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Dehn twists and Lagrangian spherical manifolds

  • Cheuk Yu MakEmail author
  • Weiwei Wu
Open Access
Article

Abstract

We study Dehn twists along Lagrangian submanifolds that are finite free quotients of spheres. We describe the induced auto-equivalences to the derived Fukaya category and explain their relations to mirror symmetry.

Mathematics Subject Classification

53D37 53D40 53D12 

1 Introduction

In his early groundbreaking papers [1, 2], Seidel studied the Dehn twist along a Lagrangian sphere and its induced auto-equivalence on the derived Fukaya category. There are often no automorphism of the mirror which induces such an auto-equivalence [3]. It turns out that this auto-equivalence 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 auto-equivalence of a given symplectomorphism.

In this paper, we investigate a new type of Dehn twist and its associated auto-equivalences.

Question 1.1

On a Fukaya category, what is the induced auto-equivalence 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 auto-equivalences, which distinguishes this question from all previous twist auto-equivalences, 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 auto-equivalence is a composition of two spherical twists. However, when \(char=2\), P becomes a \({\mathbb {P}}^n\)-object and the auto-equivalence 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 auto-equivalence decomposes into a composition of spherical twists in characteristic zero, but when one considers ground field of non-zero characteristics, such twists yield an entire family of previously unknown auto-equivalences. 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 (PE) 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

Let \((M^{2n},\omega )\) be a Liouville manifold with \(2c_1(M)=0\) and \(n \geqslant 3\). For any exact Lagrangian brane \(\mathcal {E}\in {\mathcal {F}}\), there is a quasi-isomorphism of the objects
$$\begin{aligned} \tau _P(\mathcal {E}) \simeq Cone(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } \mathcal {P}\xrightarrow {ev_{\Gamma }} \mathcal {E}) \end{aligned}$$
(1.1)
in \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\), where \(ev_{\Gamma }\) is the equivariant evaluation map, Cone is the \(A_{\infty }\) mapping cone and \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\) is the category of perfect \(A_{\infty }\) right \({\mathcal {F}}\) modules.
On cohomological level, Theorem 1.2 implies that for any Lagrangian branes \(\mathcal {E}_0, \mathcal {E}_1 \in {\mathcal {F}}\), there is a long exact sequence between the Floer cohomology groups
$$\begin{aligned}&HF^{*-1}(\mathcal {E}_0,\tau _P(\mathcal {E}_1)) \rightarrow H^*(CF(\mathcal {P},\mathcal {E}_1) \otimes _{\Gamma } CF(\mathcal {E}_0,\mathcal {P})) \\&\quad \rightarrow HF^*(\mathcal {E}_0,\mathcal {E}_1) \rightarrow HF^*(\mathcal {E}_0,\tau _P(\mathcal {E}_1)) \end{aligned}$$
It is natural to speculate that (1.1) holds on the functor level, i.e. \(\tau _P\cong Cone(\mathcal {P}\otimes _\Gamma \mathcal {P}\xrightarrow {ev_\Gamma }Id)\). Theorem 1.2 only shows this is true on the object level but doesn’t contain information on the morphisms or their compositions.

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 homological-algebraic 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 \(n=4k-1\) 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 quasi-isomorphic to applying \({\mathbb {P}}\)-twist to \(\mathcal {E}\) along P.

Remark 1.4

We would like to remark that Theorem 1.2 has the following reformulation. Under the assumption of Theorem 1.2, there is a spherical functor (see [6] for more about spherical functors)
$$\begin{aligned} {\mathcal {S}}: {\mathbb {K}}[\Gamma ]^{{{\,\mathrm{perf}\,}}} \rightarrow {\mathcal {F}}^{{{\,\mathrm{perf}\,}}} \end{aligned}$$
given by \(V \mapsto V \otimes _\Gamma \mathcal {P}\) (see Sect. 2.5). Moreover, for any \(\mathcal {E}\in {\mathcal {F}}\), we have
$$\begin{aligned} \tau _P(\mathcal {E}) \simeq {\mathcal {T}}_{{\mathcal {S}}}(\mathcal {E}) \end{aligned}$$
in \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\), where \({\mathcal {T}}_{{\mathcal {S}}}\) is the twist auto-equivalence of \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\) associated to \({\mathcal {S}}\).

1.1 Examples and outlooks

The current paper is focused on the foundations of the theory of twist auto-equivalences associated to \(\tau _P\) and is the starting point of a series of works investigating examples involving Lagrangian spherical space forms. Although we will not discuss these examples in depth, we give an overview of several forthcoming projects to give the readers an idea on the potential applications of the twist formula and its relations to existing works.
  • 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 3-folds 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 mirror-dual 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 3-spheres along a circle with derived categories of sheaves on local Calabi–Yau 3-folds 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 self-intersections, 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 well-adapted 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.

The examples mentioned above mostly involve lens spaces where the group \(\Gamma \) is a cyclic group. The algebro geometric counterparts of Dehn twists along more general spherical space forms such as Chiang Lagrangians will be investigated in future works.

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 neck-stretching 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 quasi-isomorphism 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 quasi-isomorphism (1.1), hence proving Theorem 1.2. Again, the study of relevant \(\mu ^k\)-multiplications are based on SFT and neck-stretching. The orientation is discussed in the “Appendix”.

Some notations.
  • \(\Gamma \) 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.

Standing assumption \((M,\omega )\) is a Liouville manifold with \(2c_1(M,\omega )=0\), and a fixed choice of a trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) is chosen. All Lagrangians are equipped with a \({\mathbb {Z}}\)-grading and a spin structure.

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

Let L be a closed exact Lagrangian submanifold in \((M,\omega )\) with a base point \(o_L \in L\). Let E be a finite rank local system on L with a flat connection \(\nabla \). For a path \(c:[0,1] \rightarrow L\), we denote the parallel transport from \(E_{c(0)}\) to \(E_{c(1)}\) along c with respect to the connection \(\nabla \).
$$\begin{aligned} I_c:E_{c(0)}\rightarrow E_{c(1)}. \end{aligned}$$
We use the monodromy action from \(\Gamma :=\pi _1(L)\) to \(E_{o_L}\) to endow \((E,\nabla )\) a right \(\Gamma \)-module structure. More explicitly, for \(y \in E_{o_L}\) and \(g \in \Gamma \), the right action is given by
$$\begin{aligned}&\rho : \Gamma \rightarrow End(E_{o_L}) \end{aligned}$$
(2.1)
$$\begin{aligned}&g \mapsto (y \mapsto I_{g}y). \end{aligned}$$
(2.2)
In particular, \((yg)h=I_{h}(I_{g}y)=I_{g*h}y=y(g*h)\), where \(*\) stands for concatenation of paths (i.e. g goes first). We use \(\mathcal {E}\) to denote the triple \((L,E,\nabla )\). For a Hamiltonian diffeomorphism \(\phi \in Ham(M,\omega )\), we define \(\phi ( \mathcal {E}):=(\phi (L),\phi _* E, \phi _* \nabla )\).
Let \(\mathcal {E}^i:=(L_i,E^i, \nabla ^i)\) for \(i=0,1\). A family of compactly supported Hamiltonian functions \(H=(H_t)_{t \in [0,1]}\) is called \((L_0,L_1)\)-admissible if
$$\begin{aligned} \phi ^H(L_0) \pitchfork L_1 \end{aligned}$$
(2.3)
where \(\phi ^H\) is the time one flow of the Hamiltonian vector field \(X_H=(X_{H_t})_{t \in [0,1]}\). Let \({\mathcal {X}}(L_0,L_1)\) be the set of H-Hamiltonian chord from \(L_0\) to \(L_1\) (i.e. \(x:[0,1] \rightarrow M\) such that \({\dot{x}}(t)=X_H(x(t))\), \(x(0) \in L_0\) and \(x(1) \in L_1\)). The Floer cochain complex between \(\mathcal {E}^0\) and \(\mathcal {E}^1\) is defined by
$$\begin{aligned} CF(\mathcal {E}^0,\mathcal {E}^1):=\bigoplus _{x \in {\mathcal {X}}(L_0,L_1)} Hom_{{\mathbb {K}}}(E^0_{x(0)},E^1_{x(1)}) \end{aligned}$$
(2.4)
Now, we want to introduce some notations to define the differential for \(CF(\mathcal {E}^0,\mathcal {E}^1)\) as well as the \(A_\infty \)-structure for a collection of Lagrangians with local systems.
Let \({\mathcal {R}}^{d+1}\) be the space of holomorphic disks with \(d+1\) boundary punctures. For each \(S \in {\mathcal {R}}^{d+1}\), one of the boundary punctures is distinguished and it is denoted by \(\xi _0\). The other boundary punctures are ordered counterclockwisely along the boundary and are denoted by \(\xi _1, \dots , \xi _d\), respectively. We denote the boundary component of S from \(\xi _j\) to \(\xi _{j+1}\) by \(\partial _j S\) for \(j=0,\dots ,d-1\). The boundary component from \(\xi _d\) to \(\xi _0\) is denoted by \(\partial _d S\). For \(j=1,\dots ,d\), we pick an outgoing/positive strip-like end for \(\xi _j\), which is a holomorphic embedding \(\epsilon _j: {\mathbb {R}}_{\geqslant 0} \times [0,1] \rightarrow S\) such that
$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon _j(s,0) \in \partial _{j-1} S \\ \epsilon _j(s,1) \in \partial _{j} S \\ \displaystyle \lim _{s \rightarrow \infty } \epsilon _j(s,t)=\xi _j \end{array} \right. \end{aligned}$$
(2.5)
We also pick an incoming/negative strip-like end for \(\xi _0\), which is a holomorphic embedding \(\epsilon _0: {\mathbb {R}}_{\leqslant 0} \times [0,1] \rightarrow S\) such that
$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon _0(s,0) \in \partial _{0} S \\ \epsilon _0(s,1) \in \partial _{d} S \\ \displaystyle \lim _{s \rightarrow -\infty } \epsilon _0(s,t)=\xi _0 \end{array} \right. \end{aligned}$$
(2.6)
The strip-like ends are assumed to have pairwise disjoint image and they vary smoothly with respect to S in \({\mathcal {R}}^{d+1}\).
Let \(\{\mathcal {E}^j\}_{j=0}^d\) be a finite collection of Lagrangians with local systems. For \(j=1,\dots ,d\), let \(H_j\) be a \((L_{j-1},L_j)\)-admissible Hamiltonian [see (2.3)]. We also pick a \((L_0,L_d)\)-admissible Hamiltonian \(H_0\). For each \(S \in {\mathcal {R}}^{d+1}\) and each collection \(\{H_j\}_{j=0}^d\), we pick a \(C_{cpt}^\infty (M)\)-valued one-form \(K \in \Omega ^1(S,C_{cpt}^\infty (M))\). Let \(X_K \in \Omega ^1(S,C^{\infty }(M,TM))\) be the corresponding Hamiltonian-vector-field-valued one-form. We require that
$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon _j^* X_K=X_{H_j}dt \\ X_K|_{\partial _j S}=0 \end{array} \right. . \end{aligned}$$
(2.7)
When \(d=1\), we assume that \(K(s,t)=H_{0,t}=H_{1,t}\) for all \((s,t) \in {\mathbb {R}}\times [0,1]\). We also assume that K varies smoothly with respect to S and is consistent with respect to gluing near boundary strata of the Deligne–Mumford–Stasheff compactification of \({\mathcal {R}}^{d+1}\).
Let \(J^M\) be an \(\omega \)-compatible almost complex structure that is cylindrical over the infinite end of M (see Definition 3.1). Let \({\mathcal {J}}(M,\omega )\) be the space of \(\omega \)-compatible almost complex structures J such that \(J=J^M\) outside a compact set. For \(j=0,\dots ,d\), let \(J_j=(J_{j,t})_{t \in [0,1]}\) be a family such that \(J_{j,t} \in {\mathcal {J}}(M,\omega )\) for all t. For each \(S \in {\mathcal {R}}^{d+1}\) and each collection \(\{J_j\}_{j=0}^d\), we pick a domain-dependent \(\omega \)-compatible almost complex structure \(J=(J_z)_{z \in S}\) such that
$$\begin{aligned} \left\{ \begin{array}{ll} J_z \in {\mathcal {J}}(M,\omega ) &{}\quad \text { for all }z\\ J \circ \epsilon _j(s,t)=J_{j,t} &{}\quad \text { for all }j,s,t \end{array} \right. \end{aligned}$$
(2.8)
When \(d=1\), we require that \(J=(J_{s,t})_{(s,t) \in {\mathbb {R}}\times [0,1]}=(J_t)_{t \in [0,1]}\) is independent of the s-direction. We assume that J varies smoothly with respect to S in \({\mathcal {R}}^{d+1}\) and is consistent with respect to gluing near boundary strata of the Deligne–Mumford–Stasheff compactification of \({\mathcal {R}}^{d+1}\).
Let \(x_j \in {\mathcal {X}}(L_{j-1},L_{j})\) for \(j=1,\dots ,d\) and \(x_0 \in {\mathcal {X}}(L_0,L_d)\). For \(d>1\), we define \(\mathcal {M}^{K,J}(x_0;x_d,\dots ,x_1)\) to be the space of smooth maps \(u:S \rightarrow M\) such that
$$\begin{aligned} \left\{ \begin{array}{ll} S \in {\mathcal {R}}^{d+1} \\ (du-X_K)^{0,1}=0 &{}\quad \text { with respect to }(J_z)_{u(z)} \\ u(\partial _j S) \subset L_j &{}\quad \text { for all }j \\ \displaystyle \lim _{s \rightarrow \pm \infty } u(\epsilon _j(s,t))=x_j(t) &{}\quad \text { for all }j \end{array} \right. \end{aligned}$$
(2.9)
When \(d=1\), we define \(\mathcal {M}^{K,J}(x_0;x_1)\) to be the corresponding space of maps after modulo the \({\mathbb {R}}\) action by translation in the s-coordinate. For simplicity, we may use \(\mathcal {M}(x_0;x_d,\dots ,x_1)\) to denote \(\mathcal {M}^{K,J}(x_0;x_d,\dots ,x_1)\) for an appropriate choice of (KJ).

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 \(|x_0|-\sum _{j=1}^d |x_j|+(d-2)\), where \(|\cdot |\) denotes the Maslov grading (see Sect. 3.2).

For each transversally cut out rigid element \(u \in \mathcal {M}(x_0;x_d,\dots ,x_1)\), we define
$$\begin{aligned}&\mu ^u:Hom(E^{d-1}_{x_d(0)},E^d_{x_d(1)}) \times \dots \times Hom(E^0_{x_1(0)},E^1_{x_1(1)}) \rightarrow Hom(E^0_{x_0(0)},E^d_{x_0(1)}) \nonumber \\&\mu ^u(\psi ^d,\dots ,\psi ^1)(a)= {{\,\mathrm{sign}\,}}(u) I_{\partial _d u} \circ \psi ^d \circ \dots \circ \psi ^1 \circ I_{\partial _0 u} (a) \end{aligned}$$
(2.10)
where \(\partial _d u=u|_{\partial _d S}\) for \(\partial _d S\) being equipped with the counterclockwise orientation, and \({{\,\mathrm{sign}\,}}(u) \in \{\pm 1\}\) is the sign determined by u (see “Appendix A”). Finally, we define the \(A_{\infty }\)-operation by
$$\begin{aligned}&\mu ^d:CF(\mathcal {E}^{d-1},\mathcal {E}^d) \times \dots \times CF(\mathcal {E}^0,\mathcal {E}^1) \rightarrow CF(\mathcal {E}^0,\mathcal {E}^d) \nonumber \\&\mu ^d(\psi ^d,\dots ,\psi ^1)=\sum _{u \in \mathcal {M}(x_0;x_d,\dots ,x_1), u \text { rigid}} \mu ^u(\psi ^d,\dots ,\psi ^1) \end{aligned}$$
(2.11)
The fact that the auxiliary structures can be chosen generically and consistently in the sense of [2], and that \(\{\mu ^d\}_{d\geqslant 1}\) gives rise to an \(A_\infty \) structure follows the argument in [2] line-by-line. This defines the Fukaya category \(\mathcal {F}uk(X)\) that we will use throughout.

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.

Let L be a closed exact Lagrangian and \({\mathbf {L}}\) be its universal cover with covering map \(\pi :{\mathbf {L}} \rightarrow L\). Let \(o_L \in L\) be a base point of L and we pick a lift \(o_{\mathbf {L}} \in {\mathbf {L}}\) such that \(\pi (o_{\mathbf {L}})=o_L\). We assume throughout that \(\Gamma :=\pi _1(L,o_L)\) is a finite group so that \({\mathbf {L}}\) is compact. For each \({\mathbf {q}}\in {\mathbf {L}}\), there is a unique path \(c_{\mathbf {q}}\) (up to homotopy) from \(o_{\mathbf {L}}\) to \({\mathbf {q}}\) and we identify \({\mathbf {q}}\) with the homotopy class \([\pi \circ c_{\mathbf {q}}]\). We have a left \(\Gamma \)-action on \({\mathbf {L}}\) given by
$$\begin{aligned} g {\mathbf {q}}:=g*[(\pi \circ c_{\mathbf {q}})] \end{aligned}$$
(2.12)
for \(g \in \pi _1(L,o_L)\), where \(g*[(\pi \circ c_{\mathbf {q}})]\) is a homotopy class of path from \(o_L\) to \(\pi ({\mathbf {q}})\) and we identify it as a point in \({\mathbf {L}}\). It is clear that \(h(g{\mathbf {q}})=(h*g){\mathbf {q}}\). If we pick a Morse function and a Riemannian metric on L to define a Morse cochain complex \(C^*(L)\), we can lift the function and metric to \({\mathbf {L}}\) to define a Morse cochain complex \(C^*({\mathbf {L}})\). The \(\Gamma \)-action on \({\mathbf {L}}\) induces a left \(\Gamma \)-action on \(C^*({\mathbf {L}})\). The \(\Gamma \)-invariant part of \(C^*({\mathbf {L}})\) can be identified with \(C^*(L)\), in other words,
$$\begin{aligned} C^*(L)=Rhom_{{\mathbb {K}}[\Gamma ]-mod}({\mathbb {K}},C^*({\mathbf {L}}))=(C^*({\mathbf {L}}))^\Gamma \end{aligned}$$
(2.13)
We want to discuss the analog when L is equipped with local systems.

Given a local system E on L, we use \({\mathbf {E}}=\pi ^*E\) to denote the pull-back local system. For a path \(c{:}[0,1] \rightarrow {\mathbf {L}}\), we use \(I_c\) to denote the parallel transport with respect to the pull-back flat connection on \({\mathbf {E}}\).

Let \(E^i\) be local systems on L for \(i=0,1\). We have right actions [see (2.1)]
$$\begin{aligned} \rho ^i{:}\Gamma \rightarrow End(E^i_{o_L}) \end{aligned}$$
(2.14)
for \(i=0,1\). It induces a left \(\Gamma \)-module structure on \(Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})\) by
$$\begin{aligned} \psi \mapsto g \cdot \psi :=\rho ^1(g^{-1}) \circ \psi \circ \rho ^0(g) \end{aligned}$$
(2.15)

Lemma 2.2

Let \(E^i\) be local systems on L for \(i=0,1\). Then there is a DG left \(\Gamma \)-module isomorphism
$$\begin{aligned} \Phi : CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1)) \simeq C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L}) \end{aligned}$$
(2.16)
where the differential on \(C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})\) is only the differential on the first factor, and the \(\Gamma \)-action on it is given by \(g \cdot (x\otimes \psi ):=gx \otimes g \cdot \psi \) [see (2.12) and (2.15)].

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.

For each \({\mathbf {q}}\in {\mathbf {L}}\) and both \(i=0,1\), there is a canonical identification
$$\begin{aligned} I_{c_{{\mathbf {q}}}^{-1}}: {\mathbf {E}}^i_{{\mathbf {q}}} \rightarrow {\mathbf {E}}^i_{o_{{\mathbf {L}}}} \end{aligned}$$
(2.17)
where \(c_{{\mathbf {q}}}\) is the unique (up to homotopy) path from \(o_{\mathbf {L}}\) to \({\mathbf {q}}\). Therefore, it induces a trivialization of \({\mathbf {E}}^i\). We can also trivialize \(Hom_{\mathbb {K}}({\mathbf {E}}^0,{\mathbf {E}}^1)\) using the canonical isomorphism
$$\begin{aligned}&Hom_{\mathbb {K}}({\mathbf {E}}^0_{\mathbf {q}},{\mathbf {E}}^1_{\mathbf {q}}) \rightarrow Hom_{\mathbb {K}}({\mathbf {E}}^0_{o_{\mathbf {L}}},{\mathbf {E}}^1_{o_{\mathbf {L}}})=Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L}) \end{aligned}$$
(2.18)
$$\begin{aligned}&\psi \mapsto I^1_{c_{\mathbf {q}}^{-1}} \circ \psi \circ I^1_{c_{\mathbf {q}}} \end{aligned}$$
(2.19)
Using the trivialization (2.18), (2.19), we have a graded vector space isomorphism (2.16).

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 \(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\).

For \({\mathbf {q}}_1 \otimes \psi \in C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})\),
$$\begin{aligned}&\Phi (\mu ^{\mathbf {u}}(\Phi ^{-1}({\mathbf {q}}_1 \otimes \psi ))) \end{aligned}$$
(2.20)
$$\begin{aligned}&={{\,\mathrm{sign}\,}}({\mathbf {u}}){\mathbf {q}}_0 \otimes I_{c_{{\mathbf {q}}_0}^{-1}}I_{\partial _1 {\mathbf {u}}}I_{c_{{\mathbf {q}}_1}} \psi I_{c_{{\mathbf {q}}_1}^{-1}}I_{\partial _0 {\mathbf {u}}}I_{c_{{\mathbf {q}}_0}} \end{aligned}$$
(2.21)
$$\begin{aligned}&={{\,\mathrm{sign}\,}}({\mathbf {u}}){\mathbf {q}}_0 \otimes \psi \end{aligned}$$
(2.22)
where the second equality uses the fact that \(\pi _1({\mathbf {L}})=1\). Therefore, \(\Phi \) is an isomorphism of differential graded vector spaces if we define the differential on \(C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})\) to be \(\partial _{{\mathbf {L}}}\) acting on the first factor.
Finally, we want to compare the left \(\Gamma \)-module structures. In \(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\), the action on \(\psi \in Hom_{\mathbb {K}}({\mathbf {E}}^0_{{\mathbf {q}}},{\mathbf {E}}^1_{{\mathbf {q}}})=Hom_{\mathbb {K}}(E^0_{q},E^1_q)\) is given by
$$\begin{aligned} \psi \mapsto g\psi =\psi \end{aligned}$$
(2.23)
where the last \(\psi \) lies in \(Hom_{\mathbb {K}}({\mathbf {E}}^0_{g{\mathbf {q}}},{\mathbf {E}}^1_{g{\mathbf {q}}})=Hom_{\mathbb {K}}(E^0_{q},E^1_q)\). For \({\mathbf {q}}\otimes \psi \in C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})\),
$$\begin{aligned}&\Phi (g(\Phi ^{-1}({\mathbf {q}}\otimes \psi ))) \end{aligned}$$
(2.24)
$$\begin{aligned}&=g{\mathbf {q}}\otimes I_{c_{g{\mathbf {q}}}^{-1}}I_{c_{{\mathbf {q}}}} \psi I_{c_{{\mathbf {q}}}^{-1}}I_{c_{g{\mathbf {q}}}} \end{aligned}$$
(2.25)
$$\begin{aligned}&=g{\mathbf {q}}\otimes I_{g^{-1}}\psi I_g \end{aligned}$$
(2.26)
which is exactly the one given in (2.12) and (2.15). It finishes the proof. \(\square \)

We have the following consequence of Lemma 2.2:

Lemma 2.3

Let \(E^i\) be local systems on L for \(i=0,1\). Then
$$\begin{aligned} CF(\mathcal {E}^0,\mathcal {E}^1)=Rhom_{{\mathbb {K}}[\Gamma ]-mod}({\mathbb {K}},C^*({\mathbf {L}}) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(E^0_{o_L},E^1_{o_L})) \end{aligned}$$
(2.27)

Proof

We use the notation in the proof of Lemma 2.2. Let u be a Morse trajectory from \(q_0\) to \(q_1\) contributing to \(\partial _L(q_1)\). Let \({\mathbf {q}}_0 \in {\mathbf {L}}\) be a lift of \(q_0\) and let \({\mathbf {q}}_1 \in {\mathbf {L}}\) be the corresponding lift of \(q_1\) such that u lifts to a Morse trajectory \({\mathbf {u}}\) from \({\mathbf {q}}_0\) to \({\mathbf {q}}_1\). Let \(\psi \in Hom_{\mathbb {K}}(E^0_{q_1},E^1_{q_1})\). By (2.10), we have
$$\begin{aligned} \mu ^u(\psi )={{\,\mathrm{sign}\,}}(u)I_{\partial _1u} \psi I_{\partial _0u} \end{aligned}$$
(2.28)
In the above notation, we regard u as a degenerated holomorphic strip and have suppressed the direction of parallel transport for brevity, since it should be clear from the context.
By definition, \(Hom_{\mathbb {K}}({\mathbf {E}}^0_{g{\mathbf {q}}_j},{\mathbf {E}}^1_{g{\mathbf {q}}_j}) \cong Hom_{\mathbb {K}}(E^0_{q_j},E^1_{q_j})\) for \(j=0,1\) and for all \(g \in \Gamma \). Therefore, for \(\psi \in Hom_{\mathbb {K}}({\mathbf {E}}^0_{g{\mathbf {q}}_1},{\mathbf {E}}^1_{g{\mathbf {q}}_1}) \subset CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\),
$$\begin{aligned} \mu ^{g{\mathbf {u}}}(\psi )&={{\,\mathrm{sign}\,}}(g{\mathbf {u}})I_{\partial _1 g{\mathbf {u}}} \psi I_{\partial _0 g{\mathbf {u}}} \end{aligned}$$
(2.29)
$$\begin{aligned}&={{\,\mathrm{sign}\,}}(u)I_{\partial _1u} \psi I_{\partial _0u} \end{aligned}$$
(2.30)
where \(\mu ^{g{\mathbf {u}}}\) is the term in the differential of \(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\) contributed by \(g{\mathbf {u}}\), and the second equality uses \(Hom_{\mathbb {K}}({\mathbf {E}}^0_{g{\mathbf {q}}_1},{\mathbf {E}}^1_{g{\mathbf {q}}_1}) \cong Hom_{\mathbb {K}}(E^0_{q_1},E^1_{q_1})\). The \(\Gamma \) action on the generators (\({\mathbf {q}}\otimes \psi \mapsto g{\mathbf {q}}\otimes \psi \)) and differentials (\(\mu ^u \mapsto \mu ^{g{\mathbf {u}}}\)) of \(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1))\) are free and the invariant part can be identified with \(CF(\mathcal {E}^0,\mathcal {E}^1)\) so
$$\begin{aligned} CF(\mathcal {E}^0,\mathcal {E}^1)=Rhom_{{\mathbb {K}}[\Gamma ]-mod}({\mathbb {K}},CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1)))=(CF(({\mathbf {L}},{\mathbf {E}}^0),({\mathbf {L}},{\mathbf {E}}^1)))^{\Gamma } \end{aligned}$$
(2.31)
and the result follows from Lemma 2.2. \(\square \)

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

Let E be the universal local system on L. For \(q \in L\), \(y \in E_q\), \(g \in \Gamma \) and \(c:[0,1] \rightarrow L\) such that \(c(0)=q\), we have
$$\begin{aligned} g(I_cy)=I_c(gy) \end{aligned}$$
(2.32)

Proof

Without loss of generality, let \(y \in \pi ^{-1}(q)\). We can identify y with the homotopy class \([\pi \circ c_y]\) from \(o_L\) to q. Then we have [see (2.12)]
$$\begin{aligned} g(I_cy)=g*[\pi \circ c_y]*[c]=I_c(gy) \end{aligned}$$
(2.33)
where [c] is the homotopy class of path from c(0) to c(1) that c represents. \(\square \)
Let \(\mathcal {E}=(L,E,\nabla )\). Since we have a left action on \(E_q\) for all \(q \in L\), it induces a left \(\Gamma \) action on \(CF(\mathcal {E}',\mathcal {E})\)
$$\begin{aligned} \psi \mapsto g\psi \end{aligned}$$
(2.34)
for any \(\mathcal {E}' \in Ob({\mathcal {F}})\). Similarly, for any \(\mathcal {E}' \in Ob({\mathcal {F}})\), we have the induced right \(\Gamma \) action on \(CF(\mathcal {E},\mathcal {E}')\)
$$\begin{aligned} \psi (\cdot ) \mapsto \psi (g \cdot ) \end{aligned}$$
(2.35)
As an immediate consequence of Lemma 2.5 and the definition of \(\mu ^u\) [see (2.10)], we have

Corollary 2.6

Let E be the universal local system on L. Let \(\mathcal {L}_1 \dots , \mathcal {L}_r, \mathcal {K}_1,\dots ,\mathcal {K}_s \in Ob({\mathcal {F}})\). Let \(y_j \in CF(\mathcal {K}_j,\mathcal {K}_{j+1})\) for \(j=1,\dots ,s-1\), \(x_j \in CF(\mathcal {L}_j,\mathcal {L}_{j+1})\) for \(j=1,\dots ,r-1\), \(\psi _2 \in CF(\mathcal {E},\mathcal {K}_1)\) and \(\psi _1 \in CF(\mathcal {L}_r,\mathcal {E})\), we have
$$\begin{aligned} \mu ^u(y_{s-1},\dots ,y_1,\psi _2g,\psi _1,x_{r-1},\dots ,x_1)= \mu ^u(y_{s-1},\dots ,y_1,\psi _2,g\psi _1,x_{r-1},\dots ,x_1) \end{aligned}$$
(2.36)
for all \(g \in \Gamma \), where u is an element in the appropriate moduli contributing to the \(A_{\infty }\)-structural maps of \({\mathcal {F}}\). When \(s=0\ (\)resp. \(r=0)\), we have
$$\begin{aligned}&g\mu ^u(\psi _1,x_{r-1},\dots ,x_1)= \mu ^u(g\psi _1,x_{r-1},\dots ,x_1) \text {, respectively} \end{aligned}$$
(2.37)
$$\begin{aligned}&\mu ^u(y_{s-1},\dots ,y_1,\psi _2g)= \mu ^u(y_{s-1},\dots ,y_1,\psi _2)g \end{aligned}$$
(2.38)

Remark 2.7

We offer an alternative way to understand (2.36) using \({\mathbf {L}}\) instead of \(\mathcal {E}\). For each \(q \in L\) and a lift \({\mathbf {q}}\) of q, we can view \({\mathbf {q}}\) as a point in \({\mathbf {L}}\) or as an element in \(E_q\). Therefore, we can identify the generators of \(CF(T^*_qL,\mathcal {E})\) and the generators of \(CF( \bigcup _{g \in \Gamma } T^*_{g{\mathbf {q}}}{\mathbf {L}},{\mathbf {L}})\) by
$$\begin{aligned} E_q \ni {\mathbf {q}}\mapsto {\mathbf {q}}\in T^*_{\mathbf {q}}{\mathbf {L}}\cap {\mathbf {L}}\end{aligned}$$
(2.39)
Dually, \(CF(\mathcal {E}, T^*_qL)\) can be identified with \(CF({\mathbf {L}}, \cup _{g \in \Gamma } T^*_{g{\mathbf {q}}}{\mathbf {L}})\) by
$$\begin{aligned} Hom_{\mathbb {K}}(E_q,{\mathbb {K}}) \ni {\mathbf {q}}^\vee \mapsto {\mathbf {q}}\in {\mathbf {L}}\cap T^*_{\mathbf {q}}{\mathbf {L}}\end{aligned}$$
(2.40)
The right action (2.35) on \(Hom_{\mathbb {K}}(E_q,{\mathbb {K}})\) is given by \({\mathbf {q}}^\vee g=(g^{-1}{\mathbf {q}})^\vee \), which corresponds to the right action on \({\mathbf {L}}\) by \({\mathbf {q}}g=g^{-1}{\mathbf {q}}\).

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 non-zero 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 \(h{\mathbf {q}}_2\).

Roughly speaking, one can define a Floer theory by counting \((u, {\mathbf {u}})\), where u is as in Corollary 2.6 and \({\mathbf {u}}\) is a lift of \(u|_\gamma \). This definition is explained in details in [30] and the outcome is the same as Lagrangian Floer theory with local systems. In this setting, the pair \((u,h{\mathbf {u}})\) contributes to
$$\begin{aligned} \mu ^{(u,h{\mathbf {u}})}(y_{s-1},\dots ,y_1,h{\mathbf {q}}_2,hg{\mathbf {q}}_1,x_{r-1},\dots ,x_1) \end{aligned}$$
(2.41)
and it equals to \(\mu ^{(u,{\mathbf {u}})}(y_{s-1},\dots ,y_1,{\mathbf {q}}_2,g{\mathbf {q}}_1,x_{r-1},\dots ,x_1)\). Under the identification (2.39), (2.40), it means that (when \(h=g^{-1}\))
$$\begin{aligned}&\mu ^{(u,{\mathbf {u}})}(y_{s-1},\dots ,y_1,{\mathbf {q}}_2^\vee ,g{\mathbf {q}}_1,x_{r-1},\dots ,x_1)\\&\quad =\mu ^{(u,g^{-1}{\mathbf {u}})}(y_{s-1},\dots ,y_1,(g^{-1}{\mathbf {q}}_2)^\vee ,{\mathbf {q}}_1,x_{r-1},\dots ,x_1) \end{aligned}$$
which is exactly the same as (2.36)
The rest of this subsection is devoted to the self-Floer chain complex \(CF(\mathcal {E},\mathcal {E})\) when E is the universal local system of L. Let \(R:={\mathbb {K}}[\Gamma ]\) and \(1_\Gamma \) be the unit of \(\Gamma \). For \(h \in \Gamma \), we define \(\tau _h \in Hom_{\mathbb {K}}(R,R)\) by
$$\begin{aligned} \tau _h(g)=\left\{ \begin{array}{ll} 1_\Gamma &{}\quad \text { if }g=h^{-1} \\ 0 &{}\quad \text { if } g \in \Gamma \backslash \{h^{-1}\} \end{array} \right. \end{aligned}$$
(2.42)
Note that \(R \cong E_{o_L}\) so, by Lemma 2.3, we have
$$\begin{aligned} CF(\mathcal {E},\mathcal {E})=(C^*({\mathbf {L}}) \otimes Hom_{\mathbb {K}}(R,R))^{\Gamma } \end{aligned}$$
(2.43)
as a \(\Gamma \)-module.

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. \(\mu ^2_{{\mathbf {L}}}(-,-) \otimes - \circ - \)).

Let \(\Phi _2:C^*({\mathbf {L}}) \otimes R \rightarrow (C^*({\mathbf {L}}) \otimes Hom_{\mathbb {K}}(R,R))^{\Gamma }\) be the graded vector space isomorphism given by
$$\begin{aligned} \Phi _2:x \otimes h \mapsto \sum _{g \in \Gamma } gx \otimes g \cdot \tau _h=\sum _{g \in \Gamma } gx \otimes I_{g^{-1}} \tau _h I_g \end{aligned}$$
(2.44)

Lemma 2.8

We have the following equalities
$$\begin{aligned} \Phi _2^{-1} \circ \mu ^1 \circ \Phi _2(x \otimes h)&=\partial _{{\mathbf {L}}}(x) \otimes h \end{aligned}$$
(2.45)
$$\begin{aligned} \Phi _2^{-1} \circ \mu ^2 \circ (\Phi _2(x_2 \otimes h_2),\Phi _2(x_1 \otimes h_1))&=\mu ^2_{{\mathbf {L}}}(x_2,h_2x_1) \otimes h_2h_1 \end{aligned}$$
(2.46)
As a consequence of (2.45), we have \(H^*(CF(\mathcal {E},\mathcal {E}))=H^*({\mathbf {L}}) \otimes R\) as a vector space.

Proof

For \(x \otimes h \in C^*({\mathbf {L}}) \otimes R\),
$$\begin{aligned}&\Phi _2^{-1} \circ \mu ^1 \circ \Phi _2(x \otimes h) \end{aligned}$$
(2.47)
$$\begin{aligned}&=\Phi _2^{-1}\left( \sum _{g \in \Gamma } \partial _{\mathbf {L}}(gx) \otimes I_{g^{-1}} \tau _h I_g \right) \end{aligned}$$
(2.48)
$$\begin{aligned}&=\Phi _2^{-1} \left( \sum _{g \in \Gamma } g\partial _{\mathbf {L}}(x) \otimes I_{g^{-1}} \tau _h I_g \right) \end{aligned}$$
(2.49)
$$\begin{aligned}&=\partial _{{\mathbf {L}}}(x) \otimes h \end{aligned}$$
(2.50)
where the second equality uses Corollary 2.6.
For \(x_i \otimes h_i \in C^*({\mathbf {L}}) \otimes R\), \(i=1,2\), we have
$$\begin{aligned}&\Phi _2^{-1} \circ \mu ^2 \circ (\Phi _2(x_2 \otimes h_2),\Phi _2(x_1 \otimes h_1)) \end{aligned}$$
(2.51)
$$\begin{aligned}&=\Phi _2^{-1}\left( \sum _{g_1,g_2 \in \Gamma } \mu ^2_{\mathbf {L}}(g_2x_2,g_1x_1) \otimes I_{g_2^{-1}} \tau _{h_2} I_{g_2} I_{g_1^{-1}} \tau _{h_1} I_{g_1}\right) \end{aligned}$$
(2.52)
For \(\tau _{h_2} I_{g_2} I_{g_1^{-1}} \tau _{h_1}\) and hence \(I_{g_2^{-1}} \tau _{h_2} I_{g_2} I_{g_1^{-1}} \tau _{h_1} I_{g_1}\) to be non-zero, we must have
$$\begin{aligned} 1_{\Gamma } * g_1^{-1} * g_2=h_2^{-1} \end{aligned}$$
(2.53)
and for any \(g_2\), there is a unique \(g_1\) (\(=g_2h_2\)) such that \(g_1^{-1}g_2=h_2^{-1}\). Therefore, the sum becomes
$$\begin{aligned}&\Phi _2^{-1} \left( \sum _{g_2 \in \Gamma } \mu ^2_{\mathbf {L}}(g_2x_2,g_2h_2x_1) \otimes I_{g_2^{-1}} \tau _{h_2} I_{h_2^{-1}} \tau _{h_1} I_{g_1} I_{g_2^{-1}} I_{g_2}\right) \end{aligned}$$
(2.54)
$$\begin{aligned}&=\Phi _2^{-1} \left( \sum _{g_2 \in \Gamma } g_2\mu ^2_{\mathbf {L}}(x_2,h_2x_1) \otimes I_{g_2^{-1}} \tau _{h_2} I_{h_2^{-1}} \tau _{h_1} I_{h_2} I_{g_2}\right) \end{aligned}$$
(2.55)
$$\begin{aligned}&=\Phi _2^{-1} \left( \sum _{g_2 \in \Gamma } g_2\mu ^2_{\mathbf {L}}(x_2,h_2x_1) \otimes I_{g_2^{-1}} \tau _{h_2h_1} I_{g_2} \right) \end{aligned}$$
(2.56)
$$\begin{aligned}&=\mu ^2_{{\mathbf {L}}}(x_2,h_2x_1) \otimes h_2h_1 \end{aligned}$$
(2.57)
where the second equality uses that \(\mu ^2_{\mathbf {L}}\) is \(\Gamma \)-equivariant, and the third equality uses \(\tau _{h_2} I_{h_2^{-1}} \tau _{h_1} I_{h_2}= \tau _{h_1} I_{h_2}=\tau _{h_2h_1}\). \(\square \)

2.4 Spherical Lagrangians

In this subsection, we apply the results from the previous subsections to the case that \(L=P\) such that
$$\begin{aligned}&P\text { is diffeomorphic to }S^n/\Gamma \text { for some }\Gamma \subset SO(n+1),\nonumber \\&\quad \text { so that the }\Gamma \text {-action is free, and }P\text { is spin} \end{aligned}$$
(2.58)

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

We apply the Leray spectral sequence to Lemma 2.3. The \(E_2\)-page is given by
$$\begin{aligned} E_2^{p,q}=H^p(\Gamma , H^q(S^n) \otimes _{{\mathbb {K}}} Hom_{\mathbb {K}}(\mathcal {E}^0_{o_L}, \mathcal {E}^1_{o_L})) \end{aligned}$$
where the \(\Gamma \)-action is given by \(x \otimes \psi \mapsto x \otimes g \cdot \psi \) and \(g \cdot \psi =\rho ^1(g^{-1}) \circ \psi \circ \rho ^0(g)\). As a result, we have
$$\begin{aligned} E_2^{p,q}=H^q(S^n) \otimes Ext^p_{\Gamma }(\Gamma , Hom_{\mathbb {K}}(\mathcal {E}^0_{o_L}, \mathcal {E}^1_{o_L})) \end{aligned}$$
When \({{\,\mathrm{char}\,}}({\mathbb {K}})\) does not divide \(|\Gamma |\), \({\mathbb {K}}[\Gamma ]\) is semi-simple by Maschke’s theorem. Therefore, \(Ext^p_{\Gamma }(\Gamma , Hom_{\mathbb {K}}(\mathcal {E}^0_{o_L}, \mathcal {E}^1_{o_L})) \ne 0\) only if \(p=0\). It implies that the spectral sequence degenerate at \(E_2\)-page and the result follows from the fact that \(Ext^0_{\Gamma }(\Gamma , Hom_{\mathbb {K}}(\mathcal {E}^0_{o_L}, \mathcal {E}^1_{o_L}))\) consists of \(\psi \in Hom_{\mathbb {K}}(\mathcal {E}^0_{o_L}, \mathcal {E}^1_{o_L})\) such that \(g \cdot \psi =\psi \), which is clearly \(Hom_{{\mathbb {K}}[\Gamma ]}(\mathcal {E}^0_{o_L},\mathcal {E}^1_{o_L})\). \(\square \)

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

If C is a differential graded algebra (e.g. a \({\mathbb {K}}\)-algebra with no differential), then C is viewed as an \(A_{\infty }\) algebra by
$$\begin{aligned} \mu ^1(a)&=(-1)^{|a|}\partial (a) \end{aligned}$$
(2.59)
$$\begin{aligned} \mu ^2(a_1,a_0)&=(-1)^{|a_0|}a_1a_0 \end{aligned}$$
(2.60)
and \(\mu ^k=0\) for \(k \geqslant 3\), where \(a,a_0,a_1 \in C\) and \(\partial \) is the differential of C.

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)].

Similarly, if x represents a generator of \(H^n({\mathbf {P}})\), then \(\{[x] \otimes h\}_{h \in \Gamma }\) are the degree n generators of \(H(C^*({\mathbf {P}}) \otimes R)\). It follows from (2.46) that
$$\begin{aligned} \mu ^2\left( \left[ \sum _g g{\mathbf {e}}\right] \otimes h_2,\left[ \sum _g g{\mathbf {e}}\right] \otimes h_1\right) =\left[ \sum _g g{\mathbf {e}}\right] \otimes h_2h_1 \end{aligned}$$
(2.61)
$$\begin{aligned} \mu ^2\left( [x] \otimes h_2,\left[ \sum _g g{\mathbf {e}}\right] \otimes h_1\right) =[x] \otimes h_2h_1 \end{aligned}$$
(2.62)
$$\begin{aligned} \mu ^2\left( \left[ \sum _g g{\mathbf {e}}\right] \otimes h_2,[x] \otimes h_1\right) =(-1)^{|x|}[h_2x] \otimes h_2h_1=(-1)^{|x|}[x] \otimes h_2h_1 \end{aligned}$$
(2.63)
Therefore, \(H(C^*({\mathbf {P}}) \otimes R)=H^*(S^n) \otimes _{{\mathbb {K}}} R\) as a \({\mathbb {K}}\)-algebra (see Convention 2.12). The result now follows from Lemma 2.3, 2.8 [see (2.31), (2.44)]. \(\square \)
Let \(\theta _g=1_{H^0(S^n)} \otimes g \in H^0(S^n) \otimes R\). By Lemma 2.13, we have a left \(\Gamma \)-action on \(HF(\mathcal {E},\mathcal {P})\) given by
$$\begin{aligned} x \mapsto [\mu ^2(\theta _g,x)] \end{aligned}$$
(2.64)
for any \(\mathcal {E}\in {\mathcal {F}}\). On the other hand, we have another left \(\Gamma \)-action on \(CF(\mathcal {E},\mathcal {P})\) given by (2.34), which descends to a left \(\Gamma \)-action on the cohomology \(HF(\mathcal {E},\mathcal {P})\).

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

We use the notations in the proof of Lemma 2.13. The element \(\theta _h \in H^0(S^n) \otimes R\) is represented by \(\sum _g g{\mathbf {e}}\otimes h \in C^0({\mathbf {P}}) \otimes R\). We have [see (2.44)]
$$\begin{aligned} \Phi _2\left( \sum _g g{\mathbf {e}}\otimes h\right)&=\sum _{g_2,g_1} g_2g_1{\mathbf {e}}\otimes I_{g_2^{-1}} \tau _h I_{g_2} \end{aligned}$$
(2.65)
$$\begin{aligned}&=\sum _{g} g{\mathbf {e}}\otimes I_{g^{-1}}\left( \sum _{g'} I_{g'} \tau _h I_{(g')^{-1}}\right) I_{g} \end{aligned}$$
(2.66)
Undoing the trivialization (2.16), we have
$$\begin{aligned} \Phi ^{-1}\left( \Phi _2\left( \sum _g g{\mathbf {e}}\otimes h\right) \right) =\sum _{g} g{\mathbf {e}}\otimes I_{c_{\mathbf {e}}}\left( \sum _{g'} I_{g'} \tau _h I_{(g')^{-1}}\right) I_{c_{\mathbf {e}}^{-1}} \end{aligned}$$
(2.67)
where \(c_{\mathbf {e}}:[0,1] \rightarrow {\mathbf {P}}\) is a path from \(o_{\mathbf {P}}\) to \({\mathbf {e}}\). With respect to the identification \((CF(({\mathbf {P}},{\mathbf {E}}),({\mathbf {P}},{\mathbf {E}})))^{\Gamma } =CF(\mathcal {P},\mathcal {P})\) [see (2.31)],
$$\begin{aligned}&\Phi ^{-1}\left( \Phi _2\left( \sum _g g{\mathbf {e}}\otimes h\right) \right) \nonumber \\&\quad =I_{\pi \circ c_{\mathbf {e}}}\left( \sum _{g'} I_{g'} \tau _h I_{(g')^{-1}}\right) I_{(\pi \circ c_{\mathbf {e}})^{-1}} \in Hom(E_e,E_e) \subset CF(\mathcal {P},\mathcal {P}) \end{aligned}$$
(2.68)
Without loss of generality, we can assume \(e=o_L\) so
$$\begin{aligned} \sum _{g'} I_{g'} \tau _h I_{(g')^{-1}} \in Hom(E_{o_L},E_{o_L}) \subset CF(\mathcal {P},\mathcal {P}) \end{aligned}$$
(2.69)
represents \(\theta _h\) under the isomorphism \(HF^0(\mathcal {P},\mathcal {P})=H^0(S^n) \otimes R\).
For each \(y \in \Gamma \subset E_{o_L}\), there is a unique \(g'\)(\(=h*y\)) such that \(\tau _h I_{(g')^{-1}}(y) \ne 0\). Therefore, \(\sum _{g'} I_{g'} \tau _h I_{(g')^{-1}}(y) =hy\) for all \(y \in E_{o_L}\). In particular, it means that
$$\begin{aligned} \sum _{g'} I_{g'} \tau _h I_{(g')^{-1}}= h \left( \sum _{g'} I_{g'} \tau _{1_{\Gamma }} I_{(g')^{-1}}\right) \end{aligned}$$
(2.70)
so \(\theta _{h}=h \theta _{1_{\Gamma }}\) and hence \(\mu ^2(\theta _h,\theta _{1_{\Gamma }})=(-1)^{|\theta _{1_{\Gamma }}|}\theta _h=h\theta _{1_{\Gamma }}\) as desired. \(\square \)

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

Let \(x \in HF(\mathcal {E},\mathcal {P})\). We have
$$\begin{aligned} {[}\mu ^2(\theta _g,x)]&=[\mu ^2(\mu ^2(\theta _g,\theta _{1_{\Gamma }}),x)] \end{aligned}$$
(2.71)
$$\begin{aligned}&={[}\mu ^2(g\theta _{1_{\Gamma }},x)] \end{aligned}$$
(2.72)
$$\begin{aligned}&={[}g\mu ^2(\theta _{1_{\Gamma }},x)] \end{aligned}$$
(2.73)
$$\begin{aligned}&=(-1)^{|x|}gx \end{aligned}$$
(2.74)
where the first equality uses Lemma 2.13, the second equality uses Lemma 2.14, the third equality uses Corollary 2.6 and the last equality uses that \(\theta _{1_{\Gamma }}\) is a cohomological unit. \(\square \)
Similarly, for any \(\mathcal {E}\in {\mathcal {F}}\), we have a right \(\Gamma \)-action on \(HF(\mathcal {P},\mathcal {E})\) given by
$$\begin{aligned} x \mapsto [\mu ^2(x,\theta _g)] \end{aligned}$$
(2.75)
and another right action on \(HF(\mathcal {P},\mathcal {E})\) given by (2.35). The analog of Corollary 2.16 holds, (i.e. \(\mu ^2(\theta _{1_{\Gamma }},\theta _{h})=\theta _{h}= \theta _{1_{\Gamma }}h\)) and we leave the details to readers.

Corollary 2.17

The two right \(\Gamma \)-actions (2.75) and (2.35) on \(HF(\mathcal {P},\mathcal {E})\) coincide (without additional factor of \(-1)\) for all \(\mathcal {E}\in {\mathcal {F}}\).

2.5 Equivariant evaluation

In this subsection, we want to give the definition of
$$\begin{aligned} T_{\mathcal {P}}(\mathcal {E}):=Cone(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } \mathcal {P}\xrightarrow {ev} \mathcal {E}) \end{aligned}$$
(2.76)
that arises in (1.1) in the context of Fukaya category. We will keep the exposition minimal and self-contained here.
Let \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\) be the DG category of perfect \(A_{\infty }\) right modules over \({\mathcal {F}}\). We have a cohomologically full and faithful Yoneda embedding [2, Section (2g)]
$$\begin{aligned} {\mathcal {Y}}: {\mathcal {F}}\rightarrow {\mathcal {F}}^{{{\,\mathrm{perf}\,}}} \end{aligned}$$
(2.77)
By abuse of notation, we use \(\mathcal {E}\) to denote \({\mathcal {Y}}(\mathcal {E})\) for \(\mathcal {E}\in Ob({\mathcal {F}})\).
Let P be a Lagrangian brane such that \(\pi _1(P)=\Gamma \), and \(\mathcal {P}\) be the object with underlying Lagrangian P equipped with the universal local system E. Let \(\mathcal {E}\in Ob({\mathcal {F}})\). By Corollary 2.6, we know that [see (2.35)]
$$\begin{aligned} \mu ^1_{\mathcal {F}}(\psi )g=\mu ^1_{\mathcal {F}}(\psi g) \end{aligned}$$
(2.78)
for \(\psi \in hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E})\) so \(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E})\) is a DG right \(\Gamma \)-module.
Given a DG right \(\Gamma \)-module V, we define an object \(V\otimes _\Gamma \mathcal {P}\in Ob({\mathcal {F}}^{perf})\) as follows: For every \(X \in Ob({\mathcal {F}})\), we have a cochain complex
$$\begin{aligned} (V \otimes _{\Gamma } \mathcal {P})(X):=V \otimes _{\Gamma } hom_{\mathcal {F}}(X,\mathcal {P}) \end{aligned}$$
(2.79)
where the left \(\Gamma \)-actions on \(hom_{\mathcal {F}}(X,\mathcal {P})\) is given by (2.34). By Corollary 2.6, we have
$$\begin{aligned} \left\{ \begin{array}{ll} \mu ^1_V(vg) \otimes \psi =\mu ^1_V(v)g \otimes \psi \\ v \otimes \mu ^1_{\mathcal {F}}(g\psi )=v \otimes g\mu ^1_{\mathcal {F}}(\psi ) \end{array} \right. \end{aligned}$$
(2.80)
for \(v \otimes \psi \in V \otimes _{\Gamma } hom_{\mathcal {F}}(X,\mathcal {P})\) so
$$\begin{aligned} \mu ^{1|0}: v \otimes \psi \mapsto (-1)^{|\psi |-1}\mu ^1_V(v) \otimes \psi + v \otimes \mu _{\mathcal {F}}^1(\psi ) \end{aligned}$$
(2.81)
is a well-defined differential on \(V \otimes _{\Gamma } hom_{\mathcal {F}}(X,\mathcal {P})\).
The \(A_{\infty }\) right \({\mathcal {F}}\) module structure on \( V \otimes _{\Gamma } \mathcal {P}\) is given by
$$\begin{aligned} \mu ^{1|d-1}: (v \otimes \psi , x_{d-1}, \dots ,x_1) \mapsto v \otimes \mu _{\mathcal {F}}^d(\psi ,x_{d-1},\dots ,x_1) \end{aligned}$$
(2.82)
for \(v \otimes \psi \in V \otimes _{\Gamma } hom_{\mathcal {F}}(X_d,\mathcal {P})\) and \(x_j \in hom_{\mathcal {F}}(X_{j},X_{j+1})\). The morphism \(\mu ^{1|d-1}\) is well-defined by Corollary 2.6 and we leave it to readers to check that \(\{\mu ^{1|j}\}_{j=0}^\infty \) satisfies \(A_{\infty }\) module relations [2, Equation (1.19)]. In particular, we have an \(A_{\infty }\) right \({\mathcal {F}}\) module \(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } \mathcal {P}\).
Now we want to define an \(A_{\infty }\) morphism
$$\begin{aligned} ev_\Gamma : hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } \mathcal {P}\rightarrow \mathcal {E}\end{aligned}$$
(2.83)
as follows. For \(\psi ^2 \otimes \psi ^1 \in hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } hom_{\mathcal {F}}(X_d,\mathcal {P})\) and \(x_j \in hom_{\mathcal {F}}(X_{j},X_{j+1})\), we define
$$\begin{aligned} ev_\Gamma ^d: (\psi ^2 \otimes \psi ^1, x_{d-1}, \dots ,x_1) \mapsto \mu ^{d+1}_{\mathcal {F}}(\psi ^2,\psi ^1, x_{d-1}, \dots ,x_1) \end{aligned}$$
(2.84)
The well-definedness follows from Corollary 2.6 again. The fact that \(ev_\Gamma =\{ev_\Gamma ^d\}_{d=1}\) defines an \(A_{\infty }\) morphism follows from the \(A_{\infty }\) relations of \({\mathcal {F}}\). As a consequence, we can define
$$\begin{aligned} T_{\mathcal {P}}(\mathcal {E}):=Cone(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } \mathcal {P}\xrightarrow {ev_\Gamma } \mathcal {E}) \end{aligned}$$
(2.85)
as the \(A_{\infty }\) mapping cone for the \(A_{\infty }\) morphism \(ev_\Gamma \) (see [2, Section (3e)]). In particular, for \(X \in Ob({\mathcal {F}})\), we have a cochain complex
$$\begin{aligned} T_{\mathcal {P}}(\mathcal {E})(X)=(hom_{{\mathcal {F}}}(\mathcal {P},\mathcal {E}) \otimes _{\Gamma } hom_{{\mathcal {F}}}(X,\mathcal {P}))[1] \oplus hom_{{\mathcal {F}}}(X,\mathcal {E}) \end{aligned}$$
(2.86)
with differential and multiplication given by
$$\begin{aligned} \mu ^1_{T_{\mathcal {P}}(\mathcal {E})}(\psi ^2 \otimes \psi ^1,x)&=((-1)^{|\psi ^1|-1}\mu ^1_{\mathcal {F}}(\psi ^2) \otimes \psi ^1 \nonumber \\&\quad + \psi ^2 \otimes \mu _{\mathcal {F}}^1(\psi ^1),\mu ^1_{\mathcal {F}}(x)+\mu ^2_{\mathcal {F}}(\psi ^2, \psi ^1)) \end{aligned}$$
(2.87)
$$\begin{aligned} \mu ^2_{T_{\mathcal {P}}(\mathcal {E})}((\psi ^2 \otimes \psi ^1,x),a)&=(\psi ^2 \otimes \mu ^2_{\mathcal {F}}(\psi ^1,a), \mu ^2_{\mathcal {F}}(x,a)+\mu ^3_{\mathcal {F}}(\psi ^2,\psi ^1,a)) \end{aligned}$$
(2.88)
Finally, we want to state a functorial property of \(T_{\mathcal {P}}(\mathcal {E})\).

Corollary 2.18

Let \({\mathcal {F}}_0, {\mathcal {F}}_1\) be the Fukaya categories with respect to two different sets of choices of auxiliary data. The Lagrangian branes \(\mathcal {P}, \mathcal {E}\) above will be denoted by \(\mathcal {P}_j, \mathcal {E}_j\), respectively, when we regard them as objects in \({\mathcal {F}}_j\), for \(j=0,1\). Let \({\mathcal {G}}:{\mathcal {F}}_0 \rightarrow {\mathcal {F}}_1\) be a quasi-equivalence sending \(\mathcal {P}_0\) to \(\mathcal {P}_1\) and \(\mathcal {E}_0\) to \(\mathcal {E}_1\). Then
$$\begin{aligned} {\mathcal {G}}(T_{\mathcal {P}_0}( \mathcal {E}_0)) \simeq T_{\mathcal {P}_1}(\mathcal {E}_1) \end{aligned}$$

The proof is straightforward along the same line as [2, Lemma 5.6] and is left to interested readers.

Remark 2.19

A thorough discussion of the categorical notions can be found in [31], which is the extended version of the current paper. The readers can also find an intrinsic proof of Corollary 2.18, and an explanation of Remark 1.4, in [31].

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

A cylindrical almost complex structure on the symplectization \(SY:=({\mathbb {R}}\times Y,d(e^r\alpha ))\) is an almost complex structure such that
  • 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 \(\ker (\alpha )\)

The set of cylindrical almost complex structures is denoted by \({\mathcal {J}}^{cyl}(Y,\alpha )\). If \(I \subset {\mathbb {R}}\) is an interval, we call J a cylindrical almost complex structure on \((I \times Y,d(e^r\alpha ))\) if \(J=J'|_{I \times Y}\) for some \(J' \in {\mathcal {J}}^{cyl}(Y,\alpha )\). Let \((M,\omega ,\theta )\) be a Liouville domain with a separating contact hypersurface \((Y, \alpha =\theta |_Y)\) such that \(Y \cap \partial M=\varnothing \). By the neighborhood theorem, there is a neighborhood \(N(Y) \subset M\) of Y such that we have a symplectomorphism
$$\begin{aligned} \Phi _{N(Y)}:(N(Y),\omega |_{N(Y)}) \simeq ((-\epsilon ,\epsilon ) \times Y,d(e^t \alpha )) \end{aligned}$$
(3.1)
for some \(\epsilon >0\).
Let \(J^0\) be a compatible almost complex structure on M such that \((\Phi _{N(Y)})_*(J^0|_{N(Y)})\) is cylindrical. We say that a smooth family of compatible almost complex structure \((J^{\tau })_{\tau \in [0,\infty )}\) on M is adjusted to N(Y) if
$$\begin{aligned} \left\{ \begin{array}{ll} J^\tau |_{M \setminus N(Y)}=J^0|_{M \backslash N(Y)} \text { for all } \tau \\ \text {for each }\tau \text {, we have }\Phi ^{\tau }_{N(Y)}:(N(Y),J^{\tau }|_{N(Y)}) \simeq ((-(\tau +\epsilon ),\tau +\epsilon ) \times Y, (J^\tau )') \end{array} \right. \end{aligned}$$
(3.2)
where \(\Phi ^{\tau }_{N(Y)}\) is an isomorphism of almost complex manifolds, the diffeomorphism \(\Phi ^{\tau }_{N(Y)} \circ (\Phi _{N(Y)})^{-1}\) is the identity on the Y factor, and \((J^\tau )'\) is the unique cylindrical almost complex structure such that \((J^\tau )'|_{(-\epsilon ,\epsilon ) \times Y}=(\Phi _{N(Y)})_*(J^0|_{N(Y)})\).

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

There is a variant for being adjusted to N(Y). For a fixed number \(R \geqslant 0\), we call a smooth family of compatible almost complex structure \((J^{\tau })_{\tau \in [3R,\infty )}\) on M is R-adjusted to N(Y) if (3.2) is satisfied but the property of \((J^\tau )'\) is replaced by the following conditions.
$$\begin{aligned} \left\{ \begin{array}{ll} (J^\tau )'|_{[-(\tau +\epsilon -2R),\tau +\epsilon -2R] \times Y} \text { is cylindrical for all }\tau \\ (J^\tau )'|_{(-\epsilon ,\epsilon ) \times Y}=(\Phi _{N(Y)})_*(J^0|_{N(Y)}) \text { for all }\tau \\ (J^{\tau _1})'|_{(-(\tau _1+\epsilon ),-(\tau _1+\epsilon -2R)] \times Y}= (\phi _{\tau _1,\tau _2}^-)_* (J^{\tau _2})'|_{(-(\tau _2+\epsilon ),-(\tau _2+\epsilon -2R)] \times Y} \text { for all } \tau _1,\tau _2\\ (J^{\tau _1})'|_{[\tau _1+\epsilon -2R,\tau _1+\epsilon ) \times Y}= (\phi _{\tau _1,\tau _2}^+)_* (J^{\tau _2})'|_{[\tau _2+\epsilon -2R,\tau _2+\epsilon ) \times Y} \text { for all } \tau _1,\tau _2 \end{array} \right. \end{aligned}$$
where \(\phi _{\tau _1,\tau _2}^-:(-(\tau _2+\epsilon ),-(\tau _2+\epsilon -2R)] \times Y \rightarrow (-(\tau _1+\epsilon ),-(\tau _1+\epsilon -2R)] \times Y \) and \(\phi _{\tau _1,\tau _2}^+:[\tau _2+\epsilon -2R,\tau _2+\epsilon ) \times Y \rightarrow [\tau _1+\epsilon -2R,\tau _1+\epsilon ) \times Y\) are the r-translation.

When \(R=0\), being R-adjusted 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 R-adjusted to N(Y). Let \(x_0 \in CF(L_0,L_d)\) and \(x_j \in CF(L_{j-1},L_j)\) 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 well-known so we only give a quick review and introduce necessary notations along the way.

First, \(\mathcal {T}\) is a tree with \(d+1\) semi-infinite edges and one of them is distinguished which is called the root. The other semi-infinite 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 \(j=0,\dots ,d_v-1\), 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 \(j^{th}\) semi-infinite edge adjacent to v, then \(u_v\) is asymptotic to \(x_j\) at \(f^{-1}_v(e)\).

Finally, for each \(j=1,\dots ,n_\mathcal {T}-1\), there is at least one \(v \in V(\mathcal {T})\) such that \(l_\mathcal {T}(v)=j\) and \(u_v\) is not a trivial cylinder (i.e. \(u_v\) is not a map \({\mathbb {R}}\times [0,1] \rightarrow SY\) or \({\mathbb {R}}\times S^1 \rightarrow SY\)) such that
$$\begin{aligned} u_v(s,t)=(f_r(s),f_Y(t)) \in {\mathbb {R}}\times Y \end{aligned}$$
(3.3)
for some \(f_r\), \(f_Y\)). We use \(\mathcal {M}^{J^{\infty }}(x_0;x_d,\dots ,x_1)\) to denote the set of such holomorphic buildings.

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 SFT-stretching 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.

Let
  • \(V^{core}\) 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 \(\partial \Sigma _v= \varnothing \).

In particular, we have \(V(\mathcal {T})=V^{core} \sqcup V^{\partial } \sqcup V^{int}\). Let \(\mathcal {T}^{core}\), \(\mathcal {T}^{\partial }\) and \(\mathcal {T}^{int}\) be the subgraphs of \(\mathcal {T}\), which consists of vertices \(V^{core}\), \(V^{\partial }\) and \(V^{int}\), and edges adjacent to their respective vertices (see Fig. 1 for an example). Note that these three subtrees could have overlaps.

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.\(\square \)

Fig. 1

A tree \(\mathcal {T}\) with 2 leaves. Black dots: elements in \(V^{core}\); Green dots: elements in \(V^{\partial }\); Red dots: elements in \(V^{int}\); Black tree: \(\mathcal {T}^{core} \backslash (\mathcal {T}^{\partial } \cup \mathcal {T}^{int})\); Green sugbraph: \(\mathcal {T}^{\partial } \backslash \mathcal {T}^{int}\); Red subgraph: \(\mathcal {T}^{int}\) (Color figure online)

Convention 3.6

We need to explain the convention of strip-like 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.

First assume that \(l_\mathcal {T}(v)+1=l_\mathcal {T}(v')\). If e is a boundary (resp. interior) edge, we use an outgoing/positive strip-like end (2.5) (resp. cylindrical end) for \(f_v^{-1}(e)\), where an outgoing/positive cylindrical end for \(f_v^{-1}(e)\) is a holomorphic embedding of \(\epsilon _{v,e}:\{z=s\exp (\sqrt{-1}t) \in {\mathbb {C}}| s \geqslant 1\} \rightarrow \Sigma _v\) such that \(\lim _{|z| \rightarrow \infty } \epsilon _{v,e}(z)=f_v^{-1}(e)\). With respect to coordinates given by the strip-like (resp. cylindrical) end \(\epsilon _{v,e}\), we have
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \lim _{s \rightarrow \infty } \pi _Y(u_v(\epsilon _{v,e}(s,t)))=x(Tt) \text { (resp. } \gamma (Tt) ) \\ \displaystyle \lim _{s \rightarrow \infty } \pi _{\mathbb {R}}(u_v(\epsilon _{v,e}(s,t)))=\infty \end{array} \right. \end{aligned}$$
(3.4)
for some Reeb chord \(x\ (\)resp. orbit \(\gamma )\) and some \(T>0\), where \(\pi _Y,\pi _{\mathbb {R}}\) are the projection from SY to the two factors. In this case, we call \(x\ (\)resp. \(\gamma )\) the positive asymptote of \(u_v\) at \(f_v^{-1}(e)\).
On the other hand, we use an incoming/negative strip-like end (2.6) (resp. cylindrical end) for \(f_{v'}^{-1}(e)\), where an incoming/negative cylindrical end for \(f_{v'}^{-1}(e)\) is a holomorphic embedding of \(\epsilon _{v',e}:\{z=s\exp (\sqrt{-1}t) \in {\mathbb {C}}| 0 < s \leqslant 1\} \rightarrow \Sigma _{v'}\) such that \(\lim _{|z| \rightarrow 0} \epsilon _{v',e}(z)=f_{v'}^{-1}(e)\). With respect to coordinates given by the strip-like (resp. cylindrical) end \(\epsilon _{v',e}\), we have
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \lim _{s \rightarrow 0} \pi _Y(u_v(\epsilon _{v',e}(s,t)))=x(Tt) \text { (resp. } \gamma (Tt) ) \\ \displaystyle \lim _{s \rightarrow 0} \pi _{\mathbb {R}}(u_v(\epsilon _{v',e}(s,t)))=-\infty \end{array} \right. \end{aligned}$$
(3.5)
or some Reeb chord \(x\ (\)resp. orbit \(\gamma )\) and some \(T>0\). In this case, we call \(x\ (\)resp. \(\gamma )\) the negative asymptote of \(u_{v'}\) at \(f_{v'}^{-1}(e)\).

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 strip-like end for \(f_v^{-1}(e)\) and an incoming/negative strip-like 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

Let \(P \subset (M,\omega ,\theta )\) be a Lagrangian submanifold which satisfies (2.58). In particular, \(H^1(P,{\mathbb {R}})=0\) and P is an exact Lagrangian. The round metric on \(S^n\) descends to a Riemannian metric on P. Let U be a Weinstein neighborhood of P and we identify \(\partial U\) with the set of covectors of P having a common small fixed norm. Without loss of generality, we can assume that \(\theta |_U=\theta _{T^*P}\), where \(\theta _{T^*P}\) is the standard Liouville one-form on \(T^*P\). Let \(\alpha _0:=\theta |_{\partial U}\) be the standard contact form on \(\partial U\). Eventually, we will apply Theorem 3.3 along a perturbation \((\partial U)'\) of \(\partial U\). Since \(((\partial U)',\theta |_{(\partial U)'}) \simeq (\partial U, \alpha ')\) for a perturbation \(\alpha '\) of \(\alpha _0\), we will need to understand the Reeb dynamics of \(\alpha '\). Therefore, it is helpful to explain the Reeb dynamics of \((\partial U, \alpha _0)\) first. We assume \(\Lambda _i:=L_i \cap \partial U\) are (possibly empty) unions of cospheres at points of P. There are four types of asymptotes that can appear for \(u_v\) near the punctures.
  1. 1.

    Lagrangian intersection points between \(SL_i^\pm \) and \(SL_j^\pm \) in \(SM^\pm \),

     
  2. 2.

    Reeb chords from \(\Lambda _i\) to \(\Lambda _j\) in Y for \(i \ne j\),

     
  3. 3.

    Reeb chords from \(\Lambda _i\) to itself in Y, and

     
  4. 4.

    Reeb orbits in Y

     
We want to discuss the grading for each of these types.

3.2.1 Type one

Let \(\Omega \) be the nowhere-vanishing section of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) which equals to 1 with respect to the chosen trivialization (see Standing Assumption). For a Lagrangian subspace \(V \subset T_pM\) and a choice of basis \(\{X_1, \dots , X_n\}\) of V, we define
$$\begin{aligned} Det_{\Omega }(V):=\frac{\Omega (X_1,\dots ,X_n)}{\Vert \Omega (X_1,\dots ,X_n)\Vert } \in S^1 \end{aligned}$$
(3.6)
which is independent of the choice of basis. A \({\mathbb {Z}}\)-grading of \(L_i\) is a continuous function \({\theta _{L_i}:}L_i \rightarrow {\mathbb {R}}\) such that \(e^{2\pi \sqrt{-1} \theta _{L_i}(p)}=Det_{\Omega }(T_pL_i)\) for all \(p \in L_i\).
At each transversal intersection point \(x \in L_i \cap L_j\), we have two graded Lagrangian planes \(T_xL_i\), \(T_xL_j\) inside \(T_xM\). The grading of x as a generator of \(CF(L_i,L_j)\) is given by the Maslov grading from \(T_xL_i\) to \(T_xL_j\) which is
$$\begin{aligned} |x|=\iota (T_xL_i,T_xL_j):=n+\theta _{L_j}(x)-\theta _{L_i}(x)-2Angle(T_xL_i,T_xL_j) \end{aligned}$$
(3.7)
where \(Angle(T_xL_i,T_xL_j)= \sum _{j=1}^n \beta _j\) and \(\beta _j \in (0,\frac{1}{2})\) are such that there is a unitary basis \(u_1, \dots ,u_n\) of \(T_xL_i\) satisfying \(T_xL_j=Span_{\mathbb {R}}\{e^{2\pi \sqrt{-1}\beta _j}u_j\}_{j=1}^n\). If we regard x as an element in \(CF(L_j,L_i)\), then we have \(\iota (T_xL_j,T_xL_i)=n-\iota (T_xL_i,T_xL_j)\).

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 \(|x|=n-|x^\vee |\).

Since \(SM^-=T^*P\) and \(w_2(P)=0\) and \(c_1(T^*P)=0\), there is a preferred choice of trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*SM^-)^{\otimes 2}\) such that the grading functions on cotangent fibers and the zero section are constant functions (see [29]). Without loss of generality, we can assume that the restriction to \(M^-\) of the choice of trivialization of \((\Lambda _{\mathbb {C}}^{top}T^*M)^{\otimes 2}\) we picked coincides with that of \((\Lambda _{\mathbb {C}}^{top}T^*SM^-)^{\otimes 2}\). We call that the cotangent fibers and the zero section are in canonical relative grading if the following holds:
$$\begin{aligned} CF(P,T^*_qP) \hbox { is concentrated at degree}\ 0 \end{aligned}$$
(3.8)
for all \(q \in P\).

We refer readers to [29, 2, Section 11, 12] for more about Maslov gradings.

3.2.2 Type two

In general, if we have a Reeb chord \(x=(x(t))_{t \in [0,1]}\) from \(\Lambda _0\) to \(\Lambda _1\) in a contact manifold \((Y,\alpha )\) such that \(S\Lambda _i\) are graded Lagrangians in SY for both i, we will assign a grading to x by regarding x as a Hamiltonian chord between graded Lagrangians \(S\Lambda _0\) and \(S\Lambda _1\) in the symplectic manifold SY as follows: There is an appropriate Hamiltonian H in SY that depends only on the radial coordinate r such that the Reeb vector field \(R_{\alpha }\) in Y coincides with the restriction of the Hamiltonian vector field \(X_H\) to \(\{0\} \times Y\). Let \(\phi ^H\) be the time-one flow of H. We identify \(x(t) \in Y\) with \((0,x(t)) \in SY\) so x is a H-Hamiltonian chord. We have graded Lagrangian subspaces \((\phi ^H)_*T_{x(0)}S\Lambda _0\) and \(T_{x(1)}S\Lambda _1\) in \(T_{x(1)}SY\). Let
$$\begin{aligned} K_x:=(\phi ^H)_*(T_{x(0)}S\Lambda _0) \cap T_{x(1)}S\Lambda _1 \end{aligned}$$
(3.9)
The grading |x| of x is defined to be
$$\begin{aligned} |x|=\iota ((\phi ^H)_*(T_{x(0)}S\Lambda _0)/K_x, T_{x(1)}S\Lambda _1/K_x) \end{aligned}$$
(3.10)
where the Maslov grading [see (3.7)] is computed in the symplectic vector space \(T_{x(1)}M/(K_x+ J(K_x))\). More details about Maslov gradings assigned to non-transversally intersecting graded Lagrangian subspaces can be found in [21, Section 4.1], for example.

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 \(|u|=-(n-1)\)) 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.

Let \({\mathbf {q}}_i \in {\mathbf {P}}\) be a lift of \(q_i\). Each Reeb chord x from \(\Lambda _i\) to \(\Lambda _j\) corresponds to a geodesic from \(q_i\) to \(q_j\), which can be lifted to a geodesic \({\mathbf {x}}\) from \({\mathbf {q}}_i\) to a point \({\mathbf {q}}_j \in {\mathbf {P}}\) such that \(\pi ({\mathbf {q}}_j)=q_j\). If
$$\begin{aligned} {\mathbf {q}}_j\text { is not the antipodal point of }{\mathbf {q}}_i\text { and }{\mathbf {q}}_j \ne {\mathbf {q}}_i \end{aligned}$$
(3.11)
then there is a unique closed geodesic (assumed to have length \(2\pi \)) passing through \({\mathbf {q}}_i\) and \({\mathbf {q}}_j\). Therefore, for each interval \(I_k=(k\pi ,(k+1 \pi ))\), \(k\in {\mathbb {N}}\), there is a unique geodesic from \({\mathbf {q}}_i\) to \({\mathbf {q}}_j\) with length lying inside \(I_k\). If the length of \({\mathbf {x}}\) lies in \(I_k\), then
$$\begin{aligned} |x|=-k(n-1) \end{aligned}$$
(3.12)
For generic \(q_i,q_j\), every lifts \({\mathbf {q}}_i,{\mathbf {q}}_j\) of \(q_i,q_j\) satisfies (3.11).

3.2.3 Type three

There are four kinds of Reeb chords from \(\Lambda _i\) to itself. First, if x is a Reeb chord from one connected component of \(\Lambda _i\) to a different one, then the computation of |x| reduces to the Type two (Sect. 3.2.2). For the remaining three kinds, we assume \(\Lambda _i=T^*_{q_i}P \cap \partial U\), i.e. it has exactly one connected component. Let \({\mathbf {q}}_i\) be a lift of \(q_i\) and \({\mathbf {x}}:[0,1] \rightarrow {\mathbf {P}}\) be the lift of the geodesic such that \({\mathbf {x}}(0)={\mathbf {q}}_i\), \({\mathbf {q}}_i':={\mathbf {x}}(1)\) and \(\pi ({\mathbf {q}}_i')=q_i\). The three possibilities are
  1. 1.

    \({\mathbf {q}}_i,{\mathbf {q}}_i'\) satisfy (3.11) (with \({\mathbf {q}}_i'\) replacing \({\mathbf {q}}_j\)), or

     
  2. 2.

    \({\mathbf {q}}_i'\) is the antipodal point of \({\mathbf {q}}_i\), or

     
  3. 3.

    \({\mathbf {q}}_i={\mathbf {q}}_i'\)

     
For the first case, the computation of |x| reduces to the previous one again (Sect. 3.2.2). For the second and the third cases, we have [see (3.9) for the meaning of \(K_x\)]
$$\begin{aligned}&K_x= T_{x(1)}L_i \end{aligned}$$
(3.13)
$$\begin{aligned}&|x|=\iota ((\phi ^H)_*T_{x(0)}L_i/K_x, T_{x(1)}L_j/K_x)=-k(n-1) \end{aligned}$$
(3.14)
where \(k\pi \) is the length of \({\mathbf {x}}\), and the term \(-k(n-1)\) is exactly the (integral) difference of the values of the grading functions at \((\phi ^H)_*T_{x(0)}L_i\) and \(T_{x(1)}L_i\) as graded Lagrangian planes.

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 \(\dim (S_{x})=n-1\).

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 Conley-Zehnder 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 can now define a path of symplectic matrices \((\Phi _t)_{t \in [0,1]}\) given by \(\Phi _t:=(\phi ^{R}_t)_*:\xi _{\gamma (0)} \rightarrow \xi _{\gamma (t)} \simeq \xi _{\gamma (0)}\), where \(\phi ^{R}_t\) is the time-t flow generated by \(R_{\alpha _0}\) and the last isomorphism is given by \(\Phi _\gamma \). We can assign the Robbin–Salamon index for \(\Phi _t\) as follows: first, we isotope (relative to end points) \(\Phi _t\) to a path of symplectic matrices \(\Phi _t'\) such that \(\ker (\Phi _t'-Id) \ne 0\) happens at finitely many times \(t=t_1,\dots ,t_k \) and for each \(t_j\), the crossing form \(J\frac{d}{dt}|_{t=t_j}(\Phi _t')\) is non-degnerate on \(\ker (\Phi _{t_j}'-Id)\). The signature of \(J\frac{d}{dt}|_{t=t_j}(\Phi _t')\) is denoted by \(\sigma (t_j)\) and the Robbin–Salamon index is defined by
$$\begin{aligned} \mu _{RS}(\Phi _t):=\frac{1}{2}\sigma (0) +\sum _{j=1}^k \sigma (t_j) +\frac{1}{2}\sigma (1) \end{aligned}$$
(3.15)
where \(\sigma (1)\) is defined to be zero if \(\Phi _1\) is invertible. The index is independent of the choice of \(\Phi _{t_j}'\). The Robbin–Salamon index of \(\gamma \) with respect to the trivialization \(\Phi _\gamma \) is
$$\begin{aligned} \mu _{RS}(\gamma ):=\mu _{RS}(\Phi _t) \end{aligned}$$
(3.16)
Any two choices of \(\Phi _\gamma \) satisfying the compatibility condition would give the same index.
There are two kinds of Reeb orbits \(\gamma \) in \(\partial U\), namely, contractible in U or not. We are only interested in the case that \(\gamma \) is contractible in U, which means that it can be lifted to a Reeb orbit in \(\partial {\mathbf {U}}\) that is contractible in \({\mathbf {U}}\). The lifted Reeb orbit corresponds to a geodesic loop \(l_{\gamma }\) in \({\mathbf {P}}\). The Robbin–Salamon index \(\mu _{RS}(\gamma )\) is computed in [38, Lemma 7] (the proof there can be directly generalized to all n)
$$\begin{aligned} \mu _{RS}(\gamma )=2k(n-1) \end{aligned}$$
(3.17)
where k is the covering multiplicity of \(l_{\gamma }\) with respect to the simple geodesic loop, or equivalently, \(2k\pi \) is the length of \(l_{\gamma }\).

We want to point out that \(\gamma \) lies in a Morse–Bott family \(S_{\gamma }\) of (unparametrized) Reeb orbits and \(\dim (S_{\gamma })=2n-2\).

3.3 Dimension formulae

In this section, we first review the virtual dimension formula from [37], where the domain of the pseudo-holomorphic 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 J-holomorphic map with positive asymptotes \(\{\gamma _j^+ \}_{j=1}^{s_+}\) and negative asymptotes \(\{\gamma _j^- \}_{j=1}^{s_-}\ (\)see Convention 3.6).

Then the virtual dimension of u is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u)=\,&(n-3)(2-2g-s^+-s^-)+\sum _{j=1}^{s^+} \mu _{RS}(\gamma ^+_j)\nonumber \\&-\sum _{j=1}^{s^-} \mu _{RS}(\gamma ^-_j) +\frac{1}{2}\sum _{j=1}^{s^+} \dim (S_{\gamma ^+_j}) \nonumber \\&+\frac{1}{2}\sum _{j=1}^{s^-} \dim (S_{\gamma ^-_j})+2c_1^{rel}(TX)([u]) \end{aligned}$$
(3.18)
where \(2c_1^{rel}(TX)([u])\) is the relative first Chern class computed with respect to the fixed symplectic trivializations along the Reeb orbits that we chose to compute \(\mu _{RS}\ (\)see Sect. 3.2.4).

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 \(\mu _{RS}({\overline{\Phi }}_t^{\pm ,j})=\mu _{RS}(\Phi _t^{\pm ,j})+\mu _{RS}(I_{2 \times 2})=\mu _{RS}(\Phi _t^{\pm ,j})\).

If \(\ker ({\overline{\Phi }}_1^{\pm ,j}-Id)=0\) (which is never the case) for all \(\gamma ^\pm _j\), then the index of u is given by
$$\begin{aligned} {{\,\mathrm{ind}\,}}(u)=\,&n(2-2g-s^+-s^-)+\sum _{j=1}^{s^+} \mu _{RS}(\gamma ^+_j)\nonumber \\&-\sum _{j=1}^{s^-} \mu _{RS}(\gamma ^-_j)+2c_1^{rel}(TX)([u]) \end{aligned}$$
(3.19)
If \(\ker ({\overline{\Phi }}_1^{\pm ,j}-Id) \ne 0\), then it contributes \(\dim (\ker ({\overline{\Phi }}_1^{\pm ,j}-Id))=\dim (S_{\gamma ^\pm _j})+2\) (resp. 0) to \({{\,\mathrm{ind}\,}}(u)\) when \(\gamma ^\pm _j\) is a positive (resp. negative) asymptote. However, the definition of \(\mu _{RS}\) already takes into account \(\frac{1}{2}\dim (\ker ({\overline{\Phi }}_1^{\pm ,j}-Id))\) so we have
$$\begin{aligned} {{\,\mathrm{ind}\,}}(u)&=n(2-2g-s^+-s^-)+\sum _{j=1}^{s^+} (\mu _{RS}(\gamma ^+_j)+\frac{1}{2}(\dim (S_{\gamma ^+_j})+2)) \end{aligned}$$
(3.20)
$$\begin{aligned}&\quad -\sum _{j=1}^{s^-} (\mu _{RS}(\gamma ^-_j)-\frac{1}{2}(\dim (S_{\gamma ^-_j})+2))+2c_1^{rel}(TX)([u]) \end{aligned}$$
(3.21)
$$\begin{aligned}&=n(2-2g-s^+-s^-)+\sum _{j=1}^{s^+} \mu _{RS}(\gamma ^+_j)-\sum _{j=1}^{s^-} \mu _{RS}(\gamma ^-_j) +\frac{1}{2}\sum _{j=1}^{s^+} \dim (S_{\gamma ^+_j}) \nonumber \\&\quad +\frac{1}{2}\sum _{j=1}^{s^-} \dim (S_{\gamma ^-_j})+(s^++s^-)+2c_1^{rel}(TX)([u]) \end{aligned}$$
(3.22)
Finally, to obtain the virtual dimension, we need to add the dimension of the Teichmüller space that \(\Sigma \) lies, which is \(6g-6+2(s^++s^-)\). It gives the formula (3.18). \(\square \)

We note that Lemma 3.8 still holds when \(Y^\pm =\varnothing \), where the corresponding \(s^\pm =0\).

Example 3.9

The virtual dimension of \(u:{\mathbb {C}}\rightarrow T^*S^n\) with the puncture asymptotic to a simple Reeb orbit is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u)=(n-3)(2-2(0)-1-0)+2(n-1)+\frac{1}{2}(2n-2)=4n-6 \end{aligned}$$
(3.23)
because \(c_1^{rel}(TT^*S^n)=0\). When \(n=2\), we have \({{\,\mathrm{virdim}\,}}(u)=2\) which is obtained in [38, Lemma 7].
Now, we consider the relative setting. A Lagrangian cobordism L in X is a Lagrangian such that there exists \(T>T_X\) so that \(L \cap (-\infty ,-T] \times Y^-=(-\infty ,-T] \times \Lambda ^-\) and \(L \cap [T, \infty ) \times Y^+=[T, \infty ) \times \Lambda ^+\) for some Legendrian submanifolds \(\Lambda ^\pm \) in \(Y^\pm \). Let \(L_0,L_1\) be exact Lagrangian cobordisms such that \(L_0 \pitchfork L_1\). We assume that every Reeb chord x from \(\Lambda _0^{\pm }\) to \(\Lambda _1^{\pm }\) lies in a Morse–Bott family \(S_x\) of Reeb chords. In this case, we define [see (3.9)]
$$\begin{aligned} {{\,\mathrm{mb}\,}}(x)=\dim (S_x)+1=\dim (K_x) \end{aligned}$$
(3.24)
If \(x \in L_0 \cap L_1\), we define \({{\,\mathrm{mb}\,}}(x)=0\). The reader should note that the discrepancy between mb(x) and \(dim(S_x)\) comes from the \({\mathbb {R}}\)-direction of symplectizations. As always, we assume that \(L_0,L_1\) are \({\mathbb {Z}}\)-graded so all elements in \(L_0 \cap L_1\), and all Reeb chords from \(\Lambda _0^{\pm }\) to \(\Lambda _1^{\pm }\) are graded (see Sect. 3.2).

Lemma 3.10

Let \(S \in {\mathcal {R}}^{d+1}\) be equipped with Lagrangian labels \(L_j\) on \(\partial _j S\), where each \(L_j\) is a Lagrangian cobordism. Let J be a compatible almost complex structure on X that is cylindrical on the ends. Let \(u:S \rightarrow X\) be a J-holomorphic map with positive asymptotes \(\{x_j^+ \}_{j=1}^{r^+}\) and negative asymptotes \(\{x_j^- \}_{j=1}^{r^-}\) such that \(u(\partial _j S) \subset L_j\). Assume all asymptotes are Morse–Bott, then the virtual dimension of u is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u)=n(1-r^-)+\sum _{j=1}^{r^-} (\iota (x^-_j)+{{\,\mathrm{mb}\,}}(x^-_j))-\sum _{j=1}^{r^+} \iota (x^+_j)+(r^-+r^+-3) \end{aligned}$$
(3.25)

Sketch of proof

When all \(x^\pm _j\) are Lagrangian intersection points, then the index of u is given by (see [2, Proposition 11.13])
$$\begin{aligned} {{\,\mathrm{ind}\,}}(u)=n(1-r^-)+\sum _{j=1}^{r^-} \iota (x^-_j)-\sum _{j=1}^{r^+} \iota (x^+_j) \end{aligned}$$
(3.26)
If \(x^\pm _j\) is a Reeb chord, then the intersection of the graded Lagrangian subspaces \(K_{x^\pm _j}\) is non-zero. Similar to the proof of Lemma 3.8, there are extra contributions to \({{\,\mathrm{virdim}\,}}(u)\) from the asymptotes. This time, \(x^\pm _j\) contributes \(\dim (K_{x^\pm _j})={{\,\mathrm{mb}\,}}(x^\pm _j)\) (resp. 0) to \({{\,\mathrm{ind}\,}}(u)\) when \(x^\pm _j\) is a negative (resp. positive) asymptote. The reversing of the roles of positive and negative asymptotes between here and the proof of Lemma 3.8 can be understood from the fact that in (3.19), positive asymptotes contribute positively while in (3.26), positive asymptotes contribute negatively, which in turn boils down to the reversing convention of the definition of indices between orbits and chords.
After all, we have
$$\begin{aligned} {{\,\mathrm{ind}\,}}(u)=n(1-r^-)+\sum _{j=1}^{r^-} (\iota (x^-_j)+{{\,\mathrm{mb}\,}}(x^-_j))-\sum _{j=1}^{r^+} \iota (x^+_j) \end{aligned}$$
(3.27)
The last term of (3.25) comes from the dimension of \({\mathcal {R}}^{d+1}\). \(\square \)

Again, Lemma 3.10 applies also in the case when \(Y^-=\varnothing \) or \(Y^+=\varnothing \).

Example 3.11

Let \(q_0,q_1 \in S^n\) and \(\Lambda _{i}\) be the unit cospheres at \(q_i\), and assume \(T^*_{q_i}\) and the zero section are equipped with the canonical relative grading. Let x be the shortest Reeb chord from \(\Lambda _{0}\) to \(\Lambda _1\) in the unit cotangent bundle of \(S^n\). The virtual dimension of \(u:S \rightarrow T^*S^n\) such that \(S \in {\mathcal {R}}^3\), \(u(\partial _0 S) \subset S^n\), \(u(\partial _1 S) \subset T^*_{q_0}S^n\), \(u(\partial _2 S) \subset T^*_{q_1}S^n\) with positive asymptotes \(q_0\) and x at \(\xi _1\) and \(\xi _2\), respectively, and a negative asymptote \(q_1\) at \(\xi _0\) is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u)=n(1-1)+0-0-0=0 \end{aligned}$$
(3.28)
Finally, note that the shifting on the gradings of \(T^*_{q_i}S^n\) or \(S^n\) do not change this virtual dimension (see Sect. 3.2).

Now, we combine Lemmas 3.8 and 3.10.

Lemma 3.12

Let S be a disk with \(r^++r^-\) boundary punctures and \(s^++s^-\) interior punctures. Let \(u:S \rightarrow X\) be a J-holomorphic map with positive asymptotes \(\{x_j^+ \}_{j=1}^{r^+}\) and negative asymptotes \(\{x_j^- \}_{j=1}^{r^-}\) at boundary punctures, and positive asymptotes \(\{\gamma _j^+ \}_{j=1}^{s_+}\) and negative asymptotes \(\{\gamma _j^- \}_{j=1}^{s_-}\) at interior punctures such that \(u(\partial S)\) lies in the corresponding Lagrangians determined by the boundary asymptotes. Then the virtual dimension of u is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u)= \,&(n-3)(1-s^+-s^-)+\sum _{j=1}^{s^+} \mu _{RS}(\gamma ^+_j)\nonumber \\&-\sum _{j=1}^{s^-} \mu _{RS}(\gamma ^-_j) +\frac{1}{2}\sum _{j=1}^{s^+} \dim (S_{\gamma ^+_j}) \nonumber \\&+\frac{1}{2}\sum _{j=1}^{s^-} \dim (S_{\gamma ^-_j})+2c_1^{rel}(TX)([u]) \nonumber \\&+\sum _{j=1}^{r^-} (\iota (x^-_j)+{{\,\mathrm{mb}\,}}(x^-_j))-\sum _{j=1}^{r^+} \iota (x^+_j)-(n-1)r^-+r^+ \end{aligned}$$
(3.29)

Proof

We follow the proof in [2, Proposition 11.13]. The domain S is the connected sum of a disk \(S_1\) with \(r^++r^-\) boundary punctures and a sphere \(S_2\) with \(s^++s^-\) interior punctures. Let \(u_1:S_1 \rightarrow X\) be a J-holomorphic map with positive asymptotes \(\{x_j^+ \}_{j=1}^{r^+}\) and negative asymptotes \(\{x_j^- \}_{j=1}^{r^-}\) such that \(u_1(\partial S_1)\) lies in the corresponding Lagrangians determined by the boundary asymptotes. Let \(u_2:S_2 \rightarrow X\) be a J-holomorphic map with positive asymptotes \(\{\gamma _j^+ \}_{j=1}^{s_+}\) and negative asymptotes \(\{\gamma _j^- \}_{j=1}^{s_-}\). Then, we have
$$\begin{aligned} {{\,\mathrm{ind}\,}}(u)={{\,\mathrm{ind}\,}}(u_1)+{{\,\mathrm{ind}\,}}(u_2)-2n \end{aligned}$$
(3.30)
which can be computed by (3.22) and (3.27). Finally, to get the virtual dimension, we need to add the dimension of the Teichmüller space that S lies, which is \((r^-+r^+-3)+2(s^++s^-)\). It gives the formula (3.29). \(\square \)
We want to use Lemmas 3.8 and 3.12 to derive some corollaries for the holomorphic buildings \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) obtained in Theorem 3.3. Let \(v \ne v'\) be adjacent to the edge e. If \(l_\mathcal {T}(v)+1=l_\mathcal {T}(v')\), then there is a Reeb chord x (or orbit \(\gamma \)) which is the positive asymptote of \(u_v\) at \(f_v^{-1}(e)\) and the negative asymptote of \(u_{v'}\) at \(f_{v'}^{-1}(e)\). Let \(u_{v} \#_x u_{v'}\) (resp \(u_{v} \#_\gamma u_{v'}\)) be a pseudo-holomorpic map with boundary and asymptotic conditions determined by gluing \(u_v\) and \(u_{v'}\) along x (resp. \(\gamma \)). By a direct application of Lemmas 3.8 and 3.12, we get
$$\begin{aligned} \left\{ \begin{array}{ll} {{\,\mathrm{virdim}\,}}(u_{v} \#_x u_{v'})&{}={{\,\mathrm{virdim}\,}}(u_{v})+{{\,\mathrm{virdim}\,}}(u_{v'})-\dim (S_x) \\ {{\,\mathrm{virdim}\,}}(u_{v} \#_\gamma u_{v'})&{}={{\,\mathrm{virdim}\,}}(u_{v})+{{\,\mathrm{virdim}\,}}(u_{v'})-\dim (S_\gamma ) \end{array} \right. \end{aligned}$$
(3.31)
On the other hand, if \(l_\mathcal {T}(v)=l_\mathcal {T}(v')\) so that there is a Lagrangian intersection point x which is the positive asymptote of \(u_v\) at \(f_v^{-1}(e)\) and the negative asymptote of \(u_{v'}\) at \(f_{v'}^{-1}(e)\), then we have
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_{v} \#_x u_{v'})&={{\,\mathrm{virdim}\,}}(u_{v})+{{\,\mathrm{virdim}\,}}(u_{v'})+1 \end{aligned}$$
(3.32)
where \(u_{v} \#_x u_{v'}\) is defined analogously.

3.4 Action

This subsection discuss the action of the generators. A similar discussion can be found in [32] and [27, Section 3].

Let \(L_0,L_1\) be exact Lagrangians in \((M,\omega ,\theta )\). It means that, for \(j=0,1\), there exists a primitive function \(f_j \in C^{\infty }(L_j,{\mathbb {R}})\) such that \(df_j=\theta |_{L_j}\). For a Lagrangian intersection point \(p \in CF(L_0,L_1)\), the action is
$$\begin{aligned} A(p)=f_0(p)-f_1(p) \end{aligned}$$
(3.33)
so \( A(p^\vee )=-A(p)\) (see Convention 3.7). For a contact hypersurface \((Y,\alpha =\theta |_Y) \subset (M,\omega , \theta )\) and a Reeb chord \(x:[0,l_x] \rightarrow Y\) from \(\Lambda _0=L_0 \cap Y\) to \(\Lambda _1=L_1 \cap Y\). The length of x is
$$\begin{aligned} L(x)= \int x^*\alpha =l_x \end{aligned}$$
(3.34)
and the action is
$$\begin{aligned} A(x)=L(x)+(f_0(x(0))-f_1(x(l_x)) \end{aligned}$$
(3.35)
Reeb orbits are special kinds of Reeb chords so the length and action of a Reeb orbit \(\gamma \) is
$$\begin{aligned} L(\gamma )=A(\gamma )=\int \gamma ^*\alpha \end{aligned}$$
(3.36)
We have the following action control.

Lemma 3.13

see [27](Lemma 3.3, Proposition 3.5) Let \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) be a holomorphic building obtained in Theorem 3.3. If \(u_v\) has positive asymptotes \(\{x_j^+\}_{j=1}^{r^+}\), \(\{\gamma _j^+\}_{j=1}^{s^+}\) and negative asymptotes \(\{x_j^-\}_{j=1}^{r^-}\), \(\{\gamma _j^-\}_{j=1}^{s^-}\), then
$$\begin{aligned} E_{\omega }(u_v):=\sum _{j=1}^{r^+} A(x_j^+) + \sum _{j=1}^{s^+} A(\gamma _j^+) -\sum _{j=1}^{r^-} A(x_j^-)-\sum _{j=1}^{s^-} A(\gamma _j^-) \geqslant 0 \end{aligned}$$
(3.37)
The equality holds if and only if \(u_v\) is a trivial cylinder [see (3.3)].

Since \(A(\gamma ) >0\) for any Reeb orbit \(\gamma \) and \(A(x)>0\) if x is a non-constant 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 \(v'\).

Denote \(x_{v,i}\) by the asymptotes corresponding to the vertex v. Let \(sgn(x)=0\) (resp. \(sgn(x)=1\)) if x is a positive (resp. negative) asymptote. Then
$$\begin{aligned} \begin{aligned} 0&\leqslant \sum _{v \in G} E_\omega (u_v) \\&= \sum _{v\in G}\sum _{i}(-1)^{sgn(x_{i,v})}A(x_{i,v})\\&\leqslant (-1)^{sgn(x)}A(x)+ \sum _{j} |A(x_j)|<0 \end{aligned} \end{aligned}$$
(3.38)
Here \(x_{i,v}\) runs over all asymptotes of \(u_v\), and j over all semi-infinite edges. The second inequality holds because all finite edges are cancelled between the components they connect. This concludes the lemma. \(\square \)

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 non-negative, 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

Notice that, when \(u_v\) maps to SY, the sum (3.37) becomes
$$\begin{aligned} \sum _{j=1}^{r^+} L(x_j^+) + \sum _{j=1}^{s^+} L(\gamma _j^+) -\sum _{j=1}^{r^-} L(x_j^-)-\sum _{j=1}^{s^-} L(\gamma _j^-) \end{aligned}$$
(3.39)
because the terms involving the primitive functions on the Lagrangians add up to zero.

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 non-degenerate 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 \({{\,\mathrm{critp}\,}}(f_Q)\).

Furthermore, if \(\gamma \) is the m-fold cover of a simple \(\alpha \)-Reeb orbit \(\gamma ^s\) such that \(L(\gamma )<T\), then
$$\begin{aligned} \mu ^{\alpha _0}(\gamma )+\frac{1}{2}dim(S_\gamma )\geqslant \mu _{RS}^{\alpha }(\gamma ) \geqslant \mu _{RS}^{\alpha _0}(\gamma ) - \frac{1}{2}\dim (S_{\gamma }) \end{aligned}$$
(3.40)
where \(\mu _{RS}^{\alpha }(\gamma )\), \(\mu _{RS}^{\alpha _0}(\gamma )\) are the Robbin–Salamon index of \(\gamma \) with respect to \(\alpha \) and \(\alpha _0\), respectively, and \(S_{\gamma }\) is the Morse–Bott family with respect to \(\alpha _0\) that \(\gamma \) lies.

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 \(\mu _{RS}^{\alpha }(\gamma ) \geqslant n-1\). As a result, the virtual dimension of \(u:{\mathbb {C}}\rightarrow SM^-\) with positive asymptote \(\gamma \) satisfies \({{\,\mathrm{virdim}\,}}(u) \geqslant 2n-4\).

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 \(\mu _{RS}^{\alpha _0}(\gamma )=2k(n-1)\) for some \(k>0\) and \(\dim (S_{\gamma })=2n-2\). Therefore, \(\mu _{RS}^{\alpha }(\gamma ) \geqslant n-1\) by Lemma 3.19 and \({{\,\mathrm{virdim}\,}}(u)=(n-3)+\mu _{RS}^{\alpha }(\gamma ) \geqslant 2n-4\). \(\square \)

We have a similar index calculation for Reeb chords:
$$\begin{aligned} \iota (x_0)-dim(S_{x_0})\leqslant \iota (x)\leqslant \iota (x_0)+dim(S_{x_0}). \end{aligned}$$
(3.41)
The proof is identical to Lemma 3.19 hence omitted. Let \(\Lambda _q \subset \partial U\) be the cosphere at q.

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 non-degenerate \(\alpha _0\)-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 \(\iota (x_0)=-k(n-1) \leqslant -(n-1)\) for some \(k>0\). We have \(\dim (S_{x_0})=n-1\), so if x is \(C^1\)-close to \(x_0\), then \(\iota (x) \leqslant \iota (x_0)+\dim (S_{x_0}) \leqslant -(n-1)+(n-1)=0\). 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 non-degenerate 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 non-degenerate.

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

We assume \(n \geqslant 2\). For all \(T> L(\partial U)\) and \(k \in {\mathbb {N}}\), there exists \(\delta >0\), \(f_\partial :\partial U \rightarrow {\mathbb {R}}\) and pairwise distinct \(q_1,\dots , q_k \in P\) such that \(\alpha =(1+\delta f_\partial )\alpha _0\) satisfies
  1. (1)

    every simple \(\alpha \)-Reeb orbit \(\gamma \) that is contractible in U and \(L(\gamma ) < T\) is non-degenerate and \(\mu _{RS}(\gamma ) \geqslant n-1\), and

     
  2. (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 non-degenerate, 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.

     
As a consequence, the image of the \(\alpha \)-Reeb chords x from \(\bigcup _{i=1}^k \Lambda _{q_i}\) to \(\bigcup _{i=1}^k \Lambda _{q_i}\) with \(L(x)<T\) are pairwise disjoint, and they are disjoint from the image of simple \(\alpha \)-Reeb orbits.

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 pseudo-holomorphic 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

(Neighborhood theorem) For any Reeb chord \(x \in {\mathcal {X}}_T^c\), there exists a neighborhood \(N_x\) of Im(x), an open ball \(B_x \subset {\mathbb {R}}^{2n-2}\) containing the origin, an open interval \(I_x \subset {\mathbb {R}}\) and a diffeomorphism \(\phi _{N_x}:N_x \rightarrow B_x \times I_x\) such that
$$\begin{aligned} \left\{ \begin{array}{ll} \alpha =\phi _{N_x}^*(dz+\displaystyle \sum _{i=1}^{n-1} x_i dy_i) \\ \pi _{B_x}(\phi _{N_x}(x(t)))=0 \end{array} \right. \end{aligned}$$
(3.42)
where \((x_i,y_i) \in B_x\), \(z \in I_x\) and \(\pi _{B_x}:B_x \times I_x \rightarrow B_x\) is the projection to the first factor.

Proof

It follows from a Moser’s argument. We give a sketch following [39, Theorem 2.5.1]. Since \(d \alpha \) is non-degenerate on \(T_pY/T_pIm(x)\) for all \(p \in Im(x)\), we can use exponential map with respect to an appropriate metric to find coordinates \((x_1,y_1,\dots ,x_{n-1},y_{n-1},z)\) near Im(x) such that \(Im(x)=\{x_i=y_i=0\}\) and on \(TY|_{Im(x)}\),
$$\begin{aligned} \left\{ \begin{array}{ll} \alpha (\partial _z)=1, \iota _{\partial _z} d\alpha =0 \\ \partial _{x_i}, \partial _{y_i} \in \ker (\alpha ), d\alpha =\displaystyle \sum _{i=1}^{n-1} dx_i \wedge dy_i \end{array} \right. \end{aligned}$$
(3.43)
Let \(\alpha _{{\mathbb {R}}^{2n-1},std}=dz+\sum _{i=1}^{n-1} x_i dy_i\) and \(\alpha _t=(1-t)\alpha _{{\mathbb {R}}^{2n-1},std}+t \alpha \). It follows that on \(TY|_{Im(x)}\),
$$\begin{aligned} \alpha _t=\alpha ,\qquad d\alpha _t=d\alpha \quad \text { for all }t \end{aligned}$$
(3.44)
In particular, \(\alpha _t\) is a family of contact forms in a sufficiently small neighborhood of Im(x). By Moser trick, there exists a vector field \(X_t\) near Im(x) such that the flow \(\psi _t\) satisfies \(\psi _t^* \alpha _t=\alpha _{{\mathbb {R}}^{2n-1},std}\) for all \(t \in [0,1]\) and \(X_t(p)=0\) for all \(p \in Im(x)\). We set \(\phi _{N_x}=(\psi _1)^{-1}\). \(\square \)

Remark 3.25

If we replace \(dz+\sum _{i=1}^{n-1} x_i dy_i\) by \(dz+\sum _{i=1}^{n-1} x_i dy_i+dy_1\) 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. \(\square \)

Now we may address the regularity of neck-stretching limits along \(\partial U\). We summarize various auxiliary data chosen so far.
  1. (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. (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. (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. (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 that
    $$\begin{aligned} D_{i,x}:=\Lambda _{q_i} \cap N_x \end{aligned}$$
    (3.45)
    is a disk for \(i=0,1\). Moreover, by the fact that x is non-degenerate, 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 \(x \in {\mathcal {X}}^c_T\).
     
  5. (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. (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 R-adjusted to \((Y, \alpha )\) as explained in Sect. 3 (see Remark 3.2).

     
  7. (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 that
    $$\begin{aligned} \text {for all Reeb chords }x\text { from }\Lambda _i\text { to }\Lambda _j, |A(x)|<T^{adj}\text { implies }|L(x)|<T \end{aligned}$$
    (3.46)
    Without loss of generality, we can assume \(T^{adj}\) exists and \(\sum _{j=0}^d|A(x_j)|<T^{adj}\). 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 \({\mathcal {X}}^{o}_T\).
     
For \(u_{\infty }\), we have the following regularity result.

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 \(SN_x:={\mathbb {R}}\times N_x\).

Let \(\Lambda ^{tot}=\bigcup _i \Lambda _{q_i}\) and \(S\Lambda =\bigcup _i S\Lambda _{q_i}\) where \(\Lambda _{q_i}\) are obtained in Corollary 3.23. There is a Banach manifold \({\mathcal {B}}\) consisting of maps
$$\begin{aligned} u:(\Sigma _v, \partial \Sigma _v) \rightarrow (SY, S\Lambda ^{tot}) \end{aligned}$$
(3.47)
in an appropriate Sobolev class with positive weight (see [26, 40]). Let \(U_{\Delta }\) be an appropriate Banach manifold that is dense inside \({\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\). The map
$$\begin{aligned} (u,J) \mapsto {\overline{\partial }}_J u \end{aligned}$$
(3.48)
defines a section \(\mathcal {F}\) of a bundle \(\mathcal {E}^{0,1} \rightarrow {\mathcal {B}}\times U_\Delta \) with differential
$$\begin{aligned}&D\mathcal {F}(u,J): T_u{\mathcal {B}}\times T_JU_\Delta \rightarrow \mathcal {E}^{0,1}_u \end{aligned}$$
(3.49)
$$\begin{aligned}&(\eta ,{\mathbf {Y}}) \mapsto D_u(\eta )+ {\mathbf {Y}}(u) \circ du \circ j_{\Sigma _v} \end{aligned}$$
(3.50)
where \(j_{\Sigma _v}\) is the complex structure on \(\Sigma _v\). By a choice of metric, we identify
$$\begin{aligned} T_u{\mathcal {B}}\simeq \Gamma (u^*TSY, u|_{\partial \Sigma _v}^* S\Lambda ^{tot}) \end{aligned}$$
(3.51)
where the right-hand side is the completion of the space of smooth sections in \(u^*TSY\), which takes value in \(u|_{\partial \Sigma _v}^* S\Lambda ^{tot}\) along the boundary, with respect to an appropriate Sobolev norm. On the other hand, we have \(\mathcal {E}^{0,1}_u=\Omega ^{0,1}(u^*TSY)\), where the right hand side is the completion of the space of smooth \(u^*TSY\)-valued (0, 1)-form with respect to an appropriate Sobolev norm. We want to argue \(D\mathcal {F}(u,J)\) is surjective at (uJ) using that fact that there exists a boundary asymptote \(x \in {\mathcal {X}}^c_T\) of u that appears only once among its asymptotes and \({\overline{\partial }}_J u=0\).
Suppose not, then there exists \(0 \ne l \in \mathcal {E}^{0,1}_u\) such that
$$\begin{aligned} \langle l, D\mathcal {F}(u,J)(\eta ,{\mathbf {Y}}) \rangle _{L^2, \Sigma _v} =0 \end{aligned}$$
(3.52)
for all \(\eta \in T_u {\mathcal {B}}\) and \({\mathbf {Y}}\in T_JU_\Delta \). By unique continuation principle, it suffices to show that \(l=0\) on some non-discrete set of \(\Sigma _v\) to get a contradiction.
Let \({\mathcal {R}}=u^{-1}(N_x) \subset \Sigma _v\) and we will show that for \(\eta \) supported in \({\mathcal {R}}\) and \({\mathbf {Y}}\) supported in \(SN_x\), it is sufficient to get \(l|_{\mathcal {R}}=0\). By Lemma 3.24, we can identify \(SN_x\) with \({\mathbb {R}}_r \times (B_x)_{x_i,y_i} \times (I_x)_z\). Let \({\overline{u}}=\pi _{B_x} \circ \pi _{Y} \circ u_v|_{\mathcal {R}}\). In the coordinates \(((r,z),(\{x_i\},\{y_i\}))\), we can write \(l|_{\mathcal {R}}=(l_1,l_2)\). For \(\eta =0\) and \({\mathbf {Y}}\) supported in \(SN_x\),1 (3.52) implies
$$\begin{aligned} \langle l_2, {\mathbf {Y}}({\overline{u}}) \circ d{\overline{u}} \circ j_{\Sigma _v} \rangle _{L^2, {\mathcal {R}}} =0 \end{aligned}$$
(3.53)
where \({\mathbf {Y}}\) is rz-invariant in \(SN_x\) by the definition of \({\mathcal {J}}^{cyl}(\partial U; \{N_x\}_{x \in {\mathcal {X}}^{c}_T})\) so \({\mathbf {Y}}({\overline{u}})\) is well-defined.

Lemma 3.28

It follows from (3.53) that \(l_2=0\).

Assuming Lemma 3.28, it suffices to show that \(l_1=0\). Similarly, \(l_1\) admits the unique continuation property (see [26, p. 754]) so we only need to show that \(l_1=0\) on some non-discrete set of \({\mathcal {R}}\). For \({\mathbf {Y}}=0\) and \(\eta \) supported in \({\mathcal {R}}\), (3.52) becomes
$$\begin{aligned} \langle l_1, D(\pi _{r,z}) \circ D_u \eta \rangle _{L^2, {\mathcal {R}}} =0 \end{aligned}$$
(3.54)
where \(\pi _{r,z}: SN_x \rightarrow {\mathbb {R}}_r \times (I_x)_z\) is the projection. Notice that \(J|_{T({\mathbb {R}}_r \times (I_x)_z)}\) is the standard complex structure, and \(\eta \) depends on the domain \({\mathcal {R}}\) rather than the target \(SN_x\). Therefore, we can find an interior point p of \({\mathcal {R}}\) and construct \(\eta \) appropriately supported near p to show that \(l_1=0\). The details of the construction of \(\eta \) can be found in [26, page 754].

As a result, \(l|_{\mathcal {R}}=0\) and hence \(l \equiv 0\). The existence of \({\mathcal {J}}^{cyl, reg}\) follows from applying Sard’s-Smale 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.

By the asymptotic behavior of holomorphic disks, we can assume that for sufficiently small \(\delta >0\), there exists a neighborhood \((E_0,\partial E_0) \subset ({\mathcal {R}}, \partial {\mathcal {R}})\) of \(\xi _{j_x}\) such that
  1. (i)

    \(({\overline{u}}(E_0),{\overline{u}}(\partial E_0)) \subset (B(0,2\delta ), \pi _{B_x}(D_{0,x} \cup D_{1,x}) \cup \partial B(0,2\delta ))\),

     
  2. (ii)

    \(\pi _{B_x}(D_{0,x} \cup D_{1,x}) \cap \partial B(0,2\delta )\) are two real analytic disjoint branches,

     
  3. (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.

     
Here \(B(0,2\delta )\) is a \(2\delta \)-ball centered at the origin and \(D_{i,x}\) are defined in (3.45).

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 \({\mathbb {R}}\times x\).

Let \(p_{r+1}, \dots p_s \in {\mathcal {R}}\backslash \partial {\mathcal {R}}\) be the preimages under \({\overline{u}}\) of 0 with the property that the preimage of \(\gamma _0\) under \({\overline{u}}\) intersects some neighborhood of \(p_j\) in a 1-dimensional subset. By monotonicity lemma and maximum principle, this set is also finite, and is identified with the interior intersections between u and \({\mathbb {R}}\times x\).
Fig. 2

Green region: \(U_1\); Blue region: \(U_2\); Red region: \(B(x_1,\epsilon )\); Purple region: \(B(x_2,\epsilon )\). Only \(u_0(E_0)\) hits \(U_1\) but not \(U_2\) among \(u_j(E_j)\) because unlike \(p_j\) (for \(1 \leqslant j \leqslant s\)), \(\xi _{j_x}\) is a boundary puncture (Color figure online)

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).

By monotonicity lemma and maximum principle, for \(i=1,2\) we can find a point \(x_i \in U_i \backslash (B(0,\delta ) \cup \pi _{B_x}(D_{0,x} \cup D_{1,x}))\) and small neighborhoods \(B(x_i,\epsilon )\), \(\epsilon \ll r\), such that
$$\begin{aligned} {\overline{u}}^{-1}(B(x_i,\epsilon )) \subset \bigcup _{j=0}^s E_j \end{aligned}$$
(3.55)
When a certain branch of Open image in new window , then \(x_i{{\bar{u}}}(E_j)\) for \(i=1,2\) We exclude from our list any such \(j \geqslant 1\). Note that for \(j \geqslant 1\), \(x_1 \in {\overline{u}}(E_j)\) if and only if \(x_2 \in {\overline{u}}(E_j)\).

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 anti-holomorphic) 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 bi-holomorphic identifications \(\phi _j\) of E with \(E_j\) such that \((l_2)_j(\phi _j(z))=f_E(z^{k_j})\) for \(z \in E\).

Via our choice of perturbation of the complex structure, we can choose \({\mathbf {Y}}\) to be supported in \(B(x_2, \epsilon )\). We get
$$\begin{aligned} \left\langle \sum _{j=1}^s (l_2)_j(\phi _j(z)), {\mathbf {Y}}(u_j \circ \phi _j) \circ d(u_j \circ \phi _j) \circ j_{E} \right\rangle _{L^2,E}=0 \end{aligned}$$
(3.56)
where \(j_E\) is the complex structure on E. Varying \({\mathbf {Y}}\), this implies
$$\begin{aligned} \sum _{j=1}^s (l_2)_j(\phi _j(z)) =0 \end{aligned}$$
(3.57)
We can also choose \({\mathbf {Y}}\) to be supported in \(B(x_1, \epsilon )\). We get
$$\begin{aligned}&\left\langle \sum _{j=1}^s (l_2)_j(\phi _j(z)), {\mathbf {Y}}(u_j \circ \phi _j) \circ d(u_j \circ \phi _j) \circ j_{E} \right\rangle _{L^2,E} \\&\quad +\langle (l_2)_0(z), {\mathbf {Y}}(u_0) \circ du_0 \circ j_{E_0} \rangle _{L^2,E_0}=0 \end{aligned}$$
Since the first term is 0 by (3.57), by varying \({\mathbf {Y}}\), it implies \(l_2|_{E_0}=(l_2)_0=0\) and hence \(l_2 \equiv 0\). \(\square \)

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.

Remark 3.31

There is a possible alternative approach to the above regularity results if one could generalize the work of Lazzarini [41, 42] and Perrier [43] to the SFT settings. This seems promising at least for \(SM^\pm \), but the general regularity of SY might be more difficult.

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 R-adjusted 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.

Let \(x_0 \in CF(L_0,L_d)\) and \(x_j \in CF(L_{j-1},L_j)\) for \(j=1,\dots ,d\). When T is large enough, we may assume that
$$\begin{aligned} \sum _{j=0}^d |A(x_j)|<T^{adj} \end{aligned}$$
(3.58)
Suppose that there exists a sequence \(\{\tau _k\}_{k=1}^\infty \) such that \(\lim _{k \rightarrow \infty } \tau _k =\infty \), and a sequence \(u_k \in \mathcal {M}^{J^{\tau _k}}(x_0;x_d,\dots ,x_1)\). We assume that \({{\,\mathrm{virdim}\,}}(u_k)=0\). Let \(u_\infty =\{u_v\}_{v \in V(\mathcal {T})}\) be the holomorphic building obtained in Theorem 3.3. Then we have

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

For a subtree \(G \subset \mathcal {T}\), we use \({{\,\mathrm{virdim}\,}}(G)\) to denote the virtual dimension of the map \(\#_{v \in G} u_v\), where \(\#_{v \in G} u_v\) refers to the map obtained by gluing all \(u_v\) such that \(v \in G\) along the asymptotes determined by the edges. By (3.31), (3.32) and the fact that all Reeb chords/orbits arising as asymptotes of \(u_v\) are non-degenerate, we have
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(G)= \sum _{v \in G} {{\,\mathrm{virdim}\,}}(u_v)+k_G \end{aligned}$$
(3.59)
where \(k_G\) is the number of edges that correspond to Lagrangian intersections points and connect two distinct vertices in G. By assumption, \({{\,\mathrm{virdim}\,}}(\mathcal {T})=0\). Since \(u_v\) are transversally cut out for \(v \in V^{core} \cup V^{\partial }\) (Propositions 3.27, 3.29, 3.30), we have \({{\,\mathrm{virdim}\,}}(u_v) \geqslant 0\). For \(v \in V^{int}\), we cannot address the regularity but we have the following.

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 \(2n-4>0\) (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 r-direction. 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 \({{\,\mathrm{virdim}\,}}(u_v)=n-\sum _{j=0}^{d_v} \iota (y_j)-(2-d_v) \geqslant n-2>0\). 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 neck-stretching, 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 non-degenerate.

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

For \(\mathcal {E}^0,\mathcal {E}^1 \in Ob({\mathcal {F}})\), we have cohomological level isomorphism
$$\begin{aligned} H(hom_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\mathcal {E}^0,T_{\mathcal {P}}(\mathcal {E}^1))) \simeq H(hom_{{\mathcal {F}}}(\mathcal {E}^0,\tau _P(\mathcal {E}^1))) \end{aligned}$$
(4.1)
We will only consider the case that \(\mathcal {E}^i=L_i\) are Lagrangians without local system. The proof of the general case is identical except that the notations become more involved. In slightly more geometric terms, we would like to directly construct a chain map \(\iota \) from
$$\begin{aligned} C_0:=Cone(CF(\mathcal {P}, L_1) \otimes _\Gamma CF(L_0,\mathcal {P}) \xrightarrow {ev_\Gamma } CF(L_0,L_1)) \end{aligned}$$
(4.2)
to
$$\begin{aligned} C_1:=CF(L_0, \tau _P L_1) \end{aligned}$$
(4.3)
which induces isomorphism on cohomology.
By applying a Hamiltonian perturbation, we assume \(L_0 \pitchfork L_1\), and that each connected component of \(L_i\cap U\) is a cotangent fiber in U. The cotangent fiber \(T^*_{q}P \cap U\) has \(|\Gamma |\) different lifts \(\{T^*_{g{\mathbf {q}}}{\mathbf {P}}\}_{g\in \Gamma } \cap {\mathbf {U}}\) in \({\mathbf {U}}\), where \({\mathbf {U}}\subset T^*{\mathbf {P}}\) is the universal cover of U. We assume the Dehn twist \(\tau _P\) is supported inside U and we have a commutative diagram:
where \(\pi :{\mathbf {U}}\rightarrow U\) is the covering map. As always, we assume that \(L_0,L_1\) are equipped with \({\mathbb {Z}}-\)gradings and spin structures.

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

We denote the set of generators in \(C_0\) by \({\mathcal {X}}(C_0)\), which is divided into two types \({\mathcal {X}}_a(C_0)\) and \({\mathcal {X}}_b(C_0)\):
  • \({\mathcal {X}}_a(C_0)\): generators in \(hom(\mathcal {P}, L_1)\otimes _\Gamma hom(L_0,\mathcal {P})[1]\)

  • \({\mathcal {X}}_b(C_0)\): generators in \(hom(L_0,L_1)\)

More precisely, \({\mathcal {X}}_b(C_0)=L_0 \cap L_1\) and \({\mathcal {X}}_a(C_0)\) is the set of elements of the form \([{\mathbf {q}}^\vee \otimes g{\mathbf {p}} ]\sim [{\mathbf {q}}^\vee g\otimes {\mathbf {p}} ]\sim [(g^{-1}{\mathbf {q}})^\vee \otimes {\mathbf {p}}]\), where we are using the correspondence (2.39) and (2.40). On the other hand, we denote \(L_0\cap \tau _P L_1\) by \({\mathcal {X}}(C_1)\) which is a set of generators for \(C_1\).
Let \(p \in L_0 \cap P\), \(q \in L_1 \cap P\) and \({\mathbf {p}}, {\mathbf {q}}\in {\mathbf {P}}\) be a lift of p and q, respectively. We also introduce the following notation
$$\begin{aligned} \begin{aligned}&{\mathbf {c}}_{{\mathbf {p}}, {\mathbf {q}}}:\text { the unique intersection }T^*_{\mathbf {p}}{\mathbf {P}}\cap \tau _{\mathbf {P}}(T^*_{\mathbf {q}}{\mathbf {P}})\\&c_{{\mathbf {p}},{\mathbf {q}}}:=\pi (\mathbf{c }_{ {\mathbf {p}},{\mathbf {q}}}) \text {, which is an intersection of} L_0\cap \tau _P L_1 \end{aligned} \end{aligned}$$
(4.4)

Lemma 4.2

There is a grading-preserving 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 \(\iota |_{{\mathcal {X}}_a(C_0)}\).

We define \(\iota |_{{\mathcal {X}}_a(C_0)}\) by
$$\begin{aligned} \iota |_{{\mathcal {X}}_a(C_0)}: {\mathbf {q}}^\vee \otimes {\mathbf {p}} \mapsto c_{{\mathbf {p}},{\mathbf {q}}} \end{aligned}$$
(4.5)
This map is well-defined because
$$\begin{aligned} \iota ({\mathbf {q}}^\vee g^{-1} \otimes g{\mathbf {p}})=\iota ((g{\mathbf {q}})^\vee \otimes g{\mathbf {p}})=\pi (\mathbf{c }_{ g{\mathbf {p}},g{\mathbf {q}}})= c_{{\mathbf {p}},{\mathbf {q}}} \end{aligned}$$
(4.6)
The last equality comes from the equivariance of \(\tau _{\mathbf {P}}\). It is clear that \(\iota |_{{\mathcal {X}}_a(C_0)}\) is a bijection from \({\mathcal {X}}_a(C_0)\) to the intersections of \(L_0\cap \tau _P L_1\) inside U.

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 well-known (see e.g. [1]). \(\square \)

Using Lemma 4.2, we define \({\mathcal {X}}_a(C_1)=\iota ({\mathcal {X}}_a(C_0))\) and \({\mathcal {X}}_b(C_1)=\iota ({\mathcal {X}}_b(C_0))\). We summarize our notation in Fig. 3.
Fig. 3

Generator correspondence between \(C_0\) and \(C_1\)

4.2 Overall strategy

The differentials in \(C_0\) can be divided into four types.
  • Type (A1): differentials in \(hom(L_0,\mathcal {P})\), i.e. pseudo-holomorphic strips in \(\mathcal {M}( \mathbf {p'} ; {\mathbf {p}})\)

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

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

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

Fig. 4

Types of holomorphic curves in \(C_0\)

For \(C_1\), we divide the differentials similarly, using correspondence of generators \(\iota \). Concretely, we have:
  • Type (A1\('\)): pseudo-holomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}})\);

  • Type (A2\('\)): pseudo-holomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}},{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}})\);

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

  • Type (B\('\)): pseudo-holomorphic strips in \(\mathcal {M}(x_0;x_1)\);

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

  • Type (D\('\)): pseudo-holomorphic strips in \(\mathcal {M}(c_{{\mathbf {p}},{\mathbf {q}}};x)\);

where \(x,x_0,x_1 \in {\mathcal {X}}_b(C_1)\).
Fig. 5

Types of holomorphic curves in \(C_1\)

By the discussion in Sect. 3.8, we know that for an appropriate choice of \(\{J^\tau \}\) and \(\tau \gg 1\), all the rigid \(J^\tau \)-holomorphic polygons in the moduli above are transversally cut out and they are bijective to the corresponding holomorphic buildings (Figs. 4 and 5). By studying the holomorphic buildings, we will show that there are bijective correspondences
$$\begin{aligned}&\mathcal {M}^{J^\tau }( \mathbf {p'} ; {\mathbf {p}}) \simeq \mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}}) \text { for all }{\mathbf {q}}; \end{aligned}$$
(4.7)
$$\begin{aligned}&\mathcal {M}^{J^\tau }(({\mathbf {q}}')^\vee ; {\mathbf {q}}^\vee ) \simeq \mathcal {M}^{J^\tau }(c_{{\mathbf {p}},{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}}) \text { for all }{\mathbf {p}}; \end{aligned}$$
(4.8)
$$\begin{aligned}&\mathcal {M}^{J^\tau }(x_0;x_1) \simeq \mathcal {M}^{J^\tau }(x_0;x_1); \end{aligned}$$
(4.9)
$$\begin{aligned}&\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee , {\mathbf {p}}) \simeq \mathcal {M}^{J^\tau }(x; c_{{\mathbf {p}},{\mathbf {q}}}); \end{aligned}$$
(4.10)
$$\begin{aligned}&\hbox { Type (A3}'\hbox {) and (D}'\hbox {) are empty with respect to}\ J^\tau . \end{aligned}$$
(4.11)
where the two sides of (4.9) are with respect to boundary conditions \((L_0,L_1)\) and \((L_0, \tau _P(L_1))\), respectively. In other words, for \(\tau \gg 1\), \(\iota :C_0 \rightarrow C_1\) is an isomorphism which clearly implies Proposition 4.1.

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 Neck-stretching limits of holomorphic strips and triangles

In this section, we will list all possible holomorphic buildings \(u_{\infty }=\{u_v\}_{v \in V(\mathcal {T})}\) that arises as the limit (when \(\tau \rightarrow \infty \)) of curves in the moduli discussed in Sect. 4.2. By Proposition 3.32, we know that \(u_\infty \) satisfies the following conditions
$$\begin{aligned} \begin{array}{ll} (i)\text { The total level }n_\mathcal {T}=1, \\ (ii) \text { }{{\,\mathrm{virdim}\,}}(u_v)=0 \text { for all }v. \\ (iii)\text { All compact edges in }\mathcal {T}\text { correspond to Reeb chords}.\\ (iv)\text { If }v \in V^{\partial }, \text { then }l_\mathcal {T}(v)=1\text { and }u_v\text { has exactly one boundary asymptote.} \end{array} \end{aligned}$$
(4.12)
Therefore, we assume (4.12) hold throughout this section. Recall also that Corollary 3.23 holds for our choice of \((\partial U,\alpha )\), hence the asymptotes under consideration are non-degenerate and \(mb(x)=1\) (cylindrical direction).

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

By Lemma 3.10 and \({{\,\mathrm{virdim}\,}}(u_v)=0\), we have
$$\begin{aligned} 0={{\,\mathrm{virdim}\,}}(u_v)=|x|+{{\,\mathrm{mb}\,}}(x)-2=|x|-1 \end{aligned}$$
(4.13)
Therefore, \(|x|=1\). \(\square \)

Lemma 4.4

If \(l_\mathcal {T}(v)=0\), then \(u_v\) has at least one asymptote that is not a Reeb chord.

Proof

Suppose not. Let \(y_1,\dots ,y_k\) be the asymptotes of \(u_v\) which are all positive Reeb chord. Notice that the shift of gradings for any individual boundary condition does not affect the virtual dimension of \(u_v\). Therefore we can use the canonical relative grading to compute the virtual dimension of \(u_v\). By Lemma 3.10 and Corollary 3.23, we have
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_v)=n- \sum _{j=1}^k |y_j| -(3-k) \geqslant n-3+k \geqslant n-2 >0 \end{aligned}$$
(4.14)
which contradicts the assumption (4.12) that \({{\,\mathrm{virdim}\,}}(u_v)=0\). \(\square \)

Lemma 4.5

Every generator \(c_{{\mathbf {p}},{\mathbf {q}}} \in CF(T^*_p P, \tau _P(T^*_qP))\) satisfies \(|c_{{\mathbf {p}},{\mathbf {q}}}|=n-1\) with respect to the canonical relative grading. Moreover, if \(c_{{\mathbf {p}},{\mathbf {q}}}\) is the only asymptote of a non-constant \(J^-\)-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

To see that \(|c_{{\mathbf {p}},{\mathbf {q}}}|=n-1\), it suffices to show that \(|{\mathbf {c}}_{{\mathbf {p}},{\mathbf {q}}}|=n-1\). One can compute it directly by noting that \(\tau _{\mathbf {P}}(T^*_{\mathbf {q}}{\mathbf {P}})={\mathbf {P}}[1] \# T^*_{\mathbf {q}}{\mathbf {P}}\), where \({\mathbf {P}}[1]\) is the grading shift of \({\mathbf {P}}\) by 1 and \(\#\) denotes the graded Lagrangian surgery at the point \({\mathbf {q}}\) (see [29] or [21]). Alternatively, one can see it using the Dehn twist exact sequence [1]
$$\begin{aligned} 0 \rightarrow HF^k(T^*_{\mathbf {p}}{\mathbf {P}}, \tau _{\mathbf {P}}(T^*_{\mathbf {q}}{\mathbf {P}})) \rightarrow \bigoplus _{a+b-1=k} HF^a({\mathbf {P}}, T^*_{\mathbf {q}}{\mathbf {P}})\otimes HF^b(T^*_{\mathbf {p}}{\mathbf {P}}, {\mathbf {P}}) \rightarrow 0 \end{aligned}$$
(4.15)
and the fact that the second non-trivial term is non-zero only when \(a=0\) and \(b=n\).
On the other hand, if \(c_{{\mathbf {p}},{\mathbf {q}}}\) is a negative asymptote and the remaining asymptotes are denoted by \(y_1,\dots ,y_k\), we would have (computed in canonical relative grading)
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_v)=|c_{{\mathbf {p}},{\mathbf {q}}}| - \sum _{i=1 }^k |y_i| - (2-k) \geqslant n-2>0 \end{aligned}$$
(4.16)
which contradicts to the assumption (4.12) that \({{\,\mathrm{virdim}\,}}(u_v)=0\). \(\square \)

Now, we can describe the SFT limits of various moduli.

Lemma 4.6

(Type (A1)) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a non-empty SFT limit of curves in \(\mathcal {M}^{J^\tau }( \mathbf {p'} ; {\mathbf {p}})\). Then \(\mathcal {T}\) consists of exactly two vertices \(v_1,v_2\) and
  • \(u_{v_1}\) is a \(J^-\)-holomorphic triangle with negative asymptote \(p':=\pi ({\mathbf {p}}')\) and positive asymptotes xp 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.

Fig. 6

Multiple side bubbles

Proof

Notice that, by the boundary condition P, p and \(p'\) must be asymptotes of the same \(u_v\). We call it \(u_{v_1}\). We label the other vertices of \(\mathcal {T}\) by \(v_2, \dots , v_k\) for some \(k \geqslant 0\). By boundary condition again, we know that \(v_j \in V^{\partial }\) for \(j > 1\). By (4.12), we have \(l_\mathcal {T}(v_j)=1\) for \(j >1\). Moreover, all \(v_j\) are adjacent to \(v_1\) because \(u_{v_j}\) has a negative asymptote (see Fig. 6). By Lemma 3.10 and Corollary 3.23 again,
$$\begin{aligned} 0={{\,\mathrm{virdim}\,}}(u_{v_1})=|p'|-|p|- \sum _{j=1}^k |y_j|-(1-k) \geqslant k-1 \end{aligned}$$
(4.17)
so \(k=0,1\). However, \(k \ne 0\) by boundary condition. As a result, \(k=1\) and we denote \(y_1\) by x.

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

(Type (A1\('\))) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a non-empty SFT limit of curves in \(\mathcal {M}^{J^\tau }( c_{\mathbf {p'},{\mathbf {q}}} ; c_{{\mathbf {p}},{\mathbf {q}}})\). Then \(\mathcal {T}\) consists of exactly two vertices \(v_1,v_2\) and
  • \(u_{v_1}\) 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 non-empty 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

(Type (C)) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a non-empty SFT limit of curves in \(\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee , {\mathbf {p}})\). Then \(\mathcal {T}\) consists of exactly two vertices \(v_1,v_2\) and
  • \(u_{v_1}\) 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 \(y_j\).

For \(u_{v_1}\) to be rigid, we have
$$\begin{aligned} 0=n-|{\mathbf {p}}|-|{\mathbf {q}}^\vee |-\sum _{j=2}^k |y_j|-(2-k) \geqslant n-n-0+k-2 \end{aligned}$$
so \(k \leqslant 2\). However, we have \(k \geqslant 2\) so we get \(k=2\). Moreover, the canonical relative grading of \(y_2\) is 0. \(\square \)

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 neck-stretching limit will remain the same as Type (C), because this is simply a modification of some of the strip-like ends (from outgoing to incoming, and vice versa) and does not change the behavior of the underlying curve.

Lemma 4.11

(Type (C\('\))) Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a non-empty SFT limit of curves in \(\mathcal {M}^{J^\tau }(x; c_{{\mathbf {p}},{\mathbf {q}}})\). Then \(\mathcal {T}\) consists of exactly two vertices \(v_1,v_2\) and
  • \(u_{v_1}\) 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

The argument is entirely parallel to Lemma 4.9. Let \(u_{v_1}\) be the \(J^-\)-holomorphic curve such that \(c_{{\mathbf {p}},{\mathbf {q}}}\) is an asymptote of it. Let the other asymptotes of \(u_{v_1}\) be \(y_1, \dots , y_k\). For \(u_{v_1}\) to be rigid, by Lemma 4.5,
$$\begin{aligned} 0={{\,\mathrm{virdim}\,}}(u_{v_1})=n-|c_{{\mathbf {p}},{\mathbf {q}}}|-\sum _{j=1}^k |y_j|-(2-k) \geqslant n-(n-1)-2+k=k-1 \end{aligned}$$
(4.18)
so \(k=1\) because \(u_{v_1}\) has at least one positive Reeb chord asymptote. \(\square \)

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.

We are interested in the algebraic counts of the following moduli spaces
  1. (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. (2)

    \(\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {c}}_{3,2};x_{1,2},{\mathbf {c}}_{3,1})\),

     
  3. (3)

    \(\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {c}}_{1,3};{\mathbf {c}}_{2,3},x_{1,2})\),

     
  4. (4)

    \(\mathcal {M}^{{\mathbf {J}}^-}(\varnothing ;{\mathbf {c}}_{2,1},x_{1,2})\) (Fig. 7).

     
Fig. 7

Six moduli spaces in Theorem 4.14

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.

It will be clear that we should, for \(j=1,2,3\), naturally abuse the notation to denote \({\mathbf {q}}_j \in CF(S_j,{\mathbf {P}})\), which is the only generator in the corresponding cochain complex. Let \(e, pt\in CF(S_1,S_2)\) be the minimum and maximum of the Morse function, respectively, where e represents the identity in cohomology. On the cohomological level, it is clear that \([{\mathbf {q}}_2][e]=\pm [{\mathbf {q}}_1]\) and \([e][{\mathbf {q}}_1^\vee ]=\pm [{\mathbf {q}}_2^\vee ]\). This implies the algebraic count
$$\begin{aligned} \begin{aligned}&\#\mathcal {M}({\mathbf {q}}_1; {\mathbf {q}}_2, e)=\pm 1\\&\#\mathcal {M}({\mathbf {q}}_2^\vee ; e,{\mathbf {q}}_1^\vee )=\pm 1. \end{aligned} \end{aligned}$$
(4.19)
Fig. 8

Big local model before stretch

We now apply the same argument to other cochain complexes. For \(i \ne j\), let \({\mathbf {c}}_{i,j}\in CF^*(S_i,\tau _{{\mathbf {P}}}(S_j))\) (Fig. 8). be the only generator in their corresponding complex. Again, the multiplication by [e] on \([{\mathbf {c}}_{1,3}] \) and \( [{\mathbf {c}}_{3,1}]\) yields
$$\begin{aligned}&\#\mathcal {M}({\mathbf {c}}_{1,3}; {\mathbf {c}}_{2,3}, e)=\pm 1, \end{aligned}$$
(4.20)
$$\begin{aligned}&\#\mathcal {M}({\mathbf {c}}_{3,2}; e,{\mathbf {c}}_{3,1})= \pm 1. \end{aligned}$$
(4.21)
For the case of \({\mathbf {c}}_{2,1}\in CF(S_2,\tau _{\mathbf {P}}(S_1))\), it is immediate from Seidel’s exact sequence that \({{\,\mathrm{rank}\,}}HF(S_2,\tau _{\mathbf {P}}(S_1))=1\), concentrated on degree 0. \(CF(S_2,\tau _S S_1)\) has two additional generators \(|{\mathbf {c}}_{2,1}|=n-1\) and \(|pt|=n\), which cancel each other. Therefore, one has
$$\begin{aligned} \#\mathcal {M}(pt; {\mathbf {c}}_{2,1})=\pm 1 \end{aligned}$$
(4.22)
To deduce Theorem 4.14, we perform a neck-stretching along \(\partial {\mathbf {U}}\). It means that we choose a family of almost complex structure \({\mathbf {J}}^{\tau }\) adapted to \(\partial {\mathbf {U}}\) and see how the \({\mathbf {J}}^{\tau }\)-holomorphic curves converge as \(\tau \) goes to infinity. We require that the limiting almost complex structure on \(S {\mathbf {U}}\) coincides with \({\mathbf {J}}^-\) and we denote the limiting almost complex structure outside \({\mathbf {U}}\) by \({\mathbf {J}}^+\). \(S_1\) and \(S_2\) give two fibers in \({\mathbf {U}}\), and every holomorphic curve in \(\mathcal {M}^{{\mathbf {J}}^\tau }({\mathbf {q}}_1; {\mathbf {q}}_2,e)\) will converge, in the \({\mathbf {U}}\) part, to a curve in \(\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {q}}_1;{\mathbf {q}}_2, x_{1,2})\) (see Lemma 4.9, where the direction of the strip-like ends are switched). This implies
$$\begin{aligned} (\#\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {q}}_1;{\mathbf {q}}_2, x_{1,2}))\cdot (\#\mathcal {M}^{{\mathbf {J}}^+}(x_{1,2};e))=\#\mathcal {M}^{{\mathbf {J}}^\tau }({\mathbf {q}}_1; {\mathbf {q}}_2,e)=\pm 1. \end{aligned}$$
Since all counts are integers, it follows that \(\#\mathcal {M}^{J^-}({\mathbf {q}}_1;{\mathbf {q}}_2,x_{1,2})= \pm 1\) which implies the same is true for \(\#\mathcal {M}^{J^-}({\mathbf {q}}_2^\vee ;x_{1,2},{\mathbf {q}}_1^\vee )\) and \(\#\mathcal {M}^{J^-}(\varnothing ;{\mathbf {q}}_2,x_{1,2},{\mathbf {q}}_1^\vee )\).
The same stretching argument, along with (4.20), (4.21) and (4.22) yields
$$\begin{aligned}&(\#\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {c}}_{1,3};{\mathbf {c}}_{2,3}, x_{1,2}))\cdot (\#\mathcal {M}^{{\mathbf {J}}^+}(x_{1,2};e))=\#\mathcal {M}^{{\mathbf {J}}^\tau }({\mathbf {c}}_{1,3}; {\mathbf {c}}_{2,3}, e)=\pm 1, \end{aligned}$$
(4.23)
$$\begin{aligned}&(\#\mathcal {M}^{{\mathbf {J}}^-}({\mathbf {c}}_{3,2};x_{1,2},{\mathbf {c}}_{31}))\cdot (\#\mathcal {M}^{{\mathbf {J}}^+}(x_{1,2};e))=\#\mathcal {M}^{{\mathbf {J}}^\tau }({\mathbf {c}}_{3,2};e, {\mathbf {c}}_{3,1})=\pm 1, \end{aligned}$$
(4.24)
$$\begin{aligned}&(\#\mathcal {M}^{{\mathbf {J}}^-}(\varnothing ;{\mathbf {c}}_{2,1},x_{1,2}))\cdot (\#\mathcal {M}^{{\mathbf {J}}^+}(x_{1,2},pt;\varnothing ))=\#\mathcal {M}^{{\mathbf {J}}^\tau }(pt;{\mathbf {c}}_{2,1})=\pm 1. \end{aligned}$$
(4.25)
which give the remaining algebraic counts.

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. (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. (2)

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

     
  3. (3)

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

     
  4. (4)

    \(\mathcal {M}^{J^-}(\varnothing ;c_{{\mathbf {p}},{\mathbf {q}}},x_{{\mathbf {q}},{\mathbf {p}}})\)

     
where \(x_{{\mathbf {p}}',{\mathbf {p}}}\) is the unique Reeb chord of canonical relative grading 0 from \(\Lambda _{p'}\) to \(\Lambda _{p}\) which can be lifted to a Reeb chord from \(\Lambda _{{\mathbf {p}}'}\) to \(\Lambda _{{\mathbf {p}}}\). The definition of \(x_{{\mathbf {q}},{\mathbf {q}}'}\) and \(x_{{\mathbf {q}},{\mathbf {p}}}\) are similar.

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

For \(\tau \gg 1\), the algebraic count of following moduli spaces are equal
  • \(\mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}})\), 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 \(\mathcal {M}^{J^-}(\mathbf{p }';\mathbf{p },x_{{\mathbf {p}}',{\mathbf {p}}})\).

Therefore, for \(\tau \gg 1\),
$$\begin{aligned}&\#\mathcal {M}^{J^\tau }(\mathbf{p }',\mathbf{p })\\&=\#\mathcal {M}^{J^+}(x_{{\mathbf {p}},{\mathbf {p}}'};\varnothing )\cdot \#\mathcal {M}^{J^-}(p';p,x_{{\mathbf {p}}',{\mathbf {p}}})\\&=\#\mathcal {M}^{J^+}(x_{{\mathbf {p}},{\mathbf {p}}'};\varnothing )\cdot \#\mathcal {M}^{J^-}(c_{{\mathbf {p}}',{\mathbf {q}}};c_{{\mathbf {p}},{\mathbf {q}}}, x_{{\mathbf {p}}',{\mathbf {p}}})\\&=\#\mathcal {M}^{J^\tau }(c_{{\mathbf {p}}',{\mathbf {q}}}; c_{{\mathbf {p}},{\mathbf {q}}}) \end{aligned}$$
where the second equality uses Corollary 4.15 (1) and (3). \(\square \)

Similarly, we compare the differentials of Type (A2) and (A2\('\)).

Lemma 4.17

For \(\tau \gg 1\), the algebraic count of following moduli spaces are equal
  • \(\mathcal {M}^{J^{\tau }}(c_{{\mathbf {p}},{\mathbf {q}}'};c_{{\mathbf {p}},{\mathbf {q}}})\),

  • \(\mathcal {M}^{J^\tau }({\mathbf {q}}'^\vee ;{\mathbf {q}}^\vee )\).

Proof

The proof is almost word-by-word taken from Lemma 4.16. Lemma 4.6, 4.7 and Corollary 4.15 (1) and (2) implies
$$\begin{aligned}&\#\mathcal {M}^{J^\tau }(\mathbf{q }'^\vee ;\mathbf{q }^\vee )\\&=\#\mathcal {M}^{J^+}(x_{{\mathbf {q}},{\mathbf {q}}'})\cdot \#\mathcal {M}^{J^-}(q'^\vee ;x_{{\mathbf {q}},{\mathbf {q}}'},q^\vee )\\&=\#\mathcal {M}^{J^+}(x_{{\mathbf {q}},{\mathbf {q}}'})\cdot \#\mathcal {M}^{J^-}(c_{{\mathbf {p}},{\mathbf {q}}'};x_{{\mathbf {q}},{\mathbf {q}}'}, c_{{\mathbf {p}},{\mathbf {q}}})\\&=\#\mathcal {M}^{J^\tau }(c_{{\mathbf {p}},{\mathbf {q}}'}; c_{{\mathbf {p}},{\mathbf {q}}}) \end{aligned}$$
\(\square \)

The last lemma addresses differentials of Type (C) and (C\('\)).

Lemma 4.18

For \(\tau \gg 1\), the algebraic count of following moduli spaces are equal
  • \(\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee ,{\mathbf {p}})\), 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

The strategy is still similar. Apply the same neck-stretching as in Lemmas 4.16 and 4.17, one obtains a building consisting of a triangle and a bigon for \(\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee , {\mathbf {p}})\), thanks to Lemma 4.9; and a building consisting of two bigons for \(\mathcal {M}^{J^\tau }(x; c_{{\mathbf {p}},{\mathbf {q}}})\) from Lemma 4.11. Therefore
$$\begin{aligned}&\#\mathcal {M}^{J^\tau }(x; {\mathbf {q}}^\vee , {\mathbf {p}})\\&=\#\mathcal {M}^{J^+}(x,x_{{\mathbf {q}},{\mathbf {p}}};\varnothing ) \cdot \#\mathcal {M}^{J^-}(\varnothing ; p, x_{{\mathbf {q}},{\mathbf {p}}},q^\vee )\\&=\#\mathcal {M}^{J^+}(x,x_{{\mathbf {q}},{\mathbf {p}}};\varnothing ) \cdot \#\mathcal {M}^{J^-}(\varnothing ; c_{{\mathbf {p}},{\mathbf {q}}}, x_{{\mathbf {q}},{\mathbf {p}}})\\&=\#\mathcal {M}^{J^\tau }(x; c_{{\mathbf {p}},{\mathbf {q}}}) \end{aligned}$$
where the second equality uses Corollary 4.15 (1) and (4). \(\square \)

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

For any object \(\mathcal {E}^1 \in Ob({\mathcal {F}})\), we can perform a Hamiltonian perturbation for \(\mathcal {E}^1\) to obtain another object \((\mathcal {E}^1)'\) of \({\mathcal {F}}\) such that there is a degree zero cochain \(c_\mathcal {D}\in hom^0_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))\) so that \(c_\mathcal {D}\) is a cocycle, and
$$\begin{aligned} \mu ^2(c_\mathcal {D}, \cdot ):hom^0_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\mathcal {E}^0,\tau _P((\mathcal {E}^1)')) \rightarrow hom^0_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\mathcal {E}^0,T_\mathcal {P}(\mathcal {E}^1)) \end{aligned}$$
(5.1)
is a quasi-isomorphism for all \(\mathcal {E}^0 \in Ob({\mathcal {F}})\)

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 overall strategy goes as follows. By Proposition 4.1, Corollary 2.18 and the fact that Hamiltonian isotopic objects are quasi-isomorphic, we know that
$$\begin{aligned} H(hom_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1)))=HF(\tau _P(\mathcal {E}^1),\tau _P(\mathcal {E}^1)) \end{aligned}$$
(5.2)
Our goal is to pick an appropriate non-exact degree zero cocycle \(c_\mathcal {D}\in hom^0_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))\), and check that \(\mu ^2(c_\mathcal {D},-)\) is a quasi-isomorphism for all \(\mathcal {E}^0 \in Ob({\mathcal {F}})\) [see (5.1)]. By a Hamiltonian perturbation if necessary, it suffices to check the equality for those \(\mathcal {E}^0\) such that \(L_0\) intersects \(L_1\), \((L_1)'\) and P transversally. This allows us to apply neck-stretching along \(\partial U\) to compute \(\mu ^2(c_\mathcal {D}, \cdot )\) for \(\tau \gg 1\) (see Sect. 3.8).

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(LL) 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 quasi-isomorphism (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 neck-stretching. The discussion in this section works for field \({\mathbb {K}}\) of arbitrary characteristics.

Let \(L_1'\) be a \(C^2\)-small Hamiltonian push-off of \(L_1\) such that \(L_1' \cap U\) is a union of cotangent fibers. Let \(q_1,\dots ,q_{d_{L_1}} \in CF(L_1,P)\) and \(q_1',\dots ,q_{d_{L_1}}' \in CF(L_1',P)\) be the cochain representatives of the geometric intersection points, where \(d_{L_1}=\# (P \cap L_1)=\#(P \cap L_1')\). We also number the intersection points so that \(d_P(q_i,q_i')\ll \epsilon \) in the standard quotient round metric. Let \(\Lambda _{q_i},\Lambda _{q_j'} \subset \partial U\) be the cospheres at \(q_i\) and \(q_j'\), respectively. We assume \(q_i,q_j'\) satisfy Corollary 3.23. Fix \({\mathbf {q}}_i,{\mathbf {q}}_j'\) be a lift of \(q_i,q_j'\), respectively, for all ij. Our focus will be the cochain complex
$$\begin{aligned} \mathcal {D}&:=hom_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))=(CF(\mathcal {P},\mathcal {E}^1)\nonumber \\&\quad \otimes _{\Gamma } CF(\tau _P((\mathcal {E}^1)'),\mathcal {P}))[1] \oplus CF(\tau ((\mathcal {E}^1)'),\mathcal {E}^1) \end{aligned}$$
(5.3)
which is generated by elements supported at the intersection points
$$\begin{aligned} \left\{ \begin{array}{ll} q_i^\vee \otimes \tau _P(q_j'), &{}\text { for } i,j=1,\dots ,d_{L_1}\\ c_{i,g,j}^\vee :=c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}^\vee , &{}\text { for }g \in \Gamma , i,j=1,\dots ,d_{L_1}\\ w_k, &{}\text { for }k=1,\dots ,\# (L_1' \cap L_1) \end{array} \right. \end{aligned}$$
(5.4)
The first two kinds of intersection points are inside U while \(\{w_k\}\) are outside U. Elements supported at \(c_{i,g,j}^\vee \) and \(w_k\) are given by
$$\begin{aligned} Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}},\mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}) \text { and } Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{w_k},\mathcal {E}^1_{w_k}) \end{aligned}$$
(5.5)
respectively. On the other hand, the elements supported at \(q_i^\vee \otimes \tau _P(q_j')\) are generated by
$$\begin{aligned} (\psi ^2 \otimes {\mathbf {q}}_i^\vee ) \otimes (g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1),&\text { for } \psi ^2 \in \mathcal {E}^1_{q_i}, \psi ^1 \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')},{\mathbb {K}}), g \in \Gamma \end{aligned}$$
(5.6)
Here we use the commutativity \(\pi (\tau _{\mathbf {P}}({\mathbf {q}}_j'))=\tau _P(\pi ({\mathbf {q}}_j'))=\tau _P(q_j')\).

Lemma 5.2

With respect to canonical relative grading, we have
$$\begin{aligned} \left\{ \begin{array}{ll} |q_i^\vee |=0, &{}\text {for }q_i^\vee \in hom(P,T^*_{q_i} P) \\ |\tau _P(q_j')|=1, &{}\text {for }\tau _P(q_j') \in hom(\tau _P(T^*_{q_j'} P),P) \\ |c_{i,g,j}^\vee |=1, &{}\text {for }c_{i,g,j}^\vee = \pi (\tau _{\mathbf {P}}(T^*_{g{\mathbf {q}}_j'} {\mathbf {P}}) \cap T^*_{{\mathbf {q}}_i} {\mathbf {P}}) \in hom(\tau _P(T^*_{q_j'} P),T^*_{q_i} P) \end{array} \right. \end{aligned}$$
(5.7)

Proof

The fact that \(|q_i^\vee |=0\) follows from the definition of canonical relative grading (3.8). \(|c_{i,g,j}^\vee |=1\) follows from \(|c_{i,g,j}|=n-1\) (see Lemma 4.5). Finally, from the long exact sequence
$$\begin{aligned} HF^k(P, T^*_{q_j'} P) \rightarrow HF^k(P, \tau _P(T^*_{q_j'} P)) \rightarrow HF^{k+1}(P, P) \rightarrow HF^{k+1}(P, T^*_{q_j'} P) \end{aligned}$$
(5.8)
and the fact that \(HF(P,\tau _P(T^*_{q_j'} P)) \) has rank 1, we know that \(HF^{0}(P, P) \simeq HF^{0}(P, T^*_{q_j'} P)\), and \(HF^k(P, \tau _P(T^*_{q_j'} P)) \rightarrow HF^{k+1}(P, P)\) is an isomorphism when \(k=n-1\). Therefore, \(|\tau _P(q_j')^\vee |=n-1\) and \(|\tau _P(q_j')|=n-|\tau _P(q_j')^\vee |=1\). \(\square \)
Without loss of generality, we assume that there is a unique \(w_k\) with degree 0 and we denote it by \(e_L\). All other \(w_k\) has \(|w_k|>0\). With generators understood, we now recall that the differential for element \(\psi _x\) supported at \(x=c_{i,g,j}^\vee \) or \(x=w_k\) is given by \(\mu ^1(\psi _x)=\mu ^1_{\mathcal {F}}(\psi _x)\), and for element supported at \(q_i^\vee \otimes \tau _P(q_j')\) is given by [see (2.87)]
$$\begin{aligned} \mu ^1_{\mathcal {D}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee \otimes g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1) =\,&(-1)^{|g\tau _{\mathbf {P}}({\mathbf {q}}_j')|}\mu ^1_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee ) \otimes (g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\nonumber \\&+(\psi ^2 \otimes {\mathbf {q}}_i^\vee ) \otimes \mu ^1_{\mathcal {F}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1) \nonumber \\&+\mu ^2_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1) \end{aligned}$$
(5.9)
Our focus will be put on \(\mu ^1_{\mathcal {F}}(\psi _{e_L})\) and \(\mu ^2_{\mathcal {F}}(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\).

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 push-off 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 \(\{g{\mathbf {q}}_i'\}_{g \in \Gamma }\).

We recall from (3.35) that the action of a Reeb chord x from \(\Lambda _{q_j'}\) to \(\Lambda _{q_i}\) is given by
$$\begin{aligned} A(x):= L(x) +h_j' -h_i \end{aligned}$$
(5.10)

Lemma 5.3

There is a constant \(\epsilon >0\) depending only on \(\{q_i\}_{i=1}^{d_{L_1}}\) and \(L_1\) such that when \(L_1'\) is a sufficiently small Hamiltonian push-off of \(L_1\),
  • \(A(x)> \epsilon \) 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 non-constant 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 non-shortest 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 birth-death bifurcation cancels a pair of \(J_{t_l-\epsilon }\)-triangles at time \(t_l\); and the death-birth 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 \(t_l\).

Therefore, to show that the algebraic count persists to be \(\pm 1\) crossing \(t_l\), we will analyze each bifurcation moment \(t_l\) below and prove the contribution to \(\mathcal {M}^{J_{t}}(q_i^\vee ;e_L,(q_i')^\vee )\) is zero. For simplicity we let \(l=1\). Since there are exactly two irreducible components at \(t=t_1\), one of them has to has virtual dimension 0 and the other one has dimension \(-1\). Let u denote the component of virtual dimension \(-1\) (it can be either a strip or a triangle), and we divide the possible stable \(J_{t_1}\)-holomorphic triangles into three cases:
  1. (i)

    both \(q_i^\vee \) and \((q_i')^\vee \) are asymptotes of u;

     
  2. (ii)

    exactly one of \(q_i^\vee \) and \((q_i')^\vee \) is an asymptote of u;

     
  3. (iii)

    neither of \(q_i^\vee \) nor \((q_i')^\vee \) is an asymptote of u.

     
Case (i) If both \(q_i^\vee \) and \((q_i')^\vee \) are asymptotes of u, then the last asymptote x of u must be a generator of \(CF(L_1', L_1)\) by boundary condition. Moreover, x is a degree 1 element of \(CF(L_1', L_1)\) because \({{\,\mathrm{virdim}\,}}(u)=-1\) and \(|e_L|=0\). This bifurcation contributes to a change in the algebraic count of \(\mathcal {M}^{J_t}(q_i^\vee ;e_L,(q_i')^\vee )\) by the algebraic count of rigid elements from \(\mathcal {M}^{J_{t_1}}(x;e_L)\) (when \(t>t_1\), the moduli \(\mathcal {M}^{J_{t_1}}(x;e_L)\) and \(\mathcal {M}^{J_{t_1}}(q_i^\vee ;x,(q_i')^\vee )\) glue together to give a change). However, the algebraic count of rigid elements from \(\mathcal {M}^{J_{t_1}}(x;e_L)\) is zero because \(e_L\) is a cocycle.

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. Let \(\mathcal {E}'\), \((\mathcal {E}^1)'\) be local systems supported on \(L_1\), \(L_1'\), respectively. Using the Hamiltonian push-off, we have the identifications
$$\begin{aligned} \tau _P((\mathcal {E}^1)')_{w_k} \simeq (\mathcal {E}^1)'_{w_k} \simeq \mathcal {E}^1_{w_k}, \text { and } \tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}} \simeq \mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}} \end{aligned}$$
(5.11)
for all \(w_k\) and \(c_{{\mathbf {q}}_i,g{\mathbf {q}}_j}\). In particular, we can define \(t_\mathcal {D}\) to be the identity morphism supported at the intersection underlying \(e_L\), but as a morphism, it is written as:
$$\begin{aligned} t_\mathcal {D}:=id \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{e_L}, \mathcal {E}^1_{e_L}) \subset hom_{{\mathcal {F}}^{{{\,\mathrm{perf}\,}}}}(\tau _P((\mathcal {E}^1)'),T_\mathcal {P}(\mathcal {E}^1))=\mathcal {D} \end{aligned}$$
(5.12)
We also denote \(e_{\mathcal {E}}\) as
$$\begin{aligned} e_{\mathcal {E}}:=id\in Hom\left( (\mathcal {E}^1)',\mathcal {E}'\right) . \end{aligned}$$
(5.13)
Geometrically, both \(t_\mathcal {D}\) and \(e_\mathcal {E}\) are supported at the same intersection point and represents the same identity morphism between the stalks. \(t_\mathcal {D}\) can be regarded as a chain-level preimage of the \(\mathcal {E}\) under the (Poincaré) dualized Seidel’s exact sequence, hence has no guarantee to be closed.

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

For \(L_1'\) sufficiently close to \(L_1\) and \(\tau \gg 1\), we have
$$\begin{aligned} \mu ^1(t_\mathcal {D})=\sum _{i,j,g} \psi _{c_{i,g,j}^\vee } \end{aligned}$$
(5.14)
where \(\psi _{c_{i,g,j}^\vee } \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}},\mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}})\) and
$$\begin{aligned} \left\{ \begin{array}{ll} \psi _{c_{i,g,j}^\vee }=0 \text { if }j>i \text { and } g \in \Gamma (\text {or }j=i \text { and } g \ne 1_\Gamma ) \\ \psi _{c_{i,1_\Gamma ,i}^\vee }=\pm id \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,{\mathbf {q}}_i'}},\mathcal {E}^1_{c_{{\mathbf {q}}_i,{\mathbf {q}}_i'}}) \end{array} \right. \end{aligned}$$
(5.15)

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

We argue by contradiction as before. Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a limiting holomorphic building. By boundary condition, there is \(v_1 \in V(\mathcal {T})\) such that \(q_i^\vee , \tau _P(q_j')\) are asymptotes of \(u_{v_1}\). The other asymptotes of \(u_{v_1}\) are positive Reeb chords \(y_1,\dots ,y_k\). The virtual dimension of \(u_{v_1}\) can be computed using canonical relative grading (Lemma 5.2), and is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_{v_1})=\,&n(1-0)-|q_i^\vee |-|\tau _P(q_j')|\\&-\sum _{s=1}^k |y_s|-(1-k) \geqslant n-0-1-(1-k)>0 \end{aligned}$$
because \(n \geqslant 3\). It contradicts to \({{\,\mathrm{virdim}\,}}(u_{v_1})=0\). \(\square \)

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

First, we want to argue that any \(u \in \mathcal {M}^{J^{\tau }}(c_{i,h,j}^\vee ;q_i^\vee ,\tau _P(q_j'))\) contributing to \(\mu ^2({\mathbf {q}}_i^\vee ,g\tau _P({\mathbf {q}}_j'))\) has image completely lying inside U when \(\tau \gg 1\). We argue as before. Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a limiting holomorphic building. By boundary condition, there is \(v_1 \in V(\mathcal {T})\) such that \(q_i^\vee , \tau _P(q_j')\) are asymptotes of \(u_{v_1}\). If \(c_{i,h,j}^\vee \) is not an asymptote of \(u_{v_1}\), then we get a contradiction as in Lemma 5.11. Therefore, \(u_{v_1}\) has asymptotes \(c_{i,h,j}^\vee \), \(q_i^\vee \), \(\tau _P(q_j')\) and positive Reeb chords \(y_1,\dots ,y_k\). The virtual dimension of \(u_{v_1}\) is given by
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_{v_1})=|c_{i,h,j}^\vee |-|q_i^\vee |-|\tau _P(q_j')|-\sum _{s=1}^k |y_s|+k \geqslant 1-0-1+k=k \end{aligned}$$
It means that \(k=0\) so \(u_{v_1}\) has no positive Reeb chord and the claim follows.
In particular, we can lift \(u \in \mathcal {M}^{J^{\tau }}(c_{i,h,j}^\vee ;q_i^\vee ,\tau _P(q_j'))\) to the universal cover \({\mathbf {U}}\). By considering the boundary condition, it is clear that we must have \(h=g\) for u to exist. Now, to compute the \(c_{i,g,j}^\vee \)-coefficient of \(\mu ^2({\mathbf {q}}_i^\vee ,g\tau _P({\mathbf {q}}_j'))\), we use the following model.
Fig. 9

Holomorphic triangles in \({\mathbf {U}}\)

We consider an \(A_3\)-Milnor fiber as in the proof of Theorem 4.14 but rename the objects to keep the notation aligned with the current situation. For example, we denote the Lagrangian spheres by \(S_1\) ,S and \(S_2\) such that \(S_1 \cap S_2 =\varnothing \). Let \(\tau \) be the Dehn twist along S, \(q^\vee \in CF(S,S_1)\), \(q' \in CF(S_2,S)\), \(\tau (q') \in CF(\tau (S_2),S)\), \(c^\vee \in CF(\tau (S_2),S_1)\) and \(e,f \in CF(S,S)\). We have \(|q^\vee |=0\), \(|q'|=n\), \(|\tau (q')|=1\), \(|c^\vee |=1\), \(|e|=0\) and \(|f|=n\). Consider the following commutative diagram (up to sign)All the Floer cohomology has rank 1 except that HF(SS) has rank 2. The bottom arrow gives \(\mu ^2(q,q^\vee )=f\). By the long exact sequence
$$\begin{aligned} HF^{k}(S_1,S_2) \rightarrow HF^k(S_1,\tau (S_2)) \rightarrow HF^{k+1}(S_1,S) \rightarrow HF^{k+1}(S_1,S_2) \end{aligned}$$
(5.16)
and the fact that \(HF(S_1,S_2)=0\), we know that \(HF^{n-1}(S_1,\tau (S_2)) \rightarrow HF^{n}(S_1,S)\) is an isomorphism. Since \(\tau (q')\) represents the unique (up to multiplications by a unit) non-zero class in \(HF(\tau (S_2),S)\), we know that \(\mu ^2(\tau (q'),\cdot )\) induces the isomorphism \(HF^{n-1}(S_1,\tau (S_2)) \simeq HF^{n}(S_1,S)\). Therefore, we must have \(\mu ^2(\tau (q'),c)= \pm q\).

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

For \(\tau \gg 1\), we have [see (5.6)], up to sign,
$$\begin{aligned} \mu ^2(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)=I_{[q_i \rightarrow c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}]} (\psi ^2) \otimes (\psi ^1 \circ I_{[c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'} \rightarrow \tau _{P}(q_j')]})\nonumber \\ \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}, \mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}) \qquad \end{aligned}$$
(5.17)
for \(\psi ^2 \in \mathcal {E}^1_{q_i}\) and \(\psi ^1 \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')},{\mathbb {K}})\). In particular, the right hand side is supported at the intersection point \(c_{i,g,j}^\vee \) only and the morphism \(\Phi ^{\otimes }_{i,g,j}:=\mu ^2(- \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes -)\)
$$\begin{aligned} \Phi ^{\otimes }_{i,g,j}:\mathcal {E}^1_{q_i} \otimes Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')},{\mathbb {K}}) \rightarrow Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}},\mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}) \end{aligned}$$
(5.18)
is an isomorphism.

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.

To obtain the result, it suffices to understand the parallel transport maps. Let \(u \in \mathcal {M}^{J^{\tau }}(c_{i,g,j}^\vee ;q_i^\vee ,\tau _P(q_j'))\). The contribution to \(\mu ^2(\psi ^2 \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1)\) by u is given by (up to sign)
$$\begin{aligned} (I_{\partial _2 u} \circ \psi ^2) \otimes ({\mathbf {q}}_i^\vee \circ I_{\partial _1 u} \circ g\tau _{\mathbf {P}}({\mathbf {q}}_j')) \otimes (\psi ^1 \circ I_{\partial _0 u}) \end{aligned}$$
(5.19)
Since the domain of u is contractible, u can be lifted to the universal cover and therefore the generator \(c_{i,g,j}^\vee \) uniquely determine the homotopy class of the path \(\partial _1 u\) on P (and also \(\partial _0 u\) on \(\tau _P(L_1')\) and \(\partial _2 u\) on \(L_1\), which is why the parallel transport of \(\partial _1 u\) is omitted in the statement), which is exactly the path such that \({\mathbf {q}}_i^\vee \circ I_{\partial _1 u} \circ g\tau _{\mathbf {P}}({\mathbf {q}}_j')=1\), where \(g\tau _{\mathbf {P}}({\mathbf {q}}_j')\) is regarded as an element of the universal local system at \(q_j'\) and \({\mathbf {q}}_i^\vee \) is regarded as an element of the dual of the universal local system at \(q_i\). In other words, we have \(I_{\partial _1 u}(g\tau _{\mathbf {P}}({\mathbf {q}}_j'))={\mathbf {q}}_i\). On the other hand, we have \(I_{\partial _0 u}=I_{[c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'} \rightarrow \tau _{P}(q_j')]}\) and \(I_{\partial _2 u}=I_{[q_i \rightarrow c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}]}\) so (5.19) reduces to \(I_{[q_i \rightarrow c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}]} (\psi ^2) \otimes (\psi ^1 \circ I_{[c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'} \rightarrow \tau _{P}(q_j')]})\). Now, (5.17) follows immediately from Lemma 5.12.
On the other hand, since \(I_{[q_i \rightarrow c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}]}\) and \(I_{[c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'} \rightarrow \tau _{P}(q_j')]}\) are isomorphisms from \(\mathcal {E}^1_{q_i}\) to \(\mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}\) and from \(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}}\) to \(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')}\), respectively, (5.17) clear induces the isomorphism
$$\begin{aligned} \mathcal {E}^1_{q_i} \otimes Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')},{\mathbb {K}}) \rightarrow \mathcal {E}^1_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}} \otimes Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,g{\mathbf {q}}_j'}},{\mathbb {K}}) \end{aligned}$$
(5.20)
as desired \(\square \)

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 ij, \(\psi ^{i,j}\) 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 non-trivial element supported at \(c_{i_0,g,j_0}^\vee \) for some g. Because all other \(\mu ^1(\psi ^{i,j})\) do not have non-zero element supported at \(c_{i_0,g,j_0}^\vee \), this draws a contradiction.    \(\square \)

By Corollary 5.15, we can write every degree 0 cocycle \(\beta \) of \(\mathcal {D}\) as
$$\begin{aligned} \beta =\psi _{e_L}+ \sum _{i,j} \psi _{q_i^\vee \otimes \tau _P(q_j')} \end{aligned}$$
(5.21)
where \(\psi _{x}\) is an element supported at x. Moreover, by (5.6), we can further decompose \(\psi _{q_i^\vee \otimes \tau _P(q_j')}\) as
$$\begin{aligned} \psi _{q_i^\vee \otimes \tau _P(q_j')}= \sum _{g \in \Gamma } \sum _{k=1}^{n_{i,g,j}} \psi ^2_{i,g,j,k} \otimes {\mathbf {q}}_i^\vee \otimes g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k} \end{aligned}$$
(5.22)
for some \(\psi ^2_{i,g,j,k} \in \mathcal {E}^1_{q_i}\), \(\psi ^1_{i,g,j,k} \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{\tau _P(q_j')},{\mathbb {K}})\) and \(n_{i,g,j} \in {\mathbb {N}}\).

Proposition 5.16

(Cocycle elements) For \(L_1'\) sufficiently close to \(L_1\) and \(\tau \gg 1\), there is a non-exact degree 0 cocycle \(c_\mathcal {D}\) in \(\mathcal {D}\) of the form
$$\begin{aligned} c_\mathcal {D}= t_{\mathcal {D}} +\sum _{g,k,i,j} \psi ^2_{i,g,j,k} \otimes {\mathbf {q}}_i^\vee \otimes g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k} \end{aligned}$$
(5.23)
where \( \psi ^2_{i,g,j,k}=\psi ^1_{i,g,j,k}= 0\) if either \(j>i\) or \((j=i\) and \(g \ne 1_\Gamma )\), and [see (5.18)]
$$\begin{aligned} \Phi ^{\otimes }_{i,1_\Gamma ,i}\left( \sum _k(\psi ^2_{i,1_\Gamma ,i,k} \otimes {\mathbf {q}}_i^\vee \otimes \tau _{\mathbf {P}}({\mathbf {q}}_i') \otimes \psi ^1_{i,1_\Gamma ,i,k})\right) =\pm id \end{aligned}$$
(5.24)
where \(\pm id \in Hom_{\mathbb {K}}(\tau _P((\mathcal {E}^1)')_{c_{{\mathbf {q}}_i,{\mathbf {q}}_i'}},\mathcal {E}^1_{c_{{\mathbf {q}}_i,{\mathbf {q}}_i'}})\).

Proof

Let \(\beta \) be a non-exact 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 \(\mu ^1_{hom((\mathcal {E}^1)',\mathcal {E}^1)}(\psi _{e_L})=0\).

Also, Corollary 5.15 implies that the degree zero cocycle \(\beta \) is uniquely determined by its \(\psi _{e_L}\) component (or, as a cochain of \(\mathcal {D}\), \(\mu ^1_\mathcal {D}(\psi _{e_L})\) has no \(w_k\)-components). Therefore,
$$\begin{aligned} {{\,\mathrm{rank}\,}}(H^0(\mathcal {D})) \leqslant {{\,\mathrm{rank}\,}}(H^0(hom((\mathcal {E}^1)',\mathcal {E}^1))) \end{aligned}$$
(5.25)
However, as explained in (5.2), we have
$$\begin{aligned} {{\,\mathrm{rank}\,}}(H^0(\mathcal {D}))={{\,\mathrm{rank}\,}}(HF^0((\mathcal {E}^1)',\mathcal {E}^1)) \end{aligned}$$
(5.26)
It implies that for each degree 0 cocycle \(\psi _{e_L} \in hom((\mathcal {E}^1)',\mathcal {E}^1)\), there exists \(\psi _{q_i^\vee \otimes \tau _P(q_j')}\) such that \(\psi _{e_L}+ \sum _{i,j} \psi _{q_i^\vee \otimes \tau _P(q_j')}\) is a degree 0 cocycle in \(\mathcal {D}\).

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 ijg. Therefore, we obtain the result by Propositions 5.8 and 5.14 [see (5.9)]. \(\square \)

5.2 Quasi-isomorphisms

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 \(\mathcal {E}^0 \in Ob({\mathcal {F}})\).

We assume that \(L_0 \pitchfork L_1\), \(L_0 \pitchfork \tau _P(L_1')\), and that \(L_0 \cap U\) is a union of cotangent fibers \(\bigcup _{i=1}^{d_{L_0}} T^*_{p_i} P\), where \(d_{L_0}=\#(L_0 \cap P)\). Let \({\mathbf {p}}_i\) be a choice of lift of \(p_i\) in \({\mathbf {P}}\). Let \(C_0:=hom(\mathcal {E}^0,\tau _P((\mathcal {E}^1)'))\) and \(C_1:=hom(\mathcal {E}^0,T_\mathcal {P}(\mathcal {E}^1))\). We know from Lemma 4.13 that, when \(\tau \) is large enough, there is a subcomplex \(C_0^{s} \subset C_0\) generated by generators of \(C_0\) outside U. Let \(C_0^q:=C_0 /C_0^s\) be the quotient complex, which is generated by generators of \(C_0\) inside U. Similarly, \(C_1^s:=hom(\mathcal {E}^0,\mathcal {E}^1) \subset C_1\) is a subcomplex and \(C_1^q:=C_1/C_1^s\) is the quotient complex. By definition [see (2.88)], for \(\psi \in C_0\),
$$\begin{aligned} \mu ^2(c_\mathcal {D},\psi )=\,&\mu ^2_{{\mathcal {F}}}(t_\mathcal {D},\psi )+\sum _{i,j,g,k} (\psi ^2_{i,g,j,k} \otimes {\mathbf {q}}_i^\vee ) \otimes \mu ^2_{{\mathcal {F}}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k},\psi ) \nonumber \\&+\sum _{i,j,g,k} \mu ^3_{{\mathcal {F}}}(\psi ^2_{i,g,j,k} \otimes {\mathbf {q}}_i^\vee ,g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k},\psi ) \end{aligned}$$
(5.27)
We define \(\mu ^2_s(c_\mathcal {D},-):=\mu ^2(c_\mathcal {D},-)|_{C_0^s}:C_0^s \rightarrow C_1\).

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

Note that the first and last term on the right hand side of (5.27) lie inside \(C_1^s\) as a consequence of Lemma 4.5. Therefore, it suffices to show that \(\mu ^2_{{\mathcal {F}}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k},\psi )=0\) for \(\psi \in C_0^s\). We consider the moduli \(\mathcal {M}^{J^{\tau }}(p_s;\tau _P(q_j'),y)\) where \(y \in (L_0 \cap \tau _P(L_1')) \backslash U\) and \(p_s \in L_0 \cap P\). Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a holomorphic building converging from curves in \(\mathcal {M}^{J^{\tau }}(p_s;\tau _P(q_j'),y)\). From the boundary condition, there exists \(v_1 \in V(\mathcal {T})\) such that \(p_s\) and \(\tau _P(q_j')\) are asymptotes of \(u_{v_1}\). The other asymptotes of \(u_{v_1}\) are positive Reeb chords \(y_1,\dots ,y_m\). We have
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_{v_1})=|p_s|-|\tau _P(q_j')|- \sum _{l=1}^m |y_l|-(1-m) \geqslant n-1-(1-m) \geqslant n-2>0,\qquad \end{aligned}$$
(5.28)
contradiction. Therefore, \(\mathcal {M}^{J^{\tau }}(p_s;\tau _P(q_j'),y)= \varnothing \) for \(\tau \gg 1\). \(\square \)

Lemma 5.18

For \(\tau \gg 1\), \(\mu ^2_s(c_\mathcal {D},-)=\mu ^2_{{\mathcal {F}}}(t_\mathcal {D},-)\).

Proof

By Lemma 5.17, the second term in (5.27) vanishes, so it suffices to prove that \(\mathcal {M}^{J^{\tau }}(x;q_i^\vee , \tau _P(q_j'),y)= \varnothing \) for \(\tau \gg 1\), where \(y \in (L_0 \cap \tau _P(L_1')) \backslash U\) and \(x \in L_0 \cap L_1\). Let \(u_{\infty }=(u_v)_{v \in V(\mathcal {T})}\) be a holomorphic building converging from curves in \(\mathcal {M}^{J^{\tau }}(x;q_i^\vee , \tau _P(q_j'),y)\). From the boundary condition, there exists \(v_1 \in V(\mathcal {T})\) such that \(q_i^\vee \) and \(\tau _P(q_j')\) are asymptotes of \(u_{v_1}\). The other asymptotes of \(u_{v_1}\) are positive Reeb chords \(y_1,\dots ,y_m\). We have
$$\begin{aligned} {{\,\mathrm{virdim}\,}}(u_{v_1})=&n-|q_i^\vee |-|\tau _P(q_j')|\nonumber \\&- \sum _{l=1}^m |y_l|-(1-m) \geqslant n-1-(1-m) \geqslant n-2>0 \end{aligned}$$
(5.29)
Therefore, \(\mathcal {M}^{J^{\tau }}(x;q_i^\vee , \tau _P(q_j'),y)= \varnothing \) for \(\tau \gg 1\), \(\square \)

Proposition 5.19

For \(\tau \gg 1\), \(\mu ^2_s(c_\mathcal {D},-)\) is a quasi-isomorphism.

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 quasi-isomorphism. \(\square \)

By Lemma 5.17, we know that \(\mu ^2(c_\mathcal {D},-)\) induces a chain map on the quotient complexes \(\mu _q^2(c_\mathcal {D},-):C_0^q \rightarrow C_1^q\). Since the first and last term on the right hand side of (5.27) are, by definition, lying inside \(C_1^s\), the map \(\mu _q^2(c_\mathcal {D},-)\) is given by
$$\begin{aligned} \mu ^2_q(c_\mathcal {D},\psi )=\sum _{i,j,g,k} (\psi ^2_{i,g,j,k} \otimes {\mathbf {q}}_i^\vee ) \otimes \mu ^2_{{\mathcal {F}}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') \otimes \psi ^1_{i,g,j,k},\psi ) \end{aligned}$$
(5.30)
By Proposition 5.19 and the five lemma, to show that \(\mu ^2(c_\mathcal {D},-)\) is a quasi-isomorphism, it suffices to show that \(\mu ^2_q(c_\mathcal {D},-)\) is a quasi-isomorphism.
We recall from Lemma 4.2 that there is a bijective correspondence
$$\begin{aligned} \iota : hom(\mathcal {P},L_1') \otimes _{\Gamma } hom(L_0,\mathcal {P}) \rightarrow (L_0 \cap \tau _P(L_1')) \cap U \end{aligned}$$
(5.31)
so we can write a point \(y \in (L_0 \cap \tau _P(L_1')) \cap U\) as \(c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'}:=\iota ({\mathbf {q}}_l'^\vee \otimes h{\mathbf {p}}_s)\) for some \(h \in \Gamma \) and some sl. We want to understand the moduli \(\mathcal {M}^{J^\tau }(p_m;\tau _P(q_j'),c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'})\) for various jslm, which is responsible for (part of) the operation
$$\begin{aligned} hom(\tau _P(L_1'),\mathcal {P}) \times hom(L_0,\tau _P(L_1')) \rightarrow hom(L_0,\mathcal {P}) \end{aligned}$$
(5.32)
Notice that, by switching the appropriate strip-like ends from incoming to outgoing (and vice versa) for the same holomorphic triangles, (5.32) can be dualized to
$$\begin{aligned} hom(\mathcal {P},L_0) \times hom(\tau _P(L_1'),\mathcal {P}) \rightarrow hom(\tau _P(L_1'),L_0) \end{aligned}$$
(5.33)
If we replace \(L_0\) by \(L_1\) (both of them are union of cotangent fibers in U), then we see that (5.33) has already been studied in Lemmas 5.11 and 5.12. The outcome is the following.

Lemma 5.20

For \(\tau \gg 1\), for \(\psi _{c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'}} \in C_0\) supported at \(c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'}\)
$$\begin{aligned} \mu ^2_{\mathcal {F}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j')\otimes \psi ^1_{i,g,j,k},\psi _{c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'}}) \end{aligned}$$
(5.34)
is 0 if \(l \ne j\). When \(l=j\), (5.34) becomes
$$\begin{aligned} gh{\mathbf {p}}_s \otimes (I_{[\tau _P(q_j') \rightarrow p_s]} \circ \psi ^1_{i,g,j,k} \circ I_{[c_{h{\mathbf {p}}_s,{\mathbf {q}}_j'} \rightarrow \tau _P(q_j')]} \circ \psi _{c_{h{\mathbf {p}}_s,{\mathbf {q}}_j'}} \circ I_{[p_s \rightarrow c_{h{\mathbf {p}}_s,{\mathbf {q}}_j'}]}) \end{aligned}$$
(5.35)
where all the parallel transport maps are the unique one determined by the boundary condition inside \(U\ (\)cf. Proposition 5.14).

Proof

The argument largely resembles the proof of Lemmas 5.11, 5.12 and Proposition 5.14. A neck-stretching argument as in Lemma 5.11 deduces that \(\mathcal {M}^{J^\tau }(p_m;\tau _P(q_j'),c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'})\) is not empty only if \(j=l\) and \(m=s\). The same dimension count implies that when \(j=l\), \(m=s\) and \(\tau \gg 1\), every rigid element of \(\mathcal {M}^{J^\tau }(p_m;\tau _P(q_j'),c_{h{\mathbf {p}}_s,{\mathbf {q}}_l'})\) has image inside U. The local count and the chasing of local systems from Lemma 5.12 and Proposition 5.14 applies directly to the current case because it is a computation in \({\mathbf {U}}\) about cotangent fibers and their Dehn twists. In particular, if we remove the local systems on \(L_0\) and \(\tau _P(L_1')\), we get
$$\begin{aligned} \mu ^2_{{\mathcal {F}}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') ,c_{h{\mathbf {p}}_s,{\mathbf {q}}_j'})=\mu ^2_{{\mathcal {F}}}(g\tau _{\mathbf {P}}({\mathbf {q}}_j') ,c_{gh{\mathbf {p}}_s,g{\mathbf {q}}_j'})=gh{\mathbf {p}}_s \in hom(L_0, \mathcal {P}) \end{aligned}$$
(5.36)
The parallel transport maps are uniquely determined by boundary conditions, and after chasing all of them, we get the result. \(\square \)
Let \(V=hom(\mathcal {P},(\mathcal {E}^1)') \otimes _{\Gamma } hom(\mathcal {E}^0,\mathcal {P})\) which is generated by elements of the form
$$\begin{aligned} (\Upsilon ^2 \otimes ({\mathbf {q}}_r')^\vee ) \otimes (h{\mathbf {p}}_t \otimes \Upsilon ^1) \end{aligned}$$
(5.37)
for \(h \in \Gamma \), \(r=1,\dots , d_{L_1}\), \(t=1,\dots , d_{L_0}\), \(\Upsilon ^2 \in (\mathcal {E}^1)'_{q_r'}\) and \(\Upsilon ^1 \in Hom_{\mathbb {K}}(\mathcal {E}^0_{p_t},{\mathbb {K}})\) [cf. (5.6)].
  • 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 \(h=g\).

Therefore, we have direct sum decompositions
$$\begin{aligned} V=\oplus _s V_s, \text { } V_s =\oplus _l V_{s,l}, \text { } V_{s,l}= \oplus _{g} V_{s,l,g} \end{aligned}$$
(5.38)
The bijective correspondence \(\iota \) (5.31) extends to an isomorphism, also denoted by \(\iota \), from V to \(C_0^q\) by keeping track of the (uniquely determined) parallel transport maps along Lagrangians inside U. On the other hand, there is an obvious isomorphism \(F:hom(\mathcal {P},\mathcal {E}^1) \otimes _{\Gamma } hom(\mathcal {E}^0,\mathcal {P}) \rightarrow V\) given by
$$\begin{aligned} (\Upsilon ^2 \otimes {\mathbf {q}}_l^\vee ) \otimes (h{\mathbf {p}}_s \otimes \Upsilon ^1) \mapsto (\Upsilon ^2 \otimes ({\mathbf {q}}_l')^\vee ) \otimes (h{\mathbf {p}}_s \otimes \Upsilon ^1) \end{aligned}$$
(5.39)
where we used the identification \(\mathcal {E}^1_{q_l} \simeq (\mathcal {E}^1)'_{q_l}\) by the Hamiltonian push-off. As a result, we have a composition map
$$\begin{aligned} \Theta : V\xrightarrow {\iota } (L_0 \cap \tau _P(L_1')) \cap U\xrightarrow {\mu ^2_q(c_\mathcal {D},-)} C_1^q\xrightarrow {F} V \end{aligned}$$
(5.40)
which respects a filtration on V in the following sense.

Lemma 5.21

We have
$$\begin{aligned} \left\{ \begin{array}{ll} \Theta (V_s) \subset V_s &{}\quad \text { for all }s\\ \Theta (V_{s,l}) \subset \bigoplus _{t \geqslant l} V_{s,t} &{}\quad \text { for all }s,l\\ \Theta (V_{s,l,h}) \subset V_{s,l,h}+(\bigoplus _{t > l} V_{s,t}) &{}\quad \text { for all }s,l,h \end{array} \right. \end{aligned}$$
(5.41)

Proof

Explicitly, \(\Theta \) is given by [see (5.30) and Lemma 5.20]
$$\begin{aligned}&(\Upsilon ^2 \otimes ({\mathbf {q}}_l')^\vee ) \otimes (h{\mathbf {p}}_s \otimes \Upsilon ^1) \end{aligned}$$
(5.42)
$$\begin{aligned}&\qquad \mapsto \sum _{i,g,k} (\psi ^2_{i,g,l,k} \otimes ({\mathbf {q}}_i')^\vee ) \otimes (gh{\mathbf {p}}_s \otimes R(\psi ^1_{i,g,l,k},\Upsilon ^2,\Upsilon ^1)) \end{aligned}$$
(5.43)
where \(R(\psi ^1_{i,g,l,k},\Upsilon ^2,\Upsilon ^1)\) is a term depending on \(\psi ^1_{i,g,l,k},\Upsilon ^2,\Upsilon ^1\) given by composing parallel transport maps. It is therefore clear that \(\Theta (V_s) \subset V_s\). By Propositions 5.16 and (5.27), we know that \(\psi ^2_{i,g,l,k}=0\) unless \(j \leqslant i\) so \(\Theta (V_{s,l}) \subset \oplus _{t \geqslant l} V_{s,t}\).

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 quasi-isomorphism.

Proof

Since \(\mu ^2_q\) is a chain map, it suffices to show that \(\mu ^2_q\) is bijective. We know that \(\iota \) and F are isomorphisms so it suffices to show that \(\Theta \) is surjective [see (5.40)]. By (5.38) and Lemma 5.21, it suffices to show that
$$\begin{aligned} \Theta |_{V_{s,l,h}}:V_{s,l,h} \rightarrow (V_{s,l,h}+(\oplus _{t> l} V_{s,t}))/(\oplus _{t > l} V_{s,t}) \end{aligned}$$
(5.44)
is bijective for all slh. For fixed slh, the map (5.44) can be identified with the map
$$\begin{aligned} (\mathcal {E}^1)'_{q_l'} \otimes Hom_{\mathbb {K}}(\mathcal {E}^0_{p_s},{\mathbb {K}}) \rightarrow (\mathcal {E}^1)'_{q_l'} \otimes Hom_{\mathbb {K}}(\mathcal {E}^0_{p_s},{\mathbb {K}}) \nonumber \\ \Upsilon ^2 \otimes \Upsilon ^1 \mapsto \sum _k (\psi ^2_{l,1_{\Gamma },l,k} \otimes R(\psi ^1_{l,1_{\Gamma },l,k},\Upsilon ^2,\Upsilon ^1)) \end{aligned}$$
(5.45)
By (5.24) and keeping track of the uniquely determined parallel transport maps, it is clear that (5.45) is an isomorphism. \(\square \)

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 quasi-isomorphic \(\mathcal {E}^0\), 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 quasi-isomorphism. \(\square \)

Proof of Corollary 1.3

When P is diffeomorphic to \(\mathbb {RP}^n\) and \(n=4k-1\), P is spin and can be equipped with the spin structure descended from \(S^n\). When \({{\,\mathrm{char}\,}}({\mathbb {K}}) \ne 2\), the universal local system \(\mathcal {P}\) is a direct sum of two rank 1 local systems \(\mathcal {E}^1\) and \(\mathcal {E}^2\). This is because \({\mathbb {K}}[{\mathbb {Z}}_2]\) splits when \({{\,\mathrm{char}\,}}({\mathbb {K}}) \ne 2\). Moreover, by Lemma 2.10 and Corollary 2.11, we have
$$\begin{aligned} HF^*(\mathcal {E}^i,\mathcal {E}^j)= \left\{ \begin{array}{ll} 0 &{} \text { if }i\ne j\\ H^*(S^n)\text { if }i= j \end{array} \right. \end{aligned}$$
(5.46)
so \(\mathcal {E}^1\) and \(\mathcal {E}^2\) are orthogonal spherical objects. In this case,
$$\begin{aligned} T_\mathcal {P}(\mathcal {E})\simeq&Cone(\oplus _{i=1,2} (hom_{\mathcal {F}}(\mathcal {E}^i,\mathcal {E}) \otimes \mathcal {E}^i) \xrightarrow {ev} \mathcal {E}) \end{aligned}$$
(5.47)
where ev is the evaluation map. The spherical twist to \(\mathcal {E}\) along \(\mathcal {E}^i\) is defined to be \(Cone(hom_{\mathcal {F}}(\mathcal {E}^i,\mathcal {E}) \otimes \mathcal {E}^i \xrightarrow {ev} \mathcal {E})\). A direct verification shows that (5.47) is the same as applying the spherical twist to \(\mathcal {E}\) along \(\mathcal {E}^1\) and then \(\mathcal {E}^2\). It is the same as first applying spherical twist along \(\mathcal {E}^2\) and then \(\mathcal {E}^1\) because \(\mathcal {E}^1\) and \(\mathcal {E}^2\) are orthogonal objects.
When P is diffeomorphic to \(\mathbb {RP}^n\) and \({{\,\mathrm{char}\,}}({\mathbb {K}}) = 2\), then \(H^*(P)=H^*(\mathbb {RP}^n,{\mathbb {Z}}_2)\). In this case, one can define a \({\mathbb {P}}\)-twist along P (see [5, 47]) which is an auto-equivalence on \({\mathcal {F}}^{{{\,\mathrm{perf}\,}}}\). More precisely, the algebra \(H^*(\mathbb {RP}^n,{\mathbb {Z}}_2)\) is generated by a degree 1 element instead of a degree 2 element so the \({\mathbb {P}}\)-twist along P is not exactly, but a simple variant of, the \({\mathbb {P}}\)-twist defined in [5]. To compare (1.1) with the \({\mathbb {P}}\)-twist, we note that \({\mathbb {K}}[{\mathbb {Z}}_2]\) fits into a non-split exact sequence
$$\begin{aligned} 0 \rightarrow {\mathbb {K}}\rightarrow {\mathbb {K}}[{\mathbb {Z}}_2] \rightarrow {\mathbb {K}}\rightarrow 0 \end{aligned}$$
(5.48)
it implies that \(\mathcal {P}=Cone(P[-1] \rightarrow P)\) and the morphism in the cone is the unique non-trivial one. In this case, the fact that \(T_\mathcal {P}(\mathcal {E})\) is the \({\mathbb {P}}\)-twist of \(\mathcal {E}\) along P is explained in [48, Remark 4.4]. \(\square \)

Footnotes

  1. 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, Yu-Wei 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. DMS-1128155 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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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.DPMMS, University of CambridgeCambridgeUK
  2. 2.Department of MathematicsUniversity of GeorgiaAthensGeorgia

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