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Fukaya A_\infty-structures associated to Lefschetz fibrations. II 1/2

We consider a version of the relative Fukaya category for anticanonical Lefschetz pencils. There are direct connections between the behaviour of this category and enumerative geometry: some of these are results announced here, others remain conjectural. The ultimate aim of this approach is to determine the Fukaya category of the Calabi-Yau hypersurfaces that constitute the pencil.


Introduction
This paper continues a discussion [42,43,44] of the Floer-theoretic structures associated to Lefschetz fibrations. The specific part under consideration is a linear differential equation introduced in [44], for a pair of functions (ρ, σ) of a formal parameter q: The geometric context is that we are looking at a Lefschetz fibration over CP 1 , for which the fibre represents the first Chern class of the total space (this is the case for a rational elliptic surface; or more generally, for the fibrations obtained by blowing up the base locus of an anticanonical Lefschetz pencil). The coefficients ψ, η and z (2) in (1.1) (each a function of q) are extracted from the genus zero Gromov-Witten invariants of the total space. In fact, a restriction on those Gromov-Witten invariants must be imposed: that is Assumption 2.7 below, already extensively discussed in [44].
In the application to Lefschetz fibrations (more precisely, ones obtained by blowing up the base locus of an anticanonical Lefschetz pencil), (1.4) has a straightforward meaning: it describes the q-dependence of the eigenvalues of quantum multiplication with the fibre class [44,Section 3]. On a deeper level, which is where the main motivation lies, it has been conjectured that (1.1) governs the q-dependence of canonical natural transformations on the Fukaya category of the Lefschetz fibration [44,Section 4b]; as a consequence, (1.5) would then describe a kind of mirror map for the fibre of the Lefschetz fibration. That idea was supported by a form of the Homological Mirror Symmetry conjecture, and by example computations, but lacked a direct geometric interpretation. Our task is to show that versions of (1.1)-(1.4) arise in symplectic cohomology. More precisely, we remove a fibre from the Lefschetz fibration, so as to obtain an open symplectic manifold, whose symplectic cohomology we study. To obtain the differential equations above, one replaces the functions in them by distinguished symplectic cohomology classes, and ∂ q by a connections (an operator of differentiation in q-direction) acting on symplectic cohomology. The relation between Gromov-Witten invariants and symplectic cohomology has already been the object of considerable study [6,7,3] (one can also consider Symplectic Field Theory as an intermediate theory, and then the relevant literature becomes much larger). What's new here is the appearance of connections, and their interplay both with Gromov-Witten invariants and with the algebraic structure of symplectic cohomology.
There are at least two perspectives on the material developed here. One can focus on the abstract framework surrounding the construction and properties of connections. This leads away from the specific geometric situation, towards more abstract and axiomatic (TQFT) considerations. Alternatively, one can emphasize the concrete computational aspects, in relation with Gromov-Witten theory. We will try to accomodate both tastes. The first part of the paper, consisting of Sections 2-4, presents an overview of the constructions and results, including their motivation. After that, in Sections 5-6, we operate in a greatly simplified TQFT-style framework, and explain what can be seen at that level of abstraction. In contrast, Sections 7-10 are much more geometric, and set up just enough Floer-theoretic machinery to enable us to carry out the main computations. In Section 11, we translate the outcome of those computations to symplectic cohomology, and this fairly straightforward process completes our argument.
Acknowledgments. I would like to thank Nick Sheridan for an illuminating conversation about [3]. Partial support was provided by the Simons Foundation, through a Simons Investigator award, and by NSF grant DMS-1500954.

Outline of the results
This section summarizes the Floer-theoretic constructions and results which are the main topic of the paper. Technical details will be kept to a minimum.
(2a) Connections. Symplectic cohomology [47,5] is an invariant SH * (E) of a (certain kind of) non-compact symplectic manifold (E 2n , ω E ). Our version of symplectic cohomology will be linear over a specific (characteristic 0) field K, the single-variable Novikov field with real coefficients; elements f ∈ K are formal series Note that K is closed under differentation ∂ q . We'll assume throughout that c 1 (E) = 0. This allows us to lift the usual Z/2-grading of symplectic cohomology to a Z-grading, and it also simplifies the technical aspects of the construction. (Readers who wish for more specifics about the class of symplectic manifolds under consideration are referred to Section 11a, but should keep in mind that the same ideas apply to other situations as well.) Symplectic cohomology has the structure of a BV (Batalin-Vilkovisky) algebra, which one can think of as a Gerstenhaber algebra structure (consisting of a product, denoted here by •, and an degree −1 Lie bracket) together with a BV operator ∆ (see e.g. [40,32]). It also comes with a map We like to think of B as a version of the construction from [30], hence call it a PSS map. It is a map of BV algebras, if one equips H * (E; K) with the small quantum product (denoted here by * E ), as well as the zero bracket and zero BV operator. In particular, 1 ∈ H 0 (E; K) maps to the unit for the product structure, which we denote by e ∈ SH 0 (E). We denote the image of q −1 [ω E ] under (2.2) by k, and call it the Kodaira-Spencer class (the terminology is motivated by mirror symmetry).
There is also a reduced symplectic cohomology group SH * (E) red (see e.g. [4]), which fits into a long exact sequence Based on the vanishing of the BV operator on H * (E; K), one can define a partial refinement ∆ red , which fits into a diagram As should already be obvious from the discussion so far, we do not require that [ω E ] ∈ H 2 (E; R) is trivial, or even that it vanishes on spherical homology classes (such assumptions occur in most, but not all, of the literature concerning symplectic cohomology; among the exceptions are [33,19,18]). Eventually, we will impose a vanishing condition, but one which is weaker and not expressed in terms of classical topology: Assumption 2.1. The Kodaira-Spencer class vanishes.
Whenever Assumption 2.1 holds, we want to fix a choice of bounding cochain, in the following sense. Consider the chain complex underlying symplectic cohomology. Pseudo-holomorphic curve theory yields a cocycle in that complex which represents k. By assumption, this is nullhomologous, and the desired choice is that of a cochain which bounds it. Two such choices have the same effect if they differ by a nullhomologous cocycle, so the essential amount of freedom is an affine space over SH 1 (E). A choice of bounding cochain determines a class (2.5) t ∈ SH 1 (E) red , which maps to q −1 [ω E ] under the connecting homomorphism from (2.3); and that gives rise to a class (2.6) a = ∆ red t ∈ SH 0 (E), which satisfies (2.7) ∆a = 0.
Proposition 2.2. Suppose that Assumption 2.1 holds. Then, a choice of bounding cochain determines a connection ∇ on symplectic cohomology, which is compatible with the Gerstenhaber algebra structure.
Here, by a connection, we mean an operator of differentiation in ∂ q -direction: an additive (and grading-preserving) endomorphism ∇ which satisfies (2.9) Of course, any K-vector space can be equipped with a connection, simply by differentiating the coefficients with respect to some choice of basis. What makes Proposition 2.2 nontrivial is compatibility with the Gerstenhaber algebra structure, which means that ∇ acts as a derivation for both the product and Lie bracket: (2.11) Note that (2.10) implies (2.12) ∇e = 0.
Our connection depends on the choice of bounding cochain. Changing that cochain by α affects it as follows: Consider the interaction of the connection and the BV operator, as measured by the K-linear endomorphism ∇∆ − ∆∇. From the general relation between the BV operator, the product, and the bracket, it follows that ∇∆ − ∆∇ is a (degree −1) derivation for the product. The following result sharpens that statement, by saying that this derivation is inner: The connection from Proposition 2.2 satisfies One can think of ∇ as part of a family of connections, In particular, from (2.12) one gets (2.16) ∇ c e = c a.
One of those connections, ∇ −1 , is compatible with the BV operator (but not with the product or bracket, in general): by (2.7), (2.14), and the definition (2.15), Remark 2.4. From (2.13), one can derive formulae for how a change of bounding cochain affects ∇ c . Again, the case c = −1 is of particular interest: Combining (2.17) and (2.18) yields the following: if one considers ∆ as a differential on SH * (E), then ∇ −1 induces a connection on its cohomology, and that induced connection is independent of the choice of bounding cochain.
(2b) Lefschetz fibrations. Take a proper symplectic Lefschetz fibration (2.19) p : F −→ CP 1 , and choose a smooth fibre M . We will assume throughout that c 1 (F ) is Poincaré dual to [M ]. Consider genus 0 pseudo-holomorphic curves in F which have degree k over the base of (2. 19), and which come with one marked point. The evaluation map on the moduli space of such curves gives rise to Gromov-Witten invariants which are zero if k < 0. Let's remove our fixed fibre from (2.19) to get an open symplectic manifold By assumption, c 1 (E) = 0. Our main Floer-theoretic object of study is the symplectic cohomology of E.
Lemma 2.5. The cohomology class z (1) |E lies in the kernel of (2.2). The obvious ruling of D by lines gives a contribution of q γ−1 [D] to z (1) , and that is the lowest energy term. If we assume additionally that Assumption 2.7 will therefore be satisfied. One example is the algebro-geometric pencil of Calabi-Yau hypersurfaces on CP n , n ≥ 3. There is also a useful generalization, in which one replaces H 2 (F ; R) with the subspace which is invariant under a suitable automorphism group [44,Lemma 3.3]. That generalization applies e.g. to the pencil of elliptic curves on CP 2 (giving rise to a space F which is a rational elliptic surface). Remark 2.9. As explained in [44], allowing formal bulk deformations of the symplectic form greatly extends the reach of Assumption 2.7, for appropriately twisted versions of all the Floer-theoretic structures. It would be possible to allow such deformations throughout the current paper as well, but we won't do that, in the interest of simplicity.
(2c) Main results. Returning to the main thread of our argument, we notice that, by Lemma 2.5, Assumption 2.7 implies Assumption 2.1. One can be a little more precise: Lemma 2.10. Suppose that Assumption 2.7 holds. Then, for a suitable choice of bounding cochain, the class (2.6) is given by This is fundamentally not surprising: the definitions of the two sides of (2.30) both start with the vanishing of certain other elements in symplectic cohomology, and the condition q −1 [ω E ] = ψ(z (1) |E) relates those elements. In view of (2.16), one can equivalently rewrite (2.30) as (2.31) ∇ c e = −c ψs.
We can go further, and describe how the connection acts on s: Theorem 2.11. Suppose that Assumption 2.7 holds. Then, one can choose the bounding cochain so that Lemma 2.10 is satisfied, and with the additional property that (2.32) ∇s − ψs • s + ηs + 4z (2) ψ e = 0. This is a counterpart of (1.4), with s instead of λ, and ∇ instead of ordinary differentiation.
This equation and the c = 1 case of (2.31) are counterparts of the components of (1.1), with ∇ 1 replacing ordinary differentiation. Imitating the argument there, one can eliminate s by substitution, and obtain the analogue of (1.2) for e alone: (2.36) (2) ψ 2 e = 0.
We have now fulfilled the promise, made at the beginning of the paper, that all forms (1.1)-(1.4) of the basic differential equation would occur in the context of symplectic cohomology.
3. An outline of the equivariant theory This (partly speculative) section is not strictly necessary for the rest of the paper, but expands our overall theme in a natural direction. There is an operation on S 1 -equivariant symplectic cohomology, which we'll call the Gauss-Manin connection. It exists without any additional assumptions, and is canonical [45]. Even though it is therefore somewhat different from our ∇ c , one expects the two to be related via a suitable intermediate object. More precisely, the relevant value is c = −1, as foreshadowed by Remark 2.4. We will explain this picture, and how it leads to another approach towards proving our main results. Some of the steps in this approach remain conjectural, but those (specifically Conjecture 3.3) may actually be of independent interest.
(3a) Background. Let's start in the same overall situation as in our discussion of ordinary symplectic cohomology (Section 2a). The S 1 -equivariant symplectic cohomology SH * eq (E) is a Z-graded module over K [[u]], where the formal variable u has degree 2. It fits into a long exact sequence The BV operator can be recovered by composing the middle and right map in this sequence (in the nontrivial order). There is also an equivariant version of the PSS map, which fits into a commutative diagram with the forgetful maps from (3.1): There are actually different versions of the equivariant theory, which share the basic properties mentioned so far, but otherwise behave quite differently. The one convenient for the present discussion was introduced in [2,48]; in that version, the underlying chain complex is not u-adically complete (as a consequence, vanishing of SH * (E) does not imply the corresponding result for SH * eq (E)). The main result of [2,48] is a localisation theorem, which says that the equivariant PSS map induces an isomorphism . To be clear, the geometric situation here is not exactly the same as that in which (3.3) was proved originally, but the proof goes through without any significant modifications.
The ordinary cohomology of E carries the quantum connection, which is defined in terms of the small quantum product, as follows: In spite of the name, this (in our formulation) is not a connection in the strict sense, but rather what one would get after multiplying a connection by u. It turns out that a similar operation can be defined on equivariant symplectic cohomology: There is a canonical additive endomorphism which fits into a commutative diagram This Γ is what we call the Gauss-Manin connection. For Hamiltonian Floer cohomology with "finite slope", it is constructed in [45] (where it is written as Γ q ); the same argument as in [45, Section 5f] could be used to prove its compatibilty with continuation maps which "increase the slope", yielding the theorem in the form stated above (or alternatively, one could work directly with a single Hamiltonian on E whose slope grows unboundedly).
Intuitively, (3.6) says that the Kodaira-Spencer class is the obstruction to being able to divide Γ by u in order to get an honest connection on the equivariant theory. A more precise form of this idea is the following (see [45,Remark 5.3] for further discussion): Conjecture 3.2. Suppose that Assumption 2.1 holds (and that a bounding cochain has been chosen). Then, SH * eq (E) can be equipped with a connection ∇ eq , which preserves gradings and satisfies the analogue of (2.9) for f ∈ K [[u]]. The Gauss-Manin connection is related to it by moreover, it is a lift of ∇ −1 , in the sense that there is a commutative diagram (3b) A bit more Gromov-Witten theory. We now return to the more specialized geometric situation from Section 2b. H * (F ; K) also carries a quantum product, but that no longer preserves the grading. Let's write it as which each term * (k) F has degree −2k. The terms with k < 0 are zero, while the k = 0 contribution is related to the quantum product on E by restriction: For classes x k ∈ H 2 (F ; K), one can use the divisor axiom to reduce the quantum product to one-pointed invariants. In particular, the following products can be expressed in terms of (2.20): is the cup product. More relations between Gromov-Witten invariants come from the WDVV (associativity) equation. One instance which is relevant for us is this: Gromov-Witten invariants can be generalized in various directions. The most relevant one for our purpose are relative invariants, and specifically ones with tangency conditions to M . Consider curves which intersect M in a single point, with multiplicity k, and which carry an additional marked point. These give rise to classes (3.15)z (k) ∈ H 2 (F ; K), withz (1) = z (1) . All these relative invariants can be reduced to ordinary Gromov-Witten theory, see [27] for a general discussion. We will only appeal to the simplest special case of such a reduction: This is proved by first using a degeneration argument to relatez (2) to a Gromov-Witten invariant with a gravitational descendant, and then by eliminating that descendant using the TRR relation.
(3c) The Borman-Sheridan class and enumerative geometry. Let's return to our main object of study, the symplectic cohomology of (2.21). We will use the following (highly convenient, but not absolutely indispensable) fact. One can arrange that the Conley-Zehnder indices of periodic orbits are all nonnegative, which implies that As a consequence of this and (3.1), we have We will denote by x eq the unique equivariant lift of any class x ∈ SH 0 (E). Note that in several important cases, such as the unit class e or the Borman-Sheridan class s, there are geometric constructions of equivariant lifts which do not depend on (3.18).
At this point, we are faced with an interesting puzzle. After multiplication by some power of u, any element of SH * eq (E) comes from an ordinary cohomology class, by (3.3). Applying this to the Borman-Sheridan class and its powers, we get specific cohomology classes, which ought to have some enumerative meaning. An attempt to make this more precise leads to the following: The first equation (3.19) is closely related to the discussion in Section 8b below. Concerning the somewhat less transparent (3.20), we have formulated it in a way which is convenient for our applications, but which is not necessarily the most conceptual one. To get a better picture, we should point out that the Borman-Sheridan class is the first in a sequence of classes defined by looking at thimbles with k-fold tangency to M . In the simplest nontrivial case, (3.21) is related to the pair-of-pants square of (2.22) by the formula (proved in Section 8c below) Figure 1. Holomorphic curves corresponding to terms in (3.23).
where e is the unit. (The classes s (k) have natural equivariant lifts, independent of (3.18), something which is not a priori true of s • s.) Using (3.22), as well as (3.16) and (3.14), one rewrites (3.20) as (2) )|E . Figure 1 gives a schematic picture of the terms involved. In spite of this promising simplification, we will not speculate on the infinite hierarchy of equations of which (3.19) and (3.20) should be the first two instances.

Remark 3.4.
Here is another way to see the meaning of formulae such as (3.19). Suppose that L ⊂ E \ ∂E is a closed Lagrangian submanifold (Spin, and with vanishing Maslov class). For the sake of argument, let's also assume that L bounds only constant holomorphic discs in E (for some choice of almost complex structure). Consider holomorphic discs in F with boundary on L, and which intersect M once. The boundary values of such discs gives rise to a class (3.24) w eq ∈ H eq,n (LL; K), lying in the equivariant homology of the free loop space. More precisely, we use the equivariant homology theory from [21], which localises along constant loops. Taking Poincaré duality into account, the localisation theorem says that After localisation, (3.24) contains only contributions from S 1 -invariant (stable) discs, which are holomorphic spheres with a constant disc attached. Hence, under (3.25), On the other hand, (3.24) is the image of the Borman-Sheridan class under a canonical map SH * eq (E) → H eq,n− * (LL; K) (for which see e.g. [49]). Hence, (3.26) is consistent with (3.19).
(3d) Implications for the connections ∇ c . We now combine the general theory above with Assumption 2.7. As an immediate consequence, the formulae (3.19), (3.20) take on the following form: Using (3.6) as well as (3.11) and (3.12), one arrives at the following statement for the Gauss-Manin connection: Assuming Conjecture 3.2, one would conclude that (3.29) ∇ eq e eq = ψs eq + something in the kernel of u, ∇ eq s eq = 2ψ(s • s) eq − ηs eq − 4z (2) ψe eq + something in the kernel of u 2 .
Let's suppose that we already knew that, for some undetermined constants γ ij ∈ K, (It is not hard to arrive at such a qualitative statement, using a filtration as in [29]; however, we will not explain that argument here.) Because of (3.8) and (3.18), the equivariant counterpart must then hold as well: (3.31) ∇ eq e eq = γ 10 s eq + γ 00 e eq , ∇ eq s eq = γ 21 (s • s) eq + γ 11 s eq + γ 01 e eq .
From (3.19) and (3.3), one sees that e eq and s eq generate a free rank two K[[u]]-submodule of SH * eq (E). Hence, the first expression in (3.32) is not just u-torsion, but actually zero. If we additionally suppose that dim(E) > 4, the same argument shows that (e eq , s eq , (s • s) eq ) generate a free rank three submodule, leading to the same conclusion for (3.32). The outcome would then be that (3.33) ∇ −1 e = ψs, which are the c = −1 cases of (2.31) and (2.34), respectively. By the same argument as in Section 2c (taken in reverse), Lemma 2.10 and Theorem 2.11 would then follow from this, in a way which is independent of the proofs given in this paper (conversely, the fact that we already have a proof of these results provides a strong check on Conjecture 3.3).
It is worth while reiterating the ingredients that enter into this approach to (3.33). One is Conjecture 3.3, which is a relation between equivariant symplectic cohomology and enumerative geometry (it does not require Assumption 2.7, and does not involve connections in any way). The second ingredient is the general theory of Gauss-Manin connections. Finally, one needs to know that a formula of the form (3.30) exists, which is a qualitative statement about filtrations on symplectic cohomology. Altogether, this approach is somewhat complicated, but has the advantage of separating out different parts, each of which is meaningful in itself.

Motivation and context
Like the previous one, this section is not strictly speaking needed for the rest of the paper. Its aim is to put the main results in a wider context. With that in mind, most of the arguments will just be outlined.
(4a) Homological algebra. Let A be an A ∞ -algebra over K. We denote by HH * (A, A) and HH * (A, A) its Hochschild homology and cohomology. HH * (A, A) is a Gerstenhaber algebra. HH * (A, A) is a Gerstenhaber module over that algebra, and also carries the Connes operator, an endomorphism of degree −1. The Kaledin class [23,26] is a distinguished element which describes the dependence of A on the formal variable q.
To define it, choose a connection on A as a vector space; then, the derivative of the A ∞operations with respect to that connection yields a cocycle representing (4.1).
One can generalize Hochschild (co)homology by allowing coefficients in a bimodule P (the classical case corresponds to taking P to be the diagonal bimodule A). The previously mentioned module structure extends to maps (expressing the functoriality of Hochschild homology) The choice that will be of particular interest for us is the "inverse dualizing" bimodule P = A ! [24, Definition 8.1.6]. A little confusingly, its cohomology is already itself a Hochschild cohomology group, Let's assume that A is homologically smooth (the diagonal bimodule is perfect): in that case, A ! is also perfect [24, Proposition 8.
for perfect P (one recovers (4.3) by taking P = A ⊗ K A). Assume additionally that A ! is an invertible bimodule (with respect to tensor product), and denote its inverse by (A ! ) −1 .
Then, tensor product with A ! or its inverse preserves perfect bimodules (because of their characterization as compact objects). We can therefore replace P by (A ! ) −1 ⊗ A P in (4.4), and rewrite the outcome as where the morphism space on the left hand is in the category of bimodules. We will actually use only the special case P = A.
Following [17], let's say that A is Calabi-Yau (of dimension n) if it is homologically smooth and comes with a choice of quasi-isomorphism One then gets induced isomorphisms By (4.5), the map (4.6) can also be viewed as an element ν ∈ HH −n (A, A). In those terms, the second map in (4.7) is given by letting Hochschild cohomology classes act on ν (through the module structure of Hochschild homology). Similarly, the first map is given by inserting ν into (4.2), for P = A ! , to get a map HH * +n (A, A ! ) −→ HH * (A, A ! ), and then using the isomorphism (4.3) (which is independent of the Calabi-Yau structure).
From now on, let's consider the special case when the Kaledin class (4.1) vanishes. This holds if and only if A admits an A ∞ -connection, which is defined to be a sequence of operations (4.8) with the following properties. Each ∂ d , d = 1, is K-(multi)linear, whereas ∂ 1 is a connection. Finally, the relations for a Hochschild cocycle are satisfied. An A ∞ -connection on A induces a connection ∇ (in the ordinary sense) on HH * (A, A), as well as connections ∇ −1 on HH * (A, A) and ∇ 1 on HH * (A, A ! ). These connections are compatible with all algebraic structures on those groups. In particular, ∇ satisfies analogues of (2.10) and (2.11), while ∇ −1 commutes with the Connes operator.   Similarly, compatibility with (4.2) for P = A ! means that The formula (4.10) is obviously parallel to (2.15), even though the direction of the argument is reversed. Here, we were given ∇ and ∇ ±1 with certain properties (for instance, that ∇ −1 was compatible with the Connes operator), and then derived the relation (4.10) from those; in the case of (2.15), we obtained ∇ c by an explicit modification of ∇, and any properties of those connections had to be established based on that. By using powers of the Serre functor, one could construct homological algebra counterparts of ∇ c for any integer value of c, but we will not discuss that here.
The relation between homological algebra and symplectic cohomology goes via the wrapped Fukaya category W(E). This is a Z-graded A ∞ -category defined over K (and expected to be Calabi-Yau, see [11]). In general, Fukaya categories have curvature (µ 0 ) terms. Here, we work in the framework of Lefschetz fibrations (as in Section 2b), and allow only Lefschetz thimbles as objects. In that case, one can arrange that µ 0 = 0. The previous discussion of connections on A ∞ -algebras carries over without any difficulties to the categorical case. The Hochschild (co)homology of W(E) is related to symplectic cohomology by open-closed string maps [1,11,33] SH * (E) −→ HH * (W(E), W(E)),   It makes sense to think that there should be another canonical map   Getzler's connection [15] is a canonical u-linear map which is related to the Kaledin class by a diagram Assuming that the Kaledin class vanishes, an A ∞ -connection on A induces a u-linear connection ∇ eq on HC − * (A), which satisfies and which is a lift of ∇ −1 : Suitable equivariant open-closed string maps (like those for compact manifolds studied in [12]) should relate HC − * (A) to the u-adically completed version of equivariant symplectic cohomology. This is not the version we discussed in Section 3, but it comes with a map from that version, and structures like the Gauss-Manin connection carry over by u-adically completing (on the chain level). Therefore, is still makes sense to consider (4.19) and (4.20) as algebraic counterparts of Conjecture 3.2.
(4b) Mirror symmetry. Homological mirror symmetry relates wrapped Fukaya categories to categories of coherent sheaves (on smooth open Calabi-Yau varieties). Symplectic cohomology is mirror to polyvector field cohomology. We will consider a specific algebro-geometric situation which formally corresponds to the symplectic one encountered before.
For the sake of familiarity, let's switch the coefficient field to F = C((h)), which is the algebraic closure of the Laurent series field (obtained by adjoining roots h 1/d of all orders; in symplectic geometry, such a field can be used instead of the Novikov field whenever the symplectic class is rational). Let X be a smooth variety of dimension n over F. The polyvector field cohomology (4.21) HT * (X) = H * (X, Λ * T X) has the structure of a Gerstenhaber algebra (given by the exterior product and Schouten bracket). The dependence of X on h is measured by the Kodaira-Spencer class (4.22) k ∈ H 1 (X, T X) ⊂ HT 2 (X).
If k vanishes, X admits a connection in ∂ h -direction (acting on the sheaf of functions). This induces a connection on polyvector fields (and their cohomology), which we denote by ∇; and also one on differential forms, which we denote by ∇ −1 .
Suppose that X has trivial canonical bundle, and comes with an algebraic volume form ν. This gives rise to a BV operator on (4.21), induced by the sheaf homomorphism Now consider the case where X comes with a connection, and set If we use ν to identify the differential forms and polyvector fields, as in (4.23), then Since ∇ −1 commutes with the de Rham differential, (4.25) defines a connection on polyvector fields which commutes with the BV operator. This implies that the original connection ∇ satisfies the analogue of (2.14) for the function (4.24). More generally, there are natural connections ∇ c on Λ * T X ⊗ K ⊗−c X for any c ∈ Z (reducing to the case of differential forms for c = −1). If we use the volume form to trivialize K X , these connections can be written as ∇ c = ∇ + ca, acting on polyvector fields.
The specific situation we want to consider is the following one. Start with a fibration over CP 1 which is log Calabi-Yau. By this, we mean a smooth projective varietyX 0 over C and a surjective map such that KX 0 is the pullback of O(−1). If we remove the fibre at ∞ to form (4.27) p 0 : X 0 −→ C, then X 0 carries a holomorphic volume form ν 0 (with a simple pole along the divisor we have removed). Extend constants to formX =X 0 × C F, and then remove the fibre at h −1 to get X ⊂X. This carries an algebraic volume form where f ∈ F × is an arbitrary scaling factor. Let's write l = (∂ h f )f −1 . In this situation, the connection on X is just differentiation ∂ h . The function (4.24) is therefore which obviously satisfies This equation is the mirror of (2.33). Correspondingly, the section of K −1 X given by satisfies the mirror of (2.36), which is (4.32) ∂ 2 h η + 2l∂ h η + (∂ h l + l 2 )η = 0. In fact, (4.32) has scalar solutions 1/f and h/f , of which our η was a linear combination (with coefficients in sections of K −1 X0 ). If one wrote down an equation for quotients of solutions of (4.32) in parallel with (1.5), that equation would just be vanishing of the Schwarzian S h (functions with that property are exactly the rational automorphisms of the h-line). What this means is that under mirror symmetry, the variable h here does not correspond to q, but rather to the "mirror coordinate" which is one particular solution of (1.5) (see the discussion in [44, Sections 4c and 7b]).

Configuration spaces
This section and the two following ones serve as a transition between our introductory discussion, in which results were stated with no attempt at explaining the underlying Floertheoretic ideas, and the subsequent detailed constructions. We will consider an operad-style framework in which abstract versions of the desired properties can be seen to hold, without the technicalities inherent in working with pseudo-holomorphic curves (one could call this a "photoshopped version" of the operations in Floer theory).
Setup 5.1. A disc configuration is a collection of points ζ i ∈ C together with radii r i > 0, indexed by a finite set I. We require that the closed discs of radius r i around the ζ i should be pairwise disjoint, and contained in the open unit disc centered at the origin. We allow one exception to the last-mentioned condition, called the identity configuration: this consists of a single point ζ 1 = 0 at the origin, with r 1 = 1. (When representing disc configurations graphically, we will often only draw the centers ζ i ; this causes no major issues, since the choices of r i form a contractible set.) Given two disc configurations, indexed by sets I 1 and I 2 , and a choice of i 1 ∈ I 1 , one can carry out the following gluing process. Take the second configuration, rescale it by r i1 , and then insert it into the first configuration centered at ζ i1 , replacing the point originally located there. The outcome is a configuration indexed by I = (I 1 \ {i 1 }) ∪ I 2 . As the name suggests, gluing with the identity configuration does nothing.
We will also consider families of disc configurations, parametrized by an auxiliary smooth manifold P (which may have boundary or corners). Given two families with parameter spaces P 1 and P 2 , gluing produces a family over P 1 × P 2 .
Fix some commutative coefficient ring R. The algebraic structure we will look at is a graded R-module H * with operations induced by families of configurations. More precisely, we require that for any family with closed oriented P , there is an associated R-multilinear map The basic operations obtained from this framework are: e ∈ H 0 unit, (5.3H) The geometry underlying (5.3H)-(5.5H) is shown in Figure 2 (where we always set I = {1, . . . , m}). The most interesting case is that of the bracket, which comes from a family of configurations parametrized by P = S 1 . From the general axioms, it follows that the pair-of-pants product is commutative, associative, and e is its unit; and that the bracket satisfies Remark 5.2. Our sign for the bracket [x 1 , x 2 ] departs from the standard convention for Gerstenhaber algebras by (−1) |x1| , see e.g. (5.6H). This discrepancy will also be visible later on, when we define the bracket in terms of the BV operator (6.5H).
As a reminder of how the relations (5.6H)-(5.9H) are proved, consider for instance (5.7H). By gluing, one defines families over S 1 which represent each of the terms involved; the equality involved is then given by a cobordism (a family whose parameter space P is the pair-of-pants), see Figure 3. (5b) Chain level operations. The familiar next step is to refine the framework from Section 5a by lifting it to the chain level. Here, we assume that the previous H * is the cohomology of a chain complex of R-modules, denoted by (C * , d). To any family of disc configurations over a compact P , we now associate a map If P is closed, this must be a chain map, inducing (5.1H) on cohomology. More generally, if φ is a map (5.10) (with I = {1, . . . , m} for simplicity), and ψ the sum of the operations associated to the restriction of the family to the (closed codimension 1) boundary faces of P , then This replaces the cobordism axiom. The axioms concerning disjoint union (of parameter spaces), the identity configuration, and the composition law (5.1H) remain as before. Reversing the orientation of P should switch the sign of the associated operation (up to homotopy, this follows from (5.11), but we prefer to impose a strict equality). We will also keep the property, inherent in the formulation with an arbitrary indexing set I, that permuting the inputs results in the standard Koszul signs.
Remark 5.4. For a more in-depth study of the theory, one would probably want to assume that the complex C * is cohomologically well-behaved (let's say, R is a ring of finite global dimension, and C * is a complex of projective R-modules; this would imply that it is both K-flat and K-projective). However, for our very limited purpose, this is not necessary.
The basic operations (5.3H)-(5.5H) have chain level representatives which are unique up to chain homotopy. We fix such a choice once and for all, and use the same notation for them as before. The previously stated algebraic relations will now hold up to chain homotopies, which one can think of as as secondary operations. For instance, the family from Figure 3 now gives rise to a map (5.12) Concerning such secondary operations, some uniqueness considerations are appropriate, and we use (5.12) as an example to explain them. From the viewpoint of Remark 5.3, the construction of (5.12) relies on a choice of map P → P 3 , where P is a surface with three boundary components, and where the behaviour of the map on ∂P is fixed. Within this general context there are genuinely different choices, since This corresponds to the fact that we could add a multiple of [x 3 , [x 2 , x 1 ]] or its cyclic permutations to γ, and the outcome would still satisfy (5.12). When choosing the actual family in Figure 3, we want to make sure that the point ζ 3 always lies in the half-plane to the right of ζ 2 . This restricts us to a subspace of P 3 for which the counterpart of (5.13) vanishes (in fact, that subspace is itself homotopy equivalent to a pair-of-pants). Two choices of γ constructed under this restriction differ by a nullhomotopic chain map; we say that γ has been defined Kodaira-Spencer class endomorphism r Z Z 1 "in a homotopically unique way". For future use, pick a concrete representative γ within that class.
(5c) The differentiation axiom. We will now further enrich the previous framework. At first, this will be an extremely simple modification, which adds a distinguished cohomology class; in a second step, that class will take on a more fundamental meaning.
Setup 5.5. We allow disc configurations with an optional extra point. Formally, this means that we equip them with the additional datum of some Z ⊂ C, with |Z| ≤ 1, which lies in the closed unit disc, and avoids the interior of the discs around the ζ i .
In the exceptional instance where we have just one point ζ 1 = 0 with r 1 = 1, the Z point can lie anywhere on the unit circle. Let's call this the r-family. Given a disc configuration with no Z point, gluing in the r-family corresponds to adding a Z point which runs around a circle (either the unit circle, or the r i -circles around the ζ i ).
Suppose that we have a family over P as in Setup 5.1, which means with no extra point; and we also want to exclude the identity configuration. One can then define a new family by "inserting the extra point in all possible places". The base of the new family, denoted by P • , has dim(P • ) = dim(P ) + 2. (Figure 6 shows how this looks like for the configuration underlying the pair-of-pants product; this time, it is helpful to see the radii r i , and we have drawn them as dashed circles.) Generalizing (5.1H), each family over a closed P should now give rise to a map with the same properties as before. The effect of this is to introduce a distinguished element, called the Kodaira-Spencer class Geometrically, this comes from a single configuration with a Z point (see Figure 4). We also use the r-family to define an endomorphism but that can be reduced to (5.15H), The cobordism which proves this is based on a simple idea, which is to "pull out the Z point", thinking of it as a copy of k being glued in; see Figure 5.
It is straightforward to introduce the chain level version of (5.14). As before, we use the same notation for the chain level representatives of (5.15H) and (5.16H). Then, Figure 5 gives rise to a (homotopically unique) secondary operation (When choosing the orientation of the underlying parameter space, our convention is that the "pulling out" parameter that goes from left to right in Figure 5 is the first coordinate, and the anticlockwise rotation of Z is the second coordinate.) Remark 5.6. In principle, it is possible to generalize our setup to |Z| > 1. However, one has to deal with points of Z colliding, and we prefer not to discuss that here. The first consequence of such a generalization, using only |Z| = 2, would be that [k, k] = 0. More importantly, in the context of the differentiation axiom (to be introduced next), having |Z| > 1 would allow multiple differentiation.
Assume from now on that the coefficient ring R comes with a derivation ∂. Correspondingly, we require that the graded module C * should come with a connection ∇ / in ∂-direction, which means an endomorphism (in each degree) which satisfies the analogue of (2.9), with ∂ instead of ∂ q . This is not required to be compatible with the differential: instead, we ask that Remark 5.7. An immediate consequence of (5.19) is that r induces the trivial map on H * . In spite of that, r is not nullhomotopic in general. For instance, suppose that R = C[t], with ∂ = ∂ t . Consider the chain complex with ∇ / given by the obvious choice of basis. Then, r maps the first R group identically to the second one, and that is clearly not nullhomotopic. Alternatively (anticipating some of the later discussion) one can argue that if r nullhomotopic, H * carries a connection in ∂direction; in our example, this contradicts the fact that the cohomology of (5.20) is located at t = 0.
Suppose that φ is the operation (5.10) associated to a family of configurations over a compact base P (with Z = ∅), and write φ • for the associated family over P • , as described in Setup 5.5. We then require that The two conditions (5.19) and (5.21) together are referred to as the differentiation axiom.
Remark 5.8. As a check on the internal consistency of this two-part axiom, consider the case when P is closed. It then follows from the structure of P • (see Figure 6 for an example) that On the other hand, from (5. 19) and the fact that φ is a chain map, we get This shows that if we apply the differential d to both sides of (5.21), the outcome is the same; which is not a tautology, since the argument did not use (5.21), only (5. 19).
Note that if we change the connection from ∇ / to ∇ / + ρ, then (5.18) implies that In words, the failure of the new connection to commute with the differential is measured by the Kodaira-Spencer class. We will now impose a final condition, but one which should be thought of in an entirely different way from the previous ones. Whereas the discussion so far described the setup for a general class of theories, we now specialize to instances where the Kodaira-Spencer class vanishes, something one shouldn't expect to be true in general. Figure 6. Inserting an added point, as in Setup 5.5.
Assumption 5.9. There is a θ ∈ C 1 such that Naturally in view of (5.24), the connection can then be modified to be compatible with the differential. This modified connection is Proposition 5.10. The connection on H * induced by ∇ is compatible with the Gerstenhaber algebra structure, meaning (2.10) and (2.11) hold.
Proof. For simplicity, we will consider only compatibility with the product (the proof for the bracket is parallel). Start with the family of configurations shown in Figure 6. One can "pull out" the Z point, in the same way as when defining ρ. The resulting three-dimensional parameter space (a pair-of-pants times an interval) has five codimension 1 boundary faces. The first face corresponds to the original family. Three more boundary faces appear when the additional point moves either towards the boundary of the unit disc, or to those of the discs around the ζ i ; one can arrange that this corresponds exactly to gluing in a copy of the family underlying ρ. The final boundary face occurs when we have completely "pulled out" the first point moves, and its the additional point; one can arrange that this corresponds exactly to the family from Figure  3, with the Kodaira-Spencer element glued in at the third point. Together with (5.21), this says that the following expression is a nullhomotopic map: On the other hand, using (5.12) and Assumption 5.9, we find a chain homotopy Inserting that into (5.27) yields the desired result.

Framed configuration spaces
The little disc operad can be enlarged by adding framings [14,35]. The aim of this section is to extend the previous discussion to the framed context, and to explain how that gives rise to the phenomenon encountered in (2.14).
(6a) The BV operator. The first step is to introduce geometric data which give rise to a version of the framed little disc operad. For symplectic geometers, an appropriate idea to think of would be "allowing the asymptotic markers to rotate". Setup 6.1. A framing of a disc configuration is an additional choice of τ i ∈ S 1 = R/Z, also identified with the complex number exp(2πiτ i ), for each i ∈ I. If one has an added Z point, as in Setup 5.5, that point should not be equipped with a framing. Any disc configuration can be considered as a framed one, by setting all τ i = 0. For the identity configuration, as well as for the r-family (Setup 5.5), we allow only this trivial choice of framing.
Given two framed configurations, and an input point i 1 of the first one, we always rotate the second configuration by τ i1 before gluing it into the first one. (Graphically, the τ i are represented as a tangent arrow at ζ i . Actually, in all the pictures in Sections 5-6, the complex plane is shown rotated, so that the positive real axis points upwards; for compatibility with standard drawing conventions, read this paper while lying on your side.) Suppose that we have a graded R-module H * as before, with operations (5.1H) induced by families of framed configurations (also allowing ones with a Z point). In addition to the previous (5.3H)-(5.5H) and (5.15H), we now have the BV operator ( Figure 7) which satisfies ∆e = 0, (6.2H) We will find it useful to consider a modified version of the bracket, This can be defined directly as shown in Figure 7. It is no longer graded commutative, but it commutes with the BV operator for fixed x 1 : Because of (6.3H), one can write (5.17H) equivalently as (6b) Chain level operations. One adds framings to the chain level story in the same way as before. Concerning the chain level versions of the relations (6.2H)-(6.6H), consider for instance (6.4H). If we glue together two copies of the family underlying ∆, the outcome is a family over the two-torus, with parameters τ 1 , τ 2 ∈ S 1 , but which actually depends only on τ 1 + τ 2 . Because of this, the family extends over the solid torus, and such an extension gives rise to a secondary operation As another example of chain level relations, take (6.3H). If we take one of the framed configurations in the family defining ∆, and the configuration defining k, and glue them together, the outcome is always the same. Hence, the outcome of the gluing process is a family over a circle which bounds one over the disc, and that yields a cochain To prove that, one considers the families defining ∆κ and δk, which are both parametrized by S 1 × D 2 , and have common behaviour over the boundary. One constructs a family over D 2 × D 2 whose restriction to the two boundary faces is given by those two families. This is fairly straightforward, and we will not discuss it further here.
We use the family of framed configurations from Figure 7 to define a chain level representative of [·, ·] −1 . Given this, as well as the standard chain level representative for r from Figure 4, the equality (6.9H) should be replaced by a chain homotopy (6.12) This is parallel to (5.18), and is defined by a similar family over P = [0, 1] × S 1 (Figure 8). At this point, it may make sense to briefly revisit the uniqueness question for such secondary operations. Let P fr m;q be the universal space of framed disc configurations with |I| = m and |Z| = q. This is homotopy equivalent to the ordered configuration space of m + q points times (S 1 ) m . In principle, to find a homotopy ρ −1 satisfying (6.12), one chooses a map from a surface P with two boundary components to P fr 1;1 with fixed behaviour near the boundary. Since P fr 1;1 (S 1 ) 2 , there are essentially different choices (up to cobordism or homology) of such maps; the corresponding operations differ (up to chain homotopy) by multiples of ∆(r(x)). However, in our construction (Figure 8), we impose the condition that the framing marker at the input point should not move. This narrows down the choice to a subspace of P fr 1;1 homotopy equivalent to S 1 , which singles out a homotopically unique operation (6.12).
Another property of [·, ·] −1 is (6.8H). The chain level version involves a homotopy (6.13) : Both the bracket [·, ·] −1 and the BV operator are defined by families of framed configurations over S 1 . Let's call their parameters τ 1 (for the BV operator) and τ 2 (for the bracket). If we carry out the gluing process corresponding to −∆[x 1 , x 2 ] −1 , we get a family over (S 1 ) 2 .
Changing parameters toτ 1 = τ 1 +τ 2 ,τ 2 = τ 1 yields the corresponding glued family associated to (−1) |x1|+1 [x 1 , ∆x 2 ] −1 (note that the parameter change is orientation-reversing). The outcome of this consideration is a family over [0, 1] × (S 1 ) 2 , shown schematically in Figure 9; in this case, homotopical uniqueness is obtained by the prescription that the framing marker at the first point should always point away from the second point.
(6c) The differentiation axiom in the framed context. We now impose the same differentiation axiom as in Section 5c, extended to framed disc configurations. We also impose Assumption 5.9, but now consider a different construction of connection: This time, to show that ∇ −1 is compatible with the differential, one uses (5.19) together with (6.12) and (5.25). Proof. The basic approach is the same as for Proposition 5.10. The differentiation axiom describes ∇ / ∆ − ∆∇ / in terms of a three-parameter family of framed disc configurations (with parameter space [0, 1]×(S 1 ) 2 ). One adds another parameter which "pulls out" the additional Z point. The resulting family, drawn schematically in Figure 10, shows that the following expression is a nullhomotopic map: On the other hand, using (6.13) and Assumption 5.9, we find a homotopy Inserting that into (6.15) yields the desired result.
One can associate to our choice of bounding cochain θ a cocycle where κ is the cochain from (6.11). Using Lemma 6.2, ones sees that Figure 10. The nullhomotopy (6.15).
Hence, the cohomology class of a (for which we use the same notation) is annihilated by the BV operator.
Proof. There is a chain homotopy underlying the first of the cohomology level equalities in (6.7H):  Figure 11. The operation η from (6.19). This comes with a secondary nullhomotopy The family behind (6.19) is shown in Figure 11. Note that (6.19), together with the properties of ρ, ρ −1 and κ, ensures that the left hand side of (6.20) is a chain map. The corresponding geometric fact is that the underlying families can be combined to form a family over a closed surface (a torus), as shown in Figure 12. To obtain (6.20), one has to show that this family bounds in P fr For that, it is sufficient to notice that all disc configurations in the family have the property that the framing marker (at the unique I point) points in the same direction. Now let's see what this says about connections. By definition and (6.19), Then, applying (6.20) achieves the desired result.
Combining Lemma 6.4 and Proposition 6.3 yields the following: Corollary 6.5. The connection on H * induced by ∇ satisfies (2.14).

Floer cohomology
In contrast with the formal preoccupations of the preceding sections, the next few go straight for the jugular, meaning the relation between enumerative geometry and connections on Floer cohomology. The present section sets up the basic Floer-theoretic machinery needed for that purpose.
(7a) The definition. We will work with a class of symplectic manifolds which includes the Lefschetz fibrations from Section 2b (more precisely, it includes them after possibly deforming the symplectic form). Leaving aside technical details, the key condition is that the first Chern class of our manifold is represented by a symplectic hypersurface with trivial normal bundle.
Setup 7.1. Let F be a closed connected symplectic manifold, and M ⊂ F a connected codimension two symplectic submanifold. We require that there is a map which has M = F 0 = p −1 (0) as a regular fibre, and which is a trivial symplectic fibration over some closed disc (centered at the origin) B ⊂ C ⊂ CP 1 . This means that there is a (necessarily unique) diffeomorphism which is symplectic (with respect to the product of ω M = ω F |M and some rotationally symmetric form ω CP 1 on the base), and which restricts to the inclusion on {0} × M .
We require that c 1 (F ) is Poincaré dual to F 0 , and choose a trivialization of the canonical bundle with a pole along that fibre. This means that, for some compatible almost complex structure on F , we have a complex volume form η on the complement such that pη extends to a complex volume form near F 0 .
We also want to fix a cycle representative for [ω F ]. More precisely, we choose (not necessarily symplectic) codimension two submanifolds Ω 1 , . . . , Ω k ⊂ F , which are transverse to F 0 , along with multiplicities µ 1 , . . . , µ k ∈ R, such that Ω = µ 1 Ω 1 + · · · µ k Ω k satisfies Additionally, we choose a current Θ on F , which is regular (represented by a smooth oneform) outside Ω, and such that Then, if S is a compact oriented surface with boundary, and u : S → F a map with u(∂S) ∩ Ω = ∅, we can compute its symplectic area by The auxiliary geometric data used to define Floer cohomology will be specifically chosen to be compatible with the structure of F , and in particular with (7.2).
Setup 7.2. We will use compatible almost complex structures J on F such that p is Jholomorphic over B. This means that in the partial trivialization (7.2), for some family (J M,b ) b∈B of compatible almost complex structures on M . As a consequence of this and the assumption on c 1 (F ), there are no J-holomorphic curves with negative Chern number. If one defines η using such an almost complex structure, its restriction to (7.2) is Similarly, we consider Hamiltonian functions H ∈ C ∞ (F, R) whose associated vector field X satisfies, for some α ∈ R \ Z, (Here, 2πib∂ b be the standard rotational vector field on the complex plane, normalized so that its flow is one-periodic.) An equivalent condition is that there is a Hamiltonian vector field X M on the fibre, such that Fix some α, and choose a time-dependent Hamiltonian function H = (H t ), t ∈ S 1 = R/Z, satisfying (7.10). Each one-periodic orbit of the associated vector field X = (X t ) either lies entirely inside or entirely outside F 0 ; in the second case, such orbits in fact lie outside p −1 (B), thanks to the assumption on α. From now on, suppose that the one-periodic orbits are nondegenerate, and disjoint from Ω (this is true for generic H). To any orbit x, one can associate two quantities. The first one is the action, defined in terms of (7.5) as The second one is the Conley-Zehnder index There are actually two cases to be considered here: if x lies outside F 0 , we define i(x) using η (see e.g. [38]); and if x is contained in F 0 , we define it using pη instead (in terms of (7.8), this would be db ∧ η M,b ). For the second case, we can also define an index i M (x) by working entirely inside M = F 0 (and using η M,0 ). The two indices are related by Remark 7.3. Equivalently, one can work with (7.14) which is a complex volume form on F \ F b . This yields the same index for one-periodic orbits lying outside F 0 , since one can deform η to (7.14) by moving the pole from F 0 to F b . The same holds if x is contained in F 0 , since one can also deform pη to (7.14) by multiplying it with −1/b and then moving the pole from F ∞ to F b . The advantage of (7.14) is that the pole is now disjoint from all one-periodic orbits.
Choose a family J = (J t ), t ∈ S 1 , of almost complex structures satisfying (7.7). The Hamiltonian Floer equation is where x ± are one-periodic orbits. We will consider three quantities associated to each solution u. The first is a kind of degree with respect to the map p, The second, closely related, one is the Fredholm index of the associated linearized operator D u . By a standard index formula (which becomes particularly clear from the perspective of Remark 7.3), this can be written as The third one is the energy The quantities (7.16)-(7.18) are topological, by which we mean that they are unchanged under deformations of u. In fact, their definition can be extended to maps u which do not satisfy our Cauchy-Riemann equation (but are still asymptotic to one-periodic orbits).
Let's discuss the behaviour of solutions u of (7.15) with respect to (7.2).
There, v obeys a linear equation As a consequence, ∆(log v) = 0 at all points where v(s, t) ∈ B \ {0}. By applying the maximum (and minimum) principle to log |v| = re(log v), one gets: Lemma 7.5. Take a solution of (7.15) both of whose limits x ± lie outside F 0 . If that solution enters p −1 (B \ ∂B) anywhere, it must intersect F 0 .
Solutions of (7.19) are of the form for holomorphicṽ. Consider the special case of (7.15) where x − lies in F 0 , and assume that u is not entirely contained in F 0 . In that case, we get a nontrivial solution of (7.19), defined for all s 0. In the limit s → −∞, Here, ∼ means that the quotient of the two functions converges to a nonzero (complex) constant. In terms of (7.20), m − is the order of vanishing (or pole order, if it is negative) of the functionṽ(z) at z = 0. For the corresponding case where x + lies in F 0 , one considers s → +∞ and finds that there, Since m ± is an integer but α isn't, the inequalities in (7.21), (7.22) can be equivalently rewritten as Lemma 7.6. Consider a solution of (7.15) which is not entirely contained in F 0 . Then, u −1 (F 0 ) is finite, and each point of that subset contributes positively to the intersection number u · F 0 .
Proof. By (7.20), u −1 (F 0 ) is a discrete subset. Thanks to (7.21) and (7.22), it must be compact. Hence, the intersection number is well-defined. The local contribution of each point is the same as the order of vanishing ofṽ, which is of course positive.
Lemma 7.7. Take a solution of (7.15) which is not entirely contained in F 0 . Then Proof. Let's consider the situation where x − lies in F 0 . Restrict to a region where s 0, on which v is defined. Partially compactify that region by adding a point at the limit s → −∞, and extend v by setting v(−∞) = 0. Consider the degree of the resulting map over 0. The point −∞ contributes m − (u), and the other points contribute u · F 0 . But the outcome is clearly equal to the degree over nearby points b = 0, where this time −∞ does not contribute. The other parts are similar.
Then for generic J = (J t ), the following properties hold. (i) All solutions of (7.15) are regular. (ii) Any J t -holomorphic sphere with zero Chern number avoids u(s, t) for any u such that index(D u ) ≤ 2, and any s ∈ R. (iii) Any J t -holomorphic sphere with Chern number 1 avoids x(t), for any one-periodic orbit x.
Sketch of proof. Overall, this is quite standard, using only stone age methods [10,20,28]. (i) Regularity of solutions u which remain inside F 0 is equivalent to regularity as maps to that fibre (the linearized operator D u splits, and the part in base direction is always invertible). All other solutions must leave p −1 (B), by Lemma 7.4, and outside that subset J is unconstrained. (ii) A pseudo-holomorphic sphere with zero Chern number is either contained in a single fibre F b , b ∈ B; or else, it must be disjoint from p −1 (B) by positivity of intersections. As before, one can prove generic regularity (for spheres which are not multiply covered) by considering the two cases separately. A similar remark applies to "transversality of evaluation maps". (iii) A pseudo-holomorphic sphere with Chern number 1 must intersect every fibre of p, and can never be multiply covered. Hence, transversality is again easy to show.
We now have all the basic ingredients needed for the relevant version of Hamiltonian Floer cohomology. Suppose that J has been chosen as in Lemma 7.8. We consider solutions of (7.15) with the conditions By Lemma 7.5, this means that u avoids p −1 (B \ ∂B), hence is contained in a compact subset of E. If we additionally require that index(D u ) = 1 and put an upper bound on u · Ω (or equivalently on the energy), there are only finitely many solutions, up to translation in R-direction.
Let CF * (E, H) be the graded K-vector space with one generator, of degree i(x), for each one-periodic orbit x which lies outside F 0 . The Floer differential has the form where the sum is over the kind of solution considered above. To determine the sign ± associated to each u, some additional choices are required (in the terminology of [41], a trivialization of the orientation line of each x), which we will not explain further. The cohomology HF * (E, H) of (7.26) depends only on α, up to canonical isomorphism, hence will often be referred to as HF * (E, α).
(7b) Continuation maps and the BV operator. The well-definedness of Floer cohomology is established using continuation maps [34]. Suppose that we have two choices (H ± , J ± ), either of which can be used to define HF * (E, α). To compare them, we choose an interpolating family (H C , J C ) which depends on (s, t) ∈ R × S 1 , lies in the class (7.10), and satisfies as s → ±∞. This is slightly weaker than the commonly imposed requirement that (H C , J C ) should agree with (H ± , J ± ) for ±s 0, and we need to be precise about the notion of convergence that appears. Namely, when restricted to any fixed length cylinder [s, s + σ] × S 1 , we want (H C , J C ) to converge to (H ± , J ± ) exponentially fast in any C r -topology, as s → ±∞.
The continuation map equation is the following generalization of (7.15): Since (7.19) still applies to solutions of this equations, all its consequences (Lemmas 7.4-7.7) continue to hold. So does the index formula (7.17). It is useful to distinguish between geometric and topological energy: These are related by An upper bound on u · Ω yields a bound on the topological energy. Due its exponential decay as s → ∞, there is an upper bound on ∂ s H C s,t ds ∧ dt. Hence, a bound on u · Ω also bounds the geometric energy, which gets the Gromov compactness argument off the ground. This is the counterpart of Lemma 7.8; if anything, the proof is actually simpler this time, since we have allowed perturbations of both the Hamiltonian and the almost complex structure. Note that keeping (H ± , J ± ) fixed does not constrain the value of (H C s,t , J C s,t ) at any point (s, t) ∈ R × S 1 , because the two are only related asymptotically.
As in (7.26), one counts isolated solutions of (7.28) satisfying (7.25). This leads to a chain map, the continuation map Up to chain homotopy, continuation maps are unique and well-behaved with respect to composition. For (H − , J − ) = (H + , J + ) and the constant deformation between them, the continuation map is the identity. These two properties imply that the maps on Floer cohomology induced by (7.32) are (canonical) isomorphisms.
A minor variation on the previous construction yields the BV operator, a chain map This involves a parametrized moduli space, with a single parameter τ ∈ S 1 . The auxiliary data (H ∆ , J ∆ ) depend on τ (as well as on (s, t), obviously), with the analogue of (7.27) being The appropriate version of (7.28) is an equation for pairs (τ, u): lim s→+∞ u(s, t) = x + (t + τ ).
As before, only those solutions which satisfy (7.25) are taken into account when defining (7.33). The induced map on Floer cohomology is independent of all choices, and compatible with continuation isomorphisms.
(7c) Changing the angle of rotation. The construction of continuation maps can be extended to the case where the Hamiltonians H ± satisfy (7.10) for two different angles α − ≥ α + in R \ Z. One first chooses α s , s ∈ R, such that Then, take (H C , J C ) as before, except that now H C s,t lies in the class (7.10) for α s . Consider solutions of the associated equation (7.28). The analogue of (7.19) is and the appropriate generalization of (7.20) is (7.38) v(s, t) = exp(2π α s ds)ṽ(exp(2π(s + it))).
Lemma 7.10. Consider the left hand side of (7.37) as a linear operator between suitable Sobolev completions, let's say Proof. If one writes solutions of (7.37) explicitly using (7.38), then the W 1,2 condition says that the Taylor expansion ofṽ(z) can only have nontrivial terms z −k with k ∈ [α + , α − ] ∩ Z. This matches the Fredholm index of our operator, which can be determined either directly using a suitable index formula, or by using the relation with spectral flow.
Lemma 7.4 carries over to our situation in a very special case, namely when [α + , α − ]∩Z = ∅. Lemmas 7.5-7.7 continue to hold, with the obvious adaptation that the numbers α ± should be used to give bounds for m ± (u). Lemma 7.9 also generalizes. The proof is largely as before, with one added wrinkle, which concerns solutions u of (7.28) which remain inside p −1 (B). Let's write such a solution as u = (v, w), with respect to the trivialization (7.2). By Lemma 7.10, v is a regular solution of (7.37). On the other hand, if one considers v as fixed, then w is itself a solution of a continuation-type equation in M , for which one can easily show transversality. Combining the two components then yields the desired transversality result for u. One now defines a continuation map exactly as in (7.32).
We will not give the proof of this, since it follows a standard pattern (compare e.g. [43,Lemma 3.4]). One can correlate the two choices of Hamiltonians so that both have the same one-periodic orbits. Again assuming specific choices of almost complex structures, a maximum principle argument shows that the Floer differentials coincide, and that the continuation map is essentially the identity.
(7d) Pseudo-holomorphic thimbles. We now consider maps defined on a partial compactification of the cylinder, the "thimble" Riemann surface where the added point closes up the end s → +∞. Fix α > 0, α ∈ R \ Z, and suppose that HF * (E, α) has been defined using some (H, J). Choose a nonincreasing α s , in the same sense as in (7.36), which goes from α (for s 0) to zero (for s 0). Next, choose (H thimble , J thimble ) which depend on (s, t) ∈ R × S 1 , such that H thimble s,t belongs to the class (7.10) for α s . For s → −∞, one requires this to converge to (H, J), in the same sense as before. We also require that both H thimble s,t dt (as a one-form on R × S 1 × M ) and J thimble should extend smoothly over ∞ ∈ T . The equation one considers is a version of (7.28): We have written down the Cauchy-Riemann equation at points (s, t) ∈ R×S 1 , but it extends smoothly to all of T . Lemma 7.10 has an analogue for this situation, leading to the required transversality result. Lemma 7.5-7.7, as well as the index formula (7.17) and energy formulas (7.29)-(7.31), carry over after one has removed any mention of the s → +∞ limit.
The simplest use of (7.41) is to count solutions satisfying This yields a cocycle (7.43) e ∈ CF 0 (E, H), whose Floer cohomology class is independent of all choices. A natural generalization is to introduce evaluation constraints. Let's fix a point ζ ∈ T , for concreteness say ζ = ∞. Suppose that we are given an oriented manifold K together with a proper map κ : K → E, which is transverse to the one-periodic orbits of H. Consider pairs (u, k) consisting of a solution u of (7.41), (7.42) and a point k ∈ K, such that Assuming suitably generic choices, counting solutions which satisfy this constraint yields a cocycle Generalizing slightly, one can replace K by a proper pseudo-cycle in E, let's say one with R-coefficients. Finally, one can allow proper pseudo-cycles with K-coefficients, by which we mean formal sums where the K j are proper pseudo-cycles with R-coefficients. For (7.46), one defines b K by adding up the cocycles (7.45) associated to K 0 , K 1 , . . . with corresponding weights. The cohomology class of (7.45) is independent of all choices, and represents the image of [K] ∈ H * (E; K) under the PSS map [30] (7.47) In fact, one can use this as the definition of (7.47), even though that is not necessarily the most convenient or natural approach (using Morse homology instead of pseudo-cycles would have distinct advantages, one of them being that it leads to a map on the chain level). However, here we will not actually use the entire map (7.47), but only specific instances of the classes (7.45), which explains our choice of approach.
Remark 7.12. While an extensive discussion of pseudo-cycles would be out of place here (see [28, Section 6.5]), we do want to briefly explain the notion of properness which appeared above. A proper pseudo-cycle in E is a map κ : K → E, where K is an oriented manifold, such that there exists anotherκ :K → E, with dim(K) ≤ dim(K) − 2, satisfying the following condition. If k 1 , k 2 , · · · ∈ K is a sequence of points which goes to infinity, and such that κ(k 1 ), κ(k 2 ), . . . converges, then the limit of the latter sequence lies inκ(K). In the special caseK = ∅, this would reduce to saying that the original map κ is proper. Also, if we have a pseudo-cycle in F , then its intersection with E is a proper pseudo-cycle in E. Any proper pseudo-cycle in E determines a class [K] ∈ H * (E; Z).
To get a proper pseudo-cycle with R-coefficients, one equips K with a multiplicity function (a locally constant complex-valued function). One can always think of this as a formal sum of proper pseudo-cycles K j (in the previously considered sense) with multiplicities µ j . It makes sense to consider these modulo some equivalence relation, namely: any term in (7.48) with zero multiplicity may be removed freely; two terms that contain the same pseudo-cycle may be combined by adding up their multiplicities; and the orientation of any K j may be reversed, while changing the sign of µ j at the same time.
Lemma 7.13. For any Floer cocycle b K as in (7.45), ∆b K is a coboundary.
Proof. By considering a suitable parametrized moduli problem for maps on the thimble, we will construct a Floer cochain β K satisfying The parameter space will be disc, which itself has to be compactified. While generally speaking a standard construction, setting it up requires a bit of care. Fix all the data involved in the relevant previously defined structures, namely: the definition of HF * (E, α) uses some (H, J); the BV operator uses (H ∆ , J ∆ ), where we take both asymptotics (H ± , J ± ) = (H, J) to be the same for simplicity; finally, b K uses (H thimble , J thimble ).
Part of our parameter space is a half-infinite cylinder ξ = (ρ, τ ), where ρ 0 and τ ∈ S 1 . For such a parameter value, we construct (H β ξ , J β ξ ) as follows. Consider Take a finite piece of the cylinder where s ∈ [−ρ/2 − L/2, −ρ/2 + L/2], for some constant L/2. Because of (7.34) and the translation by ρ, (7.50) is close to (H t+τ , J t+τ ) on that region (it converges uniformly exponentially fast as ρ → ∞). The same holds for (7.51), simply because of its limit. One can therefore use a partition of unity to glue the two data together to a new datum on the thimble, which we denote somewhat informally by (7.52) (H β ρ,τ,s,t , J β ρ,τ,s,t ) = (H ∆ τ,s,t , J ∆ τ,s,t )# ρ (H thimble s,t+τ , J thimble s,t+τ ) (the subscript in # ρ is the gluing length, which in our previous description we had realized by translating the first datum by ρ in s-direction). Our full parameter space is ξ ∈ (R×S 1 )∪{ξ = −∞}. We extend our choice (7.52) arbitrarily to the rest of that space.
Consider the parametrized moduli space of pairs (ξ, u, k), where k is as in (7.44). In the limit where ξ = (ρ, τ ) with ρ → ∞, we can get broken solutions (τ,ũ, u, k), where (τ,ũ) is a point in the space contributing to ∆, and (u, k) (after rotation t → t − τ ) similarly contributes to b K . Assuming transversality, such limits describe precisely the ends of the 1-dimensional parametrized moduli spaces. By a standard argument, it follows that counting isolated points of the parametrized problem yields the desired cochain (7.49).
Discussion 7.14. The map (7.47) fits into a long exact sequence and the third term HF * (E, α) red comes with a refined BV operator One defines HF * (E, α) red as the mapping torus for a suitable chain level version of B. Similarly, a nullhomotopy for the chain level version of ∆ • B gives rise to ∆ red . For our purpose, the following partial description suffices: a pair (K, ξ) consisting of a proper pseudo-cycle K and a Floer cochain ξ ∈ CF * (E, H) with gives rise to a class which maps to [K] under the boundary homomorphism in (7.53). Moreover,

The Borman-Sheridan class
This section discusses the Borman-Sheridan class and various ideas surrounding it. The geometric setup remains the same, but we will now start to make more serious use of its specific properties.
Lemma 8.1. Suppose that α > 1. Consider solutions of (7.41), (8.1). Suppose that a sequence of such solutions converges to a broken solution (ũ, u), consisting of a Floer trajectorỹ u and a solution u of (7.41). Then,ũ avoids F 0 entirely, and u again satisfies (8.1).
Proof. By definition of a broken solution, there is some one-periodic orbitx such that We know that (8.5) deg(ũ) + deg(u) = 1.
If one assumes thatx lies in F 0 , (7.24) and its analogue for thimbles say that Ordinarily, there should be an exception in (8.7) for when u is entirely contained in F 0 ; but in that case deg(u) = 0 by definition, whereas 1 − α < 0 by assumption, so the inequality still holds. Adding up (8.6) and (8.7) yields a contradiction to (8.5). We now know thatx does not lie in F 0 , and therefore By (8.5), equality must hold in (8.8) and (8.9), which yields the desired behaviour.
Remark 8.2. It is maybe instructive to look at more general broken solutions, even though (for codimension reasons) they are not relevant for our counting arguments. Suppose that a sequence of solutions of (7.41), (8.1) converges to a broken solution (ũ 1 , . . . ,ũ k , u), where the first k components are Floer trajectories. As before, we know that (8.10) deg(ũ 1 ) + · · · + deg(ũ k ) + deg(u) = 1.
Suppose first that u is not contained in F 0 . By applying Lemma 7.7 to all components, and using (7.23), one then finds that The first inequality in (8.11) is an equality iff none of our maps has a limit lying in F 0 . Assuming this holds, the second inequality is an equality iff allũ 1 , . . . ,ũ k are disjoint from F 0 , while u satisfies (8.1). The other potential situation is where, for some j ≥ 1, (ũ j+1 , . . . ,ũ k , u) are all contained in F 0 , whileũ j isn't. However, this means that If α > 1, this contradicts (8.10). Hence, we see that all broken solutions that appear as limits are built by adding Floer trajectories inside E as extra components. One can further extend this discussion, including sphere bubbles as well, which leads to a complete description of the Gromov compactification. As usual, sphere bubbling turns out to be always a codimension ≥ 2 phenomenon, hence will not occur in our applications.
Lemma 8.1 implies that (8.3) is a cocycle provided that α > 1. Its Floer cohomology class, the Borman-Sheridan class, is independent of all choices. An argument parallel to Lemma 7.13 (and which we will therefore omit) shows the following: Lemma 8.3. For α > 1, the Floer cocycle ∆s is a coboundary.
(8b) The Borman-Sheridan class and Gromov-Witten theory. The starting point for the following discussion is the Gromov-Witten invariant which is the special case k = 1 of (2.20). Our notation is as follows: We also fix a point to be used throughout the rest of the paper. Choose some almost complex structureJ as in (7.7) (one could allow a family of almost complex structures varying alongS, but that is not necessary or helpful for our purpose). Suppose that we have a map u :S → F , which is pseudo-holomorphic with respect toJ and satisfies (8.15)ũ · F 0 = 1.
Any intersection point ofũ with a fibre F b , b ∈ B, contributes positively to the intersection number, and contributes 1 iff the intersection is transverse. Hence, the part ofũ lying inside p −1 (B) is a pseudo-holomorphic section. In particular, there is a unique automorphism ofS which we can apply to achieve that From now on, we will consider mapsũ satisfying (8.15), (8.16). For generic choice ofJ, the space of such maps in any fixed homology class, with evaluation at the pointζ 3 , defines a pseudo-cycle of codimension 2 in F . If we add up the contributions of all homology classes with the usual weights qũ ·Ω , the outcome is a pseudo-cycle with K-coefficients, which represents z (1) ∈ H 2 (F ; K). Let's denote this pseudo-cycle by Z (1) .
Restricting to the open subset whereũ(ζ 3 ) / ∈ F 0 yields a proper pseudo-cycle with K-coefficients in E, denoted by Z (1) |E. Proof. By considering an appropriate parametrized moduli space, we will construct a Floer cochain bounding the cocycle (7.45) associated to Z (1) |E: The parameter is a number σ ∈ R, which appears as the position of a marked point ζ 2 = (σ, 0) on the thimble (this is in addition to our usual marked point ζ 1 = ∞). We consider the space of pairs (σ, u), where u is a solution of an equation of type (7.41), with additional conditions There are two relevant degenerations to consider. In the first one, σ → −∞, the limit is a broken solution (ũ, u), whereũ is a Floer trajectory satisfying The other component u is again a solution of (7.41), with u(ζ 1 ) ∈ F 0 . By the same argument as in Lemma 8.1, one finds that the limits ofũ must lie outside F 0 , and that deg(ũ) = 0; but then,ũ must be disjoint from p −1 (B \ ∂B) by Lemma 7.5, which is a contradiction to (8.19). Hence, there are no such limit points.
The other relevant degeneration happens when σ → +∞; the limit (u,ũ) consists of a solution u of (7.41), which for the same reason as before will be disjoint from F 0 , together with a sphere bubble which satisfies (8.15) and (when appropriately parametrized) (8.16). More precisely, the almost complex structureJ which arises is that appearing in (7.41) for the point ζ 1 ∈ T . The two components are related by the incidence requirement Hence, counting such pairs (u,ũ) just yields b Z (1) |E . From this, a standard argument shows that counting points in the parametrized moduli space leads to the desired structure (8.17).
Remark 8.5. Sphere bubbling is normally a codimension 2 phenomenon, and correspondingly, the relevant gluing problem has two parameters: a real gluing length and a circle-valued gluing angle. In the situation of Lemma 8.4, we use σ instead of the gluing length as a parameter, and since ζ 2 is constrained to lie on R × {0}, there is no gluing angle. This explains why it makes sense for sphere bubbling to yield a codimension 1 boundary stratum of our moduli space.
As an instance of Discussion 7.14, (8.17) gives rise to a class which maps to z (1) |E under the connecting map from (7.53). We also get a Floer cocycle which represents the image of (8.21) under ∆ red . Proof. The first step is to combine the ideas from Lemmas 7.13 and 8.4 (since this is a reprise of previous arguments, we will be rather light on details). Consider a moduli space as in Lemma 7.13, but with one additional parameter σ as in Lemma 8.4, again thought of as giving a marked point ζ 2 = (σ, 0). Look at maps u satisfying (8.18).
Three kinds of degenerations of such (ξ, σ, u) are relevant for us. The first one is σ → ∞, in which case one gets sphere bubbling as in Lemma 8.4. The contribution obtained by counting such configurations is β Z (1) |E . The second kind of degeneration happens when ξ = (ρ, τ ) with ρ → ∞, and simultaneously σ → −∞. This contributes zero, for the same reason as in Lemma 8.4. The final possibility is that ξ = (ρ, τ ) with ρ → ∞, while σ remains bounded (see Figure 13). The resulting broken solutions are of the form (τ, σ,ũ, u), where (τ,ũ) satisfies the equation which defines ∆, and u is a map as in (7.41), such that Figure 13. One of the limits from the proof of Proposition 8.6.
The equations for (τ,ũ) and (σ, u) are not independent, because of the appearance of τ in (8.23). Hence, the contribution from this degeneration can't be written as a combination of previously introduced ones.
To make up for this shortcoming, we introduce another moduli space. This has parameters τ 1 ∈ [0, 1], σ ∈ R and τ 2 ∈ [0, τ 1 ], and consists of (ũ, u) such that: (τ 1 ,ũ) is again a point in the moduli space defining ∆, and u is as in (8.23) except that the second part of that equation must be replaced by The space of parameter values has three boundary components. One is when τ 1 = τ 2 , which yields exactly the same contribution as in (8.23). The second boundary component τ 2 = 0 contributes ∆ν. The final boundary component appears when τ 1 = 1. In that case, (σ, τ 2 ) can be an arbitrary point in R × S 1 . Now, suppose that when defining (8.3), we have chosen the auxiliary data so that isolated solutions are transverse to F † . The count (with signs) of points at which any such solution passes through F † is 1. Hence, we get a contribution which equals (8.3) (since τ 1 is fixed to be 1, one can arrange that the only mapsũ that appear in that case are of the formũ(s, t) = x(t), which do not affect the contribution).
By combining both parametrized moduli spaces, and counting isolated points in them (more precisely, by subtracting the contribution of the second space from that of the first one), we therefore obtain a Floer cochain which bounds β Z (1) |E − ∆ν − s.
Remark 8.7. The sign conventions we have introduced in the formal TQFT context, specifically (5.1H) and (5.11), are compatible with those that arise in Floer theory from juggling determinant lines of elliptic operators (compare e.g [41,Section 11] or [51]). We do not normally dig into such details, but Proposition 8.6 may be a good point to do so, since the sign in its statement has consequences throughout the paper, for instance (2.30).
Points in the first parametrized moduli space are of the form (ξ, σ, u). Since ξ takes values in a two-dimensional space, the associated orientation does not change if we reorder the components as (σ, ξ, u). Hence, σ → +∞ corresponds to a "boundary face at infinity" which is positively oriented. The other nontrivially contributing "boundary face infinity" corresponds to writing ξ = (ρ, τ ) and sending ρ → +∞, which is again positively oriented. As in (5.11), this means that the parametrized moduli space yields a cocycle α 1 ∈ CF −1 (E, H) with where θ is the contribution from (8.23). In the second parametrized moduli space, points are of the form (τ 1 , σ, τ 2 ,ũ, u), or equivalently as far as orientation is concerned, (τ 2 , τ 1 , σ,ũ, u). The boundary value τ 1 = τ 2 is the maximal value of τ 2 , hence counts positively as in (8.25) (it is important that the ordering (τ 1 , σ) of the remaining parameters is the same as the one encountered before). On the other hand, τ 2 = 0 is the minimal value, which a priori means that the corresponding boundary points (τ 1 , σ,ũ, u) are counted negatively. However, to recover the orientation that corresponds to ∆ν, one has to reorder the entries as (τ 2 ,ũ, σ, u), and since index(Dũ) = −1 at the relevant points, that permutation introduces another sign, which cancels the previous one. The final contribution comes from τ 1 reaching its maximal value, and since the remaining parameters (σ, τ 2 ) correspond to the standard complex orientation of the cylinder, we get a positive sign. The outcome is that the second moduli space yields an α 2 ∈ CF −1 (E, H) such that  First, fix distinct points ζ 1 , ζ 2 ∈ T on the thimble, and consider solutions of (7.41) such that This leads to a cochain H) which, by a straightforward analogue of Lemma 8.1, is a cocycle if α > 2. As the notation is intended to suggest, we will show later on that the Floer cohomology class of (8.28) is the (pair-of-pants) square of the Borman-Sheridan class; see Section 10b.
The second possibility is to use only one marked point, but to ask for a tangency condition: This makes sense because, if u(ζ) ∈ F 0 holds, Dp • Du ζ is complex-linear, which means that its vanishing is a condition of (real) codimension 2. As before, counting such solutions leads to a Floer cochain, which is a cocycle if α > 2; we denote it by (8.30)s (2) ∈ CF 0 (E, H).
Proof. The basic idea is similar to that in Proposition 8.6. We introduce one parameter ρ ≥ 0. One of the marked points is fixed, let's say ζ 1 = ∞, and the other one is ζ 2 = (ρ, 0). One then considers the parametrized version of (8.27).
As ρ → ∞, the relevant limiting behaviour is sphere bubbling at ζ 1 , leading to a principal component u together with a bubble componentũ :S → F , which (for a suitable parametrization) satisfies There are in fact two topologically distinct sub-cases. One is thatũ is a constant "ghost bubble". In that case, the principal component is tangent to F 0 at ζ 1 . Counting such configurations contributess (2) . The other case is whereũ is not contained in F 0 , so that u · F 0 = 2. In that case, the principal component satisfies deg(u) = u · F 0 = 0, which means that it avoids F 0 altogether. If one counts the mapsũ, thinking of u(ζ 1 ) as a fixed (generic) point in E, the outcome is 2z (2) , where the factor of 2 comes from the possibility of changing the parametrization (8.31), exchangingζ 1 andζ 2 while keepingζ 3 fixed. Hence, the contribution of such pairs (u,ũ) is 2z (2) e. Note that other kinds of sphere bubbling, for instance whereũ is a non-constant map to F 0 , are irrelevant because they are of codimension ≥ 2.
The other potentially problematic limiting behaviour as ρ → ∞ is where the limit is of the form (ũ, u), whereũ is a Floer trajectory, and the principal component u is entirely contained in F 0 . However, that can be excluded by the same arguments as in Lemma 8.1.

A connection
We will construct, under Assumption 2.7, a connection on Floer cohomology, and show that it satisfies a version for (2.34) with c = −1; see (9.32). Eventually, one has to show that this construction fits into the formalism set out in Sections 2a and 5c; but we postpone that to Sections 10-11.
(9a) The setup. Fix some α ∈ R \ Z, and the data (H, J) required to define the chain complex CF * (E, H) underlying HF * (E, α). We denote by ∂ q the operation of differentiating Floer cochains, which means differentiating their coefficients with respect to the obvious basis. It follows from the definition of the differential (7.26) that We can assume, since this is true for generic choices of (H, J), that each solution u with index(D u ) = 1 intersects the components Ω j of Ω transversally. Let's introduce a parameter τ ∈ S 1 , which singles out a point ζ = (0, −τ ) ∈ R × S 1 . We can then consider pairs (τ, u), where u is a solution of (7.15) such that u(ζ) ∈ Ω j . For each j, counting points in the associated parametrized moduli space yields a chain map of degree 1. Adding up those chain maps, with the weights µ j from (7.4), and then multiplying the whole with q −1 , recovers (9.1).
Remark 9.1. One may wonder why, in order to obtain the count from (9.1), one has to choose orientations of the parameter space so that the marked point ζ runs negatively around the S 1 factor. This has its origin in the sign convention for the differential d. Namely, if u is a regular solution of (7.15) with index(D u ) = 1, The general theory of orientation operators associated to one-periodic points x (going back to [8], see also [9]; the terminology is borrowed from [41]) produces orientations of ker (D u ) ⊕ coker (D u ) for any u, regular or not. In (7.26) we count u with ±1 depending on whether this orientation agrees with the obvious one on the right hand side of (9.2) or not.
Having reformulated the right hand side of (9.1) as a parametrized moduli problem, we can consider it as a special case of a more general construction. Namely, take a proper pseudocycle K with K-coefficients in E, as in (7.46). Choose (H r , J r ) which depend on (τ, s, t), with asymptotics for ±s 0 as in (7.27). Consider the parametrized space of solutions of the associated equation (7.28), with Counting points in this space (with real multiplicities, since K j is a pseudo-cycle with Rcoefficients) gives a chain map r Kj . We then define Let's specialize to K = q −1 Ω|E. Suppose that we take (H r τ,s,t , J r τ,s,t ) = (H t , J t ). In that case, our previous argument shows that r q −1 Ω|E agrees with the right hand side of (9.1). If we still choose (H r , J r ) to converge to (H, J) as s → ±∞, but otherwise arbitrary, then r q −1 Ω|E is still at least chain homotopic to (9.1) (the chain homotopy is defined in a straightforward way, by adding another parameter that deforms the auxiliary data). Finally, we can choose the asymptotics (H − , J − ) and (H + , J + ) different, even with different angles α − and α + . In that case, the resulting map r q −1 Ω|E is chain homotopic to (9.1) composed with the continuation map (7.32). To make this entirely clear, let's write d ± for the two Floer differentials involved. Then we have chain homotopies These homotopies are obtained by moving the marked point ζ = (0, −τ ) towards one of the ends of the cylinder, and considering the resulting two-parameter moduli space.
From this point onwards, we will work under Assumption 2.7. Let's choose a pseudo-chain A such that To be precise, A is a pseudo-chain with K-coefficients in F ; ∼ = stands for equivalence of pseudo-cycles; and the inclusion F 0 → F is itself thought of as a pseudo-cycle. Restricting to E then yields a proper pseudo-chain A|E, which satisfies By imitating the construction of (9.4) with A|E instead of K, one gets a chain homotopy Remark 9.2. We refer to [37,22,50] for a general discussion of the relation between equivalence classes of pseudo-cycles and homology, and only consider the aspect which is immediately relevant here. A pseudo-chain (or relative pseudo-cycle) in F is a map a : A → F , where A is an oriented manifold with boundary, which satisfies the same conditions as a pseudo-cycle, and such that a|∂A is itself a pseudo-cycle. Any pseudo-cycle which represents the trivial class in H * (F ; Z) can be represented as such a boundary.
For a pseudo-cycle with R-coefficients, the same is true up to equivalence (in the sense mentioned in Remark 7.12): if its class in H * (F ; R) is trivial, the pseudo-cycle is equivalent to the boundary of a pseudo-chain with R-coefficients. Finally, for the case of K-coefficients which we have encountered above, one simply argues order by order in the Novikov variable. We have used this implicitly when asserting the existence of (9.6).
Our strategy will be to show, using the same ideas as in Lemma 8.4, that for suitable choices of α ± , the map r Z (1) |E is chain homotopic to zero. More precisely: Then there is a map (9.9) Let's explain why this is useful. From (9.5) and (9.8), we have chain homotopies (for which we now introduce the notation h + and r A|E ) We combine them with (9.9) to form (9.12) Hence, the induced map on cohomology, which we also denote by is a connection with respect to the continuation map between those Floer cohomology groups, meaning that it satisfies the cohomology level counterpart of (9.14).
Discussion 9.4. Alternatively, one could use the other homotopy from (9.5), which we now denote by h − . It satisfies This leads to another formula for a connection: which however turns out to be the same as (9.12) up to chain homotopy. This follows from the existence of a nullhomotopy To construct it, one interprets ∂ q C −C∂ q geometrically as in (9.1): it counts pairs consisting of a solution u of the continuation map equation and a point (σ, τ ) ∈ R × S 1 whose image goes through q −1 Ω|E. Now, h − and h + are obtained from the same kind of moduli problem, but where the marked point is required to lie in one of the two halves of the cylinder. Adding up the two possibilites gives a space which is very similar to that underlying ∂ q C − C∂ q : the choices of auxiliary data may not agree, but one can interpolate between them, and the associated parametrized moduli space gives rise to the homotopy in (9.18).
(9b) Constructing the nullhomotopy. This section contains the proof of Lemma 9.3. Fix α ± and (H ± , J ± ) as before. We will introduce a moduli space with two parameters (σ, τ ) ∈ (−∞, 0) × S 1 . The parameters determine the position of two marked points We choose auxiliary data (H χ , J χ ) which depend on (σ, τ, s, t), and which on the ends, are asymptotically equal to the previously chosen (H ± , J ± ). In fact, we want our data to extend smoothly to σ = 0, and there, to agree with those used to define r Z (1) |E . We impose an additional requirement on those data, which is that J χ 0,τ,0,−τ =J should be the almost complex structure used to define Z (1) . We also ask that the data should extend to the other limit σ = −∞. It is worth emphasizing what this means: as σ → −∞, the marked point ζ 2 goes to −∞, but the data (H χ , J χ ) converge uniformly over the entire cylinder (in any C r topology, with a sufficiently small exponential weight over the ends s → ±∞).
We consider solutions of the associated equation of type (7.28), with the conditions (9.20) The limiting behaviour as σ → 0 is bubbling off of a holomorphic sphere. More precisely, the relevant situation is as follows. The bubble is aJ-holomorphic sphereũ which (suitably parametrized) satisfies (8.15), (8.16). The principal component of the limit satisfies deg(u) = 0; and the two components are joined by the incidence condition u(ζ 1 ) =ũ(ζ 3 ). This is exactly what ones sees when spelling out the definition of r Z (1) |E , and has the desired codimension 1. More complicated potential degenerations can be excluded because they have higher codimension.
Among the other potential degenerations as σ → −∞, there is one which deserves a separate discussion, since it leads to the restriction on α ± in Lemma 9.3. Namely, suppose that the limit consists of three components (ũ − , u,ũ − ) with the following properties. The principal component u lies entirely in F 0 ; The other componentsũ ± are Floer trajectories for (H ± , J ± ), but do not quite play symmetrical roles:ũ − carries a marked point with an evaluation constraint, as in (9.21);ũ + does not, and is considered up to translation in s-direction. To be more specific, the counterpart of (9.21) is Ordinarily, since such a limit has three components, one would expect to be able to rule it out on the basis having of codimension 2. In this case, u originally came with an evaluation condition u(ζ) ∈ F 0 , which now holds tautologically (hence is non-transverse), which means that the dimension count has to be reconsidered. Assuming suitably generic choices, the actual dimension of the space of solutions (τ,ũ − , u,ũ + ) is where the α ± terms come from (7.13) The first term in the last line of (9.24) is the dimension of the main moduli space, and the second term is therefore the codimension. Our assumption [α + , α − ] ∩ Z = ∅ is equivalent to α − − α + > 0, which ensures codimension ≥ 2 in (9.24), enough to make such degenerations irrelevant. However, it turns out that in this case there is a simpler topological argument, which excludes their existence anyway. Namely, Lemma 7.7 and (7.23) lead to a contradiction with (9.23): We have considered only the simplest possibly limits, but more complicated ones can be disregarded for codimension reasons. The upshot is that counting isolated points in our parametrized moduli space indeed provides a map χ with the desired property (9.9), concluding our proof of Lemma 9.3.
(9c) Applying the connection to the identity element. Our next goal is: Proposition 9.5. The connection (9.12) has the property that, for ψ as in (2.26), The proof hinges on introducing an intermediate object, a cocycle (see Figure 14 for a schematic description) which will then be compared to both sides of (9.26). The construction of − is very similar to that in Lemma 9.3, which means that it uses the same parameters (σ, τ ) and associated marked points (ζ 1 , ζ 2 ), except that we are now working on the thimble surface (7.41). Accordingly, we consider auxiliary data (H − , J − ) on that surface, with limit (H, J) over the end. As before, we require that these data should extend smoothly to σ = −∞ and σ = 0. In the case of σ = 0, we impose the previous restriction that J − 0,τ,0,−τ =J should be the almost complex structure used to define Z (1) . The maps u are required to satisfy (9.20) (we only have one limit instead of two, of course). Concening the analysis of ends of one-dimensioal moduli spaces, we get bubbling off ofJ-holomorphic spheres as σ → 0. The other limit σ → −∞ does not contribute, provided that α > 1, for the same reason as in Section 9b.
To define + , one considers another moduli space of maps on the thimble, where the parameters are given by a choice of point ζ ∈ T lying in the interior of the circle {0} × S 1 (which means that either ζ = (σ, −τ ) with σ > 0, or ζ = ∞). We want data (H + , J + ) as before, extending smoothly to the case σ = 0, and which in that case match up with (H − , J − ). Moreover, we now impose the condition J + ζ,ζ =J for all ζ (to clarify: for every value of the parameter ζ, we have a family of almost complex structures varying over the thimble, and we constrain that family at one point, whose position is given by ζ). The maps u should satisfy  The first line of (9.28) is shorthand: we consider the moduli spaces such that u(ζ) goes through some component of Z (1) (because deg(u) = 0, u(ζ) automatically lies in E), and add them up with suitable multiplicities. Another way to see this is to write the moduli space as one of pairs (u,ũ), whereũ is aJ-holomorphic sphere as in (8.15), (8.16), connected to u by the incidence condition u(ζ) =ũ(ζ 3 ) (this is what we drew in Figure 14).
The important point is that at σ = 0, the two moduli spaces match. Hence, the associated contributions to d − and d + will cancel, which ensures that (9.27) is a cocycle (provided that α > 1).
Proof. As usual, the argument constructs an explicit coboundary which fits into (9.29). Even though none of the steps is surprising or difficult, the number of terms threatens to be notationally overwhelming, and we will therefore use a graphical shorthand description. The starting point is to spell out what ∇ −1 e is, which we have done in Figure 15. All the terms are obvious from (9.12) except for ∂ q e, which we interpret as thimbles with a marked point, in exact analogy with what we have done for (9.1). To construct the coboundary, one adds up contributions from four parametrized moduli spaces, whose construction is outlined in Figure 16-19, and where the contribution of the last of the four should be multiplied by ψ.
In words, what we do in the first moduli space ( Figure 16) is to take the two surfaces (a cylinder and a thimble) involved in defining c(∂ q (e)), and glue them together (introducing a gluing parameter). Moreover, once we carry out the gluing, the marked point is allowed to move freely over the glued surface, as long as it remains to the right of the circle inherited from {0} × S 1 ⊂ R × S 1 . The maps u involved are all assumed to satisfy deg(u) = 0, and therefore remain inside E. There are two different (codimension 2) degenerations as the gluing parameter goes to infinity, depending on which component of the limit the marked point ends up in, see Figure 16(i) and (ii); those two components contribute c(∂ q (e)) and h + (e), respectively, where h + is as in (9.10). There are two other relevant boundary components: one where the gluing length has reached its (arbitrarily chosen) minimum, Figure 16(iii); and one where the marked point moves as far to the left as is allowed, Figure 16(iv). In those two last cases, the resulting contribution can not be expressed in terms of previously introduced operations.
The construction of the second moduli space (Figure 17) uses the same idea as the first one, but now applied to r A|E (e), in the notation from (9.11). Since here, the evaluation at the marked point is assumed to go through the proper pseudo-chain A|E, we obtain  two boundary components corresponding to the terms in (9.7), shown in Figure 17(i) and (ii); the first of them produces the same contribution (up to sign) as one of our previous degenerations, Figure 16(iv). Another contribution comes from reaching the minimal gluing length, Figure 17(iii). The final one appears when the gluing length goes to infinity, which by design recovers r A|E (e).    The third moduli space ( Figure 18) does not involve any neck-stretching, and the marked point lies in a bounded subset of the thimble Riemann surface. As in the previous two cases, the maps involved have deg(u) = 0, hence remain inside E. The three boundary contributions appear for very straightforward reasons; two of them, Figures 18(i) and (iii), match previously encountered ones, whereas the remaining one reproduces + up to the factor of ψ.
The final moduli space ( Figure 19) can again be thought of as having been obtained by gluing together two pieces, this time those defining χ(e). Here, the maps have deg(u) = 1. Besides the limit where the gluing length goes to infinity, shown in Figure 19 χ(e), we have two other contributions to consider: one where the gluing length reaches its minimum, which yields − , and another one where the two marked points collide ( Figure  19(ii) and (iii), respectively). In fact, there is another possible limit, which is where one of the marked points goes to −∞, but its contribution is trivial, for the same reason as in the definition of (9.9) or of − .
Inspection shows that when taking all four moduli spaces into account, all codimension 1 contributions vanish except those which occur on both sides of (9.29), at least up to sign.
To confirm the correctness of the signs, one has to inspect all the orientations, something which we will omit here.
Proof. We introduce a moduli space similar to that underlying − , but with an additional parameter. The outcome will be a cochain ι satisfying wheres is cohomologous to s. The construction is shown in Figure 20, but it still makes sense to spend a little time discussing it.
Concretely, the position of the two marked points is now ζ 1 = (ρ, −τ ) and ζ 2 = (σ, −τ ), where ρ ∈ (0, ∞) is the new parameter, and σ ∈ (−∞, ρ), τ ∈ S 1 . Concerning the auxiliary data (H ι , J ι ), in the limit ρ → 0 they should agree with (H − , J − ), which gives a contribution of − . There are three other limits that one has to discuss. As usual, having the two marked points collide leads to bubbling off of a holomorphic sphere at ζ 1 . One can arrange that the resulting contribution is exactly + . Next, the limit ρ → −∞ has zero contribution, as usual.
The most interesting limiting situation is ρ → ∞, which one thinks of as follows: ζ 1 = ∞ ∈ T , whereas the position of ζ 2 ∈ R×S 1 ⊂ T can be arbitrary. We claim that this yields a cochaiñ s in the same homology class as s. To show that, one introduces another parameter space, which deforms the auxiliary data until they are no longer dependent on the position of ζ 2 . Then, what one is counting are maps u with u(ζ 1 ) ∈ F 0 , with a multiplicity which depends on the possible positions of ζ 2 . However, the signed count of such possibilities is precisely u · F † = deg(u), which is 1 by definition (we have seen the same kind of argument before, towards the end of the proof of Proposition 8.6). Hence, the multiplicity issue turns out to be trivial, and we recover the definition of s.
The combination of Lemma 9.6 and 9.7 establishes Proposition 9.5.
(9d) Applying the connection to the Borman-Sheridan element. The following result is the core of our discussion of ∇ −1 .
The proof is an adaptation of that of Proposition 9.5. Again, the two sides of (9.31) will the relevant boundary/degeneration contributions are: To define σ ± , one modifies our previous construction of ± by adding another marked point at ∞ ∈ T , whose image is required to go through F 0 , and increasing deg(u) by 1.
ζ 2 can be anywhere to the right of the circle, its image goes through A ζ 1 is fixed, its image lies on F 0 the extra degeneration, with the η factor removed: constant "ghost bubble" in F 0 Figure 22. Origin of the ηs term on the right side of (9.34).
Lemma 9.9. The cocycle (9.33) is related to the connection (9.12) by Proof. The proof follows closely that of Lemma 9.6. The modifications one has to apply to the moduli spaces involved are the same as when transforming (9.27) into (9.33), and much of the argument carries over without any essential changes. Therefore, we will mostly just do a quick review, slowing down only to focus on the new aspect of (9.34), namely the ηs term.
The starting point is the analogue of Figure 15, appropriately modified by replacing e with s; which means that the thimble component acquires an additional marked point, and is now of degree 1. The same applies to Figure 16. Note that now, since the maps will no longer remain in E, we must use q −1 Ω instead of its restriction to E. Similarly, in Figure 17, one uses A instead of its restriction to E, and hence relies on (9.6) instead of (9.7). Potentially, the additional term ηF 0 in (9.6) could cause an additional boundary contribution to appear, which would consist of thimbles with two distinct marked points (one at ∞, and another on the circle {0} × S 1 ) whose images go through F 0 . However, our maps have deg(u) = 1, hence can't have two distinct intersection points with F 0 , ruling out this contribution.
In the analogue of Figure 18, we again have to consider the impact of the extra term in (9.6). One might want to argue as before that this vanishes, but there are points in the compactification of our moduli space to which this argument does not apply. To illustrate this, the space itself is shown in Figure 22 at the top, and the problematic degenerations, which make up what we will call the "ghost bubble stratum", at the bottom. The dimension of the entire moduli space is i(x) + 2 deg(u) + 2 − 3 = i(x) + 1, and the "ghost bubble stratum" has dimension i(x). We may assume that, if ζ 2 is sufficiently close to ζ 1 , the auxiliary data involved (Hamiltonian and almost complex structures) are independent of the position of ζ 2 . In that case, and assuming suitable transversally, the situation is as follows. Given any map u with u(ζ 1 ) ∈ F 0 , the points ζ 2 sufficiently close to ζ 1 such that u(ζ 2 ) lies in a given component of A form a one-dimensional submanifold of T . That submanifold has endpoints corresponding to the various parts of ∂A, and one kind of such endpoints corresponds precisely to our "ghost bubbles". In other words, this argument shows that in the compactification of a one-dimensional moduli space, the "ghost bubbles" are ordinary boundary points. Counting such points then makes a contribution which is at least cohomologus to ηs (like thes term encountered in the proof of Lemma 9.7).
Proof. This is a modification of the proof of Lemma 9.7, entirely analogous to what we have done previously. We summarize it in Figure 23. The only part that requires extra attention is the limit where the two marked points collide. In that case, there are two possibilities, in parallel with Lemma 8.8. One is the appearance of a constant "ghost bubble", in which case (for topological reasons) the principal component u is tangent to F 0 at ζ 1 . We can think of the condition u(ζ 2 ) ∈ F † as counting the possible positions of ζ 2 , in analogy with the argument from Lemma 9.7. Since deg(u) = 2, the multiplicity is 2 this time, hence a count leads to (a cocycle cohomologous to) 2s (2) . The other possibility (shown at the bottom of Figure 23) is bubbling off of a nonconstant pseudo-holomorphic sphere, with two points on F 0 . In that case, the principal component satisfies deg(u) = 0, which by the same argument before, means that the relevant contribution is (cohomologous to) zero.
The combination of Lemma 9.9 and 9.10 establishes Proposition 9.8.
Remark 9.11. We close our discussion with a technical point, which applies to Proposition 9.8 and all earlier computations regarding specific Floer cochains of degree 0 (Proposition 8.6, Lemma 8.8, Proposition 9.5). Namely, inspection of the geometry of our Lefschetz fibrations shows that it is always possible to choose H so that CF * (E, H) is concentrated in degrees ≥ 0.
In that case, any relation in CF 0 (E, H) that holds up to coboundaries actually holds strictly.
In other words, all the zero-dimensional moduli spaces that define those coboundaries would turn out to be empty for dimension reasons. For the same reason, Lemma 8.3 would then become trivial. We have not made use of that option, to avoid giving the false impression that a specific choice of H is an important part of the construction.

Multilinear operations
Up to this point, our Floer-theoretic discussion has involved only two Riemann surfaces: the cylinder and the thimble. One can embed this into a more general TQFT structure, of which we will now recall a version (related expositions are [36,40,32]). Our main goal is to lies to the right of this circle use that structure to give a slightly different interpretation of the connection introduced in Section 9.
(10a) Riemann surfaces. We will use only genus zero surfaces, which are decorated in such a way that they produce operations with one output (in principle, one could work in greater generality; but this level is sufficient for our purpose, and allows some expository simplifications).
Setup 10.1. LetS be a compact Riemann surface isomorphic to the sphere, with marked points ζ − and ζ + i , i ∈ I, for some finite set I. Let S be the complement of the marked points. A set of tubular ends for S consists of proper holomorphic embeddings with disjoint images, for some s − , s + i ∈ R. Each embedding (10.1) should map to a punctured disc inS surrounding the corresponding marked point. By a worldsheet, we mean such a surface S together with a choice of tubular ends, as well as a one-form β satisfying and such that, for some α + i , α − ∈ R \ Z, In terms ofS, the tubular ends equip it with distinguished framings (half-lines in the tangent spaces at the marked points). These are defined as The tangent direction is the "homotopically essential" part of choosing a tubular end (the remaining choices belong to a weakly contractible space).
Setup 10.2. Given a worldsheet S, we consider pairs (K S , J S ) of the following kind. K S is a one-form on S taking values in C ∞ (F, R), with the property that for any tangent vector ξ, K S (ξ) satisfies (7.10) for the angle α = β(ξ). J S is a family of almost complex structures on F parametrized by S, all belonging to the class (7.7). Moreover, these should be compatible with the choice of tubular ends; this means that there are families (H − , J − ) and (H + i , J + i ) parametrized by S 1 , such that (exponentially fast in any C r topology) To K S , one can associate a one-form with values in (Hamiltonian) vector fields on F , which we denote by Y S . The associated inhomogeneous Cauchy-Riemann equation is In the second line, we consider Du − Y S z,u(z) : T S z → T F u(z) , and then take its complex antilinear part with respect to J S z,u(z) . The limits x + i and x − are one-periodic orbits of the Hamiltonians appearing in (10.5). The equation (10.5) generalizes the equations (7.15), (7.28) on the cylinder, as well as (7.41) on the thimble. The appropriate generalization of (7.17) is where deg(u) is as in (7.16). Similarly, instead of (7.29)-(7.31) one has Here,ũ : S → S × F is the graph of u. We can consider K S as a one-form S × F which vanishes along the tangent directions to F , and that explains the expressionũ * K S in (10.9). Similarly, the expression R S in (10.10) is a two-form on S ×F which vanishes if one contracts it with a vector tangent to F . We will not write it down explicitly; the only thing that's relevant at the moment is that R S is compactly supported, which means that ũ * R S can be bounded by a constant which is independent of u.
The appropriate generalization of Lemma 7.10 is: Consider the left hand side of (10.11) as linear operator between suitable Sobolev completions, let's say W 1,2 (S, C) → L 2 (S, Ω 0,1 S ). Then, that operator is surjective if and only if d ≥ −1, and injective if and only if d < 0.
Proof. As mentioned before, one can write β 0,1 =∂f , simply because S is an open Riemann surface. Since S is of genus zero, one can find such an f which satisfies (10.15) f ( − (s, t)) = iα − s + constant, In terms of (10.12), if v is a solution of class W 1,2 , thenṽ must extend to a holomorphic function lying in OS(Z), where This divisor has degree exactly (10.14). On the other hand, a standard index formula shows that the (complex) Fredholm index of D u is d + 1. Both statements follow from those two facts.
The counterpart of Lemma 7.4 holds only if (10.14) is negative. From (10.13) one gets generalizations of Lemmas 7.5 and 7.6. The generalization of Lemma 7.7 says that if u is not contained in F 0 , Here, m − (u) and m + i (u) are defined as in (7.21) and (7.22), respectively. The analogue of Lemma 7.9 goes as follows: Then, for a generic choice of (K S , J S ), the following holds. (i) All solutions of (10.6) are regular, with the possible exception (which applies only if (10.14) is < −1) of those lying in p −1 (B). (ii) For any z ∈ S, all J S zholomorphic spheres with zero first Chern number avoid u(z) for any solution u of (10.6) with index(D u ) ≤ 1.
The counterpart of Lemma 7.5 implies that all such solutions remain in a compact subset of E. By counting them, one obtains the chain level pair-of-pants product (10.20) • : which is independent of all choices up to chain homotopy. We will denote the surface used to define this by S pants , and the auxiliary data that enter into this construction by (K pants , J pants ). Now suppose that α + 1 = α + 2 = α > 1, so that the Borman-Sheridan cocycle is defined. We take the thimble T with the datum (H thimble , J thimble ) which underlies the definition of that cocycle, and glue one copy of it to each input end of our pair-of-pants (using the same gluing length ρ 0 both times). The outcome is a surface isomorphic to the thimble, schematically denoted by (10.21) S pants # ρ 2T.
This comes equipped with two marked points ζ 1 , ζ 2 . It also comes with a ρ-dependent auxiliary datum which, in notation analogous to (7.52), one can write as Correspondingly, one considers a parametrized moduli space of (ρ, u), where the map u is defined on (10.21) and subject to the topological conditions For a fixed value of ρ, this is the same space previously used to define (8.28). In the limit ρ → ∞, the most relevant limiting configuration consists of a triple (u,ũ + 1 ,ũ + 2 ), where u is a map on the pair-of-pants, whereas theũ + k are maps on the thimble. They obviously satisfy x + 1 = lim s→+∞ u( + 1 (s, t)) = lim s→−∞ũ + 1 (s, t), x + 2 = lim s→+∞ u( + 2 (s, t)) = lim s→−∞ũ + 2 (s, t). where ζ = +∞ is the usual marked point on the thimble.
Case 3: Neither of theũ + k is contained in F 0 . One then has (10.27) Clearly, the only possibility is as follows: neitherx + k lies in F 0 ; u·F 0 = 0 (hence u is contained in E); and deg(ũ + k ) = 1. Counting such configurations yields the pair-of-pants square s • s of the Borman-Sheridan cocycle.
Among the other limiting configurations that could occur for ρ → ∞, one deserves particular mention. Namely, suppose that we have four components (ũ − , u,ũ + 1 ,ũ + 2 ) (whose degrees of course sum up to 2). Here,ũ − is a Floer trajectory, u is a map on the pair-of-pants, andũ + k are thimbles. The interesting case occurs when u is contained in p −1 (B). Transversality can fail for such u, see Lemma 10.4, but on the other hand, u must in fact be contained in F 0 , because of Lemma 10.3 (the quantity (10.14) is 2α − 2 α − 2 < 0). Hence deg(u) = 0, and one again gets a contradiction for topological reasons: As usual, there are similar arguments for more complicated degenerations, modelled on the same principles (compare Remark 8.2). As a consequence, counting points in our parametrized moduli space, where ρ ranges over some half-infinite interval, shows the following result: Lemma 10.5. The cocycle s (2) from (8.28) is cohomologus to s • s.
(10c) A bracket. We will be concerned with a specific family of worldsheets, parametrized by τ ∈ S 1 , and where S 1 is given the opposite of the standard orientation. The Riemann surfaces and tubular ends are If one thinks in terms of the framings (10.4), we have two marked points onS = R × S 1 ∪ {±∞}, namely ζ − = −∞ and ζ + 1 = ∞, which are independent of τ , and so are their framings. The third point ζ + 2 = (0, τ ) moves around a circle, but its framing always points in −∂ s -direction, which means towards ζ − .
Fix constants (10.18), and equip S τ with a corresponding closed one-form. Assume additionally that we have fixed auxiliary data which defines the Floer cochain complexes associated to the three ends, denoted by (H − , J − ) and (H + k , J + k ). We then choose, on each S τ and smoothly depending on τ , auxiliary data (K Sτ , J Sτ ). Let's denote the entire family of such data by (K Lie , J Lie ). We consider a parametrized moduli space of maps on S τ which satisfy (10.23), and use that to define a chain map (10.30) [·, ·] −1 : CF * (E, H + 2 ) ⊗ CF * (E, H + 1 ) −→ CF * −1 (E, H − ). As usual, we use descriptive notation for the Riemann surfaces involved in this construction, writing S τ = S Lie(−1) τ . Remark 10.6. One can get an alternative picture of the same construction by rotating S τ in S 1 -direction, so as to remove its dependence on τ . After this change of coordinates, the situation looks as follows: The τ -dependence now comes from rotating the ends at ζ − and ζ + 1 . A gluing argument based on this observation provides a chain homotopy We omit the details.
Our next task is to explain what happens when one applies the bracket to a class in the image of the PSS map (7.47). Assume from that α + 2 > 0. Let K be a proper pseudo-cycle with K-coefficients in E. Recall the previously defined structures from (7.45) and (9. Proof. As usual, we will prove this by direct construction of a map (10.35) . (To simplify the notation, we have used d for the Floer differentials in both complexes involved.) This is given by a moduli space with parameters (λ, τ ) ∈ R + × S 1 . The associated Riemann surfaces are constructed as follows: for λ 0, More precisely, we start with the Riemann surface underlying the bracket (and its auxiliary data), and glue in the thimble (equipped with the auxiliary data used to define b K ) into the ζ + 2 end. The outcome (10.36) is a cylinder with one marked point (which depends on τ ), and this (with its auxiliary data) can easily be deformed to the corresponding Riemann surface in the family underlying r K , which is what we define S λ,τ to be for λ = 0. (It may be more intuitive to think of starting with the given surfaces defining r K , and where we "pull out" a neighbourhood of the marked point, thereby creating a neck that stretches as λ → ∞.) From this point onwards, the argument follows the standard pattern.
Let's specialize to K = q −1 Ω|E, and assume that α + 2 > 1. Let's introduce the notation We also re-introduce Assumption 2.7, and choose a proper pseudo-chain A as in (9.6). By the same construction as in the cocycle case (7.45), this gives rise to a Floer cochain One therefore has chain homotopies (10.40) r Proof. Explicitly, putting in all the necessary notation, the claim is that the map (10.42) is nullhomotopic. The nullhomotopy will be constructed in two parts, of which the first one is not surprising. Namely, the argument from Lemma 10.7 can be generalized to proper pseudo-chains, in which case the expression in (10.35) acquires an additional term involving the boundary of the pseudo-chain. Specializing to the pseudo-chain A|E, one gets The second part of demonstrating our claim is therefore to provide a homotopy To construct that, we take the idea of pulling out a marked point (from Lemma 10.7) and apply it to the moduli space underlying χ (from Section 9b); more precisely, the point we pull out is ζ 1 from (9.19). The outcome is a three-dimensional parameter space with one boundary component. Figure 24 shows a schematic picture of that space together with the relevant compactification (where degenerations arise; of course, this is a dimensionally reduced picture, and the actual space is an S 1 -bundle over it). Some degenerations contribute zero, for topological reasons (incidence conditions would force each of the two maps to have positive degree, in the sense of (7.16), which is impossible). The remaining ones, together with the boundary, give rise to the three terms in (10.44).
Rather than discussing the technical aspects of the construction, which offer nothing substantially new at this point, we want to focus on one fundamental point. Gluing together the surfaces involved in the expression [ν, ·] −1 yields a family of surfaces which are all cylinders with two marked points, which we denote by ζ 1 , ζ 2 (this is the same numbering as in Figure  24). Only our specific choice of framing ensures that ζ 1 always lies on the straight horizontal half-line connecting ζ + 2 to the negative end. This explains why, (10.44), it is necessarily this particular bracket [·, ·] −1 which arises.
Let's introduce the following notation for elements of CF * (E, H + 2 ): Proof. By Proposition 8.6, We have to show that the term in brackets in (10.53) is nullhomologous. That is actually an instance of a general fact, valid for any proper pseudo-chain: in the same way as in Lemma 7.13, one can find a Floer cochain β A|E satisfying Finally, we can use Lemma 10.8 to rewrite the connection (9.15) as (10.55) Remark 10.10. At this point, the similarity with the axiomatic constructions from Section 6 is already evident. The family (10.29) defining [·, ·] −1 matches that from Figure 7 (including the orientation convention for the parameter spaces). The formula (10.32) agrees exactly with that in (6.19). There is also a close similarity between (6.14) and (10.55) (the two differences being the appearance of continuation maps, which is itself a reflection of the fact that geometrically, we cannot get connections on Floer cohomology groups for fixed angle α).

Symplectic cohomology
This section (finally) formally introduces symplectic cohomology, as well as its standard formalism of operations, into our setup. Combining the abstract ideas from Sections 5c and 6c with some Floer theory from Section 10c, we carry out the construction of connections on symplectic cohomology. Finally, we need to discuss the compatibility between this and the previous material.
(11a) The definition. We consider a more general geometric situation than in Section 7a; it includes Lefschetz fibrations over C with nontrivial monodromy at infinity. This is essentially for didactic purposes: we want to show that, for the abstract theory, the precise structure of our manifolds is less relevant. We retain the Calabi-Yau assumption, but only for its technical role in pseudo-holomorphic curve theory: in principle, it could be modified or dropped, with the usual implications (loss of the Z-grading, and technical complications). Similarly, the theory could be set up in other situations, such as symplectic manifolds with contact type boundary (again, with modifications on the technical level).
Setup 11.1. Let E be an open symplectic manifold. We require that this should come with a proper map p : E → C, which is a locally trivial symplectic fibration over A = {|b| ≥ r 0 } ⊂ C, for some R. This means that every point x ∈ p −1 (A) is a regular point, with T E v x = ker (Dπ) ⊂ T E x a symplectic subspace, and that its symplectic orthogonal complement T E h x satisfies (11.1) ω E − p * ω C |T E h x = 0 for some choice of rotationally invariant symplectic form ω C on the base. In addition we will fix a sequence of annuli, called barriers, We also require that c 1 (E) should be zero, and choose a complex volume form η. Finally, we fix a (proper) cycle representative Ω for [ω E ], in the same sense as in (7.4).
Setup 11.2. We will use compatible almost complex structures J on E such that p is Jholomorphic on p −1 (W ). Similarly, we use Hamiltonian functions H ∈ C ∞ (E, R) whose associated vector field X satisfies The simplest way to achieve that is to take X|p −1 (W j ) to be the unique horizontal (meaning, lying in T E h ) lift of a j (2πib∂ b ). Indeed, for most of our discussion, it will be entirely sufficient to restrict to such choices.
As usual, much of the behaviour of the solutions of the associated Cauchy-Riemann equations comes from elementary arguments involving their projection to C. The following three Lemmas summarize those arguments (compare [42,Section 4.4], which applies the same idea in a slightly different geometric context).
Lemma 11.3. Let S be a compact connected Riemann surface, with nonempty boundary, carrying a one-form γ with dγ ≤ 0. Fix some j, and an r ∈ (r − j , r + j ). Suppose that v : S → C satisfies Suppose moreover that |v(z)| = r for z ∈ ∂S, and that v intersects the circle of radius r transversally at each boundary point. Then there is some point z ∈ S where |v(z)| < r.
Proof. One can find an r ∈ (r − j , r + j ) such that |v(z)| < r for all z ∈ ∂S, and such that v intersects the circle of radius r transversally. If the intersection is empty, we have |v(z)| < r < r + j everywhere. Otherwise, takeS to be one of the connected components of {z ∈ S : |v(z)| ≥ r}. By applying Lemma 11.3 toS, one obtains a contradiction.
Lemma 11.5. Let S be as in Setup 10.1, except that we denote its one-form by γ (instead of β), and where α − , α + i can be arbitrary real numbers. Consider a map v : S → C which satisfies (11.4) for some j, and such that v −1 (W j ) ⊂ S is compact. Define where w − = w − (v) ∈ Z is the winding number of the loop v( − (s, ·)) around the origin, for s 0. The same idea, applied to the other ends, yields similar numbers B(ζ + i ). Then, one necessarily has Moreover, equality in (11.10) can hold only if v −1 (W j \ ∂W j ) = ∅.
Proof. If |v(z)| ≤ r − j for all z ∈ S, the desired inequality (11.10) holds trivially, since both sides are zero. If |v(z)| ≥ r + j for all z ∈ S, the inequality is true because w − = i w + and, by (10.2), α − ≥ i α + i . We therefore focus on the remaining situation, where we expect a strict inequality in (11.10).
Choose some r ∈ (r − j , r + j ) such that v intersects the circle of radius r transversally, and so that the intersection is not empty. LetS = {z ∈ S : |v(z)| ≥ r}. Then, (11.8) yields (11.11) 0 To see the equality in (11.11), one cuts off parts of the cylindrical ends ofS, and then applies Stokes to the resulting compact surface. The winding number term comes from integrating −d arg(v) over the boundary circles associated to the ends, and the α ± similarly comes from integrating γ. Since 2πdγ ≤ 0 by (10.2), the desired inequality follows.
We will construct symplectic cohomology as the Floer cohomology of a single time-dependent Hamiltonian (for similar definitions, see [1,11]). More precisely: Setup 11.6. Choose numbers (11.12) 0 < a 1 ≤ a 2 ≤ · · · ∈ R \ Z, We use a time-dependent Hamiltonian H = (H t ) which, for each time t and any j, satisfies (11.3) with the given a j . As a consequence, any one-periodic orbit x of the associated Hamiltonian vector field is disjoint from p −1 (W ). We require three additional properties to hold for all x. They should be nondegenerate; disjoint from Ω; and finally, (11.13) if |p(x(t))| > r + j , the loop p(x) winds > a j times around the origin.
To show that these properties can be satisfied, let's first choose a function ψ(r), r ≥ 0, with (11.14) Take our standard rotational vector field 2πib∂ b , multiply it by ψ(|b|), and then lift that to a vector fieldX on E, as follows. On the preimage of the closed unit disc,X shold be zero, and elsewhere, it should be horizontal (a section of T E h ). These conditions determine the lift uniquely. Moreover, the resulting vector field is Hamiltonian, and we fix a Hamiltonian functionH which induces it. Because of the convexity of ψ, this certainly satisfies (11.13). Now take H to be a sufficiently small time-dependent perturbation ofH, which does not change the function on p −1 (W ). That perturbation will still satisfy (11.13), and for a generic choice, it will achieve nondegeneracy.
Lemma 11.7. Take H as in Setup 11.6 and J = (J t ) as in 11.2. Then, the following holds for all solutions u of the associated Floer equation (7.15): if |p(x + )| < r − j for some j, then |p(u)| < r + j everywhere.
Proof. By construction, the projection v = p(u) satisfies (11.4) for any j. More precisely, the Riemann surface involved is S = R × S 1 , with γ = a j dt determined by (11.12).
The previous argument, together with the fact that the one-periodic orbits are disjoint from p −1 (W j ), shows that |p(x − )| < r − j . Of course, the other limit already satisfies the same condition. We can therefore find a large compact subset S ⊂ R × S 1 , which is a Riemann surface with boundary, such that |v(z)| < r + j for all z ∈ (R × S 1 ) \ S. Applying Lemma 11.4 to the connected components of S yields the desired result.
Results like Lemma 11.7 are usually called C 0 -bounds in the literature on symplectic cohomology. Once one has obtained such a bound, one can define CF * (E, H) and its differential d as before (7.26). Symplectic cohomology is its cohomology, SH * (E) = HF * (E, H).
Remark 11.8. The Floer complex is defined as the direct sum of copies of K associated to one-periodic orbits. In particular, infinite sums of the form (11.15) x 0 q d0 + x 1 q d1 + · · · involving infinitely many distinct one-periodic orbits x i do not define Floer cochains, even if the d i go to infinity.
The well-definedness of symplectic cohomology is again established using continuation maps (7.32). Suppose that we have two collection of numbers (a ± j ) as in Setup 11.6, satisfying The actual H C will be a perturbation ofH C , but one which leaves it unchanged on p −1 (W ), and which still converges to H ± exponentially fast as s → ±∞. If u is a solution of the associated continuation map equation, then v = p(u) satisfies (11.4) for any j, with Lemma 11.7 and its proof therefore carry over without any changes, leading one to define a continuation map with the usual uniqueness and composition properties. Before continuing, we want to collect some more basic observations. Discussion 11.9. (i) For any j, denote by CF * (E, H) ≤j ⊂ CF * (E, H) the subspace generated by one-periodic orbits which satisfy |p(x)| < r − j+1 . Lemma 11.7 says that these subspaces are preserved by the differential, hence constitute an increasing filtration of the chain complex CF * (E, H). Denote their cohomology groups by HF * (E, H) ≤j . The continuation maps are compatible with this filtration (as are the homotopies that relate different choices of continuation maps).
(ii) When a − j = a + j , one can define continuation maps in either direction, which are homotopy inverses of each other. Moreover, all of this is compatible with the previously introduced filtrations. Hence, for fixed (a j ), both HF * (E, H) and HF * (E, H) ≤j are well-defined up to canonical isomorphisms, and so are the continuation maps which relate one choice (a − j ) to another one (a + j ). To signal this, we will temporarily change notation to HF * (E, a 1 , a 2 , . . . ) and HF * (E, a 1 , a 2 , . . . ) ≤j .
(iii) Suppose that there is some j * such that a + j = a − j for j ≤ j * . Then, the continuation map HF * (E, a + 1 , a + 2 , . . . ) ≤j * → HF * (E, a − 1 , a − 2 , . . . ) ≤j * is an isomorphism. In fact, by suitably correlating the choice of Hamiltonians, one can achieve that the underlying chain map is already an isomorphism.
(iv) In all the arguments above, one is free to ignore finitely many j. For instance, the sequence (a j ) only has to be eventually nondecreasing. Similarly, one can define continuation maps as long as (11.16) holds for all but finitely many j. As a particularly simple application, one can arbitrarily change finitely many a j , and HF * (E, a 1 , a 2 , . . . ) remains the same up to canonical isomorphism.
Lemma 11.10. The continuation map is always an isomorphism.
Proof. Given (a ± j ) and some j * , one can find another choice (a 0 j ) such that (11.19) pseudo-cycle K with K-coefficents, one can define cocycles (7.45), secondary cochains (7.49), and a chain map (9.4). In particular, Discussion 7.14 carries over immediately to our context, leading to reduced symplectic cohomology (2.3) and its operator ∆ red from (2.4).
We now turn to operations associated to other surfaces (in parallel with Section 10). In the case of symplectic cohomology, the ends of our surface S should come with choices of sequences (a − j ) and (a + i,j ), respectively, such that (11.21) a − j ≥ i∈I a + i,j for all j.
One also supposes that S comes with one-forms γ j , each of which satisfies dγ j ≤ 0 as well as an analogue of (10.3): Given functions H − and H + i associated to the ends, one chooses K S (in parallel with Setup 10.2) such that for each tangent vector ξ on S, K S (ξ) belongs to the class of functions (11.3) with a j = γ j (ξ). One then considers solutions of the same equation (10.6) as before (taking values in E), and gets a generalization of Lemma 11.7: Lemma 11.11. If, for some j, all x + i satisfy |p(x + i )| < r − j , then |p(u)| < r + j everywhere.
Given that, is is straightforward to equip symplectic cohomology with versions of the pair-ofpants product (10.20) and bracket (10.30). Slightly more generally, suppose that we replace the last part of (10.29) by (11.23) + τ,2 (s, t) = (0, τ ) − exp(−2π(s + it) + 2πi(c + 1)τ ), for some integer c. This gives rise to a family of bracket operations (11.24) [·, ·] c : CF * (E, H + 2 ) ⊗ CF * (E, H + 1 ) −→ CF * −1 (E, H − ). The special case [·, ·] = [·, ·] 0 yields the bracket which, with the pair-of-pants product, forms the Gerstenhaber algebra structure on symplectic cohomology. We will use the same notational convention (of omitting the superscript for c = 0) several times from now on, without specifically pointing that out. Since the only difference is the choice of framing at one end, it is not hard to produce chain homotopies which relate the brackets (11.24) for different choices of c: For the proof of Lemma 10.7, the specific choice of (11.23) is actually irrelevant. This means that the same argument produces maps (11.26)  Note that the following is a chain map (of degree codim(K) − 2): (11.27) x −→ η c (b K , x) + (−1) codim(K) (ρ c−1 K (x) − ρ c K (x)) − β K • x. Copying the argument from (6.20) in our framework (which means that, instead of the operation of inserting point configurations into each other, one has to use gluing in a Floertheoretic sense), one sees that (11.27) is nullhomotopic for c = 0; and it is straightforward to extend that to all c ∈ Z. We omit the details.
(11c) Connections on symplectic cohomology. Let's specialize to K = q −1 Ω, and use the notation (10.45), (10.46). At this point, we impose Assumption 2.1 and choose a bounding cochain, which means some θ ∈ CF 1 (E, H + 2 ) such that dθ = k. As in (10.51) and (10.52), this gives rise to classes t = [−q −1 Ω, θ] ∈ SH 1 (E) red , (11.28) a = [∆θ − κ] = ∆ red t ∈ SH 0 (E). (11.29) One has a chain homotopy as in (9.10), defined in exactly the same way. As a consequence, one can introduce connections (11.30) x] c ]. One can show (but we omit the argument, which is fairly tedious) that these connections commute with continuation maps, hence are independent of the auxiliary data used to define symplectic cohomology. Moreover, using the fact that (11.27) is nullhomotopic, it is straightforward to show the following result, which is modelled on our previous Lemma 6.4: Lemma 11.12. ∇ c x = ∇x + ca • x for all cinZ.
Our next task is to show that ∇ is compatible with the Gerstenhaber algebra structure, meaning that it satisfies (2.10) and (2.11). The proof is a translation (a Floerization) of that of Proposition 5.10. The main difficulty in carrying out such a translation is the appearance of continuation map terms in (11.30). To explain how one addresses that, we will consider just the first property (2.10), for which the technical form of the statement is as follows: Proposition 11.13. Take Floer complexes CF * (E, H + i ) (i = 1, 2, 3) underlying symplectic cohomology, with associated sequences (a + i,j ). Suppose that the bounding cochain θ lies in CF 1 (E, H + 3 ). Define CF * (E, H − ) with underlying sequence (a − j = a + 1,j + a + 2,j + 3a + 3,j ). Then, the chain map  In the expressions above, the same notation is used for related but different maps, which calls for some explanation. In (11.31), we consider Cx i ∈ CF * (E, H > i ), which are Floer cochain groups associated to sequences (a > i,j = a + i,j + a + 3,j ). Their pair-of-pants product is Cx 2 • Cx 1 ∈ CF * (E, H < ), where the Floer cochain group is associated to the sequence (a < j = a + 1,j + a + 2,j + 2a + 3,j ). One then applies a form of the connection (11.30) to get from that group to CF * (E, H − ). In (11.32), the connection which appears goes from CF * (E, H + 2 ) to CF * (E, H > 2 ), and its formula has been tweaked as in Discussion 9.4. The pair-of-pants product is the same as in (11.31), and the continuation map which is applied last goes from CF * (E, H < ) to CF * (E, H − ). The situation for (11.33) is analogous. The reason for "padding" (11.31)-(11.33) with extra copies of continuation maps is to ensure that only one kind of pair-of-pants product appears in the terms involving differentiation.
Proof of Proposition 11.13. Our claim amounts to the following nullhomotopy: In the same spirit as (9.1), the first two lines of (11.34) can be given the following geometric interpretation. Think of the disconnected Riemann surface S = S − ∪ S 0 ∪ S + 1 ∪ S + 2 , where S 0 is an open pair-of-pants, and the other components are copies of R × S 1 (see Figure 25). Let's suppose that these components come equipped with the auxiliary data that define the pair-of-pants product and continuation maps s→∞ u − (s, ·) = lim s→−∞ u 0 ( − (s, ·)), lim s→∞ u 0 ( + i (s, ·)) = lim s→−∞ u + i (s, ·). Counting such solutions defines the operation C(Cx 2 • Cx 1 ). Now suppose that our surface comes with an additional marked point ζ, which can be (11.37)        an arbitrary ζ ∈ S, or ζ = (s, t) ∈ S − with s ≥ 0, or ζ = (s, t) ∈ S + i with s ≤ 0. We require that the image of that point should go through q −1 Ω, and count solutions accordingly. That (depending on which component of S the point lies in) gives rise to four different expressions, which are exactly the four first terms of (11.34) (see again Figure 25).
continuation map continuation map continuation map The additional puncture (tubular end) could be anywwhere here. . .
. . . or here, but to the right of (or on) the dashed circle. . .
. . . or here, but to the left of (or on) the dashed circle. Figure 26. The construction of (11.38).
Without changing the statement, one can replace the last line of (11.34) with −γ(k, x 2 , x 1 ), following what we've done in (5.27). At this point, the required construction is clear: one starts from Figure 25 and "pulls out" the additional marked point, a process which in the limit yields Figure 26 with a copy of k inserted at the extra end. The associated threeparameter moduli space has other relevant boundary components, namely those where the "pulling out" process it being carried out at a point which lies on the boundary of its allowed domain (in Figures 25-26, the dashed circles). But the contributions of those precisely yield the third line of (11.34).
We omit the proof of the appropriate version of (2.11), which is parallel, and this concludes our discussion of Proposition 2.2. The Floerization of Proposition 6.3 is this: Proposition 11.14. Take Floer complexes CF * (E, H + i ) (i = 1, 2) underlying symplectic cohomology, with associated sequences (a + i,j ). Suppose that the bounding cochain θ lies in CF 1 (E, H + 2 ). Define CF * (E, H − ) with underlying sequence (a − j = a + 1,j + 2a + 2,j ). Then, the continuation map continuation map BV operator . . . or here, but to the right of (or on) the dashed circle. . .
. . . or here, but to the left of (or on) the dashed circle.
The marked point could be anywhere here. . . Figure 27. The geometric meaning of the first two lines of (11.40).
Let's use our bounding cochains to define connections (11.30) on symplectic cohomology. Then, these fit into a commutative diagram with (10.55), (11.51) HF * (E, α + 1 ) Of course, the formulae for the two connections look identical, but some explanation is still needed. As in Discussion 11.16(ii), one can arrange that the two continuation maps in (11.51) are realized by inclusions of subcomplexes. All the chain level operations that define the connection on symplectic cohomology can be set up to preserve those subcomplexes, and then (11.51) commutes trivially.
In view of this compatibility statement, Proposition 9.8 and Lemma 10.5 imply that the connection ∇ −1 on symplectic cohomology satisfies (2.34). Since we know how the connections ∇ c for different c are related (Lemma 11.12), it follows that (2.34) holds for all c, and in particular c = 0, which proves Theorem 2.11. Remark 11.17. One of the computations from Section 9, namely Proposition 9.5, has not been used here. In fact, its main role was to serve as a model for the similar, but more complicated, Proposition 9.8. Still, it is interesting to have a direct proof of that formula, since the alternative derivation (given in Section 2: one starts with (2.12), which implies (2.16), and then inserts Lemma 2.10 into that formula, leading to (2.31) for all ∇ c ) is not particularly geometric.