Functors of wrapped Fukaya categories from Lagrangian correspondences

As Lagrangian correspondences are simply Lagrangian submanifolds of the product manifold, it is natural to study first of all the wrapped Fukaya category of a product Liouville manifold and relate it to the wrapped Fukaya categories of both factors, $\mathcal{W}(M)$ and $\mathcal{W}(N)$. To understand such a relation, it is natural to setup up wrapped Floer theory using the split Hamiltonian, which is the sum of admissible Hamiltonians on both factors, i.e. Hamiltonian of the form $\pi_{M}^{*}H_{M}+\pi_{N}^{*}H_{N}$. This together with the product almost complex structure defines a version of wrapped Fukaya category of $M\times N$, which we call the split model of wrapped Fukaya category and denote by $\mathcal{W}^{s}(M\times N)$.

However, as already addressed in [Gao17], there is a technical issue: the split Hamiltonian is a priori not admissible for wrapped Floer theory, meaning that it might have defined a different category compared to the quadratic Hamiltonian on the product $M\times N$ with respect to a natural cylindrical end. This issue was resolved on the cohomology level there. In this paper, we carry out a chain-level discussion, confirming that the wrapped Fukaya category defined with respect to the split Hamiltonians is quasi-isomorphic to the one defined with respect to a quadratic Hamiltonian. Thus up to canonical quasi-equivalence, there is no ambiguity in mentioning the wrapped Fukaya category of the product manifold. This would simplify many functoriality arguments in wrapped Floer theory, and expectantly in the study of homological mirror symmetry.

As this $A_{\infty}$-functor is constructed using an action-filtration argument, it will be called the action-restriction functor.

Next, we investigate a specific class of Lagrangian correspondences, which are either products or cylindrical with respect to a natural choice of cylindrical end of the product $M^{-}\times N$. Associated to every such Lagrangian correspondence $\mathcal{L}\subset M^{-}\times N$, we would like to construct an $A_{\infty}$-functor from the wrapped Fukaya category of $M$ to that of $N$. Technically, this is not always possible. To overcome this, we include wider class of objects in the wrapped Fukaya category, which are to be introduced in section 7. Details are to be presented later, but let us first illustrate the main spirit below.

Using a quilted version wrapped Floer cohomology, we first construct an $A_{\infty}$-functor

$$\mathcal{W}(M^{-}\times N)\to(\mathcal{W}(M),\mathcal{W}(N))^{bimod}$$

from the wrapped Fukaya category of the product to the $A_{\infty}$-category of $(\mathcal{W}(M),\mathcal{W}(N))$-bimodules. By purely algebraic considerations, this gives rise to an $A_{\infty}$-functor

$$\mathcal{W}(M^{-}\times N)\to func(\mathcal{W}(M),\mathcal{W}(N)^{l-mod})$$

by applying the Yoneda embedding on the second factor $\mathcal{W}(N)$. Concretely, to each admissible Lagrangian correspondence $\mathcal{L}\subset M^{-}\times N$, we associate an $A_{\infty}$-bimodule $P_{\mathcal{L}}$ over $(\mathcal{W}(M),\mathcal{W}(N))$ whose value on a pair $(L,L^{\prime})$ is the quilted wrapped Floer cochain complex $CW^{*}(L,\mathcal{L},L^{\prime})$. Regarding $L^{\prime}\subset N$ as a testing object, with small amount of homological algebra argument we then immediately get an $A_{\infty}$-functor

(1.2)

$$\Phi_{\mathcal{L}}:\mathcal{W}(M)\to\mathcal{W}(N)^{l-mod}.$$

Then we study the geometric composition of Lagrangian correspondences, which is in general a Lagrangian immersion. In order to be able to include these immersed Lagrangian submanifolds as objects of the wrapped Fukaya category, in section 4 we define the immersed wrapped Fukaya category of $M$, denoted by $\mathcal{W}_{im}(M)$, whose objects are unobstructed proper exact cylindrical Lagrangian immersions of $M$ (with transverse self-intersections), together with bounding cochains for them. Floer theory for such Lagrangian immersions turns out to work over $\mathbb{Z}$, so that $\mathcal{W}_{im}(M)$ is an $A_{\infty}$-category over $\mathbb{Z}$ in the usual sense. We also extend this theory to cylindrical Lagrangian immersions with clean self-intersections in section 5.

The ordinary wrapped Fukaya category $\mathcal{W}(M)$ can be embedded into $\mathcal{W}_{im}(M)$ as a full sub-category. Also, the $A_{\infty}$-functors $\Phi_{\mathcal{L}}$ and $\Phi$ can be extended to the category of modules over the immersed wrapped Fukaya category

(1.3)

$$\Phi_{\mathcal{L}}:\mathcal{W}(M)\to\mathcal{W}_{im}(N)^{l-mod},$$

and

(1.4)

$$\Phi:\mathcal{W}(M^{-}\times N)\to func(\mathcal{W}(M),\mathcal{W}_{im}(N)^{l-mod}).$$

We expect that the above $A_{\infty}$-functor extends over $\mathcal{W}_{im}(M)$ in a canonical and unique way. However, as that is not one of the main subjects of this paper and involves quite a lot of technicality, we will not try to give a proof.

In a more functorial form, the assignment of $A_{\infty}$-functors to Lagrangian correspondences is functorial in the wrapped Fukaya category of the product manifold $M^{-}\times N$:

As a particular application of the construction of functors, we present a well-known and expected (but not fully established) Künneth formula for wrapped Fukaya categories, which relate $\mathcal{W}(M\times N)$ to the $A_{\infty}$-tensor product $\mathcal{W}(M)\otimes\mathcal{W}(N)$ in an appropriate sense. We also show that under the condition that both $\mathcal{W}(M)$ and $\mathcal{W}(N)$ have finite collections of split-generators, there is a quasi-equivalence between $\mathcal{W}(M\times N)$ and $\mathcal{W}(M)\otimes\mathcal{W}(N)$. These results will be discussed in a more formal way in subsection 7.9.

An especially important instance of Lagrangian correspondence comes from Liouville sub-domains $U_{0}\subset M_{0}$: the graph of the natural inclusion can be completed to a Lagrangian correspondence between $U$ and $M$. This is called the graph correspondence, denoted by $\Gamma$. As an admissible Lagrangian correspondence from $M$ to $U$, it satisfies the hypothesis of Theorem 1.2. Thus we obtain an $A_{\infty}$-functor

(1.7)

$$\Theta_{\Gamma}:\mathcal{W}(M)\to\mathcal{W}_{im}(U).$$

There is a full sub-category on which this functor takes a simpler form. First, there is the full sub-category $\mathcal{B}(M)$ consisting of exact cylindrical Lagrangian submanifolds $L$ whose primitive $f$ is locally constant near both $\partial M$ and $\partial U$ (Assumption 8.3). Then, we may further consider a full sub-category $\mathcal{B}_{0}(M)$ of $\mathcal{B}(M)$ whose objects satisfy an additional geometric condition (Assumption 8.4). Restricted to this sub-category, the functor $\Theta_{\Gamma}$ induces a functor

$$\Theta_{\Gamma}:\mathcal{B}_{0}(M)\to\mathcal{W}(N),$$

whose image lies in the ordinary wrapped Fukaya category consisting of properly embedded exact cylindrical Lagrangian submanifolds, with zero bounding cochains. On the other hand, recall that the Viterbo restriction functor is defined on this sub-category $\mathcal{B}(M)$. More detailed definitions will be given in section 8. We shall see that their linear terms agree, when the Viterbo restriction functor is further restricted to $\mathcal{B}_{0}(M)$.

It is expected that $\Theta_{\Gamma}$ and $r$ are in fact homotopic as $A_{\infty}$-functors, when restricted to the full sub-category $\mathcal{B}_{0}(M)$. Our current method of proof needs improvement in order to prove this more general statement. Moreover, they should be indeed homotopic on the bigger sub-category $\mathcal{B}(M)$ in appropriate sense, if we can manage to show that the geometric composition with the graph correspondence is Floer-theoretically equivalent to the actual restriction of Lagrangian submanifolds. Such points are to be discussed in [Gao].

Going back to the definition of the Viterbo restriction functor, we recall that it is only defined for those Lagrangian submanifolds which satisfy Assumption 8.3. However, the restriction functor $\Theta_{\Gamma}$ is defined on the whole wrapped Fukaya category, though the image of a Lagrangian submanifold is not necessarily simply its restriction to the sub-domain. However, it is conjectured that the Viterbo restriction functor can be extended to the whole wrapped Fukaya category via some kind of deformation theory, as stated in Conjecture 8.26. It is not completely known but a very interesting question to ask in what cases the extension agrees with the functor $\Theta_{\Gamma}$. This is a question for future research.

The structure of this paper is as follows. Sections 2 to 5 provide fundamental materials for the theory to work: section 2 reviews basic homological algebra regarding $A_{\infty}$-modules and presents a version of $A_{\infty}$-direct limit; section 3 recalls the definition of the wrapped Fukaya category and introduces a new class of objects necessary to establish representability of the functors; section 4 defines the immersed wrapped Fukaya category and section 5 generalizes the story to Lagrangian immersions with clean self-intersections. Section 6 studies Floer theory on product manifolds and proves the two models of wrapped Fukaya categories are equivalent. Section 7 formally sets up the framework of functoriality in Lagrangian correspondences and proves representability, with a discussion on the Künneth formula at the end. Section 8 concerns sub-domains and how the Viterbo restriction functor is related to our framework.

Most of the results about $A_{\infty}$-categories in this section are well-known, which are included for the purpose of fixing notations and conventions for $A_{\infty}$-categories. Let $k$ be a field. We shall work with $A_{\infty}$-categories that are linear over $k$. Let $Ch$ be the dg-category of chain complexes over $k$, regarded as an $A_{\infty}$-category whose higher order operations $m^{k},k\geq 3$ are all zero. Here chain complexes are not necessarily bounded.

By definition, a non-unital left $A_{\infty}$-module over $\mathcal{A}$ is a non-unital $A_{\infty}$-functor $\mathcal{A}\to Ch^{op}$, where $Ch^{op}$ is the opposite category of $Ch$. All left $A_{\infty}$-modules over $\mathcal{A}$ also form an $A_{\infty}$-category (in fact a dg-category), $\mathcal{A}^{nu-l-mod}=nu-func(\mathcal{A},Ch^{op})$.

There are also non-unital right $A_{\infty}$-modules over $\mathcal{A}$, which are functors from $\mathcal{A}^{op}$ to $Ch$. These also form a dg-category, $\mathcal{A}^{nu-r-mod}=func(\mathcal{A}^{op},Ch)$.

When $\mathcal{A}$ comes with strict units, homotopy units or cohomological units, there are also unital versions of left and right $A_{\infty}$-modules:

$$\mathcal{A}^{l-mod}=func(\mathcal{A},Ch^{op}),$$

$$\mathcal{A}^{r-mod}=func(\mathcal{A}^{op},Ch),$$

as unital $A_{\infty}$-functors.

First, we recall the non-unital version of the Yoneda embedding. The left Yoneda functor is a non-unital $A_{\infty}$-functor

(2.1)

$$\mathfrak{y}_{l}:\mathcal{A}\to\mathcal{A}^{nu-mod}$$

which sends an object $Y\in Ob\mathcal{A}$ to its left Yoneda module $\mathcal{Y}^{l}\in\mathcal{A}^{nu-mod}$, which is defined as follows. For objects $X\in Ob\mathcal{A}$,

(2.2)

$$\mathcal{Y}^{l}(X)=hom_{\mathcal{A}}(Y,X),$$

and the module structure is given by the $A_{\infty}$-structure maps of $\mathcal{A}$:

(2.3)

$$n_{\mathcal{Y}^{l}}^{d}:\hom_{\mathcal{A}}(X_{d-2},X_{d-1})\otimes\cdots\otimes\hom_{\mathcal{A}}(X_{0},X_{1})\otimes\mathcal{Y}^{r}(X_{0})\to\mathcal{Y}^{r}(X_{d-1}),$$

(2.4)

$$n_{\mathcal{Y}^{l}}^{d}(a_{d-1},\cdots,a_{1},b)=m^{d+1}_{\mathcal{A}}(a_{d-1},\cdots,a_{1},b).$$

On the level of morphisms, $\mathfrak{y}_{l}^{1}$ assigns to a morphism $c\in hom_{\mathcal{A}}(Y_{0},Y_{1})$ a pre-module homomorphism:

$$(\mathfrak{y}_{l}^{1}(c))^{d}:\hom_{\mathcal{A}}(X_{d-2},X_{d-1})\otimes\cdots\otimes\hom_{\mathcal{A}}(X_{0},X_{1})\otimes\mathcal{Y}_{1}^{l}(X_{0})\to\mathcal{Y}_{0}^{l}(X_{d-1}),$$

(2.5)

$$(\mathfrak{y}_{l}^{1}(c))^{d}(a_{d-1},\cdots,a_{1},b)=m^{d+1}_{\mathcal{A}}(a_{d-1},\cdots,a_{1},b,c).$$

Higher order terms $\mathfrak{y}_{l}^{k}$ are defined in by analogous formulas:

$$(\mathfrak{y}_{l}^{k}(c_{k},\cdots,c_{1}))^{d}:\hom_{\mathcal{A}}(X_{d-2},X_{d-1})\otimes\cdots\otimes\hom_{\mathcal{A}}(X_{0},X_{1})\otimes\mathcal{Y}_{k}^{l}(X_{0})\to\mathcal{Y}_{0}^{l}(X_{d-1}),$$

(2.6)

$$(\mathfrak{y}_{l}^{k}(c_{k},\cdots,c_{1}))^{d}(a_{d-1},\cdots,a_{1},b)=m^{d+k}_{\mathcal{A}}(a_{d-1},\cdots,a_{1},b,c_{k},\cdots,c_{1}).$$

In case $\mathcal{A}$ is cohomologically unital, the image of the Yoneda functor lies in the $A_{\infty}$-subcategory of $\mathcal{A}^{nu-l-mod}$ consisting of c-unital right $\mathcal{A}_{\infty}$-modules over $\mathcal{A}$. We denote this $A_{\infty}$-subcategory by $\mathcal{A}^{l-mod}$.

An important notion is the representability of an $A_{\infty}$-functor, in the sense of [Fuk02]. We briefly recall it here.

There is also a right Yoneda functor

(2.7)

$$\mathfrak{y}_{r}:\mathcal{A}\to\mathcal{A}^{nu-r-mod},$$

whose c-unital version becomes

(2.8)

$$\mathfrak{y}_{r}:\mathcal{A}\to\mathcal{A}^{r-mod}.$$

Similarly, a right $A_{\infty}$-module over $\mathcal{A}$ is said to be representable, if it is homotopic to a right Yoneda module $\mathcal{Y}^{r}=\mathfrak{y}_{r}(Y)$.

We will see in the following subsection that the notion of representability does not have ambiguity, up to quasi-isomorphism. That is, the representative is unique up to quasi-isomorphism.

The key point related to Yoneda embedding we want to emphasize here is that whenever $\mathcal{A}$ is c-unital, the Yoneda embedding is cohomologically fully faithful, and therefore is an $A_{\infty}$-homotopy equivalence to its image. This was first proved in [Fuk02] under the assumption that $\mathcal{A}$ is strictly unital, with a new proof given in [Sei08] in case $\mathcal{A}$ is only cohomologically unital. To see this, let $\mathcal{M}$ be any c-unital left $A_{\infty}$-module over $\mathcal{A}$ and consider the following cochain map

(2.9)

$$\lambda:\mathcal{M}(Y)\to hom_{\mathcal{A}^{mod}}(\mathcal{Y}^{l},\mathcal{M})$$

(2.10)

$$\lambda(c)^{d}(a_{d-1},\cdots,a_{1},b)=n_{\mathcal{M}}^{d+1}(a_{d-1},\cdots,a_{1},b,c).$$

Note in fact this definition also makes sense for general non-unital $\mathcal{A}_{\infty}$-modules, but we will only emphasize its importance in the c-unital case.

There is a parallel discussion for right Yoneda functors. That is, the right Yoneda functor

$$\mathfrak{y}_{r}:\mathcal{A}\to\mathcal{A}^{r-mod}$$

is cohomologically fully faithful.

For this reason, we also call the Yoneda functor the Yoneda embedding. And there is no ambiguity for the notion of representability. This allows us to make the following definition.

Let us relate the story of representable $A_{\infty}$-modules to that of $A_{\infty}$-functors. We shall be considering $A_{\infty}$-functors of the following kind

(2.16)

$$\mathcal{F}_{m}:\mathcal{A}\to\mathcal{B}^{l-mod}.$$

Such an $A_{\infty}$-functor is called a module-valued functor, which says that it suffices to verify representability on objects.

We have the following concrete criterion for representability of a module-valued functor.

A natural source of module-valued functors is given by $A_{\infty}$-bimodules. Let $\mathcal{P}$ be a left-$\mathcal{B}$ right-$\mathcal{A}$ $A_{\infty}$-bimodule, or simply called an $A_{\infty}$-bimodule over $(\mathcal{A},\mathcal{B})$. This notation is slightly misleading as we write $\mathcal{A}$ on the left and $\mathcal{B}$ on the right, but we shall keep this convention as it is fitted into Floer theory which will be discussed later on. From $\mathcal{P}$ we can define an $A_{\infty}$-functor

(2.25)

$$\mathcal{F}_{\mathcal{P}}:\mathcal{A}\to\mathcal{B}^{l-mod}$$

as follows. For each object $X\in Ob\mathcal{A}$, we set

$$\mathcal{F}_{\mathcal{P}}(X)=\mathcal{P}(X,\cdot).$$

That is, $\mathcal{F}_{\mathcal{P}}(X)$ is the left-$\mathcal{B}$ module that takes value $\mathcal{P}(X,Y)$ for each object $Y\in Ob\mathcal{B}$, and has module structure maps

(2.26)

$$n^{k}_{\mathcal{F}_{\mathcal{P}}(X)}:\hom_{\mathcal{B}}(Y_{k-1},Y_{k})\otimes\hom_{\mathcal{B}}(Y_{0},Y_{1})\otimes\mathcal{P}(X,Y_{0})\to\mathcal{P}(X,Y_{k}),$$

(2.27)

$$n^{k}_{\mathcal{F}_{\mathcal{P}}(X)}(b_{k},\cdots,b_{1},p)=n^{k,0}_{\mathcal{P}}(b_{k},\cdots,b_{1},p),$$

where $n^{k,l}_{\mathcal{P}}$ are the $A_{\infty}$-bimodule structure maps of $\mathcal{P}$. Next we define the action of $\mathcal{F}_{\mathcal{P}}$ on morphism spaces

(2.28)

$$\mathcal{F}_{\mathcal{P}}^{l}:\hom_{\mathcal{A}}(X_{l-1},X_{l})\otimes\cdots\otimes\hom_{\mathcal{A}}(X_{0},X_{1})\to\hom_{\mathcal{B}^{l-mod}}(\mathcal{F}_{\mathcal{P}}(X_{l}),\mathcal{F}_{\mathcal{P}}(X_{0}))$$

by the formula

(2.29)

$$(\mathcal{F}_{\mathcal{P}}^{l}(a_{l},\cdots,a_{1}))^{k}(b_{k},\cdots,b_{1},p)=n^{k,l}_{\mathcal{P}}(b_{k},\cdots,b_{1},p,a_{l},\cdots,a_{1}).$$

By the $A_{\infty}$-bimodule equations for $n^{k,l}_{\mathcal{P}}$, it is straightforward to verify:

All $A_{\infty}$-bimodules over $(\mathcal{A},\mathcal{B})$ form an $A_{\infty}$-category, denoted by $(\mathcal{A},\mathcal{B})^{bimod}$. The above construction of a module-valued functor $\mathcal{F}_{\mathcal{P}}$ associated to an $A_{\infty}$-bimodule $\mathcal{P}$ can be made on the level of the whole category $(\mathcal{A},\mathcal{B})^{bimod}$.

In general, it is not possible to expect $\mathcal{F}(X)$ is representable for every $X\in Ob\mathcal{A}$. But we might say that it is representable for some of the objects. This leads to the following definition.

By the criterion stated in Lemma 2.6, this definition makes sense.

In this subsection, we introduce the notion of a cyclic element, which is key to proving the existence and uniqueness of a bounding cochain under certain assumptions. One origianl formulation due to Fukaya (Proposition 3.5 of [Fuk15]), works with filtered $A_{\infty}$-algebras and filtered $A_{\infty}$-modules over the Novikov ring. In our case, we consider curved $A_{\infty}$-algebras and $A_{\infty}$-modules over $\mathbb{Z}$. There is in general no intrinsic way of making the method work for arbitrary curved $A_{\infty}$-algebras and $A_{\infty}$-modules, so additional structures are required, to be explained below.

Let $(C,m^{k})$ be a curved $A_{\infty}$-algebra over $\mathbb{Z}$ ($m^{0}\neq 0$), where $C$ is a free $\mathbb{Z}$-module (of finite or infinite rank), and $(D,n^{k})$ a left $A_{\infty}$-module over $(C,m^{k})$, where $D$ is also a free $\mathbb{Z}$-module. We need an additional condition, similar to the gappedness condition for filtered $A_{\infty}$-algebras and filtered $A_{\infty}$-modules. Although we work with usual $A_{\infty}$-algebras and $A_{\infty}$-modules, we still want to have some inductive structures, similar to those for filtered $A_{\infty}$-algebras and filtered $A_{\infty}$-modules. For us, the notion we need is a filtration which satisfies certain analogous conditions.

In order to deform a curved $A_{\infty}$-algebra to a non-curved $A_{\infty}$-algebra, we need the notion of a bounding cochain.

There is a very important class of examples for which nilpotent elements naturally exist. For example, if $C$ as a $\mathbb{Z}$-module has finitely many generators which lie in $F_{C}^{0}$, then any such generator in the strictly positive part of the filtration is nilpotent, because the $A_{\infty}$-structure maps increase the filtration. For geometric applications, we shall see that wrapped Floer cochain spaces have this property. Thus it does not harm to give a name for such a filtration.

Suppose that we have chosen a discrete filtration $F_{C}$ on $(C,m^{k})$ as well as a discrete strictly compatible filtration $F_{D}$ on $(D,n^{k})$. For simplicity, we assume that $0\in\Lambda_{D}$. Now we are going to introduce the key notion.

Note that $n^{0}(F_{D}^{\lambda}D)\subset F_{D}^{\lambda}D$ for every $\lambda$. Thus the second condition means when $n^{0}$ acts on $u$, it strictly increases the filtration.

The following result is proved in [Fuk15] in the case of filtered $A_{\infty}$-algebras and filtered $A_{\infty}$-modules (Proposition 3.5 of [Fuk15]) over the Novikov ring. We have a similar result in our case, which yields bounding cochains of ”finite type”.