fukaya a -structures associated to lefschetz brations....

Fukaya A∞-structures associated to

Lefschetz fibrations. II

Paul Seidel

Abstract. We consider the Fukaya category associated to a basis of vanishing cycles

in a Lefschetz fibration. We show that each element of the Floer cohomology of the

monodromy around ∞ gives rise to a natural transformation from the Serre functor to

the identity functor. This complements previously known constructions, in a way which

fits in well with homological mirror symmetry.

1. Introduction

This paper is part of an investigation of Floer-theoretic structures arising from Lefschetz

fibrations. Its purpose is to introduce a new piece of that puzzle; and also, to set up a wider

algebraic framework into which that piece might (conjecturally) fit, centered on a notion of

noncommutative pencil.

(1a) Symplectic geometry. It is well-known that there is a version of the Fukaya category

tailored to a Lefschetz fibration (the idea is originally due to Kontsevich, and a first rigorous

definition appears in [36]). In [45], it was pointed out that these categories always come with

an additional datum: a distinguished natural transformation from the Serre functor to the

identity functor (see [41, 8, 46, 3] for related theoretical developments, and [27, 28, 4] for

applications). Concretely, let’s fix a symplectic Lefschetz fibration

(1.1) π : E2n −→ C

with fibre M2n−2. For technical simplicity, we will impose an exactness condition on the

symplectic form. Choose a basis of vanishing paths, and let A be the associated (directed)

Fukaya A∞-algebra. A can be defined over an arbitrary coefficient field K, with the case

char(K) = 2 being technically slightly easier. In general, A is Z/2-graded, and this lifts to

a Z-grading if c1(E) = 0. Natural transformations of degree n from the Serre functor to

the identity (in the category of perfect A∞-modules over A) can be described as maps of

A∞-bimodules

(1.2) A∨[−n] −→ A,

where A∨ is the dual of the diagonal bimodule, to which we have applied an upwards shift

by n, written as [−n]. To be more precise, A∞-bimodules form a dg category, denoted here

by [A,A]. By a bimodule map (1.2), we mean an element of

(1.3) H0(hom [A,A](A∨[−n],A)) ∼= Hn(hom [A,A](A

∨,A)).1

2 PAUL SEIDEL

In these terms, the story so far can be summarized as:

Lemma 1.1. A always comes with a distinguished map (1.2).

According to [46], there are two ways to describe this map: one is in terms of the geom-

etry and Floer cohomology of Lefschetz thimbles in E; the other uses the (faithful, but

not full) functor from A to the standard Fukaya category F(M) of the fibre, following the

original proposal of [45] (to be precise, [46] only shows that the two descriptions agree up

to an automorphism of A∨). There are simple examples of Lefschetz fibrations which have

quasi-isomorphic A∞-algebras A, but are distinguished by the maps from Lemma 1.1 (this

observation is due to Maydanskiy).

The specific contribution of this paper is to introduce another geometric source of maps

of the form (1.2). The formulation involves fixed point Floer cohomology for symplectic

automorphisms φ of M (which are exact and equal the identity near the boundary). Defining

this Floer cohomology requires an auxiliary choice of perturbation, by a multiple of the Reeb

vector field near the boundary. We write the outcome as HF ∗(φ, ε), where ε denotes the

amount of perturbation (for small |ε|, this appears in [37, 29]; for the general case, see [51]

or the exposition in the body of the paper).

Theorem 1.2. Let µ : M →M be the monodromy of the Lefschetz fibration around a large

circle. There is a map

(1.4) HF ∗+2(µ, ε) −→ H∗(hom [A,A](A∨[−n],A)).

This is the outcome of a Floer cohomology computation in E modelled on [29], together with

an application of a suitable open-closed string map, similar to those in [2].

An important special case is that of a Lefschetz fibration coming from an anticanonical

Lefschetz pencil (a pencil of hypersurfaces representing the first Chern class) on a closed

symplectic manifold E. One obtains E by removing a suitable neighbourhood of one fibre of

the pencil from E, hence c1(E) = 0. One can arrange that the Reeb flow on ∂M is periodic

(say with period 1), and the monodromy µ is the right-handed boundary twist associated

to that periodic flow. We can undo the effect of that twist by a suitable choice of ε, which

results in a particularly simple formula:

(1.5) HF ∗+2(µ, ε) ∼= H∗(M ;K) for 1 < ε < 2.

In particular, the element 1 ∈ H0(M ;K) gives rise to a distinguished Floer cohomology class

of degree 2, hence by Theorem 1.2 to a distinguished map (1.2), which in general is different

from that in Lemma 1.1. We summarize the outcome:

Corollary 1.3. If (1.1) comes from an anticanonical Lefschetz pencil, there is a natural

pair of maps (1.2) (obtained through Lemma 1.1 and Theorem 1.2, respectively).

Remark 1.4. More generally, consider a Lefschetz pencil of hypersurfaces such that (1−m)

times the class of the hypersurface is c1(E), for some integer m. The associated Lefschetz

fibration still satisfies c1(E) = 0, but the appropriate generalization of (1.5) is

(1.6) HF ∗+2(µ, ε) ∼= H∗+2m(M ;K).

LEFSCHETZ FIBRATIONS 3

Hence, one ends up with one bimodule map (1.2) of degree 0 (from Lemma 1.1) and another

of degree 2m (from Theorem 1.2).

Remark 1.5. More speculatively, one could consider pencils which satisfy the Lefschetz

(nondegeneracy) condition away from one of the fibres. Removing a neighbourhood of that

fibre still results in a Lefschetz fibration, but which has more complicated geometry at infinity.

While (1.5) no longer applies, it seems likely that one can still produce a distinguished element

of HF ∗(µ, ε) in this context, but where in general (unless the singularities of the fibre we have

removed are isolated) ε needs to be large. We will not pursue this idea further here, but it

is relevant to many examples from mirror symmetry, which have a highly singular “large

complex structure limit” as a fibre.

(1b) Noncommutative geometry. The starting point for our algebraic framework is the

notion of noncommutative divisor. While the terminology is new, the concept already ap-

peared in [41], and a version was considered in [46, 23] (the last two references include an

additional cyclic symmetry condition, which corresponds more specifically to a noncommuta-

tive anticanonical divisor; we have chosen not to impose that condition here, for simplicity).

Let A be an A∞-algebra. A noncommutative divisor on A consists of an A∞-bimodule P

which is invertible (with respect to tensor product of bimodules, up to quasi-isomorphism),

together with an A∞-algebra structure on B = A ⊕ P[1], which extends the given one on

A as well as the A-bimodule structure of P. In particular, if one then considers B as an

A-bimodule, it fits into a short exact sequence

(1.7) 0→ A −→ B −→ P[1]→ 0.

As boundary homomorphism of that sequence, one gets a bimodule map

(1.8) P −→ A,

which we refer to as the first order part of the noncommutative divisor. We extend the

algebro-geometric terminology to cover two related structures:

(1.9)

(i) The A∞-algebra B is thought of as describing the divisor by itself (inde-

pendently of its relationship with the ambient space A).

(ii) There is a localization process, in which one makes (1.8) invertible by

passing to a quotient A∞-algebra. We consider this to be the abstract

analogue of taking the complement of the divisor (see [41, Section 1] for

an explanation), and write the outcome accordingly as A \B.

Generalizing the previous concept, we will introduce noncommutative pencils. The precise

definition will be given later, but note that the first order part of such a pencil is a pair

of maps (1.8). A noncommutative pencil gives rise to a family of noncommutative divisors,

parametrized by the projective line P1 = K ∪ ∞ (K is our cofficient field). In particular,

one has the fibre B∞ associated to the point z = ∞ ∈ P1, and also the complement of the

4 PAUL SEIDEL

fibre, A \B∞, both as in (1.9). Other associated structures are new to this situation:

(1.10)

(iii) One can restrict to a formal disc in P1 centered at ∞, which gives a

one-parameter formal deformation of the A∞-structure on B∞.

(iv) The A∞-algebra A \ B∞ admits a (curved) formal deformation whose

deformation parameter has degree 2, and which we would like to think of

as the noncommutative analogue of a map to the affine line. The generic

fibre of the deformation is called the associated noncommutative Landau-

Ginzburg model (borrowing language from [30]). In fact, there is a family

of such models parametrized by K, obtained by a reparametrization of

the projective line (preserving ∞).

Of course, from a purely algebraic viewpoint the point ∞ ∈ P1 is not distinguished in any

way, but in our intended applications to symplectic geometry it does play a special role,

which explains why we have singled it out in the exposition above. We will end the abstract

discussion here, but note that there are further interesting directions, notably ones involving

Hochschild and cyclic homology (such as, studying the Gauß-Manin connection for the family

of fibres Bz of the pencil).

(1c) Speculations. In the situation of Lemma 1.1, one can realize the map (1.2) as the first

order part of a noncommutative divisor, by using the embedding A → F(M), as in [45, 41].

Then, the algebraic constructions introduced above have the following meaning:

(1.11)

(i) The divisor by itself corresponds to a full subcategory of F(M) (this

holds tautologically in the approach we have taken; however, there should

be another construction of the noncommutative divisor, in terms of the

geometry of Lefschetz thimbles, and from that viewpoint the relation

with F(M) would require a strengthening of the main result of [46]).

(ii) Passing to the complement of the divisor yields a full subcategory of the

wrapped Fukaya category of the total space, W(E). This is the main

result of [3].

Let’s advance this line of thought to the point where it can include Theorem 1.2. In spite

of its tentative nature, this direction seems worth while exploring, because of the potential

implications for symplectic geometry and mirror symmetry.

Conjecture 1.6. Suppose that we have an exact symplectic Lefschetz fibration coming from

an anticanonical Lefschetz pencil. Then, the A∞-algebra A carries a canonical structure of

a noncommutative pencil, whose underlying A∞-bimodule is A∨[−n], and whose first order

data are those described in Corollary 1.3.


The appeal of the noncommutative pencil structure in this context is that it can serve as an

intermediate object between various other Fukaya A∞-structures. Consider:

(1.12)

(iii) M admits a natural closure M , which is such that M \M is a neighbour-

hood of the base locus of the pencil. Associated to that is the relative

Fukaya category, which is a formal deformation of F(M) (inverting the

deformation parameter then yields a full subcategory of the Fukaya cat-

egory F(M), see [38, 33, 49]). One conjectures that, after a change in

the deformation parameter, the relative Fukaya category corresponds to

the restriction of the noncommutative pencil to a formal disc around ∞(this implies that, after reparametrization, the relative Fukaya category

should show a rational dependence on the deformation parameter).

(iv) Let E be the manifold on which the original Lefschetz pencil was defined.

By assumption, this is monotone, hence has a Fukaya category F(E)

defined over K (and whose grading is 2-periodic). By formal analogy,

these categories should be related to the Landau-Ginzburg categories

from (1.10)(iv). More precisely, there is a full subcategory of F(E) whose

objects are exact Lagrangian submanifolds in E (this appears for instance

in [7]), and one can conjecture that this should be isomorphic to a full

subcategory of the Landau-Ginzburg category. The same extends to

bulk-deformed Fukaya category F(E, z), defined for any z ∈ K, which

should correspond to the other Landau-Ginzburg models in the family

mentioned above.

Remark 1.7. We have limited ourselves to anticanonical Lefschetz pencils (the setup of

Corollary 1.3) for the sake of concreteness, but other cases are also of interest. For instance,

along the same lines as in Conjecture 1.6, the situation from Remark 1.4 should lead to a

Z/2m-graded noncommutative pencil. The residual information given by the Z-grading of A

means that this pencil is homogeneous with respect to a circle action on P1 of weight 2m.

In particular, if m 6= 0 and K is algebraically closed, the fibres over points of the open orbit

K× ⊂ P1 are mutually isomorphic.

(1d) Homological Mirror Symmetry. The mirror (B-model) counterpart of Conjecture

1.6 is far more straightforward, and amounts to embedding ordinary algebraic geometry into

its noncommutative cousin. Suppose that we have a smooth algebraic variety X, a line

bundle L on it, and a section s ∈ Γ(X,L−1) of its dual. Let X be an A∞-algebra underlying

the bounded derived category DbCoh(X). Then, L gives rise to an invertible bimodule L,

and s to a map σ : L → X. In fact, there is a noncommutative divisor of which this is the

first order part. This fits in with mirror symmetry in the formulation of [7], where the crucial

ingredient is a choice of anticanonical divisor (see [41, Section 6], where the connection is

made explicit in a simple example).

Suppose that for X and L as before, we have a two-dimensional subspace of Γ(X,L−1). This

gives rise to the structure of a noncommutative pencil on X. Homological Mirror Symmetry

6 PAUL SEIDEL

can now be thought of as operating on the level of pencils, schematically like this:

(1.13)

Symplectic

geometry

(A-model)

Symplectic

(anticanonical)

pencil

−→

Homological algebra

(Noncommutative geometry)

Noncommutative pencil

(for the dual diagonal bimodule)

←−

Algebraic

geometry

(B-model)

Algebraic

(anticanonical)

pencil

As a consequence, one gets a unified approach to the associated equivalences of categories

(1.9)–(1.12). Via point (iv) this picture includes counts of holomorphic discs, following the

model of [7].

Acknowledgments. Conversations with Anatoly Preygel were useful at an early stage of this

project. The discussion in Remark 2.11 includes an issue suggested by Ludmil Katzarkov.

Partial support was provided by NSF grant DMS-1005288, and by the Simons Foundation

through a Simons Investigator grant. I would also like to thank Boston College, where a

large part of the paper was written, for its hospitality.

2. Homological algebra

This section is of an elementary algebraic nature. This first part is classical, and only in-

cluded to make the exposition more self-contained: it recalls the relation between Hochschild

homology (with suitable coefficients) and bimodule maps (1.8). The next part concerns non-

commutative divisors, and follows [41, Section 3] except for the terminology. We then adapt

the same ideas to noncommutative pencils.

(2a) A∞-bimodules. Let A be a graded vector space over a field K. We denote by T (A[1])

the tensor algebra over the (downwards) shifted space A[1]. The structure of an A∞-algebra

on A is given by a map

(2.1) µA : T (A[1]) −→ A[2],

which vanishes on the constants K ⊂ T (A[1]). The components of (2.1) are operations

µdA : A⊗d → A[2−d] (for d ≥ 1). We will assume that A is strictly unital, denoting the unit

by eA, and write A = A/K eA.

An A∞-bimodule over A consists of a graded vector space P and a map

(2.2) µP : T (A[1])⊗ P⊗ T (A[1]) −→ P[1],

whose components we write as µs;1;rP : A⊗s ⊗ P⊗A⊗r → P[1−r−s] (for r, s ≥ 0). All A∞-

bimodules will be assumed to be strictly unital. A∞-bimodules over A form a dg category,

which we denote by [A,A]. An element of hom [A,A](P0,P1) of degree d is given by a map

(2.3) φ : T (A[1])⊗ P0 ⊗ T (A[1]) −→ P1[d],

which factors through the projection to T (A[1]) on both sides. We denote the components

of (2.3) again by φs;1;r. The actual bimodule maps are cocycles in hom [A,A](P0,P1), which


means solutions of

(2.4) ∑i,j

(−1)|φ|(‖a1‖+···+‖ai‖) µs−j;1;iP1

(a′s, . . . ;φj;1;r−i(a′j , . . . , a

′1; p; ar, . . . , ai+1); . . . , a1)

=∑i,j

(−1)|φ|+‖a1‖+···+‖ai‖ φs−j;1;i(a′s, . . . ;µj;1;r−iP0

(a′j , . . . , a′1; p; ar, . . . , ai+1); . . . , a1)

+∑i,j

(−1)|φ|+‖a1‖+···+‖ai‖ φs;1;r−j+1(a′s, . . . ; p; ar, . . . , µjA(ai+j , . . . , ai+1), . . . , a1)

+∑i,j

(−1)|φ|+‖a1‖+···+‖ar‖+|p|+‖a′1‖+···+‖a

′i‖φs−j+1;1;r(a′s, . . . ,

µjA(a′i+j , . . . , a′i+1), . . . ; p; ar, . . . , a1).

Here, ‖a‖ = |a| − 1 is the reduced degree. For more background material concerning A∞-

bimodules, see [50, 24, 22, 25, 41].

Remark 2.1. The sign conventions for A∞-algebras follow [43]. The sign conventions

for A∞-bimodules follow [41] except that, for greater compatibility with [43], we reverse the

ordering of the entries. This applies in particular to (2.4), which agrees with [41] up to

ordering (but differs from the convention for A∞-modules in [43]). The following observation

may help address some sign issues. Given any A∞-bimodule structure, the sign change

(2.5)µs;1;rP (a′s, . . . , a

′1; p; ar, . . . , a1) 7−→

− (−1)‖a1‖+···+‖ar‖+‖a′1‖+···+‖a

′s‖µs;1;r

P (a′s, . . . , a′1; p; ar, . . . , a1),

yields another A∞-bimodule structure. The two structures are generally distinct, but isomor-

phic (by an isomorphism that acts by ±1 on each graded piece). Hence, whenever we define

some construction of A∞-bimodules, there are potentially at least two equivalent versions,

which differ by (2.5).

We need to recall a few specific operations on A∞-bimodules. The shift P[1] moves the

grading down, and affects the operations as follows:

(2.6) µs;1;rP[1] (a′s, . . . , a

′1; p; ar, . . . , a1) = −(−1)‖a1‖+···+‖ar‖µs;1;r

P (a′s, . . . , a′1; p; ar, . . . , a1).

Next, the dual of P is P∨ = Hom(P,K), with

(2.7) 〈µs;1;rP∨ (as, . . . , a1;π; a′r, . . . , a

′1), p〉 = (−1)|p|+1〈π, µr;1;s

P (a′r, . . . , a′1; p; as, . . . , a1)〉.

Given two A∞-bimodules P0 and P1, one defines the tensor product P1 ⊗A P0 to be the

graded vector space P1 ⊗ T (A[1])⊗ P0, with the differential

(2.8)

µ0;1;0P1⊗AP0

(p1 ⊗ a′t ⊗ · · · ⊗ a′1 ⊗ p0) =∑j

(−1)|p0|+‖a′1‖+···+‖a

′t−j‖µ0;1;j

P1(p1; a′t, . . . , a

′t−j+1)⊗ a′t−j ⊗ · · · ⊗ a′1 ⊗ p0

+∑i,j

(−1)|p0|+‖a′1‖+···+‖a

′i‖p1 ⊗ a′t ⊗ · · · ⊗ µ

jA(a′i+j , . . . , a

′i+1)⊗ a′i ⊗ · · · ⊗ a′1 ⊗ p0

+∑i

p1 ⊗ a′t ⊗ · · · ⊗ a′i+1 ⊗ µi;1;0P0

(a′i, . . . , a′1; p0);

8 PAUL SEIDEL

the operations (for r > 0 or s > 0)

(2.9)

µs;1;0P1⊗AP0

(a′′s , . . . , a′′1 ; p1 ⊗ a′t ⊗ · · · ⊗ a′1 ⊗ p0) =∑

j

(−1)|p0|+‖a′1‖+···+‖a

′t−j‖µs;1;j

P1(a′′s , . . . , a

′′1 ; p1; a′t, . . . , a

′t−j+1)⊗ a′t−j ⊗ · · · ⊗ a′1 ⊗ p0,

µ0;1;rP1⊗AP0

(p1 ⊗ a′t ⊗ · · · ⊗ a′1 ⊗ p0; ar, . . . , a1) =∑i

p1 ⊗ a′t ⊗ · · · ⊗ a′i+1 ⊗ µi;1;rP0

(a′i, . . . , a′1; p0; ar, . . . , a1);

and with µs;1;rP1⊗AP0

= 0 if both r and s are positive.

Example 2.2. The diagonal A∞-bimodule is the space A itself, with the operations arranged

in such a way that the induced bimodule structure on A[1] satisfies µs;1;rA[1] = µs+1+r

A . Tensor

product with the diagonal bimodule (on either side) is essentially a trivial operation. More

precisely, there are canonical quasi-isomorphisms [41, Eqn. (2.21)–(2.24)]

(2.10) A⊗A P ' P ' P⊗A A.

(2b) Hochschild homology. The Hochschild homology of A with coefficients in P, written

as HH ∗(A,P), is the homology of the complex

(2.11)CC ∗(A,P) = T (A[1])⊗ P,

∂(ad ⊗ · · · ⊗ a1 ⊗ p) =∑i,j

(−1)|p|+‖a1‖+···+‖ai‖ad ⊗ · · · ⊗ ai+j+1 ⊗ µjA(ai+j , . . . , ai+1)⊗ ai ⊗ · · · ⊗ a1 ⊗ p

+∑i,j

(−1)∗ad−i ⊗ · · · ⊗ aj+1 ⊗ µj;1;iP (aj , . . . , a1; p; ad, . . . , ad−i+1),

where ∗ = (‖ad−i+1‖+· · ·+‖ad‖)(|p|+‖a1‖+· · ·+‖ad−i‖). In spite of the subscript notation,

the grading on CC ∗(A,P) is cohomological, meaning that ∂ has degree 1.

Suppose that Q is a finite-dimensional bimodule, and P an arbitrary bimodule. In that case,

one has a canonical isomorphism of chain complexes

(2.12) CC ∗(A,Q∨ ⊗A P)∨ ∼= hom [A,A](P,Q).

In the case where Q is merely proper (has finite-dimensional cohomology), one gets a corre-

sponding quasi-isomorphism. In particular:

Lemma 2.3. Suppose that A is proper. Then, for any P there are canonical isomorphisms

H∗(hom [A,A](P,A)) ∼= HH ∗(A,A∨ ⊗A P)∨,(2.13)

H∗(hom [A,A](P,A∨)) ∼= HH ∗(A,P)∨.(2.14)


One can write down an explicit chain representative for (2.14), going from right to left [46,

Equation (3.2)]: it sends ξ ∈ CC ∗(A,P)∨ to φ ∈ hom [A,A](P,A∨), where

(2.15) 〈φs;1;r(a′s, . . . , a′1; p; ar, . . . , a1), a〉 =

∑i,j

±ξ(aj ⊗ · · · ⊗ ai+1⊗

µi;1;r+1+s−jP (ai, . . . , a1, a, a

′s, . . . , a

′1; p; ar, . . . , aj+1))

(we omit the sign, which is not particularly relevant for our purpose). This gives us a way

of showing that a given P is quasi-isomorphic to A∨, already used in [46, Proposition 3.1]:

Lemma 2.4. Assume that both A and P are proper. Take a degree −n cocycle ξ ∈ CC (A,P)∨

and its leading order term ξ0 ∈ (Pn)∨. Suppose that the map

(2.16)A⊗ P −→ K[−n],

a′ ⊗ p 7−→ ξ0(µ1;1;0P (a′; p))

induces a nondegenerate pairing between H∗(P) and Hn−∗(A). Then P is quasi-isomorphic

to A∨[−n].

(2c) Noncommutative divisors. Fix an A∞-algebra A. We say that an A∞-bimodule P

is invertible if there is a bimodule Q and quasi-isomorphisms

(2.17) Q⊗A P ' A ' P⊗A Q.

Definition 2.5. A noncommutative divisor on A, with underlying invertible bimodule P, is

an A∞-algebra structure µB on the graded vector space B = A ⊕ P[1], with the following

properties:

• A ⊂ B is an A∞-subalgebra.

• Consider B as an A-bimodule (by restriction of the diagonal bimodule). Then, the

induced A-bimodule structure on B/A = P[1] agrees with the previously given one.

Two noncommutative divisors (with the same underlying bimodule) are isomorphic if there

is an isomorphism of the associated A∞-algebras B which restricts to the identity map on

A, and induces the identity map on B/A.

Let’s discuss this in more detail. The first condition in Definition 2.5 says that if we restrict

µB to T (A[1]) ⊂ T (B[1]), it takes values in A and agrees with µA. Next, if we restrict µB

to T (A[1])⊗ P[2]⊗ T (A[1]) ⊂ T (B[1]), it has the form

(2.18)µr+s+1B (a′s, . . . , a

′1, p, ar, . . . , a1) = µs;1;r

P (a′s, . . . , a′1; p; ar, . . . , a1)

+ θs;1;r(a′s, . . . , a′1; p; ar, . . . , a1)

for some

(2.19) θ : T (A[1])⊗ P⊗ T (A[1]) −→ A,

which is the first new piece of information (not described by A and P) in the noncommutative

divisor. The A∞-associativity equation for µB implies that θ is a bimodule map (1.8),

meaning that it satisfies (2.4). We call it the first order part of the noncommutative divisor.

10 PAUL SEIDEL

Example 2.6. Given A and P, one can form the trivial extension algebra Btriv , which is

the unique noncommutative divisor whose only nontrivial components are those dictated by

µA and µP; in particular, this has θ = 0.

The next piece of the structure of a noncommutative divisor is obtained by restricting µB to

T (A[1])⊗P[2]⊗T (A[1])⊗P[2]⊗T (A[1]), and then projecting the outcome to P[3]. This can

be viewed as an element of hom [A,A](P⊗A P,P) of degree −1. It is not a cocycle: instead,

as shown in [41, Lemma 3.2], it establishes the homotopy commutativity of the diagram

(2.20) P⊗A P

θ⊗id

yy

id⊗θ

%%A⊗A P

'%%

P⊗A A

'yy

P.

Example 2.7. Let A be the exterior algebra in one variable x (|x| = 1). Given any λ ∈ K×,

one has an invertible A-bimodule Pλ, whose underlying graded vector space is the same as

A, but where the action of x on the right is multiplied by λ (while that on the left remains

the same). Consider the bimodule map (of degree 1) t : Pλ → A which sends 1 to x. Then

(2.21)

Pλ ⊗A Pλ

t⊗id−−−→ A⊗A Pλ ∼= Pλ maps 1⊗ 1 7→ x,

Pλ ⊗A Pλid⊗t−−−→ Pλ ⊗A A ∼= Pλ maps 1⊗ 1 7→ λx.

This translates straightforwardly into the A∞-world, showing that homotopy commutativity

of (2.20) is not an empty requirement in general.

Suppose from now on that the ground field K has characteristic 0. One can then put the

previous discussion on a more systematic footing, as follows. Let’s equip B with an additional

grading (called weight grading to distinguish it from the usual one), in which A has weight

0 and P[1] has weight −1. For any i > 0, let

(2.22) W ig ⊂ Hom(T (B[1]),B[1]

)be the space of maps which increase the weight by (exactly) i. Each W ig is itself a graded

vector space, hence together they form a bigraded space g. This can be equipped with a

graded Lie bracket (in the way which is standard for A∞-structures), and with a differential

which uses the A∞-structure on A and the A∞-bimodule structure on P. In these terms, a

noncommutative divisor is given by a solution of the Maurer-Cartan equation [19, 26]

(2.23) ρ ∈ g1, dρ+ 12 [ρ, ρ] = 0

(the trivial solution ρ = 0 corresponds to Example 2.6). Similarly, the notion of isomorphism

for noncommutative divisors reduces to the standard gauge equivalence for solutions of (2.23).

There is another equivalent way of formulating this, which can be convenient. Divide ρ into

its weight components ρi; for instance, ρ1 consists of θ as well as the chain homotopy for


(2.20). Then (2.23) decomposes into

(2.24) dρi + 12

∑j

[ρj , ρi−j ] = 0.

Now take the one-dimensional vector space K[−1], placed in degree 1 and weight 1, and

considered as a dg Lie algebra with trivial differential and bracket. A solution of (2.24) is

the same as an L∞-homomorphism

(2.25) K[−1] −→ g

which preserves the weight grading (more precisely, the image of 1⊗i under the i-linear part

of (2.25) should be divided by i! to get ρi; this is the usual action of L∞-homomorphisms

on Maurer-Cartan elements [21, Section 4.3]).

The cohomology of (2.22) fits into a long exact sequence

(2.26) · · · → H∗−2i(hom [A,A](P

⊗Ai+1,P))−→ H∗(W ig)

−→ H∗−2i+1(hom [A,A](P

⊗Ai,A))−→ H∗−2i+1

(hom [A,A](P

⊗Ai+1,P))→ · · ·

The first group in (2.26) is isomorphic to H∗−2i(hom [A,A](P

⊗Ai,A)), because P is invertible

(and similarly for the last group). Example 2.7 shows that the connecting map (the last map

in the sequence) is generally nontrivial, even for i = 1. By applying standard obstruction

theory, one arrives at the following:

Lemma 2.8. Suppose that P satisfies

(2.27) H l(hom [A,A](P⊗Ak,A)) = 0 for k ≥ 1 and l < 0.

Then, the structure of a noncommutative divisor is determined (up to isomorphism) by the

cohomology class [θ] ∈ H0(hom [A,A](P,A)).

We conclude this discussion by recalling one of the definitions of the complement of a non-

commutative divisor (the notion of complement appeared previously in (1.9)(ii), but without

further explanation). Take B[[u]], where the formal variable u has degree 2. Let

(2.28) D ⊂ B[[u]]

be the subspace of those formal series in u whose constant term lies in A. Following [45], we

make D into a curved A∞-algebra by extending µB u-linearly, and then adding a curvature

term µ0D = u eA (more precisely, we consider D to be defined over K, but equipped with the

complete decreasing u-adic filtration). Then, A is a (right) A∞-module over D, by pullback

under the projection D→ A. One can then define A \B as the endomorphism ring of that

module. Explicitly,

(2.29) A \B = Hom(A⊗ T (D[1]),A),

where Hom stands for the u-adically continuous maps [41, Equation (4.7)]. The identification

of this construction with a suitable categorical localization (or quotient) process is provided

by [41, Theorem 4.1]. This identification shows that up to quasi-isomorphism, A \ B only

depends on the leading order part (2.19) of the noncommutative divisor.

12 PAUL SEIDEL

(2d) Noncommutative pencils. We continue in the same framework as before, with A

(defined over a field K of characteristic 0) and an invertible bimodule P, and will use the

same weight grading on B = A ⊕ P[1]. In addition, we fix a two-dimensional vector space

V , which will be given weight grading −1. Denote the symmetric algebra by Sym∗(V ), and

the dual vector space by W = V ∨.

Definition 2.9. A noncommutative pencil on A, with underlying bimodule P, is a map

(2.30) ℘ : T (B[1]) −→ B[2]⊗ Sym∗(V )

which preserves both the ordinary grading and the one by weight, and with the following

property. For any w ∈ W , we can use the associated evaluation map Sym∗(V ) −→ K to

specialize (2.30) to a map

(2.31) µB,w : T (B[1]) −→ B[2].

The requirement is that these should be noncommutative divisors.

As before, this definition lends itself to a piecewise analysis. The leading order term has the

form (after dualization)

(2.32) W ⊗ T (A[1])⊗ P⊗ T (A[1]) −→ A.

This is a family of bimodule maps (1.8) depending linearly on w ∈W . We write them as θw.

To make things even simpler, suppose that we identify V and W with K2 by choosing dual

bases. Then, the first order part of a noncommutative pencil consists of two bimodule maps

θ0 and θ∞, corresponding to w0 = (1, 0), w∞ = (0, 1) ∈ W . Note that for general reasons,

the following diagram commutes up to homotopy:

(2.33) P⊗A P

θ0⊗id

vv

id⊗θ∞

((θ0⊗θ∞

A⊗A P

' id⊗θ∞ ((

P⊗A A

'θ0⊗idvv

P

θ∞))

A⊗A A

'

P

θ0uu

A.

This, together with the homotopy commutativity of (2.20), implies that θ0⊗θ∞ is homotopic

to θ∞ ⊗ θ0.

To approach the structure of a noncommutative pencil in deformation-theoretic terms, we

return to the bigraded dg Lie algebra g from (2.22). Take W [−1], as a vector space placed

in degree 1 and weight 1, and equip it with the trivial dg Lie structure. In analogy with

(2.25), a noncommutative pencil is the same as an L∞-homomorphism (preserving the weight

grading)

(2.34) W [−1] −→ g.

There is also an analogue of Lemma 2.8 in this context:


Lemma 2.10. If (2.27) holds, the structure of a noncommutative pencil is determined up

to isomorphism by the cohomology classes [θ0], [θ∞].

Finally, we’d like to discuss some algebraic structures that can be constructed from a non-

commutative pencil:

(2.35)

(i) For any w ∈ W one has an A∞-structure µB,w on B. If w = 0, this

reduces to the trivial extension algebra (Example 2.6). Moreover, for

any nonzero λ ∈ K, the A∞-structures µB,w and µB,λw are isomorphic

(they are related by the automorphism of B which acts trivially on A

and rescales P by λ). We write Bz for the A∞-algebra (unique up to

isomorphism) associated to z = [w] ∈ P(W ). These are the algebras

previously referred to as the fibres of the noncommutative pencil.

(ii) Parallel remarks apply to the complements A \Bz.

(iii) One can generalize the construction of fibres by allowing z to vary. For

instance, take a formal disc in P(W ), given by a map ζ : Spec(K[[q]])→P(W ). Lift this to a formal disc in W , and consider the dual algebra

homomorphism Sym∗(V )→ K[[q]]. By applying that homomorphism to

℘, one gets a map

T (B[1]) −→ B[2]⊗K[[q]].

This defines an A∞-structure on B = B[[q]]. As before, this A∞-structure

is independent (up to isomorphism) of the choice of lift. This is what

appeared in part (iii) of (1.10), for V = K2 and the disc around infinity

given by choosing ζ(q) = [q : 1] = [1 : q−1].

(iv) It remains to explain the Landau-Ginzburg construction mentioned in

(1.10)(iv). Take a noncommutative pencil with V = K2, and let v0, v∞be the basis of V dual to w0, w∞. One can write

℘d =∑i+j≤d

℘di,j ⊗ vi0vj∞.

Replace v0 by u−1t and v∞ by 1, where both t and u are formal variables

of degree 2. Let the outcome act on D[[t]], where D is as in (2.28) (and

the action of u−1 removes a power of u, killing the outcome if there are

no powers left), then add the same curvature term u eA as before. The

outcome is a formal deformation of D with parameter t, which induces a

deformation of the dga A \B∞ (this induced deformation will include a

curvature term in general).

Remark 2.11. There are many open questions about noncommutative pencils. For instance,

what is the precise relation between A \B∞ and the generic fibre of the deformation of B∞obtained by taking a formal disc near∞? In the classical algebro-geometric case, this relation

becomes particularly simple when the pencil has no base locus, but it is not a priori clear what

the correct generalization is (this is important for applications to symplectic geometry, since it

concerns the relation between the Fukaya category of an ample hypersurface and the wrapped

Fukaya category of its complement).

14 PAUL SEIDEL

Indeed, there are open questions concerning the notion of “noncommutative base locus” itself.

Thinking of this as the intersection of the fibres at 0 and ∞, it should be given by

(2.36) B0 ⊗A B∞,

which admits a filtration whose associated graded space is quasi-isomorphic to

(2.37) (P[1]⊗A P[1])⊕ P[1]⊕ P[1]⊕A.

However, to make the base locus into a noncommutative space in its own right, one would

want to construct it as an A∞-algebra, whereas (2.36) is only an A-bimodule.

(2e) Discussion of the assumptions. Throughout this section, we have required that all

A∞-structures should be Z-graded. One can actually work with Z/2-gradings throughout,

with the same results, except that Lemmas 2.8 and 2.10 then become rather vacuous (since

their assumptions will be impossible to satisfy).

Remark 2.12. This is not quite the end of this topic, as illustrated by the intermediate

situation mentioned in Remark 1.7. There, A and P are Z-graded, and so is ℘ provided

that we take V = K ⊕ K[2m]. In particular, the first order term of the pencil is then given

by bimodule maps θ0 and θ∞ which have degrees 0 and 2m, respectively. There are Z/2-

graded noncommutative divisors Bz for z ∈ K, whose first order term is θ0 + zθ∞. The

residual information provided by the original Z-grading can be expressed as an isomorphism

Bz → Bλ2mz, for any λ ∈ K×. Moreover, if we restrict to the standard formal disc near

z = ∞, the A∞-structure one gets on B is K[[q]]-linear and Z-graded if we assign degree

−2m to the deformation variable q. If one then additionally assumes that A and P are

finite-dimensional (hence have bounded degree), each homogeneous piece of B contains only

finitely many powers of q. After inverting q, one then gets a Z-graded A∞-structure on

B[q, q−1].

The other assumption we have imposed (starting midway through Section 2c) is that the

coefficient field K has characteristic 0. This can be dropped as well, at the cost of making

some formulations a little more complicated. General Maurer-Cartan theory, which means

the classification theory of solutions of (2.23), does not apply in positive characteristic.

However, a simpler obstruction theory argument suffices to prove Lemma 2.8 (and its variant,

Lemma 2.10), and that does not require any assumption on K. The second case where we

have used the assumption on K is in the definition of noncommutative pencil, which was

formulated by specializing two-variable polynomials, appearing as elements of Sym(V ), to

one-dimensional linear subspaces. In positive characteristic, such specializations may fail to

recover the original polynomial (for instance, f(x, y) = xpy− xyp vanishes on every line, for

K = Fp). Hence, the definition of noncommutative pencil would have to be reformulated as

a condition on (2.30) itself.

Finally, there is a variation on our general setup which will be important for applications.

Namely, one can replace the ground field K by the semisimple ringR = Km = Ke1⊕· · ·⊕Kem.

An A∞-algebra over R consists of a graded R-bimodule A, together with R-bimodule maps

(2.38) µdA : A⊗R A⊗R · · · ⊗R A −→ A[2−d].


This is in fact the same as an A∞-category with objects labeled by 1, . . . ,m. The strict

unit eA must lie in the diagonal part of the R-bimodule A, which means that it can be

written as eA = eA,1 + · · · + eA,m with eA,i ∈ eiAei. Similarly, an A-bimodule consists of

a graded R-bimodule P, with structure maps that are linear over R as in (2.38). The same

applies to the definition of morphisms in [A,A] and of Hochschild homology. All definitions

and results in this section then carry over without any other significant modifications.

3. Floer cohomology: background

As a toy model and starting point for our considerations, we recall the basics of fixed point

Floer cohomology. The original references are [13, 14], generalizing the case of Hamiltonian

automorphisms [15]. Here, we use a version for Liouville domains, as in [37, Section 4], [29], or

[51]. Most of the technical details are familiar and will be omitted, but our discussion includes

two topics which are more commonly encountered in the special version of Floer cohomology

known as symplectic cohomology, namely: the contribution of Reeb orbits [53, 10]; and the

BV (Batalin-Vilkovisky or loop rotation) operator [42, 47, 11].

(3a) Definition. We will work in the following setting, which benefits from strong exactness

assumptions:

Setup 3.1. Let M2n be a Liouville domain. This means that it is a compact manifold with

boundary, equipped with an exact symplectic structure ωM = dθM , such that the Liouville

vector field ZM dual to θM points strictly outwards along the boundary. In a neighbourhood

of the boundary, there is a unique function ρM such that

(3.1)

ρM |∂M = 1,

ZM .ρM = ρM .

We denote by RM the Hamiltonian vector field of ρM . This is the natural extension of

the Reeb vector field on ∂M to the collar neighbourhood given by the Liouville flow. Other

associated geometric structures are as follows:

• We consider symplectic automorphisms φ : M → M which are equal to the identity

near the boundary, and exact, in the sense that there is a function Gφ (necessarily,

locally constant near ∂M) such that

(3.2) φ∗θM − θM = dGφ.

• To perturb those automorphisms, we use Hamiltonian functions H ∈ C∞(M,R) with

(3.3) dH = ε dρM near ∂M

for some ε ∈ R (which means H = ε ρM + a locally constant function).

• Finally, we use ωM -compatible almost complex structures J such that

(3.4) θM J = dρM near ∂M .

16 PAUL SEIDEL

Fix an automorphism φ in the class introduced above. Choose an ε ∈ R with the following

property:

(3.5)εRM |∂M may not have any 1-periodic orbits (this is equivalent to saying that

ε 6= 0 and that there are no Reeb orbits of period ε).

Take a family of functions H = (Ht), t ∈ R, as in (3.3), as well as a family J = (Jt) of almost

complex structures in the class (3.4). These should have the following periodicity properties:

(3.6)

Ht+1(x) = Ht(φ(x)),

Jt+1 = φ∗Jt.

On the twisted free loop space Lφ = x : R → M : x(t) = φ(x(t + 1)) for all t, consider

the H-perturbed action functional:

(3.7) Aφ,H(x) = −∫ 1

0x∗θM +Ht(x(t)) dt −Gφ(x(1)).

Its critical points are solutions x ∈ Lφ of

(3.8) dx/dt = XH,t,

where XH is the time-dependent Hamiltonian vector field of H. Such x correspond bijectively

to fixed points x(1) of φ1H φ, where (φtH) is the Hamiltonian isotopy generated by H. On

the boundary, φtH is the Reeb flow for time εt. By assumption on ε, this implies that the

solutions of (3.8) are contained in the interior of M . Consider bounded trajectories of Floer’s

gradient flow equation for (3.7). These are maps u : R2 →M satisfying

(3.9)

u(s, t) = φ(u(s, t+ 1)),

∂su+ Jt(∂tu−XH,t) = 0,

lims→±∞ u(s, t) = x±(t),

for x± as in (3.8).

Lemma 3.2. No solution of (3.9) can reach ∂M .

Proof. This is a standard maximum principle argument, which becomes particularly simple

when formulated as follows. In a neighbourhood of ∂M , the Cauchy-Riemann equation in

(3.9) reduces to

(3.10) ∂su+ Jt(∂tu− εRM ) = 0.

If we then write u(s, t) = φεtρM (u(s, t)), where φtρM is the Hamiltonian flow of ρM , the equation

becomes homogeneous:

(3.11) ∂su+ Jt ∂tu = 0.

The almost complex structures Jt = (φεtρM )∗Jt still satisfy (3.4). Hence, at all points (s, t)

where u(s, t) is sufficiently close to ∂M , the function ρu(s, t) = ρM (u(s, t)) = ρM (u(s, t)) is

subharmonic:

(3.12) ∆ρu = ωM (∂su, ∂tu) ≥ 0.


For a generic choice of H, all solutions of (3.8) are nondegenerate. Then, each of them has

an index i(x) ∈ Z/2, which is such that (−1)i(x) is the contribution of x(1) to the Lefschetz

fixed point number of φ1H φ. Fix a field K of characteristic 2. We define the Floer cochain

space CF ∗(φ,H) to be the Z/2-graded vector space with one summand K for each x, placed

in degree i(x). For generic choice of J , the solutions of (3.9) are regular. Exclude solutions

which are constant in s, and then divide the resulting moduli spaces by translation in s-

direction. Having done that, one counts isolated points (mod 2) to obtain the coefficients of

the Floer differential, denoted by dJ .

Remark 3.3. By counting Floer trajectories with appropriate signs, one can define fixed

point Floer cohomology over an arbitrary coefficient field K; this goes back to [15]. Finally,

under additional assumptions, one can refine the Z/2-grading to a Z-grading. A convenient

way is to assume that c1(M) = 0; to choose a trivialization of the canonical bundle KM ;

and then require all symplectic automorphisms to come with gradings [35] (this gives an

absolute Z-grading on Floer cohomology, which is well-behaved with respect to various addi-

tional structures). Since these are standard considerations, we will continue our exposition

in the simplest setup (char(K) = 2 and Z/2-gradings), but take care that all properties are

formulated so that they remain correct when signs and Z-gradings are added.

Example 3.4. For sufficiently small |ε|, one has

(3.13) HF ∗(idM , ε) ∼=

H∗(M ;K) ε > 0,

H∗(M,∂M ;K) ε < 0.

There are at least two ways of proving this (and one can show that they yield the same

isomorphism). The first approach is to choose a time-independent small Morse function

H, and similarly a constant family of almost complex structures J . Then, all 1-periodic

trajectories x will be constant maps at critical points of H. Moreover, all solutions u of

(3.9) whose Fredholm index is ≤ 1 are again constant in t, hence reduce to gradient flow

lines of H. This is proved in [20, 16]. As a consequence, the Floer complex itself reduces

to the Morse complex. The second approach is to adapt [31]; this is less direct, but has the

advantage that the resulting PSS isomorphism is more easily shown to be canonical.

Some of the features that appear in Example 3.4 are instances of more general properties.

First, Floer cohomology is invariant under deformation of (φ, ε) within the range of allowed

values. In particular, if |ε| is sufficiently small (smaller than the shortest Reeb period), only

the sign of ε matters, which explains the form of the answer in (3.13). Next, the Poincare

duality between the two groups in (3.13) extends to a general duality property of Floer

cohomology:

(3.14) HF ∗(φ−1,−ε) ∼= HF 2n−∗(φ, ε)∨.

For suitably coordinated choices, this is induced by an isomorphism of Floer cochain com-

plexes, which comes from the diffeomorphism Lφ → Lφ−1 reversing the parametrization of

a loop.

Example 3.5. Suppose that the Reeb flow on ∂M is periodic, of period b. Choose a Hamil-

tonian function F which is equal to b · ρM near the boundary, and let (φtF ) be its flow. By

18 PAUL SEIDEL

construction, φ1F is the identity near the boundary. We call it the left-handed boundary twist

(see [35, Section 4] for a general discussion of such symplectic automorphisms). Note that

one can choose F to be supported arbitrarily close to ∂M . Hence, given another automor-

phism φ as in Setup 4.8, one can arrange that the flow of F commutes with φ. Consider the

diffeomorphism

(3.15)F : Lφ1

F φ −→ Lφ,

(Fx)(t) = φtF (x(t)),

which satisfies

(3.16)F∗Aφ,H = Aφ1

F φ,H,

Ht(x) = Ht(φtF (x))− F (x).

Given suitable choices of almost complex structures, this induces an isomorphism of Floer

cochain complexes, hence

(3.17) HF ∗(φ1F φ, ε− b) ∼= HF ∗(φ, ε).

In particular, by combining this with (3.13), one obtains the expression (1.5) for the Floer

cohomology of µ = φ−1F (where b = 1); except for grading issues, which we will address next.

In the situation where Floer cohomology is Z-graded, the isomorphism (3.17) is compatible

with the grading, provided that φ1F is made into a graded symplectic automorphism by using

the isotopy (φtF ) to deform it to the identity. However, since this isotopy is not constant

near ∂M , the resulting grading of φ1F may not be trivial near ∂M . In the original setup of

(1.5), we considered a Liouville manifold M obtained from a closed symplectic manifold with

trivial canonical bundle by removing a neighbourhood of a divisor. In that case, near the

boundary we have φ1F = [−2] as a graded symplectic automorphism; this explains the shift by

2 in (1.5). A version of the same phenomenon is responsible for (1.6).

(3b) Reeb orbits. We now return to the issue of how HF ∗(φ, ε) depends on the parameter

ε. It will be convenient to enlarge the given Liouville domain by some amount C > 1, forming

(3.18) M = M ∪∂M ([0, C]× ∂M).

The conical part which we have added carries the one-form θM = er(θM |∂M) (r being the

coordinate in [0, C]) and the symplectic form ωM = dθM . Note that in fact, M is isomorphic

to M with the symplectic form rescaled by eC (the isomorphism is given by the Liouville

flow for time C).

The function ρM has a natural extension, ρM (r, x) = er. Take some φ, ε, and the data (H,J)

used to define Floer cohomology. Suppose for simplicity that Ht|∂M = ε and Gφ|∂M = 0

(the general case differs from this only by some constants in the action (3.21), which do not

affect the argument for large C). One can extend φ and Gφ trivially over the cone, forming

φ and Gφ. Similarly, H has a preferred extension to M , which is εer on the cone. There is

no comparable statement for an arbitrary J , but one can assume that it has been chosen in

such a way that there is an extension over the cone satisfying the analogue of (3.4). In that

situation, the Floer cohomology does not change when passing from M to M . Indeed, the


Floer cochain complex remains the same: there are no solutions of (3.8) in M \M , and the

same maximum principle as in Lemma 3.2 shows that all Floer trajectories remain inside M

(alternatively, one can use the “integrated maximum principle” from [5, Section 7c], which

allows more freedom in the choice of almost complex structure on [0, C]× ∂M).

For our purpose, we want to look at a different extension of H. Suppose that we are given

ε > ε, again satisfying (3.5). Take a function f ∈ C∞([1, e],R) such that

(3.19)

f(a) = εa for a close to 1,

f ′(a) = ε for a close to e,

f ′′(a) ≥ 0 everywhere,

f ′′(a) > 0 if f ′(a) ∈ (ε, ε).

We extend the given H to H ∈ C∞(R× M,R) by setting

(3.20) Ht(r, x) =

εer if (r, x) ∈ [0, C − 1]× ∂M,

eC−1f(er−C+1) if (r, x) ∈ [C − 1, C]× ∂M.

In this case, besides solutions of (3.8) lying inside M , there are ones corresponding to Reeb

orbits whose period lies in the interval (ε, ε). We denote them by x. Their action is

(3.21) Aφ,H(x) = eC−1(− af ′(a) + f(a)

),

where a ∈ [1, e] is such that f ′(a) is the period of the Reeb orbit. Because of the convexity

conditions in (3.19), −af ′(a) + f(a) is strictly negative. Hence, all the actions (3.21) go to

−∞ as C → ∞. The solutions x are never nondegenerate (though they may be nondegen-

erate in the weaker Morse-Bott sense), but one can remedy that by a small time-dependent

perturbation of H, and the statement about actions will then continue to hold. Together

with an appropriate maximum principle argument, this means that for large C, we get an

inclusion of CF ∗(φ,H) as a subcomplex into CF ∗(φ, H), hence a long exact sequence

(3.22) · · · → HF ∗(φ, ε) −→ HF ∗(φ, ε) −→ Q∗ → · · ·

This kind of argument is familiar from symplectic cohomology theory [10]. In general, while

the cochain space underlying Q∗ depends only on the Reeb flow, its differential may involve

the whole of M (one encounters this phenomenon already for surfaces [10, Section 8.1]).

However, there is a better-behaved special case. Namely, assume that in the interval (ε, ε′)

there is only one Reeb period. Then, there is only one value of a which appears in (3.21),

which means that all x have the same action. This property may be destroyed by a small

perturbation which makes them nondegenerate, but it will still be the case that

(3.23) |Aφ,H(x0)−Aφ,H(x1)| < B

for some constant B which can be made arbitrarily small (in a way which is independent of

C). Then, an application of the Monotonicity Lemma shows that if C is large, any Floer

trajectory joining (x0, x1) must remain inside [0, C]× ∂M .

Example 3.6. For sufficiently small ε > 0, one gets a long exact sequence

(3.24) · · · → HF ∗(φ,−ε) −→ HF ∗(φ, ε) −→ H∗(∂M ;K)→ · · ·

20 PAUL SEIDEL

The contribution H∗(∂M ;K) from constant orbits can be computed by a Morse-Bott argu-

ment. For φ = id, this reduces to the ordinary cohomology long exact sequence for (3.13).

In the situation of Example 3.5 (Reeb flow periodic of period b), one has similar jumps when

ε crosses an integer multiple of b, but where the grading of H∗(∂M ;K) may be shifted by an

even amount.

Example 3.7. Suppose that ε and ε have the same sign, and that (up to reparametrization)

there is a unique periodic Reeb orbit of period (ε, ε), which is Morse-Bott (transversally)

nondegenerate. By the same kind of argument as in the previous example, one has

(3.25) · · · → HF ∗(φ, ε) −→ HF ∗(φ, ε) −→ H∗+k(S1; ξ)→ · · ·

where k ∈ Z depends on the Conley-Zehnder index, and ξ is a local K-coefficient system

on S1, which is trivial if the Reeb orbit is “good”, and has holonomy −1 if it is “bad” (the

terminology is taken from Symplectic Field Theory; for an explanation in a framework close

to ours, see [9, §5] or [12, §4.4]).

(3c) The BV operator. We consider a situation which generalizes that in Example 3.5,

namely a Hamiltonian flow which, for a particular time, gives rise to a symplectic automor-

phism that is the identity near the boundary.

Setup 3.8. Let F ∈ C∞(M,R) be a function whose Hamiltonian vector field XF and flow

(φtF ) have the following properties:

• XF is tangent to ∂M (hence, φtF is defined for all times);

• (φtF )∗θM = θM near ∂M (therefore, φtF also preserves ρM );

• φ = φ1F is the identity near ∂M .

In this case, the twisted free loop space admits a circle action, which combines our flow with

loop rotation: r ∈ S1 = R/Z acts by

(3.26)Rr : Lφ −→ Lφ,

(Rrx)(t) = φrF (x(t+ r)).

In classical cohomology, a circle action gives rise to an operator of degree −1, which is dual

to moving cycles around. The analogue for Floer cohomology is the BV operator

(3.27) ∆ : HF ∗(φ, ε) −→ HF ∗−1(φ, ε).

Remark 3.9. In the classical topological situation, one does not really need a circle action,

but only a loop of self-maps. Correspondingly, it is not really necessary to have φ represented

as time-one map of a Hamiltonian flow: one can allow a more general Hamiltonian path (φt)

from φ0 = id to φ1 = φ, as long as that path preserves θM near ∂M , and the associated loop

(φ−1φtφφ−1t ) is contractible (inside the group of exact symplectic automorphisms which are

the identity near the boundary). From that viewpoint, the construction looks as follows: for

each t, one has a quasi-isomorphism

(3.28) CF ∗(φ, ε) ∼= CF ∗(φtφφ−1t , ε) −→ CF ∗(φ, ε).


where the first isomorphism is conjugation by φt, and the second is the continuation map for

a suitable path from φtφφ−1t to φ. One can choose these paths to depend smoothly on t, and

to be constant if t = 0, 1; then, a parametrized version of the same construction yields a chain

homotopy between the identity map (t = 0) and the conjugation action of φ on CF ∗(φ, ε).

On the other hand, there is another source of such chain homotopies, namely loop rotation,

see e.g. [48, Equation (3.6)]. The difference between the two constructions then yields the

BV operator. However, in this degree of generality, it may lack some properties that are

familiar from the circle action case, such as (3.34).

To construct this concretely from (3.26), one proceeds as follows. Choose an auxiliary H

and J as required in the definition of CF ∗(φ,H). For r ∈ S1, define (Hr, Jr) by

(3.29)Hr,t(x) = Ht−r(φ

rF (x)),

Jr,t = (φrF )∗Jt−r.

Then, the Floer cochain complex CF ∗(φ,Hr) is canonically isomorphic to CF (φ,H), by

applying (3.26). Consider a continuation-type equation for maps u : R2 →M , depending on

an auxiliary parameter r ∈ S1:

(3.30)

u(s, t) = φ(u(s, t+ 1)),

∂su+ Jr,s,t(∂tu−XK,r,s,t) = 0,

lims→+∞ u(s, t) = x+(t),

lims→−∞ u(s, t) = φrF (x−(t+ r)).

Here, the inhomogeneous term involves the Hamiltonian vector field of a family of functions

K = Kr,s,t, which are chosen so that Kr,s,t = ερM near the boundary, and

(3.31) Kr,s,t =

Ht s 0,

Hr,t s 0.

The almost complex structures Jr,s,t must have a corresponding property. For any (r, u)

satisfying (3.30), one has an energy equality

(3.32) E(u) =

∫R×[0,1]

‖∂su‖2 ds∧ dt = Aφ,H(x−)−Aφ,H(x+) +

∫R×[0,1]

∂sKr,s,t(u) ds∧ dt.

The last term is bounded, and that ensures the necessary compactness properties. Counting

isolated points in the parametrized moduli space of solutions of (3.30) yields a degree −1

chain map

(3.33) δ : CF ∗(φ,H) −→ CF ∗−1(φ,H),

which induces (3.27). Note that, unlike the differential, (3.33) does not preserve the action

filtration of CF ∗(φ,H). However, the amount by which the action may decrease can be

made arbitrarily small, by a suitable choice of H and K.

Example 3.10. Take F = 0, so that φ = idM , and a sufficiently small ε > 0. Then,

the argument from [20, 16] shows that one can take H to be t-independent, and K to be

independent of all parameters (r, s, t). As a consequence, ∆ = 0. More generally, by using

22 PAUL SEIDEL

this fact and the isotopy invariance of Floer cohomology, one can show that ∆ = 0 whenever

F vanishes near ∂M .

Example 3.11. Suppose that F is the moment map of a Hamiltonian circle action, so

that again φ = idM . Set ε > 0 small. In that case, (3.27) is an endomorphism of ordi-

nary cohomology by (3.13), and in fact that agrees with the previously mentioned topological

construction. Informally, the reason is that the circle action (3.26) restricts to φtF on the

subspace of constant loops. This can be turned into a rigorous proof by using Morse-Bott

methods, but a better approach is to use the isomorphism from [31] (there is a similar result

with the opposite sign of ε, and cohomology rel ∂M). The simplest concrete example is to

take M = [−1, 1] × S1, with the standard symplectic structure, and the circle action which

rotates the S1-direction. In that case, we get that the BV operator is an isomorphism from

HF 1 to HF 0.

We conclude by stating (without proofs, which however are not difficult) two properties of

the BV operator. First,

(3.34) ∆2 = 0.

Secondly, recall that HF ∗(φ, ε) has the structure of a module over the ordinary cohomology

H∗(Lφ;K), via the cap product (extensively used in [53, Section 1.4], see also [52, 39]). Let’s

write the cap product simply as

(3.35) (a, b) 7−→ a · b, a ∈ HF ∗(φ, ε), b ∈ H∗(Lφ;K).

On the other hand, the circle action (3.26) induces an operator on H∗(Lφ;K) as well, which

we denote by ∆class to distinguish it from its Floer cohomology counterpart (3.27). Then,

∆class is a derivation for the classical (cup product) ring structure on H∗(Lφ;K), and it

satisfies

(3.36) ∆(a · b) = ∆(a) · b+ (−1)|a|a · (∆classb).

4. Floer cohomology: setup

Following the model of [46], we now modify the standard Floer-theoretic setup (the “closed

string” symplectic fixed point flavour from the previous section, as well as the “open string”

Lagrangian intersection flavour), to make it suitable for applications to Lefschetz fibrations.

Open-closed string maps [38, 1, 18] establish a relationship between the two flavours. As

indicated by the references above (which are only a sample of the literature), there is nothing

fundamentally new here. A particularly close parallel is formed by work (in progress) of

Abouzaid-Ganatra on split-generation criteria for Lefschetz fibrations [2].

(4a) Formalism. We begin by listing all the Floer-theoretic structures to be defined,

without any details, but giving a few remarks to aid the intuition. We will be working with

a fixed π : E2n → C, whose fibre M2n−2 is a Liouville domain. We consider a certain class


of automorphisms φ of E, which outside a compact subset take each fibre to itself. The

construction of fixed point Floer cohomology

(4.1) HF ∗(φ, δ, ε)

involves two auxiliary parameters δ, ε. One of these, ε, deals with the fact that the fibre M

has boundary, and is restricted in the same way (3.5) as before. The other one, δ, can be

any nonzero real number, and is used as a perturbation in positive imaginary direction over

the base C.

Remark 4.1. We will follow the same expository strategy as in the previous section, which

means that the actual construction of Floer cohomology is carried out over a coefficient field

K of characteristic 2, and yields a Z/2-graded vector space. In fact, one can use any field,

and the Z/2-grading can be upgraded to a Z-grading exactly in the same circumstances as in

Remark 3.3.

Example 4.2. Even though not strictly necessary for our purpose, it is instructive to see

what happens for φ = idE, which is the analogue of the PSS isomorphism (3.13) in this

context. For δ > 0 and any ε, this says that

(4.2)HF ∗(idE , δ, ε) ∼= H∗(E,E−∞;K),

HF ∗(idE ,−δ, ε) ∼= H∗(E,E+∞;K),

where

(4.3) E±∞ = x ∈ E : ±re(π(x)) 0

are the parts lying “near infinity” in positive and negative real directions. There is a diffeo-

morphism of E which maps E−∞ to E+∞, hence induces an isomorphism H∗(E,E−∞;K) ∼=H∗(E,E+∞;K) (in fact, there are two choices differing by a monodromy automorphism).

The analogue of (3.14) is

(4.4) HF ∗(φ−1,−δ,−ε) ∼= HF 2n−∗(φ, δ, ε)∨.

In Example 4.2, this reproduces an appropriate version of Poincare duality. Generalizing

another observation made there, one always has HF ∗(φ, δ, ε) ∼= HF ∗(φ,−δ, ε). This shows

that Floer cohomology is independent of δ, but only in a weak sense, since there are again

two different possible choices of the isomorphism (we will not give any more details). More

interestingly, and again following what we have seen in (4.2), HF ∗(φ, δ, ε) is independent of

the choice of ε up to canonical isomorphism (Proposition 4.19). In spite of that, we choose

to keep ε in the notation, since the existence of these isomorphisms is not quite trivial.

We consider oriented exact Lagrangian submanifolds L ⊂ E \ ∂E whose projection π(L),

outside a compact subset, is a horizontal half-line

(4.5) re(y) 0, im(y) = λ ⊂ C.

If we have two such submanifolds L0, L1, such that the corresponding numbers (4.5) satisfy

(4.6) λ0 6= λ1,

there is a well-defined Floer cohomology HF ∗(L0, L1).

24 PAUL SEIDEL

Remark 4.3. We construct HF ∗(L0, L1) as a Z/2-graded vector space over a characteristic

2 field K. This time, if one wants to introduce signs (which means, extend the definition to

arbitrary K), one needs to impose some topological restrictions on the Lagrangian subman-

ifolds, for instance asking them to come with Spin structures, see [17, Chapter 8] or [43,

Section 12]. As before, one can obtain Z-graded Floer cohomology groups by working with

graded Lagrangian submanifolds [35].

We will also use a minor generalization, which involves perturbations as in (4.1), denoted by

HF ∗(L0, L1, δ, ε). This time, ε ∈ R can be arbitrary (and the outcome is independent of it

up to canonical isomorphism, for quite straightforward reasons); and the condition (4.6) is

replaced by

(4.7) δ 6= λ1 − λ0.

Duality takes the form

(4.8) HF ∗(L1, L0,−δ,−ε) ∼= HFn−∗(L0, L1, δ, ε)∨.

Example 4.4. The appropriate version of the Lagrangian PSS isomorphism [6] says that

(4.9) HF ∗(L,L, δ, ε) =

H∗(L;K) δ > 0,

H∗(L,L∞;K) δ < 0,

where L∞ is the part at infinity of L (so H∗(L,L∞;K) is compactly supported cohomology).

As usual, (4.8) then becomes ordinary Poincare duality.

We now go beyond Lagrangian Floer cohomology, to the chain level A∞-structures. Fix

Lagrangian submanifolds L1, . . . , Lm ⊂ E \ ∂E of the same kind as before, with pairwise

different constants λ1, . . . , λm ∈ R as in (4.5) (eventually, the Lk will be a basis of Lefschetz

thimbles). To these, one can associate a (directed) A∞-algebra A over R = Km = Ke1 ⊕· · ·Kem, which is such that

(4.10) H(A) = R⊕⊕i<j

HF ∗(Li, Lj),

or equivalently

(4.11) ejH(A)ei =

HF ∗(Li, Lj) i < j,

K i = j,

0 i > j.

The nontrivial part of the algebra structure on H(A) is the triangle product on Lagrangian

Floer cohomology. Next, take an automorphism φ and constants δ, ε ∈ R, satisfying

(4.12) δ 6= λi − λj for all i, j (including i = j, which means δ 6= 0).

Generalizing [46, Section 6.3], one can associate to this an A∞-bimodule Pφ,δ,ε over A, with

(4.13) H(Pφ,δ,ε) =⊕i,j

HF ∗(φ(Li), Lj , δ, ε).


Remark 4.5. The definition of Pφ,δ,ε lacks a directedness condition parallel to that in

(4.10), (4.11). In particular, even if one takes φ = id, there is no immediate relation-

ship between Pid,δ,ε and A (however, in the context of bases of Lefschetz thimbles, Pid,δ,ε is

quasi-isomorphic to the diagonal bimodule A if one takes δ 0 [46, Corollary 6.1], and to

the shifted dual diagonal bimodule A∨[−n] for δ 0 [46, Corollary 6.2]).

We now combine the previous two ingredients. For φ as before, take (δ, ε) so that fixed point

Floer cohomology is well-defined, and in addition (4.12) holds. The open-closed string map

in this context has the form

(4.14) HH ∗(A,Pφ,δ,ε) −→ HF ∗+n(φ, δ, ε),

The simplest part of (4.14) is a homomorphism

(4.15) HF ∗(φ(Li), Li, δ, ε) −→ HF ∗+n(φ, δ, ε).

In view of (4.4), the target of our map is dual to HFn−∗(φ,−δ,−ε). Let’s write this as Floer

homology HFn−∗(φ,−δ,−ε). Similarly, for Lagrangian Floer cohomology we have (4.8). If

we use these isomorphisms, and switch the signs of both δ and ε, (4.15) becomes a map

(4.16) HF ∗(Li, φ(Li), δ, ε) −→ HF ∗(φ, δ, ε).

Let’s specialize to φ = id . Applying (4.2) and (4.9), one sees that (4.16) yields maps

(4.17)

H∗(Li;K) −→ H∗(E,E−∞;K),

H∗(Li, Li,∞;K) −→ H∗(E,E+∞;K).

Unsurprisingly, these can be identified with the classical inclusion maps in homology.

We will also need a variation of (4.14) which involves two automorphisms φk (k = 0, 1), with

their associated bimodules Pk = Pφk,δk,εk . For simplicity, we assume that ε0/δ0 = ε1/δ1(this condition could be dropped, at the cost of making the formalism later on a bit more

complicated). Set

(4.18)

φ = φ1 φ0,

δ = δ0 + δ1,

ε = ε0 + ε1.

We suppose that HF ∗(φ, δ, ε) is defined. Then, the modified version of the open-closed string

map has the form

(4.19) HH ∗(A,P1 ⊗A P0) −→ HF ∗+n(φ, δ, ε).

Remark 4.6. One can think of (4.19) as follows. There are triangle product maps

(4.20) HF ∗(φ1(Lj), Lk, δ1, ε1)⊗HF ∗(φ0(Li), Lj , δ0, ε0) −→ HF ∗(φ(Li), Lk, δ, ε).

These can be extended to an A∞-bimodule homomorphism

(4.21) P1 ⊗A P0 −→ P,

where P = Pφ,δ,ε. Using the functoriality of Hochschild homology, one can reduce (4.19) to

(4.14) for φ, by writing it as the composition

(4.22) HH ∗(A,P1 ⊗A P0) −→ HH ∗(A,P) −→ HF ∗+n(φ, δ, ε).

26 PAUL SEIDEL

However, we will prefer a direct definition of (4.19), since that is a little simpler, and we

have no use for the maps (4.21) by themselves.

(4b) Target space geometry. The first step is in setting up the geometric formalism is

to characterize exactly the classes of basic objects (symplectic manifolds, automorphisms,

Lagrangian submanifolds, Hamiltonian functions, almost complex structures) which appear.

Setup 4.7. Let E be a 2n-dimensional manifold with boundary, with an exact symplectic

form ωE = dθE, and a proper map

(4.23) π : E −→ C.

Equip the base C with its standard symplectic structure ωC = dre(y) ∧ dim(y). At any

x ∈ E, the horizontal subspace TEhx ⊂ TE is defined as the ωE-orthogonal complement of

TEvx = ker(Dπx). We assume that π is an exact symplectic fibration with singularities,

which means the following:

• There is an open subset U ⊂ E which contains ∂E and has compact complement,

such that at each x ∈ U : TEvx is a codimension 2 symplectic subspace of TE (in

particular, x is a regular point of π); and

(4.24) (ωE − π∗ωC) |TEhx = 0.

• At every point x ∈ ∂E, TEhx is tangent to the boundary.

• There is a y∗ > 0 with the following properties. All fibres Ey = π−1(y) with |y| ≥ y∗lie inside U ; and M = Ey∗ , with its induced exact symplectic structure, is a Liouville

domain (in all subsequent developments, we will assume that such a y∗ has been

fixed).

Let’s consider the implications of these assumptions. Given a symplectic fibration with

singularities, we have symplectic parallel transport maps in a neighbourhood of ∂E. More

precisely, let W ⊂M be a small open neighbourhood of ∂M . Then, parallel transport yields

a canonical embedding, whose image is an open neighbourhood of ∂E ⊂ E:

(4.25)

Θ : C×W −→ E,

Θ(C× ∂M) = ∂E,

π(Θ(y, x)) = y,

Θ(y∗, x) = x,

Θ∗ωE = ωC + ωM .

To see why that is the case, note that (4.24) implies that the symplectic connection is flat

near ∂E. Hence, the relevant part of parallel transport is independent of the choice of path,

and Θ∗ωE − ωM must be the pullback of some two-form on C; to determine that two-form,

one again appeals to (4.24).

We also have the same flatness property outside a compact subset, with the following conse-

quence. Let µ be the monodromy around the circle of radius y∗, which is an exact symplectic


automorphism of M (and restricts to the identity on W ). We will use the (holomorphic)

radial coordinates

(4.26)

ψ : R× R/Z −→ C∗,

ψ(p, q) = e2π(p+iq),

ψ∗ωC = 4π2e4πp dp ∧ dq .

Then, for p∗ so that ψ(p∗, 0) = y∗, there is a unique covering map

(4.27)

Ψ : [p∗,∞)× R×M −→ |π(x)| ≥ y∗ ⊂ E,π(Ψ(p, q, x)) = ψ(p, q),

Ψ(p∗, 0, x) = x,

Ψ(p, q + 1, x) = Ψ(p, q, µ(x)),

Ψ∗ ωE = ψ∗ωC + ωM .

The partial trivializations (4.25) and (4.27) are compatible, in the sense that

(4.28) Θ(ψ(p, q), x) = Ψ(p, q, x) for p ≥ p∗ and x ∈W .

In words, what we have observed is that (4.23) is symplectically trivial near ∂E, whereas

outside the preimage of a disc, it is modelled on the symplectic mapping torus of µ.

Setup 4.8. We consider exact symplectic automorphisms φ of E of the following kind:

• There is a symplectic automorphism φC of the base, which is the identity outside a

compact subset, such that

(4.29) π(φ(x)) = φC(π(x))

for x ∈ U . Here, U ⊂ E is an open subset as in Setup 4.7.

• If we restrict φ to a fibre Ey, |y| 0, then it is the identity near ∂Ey.

The assumption (4.29) is quite restrictive, since it implies that φ must commute with sym-

plectic parallel transport between the fibres: in combination with the other condition, this

means that the behaviour of φ near ∂E is completely determined by φC. Concretely, possibly

after making the subset W from (4.25) smaller, we have

(4.30) φ(Θ(y, x)) = Θ(φC(y), x).

Similarly, possibly after making p∗ in (4.27) bigger, we have

(4.31) φ(Ψ(p, q, x)) = Ψ(p, q, φM (x)),

where φM is an exact symplectic automorphism of M , which is the identity near ∂M . Note

that (4.31) together with the periodicity property in (4.27) imply that φM must commute

with µ (in our applications, φM will be some power of µ). All the subsequently defined

classes of geometric data (Lagrangian submanifolds, almost complex structures, Hamiltonian

functions) are chosen so that they are preserved by automorphisms φ.

Setup 4.9. We consider exact Lagrangian submanifolds L ⊂ E \ ∂E with the following

properties:

28 PAUL SEIDEL

• π|L is a proper map.

• There is a relatively compact open subset W ⊂ L and constants C, λ, such that the

restriction of π to L \W is a smooth fibration over the half-line

(4.32) re(y) ≥ C, im(y) = λ ⊂ C.

This is the precise version of the earlier statement (4.5).

Concretely, this means that there is an exact Lagrangian submanifold LM ⊂ M \ ∂M such

that for y as in (4.32),

(4.33) L ∩ Ey = Ψ((p, q)× LM ),

where p > 0 and q ∈ (− 12 ,

12 ) are such that ψ(p, q) = y.

Setup 4.10. Given constants δ, ε ∈ R, we consider functions H ∈ C∞(E,R) as follows:

• For x near ∂M ,

(4.34) H(Θ(y, x)) = HC(y) + ε ρM (x) + locally constant.

Here, ρM is the function from (3.1), and HC is a function on the base such that

HC(y) = δ re(y) outside a compact subset.

• For p 0,

(4.35) H(Ψ(p, q, x)) = δ re(ψ(p, q)) +HM (x) + locally constant.

Here, HM is a function on the fibre such that HM (x) = ε ρM (x) near ∂M .

As before, it follows that HM must be invariant under µ (in practice, we will achieve this

by setting HM = 0 on a large compact subset of M \ ∂M , outside which µ is the identity).

Note also that for such an H, the Hamiltonian vector field XH satisfies

(4.36) Dπ(XH) = δ i∂y

outside a compact subset. Moreover, near the boundary

(4.37) Θ∗XH = XHC + εRM ,

which implies that XH is boundary-parallel. Both these facts together show that the Hamil-

tonian flow of H exists for all time.

Setup 4.11. We use compatible almost complex structures J on E of the following kind:

• At points (y, x) with x sufficiently close to ∂M ,

(4.38) (Θ∗J) = JC × JM,y.

Here, JC can be any almost complex structure on C which is standard outside a

compact subset; and each JM,y is an almost complex structure on M satisfying (3.4).


r = re(y)

J+M,r

J−M,r

monodromy µ

Figure 1.

• At points (p, q, x), where p 0 and −1/2 ≤ q ≤ 1/2,

(4.39) Ψ∗J =

i× J+

M,re(ψ(p,q)) for 0 ≤ q ≤ 1/2,

i× J−M,re(ψ(p,q)) for −1/2 ≤ q ≤ 0.

Here, J±M,r are two auxiliary families of compatible almost complex structures on the

fibre, both parametrized by r ∈ R and belonging to the class (3.4), and such that

moreover,

(4.40)

J−M,r = J+

M,r if r 0,

J−M,r = µ∗J+M,r if r 0.

This implies that outside a compact subset, π : E → C is J-holomorphic. Moreover, because

of (4.39), the induced almost complex structures on the fibres Ey, |y| 0, are locally

constant under translation in imaginary direction (this is is sketched in Figure 1).

(4c) Cauchy-Riemann equations (local behaviour). We begin with some considera-

tions that are local on the Riemann surface level, introducing the relevant data and some

properties of the associated Cauchy-Riemann equation. Fix constants (δ, ε).

Setup 4.12. Let O ⊂ R× R+ be an open subset of the upper half plane, which comes with

a real-valued function bO which is constant on the boundary ∂O = O ∩ (R × 0). We write

βO = dbO. Suppose that the following additional data are given:

• A Lagrangian submanifold L ⊂ E as in Setup 4.9 (this is necessary only if O inter-

sects the real axis).

• A family J = (Js,t)(s,t)∈O of almost complex structures, as in Setup 4.11.

30 PAUL SEIDEL

• Two functions on O × E, denoted by K(∂s) and K(∂t). These should have the

following properties. For each (s, t) ∈ O, the restriction of K(∂s) to (s, t) × E

belongs to the class of functions from Setup 4.10, but where the relevant constants

are (δ βO(∂s), ε βO(∂s)). The same is true for K(∂t) and βO(∂t). Finally, K(∂s)

restricted to (s, 0)× L is zero.

We think of K(∂s) as a family of functions on E parametrized by O, so that there is an

associated family of Hamiltonian vector fields, denoted by YK(∂s). The same applies to

K(∂t), with corresponding YK(∂t). One considers maps u : O → E satisfying

(4.41)

Js,t(u)

(∂su− YK(∂s)

)= ∂tu− YK(∂t),

u(s, 0) ∈ L.

We will discuss some properties of solutions to this equation.

Lemma 4.13. Take a point (s, t) such that u(s, t) lies sufficiently close to ∂E (this is

automatically an interior point). Using (4.25), write u(s, t) = Θ(w(s, t), v(s, t)) locally near

that point. Then the function ρv(s, t) = ρM (v(s, t)) is (weakly) subharmonic at (s, t).

Proof. Locally near such a point, (4.41) decouples into inhomogeneous ∂-equations for w

and v, of which the second one has the form

(4.42) JM,s,t(v)(∂sv − βO(∂s)RM ) = ∂tv − βO(∂t)RM ,

with ρM as in (3.10). As in the proof of Lemma 3.2, writing v(s, t) = φbO(s,t)ρM (v(s, t)) reduces

(4.42) to a homogeneous Cauchy-Riemann equation.

Lemma 4.14. Take a point (s, t) such that |π(u(s, t))| is sufficiently large. Then locally

near that point, the real part of w = π(u) is harmonic (with Neumann boundary conditions,

if t = 0).

Proof. Near such a point u(s, t), the map π is holomorphic, with respect to Js,t and the

standard complex structure on the base π. Moreover, under π, the Hamiltonian vector field

of K(∂s) projects to εβO(∂s)i∂y, and correspondingly for K(∂t). Hence,

(4.43) i(∂sw − εβO(∂s)i) = ∂tw − εβO(∂t)i.

If we write w(s, t) = w(s, t) + iε bO(s, t), (4.43) says that w is holomorphic. In particular,

re(w) = re(w) is harmonic. At a boundary point (s, 0), we have βO(∂s) = 0 by assumption,

hence

(4.44) ∂tre(w) = re(∂tw − ε βO(∂t)i) = −im(∂sw) = 0,

where the last step used (4.32).

The geometric energy of a solution of (4.41) is defined as

(4.45) Egeom(u) =

∫O

|∂su− YK(∂s)|2 =

∫O

|∂tu− YK(∂t)|2.

Lemma 4.15. Consider a sequence of solutions uk of (4.41), such that:


• The energies Egeom(uk) are (finite and) bounded.

• The image of each uk is contained in the region |re(π(x))| ≤ B (for a constant B

which is independent of k).

• There are points zk ∈ O such that ‖duk‖ ≤ C|duk(zk)|+D (for constants C,D which

are independent of k), and which converge to some z∞ ∈ O as k →∞.

Then ‖duk‖ must be bounded. As a consequence, a subsequence of the (uk) converges on

compact subsets.

Proof. This is a simple Gromov compactness argument, following [46, Proposition 5.1], and

we will only give a limited amount of details. If the result is false, we may assume (after

passing to a subsequence) that Ck = |duk(zk)| → ∞. Consider the rescaled maps

(4.46) uk(z) = uk(zk + Ckz),

which satisfy ‖duk‖ ≤ ‖duk‖/|duk(zk)| ≤ C + D/|duk(zk)|. If the values uk(0) = uk(zk)

remain inside a compact subset of E, then a subsequence of uk converges to a non-constant,

finite energy Jz∞ -holomorphic map

(4.47)

u∞ : C −→ E if im(z∞) > 0, or

u∞ : (R× R+,R× 0) −→ (E,L) if im(z∞) = 0.

In either case, the image of u∞ is contained in the same region |re(π(x))| ≤ B as the

original uk. That region (with the metric induced by ωE and Jz∞) has bounded geometry.

Hence, the removable singularities theorem applies, showing that u∞ extends to a pseudo-

holomorphic sphere or disc, which are both impossible because of the exactness assumptions.

In the other situation, where uk(zk) does not remain inside a compact subset, one can

assume after passing to a subsequence that im(π(uk(zk)))→ ±∞. Then, after a translation

in imaginary direction over the base, the same argument as before applies (it is here that we

use the specific conditions on the almost complex structures indicated in Figure 1).

(4d) Floer cohomology groups. Let φ be an automorphism as in Setup 4.8. Choose

δ 6= 0 and an ε satisfying (3.5). Take a family H = (Ht) of Hamiltonian functions as in

Setup 4.10, and a family J = (Jt) of almost complex structures as in Setup 4.11, with the

periodicity property (3.6). From (4.36) and (4.37), it follows that:

Lemma 4.16. All solutions of (3.8) are contained in a compact subset of E \ ∂E.

In fact, a slightly stronger statement holds. If we think of solutions of (3.8) as fixed points

of φ1H φ, then there are no asymptotic solutions either: this means that outside a compact

subset of E \ ∂E, there is a constant c > 0 such that

(4.48) dist(x, (φ1H φ)(x)) ≥ c.

We have not specified the metric to be used in this statement, but it will hold for the

Riemannian metric obtained from any almost complex structure in our class (Setup 4.11).

32 PAUL SEIDEL

Lemma 4.17. One can choose H so that the solutions of (3.8) are nondegenerate.

To see that, one fixes some initial choice of H. One can find an open subset V ⊂ E such that

V is compact and disjoint from the boundary, and such that any solution of (3.8) satisfies

x(1/2) ∈ V . On the other hand, H can be perturbed arbitrarily on V , while remaining inside

our desired class of functions. This easily leads to the desired result; we omit the details.

Assume from now on that the nondegeneracy condition from Lemma 4.17 is satisfied. We

can then define the Z/2-graded K-vector space CF ∗(φ,H) as before.

Lemma 4.18. All solutions of (3.9) are contained in a compact subset of E \ ∂E.

Proof. Floer’s equation (3.9) is locally (on the domain) of the form (4.41) with bO = dt ,

K(∂s) = 0, K(∂t) = Ht. Consider a solution u. Since its limits are contained in the interior

of E, it follows from Lemma 4.13 that the same is true for all of u. In fact, this argument

provides a neighbourhoof of ∂E which no u can enter. Similarly, Lemma 4.14 gives a bound

|re(π(u(s, t)))| ≤ B which is uniform for all u. It remains to bound |im(π(u(s, t)))|. The

following observation, which is a consequence of (4.48), will be useful:

(4.49)There is a compact subset V ⊂ E \∂E, and a constant γ > 0, such that if u(s, 1)

lies outside that subset, then∫ 1

0|∂su(s, t)| dt ≥ γ.

Assume that there is a sequence (uk) of solutions, such that the maximal value of |im(π(uk))|goes to infinity. Because of the general exactness assumptions, we have a uniform upper

bound on the energy of uk. Take

(4.50) Ck = max|∂suk(s, t)| : (s, t) ∈ R× [0, 1]

.

After passing to a subsequence, and applying translations in s-direction, we may assume that

the points in R × [0, 1] where the maximum (4.50) is achieved form a convergent sequence.

Taking O = R× (−σ, 1 + σ) for some σ > 0, one can apply Lemma 4.15, which shows that

Ck is bounded. But then, as k →∞ there must be larger and larger intervals in s where the

condition from (4.49) applies. This contradicts the a priori bound on the energy.

Since J can be arbitrarily perturbed on any compact subset of E \ ∂E, it is easy to show

that for a suitable choice, the moduli spaces of Floer trajectories are regular (compare

Lemma 4.17). As usual, one uses them to define the Floer differential dJ . The fact that the

Floer cohomology HF ∗(φ, δ, ε) is independent of (H,J) can be shown by a continuation map

argument [32], with supplemental analytic details which are parallel to the ones above.

For the second version of Floer theory, we take two Lagrangian submanifolds L0 and L1 as

in Setup 4.9, which at infinity are fibered over half-lines (4.32) with imaginary coordinates

λ0 and λ1, respectively. We choose an arbitrary ε, and a δ satisfying (4.7). Take families

(Ht) and (Jt), this time parametrized by t ∈ [0, 1]. The counterpart of (3.7) on LL0,L1=

x : [0, 1]→ E : x(0) ∈ L0, x(1) ∈ L1 is

(4.51) AL0,L1,H(x) = −∫ 1

0x∗θE +Ht(x(t)) dt +GL1

(x(1))−GL0(x(0)),


where the GLk are functions such that dGLk = θE |Lk. The critical points are solutions of

(4.52)

x(0) ∈ L0, x(1) ∈ L1,

dx/dt = XH,t,

hence correspond to points of φ1H(L0) ∩ L1. The assumption (4.7) implies that all of them

are contained in a compact subset of E \ ∂E (which is the analogue of Lemma 4.16). The

appropriate version of Floer’s equation, for maps u : R× [0, 1]→ E, is

(4.53)

u(s, 0) ∈ L0, u(s, 1) ∈ L1,

∂su+ Jt(∂tu−XH,t) = 0,

lims→±∞ u(s, t) = x±(t).

The (simpler) analogue of Lemma 4.18 follows from [46, Proposition 5.1]. Using that as the

main ingredient, one defines the Floer cochain space CF ∗(L0, L1, H) with its differential dJ .

(4e) Independence of ε. Choose a symplectic automorphism φ as in Setup 4.8, and (δ, ε)

as required to define HF ∗(φ, δ, ε). We enlarge the given E by attaching a conical piece to

the boundary of each fibre, following the model of (3.18):

(4.54) E = E ∪∂E (C× [0, C]× ∂M),

where ∂E is identified with C× ∂M using (4.25). Extend the given φ to an automorphism

φ of E by setting it equal to φC × id [0,C]×∂M on the cone, where φC is as in (4.30). Along

the same lines as in Section 3b, it is straightforward to show that the Floer cohomology of

φ is the same as that of the original φ.

What we are interested in is the analogue of (3.22), for ε > ε. For simplicity, we assume

that the function Gφ appearing in (3.2) vanishes near ∂E. We can then extend it trivially

to a function Gφ on E. To define the Floer cohomology of φ, we used a time-dependent

Hamiltonian H, and we also assume that this satisfies (4.34) with no locally constant term

(both assumptions can be removed, at the cost of making the formulae later on slightly more

complicated). Choose a function f as in (3.19), and extend H to E by the fibrewise analogue

of (3.20):

(4.55) Ht(y, r, x) = HC,t(y) +

εer (y, r, x) ∈ C× [0, C − 1]× ∂M,

eC−1f(er+C−1) (y, r, x) ∈ C× [C − 1, C]× ∂M.

The solutions of (3.8) which do not lie inside the original E are of the form

(4.56) x(t) = (xC(t), C − 1 + log(a), x∂M (t)) ∈ C× [C − 1, C]× ∂M.

Here, xC is a solution of the corresponding equation for (φC, HC); and x∂M is a 1-periodic

orbit of f ′(a) times the Reeb vector field, where a ∈ [1, e]. The appropriate version of (3.21)

is

(4.57) Aφ,H(x) = AφC,HC(xC) + eC−1(−af ′(a) + f(a)).

As in the previously discussed situation (3.22), it follows that for large C, we get a long

exact sequence

(4.58) · · · → HF ∗(φ, δ, ε) −→ HF ∗(φ, δ, ε) −→ Q∗ → · · ·

34 PAUL SEIDEL

Consider the case when there is only one Reeb period in the interval (ε, ε), and let’s tem-

porarily ignore transversality issues. From (4.57) one gets a bound (3.23), which shows that

computations of Q∗ can be done entirely inside the subset C×[0, C]×∂M . By an appropriate

choice of almost complex structure, one can achieve that for solutions remaining inside that

subset, (3.9) decouples into equations for the components

(4.59)

v : R2 −→ C,

w : R2 −→ [0, C]× ∂M.

Correspondingly, the last term in (4.58) can be written as

(4.60) Q∗ ∼= Q∗C ⊗Q∗M ,

where the first factor is the Floer cohomology of the automorphism φC perturbed by the

Hamiltonian term HC. Recall that φC is a symplectic automorphism which is the identity

outside a compact subset. Hence (by Moser’s Lemma and the contractibility of the group

of compactly supported diffeomorphisms of the plane) we can deform it to the identity by

a compactly supported Hamiltonian isotopy; and this leaves Floer cohomology unchanged.

Similarly, we may replace the given HC by the linear function δ re(y). But since that has no

critical value, the Floer cohomology of idC perturbed by that function is zero. This shows

that Q∗C = 0, hence by (4.60) that Q∗ = 0. To complete the argument, we have to return to

the issue of transversality. One can achieve nondegeneracy of x by perturbing H within the

class of functions that are of the form HC,t(y) + HM,t(r, x) on the cone. Similarly, regularity

of solutions to Floer’s equation (inside C × [0, C] × ∂M) can be achieved within the class

of almost complex structures that respect the decomposition into the two factors C and

R× ∂M . This establishes (4.60).

Finally, the case where there is more than one Reeb period can be reduced to the previous

one by a slight perturbation of the contact one-form (which makes the set of Reeb periods

discrete) and working step-by-step. The outcome is:

Proposition 4.19. For any ε < ε satisfying (3.5), we have HF ∗(φ, δ, ε) ∼= HF ∗(φ, δ, ε).

(4f) Cauchy-Riemann equations (global behaviour). As usual, our setup involves a

specific class of Riemann surfaces. Compared to similar frameworks in the literature [42, 5,

46], the main difference is that we allow nontrivial monodromy around interior punctures

(an early example of an exposition with the same feature is [34, 44]). In other respects, the

exposition here is actually somewhat restrictive, since we want to keep the local situation as

simple as in Section 4c.

Setup 4.20. Fix a fibration π : E → C as in Setup 4.7, along with constants (ε, δ). A

worldsheet is a connected non-compact Riemann surface S, possibly with boundary, with the

following additional data:

• We assume that there is a compactification S = S ∪ Σ (to a Riemann surface with

boundary), obtained by adding a finite set of points. We divide the points of Σ into

closed string (interior) and open string (boundary) ones, and (independently, and

arbitrarily) into inputs and outputs, denoting the respective subsets by Σcl/op,in/out .


These should come with local holomorphic coordinates (tubular and strip-like ends),

of the form

(4.61)

εζ : (−∞, 0]× S1 −→ S, ζ ∈ Σcl,out ,

εζ : [0,∞)× S1 −→ S, ζ ∈ Σcl,in ,

εζ : (−∞, 0]× [0, 1] −→ S, ζ ∈ Σop,out ,

εζ : [0,∞)× [0, 1] −→ S, ζ ∈ Σop,in .

Let S → S be the universal covering. We fix lifts of (4.61), of the form

(4.62)

εζ : (−∞, 0]× R −→ S, ζ ∈ Σcl,out ,

εζ : [0,∞)× R −→ S, ζ ∈ Σcl,in ,

εζ : (−∞, 0]× [0, 1] −→ S, ζ ∈ Σop,out ,

εζ : [0,∞)× [0, 1] −→ S, ζ ∈ Σop,in .

• Write Γ ∼= π1(S) for the covering group of S → S. We want to have a homomorphism

(4.63) Φ : Γ −→ Symp(E),

which takes values in the subgroup of those symplectic automorphisms of E described

in Setup 4.8. To each point z ∈ ∂S we want to associate a Lagrangian submanifold

Lz as in Setup 4.9, in a way which is locally constant in z, and compatible with

(4.63):

(4.64) Lγ(z) = Φ(γ)(Lz).

An immediate consequence is this: given ζ ∈ Σop, we have distinguished Lagrangian

submanifolds (Lζ,0, Lζ,1) which are associated to the points (εζ(s, 0), εζ(s, 1)). By

definition, these come with real numbers (4.5), which we denote by (λζ,0, λζ,1). Sim-

ilarly, given ζ ∈ Σcl , passing from εζ(s, t + 1) to εζ(s, t) amounts to acting by an

element of Γ, which determines an automorphism φζ by (4.63).

• We also want S to carry a closed one-form βS with βS |∂S = 0, and which over each

end (4.61) satisfies

(4.65) ε∗ζβS = βζdt ,

where the βζ are constants. Set δζ = βζδ, εζ = βζε. We require that for ζ ∈ Σcl ,

εζ should satisfy (3.5), and δζ 6= 0; while for ζ ∈ Σop, the pair (Lζ,0, Lζ,1) and the

number δζ must satisfy (4.7).

Setup 4.21. A perturbation datum (JS ,KS) on S consists of the following:

• A family JS = (JS,z) of almost complex structures on E (as in Setup 4.11), para-

metrized by z ∈ S. These should be equivariant with respect to (4.63), and over

the strip-like ends they should be invariant under translation in the first variable.

Concretely, this means that there are families Jζ = (Jζ,t) such that

(4.66) JS,εζ(s,t) = Jζ,t

36 PAUL SEIDEL

If ζ ∈ Σcl , the associated family (4.66) satisfies a periodicity condition as in (3.6)

with respect to φζ .

• An inhomogeneous term KS, which is a one-form on S with values in the space

of functions on E. More precisely, for any tangent vector ξ, KS(ξ) ∈ C∞(E,R)

belongs to the class from Setup 4.10 for the constants (βS(ξ)δ, βS(ξ)ε). This fam-

ily must be equivariant with respect to (4.63), and if ξ is tangent to ∂S, then

KS(ξ)|Lz = 0. Finally, over each end, ε∗ζKS is invariant under s-translation, and

satisfies (ε∗ζKS)(∂s) = 0. In analogy with (4.66), this can be expressed as

(4.67) ε∗ζKS = Hζ,t dt ,

and if ζ ∈ Σcl , we again have (3.6).

Consider a map u : S → E which is equivariant with respect to the Γ-action on both spaces,

and which satisfies:

(4.68)

u(z) ∈ Lz for z ∈ ∂S,

(du− YKS )0,1 = 0.

Here, YKS is the one-form with values in C∞(TE) associated to KS , and we use the almost

complex structures JS,z to form the (0, 1)-part. Note that by construction, the equation

(4.68) is invariant under the Γ-action. The corresponding equation over the ends, which

means for uζ(s, t) = u(εζ(s, t)), is that

(4.69)

uζ(s, k) ∈ Lζ,k for k = 0, 1, if ζ ∈ Σop ,

uζ(s, t) = φζ(uζ(s, t+ 1)) if ζ ∈ Σcl ,

∂suζ + Jζ,t(∂tuζ −XHζ ,t) = 0.

It therefore makes sense to impose convergence conditions

(4.70) lims→±∞ uζ(s, t) = xζ(t).

The necessary compactness argument for the moduli space of solutions of (4.68) combines:

an overall energy bound; a maximum principle argument in horizontal direction over the base

(Lemma 4.14); and a translation argument in vertical direction (Lemma 4.15). This runs

parallel to the case of Floer cohomology (Lemma 4.18). Transversality is straightforward,

given the freedom to choose JS and KS . We omit the details.


Counting solutions of (4.68), (4.70) for a single Riemann surface S yields a map between

Floer cochain spaces, which induces a cohomology level map

(4.71)⊗

ζ∈Σin,op

HF ∗(Lζ,0, Lζ,1, δζ , εζ)⊗⊗

ζ∈Σin,cl

HF ∗(φζ , δζ , εζ)

⊗ζ∈Σout,op

HF ∗(Lζ,0, Lζ,1, δζ , εζ)⊗⊗

ζ∈Σout,cl

HF ∗(φζ , δζ , εζ),

of degree n(−χ(S) + |Σout,op |+ 2|Σout,cl |) (as usual, this formula for the degree holds in Z/2in general, and in Z if we have integer gradings). This is a version of the TQFT formalism

for Floer theory. For us, what is relevant is the TCFT extension, which includes families of

Riemann surfaces. Besides the gluing theory (which we regard as well-known), this has no

significantly new analytic features. We will therefore only give an outline of the construction,

focusing on what the relevant families of Riemann surfaces are in each case.

(4g) Moduli spaces. Suppose that we have (L1, . . . , Lm) with pairwise different con-

stants (λ1, . . . , λm). We choose the data necessary to define the Floer cochain complexes

CF ∗(Li, Lj , Hij) for i < j. The directed A∞-structure is obtained by considering world-

sheets S which are discs with d + 1 ≥ 3 boundary punctures. We equip the components of

∂S with boundary conditions Li0 , . . . , Lid , for i0 < · · · < id (in positive order around ∂S).

The one-forms βS are taken to be zero throughout. The data (JS ,KS) need to be chosen

so that their restriction to the strip-like ends, as in (4.66) and (4.67), reduces to the choices

previously made to define Floer theory. Finally, the choices need to depend smoothly on the

moduli of the Riemann surface, and there are conditions about their behaviour as the surface

degenerates. We will not give details, referring instead to [43, Section 9]. The outcome are

A∞-operations

(4.72) CF ∗(Lid−1, Lid , Hid−1id)⊗ · · · ⊗ CF ∗(Li0 , Li1 , Hi0i1) −→ CF ∗+2−d(Li0 , Lid , Hi0id),

which satisfy the A∞-associativity equations. Setting R = Km, there is a unique extension

of these operations to a strictly unital A∞-structure on

(4.73) A = R⊕⊕i<j

CF ∗(Li, Lj , Hij),

which is the structure we have previously described on the cohomology level (4.10).

Fix an automorphism φ. To define P = Pφ,δ,ε, one chooses Hamiltonians Hφ,ij and almost

complex structures Jφ,ij for all (i, j), and then sets

(4.74) P =⊕ij

CF ∗(φ(Li), Lj , Hφ,ij),

with µ0;1;0P the direct sum of Floer differentials. Fundamentally, the moduli spaces of Rie-

mann surfaces which define µs;1;rP are the same (Stasheff polyhedra) as for µr+1+s

A . However,

38 PAUL SEIDEL

φ

Li0

Li′0

Li1 Li2

Li′3 Li′2 Li′1

Figure 2.

the other data they carry are different, and we find it convenient to think of the Riemann

surfaces themselves in a slightly different way, namely to write them as

(4.75) S = Z \ ζ1, . . . , ζr, ζ ′1, . . . , ζ ′s,

where: Z = R× [0, 1]; the ζi ∈ R× 0 ⊂ ∂Z are increasing; and the ζ ′i ∈ R× 1 ⊂ ∂Z are

decreasing. The one-forms βS should be equal to dt on the region where |s| 0, and should

vanish sufficiently close to the ζi and ζ ′i. The boundary conditions are

(4.76) (φ(Li0), . . . , φ(Lir )) along R× 0

(as s increases, and where i0 < · · · < ir), respectively

(4.77) (Li′0 , . . . , Li′s) along R× 1

(as s decreases, and where again i′0 < · · · < i′s). The inhomogeneous terms and almost

complex structures are determined by those chosen for (4.74) over the ends where |s| 0,

and by the choices made for A on the remaining ends. As we vary over all possible S, the

outcome are operations

(4.78)

CF ∗(Li′s−1, Li′s , Hi′s−1i

′s)⊗ · · · ⊗ CF ∗(Li′0 , Li′1 , Hi′0i

′1)

⊗CF ∗(φ(Lir ), Li′0 , Hφ,iri′0)⊗

CF ∗(Lir−1, Lir , Hir−1ir )⊗ · · · ⊗ CF ∗(Li0 , Li1 , Hi0i1)y

CF ∗+1−r−s(φ(Li0), Li′s , Hφ,i0i′s),

which together constitute µs;1;rP . It may be convenient to think of (4.75) as glued together

from two half-strips S± = (s, t) ∈ S : ±t ≤ 12, each of which carries boundary conditions

taken from the Lk, but where the gluing identifies the two target spaces E using φ. This

amounts to rewriting the relevant equation (4.68) as two parts u± : S± → E, joined by

u+(s, 12 ) = φ(u−(s, 1

2 )). Mathematically, this does not change anything, but it can be useful

as an aid to the intuition (Figure 2).

Now consider surfaces of the form

(4.79) S = H \ ζ∗, ζ1, . . . , ζd

where H ⊂ C is the closed upper half-plane, from which we remove a fixed interior point

ζ∗ (say ζ∗ = i) as well as boundary points ζ1 < · · · < ζd. The one-form βS should vanish


Li1

Li0

Li2

φ

Li0 Li1 Li2

φ

Figure 3.

near the ζk, and its integral along a small (counterclockwise) loop around ζ∗ should be equal

to 1. The representation (4.63) maps that same loop to φ (Figure 3 shows two equivalent

pictures of this; in the right one, we have drawn H itself as a once-punctured disc). Over

∂H ⊂ H \ ζ∗, we can choose a section of the universal cover. Along that section we place

the boundary conditions Li0 , . . . , Lid with i0 < · · · < id, and then extend that to all of ∂S

in the unique way which satisfies (4.64). The outcome of this construction are maps

(4.80)

CF ∗(Lid−1, Lid , Hid−1id)⊗ · · · ⊗ CF ∗(Li0 , Li1 , Hi0i1)⊗ CF ∗(φ(Lid), Li0 , Hφ,idi0)y

CF ∗+n−d(φ,Hφ),

whereHφ is a new Hamiltonian, chosen (together with a corresponding Jφ) to form HF ∗(φ, δ, ε).

The (4.80) are the components of a chain map

(4.81) CC ∗(A,Pφ,δ,ε) −→ CF ∗+n(φ,Hφ),

which induces (4.14).

The construction of (4.19) is a mixture of the previous two. One considers surfaces

(4.82) S = Z \ ζ∗, ζ1, . . . , ζr, ζ ′1, . . . , ζ ′s,

where ζ∗ is a fixed interior point of Z, say ζ∗ = (0, 1/2), and the other points are as in

(4.75). Note however that this time, both ends s → ±∞ will be considered as inputs. The

map (4.63) has monodromy φ = φ1φ0 around ζ∗. This structure, as well as the choice of

boundary conditions, is indicated in Figure 4. One chooses one-forms βS such that

(4.83) βS =

−(ε0/ε) dt s 0,

(ε1/ε) dt s 0,

40 PAUL SEIDEL

Li0

Li′0

Li1 Li2

Li′3 Li′2 Li′1

φ1

φ0

Figure 4.

and which vanish near the ζk, ζ′k. As a consequence, the integral of βS along a loop around

ζ∗ is necessarily ε0/ε+ ε1/ε = 1. The outcome are maps

(4.84)

CF ∗(Li′s−1, Li′s , Hi′s−1i

′s)⊗ · · · ⊗ CF ∗(Li′0 , Li′1 , Hi′0i

′1)

⊗CF ∗(φ1(Lir ), Li′0 , Hφ0,iri′0)⊗

CF ∗(Lir−1 , Lir , Hir−1ir )⊗ · · · ⊗ CF ∗(Li0 , Li1 , Hi0i1)

⊗CF ∗(φ0(Li′s), Li0 , Hφ1,is′ i0)yCF ∗+n−r−s(φ,Hφ).

These are the components of the chain map underlying (4.19), which is analogous to (4.81):

(4.85) CC ∗(A,P1 ⊗A P0) −→ CF ∗+n(φ,Hφ).

5. Floer cohomology: computations

Having set up the general theory, we now restrict attention to two particularly simple exam-

ples of automorphisms, namely the global monodromy and its square, and partially compute

their Floer cohomology. The first computation recovers a version of the well-known con-

nection between monodromy and the Serre functor. The second computation is the main

ingredient in proving Theorem 1.2.

(5a) The strategy. We’ll start with the last-mentioned argument, since that makes a more

significant contribution to the overall discussion. In the situation of Setup 4.7, we consider

a particular automorphism φ of E obtained by rotating the part at infinity of the base twice

(the square of the global monodromy). We choose a small δ and ε (nonzero, but of arbitrary

sign for now). For suitable choices of the auxiliary data (H, J), the Floer cochain complex

has a two-step filtration

(5.1) CF ∗(φ, H) ⊃ F 1 ⊃ F 2 ⊃ 0


which satisfies

H∗(F 2) ∼= H∗(E;K) or H∗(E, ∂E;K), depending on the sign of ε,(5.2)

H∗(F 1/F 2) ∼= HF ∗+2(µ, ε)⊗H∗(S1;K).(5.3)

Here, µ is the monodromy on M . One can think of this as the initial part of the spectral

sequence from [29] for the symplectic cohomology of E (in fact, H∗(CF ∗(φ, H)/F 1) is the

Floer cohomology of µ2, but we will not need that). There is an important additional

piece of information. Namely, one can equip F 1 with a BV operator, which is a chain map

of degree −1, respecting the subspace F 2. The induced map on (5.2) vanishes; whereas

the induced map on (5.3) has the property that its image projects isomorphically to (the

Kunneth direct summand) HF ∗+2(µ, ε) = HF ∗+2(µ, ε) ⊗H0(S1;K). By inverting the last-

mentioned projection, we get an injective map HF ∗+2(µ, ε) → H∗(F 1/F 2). Because of the

compatibility between BV operator and differential, that map admits a lift to H∗(F 1). We

have therefore constructed a map

(5.4) HF ∗+2(µ, ε) −→ H∗(F 1) −→ HF ∗(φ, ε, δ).

A simpler version of this discussion applies to the global monodromy itself, which is the

automorphism φ obtained by rotating the part at infinity of the base only once. Take δ

and ε small, but now assuming that ε > 0. Again after choosing (H,J) appropriately, the

counterpart of (5.1) is a subcomplex

(5.5) CF ∗(φ,H) ⊃ F 1 ⊃ 0.

For the same reason as in (5.2),

(5.6) H∗(F 1) ∼= H∗(E;K).

In particular, the class 1 ∈ H0(E;K) yields a preferred element

(5.7) e ∈ HF 0(φ, δ, ε).

We will be interested in its image under

(5.8) HF ∗(φ, δ, ε) −→ HH n−∗(A,Pφ−1,−δ,−ε)∨ ∼= H∗−n(hom [A,A](Pφ−1,−δ,−ε,A

∨)).

Here, the first map is the dual of (4.14) (with (4.4) built in), and the second map is the general

isomorphism (2.14). We will show later (see Section 5d for the precise setup, and Lemma

5.10 for the statement) that in the framework of Lefschetz fibrations and bases of Lefschetz

thimbles, and with δ < 0, the image of e under (5.8) gives rise to a quasi-isomorphism

(5.9) Pφ−1,−δ,−ε'−→ A∨[−n].

Remark 5.1. In fact, H∗(CF ∗(φ,H)/F 1) ∼= HF ∗+1(µ, ε). As a consequence, one gets a

long exact sequence

(5.10) · · · → H∗(E;K)→ HF ∗(φ, δ, ε)→ HF ∗+1(µ, ε)→ · · ·

By (5.9), the target of (5.8) is the Hochschild cohomology of A,

(5.11) H∗−n(hom [A,A](Pφ−1,−δ,−ε,A∨)) ∼= H∗−n(hom [A,A](A

∨[−n],A∨)) ∼= HH ∗(A,A).

42 PAUL SEIDEL

One expects (5.8) to be an isomorphism, in which case (5.10) would provide a natural expla-

nation for [37, Conjecture 6.1] (there are two unpublished proofs of statements closely related

to that conjecture, by Perutz and Ma’u).

Finally, let’s return to the original φ, which by definition is (or is at least isotopic to,

depending on the particular choices made) φ2. We adjust the parameters so that (δ, ε) =

(2δ, 2ε). Using (4.19), (5.9), and Lemma 2.3, one defines a map

(5.12)HF ∗(φ, δ, ε) −→ HH n−∗(A,Pφ−1,−δ,−ε ⊗A Pφ−1,−δ,−ε)

∨

∼= HH−n−∗(A,A∨ ⊗A A∨)∨ ∼= H∗(hom [A,A](A

∨[−n],A)).

The composition of (5.4) and (5.12) yields Theorem 1.2.

(5b) A simplified setting. We temporarily switch to a topologically simpler situation,

namely that of a locally trivial symplectic fibration whose base is the cylinder T , with

ωT = dp ∧ dq and θT = p dq , in coordinates (p, q) ∈ R× S1 = T (the following discussion is

very close to that in [29, pp. 486–489]).

Setup 5.2. Take a Liouville domain M and an exact symplectic automorphism µ, with

µ∗θM = θM + dGµ, and which is the identity near the boundary. Choose g ∈ C∞(R×M,R)

such that g(q + 1, x) = g(q, µ(x))−Gµ(x) for all (q, x). Consider

(5.13) E = R2 ×M/(p, q + 1, x) ∼ (p, q, µ(x))π=(p,q)−−−−−→ T.

We equip this with the symplectic form ωE = ωT +ωM and its primitive θE = θT + θM + dg.

Choose f ∈ C∞(R,R) such that

(5.14)

f ′(p) = q− ∈ R \ Z if p 0,

f ′(p) = q+ ∈ R \ Z if p 0,

f ′′(p) ≥ 0 everywhere,

f ′′(p) > 0 at those points where f ′(p) ∈ Z.

Pull this back via (5.13) to a function F on E, and let φ = φ1F be the time-one map of its

Hamiltonian flow. Concretely,

(5.15) φ(p, q, x) = (p, q + f ′(p), x) ∼ (p, q + f ′(p)− k, µk(x)).

In particular, outside a compact subset, this covers a fixed point free rotation of the cylinder.

Choose ε > 0 satisfying (3.5), as well as a function HM which agrees with ερM near the

boundary, and which is invariant under µ. To define the Floer cohomology HF ∗(φ, ε) of

(5.15), we use the following auxiliary data.

Setup 5.3. We consider functions H ∈ C∞(E,R) such that:

• Near the boundary ∂E ∼= T × ∂M ,

(5.16) H(p, q, x) = ερM (x) +HT (p, q) + locally constant,

where HT is some compactly supported function on T .


• Outside a compact subset, Ht(p, q, x) = HM (x) + locally constant (this makes sense

because HM (x) = HM (µ(x)) by definition).

Similarly, we consider compatible almost complex structures J on E in the following class:

• Near the boundary,

(5.17) Jp,q,x = JT,p,q × JM,p,q,x

where JT is some compatible almost complex structure on T , standard outside a com-

pact subset; and JM,p,q is a family of almost complex structures on M parametrized

by (p, q) ∈ R2, satisfying JM,p,q+1 = µ∗JM,p,q, and such that (3.4) holds.

• Outside a compact subset, (5.13) is pseudo-holomorphic for J and the standard com-

plex structure on T .

The standard construction based on such choices yields a Floer cohomology group HF ∗(φ, ε),

which moreover carries a BV operator (3.27). The existence of the latter is fundamentally

due to the fact that φ is the time-one map of a Hamiltonian flow, but some attention needs

to be paid to the choices of perturbations, and the behaviour of the associated continuation

map equation. One chooses each function Kr,s,t and Jr,s,t to lie in the general class from

Setup 5.3 (noting that the Hamiltonian flow of F preserves those classes). Then, the same

argument as in Lemma 4.13 prevents solutions of (3.30) from reaching ∂E. To see that

solutions remain inside a compact subset of E, one argues by projection to T , in parallel

with Lemma 4.14.

We need a few topological observations. Since φ is isotopic to the identity, Lφ can be

identified with the ordinary free loop space Map(S1, E). In particular, we get a map

(5.18) π0(Lφ) ∼= π0(Map(S1, E)) −→ π0(Map(S1, T )) = Z.

We denote by L(k)φ the (open and closed) subset of Lφ which corresponds to k ∈ Z under

this map. This induces a splitting

(5.19) HF ∗(φ, ε) =⊕k

HF ∗(φ, ε)(k).

The k-th summand in (5.19) is nonzero only if k ∈ (q−, q+) (if that is the case and ε is small,

HF ∗(φ, ε)(k) the group denoted by HF ∗(φ, k) in [29]). Evaluation at a point induces a map

Lφ → E → T , and by pullback of H1(T ) ∼= Z a canonical class in H1(Lφ;K), which we

denote by c. Let c(k) be its restriction to H1(L(k)φ ;K). Concretely, this is Poincare dual to

the hypersurface of those x ∈ L(k)φ such that π(x(0)) ∈ R × 0 ⊂ T . The map induced by

loop rotation (3.26) satisfies

(5.20) ∆class(c(k)) = k · 1 ∈ H0(L(k)φ ;K).

Floer cohomology carries the structure of a module over H∗(Lφ;K) by (3.35). This, as well

as the BV operator, are compatible with (5.19).

44 PAUL SEIDEL

Lemma 5.4. If 1 ∈ (q−, q+), there is an isomorphism

(5.21) HF ∗(φ, ε)(1) ∼= HF ∗(µ, ε)⊗H∗(S1;K).

Under this isomorphism, the action (3.35) of c(1) corresponds to taking the cup product with

the generator of H1(S1;K). Moreover, the image of the BV operator on HF ∗(φ, ε)(1) projects

isomorphically to the Kunneth summand HF ∗(µ, ε)⊗H0(S1;K).

Proof. The isomorphism (5.21) is [29, Theorem 1.3], but we include an outline of the argu-

ment, to make the exposition more self-contained. For simplicity, suppose that the constants

in (5.14) satisfy q− ∈ (0, 1), q+ ∈ (1, 2), which implies that HF ∗(φ, ε) = HF ∗(φ, ε)(1). Sup-

pose also that

(5.22) φ1HM µ has nondegenerate fixed points.

The general case can be reduced to this one.

When defining the Floer cochain complex CF ∗(φ,H), let’s choose the Hamiltonian to be

Ht(p, q, x) = HM (x), and the almost complex structures so that projection to T is (Jt, jt)-

holomorphic for some jt. The fixed points of φ are of the form (p1, q, x), where p1 is the

unique point where f ′(p1) = 1, q can be arbitrary, and x is a fixed point of (5.22). These

points therefore appear in circles, which are transversally nondegenerate in the Morse-Bott

sense. Moreover, if u : R2 → E is a solution of the associated Floer equation, then v = π(u) :

R2 → T is a solution of

(5.23)

v(s, t) = v(s, t+ 1) + (0, f ′(re(v(s, t+ 1))),

∂sv + JT,t ∂tv = 0,

lims→±∞ v(s, t) = (p1, q±).

By an energy argument, one sees that v must be constant. It follows that all solutions u

correspond to a choice of q together with a Floer trajectory for µ (inside M). By further

restricting Jt, one can achieve that those Floer trajectories are independent of q. Standard

Morse-Bott methods now imply (5.21) (for char(K) 6= 2, this also requires a sign considera-

tion, which we omit here).

Since c is pulled back from evaluation at a point, the cap product is given (roughly speaking)

by counting Floer trajectories u with the additional condition that v(0, 0) ∈ R × 0 ⊂ T .

Using the description of all Floer trajectories which we have just given, one easily checks

that this satisfies the desired property. Finally, the statement about the BV operator follows

from our computation of the cap product and the general properties (3.34), (3.36). Namely,

by (5.20) and (3.36) we have

(5.24) ∆(a · c(1)) = ∆(a) · c(1) + (−1)|a|a.

This shows that the projection from the image of ∆ to HF ∗(µ, ε)⊗H0(S1;K) is surjective.

On the other hand, as a consequence of (3.34), the dimension of the image of ∆ can be at

most half of that of HF ∗(φ, ε)(1), hence the projection must be an isomorphism.


(5c) The filtration argument. We work with an exact symplectic fibration with singu-

larities, π : E → C. To simplify the notation, we assume that the subset U appearing in

Setup 4.7 contains all fibres Ey with |y| ≥ 1, and correspondingly that one takes y∗ = 1. We

define a symplectic automorphism φ, which belongs to the class from Setup 4.8, by setting

φ = id on the preimage of the unit disc in C, as well as

(5.25) φ(Ψ(p, q, x)) = Ψ(p, q + f(p), x)

for a suitable cutoff function:

(5.26)

f(p) = 0 for p ≤ 0,

f(p) = 2 for p 0.

Call φ the global squared monodromy. It is the time-one map of the Hamiltonian flow of the

pullback of the following function on the base:

(5.27) h(p) =∫ p−∞ 4π2e4πrf(r) dr .

In particular, φ is exact. We make the natural choice of associated function Gφ, namely

by setting it equal to zero on the preimage of the unit disc. The fixed points of φ (outside

the preimage of the unit disc) are of the form Ψ(p, q, x) where f(p) = k ∈ Z, and x is a

fixed point of µk. The action of such a point can be written as the sum of fibre and base

contributions:

(5.28) Aφ(Ψ(p, q, x)) = Aµk(x) + a(p),

where

(5.29) a(p) = −∫ p−∞ πe4πrf ′(r)dr .

At this point, we want to be more specific; suppose that in addition to (5.26),

(5.30)

f ′(p) ≥ 0 everywhere,

f ′(p1) > 0 at the point p1 where f(p1) = 1 (this point is then unique).

Then, the fixed points of φ can be listed more concretely as follows:

• All points in the preimage of the unit disc, and all points Ψ(p, q, x) where p is

sufficiently small, so that f(p) = 0, are fixed; by definition, their action is zero.

• If p1 is as in (5.30), then the points Ψ(p1, q, x) where µ(x) = x are fixed. By (5.28),

their action is

(5.31) Aφ(Ψ(p1, q, x)) = Aµ(x) + a(p1).

• Similarly, if p is sufficiently large so that f(p) = 2, and x is a fixed point of µ2, then

Ψ(p, q, x) is a fixed point of φ. The analogue of (5.31) is

(5.32) Aφ(Ψ(p2, q, x)) = Aµ2(x) + a(∞),

46 PAUL SEIDEL

where a(∞) is the value of the function a(p) for p 0 (where it becomes constant). By

a suitable choice of f , one can achieve that all actions in (5.31) are strictly negative, and

that all actions in (5.32) are strictly smaller than those in (5.31). So far, we have not

introduced any Hamiltonian perturbations (and this was reflected by our notation for action

functionals). For a fixed ε, choose a function HM supported in a neighbourhood of ∂M , and

which equals ερM near the boundary. One can extend this in a natural way to a function

on E, whose flow commutes with that of (5.27). Then, all the previous observations remain

true for the perturbed action functional: all one has to do is to replace fixed points of µk by

ones of µk φ1HM

.

Now choose a small δ 6= 0, and introduce a further Hamiltonian perturbation of our function,

in the associated class of functions (Setup 4.10), denoting the result by H. Assuming that

this perturbation is small, the previous observations about actions remain true. In other

words, the generators of CF ∗(φ, H) are divided into three kinds, and this gives rise to the

previously mentioned filtration (5.1). To see (5.2), note that F 2 arises from fixed points

close to the unit ball (whose action is very small after the perturbation). On that region,

one can arrange that φ1H φ is the time-one map of the flow of a small Hamiltonian function,

whose gradient increases in radial direction on the base. Similarly, F 1/F 2 arises from fixed

points close to those in (5.31). The difference in actions between two such points comes from

µk φ1HM

, hence is bounded independently of f . By choosing that function suitably, and

applying a Monotonicity Lemma argument, one can achieve that the differential on F 1/F 2

counts only maps whose image is contained in the part of E where (4.27) applies. This

reduces (5.3) to the situation from Lemma 5.4 (the same purpose is achieved in a different

way in [29], by a “minimum principle argument” [29, Lemma 6.1]).

To complete the construction of (5.4), we have to discuss the BV operator in this context.

Let’s impose the following additional assumption on f :

(5.33) There is a p3/2 such that f(p3/2) = 3/2, and f is constant near that point.

We can choose H so that

(5.34)

Ht(Θ(y, x)) = ερM (x) for x close to ∂M , and |y| ≤ e2πp3/2 ,

Ht(Ψ(p, q, x)) = HM (x) if p is close to p3/2.

This satisfies (4.34) as well as (4.35) (in the latter case, this is because we have only required

the condition to hold over a bounded subset of the base). Similarly, when choosing the

relevant almost complex structures Jt, we can assume that π is Jt-holomorphic (for the

standard complex structure on the base C) at any point (p, q, x) such that p is close to p3/2.

Lemma 5.5. If the endpoints of a Floer trajectory lie in the preimage of |y| ≤ e2πp3/2 in

E, then the entire trajectory will be contained in that subset.


Proof. Take any map

(5.35)

u : R2 −→ E,

u(s, t) = φ(u(s, t+ 1)),

|π(u(s, t))| < e2πp3/2 for |s| 0.

Fix some y ∈ C with |y| = e2πp3/2 . Then, the set of points (s, t) such that u(s, t) = ±y is

closed under (s, t) 7→ (s, t ± 1). Hence, after a small perturbation, it makes sense to count

all such points in R×S1 with appropriate signs. The outcome is a topological invariant of u

(as a map satisfying the conditions above). Since one can always deform u so that its entire

image lies in the unit disc, this invariant is always zero.

Now suppose that u is a Floer trajectory, and let v = π(u) : R2 → C be its projection to

the base. By assumption, v is a holomorphic map near each point where v(s, t) = ±y. For

degree reasons, the vanishing of the invariant above implies that v−1(±y) = ∅.

The importance of (5.34) is that if we restrict to the region from Lemma 5.5, then the flow

which gives rise to φ preserves H near the (fibrewise) boundary, something which would fail

for the whole of E (since the function δ im(y) is definitely not preserved by rotation in the

complex plane). By staying inside that region, one can equip F 1 with a BV operator as

in Section 3c. One can arrange that that operator preserves the action filtration up to an

arbitrarily small error, hence preserves the subspace F 2. Moreover, on that subspace it will

vanish, for essentially the same reason as in Example 3.10. On the other hand, just like in

the case of the Floer differential, the behaviour of the BV operator on F 1/F 2 reduces to the

case considered in Lemma 5.4.

Remark 5.6. Our unability to define a BV operator on the whole of HF ∗(φ, δ, ε) is not

merely due to technical issues. As an indication of its deeper meaning, let’s consider the

algebraic counterpart (1.3), and think of it in the mirror symmetry context of Section 1d,

with line bundle L = K−1X = OX(D). It then corresponds to a twisted Hochschild cohomology

group

(5.36) Ext∗X×X(O∆,O∆(D)),

where ∆ ⊂ X ×X is the diagonal. If we pass to the simpler case when X is affine, (5.36)

reduces to

(5.37) Λ∗TX(D)⊗OX K−1X∼= Ωn−∗X (2D),

which means rational differential forms with a pole of order ≤ 2 along D. This space is not

closed under exterior differentiation (the de Rham operator, which is usually the counterpart

of the BV operator). However, it contains subspaces on which the de Rham operator acts

(for instance, if D = f−1(0), take the subspace of differential forms which can be written as

α/f + β ∧ df/f2, where α, β extend regularly to D).

We quickly summarize the parallel (but much simpler) argument for the global monodromy

φ. This time, we take a function f as in (5.26) but with f(p) = 1 for p 0. The associated

automorphism φ, defined as in (5.25), has two kinds of fixed points: ones lying in the

48 PAUL SEIDEL

l4

l1

l2

l3

Figure 5.

preimage of the unit disc, and the others lying over points ψ(p, q) ∈ C where f(p) = 1. After

a small perturbation, the first kind generate the subspace F 1 from (5.5).

(5d) The Lefschetz condition. The closed string flavour of Floer cohomology does not

need any particular assumptions on the singularities of π. However, to make the open string

flavour effective, we need a source of Lagrangian submanifolds, which is naturally provided

by Lefschetz thimbles. We therefore specialize the situation as follows:

Setup 5.7. Take π : E → C as in Setup 4.7). We call it an exact symplectic Lefschetz

fibration if the following holds:

• At each regular point of π, TEvx = ker(Dπx) ⊂ TEx is a symplectic subspace.

• There are only finitely many critical points.

• There is an (integrable and ωE-compatible) complex structure IE, defined in a neigh-

bourhood of the critical point set, such that π is IE-holomorphic, and the (complex)

Hessian D2πx at each critical point is nondegenerate.

Suppose that we have a Lefschetz fibration. A basis of vanishing paths (see Figure 5) is a

collection of properly embedded half-infinite paths l1, . . . , lm ⊂ C with these properties:

• The endpoints of the lk are precisely the critical values of π.

• There is a constant C such that the parts of the lk lying in the region re(y) ≤ Care pairwise disjoint. Moreover, all the critical values of π lie inside that region.

• With C as before, there are functions ι1, . . . ιm : [C,∞]→ R which are constant near

infinity, such that ι1(C) < · · · < ιm(C), ι1(∞) > · · · > ιm(∞), and

(5.38) lk ∩ re(y) ≥ C = im(y) = ιk(re(y)).

The corresponding basis of Lefschetz thimbles consists of the unique Lagrangian submanifolds

L1, . . . , Lm ⊂ E such that π(Lk) = lk. These are automatically of the form (4.32), with

λk = ιk(∞).

Lemma 5.8. Let φ be the global monodromy. For any j, any δ < 0, and any ε,

(5.39) HF ∗(Lj , φ−1(Lj), δ, ε) ∼= K


Lj

Li

φ−1(Lj)perturbed version of

Figure 6.

Lj

Li

φ−1(Lj)perturbed version of

Figure 7.

is one-dimensional and concentrated in degree 0. Moreover, for any i < j and any δ < λi−λj,the triangle product

(5.40) HF ∗(Lj , φ−1(Lj), δ, ε)⊗HF ∗(Li, Lj) −→ HF ∗(Li, φ

−1(Lj), δ, ε)

is an isomorphism.

Proof. Let H be the Hamiltonian used to define HF ∗(Lj , φ−1(Lj), δ, ε), and φ1

H the time-one

map of the isotopy it generates. The generators of the underlying chain complex correspond

to points of Lj ∩ (φ1H)−1φ−1(Lj). After a compactly supported isotopy (see Figure 6), these

two Lagrangian submanifolds will intersect in a single point, and one easily computes its

Maslov index. The product (5.40) is obtained by counting holomorphic triangles in E which

project to the triangle in the base shaded in Figure 6. One can in principle determine those

directly, but it is easier to apply a further isotopy an indicated in Figure 7, after which the

triangle in the base can be shrunk to a point, making the computation straightforward (the

same trick is used in [28, Figure 6]).

A dual way of expressing Lemma 5.8, in the sense of (4.8), is the following:

(5.41) HF ∗(φ−1(Lj), Lj ,−δ,−ε) ∼= K[−n]

50 PAUL SEIDEL

is one-dimensional and concentrated in degree n; and the map

(5.42) HF ∗(Li, Lj)⊗HFn−∗(φ−1(Lj), Li,−δ,−ε) −→ HFn(φ−1(Lj), Lj ,−δ,−ε)

is a nondegenerate pairing.

Lemma 5.9. Take δ < 0 small, and ε > 0 small, so that (5.6) holds. Then, the resulting

element e from (5.7) has nontrivial image under the (cohomology level) map

(5.43) HF 0(φ, δ, ε) −→ HF 0(Lj , φ−1(Lj), δ, ε) ∼= HFn(φ−1(Lj), Lj ,−δ,−ε)∨.

Proof. This is quite straightforward. There is a Lagrangian parallel of the filtration intro-

duced in Section 5c. In the case of the chain complex underlying HF ∗(Lj , φ−1(Lj), δ, ε),

the entire complex is concentrated in what would be the F 1 level of the filtration, and the

corresponding part of (5.43) is the ordinary restriction map H∗(E;K) → H∗(Lj ;K), which

of course is nontrivial in degree 0.

Lemma 5.10. In the situation of Lemma 5.9, let’s choose a basis of vanishing cycles so

that the associated numbers λ1, . . . , λm satisfy δ < λi−λj for all (i, j). Let A be the directed

A∞-algebra associated to this basis, and Pφ,δ,ε the A-bimodule associated to φ. Then, the

image of e under (5.8) is a quasi-isomorphism.

Proof. This follows directly from Lemma 5.9, Lemma 5.8 (with its reformulation in dual

terms), and the purely algebraic Lemma 2.4.

References

[1] M. Abouzaid. A geometric criterion for generating the Fukaya category. Publ. Math. IHES, 112:191–240,

2010.

[2] M. Abouzaid and S. Ganatra. In preparation.

[3] M. Abouzaid and P. Seidel. Lefschetz fibration techniques in wrapped Floer cohomology. In preparation.

[4] M. Abouzaid and P. Seidel. Altering symplectic manifolds by homologous recombination. Preprint

arXiv:1007.3281, 2010.

[5] M. Abouzaid and P. Seidel. An open string analogue of Viterbo functoriality. Geom. Topol., 14:627–718,

2010.

[6] P. Albers. A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms

in Floer homology. Int. Math. Res. Not., pages Art. ID 134, 56 pages, 2008.

[7] D. Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Gokova

Geom. Topol., 1:51–91, 2007.

[8] F. Bourgeois, T. Ekholm, and Ya. Eliashberg. Effect of Legendrian surgery (with an appendix by S. Gana-

tra and M. Maydanskiy). Geom. Topol., 16:301–390, 2012.

[9] F. Bourgeois and K. Mohnke. Coherent orientations in symplectic field theory. Math. Z., 248:123–146,

2004.

[10] F. Bourgeois and A. Oancea. An exact sequence for contact- and symplectic homology. Invent. Math.,

175:611–680, 2009.

[11] F. Bourgeois and A. Oancea. The Gysin exact sequence for S1-equivariant symplectic homology. Preprint

arXiv:0909.4526, 2009.

[12] F. Bourgeois and A. Oancea. Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli

spaces. Duke Math. J., 146:71–174, 2009.


[13] S. Dostoglou and D. Salamon. Instanton homology and symplectic fixed points. In D. Salamon, editor,

Symplectic Geometry, volume 192 of LMS Lecture Note Series, pages 57–94. Cambridge Univ. Press,

1993.

[14] S. Dostoglou and D. Salamon. Self dual instantons and holomorphic curves. Annals of Math., 139:581–

640, 1994.

[15] A. Floer. Symplectic fixed points and holomorphic spheres. Commun. Math. Phys., 120:575–611, 1989.

[16] A. Floer, H. Hofer, and D. Salamon. Transversality in elliptic Morse theory for the symplectic action.

Duke Math. J., 80:251–292, 1995.

[17] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian intersection Floer theory - anomaly and

obstruction. Amer. Math. Soc., 2010.

[18] S. Ganatra. Symplectic cohomology and duality for the wrapped Fukaya category. Preprint

arXiv:1304.7312, 2013.

[19] W. Goldman and J. Millson. The deformation theory of the fundamental group of compact Kahler

manifolds. IHES Publ. Math., 67:43–96, 1988.

[20] H. Hofer and D. Salamon. Floer homology and Novikov rings. In H. Hofer, C. Taubes, A. Weinstein,

and E. Zehnder, editors, The Floer memorial volume, volume 133 of Progress in Mathematics, pages

483–524. Birkhauser, 1995.

[21] M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66:157–216, 2003.

[22] M. Kontsevich and Y. Soibelman. Notes on A∞-algebras, A∞-categories, and noncommutative geometry.

In Homological Mirror Symmetry: New Developments and Perspectives, volume 757 of Lecture Notes

in Physics, pages 153–219. Springer, 2008.

[23] M. Kontsevich and Y. Vlassopoulos. Pre-Calabi-Yau algebras and topological quantum field theories.

In Preparation.

[24] K. Lefevre-Hasegawa. Sur les A∞-categories. PhD thesis, Universite Paris 7, 2002.

[25] V. Lyubashenko and O. Manzyuk. A∞-bimodules and Serre A∞-functors. In Geometry and dynamics

of groups and spaces, volume 265 of Progr. Math., pages 565–645, Basel, 2008. Birkhauser.

[26] M. Manetti. Lectures on deformations of complex manifolds (deformations from differential graded

viewpoint). Rend. Mat. Appl. (7), 24:1–183, 2004.

[27] M. Maydanskiy. Exotic symplectic manifolds from Lefschetz fibrations. PhD thesis, MIT, 2009.

[28] M. Maydanskiy and P. Seidel. Lefschetz fibrations and exotic symplectic structures on cotangent bundles

of spheres. J. Topology, 3:157–180, 2010.

[29] M. McLean. Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map.

Selecta Math., 18:473–512, 2012.

[30] D. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc.

Steklov Inst. Math., 246:227–248, 2004.

[31] S. Piunikhin, D. Salamon, and M. Schwarz. Symplectic Floer-Donaldson theory and quantum cohomol-

ogy. In C. B. Thomas, editor, Contact and symplectic geometry, pages 171–200. Cambridge Univ. Press,

1996.

[32] D. Salamon and E. Zehnder. Morse theory for periodic solutions of Hamiltonian systems and the Maslov

index. Comm. Pure Appl. Math., 45:1303–1360, 1992.

[33] P. Seidel. Homological mirror symmetry for the quartic surface. Mem. Amer. Math. Soc., to appear.

[34] P. Seidel. Floer homology and the symplectic isotopy problem. PhD thesis, Oxford Univ., 1997. Available

from the author’s webpage.

[35] P. Seidel. Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128:103–146, 2000.

[36] P. Seidel. Vanishing cycles and mutation. In Proceedings of the 3rd European Congress of Mathematics

(Barcelona, 2000), volume 2, pages 65–85, Birkhauser, 2001.

[37] P. Seidel. More about vanishing cycles and mutation. In K. Fukaya, Y.-G. Oh, K. Ono, and G. Tian,

editors, Symplectic Geometry and Mirror Symmetry (Proceedings of the 4th KIAS Annual International

Conference), pages 429–465. World Scientific, 2001.

[38] P. Seidel. Fukaya categories and deformations. In Proceedings of the International Congress of Mathe-

maticians (Beijing), volume 2, pages 351–360. Higher Ed. Press, 2002.

[39] P. Seidel. Symplectic Floer homology and the mapping class group. Pacific J. Math., 206:219–229, 2002.

52 PAUL SEIDEL

[40] P. Seidel. Exact Lagrangian submanifolds in T ∗Sn and the graded Kronecker quiver. In Different faces

of geometry, pages 349–364. Kluwer/Plenum, 2004.

[41] P. Seidel. A∞-subalgebras and natural transformations. Homology, Homotopy Appl., 10:83–114, 2008.

[42] P. Seidel. A biased survey of symplectic cohomology. In Current Developments in Mathematics (Harvard,

2006), pages 211–253. Intl. Press, 2008.

[43] P. Seidel. Fukaya categories and Picard-Lefschetz theory. European Math. Soc., 2008.

[44] P. Seidel. Lectures on four-dimensional Dehn twists. In Symplectic 4-manifolds and algebraic surfaces,

volume 1938 of Lecture Notes in Math., pages 231–267. Springer, 2008.

[45] P. Seidel. Symplectic homology as Hochschild homology. In Algebraic Geometry: Seattle 2005, volume 1,

pages 415–434. Amer. Math. Soc., 2008.

[46] P. Seidel. Fukaya categories associated to lefschetz fibrations, I. J. Symplectic Geom., 10:325–388, 2012.

[47] P. Seidel and J. Solomon. Symplectic cohomology and q-intersection numbers. Geom. Funct. Anal.,

22:443–477, 2012.

[48] P. Seidel. Higher analogues of flux as symplectic invariants. Memoires de la Soc. Math. France, to

appear.

[49] N. Sheridan. Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space. Preprint

arXiv:1111.0632, 2011.

[50] T. Tradler. Infinity-inner-products on A-infinity-algebras. J. Homotopy Related Struct., 3:245–271, 2008.

[51] I. Uljarevic. Symplectic homology for symplectomorphism and symplectic isotopy problem. Preprint

arXiv:1404.2128, 2014.

[52] C. Viterbo. The cup-product on the Thom-Smale-Witten complex, and Floer cohomology. In H. Hofer,

C. Taubes, A. Weinstein, and E. Zehnder, editors, The Floer memorial volume, pages 609–625.

Birkhauser, 1995.

[53] C. Viterbo. Functors and computations in Floer homology with applications, Part I. Geom. Funct.

Anal., 9:985–1033, 1999.

fukaya a -structures associated to lefschetz brations....

Documents