1 Introduction↩︎

1.1 Background↩︎

A symplectic manifold is a pair \((X,\omega)\), where \(X\) is a smooth \(2n\)-dimensional manifold and \(\omega\) is a closed, non-degenerate differential \(2\)-form. Non-degeneracy of \(\omega\) means that the top-degree form \(\omega^{n}\) is nowhere vanishing on \(X\). In this article, we will focus on a particular class of symplectic manifolds, called Liouville domains. By a Liouville domain we mean a compact oriented smooth manifold \(X\) with nonempty boundary \(\partial X\) equipped with a \(1\)-form \(\lambda\) such that \(d\lambda\) is a symplectic form on \(X\). The restriction \(\lambda|_{\partial X}\) is a contact form on \(\partial X\), and the orientation induced by \(\lambda\) on \(\partial X\) agrees with the boundary orientation.

Elementary examples of Liouville domains arise from star-shaped domains \((X,\lambda_{\mathrm{std}})\) with smooth boundary in the standard symplectic vector space \((\mathbb{R}^{2n},\omega_{\mathrm{std}}:=d\lambda_{\mathrm{std}})\), where \[\label{eq:std} \lambda_{\mathrm{std}}:=\frac{1}{2}\sum_{i=1}^{n}(x_idy_i-y_idx_i).\tag{1}\] Ellipsoids provide a particularly important class of smooth star-shaped domains. Given real numbers \(0 < a_1 \le a_2 \le \cdots \le a_n < \infty\), the associated \(2n\)-dimensional ellipsoid is defined by \[E^{2n}(a_1,\dots,a_n):= \left\{ (z_1,\dots,z_n) \in \mathbb{C}^n \left\vert\sum_{i=1}^{n} \frac{\pi |z_i|^2}{a_i} \le 1 \right.\right\}.\] In particular, \[\label{eq:ball} B^{2n}(a) := E^{2n}(a,\dots,a)\tag{2}\] is the standard symplectic ball of capacity \(a>0\), or equivalently, with radius \(\sqrt{a/\pi}\).

In this paper, we establish a relation between the Gutt–Hutchings capacities [2] and the Cieliebak–Mohnke capacity [3] for Liouville domains—the latter is also known as the Lagrangian capacity. To define the latter, let \(L \subset (X,\omega)\) be a closed Lagrangian submanifold. We define the symplectic area of \(L\) by \[A_{\min}(L):=\inf_{\substack{A \in \pi_2(X,L) \\ \int_A \omega > 0}} \int_A \omega \in [0,\infty].\]

We recall some conjectures concerning the Cieliebak–Mohnke capacity, which we will study in this paper and subsequent works.

This conjecture is of particular interest to us, as it suggests that extremal Lagrangian tori may provide a natural replacement for monotone Lagrangian tori, which only exist in monotone symplectic manifolds, but we will not prove it in this paper.

The next conjecture predicts that Lagrangian embedding problems are governed by the same numerical obstruction as symplectic embedding problems.

We summarize below the progresses so far on the conjectures stated above.

• Cieliebak–Mohnke (2014) ([3], Corollary 1.3) proved Conjecture 1 for all balls and cylinders, i.e., for \[E^{2n}(a,a ,\dots, a)\text{ and } E^{2n}(a,\infty,\dots, \infty).\] Moreover, they established that every monotone Lagrangian torus in \((\mathbb{CP}^n,\omega_{\mathrm{FS}})\) is extremal (cf. [3], Corollary 1.7), which proves one direction of Conjecture 3.

• Dimitroglou-Rizell (2015) [5] gave a proof of Conjecture 2 for the \(4\)-dimensional ball \(B^{4}(1)\). His proof is essentially based on holomorphic curve techniques in dimension \(4\), e.g. positivity of the intersection and automatic transversality.

• Pereira in his PhD thesis (2022, cf. [6], Theorem 3.28) confirmed Conjecture 1 for all \(4\)-dimensional ellipsoids. In higher dimensions, he shows that the conjecture (cf. [6], Theorem 4.37) holds under the assumption that a suitable virtual perturbation scheme exists to define curve counts for linearized contact homology and moduli spaces with local tangency constraints, which are used to construct augmentations on the linearized contact homology algebra of ellipsoids (cf. Siegel [7]). Under the same assumption for the perturbation scheme, a proof of Conjecture 5 would also follow from [6].

• The first author in his PhD thesis (2025) proves Conjecture 2 in all dimensions (cf. [1], Theorem 1.7). Using similar ideas, he also confirms Conjecture 4 for a class of toric domains including all ellipsoids \(E^4(a, b)\) and cylinders \(B^{2k}(1)\times \mathbb{C}^m\) for arbitrary \(k, m\in \mathbb{N}\) (cf. [1], Theorem 1.11 and Theorem 1.8). Moreover, he proves Conjecture 3 for \(\mathbb{CP}^2\) (cf. [1], Theorem 1.18).

In this paper, we introduce a new symplectic capacity using all aspherical Lagrangian submanifolds, extending the Cieliebak–Mohnke capacity. Using the techniques from Fukaya–Irie [8]–[10] and the second author [11], which are based on chain level string topology, we establish an upper bound on the symplectic area of closed oriented aspherical Lagrangian submanifolds in a Liouville domain in terms of its Gutt–Hutchings capacities. We prove that this bound is sharp for the new capacity on convex and concave toric domains in \(\mathbb{R}^{2n}\), thereby completely settle the Conjectures 1 and 5. Furthermore, under an additional assumption on the asymptotic behavior of the Gutt–Hutchings capacities, we obtain a boundary rigidity result for extremal aspherical Lagrangians in Liouville domains, which is strong enough to confirm Conjecture 4.

A notable feature of our approach is that it works without any putative perturbative schemes, which is one of the main novelties of this paper.

1.2 Summary of results↩︎

Before stating the main results of this paper, we fix some notations and introduce a variant of the Cieliebak–Mohnke capacity using aspherical Lagrangian submanifolds.

Let \(X\) be a Liouville domain with \(c_1(X)=0\). Denote by \(\mathit{SH}^\ast(X)\) the symplectic cohomology of \(X\), and by \(\mathit{SH}^\ast_{S^1}(X)\) its \(S^1\)-equivariant version, which are both \(\mathbb{Z}\)-graded vector spaces over some field \(\mathbb{K}\). For the purpose of this paper, we shall take \(\mathbb{K}=\mathbb{R}\) to be the field of real numbers.

Recall that the \(S^1\)-equivariant symplectic cohomology \(\mathit{SH}^\ast_{S^1}(X)\) is the cohomology of the complex \[\label{eq:SC} \left(\mathit{SC}^\ast_{S^1}(X):=\mathit{SC}^\ast(X)\otimes_\mathbb{R}\mathbb{R}(\!(u)\!)/u\mathbb{R}[\![u]\!],\partial^{S^1}:=\partial+u\delta_1+u^2\delta_2+\cdots\right),\tag{3}\] where \(\mathit{SC}^\ast(X)\) is the cochain complex defining the (non-equivariant) symplectic cohomology \(\mathit{SH}^\ast(X)\), \(\partial\) is the usual Floer differential, \(\delta_1\) is the cochain level BV operator, and \(u\) is a formal variable of degree 2. The action filtration on \(\mathit{SC}^\ast(X)\) induces a filtration \(F^\bullet\) on \(\mathit{SC}^\ast_{S^1}(X)\), and the \(d\)-th Gutt–Hutchings capacity of \(X\), introduced in \(\cite{ghs}\), is defined to be \[C_d^\mathrm{GH}(X):=\inf\left\{a\left\vert\partial^{S^1}(\tilde{y})=e_X\otimes u^{-d+1}\textrm{ for some }\tilde{y}\in F^{\leq a}\mathit{SC}_{S^1}^{1-2d}(X)\right.\right\}, \nonumber\] where \(e_X\in\mathit{SC}^0(X)\) is the cochain level representative of the identity \(1\in\mathit{SH}^0(X)\).

When the choice of the symplectic form is clear, we shall simply write \(C^\mathrm{CM}(X)\) and \(C^\mathrm{AL}(X)\) for the Lagrangian capacities.

We note that \(C^{\mathrm{AL}}(X)\) defines a symplectic capacity on star-shaped domains \(X\subset\mathbb{C}^n\). Moreover, by definition \[C^{\mathrm{CM}}(X) \leq C^{\mathrm{A L}}(X).\] Our first result relates the Lagrangian capacity \(C^{\mathrm{AL}}(X)\) to Gutt–Hutchings capacities.

Our method of proving Theorem 6 also leads to a new upper bound for Lagrangian width for Lagrangian submanifolds in Liouville domains. Let \(L\) be a closed Lagrangian submanifold in a symplectic manifold \((X,\omega)\). We say that a symplectic embedding \[i:\left(B^{2n}(r),\omega_{\mathrm{std}}\right)\hookrightarrow (X,\omega)\] is relative to \(L\) if \[i^{-1}(L)=B^{2n}(r)\cap \mathbb{R}^n.\] We define the Lagrangian width of \(L\subset X\) by \[w(L;X):=\sup\left\{r\left\vert\text{there exists a symplectic embedding of }B^{2n}(r)\hookrightarrow X \text{ relative to }L\right.\right\}.\] Let \((X,\lambda)\) be a Liouville domain. For a Lagrangian submanifold \(L\subset\mathrm{int}(X)\) in the interior of \(X\), we introduce the following quantities: \[C^{\operatorname{GH}}_d(L;X):=\inf_{L\subset W\subset X}C^{\operatorname{GH}}_d(W),\textrm{ } C^{\operatorname{GH}}_{\inf}(L;X):=\inf_{d\in\mathbb{N}}\frac{C^{\operatorname{GH}}_d(L;X)}{d},\] where the infimum is taken over all Liouville subdomains \(W\subset X\) containing \(L\). Note that the quantities \(C^{\operatorname{GH}}_d(L;X)\) and \(C^{\operatorname{GH}}_{\inf}(L;X)\) are invariant under Hamiltonian isotopies of \(L\). We say that a closed Lagrangian submanifold \(L\subset\mathrm{int}(X)\) admits an exact Lagrangian cap if there exists a Liouville subdomain \(W\subset X\) such that \(L\setminus \operatorname{int}(W)\) is a non-empty exact Lagrangian and \(\lambda|_{L\setminus \operatorname{int}(W)}\) admits a primitive which vanishes along \(L\cap \partial W\); cf. [13], [14]. Our argument in Section 2 gives the following new estimate on the Lagrangian widths of \(L\), which has the advantage that in general no assumption on the displaceability of \(L\) is needed. When \(L\subset\mathrm{int}(X)\) is displaceable, it recovers the upper bound given by Borman-McLean (cf. [15], Theorem 1.1).

Using Murphy’s \(h\)-principle for loose Legendrians [16] and its extension to Lagrangian caps [14], Ekholm-Eliashberg-Murphy-Smith constructed in [17] Lagrangian embeddings of \(S^1\times S^{n-1}\) into \(\mathbb{C}^n\) for all \(n\geq 3\), which have infinite Lagrangian widths by [13]. Obviously, these Lagrangians are contained in smooth star-shaped domains whose Gutt–Hutchings capacities are finite. This shows that Corollary 2 fails in general for non-aspherical Lagrangians.

Another application of Theorem 6 is a proof of Conjecture 5.

[4], Problem 11 asks about the stability of symplectic capacities. For the Cieliebak-Mohnke capacity we can prove the following.

Our proof of Theorem 6 is based on a general result about the non-exact \(S^1\)-equivariant Viterbo functoriality for closed Lagrangian embedding \(L\hookrightarrow(X,\lambda)\) in a Liouville domain \((X,\lambda)\) with finite \(d\)-th Gutt–Hutchings capacity \(C_d^\mathrm{GH}(X)\) for some \(d\in\mathbb{N}\). Note that if we assume in addition that \(L\) is aspherical, then any such Lagrangian embedding is necessarily non-exact since the disc cotangent bundle \(D^\ast L\) has infinite \(d\)-th Gutt–Hutchings capacity for any \(d\in\mathbb{N}\). To state the result, we need to introduce some notations.

For \(a\in H_1(L;\mathbb{Z})\), denote by \(\mathcal{L}(a)L\subset\mathcal{L}L\) the subspace of the free loop space of \(L\) which consists of loops in the class \(a\). Consider the \(S^1\)-equivariant homology group \[\label{eq:StrH} \mathbb{H}^{S^1}_\ast(a):=H_{\ast+n+\mu(a)-1}^{S^1}(\mathcal{L}(a)L;\mathbb{R}),\tag{4}\] where \(\mu:H_1(L;\mathbb{Z})\rightarrow\mathbb{Z}\) is the Maslov index and the \(S^1\)-action is given by reparametrization of loops. The direct sum \[\label{eq:dirsum} \mathbb{H}_\ast^{S^1}:=\bigoplus_{a\in H_1(L;\mathbb{Z})}\mathbb{H}_\ast^{S^1}(a)\tag{5}\] carries the action filtration \[F^\Xi\mathbb{H}_\ast^{S^1}:=\bigoplus_{\lambda(a)>\Xi}\mathbb{H}^{S^1}_\ast(a), \nonumber\] which allows us to define the completion \[\label{eq:H-c} \widehat{\mathbb{H}}_\ast^{S^1}:=\varprojlim_{\Xi\rightarrow\infty}\mathbb{H}_\ast^{S^1}/F^\Xi\mathbb{H}_\ast^{S^1}.\tag{6}\] Note that \(\widehat{\mathbb{H}}_\ast^{S^1}\) is a module over \(\mathbb{R}(\!(h)\!)/h\mathbb{R}[\![h]\!]\), where \(h\) is a formal variable of degree \(-2\) coming from the chain level \(S^1\)-action induced by reparametrization of loops.

The following theorem generalizes [11], Theorem 6, which deals with the case when \(d=1\).

The relation of the above theorem with the \(S^1\)-equivariant Viterbo functoriality is as follows. Given an exact Lagrangian embedding \(L\hookrightarrow\mathrm{int}(X)\), the \(S^1\)-equivariant Viterbo functoriality gives a map \[\mathit{SH}^\ast_{S^1}(X)\rightarrow\mathit{SH}^\ast_{S^1}(D^\ast L)\xrightarrow{\cong}H_{n-\ast}^{S^1}(\mathcal{L}L;\mathbb{R})\] relating the \(S^1\)-equivariant symplectic cohomology of \(X\) to the string homology of \(L\). When the Lagrangian submanifold \(L\) is non-exact, this map needs to be corrected by deforming \(H_{n-\ast}^{S^1}(\mathcal{L}L;\mathbb{R})\) using the Maurer-Cartan element obtained by counting pseudoholomorphic discs in \(X\) with boundary on \(L\). In the above theorem, the class \(x\in\widehat{\mathbb{H}}_{-2}^{S^1}\) plays the role of the Maurer-Cartan element. Moreover, the assumption \(C_d^\mathrm{GH}(X)<\infty\) implies the existence of an \(S^1\)-equivariant cochain \(\tilde{y}\in\mathit{SC}_{S^1}^{1-2d}(X)\) satisfying the equation \(\partial^{S^1}(\tilde{y})=e_X\otimes u^{-d+1}\). Under the \(S^1\)-equivariant Viterbo functoriality, there should be a corresponding equation in the string homology \(H_{n-\ast}^{S^1}(\mathcal{L}L;\mathbb{R})\), where the role of the primitive \(\tilde{y}\) is played by \(y\in\widehat{\mathbb{H}}_{2d}^{S^1}\) in Theorem 9. However, when \(L\subset\mathrm{int}(X)\) is non-exact, such an equation also needs to be deformed by the Maurer-Cartan element \(x\), and (?? ) is the deformed equation, which holds in the completed string homology \(\widehat{H}_{n-\ast}^{S^1}(\mathcal{L}L;\mathbb{R})\).

It is clear that the assumptions of Theorem 9 are satisfied for star-shaped domains in \(\mathbb{C}^n\). There are actually many examples of non-subcritical Weinstein domains where Theorem 9 is applicable, see our discussions in Section 8.1.

The proof of Theorem 9 relies on a class of moduli spaces of solutions to certain parametrized Floer equations considered by Cohen–Ganatra [18]. The study of these Floer solutions also enables us to get geometric constraints on the extremal Lagrangian submanifolds in a class of Liouville domains defined below.

The term “spectrally convex" is inspired by the fact that all convex toric domains \(X\subset\mathbb{C}^n\) satisfy the required decay condition of Gutt–Hutchings capacities, while there are examples of non-convex star-shaped domains for which the condition fails. With the above definition we have the following.

The spectral convexity assumption of Definition 5 is satisfied for many Liouville domains, there are both subcritical and non-subcritical examples (cf. Section 8.2). In particular, we have the following, which confirms Conjecture 4 in view of Remark 8.

As a byproduct of the results obtained in this paper, especially Theorem 6 and Corollary 1, we give the first computations of the Lagrangian capacities \(C^\mathrm{CM}(X)\) and \(C^\mathrm{AL}(X)\) for many non-subcritical Weinstein domains \(X\) in Section 8.2.

This paper studies a version of Lagrangian capacity which is more general than the original definition of Cieliebak–Mohnke [3] in the sense that we use closed aspherical Lagrangians in the symplectic manifold \(X\) instead of just the Lagrangian tori. One can include more general Lagrangians in \(X\) and consider the corresponding capacity. For example, the recent work of S. Li [19] suggests that similar techniques as explored here can be used to study capacities defined using all \(L\subset X\) with \(\pi_2(L)=0\), which, however, would differ drastically from \(C^\mathrm{AL}(X)\) since there exist closed Lagrangians in \(\mathbb{C}^n\) with \(\pi_2(L)=0\) and Maslov number \(>2\). See also [3], Remark 1.6, where the authors suggest defining the Lagrangian capacity using all closed Lagrangians. In the case of a unit ball \(B^{2n}(1)\) with \(n\geq2\), the capacity defined using all closed Lagrangians is conjectured to be \(\frac{1}{2}\). On the other hand, \(C^\mathrm{CM}\left(B^{2n}(1)\right)=C^\mathrm{AL}\left(B^{2n}(1)\right)=\frac{1}{n}\).

The paper is organized as follows. In Section 2, we prove Theorems 6 by assuming the validity of Theorem 9. In Section 3, after further assuming the correctness of Theorem 10, we prove Corollaries 1, 2, 3, 4 and 5. The proof of Theorem 9 is given in Section 6, after introducing the relevant moduli spaces in Section 4 and the required \(S^1\)-equivariant de Rham chain model of the free loop space in Section 5. The proof of Theorem 10 is postponed to Section 7, since some familiarity with the moduli spaces considered in Section 4 would be helpful for understanding the argument. Finally, we discuss the examples of (non-subcritical) Liouville domains where Corollary 2 and Theorem 9 are applicable and explicitly compute the Lagrangian capacites of some of these Liouville domains in Section 8.

Acknowledgements↩︎

The possibility of applying the techniques of [11] to Liouville domains with finite higher Gutt–Hutchings capacities was first asked by Mark McLean during a talk given by the second author at Stony Brook University in 2023. The first author is grateful to Georgios Dimitroglou-Rizell and Luis Diogo for many valuable conversations, and to Kei Irie for helpful discussions during the 2024 Paris conference From Hamiltonian Dynamics to Symplectic Topology and Beyond, held in honor of Claude Viterbo.

2 Bounding the Lagrangian capacity↩︎

Assuming Theorem 9, we prove Theorem 6 in this section. We remark that due to the complexities in the constructions of the string homology classes \(x,y\in\widehat{\mathbb{H}}_\ast^{S^1}\), there are some unavoidable technical ambiguities in the proof, which will become clear once Theorem 9 is proved in Section 6. To prove Theorem 6, we will use the following fact about the free loop spaces of aspherical manifolds.

3 Proofs of Corollaries↩︎

In this section, we prove the corollaries of Theorems 6 and 10. The proof of Theorem 10 will be given in Section 7, here we assume its correctness. We start by recalling some basic notions.

Given a domain \(\Omega \subset \mathbb{R}^n_{\geq 0}\), the associated toric domain \(X_{\Omega} \subset \mathbb{C}^n\) is defined by \[X_{\Omega} := \mu^{-1}(\Omega),\] where \(\mu: \mathbb{C}^n \to\mathbb{R}^n_{\geq 0}\) is the moment map \(\mu(z_1,\dots,z_n)=\pi\big(|z_1|^2,\dots,|z_n|^2\big)\) associated to the standard Hamiltonian torus action on \(\mathbb{C}^n\). \(X_\Omega\) is equipped with the restriction of the standard symplectic form \(\omega_\mathrm{std}\) on \(\mathbb{C}^n\). Define the diagonal of \(X_{\Omega}\) by \[\delta_{\Omega}:=\sup\{a: (a,\dots,a) \in \Omega\}.\] We say that \(X_\Omega\) is a convex toric domain if the set \(\Omega\subset\mathbb{R}^n_{\geq 0}\) is convex, and it is a concave toric domain if \(\mathbb{R}^n_{\geq 0}\setminus\Omega\) is convex.

For \(\delta>0\), we denote by \(N^{2n}(\delta) \subset \mathbb{C}^n\) the non-disjoint union of cylinders of diagonal \(\delta\), which is given by \[N^{2n}(\delta) := \mu^{-1}(\Omega'),\] where \[\Omega':=\left.\left\{(x_1,\dots,x_n)\in \mathbb{R}^n_{\geq 0}\right\vert x_i \leq \delta \text{ for some } i \in \{1,\dots,n\}\right\}.\]

We start with the proof of Corollary 1, which computes the Lagrangian capacities of convex and concave toric domains and in particular confirms [3], Conjecture 1.5.

Next, we prove Corollary 5, which confirms [1], Conjecture 9.11. According to Theorem 10, it is enough to verify that ellipsoids are spectrally convex as Liouville domains in the sense of Definition 5.

We then prove Corollary 2, which gives an upper bound of Lagrangian width for aspherical Lagrangian submanifolds in terms of Gutt–Hutchings capacities.

Next, we prove Corollary 3, which relates the Cieliebak–Mohnke capacity to the embedding capacity of a \(4\)-dimensional ellipsoid, which confirms [4], Conjecture 2.

Finally, we prove Corollary 4, which establishes the stability of the Cieliebak-Mohnke capacity.

4 Cohen–Ganatra moduli spaces↩︎

We recall the Cohen–Ganatra moduli spaces from [11], Section 4.2, whose definition is inspired by closely related moduli spaces studied earlier by Cohen–Ganatra [18]. Here we will only provide the essential definitions and properties, and refer to [11] for details. See also [25], [26].

Let \({}_l\mathcal{R}_{k+1}^1\) be the moduli space of domains \[\label{eq:domain} (S;z_0,\cdots,z_k,p_1,\cdots,p_l;\ell)\tag{14}\] modulo the group of automorphisms, where \(S=D\setminus\{\zeta\}\) is a closed unit disc with an interior puncture \(\zeta\), which will serve as an input. At \(\zeta\) there is an asymptotic marker, namely a half-line \(\ell\in T_\zeta D\). \(z_0,\cdots,z_k\in\partial D\) are distinct marked points on the boundary, labeled in counterclockwise order. Moreover, there is a set of auxiliary marked points \(p_1,\cdots,p_l\in S\) lying in the interior of \(D\). After choosing the representative of an element of \({}_l\mathcal{R}_{k+1}^1\) with the puncture \(\zeta\) fixed at the origin, and the boundary marked point \(z_0\) fixed at \(1\), the points \(p_0,\cdots,p_l\) are required to be strictly radially ordered with norms in \((0,\frac{1}{2})\), i.e. \[\label{eq:radial} 0<|p_l|<\cdots<|p_1|<\frac{1}{2}.\tag{15}\] We require that the asymptotic marker \(\ell\) at the origin points toward the point \(p_l\), see Figure 1.

To describe the compactifcation \({}_l\overline{\mathcal{R}}_{k+1}^1\), which is a manifold with corners, it would be convenient to introduce the following auxiliary moduli spaces.

• \({}_l^{j,j+1}\mathcal{R}_{k+1}^1\) is the moduli space of the domains (14 ), except that the condition (15 ) is replaced with \[|p_l|<\cdots<|p_{j+1}|=|p_j|<\cdots<|p_1|<\frac{1}{2}, \nonumber\] for some \(1\leq j<l\).

• \({}_{l-1}\mathcal{R}_{k+1}^{S^1}\) is the moduli space of the same domains, but with (15 ) replaced with \[|p_l|<\cdots<|p_1|=\frac{1}{2}. \nonumber\] By forgetting the marked point \(p_1\), there is an abstract identification \({}_{l-1}\mathcal{R}_{k+1}^{S^1}\cong{}_{l-1}\mathcal{R}_{k+1}^1\times S^1\), under which the compactification \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) is abstractly modeled by \({}_{l-1}\overline{\mathcal{R}}_{k+1}^1\times S^1\). In particular, the codimension \(1\) boundary stratum \({}_{l-1}^{j,j+1}\overline{\mathcal{R}}_{k+1}^1\subset{}_{l-1}\overline{\mathcal{R}}_{k+1}^1\) for some \(2\leq j\leq l-1\) corresponds to a codimension \(1\) boundary stratum \({}_{l-1}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{S^1}\subset\partial{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\), where the \(S^1\) factor describes the situation that \(|p_1|=|p_2|=\frac{1}{2}\). Denote by \[\label{eq:forget} \pi_j^{S^1}:{}_{l-1}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{S^1}\rightarrow{}_{l-2}\overline{\mathcal{R}}_{k+1}^{S^1}\tag{16}\] the map which forgets \(p_j\).

  Note also that there is a free \(\mathbb{Z}_{k+1}\)-action on the moduli space \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) generated by the map \[\label{eq:permu} \kappa:{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\rightarrow{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1},\tag{17}\] which cyclically permutes the labels of the boundary marked points \(z_0,\cdots,z_k\). It can be shown that this \(\mathbb{Z}_{k+1}\)-action is properly discontinuous.

  For an element \(S\) of \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\), we say that \(p_1\) points at a boundary point \(z_i\), for some \(0\leq i\leq k\), if for a representative of \(S\) with \(\zeta\) fixed at the origin, the ray from \(\zeta\) to \(p_1\) points at \(z_i\). Denote by \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S_i^1}\subset{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) the codimension \(1\) locus where \(p_1\) points at \(z_i\). There is a bijection \[\tau_i:{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S_i^1}\rightarrow{}_{l-1}\overline{\mathcal{R}}_{k+1}^1 \nonumber\] defined as follows. When \(l\geq2\), \(\tau_i\) forgets the point \(p_1\) on the circle \(|z|=\frac{1}{2}\), and relabels the remaining auxiliary marked points \(p_2,\cdots,p_l\) as \(p_1,\cdots,p_{l-1}\). When \(l=1\), \(\tau_i\) is defined by cyclically permuting the boundary marked points until the original \(z_i\) is now labeled \(z_k\), and then forgetting \(p_1\). Similarly, we say that \(p_1\) points between \(z_i\) and \(z_{i+1\textrm{ mod }k}\) if for such a representative, the ray from \(\zeta\) to \(p_1\) intersects the arc in \(\partial S\) from \(z_i\) to \(z_{i+1\textrm{ mod }k}\). The (codimension \(0\)) locus in \({}_{l-1}\mathcal{R}_{k+1}^{S^1}\) where \(p_1\) points between \(z_i\) and \(z_{i+1\textrm{ mod }k}\) is denoted by \({}_{l-1}\mathcal{R}_{k+1}^{S_{i,i+1}^1}\).

Denote by \({}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1\) and \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) the compactifcations of the moduli spaces introduced above.

It is clear that we have a decomposition of \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) into its sectors \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S_{i,i+1}^1}\). To understand the situation better, we introduce a new moduli space \[{}_{l-1}\mathcal{R}_{k+1,\tau_i}^1 \nonumber\] for each \(i=0,\cdots,k\). This is the abstract moduli space of discs with boundary marked points \(z_0,\cdots,z_i,z_f,z_{i+1},\cdots,z_k\) aligned in counterclockwise order, where \(z_f\) is marked as auxiliary, one interior puncture \(\zeta\) at the origin, marked as an input, equipped with an asymptotic marker \(\ell\) pointing in the direction of \(p_l\) (or \(z_f\) if \(l=0\)), and \(l\) marked points \(p_1,\cdots,p_l\) which are strictly radially ordered in the interior of the disc of radius \(\frac{1}{2}\). Note that the compactification \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1\) of \({}_{l-1}\mathcal{R}_{k+1,\tau_i}^1\) is abstractly isomorphic to \({}_{l-1}\overline{\mathcal{R}}_{k+2}^1\), except that \(z_f\) is marked as auxiliary. Moreover, at any stratum of \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1\):

• we regard the main component (the one with the puncture \(\zeta\)) with \(k'+2\) boundary marked points as an element of \({}_{l-1}\overline{\mathcal{R}}_{k'+1,\tau_j}^1\) for some \(0\leq k'\leq k\) and \(0\leq j\leq k'\) if it has \(z_f\) as a boundary marked point, and an element of \({}_{l-1}\overline{\mathcal{R}}_{k'+2}^1\) if it does not;

• if a non-main disc component (the one without \(\zeta\)) has the boundary marked point \(z_f\), we view it as an element of \(\mathcal{R}_{k'+1,f_i}\) for some \(0\leq k'\leq k\). This is the moduli space of discs with \(k'+1\) boundary marked points, with the \(i\)-th point marked as forgotten. See [25], Appendix A.2 for its construction.

The moduli space \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}\) is related to the sector \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S_{i,i+1}^1}\subset{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) via the auxiliary-rescaling map \[\label{eq:aux-res} \phi_f^i:{}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1\rightarrow{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S_{i,i+1}^1},\tag{18}\] which, for a representative of an element of \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}\), adds a point \(p_0\) on the line segment connecting \(\zeta\) and \(z_f\) with \(|p_0|=\frac{1}{2}\) and deletes \(z_f\). Then, we relabel the marked points \(p_0,\cdots,p_{l-1}\) as \(p_1,\cdots,p_{l}\), where now \(|p_1|=\frac{1}{2}\). See Figure 2. With the appropriate choices of orientations (cf. [11], Appendix B), the map \(\phi_f^i\) is an oriented diffeomorphism.

The codimension \(1\) boundary of \({}_l\overline{\mathcal{R}}_{k+1}^1\) is covered by the natural inclusions of the following strata (cf. [11], Proposition 40): \[\label{eq:dom-bdy1} {}_j\overline{\mathcal{M}}\times{}_{l-j}\overline{\mathcal{R}}_{k+1}^1,\textrm{ }1\leq j\leq l,\tag{19}\] \[\label{eq:dom-bdy2} {}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1,\textrm{ }1\leq j\leq l-1,\tag{20}\] \[\label{eq:dom-bdy3} {}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1},\tag{21}\] \[\label{eq:dom-bdy4} {}_l\overline{\mathcal{R}}_{k_1+1}^1\textrm{ }{{}_i\times}_0\textrm{ }\overline{\mathcal{R}}_{k_2+1},\textrm{ }k_1\geq1,k_2\geq2,k_1+k_2=k+1,1\leq i\leq k_1,\tag{22}\] \[\label{eq:dom-bdy5} \overline{\mathcal{R}}_{k_1+1}\textrm{ }{{}_i\times}_0\textrm{ }{}_l\overline{\mathcal{R}}_{k_2+1}^1,\textrm{ }k_1\geq2,k_2\geq0,k_1+k_2=k+1, 1\leq i\leq k_1.\tag{23}\] In the above, \({}_j\mathcal{M}\) is the moduli space of \(j\)-point angle decorated cylinders introduced by Ganatra in [25], Section 4.3, and \({}_j\overline{\mathcal{M}}\) is its compactification. \(\mathcal{R}_{k+1}\) is the moduli space of closed unit discs \(D\) with \(k+1\) distinct marked points \(z_0,\cdots,z_k\in\partial D\) aligned in counterclockwise order, modulo the automorphism group \(\mathit{PSL}(2,\mathbb{R})\) of \(D\). The notation \({{}_i\times_0}\) means that the disc breaking happens at \(z_i\in\partial S\), with the nodal point playing the role of \(z_0\) on the disc bubble in \(\overline{\mathcal{R}}_{k_2+1}\). The strata in (19 ) come from the real blow-ups, when the modulus of the marked points \(p_{l-j+1},\cdots,p_l\) tend to \(0\), see Figure 3.

Let \((X,\lambda)\) be a Liouville domain. A function \(H_t:S^1\times X\rightarrow\mathbb{R}\) is an admissible Hamiltonian if \(H_t=H+F_t\), where \(H:X\rightarrow\mathbb{R}\) is an autonomous Hamiltonian which equals \(r^2\) on the collar neighborhood \((1-\delta,1]\times\partial X\) for some sufficiently small \(\delta>0\) and \(C^2\)-small away from the collar \((1-\delta',1]\times\partial X\) for some \(\delta'>\delta\), and a time-dependent perturbation \(F_t:S^1\times X\rightarrow\mathbb{R}\). Denote by \(\mathcal{H}(X)\) the set of admissible Hamiltonians \(H_t\) so that all the \(1\)-periodic orbits of the Hamiltonian vector field \(X_{H_t}\) are non-degenerate. An almost complex structure \(J_t:S^1\times\mathit{TX}\rightarrow\mathit{TX}\) is called contact type if \(dr\circ J_t=-\lambda\) in the collar \((1-\delta,1]\times\partial X\). Denote by \(\mathcal{J}(X)\) the set of \(d\lambda\)-compatible almost complex structure on \(X\) which are of contact type. Before defining the Cohen–Ganatra moduli spaces, we need to choose the corresponding Floer data first.

Similarly, one can define Floer data for the moduli spaces \({}_l^{j,j+1}\mathcal{R}_{k+1}^1\), \({}_{l-1}\mathcal{R}_{k+1}^{S^1}\) and \({}_{l-1}\mathcal{R}_{k+1,\tau_i}^1\). We will inductively choose the Floer data on \({}_l\overline{\mathcal{R}}_{k+1}^1\) so that certain consistency conditions are satisfied. To do so, we first choose the Floer data on \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) so that

• The Floer datum on \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\) is \(\mathbb{Z}_{k+1}\)-equivariant under the cyclic permutation map \(\kappa\) (cf. (17 )).

• On the boundary stratum \({}_{l-1}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{S^1}\subset\partial{}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\), the Floer datum is conformally equivalent to the one pulled back from \({}_{l-2}\overline{\mathcal{R}}_{k+1}^{S^1}\) via the forgetful map \(\pi_j^{S^1}\) (cf. (16 )).

By [11], Proposition 44, Cohen–Ganatra Floer datum exists. Fix such a datum, a \(1\)-periodic orbit \(x\) of \(X_{H_t}\) and a Lagrangian submanifold \(L\subset\mathrm{int}(X)\) in the interior of \(X\), define the Cohen–Ganatra moduli space \[{}_l\mathcal{R}_{k+1}^1(x,L) \nonumber\] to be the space of pairs \[\left((S,z_0,\cdots,z_k,p_1,\cdots,p_l,\ell),u\right), \nonumber\] where \((S,z_0,\cdots,z_k,p_1,\cdots,p_l,\ell)\in{}_l\mathcal{R}_{k+1}^1\) and \(u:S\rightarrow X\) is a smooth map satisfying \[\label{eq:CR} \left\{\begin{array}{l} (du-X_{H_S}\otimes\gamma_S)^{0,1}=0, \\ u(\partial S)\subset L, \\ \lim_{s\rightarrow\infty}(\varepsilon^+)^\ast u(s,\cdot)=x, \end{array}\right.\tag{24}\] where the \((0,1)\)-part in the Floer equation is taken with respect to the domain-dependent almost complex structure \(J_S\). We can decompose the moduli space \({}_l\mathcal{R}_{k+1}^1(x,L)\) according to the homotopy classes of the maps \(u\), which gives \[{}_l\mathcal{R}_{k+1}^1(x,L)=\bigsqcup_{\ring{\bar{a}}\in\pi_2(X,x,L)}{}_l\mathcal{R}_{k+1}^1(x,L,\ring{\bar{a}}), \nonumber\] where \(\pi_2(X,x,L)\) denotes the set of homotopy classes of maps \(u:S\rightarrow X\) with boundary on \(L\) and asymptotic to \(x\) at its positive puncture. We recall the following basic fact (cf. [11], Lemma 45, (ii)).

In a similar way, one can define the moduli spaces \[{}_{l-1}\mathcal{R}_{k+1}^{S^1}(x,L,\ring{\bar{a}}), \nonumber\] \[\label{eq:sector} {}_{l-1}\mathcal{R}_{k+1,\tau_i}^1(x,L,\ring{\bar{a}}),\textrm{ }0\leq i\leq k,\tag{25}\] \[{}_l^{j,j+1}\mathcal{R}_{k+1}^1(x,L,\ring{\bar{a}}),\textrm{ }1\leq j\leq l-1, \nonumber\] using the auxiliary moduli spaces of domains \({}_{l-1}\mathcal{R}_{k+1}^{S^1}\), \({}_{l-1}\mathcal{R}_{k+1,\tau_i}^1\) and \({}_l^{j,j+1}\mathcal{R}_{k+1}^1\) introduced earlier.

In general, due to the non-exactness of \(L\), the transversality of these moduli spaces cannot be achieved with standard perturbations of the domain-dependent almost complex structure \(J_S\). Instead, we need the virtual perturbation scheme developed by Fukaya–Oh–Ohta–Ono [27], [28]. More precisely, one can construct Kuranishi structures on the moduli spaces \({}_l\mathcal{R}_{k+1}^1(x,L,\ring{\bar{a}})\), \({}_{l-1}\mathcal{R}_{k+1}^{S^1}(x,L,\ring{\bar{a}})\), \({}_{l-1}\mathcal{R}_{k+1,\tau_i}^1(x,L,\ring{\bar{a}})\) and \({}_l^{j,j+1}\mathcal{R}_{k+1}^1(x,L,\ring{\bar{a}})\) so that they become oriented K-spaces.

Recall that for \(\bar{a}\in\pi_2(X,L)\) we have a moduli space \(\mathcal{R}_{k+1}(L,\bar{a})\) parametrizing pairs \(((D,z_0,\cdots,z_k),u)\), where \((D,z_0,\cdots,z_k)\in\mathcal{R}_{k+1}\) and \(u:(D,\partial D)\rightarrow(X,L)\) is a \(J\)-holomorphic map in the class \(\bar{a}\). Note that the Gromov compactification \(\overline{\mathcal{R}}_{k+1}(L,\bar{a})\) is governed combinatorially by a decorated rooted ribbon tree \((T,B)\) (cf. [9], Definition 7.18), where \(T\) is a connected tree with the set of interior vertices \(V_\mathrm{int}(T)\) and the set of interior edges \(E_\mathrm{int}(T)\), and \(B:V_\mathrm{int}(T)\rightarrow\pi_2(X,L)\) is a map which associates every interior vertex of \(T\) to a relative homotopy class in \(\pi_2(X,L)\). Denote by \(\mathcal{T}(k+1,\bar{a})\) the set of decorated rooted ribbon trees with \(k+1\) exterior vertices and \(\sum_{v\in V_\mathrm{int}(T)}B(v)=\bar{a}\). For every \((T,B)\in\mathcal{T}(k+1,\beta)\), there is an interior evaluation map \[\label{eq:ev-int} \mathrm{ev}_\mathrm{int}:\prod_{v\in V_{\mathrm{int}}(T)}\mathcal{R}_{k_v+1}(L,B(v))\rightarrow\prod_{e\in E_{\mathit{int}}(T)}L^2\tag{26}\] defined by evaluating at the endpoints of each edge \(e\in E_\mathrm{int}(T)\).

As in the case of \(\overline{\mathcal{R}}_{k+1}(L,\bar{a})\), the (disc-breaking part of the) Gromov compactifications \({}_l\overline{\mathcal{R}}_{k+1}^1(x,L,\ring{\bar{a}})\), \({}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1(x,L,\ring{\bar{a}})\), \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}(x,L,\ring{\bar{a}})\) and \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1(x,L,\ring{\bar{a}})\) are modeled on decorated rooted ribbon trees with a single puncture, see [11], Definition 46. Roughly speaking, this is a triple \((T,\ring{B},v_0)\), where \(T\) is a tree with a distinguished interior vertex \(v_0\in V_\mathrm{int}(T)\), such that the map \(\ring{B}\) associates to \(v_0\) a class in \(\pi_2(X,x,L)\), and to other interior vertices a class in \(\pi_2(X,L)\) with non-negative symplectic area. Denote by \(\mathcal{T}(k+1,\ring{\bar{a}})\) the set of decorated rooted ribbon trees with a single puncture \((T,\ring{B},v_0)\) with \(k+1\) exterior vertices and \(\sum_{v\in V_\mathrm{int}(T)}\ring{B}(v)=\ring{\bar{a}}\). Similar to (26 ), one can define interior evaluation maps on the strata in the compactified moduli space \({}_l\overline{\mathcal{R}}_{k+1}^1(x,L,\ring{\bar{a}})\).

Let \(\varepsilon>0\) be chosen so that \(2\varepsilon\) is less than the minimal symplectic area of \(J\)-holomorphic discs \(u:(D,\partial D)\rightarrow(X,L)\), where the almost complex structure \(J\) is fixed to be the one in (?? ). Choose \(U\in\mathbb{N}\) so that \[\varepsilon(U-1)\geq\left\vert A_{H_t}(x)\right\vert. \nonumber\] We summarize the properties of the compactified moduli spaces \({}_l\overline{\mathcal{R}}_{k+1}^1(x,L,\ring{\bar{a}})\), \({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}(x,L,\ring{\bar{a}})\), \({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1(x,L,\ring{\bar{a}})\) and \({}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1(x,L,\ring{\bar{a}})\) in the following theorem (cf. [11], Theorem 48 and Remark 49).

As is observed in [11], the construction of the Maurer-Cartan element (the chain \(x\in\widehat{\mathbb{H}}_{-2}^{S^1}\) in Theorem 9) in the \(S^1\)-equivariant case relies on the consideration of a moduli space of holomorphic discs that is a slight variant of \(\mathcal{R}_{k+1}(L,\bar{a})\).

For \(k\geq3\), define \[\mathcal{R}_{k+1,\vartheta}:=\left\{(D,z_0,\cdots,z_{k-1},z_k=z_0e^{i\vartheta_k})\right\}/\mathit{PSL}(2,\mathbb{R}), \nonumber\] where \(z_0,\cdots,z_k\in\partial D\) are aligned in counterclockwise order and \(\vartheta_k\in(0,2\pi)\) is a fixed constant. Note that there is an obvious embedding \(\mathcal{R}_{k+1,\vartheta}\subset\mathcal{R}_{k+1}\). For \(1\leq i\leq k\), there is a map \[\label{eq:f1} \pi_{\vartheta,i}:\mathcal{R}_{k+1,\vartheta}\rightarrow\mathcal{R}_k\tag{27}\] defined by applying the cyclic permutation \(i\) times to the boundary marked points \(z_0,\cdots,z_k\), so that \(z_{i+j\textrm{ mod }k+1}\) becomes \(z_j\) for \(0\leq j\leq k\), and then forgetting the point labeled \(z_{k+1-i}\) after the permutation. The map \((-1)^{k,i}\pi_{\vartheta,i}\) is orientation-preserving (cf. [11], Appendix B), therefore identifies \(\mathcal{R}_{k+1,\vartheta}\) as an open sector of \(\mathcal{R}_k\). By [11], Lemma 38, we actually have \(\bigsqcup_{1\leq i\leq k}(-1)^{ki}\pi_{\vartheta,i}(\mathcal{R}_{k+1,\vartheta})\) covers all but codimension \(1\) strata of \(\mathcal{R}_k\).

For \(\bar{a}\in\pi_2(X,L)\), define \[\mathcal{R}_{k+1,\vartheta}(L,\bar{a}):=\left\{(u,z_0,\cdots,z_k=z_0e^{i\vartheta_k}),[u]=\bar{a}\right\}/\mathit{PSL}(2,\mathbb{R}), \nonumber\] where \(u:(D,\partial D)\rightarrow(X,L)\) is a \(J\)-holomorphic map in the homotopy class \(\bar{a}\) and \(z_k=z_0e^{i\vartheta_k}\). Note that when \(\bar{a}=0\), we have \[\mathcal{R}_{2,\vartheta}(L,0)=\mathcal{R}_{3,\vartheta}(L,0)=\emptyset. \nonumber\] The compactification \(\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a})\) is an admissible K-space and is modeled on decorated rooted ribbon trees \((T,B)\in\mathcal{T}(k+1,\beta)\). It follows from the embedding (27 ) that \[\dim\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a})=\dim\overline{\mathcal{R}}_k(L,\bar{a})=\mu(\bar{a})+n+k-3. \nonumber\] For \(P=\{m\}\) or \([m,m+1]\) with \(m\in\mathbb{Z}_{\geq0}\), and \(a\in H_1(L;\mathbb{Z})\) satisfying \(\lambda(a)<(m-k+1)\varepsilon\), there are corner-stratified strongly smooth maps \[\mathrm{ev}_\vartheta^{\mathcal{R},P}:\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a};P)\rightarrow P\times L^{k+1} \nonumber\] and orientation-preserving isomorphisms of admissible K-spaces \[\label{eq:bd1} \partial\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a};\{m\})\cong\bigsqcup_{\substack{k_1+k_2=k+1\\1\leq i\leq k_1+1\\ \bar{a}_1+\bar{a}_2=\bar{a}}}(-1)^{\varepsilon_{13}}\overline{\mathcal{R}}_{k_1+1,\vartheta}(L,\bar{a}_1;\{m\})\textrm{ }{{}_i\times_0}\textrm{ }\overline{\mathcal{R}}_{k_2+1}(L,\bar{a}_2;\{m\}),\tag{28}\] \[\label{eq:bd4} \begin{align} &\partial\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a};[m,m+1])\cong-\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a};\{m\})\sqcup\overline{\mathcal{R}}_{k+1,\vartheta}(L,\bar{a};\{m+1\}) \\ &\sqcup\bigsqcup_{\substack{k_1+k_2=k+1\\1\leq i\leq k_1+1\\ \bar{a}_1+\bar{a}_2=\bar{a}}}(-1)^{\varepsilon_{14}}\overline{\mathcal{R}}_{k_1+1,\vartheta}(L,\bar{a}_1;[m,m+1])\textrm{ }{{}_i\times_0}\textrm{ }\overline{\mathcal{R}}_{k_2+1}(L,\bar{a}_2;[m,m+1]), \end{align}\tag{29}\] where \[\varepsilon_{13}+1=\varepsilon_{14}=(k_1-i)(k_2-1)+n+k_1, \nonumber\] and \[\label{eq:corner1} \begin{align} &\widehat{S}_r\overline{\mathcal{R}}_{k+2,\vartheta}(L,\beta;P)\cong\bigsqcup_{\substack{(T,B)\in\mathcal{T}(k+2,\beta)\\ \#E_{\mathrm{int}}(T)+d=r \\ v_0\in V_{0,\mathrm{int}}(T)}}\left(\prod_{e\in E_{\mathrm{int}}(T)}\widehat{S}_dP\times L\right)\textrm{ }{{}_\Delta\times\mathrm{ev}_\mathrm{int}}\textrm{ } \\ &\left(\prod_{v\in V_{0,\mathrm{int}}(T)\setminus\{v_0\}}\overline{\mathcal{R}}_{k_v+1}\left(L,B(v);\widehat{S}_dP\right)\times\overline{\mathcal{R}}_{k_{v_0}+1,\vartheta}\left(L,B(v_0);\widehat{S}_dP\right)\right), \end{align}\tag{30}\] where the fiber product on the right-hand side is taken over \(\prod_{e\in E_{\mathrm{int}}(T)}\left(\widehat{S}_dP\times L\right)^2\).

5 The de Rham chain model↩︎

Let \(L\) be a closed, orientable manifold of dimension \(n\). We shall recall in this section a de Rham chain model of the \(S^1\)-equivariant homology of \(\mathcal{L}L\) introduced in [11].

We start by recalling a convenient model for the space of Moore loops with marked points in \(L\), which is due to Wang [29]. Denote by \(\Pi_1L\) the fundamental groupoid of \(L\), which assigns to each ordered pair of points \((p,q)\in L^2\) the collection of equivalence classes of smooth paths from \(p\) to \(q\). Denote by \[s:\Pi_1L\rightarrow L\textrm{ and }t:\Pi_1L\rightarrow L \nonumber\] the source and the target maps, which assigns the points \(p\) and \(q\) to each equivalence class of ordered pair \((p,q)\), respectively. There is an obvious map \[\left\{(c_0,c_1)\in(\Pi_1L)^2|t(c_0)=s(c_1)\right\}\rightarrow\Pi_1L,\textrm{ }(c_0,c_1)\mapsto c_0\ast c_1 \nonumber\] induced by the concatenation of two paths. For every \(k\in\mathbb{Z}_{\geq0}\), denote by \(\mathcal{P}_{k+1}L\subset(\Pi_1L)^{k+1}\) the subspace consisting of \((c_0,\cdots,c_k)\) such that \(t(c_i)=s(c_{i+1})\) for \(0\leq i\leq k-1\). Define \[\label{eq:flsk} \mathcal{L}_{k+1}L:=\left\{(c_0,\cdots,c_k)\in\mathcal{P}_{k+1}L|t(c_k)=s(c_0)\right\}.\tag{31}\] Note that compared to the space of Moore loops with \(k+1\) marked points in \(L\) (cf. [9], Section 7 and [12], Section 4.1), the space \(\mathcal{L}_{k+1}L\) defined above has the advantage that it is a smooth oriented manifold of dimension \((k+1)n\). The spaces \(\{\mathcal{L}_{k+1}L\}_{k\geq0}\) form finite-dimensional approximations of the free loop space \(\mathcal{L}L\).

From now on, we will simply write \(\mathcal{L}_{k+1}\) for \(\mathcal{L}_{k+1}L\). These spaces are equipped with smooth evaluation maps \[\mathrm{ev}_j^\mathcal{L}:\mathcal{L}_{k+1}\rightarrow L,\textrm{ }(c_0,\cdots,c_k)\mapsto s(c_j),\textrm{ }0\leq j\leq k \nonumber\] and concatenation maps \[\label{eq:con} \begin{align} \mathrm{con}_j:&\mathcal{L}_{k+1}\textrm{ }{{}_{\mathrm{ev}_j^\mathcal{L}}\times_{\mathrm{ev}_0^\mathcal{L}}}\textrm{ }\mathcal{L}_{k'+1}\rightarrow\mathcal{L}_{k+k'}, \\ &\left((c_0,\cdots,c_k),(c_0',\cdots,c_k')\right)\mapsto \\ &\left\{\begin{array}{ll} (c_0,\cdots,c_{j-2},c_{j-1}\ast c_0',c_1',\cdots,c_{k'-1}',c_{k'}'\ast c_j,\cdots,c_k) & \textrm{if }k'\geq1, \\ (c_0,\cdots,c_{j-2},c_{j-1}\ast c_0'\ast c_j,c_{j+1},\cdots,c_k) & \textrm{if }k'=0. \end{array}\right. \end{align}\tag{32}\] It is clear from the definition that the concatenation maps are compatible with the decomposition \(\mathcal{L}_{k+1}=\bigsqcup_{a\in H_1(L;\mathbb{Z})}\mathcal{L}_{k+1}(a)\) of \(\mathcal{L}_{k+1}\) into different homotopy classes, where \(\mathcal{L}_{k+1}(a)\subset\mathcal{L}_{k+1}\) is the subspace consisting of loops \(\gamma=c_0\ast\cdots\ast c_k\in\Pi_1L\) with \([\gamma]=a\). Thus we obtain a map \[\label{eq:conca} \mathrm{con}_j:\mathcal{L}_{k+1}(a)\textrm{ }{{}_{\mathrm{ev}_j^\mathcal{L}}\times_{\mathrm{ev}_0^\mathcal{L}}}\textrm{ }\mathcal{L}_{k'+1}(a')\rightarrow\mathcal{L}_{k+k'}(a+a')\tag{33}\] for \(a,a'\in H_1(L;\mathbb{Z})\).

Let \(V\) be a smooth manifold and consider the map \(\varphi:V\rightarrow\mathcal{L}_{k+1}\). We say that \(\varphi\) is a smooth map if the map \(\varphi\) is \(C^\infty\) and the composition \(\mathrm{ev}_0^\mathcal{L}\circ\varphi:V\rightarrow L\) is a submersion. For \(N\in\mathbb{N}\), let \(\mathfrak{V}_N\) be the collection of oriented submanifolds in \(\mathbb{R}^N\), and define \(\mathfrak{V}:=\bigsqcup_{N\geq1}\mathfrak{V}_N\). Let \(\mathcal{P}(\mathcal{L}_{k+1}(a))\) denote the set of plots, i.e. pairs \((V,\varphi)\), where \(V\in\mathfrak{V}\) and \(\varphi:V\rightarrow\mathcal{L}_{k+1}(a)\) is a smooth map.

For each \(N\), consider the vector space \[\label{eq:sp} \bigoplus_{(V,\varphi)\in\mathcal{P}(\mathcal{L}_{k+1}(a))}\Omega_c^{\dim(V)-N}(V),\tag{34}\] where \(\Omega_c^\ast(V)\) is the space of compactly supported differential forms on \(V\). Denote by \(Z_N\) the subspace of (34 ) defined by \[\begin{align} &\left\{(V,\varphi,\pi_!\omega)-(V',\varphi\circ\pi,\omega)\left\vert(V,\varphi)\in\mathcal{P}(\mathcal{L}_{k+1}(a)),V'\in\mathfrak{V}, \right. \right. \\ &\left.\omega\in \Omega_c^{\dim(V')-N}(V'),\pi:V'\rightarrow V\textrm{ is a submersion}\right\}. \nonumber \end{align}\] As a graded vector space, the \(N\)-th degree de Rham chain complex of \(\mathcal{L}_{k+1}(a)\) is the quotient \[C_N^\mathrm{dR}(\mathcal{L}_{k+1}(a)):=\left(\bigoplus_{(V,\varphi)\in\mathcal{P}(\mathcal{L}_{k+1}(a))}\Omega_c^{\dim(V)-N}(V)\right)/Z_N. \nonumber\] The differential \(\partial:C_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\rightarrow C_{\ast-1}^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) is defined as \[\partial(V,\varphi,\omega):=(-1)^{|\omega|+1}(V,\varphi,d\omega). \nonumber\] It is straightforward to check that \(\partial\) is well-defined and \(\partial^2=0\). The homology of \(\left(C_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a)),\partial\right)\) is denoted by \(H_\ast^\mathrm{dR}\left(\mathcal{L}_{k+1}(a)\right)\). Unlike in the case when \(\mathcal{L}_{k+1}\) is the space of Moore loops considered in [9], [12], \(H_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) is no longer isomorphic to \(H_\ast(\mathcal{L}(a)L;\mathbb{R})\), but the usage of Wang’s finite-dimensional model of \(\mathcal{L}_{k+1}\) is justified by the following.

If one forms the total complex \[\label{eq:total} \left(C_\ast^\mathrm{dR}(a):=\prod_{k=0}^\infty C_\ast^\mathrm{dR}\left(\mathcal{L}_{k+1}(a)\right),\partial^\mathrm{tot}\right),\tag{35}\] where the differential \(\partial^\mathrm{tot}\) is given by \[\label{eq:totd} \partial^\mathrm{tot}(x)(a,k)=\left\{\begin{array}{ll} \partial x(a,0) & \textrm{if }k=0, \\ \partial x(a,k)+(-1)^{|x|}\sum_{i=0}^k(-1)^i\delta_{k,i}(x(a,k-1)) & \textrm{if }k\geq1. \end{array}\right.\tag{36}\] Then there is an isomorphism \[\label{eq:sing} H_\ast\left(C_\ast^\mathrm{dR}(a),\partial^\mathrm{tot}\right)\cong H_\ast^\mathrm{sing}(\mathcal{L}(a)L;\mathbb{R}),\tag{37}\] where the singular homology on the right-hand side is defined using the \(C^\infty\)-topology on \(\mathcal{L}(a)L\). See [9], [29].

For \(k\in\mathbb{N}\), \(k'\in\mathbb{Z}_{\geq0}\), and \(1\leq j\leq k\), define the fiber product \[\label{eq:fp} \circ_j:C_{n+d}^\mathrm{dR}(\mathcal{L}_{k+1}(a))\otimes C_{n+d'}^\mathrm{dR}\left(\mathcal{L}_{k'+1}(a')\right)\rightarrow C_{n+d+d'}^\mathrm{dR}\left(\mathcal{L}_{k+k'}(a+a')\right)\tag{38}\] by \[x\circ_jy:=(-1)^{(\dim(V)-|\omega|-n)|\omega'|}\left(V{{}_{\varphi_j}\times_{\varphi_0'}}V',\mathrm{con}_j\circ(\varphi_j\times\varphi_0'),\omega\times\omega'|_{V{{}_{\varphi_j}\times_{\varphi_0'}}V'}\right), \nonumber\] where \(\varphi_j=\mathrm{ev}_j^\mathcal{L}\circ\varphi\) and \(\varphi_0'=\mathrm{ev}_0^\mathcal{L}\circ\varphi'\). One can check that this is a chain map, and after passing to homology of the total complex (35 ), it corresponds to the Chas-Sullivan loop product under the isomorphism (37 ).

We also need the relative version of the de Rham complex \(C_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\), whose construction uses de Rham chains on \([-1,1]\times\mathcal{L}_{k+1}(a)\) relative to \(\{-1,1\}\times\mathcal{L}_{k+1}(a)\). This is defined by replacing the set of plots \(\mathcal{P}(\mathcal{L}_{k+1}(a))\) above with \(\overline{\mathcal{P}}(\mathcal{L}_{k+1}(a))\), which consists of tuples \((V,\varphi,\tau_+,\tau_-)\), where

• \(V\in\mathfrak{V}\) and \(\varphi:V\rightarrow\mathbb{R}\times\mathcal{L}_{k+1}(a)\). Write \(\varphi=(\varphi_\mathbb{R},\varphi_\mathcal{L})\), define \(V_I:=(\varphi_\mathbb{R})^{-1}(I)\) for every interval \(I\subset\mathbb{R}\).

• \(\varphi_\mathbb{R}\) and \(\varphi_\mathcal{L}\) are \(C^\infty\), and the map \(V\rightarrow\mathbb{R}\times L\) defined by \(v\mapsto\left(\varphi_\mathbb{R}(v),\mathrm{ev}_0\circ\varphi_\mathcal{L}(v)\right)\) is a submersion.

• \(\tau_+:V_{\geq1}\rightarrow\mathbb{R}_{\geq1}\times V_1\) is a diffeomorphism such that \[\varphi|_{V_{\geq1}}=(i_{\geq1}\times\varphi_\mathcal{L}|_{V_1})\circ\tau_+, \nonumber\] where \(i_{\geq1}:\mathbb{R}_{\geq1}\hookrightarrow\mathbb{R}\) is the obvious inclusion.

• \(\tau_-:V_{\leq-1}\rightarrow\mathbb{R}_{\leq-1}\times V_{-1}\) is a diffeomorphism such that \[\varphi|_{V_{\leq-1}}=(i_{\leq-1}\times\varphi_\mathcal{L}|_{V_{-1}})\circ\tau_{-1}, \nonumber\] where \(i_{\leq-1}:\mathbb{R}_{\leq-1}\hookrightarrow\mathbb{R}\) is the obvious inclusion.

Note that \(V_{\geq1}\) and \(V_{\leq-1}\) can be empty.

For any \((V,\varphi,\tau_+,\tau_-)\in\overline{\mathcal{P}}(\mathcal{L}_{k+1}(a))\) and \(N\in\mathbb{Z}\), let \(\Omega^N(V,\varphi,\tau_+,\tau_-)\) be the vector space of differential \(N\)-forms \(\omega\in\Omega^N(V)\) such that

• \(\omega|_{V_{[-1,1]}}\) is compactly supported,

• \(\omega|_{V_{\geq1}}=(\tau_+)^\ast(1\times\omega|_{V_1})\),

• \(\omega|_{V_{\leq-1}}=(\tau_-)^\ast(1\times\omega|_{V_{-1}})\).

Define the space of \(N\)-th degree relative de Rham chains to be \[\overline{C}_N^\mathrm{dR}(\mathcal{L}_{k+1}(a)):=\left(\bigoplus_{(V,\varphi,\tau_+,\tau_-)\in\overline{\mathcal{P}}(\mathcal{L}_{k+1}(a))}\Omega^{\dim(V)-N-1}(V,\varphi,\tau_+,\tau_-)\right)/\overline{Z}_N, \nonumber\] where \(\overline{Z}_N\subset\overline{C}_N^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) is generated by \[(V,\varphi,\tau_+,\tau_-,\omega)-(V',\varphi',\tau_+',\tau_-',\omega'), \nonumber\] if there exists a submersion \(\pi:V'\rightarrow V\) satisfying \(\varphi'=\varphi\circ\pi\), \(\omega=\pi_!\omega'\), and \[\tau_+\circ\pi|_{V_{\geq1}'}=(\mathrm{id}_{\mathbb{R}_{\geq1}}\times\pi|_{V_1'})\circ\tau_+', \nonumber\] \[\tau_-\circ\pi|_{V_{\leq-1}'}=(\mathrm{id}_{\mathbb{R}_{\leq-1}}\times\pi|_{V_{-1}'})\circ\tau_-', \nonumber\] where \(\mathrm{id}_{\mathbb{R}_I}\) denotes the identity map on \(\mathbb{R}_I\). The differential \[\overline{\partial}:\overline{C}_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\rightarrow\overline{C}_{\ast-1}^\mathrm{dR}(\mathcal{L}_{k+1}(a)) \nonumber\] and the fiber product \[\label{eq:fpr} \circ_j:\overline{C}_{n+d}^\mathrm{dR}(\mathcal{L}_{k+1}(a))\otimes\overline{C}^\mathrm{dR}_{n+d'}(\mathcal{L}_{k'+1}(a'))\rightarrow\overline{C}^\mathrm{dR}_{n+d+d'}(\mathcal{L}_{k+k'}(a+a'))\tag{39}\] are defined similarly as in the case of ordinary de Rham chains. We omit the details and refer the reader to [11], Section 3.1 for details.

The de Rham chain complexes \(C_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) and \(\overline{C}_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) are closely related. Consider the projections \(e_\pm:\overline{C}_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\rightarrow C_\ast^\mathrm{dR}(\mathcal{L}_{k+1}(a))\) given by \[\label{eq:e43} e_+(V,\varphi,\tau_+,\tau_-,\omega):=(-1)^{\dim(V)-1}(V_1,\varphi|_{U_1},\omega|_{U_1})\tag{40}\] and \[\label{eq:e-} e_-(V,\varphi,\tau_+,\tau_-,\omega):=(-1)^{\dim(V)-1}(V_{-1},\varphi|_{V_{-1}},\omega|_{V_{-1}}),\tag{41}\] where \(V_1\) and \(V_{-1}\) are oriented so that \(\tau_+\) and \(\tau_-\) are orientation preserving, respectively. One can show that \(e_\pm\) are quasi-isomorphisms, with the same quasi-inverse given by the obvious inclusion \[\label{eq:i} i(V,\varphi,\omega):=(-1)^{\dim(V)}(\mathbb{R}\times V,\mathrm{id}_\mathbb{R}\times\varphi,\tau_+,\tau_-,1\times\omega),\tag{42}\] so we have \[H_\ast\left(\overline{C}_\ast^\mathrm{dR}(a)\right)\cong H_\ast\left(C_\ast^\mathrm{dR}(a)\right)\cong H_\ast(\mathcal{L}(a)L;\mathbb{R}), \nonumber\] where \(\overline{C}_\ast^\mathrm{dR}(a):=\prod_{k=0}^\infty\overline{C}_\ast^\mathrm{dR}\left(\mathcal{L}_{k+1}(a)\right)\) is the relative total de Rham complex.

For the purpose of establishing the equation (?? ) in Theorem 9, we need a chain model of the string homology that carries the structures of a strict \(S^1\)-complex and an odd dg Lie algebra. Write \[C_\ast(a,k):=C^\mathrm{dR}_{\ast+n+\mu(a)+k-1}(\mathcal{L}_{k+1}(a)), \nonumber\] \[C_\ast(k):=C_{\ast+n}^\mathrm{dR}(\mathcal{L}_{k+1})=\bigoplus_{a\in H_1(L;\mathbb{Z})}C_{\ast+n+\mu(a)+k-1}^\mathrm{dR}(\mathcal{L}_{k+1}(a)). \nonumber\] The chain complexes \(\left\{C_\ast(k)\right\}_{k\geq0}\) form a non-symmetric dg operad \(\mathcal{O}_L\), with multiplication \[\mu_L:=(L',i_2\circ\phi,1)\in C_{-1}(0,2), \nonumber\] and unit \[e_L:=(L',i_0\circ\phi,1)\in C_1(0,0), \nonumber\] where the map \(i_0:L\rightarrow\mathcal{L}_1(0)\) is given by \(p\mapsto[p]=(p,p)\in\Pi_1L\), \(i_2:L\rightarrow\mathcal{L}_3(0)\) is defined by taking three copies of \(i_0\) and \(\phi:L'\rightarrow L\) is an orientation-preserving diffeomorphism. We have \[\mu_L\circ_1e_L=\mu_L\circ_2e_L=\mathrm{id}_{\mathcal{O}_L}, \nonumber\] where \[\mathrm{id}_{\mathcal{O}_L}:=(L',i_1\circ\phi,1)\in C_0(0,1) \nonumber\] is the identity, with \(i_1:L\rightarrow\mathcal{L}_2(0)\) given by taking two copies of \(i_0\).

From now on, we assume that \(L\subset X\) is a closed Lagrangian submanifold in the Liouville domain \(X\). Consider the total complex \[\label{eq:tot} \left(C_\ast:=\bigoplus_{a\in H_1(L;\mathbb{Z})}\prod_{k=0}^\infty C_\ast(a,k),\partial^\mathrm{tot}\right),\tag{43}\] where the differential \(\partial^\mathrm{tot}\) is defined as in (36 ).

\(C_\ast\) has the structure of an associative dg algebra, with the product \(\bullet:C_i\otimes C_j\rightarrow C_{i+j-1}\) given by \[\label{eq:bullet} (x\bullet y)(a,k):=\sum_{\substack{k_1+k_2=k\\a_1+a_2=a}}(-1)^{k_1(|y|+1)}(\mu_L\circ_1 x(a_1,k_1))\circ_{k_1+1}y(a_2,k_2).\tag{44}\] Using the Liouville form \(\lambda\), one can define an action filtration \[\label{eq:Xi} F^\Xi C_\ast:=\bigoplus_{\lambda(a)>\Xi}\prod_{k=0}^\infty C_\ast(a,k).\tag{45}\] We denote by \(\widehat{C}_\ast\) the corresponding completion of \(C_\ast\).

The dg operad \(\mathcal{O}_L\) also carries a cyclic structure \[\label{eq:cyc} \tau_k:C_\ast(k)\rightarrow C_\ast(k),\tag{46}\] which is induced by the cyclic permutation \[\mathcal{L}_{k+1}\rightarrow\mathcal{L}_{k+1},\textrm{ }(c_0,\cdots,c_k)\mapsto(c_1,\cdots,c_k,c_0). \nonumber\] Note that \(\tau_k^{k+1}=\mathrm{id}_C\), where \(\mathrm{id}_C\) is the identity map on \(C_\ast(k)\). This gives rise to a chain level BV operator \[\label{eq:BV-c} \begin{align} \delta_\mathrm{cyc}&:C_\ast(a,k+1)\rightarrow C_{\ast+1}(a,k), \\ (\delta_\mathrm{cyc} x)(a,k)&:=\sum_{j=1}^{k+1}(-1)^{|x|+k(j-1)}\tau_{k+1}^j\left(x(a,k+1)\right)\circ_{k+2-j}e_L. \end{align}\tag{47}\] After passing to homology and composing with the aforementioned isomorphism \(H_\ast(C_\ast)\cong H_\ast(\mathcal{L}L;\mathbb{R})\) (cf. (37 )), it coincides with the BV operator \(\Delta:H_\ast(\mathcal{L}L;\mathbb{R})\rightarrow H_{\ast+1}(\mathcal{L}L;\mathbb{R})\) defined by loop rotations. As before, \(\delta_\mathrm{cyc}\) is compatible with the decomposition (43 ) of the total complex \(C_\ast\), therefore it extends to an operator on the completion \(\widehat{C}_\ast\).

\(\mathcal{O}_L\) has the structure of a cosimplicial chain complex, with the operations \[\delta_{k,i}:C_\ast(k-1)\rightarrow C_\ast(k),\textrm{ }\sigma_{k,i}:C_\ast(k+1)\rightarrow C_\ast(k)\] for \(0\leq i\leq k\) defined by \[\label{eq:cosimp} \delta_{k,i}(x):=\left\{\begin{array}{ll} \mu_L\circ_2x & i=0, \\ x\circ_i\mu_L & 1\leq i\leq k-1, \\ \mu_L\circ_1x & i=k, \end{array}\right.\tag{48}\] \[\sigma_{k,i}(x):=x\circ_{i+1} e_L. \nonumber\] More concretely, \(\delta_{k,i}\) is induced by the map \[\begin{align} \mathcal{L}_k&\rightarrow\mathcal{L}_{k+1} \\ (c_0,\cdots,c_{k-1})&\mapsto\left\{\begin{array}{ll} \left(c_0,\cdots,c_{i-1},[s(c_i)],c_i,\cdots,c_{k-1}\right) & \textrm{if }0\leq i\leq k-1, \\ \left(c_0,\cdots,c_{k-1},[t(c_{k-1})]\right) & \textrm{if }i=k, \end{array}\right. \nonumber \end{align}\] and \(\sigma_{k,i}\) is induced by the map \[\begin{align} \mathcal{L}_{k+1}&\rightarrow\mathcal{L}_k \\ (c_0,\cdots,c_k)&\mapsto(c_0,\cdots,c_{i-1},c_i\ast c_{i+1},c_{i+2},\cdots,c_k). \nonumber \end{align}\] Recall the cosimplicial identities \[\label{eq:cod} \delta_{k+1,j}\circ\delta_{k,i}=\delta_{k+1,i}\circ\delta_{k,j-1}\textrm{ for }0\leq i<j\leq k+1,\tag{49}\] \[\label{eq:cos} \sigma_{k-1,j}\circ\sigma_{k,i}=\sigma_{k-1,i}\circ\sigma_{k,j+1}\textrm{ for }0\leq i\leq j\leq k-1,\tag{50}\] \[\label{eq:sd} \sigma_{k,j}\circ\delta_{k+1,i}=\left\{\begin{array}{ll} \delta_{k,i}\circ\sigma_{k-1,j-1} & 0\leq i<j\leq k, \\ \mathrm{id}_C & i=j,j+1, \\ \delta_{k,i-1}\circ\sigma_{k-1,j} & k+1\geq i>j+1\geq1, \end{array}\right.\tag{51}\] and the cocyclic identities \[\label{eq:td} \tau_k\circ\delta_{k,i}=\left\{\begin{array}{ll} \delta_{k,k} & i=0, \\ \delta_{k,i-1}\circ\tau_{k-1} & 1\leq i\leq k, \end{array}\right.\tag{52}\] \[\label{eq:cocy2} \tau_k\circ\sigma_{k,i}=\left\{\begin{array}{ll} \sigma_{k,k}\circ\tau_{k+1}^2 & i=0, \\ \sigma_{k,i-1}\circ\tau_{k+1} & 1\leq i\leq k. \end{array}\right.\tag{53}\]

A chain \(x\in C_\ast(k)\) is normalized if \(\sigma_{k-1,i}(x)=0\) for all \(0\leq i\leq k-1\). The normalized chains form a subcomplex \(C_\ast^\mathrm{nm}\subset C_\ast\). It follows from [12], Lemma 2.5 that the natural inclusion \(C_\ast^\mathrm{nm}\hookrightarrow C_\ast\) is a quasi-isomorphism. On the other hand, a chain \(x\in C_\ast(k)\) is degenerate if there exist \(1\leq i\leq k\) and \(y\in C_\ast(k-1)\) such that \(x=\delta_{k,i}(y)\). Degenerate chains also form a subcomplex \(D_\ast\subset C_\ast\), and the quotient \(C_\ast^\mathrm{nd}:=C_\ast/D_\ast\) is called the complex of non-degenerate de Rham chains. It follows from the Dold-Kan correspondence that the composition \[C_\ast^\mathrm{nm}\hookrightarrow C_\ast\twoheadrightarrow C_\ast^\mathrm{nd} \nonumber\] is a quasi-isomorphism, see [11], Lemma 18. It follows that there is a well-defined BV operator on \(C_\ast^\mathrm{nd}\), which we still denote by \(\delta_\mathrm{cyc}\) by abuse of notations, such that \((C_\ast^\mathrm{nd},\partial^\mathrm{tot},\delta_\mathrm{cyc})\) is a strict \(S^1\)-complex (cf. [25], Definition 2.1). In particular, \(\delta_\mathrm{cyc}^2=0\). See [11], Lemma 21.

Associated to the strict \(S^1\)-complex \((C_\ast^\mathrm{nd},\partial^\mathrm{tot},\delta_\mathrm{cyc})\) is an \(S^1\)-equivariant chain complex \[\label{eq:cpx-S1} C_\ast^{S^1}:=\left(C_\ast^\mathrm{nd}\otimes_\mathbb{R}\mathbb{R}(\!(h)\!)/h\mathbb{R}[\![h]\!],\partial^{S^1}:=\partial^\mathrm{tot}+h\delta_\mathrm{cyc}\right),\tag{54}\] where the formal variable \(h\) has degree \(-2\). There is an obvious decomposition \(C_\ast^{S^1}=\bigoplus_{a\in H_1(L;\mathbb{Z})}\prod_{k=0}^\infty C_\ast^{S^1}(a,k)\).

The action filtration (45 ) induces a similar action filtration on \(C_\ast^{S^1}\), and we denote the corresponding completion by \(\widehat{C}_\ast^{S^1}\). The homology \(H_\ast(\widehat{C}_\ast^{S^1})\) gives the vector space \(\widehat{\mathbb{H}}_\ast^{S^1}\) defined by (6 ).

We now define the chain level string bracket, which is expected to be a refinement of the string bracket introduced by Chas-Sullivan [31]. For two \(S^1\)-equivariant chains \(\tilde{x}=\sum_{d=0}^\infty x_d\otimes h^{-d}\in C_\ast^{S^1}\) and \(\tilde{y}=\sum_{d=0}^\infty y_d\otimes h^{-d}\in C_\ast^{S^1}\), define \[\label{eq:bracket-pr} \begin{align} \left\{\tilde{x},\tilde{y}\right\}(a,k)&:=\sum_{\substack{a_1+a_2=a\\k_1+k_2=k+1}}\sum_{d=0}^\infty\sum_{d_1+d_2=d}\sum_{i=1}^{k_1}\sum_{j=1}^{k_2+1}(-1)^{\maltese_{ij}^{d_2}}x_{d_1}(a_1,k_1) \\ &\circ_i\left(\tau_{k_2+1}^j(y_{d_2}(a_2,k_2+1))\circ_{k_2+2-j}e_L\right)\otimes h^{-d} \\ &-\sum_{\substack{a_1+a_2=a\\k_1+k_2=k+1}}\sum_{d=0}^\infty\sum_{d_1+d_2=d}\sum_{i=1}^{k_1}\sum_{j=1}^{k_1+1}(-1)^{\maltese_{ij}^{d_1}+(|x_{d_2}|+1)(|y_{d_1}|+1)} \\ &\left(\tau_{k_1+1}^j(y_{d_1}(a_1,k_1+1))\circ_{k_1+2-j}e_L\right)\circ_ix_{d_2}(a_2,k_2)\otimes h^{-d}, \end{align}\tag{55}\] where \[\maltese_{ij}^d=(i-1)(k_2-1)+(k_1-1)(|y_d|+k_2)+|y_d|+k_2(j-1). \nonumber\] This gives a bilinear operation on \(C_\ast^{S^1}\), which is in general not a Lie bracket. However, \(\{\cdot,\cdot\}\) becomes an odd Lie bracket after passing to Connes’ complex, which is the quotient \[\label{eq:Connes} C_\ast^\lambda:=C_\ast^\mathrm{nd}/\mathrm{im}(1-t),\tag{56}\] where \(t:C_\ast^\mathrm{nd}\rightarrow C_\ast^\mathrm{nd}\) is the cyclic operator defined by \(t_k(x):=(-1)^k\tau_k(x)\) for \(x\in C_\ast(k)\). Using the cocyclic identities (52 ) and (53 ), one can check that the differential \(\partial^\mathrm{tot}\) on \(C_\ast^\mathrm{nd}\) descends to one on \(C_\ast^\lambda\), which we will still denote by \(\partial^\mathrm{tot}\). It follows from [11], Lemma 23 that the natural projection \[\label{eq:proj} \left(C_\ast^\mathrm{nd}\otimes_\mathbb{R}\mathbb{R}(\!(h)\!)/u\mathbb{R}[\![h]\!],\partial^\mathrm{tot}+h\delta_\mathrm{cyc}\right)\rightarrow(C_\ast^\lambda,\partial^\mathrm{tot})\tag{57}\] is a quasi-isomorphism. For a cochain \(\tilde{x}\in C_\ast^{S^1}\), denote its image under the projection (57 ) by \(\underline{x}\in C_\ast^\lambda\). Then the string bracket is defined as \[\begin{align} \{\cdot,\cdot\}:C_i^\lambda\otimes C_j^\lambda&\rightarrow C_{i+j+1}^\lambda, \\ \underline{x}\otimes\underline{y}&\mapsto\underline{\left\{\tilde{x},\tilde{y}\right\}}. \nonumber \end{align}\] This is independent of the choices of the lifts \(\tilde{x}\) and \(\tilde{y}\) of \(\underline{x}\) and \(\underline{y}\).

Since the odd Lie bracket \(\{\cdot,\cdot\}\) is compatible with the decomposition of \(C_\ast^\lambda\) according to \((a,k)\in H_1(L;\mathbb{Z})\times\mathbb{Z}_{\geq0}\), it naturally extends to the completion \(\widehat{C}_\ast^\lambda\) of \(C_\ast^\lambda\) with respect to the action filtration induced from the one on \(C_\ast^\mathrm{nd}\) defined by (45 ), and equips it with the structure of a dg Lie algebra \(\left(\widehat{C}_\ast^\lambda,\partial^\mathrm{tot},\{\cdot,\cdot\}\right)\) of degree \(1\).

There is a parallel story in the relative case, where one can construct a dg Lie algebra of degree \(1\) on the Connes’ complex \(\overline{C}_\ast^\lambda=\overline{C}_\ast^\mathrm{nd}/\mathrm{im}(1-\bar{t})\) and its completion. The complexes \(C_\ast^\lambda\) and \(\overline{C}_\ast^\lambda\) are related by the chain maps \[\label{eq:ie} \underline{i}:C_\ast^\lambda\rightarrow\overline{C}_\ast^\lambda,\textrm{ }\underline{e}_\pm:\overline{C}_\ast^\lambda\rightarrow C_\ast^\lambda\tag{58}\] induced by (42 ), (40 ) and (41 ), respectively, which satisfy \(\underline{e}_+\circ\underline{i}=\underline{e}_-\circ\underline{i}=\mathrm{id}_C\), where \(\mathrm{id}_C\) is the identity map on \(C_\ast^\lambda\), and are compatible with the odd Lie brackets. The chain maps (58 ) on Connes’ complexes can also be lifted to maps \[\tilde{i}:C_\ast^{S^1}\rightarrow\overline{C}_\ast^{S^1},\textrm{ }\tilde{e}_\pm:\overline{C}_\ast^{S^1}\rightarrow C_\ast^{S^1} \nonumber\] on the corresponding \(S^1\)-equivariant complexes such that \(\tilde{e}_+\circ\tilde{i}=\tilde{e}_-\circ\tilde{i}=\mathrm{id}_C\). Here, \(\mathrm{id}_C\) denotes the identity map on \(C_\ast^{S^1}\). We refer to [11], Section 3.2 for details.

6 Proof of Theorem 9↩︎

This section is devoted to the proof of Theorem 9. Using the chain model introduced in Section 5, we first write down its chain level statement (Theorem 16) in Section 6.1. Then, we define the required chains by pushing forward the virtual fundamental chains of the moduli spaces introduced in Section 4. This is done in Section 6.2. Finally, we analyze the boundary strata of these moduli spaces to prove Theorem 16 in Section 6.3.

6.1 Chain level statements↩︎

Let \(L\subset\mathrm{int}(X)\) be a closed Lagrangian submanifold in the interior of a Liouville domain \(X\) with \(c_1(X)=0\) that is oriented and \(\mathit{Spin}\). Recall that given this data, we have constructed a dg Lie algebra \(\left(\widehat{C}_\ast^\lambda,\partial^\mathrm{tot},\{\cdot,\cdot\}\right)\) of degree \(1\) in Section 5, where \(C_\ast^\lambda\) is a quotient complex of \(C_\ast^{S^1}\), and the latter complex carries a chain map \[\label{eq:B} \begin{align} B:C_\ast^{S^1}(a,k+1)&\rightarrow C_{\ast+1}^\mathrm{nd}(a,k),\\ B(\tilde{x})(a,k)&:=\sum_{j=1}^{k+1}(-1)^{|\tilde{x}|+k(j-1)}\tau_{k+1}^j(x_0(a,k+1))\circ_{k+2-j}e_L, \end{align}\tag{59}\] which induces the marking map \(H_\ast^{S^1}(\mathcal{L}L;\mathbb{R})\rightarrow H_{\ast+1}(\mathcal{L}L;\mathbb{R})\) in string topology [31] after passing to the homology (of the total complex) and composing with the isomorphisms (37 ) and (?? ).

The aim of this subsection is to show that Theorem 9 is a consequence of the following theorem.

To show that Theorem 16 implies Theorem 9, first note that Theorem 16, (i) and (iii) imply that \[\underline{x}(0):=\sum_{k=3}^\infty\underline{x}(0,k)\in\widehat{C}_{-2}^\lambda \nonumber\] is still a Maurer-Cartan element for the odd dg Lie algebra \(\left(\widehat{C}_\ast^\lambda,\partial^\mathrm{tot},\{\cdot,\cdot\}\right)\), so we can use it to deform the differential \(\partial^\mathrm{tot}\). Define \[C_\ast^\lambda(a):=\prod_{k=0}^\infty C_\ast^\lambda(a,k), \nonumber\] which carries the deformed differential \(\partial^\mathrm{tot}_{\underline{x}(0)}:C_\ast^\lambda(a)\rightarrow C_{\ast-1}^\lambda(a)\) defined as \[\partial_{\underline{x}(0)}^\mathrm{tot}(\underline{w}):=\partial^\mathrm{tot}(\underline{w})-\left\{\underline{x}(0),\underline{w}\right\} \nonumber\] for any \(\underline{w}\in C_\ast^\lambda(a)\). It follows from [11], Lemma 30 that the deformation of the homology \(H_\ast\left(C_\ast^\lambda(a),\partial^\mathrm{tot}\right)\) induced by the “low energy" Maurer-Carten element \(\underline{x}(0)\) is trivial, i.e. we have an isomorphism \[\label{eq:iso1} H_\ast\left(C_\ast^\lambda(a),\partial^\mathrm{tot}_{\underline{x}(0)}\right)\cong H_{\ast+n+\mu(a)-1}^{S^1}(\mathcal{L}(a)L;\mathbb{R}).\qquad{(22)}\] On the other hand, consider the”high energy" part of the Maurer-Cartan element \(\underline{x}^+:=\underline{x}-\underline{x}(0)\), direct computations yield the following identities \[\label{eq:43} \partial^\mathrm{tot}_{\underline{x}(0)}(\underline{x}^+)-\frac{1}{2}\left\{\underline{x}^+,\underline{x}^+\right\}=0,\textrm{ }\partial^\mathrm{tot}_{\underline{x}(0)}(\underline{y})-\left\{\underline{x}^+,\underline{y}\right\}=\underline{z}.\tag{60}\]

To overcome the technical difficulties of simultaneous perturbations of infinitely many Kuranishi maps (cf. [9], Section 6), we will translate Theorem 16 into a statement involving finite energy chains, i.e. chains in the uncompleted complex \(C_\ast^\lambda\). From now on, we choose \(\varepsilon>0\) in the statement of Theorem 16 so that \(2\varepsilon\) is less than the minimal symplectic area of \(J\)-holomorphic discs with boundary on \(L\subset X\), see Section 4. Define \[\mathcal{F}^mC_\ast^{S^1}:=\bigoplus_{\substack{(a,k)\in H_1(L;\mathbb{Z})\times\mathbb{Z}_{\geq0}\\ \lambda(a)\geq\varepsilon(m+1-k)}}C_\ast^{S^1}(a,k), \nonumber\] which induces a similar filtration \(\mathcal{F}^\bullet\) on its quotient complex \(C_\ast^\lambda\). It is clear from the definition that the differential \(\partial^\mathrm{tot}\) and the Lie bracket \(\{\cdot,\cdot\}\) preserve the filtration \(\mathcal{F}^\bullet\) on \(C_\ast^\lambda\), i.e. \(\partial^\mathrm{tot}\mathcal{F}^m\subset\mathcal{F}^m\) and \(\left\{\mathcal{F}^m,\mathcal{F}^{m'}\right\}\subset\mathcal{F}^{m+m'}\). In the same way, one can define a filtration \(\overline{\mathcal{F}}^\bullet\) on the relative chain complexes \(\overline{C}_\ast^{S^1}\) and \(\overline{C}_\ast^\lambda\). We will define the chains \(\underline{x},\underline{y},\underline{z}\in\widehat{C}_\ast^\lambda\) in the statement of Theorem 16 as limits of sequences of finite energy chains in \(C_\ast^\lambda\) with respect to the filtration \(\mathcal{F}^\bullet\) (and similarly for the corresponding relative chains \(\underline{\bar{x}},\underline{\bar{y}},\underline{\bar{z}}\in\widehat{\overline{C}}_\ast^\lambda\)).

Note that a priori, the completions of \(C_\ast^{S^1}\) (resp. \(C_\ast^\lambda\)) with respect to the filtration \(\mathcal{F}^m\) defined above can be different from \(\widehat{C}_\ast^{S^1}\) (resp. \(\widehat{C}_\ast^\lambda\)) defined using the action filtration \(F^\Xi\) (cf. (45 )), hence the different notations. However, following a strategy of [10], we will show that the limits \(\underline{x}\), \(\underline{y}\) and \(\underline{z}\) actually lie in \(\widehat{C}_\ast^\lambda\), see Proposition 19 below. We now give the finite energy version of the chain level statement of Theorem 9.

We will show that Theorem 18 implies Theorem 16. We start with the following generalization of [11], Lemma 35.

6.2 Defining the chains↩︎

In order to define de Rham chains on our models \(\{\mathcal{L}_{k+1}\}_{k\geq0}\) of the free loop space of \(L\) using the moduli spaces introduced in Section 4, we need to have strongly smooth maps from these moduli spaces to \(\{\mathcal{L}_{k+1}\}_{k\geq0}\).

For \(\bar{a}\in\pi_2(X,L)\) and \((T,B)\in\mathcal{T}(k+1,\bar{a})\), the interior evaluation map (26 ) gives rise to a smooth map \[\mathrm{ev}_\mathrm{int}^P:\prod_{v\in V_{\mathrm{int}}(T)}P\times\mathcal{L}_{k_v+1}(\partial B(v))\rightarrow\prod_{e\in E_{\mathrm{int}}(T)}(P\times L)^2 \nonumber\] on our finite-dimensional models of the free loop space. Using the concatenation map (32 ), which is also smooth, we obtain a smooth map \[\label{eq:ev-con} \left(\prod_{e\in E_{\mathrm{int}}(T)}P\times L\right)\textrm{ }{{}_\Delta\times_{\mathrm{ev}_\mathrm{int}^P}}\textrm{ }\left(\prod_{v\in V_{\mathrm{int}}(T)}P\times\mathcal{L}_{k_v+1}(\partial B(v))\right)\rightarrow P\times\mathcal{L}_{k+1}(a).\tag{63}\] Similarly, we have a smooth map \[\label{eq:ev-con1} \left(\prod_{e\in E_\mathrm{int}(T)}P\times L\right){{}_\Delta\times_{\mathrm{ev}_\mathrm{int}^P}}\left(\prod_{v\in V_\mathrm{int}(T)}P\times\mathcal{L}_{k_v+1}\left(\partial \ring{B}(v)\right)\right)\rightarrow P\times\mathcal{L}_{k+1}(a)\tag{64}\] for any \(\ring{\bar{a}}\in\pi_2(X,x,L)\) and \((T,\ring{B},v_0)\in\mathcal{T}(k+1,\ring{\bar{a}})\).

Using Wang’s model \(\{\mathcal{L}_{k+1}\}_{k\geq0}\), it is easy to prove the following.

The following theorem is a variant of [9], Theorem 7.2, see also [11], Appendix A.2 for details.

Let \(X\) be a Liouville domain with \(c_1(X)=0\) and \(C_d^\mathrm{GH}(X)<\infty\) for some \(d\in\mathbb{N}\). It follows that there exists a cochain \(\tilde{\beta}=\sum_{l=0}^\infty\beta_{l}\otimes u^{-d-l+1}\in\mathit{SC}_{S^1}^{-2d+1}(X)\), where \(\beta_{l}\in\mathit{SC}^{2l-1}(X)\) and only finitely many cochains \(\beta_l\) are non-vanishing, such that \[\label{eq:cl} \partial^{S^1}(\tilde{\beta})=\left(\partial(\beta_0)+\sum_{l=1}^\infty\delta_l(\beta_l)\right)\otimes u^{-d+1}=e_X\otimes u^{-d+1}.\tag{65}\] Note that the terms in \(\partial^{S^1}(\tilde{\beta})\) with different powers of \(u\) necessarily vanish due to the form of the right-hand side of (65 ), which gives the first equality above.

Fix \(a\in H_1(L;\mathbb{Z})\). For a closed Lagrangian submanifold \(L\subset\mathrm{int}(X)\) that is oriented and \(\mathit{Spin}\), choose \(a_l\in H_1(L;\mathbb{Z})\) for each \(l\in\mathbb{Z}_{\geq0}\) such that \(a_l=0\) when \(\beta_l=0\) and \(\sum_{l=0}^\infty a_l=a\). For \(\ring{\bar{a}}_l\in\pi_2(X,\beta_l,L)\) with \(\partial\ring{\bar{a}}_l=a_l\), and \(P\in\left\{\{m\},[m,m+1]\right\}\) for some \(m\in\mathbb{Z}_{\geq0}\), consider the moduli spaces \[{}_l\mathcal{R}_{k+1}^1(\beta_l,L,\ring{\bar{a}}_l;P),\textrm{ }l\in\mathbb{Z}_{\geq0}.\] Let \(\ring{\bar{a}}=\sum_{l=0}^\infty\ring{\bar{a}}_{l-1}\), it follows that \(\partial\ring{\bar{a}}=a\).

Recall that we have the following data for every \(k,m,l\in\mathbb{Z}_{\geq0}\) and \(P\in\{\{m\},[m,m+1]\}\).

• Compact admissible K-spaces, where \(l\geq1\) in (69 ) and (70 ), \(l\geq2\) in (71 ): \[\label{eq:d} \overline{\mathcal{R}}_{k+1}(L,\bar{a};P),\textrm{ }\lambda(a)<(m+1-k)\varepsilon,\tag{66}\] \[\label{eq:theta} \overline{\mathcal{R}}_{k+2,\vartheta}(L,\bar{a};P),\textrm{ }\lambda(a)<(m+1-k)\varepsilon,\tag{67}\] \[\label{eq:CG} {}_l\overline{\mathcal{R}}_{k+1}^1(\beta_{l},L,\ring{\bar{a}}_{l};P),\textrm{ }\lambda(a)<(m-k-U)\varepsilon,\tag{68}\] \[\label{eq:S1} {}_{l-1}\overline{\mathcal{R}}^{S^1}_{k+1}(\beta_{l},L,\ring{\bar{a}}_{l};P),\textrm{ }\lambda(a)<(m-k-U)\varepsilon,\tag{69}\] \[\label{eq:tau} {}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1(\beta_{l},L,\ring{\bar{a}}_{l};P),\textrm{ }0\leq i\leq k,\textrm{ }\lambda(a)<(m-k-U)\varepsilon,\tag{70}\] \[\label{eq:jj} {}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1(\beta_{l},L,\ring{\bar{a}}_{l};P),\textrm{ }1\leq j\leq l-1,\textrm{ }\lambda(a)<(m-k-U)\varepsilon,\tag{71}\] and admissible CF-perturbations on these moduli spaces.

  Moreover, Kuranishi structures and CF-perturbations on (66 ), (67 ), (68 ), (69 ) and (71 ) are \(\mathbb{Z}_{k+1}\)-invariant with respect to the cyclic permutation of the boundary marked points \(z_0\cdots,z_k\). For the moduli spaces (70 ), we require that \(\mathbb{Z}_{k+1}\) acts transitively on the set \(\left\{\left({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1(\beta_l,L,\ring{\bar{a}}_l;P),\widehat{\mathcal{S}}_i^\theta\right)\right\}_{i=0}^k\) of K-spaces together with their admissible CF-perturbations.

• Admissible maps \[\label{eq:ev1} \mathrm{Ev}^\mathcal{R}:\overline{\mathcal{R}}_{k+1}(L,\bar{a};P)\rightarrow P\times\mathcal{L}_{k+1}(a),\tag{72}\] \[\label{eq:ev2} \mathrm{Ev}_\vartheta^\mathcal{R}:\overline{\mathcal{R}}_{k+2,\vartheta}(L,\bar{a};P)\rightarrow P\times\mathcal{L}_{k+2}(a),\tag{73}\] \[\label{eq:ev3} {}_l\mathrm{Ev}^\mathcal{R}:{}_l\overline{\mathcal{R}}_{k+1}^1(\beta_{l},L,\ring{\bar{a}}_{l};P)\rightarrow P\times\mathcal{L}_{k+1}(a_{l}),\tag{74}\] \[\label{eq:ev4} {}_{l-1}\mathrm{Ev}^{S^1}: {}_{l-1}\overline{\mathcal{R}}^{S^1}_{k+1}(\beta_{l},L,\ring{\bar{a}}_{l};P)\rightarrow P\times\mathcal{L}_{k+1}(a_{l}),\tag{75}\] \[\label{eq:ev5} {}_{l-1}\mathrm{Ev}_i:{}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1(\beta_{l},L,\ring{\bar{a}}_l;P)\rightarrow P\times\mathcal{L}_{k+1}(a_{l}),\tag{76}\] \[\label{eq:ev6} {}_l^{j,j+1}\mathrm{Ev}^\mathcal{R}:{}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1(\beta_{l},L,\ring{\bar{a}}_{l};P)\rightarrow P\times\mathcal{L}_{k+1}(a_{l}),\tag{77}\] such that their compositions with \(\mathrm{id}_P\times\mathrm{ev}_0^\mathcal{L}\) are corner stratified strong submersions with respect to the CF-perturbations fixed in (i).

• Isomorphisms of admissible K-spaces (?? ), (?? ), (28 ), (29 ) that are compatible with CF-perturbations fixed in (i) and the evaluation maps in (ii).

By Theorem 20, we obtain the following (relative) de Rham chains \[\label{eq:c1} x_m(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m+1-k)\varepsilon}}(-1)^{n+1}\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+1}\left(L,\bar{a};\{m\}\right)\right)\in C_{-1},\tag{78}\] \[\label{eq:c2} \bar{x}_m(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m+1-k)\varepsilon}}(-1)^{k+1}\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+1}\left(L,\bar{a};[m,m+1]\right)\right)\in\overline{C}_{-1},\tag{79}\] \[\label{eq:c3} x_{m,0}(k+1):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m+1-k)\varepsilon}}(-1)^{n+1}\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+2,\vartheta}\left(L,\bar{a};\{m\}\right)\right)\in C_{-2},\tag{80}\] \[\label{eq:c4} \bar{x}_{m,0}(k+1):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m+1-k)\varepsilon}}(-1)^k\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+2,\vartheta}\left(L,\bar{a};[m,m+1]\right)\right)\in\overline{C}_{-2},\tag{81}\] \[\label{eq:c5} y_{m,0}(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-U-k)\varepsilon}}\sum_{l=0}^\infty(-1)^{n+k+1}\mathrm{Ev}_\ast\left({}_{l}\overline{\mathcal{R}}_{k+1}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};\{m\}\right)\right)\in C_2,\tag{82}\] \[\label{eq:c6} y_{m,1}(k+1):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-U-k-1)\varepsilon}}\sum_{l=1}^\infty(-1)^{n+k+1}\mathrm{Ev}_\ast\left({}_{l-1}\overline{\mathcal{R}}_{k+2}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};\{m\}\right)\right)\in C_0,\tag{83}\] \[\label{eq:c7} \bar{y}_{m,0}(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-U-k)\varepsilon}}\sum_{l=0}^\infty\mathrm{Ev}_\ast\left({}_l\overline{\mathcal{R}}_{k+1}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\right)\in\overline{C}_2,\tag{84}\] \[\label{eq:c8} \bar{y}_{m,1}(k+1):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-U-k-1)\varepsilon}}\sum_{l=1}^\infty\mathrm{Ev}_\ast\left({}_{l-1}\overline{\mathcal{R}}_{k+2}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\right)\in\overline{C}_0,\tag{85}\] \[\label{eq:c9} z_{m}(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-1-k)\varepsilon}}(-1)^{n+k+1}\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+1}^1\left(e_X,L,\bar{a};\{m\}\right)\right)\in C_1,\tag{86}\] \[\label{eq:c10} \bar{z}_{m}(k):=-\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-1-k)\varepsilon}}\mathrm{Ev}_\ast\left(\overline{\mathcal{R}}_{k+1}^1\left(e_X,L,\bar{a};[m,m+1]\right)\right)\in\overline{C}_1.\tag{87}\] Under the natural projections \(C_\ast\rightarrow C_\ast^\mathrm{nd}\) and \(\overline{C}_\ast\rightarrow\overline{C}_\ast^\mathrm{nd}\), we get non-degenerate chains which will still denoted by \(x_m,\cdots,\bar{z}_m\) by abuse of notations. Note that in the definitions of all the de Rham chains above, we have omitted the differential form \(\hat{\omega}\), which is always taken to be \(1\), and the CF-perturbations, which are fixed earlier. The superscripts and subscripts for evaluation maps on various different moduli spaces are also omitted to simplify the notations.

Using the chains defined above, we can then form the following \(S^1\)-equivariant de Rham chains: \[\label{eq:t1} \tilde{x}_m(k):=x_{m,0}(k)\otimes1\in C_{-2}^{S^1},\tag{88}\] \[\label{eq:t2} \bar{\tilde{x}}_m(k):=\bar{x}_{m,0}(k)\otimes1\in\overline{C}_{-2}^{S^1},\tag{89}\] \[\label{eq:t3} \tilde{y}_m(k,k+1):=y_{m,0}(k)\otimes h^{-d+1}+y_{m,1}(k+1)\otimes h^{-d}\in C_{2d}^{S^1},\tag{90}\] \[\label{eq:t4} \bar{\tilde{y}}_m(k,k+1):=\bar{y}_{m,0}(k)\otimes h^{-d+1}+\bar{y}_{m,1}(k+1)\otimes h^{-d}\in \overline{C}_{2d}^{S^1},\tag{91}\] \[\label{eq:t5} \tilde{z}_m(k):=z_m(k)\otimes h^{-d+1}\in C_{2d-1}^{S^1},\tag{92}\] \[\label{eq:t6} \bar{\tilde{z}}_m(k):=\bar{z}_m(k)\otimes h^{-d+1}\in\overline{C}_{2d-1}^{S^1}.\tag{93}\] Note that for the \(y\)-type chains defined above, the \(k\) and \((k+1)\)-components are mixed. This will become natural once the \(S^1\)-equivariant differentials on \(C_\ast^{S^1}\) and \(\overline{C}_\ast^{S^1}\) are applied to these chains, as the outcome will eventually lie in the \(k\)-component.

Besides these chains that will appear in the statement of Theorem 18, there are also some auxiliary chains that we need to deal with, which are defined as follows. \[\label{eq:c11} \bar{y}^{j,j+1}_m(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-k-U)\varepsilon}}\sum_{l=2}^\infty\mathrm{Ev}_\ast\left({}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\right)\in\overline{C}_1,\tag{94}\] \[\label{eq:c12} \bar{y}_m^{S^1}(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-k-U)\varepsilon}}\sum_{l=1}^\infty\mathrm{Ev}_\ast\left({}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\right)\in\overline{C}_1.\tag{95}\] \[\label{eq:c13} \bar{y}_{m,1}^i(k):=\sum_{\substack{a\in H_1(L;\mathbb{Z})\\ \lambda(a)<(m-k-U)\varepsilon}}\sum_{l=1}^\infty\mathrm{Ev}_\ast\left({}_{l-1}\overline{\mathcal{R}}_{k+1,\tau_i}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\right)\in\overline{C}_1.\tag{96}\]

6.3 Completion of proof↩︎

By our discussions in Section 6.1, in order to prove Theorem 9, it suffices to prove Theorem 18.

Recall from Theorem 13 that the codimension one boundary of the admissible K-space \({}_{l}\overline{\mathcal{R}}_{k+1}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\) contain the following strata: \[\label{eq:st1} \bigsqcup_{0\leq j\leq l}{}_j\overline{\mathcal{M}}\left(\beta_{l},y_{j,l};[m,m+1]\right)\times{}_{l-j}\overline{\mathcal{R}}_{k+1}^1\left(y_{j,l},L,\ring{\bar{a}}_{l};[m,m+1]\right)\tag{97}\] where the \(y_{j,l}\)’s are 1-periodic orbits of \(X_{H_t}\), \[\label{eq:st2} \bigsqcup_{1\leq j\leq l-1}{}_l^{j,j+1}\overline{\mathcal{R}}_{k+1}^1\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right),\tag{98}\] and \[\label{eq:st3} {}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^1}\left(\beta_{l},L,\ring{\bar{a}}_{l};[m,m+1]\right).\tag{99}\] By analyzing these strata we obtain the following.

7 Extremal Lagrangians in spectrally convex Liouville domains↩︎

In this section, we prove Theorem 10, which asserts that extremal aspherical Lagrangians in the sense of Definition 4 lie in the boundary \(\partial X\) provided that the Gutt–Hutchings capacities of \(X\) decay fast enough in the sense of Definition 5.

8 Examples↩︎

In this section we discuss the examples (beyond star-shaped domains in \(\mathbb{C}^n\)) where the results in this paper are applicable. In Section 8.1, we explore the relations between the finiteness of Gutt–Hutchings capacities and the existence of higher dilations introduced by Zhao [26], which provides examples of Liouville domains for which Corollary 2 and Theorem 9 hold. In Section 8.2, we explicitly compute the Lagrangian capacities for some Liouville domains admitting higher dilations.

8.1 Higher dilations and Gutt–Hutchings capacities↩︎

Let \(X\) be a Liouville domain with \(c_1(X)=0\). Its completed periodic symplectic symplectic cohomology, denoted \(\widehat{\mathit{PSH}}^\ast(X)\), is the cohomology of the complex \[\label{eq:PSC} \left(\mathit{SC}^\ast(X)\otimes_\mathbb{R}\mathbb{R}(\!(u)\!),\partial^{S^1}:=\partial+u\delta_1+u^2\delta_2+\cdots\right), \nonumber\tag{112}\] where \(u\) is a formal variable of degree \(2\). Just like in the case of ordinary symplectic cohomology, there is a version of the PSS map \[\label{eq:PSS} \phi_X:H^\ast(\widehat{X};\mathbb{R})_{\otimes_\mathbb{R}}\mathbb{R}(\!(u)\!)\rightarrow\widehat{\mathit{PSH}}^\ast(X),\tag{113}\] where \(\widehat{X}\) is the Liouville manifold obtained by completing \(X\) with the cylindrical end \([1,\infty)\times\partial X\).

The following notion is introduced by J. Zhao in her thesis.

The existence of a higher dilation is related to the finiteness of the Gutt–Hutchings capacities \(C_d^\mathrm{GH}(X)\), \(d\in\mathbb{N}\) in the following sense.

It is clear that every Liouville domain \(X\) with \(\mathit{SH}^\ast(X)=0\) admits a higher dilation. This is the case of star-shaped domains in \(\mathbb{C}^n\), where most of the studies of symplectic capacities occur. Nevertheless, there are more interesting examples of Liouville domains admitting higher dilations. Let \(X=D^\ast Q\) be the unit disc cotangent bundle of some closed manifold \(Q\), it is proved in [26], Corollary 5.2.5 that \(X\) admits a higher dilation if and only if \(Q\) is rationally inessential, i.e. the fundamental class \([Q]\in H_n(Q;\mathbb{R})\) vanishes under the map \(H_n(Q;\mathbb{R})\rightarrow H_n\left(B\pi_1(Q);\mathbb{R}\right)\) induced by the classifying map. In particular, any disc cotangent bundle of a simply-connected closed manifold admits a higher-dilation, therefore having \(C_d^\mathrm{GH}(X)<\infty\) for any \(d\in\mathbb{N}\) by Proposition 21.

In another direction, Zhao proved in [26], Proposition 5.3.2 that if the Liouville manifold \(\widehat{X}\) admits a symplectic Lefschetz fibration \(\pi:\widehat{X}\rightarrow\mathbb{C}\) whose fibers admit higher dilations, then the total space \(\widehat{X}\) also admits a higher-dilation. This implies, for example, any \((A_m)\) Milnor fiber of dimension \(2n\geq6\) admits a higher dilation, since it admits a Lefschetz fibration with fiber symplectomorphic to \(T^\ast S^{n-1}\). Zhao’s argument can be extended to prove the following.

As concrete examples, consider the affine \((A_m)\) Milnor fibers \[\widehat{X}=\left\{(z_1,\cdots,z_n,z_{n+1})\in\mathbb{C}^n\times\mathbb{C}^\ast\left\vert z_1^2+\cdots+z_n^2=z_{n+1}^{m+1}-1\right.\right\}, \nonumber\] and let \(X\) be any Liouville domain whose completion gives \(\widehat{X}\). It follows from Proposition 23 that \(C_d^\mathrm{GH}(X)<\infty\) for all \(d\in\mathbb{N}\). These examples are interesting since \(\pi_1(X)\neq0\).

8.2 Lagrangian capacities of disc cotangent bundles↩︎

As examples of our discussions in Section 8.1, we explicitly compute the Lagrangian capacities \(C^\mathrm{CM}(X)\) and \(C^\mathrm{AL}(X)\) for the unit disc cotangent bundles over \(S^2\), \(S^3\), \(\mathbb{RP}^2\) and Zoll spheres. As we will point out, the computational methods presented here actually extend to many other Liouville domains in dimensions \(4\) and \(6\).

8.2.1 \(S^2\) and \(\mathbb{RP}^2\)↩︎

We first consider the case of \((D^\ast S^2,\lambda_\mathrm{can})\), where \(\lambda_\mathrm{can}\) is the canonical Liouville form on the cotangent bundle. The contact boundary \(S^\ast S^2:=\partial D^\ast S^2\) is \(\mathbb{RP}^3\) equipped with the contact form \(4\alpha_\mathrm{std}\), where \(\alpha_\mathrm{std}\) denotes the standard contact form. Since the Reeb flow on \((\mathbb{RP}^3,4\alpha_\mathrm{std})\) is periodic, we can explicitly write down the orbits, see for example [36], Appendix A. In particular, all simple orbits \(\gamma\) have symplectic action \[\mathcal{A}(\gamma)=4\int_\gamma\alpha_\mathrm{std}=2\pi.\]

By identifying \(T^\ast S^2\) with the affine quadric surface \[\widehat{X}=\left\{(z_1,z_2,z_3)|z_1^2+z_2^2+z_3^2=1\right\}\] equipped with the restriction of the standard Liouville form \(\lambda_\mathrm{std}\) on \(\mathbb{C}^3\), we get a standard Lefschetz fibration \(p_X:\widehat{X}\rightarrow\mathbb{C}\) by projecting to the \(z_1\) coordinate plane. Given any embedded closed curve \(\sigma\subset\mathbb{C}\) away from the critical values \(\pm1\) of \(p_X\), we get a Lagrangian torus \(T_\sigma\subset T^\ast S^2\) by parallel transporting the vanishing cycle. It is not hard to show that if we choose \(\sigma\) to be a curve enclosing \(\pm1\), then \(T_\sigma\subset T^\ast S^2\) is a monotone Lagrangian torus. See for example [37], Section 2.2.

To find the monotonicity constant of \(T_\sigma\) (or equivalently, the symplectic area of a Maslov \(2\) disc in \(T^\ast S^2\) with boundary on \(T_\sigma\)), consider a parametrized curve \[c(t)=\left(\sigma(t),n(t)e^{2\pi ia(t)},n(t)e^{2\pi ia(t)}\right),\] where \(t\in[0,1]\), \(n(t)>0\), \(a(t)\) is a real-valued function satisfying \(2n(t)^2e^{4\pi ia(t)}=1-\sigma(t)^2\). Then \(c(t)\subset T_\sigma\) is the lift of \(\sigma\subset\mathbb{C}\), and \(p_X|_{c(t)}\) is a degree \(1\) map. The symplectic area of any disc \(u:(D,\partial D)\rightarrow(T^\ast S^2,T_\sigma)\) with boundary on \(c(t)\) can be expressed as \[\label{eq:area1} \int_Du^\ast d\lambda_\mathrm{can}=\frac{i}{4}\int_\sigma zd\bar{z}-\bar{z}dz+2\pi\int_0^1a'(t)n(t)^2dt,\tag{115}\] where the first term is the area enclosed by \(\sigma\subset\mathbb{C}\), and the second term records the sum of areas in the other two coordinate projections.

Here we take \(\sigma\) to be the ellipse with foci at \(\pm1\) and eccentricity \(\sqrt{\frac{\sqrt{5}-1}{2}}\), see Figure 4. In this case, \(T_\sigma\subset T^\ast S^2\) coincides with the Polterovich torus \(T_\mathrm{Pol}\) (cf. [38]), which is constructed as the geodesic flow of unit covectors over the point \((1,0,0)\in S^2\). See [37], Proposition 2.10. By construction, \(T_\mathrm{Pol}\) is contained in \(D^\ast S^2\). In fact, the restriction of the Lefschetz fibration \(p_X\) to \(D^\ast S^2\) maps it to the filled ellipse, and \(T_\mathrm{Pol}\) lies in the contact boundary \(S^\ast S^2\). Let \(u:(D,\partial D)\rightarrow(D^\ast S^2,T_\mathrm{Pol})\) be a \(J\)-holomorphic disc with boundary on the curve \(c(t)\subset T_\mathrm{Pol}\) for some compatible almost complex structure \(J\). Applying the formula (115 ) we obtain \[\label{eq:area2} \int_Du^\ast d\lambda_\mathrm{can}=\pi+\frac{\pi}{2}+\frac{\pi}{2}=2\pi,\tag{116}\] because the projections of \(c(t)\) in the \(z_2\) and \(z_3\) coordinate planes are the same ellipse \(\sigma(t)\) scaled by \(\frac{1}{\sqrt{2}}\).

Our analysis above shows that the Polterovich torus \(T_\mathrm{Pol}\subset D^\ast S^2\) is an extremal Lagrangian torus. In fact, we have the following.

Similar arguments can be used to compute the Lagrangian capacities for all \(4\)-dimensional \((A_m)\) Milnor fibers \[A_{m,r}:=\left\{\left.(z_1,z_2,z_3)\in\mathbb{C}^3\right\vert z_1^{m+1}+z_2^2+z_3^2=\delta,|z_1|^2+|z_2|^2+|z_3|^2\leq r\right\},\] where \(\delta>0\) is sufficiently small and \(r>\sqrt{\delta}\). However, in this dimension it is unclear whether they admit higher dilations (or even cyclic dilations) if \(m>1\), so there may not be a Reeb orbit on the contact boundary \(\partial A_{m,r}\cong L(m+1,m)\) (equipped with \(r(n+1)\) times the standard contact form) whose symplectic action coincides with the first Gutt–Hutchings capacity \(C_1^\mathrm{GH}(A_{m,r})\).

We then compute the Lagrangian capacities of \((D^\ast\mathbb{RP}^2,\bar{\lambda}_\mathrm{can})\), where \(\bar{\lambda}_\mathrm{can}\) is the induced Liouville form under the quotient map \[\label{eq:quo} q:D^\ast S^2\rightarrow D^\ast\mathbb{RP}^2.\tag{119}\] First note that since \(\mathbb{RP}^2\) is rationally inessential, \(D^\ast\mathbb{RP}^2\) admits a higher dilation, therefore having finite Gutt-Hutchings capacities \(C_d^\mathrm{GH}(D^\ast\mathbb{RP}^2)\) for \(d\in\mathbb{N}\). The simple Reeb orbits on the contact boundary \(S^\ast\mathbb{RP}^2\), which is diffeomorphic to the lens space \(L(4,1)\) equipped with the contact form \(4\bar{\alpha}_\mathrm{std}\), have period \(\pi\), where \(\bar{\alpha}_\mathrm{std}\) is the standard contact form on \(L(4,1)\) coming from the quotient of \(S^3\). However, these Reeb orbits are not contractible, therefore cannot be the cocycle in the definition of \(C_1^\mathrm{GH}(D^\ast\mathbb{RP}^2)\) that kills the unit. Under the quotient map \(q\), the the Floer cylinder \(u:\mathbb{R}\times S^1\rightarrow D^\ast S^2\) with asymptotics \(\beta_1\) and the constant orbit \(e_X\) contributing to the equation \(\delta(\beta_1)=e_X-\partial(\beta_0)\) in the symplectic cochain complex \(\mathit{SC}^\ast(D^\ast\mathbb{RP}^2)\) now becomes a Floer cylinder in \(D^\ast\mathbb{RP}^2\) that is asymptotic to the double cover of a simple Reeb orbit on \(\left(L(4,1),4\bar{\alpha}_\mathrm{std}\right)\) at its positive puncture. It follows that \[\label{eq:C1} C_1^\mathrm{GH}(D^\ast\mathbb{RP}^2)=2\pi.\tag{120}\]

On the other hand, the Polterovich torus \(T_\mathrm{Pol}\subset D^\ast S^2\) descends to a Lagrangian torus \(T'_\mathrm{Pol}=q(T_\mathrm{Pol})\subset D^\ast\mathbb{RP}^2\) under the \(\mathbb{Z}_2\)-quotient.

Combining (120 ), Theorem 6 and Lemma 7, we obtain the following.

Note that the Lagrangian capacities for \(D^\ast S^2\) and \(D^\ast\mathbb{RP}^2\) coincide with their Gromov width, see [42], Theorem 1.1. Generalizing the example of \(D^\ast\mathbb{RP}^2\), we have the rational homology balls \(B_{p,q}\), where \(p>q>0\) and \((p,q)=1\), which arise as the quotients of the \((A_{p-1})\) Milnor fiber by the cyclic group \(\mathbb{Z}_p\). Assuming the \(B_{p,q}\)’s admit higher dilations³, then their Lagrangian capacities can be computed similarly as in the case of \(D^\ast\mathbb{RP}^2\).

8.2.2 Zoll spheres↩︎

Proposition 24 has an alternative proof based on the works [42], [43] on the symplectic embedding into disc cotangent bundles, which can be extended to compute the Lagrangian capacities of a more general classes of Liouville domains, namely unit disc cotangent bundles of Zoll spheres of resolution.

Let \(S\subset \mathbb{R}^3\) be a compact smooth surface of genus zero which is invariant under rotations about a fixed coordinate axis. We may write \(S\) in the form \[S=\left\{(\rho(x)\cos\theta,\rho(x)\sin\theta,x)\vert x\in [a,b],\; \theta\in \mathbb{R}/2\pi\mathbb{Z}\right\},\] where \(\rho:[a,b]\to[0,\infty)\) is a smooth function satisfying \(\rho(a)=\rho(b)=0\). We say that \(S\) is a Zoll sphere of revolution if all geodesics of the induced Riemannian metric are closed and have the same length.

8.2.3 \(3\)-dimensional lens spaces↩︎

Finally, we consider the higher-dimensional case and compute the Lagrangian capacity of \(D^\ast L(p,q)\), the unit disc cotangent bundle over the \(3\)-dimensional lens space \(L(p,q)\) equipped with the canonical Liouville form \(\lambda_\mathrm{can}\).

We start with the case of \(D^\ast S^3\). First observe that the minimal period of simple Reeb orbits on the contact boundary \(S^\ast S^3\cong S^2\times S^3\) is \(2\pi\). Moreover, it is known that \(D^\ast S^3\) admits a dilation and the dilation comes from a simple Reeb orbit, see [39], Example 2.5. Thus it follows that \[\label{eq:uppb} C^\mathrm{CM}(D^\ast S^3)\leq C^\mathrm{AL}(D^\ast S^3)\leq C_1^\mathrm{GH}(D^\ast S^3)\leq2\pi.\tag{123}\] The cotangent bundle \[T^\ast S^3=\left\{\left.(u,v)\in\mathbb{R}^4\times\mathbb{R}^4\right\vert |u|=1,\langle u,v\rangle=0\right\}\] with \(\lambda_\mathrm{can}=\sum_{i=1}^4v_idu_i\), where \(u=(u_1,u_2,u_3,u_4)\) and \(v=(v_1,v_2,v_3,v_4)\) can in fact be realized as the smooth affine \(3\)-fold \[\widehat{Y}=\left\{\left.(x_1,x_2,y_1,y_2,z)\in\mathbb{C}^5\right\vert x_1y_1=z+1,x_2y_2=z-1\right\}\] equipped with the restriction of the standard Liouville form \(\lambda_\mathrm{std}\) on \(\mathbb{C}^5\) up to a constant factor. To see that this is indeed the case, consider the change of variables \[\label{eq:wxy} w_1=\frac{x_1+y_1}{2\sqrt{2}},\textrm{ }w_2=\frac{x_1-y_1}{2\sqrt{2}i},\textrm{ }w_3=\frac{x_2+y_2}{2\sqrt{2}i},\textrm{ }w_4=\frac{y_2-x_2}{2\sqrt{2}},\tag{124}\] under which \(\widehat{Y}\subset\mathbb{C}^5\) is identified with the affine quadric \[\widehat{Z}=\left\{\left.(w_1,w_2,w_1,w_4)\in\mathbb{C}^4\right\vert w_1^2+w_2^2+w_3^2+w_4^2=1\right\}.\] An exact symplectomorphism \(\widehat{Z}\rightarrow T^\ast S^3\) is given by the map \[\begin{align} &(w_1,w_2,w_3,w_4)\mapsto \\ &\left(\frac{\left(\mathrm{Re}(w_1),\mathrm{Re}(w_2),\mathrm{Re}(w_3),\mathrm{Re}(w_4)\right)}{\sqrt{\sum_{k=1}^4\mathrm{Re}(w_k)^2}},\left(\mathrm{Im}(w_1),\mathrm{Im}(w_2),\mathrm{Im}(w_3),\mathrm{Im}(w_4)\right)\sqrt{\sum_{k=1}^4\mathrm{Re}(w_k)^2}\right). \nonumber \end{align}\] For the unit disc cotangent bundle \(D^\ast S^3\subset T^\ast S^3\), we have \(|v|=1\), so it follows that \[\label{eq:wuv} w_k=\sqrt{\frac{\sqrt{5}+1}{2}}u_k+i\sqrt{\frac{\sqrt{5}-1}{2}}v_k,\textrm{ }k=1,2,3,4.\tag{125}\] Combining with (124 ), we can write down an explicit symplectomorphism \(\widehat{Y}\rightarrow T^\ast S^3\), with \[x_1=\sqrt{\sqrt{5}+1}(u_1+iu_2)+i\sqrt{\sqrt{5}-1}(v_1+iv_2),\] \[y_1=\sqrt{\sqrt{5}+1}(u_1-iu_2)+i\sqrt{\sqrt{5}-1}(v_1-iv_2),\] \[x_2=\sqrt{\sqrt{5}+1}(iu_3-u_4)+\sqrt{\sqrt{5}-1}(iv_4-v_3),\] \[y_2=\sqrt{\sqrt{5}+1}(iu_3+u_4)+\sqrt{\sqrt{5}-1}(v_3+iv_4).\] In particular, \[\begin{align} z=\frac{x_1y_1+x_2y_2}{2}&=\frac{\sqrt{5}+1}{2}(u_1^2+u_2^2-u_3^2-u_4^2)-\frac{\sqrt{5}-1}{2}(v_1^2+v_2^2-v_3^2-v_4^2) \\ &+2i(u_1v_1+u_2v_2-u_3v_3-u_4v_4). \end{align}\] Using the facts \(|u|=1\), \(|v|=1\) and \(\langle u,v\rangle=0\), direct computation yields \[\label{eq:ell} \frac{\mathrm{Re}(z)^2}{5}+\frac{\mathrm{Im}(z)^2}{4}=1.\tag{126}\] In other words, under the projection \(p_Y:\widehat{Y}\rightarrow\mathbb{C}\) to the \(z\)-coordinate plane, the unit disc cotangent bundle \(D^\ast S^3\subset T^\ast S^3\) maps to the filled ellipse with foci \(\pm1\), major semiaxis \(\sqrt{5}\) and minor semiaxis \(2\), see Figure 5.

In a similar way, we can find the images of \(D^\ast S^3\) under the projections to the \(x_1\), \(y_1\), \(x_2\) and \(y_2\) coordinate planes, which turn out to be the same filled ellipse \[\label{eq:ell1} \frac{\mathrm{Re}(\zeta)^2}{2\sqrt{5}+4}+\frac{\mathrm{Im}(\zeta)^2}{2\sqrt{5}-4}\leq1,\tag{127}\] where \(\zeta\) stands for any of the variables \(x_1,y_1,x_2,y_2\).

The projection \(p_Y:\widehat{Y}\rightarrow\mathbb{C}\) to the \(z\)-coordinate plane is in fact a Morse-Bott fibration, with smooth fiber symplectomorphic to \(T^\ast T^2\), and two singular fibers at \(\pm1\) isomorphic to \((\mathbb{C}\vee\mathbb{C})\times\mathbb{C}^\ast\). If we fix a global basis \(\alpha,\beta\in H_1(T^2;\mathbb{Z})\), the vanishing cycles at \(-1\) and \(+1\) correspond to \(\alpha\) and \(\beta\), respectively. It follows from [45], Lemma 4.1 that the symplectic parallel transport is well-defined for \(p_Y\) away from the critical values. Take \(\sigma\subset\mathbb{C}\) to be a closed curve encircling \(\pm1\). Parallel transporting the zero section \(T^2\) in the smooth fibers of \(p_Y\) along \(\sigma\) we obtain a Lagrangian \(3\)-torus \(T_\sigma\subset T^\ast S^3\), which is actually monotone by [46], Lemma 2.12. We denote by \(T_\mathrm{Pol}\subset D^\ast S^3\) the monotone Lagrangian torus corresponding to the particular choice when \(\sigma\) is the ellipse (126 ), and call it the generalized Polterovich torus.

Lemma 8 shows that \(A_{\min}(T_\mathrm{Pol})=2\pi\). Combining with (123 ) we obtain the following.

It is clear that the generalized Polterovich torus \(T_\mathrm{Pol}\subset D^\ast S^3\) descends to a monotone Lagrangian torus \(T_\mathrm{Pol}'\subset D^\ast L(p,q)\) under the quotient of the \(\mathbb{Z}_p\)-action \[(x_1,y_1,x_2,y_2,z)\mapsto\left(e^{\frac{2\pi i}{p}}x_1,e^{-\frac{2\pi i}{p}}y_1,e^{\frac{2\pi iq}{p}}x_2,e^{-\frac{2\pi iq}{p}}y_2,z\right) \nonumber\] on the affine \(3\)-fold \(\widehat{Y}\cong T^\ast S^3\), where \((p,q)=1\). In fact, when \(q=1\), \(T_\mathrm{Pol}'\) can also be obtained by parallel transporting the zero section \(T^2\subset T^\ast T^2\) over the same ellipse (126 ) of a similar Morse-Bott fibration \(p_Y':T^\ast L(p,1)\rightarrow\mathbb{C}\). The critical values of \(p_Y'\) are still \(\pm1\), but the vanishing cycles are now given by \(\alpha\) and \(\alpha\pm p\beta\), respectively. By the same argument as in Lemma 7, one can show that \(A_{\min}(T_\mathrm{Pol}')=2\pi\).

On the other hand, the minimal Reeb period on the contact boundary \(S^\ast L(p,q)\cong L(p,q)\times S^2\) is \(\frac{2\pi}{p}\), but the simple Reeb orbits are not contractible. Instead, the Reeb orbit contributing to the first Gutt–Hutchings capacity is the \(p\)-th cover of a simple Reeb orbit, so we have \(C_1^\mathrm{GH}\left(D^\ast L(p,q)\right)=2\pi\), see [39], Example 3.14. Thus we have proved

More generally, one can try to compute the Lagrangian capacities of the double-bubble or even multi-bubble plumbings studied in [45] and [47]. These are Weinstein \(6\)-manifolds carrying similar Morse-Bott fibrations as \(T^\ast S^3\), therefore monotone Lagrangian tori can be constructed in the corresponding Weinstein domains by parallel transporting the \(T^2\) vanishing cycles. However, the existence of a higher-dilation is in general unclear for these Weinstein \(6\)-manifolds, despite the partial results obtained in [45], [48], [49].

Shah Faisal, Department of Mathematics, Uppsala University, 753 10 Uppsala, Sweden

E-mail address: shah.faisal@math.uu.se

Yin Li, Department of Mathematics, Uppsala University, 753 10 Uppsala, Sweden

E-mail address: yin.li@math.uu.se

References↩︎

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1. These discs exist for any convex \(d\lambda\)-compatible almost complex structure. However, their boundaries on \(L\) do not necessarily represent distinct classes in \(H_1(L;\mathbb{Z})\). For example, when \(L\) is the Chekanov–Schlenk torus [21] in \(\mathbb{C}^2\), all of these disks represent the same class in \(H_1(L;\mathbb{Z})\).↩︎

2. Recall our convention in (2 ).↩︎

3. Note however that \(2c_1(B_{p,q})\) is in general not \(0\), therefore \(\widehat{\mathit{PSH}}^\ast(B_{p,q})\) is not \(\mathbb{Z}\)-graded.↩︎