Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology
We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in  with linearized Legendrian contact homology. More precisely, if is an exact Lagrangian submanifold of an exact symplectic manifold with convex end , where is a contact manifold and is a Legendrian submanifold, and if has empty concave end, then the linearized Legendrian contact cohomology of , linearized with respect to the augmentation induced by , equals the rational SFT of . Following ideas of P. Seidel , this equality in combination with a version of Lagrangian Floer cohomology of leads us to a conjectural exact sequence which in particular implies that if then the linearized Legendrian contact cohomology of is isomorphic to the singular homology of . We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of  in terms of the resulting isomorphism.
This paper is dedicated to Oleg Viro on the occasion of his 60th birthday.
Let be a contact -manifold with contact -form (i.e., is a volume form on ). The Reeb vector field of is the unique vector field which satisfies and . The symplectization of is the symplectic manifold with symplectic form where is a coordinate in the -factor. A symplectic manifold with cylindrical ends is a symplectic -manifold which contains a compact subset such that is symplectomorphic to a disjoint union of two half-symplectizations , for some contact -manifolds , where and . We call and of the positive- and negative ends of , respectively, and and of , - and -boundaries of , respectively.
The relative counterpart of a symplectic manifold with cylindrical ends is a pair of a symplectic -manifold with a Lagrangian -submanifold (i.e., the restriction of the symplectic form in to any tangent space of vanishes) such that outside a compact subset, is symplectomorphic to the disjoint union of and , where are Legendrian -submanifolds (i.e., are everywhere tangent to the kernels of the contact forms on ). If the symplectic manifold is exact (i.e., if the symplectic form on satisfies for some -form ) and if the Lagrangian submanifold is exact as well (i.e., if the restriction satisfies for some function ), then we call the pair of exact manifolds an exact cobordism. We assume throughout the paper that is simply connected, that the first Chern class of , viewed as a complex bundle using any almost complex structure compatible with the symplectic form on , is trivial and that the Maslov class of is trivial as well. (These assumptions are made in order to have well defined gradings in contact homology algebras over . In more general cases one would work with contact homology algebras with suitable Novikov coefficients in order to have appropriate gradings.)
In  a version of rational Symplectic Field Theory (SFT) (see  for a general description of SFT) for exact cobordisms with good ends, see Section 2, was developed. (The additional condition that ends be good allows us to disregard Reeb orbits in the ends when setting up the theory. Standard contact spheres as well as -jet spaces with their standard contact structures are good.) It associates to an exact cobordism , where has components, a -graded filtered -vector space , with filtration levels, and with a filtration preserving differential . Elements in are formal sums of admissible formal disks in which the number of summands with -action below any given number is finite, see Section 2 for definitions of these notions. The differential increases -action and hence if denotes divided out by the subcomplex of all formal sums in which all disks have -action larger than , then the differential induces filtration preserving differentials with associated spectral sequences . The projection maps , give an inverse system of chain maps. The limit is invariant under deformations of through exact cobordisms with good ends and in particular under deformations of through exact Lagrangian submanifolds with cylindrical ends, see Theorem 2.1.
In this paper we will use only the simplest version of the theory just described which is as follows. Let be an exact cobordism such that is connected and without negative end, i.e., . In this case, admissible formal disks have only one positive puncture and we identify (a quotient of) with the -vector space of formal sums of Reeb chords of . Furthermore our assumptions on and vanishing of and of the Maslov class of imply that the grading of a formal disks depends only on the Reeb chord at its positive puncture. We let denote the grading of a chord . Rational SFT then provides a differential
with , and which increases action in the sense that if denotes the action of the Reeb chord and if the Reeb chord appears with non-zero coefficient in then . Furthermore, since is connected the spectral sequences have only one level and
Our first result relates to linearized Legendrian contact cohomology, see Section 3. Legendrian contact homology was introduced in [11, 4]. It was worked out in detail in special cases including -jet spaces in [8, 9, 10]. From the point of view of Legendrian contact homology an exact cobordism with good ends induces a chain map from the contact homology algebra of to that of , in particular if then the latter equals the ground field with the trivial differential. Such a chain map is called an augmentation and gives rise to a linearization of the contact homology algebra of . That is, it endows the chain complex generated by Reeb chords of with a differential . The resulting homology is called -linearized contact homology and denoted . We let be the homology of the dual complex and call it the -linearized contact cohomology of .
We say that satisfies a monotonicity condition if there are constants and such that for any Reeb chord of , . Note that if is a -jet space or the sphere endowed with a generic small perturbation of the standard contact form and if is in general position with respect to the Reeb flow then satisfies a monotonicity condition.
Let be an exact cobordism with good ends. Let denote the positive end of and assume that the -boundary of is empty. Let denote the augmentation on the contact homology algebra of induced by . Then the natural map , which takes an element in thought of as a formal sum of co-vectors dual to Reeb chords in to the corresponding formal sum of Reeb chords in , is a chain map. Furthermore if satisfies a monotonicity condition then the corresponding map on homology
is an isomorphism.
We next consider two exact cobordisms and with good ends and with the following properties: both and have empty -boundaries, if denotes the -boundary of then , and intersect transversely, and the Reeb flow of along a Reeb chord connecting to is transverse to at its endpoint. For such pairs of exact cobordisms we define Lagrangian Floer cohomology as an inverse limit of the homologies of chain complexes generated by Reeb chords between and of action at most and by points in . This Floer cohomology has a relative -grading and is invariant under exact deformations of .
Consider an exact cobordism where has empty -boundary and -boundary . Let be a slight push off of which is an extension of a small push off of along the Reeb vector field.
For any , there is a long exact sequence
where is the ordinary homology of with -coefficients. It follows in particular, that if or then and the map , induced by the maps , is an isomorphism.
The author learned about the isomorphism above, between linearized contact homology of a Legendrian submanifold with a Lagrangian filling and the ordinary homology of the filling, from P. Seidel  who explained it using an exact sequence in wrapped Floer homology [1, 13] similar to (1.1). Borrowing Seidel’s argument, we outline a proof of Conjecture 1.2 in Section 4.4 in which the Lagrangian Floer cohomology plays the role of wrapped Floer homology.
In  a duality exact sequence for linearized contact homology of a Legendrian submanifold , where for some exact symplectic manifold and where the projection of into is displaceable, was found. In what follows we restrict attention to the case . Then every compact Legendrian submanifold has displaceable projection and the duality exact sequence is the following, where denotes any augmentation and where we suppress the ambient manifold from the notation,
Here, if , then the Poincaré dual of satisfies , where is the pairing between the homology and cohomology of . Furthermore, the maps and are defined through a count of rigid configurations of holomorphic disks with boundary on with a flow line emanating from its boundary, and the map is defined through a count of rigid holomorphic disks with boundary on with two positive punctures.
In , a generic Legendrian submanifold has finitely many Reeb chords. Furthermore, if is an exact Lagrangian cobordism in the symplectization with empty -boundary then is displaceable. Hence, both Theorem 1.1 and Conjecture 1.2 give isomorphisms. Combining (1.1) and (1.2) leads to the following.
Let be an exact Lagrangian cobordism in with empty -boundary and with -boundary and let denote the augmentation on the contact homology algebra of induced by . Then the following diagram with exact rows commutes and all vertical maps are isomorphisms
Here the top row is the long exact homology sequence of , the bottom row is the duality exact sequence, the map is the map in Conjecture 1.2, the map is analogous to , and the map counts disks in the symplectization with boundary on and with two positive punctures, see Section 4.5 for details.
2. A brief sketch of relative SFT of Lagrangian cobordisms
Although we will only use the simplest version of relative SFT introduced in  in this paper, we give a brief introduction to the full theory for two reasons. First, it is reasonable to expect that this theory is related to product structures on linearized contact homology, see , in much the same way as the simplest version of the theory appears in Conjecture 1.2. Second, some of the moduli spaces of holomorphic disks that we will make use of are analogous to those needed for more involved versions of the theory.
2.1. Formal- and admissible disks
In order to describe relative rational SFT we introduce the following notation. Let be an exact cobordism with ends . Write for a compact part of obtained by cutting the infinite parts of the cylindrical ends off at some . We will sometimes think of Reeb chords of in the -boundary as lying in with endpoints on . A formal disk of is a homotopy class of maps of the -disk , with marked disjoint closed subintervals in , into , where the marked intervals are required to map in an orientation preserving (reversing) manner to Reeb chords of in the -boundary (in the -boundary) and where remaining parts of the boundary map to .
If and are exact Lagrangian cobordisms in and , respectively such that a component of the -boundary of agrees with a component of the -boundary of then these cobordisms can be joined to an exact cobordism in , where is obtained by gluing the positive end of to the corresponding negative end of . Furthermore if and are collections of formal disks of and , respectively, then we can construct formal disks in in the following way: start with a disk from , let denote the Reeb chords at its negative punctures. Attach positive punctures of disks in mapping to the Reeb chords to the corresponding negative punctures of the disk . This gives a disk with some positive punctures mapping to chords of . Attach negative punctures of disk to at . This gives a disk with some negative punctures mapping to Reeb chords in . Continue this process until there are no punctures mapping to . We call the resulting disk a formal disk in with factors from and .
Assume that the set of connected components of has been subdivided into subsets so that is a disjoint union where each is a collection of connected components of . We call the pieces of . With respect to such a subdivision, Reeb chords fall into two classes: pure with both endpoints on the same piece and mixed with endpoints on distinct pieces.
A formal disk represented by a map is admissible if for any arc in which connects two unmarked segments in that are mapped to the same piece by , all marked segments on the boundary of one of the components of map to pure Reeb chords in the -boundary.
2.2. Holomorphic disks
Let be an exact cobordism. Fix an almost complex structure on which is adjusted to its symplectic form. Let be a punctured Riemann surface with complex structure and with boundary . A -holomorphic curve with boundary on is a map such that
and such that . For details on holomorphic curves in this setting we refer to [5, Appendix B] and references therein. Here we summarize the main properties we will use. By definition, an adjusted almost complex structure is invariant under -translations in the ends of and pairs the Reeb vector field in the -boundary with the symplectization direction. Consequently, strips which are cylinders over Reeb chords as well as cylinders over Reeb orbits are -holomorphic. Furthermore, any -holomorphic disk of finite energy is asymptotic to such Reeb chord strips at its boundary punctures and to Reeb orbit cylinders at interior punctures, see [5, Section B.1]. We say that a puncture of a -holomorphic disk is positive (negative) if the disk is asymptotic to a Reeb chord strip (Reeb orbit cylinder) in the positive (negative) end of . Note that exactness of and the fact that the symplectic form is positive on -complex tangent lines imply that any -holomorphic curve has at least one positive puncture.
These results on asymptotics imply that any -holomorphic disk in with boundary on determines a formal disk. Let denote the moduli space of -holomorphic disks with associated formal disk equal to . The formal dimension of is determined by the Fredholm index of the linearized -operator along a representative of , see [5, Section 3.1].
A sequence of -holomorphic disks with boundary on may converge to a broken disk of two components which intersects at a boundary point. We will refer to this phenomenon as boundary bubbling. However, if all elements in the sequence have only one positive puncture then boundary bubbling is impossible by exactness: each component in the limit curve must have at least one positive puncture. The reason for using admissible disks to set up relative SFT is the following: in a sequence of holomorphic disks with corresponding formal disks admissible, boundary bubbling is impossible for topological reasons. As a consequence, if is a formal disk then the boundary of consists of several level -holomorphic disks and spheres joined at Reeb chords or at Reeb orbits, see .
Recall from Section 1 that we require the ends of our exact cobordisms to be good. The precise formulation of this condition is as follows. If () is a Reeb orbit in the -boundary (in the -boundary ) of then the formal dimension of any moduli space of holomorphic spheres in (in ) with positive puncture at (at ) is . Together with transversality arguments these conditions guarantee that broken curves in the boundary of , where is an admissible formal disk cannot contain any spheres, if ( if is a trivial cobordism), see [5, Lemma B.6]. In particular, in the boundary of , where satisfies these dimensional constraints and where is admissible, there can be only two level curves, all pieces of which are admissible disks, see [5, Lemma 2.5].
Under our additional assumptions ( trivial, first Chern class of and Maslov class of vanish) the grading of a formal disk depends only on the Reeb chords at its punctures. For later reference, we describe this more precisely in the case when is connected and when its -boundary is empty. Let denote the -boundary of . If is a Reeb chord of then let be any path in joining its endpoints. Since is simply connected bounds a disk . Fix a trivialization of along such that the linearized Reeb flow along is represented by the identity transformation with respect to this trivialization. Then the tangent space at the initial point of is transported to a subspace in the tangent space at the finial point of where is transverse to in the contact hyperplane at . Let denote a negative rotation along the complex angle taking to in , see [5, Section 3.1], then the Lagrangian tangent planes of along capped off with form a loop of Lagrangian subspaces in with respect to the trivialization and if denotes the moduli space of holomorphic disks in with one positive boundary puncture at which they are asymptotic to the Reeb chord strip of the Reeb chord then
see [5, p. 655]. To see that this is independent of , note that the difference of two trivializations along the disks is measured by . To see that it is independent of the path , note that the difference in the dimension formula corresponding to two different paths and is measured by the Maslov class of evaluated on the loop . Define
If and are Reeb chords of and if denotes the moduli space of holomorphic disks in with boundary on then additivity of the index gives
2.3. Hamiltonian- and potential vectors and differentials
Let be an exact cobordism and let be a formal disk of . Define the -action of as the sum of the actions
over the Reeb chords at its positive punctures. Here is the contact form in the -boundary of . Note that for generic Legendrian -boundary, , the set of actions of Reeb chords is a discrete subset of . Let denote the -graded vector space over with elements which are formal sums of admissible formal disks which contain only a finite number of summands below any given -action. The grading on is the following: the degree of a formal disk is the formal dimension of the moduli space of -holomorphic disks homotopic to the formal disk. We use the natural filtration
of , where is the number of pieces of and where the filtration level is determined by the number of positive punctures. (It is straightforward to check that an admissible formal disk has at most Reeb chords at the positive end).
We will define a differential which respects this filtration using -dimensional moduli spaces of holomorphic disks. To this end, fix an almost complex structure on which is compatible with the symplectic form and adjusted to in the ends, where is the contact form in -boundary. Assume that is generic with respect to - and -dimensional moduli spaces of holomorphic disks, see [5, Lemma B.8]. Since is invariant under translations in the ends, acts on moduli spaces , where is a formal disk of . In this case we define the reduced moduli spaces as . Let denote the vector of admissible formal disks in with boundary on represented by -holomorphic disks:
where the sum ranges over all formal disks of and where denotes the -number of points in the compact -manifold . We call and the Hamiltonian vectors of the positive and negative ends, respectively. Similarly, let denote the generating function of rigid disks in the cobordism:
where the sum ranges over all formal disks of . We call the potential vector of .
We view elements in and as sets of admissible formal disks, where the set consists of those formal disks which appear with non-zero coefficient in . Define the differential as the linear map such that if is an admissible formal disk (a generator of ) then is the sum of all admissible formal disks obtained in the following way.
Attach a positive puncture of to a negative puncture of a -disk, or
attach a negative puncture of to a positive puncture of a -disk.
If the first step was then attach -disks at remaining negative punctures of the -disk, or
if the first step was then attach -disks at remaining positive punctures of the -disk.
The fact that this is a differential is a consequence of the product structure of the boundary of the moduli space mentioned above in the case of -dimensional moduli spaces, see [5, Lemma 3.7]. Furthermore, the differential increases grading by and respects the filtration since any disk in or in has at least one positive puncture.
2.4. The rational admissible SFT spectral sequence
Fix . If denotes the subspace of formal sums of formal disks with -action at least then since holomorphic disks have positive symplectic area it follows that . If then is isomorphic to the vector space generated by formal disks of -action and there is a short exact sequence of chain complexes
The quotients form an inverse system
of graded chain complexes, where are the natural projections. Consequently, the -level spectral sequences corresponding to the filtrations
which we denote
form an inverse system as well and we define the rational admissible SFT invariant as
This is in general not a spectral sequence but it is under some finiteness conditions. The following result is a consequence of [5, Theorems 1.1 and 1.2].
Let be an exact cobordism with a subdivision into pieces. Then does not depend on the choice of adjusted almost complex structure , and is invariant under deformations of through exact cobordisms with good ends.
Any such deformation can be subdivided into a compactly supported deformation and a Legendrian isotopy at infinity. The former type of deformations are shown to induce isomorphisms of in [5, Theorem 1.1.].
To show that the later type of deformation induces an isomorphism, we note that it gives rise to an invertible exact cobordism, see [5, Appendix A], and use the same argument as in the proof of [5, Theorem 1.2.] as follows. Let be the exact cobordism of the Legendrian isotopy at infinity and let be its inverse cobordism. We use the symbol to denote the result of joining two cobordisms along a common end. Consider first the cobordism
Since this cobordism can be deformed by a compact deformation to we find that the composition of the maps and is chain homotopic to identity. Hence is injective on homology. Consider second the cobordism
Since this cobordism can be deformed to we find similarly that there is a map such that is chain homotopic to the identity on hence is surjective on homology as well. ∎
2.5. A simple version of rational admissible SFT
As mentioned in Section 1, in the present paper, we will use the rational admissible spectral sequence in the simplest case: for where has only one component. Since there is only one piece, the spectral sequence has only one level and
is the invariant we will compute. To simplify things further we will work not with the chain complex as described above but with the quotient of it obtained by forgetting the homotopy classes of formal disks. We view this quotient, using our assumption trivial, as the space of formal sums of Reeb chords of the -boundary of . Further, our assumptions and vanishing Maslov class of implies that the grading descends to the quotient, see (2.1). For simplicity we keep the notation and for the corresponding quotients.
3. Legendrian contact homology, augmentation, and linearization
In this section we will define Legendrian contact homology and its linearization. We work in the following setting: is an exact cobordism with good ends, the -boundary of is empty and the -boundary of will be denoted .
Recall that the assumption on good ends allows us to disregard Reeb orbits. Furthermore, our additional assumptions on , trivial and first Chern class and Maslov class trivial, allows us to work with coefficients in and still retain grading.
3.1. Legendrian contact homology
Assume that is generic with respect to the Reeb flow on . If is a Reeb chord of , let be as in (2.1).
The DGA of is the unital algebra over generated by the Reeb chords of . The grading of a Reeb chord is .
The contact homology differential is the map which is linear over , which satisfies Leibniz rule, and which is defined as follows on generators:
where is a Reeb chord and is a word of Reeb chords. (For notation, see (2.2).)
We give a brief explanation of why in Definition 3.2 is a differential, i.e., why . Consider the boundary of the -dimensional moduli space (which become -dimensional after the -action has been divided out) of holomorphic disks with one positive puncture at . As explained in Section 2.2, the boundary of such a moduli space consists of two level curves with all components except two Reeb chord strips. Since these configurations are exactly what is counted by and since they correspond to the boundary points of the compact -manifold , we conclude that .
An augmentation of is a chain map , where is equipped with the trivial differential.
Given an augmentation , define the algebra isomorphism by letting
for each generator . Consider the word length filtration of ,
The differential respects this filtration: . In particular, we obtain the -linearized differential
The -linearized contact homology is the -vector space
For simpler notation below we write
and think of it as the graded vector space generated by the Reeb chords of . Furthermore, the augmentation will often be clear from the context and we will drop it from the notation and write the differential as
Consider an exact cobordism with -boundary and -boundary . Define the algebra map by mapping generators of as follows
where is a word of Reeb chord of , where denotes the moduli space of holomorphic disks in with boundary on , with positive puncture at and negative punctures at . An argument completely analogous to the argument above showing , looking at the boundary of -dimensional moduli spaces shows that , where is the differential on , i.e., that is a chain map. Consequently, if is an augmentation then so is . In particular if and then with the trivial differential and is an augmentation of .
3.2. Proof of Theorem 1.1
If denotes the subspace of generated by Reeb chords of action then is a subcomplex of . By definition, the map which takes a Reeb chord viewed as a generator of to the dual of in the co-chain complex of is an isomorphism intertwining the respective differentials. To prove the theorem it thus remains only to show that the monotonicity condition implies for large enough. This is straightforward: if then
4. Lagrangian Floer homology of exact cobordisms
In this section we introduce a Lagrangian Floer cohomology of exact cobordisms. It is a generalization of the two copy version of the relative SFT of an exact cobordism , to the case when each piece of is embedded but . To prove that this theory has desired properties we will use a mixture of results from Floer homology of compact Lagrangian submanifolds and the SFT framework explained in Section 2. After having set up the theory we state a conjectural lemma about how moduli spaces of holomorphic disks with boundary on , where is a small perturbation of , can be described in terms of holomorphic disks with boundary on and a version of Morse theory on . We then show how Conjecture 1.2 and Corollary 1.3 follow from this conjectural description.
4.1. The chain complex
Let be a simply connected exact symplectic cobordism with and with good ends. Let and be exact Lagrangian cobordisms in with empty negative ends and with trivial Maslov classes. In other words, and are exact cobordisms with empty -boundaries. Let the -boundaries of and be and , respectively.
as follows. The summand is the -vector space of formal sums of Reeb chords which start on and end on . The summand is the -vector space generated by the transverse intersection points in .
In order to define grading and a differential on we will consider the following three types of moduli spaces. The first type was considered already in (2.2), if is a Reeb chord and is a word of Reeb chords we write
for the moduli space of holomorphic curves in the symplectization with boundary on , with positive puncture at and negative punctures at .
The second kind is the standard moduli spaces for Lagrangian Floer homology, if and are intersection points of and we write
for the moduli space of holomorphic disks in with two boundary punctures at which the disks are asymptotic to and , with boundary on , and which are such that in the orientation on the boundary induced by the complex orientation the incoming boundary component at maps to .
Finally, the third kind is a mixture of these, if is a Reeb chord connecting to and if is an intersection point of and we write for the moduli space of holomorphic disks in with two boundary punctures at one the disk has a positive puncture at and at the other the disks is asymptotic to .
If and are both Reeb chord generators then we define the grading difference between and to equal the formal dimension . If is a Reeb chord- or an intersection point generator and if is an intersection point generator then we take the grading difference between and to equal . Our assumptions on the exact cobordism guarantee that this is well defined.
Write , , and . Define the differential , using the decomposition to be given by the matrix
where , , and