Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.


Introduction
In this article we consider a closed, connected symplectic manifold (M, ω), which is integral, that is, [ω] ∈ H 2 (M ; Z). Furthermore, let L ⊂ M be a closed Lagrangian submanifold. Throughout this article assume that the pair (M, L) is symplectically aspherical (see equations (2.1) and (4.33) for the definition). Pre-quantization spaces and Bohr-Sommerfeld pairs naturally arise in geometric quantization theory. Both notions appear in various places in the literature. For the Lagrangian case of Bohr-Sommerfeld we refer the reader for instance to Eliashberg-Hofer-Salamon [EHS95], Eliashberg-Polterovich [EP00], and Ono [Ono96].
To a Bohr-Sommerfeld pair (E, α) for (M, ω, L) we naturally associate a Legendrian submanifold L in a pre-quantization space of (M, ω) as follows. The hyperplane distribution ξ := ker α restricted to the unit circle bundle Σ of E is a contact structure on Σ. Condition (2) in Definition 1.1 implies that L lifts to a Legendrian submanifold L of ( Σ, ξ). The group Z/2 acts on ( Σ, ξ, L) by e → −e. The quotient is denoted by (Σ, ξ, L). We note that L is diffeomorphic to L. This is not the case if we don't divide out by the Z/2-action.
Given a positive, autonomous Hamiltonian function H ∈ C ∞ (M ) on the base M we denote by α H the contact form on (Σ, ξ) which is induced by the S 1 -invariant contact form 1 N H α on Σ. We denote by R L (H) the set of Reeb chords of the triple (Σ, α H , L) and by R 1 L (H) the set of Reeb chords of period strictly less than 1. 1 The set of contractible intersection points L ∩ φ 1 H (L) of L and its image under the time-1-map φ 1 H of the Hamiltonian flow of H is denoted by P L (H). 2 The close connection between Reeb chords and Lagrangian intersection points was already fruitfully applied in the work of Eliashberg-Hofer-Salamon [EHS95], Givental [Giv89,Giv90a,Giv90b], and Ono [Ono96].
Our first main result gives a lower bound on the number of Reeb chords of period less than 1 in terms of the number of Hamiltonian chords of period equal to 1. The proof uses the observation that Reeb chords are in 1-1 correspondence to Hamiltonian chords with quantized action, see Proposition 5.15. Our result shows that in a certain sense the time-one dynamics "remembers the past" as phrased by Leonid Polterovich.
We recall that a subset of a topological space is called generic if it is a countable intersection of open and dense sets. It follows from Baire's theorem that generic subsets of C ∞ (M ) are dense. To a Hamiltonian function H : M −→ R we assign the following finite data set where A H is the action functional (see equation (2.4)) and µ L Maslov is the Maslov index as defined in [RS93]. Remark.
• In Section 5 we introduce the two notions of a huge and a non-resonant Hamiltonian function. Moreover, we define the wiggliness W(D(H)) ∈ N of a Hamiltonian function. Then in Theorem A we have N (D(H)) = W(D(H)) and C(D(H)) is so that H + C is huge. In fact, any Hamiltonian function H becomes huge after adding a sufficiently large constant. Moreover, the wiggliness of a Hamiltonian function H is large if H has 1-periodic orbits with small but non-zero difference in action values. Finally, the nonresonancy condition is the generic property appearing in Theorem A. It guarantees that the action functionals detecting intersection points and Reeb chords are Morse. • We point out that Reeb dynamics of α H+c (in particular the number #R L (H + c)) is sensitive to adding constants c while P L (H) is unaffected. • In fact, the period of the Reeb chords found in Theorem A is bounded below by a constant τ (H) > 0 depending on the wiggliness and the local behavior of H near L. Moreover, we get information on the action of the Reeb chords. We refer the reader to Theorem 5.21 for the full statement. • We note that the Bohr-Sommerfeld property is stable under taking tensor powers. In particular, whenever there exists a Bohr-Sommerfeld pair (E, α) for (M, ω, L) then a suitable high tensor power of (E, α) will satisfy the assumption of Theorem A. • The same techniques used to prove Theorem A can be adapted to obtain an analogue of Theorem A for the number of closed Reeb orbits in terms of the number of contractible fixed points. In the periodic case multiple covers of a Reeb orbit contribute to the count. However, it should be possible to use the information on action, period, and index to get estimates for the number of geometrically distinct Reeb orbits. This will be treated in the future.
Floer's theorem gives a lower bound for P L (H) in topological terms of L. Thus, we obtain Theorem B is proved as Proposition 3.20 (periodic case) and Proposition 5.7 (Lagrangian case). Theorem A follows from the Lagrangian version of Theorem B in the following way. If H is positive and autonomous then there exists a compact perturbation of the quadratic lift of H such that the action functional of this perturbation detects Reeb orbits resp. chords. Theorem A follows then from Theorem B together with the invariance of Floer homology under compact perturbations. The factor 1 2 in Theorem A is due to the Z/2-symmetry which was divided out to obtain the space Σ from Σ.
Remark 1.5. The periodic case of Theorem B could be used to prove a periodic version of Theorem A. Unfortunately, as such it's not very interesting because Reeb orbits can be iterated and iterates potentially contribute to the set R 1 (H). However, since we have additional information about period and action of the Reeb orbits a refined analysis should lead also to non-trivial estimates in the periodic case. We plan to treat this in the future. Organization of the article. In Section 2 we review the construction of classical Floer homology. In Section 3 we construct Floer homology for negative line bundles in the periodic case. In Subsection 3.1 we describe the symplectic geometry of negative line bundles and introduce the notion of strongly nondegenerate Hamiltonian function. The necessary C 0 -estimates are proved in Subsection 3.2. We show a subharmonic estimate which generalizes the known results in symplectic homology. In Subsection 3.3 we compare the indices of the action functional of classical mechanics on the base and on the total space of the bundle. We define the new Floer homology and corresponding continuation homomorphism in Subsection 3.4. Theorem B is proved as Proposition 3.20 and Proposition 5.7. In Section 4 we treat the construction of Floer homology of negative line bundles in the Lagrangian case. For this we extend the previously proved C 0 -estimates to Lagrangian boundary conditions by a reflection argument. Section 5 contains the applications to Hamiltonian/Reeb chords. Theorem A is a special case of Theorem 5.21. In Appendix A we prove that being non-resonant is a generic property in dimensions higher than 2. In Appendix B we prove a Poincaré-type theorem for the local behavior of Hamiltonian chords. In Appendix C we prove a Morse condition for the perturbed action functional. Finally, in Appendix D we collect some well-known facts about holonomy of tensor products of line bundles.

Floer homology for closed symplectic manifolds
2.1. The periodic case. In this section we briefly recall Floer's construction of his semiinfinite dimensional Morse homology on the free loop space. We follow closely Dietmar Salamon's lecture notes [Sal99]. Let (M, ω) be a closed connected symplectic manifold. We assume for simplicity that (M, ω) is symplectically aspherical, that is is called the Hamiltonian vector field of H. We denote by L the set of smooth, contractible 1-periodic loops in M . The subset of contractible 1-periodic orbits of X H is denoted by (2.3) Elements x ∈ P 1 (H) will also be referred to as (contractible) 1-periodic orbits of H. They are the critical points of the action functional of classical mechanics A H : L −→ R defined by for all x ∈ P 1 (H). This is implied by the requirement that graph(ϕ 1 H ) intersects the diagonal in M × M transversally. However, the latter condition is stronger since it implies (2.5) for all periodic orbits rather than only for contractible ones. Contractible periodic orbits of a nondegenerate Hamiltonian function are isolated. Thus, #P 1 (H) < ∞ since M is closed. To each periodic orbit x ∈ P 1 (H) the Conley-Zehnder index µ CZ (x; H) ∈ Z is assigned. This is well-defined as an integer due to the symplectic asphericity of (M, ω). The Conley-Zehnder index is normalized so that for a C 2 -small Morse function f we have (2.6) For a nondegenerate Hamiltonian function H Floer's complex (CF * (H), ∂(J, H)) is defined as follows. CF k (H) is generated over the field Z/2 by all periodic orbits with Conley-Zehnder index equal to k (2.7) To define the differential ∂(J, H) we choose an S 1 -family of ω-compatible almost complex structures J = J(t, ·) and consider solutions to Floer's equation, that is, maps u :   Counting elements of zero dimensional moduli space defines the differential ∂ = ∂(J, H) (2.14) The previous theorems imply that the boundary operator ∂ is well-defined and satisfies  . We recall the following well-known energy identity.
2.2. The relative case. Historically, the relative case of Floer homology was treated in fact before the absolute case in Floer's seminal article [Flo88].
As before (M, ω) is a closed connected symplectic manifold. Let L ⊂ M be a closed connected Lagrangian submanifold which is symplectically aspherical, that is µ Maslov | π 2 (M,L) = 0 and ω| π 2 (M,L) = 0 . (2.21) We denote by I the interval [0, 1] and let H : I × M −→ R be a smooth Hamiltonian function. In this case the action functional A H is defined on the space of contractible paths We denote D 2 + := {z ∈ D 2 | Im(z) ≥ 0}. Then for each x ∈ P we can choose a map x : D 2 + −→ M satisfyingx(e πit ) = x(t) andx(D 2 + ∩ R) ⊂ L. As in the periodic case the action functional of classical mechanics A H : P −→ R is defined by (2.23) This definition is independent of the choice ofx since L is symplectically aspherical. The set P 1 L (H) of critical points of A H are called Hamiltonian chords, i.e. P 1 There is an injective map from P 1 L (H) into the set of intersection points L ∩ ϕ 1 H (L) given by the evaluation map x → x(1). By symplectic asphericity the Maslov index µ Maslov induces a well-defined map which we denote by x → µ Maslov (x; H). Here, we use the Maslov index µ Maslov with the following normalization. For C 2 -small functions f whose restriction to L is Morse there is a 1:1 correspondence between the critical points Crit(f | L ) and Hamiltonian chords P 1 L (f ). Then the Maslov index is normalized by holds for all x ∈ P 1 L (H). For nondegenerate H the action functional A H is Morse. In this case we define Floer's complex (CF * (H; L), ∂(J, H)) as follows. The set CF k (H; L) is generated over the field Z/2 by all Hamiltonian chords with Maslov index k where k ∈ Z or k ∈ 1 2 + Z according to dim L = even or dim L = odd. To define the differential we consider the moduli space M L (x − , x + ; J, H) of perturbed holomorphic strips, that is, the set of solutions u : As in the periodic case blowing-up of derivatives in the interior leads to bubbling-off of holomorphic spheres. In addition, blowing-up of derivatives might occur at the boundary of the strip. This gives rise to bubbling-off of homomorphic disks with boundary on the Lagrangian submanifold L. Both of these phenomena are excluded by symplectic asphericity. 3. Floer homology for negative line bundles -the periodic case 3.1. Negative line bundles. As in Section 2.1 we assume that the symplectic manifold (M, ω) is closed, connected and symplectically aspherical. Moreover, we require the symplectic form to be integral, i.e.
(3.1) Therefore, for each N ∈ N we can choose a complex line bundle We continue to use the convention S 1 = R/Z. In particular, the Lie algebra equals R. With this convention the action of S 1 on the bundle E N is given by On E N we define a symplectic form Ω as follows. We choose a Hermitian connection 1-form α on E N \ M whose curvature F α = dα satisfies Furthermore, we fix the function f (r) = πr 2 + 1 N . Abbreviating r = ||e|| the following 2-form is a symplectic form on E N . We note that this is well-defined and satisfies Ω| M = ω since f ′ (0) = 0. Furthermore, on E N \ M the symplectic form can be written as Ω = d f (r)α . The vector field defined on E N \ M is a Liouville vector field for Ω, that is L X Ω = Ω, or equivalently f (r)α = ι X Ω. Here L denotes the Lie derivative. In particular, for all c > 1 N the manifold is of contact type. If we consider the canonical variable ρ = ln f (r) the Liouville vector field can be written as X = ∂ ∂ρ . (3.7) We note that the positive part of the symplectization of Σ c embeds into E N whereas the negative part only embeds partially. For a nondegenerate Hamiltonian function H : The connection 1-form α induces a natural splitting of T E N into horizontal and vertical subspaces where R is the unique vertical vector field satisfying α(R) = 1. We note that R restricts to the Reeb vector field of the contact manifold Σ c . Moreover, the projection of a 1-periodic solution of X b H is a 1-periodic solution of X H . Remark 3.1. For notational convenience we do not record the integer N in the notation of the function f , the symplectic form Ω, the lift H, etc.. Moreover, the above construction is canonical in the sense that  with respect to the splitting

maps horizontal vectors on horizontal vectors.
Proof. Since α is an Hermitean connection form dr vanishes on horizontal lifts, where r denotes the radial coordinate. Equations (3.10a) and (3.10b) imply that φ τ b H preserves the radial coordinate r. More precisely, we have the equality where a ∈ R >0 acts by multiplication in the fiber. This immediately implies that Dφ τ b H preserves the vector field ∂ ∂r and thus X according to equation (3.5). Thus, we conclude preserves the radial coordinate and dr vanishes on horizontal lifts we know that orbits are either entirely contained in the zero-section M or do not intersect M at all.
gives rise to a contact manifold ( Σ, α). By definition ( Σ, α) admits a canonical S 1 -action. Any S 1 -invariant contact form on Σ with the same co-orientation is of the form α H = 1 N H α for some autonomous, positive and S 1 -invariant function H : Σ −→ (0, ∞) which we identify with a function H : We recall that for a Hamiltonian function H on the base M we define in equation(3.8) the fiber-wise quadratic lift H to E N . The following lemma establishes a relationship between the Reeb vector field R H of α H and the Hamiltonian vector field of H.
Lemma 3.5. We fix a bundle p : (1) Assuming that H is nondegenerate, the following are equivalent.
(a) H is nondegenerate.
(c) All periodic orbits of H are contained in the zero-section M (and then are necessarily periodic orbits of H).
(2) Moreover, if there exists a 1-periodic solution e of X b H which is not contained in the zero-section M then all orbits z · e obtained by fiber-wise multiplication by z ∈ C are 1-periodic solutions of X b H . In particular, in both, the degenerate and the nondegenerate case.
Proof. Let e(t) be a 1-periodic solution of X b H and set parallel transport with respect to α along the path x and by P −t x its inverse. Let us assume that e(0) lies not in the zero-section. Since e is 1-periodic we conclude that the angle ∠ e(0), e(1) ∈ Z (due to our convention S 1 = R/Z).
We will compute this angle in two steps. We consider e 0 (t) : . Then the angle between e 0 (1) and P 1 x (e 0 (1)) = e(1) is given by the holonomy which equals, see where we chooseē : where i · e 0 (t) is multiplication by i ∈ C in the fiber E N x(0) . Thus, the angle ∠ e 0 (0), e 0 (1) between e 0 (0) = e(0) and e 0 (1) equals Then ∠ e(0), e(1) ∈ Z is equivalent to Before we prove the Lemma we observe that given x ∈ P 1 (H) and e 0 ∈ E x(0) the following path solves the ODEė (t) = X b H (t, e(t)) . (3.25) Moreover, e(1) = e(0) if and only if A H (x) ∈ 1 N Z by the computation above. We now prove part (2) of the Lemma. From equation (3.21) it is apparent that if e is a 1-periodic solution of X b H then so is z · e for any z ∈ C. In particular, if e does not lie in the zero section, by multiplication with z ∈ C we can fill the entire fibres over p(e) with periodic orbits. Since the action functional is constant on this critical manifold part (2) follows.
Let us prove part (1). (b) implies (c): We show that not (c) implies not (b). If there exists a 1-periodic orbit e ∈ P 1 ( H) not lying in the zero-section then the above discussion shows that A H (p(e)) ∈ 1 N Z. This implies not (b).
H (e) ∈ 1 N Z. As we concluded above this implies that the fibers over p(e) are filled entirely by 1-periodic orbits. This clearly shows that H is degenerate.
where the angle β = N · A b H (x) by the considerations from above. Since H is assumed to be nondegenerate Dφ H (x) has no eigenvalue equal to 1. Hence, Dφ b H (x) has an eigenvalue equal to 1 if and only if β = N · A b H (x) ∈ Z. (b) implies (c): We assume not (b) and (c). In particular, there exists x ∈ P 1 ( H) which is entirely contained in M and satisfies A H (x) ∈ 1 N Z. As we observed above the latter implies that x can be lifted via (3.24) to a loop e ∈ P 1 ( H) for any e 0 ∈ E x(0) . This clearly contradicts (c) and concludes the proof of the Lemma.
Remark 3.6. Let H : M −→ R be autonomous and g : R −→ R a smooth function. We consider a 1-periodic orbit e of H g := g( H), that is e solveṡ Since H is autonomous we compute In particular, g ′ H(e) is constant. This implies that the projection x(t) = p(e(t)) is (after reparametrization) a periodic orbit of H with period g ′ ( H(e)). In case that e is not contained in the zero section M the proof of Lemma 3.5 shows that Let H be a Hamiltonian function and c ∈ R then we denote by H c (t, x) := H(t, x) + c.
Definition 3.7. For a fixed N we call a nondegenerate Hamiltonian function H : S 1 × M −→ R strongly nondegenerate if the action spectrum of A H and 1 N Z are disjoint.
Corollary 3.8. If H is strongly nondegenerate then H : Proof. Both corollaries follow from Lemma 3.5 by noting that the number of critical values of A H for nondegenerate H is finite.
3.2. Convexity. In this section we prove a convexity result for a class of Hamiltonian functions in the symplectization of a contact manifold (Σ, ξ). We assume that ξ arises as the kernel of a contact form α. Then the symplectization can be written as (R × Σ, Ω := d(e ρ α)).
The symplectization admits the natural Liouville vector field X = ∂ ∂ρ which induces the flow φ t X (ρ, x) = (ρ + t, x). Furthermore, the Reeb vector field of α is denoted by R. We recall that it is uniquely defined by the properties α(R) = 1 and ι R dα = 0.
We denote by J Σ the space of almost complex structures J on R×Σ satisfying the following properties (1) J is invariant under the Liouville flow φ t X , (2) J(ξ) = ξ and is compatible with the fiber-wise symplectic structure dα on ξ, (3) J(X) = R. Such a J induces the Riemannian metric g(·, ·) = Ω(·, J·) on R × Σ. If we define the function f ∈ C ∞ (R × Σ) by f (ρ, x) := e ρ we obtain ∇f = X and g(X, X) = f . (3.30) We define the following class of Hamiltonian functions We point out that the Hamiltonian functions H as defined in Section 3.1 belong to H Σ .
Remark 3.10. If we extend H ∈ H Σ to R × Σ independently of the R-variable we obtain a φ X -invariant function which we denote by H again.
Proposition 3.11. Let U be some open subset of C and H ∈ H Σ . We consider a map u ∈ C ∞ (U, R × Σ) solving Floer's equation for a smooth family J(s, t) ∈ J Σ . Then (3.35) Therefore, L X Ω = Ω, Cartan's formula and ( * ) implies (3.36) Remark 3.12. If we consider s-dependent families H(s, t, x) where H(s, ·, ·) ∈ H Σ for all s then the result of the last Proposition is modified to Proof. This follows directly from equation (3.26) and the product property of the Conley-Zehnder index (see [Sal99, section 2.4]) by noting that µ CZ (e 2πiβt , t ∈ [0, 1]) = 2⌊β⌋ + 1.
Remark 3.16. The above index formula reflects the following symmetry breaking. If H : This is due to the fact that in the latter case H is negative quadratic in the fiber direction whereas in the former case it is positive quadratic.
Floer homology can now be defined as in Section 2.1, since (E N , Ω) is symplectically aspherical and convex at infinity. More precisely, by Corollary 3.13 all solutions of Floer's equation are contained within the compact set K, compare Remark 2.2. Thus, we obtain a complex (CF N * (H), ∂ N ). Definition 3.17. For a strongly nondegenerate H we set To extend the definition of continuation homomorphisms m(H 1 , H 0 ) from Section 2.1 to the current setting we need to ensure that the convexity at infinity applies to the moduli spaces M(x − , x + ; H s ) for a 1-parameter family H s . According to Corollary 3.13 this is the case if The following Proposition is proven as in the closed case, see for instance [Sal99,FH94].
where P 1 (H) is the set of contractible 1-periodic orbits of the Hamiltonian vector field of H.
Proof. This is an application of the index formula in Proposition 3.15. Since H is nondegenerate the set P 1 (H) is finite. Thus, we can choose an arbitrarily small c such that A H c has only irrational critical values. Now we choose N so large that for all x, y ∈ P 1 (H) with holds. This is possible according to Proposition 3.15. This implies that the boundary operator in Floer's complex vanishes, since action along gradient flow lines strictly decreases and the boundary operator is of degree −1.   • Since L is a Lagrangian submanifold and the curvature of E equals F α = N ω, the bundle (E| L , α| L ) is flat over L and thus, the holonomy homomorphism hol α| L : π 1 (L) −→ S 1 is well-defined. • If (E, α) is a Bohr-Sommerfeld pair for (M, ω, L) then so is E ⊗k , α ⊗k for any k ∈ N, and N (E ⊗k , α ⊗k ) = kN (E, α). We refer the reader to Proposition D.1 in the appendix for further details.
In the following we give two existence criteria for Bohr-Sommerfeld pairs, see Corollary 4.5 and Theorem 4.7. Proof. First we choose a connection 1-form α satisfying dα = F α = ω. The last equation determines α up to adding p * τ where τ ∈ Ω 1 (M ) is closed. Since L is Lagrange we conclude F α | L = 0, that is, the bundle E| L is flat, thus is defined. Since E| L is trivializable we can choose a connection 1-form α L on E| L with vanishing curvature and trivial holonomy. In particular, where β ∈ Ω 1 (L). Due to the vanishing of the curvature of both connections we conclude dβ = 0. Since by assumption i 1 : . By construction, we have β − τ | L = df for some function f : L −→ R. We extend f tof : M −→ R and set τ := τ + df . In particular, β = τ | L holds. We define α := α − p * τ .
(4.3) We notice that F ie α | L = 0 and has trivial holonomy.
Remark 4.4. In the above proof we construct a connection α with trivial holonomy. In particular, the bundle E| L −→ L is canonically trivialized via the parallel transport of α. (4.5) Note that the fixed point set of an anti-symplectic involution is a (maybe empty) Lagrangian submanifold of (M, ω). Since R * ω = −ω it follows that Hence as in the proof of Corollary 4.5 there exists n ∈ N such that We set E := E ⊗N ω with N := 2n. Thus, we obtain a complex line bundle p : E −→ M . We claim that the involution R now extends naturally to an S 1 -invariant involution R E of the bundle E with the property This is the content of Proposition 4.8 below. Assuming this fact we complete the proof of the theorem. We choose a connection 1-form α 0 on E satisfying We note that To compute the holonomy of α on the Lagrangian submanifold FixR we pick γ ∈ C ∞ (S 1 , FixR). Furthermore, we choose a loop e ∈ C ∞ (S 1 , E \ M ) satisfying p • e = γ. Then the holonomy of α along γ is given by (4.13) (4.14) Since γ takes values in FixR we have R • γ = γ, thus we finally conclude This implies that hol α (γ) ∈ {0, 1 2 } ⊂ S 1 . Hence the tuple (E, α) is a Bohr-Sommerfeld pair for (M, ω, FixR).
It remains to prove the following proposition.
The gauge group C ∞ (M, S 1 ) acts on solutions of (4.20) in the following way. Let γ ∈ C ∞ (M, S 1 ) and ψ be a solution of (4.20) then the map ψ γ : F −→ F defined by is also a solution of (4.21). Let ρ γ ∈ C ∞ (M, S 1 ) be as in (4.21) the obstruction for ψ γ to be an involution. Proof of the Lemma. For σ ∈ F and x = p(σ) ∈ M we compute This implies (4.26).
If F is a principal S 1 -bundle over M , then the tensor product of F with itself is given by (4.28) Here F × M F is the fiber product of F with itself over M . This is a principal torus bundle over M . Dividing out the antidiagonal∆ = (g, −g) : g ∈ S 1 ⊂ T 2 we obtain a principal S 1 -bundle over M again. A smooth map ψ : F −→ F satisfying (4.20) induces a map ψ 2 : F 2 −→ F 2 defined by Note that ψ 2 satisfies (4.20) for F 2 again. Let ρ 2 ∈ C ∞ (M, S 1 ) be the obstruction of ψ 2 to being an involution. Note that (4.30) Lemma 4.11. The map Ψ = (ψ 2 ρ = ρ.ψ 2 is an involution on F 2 . Proof of the Lemma. Using (4.22) and (4.26) we compute for x ∈ M This proves the Lemma.
To finish the proof of Proposition 4.8 we note that the complex line bundle E is by definition (4.32) Then the property p • R E = R • p follows immediately from the fact that Ψ is a solution of (4.20). This finishes the proof of Proposition 4.8.
Example 4.12. The Clifford torus T n ∈ CP n is the fixed point set of an anti-symplectic involution given by [z 0 : . . . : z n ] → [ z 0 |z 0 | 2 : . . . : zn |zn| 2 ]. Thus, Theorem 4.7 applies and T n is a Bohr-Sommerfeld Lagrangian, even though the simpler homological condition of Corollary 4.5 does not apply.

Definition of Floer Homology.
We are considering an integral, closed, symplectically aspherical symplectic manifold (M, ω). Furthermore, we assume that L ⊂ M is a closed Lagrangian submanifold which is symplectically aspherical: µ Maslov | π 2 (M,L) = 0 and ω| π 2 (M,L) = 0 . (4.33) Let p : E N −→ M be a complex line bundle and α a connection 1-form as in Section 3.1, that is c 1 (E) = −N [ω]. We assume that (E N , α) is a Bohr-Sommerfeld pair for (M, ω, L). By definition the power of (E N , α) is N . We fix an identification of a fiber E N x ∼ = C for some x ∈ L. Since the holonomy of α| L takes only values in {±1} parallel transport along any loop in L starting at x will map R ⊂ C ∼ = E N x into itself. Thus, parallel transport along paths in L defines a R-vector bundle L N over L. We obtain a non-compact Lagrangian submanifold L N ⊂ (E N , Ω) satisfying µ Maslov | π 2 (E N ,L N ) = 0 and Ω| π 2 (E N ,L N ) = 0 .
At the point (s, 0) we compute The last equality follows from the fact that both ∂ s u and X are tangent to the Lagrangian submanifold L N . The above computation implies F ∈ W 2,2 loc , thus the maximum principle applies to F : [−1, 1] × R −→ R. We conclude  (1) Assuming that H is nondegenerate, the following are equivalent.
(a) H is nondegenerate. (2) Moreover, if there exists a 1-periodic chord e of X b H which is not contained in the zero-section M then all chords r · e obtained by fiber-wise multiplication by r ∈ R are 1-periodic chords of X b H . In particular, in the degenerate and the nondegenerate case.
Proof. The proof is analogous to the proof of Lemma 3.5. The only modification is in the computation of the angle. In the relative case the Bohr-Sommerfeld condition (see Definition 4.1) is crucial. We denote x(t) := p(e(t)) We consider a Hamiltonian chord e ∈ P L N ( H) and choose a path γ : [0, 1] −→ L in L such that γ(0) = x(1) and γ(1) = x(0). This is possible since, by definition, [e] = 0 ∈ π 1 (E N , L N ). We study the parallel transport along the loop x#γ and consider the angle ∠ e(0), P 1 γ (e(1)) . We note that by the Bohr-Sommerfeld condition P 1 γ (e(1)) ∈ L N x(0) . In particular, we have ∠ e(0), P 1 γ (e(1)) ∈ 1 2 Z. On the other hand, as in the proof of Lemma 3.5, the angle computes to ∠ e(0), P 1 γ (e(1)) = N A H (x) ∈ 1 2 Z . (4.39) The remaining assertions follow as in the proof of Lemma 3.5.
Proof. This follows as in the proof of Proposition 3.15 and Lemma 4.14. We recall that in the absolute case HF N * (H) has been defined in Section 3.4.

Applications to Hamiltonian chords
In this section we continue to assume that (M 2n , ω) is a closed, connected, integral symplectic manifold and L ⊂ M a closed, connected, symplectically aspherical Lagrangian submanifold. Moreover, we assume that (E N , α) is a Bohr-Sommerfeld pair for (M, ω, L). We recall from the introduction. From now on we will only consider contractible Hamiltonian chords.
(1) If (x, τ ) is a Hamiltonian chord for H then also for H + c for any constant c ∈ R. Furthermore, where H c := H + c. Proof. We first recall the following inequalities: from which we obtain the following string of inequalities: Since the set P L (H) is finite the following quantities are well-defined (5.8) Obviously α > 0, therefore there exists N 0 ∈ N with We estimate for N ≥ N 0 and x, ξ ∈ P L (H) with A H (x) = A H (ξ) using Proposition 4.15 Thus, the wiggliness W(H) ≤ N 0 , thus finite. The second assertion is obvious from the definition. where the critical pointx of H| L corresponds to the Hamiltonian chord x. If H takes only positive values then − 1 N < A ǫH (x) < 0. Proof. Let p be a critical point of H| L . In case that dH(p) = 0 we are done, otherwise there exists a coordinate chart χ : V −→ U ⊂ R 2n with the following properties, where the coordinates on R 2n are denoted by (x 1 , . . . , x n , y 1 , . . . , y n ).
• χ(p) = 0 and ∃a i = 0 such that The unique (and nondegenerate) chord (x ǫ , y ǫ ) := (x ǫ 1 (t), . . . , x ǫ n (t), y ǫ 1 (t), . . . , y ǫ n (t)) of period ǫ is given by We first prove equation (5.15). In a Weinstein neighborhood of L we write H = H • π + h where π : T * L −→ L is the projection. The equality of the Maslov index and the Morse index can be seen by choosing a homotopy from H = H • π + h to H • π and noting that the Hamiltonian chords of ǫH • π are exactly the critical points for the Morse function H| L . Now assume in addition that H takes only positive values. We set c := H(p) > 0 and denote ξ ǫ (t) := χ −1 (x ǫ , y ǫ ). Then the above formulas imply Hence, for sufficiently small ǫ > 0, the action will satisfy the claimed inequality.
is called a N-quantized Hamiltonian chord. We denote the set of N -quantized chords with period less or equal than τ 0 by P q L (H; τ 0 , N ). The interest in quantized Hamiltonian chords comes from the relation to Reeb chords which we explain next. We recall that (E N , α) is a Bohr-Sommerfeld pair for (M, ω, L).
Lemma 5.13. We denote by ( Σ N , ξ) the contact manifold obtained from the S 1 -bundle of E N together with its horizontal plane field distribution induced by α. Then L N := L N ∩ Σ N is a Legendrian submanifold of ( Σ N , ξ).
Proof. Using the contact form α we decompose T e E N = T h e E N ⊕ T v e E N into horizontal and vertical part. Then vertical part T v e E N is spanned by the vectors R e and X e where R is the infinitesimal generator of the S 1 -action and X the Liouville vector field, see Section 3.1. Using the canonical identification of T h e E N ∼ = T p(e) M the definition of L N immediately implies T e L N ∼ = T p(e) L ⊕ RX e (5.23) and thus T e L N ∼ = T p(e) L (5.24) implies the claim.
Remark 5.14. If the first Stiefel-Whitney class w 1 (L N ) ∈ H 1 (L; Z/2) of L N vanishes then the Legendrian submanifold L has two connected components, otherwise one.
The group Z/2 acts on ( Σ, ξ, L) by e → −e. The quotient is denoted by (Σ, ξ, L). In particular, L is diffeomorphic to L.
As explained above Lemma 3.4 (where Σ is denoted by Σ etc.) every positive, autonomous Hamiltonian function H ∈ C ∞ (M ) gives rise to a S 1 -invariant contact form α H = 1 N H α on Σ inducing the same contact structure ξ. Since α H is S 1 -invariant it descends to a contact form on (Σ, ξ) which we denote by α H again. (1) H is strongly nondegenerate for N .
The following lemma provides a sufficient condition for the number of N -quantized chords to be finite. We point out that we assume do not assume the Bohr-Sommerfeld condition for L.
Lemma 5.18. We assume that L ⊂ (M, ω) is a closed, aspherical Lagrangian submanifold and that H : M −→ (0, ∞) is a positive Hamiltonian function satisfying the transversality conditions Dϕ τ H (T x(0) L) ⋔ T x(τ ) L, X H (x(0)) ∈ T x(0) L, and X H (x(τ )) ∈ T x(τ ) L for all Nquantized chords. Then the set P q L (H; τ 0 , N ) of N -quantized chords with period less or equal than τ 0 is finite.
Proof. The proof of this lemma is contained in the appendix, where this is Lemma B.5.
Remark 5.19. We point out that a priori condition (2) in the non-resonancy definition implies that a quantized chord (x, τ ) is isolated only in the set of τ -periodic chords and not necessarily in the set of all chords. The latter assertion is provided by the previous Lemma under the additional assumptions that the Hamiltonian function H satisfies X H (x(0)) ∈ T x(0) L, X H (x(τ )) ∈ T x(τ ) L, and H > 0. Without the assumption H > 0 quantized chords need not be isolated. We give a counterexample in the appendix, see Example B.7. for all x ∈ P(ǫH). We recall that the set of N -quantized chords with period less or equal than τ 0 is denoted by P q L (H; τ 0 , N ). We choose a function g : R ≥0 −→ R ≥0 satisfying (1) g ′ (ρ) = ǫ for ρ ≤ max H + δ where ǫ is chosen as above, (2) g(ρ) = ρ for ρ ≥ max H + 2δ, (3) g ′′ (ρ) > 0 for all ρ ∈ (max H + δ, max H + 2δ).
We claim that ǫ < τ i < 1. By properties (1) and (3) the inequality ǫ < τ i follows immediately. By definition we have τ i ≤ 1. In case τ i = 1 we conclude from equation (5.27) that X b Hg (e) = X b H (e). But by assumption H is non-resonant, in particular strongly nondegenerate, thus there are no Hamiltonian chords of X b H not lying in the zero-section M . The analog of Remark 3.6 in the relative case shows We set x i (t) = p e i (t/τ i ) (5.33) and note that x i is a Hamiltonian chord of H and has period τ i , that is (x i , τ i ) ∈ C(H).
It remains to show that A H (x, τ ) ∈ a min (H) − ||H||, a max (H) + max H for the Nquantized chords found above. This is done in two steps.
Let ξ ∈ P 1 L (H) be a 1-periodic chord of H. Then by Proposition 5.7 the chord ξ defines a non-vanishing homology class [ξ] ∈ HF N (H; L). Its image under the continuation isomorphism m( H g , H) : HF N (H; L) −→ HF( H g ; L N ) can be represented as a formal sum k ξ i=1 [y ξ i ] where y ξ i ∈ P L ( H g ). We first estimate the action value of y ξ i from above in terms of the action value of ξ. For this we interpolate between H and H g via the homotopy K s := β(s) H + (1 − β(s)) H g , where β(s) : R −→ [0, 1] is a smooth monotone cut-off function satisfying β(s) = 1 for s ≤ 0 and β(s) = 0 for s ≥ 1. According to Lemma 2.5 we have the following inequality for u ∈ M(ξ, y ξ i ; K s ) For simplicity we abbreviate y = y ξ i and denote the induced quantized chord by x(t) := p(y(t/τ )), where τ = g ′ ( H(y)). We want to find an upper bound on the action value A H (x, τ ) in terms of A b Hg (y). This is achieved as follows.
If we combine this with the two previous inequalities we obtain The lower bound on A H (x, τ ) is derived similarly by interchanging the roles of H and H g . This leads to: The last inequality follows from the fact that the function g −id : R ≥0 −→ R ≥0 is monotone decreasing and thus the function H g (e)− H(e) = (g−id)( H(e)) is maximal at min H = min H. From the inequality H g (min H) ≤ max H + 2δ − 1 m (||H|| + 2δ) (see figure 1) we conclude In the second last inequality we used again the definition of δ. Finally, we estimate Hg (y) = H g (y) − g ′ ( H(y)) H(y) and conclude (5.38)

A counterexample
We consider a closed, symplectically aspherical, and integral symplectic manifold (M, ω) which contains a Lagrangian sphere L of dimension at least 2. As Paul Biran explained to us there are plenty of examples, see Example 6.1. According to Corollary 4.5 there exists a Bohr-Sommerfeld pair (E, α) for (M, ω, L) of power N = 1.
On L we choose a Morse function f : L −→ R with two critical points. We extend f to a function H : M −→ R. After a perturbation we can achieve that H is non-resonant, see Theorem 5.20. Moreover, if we choose the perturbation small enough, we may assume that H| L still has exactly two critical points. After adding a suitable constant H takes only positive values.
We claim that ǫ 0 H has no 1-quantized chords. Hamiltonian chords of ǫ 0 H of period τ are Hamiltonian chords of τ ǫ 0 H of period 1. Thus, for 0 < τ < 1 we have to compute the Hamiltonian chords of ǫH for some 0 < ǫ < ǫ 0 . From above we know that all of these have action values in the interval (−1, 0), thus none of them is 1-quantized. In particular, R 1 L (H) = ∅. This shows that in general the estimate (1.2) in Theorem A fails. Example 6.1 (Paul Biran). We take any projective algebraic manifold M with π 2 (M ) = 0. Inside M we choose a sufficiently high degree hyperplane section Σ such that there exists a Lefschetz pencil inside M whose generic fiber is symplectomorphic to Σ. Since π 2 (M ) = 0 the Lefschetz pencil necessarily has singularities, see [Bir02, Section 5.1]. Thus, the vanishing cycles will give rise to Lagrangian spheres in Σ. By the Lefschetz hyperplane theorem π 2 (Σ) = 0 if dim R (Σ) ≥ 6.
Remark 6.2. Choosing ǫ 0 such that − 1 2 < A ǫH (x ǫ ± ) < 0 and min ǫH ≤ 1 2 for all 0 < ǫ ≤ ǫ 0 the argument from above shows that the function µ introduced in Remark 1.4 satisfies µ(c) = 0 ∀c ≤ 0 . (6.1) Appendix A. Being non-resonant is a generic property In this appendix we prove Theorem 5.20 asserting that on a symplectic manifold (M, ω) of dimension dim M ≥ 4 a generic autonomous Hamiltonian function is non-resonant (see Definition 5.16). We first prove the following lemma.  Step 1: Genericity of property (1) in definition 5.16.
Since C ∞ is not a Banach space we first work in the C k category and then deduce the C ∞ case by a standard argument due to Taubes is a Banach manifold. In order to prove this, we interpret M as zero-set of a section s in a Banach space bundle E k −→ B k as follows.
0 periodic, since x solves the autonomous ODEẋ = X H (x). In particular, (x, τ ) solves problem (A.1), unless x is a constant map. Since by assumption H ∈ H k we are left with the case x(t) = x 0 ∈ L is constant. Thus, the Hamiltonian function H has a critical point at x 0 ∈ L. This contradicts the second condition in Lemma A.2.
Thus, the chord x is injective. Therefore, for all η ∈ E k (H,x) there exists a functionĤ defined in a neighborhood of x such that XĤ(x(t)) = η(t), hence D (H,x) (Ĥ, 0) = η is surjective.
This shows that the space M is a Banach manifold. To prove that A H is Morse for generic H ∈ H k we consider the projection π = pr 1 : M −→ H k . We will show below that it is equivalent for H to be a regular value of π and for A H to be Morse. Thus, by the Sard-Smale Theorem, the action functional A H is Morse for a generic Hamiltonian function H ∈ H k .
We now show the following equivalence: H is a regular value of π iff A H is Morse. For (H, x) ∈ M let π(H, x) := H be a regular value of the projection, that is, ∀Ĥ ∈ H k × C k there existsx ∈ Γ k−1 (x * T M ) such that (Ĥ,x) ∈ T (H,x) M. In particular, Since -open and dense set of Hamiltonian functions. We now deduce the C ∞ assertion from the C k case. Using that being Morse is an C k -open and dense condition and that C ∞ is dense in C k we can find for any H ∈ C ∞ a sequence H Step 2: Genericity of property (2) in definition 5.16.
For τ > 0 we set X k τ := C k ([0, τ ], M ). We fix P ∈ N and define is a (trivial) bundle over (0, P ) and set The reason why we define the bundle X k P only over (0, P ) rather than over (0, ∞) is that sequences of chords of bounded period τ converge according to Arzela-Ascoli. This will be used below in order to apply Taubes' procedure. The tangent space of this Banach manifold B k P is given by For m ∈ 1 2N Z the zero-set of the section s m : In order to show that M(m, P ) is a Banach manifold we show that the operator is surjective along the zero-section. Given (η, r) ∈ E k (H,x,τ ) = Γ k−1 (x * T M ) × R we proved in Step (1) that there exists (Ĥ,x) such that In fact, since x is injective, we are free to choosex = 0. In light of the boundary condition x(τ ) +τẋ(τ ) ∈ T x(τ ) L this then forcesτ = 0. After setting (1) The vertical differential Definition B.2. In the situation of the above Proposition we denote the induced Hamiltonian chords byx Remark B.3. The corresponding statement of the above proposition in the periodic case was known to Poincaré and is proved Chapter 4.1 of the book [HZ94]. More precisely, in Proposition 2 in Chapter 4.1 of [HZ94] it is proved that the above family x H (s) can be chosen to be parameterized by energy, that is H(x H (s)) = H(x) + s. We point out that this stronger assertion does not hold in the relative case, in general, as Example B.7 shows.
To prove Proposition B.1 we need the following In particular, T x L ⋔ T x Σ, where Σ = H −1 (H(x)) is the level set through x.
Proof. We assume by contradiction that 0 = dH(x)ξ = ω(X H (x), ξ) ∀ξ ∈ T x L . H (x) | x ∈ U, t ∈ (−ǫ, τ + ǫ)} the function H| V has only regular values. To prove the proposition we follow closely the proof of Proposition 2 in Chapter 4.1 of [HZ94]. Due to the assumption X H (x(0)) ∈ T x(0) L and X H (x(τ )) ∈ T x(τ ) L we can choose two local hypersurface Σ i ⊂ M , i = 0, 1 in a neighborhood U 0 of x and U 1 of x τ with the property T x Σ 0 ⊕ < X H (x) >= T x M and L ∩ U 0 ⊂ Σ 0 T xτ Σ 1 ⊕ < X H (x τ ) >= T xτ M and L ∩ U 1 ⊂ Σ 1 . Dφ τ H (x) is nondegenerate since it is a symplectic transformation. Thus, dψ(x) is nondegenerate. We choose local coordinates on Σ i such that the Lagrangian submanifold L corresponds to R n ⊕ {0} ⊂ R 2n−1 in both coordinate systems. We denote the map ψ in local coordinates by ψ : R 2n−1 −→ R 2n−1 (B.10) and assume that ψ(0) = 0. With respect to the splitting R 2n−1 = R n ⊕ R n−1 we write ψ(x 1 , x 2 ) = ψ 1 (x 1 , x 2 ), ψ 2 (x 1 , x 2 ) , (B.11) and abbreviate d ψ(0) = ∂ x 1 ψ 1 ∂ x 2 ψ 1 ∂ x 1 ψ 2 ∂ x 2 ψ 2 =: A B C D . (B.12) We claim that ∂ x 1 ψ 2 has full rank. Indeed, from the transversality Dφ τ H (T x L) ⋔ T xτ L it follows (in local coordinates) that for all a = 0 ∈ R n . Since F 1 is a 1 × n-matrix the above inequality readily implies that dim ker C = 1. Hence, C = ∂ x 1 ψ 2 has full rank. This implies that locally ψ −1 that the convergent subsequence is non-constant. Because all (x ν , τ ν ) are N -quantized we have ∂ ∂s s=0 A H (x H (s), τ H (s)) = 0 . Example B.7. We construct an example of two Lagrangian submanifolds and a Hamiltonian function such that the Hamiltonian vector field intersects both Lagrangian submanifolds transversely. Moreover, the Hamiltonian flow has a one-parametric family of Hamiltonian chords of constant energy. In particular, this family cannot be parameterized by energy.
We only sketch the proof: We think of a connection α in E as an S 1 -invariant hyperplane distribution H E which is transversal to the infinitesimal generator of the S 1 -action. We construct E ⊗ F. The fiber product E × M F of E and F is defined as follows which is a principal T 2 /∆ ∼ = S 1 -bundle. We denote by H E resp. H F the hyperplane distributions on E resp. F. Then H E× M F := dp −1 E (H E ) ∩ dp −1 F (H F ) (D.7) is a T 2 -invariant codimension-2-distribution which is transversal to the infinitesimal generators of the torus action. In particular, H E× M F descends to connection H E⊗F on E ⊗ F. To compute the holonomy we recall that for a loop γ ∈ C ∞ (S 1 , M ) and e ∈ E γ(0) the holonomy hol α (γ) ∈ S 1 is determined by P α γ (e) = hol α (γ).e (D.8) where P α γ : E γ(0) −→ E γ(0) denotes the parallel transport along γ with respect to the connection α and g.e denotes the S 1 -action. We observe on E × M F that P α× M β γ (e, f ) = (P α γ (e), P β γ (f )) = (hol α (γ).e, hol β (γ).f ) ∈ (E × M F) γ(0) . (D.9) Thus, hol α⊗β = hol α + hol β holds. Statement (2) about the holonomy is proved analogously.
We construct E * . We recall that E is a compact manifold with a free S 1 -action ψ : S 1 ×E −→ E. We define ψ * : S 1 × E −→ E by ψ * (g, e) := ψ(−g, e). Then E * is the principal S 1 -bundle with total space E and action ψ * . Moreover, the connection H E * = H E . (D.10)