# Homological Mirror Symmetry for local SYZ singularities

Mohammed Abouzaid  and  Zack Sylvan

###### Abstract.

Gross and Siebert identified a class of singular Lagrangian torus fibrations
which arise when smoothing toroidal degenerations, and which come in pairs
that are related by mirror symmetry. We identify an immersed Lagrangian in
each of these local models which supports a moduli space of objects that is
isomorphic to the mirror space, and prove a homological mirror statement along
the way.

The first author was supported by NSF grant DMS-1308179, DMS-1609148, and
DMS-1564172, as well as the Simons Foundation through its “Homological Mirror
Symmetry” Collaboration grant.

###### Contents

  1. 1 Introduction
  2. 2 Mirrors to points
  3. 3 Homological Mirror Symmetry
  4. A Compatibility of Künneth maps with cup functors

##  1\. Introduction

For each pair $(m,n)$ of integers, consider the smooth hypersurface

(1.0.1)
$X_{m,n}\equiv\\{(x_{0},\ldots,x_{m},y_{1},\ldots,y_{n})\in{\mathbb{C}}^{m+1}\times\left({\mathbb{C}}^{*}\right)^{n}|\prod
x_{i}=1+\sum y_{j}\\},$

which we equip with the restriction of the standard Kähler form on
${\mathbb{C}}^{m+1}\times\left({\mathbb{C}}^{*}\right)^{n}$.

######  Example 1.1.

For $m=0$, we obtain $({\mathbb{C}}^{*})^{n}$. For $m=1$, we obtain the conic
bundle

(1.0.2) $x_{0}x_{1}=1+\sum y_{i}.$

For $n=1$, we obtain the hypersurface $\prod x_{i}=1+y$ in
${\mathbb{C}}^{m}\times{\mathbb{C}}^{*}$, which is isomorphic to the
complement in ${\mathbb{C}}^{m}$ of the hypersurface $\prod x_{i}=1$.

It is well-known that $\left({\mathbb{C}}^{*}\right)^{n}$ is self-mirror,
hence in our notation above that $X_{0,n}$ is mirror to $X_{m,0}$. The case
$n=1$ or $m=1$ was extensively studied in [8], where it was argued that a
version of SYZ mirror symmetry holds for these pairs. There are by now several
proofs of Homological mirror symmetry of other special cases, starting with
the one given by Seidel for the Fukaya category of $X_{1,1}$ [26], and
including the one by Chan, Pomerleano and Ueda [10] for $X_{1,2}$, and
recently by Pomerleano for $X_{1,n}$ [25] (there is also a literature on the
sheaf theoretic side, which is reviewed in [14], where the next result is also
independently proved). The first result of this paper is an extension to the
general case:

######  Theorem 1.2.

The wrapped Fukaya category of $X_{m,n}$ is equivalent to the (derived)
category of coherent sheaves on $X_{n,m}$.

Since $X_{n,m}$ is affine, the category of coherent sheaves is equivalent to
the category of modules over the corresponding commutative ring. This means
that we expect the existence of a Lagrangian in $X_{m,n}$ whose wrapped Floer
cohomology is isomorphic to this ring; this turns out to be the positive real
locus. The above theorem thus follows from a computation of the wrapped Floer
cohomology of this Lagrangian, and a proof that it generates the Fukaya
category.

######  Remark 1.3.

With some modification, we can extend our setup to the case where $m$ is
allowed to equal $-1$, which would then include the statement that the Fukaya
category of the hypersurface $1+\sum y_{i}=0$ in $({\mathbb{C}}^{*})^{n}$
(i.e. Mikhalkin’s pair of pants [23]) is equivalent to the derived category of
coherent sheaves on the hypersurface $\prod x_{i}=0$ in ${\mathbb{C}}^{n}$
(see [22]). We shall in fact use this result as a building block of our proof.

The main goal of this paper is to obtain an understanding of mirror symmetry
for these manifolds simultaneously at categorical (HMS) and the geometric
(SYZ) levels: we thus seek to realise all skyscraper sheaves of points on one
side as arising from (immersed) Lagrangians on the other side equipped with
bounding cochains.

Our starting point is a variant of mirror symmetry for pairs of pants in
dimension $1$ (c.f. [7]): the hypersurface

(1.0.3)
$H_{1}\equiv\\{1+y_{1}+y_{2}=0\\}\subset\left({\mathbb{C}}^{*}\right)^{2}$

is mirror to the union of the coordinate lines in ${\mathbb{C}}^{2}$.
Following Seidel’s work [29], we consider an immersed Lagrangian $L_{1}$ in
the pair of pants, shown in Figure 1 as the complement of the origin and the
point $1$ in ${\mathbb{C}}$.

$\ast$$0$$\ast$$1$$s$$t$$c$ Figure 1. Seidel’s immersed Lagrangian

As we discuss in Section 2, by identifying an open subset of $X_{m,n}$ which
is biholomorphic to $({\mathbb{C}}^{*})^{m}\times H^{n}_{1}$ we obtain from
Seidel’s construction a Lagrangian

(1.0.4) $L_{m,n}\cong T^{m}\times L_{1}^{n}\subset({\mathbb{C}}^{*})^{m}\times
H^{n}_{1}\subset X_{m,n}.$

Our main result is the following:

######  Theorem 1.4.

Every skyscraper sheaf of a point in $X_{n,m}$ is mirror to an object of the
Fukaya category of $X_{m,n}$ supported on $L_{m,n}$.

######  Remark 1.5.

The construction of $L_{m,n}$ is not unique. As the reader may see by
consulting Equation (2.1.3), the embedding of $({\mathbb{C}}^{*})^{m}\times
H^{n}_{1}$ that we use depends on an ordering of the coordinates. One can
construct a more symmetric immersed Lagrangian as follows: consider the
projection $\pi\colon\thinspace X_{m,n}\to{\mathbb{C}}$ given by
$\prod_{i=0}^{m}x_{i}$, which equals $1+\sum_{j=1}^{n}y_{j}$ by definition.
The fibre at a point $\lambda$ is the product of the hypersurfaces
$\prod_{i=0}^{m}x_{i}=\lambda$ and $\sum_{j=1}^{n}y_{j}=\lambda-1$. Since both
function are equivariant, we see that the only critical points are $0$ and
$1$, and that the fibres away from these points are equivalent to the product
of a torus $\left({\mathbb{C}}^{*}\right)^{m}$ with an $n-1$ dimensional pair
of pants. In [31], Sheridan constructed an immersed Lagrangian in $H_{n-1}$,
which contains all the information about the mirror. We expect that it is
possible to prove an analogue of Theorem 1.4 for the Lagrangian obtained by
parallel transport of the product of $T^{m}$ with Sheridan’s Lagrangians,
along the curve in the base given by Figure 1.

The most delicate part of Theorem 1.4 is to make sense of the Floer theory of
$L_{m,n}$, because it bounds holomorphic discs. The space of objects of the
Fukaya category supported on an immersed Lagrangian arises as the zero locus
of a function (the Maurer-Cartan function) on the _deformation space_ , which
can be described from the product of the space of local systems (a copy of
$\left({\mathbf{k}}^{*}\right)^{m+n}$), together with a coordinates for each
self-intersection of degree $1$. The constraint on the Maurer-Cartan function
enforces the vanishing of the weighted count of holomorphic discs with one
output.

Instead of studying the full deformation space, which is quite large in our
situation, we will consider a distinguished subspace which is a product of
$\left({\mathbf{k}}^{*}\right)^{m}$ (corresponding to the $T^{m}$ factors)
with ${\mathbf{k}}^{2n}$ (corresponding to the self intersection points $s$
and $t$ shown in Figure 1). It will turn out that the Maurer-Cartan function
vanishes identically on this subspace, but we will see the equation of the
mirror arise from the equation required for the resulting object to be non-
trivial. A difficulty is caused by the fact that these objects correspond to a
direct sum of $2^{n}$ copies of skyscraper sheaves, requiring us to pass to
the idempotent closure of the Fukaya category in order to obtain the desired
correspondence.

######  Remark 1.6.

It is illuminating to consider the mirror correspondence for some other
Lagrangians as well. Let $L_{1}^{s}$ and $L_{1}^{t}$ denote either component
of the Lagrangians shown in Figure 2 (more precisely, we should take the exact
representatives of these Lagrangians), which are obtained from $L_{1}$ by
surgery at the intersection points labelled $s$ and $t$. For each $0\leq i\leq
n$, we can then consider the Lagrangians

(1.0.5) $L_{m,n}^{i}\equiv T^{m}\times(L_{1}^{s})^{i}\times(L_{1}^{t})^{n-i}.$

The analogous result to Theorem 1.4 is that each skyscraper sheaf of a point
of $X_{m,n}$ with the property that the coordinates $x_{k}$ for $k\neq i$ do
not vanish, admits a mirror which is supported on $L_{m,n}^{i}$ (c.f. Remark
2.19).

One can more directly describe the Hamiltonian isotopy class of these
Lagrangians as follows: assume that $i\neq 0$, and consider a torus in
$H_{n-1}$ with $1\sim|y_{i}|$, and $|y_{k}|\ll 1$ for $k\neq i$ (more
precisely, under a tropical deformation of the equation for $H_{n-1}$), which
we may assume to be an exact Lagrangian by an appropriate deformation for the
primitive. The parallel transport of the product of this Lagrangian with the
real torus in $({\mathbb{C}}^{*})^{m}$ along the curve $L_{1}^{t}$ in the base
corresponds to $L_{m,n}^{i}$. For $i=0$, one can use the fact that
$\left({\mathbb{C}}^{*}\right)^{n}$ retracts to the fibre of $1+\sum y_{i}$
over a disc bounding $L_{1}^{s}$, to see that the parallel transports of these
Lagrangians along $L_{1}^{s}$ are Hamiltonian isotopic, and correspond to the
Lagrangian $L_{m,n}^{0}$.

$\ast$$1$$\ast$$0$$L_{1}^{s}$$\ast$$0$$\ast$$1$$L_{1}^{t}$ Figure 2. The
Lagrangians $L^{s}_{1}$ and $L^{t}_{1}$.

### Acknowledgements

We would like to thank Sasha Efimov and Alex Perry for help with the results
of Section 3.1, and Yanki Lekilli for useful correspondence about the results
of [22].

##  2\. Mirrors to points

In this section, we construct in each manifold $X_{m,n}$ a Lagrangian
$L_{m,n}$, which will play a key role in our SYZ interpretation of mirror
symmetry in the sense that we will identify a subspace of its space of non-
trivial bounding cochains with a copy of the mirror space.

###  2.1. Divisors and iterated fibrations on $X_{m,n}$

Recall that

(2.1.1)
$X_{m,n}\equiv\\{(x_{0},\ldots,x_{m},y_{1},\ldots,y_{n})\in{\mathbb{C}}^{m+1}\times\left({\mathbb{C}}^{*}\right)^{n}|\prod
x_{i}=1+\sum y_{j}\\}.$

We introduce a collection of maps

(2.1.2) $q_{j}\colon\thinspace X_{m,n}\to{\mathbb{C}}$

indexed by $1\leq j<n$, and which are expressed in terms of the
${\mathbb{C}}^{*}$-coordinates via the formula

(2.1.3)
$q_{j}(x_{0},\ldots,x_{m},y_{1},\ldots,y_{n})=1+y_{1}/y_{j+1}+\cdots+y_{j}/y_{j+1}.$

We define $q_{n}$ to be the projection map to ${\mathbb{C}}$ discussed in the
introduction. We note that this last map factors through the projection map to
${\mathbb{C}}^{m+1}$:

(2.1.4) ${X_{m,n}}$${{\mathbb{C}}^{m+1}}$${{\mathbb{C}}.}$

We introduce the divisor $D_{j}\subset X_{m,n}$, given as the inverse image of
$1$ under $q_{j}$:

(2.1.5) $D_{j}\equiv q_{j}^{-1}(1).$

######  Lemma 2.1.

The divisor $D_{j}$ agrees with the inverse image of $0$ under $q_{j-1}$.

###### Proof.

The divisor $D_{j}$ is the solution to the equation $\sum_{j=1}^{j}y_{j}=0$,
which holds if and only the quotient by $y_{j}$ vanishes, which gives the $0$
locus of $q_{j-1}$. ∎

The above Lemma suggests introducing an additional divisor

(2.1.6) $D_{n+1}\equiv q^{-1}_{n}(0),$

which is the normal crossing divisor $\prod x_{i}=0$.

Let $D$ denote the union of the divisors $\\{D_{j}\\}_{j=1}^{n+1}$. For the
next result, it is convenient to recall that the function $q_{n}$ is the
product of the coordinates $x_{i}$:

######  Lemma 2.2.

The maps $(x_{1},\dotsc,x_{m},q_{1},\dotsc,q_{n})$ define coordinates on
$X_{m,n}\setminus D$ identifying this space with a product of
$({\mathbb{C}}^{*})^{m}$ with ($1$-dimensional) pairs of pants:

(2.1.7) ${X_{m,n}\setminus D}$${({\mathbb{C}}^{*})^{m}\times
H^{n}_{1}}$${X_{m,n}}$${{\mathbb{C}}^{m}\times{\mathbb{C}}^{n}.}$$\scriptstyle{\cong}$

∎

This product decomposition allows us to construct our desired Lagrangian
$L_{m,n}$ as a product

(2.1.8) $L_{m,n}\cong T^{m}\times L_{1}^{n},$

where $L_{1}$ is Seidel’s Lagrangian in the pair of pants, and
$T^{m}\subset({\mathbb{C}}^{*})^{m}$ is the locus where the norm of the
coordinates is $1$. As in [29], we equip $L_{1}$ (and the standard circle in
${\mathbb{C}}^{*}$) with the bounding $\operatorname{Spin}$ structure, which
determines a $\operatorname{Spin}$ structure on $L_{m,n}$.

######  Remark 2.3.

We have used the ordering of the coordinates to construct this splitting. The
Lagrangian $L_{m,n}$ depends on this splitting. However, the outcome of our
work will imply that the Maurer-Cartan spaces which we will associate to these
Lagrangians are canonically isomorphic, because they can be identified with
the spectrum of the commutative ring $SH^{0}(X_{m,n})$.

###  2.2. Symplectic and holomorphic volume forms

Since $X_{m,n}$ is an affine variety, it is equipped with a Liouville
structure which is canonical up to symplectomorphism [12, 30], and is induced
by pulling back the standard symplectic form in ${\mathbb{C}}^{N}$ under a
proper holomorphic embedding. On any open subset of $X_{m,n}$, we may arrange
for the symplectic form to agree with a preferred choice, and we shall use
this flexibility to ensure that it agrees with the product symplectic form on
$({\mathbb{C}}^{*})^{m}\times H^{n}_{1}$ away from a neighbourhood of the
divisor $D$. In this way, $L_{m,n}$ is evidently a Lagrangian submanifold,
which is exact for an appropriate choice of primitive.

The natural holomorphic volume form on $X_{m,n}$ is given by

(2.2.1) $\Omega_{m,n}\equiv\frac{dx_{0}\wedge\cdots\wedge dx_{m}\wedge
d\log(y_{1})\wedge\cdots\wedge d\log(y_{n})}{d\left(\prod x_{i}-\sum
y_{j}\right),}$

which by definition satisfies

(2.2.2) $d\left(\prod x_{i}-\sum
y_{j}\right)\wedge\Omega_{m,n}=dx_{0}\wedge\cdots\wedge dx_{m}\wedge
d\log(y_{1})\wedge\cdots\wedge d\log(y_{n})$

in the normal bundle to $X_{m,n}$. For the next statement, we identify $H_{1}$
with the complement of the origin in ${\mathbb{C}}\setminus\\{1\\}$, which we
equip with the holomorphic volume form

(2.2.3) $\Omega_{1}\equiv\frac{dw}{w-1}=d\log(w-1).$

######  Lemma 2.4.

The restriction of $\Omega_{m,n}$ to the complement of $D$ agrees with the
wedge product of the forms $d\log x$ on the ${\mathbb{C}}^{*}$ factors with
the forms $\Omega_{1}$ on the $H_{1}$ factors.

###### Proof.

Note first that

(2.2.4) $\displaystyle d\left(\prod x_{i}-\sum y_{j}\right)\wedge d\log
x_{1}\wedge\dotsb$ $\displaystyle\wedge d\log
x_{m}\,\wedge\,\Omega_{1,1}\wedge\dotsb\wedge\Omega_{1,n}$
$\displaystyle=dx_{0}\wedge\dotsb\wedge
dx_{m}\wedge\Omega_{1,1}\wedge\dotsb\wedge\Omega_{1,n},$

since the $\Omega_{1,j}$ terms depend only on the $y$ coordinates and there
are already $n$ of them. Now,

$\displaystyle\Omega_{1,1}$ $\displaystyle\wedge\dotsb\wedge\Omega_{1,n}$
$\displaystyle=\left(d\log\frac{y_{1}}{y_{2}}\right)\wedge\dotsb\wedge\left(d\log\frac{y_{1}+\dotsb+y_{n-1}}{y_{n}}\right)\wedge\left(d\log(y_{1}+\dotsb+y_{n})\right)$
$\displaystyle=\left(\frac{dy_{1}}{y_{1}}-\frac{dy_{2}}{y_{2}}\right)\wedge\dotsb\wedge\left(\frac{dy_{1}+\dotsb+dy_{n-1}}{y_{1}+\dotsb+y_{n-1}}-\frac{dy_{n}}{y_{n}}\right)\wedge\frac{dy_{1}+\dotsb+dy_{n}}{y_{1}+\dotsb+dy_{n}}$
$\displaystyle=\begin{vmatrix}\frac{1}{y_{1}}&\frac{1}{y_{1}+y_{2}}&\cdots&\frac{1}{y_{1}+\dotsb+y_{n}}\\\
-\frac{1}{y_{2}}&\frac{1}{y_{1}+y_{2}}&\cdots&\frac{1}{y_{1}+\dotsb+y_{n}}\\\
0&-\frac{1}{y_{3}}&\cdots&\frac{1}{y_{1}+\dotsb+y_{n}}\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\frac{1}{y_{1}+\dotsb+y_{n}}\end{vmatrix}dy_{1}\wedge\dotsb\wedge
dy_{n},$

and the determinant equals $\prod\frac{1}{y_{j}}$ by Gaussian elimination. ∎

Since the Lagrangian $L_{1}\subset H_{1}$ is graded with respect to
$\Omega_{1}$, we immediately conclude:

######  Corollary 2.5.

The Lagrangian $L_{m,n}$ is graded with respect to the holomorphic volume form
$\Omega_{m,n}$. ∎

###  2.3. Bounding cochains

In order to associate to an immersed Lagrangian an object of the Fukaya
category with coefficients in a commutative ring $\mathbf{k}$, one needs to
choose an element of its deformation space, i.e. a local system as well as a
linear combination of degree $1$ generators of the self-Floer cochains, with
the property that the associated weighted count of holomorphic discs with one
output vanishes. The weighted count is encoded by a function, with value the
Floer cochains of degree $2$, defined on the deformation space. The simplest
instance of this deformation space occurs when restrict attention to local
systems of rank $1$ over $L$, in which case it consists of a product of copies
of $\mathbf{k}$ indexed by the degree $1$ generators of the self-Floer
cochains which are associated to self-intersections, with copies of
$\mathbf{k}^{*}$ indexed by the generators of first homology; we refer to this
function as the _potential function_ and its zero locus as the space of
bounding cochains. A further natural condition to impose is that the resulting
object be a non-trivial object of the Fukaya category, and we shall call the
corresponding space the space of non-trivial bounding cochains.

We shall not perform an exhaustive analysis of the space of bounding cochains
on $L_{m,n}$, as this will not be necessary for the purpose of understanding
mirror symmetry. Instead, we shall focus our attention on a specific subspace
which will turn out to be naturally identifiable with the mirror space.

######  Remark 2.6.

The usefulness of restricting to a subspace of the space of bounding cochains
is already apparent in the case of the $1$-dimensional pair of pants $H_{1}$;
a priori, the potential function is defined on a product
${\mathbf{k}}_{u}\times{\mathbf{k}}_{v}\times{\mathbf{k}}^{*}_{\mu}$, and it
is not too difficult to compute that the potential function is given by
$uv(1-\mu)$. The set of non-trivial objects is further constrained by imposing
the equation $uv=0$. The outcome is that the space of non-trivial bounding
cochains is a product of ${\mathbf{k}}^{*}$ with the union of the coordinate
lines in ${\mathbf{k}}_{u}\times{\mathbf{k}}_{v}$. In terms of the mirror
space $\Pi_{1}$, this can be explained as follows: the moduli space
parametrised by this Maurer-Cartan space is that of triples
$(p_{1},p_{2},\mu)$, with $p_{i}\in Y$ and $\mu\cdot p_{1}=p_{2}$. Thus,
specialising to $\mu=1$ recovers a copy of $\Pi_{1}$ embedded diagonally.

Restricting this space will prove even more useful in higher dimension,
because $L_{1}$ has self-intersections of negative degree, which causes
$CF^{1}(L_{m,n})$ to be quite large.

####  2.3.1. A distinguished subspace of the deformation space

Let us fix some notation: Seidel’s Lagrangian $L_{1}$ has three self-
intersections; we denote the corresponding pairs of generators of the self-
Floer complex of $L_{1}$ by $(s,\partial_{s})$, $(t,\partial_{t})$, and
$(c,\partial_{c})$; our notational convention is that the elements $(s,t,c)$
have even degree, and the elements $(\partial_{s},\partial_{t},\partial_{c})$
have odd degree.

$c$$\ast$$0$$\ast$$1$$s$$t$$c$$\ast$$0$$\ast$$1$$s$$t$$c$$\ast$$0$$\ast$$1$$s$$\ast$$0$$\ast$$1$
Figure 3. The four holomorphic discs with allowed inputs and one output (the
reflection of each disc across the horizontal axis may also contribute).

The easiest way to determine the degrees of these generators is as follows:
observe that our holomorphic volume form on $H_{1}$ extends meromorphically to
${\mathbb{P}}^{1}$, with simple poles at $1$ and $\infty$. Since the generator
$c$ is the output of a pair of genuine teardrops passing through $0$, one of
which is shown on the upper right of Figure 3, and the holomorphic volume form
extends without singularities to the origin, this generator has degree $2$
(the other teardrop is obtained by vertical reflection). The generator $s$
bounds a pair of teardrops passing through the puncture at $1$, so it has
degree $0$. Similarly, $t$ bounds a pair of teardrops surrounding the puncture
at infinity, so it also has degree $0$. The degree of the Poincaré dual
generators are therefore determined to be $1$, $1$, and $-1$.

Floer generators on the product $L_{1}^{n}$ are given as tensor products of
generators on each component. Write $s_{j}$ for the generator

(2.3.1) $\underbrace{e\otimes\dotsm\otimes e}_{j-1}\otimes
s\otimes\underbrace{e\otimes\dotsm\otimes e}_{n-j},$

where $e\in CF^{0}(L_{1})$ is the Morse unit. Similarly, write
$\partial_{s_{j}}$, $t_{j}$, $\partial_{t_{j}}$, $c_{j}$, and
$\partial_{c_{j}}$ for the other simple self-intersections.

The torus $T^{m}\subset({\mathbb{C}}^{*})^{m}$ has a natural splitting. We
call the corresponding monodromy generators for rank-$1$ local systems
$\\{\partial_{\zeta_{i}}\\}_{i=1}^{m}$. We view this as a
${\mathbf{k}}^{*}$-coordinate on the space of branes supported on $L_{m,n}$.

We now distinguish a subspace $\mathrm{Def}_{m,n}$ of the deformation space of
$L_{m,n}$ on which we will show that the potential function vanishes. This
space is given by

(2.3.2)
$\mathrm{Def}_{m,n}\equiv\\{\sum_{i=1}^{m}\rho_{i}\partial_{\zeta_{i}}+\sum_{j=1}^{n}\left(u_{j}\partial_{s_{j}}+v_{j}\partial_{t_{j}}\right)|\rho_{i}\in{\mathbf{k}}^{*}\textrm{
and }u_{j},v_{j}\in{\mathbf{k}}\\}.$

####  2.3.2. Discs with boundary on $L_{m,n}$

Our plan to compute the Maurer–Cartan formula is to reduce the count of discs
in $X_{m,n}$ to disc counts in ${\mathbb{C}}$. To that end, we use the
following result, which uses the projection map which appears in Lemma 2.2:

######  Lemma 2.7.

A holomorphic map $u\colon D^{2}\to X_{m,n}$ with boundary on $L_{m,n}$ is
determined by its projection

(2.3.3) $(x_{1},\dotsc,x_{m},q_{1},\dotsc,q_{n})\circ u$

to ${\mathbb{C}}^{m}\times{\mathbb{C}}^{n}$, which is a holomorphic disc with
boundary on the product $T^{m}\times L_{1}^{n}$.

###### Proof.

Note that $L_{m,n}$ lives away from $D$, so a holomorphic polygon $u$ with
boundary on $L_{m,n}$ cannot live entirely in $D$. By unique continuation, $u$
intersects $D$ at only finitely many points, which implies that $u$ is
determined on a dense open subset by the projection.

∎

Since we are considering a product Lagrangian in
${\mathbb{C}}^{m}\times{\mathbb{C}}^{n}$, all holomorphic discs split. The
${\mathbb{C}}^{m}$-component is a disc with boundary on the Clifford torus,
which is well-understood. Let us analyze the possible discs in the remaining
${\mathbb{C}}$-components. These are discs with boundary on $L_{1}$, and the
assumption that they contribute to the Maurer–Cartan function on
$\mathrm{Def}_{m,n}$ means that all inputs are either $e$, $s$, or $t$. The
next statement is easiest to understand while referring to Figure 3:

######  Lemma 2.8.

Any nonconstant holomorphic disc $u\colon D^{2}\to{\mathbb{C}}$ with boundary
on $L_{1}$ with zero or one output and with inputs from $\\{s,t\\}$
corresponds to one of the following possibilities:

Green:

A 3-gon in the complement of $0$ and $1$.

Yellow:

A 1-gon intersecting $0$ with output on $c$.

Pink:

A _branched cover_ of a 1-gon intersecting $1$ with one corner at $s$.

Blue:

A 4-gon intersecting $1$ with two corners at $s$, one corner at $t$, and one
corner at $c$.

∎

We are now prepared to classify the possible discs contributing to the
Maurer–cartan equation. We note that the following result in particular
asserts that the Blue disc appearing above does not contribute to the desired
count:

######  Proposition 2.9.

Any holomorphic disc $u$ contributing to the Maurer–Cartan function $\mu^{0}$
on $\mathrm{Def}_{m,n}$ is nonconstant in at most two of the factors

(2.3.4) ${\mathbb{C}}_{q_{1}},\dotsc,{\mathbb{C}}_{q_{n}},{\mathbb{C}}^{m}$

and those factors must be consecutive in the above ordering. These discs come
in one of three types.

  1. (1)

$u$ does not intersect $D$. It has one nonconstant factor, and that factor is
a Green disc from Lemma 2.8.

  2. (2)

$u$ intersects $D$, and it has two nonconstant $q$ factors. The first of those
factors is a Yellow disc in ${\mathbb{C}}_{q_{j}}$, and the second is a
_simple_ Pink disc in ${\mathbb{C}}_{q_{j+1}}$.

  3. (3)

$u$ intersects $D$, and its first nonconstant factor is a Yellow disc in
${\mathbb{C}}_{q_{n}}$. Its ${\mathbb{C}}^{m}$ component is either constant or
a Maslov $2$ disc with boundary on the Clifford torus.

###### Proof.

A disc which contributes to $\mu^{0}$ must have a degree $2$ generator as its
output. Lemma 2.8 implies that this output must be the product of the
generator $c$ in one factor with the identity in the others. We can then
consider the cases according to the intersection number with $D$.

First, if the intersection number vanishes, then the only possibility is a
Green disc in one factor with all other factors constant. Next, if the
intersection number is $1$, and the component meeting $u$ is $D_{j}$, then
Lemma 2.1 implies that $u$ must agree with the Yellow disc in the factor
labelled by $q_{j-1}$. Since the Yellow disc has an output labelled $c$, the
output of any other factor must have degree $0$, i.e. be a marked point along
the smooth part. There are now three situations to consider:

  1. (1)

if $k<n$, then by the definition of $D_{j}$, the projection of $u$ to the
$q_{j}$ factor must be non-trivial, and pass through the point $1$. Given the
above constraint that its output must have degree $0$, this second factor must
be a Pink disc.

  2. (2)

if $j=n$, and $u$ meets the component of the normal crossing divisor
corresponding to $x_{0}=0$, then the projection to ${\mathbb{C}}^{m}$ must be
trivial.

  3. (3)

if $j=n$ and $u$ meets the component of the normal crossing divisor
corresponding to $x_{i}=0$ for $i\neq 0$, then the projection to
${\mathbb{C}}^{m}$ is a disc of index $2$ in the factor labelled by $i$.

We preclude the possibility of a contribution from discs whose intersection
number with $D$ is greater than $1$ as follows: on the one hand, there cannot
be any factor in which there is a multiply-covered disc, because this would
imply a contribution of a disc with input the Poincaré dual class to $c$,
which contradicts our constraints about the inputs. On the other hand, if $u$
intersects more than one component of $D$, then there is more than one factor
in which the output is labelled by $c$, contradicting the fact that the output
must have total degree $2$. ∎

Clearly, in the first case above, the disc exists and is uniquely defined. In
the second case, we have a map

(2.3.5) $u\colon D^{2}\to{\mathbb{C}}\times T,$

where $T$ is either ${\mathbb{C}}$ or ${\mathbb{C}}^{m}$. We wish to prove
existence and uniqueness in this case as well.

######  Lemma 2.10.

Let $\Sigma$ be a holomorphic disc with at least one boundary marked points
and one interior marked point. For any collection of discs

(2.3.6) $u_{j}\colon\Sigma\to{\mathbb{C}}_{q_{j}},\qquad
v\colon\Sigma\to{\mathbb{C}}^{m}$

as in the second or third case of Proposition 2.9, mapping the interior marked
point to the divisor $D$, there is a unique lift

(2.3.7) $u\colon\Sigma\to X_{m,n}.$

Moreover, the corresponding holomorphic disc with boundary on $L_{m,n}$ is
regular.

###### Proof.

The interior marked point $z$ is the only point which could possibly map to
$D$, so we know that there is a uniquely defined map
$u_{0}\colon\Sigma\setminus z\to X_{m,n}\setminus D$. By the removable
singularities theorem, it suffices to show that the image of $u_{0}$ is
bounded away from the ends of
${\mathbb{C}}^{m+1}\times\left({\mathbb{C}}^{*}\right)^{n}$. Concretely, this
amounts to checking that the functions $x_{i}\circ u_{0}$, $y_{j}\circ u_{0}$,
and $\frac{1}{y_{j}\circ u_{0}}$ are each bounded. To that end, note that
$x_{1}\circ u_{0}$ through $x_{m}\circ u_{0}$ are bounded by assumption, while
the others we can bound by formula. Indeed,

$\displaystyle x_{0}$ $\displaystyle=\frac{q_{n}}{x_{1}\dotsm x_{m}}$
$\displaystyle y_{j}$
$\displaystyle=\begin{cases}\frac{q_{n}-1}{q_{n-1}}&\text{ if }j=n\\\
\frac{q_{j}-1}{q_{j-1}}y_{j+1}&\text{ if }1<j<n\\\
\left(q_{1}-1\right)y_{2}&\text{ if }j=1.\end{cases}$

Let us examine each of these formulas in turn. The formula for $x_{0}$ gives

(2.3.8) $x_{0}\circ u_{0}=\frac{u_{n}}{\prod v_{i}}.$

The denominator is either a nonzero constant or we are in case 3 of
Proposition 2.9, in which case at most one $v_{i}$ is allowed a single simple
zero by the Maslov 2 assumption. If this happens, then that zero is an
intersection point with $D$, which by assumption means that it occurs at the
marked point and that $u_{n}$ also has a simple zero at the marked point. The
situation for $y_{j}$ in case 2 is similar, using the fact that $u_{j}-1$ is
allowed has simple $0$ precisely when $u_{j-1}$ has a simple $0$.

To prove regularity, observe that since the inputs have Maslov index $1$ and
the output has Maslov index $2$, the holomorphic disc under consideration has
Fredholm index equal to the rank of the space of holomorphic vector fields on
the domain, which is $1$ in case 2 (because there is one input and one
output), and $2$ in case 3 (because there is no input). It thus suffices to
show that the kernel of the linearisation of the $\overline{\partial}$
operator (which are holomorphic vector field on the domain valued in the
tangent space of the fibre) consists precisely of such vector fields in the
domain. This follows from the same analysis that yields the above description
of the space of holomorphic discs: such vector fields split as a product away
from the marked point, but the matching condition along the marked point shows
that the vector fields must move the marked points in the same way, so they
either correspond to an automorphism of the domain, or to an infinitesimal
deformation. No such deformations exist because the curves in each factor are
unique. ∎

####  2.3.3. Proof that the curvature vanishes

We conclude this section with the following computation:

######  Lemma 2.11.

Every element of $\mathrm{Def}(L_{m,n})$ is a bounding cochain.

###### Proof.

We consider the contributions of the three cases in Proposition 2.9, which we
will see vanish by symmetry.

#### Case 1

The only holomorphic polygons in any component which don’t intersect $D$ are
constants and the 3-gons of the form appearing in Figure 3 and their
reflections. The contribution of these curves to $\mu_{0}$ is given by:

(2.3.9) $\sum_{j=1}^{n}\left(u_{j}v_{j}\cdot c_{j}-u_{j}v_{j}\cdot
c_{j}\right)=0.$

The sign is determined by the computation of the product in [29]: given a
choice of orientation of the Lagrangian, together with a distinguished marked
point, the sign is given as in [28, 3] by a combinatorial count depending on
the difference between the orientation of the Lagrangian and the boundary
orientation of the disc along each segment except the one which ends at the
output with respect to the counterlockwise orientation, together with a factor
of $\pm 1$ according to whether the marked point lies on the boundary of the
disc. In our case, the boundary of $L_{1}$ is partitioned between the two
possible Green discs, so only one of them can contain the marked point, which
implies that the sign contributions are opposite.

#### Case 2

These discs have input $s_{j+1}$, and output $c_{j}$. A choice of point in the
factor of $L_{m,n}$ labelled by $q_{j+1}$, representing the unit in the
cohomology of $S^{1}$ distinguishes one of the two Pink discs as contributing
to the count (since the union of the boundaries of these two discs together
bijectively cover the complement of the self-intersections). This means that
the contributions in this case are given by

(2.3.10) $\sum_{j=1}^{n}\left(u_{j+1}\cdot c_{j}-u_{j+1}\cdot c_{j}\right)=0,$

with the two terms coming from the two possible Yellow discs in the factor
labelled by $q_{j}$. The sign is determined as above by the contribution in
each factor.

#### Case 3

In this case, there are again two discs corresponding to the two possible
Yellow discs in the factor labelled by $q_{j}$, and their contribution is

(2.3.11) $\left(1+\sum_{i=1}^{m}\rho_{i}\right)\cdot
c_{n}-\left(1+\sum_{i=1}^{m}\rho_{i}\right)\cdot c_{n}=0.$

In the above expression, the terms which equal $\pm 1\cdot c_{n}$ arise from
the two discs which meet the $x_{0}=0$ component of the normal crossing
divisor $\prod x_{i}=0$, and the terms $\pm\rho_{i}\cdot c_{n}$ from the discs
with meet the component $x_{i}=0$ for $i\neq 0$.

∎

###  2.4. Floer homology with the $0$-section

The outcome of Lemma 2.11 is that every element of $\mathrm{Def}_{m,n}$
defines an object of the Fukaya category. We shall eventually identify the
quasi-isomorphism class of these objects; in this section, we shall, as a
first step, show that they do not vanish by computing the Floer cohomology
with a non-compact Lagrangian which is mirror to the structure sheaf:

######  Definition 2.12.

The _zero section_ of $X_{m,n}$ is the real Lagrangian $L^{+}_{m,n}$ given by
the conditions

(2.4.1) $x_{i},y_{j}\in(0,\infty).$

To simplify the notation, we shall usually write $L^{+}$ for this Lagrangian,
omitting the subscripts. In order for $L^{+}$ to define an object of the
Fukaya category, we must show that it is well-behaved at infinity:

######  Lemma 2.13.

The inclusion of $L^{+}$ in $X_{m,n}$ is proper.

###### Proof.

Any limit point of $L^{+}$ in $X_{m,n}$ must have some coordinate $x_{i}$ that
vanishes. By the defining equation of $X_{m,n}$, this implies that $1+\sum
y_{j}=0$, which means that some coordinate $y_{j}$ must be negative,
contradicting the assumption that $0<y_{j}$. ∎

One also needs to ensure that $L^{+}$ is invariant under the Liouville flow of
the chosen primitive for the symplectic form; this follows immediately from
choosing the primitive to be real.

We note that $L^{+}$ is contained in the complement of the divisor $D$, i.e.
in the open subset $({\mathbb{C}}^{*})^{m}\times H^{n}_{1}$ of $X_{m,n}$, and
agrees with the product of the lines $0<x_{i}$ and $1<q_{j}$ in these
coordinates. In drawing the figures, we shall describe $L^{+}$ as a product of
vertical arcs emanating from the marked point labelled $1$; this can be
achieved by deforming $L^{+}$ by Hamiltonian isotopy (deforming the primitive
as well to preserve the fact that $L^{+}$ is both a product of Lagrangians and
is conical).

The factors of the Lagrangians $L^{+}$ and $L_{m,n}$ corresponding to the
coordinates $q_{i}$ intersect transversely at two points which we label
$a_{i}$ and $b_{i}$, while the factors in $({\mathbb{C}}^{*})^{m}$ meet at one
point. For an appropriate choice of grading, the degree of $a_{i}$ is $0$ and
that of $b_{i}$ is $1$.

$a$$b$$\ast$$0$$\ast$$1$$a$$b$$s$$t$$\ast$$0$$\ast$$1$$s$$\ast$$0$$\ast$$1$$a$$s$$\ast$$0$$\ast$$1$$b$$s$$t$$\ast$$0$$\ast$$1$$a$$b$
Figure 4. The holomorphic discs that can a priori arise in each factor when
computing in the differential of the Floer complex of the pair $L_{m,n}$ and
$L^{+}_{m,n}$.

We now state the analogue of Lemma 2.8, where we now refer to Figure 4:

######  Lemma 2.14.

Any nonconstant holomorphic map
$u\colon{\mathbb{R}}\times[0,1]\to{\mathbb{C}}$ mapping
${\mathbb{R}}\times\\{0\\}$ to $L_{1}$ and ${\mathbb{R}}\times\\{1\\}$ to
$L^{+}_{m,n}$, and with inputs along $L_{1}$ labelled by the self-
intersections $\\{s,t\\}$ of $L_{1}$ has image given by one of the following
four possibilities:

Green:

A 4-gon in the complement of $0$ and $1$, with inputs $\\{a,t,s\\}$ and output
$b$.

Yellow:

A bi-gon intersecting $0$ with input $a$ and output $b$.

Pink:

A 1-gon intersecting $1$ with (i) input $\\{a,s\\}$ and output $a$ or (ii)
input $\\{b,s\\}$ and output $\\{b\\}$.

Blue:

A 5-gon intersecting $1$ with inputs $\\{a,s,t,s\\}$ and output $\\{b\\}$.

∎

We then have the analogue of Proposition 2.9

######  Proposition 2.15.

Any holomorphic disc $u$ contributing to the differential in the Floer complex
of $L_{m,n}$ equipped with a bounding cochain in $\operatorname{Def}_{m,n}$
and $L^{+}_{m,n}$ is nonconstant in at most two of the factors

(2.4.2) ${\mathbb{C}}_{q_{1}},\dotsc,{\mathbb{C}}_{q_{n}},{\mathbb{C}}^{m}$

and those factors must be consecutive in the above ordering. These discs come
in one of three types.

  1. (1)

$u$ does not intersect $D$. It has one nonconstant factor, and that factor is
a Green disc from Lemma 2.14.

  2. (2)

$u$ intersects $D$, and it has two nonconstant $q$ factors. The first of those
factors is a Yellow disc in ${\mathbb{C}}_{q_{j}}$, and the second is a Pink
curve in ${\mathbb{C}}_{q_{j+1}}$.

  3. (3)

$u$ intersects $D$, and its first nonconstant factor is a Yellow disc in
${\mathbb{C}}_{q_{n}}$. Its ${\mathbb{C}}^{m}$ component is either constant or
has Maslov index $2$, in which case it is non-trivial in exactly one
component.

###### Proof.

The argument is entirely analogous to the one for Proposition 2.9; we shall
only remind the reader that the Blue discs do not contribute because there
must be an accompanying Yellow disc in the previous factor, which implies that
the difference between the degrees of the input and the output generators of
the Floer cochains of $L_{m,n}$ and $L^{+}_{m,n}$ is $2$, so that this
homotopy class does not contribute to the differential. ∎

There is an existence and regularity statement for these discs which is
exactly analogous to Lemma 2.10, which gives the following result,
corresponding to Lemma 2.11:

######  Lemma 2.16.

The differential in the Floer cochains of $L_{m,n}$ and $L^{+}_{m,n}$
associated to an element of $\mathrm{Def}_{m,n}$ vanishes if and only if

(2.4.3) $\displaystyle u_{j}v_{j}-u_{j+1}$ $\displaystyle=0\textrm{ for }0\leq
j<n$ (2.4.4) $\displaystyle u_{1}\cdot\prod_{j=1}^{n}v_{j}$
$\displaystyle=1+\sum_{i=1}^{m}\rho_{i}.$

###### Proof.

We compute that the differential is given by

(2.4.5) $a_{j}\mapsto\begin{cases}\left(u_{j}v_{j}-u_{j+1}\right)\cdot
b_{j}&\textrm{ if }j<n\\\
\left(u_{n}v_{n}-\left(1+\sum_{i=1}^{m}\rho_{i}\right)\right)\cdot
b_{n}&\textrm{ if }j=n,\end{cases}$

where the specific sign is achieved by placing the marked point on $L_{1}$,
which records the fact that we are using the bounding $\operatorname{Spin}$
structure, along the upper segment connecting $s$ and $t$. ∎

Define $\mathcal{M}_{m,n}\subset\mathrm{Def}_{m,n}$ to consist of those
bounding cochains on which the differential in the Floer complex between
$L^{+}_{m,n}$ and $L_{m,n}$ identically vanishes.

######  Remark 2.17.

In fact, one can prove from the above description of the differential and from
the fact that $L^{+}_{m,n}$ corresponds under mirror symmetry to the structure
sheaf, which we prove in Section 3 below, that $\mathcal{M}_{m,n}$ corresponds
to the set of elements of $\mathrm{Def}_{m,n}$ whose associated object of the
Fukaya category is non-trivial.

######  Corollary 2.18.

The correspondence $x_{0}=u_{1}$, $x_{i}=v_{i}$ for $1\leq i\leq n$, and
$y_{j}=\rho_{j}$ for $1\leq j\leq m$ defines an isomorphism of
$\mathcal{M}_{m,n}$ with $X_{n,m}$. ∎

######  Remark 2.19.

Returning to Remark 1.6, recall that the Lagrangian $L_{m,n}^{i}$ is the
product of $T^{m}$ with the Lagrangians obtained by performing Polterovich
surgery on $L_{m,n}$ at $s$ for coordinates labelled by $0\leq k\leq i$, and
at $t$ for coordinates labelled by $i<k\leq n$. The surgery triangle [13] of
Fukaya, Oh, Ohta and Ono identifies the isomorphism class in the Fukaya
category of this surgery (with an appropriate local system) with the result of
equipping $L_{m,n}$ with a bounding cochain with $u_{0},\ldots,u_{i}\neq 0$,
and $v_{i+1},\ldots,v_{n}\neq 0$. Using the above correspondence, together
with Equation (2.4.3), this is the same as setting $x_{0}x_{1}\cdots x_{k}\neq
0$ for $k<i$, and $x_{k}\neq 0$ for $k>i$, which corresponds to setting all
coordinates except for $x_{i}$ to be non-zero as asserted.

We end this section by noting that the equivalence in Corollary 2.18 is at
this stage just an abstract isomorphism, and we do not know yet that it is
induced by the mirror equivalence. We shall presently see that we have a
natural map between these spaces. To construct it, we need the following basic
Lemma:

######  Lemma 2.20.

Let $R$ be the ring of functions on a smooth affine variety over a field
$\mathbf{k}$, and let

(2.4.6) $M_{+}\to M\to M_{0}$

be a short exact sequence of $A_{\infty}$ graded $R$-modules which are finite
over $\mathbf{k}$, and with trivial differential. If $M_{0}$ is supported in
degree $0$, and $M_{+}$ is supported in strictly positive degrees, then
$M_{0}$ is a summand of $M$.

###### Proof.

The module $M$ is classified by an element of
$\operatorname{Ext}^{1}(M_{0},M_{+})$, but the fact that $\operatorname{Ext}$
groups of coherent sheaves are supported in non-negative degrees, and our
support conditions imply that the only contribution comes from a map of
sheaves, which would correspond to a differential in $M$. ∎

######  Corollary 2.21.

The mirror functor defines a map

(2.4.7) $\mathcal{M}_{m,n}\to X_{n,m}$

which assigns to $b\in\mathcal{M}_{m,n}$ the support of the skyscraper sheaf
associated to the degree $0$ component of the Floer group
$HF^{*}\left((L_{m,n},b),L^{+}_{m,n}\right)$. ∎

###  2.5. The mirror map via symplectic cohomology

Our goal at this stage is to show that the mirror map in Equation (2.4.7) is
an isomorphism of varieties. Since the varieties are affine, it is easiest to
state this in terms of rings of functions: let $\Gamma(X_{n,m})$ denote the
ring of functions on $X_{n,m}$, which we know by homological mirror symmetry
(i.e. the results of Section 3) to be isomorphic to the self-Floer cohomology
of $L^{+}_{m,n}$, as well as to the degree $0$ part of the symplectic
cohomology $SH^{0}(X_{m,n})$. Let $\Gamma(\mathcal{M}_{m,n})$ denote the ring
of functions on $\mathcal{M}_{m,n}$. Floer theory defines a closed-open map,
that we call the _mirror map_ ,

(2.5.1) $SH^{0}(X_{m,n})\to\Gamma(\mathcal{M}_{m,n}).$

The fact that this agrees with the mirror map defined using Lagrangian Floer
cohomology is given by the homotopy in Figure 5.

$L_{m,n}$$L^{+}_{m,n}$$SH^{*}(X_{m,n})$$L_{m,n}$$L^{+}_{m,n}$$L_{m,n}$$L^{+}_{m,n}$
Figure 5. The action of symplectic cohomology on Floer cohomology

######  Proposition 2.22.

The mirror map in Equation (2.5.1) is an isomorphism.

######  Remark 2.23.

Both the statement and the proof of this result only involve compact
Lagrangians. This gives us flexibility in our choice of model for symplectic
cohomology classes as we shall not need to construct the closed-open map for
general (non-compact) Lagrangians, which usually involves more delicate
arguments to ensure compactness of moduli spaces of holomorphic curves.

Our proof of this result will be slightly roundabout: as discussed in
Corollary 2.18, we have natural functions $u_{1}$, $v_{j}$ for $1\leq j\leq
n$, and $\rho_{i}$ for $1\leq i\leq m$, on $\mathcal{M}_{m,n}$. The key
technical result we shall prove is:

######  Lemma 2.24.

Each of the functions $u_{1}$, $v_{j}$ for $1\leq j\leq n$, and $\rho^{\pm
1}_{i}$ for $0\leq i\leq m$ lifts to $SH^{0}(X_{m,n})$ under the mirror map.

Before proving this (lifting) result, we explain how it implies the main
result of this section:

###### Proof of Proposition 2.22.

We use the fact that $SH^{0}(X_{m,n})$ and $\Gamma(\mathcal{M}_{m,n})$ are
abstractly isomorphic Noetherian rings. Lemma 2.24 implies that the mirror map
is surjective. A surjective self-map of a Noetherian ring cannot have a non-
trivial kernel, as the inverse images of the kernel under iterates of the map
would give rise to an infinite ascending chain of ideals. We conclude that the
kernel is trivial, hence that it is an isomorphism.

∎

####  2.5.1. Construction of the lifts

The purpose of this section is to prove Lemma 2.24; the proof will make use of
the fact that, in a partial compactification $\bar{X}$ of a symplectic
manifold $X$ by normal crossing divisors, whose total space is appropriately
convex at infinity (e.g. which intersects each curve positively), each
component of the divisor at infinity defines an element of $SH^{*}(X)$. The
idea goes back to work of Seidel [27], and has been repeatedly used in the
literature, with the most complete current account given in [18]. We shall in
particular use the fact, proved in loc. cit., that given a compact Lagrangian
$L\subset X$ underlying an object of the Fukaya category, the image of the
element of $SH^{*}(X)$ associated to a divisor in $\bar{X}$ in the Floer
homology of $L$, is given by the count of discs in $\bar{X}$ with boundary on
$L$, whose intersection number with the divisor is $1$.

At this stage, we complete the proof of the existence of lifts of the desired
elements to symplectic cohomology, which will feature a choice of a different
partial compactification for each case:

###### Proof of Lemma 2.24.

For the element $u_{1}$, we consider the space $\bar{X}_{m,n}^{u_{1}}$ which
is the closure of $X_{m,n}$ in

(2.5.2)
${\mathbb{C}}_{y_{1}}\times\prod_{i=2}^{n}{\mathbb{C}}^{*}_{y_{i}}\times{\mathbb{C}}^{m+1}_{x},$

i.e. were we allow the variable $y_{1}$ to vanish. The maps $q_{j}$ extend to
$\bar{X}_{m,n}^{u_{1}}$, and the divisor $\bar{X}_{m,n}^{u_{1}}\setminus
X_{m,n}$ is a copy of $X_{m,n-1}$ (i.e. given by the equation $\prod
x_{i}=1+\sum y_{j}$, where the coordinates $y_{j}$ range from $2$ to $n$). The
map $q_{1}$ is given by $1+\frac{y_{1}}{y_{2}}$, so that the divisor at
infinity projects to $1$ (and is in fact its entire inverse image). The count
of holomorphic discs whose intersection number with this divisor is $1$
corresponds to holomorphic discs which in the $q_{1}$ factor have a simple
$1$, and are constant in all other factors. With boundary conditions
$L_{m,n}$, the only discs which could contribute to the closed-open map are
the two Pink discs, one of which is shown in the bottom left of Figure 3.
Requiring this disc to pass through a chosen marked point on $L_{m,n}$
representing the unit on cohomology, we find a single contribution to this
count, which is regular, and the fact that it has exactly one corner at $s$
shows that this divisor lifts the function $u_{1}$ to symplectic cohomology.

We define the partial compactification $\bar{X}_{m,n}^{v_{j}}$ for $1\leq
j\leq n-1$ analogously by allowing the coordinate $y_{j+1}$ to vanish. The map
$q_{j}$ again extends to $\bar{X}_{m,n}^{v_{j}}$, but now with target
${\mathbb{C}}{\mathbb{P}}^{1}$; the divisor $\bar{X}_{m,n}^{v_{j}}\setminus
X_{m,n}$ maps to infinity. All other maps $q_{j^{\prime}}$ for $j\neq
j^{\prime}$ are still well-defined with target ${\mathbb{C}}$. The count of
discs which meet the divisor at infinity once now corresponds to holomorphic
discs which in the $q_{j}$ factor pass through infinity once, and are constant
in all other factors; requiring them to pass through a marked point
representing $1$, there is a unique such disc, which moreover has one corner
at $t$. This divisor thus lifts the function $v_{j}$ to symplectic cohomology.

Next, we define the partial compactification $\bar{X}_{m,n}^{v_{n}}$ by adding
the product $\left({\mathbb{C}}^{*}\right)^{m}\times H_{n-1}$ as a fibre of
$q_{n}$ over infinity; note that this construction makes sense as an algebraic
variety because the complement of the fibres $q_{n}^{-1}(0)$ and $q^{-1}(1)$
is isomorphic to

(2.5.3) $\\{\prod x_{i}=1\\}\times\\{\sum
y_{j}=-1\\}\times{\mathbb{C}}\setminus\\{0,1\\}$

with isomorphism over $\lambda\in{\mathbb{C}}\setminus\\{0,1\\}$ given by
scaling the variable $x_{0}$ by $\lambda$ and $(y_{1},\ldots,y_{j})$ by
$1-\lambda$. The remaining functions $q_{i}$ also extend, because they are
preserved by the trivialisation. The discs whose intersection number with
$\bar{X}_{m,n}^{v_{n}}\setminus X_{m,n}$ is $1$ are again constant in all
factors except the $q_{n}$ factor, in which they pass through infinity once;
those with boundary conditions on $L_{m,n}$, and passing through the marked
point associated to the unit have one corner at $t$, hence lift $v_{n}$ to
symplectic cohomology.

In order to lift the functions $\rho^{-1}_{i}$ for $1\leq i\leq m$, we
consider the toric inclusion

(2.5.4)
${\mathbb{C}}^{2}_{x_{0},x_{i}}\subset\mathrm{Tot}_{{\mathbb{C}}{\mathbb{P}}^{1}}(\mathcal{O}(-1))$

into the total space of the (toric) line bundle over the projective space
${\mathbb{C}}{\mathbb{P}}^{1}_{x_{i}}$, which over ${\mathbb{C}}^{1}_{x_{i}}$
identifies the $x_{0}$ factor with the fibre. The map $\prod x_{i}$ on
${\mathbb{C}}^{m}$ extends as a torically equivariant function from the
product of this total space with ${\mathbb{C}}^{m-2}$ to ${\mathbb{C}}$, so we
can define a partial compactification

(2.5.5)
$\bar{X}_{m,n}^{\rho_{i}^{-1}}\subset\mathrm{Tot}_{{\mathbb{C}}{\mathbb{P}}^{1}}(\mathcal{O}(-1))\times{\mathbb{C}}^{m-1}\times\left({\mathbb{C}}^{*}\right)^{n}$

of $X_{m,n}$ to be the hypersurface where the function equals $1+\sum y_{i}$.
The divisor at infinity is a copy of $X_{m-1,n}$, corresponding to the
inclusion

(2.5.6)
${\mathbb{C}}\subset\mathrm{Tot}_{{\mathbb{C}}{\mathbb{P}}^{1}}(\mathcal{O}(-1))$

as the fibre over $x_{i}=\infty$. The maps $q_{j}$ are still well-defined, as
are the maps $x_{i^{\prime}}$ for $i\neq i^{\prime}$. The projection map to
the base ${\mathbb{C}}{\mathbb{P}}^{1}_{x_{i}}$ of the line bundle extends the
projection map to ${\mathbb{C}}_{x_{i}}$; the divisor at infinity maps to
infinity. The Lagrangian $L_{m,n}$ projects to the standard circle in this
factor, which bounds one disc whose winding number around the boundary is the
opposite of the usual orientation of the circle factors of the Clifford torus
(given as the boundary of the disc in ${\mathbb{C}}$), so we obtain a lift of
$\rho_{i}^{-1}$ to symplectic cohomology.

Finally, we consider the inclusion

(2.5.7)
${\mathbb{C}}^{2}_{x_{0},x_{i}}\subset\mathrm{Tot}_{{\mathbb{C}}{\mathbb{P}}^{1}}(\mathcal{O}(-1))$

so that $x_{0}$ now corresponds to the coordinate on the base, and $x_{i}$ to
the coordinate on the fibre away from infinity. We obtain a partial
compactification of $X_{m,n}$ which we denote $\bar{X}_{m,n}^{\rho_{i}}$
exactly as above. The function $x_{i}$ extends as a toric map

(2.5.8)
$\mathrm{Tot}_{{\mathbb{C}}{\mathbb{P}}^{1}}(\mathcal{O}(-1))\to{\mathbb{C}}$

which vanishes on the divisor at infinity, so that this map also extends to
$\bar{X}_{m,n}^{\rho_{i}}$, taking the divisor at infinity to $0$. We can then
identify the count of discs in ${\mathbb{C}}$ passing once through the origin
with the count of discs passing through this divisor, which now lifts the
function $\rho_{i}$ to symplectic cohomology. ∎

##  3\. Homological Mirror Symmetry

The goal of this section is to prove that Homological mirror symmetry holds
for the pairs $X_{m,n}$ and $X_{n,m}$ in the sense of Theorem 1.2. Our
strategy starts by noting that the total space of $X_{m,n}$ can be obtained by
completing a domain obtained by gluing symplectic domains that yield
$H_{n-1}\times{\mathbb{C}}^{m+1}$ and $\left({\mathbb{C}}^{*}\right)^{n+m}$
under completion, along the common hypersurface
$H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$. Mirror statements for the
partially wrapped Fukaya categories of the pairs
$\left({\mathbb{C}}^{m+1},\left({\mathbb{C}}^{*}\right)^{m}\right)$ and
$\left(\left({\mathbb{C}}^{*}\right)^{n},H_{n-1}\right)$ are by now well-
established, and our proof will combine these results with various structural
properties of Fukaya categories and categories of coherent sheaves.

More precisely, the following is the key technical result that we establish,
where we write $\Pi_{n-1}$ for the union of the coordinate hyperplanes
$\prod_{i=0}^{n-1}x_{i}=0$ in ${\mathbb{C}}^{n}$:

######  Proposition 3.1.

There is a commutative diagram of functors

(3.0.1)
${\mathcal{F}(H_{n-1}\times{\mathbb{C}}^{m+1},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\mathcal{F}(\left({\mathbb{C}}^{*}\right)^{n+m},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\operatorname{Coh}(\Pi_{n-1}\times
H_{m-1})}$${\operatorname{Coh}(\Pi_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\operatorname{Coh}({\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m})}$

in which the vertical maps are equivalences, the top horizontal maps are cup
(Orlov) functors, and the bottom vertical maps are respectively pullback and
pushforward functors.

In order to understand the relevance of the above result to Theorem 1.2, we
need a basic geometric construction:

######  Lemma 3.2.

The projection map

(3.0.2) $X_{n,m}\to{\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}$

associated to forgetting the variable $x_{0}$ induces an isomorphism between
$X_{n,m}$ and the complement of the proper transform of the hypersurface
$\Pi_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$ in the blowup of
${\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}$ along
$\Pi_{n-1}\times H_{m-1}$.

###### Proof.

The codimension $2$ subvariety $\Pi_{n-1}\times H_{m-1}$ is the complete
intersection associated to the two functions $f=\prod x_{i}$ and $g=1+\sum
y_{j}$. The blowup is the subvariety of
${\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}\times{\mathbb{P}}^{1}_{[X_{0},Z]}$
given by

(3.0.3) $X_{0}\cdot\prod x_{i}=Z\cdot\left(1+\sum y_{j}\right).$

The proper transform of the subvariety
$\Pi_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$ is given in these equations
by setting $Z=0$. The complement of this divisor has a standard coordinate
$x_{0}$, with respect to which we obtain our equation for $X_{n,m}$ as a
subvariety of ${\mathbb{C}}^{n+1}\times\left({\mathbb{C}}^{*}\right)^{m}$. ∎

In the next part of this section, we explain how Proposition 3.1 implies
Theorem 1.2. Afterwards, we explain the proof of the Proposition.

###  3.1. Gluing coherent sheaves and Fukaya categories

Let $X$ be a smooth variety and $Z\subset X$ a locally complete intersection
subvariety of codimension $2$. Let $X_{Z}$ denote the blowup of $X$ along $Z$
(i.e. along the ideal of functions which vanish to first order at $Z$). The
key input for our computations is Orlov’s formula [24] for the derived
category of a blowup, which asserts that the derived category of the blowup
has a semi-orthogonal decomposition

(3.1.1) $D^{b}\operatorname{Coh}X_{Z}\cong\langle
D^{b}\operatorname{Coh}Z,D^{b}\operatorname{Coh}X\rangle,$

where the embedding of $D^{b}\operatorname{Coh}Z$ in
$D^{b}\operatorname{Coh}X_{Z}$ is given by composing the pull back to the
exceptional divisor $E\subset X_{Z}$, with tensoring with the line bundle
$\mathcal{O}(-1)$, and then with push forward to $X_{Z}$, while the embedding
of $D^{b}\operatorname{Coh}X$ in $D^{b}\operatorname{Coh}X_{Z}$ is given by
pullback along the projection. Moreover, the defining bimodule is the
pushforward map from $Z$ to $X$.

######  Remark 3.3.

Orlov’s result was originally formulated only for smooth subvarieties, but is
well-known among experts that it holds as well in the local complete
intersection case (c.f. [20, Theorem 3.4]).

We now state a result which allows us to formulate a mirror to the blowup
operation:

$F\times A_{2}$$E_{1}$$E_{2}$$E_{1}\cup_{F}\left(F\times
A_{2}\right)\cup_{F}E_{2}$ Figure 6. The gluing procedure represented in the
stopped formalism.

######  Lemma 3.4 (Corollary 3.11 of [32]).

Let $E_{1}$ and $E_{2}$ be Liouville sectors with isomorphic Weinstein stop
$F$. Let $E_{1}\cup_{F}\left(F\times A_{2}\right)\cup_{F}E_{2}$ denote the
Weinstein sector obtained by gluing $E_{1}$, $E_{2}$, and $F\times A_{2}$
along $F$. Then the Fukaya category of $E_{1}\cup_{F}\left(F\times
A_{2}\right)\cup_{F}E_{2}$ admits a semi-orthogonal decomposition

(3.1.2) $\langle\mathcal{F}(E_{2}),\mathcal{F}(E_{1})\rangle,$

with defining bimodule given by the graph of the composition

(3.1.3)
$\cup_{E_{1}}\circ\cap_{E_{2}}^{L}\colon\thinspace\mathcal{F}(E_{2})\to\mathcal{F}(F)\to\mathcal{F}(E_{1})$

of the left cap functor for $E_{2}$, and the cup functor for $E_{1}$, which
assigns to a pair $(L,K)$ of Lagrangians in $E_{2}$ and $E_{1}$ the Floer
cohomology group

(3.1.4) $HF^{*}(\cup_{E_{1}}\circ\cap_{E_{2}}^{L}L,K).$

###### Proof.

Corollary 3.11 of [32] gives the desired semi-orthogonal decomposition with
gluing bimodule

(3.1.5)
$\Gamma^{\dagger}(\cup_{E_{1}})\otimes_{\mathcal{F}(F)}\Gamma(\cup_{E_{2}}),$

where the graph bimodules above are given by
$\Gamma^{\dagger}(\cup_{E_{1}})(K_{E_{1}},L_{F})=\hom_{E_{1}}(\cup_{E_{1}}L_{F},K_{E_{1}})$
and
$\Gamma(\cup_{E_{2}})(K_{F},L_{E_{2}})=\hom_{E_{2}}(L_{E_{2}},\cup_{E_{2}}K_{F})$.
Using the adjunction
$\Gamma(\cup_{E_{2}})\cong\Gamma^{\dagger}(\cap_{E_{2}}^{L})$, we get the
desired formula

(3.1.6)
$\displaystyle\Gamma^{\dagger}(\cup_{E_{1}})\otimes_{\mathcal{F}(F)}\Gamma(\cup_{E_{2}})$
$\displaystyle\cong\Gamma^{\dagger}(\cup_{E_{1}})\otimes_{\mathcal{F}(F)}\Gamma^{\dagger}(\cap_{E_{2}}^{L})$
(3.1.7)
$\displaystyle\cong\Gamma^{\dagger}(\cup_{E_{1}}\circ\cap_{E_{2}}^{L}).$

∎

In order to see the relevance of this result for mirror symmetry, we return to
the context of Orlov’s decomposition, and consider a smooth hypersurface
$H\subset X$ containing $Z$. The following result follows from the adjunction
between restriction and pushforward, and the functoriality of restriction,
which allows us to express the pushforward map from $Z$ to $X$ as a
composition of the pushforward from $H$ to $X$, and the pushforward from $Z$
to $H$ which is the adjoint of restriction from $H$ to $Z$:

######  Corollary 3.5.

There is a commutative diagram

(3.1.8)
${D^{b}\operatorname{Coh}(Z)}$${D^{b}\operatorname{Coh}(H)}$${D^{b}\operatorname{Coh}(X)}$${\mathcal{F}(E_{1})}$${\mathcal{F}(F)}$${\mathcal{F}(E_{2})}$

in which the vertical map are equivalences, the top maps are given by
pushforward and pullback, and the bottom maps are Orlov cup functors induces
an equivalence

(3.1.9)
$D^{b}\operatorname{Coh}(X_{Z})\cong\mathcal{F}(E_{1}\cup_{F}\left(A_{2}\times
F\right)\cup_{F}E_{2}).$

∎

In the example at hand, we are interested not in the blowup, but in the
complement $X_{Z}\setminus H$ of the proper transform of $H$ in $X_{Z}$. The
derived category of this complement is the quotient category

(3.1.10) $D^{b}\operatorname{Coh}(X_{Z}\setminus H)\cong
D^{b}\operatorname{Coh}X_{Z}/D^{b}\operatorname{Coh}H.$

In order to express the functor $D^{b}\operatorname{Coh}H\to
D^{b}\operatorname{Coh}X_{Z}$ associated to the proper transform in terms of
Orlov’s semi-orthogonal decomposition, note that the functor associated to
pushing forward from $H$ to $X$ then pulling back to $X_{Z}$ is represented by
the composite $D^{b}\operatorname{Coh}H\to D^{b}\operatorname{Coh}X\to
D^{b}\operatorname{Coh}X_{Z}$ corresponding to the second piece of the semi-
orthogonal decomposition. This functor geometrically corresponds to the
pullback from $H$ to its total transform. Since the total transform decomposes
as the union of the exceptional divisor and the proper transform, we are led
to consider the composite of the restriction functor
$D^{b}\operatorname{Coh}H\to D^{b}\operatorname{Coh}Z$, with pullback from $Z$
to $E$, tensoring with the natural line bundle $\mathcal{L}$ on $E$, then
pushing forward to $X_{Z}$, which corresponds to the first piece of Orlov’s
decomposition. The line bundle $\mathcal{L}$ arises from the projectivisation
of the normal bundle of $Z$ in $X$, so the section defining $TH|_{Z}$ yields a
natural transformation in the diagram of functors

(3.1.11)
${D^{b}\operatorname{Coh}H}$${D^{b}\operatorname{Coh}Z}$${D^{b}\operatorname{Coh}X}$${D^{b}\operatorname{Coh}X_{Z}.}$

The cone of this natural transformation represents the pushforward map on the
proper transform of $H$. This can be expressed in terms of the semi-orthogonal
decomposition as follows: the adjunction between pullback and pushforward
yields a natural transformation from the composition of restricting from $H$
to $Z$ and pushing forward from $Z$ to $H$, to the identity on
$D^{b}\operatorname{Coh}H$. Composing with the pushforward from $H$ to $X$
then yields the desired natural transformation between the two functors from
$D^{b}\operatorname{Coh}H$ to $\langle
D^{b}\operatorname{Coh}Z,D^{b}\operatorname{Coh}X\rangle$ described above. We
now formulate the mirror Floer theoretic construction:

######  Lemma 3.6 (Proposition 3.12 of [32]).

With respect to the semi-orthogonal decomposition in Equation (3.1.2), the
stop-removal functor

(3.1.12) $\mathcal{F}(E_{1}\cup_{F}\left(A_{2}\times
F\right)\cup_{F}E_{2})\to\mathcal{F}(E_{1}\cup_{F}E_{2})$

is obtained by localisation at the natural transformation
$\cup_{E_{1}}\circ\eta_{E_{2}}$ (where
$\eta_{E_{2}}\colon\thinspace\cap_{E_{2}}^{L}\circ\cup_{E_{2}}\Rightarrow
1_{\mathcal{F}(F)}$ is the co-unit of the cap-cup adjunction on $E_{2}$)
between the two functors

(3.1.13)
${\mathcal{F}(F)}$${\mathcal{F}({E_{1}})}$${\mathcal{F}(E_{2})}$${\langle\mathcal{F}(E_{2}),\mathcal{F}(E_{1})\rangle.}$

###### Proof.

In the presentation
$\Gamma^{\dagger}(\cup_{E_{1}})\otimes_{\mathcal{F}(F)}\Gamma(\cup_{E_{2}})$
of the gluing bimodule, Proposition 3.12 of [32] asserts that stop removal is
given by inverting a natural transformation which, at the level of objects, is
given by
$T_{L_{F}}=\operatorname{id}_{\cup_{E_{1}}L_{F}}\otimes\operatorname{id}_{\cup_{E_{2}}L_{F}}$.
Since localisation only depends on the cohomology-level natural
transformation, it is enough to show that the quasi-isomorphisms (3.1.6) sends
$T_{L_{F}}$ to $\cup_{E_{1}}((\eta_{E_{2}})_{L_{F}})$. For this, we compute

(3.1.14)
$T_{L_{F}}\mapsto\operatorname{id}_{\cup_{E_{1}}L_{F}}\otimes(\eta_{E_{2}})_{L_{F}}\\\
\mapsto\mu^{2}(\operatorname{id}_{\cup_{E_{1}}L_{F}},\cup_{E_{1}}\circ(\eta_{E_{2}})_{L_{F}})=\cup_{E_{1}}\circ(\eta_{E_{2}})_{L_{F}}.$

∎

At this stage, we are ready to deduce our homological mirror statement from
the main result of this section

###### Proof of Theorem 1.2.

Proposition 3.1 asserts that the assumptions of Corollary 3.5 hold, yielding a
comparison between the derived category of the blowup of
${\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}$ along
$\Pi_{n-1}\times H_{m-1}$ with the Fukaya category of $X_{m,n}$ stopped at
$H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$. The description of the
derived category of $X_{Z}\setminus H$ and of the Fukaya category of $X_{m,n}$
as localisations then yields the desired result. ∎

######  Remark 3.7.

A more efficient way to summarise the results of this section is to consider
the diagram

(3.1.15)
${D^{b}\operatorname{Coh}H}$${D^{b}\operatorname{Coh}Z}$${D^{b}\operatorname{Coh}X}$${D^{b}\operatorname{Coh}(X_{Z}\setminus
H).}$

of categories. This can be shown to be a pushout diagram, and there are
analogous results for the Fukaya categories of Liouville sections glued along
a common stop (c.f. Proposition 9.31 of [17]). This formulation is used in
[15].

###  3.2. A brief review of mirror symmetry for the three base cases

We begin the proof of Proposition 3.1 by specifying our description of the
three vertical maps in Diagram (3.0.1).

One simplifying point is that the three varieties being considered on the
coherent sheaf side are all affine, so that their derived categories of
coherent sheaves are modelled by the category of complexes of free modules
over the underlying ring of functions, with finitely generated total
cohomology. We shall thus construct the desired functors by identifying
Lagrangians in each of the three Liouville sections being considered, which
are mirror to the structure sheaf. The discussion is further simplified by the
intrinsic formality of algebras which are supported in degree $0$, i.e. they
are determined up to $A_{\infty}$ equivalence by the cohomological product.

We begin by considering the functor

(3.2.1)
$\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})\to\operatorname{Coh}(\Pi_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}).$

On the symplectic side, we introduce the positive real Lagrangian

(3.2.2) $L^{+}_{n-1}\times{\mathbb{R}}_{+}^{m}\subset
H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m},$

i.e. the product of the set $L^{+}_{n-1}$ of solutions to $\sum y_{j}=1$ for
$y_{j}\in{\mathbb{R}}_{+}$ with the set ${\mathbb{R}}_{+}^{m}$ of solutions to
$\prod x_{i}=1$, again with $x_{i}\in{\mathbb{R}}_{+}$.

######  Lemma 3.8.

The self-Floer cohomology of $L^{+}_{n-1}\times{\mathbb{R}}_{+}^{m}$ is
isomorphic to the ring of functions on
$\Pi_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$, and the induced Yoneda
functor

(3.2.3)
$\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})\to\operatorname{mod}_{{\mathbf{k}}[x_{j},y_{i}^{\pm}]/\prod
x_{i}}$

is a fully faithful embedding onto the subcategory corresponding to the
derived category of coherent sheaves.

###### Proof.

By the Künneth formula [17], we have a commutative diagram

(3.2.4)
${\mathcal{F}(H_{n-1}\times({\mathbb{C}}^{*})^{m})}$${\mathcal{F}(H_{n-1})\otimes\mathcal{F}(({\mathbb{C}}^{*})^{m})}$${\operatorname{mod}_{CF^{*}(L^{{\mathbb{R}}}_{n-1}\times{\mathbb{R}}_{+}^{m})}}$${\operatorname{mod}_{CF^{*}(L_{n-1}^{\mathbb{R}})}\otimes\operatorname{mod}_{CF^{*}({\mathbb{R}}_{+}^{m})},}$$\scriptstyle{\cong}$$\scriptstyle{\mathcal{Y}_{F}}$$\scriptstyle{\mathcal{Y}_{\mathrm{prod}}}$$\scriptstyle{\cong}$

in which both vertical arrows are given by testing against the object
$L_{n-1}^{\mathbb{R}}\times{\mathbb{R}}_{+}^{m}$. It thus suffices to show
that the functor $\mathcal{Y}_{\mathrm{prod}}$ is fully faithful, which
amounts to showing that its two components are individually fully faithful.
This is automatic for the second component, since ${\mathbb{R}}_{+}^{m}$
generates $\mathcal{F}(({\mathbb{C}}^{*})^{m})$, so it is enough to see that
the first component

(3.2.5)
$\mathcal{F}(H_{n-1})\to\operatorname{mod}_{CF^{*}(L_{n-1}^{\mathbb{R}})}$

is fully faithful.

We shall appeal to [22], in which it is proved that for a particular graded
structure on $H_{n-1}$, there is an equivalence

(3.2.6)
$\mathcal{F}(H_{n-1})\cong\operatorname{Coh}({\mathbf{k}}[x_{1},\dotsc,x_{n}]/(x_{1}\dotsm
x_{n}))$

sending a particular Lagrangian disc $L_{LP}$ to the rank $1$ free module.
Using the fully faithful embedding
$\operatorname{Coh}\hookrightarrow\operatorname{mod}$, this reduces our task
to showing that $L_{n-1}^{\mathbb{R}}$ is isotopic to $L_{LP}$ and that our
grading choice agrees with theirs.

We’ll begin by recalling the construction of $L_{LP}$, which takes place in
$H_{n-1}^{sym}=\operatorname{Sym}^{n-1}\Sigma$, where
$\Sigma={\mathbb{C}}\setminus\\{p_{1},\dotsc,p_{n}\\}$. Varying the position
of the points $p_{i}$ results in a deformation of Stein manifolds, so to
construct a Liouville identification $H_{n-1}^{sym}\sim H_{n-1}$ we may use
whichever $p_{i}$ we like. In this case, a convenient choice is to take

(3.2.7) $p_{1}=0\ll-p_{2}\ll\cdots\ll-p_{n-1}\ll 1,$

and $p_{n}=1$, where all $p_{i}$ are (negative) real and “$\ll$” is taken to
make the diagonal approximation

(3.2.8) $\begin{vmatrix}1&0&\cdots&0\\\ 1&p_{2}&\cdots&p_{2}^{n-1}\\\
1&p_{3}&\cdots&p_{3}^{n-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&1&\cdots&1\end{vmatrix}\approx p_{2}p_{3}^{2}\dotsm p_{n-1}^{n-2}\neq 0$

hold. Note that, under the identification
$\operatorname{Sym}^{n-1}{\mathbb{C}}\cong{\mathbb{C}}^{n-1}$ given by

(3.2.9) $\displaystyle(a_{1},\dotsc,a_{n-1})$
$\displaystyle\mapsto\prod(x-a_{i})$ $\displaystyle=\sum
z_{k+1}x^{k}\mapsto(z_{1},\dotsc,z_{n-1}),$

$H_{n-1}^{sym}$ is sent to the complement of the affine hyperplanes

(3.2.10)
$\left\\{p^{n-1}_{i}+\sum_{k=0}^{n-2}p_{i}^{k}z_{k+1}=0\right\\}_{i},$

and the nonvanishing of the determinant in Equation (3.2.8) asserts precisely
that these hyperplanes are in general position.

The Lagrangian $L_{LP}\subset H_{n-1}^{sym}$ is the product of the real arcs

(3.2.11)
$(-\infty,p_{n-1})\times(p_{n-1},p_{n-2})\times\dotsb\times(p_{2},0).$

Under the identification (3.2.9), $L_{LP}$ lands in the positive real locus
${\mathbb{R}}_{+}^{n-1}$.

To identify $H_{n-1}^{sym}$ with our preferred presentation

(3.2.12) $H_{n-1}={\mathbb{C}}^{n-1}\setminus\left\\{1+\sum z_{k}=0\right\\},$

we will deform the hyperplane loci (3.2.10) to the loci comprising the
coordinate hyperplanes together with $\\{1+\sum z_{k}=0\\}$, or equivalently
deform the matrix

(3.2.13) $M_{0}=\begin{pmatrix}1&0&\cdots&0&0\\\
1&p_{2}&\cdots&p_{2}^{n-2}&p_{2}^{n-1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
1&p_{n-1}&\cdots&p_{n-1}^{n-2}&p_{n-1}^{n-1}\\\ 1&1&\cdots&1&1\end{pmatrix}$

to the matrix

(3.2.14) $M_{1}=\begin{pmatrix}1&0&\cdots&0&0\\\ 0&-1&\cdots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&(-1)^{n-2}&0\\\
1&1&\cdots&1&1\end{pmatrix}$

through nonsingular matrices $M_{t}$. This we can do by convex interpolation,
since we have chosen the $p_{i}$ to satisfy (3.2.8).

Let us examine the effect of this deformation on the Lagrangian $L_{LP}$.
Because all the coefficients are real, $L_{LP}$ is a connected component of
$H_{n-1}^{sym}\cap{\mathbb{R}}^{n-1}$. As we follow the deformation, we obtain
a family of Lagrangians given by the corresponding connected component of the
real locus. This family is induced by an isotopy because the hyperplane
configuration that we are considering is generic. To see that the image of
$L_{LP}$ under this isotopy is $L_{n-1}^{\mathbb{R}}={\mathbb{R}}_{+}^{n-1}$,
it thus suffices to pick a point $z\in L_{LP}$, which belongs to
${\mathbb{R}}_{+}^{n-1}$, and avoids the hyperplane configuration throughout
the path. Choosing the point so that

(3.2.15) $1\ll z_{1}\ll\cdots\ll z_{n-1}$

ensures that all the sign conditions on the components of
$M\begin{pmatrix}z\\\ 1\end{pmatrix}$ remain positive, so convex interpolation
will preserve these inequalities, yielding the desired result.

Finally, to see that our gradings agree with those of [22], we note that the
space of gradings is an affine space over the integral first cohomology of
$H_{n-1}$, which has rank $n$. Since $L_{n-1}$ is contractible, we may
identify the first homology of $H_{n-1}$ relative $L_{n-1}$ with the first
homology of $H_{n-1}$. Inspecting the generators of the Floer complex of
$H_{n-1}$ given in [22], we see that the generators of the self-Floer
cohomology of $L_{n-1}$ corresponding to the functions $x_{i}$ span the first
homology. This means that any two grading structures on $H_{n-1}$ agree if the
induced gradings on the self-Floer cohomology of $L_{n-1}$ are both supported
in degree $0$ (c.f. [22, Corollary 3.2.5]). This is the case for the gradings
in [22] by their computation, and for our chosen gradings by noting that, if
we compactify $H_{n-1}$ as the hyperplane $\sum_{i=0}^{n}Y_{i}=0$ in
${\mathbb{C}}{\mathbb{P}}^{n}$, the volume form in Equation (2.2.1) has a pole
of order $1$ along the hypersurfaces $\\{Y_{i}=0\\}$ for $i\neq 0$ (but not
for $Y_{0}=0$). Since the closure of the Lagrangian $L_{n-1}$ in this
compactification only meets the divisors along which the form has a pole of
order $1$, we conclude that the Floer cohomology is supported in bounded
degrees (c.f. [25]), hence necessarily in degree $0$ since any other choice of
grading yields Floer groups supported in unbounded degrees. ∎

Next, we consider the left vertical arrow of Diagram (3.0.1). We associate to
the potential

(3.2.16)
$\prod_{i=0}^{m}x_{i}\colon\thinspace{\mathbb{C}}^{m+1}\to{\mathbb{C}}$

a Liouville sector by restricting to the inverse image of the translate of the
left half plane by a sufficiently large positive integer. The fibre of this
Liouville sector is identified with $\left({\mathbb{C}}^{*}\right)^{m}$, and
we simplify the notation by writing

(3.2.17) $\ell={\mathbb{R}}_{+}^{m}\subset\left({\mathbb{C}}^{*}\right)^{m}.$

The next result is a computation of the self Floer cohomology of the image of
this Lagrangian under the cup functor; since this computation is tied to the
formulation of the cup functor itself, we include both in the statement:

######  Lemma 3.9 ([6]).

There is a commutative diagram of $A_{\infty}$ algebra maps

(3.2.18)
${CF^{*}\left(\cup\ell\right)}$${CF^{*}(\ell)}$${{\mathbb{Z}}[y^{\pm}_{1},\ldots,y_{m}^{\pm}]/\left(1+\sum_{i=1}^{m}y_{i}\right)}$${{\mathbb{Z}}[y^{\pm}_{1},\ldots,y_{m}^{\pm}]}$

in which the vertical arrows are quasi-isomorphisms.

###### Sketch of proof:.

We start with some general considerations [9, 33]: since the cup functor is
spherical, there is a closed element $\tau_{\cup}\in CF^{*}(\ell)$ so that we
have a commutative diagram

(3.2.19)
${HF^{*}(\ell)}$${HF^{*}\left(\cup\ell\right)}$${\mathrm{Cone}(CF^{*}(\ell)\to
CF^{*}(\ell)),}$

where the morphism in the cone is multiplication by $\tau_{\cup}$, and the
vertical map is an isomorphism. In our case, the Floer cohomology
$HF^{*}(\ell)$ is supported in degree $0$, and is an algebra with no zero
divisors, so that the above diagram simplifies to

(3.2.20)
${HF^{0}\left(\cup\ell\right)}$${HF^{0}(\ell)}$${HF^{0}(\ell)/(\tau_{\cup})}$

at the cohomological level.

In the formalism of [33] the element $\tau_{\cup}$ is characterised implicitly
in terms of the existence of an adjoint to the cup functor. However, the map
$HF^{0}(\ell)\to HF^{0}\left(\cup\ell\right)$ is explicitly given as the
isomorphism between the self-Floer cohomology of $\ell$ in the fibre and the
self-Floer cohomology of its product with a cotangent fibre in a neighbourhood
of the boundary modelled after $F\times T^{*}[0,1]$, followed by the
continuation map from local to global wrapping. This isomorphism is natural,
which allows us to replace $\ell$ (which we recall is the positive real locus
in our situation) with the compact Lagrangian
${\mathbb{T}}^{m}\subset\left({\mathbb{C}}^{*}\right)^{m}$, which corresponds
to the $0$-section of the cotangent bundle under the symplectomorphism
$\left({\mathbb{C}}^{*}\right)^{m}\cong T^{*}{\mathbb{T}}^{m}$. The reason to
pass to a compact Lagrangian is to ensure that conical Lagrangians can be
produced by parallel transport along arcs in the base of our chosen Landau-
Giznburg model.

More precisely, we replace the cotangent fibre by the $0$-section, equipped
with the local system corresponding to the group ring (c.f. [4, 5]). These
objects of the Fukaya category are not isomorphic, but the Floer homology from
$\ell$ to this local system is a free module of rank-$1$ over
$HF^{*}(\ell,\ell)$; choosing the intersection point between the fibre and the
$0$-section to be the base point fixes the isomorphism

(3.2.21) $HF^{*}(\ell)\cong{\mathbb{Z}}[y^{\pm}_{1},\ldots,y_{m}^{\pm}].$

Figure 7. Computing the cup functor on fibrewise compact Lagrangians in
Liouville sectors associated to Landau-Ginzburg models: the endomorphisms of
the image of the cup functor are given by the Floer homology groups of the
orange and blue Lagrangians. The black circle represents the region where
there may be some critical points.

The above considerations reduce the problem to computing the element
$\tau_{\cup}$ for the torus, equipped with its universal local system, which
we do indirectly. We can decompose the contributions by intersection number
with the components of the fibre: since $\tau_{\cup}$ is a count of strips as
in Figure 7, and the fibre at $0$ consists of complex hypersurfaces, we can
decompose $\tau_{\cup}$ as a sum of contributions indexed by the components of
this fibre. The count of curves passing through the irreducible component
$\\{x_{i}=0\\}$ can be identified with a count of discs in the product of
${\mathbb{C}}_{x_{i}}$ with ${\mathbb{C}}^{*}_{x_{j}}$ for $j\neq i$. Note
that the restriction of the potential function $\prod_{i}x_{i}$ to this open
subset is a holomorphically trivial fibration with fibre
$\left({\mathbb{C}}^{*}\right)^{m}$. Deforming the symplectic form to the
standard one, we see that the count of discs in this relative homotopy class
is $1$, hence that the element $\tau_{\cup}$ is a sum of $m+1$ terms; the
expression $1+\sum_{i=1}^{m}y_{i}$ arises by observing that the different
trivialisation differ by their winding number (we use the component
$\\{x_{0}=0\\}$ as the reference component, in which case $y_{i}$ is the
winding number of $x_{i}/x_{0}$).

Finally, having computed the cup functor map on cohomology, we note all the
higher terms of an $A_{\infty}$ homomorphism of graded $A_{\infty}$ algebras
that are supported in degree $0$ must vanish for degree reasons. This implies
that our cohomological computation yields the desired cochain level statement.
∎

######  Corollary 3.10.

The self-Floer cohomology of the Lagrangian
$\cup\left(L^{+}_{n-1}\times{\mathbb{R}}_{+}^{m}\right)$, as an object of the
Fukaya category of $H_{n-1}\times{\mathbb{C}}^{m+1}$ stopped at
$H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$, is isomorphic to the ring of
functions on $\Pi_{n-1}\times H_{m-1}$, and the induced Yoneda functor

(3.2.22)
$\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})\to\operatorname{mod}_{{\mathbf{k}}[x_{j},y_{i}^{\pm}]/(\prod
x_{j},1+\sum y_{i})}$

is a fully faithful embedding onto the subcategory corresponding to the
derived category of coherent sheaves.

###### Proof.

Arguing as in the proof of Lemma 3.8, we can appeal to the Künneth theorem and
reduce the problem to showing that the two component Yoneda functors are fully
faithful. The first factor is the same as in Lemma 3.8, and using the above
result reduces the problem for the second factor to showing that $\cup\ell$
split-generates the Fukaya category. This follows from stop removal [32, 17],
the fact that $\ell$ generates the Fukaya category of the fibre, and the fact
that $\mathcal{F}({\mathbb{C}}^{m+1})$ is trivial. ∎

Finally, we consider the other side of this diagram. The next result can be
extracted from work of Hanlon and Hicks (c.f. [19, Theorem 6.2]).

######  Lemma 3.11.

The self-Floer cohomology of the positive real locus of
$\left({\mathbb{C}}^{*}\right)^{n+m}$, stopped at
$H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m}$ is isomorphic to the ring of
functions on ${\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}$. This
Lagrangian generates this partially wrapped category, so that the induced
Yoneda functor

(3.2.23)
$\mathcal{F}(\left({\mathbb{C}}^{*}\right)^{n}\times\left({\mathbb{C}}^{*}\right)^{m},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})\to\operatorname{mod}_{{\mathbf{k}}[x_{j},y_{i}^{\pm}]/1+\sum
y_{i}}$

is a fully faithful embedding onto the subcategory corresponding to the
derived category of coherent sheaves.

###### Proof.

The positive real locus is the unique handle for a Weinstein presentation both
of ${\mathbb{C}}^{*}$, and of the Weinstein sector corresponding to the
potential $x$ on ${\mathbb{C}}^{*}$. By Gammage-Shende [16], the sector
associated to a product potential is isomorphic to the product of the sectors
corresponding to each factor. In particular, the product of the positive real
loci is the unique handle for a Weinstein presentation of the pair
$(\left({\mathbb{C}}^{*}\right)^{n}\times\left({\mathbb{C}}^{*}\right)^{m},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})$,
hence generates the corresponding partially wrapped category. The computation
of the self-Floer cohomology then follows from the Künneth formula. ∎

######  Remark 3.12.

In the above proof, we use the fact that the co-cores of a Weinstein sector
generates its Fukaya category. The statement that appears in the literature
(c.f. [11, Theorem 1.2] and [17, Theorem 1.10]), is that co-cores generate
together with linking discs of the Legendrian stop. This distinction can be
eliminated by choosing the Weinstein structure near the boundary to have a
local minimum, rather than to be modelled after a cotangent bundle. Figure 8
shows how to make this replacement for a given Weinstein presentation. To
conclude that the co-cores of an arbitrary Weinstein structure which is
modelled near the boundary by the right side of Figure 8, we can appeal to
[17, Proposition 1.29] and [21, Proposition 3.3] which together imply that the
category generated by the co-cores does not depend on the choice of Weinstein
presentation.

Figure 8. The vertical (orange) line represents a linking disc on the left,
and the corresponding co-core on the right.

###  3.3. Compatibility of mirror symmetry with restriction

Our goal in this section is to explain the commutativity of the left square in
Diagram (3.0.1). As a starting point, recall that the pullback map on derived
categories of coherent sheaves of affine varieties associated to a ring map
$R\to S$ can be expressed on (complexes of) modules as a tensor product with
$S$, considered as an $R-S$-bimodule. Lemma 3.8 and Corollary 3.10 thus reduce
the problem to proving the commutativity of the diagram

(3.3.1)
${\mathcal{F}(H_{n-1}\times{\mathbb{C}}^{m+1},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\operatorname{mod}_{CF^{*}(\cup\left(L^{{\mathbb{R}}}_{n-1}\times{\mathbb{R}}_{+}^{m}\right))}}$${\operatorname{mod}_{CF^{*}(L^{{\mathbb{R}}}_{n-1}\times{\mathbb{R}}_{+}^{m})}}$

where the bottom horizontal map is tensoring with $CF^{*}(\cup
L^{{\mathbb{R}}}_{n-1}\times{\mathbb{R}}_{+}^{m}))$ over
$CF^{*}(L^{{\mathbb{R}}}_{n-1}\times{\mathbb{R}}_{+}^{m})$, as long as we can
compute that this map agrees with the mirror map:

(3.3.2)
${\mathbb{Z}}[x_{1},\ldots,x_{n},y^{\pm}_{1},\ldots,y_{m}^{\pm}]/\left(\prod
x_{j}\right)\to\\\
{\mathbb{Z}}[x_{1},\ldots,x_{n},y^{\pm}_{1},\ldots,y_{m}^{\pm}]/\left(\prod
x_{j},1+\sum_{i=1}^{m}y_{i}\right).$

Lemma 3.9 computes this map if we ignore the $H_{n-1}$ factor. In order to
obtain the desired computation, we therefore need a compatibility statement
between cup functors associated to Liouville sectors and Künneth theorems for
wrapped Floer cohomology. Because of intrinsic formality, the required
statement are at the cohomological level, which greatly simplifies the proof.
For the statement, let $M$ be a Liouville sector with fibre $F$, and let
$\ell\subset F$ be a properly embedded exact Lagrangian. We write
$\cup\ell\subset M$ for the image of $\ell$ under the (Orlov) cup functor. Let
$L\subset H$ be an exact Lagrangian in a Liouville domain. The following
result is proved in Appendix A:

######  Lemma 3.13.

The cup functors fit in a commutative diagram

(3.3.3) ${HF^{*}(\ell\times L)}$${HF^{*}\left(\cup(\ell\times
L)\right)}$${HF^{*}(\ell)\otimes
HF^{*}(L)}$${HF^{*}\left(\cup\ell\right)\otimes HF^{*}(L).}$

We shall now prove the commutativity of Diagram (3.3.1):

######  Lemma 3.14.

The left square in Diagram (3.0.1) commutes.

###### Proof.

Write $L$ for the object
$L_{n-1}^{\mathbb{R}}\times{\mathbb{R}}_{+}^{m}\in\mathcal{F}(H_{n-1}\times({\mathbb{C}}^{*})^{m})$,
let $K$ be an arbitrary object of
$\mathcal{F}(H_{n-1}\times({\mathbb{C}}^{*})^{m})$, and write $\tilde{K}$ for
its image in
$\operatorname{mod}_{CF^{*}(L_{n-1}^{\mathbb{R}}\times{\mathbb{R}}_{+}^{m})}$.
Write $\tilde{K}=\operatorname*{colim}_{D}\tilde{L}$, where
$L=L_{n-1}^{\mathbb{R}}\times{\mathbb{R}}_{+}^{m}$ and $\tilde{L}$ is $L$ as a
module over itself. We may arrange that $D$ is a finite diagram of filtered
diagrams. Now, because the right vertical arrow of Diagram (3.3.1) is fully
faithful, the colimit diagram
$D_{+}\to\operatorname{mod}_{CF^{*}(L_{n-1}^{\mathbb{R}}\times{\mathbb{R}}_{+}^{m})}$
lifts to a diagram $D_{+}\to\mathcal{F}(H_{n-1}\times({\mathbb{C}}^{*})^{m})$,
and the lift is a colimit diagram because it satisfies the universal property.
In other words, $K$ is a finite colimit of filtered colimits of $L$, and this
colimit is preserved by the right arrow.

Next, note that both horizontal arrows are spherical functors [9, 33], so in
particular they are left adjoints. This means they preserve the colimits
$K=\operatorname*{colim}_{D}L$ and
$\tilde{K}=\operatorname*{colim}_{D}\tilde{L}$. Using the fact that the left
vertical map is a fully faithful embedding as above, we see that all arrows
preserve the colimit $K=\operatorname*{colim}_{D}L$.

Now, let $K^{\prime}$ be another object of
$\mathcal{F}(H_{n-1}\times({\mathbb{C}}^{*})^{m})$. Following the above, we
can write $K^{\prime}=\operatorname*{colim}_{D^{\prime}}L$, and this colimit
is also preserved by all arrows in the diagram. We are thus interested in
seeing that (3.3.1) commutes for the morphism space $\hom(K,K^{\prime})$.
Since

(3.3.4)
$\hom(K,K^{\prime})=\hom(\operatorname*{colim}_{D}L,\operatorname*{colim}_{D^{\prime}}L)=\lim_{D}\hom(L,\operatorname*{colim}_{D^{\prime}}L),$

and similarly in the other three categories, it is enough to show that the
diagram commutes for morphism spaces of the form
$\hom(L,\operatorname*{colim}_{D^{\prime}}L)$. To do this, we would again like
to pull the colimit out of the $\hom$ complex. In the lower (module)
categories that is possible because $\tilde{L}$ and its image in the lower
left are compact objects. In the upper (Fukaya) categories, we use the
tautological maps of cochain complexes

(3.3.5)
$\operatorname*{colim}_{D^{\prime}}\hom(L,L)\to\hom(L,\operatorname*{colim}_{D^{\prime}}L)$

(and similarly with $\cup L$) which become quasi-isomorphisms under the
vertical arrows. On the other hand, the vertical arrows are fully faithful, so
the map (3.3.5) is already a quasi-isomorphism. This reduces commutativity of
Diagram (3.3.1) to its commutativity on $L$, which is the Yoneda lemma. ∎

###  3.4. Compatibility of mirror symmetry with pushforward

We now turn our attention to the right square in Diagram (3.0.1). Since the
vertical maps in this diagram are equivalences, it suffices to prove that the
diagram

(3.4.1)
${\mathcal{F}(H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\mathcal{F}(\left({\mathbb{C}}^{*}\right)^{n}\times\left({\mathbb{C}}^{*}\right)^{m},H_{n-1}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\operatorname{Coh}(\\{\prod_{i=0}^{n-1}x_{i}=0\\}\times\left({\mathbb{C}}^{*}\right)^{m})}$${\operatorname{Coh}({\mathbb{C}}^{n}\times\left({\mathbb{C}}^{*}\right)^{m}),}$

given by the adjoint maps commutes. On the coherent sheaf side, the map is
again given by restriction, so our proximate goal is to describe the adjoint
of the cup functor on Fukaya categories.

######  Lemma 3.15.

If $M$ is a Liouville sector with fibre $F$, which is swappable in the sense
of [33], the adjoint to the cup functor $\mathcal{F}(F)\to\mathcal{F}(M)$ is
given by a composition

(3.4.2) $\mathcal{F}(M)\to\mathcal{F}(M^{(2)})\to\mathcal{F}(A_{1}\times
F)\cong\mathcal{F}(F),$

where the first arrow is the inclusion of $M$ in a Liouville section $M^{(2)}$
obtained by doubling the stop, and the second arrow is a Viterbo restriction
functor.

###### Proof.

We start with the graph bimodule $\Gamma(\cup)(\ell,L)=CW^{*}(L,\cup\ell)$,
illustrated as the first picture of Figure 9. We will apply a sequence of
bimodule quasi-isomorphisms to obtain a formula for the adjoint graph of the
left adjoint $\cap^{L}$.

$\cup\ell$$L$$\cup_{-}\ell$$\mathbf{S}_{\phi}\cup_{+}\ell$ Figure 9. The
vertical (orange) line represents the cup functor. The (light blue) region in
the top left is the projection of an arbitrary Lagrangian $L$ underlying an
object in $\mathcal{F}(M)$.

First, we double the stop $\sigma$ to obtain a new sector $M^{(2)}$ with stops
$\sigma^{+}$ and $\sigma^{-}$, which are positive and negative pushoffs of
$\sigma$, respectively. In purely sectorial language, this is equivalent to
gluing on a copy of $F\times A_{2}$, where $F$ is the fiber of $\sigma$ and
$A_{2}$ is ${\mathbb{C}}$ with three stops. By [33, Cor. 3.11, see also proof
of Thm. 1.3], we have an equivalence

(3.4.3)
$\Gamma(\cup)\cong(\cup_{\sigma_{-}},i_{M})^{*}\Delta_{\mathcal{F}(M^{(2)})},$

where $i_{M}$ is the inclusion of $M$ into $M^{(2)}$ given by the gluing
decomposition. This situation is illustrated in the second picture (upper
right) of Figure 9.

Next, by sphericality of $\cup$, we have a semiorthogonal decomposition

(3.4.4) $\mathcal{F}(M^{(2)})=\langle\mathcal{F}(M),\mathcal{F}(F)\rangle$

with $\mathcal{F}(F)$ the image of $\cup_{\sigma^{+}}$ and $\mathcal{F}(M)$
the second orthogonal to the image of $i_{M}$. We are free to represent
$i_{M}$ in these coordinates, see the third picture of Figure 9.

Finally, we apply the wrap-once deformation $\phi\colon M^{(2)}\to M^{(2)}$
coming from the swap of $\sigma$, as shown in the last picture of Figure 9.
Algebraically, this corresponds to taking second orthogonals, which means that
we have an equivalence of bimodules

$\displaystyle(\cup_{\sigma_{-}},i_{M})^{*}\Delta_{\mathcal{F}(M^{(2)})}$
$\displaystyle\cong(\mathbf{S}_{\phi}\circ\cup_{\sigma_{-}},\mathbf{S}_{\phi}\circ
i_{M})^{*}\Delta_{\mathcal{F}(M^{(2)})}$
$\displaystyle\cong(\cup_{\sigma^{+}},\mathbf{S}_{\phi}\circ
i_{M})^{*}\Delta_{\mathcal{F}(M^{(2)})},$

where $\mathbf{S}_{\phi}$ is the autoequivalence of $\mathcal{F}(M^{(2)})$
induced by $\phi$. Here, the positivity of the wrapping $\phi$ is expressed by
the isomorphism
$\mathbf{S}_{\phi}\circ\cup_{\sigma_{-}}\cong\cup_{\sigma_{+}}$; a negative
wrapping would involve the monodromy. With $i_{M}$ expressed in the
coordinates (3.4.4), we see obtain a decomposition

(3.4.5) $\mathbf{S}_{\phi}\circ i_{M}\cong\mathrm{cone}(F_{M}\to F_{-})$

with $F_{-}$ landing in the image of $\sigma^{-}$ and $F_{M}$ landing in the
image of $i_{M}$. In particular, $F_{M}$ is left-orthogonal to
$\cup_{\sigma^{+}}$, so Viterbo restriction $V$ to a neighborhood $U\cong
F\times T^{*}[0,1]$ of $\sigma^{-}\cup\sigma^{+}$ gives an equivalence

$\displaystyle(\cup_{\sigma^{+}},\mathbf{S}_{\phi}\circ
i_{M})^{*}\Delta_{\mathcal{F}(M^{(2)})}$
$\displaystyle\cong(\cup_{\sigma^{+}},F_{-})^{*}\Delta_{\mathcal{F}(M^{(2)})}$
$\displaystyle\cong(\cup_{\sigma^{+}},VF_{-})^{*}\Delta_{\mathcal{F}(F\times
T^{*}[0,1])}$
$\displaystyle\cong(\operatorname{id}_{F},\Sigma^{-1}VF_{-})^{*}\Delta_{\mathcal{F}(F)}$
$\displaystyle=\Gamma^{\dagger}(\Sigma^{-1}VF_{-})$
$\displaystyle\cong\Gamma^{\dagger}(\Sigma^{-1}V\mathbf{S}_{\phi}i_{M}).$

Here, $\Sigma$ is the stabilization equivalence
$\mathcal{F}(F)\cong\mathcal{F}(F\times T^{*}[0,1])$, and we have used the
fact that $V\circ i_{M}=0$. ∎

We now perform the final computation of this section, which is in some sense
the analogue of Lemma 3.9:

######  Lemma 3.16.

There is a commutative diagram of $A_{\infty}$ algebra maps

(3.4.6)
${CF^{*}(\cap\left({\mathbb{R}}_{+}^{n}\times{\mathbb{R}}_{+}^{m}\right))}$${CF^{*}({\mathbb{R}}_{+}^{n}\times{\mathbb{R}}_{+}^{m})}$${{\mathbb{Z}}[x_{1},\ldots,x_{n},y^{\pm}_{1},\ldots,y_{m}^{\pm}]/\left(\prod
x_{j}\right)}$${{\mathbb{Z}}[x_{1},\ldots,x_{n},y^{\pm}_{1},\ldots,y_{m}^{\pm}]}$

in which the vertical arrows are quasi-isomorphisms.

###### Sketch of proof.

By intrinsic formality, it suffices to compute the maps at the level of
cohomology. An argument analogous to that of Proposition A.1 shows that the
Künneth isomorphism commutes with Viterbo restriction as the cohomological
level, so it suffices to show that we have a commutative diagram

(3.4.7)
${HF^{*}(\cap{\mathbb{R}}_{+}^{n})}$${HF^{*}({\mathbb{R}}_{+}^{n})}$${{\mathbb{Z}}[x_{1},\ldots,x_{n}]/\left(\prod
x_{j}\right)}$${{\mathbb{Z}}[x_{1},\ldots,x_{n}]}$

It is convenient at this stage to use the model of $H_{n-1}$ given as the
$n-1$${}^{\textrm{st}}$ symmetric product of ${\mathbb{C}}$ punctured at $n$
points, following [22]. In this model, we have an isomorphism

(3.4.8)
$HF^{*}(\cap{\mathbb{R}}_{+}^{n})\cong{\mathbb{Z}}[z_{1},\ldots,z_{n}]/\left(\prod
z_{j}\right),$

moreover, each generator $z_{j}$ is localised near the $j$${}^{\textrm{th}}$
puncture. In particular, the inclusion of the product of small neighbourhoods
of all but the $k$${}^{\textrm{th}}$ puncture yields the embedding of a
Liouville subdomain

(3.4.9) $\Omega_{k}\to H_{n-1}$

and the induced Viterbo restriction map is

(3.4.10) ${\mathbb{Z}}[z_{1},\ldots,z_{n}]/\left(\prod
z_{j}\right)\to{\mathbb{Z}}[z^{\pm}_{1},\ldots,z_{k-1}^{\pm},z_{k},z_{k+1}^{\pm},\cdots,z^{\pm}_{n}]/z_{k}.$

By considering a tropical degeneration of the pair of pants, as in [2], we see
that the composite embedding

(3.4.11) $\Omega_{k}\to H_{n-1}\to({\mathbb{C}}^{*})^{n}$

is Liouville isotopic to an embedding

(3.4.12) $\Omega_{k}\to({\mathbb{C}}^{*})^{n-1}\to({\mathbb{C}}^{*})^{n}.$

This identifies the composite

(3.4.13) $HF^{*}({\mathbb{R}}_{+}^{n})\to HF^{*}(\cap{\mathbb{R}}_{+}^{n})\to
HF^{*}({\mathbb{R}}_{+}^{n-1})$

as the map taking $x_{i}$ to $z_{i}$. We conclude the desired result using the
fact that the map

(3.4.14) ${\mathbb{Z}}[z_{1},\ldots,z_{n}]/\left(\prod
z_{j}\right)\to\bigoplus_{k}{\mathbb{Z}}[z^{\pm}_{1},\ldots,z^{\pm}_{k-1},z_{k+1}^{\pm},\ldots,z^{\pm}_{n}].$

is injective. ∎

##  Appendix A Compatibility of Künneth maps with cup functors

The purpose of this appendix is to prove the compatibility of Künneth
isomorphisms with cup functors. To formulate the desired result, consider a
Liouville sector $M_{1}$, a Liouville domain $M_{2}$, and embedded Lagrangians
$L_{i},L^{\prime}_{i}\subset M_{i}$ which are invariant under the respective
Liouville flows:

######  Proposition A.1.

There is a cohomological Künneth isomorphism

(A.0.1) $HF^{*}(L_{1},L^{\prime}_{1})\otimes HF^{*}(L_{2},L^{\prime}_{2})\cong
HF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times L^{\prime}_{2})$

which intertwines the cup functors in the sense that, if $\sigma_{1}$ is a
boundary component of $M_{1}$, and $L^{\prime}_{1}=\cup\ell^{\prime}_{1}$,
then the following diagram commutes

(A.0.2) ${HF^{*}(\ell_{1},\ell^{\prime}_{1})\otimes
HF^{*}(L_{2},L^{\prime}_{2})}$${HF^{*}(\ell_{1}\times
L_{2},\ell^{\prime}_{1}\times
L^{\prime}_{2})}$${HF^{*}(L_{1},L^{\prime}_{1})\otimes
HF^{*}(L_{2},L^{\prime}_{2})}$${HF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2}).}$

There are three technical restrictions in the above theorem, which are all
made for convenience: the restriction to cohomological information, to the
setting where $M_{2}$ is a Liouville manifold, and to the case where all
Lagrangians are globally invariant under the Liouville flow. The first allows
us to avoid the technicalities of chain-level constructions, the second to
avoid discussing Liouville sectors with corners, and the last ensures that
products are conical at infinity.

######  Remark A.2.

We may replace the condition that the Lagrangians are globally conical by the
condition that they be conical at infinity by applying the following trick
from [1]: take the product of $M_{i}$ with $T^{*}S^{1}$, the product of
$L_{i}$ and $L^{\prime}_{i}$ with different cotangent fibres, and then equip
the product $M_{i}\times T^{*}S^{1}$ with a Liouville structure for which
these Lagrangians are globally conical. The Floer cohomology groups in
$M_{i}\times T^{*}S^{1}$ decompose according to the winding number along the
circle, and those of winding number $0$ agree with the Floer groups in
$M_{i}$. Alternatively, we could take the product with the cotangent bundle of
the interval, considered as a Liouville sector, at the expense of requiring
additional explanation for how to address the appearance of corners.

###  A.1. Künneth isomorphism

We associate to a Liouville sector $M$ its horizontal completion

(A.1.1) $\hat{M}\equiv M\cup\sigma\times T^{*}(-\infty,0]/\sim,$

where $\sigma=\partial M/{\mathbb{R}}$ is the stop of $M$.

We equip $\hat{M}$ with a Liouville form $\hat{\lambda}$ whose restriction to
$\sigma\times T^{*}(-\infty,0]$ is the product of the Liouville form on
$\sigma$ with a Liouville form on $T^{*}(-\infty,0]$ whose flow is shown in
Figure 10. The boundary projection $\pi\colon\nu\partial M\to T^{*}[0,1)$
extends to a surjective projection

(A.1.2) $\hat{\pi}\colon\widehat{\nu\partial M}\to(-\infty,1).$

Figure 10. The chosen Liouville flow on $T^{*}(-\infty,0]$. The black dots
represent the vanishing of the vector field on the subset of the $0$-section
given by the interval $[-1,0]$.

Since $\hat{M}$ is a complete Liouville manifold, we can construct a wrapped
Fukaya category $\mathcal{W}(\hat{M})$ using linear Hamiltonians and contact
type almost complex structures [1] such that:

  * •

all Lagrangians are transverse to the locus
$\hat{\pi}^{-1}\left((-1,1)\right)$ living over the zero section, and

  * •

the Hamiltonian vector field $X_{H}$ of the wrapping Hamiltonians is tangent
to each fiber $\hat{\pi}^{-1}(q)$ for $q\in(-1,1)$, and

  * •

The almost complex structure fixes each of the above fibers.

The wrapped Fukaya category of the sector $M$ is defined to be the non-full
subcategory $\mathcal{W}(M)\subset\mathcal{W}(\hat{M})$, where

  * •

ahe Lagrangians are required to live in the interior of $M$, and

  * •

the morphisms are generated by Hamiltonian chords supported in $M$.

That this is a subcategory follows from positivity of intersections of
$H$-perturbed holomorphic disks with the fiber $\hat{\pi}^{-1}(0)$. This is
the linear version of the category $\mathcal{W}_{\mathrm{quad}}(M)$ in [33].

####  A.1.1. Hamiltonians on product sectors

We are interested in the situation where $M$ is a product $M_{1}\times M_{2}$,
where we would like to identify on the nose the Fukaya algebras associated to
split and cylindrical Floer data.

Fix a bump function $a\colon{\mathbb{R}}\to[0,1]$ which vanishes for $r\leq
2$, equals $1$ for $r\in[\beta_{-},\beta_{+}]$, and vanishes again for $r$
sufficiently large. We will construct Hamiltonians on $M_{1}\times M_{2}$
which are linear on a large cylinder and split outside a compact set by
rescaling via the function $h(r)=a(r)\log(r)$.

$0$$2$$0$$2$$H_{1}$$H_{2}$ Figure 11. The decomposition of
$\hat{M}_{1}\times\hat{M}_{2}$

Concretely, let $H_{1}$ and $H_{2}$ be linear exhausting positive Hamiltonians
on $M_{1}$ and $M_{2}$, respectively, so that

(A.1.3) $0\leq Z_{i}H_{i}\leq H_{i},$

where the second inequality is an equality whenever $\varepsilon\leq H_{i}$
(i.e. $H_{i}$ is linear outside a compact set). Define

(A.1.4) $H_{12}(x_{1},x_{2})=e^{h(H_{1})}\cdot
H_{2}\left(\phi_{Z_{2}}^{-h(H_{1})}x_{2}\right)+e^{h(H_{2})}\cdot
H_{1}\left(\phi_{Z_{1}}^{-h(H_{2})}x_{1}\right).$

This construction satisfies the following key properties:

  1. (1)

Provided $\varepsilon<1$ and $\beta_{+}-\beta_{-}$ is sufficiently large
relative to $\frac{1}{\varepsilon}$, $H_{12}$ is linear along
$H_{12}^{-1}(\beta_{+})$.

  2. (2)

The region where $H_{12}$ fails to agree with $H_{1}+H_{2}$ is compact, and is
disjoint from the locus where
$\varepsilon<\frac{H_{2}}{H_{1}}<\frac{1}{\varepsilon}$.

Henceforth, we will assume $\varepsilon<1$, and in fact we will shrink
$\varepsilon$ to further improve the behavior of $H_{12}$.

Our goal is to choose $H_{1}$ and $H_{2}$ so that all chords of $H_{12}$ live
in the region $H_{12}<\beta_{+}$. To that end, note that the region where
$H_{12}$ is not split decomposes into two disjoint regions, where
$H_{1}\geq\operatorname{max}\left\\{2,\frac{H_{2}}{\varepsilon}\right\\}$ and
where
$H_{2}\geq\operatorname{max}\left\\{2,\frac{H_{1}}{\varepsilon}\right\\}$, see
Figure 11. These two regions are essentially identical, so we will focus on
the first.

Let us compute $X_{H_{12}}$. First, the $M_{1}$ component of $dH_{12}$ is
given by

(A.1.5)
$\left[e^{h(H_{1})}\cdot\left(H_{2}-Z_{2}H_{2}\right)\left(\phi_{Z_{2}}^{-h(H_{1})}x_{2}\right)\cdot
h^{\prime}(H_{1})\right]dH_{1}\\\ +e^{h(H_{2})}\cdot dH_{1}\circ
d\phi_{Z}^{-h(H_{2})}$

and similarly with the $M_{2}$ component. In the region
$H_{1}\geq\operatorname{max}\left\\{2,\frac{H_{2}}{\varepsilon}\right\\}$,
this simplifies to

(A.1.6)
$\left[1+e^{h(H_{1})}h^{\prime}(H_{1})\cdot\left(H_{2}-Z_{2}H_{2}\right)\left(\phi_{Z_{2}}^{-h(H_{1})}x_{2}\right)\right]dH_{1},$

while the $M_{2}$ component simplifies to

(A.1.7) $e^{h(H_{1})}\cdot dH_{2}\circ d\phi_{Z}^{-h(H_{1})}.$

Let us examine Equaction (A.1.6). The coefficient
$e^{h(H_{1})}h^{\prime}(H_{1})$ is globally bounded independently of
$\varepsilon$. By shrinking $\varepsilon\rightsquigarrow\varepsilon^{\prime}$
and correspondingly rescaling
$H_{i}\rightsquigarrow\frac{\varepsilon^{\prime}}{\varepsilon}H_{i}\circ\phi_{Z_{i}}^{\log\frac{\varepsilon}{\varepsilon^{\prime}}}$,
we can make $\lVert H_{2}-Z_{2}H_{2}\rVert_{C^{0}}$ as small as we like, so
that Equation (A.1.6) approaches $dH_{1}$. Since $X_{H_{1}}$ has no chords in
the linear region $H_{1}>\varepsilon$, neither will $H_{12}$ once
$\varepsilon$ is small enough.

After further shrinking $\varepsilon$, the same argument shows that we can
arrange that $mX_{H_{12}}$ also has no chords in the region where
$H_{12}\geq\beta_{+}$. We have proved

######  Lemma A.3.

Let $H_{1}$ and $H_{2}$ be linear exhausting positive Hamiltonians on $M_{1}$
and $M_{2}$ which have no integer length chords from $\partial^{\infty}L_{i}$
to $\partial^{\infty}L^{\prime}_{i}$, and which satisfy Equation (A.1.3). Let
$m\in{\mathbb{N}}$. Then, for any sufficiently small $\varepsilon>0$, there
are linear exhausting positive Hamiltonians $H_{1}^{(m)}$ on $M_{1}$ and
$H_{2}^{(m)}$ on $M_{2}$, together with a positive exhausting (non-linear)
Hamiltonian $H_{12}^{(m)}$ on $M_{1}\times M_{2}$ such that:

  1. (1)

$H_{i}^{(m)}$ is a rescaling of $H_{i}$ which is linear in the region where it
is bigger than $\varepsilon$, and satisfies $H_{i}^{(m)}\leq H_{i}$
everywhere.

  2. (2)

$H_{12}^{(m)}$ is linear along ${H_{12}^{(m)}}^{-1}(\beta_{+})$.

  3. (3)

$H_{12}^{(m)}$ agrees with $H_{1}^{(m)}+H_{2}^{(m)}$ except on the regions
$\operatorname{max}\left\\{2,\frac{H_{2}^{(m)}}{\varepsilon}\right\\}\leq
H_{1}^{(m)}\leq\beta_{\operatorname{max}}$ and
$\operatorname{max}\left\\{2,\frac{H_{1}^{(m)}}{\varepsilon}\right\\}\leq
H_{2}^{(m)}\leq\beta_{\operatorname{max}}$. Here, $\beta_{\operatorname{max}}$
is a value above which the bump function $a$ vanishes.

  4. (4)

For $j=1,\dotsc,m$, all time $1$ chords of $jX_{H_{12}^{(m)}}$ from
$L_{1}\times L_{2}$ to $L^{\prime}_{1}\times L^{\prime}_{2}$ are split. In
particular, they lie in the region where both $H_{1}^{(m)}$ and $H_{2}^{(m)}$
are less than $\varepsilon$.

∎

Fix once and for all the model Hamiltonians $H_{i}$. Consider a sequence
$(m,\varepsilon_{m})_{m\in{\mathbb{N}}}$ and corresponding Hamiltonians
$H_{1}^{(m)}$, $H_{2}^{(m)}$, and $H_{12}^{(m)}$. We consider the homotopy-
commutative diagram

(A.1.8) ${CF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};H_{12}^{(1)})}$${CF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};H_{12}^{(2)})}$${CF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};H_{12}^{(3)})}$${\cdots}$${CF^{*}(L_{1}\times
L_{2},L^{\prime}_{1}\times L^{\prime}_{2};2H_{12}^{(2)})}$${CF^{*}(L_{1}\times
L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};2H_{12}^{(3)})}$${\cdots}$${CF^{*}(L_{1}\times
L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};3H_{12}^{(3)})}$${\cdots}$${\cdots}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

in which the vertical maps are continuation maps associated to increasing
slope [1]. The horizontal maps are constructed by the homotopy method [13]:
since all chords of $jH_{12}^{(m)}$ are split, they are products of
$X_{H_{1}^{(m)}}$-chords and $X_{H_{2}^{(m)}}$-chords of length $j$, which are
rescalings of length $j$ chords of $X_{H_{1}}$ and $X_{H_{2}}$. In particular,
further rescaling $H_{1}$ and $H_{2}$ by shrinking $\varepsilon$ doesn’t
change the length $j$ chords. The formula in Equation (A.1.4) thus gives a
family $H_{12}^{t}$ which interpolates between $H_{12}^{(m)}$ and
$H_{12}^{(m+1)}$ with no birth/death of chords. In particular, we obtain
canonical identifications of the generators of $CF^{*}(L_{1}\times
L_{2},L^{\prime}_{1}\times L^{\prime}_{2};H_{12}^{t})$ for varying $t$. The
homotopy method quasi-isomorphisms are then obtained by counting time-ordered
arrangements of exceptional index $0$ strips, where the generators at
different times are identified as above (see Figure 12). For homotopy-
commutativity, one similarly counts time-ordered arrangements of index $-1$ in
which one of the strips is an exceptional continuation map, exactly as in the
construction of the linear term of the Viterbo restriction map in [1].

$t=0$$t=1$$x_{0}$$x_{t_{0}}$$y_{t_{0}}$$t_{0}$$y_{t_{1}}$$y^{\prime}_{t_{1}}$$t_{1}$$y^{\prime}_{t_{2}}$$z_{t_{2}}$$t_{2}$$z_{1}$
Figure 12. A representation of the time-ordered strip count in the homotopy
method, consisting of three rigid solutions to the Floer equation, occuring at
times $t_{0}<t_{1}<t_{2}$. The labels $x_{t}$, $y_{t}$, $y^{\prime}_{t}$, and
$z_{t}$ are those for the time-$1$ chords of $H_{t}$.

######  Lemma A.4.

The colimit of the homology groups in Diagram (A.1.8) computes wrapped Floer
cohomology in the completion of $M_{1}\times M_{2}$.

###### Proof.

Write $\tilde{H}_{12}^{(m)}$ for the linear extrapolation of $H_{12}^{(m)}$,
which one can construct by replacing $\beta_{+}$ with infinity. Note that, for
$j=1,\dotsc,m$, $CF^{*}(L_{1}\times L_{2};jH_{12}^{(m)})$ and
$CF^{*}(L_{1}\times L_{2};\tilde{j}H_{12}^{(m)})$ are canonically isomorphic.
Indeed, they have the same chords, and by the integrated maximum principle [1]
along ${H_{12}^{(m)}}^{-1}(\beta_{+})$, all holomorphic disks live in the
region where the Hamiltonians agree.

We begin by observing that

(A.1.9)
$H_{12}^{(m)}\geq\tilde{H}_{12}^{(m)}\geq\operatorname{max}(H_{1}^{(m)},H_{2}^{(m)})$

globally. The right hand side is a (non-smooth) linear exhausting positive
Hamiltonian which is eventually $m$-independent outside an arbitrarily small
neighborhood of the skeleton of $M_{1}\times M_{2}$. In particular,
$m\tilde{H}_{12}^{(m)}$ has slope tending to infinity with $m$, so

(A.1.10) $\lim_{\rightarrow}HF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};m_{i}\tilde{H}_{12}^{(m_{i})})$

computes the wrapped Floer cohomology of $L_{1}\times L_{2}$ and
$L^{\prime}_{1}\times L^{\prime}_{2}$, where the values of $m_{i}$ appearing
in the colimit are chosen so that
$m_{i+1}H_{12}^{(m_{i+1})}>m_{i}H_{12}^{(m_{i})}$ outside a compact set.

It now suffices to compare these continuation maps with the maps obtained from
Diagram (A.1.8). First we use the integrated maximum principle again to see
that the colimit in Equation (A.1.10) is isomorphic to the colimit

(A.1.11) $\lim_{\rightarrow}HF^{*}(L_{1}\times L_{2},L^{\prime}_{1}\times
L^{\prime}_{2};m_{i}H_{12}^{(m_{i})}),$

where we recall that $H_{12}^{(m_{i})}$ is split at infinity. Since the
vertical maps are continuation maps, their compatibility with composition
reduces the problem to showing that the (inverse) of the horizontal maps
agrees on cohomology with continuation maps, which are well-defined because
the functions $H_{12}^{t}$ decrease with $t$ as a consequence of Equation
(A.1.3).

$t=0$$t=1$$x_{0}$$y_{t_{0}}$$t_{0}$$y_{t_{1}}$$z_{t_{1}}$$t_{1}$$z_{1}$ Figure
13. A representation of the time-ordered strip count in the comparison between
the homotopy method and the continuation map, consisting of a rigid solutions
to the continuation map from $t=0$ to $t=t_{0}$, followed by a solution to the
Floer equation, occuring at some time $t_{0}<t_{1}$.

This is achieved by using the composition of continuation maps from
$mH_{12}^{(n)}$ to $mH_{12}^{(t)}$, followed by homotopy from $mH_{12}^{(t)}$
to $mH_{12}^{(m)}$, associated to $m\leq t\leq n$ (see Figure 13). The moduli
space of solutions to these equations is compact by an application of the
maximum principle in the two factors, and the associated parametrised moduli
space defines the desired homotopy.

∎

Finally, we complete this appendix by proving it main result:

###### Proof of Proposition A.1.

In the setting of Hamiltonian wrapping, the cup functor
$\cup_{\sigma}\colon\mathcal{F}(\sigma)\to\mathcal{F}(M)$ sends a Lagrangian
$\ell\subset\sigma$ to its product with a fibre $\ell\times
T_{\frac{1}{2}}^{*}[0,1)\subset M$. At the level of morphisms, it sends
$CW^{*}(\ell,\ell^{\prime})$ to the piece of $CW^{*}(\ell\times
T_{\frac{1}{2}}^{*}[0,1),\ell^{\prime}\times T_{\frac{1}{2}}^{*}[0,1))$ living
along the zero section of $[0,1)$ [33, Construction 2.18]. The assumptions on
our wrapping Hamiltonians imply that this piece is a subalgebra and that it is
isomorphic to $CW^{*}(\ell,\ell^{\prime})$ itself.

To prove the proposition, it is enough to check it for finite slope. Since all
of the Hamiltonians $H_{12}$, $\tilde{H}_{12}$, and $H_{1}+H_{2}$ preserve the
fibre over $(q=\frac{1}{2},p=0)$ near the stop of $M_{1}\times M_{2}$, we
obtain a diagram

(A.1.12) ${CF^{*}(\ell_{1}\times L_{2},\ell^{\prime}_{1}\times
L^{\prime}_{2};j\tilde{H}_{12}^{(m)})}$${CF^{*}(\cup\ell_{1}\times
L_{2},\cup\ell^{\prime}_{1}\times
L^{\prime}_{2};j\tilde{H}_{12}^{(m)})}$${CF^{*}(\ell_{1}\times
L_{2},\ell^{\prime}_{1}\times
L^{\prime}_{2};jH_{12}^{(m)})}$${CF^{*}(\cup\ell_{1}\times
L_{2},\cup\ell^{\prime}_{1}\times
L^{\prime}_{2};jH_{12}^{(m)})}$${CF^{*}(\ell_{1}\times
L_{2},\ell^{\prime}_{1}\times
L^{\prime}_{2};jH_{1}^{(m)}+jH_{2}^{(m)})}$${CF^{*}(\cup\ell_{1}\times
L_{2},\cup\ell^{\prime}_{1}\times
L^{\prime}_{2};jH_{1}^{(m)}+jH_{2}^{(m)})}$$\scriptstyle{=}$$\scriptstyle{\cup}$$\scriptstyle{=}$$\scriptstyle{\cong}$$\scriptstyle{\cup}$$\scriptstyle{\cong}$$\scriptstyle{\cup}$

where we have abused notation and referred to the Hamiltonians in the fibre
and the total space in the same way. The top vertical arrows are the canonical
isomorphisms above, while the bottom vertical arrows are continuation quasi-
isomorphisms coming from the fact that
$H_{12}^{(m)}-(H_{1}^{(m)}+H_{2}^{(m)})$ is compactly supported. The top
square commutes because all arrows are identifications of (sub)complexes. The
bottom square commutes because the strips contributing to the continuation map
which start in the fibre must stay in the fibre, for the same reason that the
generators living in the fibre form a subcomplex.

Since the right-hand cup functor agrees with
$\cup_{\sigma_{1}}\times\operatorname{id}_{\mathcal{F}(M_{2})}$, the result
follows. ∎

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