Low-area Floer theory and non-displaceability
We introduce a new version of Floer theory of a non-monotone Lagrangian submanifold which only uses least area holomorphic disks with boundary on it. We use this theory to prove non-displaceability theorems about continuous families of Lagrangian tori in the complex projective plane and other del Pezzo surfaces.
In this paper, we introduce a new non-displaceability criterion for Lagrangian submanifolds of a given compact symplectic (four-)manifold. The criterion is based on a Hamiltonian isotopy invariant of a Lagrangian submanifold constructed using -holomorphic disks of lowest area with boundary on the Lagrangian. We apply our criterion to prove non-displaceability results for Lagrangian tori in and in other del Pezzo surfaces, for which there is no clear alternative proof using standard results in Floer theory. We shall focus on dimension four in line with our intended applications, although we do have a clear vision of a higher-dimensional setup to which our methods generalise—it is mentioned briefly in Subsection 1.5.
1.1. Challenges in Lagrangian rigidity
A classical question in symplectic topology, originating from Arnold’s conjectures and still inspiring numerous advances in the field, is to understand whether two given Lagrangian submanifolds , are (Hamiltonian) non-displaceable, meaning that there exists no Hamiltonian diffeomorphism that would map to a Lagrangian disjoint from . It is sometimes referred to as the Lagrangian rigidity problem, and the main tool to approach it is Floer theory. Historically, most applications of Floer theory were focused on monotone (or exact) Lagrangians, as in those cases it is foundationally easier to set up, and usually easier to compute.
More recent developments have given access to non-displaceability results concerning non-monotone Lagrangians. One of such developments is called Floer cohomology with bulk deformations, introduced by Fukaya, Oh, Ohta and Ono [26, 27]. Using bulk deformations, the same authors  found a continuous family of non-displaceable Lagrangian tori in , indexed by . (When we say that a Lagrangian is non-displaceable, we mean that it is non-displaceable from itself.) For some other recent methods, see [2, 10, 56].
To be able to observe such “rigidity for families” phenomena, it is essential to consider non-monotone Lagrangian submanifolds, as spaces of monotone ones up to Hamiltonian isotopy are discrete, on compact symplectic manifolds.
It is easy to produce challenging instances of the displaceability problem which known tools fail to answer. For example, consider the 2:1 cover branched along a conic curve. Taking the images of the above mentioned tori under this cover, we get a family of Lagrangian tori in the complex projective plane denoted by and indexed by — see Section 3 and [57, Section 3] for details. It turns out that the tori have trivial bulk-deformed Floer cohomology for any bulk class , as we check in Proposition 8. While one can show that the tori are displaceable when , the following remains to be a conjecture.
For each , the Lagrangian torus is Hamiltonian non-displaceable.
Motivated by this and similar problems, we introduce a new approach, called low-area Floer theory, to solve rigidity problems concerning some non-monotone Lagrangians. Let us list some application of this technique.
For each , the torus is Hamiltonian non-displaceable from the monotone Clifford torus .
An interesting detail of the proof, originating from Lemma 7(ii), is that we use coefficients for our Floer-theoretic invariants, and it is impossible to use a field, or the group , instead. To place this into context, recall that conventional Floer cohomology over finite fields can detect non-displaceable monotone Lagrangians unseen by characteristic zero fields: the simplest example is , see e.g. ; a more sophisticated example, where the characteristic of the field to take is not so obvious, is the Chiang Lagrangian studied by Evans and Lekili , see also J. Smith . However, there are no examples in conventional Floer theory that would require working over a torsion group which is not a field.
In and in with a monotone symplectic form, there exists a one-parametric family of Lagrangian tori which are non-displaceable from the standard monotone toric fibre.
The next result exhibits a two-parametric family of non-displaceable Lagrangian tori in symplectic del Pezzo surfaces. By a symplectic del Pezzo surfaces we mean monotone symplectic 4-manifolds, whose classification follows from a series of works [37, 39, 50, 51, 52, 41, 42]. Recall that their list consists of blowups of at points, and of . By the correspondence between monotone symplectic 4-manifolds and complex Fano surfaces we will omit the term “symplectic”, and call them del Pezzo surfaces from now on.
Let be a del Pezzo surface and be a pair of Lagrangian spheres with homological intersection . Then, for some , there exist two families of Lagrangian tori indexed by :
lying in a neighbourhood of the sphere resp. , and such that is non-displaceable from for all as above.
In our construction, any two different tori in the same family will be disjoint, and the same will hold for the .
Recall that pairs of once-intersecting Lagrangian spheres exist inside -blowups of when . For example, one can take Lagrangian spheres with homology classes and where are three distinct exceptional divisors, as explained in [46, 21]. These spheres can also be seen from the almost toric perspective : on an almost toric base diagram for the corresponding symplectic del Pezzo surface, these Lagrangian spheres projects onto the segment connecting two nodes on the same monodromy line. For homology reasons, blowups of at 2 or less points do not contain such pairs of spheres.
1.2. Lagrangian rigidity from low-area Floer theory
We call a symplectic manifold monotone if and give positively proportional classes in . Similarly, a Lagrangian is called monotone if the symplectic area class and the Maslov class of are positively proportional in . It is quite common to use a definition where the proportionality is only required over or ; we stick to the homological version for convenience.
Floer theory for monotone Lagrangians has abundant algebraic structure, a particular example of which are the open-closed and closed-open string maps. There is a non-displaceability criterion for a pair of monotone Lagrangians formulated in terms of these string maps; it is due to Biran and Cornea and will be recalled later. Our main finding can be summarised as follows: it is possible define a low-area version of the string maps for non-monotone Lagrangians, and prove a version of Biran-Cornea’s theorem under an additional assumption on the areas of the disks involved. This method can prove non-displaceability in examples having no clear alternative proof by means of conventional Floer theory for non-monotone Lagrangians. We shall focus on dimension 4, and proceed to a precise statement of our theorem.
Fix a ring of coefficients; it will be used for all (co)homologies when the coefficients are omitted. (The coefficient ring does not have to include a Novikov parameter in the way it is done in classical Floer theory for non-monotone manifolds; rings like are good enough for our purpose.) Let be two orientable, not necessarily monotone, Lagrangian surfaces in a compact symplectic four-manifold .
assuming this minimum exists. This is the least positive area of topological Maslov index 2 cycles with boundary on . (For example, we currently do not allow the above set of areas to have infimum equal to .) Also, denote by the next-to-the-least such area:
We assume that the minimum exists, including the possibility ; the latter is the case when is monotone.
Fix a tame almost complex structure and a point . Let be the images of all -holomorphic Maslov index 2 disks of area such that and whose boundary is non-zero in (their number is finite, by Gromov compactness ). Assume that
and the disks are regular with respect to . Recall that by convention, the above equality needs to hold over the chosen ring . Then let
be any element whose image under the map equals . We call this class the low-area string invariant of . Observe that it is defined only up to the image
i.e. up to , compare Remark 2.2. In the cases of interest, we will have ; but regardless of this, we prove in Lemma 2 that is independent of the choices we made, up to . Finally, consider instead of and define the numbers and analogously to and , respectively. Let be a point on .
Assume that Condition (1.3) holds for and , the minima , , , exist, and . Suppose that and the homological intersection number below is non-zero over :
Then and are Hamiltonian non-displaceable from each other.
Above, the dot denotes the intersection pairing . We refer to Subsection 2.1 for a comparison with Biran-Cornea’s theorem in the monotone setup, and for a connection of with the classical open-closed string map. Note that the above intersection number is well-defined due to the hypothesis .
The condition is in fact totally non-restrictive, due to the following two “lower index” non-displaceability criteria. First, if , then and are non-displaceable for topological reasons. Second, if
one can show that and are non-displaceable by a variation on Theorem 5 whose proof can be carried analogously. Finally, if the conditions of the previous two criteria fail, the intersection number is well-defined, and the reader can check that the proof of Theorem 5 still applies.
Our proof of Theorem 5 uses the idea of gluing holomorphic disks into annuli and running a cobordism argument by changing the conformal parameter of these annuli. This argument has been prevously used in Abouzaid’s split-generation criterion  and in Biran-Cornea’s theorem [7, Section 8.2.1]. We follow the latter outline with several important modifications involved. The condition , which does not arise when both Lagrangians are monotone (), is used in the proof when the disks and are glued to an annulus of area ; the condition makes sure higher-area Maslov index 2 disks on cannot bubble off this annulus. This condition, for example, translates to in Theorem 2. Additionally, we explain how to achieve transversality for the annuli by domain-dependent perturbations – although arguments of this type appeared before in the context of Floer theory [45, 1, 47, 19, 12], we decided to explain this carefully.
Our proof only uses classical transversality theory for holomorphic curves, as opposed to virtual perturbations required to set up conventional Floer theory for non-monotone Lagrangians; compare .
There is another (weaker) widely used notion of monotonicity of or , where is required to be proportional to resp. Maslov class of only when restricted to the image of resp. under the Hurewicz map. It is possible to use this definition throughout the paper; moreover, the numbers and , see (1.1), (1.2), can be defined considering . All theorems still hold as stated, except for small adaptations in the statement of Lemma 7, e.g. requiring .
Next, we shall need a technical improvement of our theorem. Fix a field , and choose an affine subspace
The field and the ring appearing earlier play independent roles in the proof, and need not be the same.
Consider all affine subspaces parallel to ; they have the form where . For each such affine subspace, select all holomorphic disks among the whose boundary homology class over belongs to that subspace and are non-zero. Also, assume that the boundaries of the selected disks cancel over . This cancellation has to happen in groups for each affine subspace of the form . The stated condition can be rewritten as follows:
to be any element whose image under equals
Note that here we only use the disks whose boundary classes belong to the subspace and ignore the rest. Again, is well-defined up to . The same definitions can be repeated for another Lagrangian submanifold .
Let and be orientable Lagrangian surfaces and , affine subspaces. Assume that Condition (1.4) holds for and , the minima , , , exist, and . Suppose that and the homological intersection number below is non-zero over :
Then and are Hamiltonian non-displaceable.
We point out that, as in Remark 1.3, the condition is in fact non-restrictive; however we will not use this observation.
When or is monotone, we shall drop the subscript low from our notation for .
1.3. Computing low-area string invariants
There is a natural setup for producing Lagrangian submanifolds whose least-area holomorphic disks will be known. Let us start from an orientable monotone Lagrangian disjoint from the zero section, and for which we know the holomorphic Maslov index 2 disks and therefore can compute our string invariant. We are still restricting to the 4-dimensional setup, so that . Next, let us apply fibrewise scaling to in order to get a family of monotone Lagrangians indexed by the parameter ; we choose the parameter to be equal to the areas of Maslov index 2 disks with boundaries on . (The scaling changes the area but not the enumerative geometry of the holomorphic disks.) The next lemma, explained in Section 3, follows from an explicit knowledge of holomorphic disks; recall that we drop the low subscript from the string invariants as we are in the monotone setup.
Similarly, there are monotone Lagrangian tori , indexed by , which bound Maslov index 2 disks of area and satisfy Condition (1.3) over , such that:
In both cases, the tori are pairwise disjoint; they are contained inside any given neighbourhood of the zero-section for small enough .
Note that is non-orientable so it only has fundamental class over , however the class modulo also exists.
Now suppose itself admits a Lagrangian embedding into some monotone symplectic manifold (We do not require that be monotone.) By the Weinstein neighbourhood theorem, this embedding extends to a symplectic embedding for a neighbourhood of the zero-section. Possibly by passing to a smaller neighbourhood, we can assume that is convex. By construction, the Lagrangians will belong to for small enough :
(The precise value of depends on the size of the available Weinstein neighbourhood.) We define the Lagrangians
which are generally non-monotone in (even if were monotone).
Consider the induced map The next lemma explains that, for sufficiently small , the low-area string invariants for the are the -images of the ones for the . We also quantify how small needs to be.
In the above setup, suppose that the image of the inclusion-induced map is -torsion, . Let be a Lagrangian embedding into a monotone symplectic manifold . Assume that is scaled in such a way that the area class in is integral, and for some . Assume that
There is a tame almost complex structure on such that all area- holomorphic Maslov index 2 disks in with boundary on belong to , and establishes a 1-1 correspondence between them and the holomorphic disks in with boundary on .
In particular, when (i) and (ii) apply and satisfy Condition (1.3), the following identity holds in :
Similarly, if satisfy Condition (1.4) then:
A proof is found in Section 2. To give a preview, part (i) is purely topological and part (ii) follows from a neck-stretching argument.
The above constructions and proofs work for any Liouville domain taken instead of . For example, there is another class of Liouville domains containing interesting monotone Lagrangian tori: these domains are rational homology balls whose skeleta are the so-called Lagrangian pinwheels. The embeddings of Lagrangian pinwheels in have beed studied in , and using such embeddings we can employ the above construction and produce non-monotone tori in which are possibly non-displaceable. In the language of almost toric fibrations on constructed in , these tori live above the line segments connecting the baricentre of a moment triangle to one of the three nodes. See also Subsection 1.5 for a short discussion of higher dimensions.
1.4. Applications to non-displaceability
Now that we have explicit calculations of the low-area string invariants available, we can start applying our main non-displaceability result. Our first application is to prove Theorem 4.
Proof of Theorem 4.
Let be a Lagrangian sphere in a del Pezzo surface with an integral symplectic form. For concreteness, we normalise the symplectic form to make it primitive integral (it integrates by over some integral homology class). Let us define the Lagrangian tori as in (1.8), using the monotone tori which appeared in Lemma 7(i), and the Lagrangian embedding . The tori are indexed by for some . Define the tori indexed by analogously, using instead of .
After decreasing and if required, we see that the conditions from Lemma 8(i,ii) are satisfied. Here and depending on the del Pezzo surface, by the normalization of our symplectic form. Therefore by Lemma 8 and Lemma 7(i) we have over :
whenever . Finally,
So Theorem 6 implies that is non-displaceable from , for small . ∎
We will prove Theorem 2 in Section 3. In fact, we shall see that the tori appearing in Theorem 2 can be obtained via Formula (1.8) using the monotone tori from Lemma 7(ii), and the Lagrangian embedding described in Section 3.1. We could have taken this as a definition, but our actual exposition in Section 3 is different: we introduce the tori in a more direct and conventional way, and subsequently use the existing knowledge of holomorphic disks for them to prove Lemma 7(ii).
1.5. Higher-dimensional versions
One can state generalisations of Theorems 5 and 6 to higher dimensions. We shall omit the precise statements and instead mention a major class of potential examples where the low-area string invariants are easy to define, and which hopefully makes the details clear. The setup is as in Subsection 1.3: one starts with a monotone Lagrangian submanifold in a Liouville domain rescaled so as to stay close to the skeleton of . Then one takes a symplectic embedding into some symplectic manifold .
Consider the composite Lagrangian embedding which is not necessarily monotone. Provided that the symplectic form on is rational and the image of is torsion, the classes in whose symplectic area is below some treshold (depending on how close we scale to the skeleton of ) lie in the image of and therefore behave as if were monotone (meaning that their area is proportional to the Maslov index). This makes it possible to define low-area string invariants for . An analogue of Lemma 7 can easily be established as well; the outcome is that the low-area string invariants of in are obtained as the composition
where is the Floer homology of in , and is the classical monotone closed-open string map (see Section 2 for references). In this setup, it is most convenient to define as the above composition. Observe that this setup does not restrict to the use of Maslov index 2 disks, and also allows higher-index disks.
Let us provide a statement which is similar to Theorem 4.
Theorem 9 (sketch).
Let and be monotone Lagrangian submanifolds in Liouville domains, and these inclusions are -torsion. Let be a monotone symplectic manifold, and be symplectic embeddings. Suppose there are classes , such that the following pairing in is non-zero:
By the above pairing, we mean the composition of the intersection product with the projection: .
Let be the Lagrangians obtained by scaling towards the skeleton within their Liouville domains, and then embedding them into via . Then are Hamiltonian non-displaceable from each other for sufficiently small .∎
We remark that the above theorem does not incorporate the modification we made in Theorem 5, namely to only consider disks with homologically non-trivial boundary; we also have not explored the possibility to re-organise disks in groups like in Theorem 1.
The idea of proof is to establish a version of Lemma 7 saying that defined as the above composition coincides with low-area disk counts on the actual Lagrangian . The key observation is that for any interval , one can scale sufficiently close to the skeleton so that all elements in of Maslov index within and sufficiently low area, come from . This can be improved to for holomorphic disks by taking , because holomorphic disks of Maslov index generically do not exist. Given this, one shows that the proof Biran-Cornea’s theorem can be carried by only using low-area curves; this forces all the possible bubbling to happen in the monotone regime. The details are left to the reader.
Currently, there seems to be a lack of interesting computations of open-closed maps for monotone Lagrangians in higher-dimensional Liouville domains—this prevents us of from providing some concrete applications of the higher-dimensional story. We believe such applications will become available in the future.
Structure of the article
We are grateful to Paul Biran, Georgios Dimitroglou Rizell, Kenji Fukaya, Yong-Geun Oh, Kaoru Ono, and Ivan Smith for useful discussions and their interest in this work; we are also thankful to Denis Auroux, Dusa McDuff and the referees for helping us spot some mistakes in the previous version of the article, and suggesting exposition improvements.
Dmitry Tonkonog is funded by the Cambridge Commonwealth, European and International Trust, and acknowledges travel funds from King’s College, Cambridge. Part of this paper was written during a term spent at the Mittag-Leffler Institute, Sweden, which provided an excellent research environment and financial support for the stay.
Renato Vianna is funded by the Herchel Smith Postdoctoral Fellowship from the University of Cambridge.
2. The non-displaceability theorem and its discussion
2.1. The context from usual Floer theory
We start by explaining Biran-Cornea’s non-displaceability criterion for monotone Lagrangians and its relationship with Theorems 5 and 6. We assume that the reader is familiar with the language of pearly trajectories to be used here, and shall skip the proofs of some facts we mention if we do not use them later.
Recall that one way of defining the Floer cohomology of a monotone Lagrangian uses the pearl complex of Biran and Cornea [6, 7, 8]; its differential counts pearly trajectories consisting of certain configurations of Morse flowlines on interrupted by holomorphic disks with boundary on . A remark about conventions: Biran and Cornea write instead of ; we do not use the Novikov parameter, therefore the gradings are generally defined modulo 2.
Also recall that the basic fact—if , then is non-displaceable,—has no intrinsic proof within the language of pearly trajectories. Instead, the proof uses the isomorphism relating to the (historically, more classical) version of Floer cohomology that uses Hamiltonian perturbations. Nevertheless, there is a different non-displaceability statement whose proof is carried out completely in the language of holomorphic disks. That statement employs an additional structure, namely the maps
defined by counting suitable pearly trajectories in the ambient symplectic manifold . These maps are frequently called the open-closed and the closed-open (string) map, respectively; note that Biran and Cornea denote them by , . The statement we referred to above is the following one.
Theorem 1 ([8, Theorem 2.4.1]).
For two monotone Lagrangian submanifolds of a closed monotone symplectic manifold , suppose that the composition
does not vanish. Then and are Hamiltonian non-displaceable. ∎
In this paper we restrict ourselves to dimension four, so let us first discuss the monotone setting of Theorem 1 in this dimension. Recalling that a del Pezzo surface has , we see that there are three possible ways for (2.1) not to vanish.
To explain this, it is convenient to pass to chain level: let be the Floer chain complex of (either the pearl complex or the Hamiltonian Floer complex) and be the Morse (or Hamiltonian Floer) chain complex of , both equipped with Morse -gradings. For the definition of the closed-open maps, see [6, 8] in the pearl setup, and [48, 44] among others in the Hamiltonian Floer setup. Below we will stick to the setup with pearls.
First, we can consider the topological part of (2.1):
In this case, as indicated by the labels, the relevant string maps necessarily factor through and are topological, i.e. involve pearly trajectories containing only constant Maslov index 0 disks. The composition above computes the homological intersection inside , where . If , then and are topologically non-displaceable; otherwise we proceed to the next possibility; compare Remark 1.3. Observe that we are using the cohomological grading convention: pearly trajectories of total Maslov index contribute to the degree part of , and to the degree part of on cochain level.
The second possibility for not to vanish is via the contribution of pearly trajectories whose total Maslov index sums to two; the relevant parts of the string maps factor as shown below. In the examples we are aiming at, we are going to have , so the parts below will vanish on homology level.
The remaining part of breaks as a sum of three compositions factoring as follows:
The labels here indicate the total Maslov index of holomorphic disks present in the corresponding pearly trajectories; this time the parts are isomorphisms on homology. Therefore, to compute we need to know the Maslov index 4 disks. We wish to avoid this, keeping in mind that in our examples we will know only the Maslov index 2 disks.
To this end, we perform the following trick to “single out” the Maslov index 2 disk contribution in the diagram above. Let us modify the definition of , by only considering -holomorphic disks whose boundary is non-zero in or . Denote this modified map by and consider the composition:
Here the modified maps , by definition count pearly trajectories contributing to the middle row of (2.2), i.e. containing a single disk, of Maslov index 2, with the additional condition that the boundary of that disk is homologically non-trivial. The superscript reflects that we are only considering Maslov index 2 trajectories, ignoring the Maslov index 0 and 4 ones; the condition about non-zero boundaries is not reflected by our notation. We claim that if composition (2.3) does not vanish, then are non-displaceable.
This modified non-displaceability criterion we have just formulated is the specialisation of Theorem 5 to the case when both Lagrangians are monotone. Indeed, if both are monotone and , then
if and only if the composition (2.3) is non-zero; compare Lemma 3. A proof can also be traced using the original approach [7, Theorem 8.1]—see the proof of Theorem 5 in Subsection 2.2 for further details.
Speaking of our second result, Theorem 6, in the monotone case it corresponds to another refinement of Biran-Cornea’s theorem which does not seem to have appeared in the literature. Note that this refinement is not achieved by deforming the Floer theories of and by local systems.
Recall that, for a two-dimensional monotone Lagrangian equipped with the trivial local system, we have
if and only if , and in the latter case . Indeed, computes the Poincaré dual of the Floer differential where is the generator of ; if we pick a perfect Morse function on , then is geometrically realised by its maximum. If , then by duality the unit is not hit by the differential, hence . For a non-monotone , the condition from Equation (1.3) above is a natural low-area version of the non-vanishing of Floer cohomology.
Recall that is well-defined up to as explained in Section 1. This is analogous to a well known phenomenon in monotone Floer theory: recall that there is no canonical identification between and , even when they are abstractly isomorphic [8, Section 4.5]. In particular, is only -graded and the element corresponding to the degree 2 generator of is defined up to adding a multiple of the unit . Recall that is dual to , and this matches with the fact that , as well as , is defined up to .
Charette  defined quantum Reidemeister torsion for monotone Lagrangians whose Floer cohomology vanishes. While it is possible that his definition generalises to the non-monotone setting, making our tori valid candidates as far as classical Floer theory is concerned, it is shown in [14, Corollary 4.1.2] that quantum Reidemeister torsion is always trivial for tori.
2.2. Proof of Theorem 5
Our proof essentially follows [7, Theorem 8.1] with the following differences: we check that certain unwanted bubbling, impossible in the monotone case, does not occur in our setting given that ; we include an argument which “singles out” the contribution of Maslov index 2 disks with non-trivial boundary from that of Maslov index 4 disks; and relate the string invariants , defined in Section 1 to the ones appearing more naturally in pearly trajectory setup. We assume that the reader is familiar with the setup moduli spaces of pearly trajectories .
Under the assumptions of Subsection 1.2, the string invariants and are independent of the choice of and the marked point .
First, we claim that for a generic 1-parametric family of almost complex structures, will not bound holomorphic disks of Maslov index . Indeed, for simple disks this follows for index reasons (recall that ); next, non-simple disks with must have an underlying simple disk with by the decomposition theorem of Kwon and Oh  and Lazzarini , so the non-simple ones do not occur as well.
Therefore, the only way disks with and area can bubble is into a stable disk consisting of and disks; the latter disk must have positive area less than . However, such disks do not exist by Condition (1.1). We conclude that Maslov index 2, area disks cannot bubble as we change , and because the string invariants are defined in terms of these disks, they indeed do not change. A similar argument (by moving the point along a generic path) shows the independence of . ∎
Suppose there exists a Hamiltonian diffeomorphism such that , and redenote by , so that .
Pick generic metrics and Morse functions on . We assume that the functions are perfect (it simplifies the proof, but is not essential); such exist because are two-dimensional and orientable. Consider the moduli space of configurations (“pearly trajectories”) of the three types shown in Figure 1, with the additional condition that the total boundary homology classes of these configurations are non-zero both in and . (By writing “total” we mean that if the configuration’s boundary on a single Lagrangian has two components, their sum must be non-zero.)
Here are the details on the pearly trajectories from Figure 1 that we use to define :
The Maslov index and the area of each curve is prescribed in the figure;
The conformal parameter of each annulus is allowed to take any value . Recall that the domain of an annulus with conformal parameter can be realised as ;
Every flowline has a time-length parameter that can take any value . The configurations with a contracted flowline (i.e. one with ) correspond to interior points of , because gluing the disk to the annulus is identified with becoming negative;
The annulus has two marked points, one on each boundary component, that are fixed in the domain. This means that if we identify an annulus with conformal parameter with , then the two marked points can be e.g. and ;
The disks also have marked points as shown in the figure. Because the disks are considered up to reparametrisation, the marked points can also be assumed to be fixed in the domain;
The non-vanishing of total boundary homology classes (stated above) must be satisfied.
Recall that the Fredholm index of unparametrised holomorphic annuli without marked points and with free conformal parameter equals the Maslov index. Computing the rest of the indices, one shows that is a smooth 1-dimensional oriented manifold [7, Section 8.2], assuming is regular. The regularity of the annuli can be achieved by a small domain-dependent perturbation of the -holomorphic equation; we give a detailed discussion of it in the next subsection. Now, we continue with the proof assuming the regularity of the annuli (and hence of , because the transversality for disks is classical).
The space can be compactified by adding configurations with broken flowlines as well as configurations corresponding to the conformal parameter of the annulus becoming or . We describe each of the three types of configurations separately and determine their signed count.
(i) The configurations with broken flowlines are shown in Figure 2. As before, they are subject to the condition that the total boundary homology classes of the configuration are non-zero both in and . The annuli have a certain conformal parameter and the breaking is an index 1 critical point of [7, Section 8.2.1, Item (a)].
The count of the sub-configurations consisting of the disk and the attached flowline vanishes: this is a Morse-theoretic restatement Condition (1.3) saying that . Hence (by the perfectness of the ) the count of the whole configurations in Figure 2 also vanishes, at least if we ignore the condition of non-zero total boundary. Separately, the count of configurations in Figure 2 whose total boundary homology class is zero either in or , also vanishes. Indeed, suppose for example that the disk in Figure 2 (left) has boundary homology class and the lower boundary of the annulus has class ; then the count of the configurations in the figure with that disk and that annulus equals the homological intersection , since is an oriented surface. We conclude that the count of configurations in the above figure whose total boundary homology classes are non-zero, also vanishes.
(ii) The configurations with contain a curve whose domain is an annulus with a contracted path connecting the two boundary components. The singular point of this domain must be mapped to an intersection point , so these configurations do not exist if [7, Section 8.2.1, Item (c)].
(iii) The configurations with correspond to an annulus breaking into two disks, one with boundary on and the other with boundary on [7, Section 8.2.1, Item (d)]. One of the disks can be constant, and the possible configurations are shown in Figure 3.
In fact, there is another potential annulus breaking at that we have ignored: the one into a Maslov index 4 disk on one Lagrangian and a (necessarily constant) Maslov index 0 disk on the other Lagrangian, see Figure 4. This broken configurations cannot arise from the configurations in by the non-zero boundary condition imposed on the elements of this moduli space. The fact that a Maslov index 0 disk has to be constant is due to the generic choice of .
In the right-most configuration in Figure 3, forget the disk so that one endpoint of the -flowline becomes free; let be the singular 2-chain on swept by these endpoints. In other words, for each disk , consider the closure of
oriented so that the component of corresponding to has the same orientation as . Then .
We claim that on chain level. Indeed, the boundary corresponds to zero-length flowlines that sweep , and to flowlines broken at an index 1 critical point of , shown below:
The endpoints of these configurations sweep the zero 1-chain. Indeed, we are given that so the algebraic count of the appearing index 1 critical points represents a null-cohomologous Morse cocycle, therefore this count equals zero by perfectness of . It follows that . Similarly, define the 2-chain on , , by forgetting the disk in the second configuration of type (iii) above, and repeating the construction. It follows that the homology class from Subsection 1.2 can be represented by the cycle
and similarly can be represented by . This intersection number can be expanded into four chain-level intersections:
The last summand vanishes because , and the other summands correspond to the three configurations of type (iii) pictured earlier. ∎
Note that the equality between the intersection number and the count of the boundary points of holds integrally, i.e. with signs. This follows from the general set-up of orientations of moduli spaces in Floer theory, which are consistent with taking fibre products and subsequent gluings, see e.g. [1, Appendix C] for the case most relevant to us. For example, in our case the signed intersection points between a pair of holomorphic disks can be seen as the result of taking fibre product along evaluations at interior marked points; therefore these intersection signs agree with the orientations on the moduli space of the glued annuli.
If the moduli space is completed by the above configurations (i)—(iii), it becomes compact. Indeed, by the condition , Maslov index 2 disks on with area higher than cannot bubble. Disks of Maslov index cannot bubble (for finite ) on either Lagrangian because the rest of the configuration would contain an annulus of Maslov index passing through a fixed point on the Lagrangian, and such configurations have too low index to exist generically (the annuli can be equipped with a generic domain-dependent perturbation of , hence are regular). Similarly, holomorphic disks of Maslov index cannot bubble as they do not exist for generic perturbations of the initial almost complex structure . (This is true for simple disks by the index formula, and follows for non-simple ones from the decomposition theorems [35, 33], as such disks must have an underlying simple disk with .) Side bubbles of Maslov index 2 disks (not carrying a marked point with a or a constraint) cannot occur because the remaining Maslov index 2 annulus, with both the and constraints, would not exist generically. Finally, as usual, sphere bubbles cannot happen in a 1-dimensional moduli space because such bubbling is a codimension 2 phenomenon in the monotone case.
2.3. Transversality for the annuli
In the proof of Theorem 5, we mentioned that in order to make the annuli appearing in the moduli space regular, we need to use a domain-dependent perturbation of the -holomorphic equation on those annuli. We wish to make this detail explicit.
First, let us recall the moduli space of domains used to define , see Figure 1, and its compactification. Recall that the annuli in were allowed to have free conformal parameter
and the limiting cases , were included in the compactifcation of as explained in the proof above.
Now, for each conformal parameter , we pick some closed disk inside the corresponding annulus:
depending smoothly on , and disjoint from the regions where the domains degenerate as or . To see what the last condition means, recall that the annulus used for defining has two fixed boundary marked points; we are assuming they are the points and . With these marked points, the annulus has no holomorphic automorphims. We can then uniformise the family of annuli as shown in Figure 5. To do so, we start with an annulus of a fixed conformal parameter, say . The annuli with are obtained from the annulus by performing a slit along a fixed line segment connecting the two boundary components. The annuli for are obtained by stretching the conformal structure of the annulus in a fixed neighbourhood of some core circle . In this presentation, all annuli have a common piece of domain away from a neighbourhood of , and it is notationally convenient to choose to be a fixed closed disk inside that common domain, for each . See Figure 5.
Next, let denote the space of compatible almost complex structures on . Let be the almost complex structure we have been using in the proof of Theorem 5—namely, we are given that the relevant -holomorphic disks are regular.
Now pick some domain-dependent almost complex structure
Using the above presentation where all the are identified with a fixed disk , it is convenient to take to be the same domain-dependent almost complex structure
for all , such that near .
In our (modified) definition of , we use the following domain-dependent perturbation of the -holomorphic equation for each of the appearing annuli:
Here is the complex structure on . Observe that (2.4) restricts to the usual -holomorphic equation, with the constant , away from . For this reason, the equation is -holomorphic in the neighbourhoods of the nodal points formed by: domain degenerations as and , and the side bubbling of holomorphic disks. The standard gluing and compactness arguments work in this setting, compare e.g. Sheridan [47, Proof of Proposition 4.3]; therefore, the and compactifications of from the proof of Theorem 5 are still valid in our perturbed case, as well as the fact that the flowline length 0 configurations in Figure 1 (middle and right) are interior points of . (This is a very hands-on version of the general notion of a consistent choice of perturbation data in the setup of by Seidel , see also , except that we are not using a Hamiltonian term in our equation.) We keep all disks appearing in to be -holomorphic, without any perturbation.
It is well known that the solutions to (2.4) are regular for a Baire set of , or equivalently for a Baire set of ’s [19, Lemma 4.1, Corollary 4.4]. The fact that a perturbation in a neighbourhood of a point is sufficient follows from the unique continuation principle for Cauchy-Riemann type equations and is contained in the statement of [19, Lemma 4.1].
In particular, there is a sequence converging to the constant :
such that the annuli solving (2.4) with respect to are regular for each .
The rest of the proof of Theorem 5 requires one more minor modification. Looking at Figure 3 (left), one of the holomorphic disks now carries a domain-dependent perturbation supported in the subdomain inherited from the annulus, compare Figure 5 (right). (Which of the two disks carries this perturbation depends on which side of the core circle the was in the annulus.) We note that the disks in Figure 3 (middle and right) do not carry a perturbation, as they arise as side bubbles from the annuli. We claim that for large enough , the count of the configurations in Figure 3 (left) where one of the disks carries an above perturbation, equals to the same count where the disks are purely -holomorphic. Indeed, it follows from the fact that , using continuity and the regularity of the unperturbed -holomorphic disks. This allows us to use Lemma 3 and conclude the proof.
2.4. Proof of Theorem 6
This is a simple modification of the proof of Theorem 5, so we shall be brief. The idea is to redefine the moduli space by considering only those configurations in Figure 1 whose total boundary homology classes in resp. belong to the affine subspace resp. .
The only difference in the proof arises when we argue that configurations of type (i) cancel, see Figure 2. At that point of the above proof, we used Condition (1.3); now we need to use Condition (1.4) instead. Let us consider configurations as in the left part of Figure 2. Assume that the area annulus in the figure has boundary homology class on . Then the area disk of the same configuration has boundary class belonging to the affine subspace ; this is true because the total boundary homology class has to lie in . By a Morse-theoretic version of Condition (1.4), the count of such area disks with the attached flowlines (asymptotic to index 1 critical points) vanishes. The rest of the proof goes without change.∎
2.5. Adjusting the area conditions
We would like to point out that the area restrictions in Theorems 5 and 6 can be weakened at the expense of requiring one of the two Lagrangians be monotone. In the setup of Section 1.2, suppose that is monotone, so that and is the monotonicity constant of , i.e. for . Below is a modified version of Theorem 5, which differs by the fact that the numbers become redefined compared to those from Section 1.2. We are still using a coefficient ring for homology.
Suppose are orientable Lagrangian surfaces, and is monotone. Fix any tame almost complex structure , and let be the moduli space of holomorphic disks in the homology class through a fixed point in . Define