2.1 Trees

First we introduce terminology for trees. Given a tree $\Gamma $ , the set of edges $\operatorname {Edge}(\Gamma )$ is equipped with head and tail maps

$$\begin{align*}h, t : \operatorname{Edge}({\Gamma}) \to \operatorname{Vert}({\Gamma}) \cup \{ \infty \}. \end{align*}$$

The valence of any vertex $v \in \operatorname {Vert}({\Gamma })$ is the number

$$\begin{align*}|v | = \# \{ e \in h^{-1}(v) \cup t^{-1}(v) \} \end{align*}$$

of edges meeting the vertex v. An edge $e \in \operatorname {Edge}({\Gamma })$ is

• • combinatorially finite if $\infty \notin \{ h^{-1}(e), t^{-1}(e) \}$ ,

• • semi-infinite or a leaf if $\{ h^{-1}(e), t^{-1}(e) \} = \{v , \infty \}$ for some $v \in \operatorname {Vert}({\Gamma })$ , and

• • infinite if $h(e) = t(e) = \infty $ .

Denote

$$\begin{align*}{\operatorname{Edge}}_{\mathrm{\operatorname{fin}}}(\Gamma)\ \mathrm{resp.}\ {\operatorname{Edge}}_\rightarrow(\Gamma) \subset {\operatorname{Edge}}(\Gamma) \end{align*}$$

the set of finite resp. semi-infinite edges, that is, leaves.

For now, we assume that trees are rooted, which means that when $\operatorname {Vert}(\Gamma ) \neq \emptyset $ there is a distinguished vertex $v_{\mathrm {root}} \in \operatorname {Vert}(\Gamma )$ called the root and a distinguished semi-infinite edge $e_{\mathrm {out}} \in {\operatorname {Edge}}_\rightarrow (\Gamma )$ with $t(e_{\mathrm {out}}) = v_{\mathrm {root}}$ called the output. All edges are then oriented towards the output. This will suffice for defining the Fukaya category. Later on, we will consider not-necessarily rooted trees.

There is a special tree which does not have vertices: a vertex-free tree is a tree $\Gamma $ with $\mathrm {Vert}(\Gamma ) = \emptyset $ with one infinite edge. However, we set ${\operatorname {Edge}}_{\rightarrow } = \{ e_{\mathrm {in}}, e_{\mathrm {out}}\}$ , the incoming and the outgoing ends of the vertex-free tree. In any case, for a tree $\Gamma $ , denote by $ {\operatorname {Edge}}_{\mathrm {in}}(\Gamma )$ resp. $ {\operatorname {Edge}}_{\mathrm {out}}(\Gamma )$ the incoming and outgoing leaves.

Our trees will be composed of two parts corresponding to the sphere and disk vertices. We colour these vertices black and white, respectively, and call the resulting structure a two-coloured tree.

We distinguish between boundary and interior leaves and disk and sphere components. Objects related to the rooted subtree (which are usually related to disks and boundary insertions) are labelled with $\circ $ , while the corresponding notions related to the complement of the rooted subtree (which are related to spheres and interior insertions) are labelled with $\bullet $ . For example, we denote by

$$\begin{align*}\mathrm{Edge}_\circ(\Gamma):= \mathrm{Edge}(\Gamma_\circ) \subset \mathrm{Edge}(\Gamma) \end{align*}$$

the set of boundary edges and we used above

$$\begin{align*}\mathrm{Edge}_\bullet(\Gamma):= \mathrm{Edge}(\Gamma) \setminus \mathrm{Edge}_\circ(\Gamma) \end{align*}$$

the set of interior edges. Semi-infinite edges are also called leaves and we denote

$$\begin{align*}\mathrm{Leaf}_\circ(\Gamma):={\operatorname{Edge}}_\rightarrow(\Gamma) \cap \operatorname{Edge}_\circ(\Gamma),\ \mathrm{ Leaf}_\bullet(\Gamma):={\operatorname{Edge}}_\rightarrow(\Gamma) \cap \operatorname{Edge}_\bullet(\Gamma). \end{align*}$$

A moduli space of metric trees is obtained by allowing the finite edges on the disk part to acquire lengths.

Definition 2.2. Let $\Gamma $ be a two-coloured tree. A metric on $\Gamma $ is a non-negative function on the space of finite boundary edges

$$\begin{align*}\ell: {\operatorname{Edge}}_{\mathrm{\operatorname{fin}}} (\Gamma_\circ) \to [0, +\infty). \end{align*}$$

A metric type on $\Gamma $ , denoted by $\underline {\smash {\ell }}$ , is the associated decomposition

$$\begin{align*}{\operatorname{Edge}}_{\mathrm{\operatorname{fin}}}(\Gamma_\circ) = {\operatorname{Edge}}_0( \Gamma_\circ) \sqcup {\operatorname{Edge}}_+( \Gamma_\circ) \end{align*}$$

corresponding to edges with zero or positive lengths.Footnote ³

To compactify the set of gradient segments we allow the lengths of the edges to go to infinity and break. A broken metric tree is obtained from a finite collection of metric trees by gluing outputs with inputs as follows: Given two metric trees $(\Gamma _1, \ell _1)$ and $(\Gamma _2, \ell _2)$ with specified leaves $e_1 \in \operatorname {Leaf}(\Gamma _1)$ and $e_2 \in \operatorname {Leaf}(\Gamma _2)$ , let $\overline {\Gamma }_1$ resp. $\overline {\Gamma }_2$ denote the space obtained by adding a point $\infty _1$ resp. $\infty _2$ at the open end of $e_1$ resp. $e_2$ . The space

(2.1) $$ \begin{align} \Gamma := \overline{\Gamma}_1 \cup_{\infty_1 \sim \infty_2} \overline{\Gamma}_2 \end{align} $$

is a broken metric tree, the point $\infty _1 \sim \infty _2$ being called a breaking. To obtain a well-defined root for the glued tree we require that exactly one of $e_1$ and $e_2$ is the output. See Figure 2.

Figure 2

Creating a broken tree.

In general, a broken metric tree $\Gamma $ is obtained from broken metric trees $\Gamma _1, \Gamma _2$ as in (2.1) in such a way that the resulting space $\Gamma $ is connected and has no non-contractible loops, that is, $\pi _0(\Gamma )$ is a point and $\pi _1(\Gamma )$ is the trivial groupFootnote ⁴. We think of the gluing points as breakings rather than vertices, so that there are no new vertices in the glued tree $\Gamma $ .

In order to obtain Fukaya algebras with strict units, we wish for our moduli spaces to admit forgetful maps. For this we introduce weightings on certain edges, as in for example Ganatra [Reference GanatraGan12, Section 10.5].

2.2 Treed disks

The domains of treed holomorphic disks are unions of disks, spheres and lines, rays and line segments. A disk is a bordered Riemann surface biholomorphic to the complex unit disk $\mathbb {D} = \{ z \in \mathbb {C} | \Vert z \Vert \leq 1 \}$ . The automorphism group of $\mathbb {D}$ is $ \operatorname {Aut} (\mathbb {D}) \cong PSL(2,\mathbb {R})$ . A nodal disk with a single boundary node is a topological space S obtained from a disjoint union of disks $S_1,S_2$ by identifying pairs of boundary points $w_{12} \in S_1, w_{21} \in S_2$ on the boundary of each component so that

(2.3) $$ \begin{align} S = S_1 \cup_{w_{12} \sim w_{21}} S_2 .\end{align} $$

See Figure 3. The image of $w_{12}, w_{21}$ in the space S is the nodal point.

Figure 3

Creating a nodal disk.

A nodal disk S with multiple nodes $w_{ij}, i,j \in \{ 1,\ldots , k\}, i \neq j$ is obtained by repeating this construction (2.3) with $S_1,S_2$ nodal disks with fewer nodes, and $w_{12}, w_{21}$ distinct from the other nodes. More generally we allow boundary and interior markings. For an integer $d \ge 0$ a nodal disk with $d+1$ boundary markings is a nodal disk S equipped with a finite ordered collection of points $\underline {\smash {x}} = (x_0,\ldots ,x_d)$ on the boundary $\partial S$ , disjoint from the nodes, in anticlockwise cyclic order around the boundary $\partial S$ . A $(d+1)$ -marked nodal disk $(S,\underline {\smash {x}})$ is stable if each component $S_v$ has at least three special (nodal or marked) points, or equivalently the group $ \operatorname {Aut} (S,\underline {\smash {x}})$ of automorphisms of S leaving $\underline {\smash {x}}$ pointwise fixed is trivial. The moduli space of $(d+1)$ -marked stable disks $ [(S,\underline {\smash {x}})]$ forms a compact cell complex, isomorphic as a cell complex to the associahedron from Stasheff [Reference StasheffSta63, Reference StasheffSta70].

More complicated configurations involve spherical components. A marked sphere is a complex surface biholomorphic to the projective line $S^2 \cong \mathbb {P}^1$ together with a distinct ordered list of markings $z_1, \ldots , z_k\in S^2$ . A nodal disk S with a single interior node $w \in S$ is defined similarly to that of a boundary node by using the construction (2.3), except in this case S is obtained by gluing together a nodal disk $S_1$ with a marked sphere $S_2$ with $w_{12}, w_{21}$ points in the interior $\operatorname {int}(S)$ .

General treed disks are defined as in Oh [Reference OhOh93], Cornea-Lalonde [Reference Cornea and LalondeCL06], Biran-Cornea [Reference Biran and CorneaBC07, Reference Biran and CorneaBC09] and Seidel [Reference SeidelSei11].

The moduli spaces of stable weighted treed disks are naturally cell complexes. Suppose $\Gamma $ is a stable domain type with $d(\circ )$ boundary inputs and $d(\bullet )$ interior leaves. Let ${\mathcal {M}}_\Gamma $ denote the set of all isomorphism classes of treed disks of type $\Gamma $ , with its natural topology induced by embedding in the product of the moduli space of stable trees and stable disks. The space ${\mathcal {M}}_\Gamma $ is a manifold of dimension

Denote

$$\begin{align*}\overline{\mathcal M}_{\Gamma} = \bigsqcup_{\Gamma' \preceq \Gamma,\ \Gamma'\ {\rm stable}} {\mathcal M}_{\Gamma'}. \end{align*}$$

As in the definition of Gromov convergence of pseudoholomorphic curves, there is a natural way to endow $\overline {\mathcal {M}}_\Gamma $ a compact Hausdorff topology that agrees on the manifold topology on each stratum ${\mathcal {M}}_{\Gamma '}$ , so that $\overline {\mathcal {M}}_\Gamma $ is a cell complex with ${\mathcal {M}}_\Gamma $ equal to the top cell.

Remark 2.6. The moduli spaces of weighted treed disks are related to unweighted moduli spaces by taking products with intervals: If $\Gamma $ has at least one vertex and $\Gamma '$ denotes the domain type obtained from $\Gamma $ by setting the weights $w (e)$ to zero and the output $e_{\mathrm {out}}$ of $\Gamma $ is unweighted then

If the outgoing edge $e_{\mathrm {out}}$ is weighted then

because of the way we define isomorphism of weighted types (see Definition 2.3). Figure 5 illustrates a one-dimensional moduli space with weighted output and its boundary strata.

Figure 5

A one-dimensional moduli space of weighted treed disks with all three semi-infinite edges weighted.

In general, moduli spaces of stable curves only admit universal curves in an orbifold sense. In the setting here, orbifold singularities are absent and the moduli spaces of stable treed disks admit honest universal curves. For any stable domain type $\Gamma $ let $\overline {\mathcal {U}}_\Gamma $ denote the universal treed disk (or called the universal curve) consisting of isomorphism classes of pairs $(C,z)$ where C is a treed disk of type $\Gamma $ and z is a point in C, possibly on a disk component $S_v \cong \{ |z| \leq 1 \}$ , a sphere component $S_v \cong \mathbb {P}^1$ , or one of the edges e of the tree part $T \subset C$ (the infinities of semi-infinite edges are allowed). The map

$$\begin{align*}\pi_\Gamma: \overline{{\mathcal{U}}}_\Gamma \to \overline{{\mathcal{M}}}_\Gamma, \quad [C,z] \to [C] \end{align*}$$

is the universal projection. Moreover, for each $[C]\in \overline {\mathcal M}_\Gamma $ represented by C, the fibre $\pi _\Gamma ^{-1}([C])$ is homeomorphic to C. In case $\Gamma $ has no vertices we define $\overline {\mathcal {U}}_\Gamma $ to be the real line, considered as a fiber bundle over the point $\overline {\mathcal {M}}_\Gamma $ .

Figure 6

Treed disks with interior leaves.

We introduce notation for particular subsets of the universal curves. First, for each vertex $v \in \mathrm {Vert}(\Gamma )$ , let

$$\begin{align*}\overline{\mathcal{U}}_{\Gamma, v} \subset \overline{\mathcal{U}}_\Gamma \end{align*}$$

denote the closed subset corresponding to points on the surface component $S_v$ and on the semi-infinite edges attached to v. For each boundary edge $e \in {\operatorname {Edge}}(\Gamma _\circ )$ , let

$$\begin{align*}\overline{\mathcal{U}}_{\Gamma, e} \subset \overline{\mathcal{U}}_{\Gamma} \end{align*}$$

the closed subset corresponding to points on the tree component $T_e$ . Denote

(2.4) $$ \begin{align} \overline{\mathcal{S}}_\Gamma:= \bigcup_{v \in \mathrm{Vert}(\Gamma)} \overline{\mathcal{U}}_{\Gamma, v} \end{align} $$

and

(2.5) $$ \begin{align} \overline{\mathcal{T}}_\Gamma:= \bigcup_{e \in {\operatorname{Edge}}(\Gamma_\circ)} \overline{\mathcal{U}}_{\Gamma, e}. \end{align} $$

Moreover, for each subtree $\Pi \subset \Gamma $ (not necessarily containing the root), denote by

$$\begin{align*}\overline{\mathcal{U}}_{\Gamma, \Pi} \subset \overline{\mathcal{U}}_{\Gamma} \end{align*}$$

the set of points z on components $S_v,T_e$ corresponding to vertices v and edges e of $\Pi $ . There is a contraction map $\overline {\mathcal {U}}_{\Gamma , \Pi } \to \overline {\mathcal {U}}_{\Pi }$ contracting edges not in $\Gamma $ . In particular, for the disk part $\Gamma _\circ $ , one has $ \overline {\mathcal {U}}_{\Gamma , \Gamma _\circ } \subset \overline {\mathcal {U}}_{\Gamma }.$ Lastly, for $\Pi \preceq \Gamma $ , one has a boundary stratum $ \overline {\mathcal {U}}_{\Pi } \subset \overline {\mathcal {U}}_\Gamma .$

The boundary of each treed disk is divided up into parts between the boundary inputs. Denote by $(\partial C)_i$ the component of $\partial C$ between the i-th and $i+1$ -st leaves, in cyclic order. Similarly, denote i-th boundary part of the universal curve $ \partial _i \overline {\mathcal {U}}_\Gamma \subset \overline {\mathcal {U}}_\Gamma $ .

2.3 Rational branes

In this subsection, we specify assumptions and additional data on the Lagrangian submanifolds in our construction. Let $(X, \omega )$ be a compact symplectic manifold. Let ${\mathcal L} = \{ L_1, \ldots , L_m\}$ be a collection of embedded Lagrangian submanifolds in $(X, \omega )$ . Denote the support of ${\mathcal L}$ to be

$$\begin{align*}|{\mathcal L}|:= \bigcup_{L \in {\mathcal L}} L \subset X. \end{align*}$$

By a brane we will mean a flat $\Lambda ^\times $ bundle ${\widehat {L}}$ over L, where L is an (oriented, spin, embedded) Lagrangian submanifold. By a weakly unobstructed brane we will mean a pair ${\boldsymbol {L}} = ({\widehat {L}}, b)$ , where ${\widehat {L}}$ is a brane and b is a Maurer-Cartan element of the curved $A_{\infty }$ algebra of ${\widehat {L}}$ .

From now on, we assume that $[\omega ]$ is integral, for simplicity, and $\mathcal {L}$ is rational:

2.3.1 Bulk deformation

Bulk deformations used in this paper are linear combinations of disjoint embedded closed submanifolds $\mathfrak {b}_1,\ldots , \mathfrak {b}_N \subset X$ denoted by

$$\begin{align*}\mathfrak{b} = \sum_{i=1}^N c_i \mathfrak{b}_i,\ c_i \in \Lambda\setminus \{0\}. \end{align*}$$

We assume that all $\mathfrak {b}_i$ are oriented and have even and positive codimensions. The support of $\mathfrak {b}$ is the union

$$\begin{align*}|\mathfrak{b}|:= \bigcup_{i = 1}^N \mathfrak{b}_i. \end{align*}$$

Remark 2.9. There will be no essential difference but only notational complexities if we allow bulk deformation to be pseudocycles rather than closed submanifolds.

2.3.2 Donaldson hypersurface

The Cieliebak-Mohnke scheme relies on the existence of Donaldson hypersurfaces, defined as follows. Given a rational symplectic manifold $(X, \omega )$ a Donaldson hypersurface is a compact codimension two symplectic submanifold $D \subset X$ whose Poincaré dual is a multiple $k[\omega ]$ of $[\omega ]$ . The positive integer k is called the degree of the Donaldson hypersurface.

Lemma 2.10 [c.f. Charest-Woodward [Reference Charest and WoodwardCW17, Section 3.1], [Reference Cieliebak and MohnkeCM07, Lemma 8.7]]

Let J be an $\omega $ -compatible almost complex structure on X such that all Lagrangians in $\mathcal {L}$ are totally real. For $l \in {\mathbb N}$ sufficiently large there exist a sequence of degree l Donaldson hypersurfaces $D = D_l \subset X$ disjoint from $|\mathcal {L}|$ with the following properties.

1. (a) $\mathcal {L}$ is exact in the complement $X - D_l$ .

2. (b) For each l, there is a tamed almost complex structure $J_0 \in {\mathcal J}_{\mathrm {tame}}(X, \omega )$ making $D_l$ almost complex such that all non-constant $J_0$ -holomorphic spheres in X intersect $D_l$ at finite but at least three points and all non-constant $J_0$ -holomorphic disks with boundary in $|\mathcal {L}|$ intersect $D_l$ in the interior.

3. (c) $D_l$ is transverse to each component of $\mathfrak {b}$ .

Proof. The construction is an extension of the original constructions [Reference DonaldsonDon96] [Reference AurouxAur97] [Reference Auroux, Gayet and MohsenAGM01] [Reference Pascaleff and TonkonogPT20, Theorem 3.3]. Let $\hat X \to X$ be a line bundle whose curvature is the symplectic form (up to a factor of $2\pi /i $ ). The argument of [Reference Pascaleff and TonkonogPT20, Theorem 3.3] (which is purely local and so applies to the cleanly intersecting Lagrangian considered here) gives the existence of an approximately J-holomorphic section

$$\begin{align*}s: X \to \hat{X}^l \end{align*}$$

of some tensor power $\hat {X}^l$ and so that the restriction of s to each $L \in \mathcal {L}$ is close to the given flat section on L. One obtains a symplectic hypersurface as the zero-set:

$$\begin{align*}D = s^{-1}(0). \end{align*}$$

The connection one-form $\alpha $ in the trivialization provided by s provides a primitive for the symplectic form $\omega $ , and the fact that s is approximately flat on $\mathcal {L}$ implies that the integral of $\alpha $ over any loop in $|\mathcal {L}|$ vanishes, so that $\mathcal {L}$ is exact; see [Reference Cieliebak and MohnkeCM07, Theorem 8.1] and the modification in [Reference Charest and WoodwardCW17, Theorem 3.6]. By Cieliebak-Mohnke [Reference Cieliebak and MohnkeCM07, Corollary 8.16], for sufficiently generic tamed almost complex structures J, each non-constant J-holomorphic sphere $u: \mathbb {P}^1 \to D $ is not contained in $D $ and intersects D in at least three points:

$$\begin{align*}\# u^{-1}(D) \ge 3. \end{align*}$$

On the other hand, since $\mathcal {L}$ is exact in the complement of D, each non-constant pseudoholomorphic disk $u: \mathbb {D} \to X$ with boundary in $\mathcal {L}$ intersects D in at least one interior point. Transversality of D to the bulk deformation $\mathfrak {b}$ follows as in [Reference Cieliebak and MohnkeCM07, Corollary 5.8].

Remark 2.11. The notion of approximately holomorphic can be made more precise as follows: A sequence of sections $s_l$ of $\hat {X}^l$ (for large l) is said to be asymptotically holomorphic with respect to the given connections and almost-complex structure if the following bounds hold: There exists a constant $C>0$ such that, for all l and at every point of X,

$$\begin{align*}|s_l| + | \nabla s_l | +| \nabla^2 s_l | \leq C, \quad | \overline{\partial} s_l| + | \nabla ( \overline{\partial} s_l) | \leq C l^{-\frac{1}{2}} \end{align*}$$

where the norms of the derivatives are evaluated with respect to the metrics defined by the rescaled two-form $l \omega $ . Such a sequence is said to be uniformly transverse to $0$ with constant $\eta $ if the derivative of $s_l$ is non-zero whenever $|s_l(x)| < \eta $ and has a right inverse bounded by $\eta ^{-1}$ , as in [Reference Auroux, Gayet and MohsenAGM01, Definition 1]. Donaldson’s construction shows the existence of asymptotically holomorphic sections uniformly transverse to the zero section, using sequences of asymptotically holomorphic sections concentrated near a point.

2.4 Perturbations

We consider domain-dependent perturbation data defined on the universal curves. We first define a condition called locality, which our perturbation data will be required to satisfy. A similar condition plays an important role in Cieliebak-Mohnke’s approach [Reference Cieliebak and MohnkeCM07].

Notation 2.12. Let $\Gamma = (\Gamma , {\underline {\smash {\ell }}}, {\underline {\smash {\operatorname {wt}}}})$ be a stable domain type. Recall that $\Gamma _\circ $ is the subtree corresponding to disk components and boundary edges. For each spherical vertex ${v \in \operatorname {Vert}(\Gamma )\setminus \mathrm {Vert}(\Gamma _\circ )=: \mathrm {Vert}_\bullet (\Gamma )}$ , let $\Gamma (v)$ denote the subtree of $\Gamma $ consisting of the vertex v and all edges e of $\Gamma $ meeting v. Let

(2.6) $$ \begin{align} \pi: \pi_1 \times \pi_2: {\mathcal{U}}_\Gamma \to {\mathcal{M}}_{\Gamma_\circ} \times {\mathcal{U}}_{\Gamma(v)} \end{align} $$

be the product of maps, where $\pi _1$ is given by projection followed by the forgetful morphism and $\pi _2$ is the contraction $C \to S_v$ .

2.4.1 Supports of perturbations

In this section, we construct open sets where the perturbations are required to vanish. Let $\overline {\mathcal {S}}_\Gamma $ and $\overline {\mathcal {T}}_\Gamma $ be the universal surface and tree from (2.4) and (2.5).

Lemma 2.15. For all stable combinatorial types $\Gamma $ , there exist collections of open subsets (where the complex structure J, the Hamiltonian perturbations H, or the Morse functions F will be fixed)

$$\begin{align*}\overline{\mathcal{S}}_{\Gamma, J} \subset \overline{\mathcal{S}}_\Gamma,\ \ \overline{\mathcal S}_{\Gamma, H}\subset \overline{\mathcal S}_\Gamma, \quad \overline{\mathcal{T}}_{\Gamma, F} \subset \overline{\mathcal{T}}_{\Gamma} \end{align*}$$

satisfying the following properties.

1. (a) The open set $\overline {\mathcal {S}}_{\Gamma , J}$ intersects with any fiber $C = S \cup T \subset \overline {\mathcal {U}}_\Gamma $ at a neighbourhood of all special points on the surface part so that for all $v \in \operatorname {Vert}(\Gamma )$ , the complement of $\overline {\mathcal {S}}_{\Gamma , J}$ has non-empty intersection with $S_v$ ;

2. (b) The intersection of $\overline {\mathcal S}_{\Gamma , H}$ with each fibre $C = S \cup T \subset \overline {\mathcal U}_\Gamma $ contains all spherical components and a neighbourhood of all nodal points. Moreover, the complement of $\overline {\mathcal S}_{\Gamma , H} \cap C$ has a non-empty intersection with each disk component $S_v \subset S$ .

3. (c) The open set $\overline {\mathcal {T}}_{\Gamma , F}$ is a neighbourhood of the locus corresponding to infinities of semi-infinite edges in all degenerations $\Pi \prec \Gamma $ .

4. (d) If $\Gamma $ is separated by a breaking into two subtrees $\Gamma _1$ and $\Gamma _2$ , then $\overline {\mathcal {S}}_{\Gamma , *}$ resp. $\overline {\mathcal {T}}_{\Gamma , *}$ is the product

   $$\begin{align*}\overline{\mathcal{S}}_{\Gamma_1, *}\boxtimes \overline{\mathcal{S}}_{\Gamma_2, *}\ \mathrm{resp.}\ \overline{\mathcal{T}}_{\Gamma_1, *}\boxtimes \overline{\mathcal{T}}_{\Gamma_2, *} \end{align*}$$
   where
   $$ \begin{align*}\overline{\mathcal{S}}_{\Gamma_1, *}\boxtimes \overline{\mathcal{S}}_{\Gamma_2, *} = \pi_1^{-1} (\overline{\mathcal{S}}_{\Gamma_1, *}) \times \pi_2^{-1}( \overline{\mathcal{S}}_{\Gamma_2, *})\end{align*} $$
   etc.

5. (e) The characteristic functions of $\overline {\mathcal {S}}_{\Gamma , *}$ and $\overline {\mathcal {T}}_{\Gamma , *}$ , viewed as maps from $\overline {\mathcal {U}}_\Gamma $ to $\{0, 1\}$ , are local maps.

The proof is left to the reader. We need to specify certain Banach space norms on perturbation data. After taking away the open sets $\overline {\mathcal {S}}_{\Gamma , J}$ and $\overline {\mathcal {T}}_{\Gamma , F}$ , the surface part and the tree part of the universal curve $\overline {\mathcal {U}}_\Gamma $ , the complements

$$\begin{align*}\overline{\mathcal{S}}_{\Gamma} \setminus \overline{\mathcal{S}}_{\Gamma, J}\ \mathrm{resp.}\ \overline{\mathcal S}_\Gamma \setminus \overline{\mathcal S}_{\Gamma, H}\ \mathrm{resp.}\ \overline{\mathcal{T}}_{\Gamma} \setminus \overline{\mathcal{T}}_{\Gamma, F} \end{align*}$$

are smooth manifolds. To measure the norms of smooth functions we choose Riemannian metrics on these complements in a way that the metrics are local functions on the universal curve and respect degeneration of curves. We omit the details.

2.4.2 Spaces of almost complex structures

In this section, we introduce domain-dependent perturbations and their spaces. The Cieliebak-Mohnke method in [Reference Cieliebak and MohnkeCM07] provides for each energy bound $E>0$ an open neighbourhood ${\mathcal J}_{\mathrm {tame}}^E(X, \omega )$ of the base almost complex structure $J_0$ consisting of almost complex structures J for which all non-constant J-holomorphic spheres $u: S^2 \to X$ of energy at most E intersect the Donaldson hypersurface D at finite but at least three points, that is, $\# u^{-1}(D) \ge 3$ . On the other hand, the domain types, especially the numbers of interior markings ${\underline {\smash {z}}}$ provide a priori bounds for energy, which allow us to define suitable spaces of perturbations.

Notation 2.16. Let $\Gamma $ be a stable domain type. Let

(2.7) $$ \begin{align} E(\Gamma):= \frac{\# \mathrm{Leaf}_\bullet(\Gamma) + 1}{k} \in \mathbb{Q} \end{align} $$

where k is the degree of the Donaldson hypersurface D.

We define suitable spaces of almost complex structures that do not allow holomorphic spheres in the Donaldson hypersurface. Assume that $ J_0 \in {\mathcal J}_{\mathrm {tame}}(X, \omega ) $ is a base almost complex structure satisfying the conditions in Proposition 2.10.

Lemma 2.17 [Reference Cieliebak and MohnkeCM07, Corollary 8.16]

For any $E>0$ , there exists an open neighbourhood ${\mathcal J}_{\mathrm {tame}}^E(X, \omega ) \subset {\mathcal J}_{\mathrm {tame}}(X, \omega )$ of $J_0$ in the $C^\infty $ -topology satisfying the following property: For every $J \in {\mathcal J}_{\mathrm {tame}}^E(X, \omega )$ , all non-constant J-holomorphic spheres with energy at most E intersect D in finitely many but at least three points.

2.4.3 The space of domain-dependent perturbations

Definition 2.18. A perturbation datum for a stable domain type $\Gamma $ is a collection ${P_\Gamma = (J_\Gamma , H_\Gamma , F_\Gamma , M_\Gamma )}$ consisting of

1. (a) A domain-dependent almost complex structure

   $$\begin{align*}J_\Gamma: \overline{\mathcal{S}}{}_\Gamma \to \mathcal{J}_{\mathrm{tame}} (X, \omega) \end{align*}$$
   satisfying the following conditions.

   1. (i) For any vertex $v \in \mathrm {Vert} (\Gamma )$ , let $\Gamma _{(v)}$ be the maximal subtree containing v which has no boundary edges with positive length. Then

      $$\begin{align*}J_\Gamma( \overline{\mathcal U}_{\Gamma, v}) \subset \mathcal{J}_{\mathrm{tame}}^{E(\Gamma_{(v)})}(X, \omega). \end{align*}$$
      Here $E(\Gamma )$ is the energy bound defined by (2.7).

   2. (ii) $J_\Gamma $ is equal to the base almost complex structure $J_0$ over the open subset $\overline {\mathcal {S}}_{\Gamma , J}$ and in a fixed neighbourhood of $D \cup \{p\}$ .

   The last condition implies that $J_\Gamma $ extends canonically to a map on $\overline {\mathcal U}_\Gamma $ .

2. (b) A domain-dependent Hamiltonian perturbation

   $$\begin{align*}H_\Gamma: \overline{\mathcal S}_\Gamma \to \Gamma( T(\overline{\mathcal S}_{\Gamma})^* / T \overline{\mathcal M}_\Gamma)^*)\otimes C^\infty(X). \end{align*}$$
   that is zero over the open set $\overline {\mathcal S}_{\Gamma , H}$ . Here $T(\overline {\mathcal S}_{\Gamma } / \overline {\mathcal M}_\Gamma )$ is the vertical tangent bundle, which is a smooth vector bundle away from nodal points. The last condition implies that $H_\Gamma $ extends canonically to a map defined over $\overline {\mathcal U}_\Gamma $ .

3. (c) A domain-dependent smooth function

   $$\begin{align*}F_\Gamma: \overline{\mathcal{T}}_\Gamma \times \left(\bigsqcup_{(L, L') \in {\mathcal{L}}^2} (L \cap L' ) \right) \to {\mathbb R} \end{align*}$$
   that is zero over the open set $\overline {\mathcal T}_{\Gamma , F}$ .

4. (d) A domain-dependent perturbation of the evaluation map given by a collection of continuous maps for the interior inputs

   $$\begin{align*}M_{\Gamma, e}: \overline{\mathcal{M}}_\Gamma \to \mathrm{Diff}(X)\ \forall e \in \mathrm{Leaf}(\Gamma) \end{align*}$$
   that are smooth in the interior ${\mathcal {M}}_\Gamma $ (with respect to the manifold structure of ${\mathcal {M}}_\Gamma $ ). Each $M_{\Gamma , e}$ can be viewed as a map from the universal curve by pullback via $\overline {\mathcal {U}}_\Gamma \to \overline {\mathcal {M}}_\Gamma $ .

Moreover, the tuple $P_\Gamma = (J_\Gamma , H_\Gamma , F_\Gamma , M_\Gamma )$ can be viewed as a map from $\overline {\mathcal {U}}_{\Gamma }$ to a certain set. We require that this map be a local map (see Definition 2.13).

We take perturbations in a small-in-the-Floer-norm neighbourhood of the base perturbation. Given a sequence of positive numbers $\epsilon = (\epsilon _i)_{i=1}^\infty $ Floer’s (complete) $C^\epsilon $ -norm on functions on a Riemannian manifold is defined by

$$\begin{align*}\| f \|_{{C^\epsilon}}:= \sum_{i=0}^\infty \epsilon_i \| \nabla^i f \|_{C^0}. \end{align*}$$

For a suitably chosen sequence $\epsilon $ , in all dimensions the space of smooth functions with finite $C^\epsilon $ -norms contains bump functions of arbitrary small supports (see [Reference FloerFlo88]). For each stable $\Gamma $ , there is a base perturbation datum in which $J_\Gamma $ is the base almost complex structure $J_0$ specified by Lemma 2.10, $H_\Gamma = 0$ , $F_\Gamma $ is the given Morse function, and $M_\Gamma = \mathrm {Id}_X$ . The tangent space of ${\mathcal J}_{\mathrm {tame}}(X, \omega )$ at $J_0$ is

$$\begin{align*}T_{J_0} {\mathcal J}_{\mathrm{tame}}(X, \omega) = \{ \zeta \in \mathrm{End}(TX) \ |\ J_0 \xi + \xi J_0 = 0 \}. \end{align*}$$

For $\delta>0$ sufficiently small we identify the $\delta $ -neighbourhood of $J_0$ in ${\mathcal J}_{\mathrm {tame}}(X, \omega )$ with respect to the $C^0$ -norm with the $\delta $ -ball of the tangent space $T_{J_0} {\mathcal J}_{\mathrm {tame}}(X, \omega )$ . Then a domain-dependent almost complex structure $J_\Gamma : \overline {\mathcal {S}}_{\Gamma } \to {\mathcal J}_{\mathrm {tame}}(X, \omega )$ that is $C^0$ -close to the base $J_0$ can be viewed as a vector in the linear space $C^\infty ( \overline {\mathcal {S}}_\Gamma \setminus \overline {\mathcal {S}}_{\Gamma , J}, T_{J_0} {\mathcal J}_{\mathrm {tame}}(X, \omega ))$ so one can measure its norms. Similarly, a domain-dependent diffeomorphism $M_{\Gamma , e}: X \to X$ that is $C^0$ -close to the identity can be identified with a $C^0$ -small vector field on X, denoted by $M_{\Gamma , e} - \mathrm {Id}_X$ . On the other hand, $H_\Gamma $ and $F_\Gamma $ are elements of certain vector spaces. For each stable $\Gamma $ , define

(2.8) $$ \begin{align} {\mathcal P}_\Gamma:= \Big\{ P_\Gamma = (J_\Gamma, H_\Gamma, F_\Gamma, M_\Gamma)\ |\ \|J_\Gamma - J_0\|_{C^\epsilon} + \| H_\Gamma \|_{C^\epsilon} + \| F_\Gamma \|_{C^\epsilon} + \| M_\Gamma - \mathrm{Id}_X \|_{C^\epsilon} < \infty \Big\}. \end{align} $$

This set with the $C^\epsilon $ -norm is a separable Banach manifold (in fact an open set of a separable Banach space).

Once a perturbation datum for a stable domain type is fixed we obtain perturbations for not-necessarily stable types as follows. Let C be a treed disk of domain type $\Gamma $ (not necessarily stable). Let $C^{st}$ denote its stabilization, obtained by collapsing unstable components of S and gluing the associated edges. The stabilization $C^{st}$ is naturally identified with a fiber of the universal curve $\mathcal {U}_{\Gamma ^{st}}$ for the type $\Gamma ^{st}$ . Via the stabilization map $C \to C^{st}$ the perturbation data $P_{\Gamma ^{st}}$ pulls back to perturbation data $P_{\Gamma }$ for $\Gamma $ .

2.5 Holomorphic treed disks

Holomorphic treed disks are combinations of holomorphic disks and gradient flow segments. We first state the assumptions on the boundary conditions. Let $(X, \omega )$ be a compact symplectic manifold. For each pair of Lagrangians $(L, L') \in \mathcal {L}^2$ (including the case $L = L'$ ) let

$$\begin{align*}F_{L, L'}: L \cap L' \to \mathbb{R}. \end{align*}$$

be a Morse function on the clean intersection. Its critical points will be asymptotic constraints for gradient rays. In order to obtain strict units, we expand the set of critical points as follows. Define

(2.9) $$ \begin{align} {\mathcal I}(L, L') = \left\{ \begin{array}{@{}ll} \mathrm{crit}(F_{L, L'}),\ &\ L \neq L',\\ \mathrm{crit}(F_{L, L'}) \cup {\mathcal I}_L^{hu},\ &\ L = L'. \end{array}\right. \end{align} $$

where

(2.10)

Interior labelling data provide constraints of maps at interior markings. The stabilizing divisor $D \subset X$ which intersects each $\iota _i$ transversely. Denote

(2.11) $$ \begin{align} {\mathcal I}_{X}:= \mathcal{I}_{X,\operatorname{Stab}} \sqcup \mathcal{I}_{X,\operatorname{Bulk}} \sqcup {\mathcal I}_{X,\operatorname{Mix}} \end{align} $$

where

$$ \begin{align*} \mathcal{I}_{X,\operatorname{Stab}} &:= \{ (D, 1), (D, 2) \}, \\ \mathcal{I}_{X,\operatorname{Bulk}} &:= \{ {\mathfrak b}_i\ |\ i = 1, \ldots, N \}, \\ {\mathcal I}_{X,\operatorname{Mix}} &:= \{ D \cap {\mathfrak b}_i\ |\ i = 1, \ldots, N\} \cup \{ {\mathfrak b}_i \cap {\mathfrak b}_j\ |\ i \neq j \}. \end{align*} $$

which will be used to label all possibly interior constraints (eventually corresponding to whether each interior leaf $T_e$ corresponds to a Morse trajectory, intersection with stabilizing divisor D, or intersection with the bulk deformation $\mathfrak {b}$ .) The elements $(D, 1)$ , $(D, 2)$ will indicate tangency order to the stabilizing divisor, so a map $u: C \to X$ with constraint $z_e \in C$ of type $(D,1)$ has $u(z_e) \in D$ resp. of type $(D, 2)$ has $u(z_e)$ and the normal derivatives of u at $z_e$ vanish.

Perturbed treed holomorphic disks are defined by allowing the almost complex structure, and Morse function to vary in the domain. Let $\Gamma $ be a domain type (not necessarily stable). Let $\Gamma ^{\mathrm {st}}$ be the stabilization of $\Gamma $ (which is not empty). Let C be a treed disk of type $\Gamma $ and $C^{\mathrm {st}}$ its stabilization which is of type $\Gamma ^{\mathrm {st}}$ . Suppose we are given a perturbation datum $P_{\Gamma ^{\mathrm {st}}}$ for type $\Gamma ^{\mathrm {st}}$ . On each surface part $S_v$ of C, $P_{\Gamma ^{\mathrm {st}}}$ induces a domain-dependent almost complex structure $J_v$ , and a domain-dependent Hamiltonian function $H_v$ . On each tree part $T_e$ of C, $P_{\Gamma ^{\mathrm {st}}}$ induces a domain-dependent function

$$\begin{align*}F_{e}: T_e \times \bigsqcup_{(L_-, L_+)\in {\mathcal L}^2} (L_- \cap L_+) \to {\mathbb R}. \end{align*}$$

These data allow one to define the equations on each component. For each surface component $S_v$ and a smooth map $u_v: S_v \to X$ , define

$$ \begin{align*} \overline\partial_{J_v, H_v} u_v:= ({\operatorname{d}}_{H_v} u_v)^{0,1}& :=({\operatorname{d}} u_v + X_{H_v}(u_v) )^{0,1}\\ & := \frac{1}{2} ( {\operatorname{d}} u_v + J_v \circ {\operatorname{d}} u_v \circ j_v) + (X_{H_v}(u_v))^{0,1}\in \Omega^{0,1}(S_v, u_v^* TX). \end{align*} $$

We say that $u_v$ is $(J_v, H_v)$ -holomorphic if $\overline \partial _{J_v, H_v} u_v = 0$ . For each tree component $T_e$ and a smooth map

$$\begin{align*}u_e: T_e \to \bigsqcup_{(L_-,L_+)} L_- \cap L_+ \end{align*}$$

we say that $u_e$ is a perturbed negative gradient segment if

$$\begin{align*}u_e'(s) + \nabla F_{e}(s, (u_e(s))) = 0. \end{align*}$$

Isomorphisms of perturbed treed holomorphic disks are defined in a way similar to that for stable pseudoholomorphic maps. A perturbed treed holomorphic disk is called stable if its automorphism group is finite, or equivalently

1. (a) every sphere component $u_v: S_v \to X$ with ${\operatorname {d}} u_v = {\operatorname {d}}_{H_v} u_v \equiv 0$ has at least three special points, and

2. (b) every disk component $u_v: S_v \to X$ with ${\operatorname {d}}_{H_v} u_v \equiv 0$ either has at least three boundary special points, or one boundary special point and one interior special point, or at least two interior special points.

3. (c) over each infinite edge $T_e \subset C$ the map $u_e: T_e \to L_e$ is non-constant.

Given a map type ${\mathbb {\Gamma }} = (\Gamma , \underline {\smash {x}}, \underline {\smash {\beta }}, \underline {\smash {\lambda }})$ , denote by $ {\mathcal {M}}_{\mathbb {\Gamma }}(P_{\Gamma ^{\mathrm {st}}}) $ the set of isomorphism classes of stable $P_{\Gamma ^{\mathrm {st}}}$ -perturbed adapted treed holomorphic disks of map type $\mathbb {\Gamma }$ . One can also define a Gromov topology and compactify the moduli spaces (we omit the details). We only consider the compactification for the case $\Gamma $ is stable. In this case, the Gromov compactification is

(2.13) $$ \begin{align} \overline{\mathcal{M}}_{\mathbb{\Gamma}}(P_\Gamma):= \bigsqcup_{\mathbb{\Pi} \preceq \mathbb{\Gamma}} {\mathcal{M}}_{\mathbb{\Pi}} \left(P_\Gamma|_{\overline{\mathcal{U}}_{\Pi^{\mathrm{st}}}} \right). \end{align} $$

Here the partial order $\mathbb {\Pi } \preceq {\mathbb {\Gamma }}$ naturally extends the partial order $\Pi \preceq \Gamma $ among domain types (see Remark 2.5 and below).

Definition 2.21 (Partial order among map types)

Let ${\mathbb {\Gamma }} = (\Gamma , \underline {\smash {x}}, {\underline {\smash {\beta }}}, {\underline {\smash {\lambda }}})$ and ${\mathbb {\Gamma }}' = (\Gamma ', \underline {\smash {x}}', {\underline {\smash {\beta }}}', {\underline {\smash {\lambda }}}')$ be two map types. We write ${\mathbb {\Gamma }}' \preceq {\mathbb {\Gamma }}$ if $\Gamma ' \preceq \Gamma $ (which induces a morphism $\psi : \Gamma ' \to \Gamma $ and a natural identification $\mathrm {Leaf}_\bullet (\Gamma ) \cong \mathrm {Leaf}_\bullet (\Gamma ')$ with respect to which $\underline {\smash {x}} = \underline {\smash {x}}'$ ) and

$$\begin{align*}\beta(v) = \sum_{v' \in \psi^{-1}(v)} \beta'(v'); \end{align*}$$

moreover, for each interior leaf $e\in \mathrm {Leaf}_\bullet (\Gamma )$ with corresponding $e'\in \mathrm {Leaf}_\bullet (\Gamma ')$ , one has $X_{e'} \subset X_e$ .

The composition laws of Fukaya algebras rely on the following relation among perturbation data.

Definition 2.22 Coherent perturbations)

A collection of perturbation data

$$\begin{align*}{\underline{\smash{P}}}:= ( P_\Gamma)_{\Gamma} \end{align*}$$

for all stable domain types $\Gamma $ are called a coherent system of perturbation data if the following conditions are satisfied.

1. (a) (Cutting-edges axiom) If a (boundary) breaking separates $\Gamma $ into $\Gamma _1$ and $\Gamma _2$ then $P_\Gamma $ is the product of the perturbations $P_{\Gamma _1},P_{\Gamma _2}$ under the isomorphism

   $$\begin{align*}\overline{\mathcal{U}}_\Gamma \simeq \pi_1^* \overline{\mathcal{U}}_{\Gamma_1} \cup \pi_2^* \overline{\mathcal{U}}_{\Gamma_2}. \end{align*}$$

2. (b) (Degeneration axiom) If $\Gamma '\prec \Gamma $ , then the restriction of $P_\Gamma $ to $\overline {\mathcal {U}}_{\Gamma '}$ is equal to $P_{\Gamma '}$ . Notice that these degenerations include the case that the weight $\operatorname {wt}(e) $ on one weighted edge e limits to $0$ or $1$ (see Remark 2.5).

3. (c) (Forgetful axiom) For a forgettable boundary input $e \in \operatorname {Edge}_\circ (\Gamma )$ , let $\Gamma _e$ be the domain type obtained from $\Gamma $ by forgetting e and stabilizing. Then $P_{\Gamma }$ is equal to the pullback of $P_{\Gamma _e}$ via the contraction $\overline {\mathcal {U}}_{\Gamma } \to \overline {\mathcal {U}}_{\Gamma _e}$ .

2.6 Transversality

In this subsection we regularize the moduli spaces used in our construction. We first review very briefly the Fredholm theory associated to treed holomorphic maps. Let ${\mathbb {\Gamma }}$ be a map type. We specify Sobolev constants $k \in \mathbb {N}$ and $p>0$ and a decay constant $\delta>0$ with $kp> 2$ and $\delta $ sufficiently small. The set $ {\mathcal B}^{k,p, \delta }(C, {\mathbb {\Gamma }}) $ of maps of type ${\mathbb {\Gamma }}$ has the structure of a Banach manifold. In the case without branch changes in the boundary condition, an element $u \in {\mathcal B}^{k,p,{\delta }}(C, {\mathbb {\Gamma }})$ is defined as in Definition 2.20 without requiring the holomorphic curve and gradient flow equations and instead requiring u to be of class $W^{k,p}_{\mathrm {loc}}$ over each surface or tree component. In the case with branch changes, that is, for each disk component $S_v \cong {\mathbb D} \subset C$ with a boundary node or marking $z \in \partial S_v$ with two sides of z are labelled by two different Lagrangian submanifolds, then we require the map u is of class $W^{k, p, \delta }$ with respect to a cylindrical type metric, which means that it differs from a map constant near infinity by exponentiation of a section of class $W^{k,p,\delta }$ . If the two sides of z are labelled by the same Lagrangian submanifold (with possibly different branes structures), then we alternatively require that the map is of class $W^{k, p}$ with respect to the smooth metric. Tangency conditions for a maximal order m as in (2.11) are defined for k sufficiently large. Choose a perturbation datum $P_\Gamma = (J_\Gamma , H_\Gamma , F_\Gamma , M_\Gamma )$ . Over the Banach manifold ${\mathcal B}^{k,p, \delta }(C, {\mathbb {\Gamma }}) $ there is a Banach vector bundle ${\mathcal E}^{k-1,p,{\delta }}(C, {\mathbb {\Gamma }})$ of $0,1$ -forms of class $k-1,p$ so that the defining equations of Definition 2.20 provide a section

$$\begin{align*}{\mathcal F}: {\mathcal B}^{k,p,{\delta}}(C, {\mathbb{\Gamma}} ) \to {\mathcal E}^{k-1,p,{\delta}}(C, {\mathbb{\Gamma}}) \end{align*}$$

combining the perturbed Cauchy-Riemann operators on the surface parts and gradient flow operators on the edges. To include the variations of the domains, one takes an open neighbourhood $\mathcal {M}^i_\Gamma \subset {\mathcal {M}}_\Gamma $ of $[C]$ over which the universal curve $\mathcal {U}_\Gamma $ has a trivialization

$$\begin{align*}\mathcal{U}_\Gamma|_{{\mathcal{M}}_\Gamma^i} \cong {\mathcal{M}}_\Gamma^i \times C. \end{align*}$$

The linearization of the map ${\mathcal F}$ at a perturbed treed holomorphic disk $(C, u)$ is a Fredholm operator

$$\begin{align*}\tilde{D}_u : T_{(u, \partial u)} {\mathcal B}^{k,p,{\delta}}(C, {\mathbb{\Gamma}}) \times T_{[C]} {\mathcal{M}}_\Gamma \to {\mathcal E}^{k-1,p,{\delta}}(C, {\mathbb{\Gamma}})|_{u}. \end{align*}$$

Since the Lagrangians are always totally real with respect to the domain-dependent almost complex structures, the linearized operator is Fredholm. Its index can be calculated using Riemann-Roch for surfaces with boundary and gives the expected dimension of the moduli space

$$\begin{align*}\mathrm{ind}(\Gamma) := \mathrm{dim} {\mathcal{M}}_{\mathbb{\Gamma}}(P_\Gamma) = \operatorname{Ind}(\tilde{D}_u) = \mathrm{dim} {\mathcal{M}}_\Gamma + \mu({\underline{\smash{\beta}}}) + i(\underline{\smash{x}}) - i({\underline{\smash{\lambda}}}) \end{align*}$$

where $\mu ({\underline {\smash {\beta }}})$ is the total Maslov index of the disk class, $i(\underline {\smash {x}})$ is the sum of Morse indices of asymptotic constraints, and $i ({\underline {\smash {\lambda }}})$ is the effect of interior constraints. For example, if $\Gamma $ has k interior leaves, all of which are labelled by $(D, 1)$ , then $i({\underline {\smash {\lambda }}}) = 2k$ .

Following Cieliebak-Mohnke [Reference Cieliebak and MohnkeCM07], we introduce a collection of map types for which transversality can be achieved by domain-dependent perturbations.

We will need certain forgetful maps to treat crowded configurations. Let $\Gamma $ be a stable domain type. Choose a subset

$$\begin{align*}W \subset \mathrm{Vert}_\bullet (\Gamma) = \mathrm{Vert}(\Gamma) \setminus \mathrm{Vert}(\Gamma_\circ) \end{align*}$$

of spherical vertices. Define $\Gamma _W$ to be the domain type obtained by the following operation: For each connected component $W_i \subset W$ , remove all interior leaves except the one with the largest labelling on $W_i$ , and stabilize the remaining configuration. The set W descends to a (possibly empty) subset $W' \in \mathrm {Vert}_{\circ }(\Gamma _W)$ . A consequence of the locality condition on the perturbation data is that each $P_\Gamma \in {\mathcal P}_\Gamma $ descends to a perturbation datum $P_{\Gamma _W} \in {\mathcal P}_{\Gamma _W}$ whose restriction to surface components $S_v$ for $v \in W'$ equals to the base almost complex structure $J_0$ and the zero Hamiltonian perturbation.Footnote ⁸ Let $ {\mathcal P}_{\Gamma _W, W'} \subset {\mathcal P}_{\Gamma _W} $ be the subset of perturbations that agree with the base almost complex structure $J_0$ and the zero Hamiltonian over surface components corresponding to vertices in $W'$ . This forgetful construction gives a smooth map of Banach manifolds

(2.14) $$ \begin{align} {\mathcal P}_{\Gamma} \to {\mathcal P}_{\Gamma_W, W'}. \end{align} $$

Indeed, this is a surjective map and essentially a projection, hence admits a smooth right inverse.

The main result of this section is the regularity of moduli spaces for uncrowded map types and the selection of a coherent collection of perturbation data.

Proof. The proof is an induction on the possible domain types according to the partial order (see Remark 2.5). First we introduce an equivalence relation among stable domain types. We write $\Gamma \sim \Pi $ for the equivalence relation generated by

$$\begin{align*}\Pi \preceq \Gamma\ \mathrm{and}\ \rho|_{\Pi_\circ}: \Pi_\circ \to \Gamma_\circ\ \mathrm{is\ an\ isomorphism}; \end{align*}$$

that is, if roughly they have isomorphic disk part. Here $\rho $ is the tree map induced from the partial order relation $\Pi \preceq \Gamma $ (see Remark 2.5). Let $[\Gamma ]$ denote the equivalence class of $\Gamma $ . The partial order relation among domain types descends to an equivalence relation among their equivalence classes.

The inductive step is the following. Fix an equivalence class $[\Gamma ]$ . Suppose we have chosen strongly regular perturbation data $P_\Pi $ for all stable domain types $\Pi $ with $[\Pi ] \prec [\Gamma ]$ as well as domain types with strictly fewer boundary inputs or the same number of boundary inputs but strictly fewer interior leaves, such that the chosen collection is coherent in the sense of Definition 2.22.

We prove the following sublemma.

Proof of the sublemma

Let $\mathcal {M}^i_\Gamma $ be a subset of $\mathcal {M}_\Gamma $ over which the universal curve $\mathcal {U}_\Gamma $ is trivial, and $\mathcal {U}_\Gamma ^i$ the restriction of $\mathcal {U}_\Gamma $ to $\mathcal {M}^i_\Gamma $ . For each uncrowded map type ${\mathbb {\Gamma }}$ with underlying domain type $\Gamma $ , consider the universal moduli space

$$\begin{align*}\mathcal{M}^{i,{\operatorname{univ}}}_{\mathbb{\Gamma}}({\mathcal P}_\Gamma^*) = \{ ([u: C \to X], P_\Gamma) | P_\Gamma \in {\mathcal P}_\Gamma^*,\ C \subset \mathcal{U}_\Gamma^i, \ [u] \in {\mathcal{M}}_{\mathbb{\Gamma}}(P_\Gamma) \}. \end{align*}$$

of maps with domain in $\mathcal {U}_\Gamma ^i$ together with a perturbation datum $P_\Gamma $ . By the Sard-Smale theorem, this sublemma can be proved once we show the regularity of the local universal moduli space. Suppose this is not the case, so that for some $(u, P_\Gamma )$ the linearization of the defining equation of the universal moduli is not surjective.

Then there exists a non-zero section $\eta $ in the $L^2$ -orthogonal complement of the image of the linearized operator, or equivalently, in the kernel of the formal adjoint of the linearized operator. By elliptic regularity, $\eta $ is actually smooth. We will derive a contradiction by showing that each component of $\eta $ vanishes identically on that component.

Step One. The form $\eta $ vanishes on any non-constant sphere component $u_v: S_v \to X$ . By our assumption on the domain-dependent Hamiltonian perturbation $H_\Gamma $ (see Lemma 2.15 and Definition 2.18), $H_\Gamma $ vanishes on spherical components. Since the support of the perturbation $J_{\Gamma , v}$ has non-zero intersection with $S_v$ , the restriction $\eta _v$ of $\eta $ to $S_v$ must vanish over a non-empty open set of $S_v$ . The unique continuation principle for first-order elliptic equation implies that $\eta _v$ vanishes identically.

Step Two. The form $\eta $ vanishes on any disk component $u_v: S_v \cong {\mathbb D} \to X$ corresponding to the vertex $v \in \mathrm {Vert}(\Gamma _\circ )$ . Suppose that ${\operatorname {d}}_{H_v} u_v$ is not identically zero. Then ${\operatorname {d}}_{H_v} u_v$ is non-zero over a non-empty open subset $U \subset S_v$ with $u_v(U)$ is disjoint from the neighbourhood of D where $J_v \equiv J_0$ . By orthogonality to images of deformation of $J_v$ over U, $\eta _v$ is zero over U. Unique continuation principle shows that $\eta _v$ vanishes identically. Suppose $u_v$ is covariantly constant, i.e., $d_{H_v} u_v \equiv 0$ . Then $u_v(S_v)$ has a non-empty intersection with the neighbourhood of $|\mathcal {L}|$ where one can perturb the Hamiltonian $H_v$ . This again shows that $\eta _v$ vanishes on a non-empty open subset of $S_v$ , and so vanishes identically on $S_v$ by the unique continuation principle.

Step Three. The form $\eta $ vanishes on each edge $T_e$ with positive length $\ell (e)> 0$ . First, for an edge $T_e$ with positive or infinite length $\ell (e) \in \{ 0, \infty \}$ , if the gradient segment $u_e: T_e \to L_e$ is mapped into a positive dimensional target $L_e$ , then since the support of the perturbation $F_{e}$ is non-empty, it also follows that the restriction $\eta _e$ to $T_e$ vanishes identically. If $L_e$ is zero-dimensional, then by definition $\eta _e \equiv 0$ .

Step Four. The form $\eta $ vanishes on each ghost sphere component $S_v \cong \mathbb {P}^1, {\operatorname {d}} u_v = 0$ . Let $u_v: S_v \to X$ be a constant map with value $x_v \in X$ . For any domain-dependent almost complex structure, the linear map

$$\begin{align*}\overline\partial_{J_v}: \Omega^0(S_v, T_{x_v} X) \to \Omega^{0,1}( {\mathbb P}^1, T_{x_v} X) \end{align*}$$

is surjective with kernel equal to the finite dimensional subspace of constant vector fields $\xi $ on $S_v$ . However, there might be constraints coming from special points on this component. For this we use the uncrowdedness condition. Consider a maximal ghost sphere tree $W \subset \mathrm {Vert}(\Gamma )$ . By the above argument for ghost disk components, we may assume that W contains only spherical vertices $v \in \operatorname {Vert}_{\bullet }(\Gamma )$ . There is at most one special point $z_e$ on the corresponding curve $C_W \subset C$ which is constrained by $(D, m)$ ; this condition puts a two-dimensional constraint on a constant vector field $\xi $ restricted to $C_W$ . For any other interior marking $z_e$ , one can use the deformation of the diffeomorphism $M_{\Gamma , e}$ to allow variations in $\xi $ while preserving the constraints. For any node connecting $C_W$ to a non-constant component, the constraints are transversely cut out using the fact that the linearized operator is surjective on deformations on the adjacent non-constant components even vanishing at the node. Thus the form $\eta $ vanishes on components in W.

Next we construct a comeager subset of strongly regular perturbations. For any subset $W \subset \mathrm {Vert}_\bullet (\Gamma )$ , consider the domain type $\Gamma _W$ with a descent subset $W' \subset \mathrm {Vert}_\bullet (\Gamma _W)$ . The trees $\Gamma _W$ and $\Gamma $ have isomorphic disk parts. Hence the pre-chosen perturbations provides a subset ${\mathcal P}_{\Gamma _W, W'}^* \subset {\mathcal P}_{\Gamma _W, W'}$ consisting perturbations whose values are fixed precisely over strata $\Pi '$ with $[\Pi '] \prec [\Gamma _W]$ . Moreover, the forgetful map (2.14) restricts to a forgetful map

$$\begin{align*}\pi_W: {\mathcal P}_\Gamma^* \to {\mathcal P}_{\Gamma_W, W'}^* \end{align*}$$

which has a right inverse given by pullback. By the same argument as the proof of the above sublemma, there is a comeager subset ${\mathcal P}_{\Gamma _W, W'}^{*, \mathrm {reg}}$ consisting of perturbations $P_{\Gamma _W}$ that regularize moduli spaces ${\mathcal {M}}_{{\mathbb {\Gamma }}_W}(P_{\Gamma _W})$ for map types ${\mathbb {\Gamma }}_W$ that are ghost on surface components corresponding to vertices in $W'$ . Define

$$\begin{align*}{\mathcal P}_{\Gamma}^{*, \mathrm{s.reg}}:= \bigcap_{W \subset \mathrm{Vert}_\bullet(\Gamma)} \pi_W^{-1} ( {\mathcal P}_{\Gamma_W, W'}^{*, \mathrm{reg}}). \end{align*}$$

This subset is still comeager and all its elements are strongly regular.

Lastly, we choose perturbations for each strata extending the pre-chosen perturbations on lower-dimensional strata. We define smaller comeager subsets ${\mathcal P}_\Gamma ^{**}$ inductively as follows. If $\Gamma $ is a smallest element of the equivalence class $[\Gamma ]$ , then define ${\mathcal P}_{\Gamma }^{**}:= {\mathcal P}_{\Gamma }^{*, \mathrm {s.reg}}$ . Suppose for a general $\Gamma $ in $[\Gamma ]$ one has defined ${\mathcal P}_{\Gamma '}^{**}$ for all $\Gamma ' \prec \Gamma $ with $[\Gamma '] = [\Gamma ]$ . Define

$$\begin{align*}{\mathcal P}_{\Gamma}^{**}:= {\mathcal P}_{\Gamma}^{*, \mathrm{s.reg}} \cap \bigcap_{\substack{\Gamma' \prec \Gamma \\ [\Gamma'] = [\Gamma]}} \pi_{\Gamma, \Gamma'}^{-1}( {\mathcal P}_{\Gamma'}^{**}). \end{align*}$$

Here $\pi _{\Gamma , \Gamma '}: {\mathcal P}_\Gamma \to {\mathcal P}_{\Gamma '}$ is the map defined by restricting to boundary strata. Then we have defined ${\mathcal P}_{\Gamma }^{**}$ for all $\Gamma $ in this equivalence class. The equivalence class $[\Gamma ]$ has a unique maximal element $\Gamma _{\mathrm {max}}$ . Choose an arbitrary perturbation $P_{\Gamma _{\mathrm { max}}}\in {\mathcal P}_{\Gamma _{\mathrm {max}}}^{**}$ . By boundary restriction this choice induces $P_\Gamma $ for all $\Gamma $ in this equivalence class. By construction, all these $P_\Gamma $ extend the existing perturbations on lower-dimensional strata. By induction, one obtains the claimed collection ${\underline {\smash {P}}}$ .

For later use, we note that the introduction of small Hamiltonian perturbations does not produce pseudoholomorphic maps of negative area:

2.7 Boundary strata

In this section, we describe the boundary of the moduli spaces we use, which are those of expected dimension at most one. Fix a coherent collection of strongly regular perturbation data ${\underline {\smash {P}}} = (P_\Gamma )$ and abbreviate all moduli spaces ${\mathcal {M}}_{\mathbb {\Gamma }}(P_\Gamma )$ by ${\mathcal {M}}_{\mathbb {\Gamma }}$ .

Remark 2.31. As in [Reference Wehrheim and WoodwardWW] the determinant lines of the linearized operators become equipped with orientations induced by relative spin structures. In particular, if all strata of ${\mathcal {M}}(\underline {\smash {x}})_0$ are regular then there is a map

$$\begin{align*}\epsilon: {\mathcal M}_{d, 1}(\underline{\smash{x}})_0 \to \{\pm 1\}. \end{align*}$$

The following lemma classifies types of topological boundaries of one-dimensional moduli spaces.

Sketch of proof.

It suffices to check sequential compactness. Let $(C_\nu , u_\nu )$ be a sequence of treed holomorphic disks representing a sequence of points in ${\mathcal {M}}_{\mathbb {\Gamma }}(P_\Gamma )$ . By a combination of Gromov compactness for (pseudo)holomorphic disks and compactness of the moduli space of gradient trajectories, there is a subsequence (still indexed by $\nu $ ) that converges to a limiting treed holomorphic disk $(C_\infty , u_\infty )$ of certain map type $\mathbb {\Pi }$ . We first claim that the domain type $\Pi $ is stable. Suppose on the contrary it is not the case. Then, there is either a domain-unstable disk or a domain-unstable sphere component $u_{\infty ,v}: S_v \to X, v \in \operatorname {Vert}(\mathbb {\Pi })$ . By the stability condition, $u_v$ must be a non-constant map. Moreover, $u_v$ is pseudoholomorphic with respect to a constant tamed almost complex structure ${J = J_{\Pi ^{\mathrm {st}}} (z)}$ on X, where $z \in C_\infty ^{\operatorname {st}}$ is the image of $S_v$ in the stabilization. On the other hand, there exists a unique maximal subtree $\Gamma _{(v)} \subset \Gamma $ which contains no boundary edge with positive edge, such that the bubble $u_v$ comes from energy blow up on components belonging to $\Gamma _{(v)}$ . Therefore, by the conditions of the perturbation data (see (a) of Definition 2.18), one has

$$\begin{align*}J \in {\mathcal J}_{\mathrm{tame}}^{E(\Gamma_{(v)})} (X, \omega). \end{align*}$$

Since the convergence preserves the total energy, the disk or the sphere has energy at most $E(\Gamma _{(v)})$ . By Lemma 2.17, the J-holomorphic map $u_v: S_v \to X$ is not contained in D and must intersect D in at least three points resp. one point in the sphere resp. disk case. Topological invariance of intersection numbers, as in Cieliebak-Mohnke [Reference Cieliebak and MohnkeCM07], implies that for $\nu $ sufficiently large, $u_v$ intersects with D in at least three resp. one nearby point. Since the type ${\mathbb {\Gamma }}$ is essential, all these intersection points are marked points labelled by $(D, 1)$ . The convergence implies that the intersection points of $u_\infty $ with D must all be marked points, contradicting the assumption that the domain of the domain $S_v$ is unstable. Therefore, the domain type $\Pi $ is stable. Since $\Pi \preceq \Gamma $ , the perturbation datum $P_\Gamma $ induces by restriction a perturbation datum $P_\Pi $ so that $[(C_\infty , u_\infty )] \in {\mathcal {M}}_{{\mathbb {\Pi }}}(P_\Pi ).$

Next we show that type of the limiting map constructed in the previous paragraph is uncrowded. Suppose this is not the case, then let $W \subset \mathrm {Vert}_{\circ }(\Pi )$ be the (non-empty) set of ghost sphere components. By the locality property, the perturbation data $P_\Pi $ descends to a perturbation $P_{\Pi _W}$ which is equal to $J_0$ over $W'$ . The limiting configuration $[(C_\infty , u_\infty )]$ then descends to an element

$$\begin{align*}[(C', u')] \in {\mathcal{M}}_{{\mathbb{\Pi}}_W}(P_{\Gamma_W}). \end{align*}$$

Since $\mathbb {\Pi }_W$ is uncrowded, the above moduli space is regular and non-empty. However, similar to the argument of [Reference Cieliebak and MohnkeCM07], the reduction drops the expected dimension by at least two. This contradiction shows that $\mathbb {\Pi }$ must be uncrowded.

Finally, we claim $\mathbb {\Pi }$ has no sphere components. Indeed, each sphere component will drop the dimension of the domain moduli space by two and $\Gamma $ has no sphere components. It follows from the dimension formula for ${\mathcal {M}}_\Gamma $ that when $\mathrm { dim} {\mathcal {M}}_{\mathbb {\Gamma }} = 0$ , $\Pi $ must be identical to $\Gamma $ and hence ${\mathcal {M}}_{\mathbb {\Gamma }}$ is compact. When $\mathrm {dim} \ {\mathcal {M}}_{\mathbb {\Gamma }} = 1$ , the only possibly types of $\Pi $ are described in the above list.

Moreover, we distinguish the boundary strata as either true or fake boundary components. The true boundaries are those corresponding to edge breaking and weight changing to zero or one, while the fake boundaries are those corresponding to disk bubbling or edges shrinking to zero. For one-dimensional moduli strata ${\mathcal {M}}_{d, 1} (\underline {\smash {x}})_1$ , define

$$\begin{align*}\overline{\mathcal{M}}_{d, 1} (\underline{\smash{x}})_1 = \bigcup_{{\mathbb{\Gamma}}}\overline{\mathcal{M}}_{\mathbb{\Gamma}} \end{align*}$$

to be the union of all compactified moduli space of expected dimension one while identifying fake boundaries. Standard gluing constructions (gluing disks or gradient lines) show that $\overline {\mathcal {M}}_{d, 1} (\underline {\smash {x}})_1$ a topological 1-manifold with boundary and its cutoff at any energy level (indexed by the number of interior markings) is compact. The boundaries are strata corresponding to edge breaking and weight changing to zero or one.

3.1 Composition maps

In this section, we apply the transversality results above to construct Lagrangian Floer theory. In the Morse model the generators of the Lagrangian Floer cochains are critical points of a Morse function on the Lagrangian intersection, assumed clean. Recall for each pair $(L, L') \in \mathcal {L}^2$ , the intersection $L\cap L'$ is a smooth manifold. We have chosen a Morse function

$$\begin{align*}F_{L, L'}: L \cap L' \to {\mathbb R}. \end{align*}$$

Consider the set of all Lagrangian branes supported on $\mathcal {L}$ , i.e.,

$$\begin{align*}\widehat {\mathcal L}:= \big\{ {\widehat{L}} \to L \ |\ L \in {\mathcal L} \big\} \end{align*}$$

For each pair of branes $({\widehat {L}}, {\widehat {L}}')$ , denote the set of critical points on the intersection

$$\begin{align*}{\mathcal I}( {\widehat{L}}, {\widehat{L}}' )= \left\{ \begin{array}{@{}cc} \mathrm{crit} ( F_{L, L'}),\ &\ {\widehat{L}} \neq {\widehat{L}}',\\ \mathrm{crit} ( F_{L, L'} ) \cup {\mathcal I}_{{\widehat{L}}}^{hu},\ &\ {\widehat{L}} = {\widehat{L}}', \end{array} \right. \end{align*}$$

where ${\mathcal I}_{{\widehat {L}}}^{hu}$ contains

and

for all connected component c of L (see (2.9) and (2.10)). The ‘Morse indices’ of the extra generators are defined by

3.1.1 Gradings

In order to obtain graded Floer cohomology groups a grading on the set of generators is defined as follows. Let $N \in \mathbb {Z}$ be an even integer and let

(3.1) $$ \begin{align} \pi^N: \operatorname{Lag}^N(X) \to \operatorname{Lag}(X) \end{align} $$

be an N-fold Maslov cover of the bundle of Lagrangian subspaces as in Seidel [Reference SeidelSei00]; we always assume that the induced $2$ -fold cover $\operatorname {Lag}^2(X) \to \operatorname {Lag}(X)$ is the bundle of oriented Lagrangian subspaces. A $\mathbb {Z}_N$ -grading of $L \in \mathcal {L}$ is a lift

(3.2)

where the horizontal arrow is the map assigning to each $x \in L$ its tangent space. Given such a grading, there is a natural $\mathbb {Z}_N$ -valued map

$$\begin{align*}\mathcal{I}( {\widehat{L}}_0, {\widehat{L}}_1) \to \mathbb{Z}_N, \quad x \mapsto |x| \end{align*}$$

defined as follows. Recall that for each pair of paths $\lambda _0, \lambda _1: [a, b] \to \operatorname {Lag}( T_x X)$ , there is a Maslov index

(3.3) $$ \begin{align} \mu(\lambda_0, \lambda_1) \in \frac{1}{2} {\mathbb Z} \end{align} $$

which is an integer if and only if $\mathrm {dim} (\lambda _0(a) \cap \lambda _1(a)) \equiv \mathrm {dim} (\lambda _0(b) \cap \lambda _1(b))\ \mathrm {mod}\ 2$ . For any $x \in {\mathcal I}({\widehat {L}}_0, {\widehat {L}}_1)$ , choose two paths

$$\begin{align*}\tilde \lambda_0, \tilde \lambda_1: [a, b] \to \operatorname{Lag}^N(T_x X) \end{align*}$$

such that $\tilde \lambda _0(a) = \tilde \lambda _1(a)$ and

$$ \begin{align*} &\ \tilde \lambda_0(b) = \phi_{L_0}^N( T_x L_0),\ &\ \tilde \lambda_1(b) = \phi_{L_1}^N( T_x L_1) \end{align*} $$

with notation from (3.2). Define

$$\begin{align*}|x|:= \frac{n}{2} - \mu( \pi^N (\tilde\lambda_0), \pi^N(\lambda_1)) + \frac{1}{2} \mathrm{dim} (L_0 \cap L_1) - \mathrm{index}_{\mathrm{Morse}}(x) \in {\mathbb Z}/N{\mathbb Z}. \end{align*}$$

with notation from (3.3) and (3.1). For example, when $L_0 = L_1$ and x is an ordinary critical point, then $|x|$ is the dimension of the stable manifold of x under the negative gradient flow.

3.1.2 Weighted counting

The moduli space of holomorphic disks is non-compact, and to remedy this the structure maps of the Fukaya algebra are defined over Novikov rings in a formal variable. The Floer cochain space is a free module over generators corresponding to Morse critical points and the two additional generators from (2.10) necessary to achieve strict units. Given two branes ${\widehat {L}}, {\widehat {L}}'$ let

$$\begin{align*}CF^{\bullet} ({\widehat{L}}, {\widehat{L}}' ) = \bigoplus_{ x \in \mathcal{I} ({\widehat{L}}, {\widehat{L}}' ) } \operatorname{Hom}({\widehat{L}}_x \otimes_{\Lambda^\times} \Lambda,{\widehat{L}}_x' \otimes_{\Lambda^\times} \Lambda) \end{align*}$$

the sum of space of linear maps between the fibers of the local systems. The space of Floer cochains is naturally $\mathbb {Z}_N$ -graded by

$$\begin{align*}CF^{\bullet} ({\widehat{L}}, {\widehat{L}}') = \bigoplus_{k \in \mathbb{Z}_N} CF^k ({\widehat{L}}, {\widehat{L}}'), \quad CF^k ({\widehat{L}}, {\widehat{L}}') = \bigoplus_{x \in \mathcal{I}^k({\widehat{L}}, {\widehat{L}}')} \Lambda x.\end{align*}$$

The q-valuation on $\Lambda $ extends naturally to $CF^{\bullet } ( {\widehat {L}}, {\widehat {L}}')$ :

$$\begin{align*}\operatorname{val}_q: CF^{\bullet} ( {\widehat{L}}, {\widehat{L}}' ) - \{ 0 \} \to \mathbb{R}, \quad \sum_{x \in \mathcal{I}( {\widehat{L}}, {\widehat{L}}' )} c(x) x \mapsto \min( \operatorname{val}_q(c(x))) .\end{align*}$$

The local systems contribute to the coefficients of the composition maps in the expected way. For any holomorphic treed disk $u: C \to X$ with boundary in some collection ${{\underline {\smash {\widehat {L}}}} = ({\widehat {L}}_0, \ldots , {\widehat {L}}_d)}$ and mapping the corners to $x_0,\ldots , x_d$ , generators

$$\begin{align*}a_i \in \operatorname{Hom}({\widehat{L}}_{i-1,x_i} \otimes_{\Lambda^\times} \Lambda,{\widehat{L}}_{i,x_i} \otimes_{\Lambda^\times} \Lambda) \end{align*}$$

denote by

(3.4) $$ \begin{align} y(u) = T_{d-1} a_{d-1} \ldots a_1 T_0 a_0 T_0 \in \operatorname{Hom}({\widehat{L}}_{0,x_0} \otimes_{\Lambda^\times} \Lambda,{\widehat{L}}_{d,x_0} \otimes_{\Lambda^\times} \Lambda) \end{align} $$

the product of parallel transports $T_i$ along the restrictions $u_v | (\partial C)_i$ . For more complicated treed disks the holonomy is defined recursively starting with the components furthest away from the root.

Definition 3.1. Fix a coherent collection of strongly regular perturbation ${\underline {\smash {P}}} = (P_\Gamma )_\Gamma $ (whose existence is provided by Theorem 2.26). For each $d \geq 0$ define higher composition maps

$$\begin{align*}m_d: CF^{\bullet} ( {\widehat{L}}_{d-1}, {\widehat{L}}_d) \otimes \ldots \otimes CF^{\bullet} ( {\widehat{L}}_0, {\widehat{L}}_1) \to CF^{\bullet} ( {\widehat{L}}_0, {\widehat{L}}_d )[2-d] \end{align*}$$

on generators by the weighted count of treed disks by

(3.5) $$ \begin{align} m_d(a_d, \ldots, a_1)= \varepsilon + \sum_{x_0 \in {\mathcal I}({\widehat{L}}_0, {\widehat{L}}_d)} \sum_{u \in {\mathcal{M}}_{d, 1} ( x_0, \ldots, x_d)_0} (-1)^{\heartsuit} \operatorname{wt}(u) \end{align} $$

where $\varepsilon = 0$ unless when $d = 1$ , $\widehat{L}_0 = \widehat{L}_1$ , , we require ,

(3.6) $$ \begin{align} \operatorname{wt}(u) := c(u,\mathfrak{b})p(u) y(u) q^{A(u)} o(u) d(u)^{-1}\end{align} $$

are defined as follows:

• • if the domain type of u is $\Gamma $ , then

  (3.7) $$ \begin{align} d(u):= ( k A(u) )! = (\# \mathrm{Leaf}_\bullet(\Gamma))! \end{align} $$
  which is the number of permutations of interior markings $z_e$ mapped into D;

• • the coefficient $c(u,\mathfrak {b})$ is a product of coefficients $c_i$ of the bulk deformation, with product taken over interior leaves mapping to $\mathfrak {b}$ ,

• • the coefficient $p(u)$ is the coefficient $p_i$ of the multivalued perturbation $P_\Gamma $ of (5.3) evaluated at the branch containing u, and

• • $y(u)$ is the holonomy of the local system as defined in (3.4);

• • the exponent $A(u)$ is area, equal to the (non-negative) perturbed energy of the map u up to a curvature term explained in [Reference McDuff and SalamonMS04, Chapter 8];

• • the sign $o(u)$ arises from the choice of coherent orientations and the overall sign $\heartsuit $ is given by

  (3.8) $$ \begin{align} \heartsuit = {\sum_{i=1}^d i|x_i|} .\end{align} $$

We first define a curved $A_{\infty }$ category with infinitely many objects supported on the given Lagrangian submanifolds. Later, we will consider a modified definition of the objects so that the $A_{\infty }$ category is flat.

Theorem 3.2. For any strongly regular coherent perturbation system ${\underline {\smash {P}}} = (P_\Gamma )$ the maps $(m_d)_{d \ge 0}$ define a (possibly curved) $A_{\infty }$ category $\operatorname {Fuk}_{\mathcal {L}}^\sim (X, \mathfrak {b})$ with

1. (a) the set of objects given by $\mathrm {Ob} \left ( \operatorname {Fuk}_{\mathcal {L}}^\sim (X, \mathfrak {b}) \right ):= \widehat {\mathcal L}$ ,

2. (b) the set of morphisms from ${\widehat {L}}$ to ${\widehat {L}}'$ given by $\mathrm {Hom}({\widehat {L}}, {\widehat {L}}'):= CF^{\bullet }( {\widehat {L}}, {\widehat {L}}')$ ,

3. (c) the composition maps $(m_d)_{d\geq 0}$ defined by Definition 3.1, and

4. (d) for each object ${\widehat {L}}$ the strict unit equal to .

Furthermore, for sufficiently small Hamiltonian perturbations the composition maps are defined over the Novikov ring $\Lambda _{\ge 0}$ .

Proof. We must show that the composition maps $m_d$ satisfy the $A_{\infty }$ -associativity equations (1.7). Up to sign the relation (1.7) follows from the description of the boundary in Lemma 2.32 of the one-dimensional components. The strict unit axiom follows in the same way as in [Reference Charest and WoodwardCW22], by noting that by definition for any edge the perturbation data is pulled back under the morphism of universal moduli spaces forgetting e and stabilizing (whenever such a map exists). For sufficiently small Hamiltonian perturbations (where small-ness is in the Floer norm, and the constant depends on an area bound) each perturbed-holomorphic map therefore has non-negative area, by Lemma 2.29. Hence the structure coefficients of the composition maps take values in the Novikov ring.

Remark 3.3. The $A_{\infty }$ homotopy type of $\operatorname {Fuk}_{\mathcal {L}}^\sim (X,\mathfrak {b})$ (as a curved $A_{\infty }$ algebra with curvature with positive q-valuation over the Novikov ring $\Lambda _{\ge 0}$ ) is independent of the choice of almost complex structures, stabilizing divisors, perturbations and depends only on the isotopy class of bulk deformation. The argument uses a moduli space of quilted disks with seams labelled by the diagonal, as in [Reference Charest and WoodwardCW22, Section 5.5]. Suppose first that the Donaldson hypersurface is fixed. Let $J_{\Gamma ,t}$ be an isotopy of almost complex structures, and $\mathfrak {b}_t $ be an isotopy of cycles $\mathfrak {b}_0$ to $\mathfrak {b}_1$ . Requiring that the markings map to $\mathfrak {b}_t$ , the treed disks $u| S_v$ are $J_{\Gamma ,t}$ -holomorphic on components at distance $d(v) = 1/(1-t) - 1/t$ as in Equation (A.2) produces a moduli space fibered over the multiplihedron as in [Reference Charest and WoodwardCW22, Section 5.5] producing a homotopy equivalence given by a collection of maps

$$\begin{align*}\phi_d: CF^{\bullet} ( {\widehat{L}}_{d-1}, {\widehat{L}}_d)_0 \otimes \ldots \otimes CF^{\bullet} ({\widehat{L}}_0, {\widehat{L}}_1)_0 \to CF^{\bullet} ( {\widehat{L}}_0, {\widehat{L}}_d )_1[1-d] \end{align*}$$

as in Seidel [Reference SeidelSei08b, Section 1d] where the groups $CF^{\bullet } ({\widehat {L}}_k,{\widehat {L}}_{k+1})_t$ are the morphism spaces for the categories defined using the data for the given value t in the family. The independence from the choice of Donaldson hypersurface is shown in the appendix. We do not address the question of invariance under Hamiltonian isotopy of Lagrangians and the relation to the Fukaya categories defined by other methods, and dependence only on the cobordism class of the cycle $\mathfrak {b}$ , and so the homology class $[\mathfrak {b}]$ .

3.1.3 Maurer-Cartan equation, potential function and Floer cohomology

Floer cohomology is defined for projective solutions to the Maurer-Cartan equation. For each ${\widehat {L}} \in \mathrm {Ob}( \operatorname {Fuk}_{\mathcal {L}}^\sim (X, {\mathfrak b}))$ , the element

$$\begin{align*}m_0 (1) \in CF^{\bullet} ( {\widehat{L}}, {\widehat{L}} ) \end{align*}$$

is the curvature of the Fukaya algebra $CF^{\bullet } ( {\widehat {L}}, {\widehat {L}} )$ . Its q-valuation $ \operatorname {val}_q(m_0(1))$ is positive because the bulk deformation $\mathfrak {b}$ and the Lagrangian submanifolds do not intersect. The Fukaya algebra $CF^{\bullet } ( {\widehat {L}}, {\widehat {L}})$ is called flat if $m_0(1)$ vanishes and projectively flat if $m_0(1)$ is a multiple of the identity

.Footnote ⁹ Consider the sub-space of $CF^{\bullet } ( {\widehat {L}}, {\widehat {L}})$ consisting of elements with positive q-valuation with notation from (1.4):

$$\begin{align*}CF^{\bullet} ( {\widehat{L}}, {\widehat{L}})_+ = \bigoplus_{x \in \mathcal{I} ({\widehat{L}}, {\widehat{L}}) } \Lambda_{>0} x. \end{align*}$$

Define the Maurer-Cartan map

$$\begin{align*}\mu : CF^{\mathrm{odd}} ( {\widehat{L}}, {\widehat{L}})_+ \to CF^{\bullet} ( {\widehat{L}}, {\widehat{L}} ), \quad b \mapsto m_0(1) + m_1(b) + m_2(b,b) + \ldots. \end{align*}$$

Let $MC ({\widehat {L}} )$ denote the space of weak solutions to the Maurer-Cartan space

The value $W(b)$ for $b \in MC ({\widehat {L}} )$ defines the disk potential

$$\begin{align*}W: MC ({\widehat{L}}) \to \Lambda. \end{align*}$$

Definition 3.4 (Weakly unobstructed branes and Floer cohomology)

1. (a) A weakly unobstructed brane is a triple ${\boldsymbol {L}} = ({\widehat {L}}, b)$ , where ${\widehat {L}} \in \widehat {\mathcal L}$ and $b \in MC({\widehat {L}})$ .

2. (b) The set of all weakly unobstructed branes supported on $\mathcal {L}$ is denoted by

   $$\begin{align*}MC( \mathcal{L} ):= \Big\{ {\boldsymbol{L}} = ({\widehat{L}}, b)\ |\ {\widehat{L}} \in \widehat{\mathcal L},\ b \in MC({\widehat{L}}) \Big\}. \end{align*}$$

3. (c) The Floer cohomology of a weakly unobstructed brane ${\boldsymbol {L}}$ is

   $$\begin{align*}HF^{\bullet} ( {\boldsymbol{L}}, {\boldsymbol{L}}):= \mathrm{ker} (m_1^b)/ \mathrm{im}(m_1^b) \end{align*}$$
   where $m_1^b$ is defined by
   $$\begin{align*}m_1^b: CF^{\bullet}( {\widehat{L}}, {\widehat{L}} ) \to CF^{\bullet} ( {\widehat{L}}, {\widehat{L}} ),\ m_1^b(a) = \sum_{k_1, k_2\geq 0} m_1( \underbrace{b, \ldots, b}_{k_1}, a, \underbrace{b, \ldots, b}_{k_2}). \end{align*}$$

3.1.4 Flat $A_{\infty }$ category and spectral decomposition

Given a curved $A_{\infty }$ category, flat $A_{\infty }$ categories are obtained by restricting to particular values of the curvature.

Definition 3.5 (Flat $A_{\infty }$ category)

Let ${\mathcal F}^\sim $ be a strictly unital curved $A_{\infty }$ category over $\Lambda $ . The flat $A_{\infty }$ category ${\mathcal F}$ associated to ${\mathcal F}^\sim $ is defined as the disjoint union

$$\begin{align*}{\mathcal F}^\flat:= \bigsqcup_{w\in \Lambda} {\mathcal F}_w \end{align*}$$

where each component ${\mathcal F}_w$ has the set of objects

$$\begin{align*}\mathrm{Ob} ({\mathcal F}_w):= \left\{ {\boldsymbol{L}} = ({\widehat{L}}, b)\ |\ {\widehat{L}} \in \mathrm{Ob}({\mathcal F}^\sim),\ b \in MC({\widehat{L}}),\ W_{\widehat{L}} (b) = w \right\}, \end{align*}$$

the set of morphisms

$$\begin{align*}\mathrm{Hom} ( {\boldsymbol{L}}, {\boldsymbol{L}}' ):= \mathrm{Hom} ({\widehat{L}}, {\widehat{L}}'), \end{align*}$$

and the higher compositions for $d \geq 1$

$$\begin{align*}m_d(a_d,\ldots,a_1) = \sum_{k_0,\ldots, k_d \ge 0} m_{d + k_0 + \ldots + k_d}(\underbrace{b_d,\ldots, b_d}_{k_d} , a_d, \ldots, \underbrace{b_1,\ldots, b_1}_{k_1}, a_1,\underbrace{b_0,\ldots, b_0}_{k_0}). \end{align*}$$

Define $m_0 = 0$ . If $\mathrm {Ob}({\mathcal F}_w) \neq \emptyset $ , then we say ${\mathcal F}_w$ is an eigen-subcategory of ${\mathcal F}$ .

Proposition 3.6. For any $w \in \Lambda $ , the category ${\mathcal F}_w$ is a flat strictly unital $A_{\infty }$ category as defined in (1.5).

Proof. The flatness condition $m_0(1) = 0$ holds by definition. The $A_{\infty }$ relation

(3.9) $$ \begin{align} & 0 = \sum m_{d -i + 1} (\underbrace{b_d,\ldots, b_d }_{i_d} , {a}_{d} , \ldots, , b_{j+i} \nonumber\\ &\qquad m_i( \underbrace{b_{j+i}, \ldots, b_{j+i}}_{k_{j+i}}, {a}_{j+i},\ldots, {a}_{j+1}, \underbrace{b_{j+1},\ldots, b_{j+1}}_{k_j}) , \nonumber\\ & \qquad\qquad\qquad\ldots, b_{j+1}, {a}_{j+1}, \ldots, \underbrace{b_1,\ldots, b_1}_{i_1}, {a}_1, \underbrace{b_0,\ldots, b_0}_{i_0} ) \end{align} $$

follows from the $A_{\infty }$ relation for ${\mathcal F}^\sim $ , the strict identity relation, and the inclusion

In the case of the bulk deformed curved Fukaya category $\operatorname {Fuk}_{\mathcal L}^\sim (X, {\mathfrak b})$ , we have the associated flat category.

Definition 3.7. Define a flat $A_{\infty }$ category

$$\begin{align*}\operatorname{Fuk}_{\mathcal{L}}^\flat(X,\mathfrak{b}):= \bigsqcup_{w \in \Lambda} \operatorname{Fuk}_{\mathcal{L}}(X,\mathfrak{b})_w \end{align*}$$

whose set of objects is the disjoint union of all objects in the eigen-subcategories, and the space of morphisms between objects in different eigen-subcategories is the zero vector space. More generally, given a subset ${\mathfrak L} \subset MC(\mathcal {L})$ , denote by

$$\begin{align*}\operatorname{Fuk}_{\mathfrak L}^\flat(X, \mathfrak{b}) \end{align*}$$

be the full $A_{\infty }$ -subcategory with the set of objects equal to ${\mathfrak L}$ .