^{0}$ is zero-dimensional only when $k=2$ , and in this case $ev_{0}^{0}$ is the identity map. Equation (6) for $k\geqslant 2$ follows.◻

2.3 Algebraic framework

Before presenting further applications, we recall some definitions pertaining to abstract $A_{\infty }$ categories. In general, a curved  $A_{\infty }$ category ${\mathcal{A}}$ consists of the following data:

1. – a collection of objects $\operatorname{Ob}{\mathcal{A}}$ ;

2. – for each pair of objects $A,B\in \operatorname{Ob}{\mathcal{A}}$ , a graded normed $\unicode[STIX]{x1D6EC}$ vector space $\mathit{Hom}_{{\mathcal{A}}}(A,B)$ ;

3. – for each $(k+1)$ -tuple of objects $A_{0},\ldots ,A_{k},k\geqslant 0$ , a multilinear map

   $$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}^{{\mathcal{A}}}:\bigotimes _{i=1}^{k}\mathit{Hom}_{{\mathcal{A}}}(A_{i-1},A_{i})\rightarrow \mathit{Hom}_{{\mathcal{A}}}(A_{0},A_{k})[2-k].\end{eqnarray}$$

The maps $\unicode[STIX]{x1D707}_{k}^{{\mathcal{A}}}$ must satisfy $\Vert \unicode[STIX]{x1D707}_{k}^{{\mathcal{A}}}\Vert \leqslant 1$ with strict inequality for $k=0$ , as well as the $A_{\infty }$ relations (3). For example, we can take the objects to be Lagrangian submanifolds, define $\mathit{Hom}(L_{0},L_{1})=\mathit{CF}^{\ast }(L_{0},L_{1})$ , and define the maps $\unicode[STIX]{x1D707}_{k}^{{\mathcal{A}}}$ as in § 2.1. An $A_{\infty }$  category is a curved $A_{\infty }$ category with $\unicode[STIX]{x1D707}_{0}=0$ .

A curved $A_{\infty }$  functor $f:{\mathcal{A}}\rightarrow {\mathcal{B}}$ consists of a map $f:\operatorname{Ob}{\mathcal{A}}\rightarrow \operatorname{Ob}{\mathcal{B}}$ along with multilinear maps

$$\begin{eqnarray}f_{k}:\bigotimes _{i=1}^{k}\mathit{Hom}_{{\mathcal{A}}}(A_{i-1},A_{i})\rightarrow \mathit{Hom}_{{\mathcal{B}}}(f(A_{0}),f(A_{k}))[1-k]\end{eqnarray}$$

for each $k+1$ tuple of objects $A_{0},\ldots ,A_{k}\in \operatorname{Ob}{\mathcal{A}}$ , for $k\geqslant 0$ . The maps $f_{k}$ must satisfy $\Vert f_{k}\Vert \leqslant 1$ with strict inequality for $k=0$ , as well as the relation

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ l \\ m_{1}+\cdots +m_{l}=k}}\unicode[STIX]{x1D707}_{l}^{{\mathcal{B}}}(f_{m_{1}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{m_{1}}),\ldots ,f_{m_{l}}(\unicode[STIX]{x1D6FC}_{k-m_{l}+1},\ldots ,\unicode[STIX]{x1D6FC}_{k}))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ k_{1}+k_{2}=k+1 \\ 1\leqslant i\leqslant k_{1}}}(-1)^{\star }f_{k_{1}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{i-1},\unicode[STIX]{x1D707}_{k_{2}}^{{\mathcal{A}}}(\unicode[STIX]{x1D6FC}_{i},\ldots ,\unicode[STIX]{x1D6FC}_{i+k_{2}-1}),\ldots ,\unicode[STIX]{x1D6FC}_{k}),\nonumber\end{eqnarray}$$

where $\star =q-1+\sum _{j=1}^{q-1}|\unicode[STIX]{x1D6FC}_{j}|$ . The sum on the left-hand side converges because $\Vert f_{0}\Vert <1$ . An $A_{\infty }$ functor is a curved $A_{\infty }$ functor with $f_{0}=0$ .

Let ${\mathcal{A}}$ be a curved $A_{\infty }$ category and $A\in \operatorname{Ob}{\mathcal{A}}$ . A Maurer–Cartan element or bounding cochain [Reference Fukaya, Oh, Ohta and OnoFOOO09] for $A$ is an element $b\in \mathit{Hom}_{{\mathcal{A}}}(A,A)$ with $\Vert b\Vert <1$ such that

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\unicode[STIX]{x1D707}_{k}(b^{\otimes k})=0.\end{eqnarray}$$

Let $f:{\mathcal{A}}\rightarrow {\mathcal{B}}$ be a curved $A_{\infty }$ functor. If $b$ is a bounding cochain for $A\in \operatorname{Ob}{\mathcal{A}}$ , then

$$\begin{eqnarray}f_{\ast }(b)=\mathop{\sum }_{k=0}^{\infty }f_{k}(b^{\otimes k})\end{eqnarray}$$

is a bounding cochain for $f(A)\in \operatorname{Ob}{\mathcal{B}}$ .

To a curved $A_{\infty }$ category ${\mathcal{A}}$ , we associate the $A_{\infty }$ category $\widehat{{\mathcal{A}}}$ defined as follows. An object of $\widehat{{\mathcal{A}}}$ is a pair of an object $A\in \operatorname{Ob}{\mathcal{A}}$ and a bounding cochain $b$ for $A$ . For $(A_{i},b_{i})\in \operatorname{Ob}\widehat{{\mathcal{A}}}$ we define

$$\begin{eqnarray}\mathit{Hom}_{\widehat{{\mathcal{A}}}}((A_{i},b_{i}),(A_{j},b_{j}))=\mathit{Hom}(A_{i},A_{j}).\end{eqnarray}$$

For $k\geqslant 1$ and $\unicode[STIX]{x1D6FC}_{i}\in \mathit{Hom}_{\widehat{{\mathcal{A}}}}((A_{i-1},b_{i-1}),(A_{i},b_{i}))$ , we define

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}^{\widehat{{\mathcal{A}}}}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\mathop{\sum }_{\mathop{\sum }_{i}m_{i}+k=l}\unicode[STIX]{x1D707}_{l}^{{\mathcal{A}}}(b_{0}^{\otimes m_{0}}\otimes \unicode[STIX]{x1D6FC}_{1}\otimes b_{1}^{\otimes m_{1}}\otimes \unicode[STIX]{x1D6FC}_{2}\otimes \cdots \otimes \unicode[STIX]{x1D6FC}_{k}\otimes b_{k}^{\otimes m_{k}}).\end{eqnarray}$$

To a curved ${\mathcal{A}}_{\infty }$ functor $f:{\mathcal{A}}\rightarrow {\mathcal{B}}$ , we associate the $A_{\infty }$ functor $\hat{f}:\widehat{{\mathcal{A}}}\rightarrow \widehat{{\mathcal{B}}}$ defined as follows. The map $\hat{f}:\operatorname{Ob}\widehat{{\mathcal{A}}}\rightarrow \operatorname{Ob}\widehat{{\mathcal{B}}}$ is given by

$$\begin{eqnarray}\hat{f}((A,b))=(f(A),f_{\ast }(b)).\end{eqnarray}$$

For $k\geqslant 1$ and $\unicode[STIX]{x1D6FC}_{i}\in \mathit{Hom}_{\widehat{{\mathcal{A}}}}((A_{i-1},b_{i-1}),(A_{i},b_{i}))$ , we define

$$\begin{eqnarray}\hat{f}_{k}(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{k})=\mathop{\sum }_{\mathop{\sum }_{i}m_{i}+k=l}f_{l}(b_{0}^{m_{0}}\otimes \unicode[STIX]{x1D6FC}_{1}\otimes \cdots \otimes \unicode[STIX]{x1D6FC}_{k}\otimes b_{k}^{\otimes m_{k}}).\end{eqnarray}$$

Let ${\mathcal{A}}$ be an $A_{\infty }$ category. The associated cohomological category $H({\mathcal{A}})$ has the same objects, its morphism spaces are given by

$$\begin{eqnarray}\mathit{Hom}_{H({\mathcal{A}})}(A,B)=H^{\ast }(\mathit{Hom}_{{\mathcal{A}}}(A,B),\unicode[STIX]{x1D707}_{1}^{{\mathcal{A}}}),\end{eqnarray}$$

and the composition of morphisms is given by

(7) $$\begin{eqnarray}[a_{2}]\circ [a_{1}]=(-1)^{|a_{1}|}[\unicode[STIX]{x1D707}_{2}(a_{1},a_{2})].\end{eqnarray}$$

Composition of morphisms is associative because of (3) for $k=3$ . If we drop the sign in (7), composition is no longer associative. Rather, we obtain an $A_{\infty }$ category, called the cohomological $A_{\infty }$ category, which has all operations zero except for $k=2$ . An $A_{\infty }$ functor $f:{\mathcal{A}}\rightarrow {\mathcal{B}}$ induces a functor $H(f):H({\mathcal{A}})\rightarrow H(B)$ . Suppose $H({\mathcal{A}}),H({\mathcal{B}})$ , have identity morphisms. Then the functor $f$ is called a quasi-equivalence if $H(f)$ is an equivalence. An $A_{\infty }$ category is called formal if it is $A_{\infty }$ equivalent to its cohomological $A_{\infty }$ category.

A paradigmatic example of quasi-equivalent $A_{\infty }$ categories is the following. Let $(M,\unicode[STIX]{x1D714})$ be as in § 2.1 and let ${\mathcal{L}}$ be a collection of graded spin compact Lagrangian submanifolds $L_{i}\subset M,\,i\in A$ , that intersect cleanly. For $j=0,1$ , let ${\mathcal{A}}_{j}$ denote the curved $A_{\infty }$ category associated to ${\mathcal{L}}$ using the almost complex structure $\unicode[STIX]{x1D6E9}_{j}\in {\mathcal{J}}_{\unicode[STIX]{x1D714}}$ . Since ${\mathcal{J}}_{\unicode[STIX]{x1D714}}$ is contractible, one can always find a path $\unicode[STIX]{x1D6E9}_{t}\in {\mathcal{J}}_{\unicode[STIX]{x1D714}},t\in [0,1]$ , from $\unicode[STIX]{x1D6E9}_{0}$ to $\unicode[STIX]{x1D6E9}_{1}$ . To such a path one can associate a curved $A_{\infty }$ functor $f^{\unicode[STIX]{x1D6E9}}:{\mathcal{A}}_{0}\rightarrow {\mathcal{A}}_{1}$ such that the $A_{\infty }$ functor $\hat{f}^{\unicode[STIX]{x1D6E9}}:\widehat{{\mathcal{A}}}_{0}\rightarrow \widehat{{\mathcal{A}}}_{1}$ is a quasi-equivalence. Thus, one can associate an $A_{\infty }$ category $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ to ${\mathcal{L}}$ that, up to quasi-equivalence, depends only on the symplectic form $\unicode[STIX]{x1D714}$ . We call $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ the Fukaya category of ${\mathcal{L}}$ .

Similarly, let $\unicode[STIX]{x1D711}_{i,t}:M\rightarrow M,i\in A,t\in [0,1]$ , be a collection of Hamiltonian flows such that the Lagrangian submanifolds $\unicode[STIX]{x1D711}_{i,1}(L_{i})$ intersect cleanly. Let ${\mathcal{L}}^{\unicode[STIX]{x1D711}}$ denote the collection of Lagrangians $\unicode[STIX]{x1D711}_{i,1}(L_{i}),i\in A$ . Then we have a curved $A_{\infty }$ functor $f^{\unicode[STIX]{x1D711}}:{\mathcal{A}}_{{\mathcal{L}}}\rightarrow {\mathcal{A}}_{{\mathcal{L}}^{\unicode[STIX]{x1D711}}}$ such that the $A_{\infty }$ functor $\hat{f}^{\unicode[STIX]{x1D711}}:\widehat{{\mathcal{A}}}_{{\mathcal{L}}}\rightarrow \widehat{{\mathcal{A}}}_{{\mathcal{L}}^{\unicode[STIX]{x1D711}}}$ is a quasi-equivalence. Thus, one sees that up to quasi-equivalence, the Fukaya category $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ depends only on the Hamiltonian isotopy classes of the Lagrangian submanifolds $L_{i},i\in A$ .

It follows that the Fukaya category $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ is well defined even when the Lagrangians $L_{i}$ do not intersect cleanly. Indeed, it is always possible to choose $\unicode[STIX]{x1D711}_{i,t}$ such that the Lagrangians $\unicode[STIX]{x1D711}_{i,1}(L_{i})$ intersect cleanly, so we may define $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}:=\widehat{{\mathcal{A}}}_{{\mathcal{L}}^{\unicode[STIX]{x1D711}}}$ . Any two choices of Hamiltonian flows $\unicode[STIX]{x1D711}_{i,t}$ differ by a Hamiltonian flow, so the preceding discussion shows that, up to quasi-equivalence, the definition does not depend on the choice of $\unicode[STIX]{x1D711}_{i,t}$ .

2.4 Locality

We return to the setting where $(M,I,J,K,g)$ is a hyperkähler manifold, $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{J}$ and ${\mathcal{L}}$ is a collection of graded spin compact $I$ -holomorphic Lagrangians $L_{i}\subset M,i\in A$ , with $A$ finite now. We do not assume the Lagrangians $L_{i}$ intersect cleanly. Let $\unicode[STIX]{x1D711}_{i,t}:M\rightarrow M,\,i\in A,t\in [0,1]$ , be a collection of real analytic Hamiltonian flows such that the Lagrangian submanifolds $\unicode[STIX]{x1D711}_{i,t}(L_{i})$ intersect cleanly for $t\in (0,1]$ . Let ${\mathcal{L}}_{t}$ denote the collection of Lagrangian submanifolds $\unicode[STIX]{x1D711}_{i,t}(L_{i}),i\in A$ . The following is a special case of Theorem 1.7.

Corollary 2.4. For all $\unicode[STIX]{x1D6E9}\in R_{I}\,\cap \,{\mathcal{J}}_{\unicode[STIX]{x1D714}}$ except a countable set, we have the following. Choose $E_{0}\geqslant 0,K\in \mathbb{Z}_{{\geqslant}0}$ , and an open $V\subset M$ such that

$$\begin{eqnarray}\mathop{\bigcup }_{i\in A}L_{i}\subset V.\end{eqnarray}$$

Then, there exists $\unicode[STIX]{x1D700}>0$ such that for all

$$\begin{eqnarray}\displaystyle & \displaystyle k\in \mathbb{Z}\,\cap \,[0,K],\quad {\mathcal{I}}\subset A^{k+1}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle E\in [0,E_{0}],\quad t\in [0,\unicode[STIX]{x1D700}],\quad (u,\vec{z})\in \widetilde{{\mathcal{M}}}({\mathcal{I}},E,{\mathcal{L}}_{t}), & \displaystyle \nonumber\end{eqnarray}$$

we have

$$\begin{eqnarray}u(D\setminus \mathbf{z})\subset V.\end{eqnarray}$$

As a consequence of Corollary 2.4 and the discussion in § 2.3, we obtain the following locality statement for $\widetilde{{\mathcal{A}}}_{{\mathcal{L}}}$ .

Corollary 2.5. Choose $E_{0},K$ and $V$ , as in Corollary 2.4. Then $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ is quasi-equivalent to an $A_{\infty }$ category $\widehat{{\mathcal{B}}}$ with the same objects, such that there exist operations

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}^{E_{0}},\unicode[STIX]{x1D707}_{k}^{V}:\bigotimes _{j=1}^{k}\mathit{Hom}_{\widehat{{\mathcal{B}}}}(L_{i_{j-1}},L_{i_{j}})\rightarrow \mathit{Hom}_{\widehat{{\mathcal{B}}}}(L_{i_{0}},L_{i_{k}})[2-k],\end{eqnarray}$$

with

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{k}^{\widehat{{\mathcal{B}}}}=\unicode[STIX]{x1D707}_{k}^{E_{0}}+\unicode[STIX]{x1D707}_{k}^{V},\quad \Vert \unicode[STIX]{x1D707}_{k}^{E_{0}}\Vert <e^{-E_{0}},\end{eqnarray}$$

and $\unicode[STIX]{x1D707}_{k}^{V}$ only depends on the geometry of $V$ . In fact, we may take $\widehat{{\mathcal{B}}}=\widehat{{\mathcal{A}}}_{{\mathcal{L}}_{t}}$ with $t<\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D700}$ as in Corollary 2.4.

2.5 Formality

Let $(M,I,J,K,g)$ be a hyperkähler manifold, take $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{J}$ and let ${\mathcal{L}}$ be a collection of graded spin compact $I$ -holomorphic Lagrangians $L_{i}\subset M,i\in A$ , with $A$ countable. By Corollary 2.1, we may choose $\unicode[STIX]{x1D6E9}\in R_{I}\,\cap \,{\mathcal{J}}_{\unicode[STIX]{x1D714}}$ with respect to which $L_{i}$ is tautologically unobstructed for $i\in A$ .

The following question, suggested by I. Smith, is natural in light of Corollary 2.8. Let $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ denote the Fukaya category of ${\mathcal{L}}$ as in § 2.3.

Behrend and Fantechi [Reference Behrend and FantechiBF09, Conjecture 5.8] formulate a similar question concerning the degeneration of the spectral sequence computing the virtual de Rham cohomology of the intersection of two holomorphic Lagrangians.

Abouzaid and Smith [Reference Abouzaid and SmithAS16] prove a formality theorem for $\widehat{{\mathcal{A}}}_{{\mathcal{L}}}$ when $M$ belongs to a certain family of Nakajima quiver varieties and ${\mathcal{L}}$ is the collection of Lagrangians introduced in [Reference Seidel and SmithSS06]. Their proof does not make use of the hyperkähler structure on $M$ . It would be interesting to find an alternative proof that does.