Non-displaceable Lagrangian links in four-manifolds

Let $\omega$ denote an area form on $S^2$. Consider the closed symplectic 4-manifold $M=(S^2\times S^2, A\omega \oplus a \omega)$ with $0<a<A$. We show that there are families of displaceable Lagrangian tori $L_{0,x},\, L_{1,x} \subset M$, for $x \in [0,1]$, such that the two-component link $L_{0,x} \cup L_{1,x}$ is non-displaceable for each $x$.

If L is not displaceable, then it is said to be (Hamiltonian) 'non-displaceable'. Understanding when a Lagrangian submanifold L is displaceable is a central question in symplectic topology; because of connections to dynamics and integrable systems, the case in which L is a Lagrangian torus is especially classical. A Lagrangian L is stably non-displaceable if L × S 1 ⊂ M × T * S 1 is non-displaceable, where S 1 ⊂ T * S 1 denotes the zero-section.
An equator S 1 eq ⊂ S 2 is manifestly non-displaceable, for area considerations. In higher dimensions, a sufficient way to prove that a Lagrangian L is non-displaceable is to show that the Floer cohomology HF(L, L) = 0. Since Floer cohomology behaves well under taking products, it follows that S 1 eq is stably non-displaceable, and that a product of equators in (S 2 × S 2 , Aω ⊕ A ω) is non-displaceable, for any areas A, A ∈ R >0 . Consider now two disjoint circles L 0 , L 1 ⊂ S 2 such that the complement of L 0 L 1 comprises two discs each of area B and a cylinder of area C. Area considerations again show that L 0 L 1 is nondisplaceable when C < 2B. In this case, the individual L i ⊂ S 2 are displaceable (by rotation of the sphere), hence have vanishing Floer cohomology, and therefore HF(L 0 L 1 , L 0 L 1 ) = 0 also vanishes. Underscoring this, Polterovich made the remarkable observation [Pol01] that L 0 L 1 is in fact stably displaceable, i.e. the Lagrangian link (L 0 × S 1 ) (L 1 × S 1 ) ⊂ S 2 × T * S 1 is displaceable. (The proof uses a Lagrangian suspension argument, and is recalled in Lemma 1.11 below.) Since the Lagrangians are compact, when a displacing Hamiltonian isotopy exists, it can be chosen to be the identity outside a compact set in the cylinder factor T * S 1 . It follows a fortiori that, if one fixes A 0 sufficiently large, then (L 0 × S 1 eq ) (L 1 × S 1 eq ) ⊂ (S 2 × S 2 , (2B + C) ω ⊕ A ω) is displaceable. In contrast, our main theorem is as follows.
Proof For any B close to A and C > 0 close to 0 such that 2B + C = 2A and B − C > a, we get a non-displaceable Lagrangian link L 0 L 1 ⊂ (M, 2Aω ⊕ 2aω) by Theorem 1.1. We can vary B to get a family of non-displaceable Lagrangian links.
As far as we know, this is the first example in higher dimensions (where area considerations do not pertain) of a non-displaceable Lagrangian link whose constituent components are displaceable. Figure 1: Displaceable (left) but non-displaceable (right) if 0 < a < B − C Remark 1.3 Floer theory for non-monotone symplectic manifolds such as M , and its Hamiltonian invariance, rely on virtual perturbation technology. Our proof uses rather formal properties of bulk deformation theory and bulk-deformed A ∞ -algebras for Lagrangian tori in symplectic orbifolds, and could presumably be established in any given framework. A complete development in the language of Kuranishi spaces has been given by Fukaya, Oh, Ohta and Ono [FOOO09a,FOOO09b,FOOO18,MTFJ19]; their work was extended to the setting of orbifolds by Cho and Poddar [CP14]. A reader uncomfortable with Kuranishi spaces should take Theorem 2.3 and Theorem 2.8 as axioms.

Idea of proof
We will write S 2 α for the sphere of area α. Let X := Sym 2 (M) be the 2-fold symmetric product of M , i.e. the quotient of M × M by the Z/2 action which exchanges the factors. By definition, X is equipped with the structure of a symplectic orbifold, where the set of orbifold points is precisely the image of the diagonal. The product Lagrangian L 0 × L 1 lies away from the diagonal, so it descends to a smooth Lagrangian submanifold in X, which we denote by Sym(L). Hamiltonian functions and Hamiltonian diffeomorphisms make sense in a symplectic orbifold. Moreover, given a Hamiltonian function H ∈ C ∞ (M × [0, 1]) and (z 1 , z 2 ) ∈ M × M , the function H(z 1 , t) + H(z 2 , t) is Z/2-invariant and hence induces a function Sym(H) ∈ C ∞ (X × [0, 1]) defined by Remark 1.4 Any Hamiltonian function on a symplectic orbifold admits a smooth lift to local uniformization charts, so any smooth lift near an orbifold point is invariant under the corresponding isotropy group. It follows that every Hamiltonian diffeomorphism of a symplectic orbifold preserves the orbifold strata (this is obvious for elements in the image of (2)).
An immediate consequence of the existence of the map (2) is that: Thus, Theorem 1.1 will be a consequence of Lemma 1.5, via: Theorem 1.6 Under the assumptions of Theorem 1.1, Sym(L) is non-displaceable.
The proof of Theorem 1.6 uses a bulk deformed superpotential argument (cf. [FOOO11,CP14]), and should be compared with the main result of [FOOO12] which proved the existence of continuum families of non-displaceable Lagrangian tori T 2 ⊂ (S 2 × S 2 , ω ⊕ ω). The calculation of the superpotential is motivated by the 'tropical-holomorphic' correspondence, which relates holomorphic curves in the total space of a Lagrangian torus fibration with tropical curves in the base; the actual computation appeals to the 'tautological' correspondence, which relates holomorphic discs in Sym(M) with holomorphic branched covers of discs mapping to M itself. In a little more detail, the main ideas can be summarised as follows. Each L i bounds 4 families of Maslov 2 discs, say in classes β j i , for j = 1, 2, 3, 4. By (a rather trivial instance of) the tautological correspondence, the disjoint union of a disc in class β j i and a constant map from a disc to L 1−i lifts to a Maslov 2 holomorphic disc in X with boundary on Sym(L). Keeping track of their areas, these 8 families of discs contribute the following terms to the superpotential of Sym(L): The function (3) has no critical point in the units of Novikov ring, but the Laurent polynomial x −1 2 + y −1 2 + x 2 + y 2 does. We introduce a bulk deformation such that certain holomorphic annuli in M with boundary on L contribute to the bulk deformed superpotential of Sym(L). The tropical picture of one of these annuli is depicted in Figure 2, where p 0 and p 1 are the projection of L 0 and L 1 under the Log map; they have been slightly perturbed to ease visualisation. Remark 1.7 The product annulus given by the annulus bound by L 0 L 1 in the S 2 2B+C factor and a constant in the S 2 2a factor is not directly helpful to prove non-displaceability of L (for instance any argument only using such annuli would translate to S 2 2B+C × T * S 1 ). The annuli we use are different, and project onto a disc in the S 2 2a factor. In particular, the annuli which contribute to the superpotential have Maslov index 2, while the 'small' visible annuli have Maslov index 0.
Remark 1.8 There is not yet a general 'tropical-holomorphic' correspondence for holomorphic curves with Lagrangian boundary conditions. Note that without some hypotheses, analytic curves in (C * ) 2 do not Gromov-Hausdorff converge to tropical curves under the rescaled logarithm maps, cf. [MN15]; for instance the curve (z, e z ) ⊂ (C * ) 2 has logarithmic limit set containing an interval (while the logarithmic limit set of an algebraic curve is a finite set). In the sequel, we will give explicit constructions of the holomorphic annuli we require in Section 3.3, but we believe it is helpful to explain how these constructions were motivated by tropical analogues; those are given first, in Section 3.2.
After appropriate deformation, we can make the two lowest order terms of the superpotential to be This is possible only when B − a − C > 0, which leads to the corresponding assumption in Theorem 1.1. The leading order term equations of (4) in the sense of [FOOO11] are which admit 6 solutions of the form x 2 = y 2 = 0 and x 1 = y 1 = 0. We will indeed prove that there are at least 6 critical points for the appropriate bulk deformed superpotential. Combining this existence result for critical points with the general machinery developed in [FOOO09a,FOOO09b,CP14], we obtain Theorem 1.6.
Remark 1.9 The Lagrangian Sym(L) lies in the smooth locus of X. It can therefore be lifted to a Lagrangian in Hilb 2 (M), the Hilbert scheme of zero dimensional subschemes of length 2 in M . Our argument can be applied to show that Sym(L) is non-displaceable in Hilb 2 (M) (see Remark 4.6).
Remark 1.10 It is natural to ask for examples of Lagrangian links with more than two components. For a trivial source of examples, one can take L 0 , . . . , L d to be parallel circles in S 2 2B+(d−1)C such that, for i = 1, . . . , d , L i−1 and L i are the respective boundary components of an area C cylinder which does not meet any other L j . The corresponding d i=0 L i is non-displaceable in S 2 2B+(d−1)C ×S 2 a for appropriate a > 0. However, there are proper subsets of the link {L i } which are already non-displaceable.
It would be more interesting to find a 'Borromean' example, i.e. a non-displaceable (d + 1)-component Lagrangian link such that any d -component sublink is displaceable. We hope to give such examples, based on a more systematic formulation of the tropical-holomorphic correspondence, in a sequel.

Stable displaceability
For completeness, we recall the statement and proof of Polterovich's stable displaceability result, mentioned previously. The argument presented here is a modification of Example 6.3.C in [Pol01].
where R × S 1 is identified with T * S 1 . Since H(p) = 0 for all p ∈ L 0 L 1 , the two Lagrangian suspensions are disjoint.
Acknowledgements. This project was influenced by a set of related questions raised to the authors by Emmanuel Opshtein. We are grateful to Pierrick Bousseau for discussions of his ongoing work on scattering diagrams [Bou19], and to Mohammed Abouzaid and Kaoru Ono for their interest. The authors are partially supported by a Fellowship from EPSRC.

Superpotentials and bulk deformation
We summarise the theory of superpotentials for Lagrangian Floer cohomology, to establish notation which will recur later in the paper. More details can be found in [FOOO09a,FOOO09b,FOOO10,FOOO11].

Lagrangian Floer theory on symplectic manifolds
Let M • k+1,l be the moduli space of closed unit discs S with k + 1 boundary marked points z 0 , . . . , z k , ordered counterclockwise, and l interior marked points z + 1 , . . . z + l . We denote the tuples (z 0 , . . . , z k ) and (z + 1 , . . . z + l ) by z and z + , respectively. The moduli space M • k+1,l can be compactified by semi-stable nodal curves, and we denote the compactification by M k+1,l . By slight abuse of notation, a typical element in M k+1,l is also denoted by (S, z, z + ).
be the Novikov ring and Λ + be its maximal ideal. Let C(L; C) be the singular cochain complex of L 1 , and C(L; Λ 0 ) be the completion of C(L; C) ⊗ C Λ 0 with respect to the R-filtration on Λ 0 .
By linear extension, we get a map Given Q := s j=1 (g j : Q → X) ∈ C * (X, Λ + ) representing b, we define q β,l,k (Q ⊗l ; P) = q β,l,k (Q ⊗l ; P) (14) One can again use homological perturbation to pass to (co)homology, obtaining maps We can define m b k,β similarly by replacing q l,k in (17) with q β,l,k . The collection of maps defines a filtered A ∞ algebra.

Lagrangian Floer theory on symplectic orbifolds
We need a generalization of Theorem 2.3 to the case that X is an effective symplectic orbifold, but L ⊂ X is assumed to be disjoint from the orbifold strata, i.e. L ⊂ X reg is contained in the locus of smooth points of X . This generalization is carried out in [CP14]. For background on symplectic orbifolds, and in particular their symplectic forms and (contractible) spaces of compatible almost complex structures, we refer the reader to [CR02].
In the orbifold case, the construction of the A ∞ -structure goes through without substantive changes provided we consider J -holomorphic stable maps from M k+1,l for which all irreducible components are smooth. More interestingly, in the orbifold case there are new bulk deformation directions for the superpotential coming from cycles in other components of the inertia stack IX of X , as probed by J -holomorphic stable orbifold discs. This flexibility increases the chance of finding a bulk deformed superpotential that has a critical point. We briefly recall how orbifold discs enter the story.
An orbifold disc with k + 1 boundary marked points and l interior marked points is a tuple (S, z, z + , m), where (S, z, z + ) ∈ M • k+1,l and m = (m 1 , . . . , m l ) is a tuple of positive integers. We give the unique orbifold structure on S such that the set of orbifold points is contained in z + , and for each j = 1, . . . , l, there is a disc neighborhood U j of z + j which is uniformized by the branched covering map z → z m j . When m j = 1, z + j is a smooth point. For a fixed m, we denote the moduli of such orbifold discs by M • k+1,l,m . It can be compactified by semi-stable nodal orbifold curves M k+1,l,m . The union of M k+1,l,m over all possible m is denoted by M k+1,l . By abuse of notation, a typical element in M k+1,l is also denoted by (S, z, z + , m).
Definition 2.4 A J -holomorphic stable map from (S, z, z + , m) to (X, L) is a pair (u, ξ) such that (1) u : (S, z, z + , m) → X is a J -holomorphic map (i.e. J -holomorphic on each irreducible component) and u(∂S) ⊂ L; (2) u is a good smooth orbifold map and ξ is an isomorphism class of compatible system; (3) the group homomorphism induced by ξ at each orbifold point is injective; (4) the set of φ : S → S satisfying the following properties is finite: Remark 2.5 Under the classical differential-geometric definitions of orbifold, orbibundle etc as in [Sat56,Sat57], a smooth orbifold map does not give sufficient information to define the pull-back of an orbifold vector bundle. (This issue arises for maps into the orbifold strata X\X reg . It can be traced to the fact that orbifolds should form a 2-category rather than a 1-category. A better formulation of orbifolds as groupoids avoids these difficulties, see [Ler10]; however, these are not the approaches taken by the references for Floer theory on orbifolds.) The extra information required to define the pullback is a choice of a 'compatible system' in the sense of [CR02,CR04]. It is possible that a smooth orbifold map has no compatible system, or has more than one isomorphism class of compatible systems. A smooth orbifold map that has a compatible system is called good. A compatible system induces a group homomorphism between the isotropy groups G x → G u(x) , and the third condition in Definition 2.4 requires that this map be injective. A useful fact is that when S is irreducible and u −1 (X reg ) is connected and dense, then u is good with a unique choice of isomorphism class of compatible system.
Let IX be the inertia orbifold of X . We denote the index set of the inertia components by T = {0} ∪ T , and for g ∈ T , we write X g for the corresponding component, with X 0 = X . Elements x ∈ X g are written as (x, g).
Given a J -holomorphic stable map ((S, z, z + , m), u, ξ), the group homomorphism at z + j induced by ξ determines a conjugacy class g in G u(z + j ) . We define ev for all j = 1, . . . , l.
We refer readers to [CP14] for the Fredholm theory and dimension formulae in the orbifold setting.
The upshot is that we can define m k,β and m k without any changes to obtain a filtered A ∞ algebra (C(L, Λ 0 ), {m k } ∞ k=0 ). As in the manifold case, bulk deformation in the orbifold case is defined using the fiber product between M k+1,l (L, J, β, x) and appropriate cycles under the evaluation maps ev + j . The crucial difference is that the codomain of ev + j is now the inertia stack IX , so the cycles used to cut down the image of evaluation are taken in IX instead of in X . For example, if x(j) = 0 for all j, then a stable map ((S, z, z + , m), u, ξ) in M k+1,l (L, J, β, x) must have m j = 1 for all j, because the local group homomorphism induced by ξ is required to be injective. This forces all the points z + j to be smooth points, and bulk insertions by cycles in X x(j) = X 0 = X are defined as before. If x(j) = 0 for some j, then J -holomorphic orbifold discs can contribute to the moduli M k+1,l (L, J, β, x), and hence to the deformed filtered A ∞ structure.
In this paper, the only cycle in X g for g = 0 that we will consider is the fundamental cycle [X g ]. Let H be the C-vector space generated by H * (X; C) and [X g ] for all g = 0. Let H ⊗ Λ + denote the completion of the tensor product with respect to the R-filtration.
. Weak bounding cochains for m b and the bulk-deformed superpotential are defined as in Definition 2.2. Most importantly, the exact analogue of Theorem 2.3 holds in the orbifold setting [CP14].
The conclusion implies in particular that L cannot be displaced by Hamiltonian isotopies in X reg .
Remark 2.9 The paper [CP14] restricts to bulk deformations by toric-invariant cycles in a toric orbifold X . However, their formalism is parallel to [FOOO09b] and applies to any bulk class.
The advantage of using toric-invariant bulks and a toric-invariant Lagrangian L is that the inclusion H 1 (L, Λ 0 )/H 1 (L, 2π √ −1Z) ⊂M weak (L, m b ) then holds automatically, once one has built a T n -equivariant Kuranishi structure on the moduli space of discs. In contrast, we will prove that H 1 (L, Λ 0 )/H 1 (L, 2π √ −1Z) ⊂M weak (L, m b ) directly (see Lemma 3.3 and 3.15).

Classification of holomorphic orbifold discs
The two-sphere S 2 α has a standard Lagrangian torus fibration arising from a Hamiltonian circle action with moment map image an interval of length α. There is a corresponding Hamiltonian T 2 -action on M with moment map image a rectangle of side lengths 2B + C and 2a; the Lagrangians L i ⊂ M of Theorem 1.1 can be taken to be fibres of the Lagrangian fibration.
We now take X = X = Sym 2 (M), and take L to be the (symmetric) product of two distinct Lagrangian torus fibers L and L in M . In particular, we have L ⊂ X reg . The indexing set T is a singleton; we denote the unique element in T by 1, so that IX = X 0 ∪ X 1 , where X 0 = X and X 1 = M . This section is devoted to the discussion of J -holomorphic stable orbifold discs mapping to (X, L).

Tautological correspondence
Let J M and J X be the canonical complex structures on M and X, respectively. There is a wellknown bijective correspondence (the 'tautological' correspondence) between isomorphism classes of J X -holomorphic maps u : S → X and isomorphism classes of pairs (v, π Σ ), where π Σ : Σ → S is a 2 to 1 branched covering and v : Σ → M is a J M -holomorphic map (Σ is possibly disconnected). Bijective correspondences of this form have been used in [Lip06, Aur10, MS19] etc.
The correspondence is defined as follows. Given a J X -holomorphic map u : S → X, we define Σ to be the fiber product between u and the quotient M × M → X, so that we have the pull-back diagram We define v := π 1 • V , where π 1 : M × M → M is the projection to the first factor. Conversely, given a pair (v, π Σ ), the corresponding u is defined by u(z) = v(π −1 Σ (z)) ∈ X.
Lemma 3.1 If (v, π Σ ) is obtained tautologically from a J X -holomorphic maps u : S → X with boundary on L, then Σ has 1 or 2 connected components. If Σ has 2 components, then each component is a disc.
Moreover, ∂Σ has 2 connected components, mapped under v to L and L respectively.
Proof Since π Σ is a 2 to 1 branched covering, Σ has either 1 or 2 connected components and ∂Σ has either 1 or 2 connected components. Moreover, if Σ has 2 components, then π Σ is an unbranched covering so each component of Σ is a disc.
Since u(∂S) ⊂ L and L ∩ L = ∅, continuity implies that ∂Σ cannot be connected, so it has 2 connected components. Again since u(∂S) ⊂ L, we know that the 2 boundary components of Σ are mapped under v to L and L , respectively.
Note that if S is a smooth disc (i.e. it is irreducible and has no orbifold point) and u is a J X -holomorphic stable map in the sense of Definition 2.4, then the condition that u is a smooth orbifold map implies that it admits a liftũ : S → M × M (because a smooth orbifold map is a map that admits a smooth lift to the uniformization charts, and hence a smooth lift to orbifold universal covers). That in turn implies that Σ = S S, π Σ : Σ → S is the trivial 2-fold covering and V =ũ (ι •ũ) : Σ → M × M , where ι : M × M → M × M is the involution swapping the two factors. Thus, when S is an orbifold disc and u is a J X -holomorphic stable map in the sense of Definition 2.4, the critical values of π Σ are precisely the orbifold points of S. Moreover, the corresponding m j at each orbifold point is necessarily 2, and the images of the evaluation maps ev + j at these orbifold points necessarily lie in X 1 ⊂ IX. In other words, when we study the filtered A ∞ structure m k,β on H(L, Λ 0 ), it is sufficient to consider the moduli spaces of J M -holomorphic maps from Σ := (S, z) (S, z) to M , where (S, z) is a semi-stable nodal disc with k +1 boundary marked points and no interior marked points. In contrast, if we introduce a bulk deformation b = b smooth + b orb [X 1 ] such that b smooth ∈ H * (X, Λ + ) and b orb ∈ Λ + , then the construction of the maps m b k,β will involve additional J X holomorphic stable maps ((S, z, z + , m), u, ξ). The orbifold points of these stable maps are precisely the interior points that are constrained to lie in [X 1 ], which translates to the fact that the number of critical points of π Σ is precisely the number of times b orb [X 1 ] contributes.
With this understood, we can now explain how to compute the virtual dimension of M k+1,l (L, J, β, x) by computing the virtual dimension of the pairs (v, π Σ ). First, the type x determines m as follows: m j = 2 if and only if x(j) = 1, and m j = 1 otherwise. For this fixed m, and for each element (S, z, z + , m) in M k+1,l,m , there is a unique double branched covering π Σ : Σ → S for which the critical values of π Σ are exactly {z + j |m j = 2}. It follows that the moduli of covers π Σ is in bijective correspondence with (S, z, z + , m), and hence has dimension dim(M k+1,l,m ) = k + 1 + 2l − 3 = k + 2l − 2.

Tropical picture
To compute the superpotential W b of L for b as in Lemma 3.3, it is sufficient to classify Maslov index two J M -holomorphic curves with boundary on L L . Since L and L are product Lagrangians and J M is a product complex structure, we can study J M -holomorphic curves with boundary on L L by projecting to the two P 1 factors of M . In other words, we want to classify holomorphic maps v = (v 1 , v 2 ) for v 1 , v 2 : Σ → P 1 with boundary on the respective projections of L and L such that the sum of the Maslov indices of v 1 and v 2 is 2. This classification is undertaken in Section 3.3, but in preparation, we find it helpful to give a tropical picture which motivates the result.
Let Log : (C * ) 2 → R 2 be Log(z 1 , z 2 ) = (log |z 1 |, log |z 2 |). Let p = Log(L ) and p = Log(L ) be two points in R 2 . The expected paradigm is (see [Gro10,GPS10]): Philosophy 3.4 The tropicalization of a connected J M -holomorphic curve with boundary on L L should give a 'broken' tropical curve γ with boundary on p ∪p . That is, γ is the image of a continuous map h from a connected weighted finite graph without bivalent vertices Γ to R 2 such that: (1) if v is a vertex of Γ such that h(v) = p (or p ), then v is univalent; (2) for every edge e of Γ, h| e is an embedding and h(e) is a line segment of rational slope; (3) at every vertex v of Γ such that h(v) ∈ {p , p }, the balancing condition 2 holds; (4) if h(e) has infinite length (i.e. is an unbounded edge), then the primitive direction of h(e) belongs to {(±1, 0), (0, ±1)}.
Vertices and edges of γ are defined to be the images of the vertices and edges of Γ. The Maslov index of γ is defined to be twice the number of unbounded edges of γ (counted with multiplicity). The genus of γ is defined to be the rank of H 1 (Γ).
Example 3.5 Suppose the x-coordinate of p is smaller than that of p . Then there is a genus 0 broken tropical curve γ with 3 edges e 0 , e 1 , e 2 such that e 0 is adjacent to p with primitive direction (1, −1), e 1 is adjacent to p with primitive direction (−1, −1) and e 2 is a multiplicity 2 unbounded edge whose primitive direction is (0, 1) (see Figure 3). In our situation, due to Lemma 3.1 (or more specifically (29)), we are interested in two cases: (1) unions of two broken tropical curves γ and γ , such that p is a vertex of γ but not γ ; and p is a vertex of γ but not γ (2) broken tropical curves γ such that both p and p are vertices of γ .
It will be useful to impose the following 'tropical general position' assumption.
Assumption 3.6 The slope of the straight line joining p and p is irrational (∞ is regarded as rational).
Lemma 3.7 If Assumption 3.6 is satisfied, then there is no non-constant Maslov index zero broken tropical curve with boundary on p ∪ p .
Proof By definition, a Maslov index zero broken tropical curve admits no unbounded edges. If such a tropical curve is not a constant, the balancing condition shows that it must be a straight line with rational slope joining p and p , which does not exist by assumption.
Remark 3.8 We define the Maslov index of a union of broken tropical curves to be the sum of the Maslov indices of the components. Therefore, by Lemma 3.7, if the union has Maslov index 2 then it is composed of exactly one Maslov 2 broken tropical curve and some number of constant tropical curves.
We now study the possible Maslov index two broken tropical curves with boundary on p ∪ p .
Lemma 3.9 Let γ be a Maslov 2 broken tropical curve with boundary on p ∪ p .
(2) If p ∪ p ∈ γ and e 0 , e 1 , e 2 are edges such that e 0 is adjacent to p , e 1 is adjacent to p and e 2 is the unbounded edge (necessarily) with multiplicity one, then the sum of the weighted directions of e 0 , e 1 , e 2 is 0.
Proof Since γ has Maslov index 2, it has exactly one unbounded edge, necessarily of multiplicity 1. Statement (1) follows. If p ∪ p ∈ γ , then clearly the edges e 0 , e 1 , e 2 are distinct. The sum of the weighted directions of e 0 , e 1 , e 2 is the sum of the balancing conditions at all vertices of γ other than p and p . Therefore, it must vanish.
We need to further analyse case (2) in Lemma 3.9. Suppose from now on that p is in the third quadrant and p is in the first quadrant of R 2 .
Lemma 3.10 Suppose that Assumption 3.6 is satisfied. Let γ be a broken tropical curve as in case (2) of Lemma 3.9. If e 2 has direction (1, 0), e 0 has weighted direction (p, q) and e 1 has weighted direction (−(p + 1), −q), then we have where m is the slope between the line joining p and p . Conversely, for every pair of integers (p, q) such that (35) is satisfied, there is a unique genus 0 broken tropical curve with boundary on p , p , consisting of the edges e 0 , e 1 , e 2 with weighted directions (p, q), (−(p + 1), −q) and (1, 0), respectively.
Proof We leave the proof as an exercise. See Figure 4.  Similarly, we have Lemma 3.11 Suppose that Assumption 3.6 is satisfied. Let γ be a broken tropical curve as in case (2) of Lemma 3.9. If e 2 has direction (0, 1), e 0 has weighted direction (p, q) and e 1 has weighted direction (−p, −(q + 1)), then we have where m is the slope between the line joining p and p . Conversely, for every pair of integers (p, q) such that (36) is satisfied, there is a unique genus 0 broken tropical curve with boundary on p , p , consisting of the edges e 0 , e 1 , e 2 with weighted directions (p, q), (−p, −(q + 1)) and (0, 1), respectively.
Example 3.12 When 0 < m < 1, the tropical curve in Lemma 3.11 with the smallest p has p = 1 and q = 0 (see Figure 2 with p 0 and p 1 understood as p and p , respectively).
Remark 3.13 There are higher genus Maslov 2 broken tropical curves with boundary on p ∪ p (see Figure 5). However, we will see that they only contribute higher order terms (in the adic filtration) to the bulk deformed superpotential, which will mean we do not need a classification of these curves to prove existence of critical points for the superpotential.

Maslov two holomorphic curves
The tropical picture is heuristic, for two reasons. First, we have not justified that Log t -image of a J Mholomorphic curve with boundary on L ∪ L converges to a broken tropical curve. More importantly, given a broken tropical curve γ , we have not proved that there is a J M -holomorphic curve with boundary on L and L whose tropicalization is γ . In this section, we use the tropical picture as a guide to help us locate and study holomorphic curves of Maslov index two.
We write L = L 1 × L 2 and L = L 1 × L 2 . We identify M with P 1 × P 1 in such a way that L = {|z| = r 1 } × {|w| = r 2 } and L = {|z| = r 1 } × {|w| = r 2 }. We assume that r 1 < 1 < r 1 and r 2 < 1 < r 2 , which correspond to the assumption that p and p are in the in the third respectively first quadrants. Let C i denote the cylinder bound by L i and L i , for i = 1, 2. Moreover, we assume that r 1 r 1 < r 2 r 2 (37) which corresponds to the slope condition 0 < m < 1, cf. Example 3.12.
Assumption 3.6 translates to the following.
By the boundary conditions, neither v 1 nor v 2 can be a constant map. Therefore, we must have that v 1 and v 2 surject onto the cylinders C 1 and C 2 , respectively. This is a contradiction because it is well-known that no Σ can simultaneously surject onto two annuli {r 1 ≤ |z| ≤ r 1 } and {r 2 ≤ |w| ≤ r 2 } which satisfy Assumption 3.14.
We present a proof here for the sake of completeness.
Taking the derivative of (39), we get By integrating that identity over a closed curve parallel to ∂ 0 and applying the residue theorem, we see that α is rational. This contradicts Assumption 3.14.
(1) If Σ is disconnected with connected components Σ 0 and Σ 1 , then one of v| Σi , say v| Σ 1 , is a constant. Then for Σ 0 , one v i | Σ 0 is a degree one map to a disc and the other is a constant.
(2) If Σ is connected, then either v 1 or v 2 is a (possibly unramified) branched covering of a cylinder, and the other surjects to a disc.
Proof If Σ has two connected components Σ 0 and Σ 1 , then ∂Σ i is connected. Therefore, if v i | Σ j is not a constant, then µ(v i ) ≥ 2. That means that 3 of the 4 maps {v i | Σ j } i,j have Maslov zero and are hence constant. Suppose v i | Σ 0 is not a constant; then it is either a branched covering of a disc or it surjects onto P 1 . However, if it surjects to P 1 , then it must have µ(v i ) ≥ 4. Since µ(v i ) = 2, we know that v i | Σ 0 has degree 1 and surjects to a disc. Now asssume that Σ is connected with 2 boundary components. Without loss of generality, we assume that µ(v 1 ) = 0 and µ(v 2 ) = 2. Since µ(v 1 ) = 0, it has to surject onto the cylinder bounded by L 1 and L 1 . On the other hand, µ(v 2 ) = 2 implies that the image of v 2 is not the entire P 1 and not the cylinder bounded by L 2 and L 2 . Therefore, v 2 surjects to a disc.
Let v be a holomorphic curve as in case (2) of Lemma 3.16, such that v 2 surjects to a disc D. Let D • be the interior of D. By the Lagrangian boundary conditions for v, there are two possibilities: (1) D • is the complement of L 2 that contains L 2 ; (2) D • is the complement of L 2 that contains L 2 .
By the obvious symmetry, it is sufficient to analyse the first situation. In this case, we have for some integers p ≥ 1 and q ≥ 0, where the relative classes are in H 2 (S 2 2B+C , L 1 ∪ L 1 ) and H 2 (S 2 2a , L 2 ∪ L 2 ), respectively. Let A p,q ∈ H 2 (M, L ∪ L ) be the class characterized by the property that the projection to the two P 1 factors are p[C 1 ] and [D] for some integers p ≥ 1 and q ≥ 0.
Lemma 3.17 There exists a J M -holomorphic curve v : Σ → M with boundary on L ∪ L in class A 1,0 . Moreover, for a generic pair of points q ∈ L and q ∈ L , the algebraic count of unparametrized holomorphic curves in A 1,0 such that q , q ∈ v(∂Σ) is ±1.
Proof First, we prove the existence of v. Let Σ be the annulus C 1 and define v 1 : Σ → C 1 ⊂ P 1 to be the inclusion map.
By rescaling, we identify D with the unit disc, L 2 with the unit circle and L 2 with the circle of radius r 0 := r 2 r 2 . Let l L 2 be a closed arc (usually called a 'slit' in the literature, cf. [Ahl78, Chapter 6], [Neh52, Chapter 7]) and define D l := D • \ l. In Lemmas 3.18 and 3.19, we provide proofs of the following classical facts: (1) For every 0 < r < r 0 , there is a biholomorphism from {r < |z| < 1} to D l for some l ⊂ L 2 .
Moreover, the biholomorphism can be smoothly extended up to its closure.
(2) Given two slits l 1 , l 2 ⊂ L 2 , the domain D l 1 is conformally isomorphic to D l 2 if any only if l 1 and l 2 are of the same length.
By assumption, Σ = { r 1 r 1 ≤ |z| ≤ 1} and r 1 r 1 < r 2 r 2 = r 0 (see (37)). Therefore, we have a holomorphic map v 2 : Σ → S 2 2a such that the two boundary components are mapped to L 2 and l ⊂ L 2 for some l, respectively. This proves the existence.
Conversely, let v = (v 1 , v 2 ) be a holomorphic curve in class A 1,0 . Since v 1 is a degree one map with the boundary components of Σ going to L 1 and L 1 , respectively, v 1 must be a biholomorphism. Therefore, we can identify Σ with C 1 via v 1 .
On the other hand, v 2 is a degree 1 map onto the disc D. Therefore v 2 | Σ\∂Σ is a biholomorphism, so v 2 (Σ \ ∂Σ) is D l for some l. This gives a complete classification of holomorphic lifts of γ . Now, let q = (q 1 , q 2 ) ∈ L and q = (q 1 , q 2 ) ∈ L . By identifying Σ with the annulus bound between L 1 and L 1 , we can suppose that q 1 , q 1 ∈ ∂Σ. To prove the last statement, it suffices to show that the algebraic count of v 2 such that v 2 (q 1 ) = q 2 and v 2 (q 1 ) = q 2 is ±1.
We now address the two classical facts used in the proof of Lemma 3.17.
Lemma 3.18 Let r 1 ∈ (0, r 0 ). There is a biholomorphism from A := {r 1 < |z| < 1} to D l for some l ⊂ L 2 . Moreover, the biholomorphism can be smoothly extended up to its closure.
Proof Let a ∈ A and G(z, a) be the Green's function, i.e. the unique function determined by the conditions: (1) G(z, a)| ∂A = 0 (2) G(z, a) + log |z − a| is harmonic and smooth everywhere on A. We denote the inner and outer boundary components of ∂A by ∂ 0 and ∂ 1 . Let w i for i = 0, 1 be the corresponding harmonic measures. That is, w i is the unique smooth harmonic function on A such that w i | ∂i = 1 and w i | ∂A\∂i = 0. More explicitly, w 0 (z) = log(|z|) log(r 1 ) and w 1 = 1 − w 0 . Since ∆G(z, a) = −2πδ a , for any harmonic function w : A → R, we have the identity (see e.g. [Neh52, where ∂ ∂n refers to the outward normal derivative. When w = 1, it gives Therefore, we have β := − ∂ 1 ∂G(z,a) ∂n = 2πw 1 (a) ∈ (0, 2π). On the other hand, a direct calculation gives For c = (2π−β) log(r 1 ) 2π , we have That means that the harmonic conjugate H(z) of −G(z, a) + cw 0 is a multi-valued function with period 2π on ∂ 1 and period 0 on ∂ 0 . The map F(z) = e −G(z,a)+cw 0 +iH(z) is therefore a single valued holomorphic function which maps a to the origin, ∂ 0 to {|z| = e c } with degree 0 and ∂ 1 to {|z| = 1} with degree 1. By a routine argument (see e.g. [Ahl78, Theorem 10 of Section 5 of Chapter 6]), one can check that F is a biholomorphism from A to the slit domain D \ {F(∂ 0 )}. Since 0 < β < 2π , we have log(r 1 ) < c < 0. On the other hand, for any value β 0 between 0 and 2π , there is a unique a (up to automorphism of A) such that β = 2πw 1 (a) = β 0 . Since r 1 < r 0 , we can pick the a such that the corresponding c is log(r 0 ), so F(∂ 0 ) ⊂ L 2 . Finally, the boundary components of A are smooth analytic curves so F can be extended smoothly up to the boundary.
Lemma 3.19 Given two slits l 1 , l 2 ⊂ L 2 , the domain D l 1 is conformally isomorphic to D l 2 if and only if l 1 and l 2 are of the same length.
Proof By the classification of multi-connected domains, D l 1 is biholomorphic to A := {r 1 < |z| < 1} for some r 1 . Let F 1 : A → D l 1 be a biholomorphism such that the smooth extension of F 1 maps {|z| = 1} to the unit circle. Let a 1 ∈ A be the point such that F 1 (a 1 ) = 0. Then log |F 1 (z)| is a harmonic function which is smooth everywhere except having a log pole at a 1 . Moreover, it maps {|z| = 1} to 0 and {|z| = r 1 } to log(r 0 ). These conditions uniquely characterise log |F 1 (z)|, and so we have log |F 1 (z)| = −G(z, a 1 ) + log(r 0 )w 0 (50) where G(z, a 1 ) and w 0 are the Green's function and harmonic measure introduced in the proof of Lemma 3.18.
It follows that if D l 2 is biholomorphic to D l 1 and hence to A, then the biholomorpism F 2 : A → D l 2 satisfies log |F 2 (z)| = −G(z, a 2 ) + log(r 0 )w 0 (51) for the a 2 ∈ A such that F 2 (a 2 ) = 0.
Moreover, we know the periods of the harmonic conjugates of (50) and (51) are the same, namely 2π on the outer boundary and 0 on the inner boundary. By the proof of Lemma 3.18 (see the second last paragraph), we know that there is an automorphism of A which sends a 1 to a 2 . It follows that l 1 and l 2 are of the same length.
Remark 3.20 For a circular cylinder {1/r ≤ |z| ≤ r}, any involution swapping the boundary components belongs to the family (z → e iθ z ) θ∈[0,2π] . Consequently, there is a 1-dimensional family of 2 to 1 branched coverings of the cylinder over the unit disc. Let v be as in Lemma 3.17, and let z , z ∈ ∂Σ be such that v(z ) = q and v(z ) = q . If we choose the involution of Σ which exchanges z and z , the corresponding 2 to 1 branched covering of the unit disc together with v will tautologically correspond to a J X -holomorphic orbifold disc u : S → X such that [q , q ] ∈ u(∂S). This means that u will contribute to m b,b k whenever b orb = 0, and the algebraic count of the relevant moduli containing the solution u is ±1.
On the other hand, we can prove the non-existence of maps v as in (42) when p = 1 and q > 0.
Lemma 3.21 There is no holomorphic curve with boundary on L ∪ L in class A 1,q for q > 0.
Proof Suppose that such a curve exists. The hypothesis p = 1 implies that v 1 is a biholomorphism to C 1 . Therefore, Σ is a cylinder; we identify it with { r 1 r 1 ≤ |z| ≤ 1}. As before, we identify D with the unit disc and L 2 with {|z| = r 0 < 1}.
Remark 3.22 Lemma 3.21 corresponds to the fact that in Lemma 3.11, there is no broken tropical curve with p = 1 and q > 0 when 0 < m < 1.
By symmetry, the analogues of Lemma 3.17 and 3.21 hold when v is a holomorphic curve as in case (2) of Lemma 3.16 for which v 2 surjects to the complementary component of L 2 that contains L 2 .

Concluding the proof
So far, we have been using Assumption 3.14. However, the fibres L 0 and L 1 in Theorem 1.1 do not satisfy this assumption, because their second factors coincide. Therefore, we cannot apply the previous results directly to Sym(L). We remedy this issue by applying what is commonly referred to as 'Fukaya's trick'.
More precisely, without loss of generality we may assume that L and L are close enough to L 0 and L 1 such that there is a diffeomorphism Φ : M → M satisfying (1) Φ(L) = L ∪ L , and Tautologically, the pull-back of J M -holomorphic curves with boundary on L ∪ L under Φ are Φ * J M -holomorphic curves with boundary on L, and vice versa. Therefore, we can apply the previous discussion to study the superpotential of Sym(L) with respect to the symmetric product of (the integrable complex structure) Φ * J M on X.
Remark 4.2 The coordinates (x 1 , x 2 , y 1 , y 2 ) are obtained as follows. We have identified L with {|z| = r 1 } × {|w| = r 2 } ⊂ C 2 . We apply the involution w → 1/w to the second factor to identify L with {|z| = r 1 } × {|w| = 1/r 2 } ⊂ C 2 . In this description, we give each factor the induced orientation from C, and we use these orientations to define x 1 , x 2 . The trivialization of TL that we use to orient the moduli spaces of holomorphic discs is the product of the translation-invariant trivializations in the factors of this chart C 2 , compare to [Cho04]. Similarly, we apply z → 1/z to the first factor to identify L with {|z| = 1/r 1 } × {|w| = r 2 } ⊂ C 2 and take the induced orientation and translation trivialization on the factors.
We have arranged that (z, w) → (1/z, 1/w) maps L to L , and by construction this map respects the orientations and trivializations chosen.
From the discussion above, we conclude that the part of the superpotential of Sym(L) arising from the contributions of smooth discs is given by precisely, the algebraic count of unparametrized holomorphic curves in each of the classes β 1,2 +δ 1 +δ 2 and β 2,2 + δ 1 + δ 2 is ±1, and the two counts have the same sign by the symmetry of our choice of orientation and trivialization (see Remark 4.2, and also Remark 4.5). The contribution of these curves to W b orb [X 1 ] is therefore given by (66) (see Remark 3.20).
From now on, we take b orb such that b 2 orb 2 = T B−a−C (67) Therefore, (66) becomes ±T B (x 1 y 1 ) −1 (x 2 + y 2 ) which is of the same order as the term T B (x 1 + y 1 ) in W smooth . Write the toric boundary divisor of M as S 1 + S 2 , where S 1 (resp. S 2 ) is the sum of the two spheres with smaller (resp. larger) area. Let D S 1 be the image of S 1 × M ⊂ M × M in X under the quotient map (i.e. this is the divisor of pairs of points at least one of which lies in S 1 ). It defines a cycle [D S 1 ] in H 6 (X 0 ) = H 6 (X). Let b = b 1 [D S 1 ] + b orb [X 1 ] with b 1 ∈ Λ + . To conclude the proof of Theorem 1.6 (and hence Theorem 1.1), it suffices to find an appropriate b 1 such that W b has a critical point in (Λ 0 \ Λ + ) 4 .
Lemma 4.4 There exists b 1 ∈ Λ + such that W b has 6 critical points with x 1 = y 1 and x 2 = y 2 . Proof For simplicity of notation, we assume the sign in (66) is positive. The other case is similar. Let G be the discrete monoid in R >0 generated by B − a − C and C. Let I G ⊂ Λ + be the following ideal: Note that the coefficients of (65) lie inside T a I G ∩ T B I G .
Remark 4.5 Working with a trivialization of TL such that the two lowest order families of annuli in (66) contribute with different signs, one can check that W b will still have critical points for some appropriate b. However, the coefficients in b of the two components of D S 1 will then be different. a J Hilb 2 (M),Φ -holomorphic disc u : S → Hilb 2 (M) such that u has no sphere components mapping entirely into D HC . In this case, the virtual dimensions of the moduli containing u and u are the same.
With this partial correspondence between holomorphic discs, we can partially compute the superpotential of Sym(L) from the superpotential of Sym(L). First of all, Sym(L) does not bound any J Hilb 2 (M),Φ -holomorphic disc u with Maslov index < 2, because otherwise u = π HC • u would be a J X,Φ -holomorphic disc bound by Sym(L) with too small virtual dimension. Secondly, the lower order terms of the superpotential of Sym(L) and Sym(L) can be identified by the (partial) correspondence. More precisely, a holomorphic disc u with area A which hits the orbifold locus at finitely many points (with algebraic intersection h in total) lifts to a holomorphic disc u of area A − hr for some r > 0 depending only on ω Hilb 2 (M) . We can use the bulk cT R [D HC ], for some appropriate R > 0 and c ∈ C * , to compensate for the bulk b orb [X 1 ] and the area lost. On the other hand, D S 1 lifts to a smooth divisor D in Hilb 2 (M). We can take b = b 1 [D] + cT R [D HC ] to deform the superpotential of Sym(L) in such a way that the leading term equations of the bulk deformed superpotentials of Sym(L) and Sym(L) agree. The rest of the argument then follows on similar lines to that given previously.