Abstract
We study Dehn twists along Lagrangian submanifolds that are finite free quotients of spheres. We describe the induced autoequivalences to the derived Fukaya category and explain their relations to mirror symmetry.
Mathematics Subject Classification
53D37 53D40 53D121 Introduction
In his early groundbreaking papers [1, 2], Seidel studied the Dehn twist along a Lagrangian sphere and its induced autoequivalence on the derived Fukaya category. There are often no automorphism of the mirror which induces such an autoequivalence [3]. It turns out that this autoequivalence of the mirror, called a spherical twist, can be described purely categorically and there are a lot of generalizations of spherical twists and spherical objects, including \({\mathbb {P}}\)twist, family twist [4], etc.
Many of these generalizations are also motivated by the corresponding symplectomorphisms associated to Lagrangian objects. For example, Lagrangian Dehn twists along spheres can be easily generalized to submanifolds whose geodesics are all closed with the same period. When the Lagrangian submanifold is a complex projective space, Huybrechts and Thomas conjectured that the resulting symplectomorphism induces a \({\mathbb {P}}\)twist in the Fukaya category [5]. However, in most cases, this is still conjectural. Recently, the authors made progress on Huybrechtz–Thomas conjecture by showing that Dehn twists along Lagrangian projective spaces yields a mapping cone operation predicted in the form of \({\mathbb {P}}\)twists on the Fukaya category. In general, it is still very difficult to compute the autoequivalence of a given symplectomorphism.
In this paper, we investigate a new type of Dehn twist and its associated autoequivalences.
Question 1.1
On a Fukaya category, what is the induced autoequivalence of the Dehn twist along a spherical Lagrangian, i.e. a Lagrangian submanifold P whose universal cover is \(S^n\)?
A particularly interesting feature of these twist autoequivalences, which distinguishes this question from all previous twist autoequivalences, is its sensitivity to the characteristic of the ground field.
Consider the basic example of \(P=\mathbb {RP}^n\). In characteristic zero, P is a spherical object in the Fukaya category. In Corollary 1.3 we show that the induced autoequivalence is a composition of two spherical twists. However, when \(char=2\), P becomes a \({\mathbb {P}}^n\)object and the autoequivalence is a \({\mathbb {P}}\)twist as defined in [5]. Indeed, given a spherical Lagrangian that is a more complicated quotient of a sphere, its twist autoequivalence decomposes into a composition of spherical twists in characteristic zero, but when one considers ground field of nonzero characteristics, such twists yield an entire family of previously unknown autoequivalences. We hope this result contributes to the increasing interests in studying derived categories and Fukaya categories of finite characteristics.
To explain our result, let \({\mathbb {K}}\) be a field of any characteristic and \(\Gamma \subset SO(n+1)\) be a finite subgroup for which there exists \({\widetilde{\Gamma }} \subset Spin(n+1)\) such that the covering homomorphism \(Spin(n+1) \rightarrow SO(n+1)\) restricts to an isomorphism \({\widetilde{\Gamma }} \simeq \Gamma \). Let P be a Lagrangian submanifold that is diffeomorphic to \(S^n/\Gamma \) in a Liouville manifold \((M,\omega )\) with \(2c_1(M,\omega )=0\). Pick a Weinstein neighborhood U of P and take the universal cover \({\mathbf {U}}\) of U. The preimage of P is a Lagrangian sphere \({\mathbf {P}}\) in \({\mathbf {U}}\). We can pick a parametrization to identify \({\mathbf {P}}\) with the unit sphere in \({\mathbb {R}}^{n+1}\), and the deck transformation with \(\Gamma \subset SO(n+1)\). Then we can define the Dehn twist \(\tau _{{\mathbf {P}}}\) along \({\mathbf {P}}\) in \({\mathbf {U}}\). Since \(\tau _{{\mathbf {P}}}\) is defined by geodesic flow with respect to the round metric on \({\mathbf {P}}\) and the antipodal map lies in the center of \(SO(n+1)\), \(\tau _{{\mathbf {P}}}\) is \(\Gamma \)equivariant and descends to a symplectomorphism \(\tau _P\) in U. We call \(\tau _P\) the Dehn twist along P.
We equip P with the induced spin structure from \(S^n\) and with the universal local system E corresponding to the canonical representation of \(\Gamma :=\pi _1(P)\) to \({\mathbb {K}}[\Gamma ]\). The pair (P, E) defines an object \(\mathcal {P}\) in the compact Fukaya category \({\mathcal {F}}\). For any Lagrangian brane (i.e. an exact Lagrangian submanifold with a choice of grading, spin structure and local system) \(\mathcal {E}\) in \((M,\omega )\), we have a left \(\Gamma \)module structure on \(hom_{{\mathcal {F}}}(\mathcal {E},\mathcal {P})\) and a right \(\Gamma \)module structure on \(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E})\). Our main result is
Theorem 1.2
For the precise definition of \(\mathcal {P}\) and the equivariant evaluation map \(ev_\Gamma \), readers are referred to Sect. 2.5. Roughly, \(\mathcal {P}\) should be thought of as a homologicalalgebraic incarnation of the immersed Lagrangian represented by the universal cover \(S^n\rightarrow P\). The equivariant evaluation is an adaption of the usual evaluation in this context. Our main theorem has the following consequence when \(P=\mathbb {RP}^n\).
Corollary 1.3
If P is diffeomorphic to \(\mathbb {RP}^{n}\) for \(n=4k1\) and \({{\,\mathrm{char}\,}}({\mathbb {K}}) \ne 2\), then there are two orthogonal spherical objects \(P_1, P_2\in {\mathcal {F}}\) coming from equipping P with different rank one local systems, and \(\tau _P(\mathcal {E})\cong \tau _{P_1}\tau _{P_2}(\mathcal {E})\).
If \(P=\mathbb {RP}^{n}\) for n odd and \({{\,\mathrm{char}\,}}({\mathbb {K}}) =2\), then P is a \({\mathbb {P}}\)object and \(\tau _P(\mathcal {E})\) is quasiisomorphic to applying \({\mathbb {P}}\)twist to \(\mathcal {E}\) along P.
Remark 1.4
1.1 Examples and outlooks

In an upcoming paper [7], the first author and Ruddat construct Lagrangian embeddings of graph manifolds (e.g. spherical space forms) systematically in some Calabi–Yau 3folds using toric degenerations and tropical curves. Previous constructions in smooth toric varieties and open Calabi–Yau manifolds using tropical curves can be found in [8] and [9], respectively.
Lagrangian spherical space forms have been studied in some physics literature (see e.g. [10]) and Dehn twists along them can be realized as the monodromy around a special point in the complex moduli. Our study in this paper can be viewed as the mirrordual of the intensive study of monodromy actions on the derived category of coherent sheaves in the stringy Kähler moduli space ([4, 11, 12, 13, 14], etc).

Hong, Lau and the first author study the local mirror symmetry in all characteristics in a subsequent paper [15] when two lens spaces P, \(P'\) are plumbed together. In this case, the lens spaces can be identified with fat spherical objects in the sense of Toda [16] in certain characteristics. This shows that Dehn twists along lens spaces are mirror to fat spherical twists in this case.
Independently, in the upcoming work [17], Evans, Smith and Wemyss relate Fukaya categories of plumbings of 3spheres along a circle with derived categories of sheaves on local Calabi–Yau 3folds containing two floppable curves. Both Lens space twists and fat spherical twists naturally arise in specific characteristics in that setting.

In principle, Theorem 1.2 can be deduced from the Lagrangian cobordism formalism [18, 19, 20]. There are several additional ingredients that need to be taken into account, though. In the most naive attempt, similar to [21], one needs to use an immersed Lagrangian cobordism that does not have clean selfintersections, which would not even have Gromov compactness on holomorphic disks. A fix could be to generalize the bottleneck immersed cobordism [21] to the categorical level, which should yield the desired mapping cone relation.
Note that this bottleneck immersed formalism is different from the ongoing work of Biran and Cornea on the immersed Lagrangian cobordism, but their framework should also enter the picture. We have not adopted this approach since the relevant tools are still under construction, but such an alternative approach should be of independent interest and yields a functor level statement mentioned below Theorem 1.2.

Another possible approach to Theorem 1.2, explained to us by Ivan Smith, is to realize the Dehn twists as the monodromy in certain symplectic fibrations and apply the Ma’u–Wehrheim–Woodward quilt formalism [22]. This point of view is particularly welladapted to the case of \(P=\mathbb {RP}^n\). In this case, \(\tau _P\) can be realized as the monodromy of a Morse–Bott Lefschetz fibration, and one could try using the techniques developed by Wehrheim and Woodward in [23]. When P is a general spherical space form, the symplectic fibration is no longer Morse–Bott and more technicalities will be involved. Carrying out this approach would be of independent interest, and it provides another possible approach to the functor version of Theorem 1.2.
1.2 Sketch of the proof
The proof of Theorem 1.2 occupies the rest of this paper. Here we give a roadmap of the proof, along with a summary of each section in the paper.
In Sect. 2, we review Lagrangian objects with local systems in the Fukaya categories. When the underlying Lagrangian has finite fundamental group, we introduced its universal local system and regard it as the immersed object coming from the universal cover the the Lagrangian. This gives the object \(\mathcal {P}\) in Theorem 1.2 when the underlying Lagrangian is a finite quotient of \(S^n\). We also define the equivariant evaluation map in (1.1).
Section 3 contains most technical tools we will need from symplectic field theory and gradings, where the main new ingredient is an adaption of [24, 25, 26, 27], which shows the regularity of various holomorphic curves that we will encounter later.
In Sect. 4, we apply symplectic field theory to understand the holomorphic curves contributing to the Floer differentials, and prove a cohomological version of Theorem 1.2, that is, Proposition 4.1. To achieve this, we first give an identification of generators on both sides by geometrically identifying the intersections, then apply neckstretching around \(\mathcal {P}\) to holomorphic curves (triangles and strips) involved in both sides of (5.2). We prove, by studying the resulting configuration, that the limiting curves in the complement of U are identical for the corresponding differentials under our earlier identification of the generators. In other words, we show that the two cochain complexes are indeed isomorphic when the neck is stretched long enough.
In Sect. 5, we prove the categorical version by constructing an appropriate degree zero cocycle between the objects on the two sides of (1.1), which induces the quasiisomorphism in (1.1) (and hence finish the proof of Theorem 1.2). This cocycle \(c_\mathcal {D}\) lives in \(\mathcal {D}\), which is defined in (5.3). Geometrically, we perturb the object \(L_1\) to a nearby copy \(L_1'\) and consider its intersection with the union of \(L_1\) and P, which consist the generators of \(\mathcal {D}\). There is an intersection between \(L_1\) and \(L_1'\) that represents that fundamental cycle \(e_L\), which is intact after the Dehn twist because it is away from the support. We pursue the naive idea that, this intersection (denoted as \(t_\mathcal {D}\) when considered as a cochain in \(\mathcal {D}\)) should be the cocycle we are looking for in \(\mathcal {D}\). Unfortunately, \(t_\mathcal {D}\) is not closed. However, we show that its differential has the form of an upper triangular matrix in Proposition 5.8. To supplement this fact, we computed the differentials from degree zero cochains that that supported at intersections between \(L_1'\cap P\). We then correct \(t_\mathcal {D}\) by considering the multiplications of terms from the term \(CF(\mathcal {P},\mathcal {E}^1)\otimes _{\Gamma } CF(\tau _P((\mathcal {E}^1)'),\mathcal {P})\) and prove that one can find a cocycle \(c_\mathcal {D}\) in the form of Proposition 5.16. A further study in the multiplications involving \(c_\mathcal {D}\) shows it indeed induces a quasiisomorphism (1.1), hence proving Theorem 1.2. Again, the study of relevant \(\mu ^k\)multiplications are based on SFT and neckstretching. The orientation is discussed in the “Appendix”.

\(\Gamma \) is a finite group.

P is a Lagrangian submanifold diffeomorphic to \(S^n/\Gamma \) for some \(\Gamma \subset SO(n+1)\) and P is spin (see Remark 2.9).

\({\mathbf {L}}\) is the universal cover of L and \(\pi :{\mathbf {L}}\rightarrow L \) (or \(\pi :T^*{\mathbf {L}}\rightarrow T^*L\)) is the covering map. In particular, \(\mathbf P\) is the universal cover of P.

\({\mathbf {p}}\in {\mathbf {L}}\) is a lift of \(p \in L\).

\(c_{{\mathbf {p}},{\mathbf {q}}}\) is the geometric intersection \(\pi ( T^*_{{\mathbf {p}}} {\mathbf {P}}\cap \tau _{{\mathbf {P}}}(T^*_{{\mathbf {q}}} {\mathbf {P}})) \in T^*P\) [see (4.5)].

\(\mathcal {P}\) denotes P equipped with the universal local system, and \(\mathcal {E}\) is a Lagrangian equipped with some local system.
2 Floer theory with local systems
In this section, we discuss the Floer theory for Lagrangians with local systems in the spirit of [28]. In Sect. 2.1, we review the definition of the Fukaya category. Universal local systems are introduced in Sects. 2.2 and 2.3, accompanied with some algebraic results surrounding this notion. These results might be known to some very experts but were not found in the literature to the best of the authors’ knowledge. We have intentionally spelled them out in the most explicit way in our capability, with in mind its comparison with immersed Floer theory, from which some readers could find independent interest. These preliminary results enable us to explain the object \(\mathcal {P}\) in Sect. 2.4 and the evaluation map in Sect. 2.5. Discussions about gradings can be found in [29, 2, Section 11,12].
2.1 Fukaya categories with local systems
Remark 2.1
In Sect. 3, we will encounter situations where \(K \equiv 0\) and J is a domain independent almost complex structure. In these cases, J has to be chosen carefully to achieve regularity, so we will emphasize J and denote the moduli by \(\mathcal {M}^J(x_0;x_d,\dots ,x_1)\) therein.
When every element in \(\mathcal {M}(x_0;x_d,\dots ,x_1)\) is transversally cut out, \(\mathcal {M}(x_0;x_d,\dots ,x_1)\) is a smooth manifold of dimension \(x_0\sum _{j=1}^d x_j+(d2)\), where \(\cdot \) denotes the Maslov grading (see Sect. 3.2).
2.2 Unwinding local systems
The goal of this subsection is to give a computable presentation of \(CF(\mathcal {E}^0,\mathcal {E}^1)\), where \(\mathcal {E}^i\) are local systems of the same underlying Lagrangian. In particular, the identification (2.16) and (2.27) will be used frequently later.
Given a local system E on L, we use \({\mathbf {E}}=\pi ^*E\) to denote the pullback local system. For a path \(c{:}[0,1] \rightarrow {\mathbf {L}}\), we use \(I_c\) to denote the parallel transport with respect to the pullback flat connection on \({\mathbf {E}}\).
Lemma 2.2
Proof
We use the Morse model to compute the Floer cochain complex. Let \(C^*(L)\) be a Morse cochain complex and \(C^*({\mathbf {L}})\) be its lift. We use \(\partial _L\) and \(\partial _{{\mathbf {L}}}\) to denote the differential of \(C^*(L)\) and \(C^*({\mathbf {L}})\), respectively.
To compare the differential on both sides of (2.16), let \({\mathbf {u}}\) be a Morse trajectory from \({\mathbf {q}}_0\) to \({\mathbf {q}}_1\) contributing to \(\partial _{\mathbf {L}}\) and hence the differential of \(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\).
We have the following consequence of Lemma 2.2:
Lemma 2.3
Proof
2.3 The universal local system
In this subsection, we introduce the universal local system and hence, in particular, the object \(\mathcal {P}\) in Theorem 1.2. Some elementary properties of the universal local system will also be given. Let us start from a general discussion of universal local systems.
Definition 2.4
(Universal local system) The universal local system E on L is a local system that is uniquely determined by the following conditions: As a vector space, \(E_q={\mathbb {K}}\langle \pi ^{1}(q) \rangle \) for \(q \in L\). For any \(y \in \pi ^{1}(q)\) and \(c:[0,1] \rightarrow L\) such that \(c(0)=q\), the parallel transport of E satisfies \(I_c(y)={\mathbf {c}}(1)\), where \({\mathbf {c}}:[0,1] \rightarrow {\mathbf {L}}\) is the unique path such that \(\pi \circ {\mathbf {c}}=c\) and \({\mathbf {c}}(0)=y\).
As usual, we have the monodromy right \(\Gamma \)action \(\rho \) on \(E_{o_L}\) (2.1). On top of that, we can use the left \(\Gamma \) action on \({\mathbf {L}}\) (2.12) to induce (by extending it linearly) a left \(\Gamma \) action on \(E_q\) for all \(q \in L\). These two actions on \(E_{o_L}\) commute and in general, we have
Lemma 2.5
Proof
Corollary 2.6
Remark 2.7
Now, we want to make connection with Corollary 2.6.
For simplicity, we assume that \(K_1\) and \(L_1\) are Lagrangians without local systems and \(\psi _1 ={\mathbf {q}}_1 \in E_{q_1}\), \(\psi _2={\mathbf {q}}_2^\vee \in Hom_{\mathbb {K}}(E_{q_2},{\mathbb {K}})\). Let \(\gamma \) be \(\partial _{r+1}S\), which is the component of \(\partial S\) with label L.
Since the parallel transport of E can be identified with moving the points in \({\mathbf {L}}\), for \(\mu ^u\) to be nonzero and contribute to the RHS of (2.36), there is exactly one \(g \in \Gamma \) and one lift of \(u_\gamma \), which is denoted by \({\mathbf {u}}:\gamma \rightarrow {\mathbf {L}}\), such that \({\mathbf {u}}\) goes from \(g{\mathbf {q}}_1\) to \({\mathbf {q}}_2\). For each \(h \in \Gamma \), the maps \(h {\mathbf {u}}:\gamma \rightarrow {\mathbf {L}}\) are the other lifts of \(u_\gamma \) and \(h{\mathbf {u}}\) goes from \(hg{\mathbf {q}}_1\) to \(h{\mathbf {q}}_2\).
In particular, we have \(\mu ^1,\mu ^2\) on \((C^*({\mathbf {L}}) \otimes Hom_{\mathbb {K}}(R,R))^{\Gamma }\) inherited from \(CF(\mathcal {E},\mathcal {E})\). In Lemma 2.2, we proved that \(\mu ^1\) coincides with the Morse differential \(\partial _{\mathbf {L}}\) on the first factor. The same line of argument can prove that \(\mu ^2\) coincides with the Floer multiplication on \(C^*({\mathbf {L}})\) tensored with the composition in \(Hom_{\mathbb {K}}(R,R)\) (i.e. \(\mu ^2_{{\mathbf {L}}}(,) \otimes  \circ  \)).
Lemma 2.8
Proof
2.4 Spherical Lagrangians
Remark 2.9
A finite free quotient of a sphere \(S^n/\Gamma \) is spin if and only if there exists \({\widetilde{\Gamma }} \subset Spin(n+1)\) such that the covering homomorphism \(Spin(n+1) \rightarrow SO(n+1)\) restricts to an isomorphism \({\widetilde{\Gamma }} \simeq \Gamma \).
First, we apply the discussion from Sect. 2.2.
Lemma 2.10
Let \(E^i\) be local systems on P for \(i=0,1\). If \({{\,\mathrm{char}\,}}({\mathbb {K}})\) does not divide \(\Gamma \), then \(HF(\mathcal {E}^0,\mathcal {E}^1)=H^*(S^n)\otimes Hom_{{\mathbb {K}}[\Gamma ]}(\mathcal {E}^0_{o_L},\mathcal {E}^1_{o_L})\) as a \({\mathbb {K}}\)vector space.
Proof
Corollary 2.11
Let \(\mathcal {E}^0\) be any local system on P corresponding to an irreducible representation of \(\Gamma \). If \({{\,\mathrm{char}\,}}({\mathbb {K}})\) does not divide \(\Gamma \), then \(HF(\mathcal {E}^0,\mathcal {E}^0)=H^*(S^n)\).
Proof
It follow from Lemma 2.10 and Schur’s lemma \(Hom_{{\mathbb {K}}[\Gamma ]}(\mathcal {E}^0_{o_L},\mathcal {E}^0_{o_L})={\mathbb {K}}\).
Notice that, the ring structure is also determined uniquely by dimension and degree reason. \(\square \)
Now, we want to compute the cohomological endomorphism algebra structure of the universal local system on P using Lemma 2.8. Since the universal local system on P plays a distinguished role in the paper, we denote it by \(\mathcal {P}\). We define \(\mu ^1,\mu ^2\) on \(C^*({\mathbf {P}}) \otimes R\) by (2.45) and (2.46), respectively. By (2.45), we know that \(H^*(C^*({\mathbf {P}}) \otimes R)\) is given by \(H^*({\mathbf {P}}) \otimes R\). We are going to determine the algebra structure in the next lemma. Before that, we recall a convention
Convention 2.12
Lemma 2.13
Let \(\mathcal {P}\) be the universal local system on P and \(R:={\mathbb {K}}[\Gamma ]\). Then the Floer cohomology \(HF(\mathcal {P},\mathcal {P})=H^*(S^n) \otimes _{{\mathbb {K}}} R\) as a \({\mathbb {K}}\)algebra, where the ring structure on the right is the product of the standard ring structure.
Proof
Pick a Morse model such that \(C^*(P)\) has only one degree 0 generator e and one degree n generator f. The corresponding Morse complex \(C^*({\mathbf {P}})\) has \(\Gamma \) degree 0 generator \(\{g{\mathbf {e}}\}_{g \in \Gamma }\) and \(\Gamma \) degree n generator \(\{g{\mathbf {f}}\}_{g \in \Gamma }\). It is clear that \(\sum _g g{\mathbf {e}}\) represents the unit of \(H^0({\mathbf {P}})\). Therefore, \(\{[\sum _g g{\mathbf {e}}] \otimes h\}_{h \in \Gamma }\) are the degree 0 generators of \(H(C^*({\mathbf {P}}) \otimes R)\) [see the correspondence of (2.44) (2.45)].
Lemma 2.14
When \(\mathcal {E}=\mathcal {P}\), the two left \(\Gamma \)actions (2.64) and (2.34) on \(\theta _{1_\Gamma } \in HF(\mathcal {P},\mathcal {P})\) coincide.
Proof
Remark 2.15
From the proof of Lemma 2.14, we see that the identity morphism at \(E_{o_L}\) represents the cohomological unit. It is in general true that if one picks a Morse cochain complex for a Lagrangian submanifold L such that there is a unique degree 0 generator \(e_L\) representing the cohomological unit of \(C^*(L)\), then the identity morphism of \(E_{o_{L}}\) is a cohomological unit of \(CF(\mathcal {E},\mathcal {E})\), where \(\mathcal {E}\) is a local system on L.
Corollary 2.16
The two left \(\Gamma \)actions (2.64) and (2.34) on \(HF^k(\mathcal {E},\mathcal {P})\) coincide, up to \((1)^k\), for all \(\mathcal {E}\in {\mathcal {F}}\).
Proof
2.5 Equivariant evaluation
Corollary 2.18
The proof is straightforward along the same line as [2, Lemma 5.6] and is left to interested readers.
3 Symplectic field theory package
The main goal of this section is to derived the regularity results (Propositions 3.27, 3.29 and 3.30) we need for the later sections. The main ingredient is a trick given in [25], combined with many special features of our setup. For clarity, we recall and specialize some generalities from symplectic field theory to our context, introducing notations that will be used specifically in our proof. This consists the main contents from Sects. 3.1 to 3.5.
The regularity results in this section allow us to establish Proposition 3.32 in Sect. 3.7, which gives us enough control on the bubbling of the moduli of maps we need in Sects. 4 and 5.
For more general backgrounds in symplectic field theory, readers are referred to [24, 25, 27, 32, 33] etc.
3.1 The set up
Let \((Y,\alpha )\) be a contact manifold with a contact form \(\alpha \).
Definition 3.1

J is invariant under \({\mathbb {R}}\) action

\(J(\partial _r)=R_{\alpha }\), where \(R_{\alpha }\) is the Reeb vector field of \(\alpha \)

\(J(\ker (\alpha ))=\ker (\alpha )\)

\(d\alpha (\cdot , J\cdot )_{\ker (\alpha )}\) is a metric on \(\ker (\alpha )\)
Let \(M^\) be the Liouville domain in M bounded by Y and \(M^+=M \backslash (M^ \backslash \partial M^)\). Let \(SM^\) and \(SM^+\) be the positive and negative symplectic completion of \(M^\) and \(M^+\), respectively. Given \((J^{\tau })_{\tau \in [0,\infty )}\), there is a unique almost complex structure \(J^\), \(J^Y\) and \(J^+\) on \(SM^\), SY and \(SM^+\), respectively, such that \((M^,J^\tau _{M^})\), \((N(Y),J^{\tau }_{N(Y)})\) and \((M^+,J^\tau _{M^+})\) converges to \((SM^,J^)\), \((SY,J^Y)\) and \((SM^+,J^+)\), respectively, as \(\tau \) goes to infinity. More details about this splitting procedure can be found in [32, Section 3].
Remark 3.2
When \(R=0\), being Radjusted to N(Y) is the same as being adjusted to N(Y). For \(R > 0\), we can also define \(J^\pm , J^Y\) accordingly.
In this case, \(J^+\) (resp. \(J^\)) are cylindrical over the end \((\infty ,2R] \times \partial M^+ \subset SM^+\) (resp. \([2R,\infty ) \times \partial M^ \subset SM^\)).
Let L be a Lagrangian submanifold in M such that \(L \cap N(Y)=(\epsilon ,\epsilon )\times \Lambda \) for some (possibly empty) Legendrian submanifold \(\Lambda \). Let \(L^\pm :=L \cap M^\pm \). We define \(SL^=L^ \cup ({\mathbb {R}}_{\geqslant 0} \times \Lambda ) \subset SM^\) and \(SL^+=L^+ \cup ({\mathbb {R}}_{\leqslant 0} \times \Lambda ) \subset SM^+\) which are the cylindrical extensions of \(L^\) and \(L^+\) with respect to the symplectic completion. We denote \({\mathbb {R}}\times \Lambda \subset SY\) by \(S\Lambda \).
The main ingredient we needed from [32] is the following compactness result in symplectic field theory.
Theorem 3.3
([32] Theorem 10.3 and Section 11.3; see also [33]) Let \(L_j\), \(j=0,\dots ,d\) be a collection of embedded exact Lagrangian submanifolds in M such that \(L_i \pitchfork L_j\) for all \(i \ne j\). Let \((Y,\alpha )\subset M\) be a contact type hypersurface and \((N(Y),\omega _{N(Y)})\cong ((\epsilon ,\epsilon )\times Y, d(e^r\alpha ))\) be a neighborhood of Y such that \(L_i\cap N(Y)=(\epsilon ,\epsilon )\times \Lambda _i\) for some (possibly empty) Legendrian submanifold \(\Lambda _i\) of Y.
Let \(J^\tau \) be a smooth family of almost complex structures Radjusted to N(Y). Let \(x_0 \in CF(L_0,L_d)\) and \(x_j \in CF(L_{j1},L_j)\) for \(j=1,\dots ,d\). If there exists a sequence \(\{\tau _k\}_{k=1}^\infty \) such that \(\lim _{k \rightarrow \infty } \tau _k =\infty \), and a sequence \(u_k \in \mathcal {M}^{J^{\tau _k}}(x_0;x_d,\dots ,x_1)\), then \(u_k\) converges to a holomorphic building \(u_\infty =\{u_v\}_{v \in V(\mathcal {T})}\) in the sense of [32].
We remark that each \(J^\tau \) above is a domain independent almost complex structure (see Remark 2.1) and we do not need to assume \(u_k\) to be transversally cut out to apply Theorem 3.3.
The rest of this subsection is devoted to the description/definition of \(u_\infty =\{u_v\}_{v \in V(\mathcal {T})}\) in Theorem 3.3. The definition is quite wellknown so we only give a quick review and introduce necessary notations along the way.
First, \(\mathcal {T}\) is a tree with \(d+1\) semiinfinite edges and one of them is distinguished which is called the root. The other semiinfinite edges are ordered from 1 to d and called the leaves. Let \(V(\mathcal {T})\) be the set of vertices of T. For each \(v \in V(\mathcal {T})\), we have a punctured Riemannian surface \(\Sigma _v\). If \(\partial \Sigma _v \ne \varnothing \), there is a distinguished boundary puncture which is denoted by \(\xi ^v_0\). After filling the punctures of \(\Sigma _v\), it is a topological disk so we can label the other boundary punctures of \(\Sigma _v\) by \(\xi ^v_1, \dots \xi ^v_{d_v}\) counterclockwise along the boundary, where \(d_v+1\) is the number of boundary punctures of \(\Sigma _v\). Let \(\partial _{j} \Sigma _v\) be the component of \(\partial \Sigma _v\) that goes from \(\xi ^v_j\) to \(\xi ^v_{j+1}\) for \(j=0,\dots ,d_v1\), and \(\partial _{d_v} \Sigma _v\) be the component of \(\partial \Sigma _v\) that goes from \(\xi ^v_{d_v}\) to \(\xi ^v_{0}\). If \(\partial \Sigma _v = \varnothing \), then \(\Sigma _v\) is a sphere after filling the punctures.
There is a bijection \(f_v\) from the punctures of \(\Sigma _v\) to the edges in \(\mathcal {T}\) adjacent to v. Moreover, \(f_v(\xi ^v_0)\) is the edge closest to the root of \(\mathcal {T}\) among edges adjacent to v. If \(v,v'\) are two distinct vertices adjacent to e, then \(f_v^{1}(e)\) and \(f_{v'}^{1}(e)\) are either both boundary punctures or both interior punctures. We call e a boundary edge (resp. an interior edge) if \(f_v^{1}(e)\) is a boundary (resp. an interior) puncture. We can glue \(\{\Sigma _v\}_{v \in V(\mathcal {T})}\) along the punctures according to the edges and \(\{f_v\}_{v \in V(\mathcal {T})}\) (i.e. \(\Sigma _v\) is glued with \(\Sigma _{v'}\) by identifying \(f_v^{1}(e)\) with \(f_{v'}^{1}(e)\) if \(v,v'\) are two distinct vertices adjacent to e). After gluing, we will get back S, the domain of \(u_k\), topologically. Therefore, there is a unique way to assign Lagrangian labels to \(\partial \Sigma _v\) such that it is compatible with gluing and coincides with that on \(\partial S\) after gluing all \(\Sigma _v\) together. We denote the resulting Lagrangian label on \(\partial _j \Sigma _v\) by \(L_{v,j}\).
There is a level function \(l_\mathcal {T}:V(\mathcal {T}) \rightarrow \{0,\dots , n_\mathcal {T}\}\) for some positive integer \(n_\mathcal {T}\). If \(l_\mathcal {T}(v)=0\), then \({u_v:} \Sigma _v \rightarrow SM^\) is a \(J^\)holomorphic curve such that \(u_v({\partial _j \Sigma _v}) \subset SL_{v,j}^\). If \(l_\mathcal {T}(v)=1,\dots , n_\mathcal {T}1\), then \(u_v : \Sigma _v \rightarrow SY\) is a \(J^Y\)holomorphic curve such that \(u_v({\partial _j \Sigma _v}) \subset S\Lambda _{v,j}\). If \(l_\mathcal {T}(v)=n_\mathcal {T}\), then \(u_v : \Sigma _v \rightarrow SM^+\) is a \(J^+\)holomorphic curve such that \(u_v({\partial _j \Sigma _v}) \subset SL_{v,j}^+\).
If \(v \ne v'\) are adjacent to the same edge e in \(\mathcal {T}\), then \(l_\mathcal {T}(v)l_\mathcal {T}(v') \leqslant 1\). If \(l_\mathcal {T}(v)+1=l_\mathcal {T}(v')\) and e is a boundary (resp. interior) edge, then there is a Reeb chord (resp. orbit) which is the positive asymptote of \(u_v\) at \(f^{1}_v(e)\), and the negative asymptote of \(u_{v'}\) at \(f^{1}_{v'}(e)\) (see Convention 3.6). If \(l_\mathcal {T}(v)=l_\mathcal {T}(v')\), then e is necessarily a boundary edge, \(l_\mathcal {T}(v)=l_\mathcal {T}(v') \in \{0,n_\mathcal {T}\}\) and \(u_v,u_{v'}\) converges to the same Lagrangian intersection point at \(f^{1}_v(e),f^{1}_{v'}(e)\), respectively. If e is the \(j^{th}\) semiinfinite edge adjacent to v, then \(u_v\) is asymptotic to \(x_j\) at \(f^{1}_v(e)\).
Remark 3.4
From this point on, Theorem 3.3 will play a major role in analyzing holomorphic curves.
It is important to note that, the domain of a holomorphic building under our consideration can always be glued up into a smooth disk with boundary, which is the domain for \(J^\tau \) when \(\tau <\infty \).
For our application, we assume every holomorphic disks \({u:}\Sigma \rightarrow M\) which undergoes an SFTstretching process must have pairwisely distinct Lagrangian boundary conditions on different components of \(\partial \Sigma \) when \(\tau <\infty \) throughout the rest of the paper. The reason we impose this condition is because we use a perturbation scheme in defining the Fukaya category, therefore, Lagrangian boundary conditions on two different connected components of \(\partial \Sigma \) are never the same Lagrangian. This will play a key role in our configuration analysis of the buildings.

\(V^{core}\) be the set of vertices \(v \in V(\mathcal {T})\) such that more than one Lagrangian appears in the Lagrangian labels of \(\partial \Sigma _v\).

\(V^{\partial }\) be the set of vertices \(v \in V(\mathcal {T})\) such that there is only one Lagrangian appears in the Lagrangian labels of \(\partial \Sigma _v\).

\(V^{int}\) be the set of vertices \(v \in V(\mathcal {T})\) such that \(\partial \Sigma _v= \varnothing \).
Lemma 3.5
The graphs \(\mathcal {T}^{(1)}:=\mathcal {T}^{core} \backslash \mathcal {T}^{int} \) and \(\mathcal {T}^{(2)}:=(\mathcal {T}^{core} \cup \mathcal {T}^{\partial }) \backslash \mathcal {T}^{int}\) are planar trees. In particular, they are connected.
Proof
Let G be a minimal subtree of \(\mathcal {T}\) containing \(\mathcal {T}^{(1)}\). If there is a vertex v in G such that \(v \in V^{int}\), then it would imply that S, the domain of \(u_k\), is not a disk. If there is a vertice v in G such that \(v \in V^{\partial }\), then it would imply that there is a Lagrangian that appears more than once in the Lagrangian label of \(\partial S\). Both of these situations are not possible.
Similarly, let \(G'\) be the smallest subtree of \(\mathcal {T}\) containing \(\mathcal {T}^{(2)}\). If there is a vertice v in \(G'\) such that \(v \in V^{int}\), then it would imply that S is not a disk and we get a contradiction.
As a result, \(G=\mathcal {T}^{(1)}\) and \(G'=\mathcal {T}^{(2)}\) so both \(\mathcal {T}^{(1)}\) and \(\mathcal {T}^{(2)}\) are trees.
The fact that \(\mathcal {T}^{(1)}\) and \(\mathcal {T}^{(2)}\) are planar follows from the fact that we can order the boundary punctures of \(\Sigma _v\), for \(v \in V^{core} \cup V^{\partial }\), in a way that is compatible with the boundary orientation.\(\square \)
Convention 3.6
We need to explain the convention of striplike ends and cylindrical ends we use for punctures of \(\Sigma _v\). Let e be an edge in \(\mathcal {T}\) and \(v \ne v'\) are the vertices adjacent to e.
If \(l_\mathcal {T}(v)=l_\mathcal {T}(v')\) and, say v is closer to the root of \(\mathcal {T}\) than \(v'\), then we use an outgoing/positive striplike end for \(f_v^{1}(e)\) and an incoming/negative striplike end for \(f_{v'}^{1}(e)\). Similarly, the intersection point that they are asymptotic to is the positive asymptote of \(u_v\) at \(f_v^{1}(e)\) and the negative asymptote of \(u_{v'}\) at \(f_{v'}^{1}(e).\)
3.2 Gradings
 1.
Lagrangian intersection points between \(SL_i^\pm \) and \(SL_j^\pm \) in \(SM^\pm \),
 2.
Reeb chords from \(\Lambda _i\) to \(\Lambda _j\) in Y for \(i \ne j\),
 3.
Reeb chords from \(\Lambda _i\) to itself in Y, and
 4.
Reeb orbits in Y
3.2.1 Type one
Convention 3.7
For a generator \(x \in CF(L_i,L_j)\), we use \(x^\vee \) to denote the generator of \(CF(L_j,L_i)\) which represents the same intersection point as x. Therefore, we have \(x=nx^\vee \).
We refer readers to [29, 2, Section 11, 12] for more about Maslov gradings.
3.2.2 Type two
Now, we go back to our situation and assume x is a Reeb chord from \(\Lambda _i\) to \(\Lambda _j\) in \((\partial U,\alpha _0)\). Since \(L_i\) is graded, \(S\Lambda _i\) has a grading function in \(S(\partial U)\) inherited from \(L_i\). The computation of x is done in the literature (e.g. [34, 35], where they indeed proved \(HW(T_{{\mathbf {q}}_i})\cong k[u]\) for \(u=(n1)\)) and we recall it here.
Without loss of generality, we assume \(\Lambda _i\) and \(\Lambda _j\) are connected. Let \(q_i,q_j \in P\) be such that \(T^*_{q_i}P \cap \partial U=\Lambda _i\) and \(T^*_{q_j}P \cap \partial U=\Lambda _j\). We equip the cotangent fibers and P with the canonical relative grading [see (3.8)]. The grading functions of \(L_i\) and \(L_j\) differs from the grading functions of \(T^*_{q_i}P\) and \(T^*_{q_j}P\) near \(\Lambda _i\) and \(\Lambda _j\), respectively, by an integer. In the following, we will assume the grading functions coincide and the actual x can be recovered by adding back the integral differences of the grading functions.
3.2.3 Type three
 1.
\({\mathbf {q}}_i,{\mathbf {q}}_i'\) satisfy (3.11) (with \({\mathbf {q}}_i'\) replacing \({\mathbf {q}}_j\)), or
 2.
\({\mathbf {q}}_i'\) is the antipodal point of \({\mathbf {q}}_i\), or
 3.
\({\mathbf {q}}_i={\mathbf {q}}_i'\)
We want to point out that in the second and third cases x lies in a Morse–Bott family \(S_{x}\) of Reeb chords from \(\Lambda _i\) to itself and \(\dim (S_{x})=n1\).
3.2.4 Type four
Reeb orbits of \(\partial U\) are graded by the Robbin–Salamon index [36] (see also [37, Section 5]), which is a generalization of the ConleyZehnder index to the degenerated case. To define the Robbin–Salamon index of a Reeb orbit \(\gamma \), we need to pick a symplectic trivialization \(\Phi _\gamma \) of \(\xi \) along \(\gamma \) subject to the following compatibility condition: together with the obvious trivialization of \({\mathbb {R}}\langle \partial _r,R_{\alpha _0} \rangle \), \(\Phi _\gamma \) gives a symplectic trivialization of TM along \(\gamma \), and hence a trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) along \(\gamma \). The compatibility condition is that the induced trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) along \(\gamma \) coincides with the trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) we picked in the beginning of Sect. 2. One may show that there is \(\Phi _\gamma \) satisfying the compatibility condition.
We want to point out that \(\gamma \) lies in a Morse–Bott family \(S_{\gamma }\) of (unparametrized) Reeb orbits and \(\dim (S_{\gamma })=2n2\).
3.3 Dimension formulae
In this section, we first review the virtual dimension formula from [37], where the domain of the pseudoholomorphic map only has interior punctures. Then we consider the case that the domain only has boundary punctures, and finally obtain the general formula by gluing.
Let \((Y^{\pm },\alpha ^{\pm })\) be contact manifolds with contact forms \(\alpha ^\pm \). We assume that every Reeb orbit \(\gamma \) of \(Y^\pm \) lies in a Morse–Bott family \(S_{\gamma }\) of (unparametrized) Reeb orbits. Let \((X,\omega _X)\) be a symplectic manifold such that there exists a compact set \(K_X \subset X\) and \(T_X \in {\mathbb {R}}_{>0}\) so that \((X \backslash K_X, \omega _X_{X \backslash K_X})\) is the disjoint union of the ends \(([T_X,\infty ) \times Y^+, d(e^r\alpha ^+))\) and \(((\infty ,T_X] \times Y^, d(e^r\alpha ^))\). In this case, we have
Lemma 3.8
([37], Corollary 5.4) Let \(\Sigma \) be a punctured Riemannian surface of genus g and \(\partial \Sigma =\varnothing \). Let J be a compatible almost complex structure on X that is cylindrical over the ends. Let \(u:\Sigma \rightarrow X\) be a Jholomorphic map with positive asymptotes \(\{\gamma _j^+ \}_{j=1}^{s_+}\) and negative asymptotes \(\{\gamma _j^ \}_{j=1}^{s_}\ (\)see Convention 3.6).
Sketch of proof
As explained in Sect. 3.2.4, the trivialization \(\Phi _{\gamma ^\pm _j}\) of \(\xi \) along \(\gamma ^\pm _j\) determines a path of symplectic matrices \(\Phi _t^{\pm ,j}\). We can trivialize TX along \(\gamma ^\pm _j\) using \(\Phi _{\gamma ^\pm _j}\) by adding the invariant directions \(\partial _r, R_{\alpha }^\pm \). The corresponding path of symplectic matrices become \({\overline{\Phi }}_t^{\pm ,j}=\Phi _t^{\pm ,j} \oplus I_{2 \times 2}\), where \(I_{2 \times 2}\) is the 2 by 2 identity matrix. By additivity property of \(\mu _{RS}\), we have \(\mu _{RS}({\overline{\Phi }}_t^{\pm ,j})=\mu _{RS}(\Phi _t^{\pm ,j})+\mu _{RS}(I_{2 \times 2})=\mu _{RS}(\Phi _t^{\pm ,j})\).
We note that Lemma 3.8 still holds when \(Y^\pm =\varnothing \), where the corresponding \(s^\pm =0\).
Example 3.9
Lemma 3.10
Sketch of proof
Again, Lemma 3.10 applies also in the case when \(Y^=\varnothing \) or \(Y^+=\varnothing \).
Example 3.11
Now, we combine Lemmas 3.8 and 3.10.
Lemma 3.12
Proof
3.4 Action
This subsection discuss the action of the generators. A similar discussion can be found in [32] and [27, Section 3].
Lemma 3.13
Since \(A(\gamma ) >0\) for any Reeb orbit \(\gamma \) and \(A(x)>0\) if x is a nonconstant Reeb chord such that \(x(0)=x(l_x)\), a direct consequence of Lemma 3.13 is
Corollary 3.14
If \(u_v:\Sigma _v \rightarrow SM^+\) has only negative asymptotes, then at least one of the asymptotes is not a Reeb orbit nor a Reeb chord x such that \(x(0)=x(l_x)\).
Lemma 3.15
Let \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) be a holomorphic building obtained in Theorem 3.3. Let \(x_j\) be the boundary punctures corresponding to the leaves and root edges of \(\mathcal {T}\) If \(\sum _{j=0}^d A(x_j) <T\), then for every \(v \in V(\mathcal {T})\), the action of every asymptote of \(u_v\) lies in \([T,T]\).
Proof
Let us assume the contrary. Then there is an asymptote of \(u_v\) with action lying outside \([T,T]\). We assume that this is a boundary asymptote and denote it by x. The case for interior asymptote is identical. If \(A(x)>T\) (resp. \(A(x)<T\)), we pick \(v' \in V(\mathcal {T})\) (which might be v itself) such that x is a negative (resp. positive) asymptote of \(u_{v'}\). Let e be the edge in \(\mathcal {T}\) corresponds to this asymptote. Let G be the subtree of \(\mathcal {T}\backslash \{e\}\) containing \(v'\).
Lemma 3.16
Let \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) be a holomorphic building obtained in Theorem 3.3. If \(v \in V^{\partial } \cup V^{int}\), then only the action of the asymptote of \(u_v\) that corresponds to the edge \(e_v\) closest to the root of \(\mathcal {T}\) contributes positively to \(E_{\omega }(u_v)\).
Proof
Let \(G_v\) be the subtree of \(\mathcal {T}\backslash \{e_v\}\) containing v. We apply induction on the number of vertices in \(G_v\).
If \(G_v\) has only one vertex, then \(0<E_{\omega }(u_v)\) is only contributed by the asymptote corresponds to \(e_v\) so the base case is done.
Now we consider the general case. Let e be an edge in \(G_v\) (so \(e \ne e_v\)). Let \(v' \ne v\) be the other vertex adjacent to e so \(v' \in V^{\partial } \cup V^{int}\) by Lemma 3.5. By induction on \(G_{v'}\), we know that the asymptote corresponding to e contributes positively to \(E_{\omega }(u_{v'})\) and hence negatively to \(E_{\omega }(u_{v})\). Finally, for \(E_{\omega }(u_{v})\) to be nonnegative, we need to have at least one term which contributes positively to \(E_{\omega }(u_{v})\). This can only be contributed by the asymptote corresponding to \(e_v\). \(\square \)
Lemma 3.17
(Distinguished asymptote) Let \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) be a holomorphic building obtained in Theorem 3.3. If \(\partial \Sigma _v \ne \varnothing \) and \(u_v\) is not a trivial cylinder [see (3.3)], then there is a boundary asymptote x of \(u_v\) that appears only once among all the asymptotes \(\{x_i^\pm \}\) of \(u_v\).
Proof
By Lemma 3.16, when \(v \in V^{\partial }\), the asymptote of \(u_v\) at \(\xi ^v_0\) is the only asymptote that contributes positively to energy and hence appears only once among the asymptotes of \(u_v\).
Now, we consider \(v \in V^{core}\). If there are more than two Lagrangians appearing in the Lagrangian labels of \(\partial \Sigma _v\), say, \(\partial _j S\) and \(\partial _{j+1} S\) are labelled by \(L_{k_1}\) and \(L_{k_2}\), respectively, for \(k_1 \ne k_2\), then the asymptote of \(u_v\) at \(\xi ^v_{j+1}\) can only appear once among the asymptotes of \(u_v\), by Lagrangian boundary condition reason.
If there are exactly two Lagrangians appearing in the Lagrangian labels of \(\partial \Sigma _v\), then there are exactly two j such that the Lagrangian labels on \(\partial _{j} S\) and \(\partial _{j+1} S\) are different. Let the two j be \(j_1\) and \(j_2\). It is clear that \(f_v(\xi _{j_1+1}^v)\) and \(f_v(\xi _{j_2+1}^v)\) are the only two edges in \(\mathcal {T}^{core} \backslash (\mathcal {T}^{\partial } \cup \mathcal {T}^{int})\) that are adjacent to v. Therefore, by our first observation, the action of the asymptotes corresponding the other edges of v contributes negatively to \(E_{\omega }(u_v)\).
If \(u_v\) converges to the same Reeb chord at \(\xi _{j_1+1}^v\) and \(\xi _{j_2+1}^v\), then one of it must be a positive asymptote and the other is a negative asymptote by Lagrangian boundary condition. Therefore, the contribution to \(E_{\omega }(u_v)\) by this same asymptote cancels.
Similarly, if \(u_v\) converges to the same Lagrangian intersection point at \(\xi _{j_1+1}^v\) and \(\xi _{j_2+1}^v\), then the contribution to \(E_{\omega }(u_v)\) by this same asymptote cancels because of the order of the Lagrangian boundary condition. As a result, we have \(E_{\omega }(u_v) \leqslant 0\) which happens only when \(u_v\) is a trivial cylinder [see (3.3)], by Lemma 3.13. \(\square \)
Remark 3.18
3.5 Morsification
We come back to our focus on \(U=T^*P\), where P satisfies (2.58). We will need to use a perturbation of the standard contact form \(\alpha _0\) on \(\partial U\) to achieve transversality later. In this section, we explain how the action and index of the Reeb chord/orbit are changed under such a perturbation.
As explained in Sect. 3.2, \((\partial U,\alpha _0)\) is foliated by Reeb orbits. The quotient of \(\partial U\) by the Reeb orbits is an orbifold, which is denoted by \(Q_{\partial U}\). We can choose a Morse function \(f_Q:Q_{\partial U} \rightarrow {\mathbb {R}}\) compatible with the strata of \(Q_{\partial U}\) and lifts \(f_Q\) to a \(R_{\alpha _0}\)invariant function \(f_\partial :\partial U \rightarrow {\mathbb {R}}\) (see [37, Section 2.2]). Let \({{\,\mathrm{critp}\,}}(f_Q)\) be the set of critical points of \(f_Q\). Let \(\alpha =(1+\delta f_{\partial })\alpha _0\), which is a contact form for \(\delta  \ll 1\). Let \(L(\partial U)\) be the length of a generic simple Reeb orbit of \((\partial U, \alpha _0)\).
Lemma 3.19
([37] Lemma 2.3) For all \(T> L(\partial U)\), there exists \(\delta >0\) such that every simple \(\alpha \)Reeb orbit \(\gamma \) with \(L(\gamma )<T\) is nondegenerate and is a simple \(\alpha _0\)Reeb orbit. Moreover, the set of simple \(\alpha \)Reeb orbits \(\gamma \) with \(L(\gamma )<T\) is in bijection to \({{\,\mathrm{critp}\,}}(f_Q)\).
Proof
The first statement follows from [37, Lemma 2.3].
For the second statement, we need to compare the path of symplectic matrices \(\Phi _t^{\alpha }\), \(\Phi _t^{\alpha _0}\) corresponding to \(\alpha \) and \(\alpha _0\), respectively. We can isotope the Poincare return map \(\Phi _t^{\alpha _0}\) relative to end points, by changing the trivialization, to \(\widetilde{\Phi }_t^{\alpha _0}\) such that \(\ker (\widetilde{\Phi }_t^{\alpha _0}Id) \ne 0\) only happens at finitely many \(t \in [0,1]\), where all such t contribute transversely. For a fixed T, we can choose \(\delta \) sufficiently small such that \(\Phi _t^{\alpha }\) and \(\widetilde{\Phi }_t^{\alpha _0}\) are arbitrarily close but with \(\ker (\widetilde{\Phi }_t^{\alpha }(1)Id) \ne 0\). As a result, only the last contribution to \(\mu _{RS}^{\alpha _0}(\gamma )\) at \(t=1\) may not persist [see (3.15)] and we obtain the result. \(\square \)
Corollary 3.20
For all \(T> L(\partial U)\), there exists \(\delta >0\) such that every \(\alpha \)Reeb orbit \(\gamma \) with \(L(\gamma )<T\) and being contractible in U has \(\mu _{RS}^{\alpha }(\gamma ) \geqslant n1\). As a result, the virtual dimension of \(u:{\mathbb {C}}\rightarrow SM^\) with positive asymptote \(\gamma \) satisfies \({{\,\mathrm{virdim}\,}}(u) \geqslant 2n4\).
Proof
The underlying simple Reeb orbit \(\gamma ^s\) of \(\gamma \) must have \(L(\gamma ^s)<T\) so it is also a \(\alpha _0\)Reeb orbit, by Lemma 3.19. Since \(\gamma \) is contractible in U, by the explanation in Sect. 3.2.4, we have \(\mu _{RS}^{\alpha _0}(\gamma )=2k(n1)\) for some \(k>0\) and \(\dim (S_{\gamma })=2n2\). Therefore, \(\mu _{RS}^{\alpha }(\gamma ) \geqslant n1\) by Lemma 3.19 and \({{\,\mathrm{virdim}\,}}(u)=(n3)+\mu _{RS}^{\alpha }(\gamma ) \geqslant 2n4\). \(\square \)
Lemma 3.21
There exists \(f_Q\) such that for all \(T> L(\partial U)\), there exists \(\delta >0\) such that every \(\alpha \)Reeb chord x from \(\Lambda _{q_1}\) to \(\Lambda _{q_2}\) with \(L(x)<T\) has \(x \leqslant 0\) in the canonical relative grading. Here, we allow \(q_1=q_2\).
Moreover, if \(q_i\) are in relatively generic position on P, for each lift \({\mathbf {q}}_i\) of \(q_i\), there is exactly one such chord \(x_{\mathbf {q_1},\mathbf {q_2}}\) with \(x_{\mathbf {q_1},\mathbf {q_2}}=0\) in canonical relative grading such that \(x_{\mathbf {q_1},\mathbf {q_2}}\) can be lifted to a Reeb chord from \(\Lambda _{{\mathbf {q}}_1}\) to \(\Lambda _{{\mathbf {q}}_2}\).
Proof
For the first statement, when \(\delta >0\) is sufficiently small, x is \(C^1\)close to a \(\alpha _0\)Reeb chord from \(\Lambda _q\) to itself. Recall from Sect. 3.2.3 that, a nondegenerate \(\alpha _0\)Reeb chord \(x_0\) from \(\Lambda _q\) to itself has \(\iota (x_0) \leqslant 0\). Therefore, if x is \(C^1\)close to \(x_0\), then \(\iota (x) \leqslant 0\).
On the other hand, a degenerated \(\alpha _0\)Reeb chord \(x_0\) from \(\Lambda _q\) to itself has \(\iota (x_0)=k(n1) \leqslant (n1)\) for some \(k>0\). We have \(\dim (S_{x_0})=n1\), so if x is \(C^1\)close to \(x_0\), then \(\iota (x) \leqslant \iota (x_0)+\dim (S_{x_0}) \leqslant (n1)+(n1)=0\). The first inequality comes from (3.41).
For the second statement, we only need to notice that \(x_{\mathbf {q_1},\mathbf {q_2}}=0\) if an only if the chord is the lift of (a perturbation of) the unique geodesic between \(\mathbf {q_1}\) and \(\mathbf {q_2}\) with length less than \(\pi \) from (3.12). \(\square \)
Note that, we do not need to assume x is nondegenerate in Lemma 3.21.
After choosing \(\alpha \) in Lemma 3.19, there are only finitely many simple Reeb orbits of length less than T. They correspond to finitely many geodesic loops in P. Therefore, for generic (on the complement of the geodesic loops) \(q \in P\), \(\Lambda _q\) does not intersect with simple Reeb orbits of length less than T. Moreover, for generic perturbation of \(f_Q\), we can achieve the following:
Lemma 3.22
We assume \(n \geqslant 2\). For generic \(C^2\)small perturbation of \(f_Q\) away from \({{\,\mathrm{critp}\,}}(f_Q)\ (\)such that the set \({{\,\mathrm{critp}\,}}(f_Q)\) is unchanged), every \(\alpha \)Reeb chord x from \(\Lambda _q\) to itself with \(L(x) <T\) satisfies \(x(t) \notin \Lambda _q\) for \(t \in (0,L(x))\). Moreover, we can assume every such x is nondegenerate.
Proof
Mike Usher has pointed out the following proof to the authors. Assume a chord x has interior insection \(x(t_i)\), \(i=1,\ldots ,k\) with \(\Lambda _q\), then we may now choose a contactomorphism \(\tau \) with small \(C^2\)norm supported near \(x(t_i)\), which pushes \(x(t_i)\) off \(\Lambda _q\) for all i, and consider the contact form \(\tau _*\alpha \). Since we did not change the contact structure, \(\Lambda _q\) remains Legendrian and the perturbation on the contact form is by a function f supported near \(x(t_i)\). \(\tau (x(t))\) is then a Reeb chord with no interior intersection with \(\Lambda _q\), and from the transversality assumption and argument above, there is no new chords created. The induction on the number of chords concludes the lemma. \(\square \)
Corollary 3.23
 (1)
every simple \(\alpha \)Reeb orbit \(\gamma \) that is contractible in U and \(L(\gamma ) < T\) is nondegenerate and \(\mu _{RS}(\gamma ) \geqslant n1\), and
 (2)
every \(\alpha \)Reeb chord x from \(\bigcup _{i=1}^k \Lambda _{q_i}\) to \(\bigcup _{i=1}^k \Lambda _{q_i}\) with \(L(x)<T\) is nondegenerate, satisfies \(x(t) \notin \bigcup _{i=1}^k \Lambda _{q_i}\) for \(t \in (0,L(x))\) and \(x \leqslant 0\) with respect to canonical relative grading.
Proof
After choosing \(\delta , f_Q\) such that (1) is satisfied by Lemma 3.19 and Corollary 3.20, and we can apply Lemma 3.22 to \(\bigcup _{i=1}^k \Lambda _{q_i}\). Since the perturbation is arbitrarily \(C^2\)small, we have \(x \leqslant 0\) by Lemmas 3.21 and (3.12). \(\square \)
In the rest of the paper, we always choose a contact form \(\alpha \) on \(\partial U\) such that Corollary 3.23 holds, we denote the set of simple \(\alpha \)Reeb orbit \(\gamma \) with \(L(\gamma ) < T\) by \({\mathcal {X}}^o_T\). Similarly, we denote the set of \(\alpha \)Reeb chord x from \(\bigcup _{i=1}^k \Lambda _{q_i}\) to \(\bigcup _{i=1}^k \Lambda _{q_i}\) with \(L(x)<T\) by \({\mathcal {X}}^c_T\).
3.6 Regularity
In this section, we address the regularity of curves \(u_v\) in the holomorphic buildings obtained in Theorem 3.3 for \(v \in V^{core} \cup V^{\partial }\). We adapt the techniques developed in [24, 25, 26] and [27]. We borrow the observation made in [24]: if there is an asymptote that only appears once among the boundary asymptotes of a pseudoholomorphic curve as proved in Lemma 3.17, then one can achieve regularity by perturbing J near the asymptote.
The main difference of our situation is that, we do not work in a contact manifold that is a contactization of an exact symplectic manifold, hence we don’t have a projection of holomorphic curve as in [24, 25]. We remedy the situation by localizing to a neighborhood of the Reeb chord.
We first explain the space of almost complex structure we use. In what follows, we always assume that a contact form \(\alpha \) on \(\partial U\) is chosen such that Corollary 3.23 is satisfied.
Lemma 3.24
Proof
Remark 3.25
If we replace \(dz+\sum _{i=1}^{n1} x_i dy_i\) by \(dz+\sum _{i=1}^{n1} x_i dy_i+dy_1\) in Lemma 3.24, the lemma still holds.
Corollary 3.26
Let \(B_x\) be one chosen in Lemma 3.24 or Remark 3.25. If \(J'\) is a compatible almost complex structure on \(B_x\), then there is a cylindrical almost complex structure J on the symplectization \({\mathbb {R}}\times N_x\) such that \((\pi _{B_x} \circ \pi _Y)_* \circ J(v)=J' \circ (\pi _{B_x} \circ \pi _Y)_*(v)\) for all \(v \in \xi \).
Proof
We can use the symplectic decomposition \(T_{(r,z)}({\mathbb {R}}\times N_x)={\mathbb {R}}\langle \partial _r, R_\alpha \rangle \oplus \xi _z\) and the isomorphism \((\pi _{B_x})_*: \xi _z \simeq T_{\pi _{B_z}(z)}B_x\) to define J such that \(J(\partial _r)=R_\alpha \) and \(J(v)=((\pi _{B_x}\circ \pi _Y)_*)^{1} \circ J' \circ (\pi _{B_x} \circ \pi _Y)_*(v)\) for \(v \in \xi _z\). One can check that J is a cylindrical almost complex structure. \(\square \)
 (1)
Let \(Y \subset (M,\omega ,\theta )\) be a perturbation of \(\partial U\) such that \((Y,\theta _Y) \cong (\partial U, \alpha )\). By abuse of notation, we denote \(\theta _Y\) by \(\alpha \).
 (2)
For the T chosen in Corollary 3.23, there are finitely many Reeb orbits or Reeb chord from \(\bigcup _j \Lambda _{q_j}\) to \(\bigcup _j \Lambda _{q_j}\) with length less than T. Moreover, the simple Reeb orbits \({\mathcal {X}}^{o}_T\) and the Reeb chords \({\mathcal {X}}^{c}_T\) have pairwise disjoint images.
 (3)
For each \(x \in {\mathcal {X}}^{c}_T \), we pick a neighborhood \(N_x\) of Im(x) using Remark 3.25. We assume that all these neighborhoods are pairwise disjoint and disjoint from the Reeb orbits of \(\alpha \).
 (4)Let \(x \in {\mathcal {X}}^{c}_T\), \(x(0) \in \Lambda _{q_0}\) and \(x(L(x)) \in \Lambda _{q_1}\). By Corollary 3.23, for sufficiently small \(N_x\), we can assume thatis a disk for \(i=0,1\). Moreover, by the fact that x is nondegenerate, we know that \(\pi _{B_x}(D_{0,x})\) and \(\pi _{B_x}(D_{1,x})\) are transversally intersecting Lagrangians. There exists a compatible \(J_{B_x}\) on \(B_x\) such that \(J_{B_x}\) is integrable near the origin. By possibly perturbing \(\Lambda _{q,i}\), or equivalently perturbing \(\alpha \), we can assume that \(\pi _{B_x}(D_{i,x})\) are real analytic submanifolds near origin for all x. We fix a choice of \(J_{B_x}\) for each \(x \in {\mathcal {X}}^c_T\).$$\begin{aligned} D_{i,x}:=\Lambda _{q_i} \cap N_x \end{aligned}$$(3.45)
 (5)
Let \({\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\) be the space of \(J \in {\mathcal {J}}^{cyl}(\partial U)\) such that J is \(R_{\alpha }\)invariant in \(N_x\) and there is a compatible almost complex structures \(J'\) on \(B_x\) so that \(J'=J_{B_x}\) near the origin and \((\pi _{B_x} \circ \pi _Y)_* \circ J(v)=J' \circ (\pi _{B_x} \circ \pi _Y)_*(v)\) for all \(v \in \xi \). By Corollary 3.26, we know that \({\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T } )\ne \varnothing \).
 (6)
We define N(Y) as in (3.1). We can pick \(J^0\) such that \((\Phi _{N(Y)})_*J^0_{N(Y)} \in {\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\). Let \(\{J^{\tau }\}_{\tau \in [3R,\infty )}\) be a smooth family Radjusted to \((Y, \alpha )\) as explained in Sect. 3 (see Remark 3.2).
 (7)We define N(Y) as in (3.1). We can pick \(J^0\) Let \(\{L_j\}_{j=0}^d\) be a collection of Lagrangians satisfying the assumptions of Theorem 3.3. Moreover, we assume that \(\Lambda _j=\bigcup _{i=1}^{c_j} \Lambda _{q_{k_{j,i}}}\) for some \(q_{k_{j,i}}\) in Corollary 3.23. If T was chosen sufficiently large, there exists \(0<T^{adj}<T\) (depending only on the primitives of \(\{L_j\}\), see Sect. 3.4) such thatWithout loss of generality, we can assume \(T^{adj}\) exists and \(\sum _{j=0}^dA(x_j)<T^{adj}\). Applying Theorem 3.3 and Lemma 3.15, we get a holomorphic building \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) such that all the asymptotes of \(u_v\) are either Lagrangian intersection points, Reeb chords in \({\mathcal {X}}^{c}_T\) or multiple cover of Reeb orbits in \({\mathcal {X}}^{o}_T\).$$\begin{aligned} \text {for all Reeb chords }x\text { from }\Lambda _i\text { to }\Lambda _j, A(x)<T^{adj}\text { implies }L(x)<T \end{aligned}$$(3.46)
Proposition 3.27
(Regularity for intermediate level components) There is a residual set \({\mathcal {J}}^{cyl, reg} \subset {\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\) such that if (the cylindrical extension of\()\ (\Phi _{N(Y)})_*J^0_{N(Y)}\) lies in \({\mathcal {J}}^{cyl, reg}\), then for \(v \in V^{core} \cup V^{\partial }\) and \(l_\mathcal {T}(v) \in \{1,\dots , n_\mathcal {T}1\}\), the \(J^Y\)holomorphic curve \(u_v\) is transversally cut out.
Proof
By Lemma 3.17, \(u_v\) has a boundary asymptote x that appears only once among its asymptotes. We want to show that transversality can be achieved by considering variation of almost complex structures in \(SN_x:={\mathbb {R}}\times N_x\).
Lemma 3.28
It follows from (3.53) that \(l_2=0\).
As a result, \(l_{\mathcal {R}}=0\) and hence \(l \equiv 0\). The existence of \({\mathcal {J}}^{cyl, reg}\) follows from applying Sard’sSmale theorem to the projection \(\mathcal {F}^{1}(0) \rightarrow U_\Delta \). \(\square \)
Proof of Lemma 3.28
The proof is the same as [25, Lemma 4.5(1)]. For readers’ convenience, we will recall the proof using our notation.
By the definition of \({\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\), \({\overline{u}}\) is a \(J'\)holomorphic curve for some compatible almost complex structure \(J'\) on \(B_x\) such that \(J'=J_{B_x}\) near origin. Moreover, exactly one boundary puncture, denoted by \(\xi _{j_x}\), of \({\mathcal {R}}\) is mapped to the origin by our choice of x.
 (i)
\(({\overline{u}}(E_0),{\overline{u}}(\partial E_0)) \subset (B(0,2\delta ), \pi _{B_x}(D_{0,x} \cup D_{1,x}) \cup \partial B(0,2\delta ))\),
 (ii)
\(\pi _{B_x}(D_{0,x} \cup D_{1,x}) \cap \partial B(0,2\delta )\) are two real analytic disjoint branches,
 (iii)
\({\overline{u}}(\partial E_0)\) contains two regular oriented curves \(\gamma _0\subset D_{0,x}\), \(\gamma _1\subset D_{1,x}\) in \(B(0,2\delta )\), respectively.
To prove \(l_2\) is zero we consider the variation of \(J'\) near a point on \(\gamma _0\). To this end, we need to keep track of other parts of \({\mathcal {R}}\) that map onto \(\gamma _0\).
Let \(p_1 \dots , p_r \in \partial {\mathcal {R}}\) be the preimages under \({\overline{u}}\) of 0 with the property that one of the components of the punctured neighborhood of \(p_j\) in \(\partial {\mathcal {R}}\) maps to \(\gamma _0\). This set is finite and is identified with the set of boundary intersections between u and \({\mathbb {R}}\times x\).
For \(1 \leqslant j \leqslant s\), let \(E_j \subset {\mathcal {R}}\) denote the connected coordinate neighborhood of \({\overline{u}}^{1}(B(0,2\delta ))\) near \(p_j\). Let \(U_1={\overline{u}}(E_0)\) and \(U_2\) be the Schwartz reflection of \(U_1\) through \(\gamma _1\) (see Fig. 2).
To simplify notation, we continue to index this possibly shortened list by \(1 \leqslant j \leqslant s\).
For \(1 \leqslant j \leqslant r\), we double the domain \(E_j\) through its real analytic boundary \(\partial E_j\). We also double the local map \({\overline{u}}_{E_j}\). We continue to denote the open disk by \(E_j\). For \(0 \leqslant j \leqslant s\), let \(u_j={\overline{u}}_{E_j}\). We can also double (for \(1 \leqslant j \leqslant r\)) the cokernel element \(l_2\) (which is antiholomorphic) locally and define (for \(0 \leqslant j \leqslant s\)) \((l_2)_j=l_2_{E_j}\).
There exists a disk \(E \subset {\mathbb {C}}\) and a map \(f_E\) defined on E such that for \(1 \leqslant j \leqslant s\), there exists positive integers \(k_j\) and biholomorphic identifications \(\phi _j\) of E with \(E_j\) such that \((l_2)_j(\phi _j(z))=f_E(z^{k_j})\) for \(z \in E\).
Next, we need to address the regularity when \(u_v\) lies in the top/bottom level of \(u_\infty \). We will explain the case that \(l_\mathcal {T}(v)=n_\mathcal {T}\) (i.e. top level) in details and the other case is similar.
Let \(J_{M^+}\) be a compatible almost complex structure of \(SM^+\) such that it is integrable near \(SL_i^+ \pitchfork SL_j^+\), \(i \ne j\). We assume that \(SL_i^+,SL_j^+\) are real analytic near \(SL_i^+ \pitchfork SL_j^+\).
For \(J^Y \in {\mathcal {J}}^{cyl}(Y,\alpha )\), we let \({\mathcal {J}}^{+}(SM^+)\) to be the set of compatible almost complex structure J such that \(J=J_{M^+}\) near \(\bigcup _{i \ne j} SL_i^+ \cap SL_j^+\) and there exists \(R>0\) so that \(J^+_{(\infty ,R] \times \partial M^+}=J^Y_{(\infty ,R] \times Y}\).
Proposition 3.29
(Regularity for \(M^+\)components) There is a residual set \({\mathcal {J}}^{+, reg} \subset {\mathcal {J}}^{+}(SM^+)\) such that if \(J^+ \in {\mathcal {J}}^{+, reg}\), then for \(v \in V^{core} \cup V^{\partial }\) and \(l_\mathcal {T}(v)=n_\mathcal {T}\), the \(J^+\)holomorphic curve \(u_v\) is transversally cut out.
Proof
By Lemma 3.17, \(u_v\) has a boundary asymptote x that appears only once among its asymptotes. If the distinguished asymptote of \(u_v\) is a Lagrangian intersection point, then we can apply the argument in [25, Lemma 4.5(1)] or Lemma 3.28 again to achieve the regularity of \(u_v\). If the distinguished asymptote of \(u_v\) is a Reeb chord, we denote the corresponding puncture by \(\xi _{j_x}^v\). By the asymptotic behavior of \(u_v\), for a sufficiently large R, the preimage of a small neighborhood of \((\infty ,R] \times Im(x)\) under \(u_v\) is a neighborhood of \(\xi _{j_x}^v\). Therefore, we can find a somewhere injectivity point near \(\xi _{j_x}^v\). Similar to the situation in SY, we can perturb J in \(SM^+\) as long as J is cylindrical outside a compact set. Therefore, we can use the somewhere injectivity point to achieve regularity (see [27, Proposition 4.19] for exactly the same argument). \(\square \)
Similarly, one define \({\mathcal {J}}^{}(SM^)\) analogously and we have
Proposition 3.30
There is a residue set \({\mathcal {J}}^{, reg} \subset {\mathcal {J}}^{}(SM^)\) such that if \(J^ \in {\mathcal {J}}^{, reg}\), then for \(v \in V^{core} \cup V^{\partial }\) and \(l_\mathcal {T}(v)=0\), the \(J^\)holomorphic curve \(u_v\) is transversally cut out.
3.7 No side bubbling
We can now summarize the previous discussion on \(u_{\infty }\) and draw geometric conclusions in this section.
Let \(L_j\), \(j=0,\dots ,d\) be a collection of embedded exact Lagrangian submanifolds in \((M,\omega ,\theta )\) such that \(L_i \pitchfork L_j\) for all \(i \ne j\). Let P be a Lagrangian such that (2.58) is satisfied (P can be one of the \(L_j\)). Let U be a Weinstein neighborhood of P and we assume that \(\theta _U\) coincides with the canonical Liouville one form on \(T^*P\). For \(T \gg 1\), we pick \(\alpha \) satisfying Corollary 3.23 and \(T^{adj}\) satisfying (3.46).
Let Y be a perturbation of \(\partial U\) such that \((Y,\theta _Y) \cong (\partial U,\alpha )\). We denote \(\theta _Y\) by \(\alpha \). We have a neighborhood \(\Phi _{N(Y)}:(N(Y),\omega _{N(Y)})\cong ((\epsilon ,\epsilon )\times Y, d(e^r\alpha ))\) of Y. We assume that \(L_j\cap N(Y)=(\epsilon ,\epsilon )\times \Lambda _{j}\) where \(\Lambda _j=\bigsqcup \Lambda _{q_{j_m}}=T^*_{q_{j_m}}P \cap Y\) for some \(q_{j_m} \in P\) in Corollary 3.23.
Let \(J^\tau \) be a smooth family of almost complex structures Radjusted to N(Y), such that \(J^Y \in {\mathcal {J}}^{cyl,reg}\), where \({\mathcal {J}}^{cyl,reg}\) is obtained in Proposition 3.27. We also assume that \(J^\pm \in {\mathcal {J}}^{\pm ,reg}\), where \({\mathcal {J}}^{\pm ,reg}\) is obtained in Propositions 3.29, 3.30.
Proposition 3.32
(No side bubbling) If \(n\geqslant 3\), then \(V^{int}=\varnothing \) and \(n_\mathcal {T}=1\). Moreover, if \(v \in V^{\partial }\), then \(u_v\) is a rigid \(J^+\)holomorphic map with exactly one boundary asymptote which is negative and goes to a Reeb chord.
Proof
Lemma 3.33
For each connected component G of \(\mathcal {T}^{int}\), we have \({{\,\mathrm{virdim}\,}}(G) >0\).
Proof
Let \(v \in G\) be the vertex closest to the root. By 3.16, we have a distinguished interior puncture \(\eta ^0 \in \Sigma _v\) which contributes positively to \(E_\alpha (u_v)\). Let \(\gamma ^0\) be the Reeb orbit that \(u_v\) is asymptotic to at \(\eta ^0\). Since \(A(\gamma ^0)=L(\gamma ^0)>0\), \(\gamma ^0\) must be a positive asymptote of \(u_v\).
Notice that, by Corollary 3.14, there is no \(v \in G\) such that \(u_v\) maps to \(SM^+\). Therefore, \(\#_{v \in G} u_v\) is a topological disk in \(SM^\) so \(\gamma ^0\) is contractible in U. Moreover, \({{\,\mathrm{virdim}\,}}(G)\) is determined by \(\gamma ^0\) and it is given by \(2n4>0\) (see Corollary 3.20). \(\square \)
By combining (3.59), \({{\,\mathrm{virdim}\,}}(\mathcal {T})=0\), \({{\,\mathrm{virdim}\,}}(u_v) \geqslant 0\) for \(v \in V^{core} \cup V^{\partial }\) and Lemma 3.33, we conclude that \(V^{int}= \varnothing \), \(k_G=0\) and \({{\,\mathrm{virdim}\,}}(u_v)=0\) for all v.
Notice that if \(u_v\) is not a trivial cylinder but \(l_\mathcal {T}(v) \notin \{0, n_\mathcal {T}\}\), then \({{\,\mathrm{virdim}\,}}(u_v) \geqslant 1\) because one can translate \(u_v\) along the rdirection. Therefore, all intermediate level curves are trivial cylinders so \(n_\mathcal {T}=1\). The last thing to show is that if \(v \in V^{\partial }\), then \(l_\mathcal {T}(v)=1\) and \(u_v\) has only one boundary asymptote.
We argue by contradiction. Suppose \(l_\mathcal {T}(v)=0\). Due to the boundary condition, all asymptotes of \(u_v\) are Reeb chords \(y_0,\dots ,y_{d_v}\). Inside \(SM^\), we can compute the index of Reeb chords using the canonical relative grading. By Corollary 3.23, we have \(\iota (y_j) \leqslant 0\) for all j. It means that \({{\,\mathrm{virdim}\,}}(u_v)=n\sum _{j=0}^{d_v} \iota (y_j)(2d_v) \geqslant n2>0\). This is a contradiction so \(l_\mathcal {T}(v)=1\) for all \(v \in V^{\partial }\).
Finally, if there exists \(v \in V^{\partial }\) such that \(u_v\) has more than one boundary asymptote, then by the fact that \(\mathcal {T}\) is a tree, we must have \(v \in V^{\partial }\) such that \(l_\mathcal {T}(v)=0\). This is a contradiction so we finish the proof of Proposition 3.32. \(\square \)
3.8 Gluings in SFT
To conclude our discussion on generalities of neckstretching, we recall the following gluing theorem for SFT, which will play an important role in our proof.
Theorem 3.34
Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})} \in \mathcal {M}^{J^{\infty }}(x_0;x_d,\dots ,x_1)\) be a holomorphic building such that \(u_v\) is transversally cut out for all v and \({{\,\mathrm{virdim}\,}}(u_{\infty })=0\). Assume also that all asymptotic Reeb chords are nondegenerate.
Then for any small neighborhood \(N_{u_{\infty }}\) of \(u_{\infty }\) in an appropriate topology, there exists \(\Upsilon >0\) sufficiently large such that for each \(\tau >\Upsilon \), there is a unique \(u^{\tau } \in \mathcal {M}^{J^{\tau }}(x_0;x_d,\dots ,x_1)\) lying inside \(N_{u_{\infty }}\). Moreover, \(u^\tau \) is regular and \(\{u^\tau \}_{\tau \in [\Upsilon ,\infty )}\) converges in SFT sense to \(u_{\infty }\) as \(\tau \) goes to infinity.
A nice account for the SFT gluing results can be found in “Appendix A” of [44]. In the presence of conical Lagrangian boundary conditions as in above, see also [25, Proposition 4.6] and [24, Section 8]. Theorem 3.34 is essentially the same as Proposition 4.6 in [25], except our contact manifold is not \(P\times {\mathbb {R}}\). But this is not a concern for the gluing argument because the argument involves local analysis on a neighborhood of the holomorphic building, which is not affected by the global topology.
The typical application of Proposition 3.32 and Theorem 3.34 goes as follows. Given a collection of Lagrangians such that the assumption of Theorem 3.3 is satisfied, we want to determine the signed count of rigid elements in \(\mathcal {M}^{J^{\tau }}(x_0;x_d,\dots ,x_1)\) for some large \(\tau \). When \(d=1\) (resp. \(d=2\)), the signed count is responsible to the Floer differential (resp. Floer multiplication). If we pick \(u_k \in \mathcal {M}^{J^{\tau _k}}(x_0;x_d,\dots ,x_1)\) such that \(\lim _{k \rightarrow \infty } \tau _k=\infty \), we get a holomorphic building \(u_\infty \) by Theorem 3.3. By Proposition 3.32, \(u_\infty \) satisfies the assumption of Theorem 3.34. Therefore, for sufficently large \(\tau \), \(\mathcal {M}^{J^{\tau }}(x_0;x_d,\dots ,x_1)\) is in bijection to \(\mathcal {M}^{J^{\infty }}(x_0;x_d,\dots ,x_1)\). Moreover, all elements in \(\mathcal {M}^{J^{\tau }}(x_0;x_d,\dots ,x_1)\) are transversally cut out. It means that the Floer differential (resp. Floer multiplication) can be computed by determining \(\mathcal {M}^{J^{\infty }}(x_0;x_d,\dots ,x_1)\), which is exactly what we will do in the following section.
4 Cohomological identification
Let P be a Lagrangian such that (2.58) is satisfied and \(\mathcal {P}\) be the universal local system on P. We pick a parametrization of P so that \(\tau _P\) can be defined. In this section, we want to prove that
Proposition 4.1
Our strategy is to study directly the Floer cochain complexes from both sides of (4.1). Section 4.1 gives a geometric correspondence between the generators from the two sides, and Sect. 4.3 will study the SFT limits of involved holomorphic strips and triangles. Section 4.4 use a local model to compute several key contribution of moduli spaces in the SFT limits, which eventually leads to the matching of differentials of (4.1) in Sect. 4.5. Due to the heaviness of notation and length of our proof, we also included a more technical guide in Sect. 4.2, in hope of keeping the readers on board.
4.1 Correspondence of intersections

\({\mathcal {X}}_a(C_0)\): generators in \(hom(\mathcal {P}, L_1)\otimes _\Gamma hom(L_0,\mathcal {P})[1]\)

\({\mathcal {X}}_b(C_0)\): generators in \(hom(L_0,L_1)\)
Lemma 4.2
There is a gradingpreserving bijection \(\iota : {\mathcal {X}}(C_0) \rightarrow {\mathcal {X}}(C_1)\).
Proof
First, there is an obvious graded identification between \({\mathcal {X}}_b(C_0)\) and the intersections of \(L_0\cap \tau _P L_1\) outside U, so we only need to explain how to define \(\iota _{{\mathcal {X}}_a(C_0)}\).
To see that \(\iota _{{\mathcal {X}}_a(C_0)}\) preserves the grading, we only need to observe that \(\pi \) interwines the canonical trivialization of \((\Lambda ^{\otimes top}_{{\mathbb {C}}}(T^*{\mathbf {U}}))^{\otimes 2}\) and \((\Lambda ^{\otimes top}_{{\mathbb {C}}}(T^*U))^{\otimes 2}\) so the computation reduces to the case that \(P=S^n\), which is wellknown (see e.g. [1]). \(\square \)
4.2 Overall strategy

Type (A1): differentials in \(hom(L_0,\mathcal {P})\), i.e. pseudoholomorphic strips in \(\mathcal {M}( \mathbf {p'} ; {\mathbf {p}})\)

Type (A2): differentials in \(hom(\mathcal {P}, L_1)\), i.e. pseudoholomorphic strips in \(\mathcal {M}(({\mathbf {q}}')^\vee ; {\mathbf {q}}^\vee )\)

Type (B): differentials in \(hom(L_0, L_1)\), i.e. pseudoholomorphic strips in \(\mathcal {M}(x_0;x_1)\)

Type (C): differentials from the evaluation map, i.e. pseudoholomorphic triangles in \(\mathcal {M}(x; {\mathbf {q}}^\vee , {\mathbf {p}})\)

Type (A1\('\)): pseudoholomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}})\);

Type (A2\('\)): pseudoholomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}},{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}})\);

Type (A3\('\)): pseudoholomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}}',{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}})\) that are not in Type(A1\('\)) and (A2\('\));

Type (B\('\)): pseudoholomorphic strips in \(\mathcal {M}(x_0;x_1)\);

Type (C\('\)): pseudoholomorphic strips in \(\mathcal {M}(x; c_{{\mathbf {p}},{\mathbf {q}}})\);

Type (D\('\)): pseudoholomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}},{\mathbf {q}}};x)\);
In the following subsections, we ignore the sign and only consider the case that \({{\,\mathrm{char}\,}}({\mathbb {K}})=2\). The complete proof of Proposition 4.1, where orientation of moduli is taken into account, will be given in “Appendix A”.
4.3 Neckstretching limits of holomorphic strips and triangles
Lemma 4.3
In the case (iv) of (4.12), let \(v \in V^{\partial }\) and x be the negative asymptote of \(u_v\). Then \(x=1\).
Proof
Lemma 4.4
If \(l_\mathcal {T}(v)=0\), then \(u_v\) has at least one asymptote that is not a Reeb chord.
Proof
Lemma 4.5
Every generator \(c_{{\mathbf {p}},{\mathbf {q}}} \in CF(T^*_p P, \tau _P(T^*_qP))\) satisfies \(c_{{\mathbf {p}},{\mathbf {q}}}=n1\) with respect to the canonical relative grading. Moreover, if \(c_{{\mathbf {p}},{\mathbf {q}}}\) is the only asymptote of a nonconstant \(J^\)holomorphic map \(u_v: \Sigma _v \rightarrow SM^=T^*P\) that is not a Reeb chord, then \(c_{{\mathbf {p}},{\mathbf {q}}}\) must be positive as an asymptote of \(u_v\).
Proof
Now, we can describe the SFT limits of various moduli.
Lemma 4.6

\(u_{v_1}\) is a \(J^\)holomorphic triangle with negative asymptote \(p':=\pi ({\mathbf {p}}')\) and positive asymptotes x, p where x is a Reeb chord with \(x=0\) in the canonical relative grading;

\(v_2 \in V^{\partial }\) so, by Lemma 4.3, \(u_{v_2}\) is a \(J^+\)holomorphic curve with one negative asymptote x such that \(x=1\) in the actual grading.
Proof
Finally, to compute x in the canonical relative grading, we just need to make a grading shift so that \(p'p=1\) on \(T_{p'}^*P\). It gives \(x=0\) in the canonical relative grading. \(\square \)
Similarly, we have.
Lemma 4.7

\(u_{v_1}\) is a \(J^\)holomorphic triangle with negative asymptote \(c_{\mathbf {p'},{\mathbf {q}}}\) and positive asymptotes \(x,c_{{\mathbf {p}},{\mathbf {q}}}\) where x is a Reeb chord with \(x=0\) in the canonical relative grading;

\(v_2 \in V^{\partial }\) so \(u_{v_2}\) is a \(J^+\)holomorphic curve with one negative asymptote x such that \(x=1\) in the actual grading.
We omit the corresponding statements for type (A2) and (A2\('\)) because of the similarity. Next we consider
Lemma 4.8
(Type (B), (B\('\))) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a nonempty SFT limit of curves in \(\mathcal {M}^{J^\tau }(x_0;x_1)\). Then \(\mathcal {T}\) consists of exactly one vertex v and \(l_\mathcal {T}(v)=1\).
Proof
If \(\mathcal {T}\) has a vertex v such that \(l_\mathcal {T}(v)=0\), then all the asymptotes of v are Reeb chords which contradicts to Lemma 4.4. Therefore, \(l_\mathcal {T}(v)=1\) for all \(v \in V(\mathcal {T})\) and it holds only when \(\mathcal {T}\) consists of exactly one vertex. \(\square \)
Lemma 4.9

\(u_{v_1}\) is a \(J^\)holomorphic triangle with positive asymptotes \(y,{\mathbf {q}}^\vee ,{\mathbf {p}}\), where y is a Reeb chord with \(y=0\) in the canonical relative grading;

\(u_{v_2}\) is a \(J^+\)holomorphic curve with two negative asymptotes x and y.
Proof
Again, we use the same argument as in the proof of Lemma 4.6. There is \(v_1 \in \mathcal {T}\) such that \(u_{v_1}\) is a holomorphic polygon and \({\mathbf {q}}^\vee \), \({\mathbf {p}}\) are asymptotes of \(u_{v_1}\). All other vertices are adjacent to \(v_1\): otherwise, there will be components in \(T^*\mathbf{P }\) with only Reeb asymptotes, contradicting Lemma 4.4. Denote these vertices by \(v_2, \dots , v_k\). There is exactly one \(j>1\) (say \(j=2\)) such that \(v_j \notin V^{\partial }\) and x is an asymptote of \(u_{v_j}\). For \(\mathcal {T}\) to be a tree, \(u_{v_2}\) has exactly one negative Reeb chord asymptote, which is denoted by \(y_2\).
Let the negative asymptote for \(u_{v_j}\) (for \(j>2\)) be \(y_j\).
Remark 4.10
Later on, we will also make use of the moduli space \(\mathcal {M}^{J^\tau }({\mathbf {p}}^\vee ;x^\vee ,{\mathbf {q}}^\vee )\). The shape of neckstretching limit will remain the same as Type (C), because this is simply a modification of some of the striplike ends (from outgoing to incoming, and vice versa) and does not change the behavior of the underlying curve.
Lemma 4.11

\(u_{v_1}\) is a \(J^\)holomorphic bigon with positive asymptotes \(y,c_{{\mathbf {p}},{\mathbf {q}}}\), where y is a Reeb chord with \(y=0\) in the canonical relative grading;

\(u_{v_2}\) is a \(J^+\)holomorphic curve with two negative asymptotes x and y.
Proof
Our final task is to show that type (A3\('\)) and (D\('\)) are empty for \(\tau \gg 1\).
Lemma 4.12
(Type (A3\('\))) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a SFT limit of curves in \(\mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}})\) that are not in Type(A1’) and (A2’). Then \(u_{\infty }\) is empty.
Proof
There is \(v \in V(\mathcal {T})\) such that \(c_{{\mathbf {p}}',{\mathbf {q}}'}\) is a negative asymptote of \(u_v\). By boundary condition, \(c_{{\mathbf {p}},{\mathbf {q}}}\) cannot be an asymptote of \(u_v\). The existence of \(u_v\) violates Lemma 4.5. \(\square \)
By Lemma 4.5 again, we have.
Lemma 4.13
(Type (D\('\))) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a SFT limit of curves in \(\mathcal {M}^{J^\tau }(c_{{\mathbf {p}},{\mathbf {q}}};x)\). Then \(u_{\infty }\) is empty.
4.4 Local contribution
In this section, we will determine the algebraic count of some moduli of rigid \(J^\)holomorphic curves in \(SM^=T^*P\), using a cohomological counting argument.
Let \(q_1,q_2,q_3 \in P\) be three generic points such that \(\bigcup _i \Lambda _{q_i}\) satisfies Corollary 3.23. Let \({\mathbf {q}}_i \in {\mathbf {P}}\) be a lift of \(q_i\) for \(i=1,2,3\). Let \({\mathbf {J}}^\) be the almost complex structure on \(T^*{\mathbf {P}}\) that is lifted from \(J^\). Since the contact form \(\theta _{\partial {\mathbf {U}}}\) equals to the lift of \(\alpha =\theta _{\partial U}\), by Lemma 3.21, there is a unique Reeb chord \(x_{i,j}\) from \(\Lambda _{{\mathbf {q}}_i}\) to \(\Lambda _{{\mathbf {q}}_j}\) such that \(x_{i,j}=0\) in the canonical relative grading. Let \({\mathbf {q}}_i \in CF(T^*_{{\mathbf {q}}_i} {\mathbf {P}}, {\mathbf {P}})\) and \({\mathbf {c}}_{i,j} \in CF(T^*_{{\mathbf {q}}_i} {\mathbf {P}}, \tau _{{\mathbf {P}}}(T^*_{{\mathbf {q}}_j} {\mathbf {P}}))\) be the chains represented by the unique geometric intersection in the respective chain complexes.
 (1)
\(\mathcal {M}^{{\mathbf {J}}^}({\mathbf {q}}_1;{\mathbf {q}}_2,x_{1,2})\), \(\mathcal {M}^{{\mathbf {J}}^}(q_{2}^\vee ;x_{1,2},{\mathbf {q}}_1^\vee )\) and \(\mathcal {M}^{{\mathbf {J}}^}(\varnothing ;{\mathbf {q}}_2, x_{1,2}, {\mathbf {q}}_1^\vee )\),
 (2)
\(\mathcal {M}^{{\mathbf {J}}^}({\mathbf {c}}_{3,2};x_{1,2},{\mathbf {c}}_{3,1})\),
 (3)
\(\mathcal {M}^{{\mathbf {J}}^}({\mathbf {c}}_{1,3};{\mathbf {c}}_{2,3},x_{1,2})\),
 (4)
\(\mathcal {M}^{{\mathbf {J}}^}(\varnothing ;{\mathbf {c}}_{2,1},x_{1,2})\) (Fig. 7).
Theorem 4.14
The algebraic count of the above moduli spaces are all \(\pm 1\).
Proof of Theorem 4.14
We will apply SFT stretching on the the following “big local model”.
Consider an \(A_3\) Milnor fiber consisting of the plumbing of three copies of \(T^*S^n\). We denote the Lagrangian spheres by \(S_1\), \({\mathbf {P}}\) and \(S_3\), respectively, where \(S_1 \cap S_3= \varnothing \). We can identify a neighborhood of \({\mathbf {P}}\) with \({\mathbf {U}}\). By Hamiltonian isotopy if necessary, we assume that \({\mathbf {U}}\cap S_j\) is a pair of disjoint cotangent fibers for \(j=1,3\). We perturb \(S_1\) to \(S_2\) by a perfect Morse function, so that \({\mathbf {U}}\cap S_2\) is another cotangent fiber.
Finally, notice that even though \(S_2\) is obtained by a perturbation of \(S_1\), we can actually Hamiltonian isotope \(S_2\) so that \(S_2 \cap P\) is the preassigned \(q_2\) and there is no new intersection between \(S_2\) and \(S_1,S_3\) being created during the isotopy. With this choice of \(S_2\) and the stretching argument explained above, Theorem 4.14 follows. \(\square \)
One may define the analogous moduli spaces similarly on \(T^*P\) for cotangent fibers \(T^*_{q_i} P\). By equivariance, every rigid \(J^\)holomorphic curve lifts to \(\Gamma \) many rigid \({\mathbf {J}}^\)holomorphic curves and every rigid \({\mathbf {J}}^\)holomorphic curve descends to a rigid \(J^\)holomorphic curve.
With this understood, we have.
Corollary 4.15
The algebraic count of the following moduli spaces are \(\pm 1\).
 (1)
\(\mathcal {M}^{J^}(p';p,x_{{\mathbf {p}}',{\mathbf {p}}})\), \(\mathcal {M}^{J^}(q'^\vee ;x_{{\mathbf {q}},{\mathbf {q}}'},q^\vee )\) and \(\mathcal {M}^{J^}(\varnothing ;p,x_{{\mathbf {q}},{\mathbf {p}}},q^\vee )\),
 (2)
\(\mathcal {M}^{J^}(c_{{\mathbf {p}},{\mathbf {q}}'};x_{{\mathbf {q}},{\mathbf {q}}'},c_{{\mathbf {p}},{\mathbf {q}}})\)
 (3)
\(\mathcal {M}^{J^}(c_{{\mathbf {p}}',{\mathbf {q}}};c_{{\mathbf {p}},{\mathbf {q}}},x_{{\mathbf {p}}',{\mathbf {p}}})\)
 (4)
\(\mathcal {M}^{J^}(\varnothing ;c_{{\mathbf {p}},{\mathbf {q}}},x_{{\mathbf {q}},{\mathbf {p}}})\)
4.5 Matching differentials
We now are ready to prove Proposition 4.1. The first lemma relates algebraic counts of differentials of Type (A1) and (A1\('\)).
Lemma 4.16

\(\mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}})\), differentials in \(hom(L_0,\tau _P (L_1))\) from \(c_{{\mathbf {p}},{\mathbf {q}}}\) to \(c_{{\mathbf {p}}',{\mathbf {q}}}\),

\(\mathcal {M}^{J^\tau }({\mathbf {p}}';{\mathbf {p}})\), differentials in \(hom(L_0, \mathcal {P})\) from \({\mathbf {p}}\) to \({\mathbf {p}}'\)
Proof
To prove the lemma, we look at the SFT limit of these moduli when \(\tau \) goes to infinity. Let \(u_\infty ^1\) and \(u_{\infty }^2\) be a limiting holomorphic building from curves in \(\mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}})\) and \(\mathcal {M}^{J^\tau }({\mathbf {p}}';{\mathbf {p}})\), respectively. Lemmas 4.6 and 4.7, \(u_{\infty }^i\) consist of a \(J^\)holomorphic curve \(u_{v_1}^i\) and a \(J^+\)holomorphic curve \(u_{v_2}^i\). Moreover, \(u_{v_2}^i\) lies in \(\mathcal {M}^{J^+}(x_{{\mathbf {p}},{\mathbf {p}}'};\varnothing )\) for both i. On the other hand, \(u_{v_1}^1\) lies in \(\mathcal {M}^{J^}(c_{{\mathbf {p}}',{\mathbf {q}}};c_{{\mathbf {p}},{\mathbf {q}}}, x_{{\mathbf {p}}',{\mathbf {p}}})\) and \(u_{v_1}^2\) lies in \(\mathcal {M}^{J^}(\mathbf{p }';\mathbf{p },x_{{\mathbf {p}}',{\mathbf {p}}})\).
Similarly, we compare the differentials of Type (A2) and (A2\('\)).
Lemma 4.17

\(\mathcal {M}^{J^{\tau }}(c_{{\mathbf {p}},{\mathbf {q}}'};c_{{\mathbf {p}},{\mathbf {q}}})\),

\(\mathcal {M}^{J^\tau }({\mathbf {q}}'^\vee ;{\mathbf {q}}^\vee )\).
Proof
The last lemma addresses differentials of Type (C) and (C\('\)).
Lemma 4.18

\(\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee ,{\mathbf {p}})\), for some \(x\in CF^*(L_0,L_1)\) represented by an intersection outside U,

\(\mathcal {M}^{J^\tau }(x; c_{{\mathbf {p}},{\mathbf {q}}})\).
Proof
As the end product of this section, we have.
Proof of Proposition 4.1
For \(\tau \gg 1\), the differential on \(C_0\) and \(C_1\) can be identified by Lemmas 4.16, 4.17, 4.8, 4.18 and 4.5. \(\square \)
The proof of Proposition 4.1 when \({{\,\mathrm{char}\,}}({\mathbb {K}}) \ne 2\) is given in “Appendix A”.
5 Categorical level identification
In this section, we want to prove Theorem 1.2 by showing the following:
Theorem 5.1
In particular, \(\tau _P(\mathcal {E}^1) \simeq \tau _P((\mathcal {E}^1)') \simeq T_\mathcal {P}(\mathcal {E}^1)\) as perfect \(A_{\infty }\) right \({\mathcal {F}}\)modules.
The discussion in this section works for fields \({\mathbb {K}}\) of arbitrary characteristics, even though we didn’t pay exclusive attention to signs.
Again, let us give a sketch of this section in hope of rescuing discouraged readers from the daunting details and notations. As pointed out in the introduction, we will pursue the generator that comes from L and the Dehn twist of a perturbation of L, which represents the fundamental class of CF(L, L) before the Dehn twist. This is not a cocycle in \(\mathcal {D}\), and we computed its differential in 5.1.1. To offset them, we use the tensor product component in \(\mathcal {D}\), whose differential, as a product in the Fukaya category, is computed in 5.1.2, which eventually yields the desired cocycle \(c_\mathcal {D}\). After studying more of the \(A_\infty \)structure, we verify \(c_\mathcal {D}\) gives the desired quasiisomorphism (1.1).
The reader should note that we postpone all issues of orientations to the “Appendix”, but as it turns out, the content in this section depends on analysis of signs minimally.
5.1 Hunting for degree zero cocycles
To find a degree zero cocycle, we need to first analyze the differential of \(hom_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))\) by neckstretching. The discussion in this section works for field \({\mathbb {K}}\) of arbitrary characteristics.
Lemma 5.2
Proof
5.1.1 Computing \(\mu ^1_{\mathcal {F}}(\psi _{e_L})\)
Let \({h:}L_1 \rightarrow {\mathbb {R}}\) be a smooth function such that \(dh=\theta _{L_1}\). We define \(h_i:=h_{\Lambda _{q_i}}\) which are constants because \(L_1\) is cylindrical near \(\Lambda _{q_i}\). Hamiltonian pushoff induces \(h'{:}L_1' \rightarrow {\mathbb {R}}\) such that \(dh'=\theta _{L_1'}\) and \(h_i'{:}=h'_{\Lambda _{q_i'}}\) are constants. By possibly reordering the index set of i, we assume that \(h_1 \leqslant h_2 \leqslant \cdots \leqslant h_d\). For each i, by relabelling if necessary, we also assume that \(q_i'\) is the closest to \(q_i\) among points in \(\{q_j'\}_{j=1}^{d_{L_1}}\), and \({\mathbf {q}}_i'\) is the closest to \({\mathbf {q}}_i\) among points in \(\{g{\mathbf {q}}_i'\}_{g \in \Gamma }\).
Lemma 5.3

\(A(x)> \epsilon \) if x is a Reed chord from \(\Lambda _{q_j'}\) to \(\Lambda _{q_i}\) and \(j>i\), and

\(A(x)> \epsilon \) if x is a Reed chord from \(\Lambda _{q_i'}\) to \(\Lambda _{q_i}\) but not the shortest one.
Proof
There is a constant \(\epsilon >0\) depending only on \(\{q_i\}_{i=1}^{d_{L_1}}\) and \(L_1\) such that \(L(x) > 3\epsilon \) if x is either a Reeb chord from \(\Lambda _{q_j}\) to \(\Lambda _{q_i}\) and \(i \ne j\), or it is a nonconstant Reeb chord from \(\Lambda _{q_i}\) to itself. We can choose a small Hamiltonian perturbation such that \(L(x) > 2\epsilon \) if either x is a Reeb chord from \(\Lambda _{q_j'}\) to \(\Lambda _{q_i}\), or a nonshortest Reeb chord from \(\Lambda _{q_i'}\) to \(\Lambda _{q_i}\). If \(j \geqslant i\), we have \(h_j \geqslant h_i\) so we can assume the Hamiltonian chosen is small enough such that \(h_j' h_i > \epsilon \) and therefore \(A(x)=L(x) +h_j' h_i > \epsilon \) in both cases listed in the lemma. \(\square \)
For each i, we denote the shortest Reeb chord from \(\Lambda _{q_i'}\) to \(\Lambda _{q_i}\) by \(x_{i',i}\). In regards to the canonical relative grading, we have \(x_{i',i}=0\). Since \({\mathbf {q}}_i'\) is the closest to \({\mathbf {q}}_i\) among points in \(\{g{\mathbf {q}}_i'\}_{g \in \Gamma }\), if we lift the Reeb chord \(x_{i',i}\) to a Reeb chord starting from \(\Lambda _{{\mathbf {q}}_i'}\), then it ends on \(\Lambda _{{\mathbf {q}}_i}\).
The following Lemmas (5.4, 5.5 and 5.6) concern some moduli of rigid bigon with input being \(e_L\). We start with the case when the output lies outside U.
Lemma 5.4
For \(\tau \gg 1\), rigid elements in \(\mathcal {M}^{J^{\tau }}(w_k;e_L)\) with respect to boundary conditions \((\tau _P(L_1'),L_1)\) and \((L_1',L_1)\ (\)i.e. they contribute to the differential in \(CF(\tau _P(L_1'),L_1)\) and \(CF(L_1',L_1))\), respectively, can be canonically identified.
Proof
By the same reasoning as in Lemma 4.8, as \(\tau \) goes to infinity, the holomorphic building \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) consists of exactly one vertex v and \(u_v\) maps to \(SM^+\). The result follows. \(\square \)
In Lemma 5.3, the \(\epsilon \) is independent of perturbation. Therefore, we can choose a perturbation such that the action of \(e_L\) in \(hom(L_1',L_1)\) (and hence in \(hom(\tau (L_1'),L_1)\)) is less than \(\epsilon \). In this case, we have
Lemma 5.5
Let \(\epsilon \) satisfy Lemma 5.3. If \(A(e_L) < \epsilon \), then for all \(j>i\) and \(g \in \Gamma \) (or \(j=i\) and \(g \ne 1_\Gamma \)), there is no rigid element in \(\mathcal {M}^{J^{\tau }}(c_{i,g,j}^\vee ;e_L)\) for \(\tau \gg 1\).
Proof
Suppose not, then we will have a holomorphic building \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) as \(\tau \) goes to infinity. Let \(u_{v_1}\) be the \(J^\)holomorphic curve such that \(c_{i,g,j}^\vee \) is an asymptote of \(u_{v_1}\). One can argue as in Lemma 4.11 to show that \(u_{v_1}\) has exactly one positive Reeb chord asymptote x. Moreover, x can be lifted to a Reeb chord from \(\Lambda _{g{\mathbf {q}}_j'}\) to \(\Lambda _{{\mathbf {q}}_i}\) by boundary condition. When \(j>i\) and \(g \in \Gamma \) (or \(j=i\) and \(g \ne 1_\Gamma \)), we have \(A(x)> \epsilon \) by Lemma 5.3. Since \(A(e_L) < \epsilon \) by assumption, we get a contradiction by Lemma 3.15. \(\square \)
Lemma 5.6
For \(L_1'\) sufficently close to \(L_1\) and \(\tau \gg 1\), the algebraic count of rigid elements in \(\mathcal {M}^{J^{\tau }}(c_{i,1_\Gamma ,i}^\vee ;e_L)\) is \(\pm 1\).
Proof
Similar to previous discussions, every limiting holomorphic building \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) from strips in \(\mathcal {M}^{J^{\tau }}(c_{i,1_\Gamma ,i}^\vee ;e_L)\) consists of two vertices (see Lemma 4.11). By boundary condition, the bottom level curve \(u_{v_1}\) lies in \(\mathcal {M}^{J^}(c_{i,1_\Gamma ,i}^\vee ; x_{i',i})\), which has algebraic count \(\pm 1\) by Corollary 4.15(4). Therefore, it suffices to determine the algebraic count of \(\mathcal {M}^{J^+}(x_{i',i};e_L)\).
We consider the rigid elements in the moduli \(\mathcal {M}(q_i^\vee ;e_L,(q_i')^\vee )\) for a compatible almost complex structure J, which is responsible to the \(q_i^\vee \)coefficient of \(\mu ^2(e_L,(q_i')^\vee )\) for the operation \(\mu ^2(\cdot ,\cdot ):hom(L_1',L_1) \times hom(P,L_1') \rightarrow hom(P,L_1)\). Therefore, it has algebraic count \(\pm 1\) with respect to J when \(L_1'\) is \(C^2\)close to \(L_1\).
Next, we will use a cascade (homotopy) type argument which goes back to Floer and argue that the algebraic count of \(\mathcal {M}^{J^\tau }(q_i^\vee ;e_L,(q_i')^\vee )\) is \(\pm 1\) for all \(\tau <\infty \). The difficulty lies in that neither \(q_i^\vee \) or \((q_i')^\vee \) is a cocycle, so the cohomological arguments would not work here. A detailed account for a cascade (homotopy) type argument involving higher multiplications can be found in, for example, [45] (see also [2, 46, Section 10e]).
Let us recall the overall strategy of the cascade argument tailored for our situation. Pick a path of compatible almost complex structures \((J_t)_{t \in [0,\infty )}\) from J to \(J^\tau \) for some finite time \(\tau \). For a generic path of almost complex structure \((J_t)_{t \in [0,\infty )}\), there are finitely many \(0<t_1<\dots<t_k <1\) such that there exists \(J_{t_l}\) stable maps with input \(e_L,(q_i')^\vee \), output \(q_i^\vee \) and consisting of two components. In our case, they consist of a \(J_{t_l}\)holomorphic triangle and a bigon, respectively. Moreover, one of the components must be of virtual dimension 0, and the other one is of dimension \(1\). In this case, we say a bifurcation occurs at \(t_l\), and denote the component of virtual dimension \(1\) as u.
If a bifurcation occurs at \(t_l\), then \(\mathcal {M}^{J_{t}}(q_i^\vee ;e_L,(q_i')^\vee )\) has the same diffeomorphism type when \(t\in (t\epsilon ,t_l)\) for some small \(\epsilon >0\). The birthdeath bifurcation cancels a pair of \(J_{t_l\epsilon }\)triangles at time \(t_l\); and the deathbirth bifurcation creates a pair of \(J_{t_l+\epsilon }\)triangles at the time \(t_l\). In either case, there is a pair of stable \(J_{t_l}\)stable triangles. When t approaches \(t_l\) from the right, we get the corresponding cobordisms. The change of algebraic count from \(\mathcal {M}^{J_{t_l\epsilon }}(q_i^\vee ;e_L,(q_i')^\vee )\) to \(\mathcal {M}^{J_{t_l+\epsilon }}(q_i^\vee ;e_L,(q_i')^\vee )\) is called the contribution to \(\mathcal {M}^{J_{t}}(q_i^\vee ;e_L,(q_i')^\vee )\) by the bifuration at time \(t_l\).
 (i)
both \(q_i^\vee \) and \((q_i')^\vee \) are asymptotes of u;
 (ii)
exactly one of \(q_i^\vee \) and \((q_i')^\vee \) is an asymptote of u;
 (iii)
neither of \(q_i^\vee \) nor \((q_i')^\vee \) is an asymptote of u.
Case (ii) If exactly one of \(q_i^\vee \) and \((q_i')^\vee \) is an asymptote of u, then P is a Lagrangian boundary condition of one of the component of \(\partial \Sigma _u\), where \(\Sigma _u\) is the domain of u. By this boundary component, there is another point \(q_j\) or \(q_j'\) for some \(j \ne i\) which is an asymptote of u. Since there is a lower bound between the distance from \(q_i\) (or \(q_i'\)) to \(q_j\) (or \(q_j'\)) for \(j \ne i\), we can apply monotonicity Lemma at an appropriate point in \(Im(u) \cap P\) to get a constant \(\delta >0\) depending only on \(\{q_i\}_{i=1}^{d_{L_1}}\) but not \(L_1'\) such that the energy \(E_\omega (u) > \delta \). If we chose \(L_1'\) to be sufficiently close to \(L_1\) such that \(A(e_L)+A((q_i')^\vee )A(q_i^\vee )< \delta \), then for u to contribute to a change of algebraic count of \(\mathcal {M}^{J_t}(q_i^\vee ;e_L,(q_i')^\vee )\), u has to be glued with a rigid \(J_{t_1}\)holomorphic curve of negative energy, which does not exist.
Case (iii) If none of \(q_i^\vee \) and \((q_i')^\vee \) are asymptotes of u, then u is a bigon with one asymptote being \(e_L\) and the other asymptote, denoted by x, being a generator of \(CF(L_1', L_1)\). Moreover, \(x=0\) because \({{\,\mathrm{virdim}\,}}(u)=1\). It is a contradiction because \(e_L\) is the only generator of \(CF(L_1', L_1)\) with degree 0 and constant maps have virtual dimension 0.
As a result, no bifurcation can possibly contribute to a change to the algebraic count and \(\#\mathcal {M}^{J^{\tau }}(q_i^\vee ;e_L,(q_i')^\vee )=\pm 1\) for all \(\tau \). By letting \(\tau \) go to infinity, the argument in Lemma 4.9 implies that the limiting holomorphic building \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) consist of two vertices. Moreover, we have \(u_{v_1} \in \mathcal {M}^{J^}(q_i^\vee ;x_{i',i},(q_i')^\vee )\) and \(u_{v_2} \in \mathcal {M}^{J^+}(x_{i',i};e_L)\). It implies that the algebraic count of rigid element in \(\mathcal {M}^{J^+}(x_{i',i};e_L)\) is \(\pm 1\). The proof finishes. \(\square \)
Remark 5.7
The fact that the algebraic count of \(\mathcal {M}^{J^+}(x_{i',i};e_L)\) is \(\pm 1\) will be used in Proposition 5.8 again.
Let us take local systems on the Lagrangians into account. Since \(\pi _1(U \cap L_1)=1\), we can identify stalks of the local system \(\mathcal {E}^1_p\) over each \(p\in U\cap L_1\) using the flat connection (equivalently, assume the connection is trivial in \(U \cap L_1\)). Similary, identify all \((\mathcal {E}^1)'_{p'}\) for \(p' \in U \cap L_1'\). This also induces an identification of stalks on \(\tau _P(T^*_qP)\), since local systems therein are pushforwards of the ones over a fiber.
We can now summarize the previous lemmss.
Proposition 5.8
Proof
By Lemma 5.4 and the fact that \(e_\mathcal {E}\) is a cocycle in \(CF((\mathcal {E}^1)', \mathcal {E}^1)\), we know that \( \mu ^1(t_\mathcal {D})=\sum _{i,j,g} \psi _{c_{i,g,j}^\vee }\). The fact that \(\psi _{c_{i,g,j}^\vee }=0\) if \(j>i\) and \(g \in \Gamma \) (or \(j=i\) and \( g \ne 1_\Gamma \)) follows from Lemma 5.5. Finally, to see that \(\psi _{c_{i,1_\Gamma ,i}^\vee }=id\) we need to understand the moduli \(\mathcal {M}^{J^{\tau }}(c_{i,1_\Gamma ,i}^\vee ;e_L)\) and the parallel transport maps given by the rigid elements in it.
Consider the holomorphic building when \(\tau =\infty \), we have two components \(u_1 \in \mathcal {M}^{J^}(q_i^\vee ;x_{i',i},(q_i')^\vee )\) and \(u_2 \in \mathcal {M}^{J^+}(x_{i',i};e_L)\) by Lemma 4.9 and Remark 4.10. When \(L_1'\) is sufficiently \(C^2\)close to \(L_1\), the action of \(u_1,u_2\) can be as small as we want. It implies that, by monotonicity lemma, \(u_2\) lies in a Weinstein neighborhood of \(L_1\).
It in turn implies that, for each strip \(u_2\) in the limit, the associated output is \(\psi _{i',i}=\pm id\) when the input at the point \(e_L\) is \(t_\mathcal {D}\) (the sign of \(\psi _{i',i}\) supported on \(x_{i',i}\) depends on the sign of \(u_2\)). This is because we have identified the stalks of \(\mathcal {E}^1\) and \((\mathcal {E}^1)'\) at the point \(e_L\), and the associated parallel transports \(I_{\partial _0 u}\) and \(I_{\partial _1 u}\) on their respective boundary conditions are inverse to each other (in fact, the strip itself provides an isotopy after projecting to \(L_1\) in the Weinstein neighborhood). Since we have proved that the algebraic count of \(\mathcal {M}^{J^+}(x_{i',i};e_L)\) is \(\pm 1\) (see Remark 5.7), the associated output by all elements in \(\mathcal {M}^{J^+}(x_{i',i};e_L)\) is \(\pm id\), when the input at \(e_L\) is \(t_\mathcal {D}\).
To get the proposition, we now replace \(u_1\) by \(u_1'\in \mathcal {M}^{J^{\tau }}(c_{i,1_\Gamma ,i}^\vee ;x_{i',i})\). As explained earlier, we have identified the fibers of the local systems of \(\mathcal {E}^1\) and \(\tau _P(\mathcal {E}^1)'\) at \(c_{{\mathbf {q}}_i,{\mathbf {q}}_i'}\). Since the parallel transports of \(\mathcal {E}^1\) and \(\tau _P(\mathcal {E}^1)'\) inside U are trivial, if the input at \(x_{i',i}\) is \(\pm id\), so is the output. By Lemma 5.6, the algebraic count of \(\mathcal {M}^{J^{\tau }}(c_{i,1_\Gamma ,i}^\vee ;x_{i',i})\) is \(\pm 1\) and each strip contributes \(\pm id\) (and the sign of \(\pm id\) only depends on the sign of the strip), therefore, the total countribution is \(\pm id\), as desired. \(\square \)
Remark 5.9
In summary, when \(L_1'\) is sufficiently close to \(L_1\), \(e_L\) being a cohomological unit is responsible for the algebraic count of \(\mathcal {M}^{J^\tau }(q_i^\vee ;e_L,(q_i')^\vee )\) being \(\pm 1\) and hence the \(q_i^\vee \)coefficient of \(\mu ^2(e_L,(q_i')^\vee )\) being \(\pm 1\). On the other hand, \(e_\mathcal {E}\) being a cohomological unit is responsible for the \({\mathbf {q}}_i^\vee \)coefficient of \(\mu ^2(e_\mathcal {E},({\mathbf {q}}_i')^\vee )\) being 1. Lemma 5.6 and Proposition 5.8 are obtained by replacing the bottom level curves at the SFT limit.
5.1.2 Computing \(\mu ^2_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\)
Next, we want to study \(\mu ^1_\mathcal {D}((\psi ^2 \otimes {\mathbf {q}}_i^\vee ) \otimes (g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1))\) [see (5.6), (5.9)]. In particular, we want to focus on the term \(\mu ^2_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\) so we need to discuss the moduli \(\mathcal {M}(c_{i,g,j}^\vee ;q_i^\vee ,\tau _P(q_j'))\) and \(\mathcal {M}(w_k;q_i^\vee ,\tau _P(q_j'))\).
Lemma 5.10
For \(\tau \gg 1\), there is no rigid element in \(\mathcal {M}^{J^\tau }(w_k;q_i^\vee ,\tau _P(q_j'))\).
Proof
Lemma 5.11
For \(\tau \gg 1\), there is no rigid element in \(\mathcal {M}^{J^\tau }(c_{\bar{i},{{\bar{g}}},{{\bar{j}}}}^\vee ;q_i^\vee ,\tau _P(q_{j}'))\) unless \(c_{\bar{i},{{\bar{g}}},{{\bar{j}}}} = c_{i,g,j}\) for some \(g \in \Gamma \).
Proof
Assume \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a limiting holomorphic building. If \(c_{{{\bar{i}}},{{\bar{g}}},{{\bar{j}}}} \ne c_{i,g,j}\) for all \(g \in \Gamma \), then \(c_{{{\bar{i}}},{{\bar{g}}},{{\bar{j}}}} \notin T^*_{q_i}P \cap \tau _P(T^*_{q_j'} P)\). By boundary condition, there is \(v_1 \in V(\mathcal {T})\) such that \(q_i^\vee , \tau _P(q_j')\) are asymptotes of \(u_{v_1}\) but \(c_{{{\bar{i}}},{{\bar{g}}},{{\bar{j}}}}^\vee \) is not an asymptote of \(u_{v_1}\). Therefore, all other asymptotes of \(u_{v_1}\) are positive Reeb chords and we get a contradiction as in Lemma 5.10. \(\square \)
The following lemma computes the \(\mu ^2\) map with trivial local systems on \(L_1\) and \(L_1'\).
Lemma 5.12
For \(\tau \gg 1\), the \(c_{i,h,j}^\vee \)coefficient of \(\mu ^2({\mathbf {q}}_i^\vee ,g\tau _P({\mathbf {q}}_j'))\) is \(\pm 1\) when \(h=g\) and is 0 when \(h \ne g\). Here \(\mu ^2: hom(\mathcal {P},L_1) \times hom(\tau _P(L_1'),\mathcal {P}) \rightarrow hom(\tau _P(L_1'),L_1)\) is the multiplication (Fig. 9).
Proof
By the associativity of cohomological multiplication, we have \(\mu ^2(\tau (q'),\mu ^2(c,q^\vee ))=\pm f\). It implies that \(\mu ^2(c,q^\vee )=\pm \tau (q')^\vee \). Dually, we have \(\mu ^2(q^\vee ,\tau (q'))=\pm c^\vee \) (it amounts to changing the asymptote c from outgoing end to incoming end, and \(\tau (q')\) from incoming end to outgoing end).
Since each \(u \in \mathcal {M}^{J^{\tau }}(c_{i,h,j}^\vee ;q_i^\vee ,\tau _P(q_j'))\) can be lifted to \({\mathbf {U}}\), there is a sign preserving bijective correspondence \(\mathcal {M}^{J^{\tau }}(c_{i,h,j}^\vee ;q_i^\vee ,\tau _P(q_j')) \simeq \mathcal {M}(c^\vee ;q^\vee ,\tau (q'))\) so we get the result. \(\square \)
Remark 5.13
There is an alternative geometric argument as follows. When the fibers corresponding \(S_1\) and \(S_2\) in the proof of Lemma 5.12 are fibers of antipodal points. The moduli computing \(c^\vee \)coefficient of \(\mu ^2(q^\vee ,\tau (q'))\) is the constant map to the point \(S_1 \cap S\). One can check that this constant map is regular so the algebraic count is \(\pm 1\). In the more general case, where \(S \cap S_2 \) is not the antipodal point of \(S_1 \cap S\), one can apply a homotopy type argument to conclude Lemma 5.12.
Now we enrich the statement of Lemma 5.12 by adding the local system on \(L_1\) and \(L_1'\) into consideration. Take the universal cover \({\mathbf {U}}\) of the neighborhood of P, there is a unique path (up to homotopy) in \(\tau _{\mathbf {P}}(T^*_{g{\mathbf {q}}_j'}{\mathbf {P}})\) from \({\mathbf {c}}_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}\) to \(g\tau _{{\mathbf {P}}}({\mathbf {q}}_j')\). It descends to the unique path (up to homotopy) in \(\tau _P(T^*_{q_j'}P)\) from \(c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}\) to \(\tau _{P}(q_j')\), which we denote by \([c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'} \rightarrow \tau _{P}(q_j')]\). Similarly, there is a unique path (up to homotopy) in \(T^*_{q_i}P\) from \(q_i\) to \(c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}\), which we denote by \([q_i \rightarrow c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}]\). Then we have
Proposition 5.14
Note that the parallel transport from \(\tau _P(q_j')\) to \(q_i\) in the statement was omitted for a reason that will become clear from the proof.
Proof
By Lemmas 5.10, 5.11 and 5.12, we already know that \(\mu ^2(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\) is supported at the intersection point \(c_{i,g,j}^\vee \). Moreover, as explained in the proof of Lemma 5.12, the rigid elements contributing to \(\mu ^2({\mathbf {q}}_i^\vee ,g\tau _P({\mathbf {q}}_j'))\) lie completely inside U.
With these preparation, we go back to the study of the degree zero cocycles of \(\mathcal {D}\).
Corollary 5.15
For \(L_1'\) sufficiently close to \(L_1\) and \(\tau \gg 1\), every degree 0 class in \(H^0(\mathcal {D})\) admits a cochain representative \(\beta \) which is a sum of elements supported at \(e_L\) and \(\{q_i^\vee \otimes \tau (q_j')\}_{i,j}\) only. Moreover, the term of \(\beta \) supported at \(e_L\) cannot be zero unless \(\beta =0\).
Proof
Every degree 0 cocycle in \(\mathcal {D}\) is a sum of elements supported at \(e_L\), \(\{c_{i,g,j}^\vee \}_{i,j,g}\) and \(\{q_i^\vee \otimes \tau (q_j')\}_{i,j}\) because \(w_k \ne 0\) for \(w_k \ne e_L\). Let \(\beta \) be a degree 0 cocycle which represents a class \([\beta ]\). By Proposition 5.14, we can eliminate the terms of \(\beta \) supported at \(c_{i,g,j}^\vee \) by adding the \(\mu ^1_\mathcal {D}\)differentials of certain cochains supported at \(q_i^\vee \otimes \tau (q_j')\). Note that the term of \(\beta \) supported at \(c_{i,g,j}^\vee \) themselves might not be exact because \(\mu ^1((\psi ^2 \otimes {\mathbf {q}}_i^\vee ) \otimes (g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1))\) involves more than just \(\mu ^2_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee , g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\) [see (5.9)], but the remainder terms cannot have \(c^\vee _{i,g,j}\)components.
Therefore, we have a cochain \(\beta '\) cohomologous to \(\beta \) such that \(\beta '\) is a sum of elements supported at \(e_L\) and \(\{q_i^\vee \otimes \tau (q_j')\}_{i,j}\) only.
Now, suppose the term of \(\beta '\) supported at \(e_L\) is 0. We write \(\beta '=\sum _{(i,j)}\psi ^{i,j}\), where, for all i, j, \(\psi ^{i,j}\) is an element supported at \(q_i^\vee \otimes \tau _P(q_j')\). If \(\psi ^{i_0,j_0} \ne 0\) for some \(i_0,j_0\), then by the isomorphism statement in Proposition 5.14, the terms of \(\mu ^1(\beta ')\) must contain a nontrivial element supported at \(c_{i_0,g,j_0}^\vee \) for some g. Because all other \(\mu ^1(\psi ^{i,j})\) do not have nonzero element supported at \(c_{i_0,g,j_0}^\vee \), this draws a contradiction. \(\square \)
Proposition 5.16
Proof
Let \(\beta \) be a nonexact degree 0 cocycle of \(\mathcal {D}\) [which exists from (5.2)]. We write \(\beta \) in the form (5.21). Note that \(\psi _{e_L}\) can be geometrically identified as an element of \(hom((\mathcal {E}^1)',\mathcal {E}^1)\). Lemma 5.4 implies that, for \(\mu ^1_{\mathcal {D}}(\beta )=0\), we must have \(\mu ^1_{hom((\mathcal {E}^1)',\mathcal {E}^1)}(\psi _{e_L})=0\).
In particular, we can take \(\psi _{e_L}=t_\mathcal {D}\). For \(\mu ^1(t_\mathcal {D}+\sum _{i,j} \psi _{q_i^\vee \otimes \tau _P(q_j'))})\) to be zero, the terms of it supported at \(c_{i,g,j}^\vee \) must be zero for all i, j, g. Therefore, we obtain the result by Propositions 5.8 and 5.14 [see (5.9)]. \(\square \)
5.2 Quasiisomorphisms
Let \(c_\mathcal {D}\) be the degree 0 cocycle obtained from Proposition 5.16. In this section, we are going to study the map (5.1) for \(\mathcal {E}^0 \in Ob({\mathcal {F}})\).
Lemma 5.17
For \(\tau \gg 1\), the image of \(\mu ^2_s(c_\mathcal {D},)\) is contained in \(C_1^s\). Therefore, \(\mu ^2_s(c_\mathcal {D},):C_0^s \rightarrow C_1^s\) is a chain map.
Proof
Lemma 5.18
For \(\tau \gg 1\), \(\mu ^2_s(c_\mathcal {D},)=\mu ^2_{{\mathcal {F}}}(t_\mathcal {D},)\).
Proof
Proposition 5.19
For \(\tau \gg 1\), \(\mu ^2_s(c_\mathcal {D},)\) is a quasiisomorphism.
Proof
For \(y \in (L_0 \cap \tau _P(L_1')) \backslash U\) and \(x \in L_0 \cap L_1\), the proof of Lemma 4.8 implies that all rigid elements in \(\mathcal {M}^{J^{\tau }}(x;e_L,y)\) have their image completely outside U.
As a result, the computation of \(\mu ^2_s(c_\mathcal {D},)=\mu ^2_{{\mathcal {F}}}(t_\mathcal {D},)\) picks up exactly the same holomorphic triangles that contributes to \(\mu ^2_{\mathcal {F}}(e_{\mathcal {E}},): C_0^s\cong hom(\mathcal {E}^0,(\mathcal {E}^1)') \rightarrow hom(\mathcal {E}^0,\mathcal {E}^1)\cong C_1^s\) via the tautological identification between \(e_{\mathcal {E}}\) and \(t_\mathcal {D}\) [see (5.12) and the paragraph after it]. Since \(e_{\mathcal {E}}\) is the cohomological unit, \(\mu ^2_s(c_\mathcal {D},)\) is also a quasiisomorphism. \(\square \)
Lemma 5.20
Proof

For \(s=1,\dots ,d_{L_0}\), let \(V_s\) be the subspace generated by elements in (5.37) such that \(t=s\).

For \(s=1,\dots ,d_{L_0}\) and \(l=1,\dots ,d_{L_1}\), let \(V_{s,l}\) be the subspace of \(V_s\) generated by elements in (5.37) such that \(r=l\).

For \(s=1,\dots ,d_{L_0}\), \(l=1,\dots ,d_{L_1}\) and \(g \in \Gamma \), let \(V_{s,l,g}\) be the subspace of \(V_{s,l}\) generated by elements in (5.37) such that \(h=g\).
Lemma 5.21
Proof
When \(i=l\), \(\psi ^2_{i,g,l,k} \ne 0\) only if \(g=1_\Gamma \) (by Proposition 5.16). Therefore, \(\Theta (V_{s,l,h}) \subset V_{s,l,h}+(\oplus _{t > l} V_{s,t})\) \(\square \)
Proposition 5.22
\(\mu ^2_q\) is a quasiisomorphism.
Proof
Concluding the proof of Theorem 1.2, 5.1 For each \(\mathcal {E}^1 \in Ob({\mathcal {F}})\), we apply Proposition 5.16 to find a degree 0 cocycle \(c_\mathcal {D}\in hom^0_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))\). Given any object \((\mathcal {E}^0)' \in Ob({\mathcal {F}})\), we consider a quasiisomorphic \(\mathcal {E}^0\), which is a Hamiltonian isotopic copy and the underlying Lagrangian \(L_0\) intersects transversally with \(L_1,\tau _P(L_1')\) and \(L_0\cap U\).
Propositions 5.19 and 5.22, together with the five lemma, then conclude that (5.1) is a quasiisomorphism. \(\square \)
Proof of Corollary 1.3
Footnotes
 1.
\({\mathbf {Y}}\) vanishes along \(\partial _r,\partial _z\) and takes values in \(\partial _{x_i}, \partial _{y_i}\).
Notes
Acknowledgements
The authors thank Richard Thomas for his interest in real projective space twist in our previous work [21], which motivate our investigation on Dehn twist along general spherical space forms in this project. The first author is deeply indepted to Paul Seidel for many insightful conversations and encouragement, and this work would be impossible without his support. The authors are very grateful to Ivan Smith for many helpful discussions, which in particular, greatly simplified the technical difficulties involved in this project. Discussion with Mohammed Abouzaid, Matthew Ballard, YuWei Fan, Sheel Ganatra, Mauricio Romo, Ed Segal, Zachary Sylvan and Michael Wemyss have influenced our understanding on spherical functors and we thank all of them. We also thank Ailsa Keating, Nick Sheridan, Michael Usher and Jingyu Zhao for helpful communications. We also thank the anonymous referee for many useful feedbacks on an earlier draft, which greatly improved the exposition. C. Y. M was supported by the National Science Foundation under Agreement No. DMS1128155 and by EPSRC (Establish Career Fellowship EP/N01815X/1), and W. W. is partially supported by Simons Collaboration Grant 524427. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation nor Simons Foundation.
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