Homological mirror symmetry for Batyrev mirror pairs

Sheel Ganatra, Andrew Hanlon, Jeff Hicks, Daniel Pomerleano, and Nick Sheridan

We prove Kontsevich's homological mirror symmetry conjecture for a large class of mirror pairs of Calabi–Yau hypersurfaces in toric varieties. These mirror pairs were constructed by Batyrev from dual reflexive polytopes. The theorem holds in characteristic zero and in all but finitely many positive characteristics.

§ INTRODUCTION

The earliest constructions of conjectural mirror pairs concern combinatorially defined Calabi–Yau subvarieties of toric varieties. Batyrev conjectured that, if $\Delta$ and $\Delta^*$ are dual reflexive polytopes, then the (crepant resolutions of) anticanonical hypersurfaces in the corresponding toric varieties should be mirror to each other <cit.>. Batyrev's work generalized Greene–Plesser's earlier construction of conjectural mirror pairs of hypersurfaces in quotients of weighted projective spaces <cit.>; and it was subsequently generalized by Batyrev–Borisov's construction of conjectural mirror pairs of complete intersections in toric varieties <cit.>, which in turn was generalized by Gross–Siebert's construction of conjectural mirror pairs of toric degenerations <cit.>. Various other related constructions have also extended mirror symmetry beyond the Calabi–Yau setting.

Kontsevich's homological mirror symmetry conjecture <cit.> posits an equivalence between natural algebro-geometric and symplectic categories associated to mirror pairs. Although homological mirror symmetry has been highly influential in algebraic and symplectic geometry, it has been proved only for a small number of cases in the original setting of mirror pairs of compact Calabi–Yau varieties. Namely, homological mirror symmetry has been proved for elliptic curves in <cit.>, higher-dimensional abelian varieties in <cit.>, quartics in $\C\mathbb{P}^3$ in <cit.>, Calabi–Yau hypersurfaces in higher dimensional projective spaces in <cit.>, and for Greene–Plesser mirror pairs, together with certain Batyrev–Borisov mirror pairs (`generalized Greene–Plesser mirror pairs', which are complete intersections in quotients of products of weighted projective spaces) in <cit.>. In the present work, we prove homological mirror symmetry for a large new class of Batyrev mirror pairs. One noteworthy feature of our result is that it holds not only in characteristic zero, but also in all but finitely many positive characteristics.

The rest of the introduction is organized as follows: in <ref>, we introduce the toric data on which our construction of Batyrev mirror pairs depends; and in <ref>, we construct the symplectic and algebraic sides of the mirror pair. We then state the homological mirror symmetry conjecture for Batyrev mirrors in <ref> and our HMS statement for Batyrev mirrors (<ref>) in  <ref>. <Ref> gives an overview of the proof of <ref>; we also include a table of notation for the reader's convenience.

§.§ Toric data

We start by introducing some notation used in the Batyrev construction, following <cit.>.

* $\Bbbk$ be a field;

* $M \cong \Z^n$ be a lattice;

* $\Delta \subset M_\R$ a reflexive polytope, $\Delta^* \subset M^*_\R$ its polar dual;

* $\Sigma$ be the fan in $M^*_\R$ dual to $\Delta$ (its rays point along the vertices of $\Delta^*$), and $\Sigma^*$ the fan in $M_\R$ dual to $\Delta^*$;

* $Y$ be the toric $\C$-variety corresponding to $\Sigma$, and $Y^*$ the toric $\Bbbk$-variety corresponding to $\Sigma^*$.

Note that there is a toric line bundle $\cL_{\Delta^*} \to Y^*$, with a basis of sections indexed by the set $\Xinonzero := \Delta^* \cap M^*$.
We denote the section corresponding to $\vec{p} \in \Xinonzero$ by $z^{\vec{p}}$; we similarly denote the section of $\cL_{\Delta} \to Y$ corresponding to $\alpha \in A:=\Delta \cap M$ by $z^\alpha$.

Let $\Xizero \subset \Xinonzero$ be the subset of lattice points lying on a face of $\Delta^*$ which has codimension $\ge 2$.
Let $\kaehl \in (\R_{>0})^{\Xizero}$ and $K \subset \R$ be a subgroup which contains $\kaehl_{\vec{p}}$ for all $\vec{p} \in \Xizero$.
Our construction will depend on the data $(\Bbbk,M,\Delta,\kaehl,K)$.

§.§ Symplectic construction

Consider the following sections of $\cL_{\Delta} \to Y$:

$$W_0 := z^{\vec{0}},\qquad W_1 := \sum_{\alpha \in A \setminus \{0\}}  z^{\alpha}.$$

If $\Sigma^*$ is simplicial, then for sufficiently large $t \in \R_{>0}$, the hypersurface

$$X_t = \{tW_0 = W_1\} \subset Y$$

intersects each toric stratum smoothly (this is proved using ideas from tropical geometry, as we explain in Section <ref>).
When $Y$ is not smooth, the hypersurface $X_t$ need not be smooth; we need therefore to take a crepant resolution of $Y$.
The data for this is determined by $\kaehl$.

Let $\psi_\kaehl: M^*_\R \to \R$ be the largest convex[We use the convention that $f(x)=x^2$ is convex, in contrast to some of the literature on toric varieties (and <cit.>).] function such that $\psi_\kaehl(t \cdot \vec{p}) \le t\cdot \kaehl_{\vec{p}}$ for all $t \in \R_{\ge 0}$ and all $\vec{p} \in \Xizero$.
The decomposition of $M^*_\R$ into domains of linearity of $\psi_\kaehl$ induces a fan $\Sigma_\kaehl$.

We say that the MPCP condition holds if $\Sigma_\kaehl$ is a projective simplicial refinement of $\Sigma$ whose rays are generated by the elements of $\Xizero$.

MPCP stands for Maximal Projective Crepant Partial desingularization (see <cit.>).
In the language of <cit.>, $\Sigma_\kaehl$ is called a simplified projective subdivision of $\Sigma$, and the MPCP condition holds if and only if $\kaehl$ lies in the interior of a top-dimensional cone $\text{cpl}(\tilde{\Sigma}_\kaehl)$ of the secondary fan (or Gelfand–Kapranov–Zelevinskij decomposition) associated to $\Xizero \subset M_\R$, where $\Sigma_\kaehl$ is a projective simplicial refinement of $\Sigma$ whose rays are generated by $\Xizero$.

The morphism of fans $\Sigma_\kaehl \to \Sigma$ induces a toric morphism corresponding to the resolution $Y_\kaehl \to Y$.
Let $X_{t,\kaehl} \subset Y_\kaehl$ be the proper transform of $X_t$.
The intersection of $X_{t,\kaehl}$ with each toric stratum is smooth; so if the MPCP condition holds, then $X_{t,\kaehl}$ is a maximal projective crepant partial desingularization of $X_t$ (hence the name of the condition).

Observe that even if the MPCP condition holds, $Y_\kaehl$ may have finite quotient singularities since $\Sigma_\kaehl$ is only assumed to be simplicial.
Therefore, $X_{t,\kaehl}$ may also have finite quotient singularities.
We would like to understand when $X_{t,\kaehl}$ is in fact smooth.
Let $Y_{\kaehl,0} \subset Y_\kaehl$ denote the pre-image of the toric fixed points of $Y$.
Observe that $X_{t,\kaehl}$ avoids $Y_{\kaehl,0}$.

We say that the MPCS condition holds if the MPCP condition holds, and, furthermore, $Y_\kaehl$ is smooth away from $Y_{\kaehl,0}$.
MPCS stands for Maximal Projective Crepant Smooth desingularization.

We remark that the MPCS and MPCP conditions are equivalent when $\dim_\C(Y_\kaehl) \le 4$ (see <cit.>).
If the MPCS condition holds, then $X_\kaehl$ avoids the non-smooth locus of $Y_\kaehl$, and if $\Sigma^*$ is simplicial, then $X_{t,\kaehl}$ is in fact smooth (again using tropical geometry), and Calabi–Yau by <cit.>.

The simplicial fan $\Sigma_\kaehl$ determines a subdivision of $\Delta^*$ into simplices.
We say that the condition on characteristic is satisfied if $\charac(\Bbbk)$ does not divide

* the affine volume of any of the simplices in the subdivision of $\Delta^*$ determined by $\Sigma_\kaehl$;[We normalize the affine volume so that the minimum possible non-zero affine volume of a lattice simplex is $1$.]

* the order of the finite abelian group $M^*/L$, where $L$ is the subgroup generated by $P$.

We denote the toric boundary divisor of $Y_\kaehl$ by $D^Y$.
Note that it has irreducible components $D^Y_{\vec{p}}$ indexed by $\vec{p} \in \Xizero$.
Let $D^Y_\kaehl := \sum_{\vec {p} \in \Xizero} \kaehl_{\vec{p}} \cdot D^Y_{\vec{p}}$ be the toric $\R$-Cartier divisor with support function $\psi_\kaehl$.
Because $\psi_\kaehl$ is strictly convex (by our assumption that its domains of linearity are the cones of the fan $\Sigma_\kaehl$), this divisor is ample, so the first Chern class of the corresponding line bundle is represented in de Rham cohomology by a toric orbifold Kähler form $\omega^Y_\kaehl$ (see the discussion in <cit.>).
We denote its restriction to $X_{t,\kaehl}$ by $\omega_\kaehl$.
Because $X_{t,\kaehl}$ avoids the non-smooth locus of $Y_\kaehl$, $\omega_\kaehl$ is an honest Kähler form.
Its cohomology class is Poincaré dual to $\sum_{\vec {p} \in \Xizero} \kaehl_{\vec{p}} \cdot [D_\vec{p}]$, where $D_\vec{p} := X_{t,\kaehl} \cap D^Y_{\vec{p}}$.

We now introduce the following notation: for any face $S$ of $\Delta$ (or $\Delta^*$), we define $i(S)$ to be the number of interior points on that face.  Furthermore, we define $S^*$ to be the face of $\Delta^*$ (or $\Delta$) which is dual to $S$.

For $\vec{p} \in \Xinonzero$, let $S_\vec{p}$ denote the face of $\Delta^*$ containing $\vec{p}$ in its interior. Then

$$\#(\text{connected components of
$D_{\vec{p}}$}) = \left\{
\begin{array}{rl} 0 & \text{if $\dim(S^*_{\vec{p}}) = 0$}\\
i(S^*_{\vec{p}}) & \text{if $\dim(S^*_{\vec{p}}) = 1$}\\
1 & \text{otherwise.}
\end{array} \right.  $$

The map $Y_\kaehl \to Y$ blows $D^Y_{\vec{p}}$ down to the stratum $Y_{\vec{p}}$ corresponding to $S^*_{\vec{p}}$. Then, $D_{\vec{p}}$ is the preimage of $X \cap Y_{\vec{p}}$. This is an ample hypersurface and therefore connected unless $Y_{\vec{p}}$ is 0-dimensional (in which case it is empty) or $1$-dimensional (in which case it is a union of $i(S^*_{\vec{p}})$ points).
The preimage of a connected set under a toric morphism is connected, so the result follows.

We say that the connectedness condition is satisfied if for every one-dimensional face $S^*$ of $\Delta^*$, either $i(S^*) = 0$ (i.e., $S^*$ has affine length $1$) or $i(S) = 0$.

§.§ Algebraic construction

We work over a universal Novikov field:

\begin{equation} \Lambda_{\Bbbk,K} := \left\{ \sum_{j=0}^\infty c_j \cdot q^{\kaehl_j}: c_j \in \Bbbk, \kaehl_j \in K, \lim_{j \to \infty} \kaehl_j = +\infty\right\}.\end{equation}

As $\Bbbk$ and $K$ will be fixed throughout the paper, we usually omit them from the notation. Note that $\Lambda$ is a field extension of $\Bbbk$, and if $\Bbbk$ is algebraically closed and $K$ is divisible, it is algebraically closed. Moreover, it has a valuation

\begin{align}
\val: \Lambda & \to K \cup \{\infty\} \\
\label{eqn:valdef} \val\left(\sum_{j=0}^\infty c_j \cdot q^{\kaehl_j}\right) &:= \min_j\{\kaehl_j: c_j \neq 0\}.
\end{align}

We consider sections of $\cL_{\Delta^*} \to Y^*$:

\begin{equation}
\label{eqn:Wb}
W_\dm(z) := -z^{\vec{0}} + \sum_{\vec{p} \in \Xizero} \dm_{\vec{p}} \cdot z^{\vec{p}},
\end{equation}

for $\dm =(\dm_{\vec{p}}) \in \mathbb{A}^{\Xizero}_\Lambda$.
We define $X^*_\dm := \{W_\dm = 0\} \subset Y^*_\Lambda$.

Observe that we have a correspondence

\begin{equation}
\text{monomial $z^{\vec{p}}$ of $W_\dm$} \leftrightarrow \text{divisor $D_\vec{p} \subset X$}.
\end{equation}

This correspondence is called the `monomial–divisor mirror map' (see <cit.>).

§.§ HMS: conjecture

On the $B$-side of mirror symmetry, we consider the $\Lambda$-linear $\Z$-graded dg category of coherent complexes $D^b_{dg}Coh(X^*_\dm)$.[By the results of <cit.>, dg-enhancements of the derived category of coherent sheaves are unique up to quasi-equivalence.] On the $A$-side of mirror symmetry, we consider (a version of) the Fukaya category $\fuk(X_{t,\kaehl},\omega_\kaehl).$ We note that most definitions of the Fukaya category work over ground fields of characteristic zero, whereas we consider mirror symmetry over fields of arbitrary characteristic. We give more details about the version of the Fukaya category we are working with below and in  <ref>. For now, we simply recall that the Fukaya category may be curved, i.e., it may have non-vanishing $\mu^0$ and therefore not be an honest $A_\infty$ category.
Therefore, we consider the version whose objects are bounding cochains on objects of the Fukaya category, which we denote by $ \fuk(X_{t,\kaehl},\omega_\kaehl)^\bc$.
It is a $\Lambda$-linear $\Z$-graded cohomologically unital $A_\infty$ category, whose idempotent completed pre-triangulated closure we denote $\Perf\fuk(X_{t,\kaehl},\omega_\kaehl)^\bc$.
One part of Kontsevich's homological mirror symmetry conjecture for Batyrev mirrors then reads:

Suppose $\Sigma^*$ is simplicial. There is a quasi-equivalence of $\Lambda$-linear $\Z$-graded $A_\infty$ categories

\begin{equation}
\Perf \fuk(X_{t,\kaehl},\omega_\kaehl)^\bc \simeq D^b_{dg}Coh(X^*_{b(\kaehl)}),
\end{equation}

for some $b(\kaehl) \in \mathbb{A}^{\Xizero}_\Lambda$ with $\val(b(\kaehl)_{\vec{p}}) = \kaehl_{\vec{p}}$.

§.§ HMS: statement

The version of the Fukaya category that we work with is the relative Fukaya category, $\fuk(X_{t,\kaehl},D,\kaehl;\Lambda_{\Bbbk,K})$, which is constructed in <cit.>.

Suppose that $\operatorname{dim}(X_{t,\kaehl}) \geq 3$, the condition on characteristic is satisfied, $\Sigma^*$ is smooth, and the MPCS and connectedness conditions hold. Then:

*  There is a fully-faithful embedding $D^b_{dg}Coh(X^*_{b(\kaehl)})\hookrightarrow \Perf \fuk(X_{t,\kaehl},D,\kaehl;\Lambda_{\Bbbk,K})^\bc.$

*  Suppose the relative Fukaya category $\fuk(X_{t,\kaehl},D,\kaehl;\Lambda_{\Bbbk,K})^{\bc}$ satisfies the hypotheses enumerated in <cit.>. Then Conjecture <ref> holds with $\fuk(X_{t,\kaehl},\omega_\kaehl)=\fuk(X_{t,\kaehl},D,\kaehl;\Lambda_{\Bbbk,K}).$

Various hypotheses appear in Theorem <ref> that are not present in Conjecture <ref>. We now provide a short discussion of these hypotheses and where they are used in our proof.

Theorem <ref> holds for any version of the Fukaya category which satisfies the hypotheses enumerated in <cit.>. These hypotheses are expected to hold in whatever version of the Fukaya category one works with (e.g., the general version defined in <cit.>).
It is shown in <cit.> that the relative Fukaya category satisfies some of the required properties; the remaining ones will be verified in forthcoming work.
These remaining hypotheses are only needed in the proof of split-generation of the Fukaya category by the image of the embedding from <ref> (<ref>).

One might hope that the MPCS condition could be replaced by the weaker MPCP condition in Theorem <ref> and that the proof would go through with minimal changes.
In that case, $(X_{t,\kaehl},\omega_\kaehl)$ would be a symplectic orbifold, so the definition of the Fukaya category would need to be adjusted accordingly (compare to <cit.>).

If the connectedness condition does not hold, then we need something more to prove Conjecture <ref>.
That is because there is more than one component of the divisor $D$ corresponding to $\vec{p}$, leading to more than one direction we can deform $\omega_\kaehl$ and more than one class in symplectic cohomology of $X_{t,\kaehl} \setminus D$ corresponding to $\vec{p}$.
(Deformations in these directions are not ambient: an ambient Kähler class would have the same coefficient for each component of $D$ corresponding to $\vec{p}$.)
On the other hand, there is only one corresponding direction in which we can deform $X^*_\dm$ by deforming $\dm$: we can only deform the coefficient in front of $z^{\vec{p}}$.
Batyrev shows that the space of deformations of $X^*_\dm$ does have the right dimension, but it must be that some deformations cannot be embedded in $Y^*$.

Smoothness of the fan $\Sigma^*$ is employed most essentially in three places:

* To match various quasi-equivalences of categories, we use the classification of derived autoequivalences of the derived category of smooth Fano varieties from <cit.>.

* The Floer-theoretic proofs of homological mirror symmetry for toric varieties due to <cit.> work under the assumption that $\Sigma^*$ is smooth. We note that the results from <cit.> obtained using microlocal sheaves only require $\Sigma^*$ to be simplicial.

* The assumption is used in the proof of B-side versality in <ref> (which is an ingredient in the proof of <ref>).

Smoothness of $\Sigma^*$ is also used in various other places to simplify arguments (e.g., <ref>,
<ref>, and <ref>). However, in these latter cases, there  exist straightforward ways to get around the smoothness assumption.

We can refine Conjecture <ref> by specifying that the mirror map $b(\kaehl)$ should be given by the formulae in <cit.>.
We can prove this refined version using the results announced in <cit.>, cf. <cit.>.
By taking care to work throughout over $\Z$, rather than over the coefficient field $\Bbbk$ as we have done, we expect that it would be possible to prove the `integrality of mirror maps' conjecture for the mirror pairs considered in this paper, as was done in the case of Greene–Plesser mirror pairs in <cit.>.
However, we have not done this.

§.§ Overview of proof

$M$     lattice        $M^*$     dual lattice

$\Delta \subset M_\R$     reflexive polytope        $\Delta^* \subset M^*_\R$     dual reflexive polytope

$\Sigma \subset M^*_\R$     fan dual to $\Delta$        $\Sigma^* \subset M_\R$     fan dual to $\Delta^*$

$Y$     toric variety corr. to $\Sigma$        $Y^*$     toric variety corre. to $\Sigma^*$

$\cL_\Delta \to Y$     line bundle corr. to $\Delta$         $\cL_{\Delta^*} \to Y^*$     line bundle corr. to $\Delta^*$

$A$    $\Delta\cap M$       $\Xinonzero$     $\Delta^* \cap M^*$, a basis for $\Gamma(\cL_\Delta)$.

$z^\alpha$     section corr. to $\alpha\in A$        $z^\vec{p}$     section corr. to $p\in \Xinonzero$

$\Xizero \subset \Xinonzero$     points on codim $\ge 2$ faces

$X_t \subset Y$     (singular) CY hypersurface         $X^*_\dm \subset Y^*$     CY hypersurface corr. to $\dm \in \mathbb{A}^{\Xizero}$

$\Sigma_\kaehl \to \Sigma$     refinement det. by $\kaehl \in \R_{>0}^{\Xizero}$

$Y_\kaehl \to Y$     resolution

$X_{t,\kaehl} \subset Y_\kaehl$     proper transform

$D \subset X_{t,\kaehl}$     $X_{t,\kaehl} \cap \partial Y_\kaehl$

$\fo$     Liouville dom. corr. to $X_{t, \kaehl}\setminus D$        $\partial Y^*$     toric boundary

$(\Ldom,\fo)$     Liouville pair        $i:\partial Y^* \hookrightarrow Y^*$     inclusion of toric boundary

$\ess$     Liouville sector corr. to $(B, \fo)$        $Y^*$

$\iota: \eff \times \C_{Re \ge 0} \hookrightarrow \Sprime$     inclusion of subsector

$\cup: \mathcal W(\eff)\to \mathcal W(\Sprime) $     Cup functor        $i_!: D^bCoh(\partial Y^*)\to D^bCoh(Y^*)$     Shriek pushforward along inclusion

$\cW(\Sprime)$     Wrapped Fukaya cat. of $\Sprime$        $D^bCoh(Y^*)$

$\cF(\eff)$     Fukaya cat. of compact Lag.

$\mathcal P(\Sprime) \subset \cW(\Sprime)$     Subcat. of arclike Lagrangians

$\cW(\Sprime)^{M^*}, \cW(\eff)^{M^*}, ...$     $M^*$- graded categories        $D^b_{dg}Coh(Y^*)^{M^*} , D^b_{dg}Coh(\partial Y^*)^{M^*}, ...$

$\cap: \mathcal P(\Sprime)\to \mathcal F(\eff)$     Geometric cap functor
$\tilde{L}_i \subset \Sprime$     ob. of $\cW(\Sprime)$ corr. to $i\Delta$        $\tilde{\cO}(i)$     line bun. on $Y^*$ corr. to $i\Delta$

$\mafuk$     cat. of trop. Lag. sections        $\Pic_{dg}(Y^*)$

$L_i = \partial \tilde{L}_i$     ob. of $\cF(F) = \cF(X_{t,\kaehl} \setminus D)$        $\cO(i)$     line bun. on $\partial Y^*$

$\partial \mafuk$     subcategory of boundaries        $\Pic_{dg}(\partial Y^*)$

$R_A$     coefficient ring of $\cF(X_{t,\kaehl},D)$       $R_B$    $\Bbbk[[\nov_{\vec{p}}]]_{\vec{p} \in P}$

$\cA_0$     subcat. of $\cF(X_{t,\kaehl} \setminus D)$         $\cB_0$     subcat. of $D^b_{dg}Coh(X^*_0) = D^b_{dg}Coh(\partial Y^*)$

$\cA_R$     subcat. of $\cF(X_{t,\kaehl},D)$         $\cB_R$     subcat. of $D^b_{dg}Coh(X^*_R)$

Notation, as well as a rough identification of constructions under the mirror correspondence.

The proof of <ref> proceeds as follows.
On the A-side, we construct certain compact Lagrangians $L_i$ in $X_{t,\kaehl} \setminus D$, and let $\cA_R \subset \fuk(X_{t,\kaehl},D)$ denote the full subcategory with these objects. On the B-side, we let $\mathcal{O}(i)$ denote the restriction to $X_r^*$ of the $i$-th power of the anticanonical bundle on $Y^*_{\Lambda}$ and consider the full subcategory $\cB_R \subset D^b_{dg}Coh(X^*_r)$ with objects $\mathcal{O}(i)$.
These categories are deformations of the $\Bbbk$-linear categories $\cA_0 \subset \fuk(X_{t,\kaehl} \setminus D)$ and $\cB_0 \subset D^b_{dg}Coh(\partial Y^*)$ with the corresponding objects.

The bulk of the paper is devoted to constructing a quasi-isomorphism $\cA_0 \simeq \cB_0$.
Assume for the moment that we have constructed this quasi-isomorphism.
We then apply the versality result from <cit.> to upgrade it to a curved filtered quasi-isomorphism $\Psi^*\cB_R \simeq \cA_R$, where $\Psi$ is an appropriate mirror map.
In order to verify the versality hypotheses on the $A$-side, we need to make certain computations in the symplectic cohomology of $X_{t,\kaehl} \setminus D$, which are accomplished using the results of <cit.> (see <ref>).
The versality hypotheses on the $B$-side are more routine (see <ref>).

Specializing to appropriate $\Lambda$-points allows us to identify the  subcategory of $D^b_{dg}Coh(X^*_r)$ with objects $\mathcal{O}(i)$, and the subcategory of $\Perf \fuk(X_{t,\kaehl},D,\kaehl;\Lambda_{\Bbbk,K})^\bc$ with objects $L_i$.
Because $X^*_r$ is smooth, the objects $\mathcal{O}(i)$ split-generate; by the `automatic generation criterion' of <cit.>, the objects $L_i$ also split-generate.
This completes the proof of <ref> (see <ref>).

Now, let us explain how the quasi-isomorphism $\cA_0 \simeq \cB_0$ is constructed.
We start from the homological mirror symmetry result proved by Gammage–Shende in <cit.>, building on work of Kuwagaki <cit.> and Ganatra–Pardon–Shende <cit.>.
Their work produces a commutative diagram

\begin{equation}\label{eq:GS_HMS}
\begin{tikzpicture}[xscale=1.5, yscale=.75,baseline=-1.125cm]
\node (v1) at (0,0) {$\Perf \cW(X_{t,{\kaehl}} \setminus D)$};
\node (v2) at (3, 0) {$D^b_{dg}Coh(\partial Y^*) $};
\node (v3) at (0,-3) {$\Perf \cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$};
\node (v4) at (3, -3) {$D^b_{dg}Coh(Y^*)$};
\draw[<->] (v1) -- (v2);
\draw[<->] (v3) -- (v4);
\draw[->] (v1) -- node[midway, left, fill=white]{$\cup$}(v3);
\draw[->] (v2) -- node[midway, right, fill=white]{$i_!$}(v4);
\end{tikzpicture}
\end{equation}

in which $\mathfrak{f}_{\Sigma^*}$ is the boundary at infinity of the Fang–Liu–Treumann–Zaslow skeleton, the horizontal functors are quasi-equivalences, the left vertical arrow is the so-called `cup functor' or `Orlov functor', the right vertical arrow is the shriek pushforward along the inclusion $i:\partial Y^* \hookrightarrow Y^*$, and the diagram commutes up to natural quasi-isomorphism.
In order to place ourselves in a setting where (<ref>) holds, we first need to deform the affine variety $X_{t,\kaehl} \setminus D$ and its symplectic structure by `tailoring' it, as in <cit.>, to a Weinstein manifold whose Lagrangian skeleton is computed in <cit.>.
This is accomplished in <ref>, using general results from <ref>.

One might think that the subcategory $\cA_0 \subset \cW(X_{t,\kaehl} \setminus D)$ could then be defined to be the category mirror to the subcategory $\cB_0 \subset D^bCoh_{dg}(\partial Y^*)$ under the quasi-equivalence from (<ref>).
Unfortunately, life is not so easy: a priori, the mirror to this subcategory may consist of non-compact Lagrangians, whereas the objects of the relative Fukaya category $\fuk(X_{t,\kaehl},D)$ are compact Lagrangians (at least as it has been defined to date, cf. <cit.>).
Thus, our main task is to show that the objects mirror to $\mathcal{O}(i)$ are representable by compact Lagrangians.
We do this by constructing a functor $\cap$ which is a `partial adjoint' to $\cup$, and which by construction takes values in the subcategory of compact Lagrangians.
It is `partial' because it is only defined on a certain subcategory $\cP \subset \Perf \cW(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})$ consisting of `arclike' Lagrangians.

The construction of $\cap$ is carried out using a new result in the theory of partially wrapped Fukaya categories, proved in <ref>, which may be of independent interest, and which we now summarize.
Let $j:S_0 \to S_1$ be an inclusion of Liouville sectors, inducing a functor $j_*:\cW(S_0) \to \cW(S_1)$ in accordance with <cit.>.
Given a Lagrangian $L \subset S_1$, one can form its Yoneda module $\cY_L$, and pull it back via $j_*$ to a module over $\cW(S_0)$.
In general, this pullback module need not be representable, i.e., quasi-isomorphic to the Yoneda module of some geometric Lagrangian in $S_0$.
The general form of the result proved in <ref> gives a criterion under which the pullback module is representable: if $L$ is `arclike' with respect to the inclusion $j$, then we can form a Lagrangian called the `negative pushoff in $S_0$ of $L \cap S_0$'; and we prove that its Yoneda module is isomorphic to the pullback of the Yoneda module of (the negative pushoff in $S_1$ of) $L$ under $j_*$.
The functor $\cup$ is defined to be the pushforward functor associated to the inclusion of a standard neighbourhood of the boundary of a Liouville sector. Applying our general result in this case gives the construction of $\cap$ on the subcategory of arclike Lagrangians (see <ref>).

Now, let $\tilde{\cO}(i)$ denote the $i$-th power of the anticanonical bundle on $Y^*$.
We have $i^* \tilde{\cO}(i) = \cO(i)$.
Therefore, if the objects $\tilde L_i$ of $\cW(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})$ mirror to $\tilde{\cO}(i)$ were arclike, then we could define $\cA_0$ to have objects $L_i = \cap \tilde L_i$, which are compact.
Because the homological mirror equivalence matches $\cap$ with $i^*$ (by the uniqueness of adjoints), it would necessarily match $\cA_0$ with $\cB_0$ as required.

Unfortunately, we're still not done: under Kuwagaki's construction of the homological mirror equivalence which fits in as the top horizontal arrow in (<ref>), there is no reason for the objects mirror to $\tilde \cO(i)$ to be arclike.
However, there is a different approach to homological mirror symmetry for Fano toric varieties, due to Abouzaid <cit.>, Hanlon <cit.>, and Hanlon–Hicks <cit.>.
They give a different construction of a mirror equivalence $\Perf \cW(T^* M^*_{S^1},\mathfrak{f}_{\Sigma^*}) \simeq D^b_{dg}Coh(Y^*)$ under which the objects mirror to $\tilde \cO(i)$ (and all other toric line bundles) are arclike.
In order to make use of this, we need to show that the homological mirror equivalence constructed by Abouzaid–Hanlon–Hicks agrees with that constructed by Kuwagaki–Ganatra–Pardon–Shende.
We prove this in <ref>, using the classification of derived autoequivalences of smooth Fano varieties due to Bondal–Orlov.

This completes the construction of the quasi-isomorphism $\cA_0 \simeq \cB_0$, and hence the proof of <ref>: we take $L_i$ to be the objects $\cap \tilde L_i$, where $\tilde L_i$ are the arclike Lagrangians mirror to $\tilde \cO(i)$ under the Abouzaid–Hanlon–Hicks mirror equivalence.
We still need to verify that this equivalence respects certain gradings which are needed in order to apply the versality result; this is done in <ref>, using results from <ref>.

§.§ Acknowledgements

We are very grateful to Danil Koževnikov for pointing out an error in an earlier version; and to Tatsuki Kuwagaki and Alexander Polishchuk for helpful conversations.
S.G. was supported by NSF grant CAREER DMS-2048055 and a Simons Fellowship (award number 1031288), and would like to thank UCLA and Caltech for their hospitality during visits in which some of this work was completed.
A.H. was supported by the Simons Foundation (Grant Number 814268 via the Mathematical Sciences Research Institute, now renamed to SLMath) and NSF grant DMS-2404882.
J.H. is supported by an EPSRC Postdoctoral Fellowship (project reference: EP/V049097/1). D.P. was partially funded by the Simons Collaboration on Homological Mirror Symmetry (award number 652299) and NSF grant
DMS-2306204 while working on this project.
N.S. is supported by a Royal Society University Research Fellowship, an ERC Starting Grant (award number 850713 – HMS), the Simons Collaboration on Homological Mirror Symmetry (award number 652236), the Leverhulme Prize, and a Simons Investigator award (award number 929034).

§ SYMPLECTIC SETUP

We would like to study the Fukaya category of $X_{t, \kaehl}\setminus D = \{tW_0 = W_1\}$ by viewing it as the fiber over $t$ of the potential $W_1/W_0$ on $M^*_{\C^*}$.
Following <cit.>, we replace the spaces $M^*_{\C^*}$ and $X_{t, \kaehl}\setminus D$ with $\Ldom$, a Liouville domain, and $\fo \subset \partial \Ldom$, a Liouville hypersurface in its boundary.
We define a Liouville sector $\ess$ by removing a neighbourhood of $\fo$ from $\Ldom$.

This construction occurs in several steps. <Ref> describes the Liouville structure on $X_{t, \kaehl}\setminus D$. <Ref> deforms the hypersurface to its tropical limit while preserving its smooth toric compactification. <Ref> shows that this tropical limit has skeleton described by  <cit.>. Finally, <Ref> describes the construction of the sector $\ess$ and its relation to $\fo$.

§.§ Relative Kähler manifold

Assume the MPCS condition holds; then we have a complex manifold $X_{t,\kaehl}$ embedded inside the complex orbifold $Y_\kaehl$.
Furthermore, we have an orbifold simple normal crossings divisor $D^Y \subset Y_\kaehl$,[Everything in <cit.> is written for manifolds, rather than orbifolds. Nevertheless, we make use of the corresponding notions for orbifolds, because it makes the exposition clearer. It would be straightforward to remove the orbifold language from any argument in which it appears because we essentially only work with the symplectic manifolds $X_{t,\kaehl}$ and $Y_\kaehl \setminus D^Y$, both of which avoid the orbifold locus.] whose intersection with $X_{t,\kaehl}$ is a simple normal crossings divisor $D \subset X_{t,\kaehl}$.
In the terminology of <cit.>, $(X_{t,\kaehl},D)$ is a sub-$\mathsf{snc}$ pair of $(Y_\kaehl,D^Y)$.

We recall the notion of a relative Kähler form on a $\mathsf{snc}$ pair $(X_{t,\kaehl},D)$ from <cit.>: it consists of a Kähler form on $X_{t,\kaehl}$, together with a Kähler potential on the complement of $D$, having a prescribed form near $D$.
It determines `linking numbers', which are strictly positive real numbers associated with the components of $D$.
One may generalize this to the notion of an orbifold relative Kähler form on $(Y_\kaehl,D^Y)$.

Note that $Y_\kaehl \setminus D^Y \cong M^*_{\C^*} = M^*_\R \times M^*_{S^1}$.
A choice of basis for $M^*$ determines coordinates $x_j$ on $M^*_\R$, and $\theta_j$ on $M^*_{S^1}$, so that the functions $z_j = \exp(x_j+i\theta_j)$ are holomorphic.

We introduce a restricted class of relative Kähler forms on $(Y_\kaehl,D^Y)$ in Definition <ref>.
These have the property that the potential $h^Y$ on $M^*_{\C^*}$ is pulled back from some $\varphi^Y:M^*_\R \to \R$, called a compactification Kähler potential.
The MPCS condition implies that the divisor $D^Y_\kaehl$ is ample, and so by <Ref> there exists a compactification Kähler potential $\varphi^Y$ with corresponding linking numbers $\kaehl_{\vec{p}}$.
We obtain a relative Kähler form $(\omega^X,h^X)$ on $(X_{t,\kaehl},D)$ by restricting this one.

We endow $Y_\kaehl$ with a meromorphic volume from whose restriction to $Y_\kaehl \setminus D^Y = M^*_{\C^*}$ is

$$\Omega^Y|_{M^*_{\C^*}} = d\log z_1 \wedge \ldots \wedge d\log z_n.$$

Observe that $X_{t,\kaehl} \setminus D  \subset M^*_{\C^*}$ is cut out by the equation $W_1/W_0 = t$.
Therefore, we have a holomorphic volume form $\Omega^Y/d(W_1/W_0)$ on $X_{t,\kaehl} \setminus D$, and this extends to a holomorphic volume form $\Omega^X$ on $X_{t,\kaehl}$.

§.§ Liouville domain

Associated to the relative Kähler manifold $(X_{t,\kaehl},D,h)$ from Section <ref>, we have an exact symplectic manifold $\cH_{t,0} := X_{t,\kaehl} \setminus D \subset M^*_{\C^*}$.
The Liouville one-form is $\lf^X = -d^ch^X$.
We denote its Liouville completion by $\widehat{\cH}_{t,0}$ (this is well-defined because the Liouville vector field applied to $h^X$ is positive near $\infty$, cf. Lemma <ref>).

We wish to make computations in the exact Fukaya category of the Liouville manifold $\widehat{\cH}_{t,0}$.
To do this, we deform $\cH_{t, 0}$ to a Liouville manifold with a combinatorially described skeleton.

The first step is to `tailor' the affine variety $\cH_{t,0}$, following <cit.>.
Recall that $A = \Delta \cap M$.
We define functions $(\nu,c): A\to \Z \times \{\pm 1\}$ which are equal to $(1,-1)$ for $\alpha = 0$, and $(0,+1)$ otherwise.
Then $\cH_{t,0} \subset M^*_{\C^*}$ is the vanishing locus of the function

$$W_{t,0}(z) = \sum_{\alpha \in A} c(\alpha)\cdot t^{\nu(\alpha)} \cdot z^{\alpha}.$$

We define the corresponding tropical amoeba $\Pi \subset M^*_\R$ to be the locus of discontinuity of the piecewise-linear function $M^*_\R \to \R$ sending

$$ x \mapsto \max_{\alpha \in A} \left(-\nu(\alpha) + \langle \alpha,x\rangle\right).$$

Smoothness of $\Sigma^*$ (which is a hypothesis of Theorem <ref>) is equivalent to the statement that the subdivision of $\Delta$ induced by $\nu$ is a unimodular triangulation, i.e., it is a subdivision into simplices of the minimal possible volume.

The components of $M_\R^* \setminus \Pi$ are in one-to-one correspondence with $A$.
We let $C_\alpha$ denote the closure of the component indexed by $\alpha \in A$. The only interior lattice point of $\Delta$ is $0$, so the only bounded component is $C_0$, which is equal to $\Delta^*$.

We introduce the map

\begin{align*}
\Log_t: M^*_{\C^*} & \to M^*_\R \\
\Log_t(x,\theta) &:= \frac{1}{\log t} x.
\end{align*}

The image $\cA_t := \Log_{t}(W_t^{-1}(0))$ is referred to as the amoeba. As shown in <cit.>, the Gromov-Hausdorff limit of $\cA_t$ as $t$ goes to infinity is $\Pi$.

Following <cit.>, we can further tailor this convergence by considering an interpolating family of potentials. Namely, we set

$$ W_{t,s} = \sum_{\alpha \in A} (1 - s\psi_\alpha (t, x)) \cdot c(\alpha) \cdot t^{\nu(\alpha)} \cdot z^\alpha $$

for $t > 0$ and $s \in [0,1]$ and where $\psi_\alpha$ are appropriately chosen cutoff functions (we recall that $x_i = \log |z_i|$). In particular, for large $t$, $\psi_\alpha(t,x/\log t)$ is equal to $0$ for $x$ in a neighbourhood of $ C_\alpha$, and $1$ outside of a larger neighbourhood of $C_\alpha$. In fact, we can use explicitly defined cutoff functions as in <cit.>.

We now consider the hypersurfaces $\mathcal{H}_{t,s} = W_{t,s}^{-1}(0)$.

For any $\delta>0$, the family $(\cH_{t,s})_{s \in [0,1]}$ is $\delta$-approximately holomorphic (in the sense of Definition <ref>) for $t$ sufficiently large.

Follows from the proof of <cit.>, together with <cit.>.

A smooth toric compactification of a submanifold $\cH \subset M^*_{\C^*}$ is a simplicial fan $\Sigma_{\cH}$ and smooth submanifold $X_{\cH} \subset Y_{\Sigma_\cH}$, such that $\cH = X_{\cH} \cap M^*_{\C^*}$, $X_{\cH}$ avoids the orbifold points, and the intersection of $X_{\cH}$ with every stratum of the toric boundary divisor is transverse.

For sufficiently large $t>0$, the family $(\cH_{t,s})_{s \in [0,1]}$ admits a family of smooth toric compactifications.

Note that it follows from <Ref> that $\cH_{t,s}$ is a smooth submanifold of $M^*_{\C^*}$ for sufficiently large $t>0$.

We now consider the closure $X_{t,s}$ of $\cH_{t,s}$ in $Y_\kaehl$.
For each cone $\sigma$ of $\Sigma_\kaehl$, such that $X_{t,s}$ intersects the corresponding torus orbit $Y_\sigma$, the latter admits a Zariski neighbourhood with coordinates $(w_\sigma,z_\sigma) \in \C^{\dim \sigma} \times (\C^*)^{\dim Y - \dim \sigma}$.
From the construction of the cutoff functions $\psi_\alpha$ in <cit.>, it is straightforward to check that in a neighbourhood of $Y_\sigma$ they are independent of $w_\sigma$; in other words, there exist cutoff functions $\psi^\sigma_\alpha(t,z_\sigma)$ such that $\psi_\alpha(t,(w_\sigma,z_\sigma)) = \psi^\sigma_\alpha(t,z_\sigma)$.
As a result, in this neighbourhood of $Y_\sigma$, we have $X_{t,s} = \left(W_{t,s}^{\sigma}\right)^{-1}(0)$ where

\[ W_{t,s}^{\sigma}(w_\sigma,z_\sigma) = \sum_{\alpha \in \Delta}  (1 - s\psi^\sigma_\alpha (t, z_\sigma)) \cdot c(\alpha) \cdot t^{\nu(\alpha)} \cdot (w_\sigma,z_\sigma)^\alpha\]

is a smooth function.

The toric morphism $Y_\kaehl \to Y$ blows $Y_\sigma$ down to $Y_{\hat \sigma}$, where $\hat \sigma$ is the smallest cone of $\Sigma$ containing $\sigma$.
Let $\Delta_\sigma$ denote the face of $\Delta$ dual to $\hat \sigma$.
Then $z^\alpha$ vanishes along $Y_\sigma$ for $\alpha \notin \Delta_\sigma$.
To show that $X_{t,s}$ is smooth in a neighbourhood of $Y_\sigma$, and intersects it transversely, it then suffices to show that

\[ X_{t,s} \cap Y_\sigma = \sum_{\alpha \in \Delta_\sigma}  (1 - s\psi^\sigma_\alpha (t, z_\sigma)) \cdot z_\sigma^\alpha\]

is smooth.
However, since $\Delta_\sigma$ is unimodular, this is simply the product of a tailored pair of pants with $(\C^*)^j$ for some $j$, which is smooth.

We now choose a potential $\varphi:M^*_\R \to \R$ which is homogeneous of weight $2$ and adapted to $\Delta^*$ in the sense of <cit.>, which induces a Liouville manifold structure on $M^*_{\C^*}$.
By restricting, we then have a Liouville manifold structure on $\cH_{t,1}$, which has well-defined Liouville completion $\widehat{\cH}_{t,1}$ by Lemma <ref>.

For sufficiently large $t>0$, we have a Liouville isomorphism $\widehat{\cH}_{t,0} \cong \widehat{\cH}_{t,1}$.

The Liouville isomorphism is constructed in two steps.
First, we fix $\cH_{t,0}$ and linearly interpolate between the potential $\varphi^Y$ coming from the toric compactification of $\cH_{t,0}$, and the potential $\varphi$ which is homogeneous of weight $2$ and adapted to $\Delta^*$.
By Lemma <ref>, the latter also has a well-defined Liouville completion; and this completion is Liouville isomorphic to that of $\cH_{t,0}$ by Lemma <ref>.

The second step is to deform $\cH_{t,0}$ to $\cH_{t,1}$ using the family $\cH_{t,s}$, equipped with the fixed homogeneous potential $\varphi$.
This deformation induces a Liouville isomorphism by Lemma <ref> (whose hypotheses are verified by Lemmas <ref> and <ref>).

§.§ Skeleton

The differential of the potential $\varphi:M^*_\R \to \R$ (which we recall is homogeneous of weight $2$ and adapted to $\Delta^*$) defines a diffeomorphism

\begin{align*}
\Phi:M^*_\R &\xrightarrow{\sim} M_\R,\\
\Phi(x) &= d\varphi_x.
\end{align*}

We use it to identify

\begin{equation}\label{eq:Leg_trans}
M^*_{\C^*} = M^*_\R \times M^*_{S^1} \xrightarrow{(d\varphi,\id)} M_\R \times M^*_{S^1}  = T^* M^*_{S^1}     ,
\end{equation}

which fits into a commutative diagram

\begin{equation}\label{eq:Leg_trans_cd} \begin{tikzcd}  M^*_{\C^*} \arrow[d, "\Log"] \arrow[r,"\eqref{eq:Leg_trans}"] & T^*M^*_{S^1}  \arrow[d, "\pi"]& \\
M^*_\R \arrow[r, "\Phi"] & M_\R
\end{tikzcd}
\end{equation}

where $\pi$ is the natural projection.
This identification identifies the Liouville one-form $-d^c \varphi$ on $M^*_{\C^*}$ with the standard Liouville one-form on the cotangent bundle.

To be more concrete, let us choose a basis for $M^*$, inducing coordinates $(x_1,\ldots,x_n)$ on $M^*_\R$, $(\theta_1,\ldots,\theta_n)$ on $M^*_{S^1}$, and $(u_1,\ldots,u_n)$ on $M_\R$.
With respect to these coordinates, we have

$$\Phi(x_1,\ldots,x_n) = \left(\frac{\partial \varphi}{\partial x_1},\ldots,\frac{\partial \varphi}{\partial x_n}\right)$$

$$-d^c \varphi = \sum_{j=1}^n \frac{\partial \varphi}{\partial x_j} d\theta_j = \sum_{j=1}^n u_j d\theta_j.$$

Associated to the fan $\Sigma^* \subset M_\R$ of $Y^*$ is a combinatorially-defined (negative) FLTZ skeleton

\begin{equation} \label{lsigma}
{\mathbb L}_{\Sigma^*} = \bigcup_{\sigma \in \Sigma^*} \sigma \times \sigma^\perp \subset M_\R \times M^*_{S^1}= T^*M^*_{S^1},
\end{equation}

with boundary at infinity

\begin{equation} \label{fsigma}
\mathfrak{f}_{\Sigma^*} = \left( {\mathbb L}_{\Sigma^*} \setminus \left(\{0\} \times M^*_{S^1}\right) \right)/\R \subset \partial_\infty (T^*M^*_{S^1}).
\end{equation}

For $t \gg 0$, the complement of the tailored amoeba $\Log_t(\cH_{t,1})$ has a unique bounded component, whose closure we denote by $\mathcal{C}_0$.
By <cit.>, $\cC_0$ can alternatively be characterized as the sublevel set of the function $H_{\mathcal{C}_0}: M^*_\R\to \R_{\geq 0}$:

\begin{equation}
1 \ge H_{\mathcal{C}_0}:=\sum_{\alpha\in A \setminus \{0\}} (1-\psi_\alpha(t,x))t^{\langle\alpha, x \rangle -1}
\label{eq:cutoffRealFunction}
\end{equation}

The region $\mathcal C_0$ is not necessarily convex. However, it is star-shaped:

For $t$ sufficiently large, there exists $\eps>0$ so that $H_{\mathcal C_0}$ is radially increasing, that is,

\[  dH_{\mathcal{C}_0}(x  \partial_x)  > 0\]

at all points $x$ such that $1-\eps\leq H_{\mathcal C_0}(x)\leq 1$ where $x\partial_x$ is the radial vector field on $M_\R^*$.

By combining Propositions 1.12(3) and 1.13(2) of <cit.>, we can choose $t$ sufficiently large so that the outward unit conormal bundle on $\mathcal{C}_0$ is arbitrarily close in the unit conormal bundle of $M_\R^*$ to the Legendrian given by the union of the outward unit conormals to the strata of $C_0$. The vector field $x \partial_x$ pairs positively with the latter as $C_0$ is a convex polytope containing the origin. The result then follows from the compactness of the level sets of $H_{\mathcal{C}_0}$.

For $t$ sufficiently large, $\Log_{t}^{-1}(\mathcal{C}_0)$ is a Liouville domain completing to $M^*_{\C^*}$.

By the commutative diagram (<ref>), $\Log_t^{-1}(\cC_0) = \pi^{-1}(\Phi^{-1}(\log t \cdot \cC_0))$.
The Liouville vector field for the standard Liouville one-form on $T^*M^*_{S^1}$ is equal to the radial vector field $u \partial_u$.
Thus it suffices to show that $\Phi^{-1}(\log t \cdot \cC_0)$ is star-shaped.
As $\varphi$ is homogeneous of weight $2$ outside a small compact set, and in particular near $\log t \cdot \partial \cC_0$, we have that $\Phi_* (x \partial_x)$ is (pointwise) positively proportional to $u \partial_u$ in that region (cf. <cit.>).
Thus it suffices to check that $\log t \cdot \cC_0$ is star-shaped, which is true by Proposition <ref>.

We then have the following result of <cit.>, which is a generalization of <cit.>:

For $t$ sufficiently large, the intersection $\Lambda_{\Sigma^*} \coloneqq {\mathbb L}_{\Sigma^*} \cap \partial \Log_{t}^{-1}(\mathcal{C}_0)$ is the skeleton of $\mathcal{H}_{t,1}$.

First, <cit.> shows that for $t$ sufficiently large, the skeleton of $\mathcal{H}_{t,1}$ is a subset of $\partial \Log_{t}^{-1}(\mathcal{C}_0)$. Next, <cit.> identifies the projection of the skeleton of $\mathcal{H}_{t,1}$ to $\partial_\infty T^*M^*_{S^1}$ with $\mathfrak{f}_{\Sigma^*}$. As this is also true for $\Lambda_{\Sigma^*}$ and both are contained in $\partial \Log^{-1}_t (\mathcal{C}_0)$, they must coincide.

§.§ Liouville sector

From this point forward, we assume that $t$ is chosen sufficiently large so that all the statements in the previous section hold. The following result is due to <cit.>:

The Liouville vector field $Z = u \partial_u$ is nowhere tangent to $\mathcal{H}_{t,1}$ in a neighbourhood of $\Lambda_{\Sigma^*}$.

The component of $Z$ that is symplectically orthogonal to $\mathcal{H}_{t,1}$ is positively proportional to the Hamiltonian vector field of $\Im(W_{t,1})$ along $\Log_t^{-1}(\cC_0)$ (and hence along the skeleton, by Proposition <ref>) by <cit.>, and thus nonvanishing in a neighbourhood of the skeleton.

We now follow <cit.> to construct a sutured Liouville domain (in the sense of <cit.>) from $(\Log_{t}^{-1}(\mathcal{C}_0), \Lambda_{\Sigma^*})$. We repeat the argument for clarity and later reference.

There exists a Liouville subdomain $\Ldom \subset M^*_{\C^*}$ completing to $M^*_{\C^*}$, together with a Liouville subdomain $\fo \subset \cH_{t,1}$ completing to $\widehat{\cH}_{t,1}$, such that $\fo \subset \partial \Ldom$.

The Liouville flow gives an identification $M^*_{\C^*} \setminus M^*_{S^1} \simeq \partial \Log_t^{-1}(\mathcal{C}_0) \times \R$, which identifies $\partial \Log_t^{-1}(\cC_0)$ with $\partial \Log_t^{-1}( \cC_0) \times \{0\}$.
We choose a Liouville subdomain $\fo \subset \mathcal{H}_{t,1}$, completing to $\cH_{t,1}$, which is a neighbourhood of $\Lambda_{\Sigma^*}$ on which the Liouville vector field is not tangent to $\cH_{t,1}$.
Then, the projection of $\fo$ to $ \partial \Log_t^{-1}( \cC_0)$ is an immersion, and the projection of $\Lambda_{\Sigma^*}$ is an embedding; so by shrinking $\fo$ further, we may arrange that the projection of $\fo$ is also an embedding.

Thus, $\fo$ is identified with the graph of a function on a hypersurface in $ \partial \Log^{-1}_t( \mathcal{C}_0)$.
Extending the function to all of $\partial \Log_t^{-1}$ and taking its graph produces the boundary of a new Liouville domain $\Ldom$ which contains $\fo$ in its boundary.

From the sutured Liouville domain $(\Ldom,\fo)$, we construct a Liouville sector $\ess \subset M^*_{\C^*}$, in the sense of <cit.>, in accordance with <cit.>.
We briefly recall the construction.
We apply the Reeb flow on $\partial \Ldom$ to obtain a neighbourhood of the form $\fo \times \R_{|t| < \eps} \subset \partial \Ldom$ and set $\ess \subset M^*_{\C^*}$ to be the complement of the set $\R_{>0} \times \R_{|t| < \eps} \times \fo$ where the first coordinate is the flow coordinate for the Liouville vector field from $\fo \times \R_{|t| < \eps} \subset \partial \Ldom$. The $1$-defining function $\aye$ is given (on the boundary of the cornered sector) by the linear extension of $-t$.
By construction, we have that there is a small neighbourhood $U$ of $\fo \subset \partial \ess$ on which

\begin{equation}
\label{eq:IandWonFiber} \fo = \aye^{-1}(0) \cap U = \left(\Im \, W_{t,1} \right)^{-1}(0) \cap U.
\end{equation}

The boundary can then be smoothed as in <cit.>.
By <cit.>, the sector embedding $\ess \hookrightarrow (M^*_{\C^*}, \fo)$ presents the pair $(M^*_{\C^*}, \fo)$, where $\fo$ here denotes the projection to $\partial_{\infty} M^*_{\C^*}$ of $\fo$, as the stopped convex completion of the sector $\ess$, that is, as the inclusion of $\ess$ into the pair $(\bar{\ess}, \fo)$ where $\bar{\ess}$ is the convex completion in the sense of <cit.>, and $\fo$ is the copy of $\fo$ lying at $-\infty$ in the completion region $\eff \times \C_{\Re \leq 0}$, where $\eff:= \widehat{\fo}$ is the Liouville completion of $\fo$.

For the construction in <Ref>, we will need one further geometric feature of the sector $\ess$.

We may choose the pair $(\Ldom, \fo)$ such that there exists a neighbourhood $V$ of ${\mathbb L}_{\Sigma^*}$ and an extension of the $1$-defining function $\aye$ to a cylindrical neighbourhood $\Nbd^Z(\partial \ess)$ of $\partial \ess$ in $M^*_{\C^*}$ such that

$$ \aye^{-1}(0) \cap V \cap \Nbd^Z(\partial \ess) = \left(\Im W_{t,1}\right)^{-1}(0) \cap V \cap \Nbd^Z(\partial \ess).$$

We have $Z = Z_{\fo} + Z^\perp$, where $Z_{\fo}$ is the Liouville vector field of $F$ and $Z^\perp$ is symplectically orthogonal to $T\fo$.
Let $g = \Im W_{t,1}$; then we have that $X_g$ is positively proportional to $Z^\perp$ along $\Lambda_{\Sigma^*}$ by <cit.>.
If $R$ is the Reeb vector field of $\partial B$, then we have

\begin{align*}
dg(R) &= \omega(X_g,R) \\
&\propto \omega(Z^\perp,R) \\
& = \omega(Z,R) \quad \text{as $\omega(Z_{\fo},R) = 0$} \\
& = \lf(R) = 1
\end{align*}

along $\Lambda_{\Sigma^*}$.
It follows that $dg(R)>0$ in a neighbourhood of $\Lambda_{\Sigma^*}$ in $\partial B$.
Hence, by shrinking $\fo$ and choosing $\eps$ sufficiently small, we may arrange that $dg(R) > 0$ along $\fo \times \R_{|t|< \eps}$.
It follows that the map

$$\fo \times \R_{|t|<\eps} \xrightarrow{(\id, g)} \fo \times \R_g$$

is a diffeomorphism onto its image.
As $g(f,0) = 0$, we may write $t = t(f,g)$ as a function of $f \in \fo$, $g \in \R$, with $t(f,0) = 0$.

We now choose $\Nbd^Z(\partial \ess)$, so that there is a smooth retraction $\pi:\Nbd^Z(\partial \ess) \to \partial \ess$.
We choose $V$ so that $V \cap \Nbd^Z(\partial \ess) = \pi^{-1}(F \times \R_{|t|< \eps})$.
We define $\pi_F: V \cap \Nbd(\partial \ess) \to \fo$ to be the composition of $\pi$ with the projection to $\fo$.
We now extend the defining function $\aye$ to $V \cap \Nbd^Z(\partial \ess)$ by setting it equal to $\aye(x) = -t(\pi_F(x),g(x))$.
Because $\aye = -t$ along $\fo \times \R_{|t|<\eps}$ by construction, this function extends $\aye$; it also has the property that $\aye=0$ if and only if $g=0$.
We now extend $\aye$ arbitrarily over the rest of $\Nbd^Z(\partial \ess)$.

§ (PARTIALLY) WRAPPED FUKAYA CATEGORIES, CUP AND CAP FUNCTORS

§.§ (Partially) wrapped Fukaya categories

Recall from <cit.> the construction of the partially wrapped Fukaya category $\W(X,\mathfrak{f})$ of a Liouville sector $X$ stopped along a closed subset $\mathfrak{f} \subset (\partial_{\infty} X)^{\circ}$; when $\mathfrak{f} = \emptyset$ we write $\W(X):=\W(X,\emptyset)$. Objects are cylindrical exact Lagrangians $L$ in $X$ disjoint from $\partial X$ and $\mathfrak{f}$. As described in <cit.>, by equipping $X$ and then its Lagrangians with grading/orientation data, one can ensure the resulting category is $\Z$-graded and defined over a ring of arbitrary characteristic.

In particular, to the Liouville sector $\ess$ introduced in Section <ref>, together with the holomorphic volume form $\Omega_{\ess}:= \Omega^Y|_{\ess}$, we have a $\Z$-graded $\Z$-linear partially wrapped Fukaya category $\cW(\ess)$ whose objects are cylindrical exact Lagrangians $L$ disjoint from $\partial \ess$, equipped with spin structures and gradings relative to $\Omega_{\ess}$.
To be precise about the latter choice, we have a squared phase function $L \to S^1$ defined by the argument of $\Omega_{\ess}(v_1 \wedge \ldots \wedge v_{n+1})^2$, where $(v_1,\ldots,v_n)$ is a basis of $TL$; and the grading of $L$ consists of a lift of this squared phase function to $\R$. Note that there are natural maps

\begin{equation} \label{eq:wrappedstoskel}
\mathcal{W}(\ess) \to \mathcal{W}(M^*_{\C^*}, \fo) \to \mathcal{W}(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*} )
\end{equation}

which are quasi-equivalences as shown in <cit.> (using the fact that $(M^*_{\C^*}, \fo)$ is the stopped convex completion of $\ess$). These quasi-equivalences make it clear that $\mathcal{W}(\ess)$ does not depend on any of the choices involved in constructing $\ess$.

The approach taken to gradings and orientations in <cit.> is equivalent, but formulated in a slightly different language.
The grading/orientation data we consider for $\ess$ consist of the constant map $\ess \to K(\Z/2,2)$ (we consider spin structures, rather than relative spin structures), together with the map $\mathrm{LGr}(\ess) \to S^1$ given by the squared phase map.
It is easy to see from the definitions how our choice of spin structure and grading on a Lagrangian $L$ induces a choice of grading/orientation data on $L$ in the sense of <cit.>. Moreover, our choice of grading/orientation conventions is compatible with that made in <cit.>, as the tautological polarization of $\ess \subset T^*M^*_{S^1}$ is a trivial bundle (agreeing with our choice of trivial map to $K(\Z/2,2)$) and has phase $0$ with respect to our choice of holomorphic volume form.

We also have a $\Z \oplus M^*$-graded version $\cW(\ess)^{M^*}$, whose objects are objects of $\cW(\ess)$ equipped with an additional lift of $L$ to the universal cover of $\ess$ (an `anchoring', in the terminology of <cit.>).
The additional grading then arises from $\pi_1(\ess) = H_1(\ess) = M^*$ (cf. <cit.>).
There is a fully faithful functor $\cW(\ess)^{M^*} \to \cW(\ess)$ which forgets the anchoring on objects, and the $M^*$-grading on morphisms. We can, of course, apply the same construction to obtain $Z\oplus M^*$-graded categories $\cW(M^*_{\C^*})^{M^*}$ and $\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})^{M^*}$ and a $\Z \oplus M^*$-graded analog of (<ref>).

Associated to the (finite-type, convex) exact symplectic manifold $\cH_{t,0}$, together with the holomorphic volume form $\Omega_{\cH_{t,0}}:=\Omega^X|_{\cH_{t,0}}$, we have a $\Z$-graded $\Z$-linear wrapped Fukaya category $\cW(\widehat{\cH}_{t,0})$, whose objects are cylindrical exact Lagrangians equipped with spin structures and gradings relative to $\Omega_{\cH_{t,0}}$.
As before, we have a $\Z \oplus M^*$-graded version $\cW(\widehat{\cH}_{t,0})^{M^*}$, whose objects are objects of $\cW(\widehat{\cH}_{t,0})$ equipped with a lift of the composition $L \to \widehat{\cH}_{t,0} \hookrightarrow M^*_{\C^*}$ to the universal cover of $M^*_{\C^*}$.

To simplify notation, we will typically write $\cW(\cH_{t,0})$ for the wrapped category $\cW(\widehat{\cH}_{t,0})$. More generally, we will apply the same convention when discussing wrapped categories of other incomplete exact symplectic manifolds that arise in our arguments (which we may do because in all such cases, the completion is well-defined).

As in <Ref>, our choice of grading/orientation data on $\cH_{t,0}$ can be interpreted in the language of <cit.>. From the equation $\Omega_{\cH_{t,0}} \wedge dW_{t,0} = \Omega_{M^*_{\C^*}}$, we have that our grading/orientation data on $\cH_{t,0}$ agrees with the grading/orientation data induced by the stable polarization of $\cH_{t,0}$ coming from the tautological polarization of $T^*M^*_{S^1}$ and the trivialization of the normal bundle of $\cH_{t,0}$ by $dW_{t,0}$.

As it will play a central role going forward, we introduce the abbreviated notation $\eff = \widehat{\fo} = \widehat{\cH}_{t,1}$. We can then similarly associate to $\eff$ a $\Z$-graded $\Z$-linear  wrapped Fukaya category $\cW(\eff)$ and a $\Z \oplus M^*$-graded version $\cW(\eff)^{M^*}$.
However, $\fo$ is not a holomorphic submanifold, and thus it is more natural to induce grading/orientation data on $\eff$ directly from a stable polarization.
Indeed, as a Liouville hypersurface of $\Ldom \subset M^*_{\C^*}$, $\fo$ has a natural stable polarization induced by the tautological polarization of $T^*M^*_{S^1} \simeq M^*_{\C^*}$ and the trivialization of the normal bundle of $\fo$ by the pair $(Z^\perp, R_\varphi)$ where $Z^\perp$ is the projection of the Liouville vector field on $M^*_{\C^*}$ to the symplectic orthogonal of $\fo$ and $R_\varphi$ is the Reeb vector field on $\partial \Ldom$. Equivalently, the trivialization of the normal bundle can be seen as coming from a standard neighbourhood of $\partial \ess$ produced in <cit.>.

In fact, the wrapped Fukaya categories of $\cH_{t,0}$ and $\eff$ are quasi-equivalent:

There is a $\Z \oplus M^*$-graded quasi-equivalence

$$ \cW(\cH_{t,0})^{M^*} \simeq \cW(\eff)^{M^*} $$

for any sufficiently large $t$.

From <Ref>, we have a Liouville isomorphism $\widehat{\cH}_{t,0} \simeq \widehat{\cH}_{t,1} = \eff$, resulting in an $M^*$-graded quasi-equivalence. It remains only to check that the Liouville isomorphism respects our chosen grading/orientation data. To show this, we work in terms of stable polarizations (see <Ref> and  <Ref>).

Through the deformation from $\cH_{t,0}$ to $\cH_{t,1}$ via the family $\cH_{t,s}$, we have a homotopy of grading/orientation data via the stable polarization of $\cH_{t,s}$ obtained from the tautological polarization of $T^*M^*_{S^1}$ and the trivialization of the normal bundle to $\cH_{t,s}$ by $dW_{t,s}$. We now must compare this grading/orientation data on $\cH_{t,1}$ with that on $\eff$. However, <cit.> shows that on a subset of $\cH_{t,1}$ homotopy equivalent to both $\eff$ and $\cH_{t,1}$ the Hamiltonian vector field $\vX_{\Im \, W_{t,1}}$ of $\Im \, W_{t,1}$ is positively proportional to $Z^\perp$. Thus, the oriented frames $\left(Z^\perp, R_\varphi\right)$ and $\left(\vX_{\Im \, W_{t,1}}, \vX_{\Re \, W_{t,1}}\right)$ cannot rotate relative to each other around any cycle, i.e., they induce homotopic trivializations of the normal bundle.

The equivalence of the grading/orientation data in the proof of <ref> and part of the isomorphism of Liouville structures have recently also been proved in <cit.>.

§.§ Cup functor

By virtue of the fact that $\ess \subset (M^*_{\C^*}, \fo)$ is the inclusion into the stopped convex completion of $\ess$, the complement of $\ess$ can be identified with the stopped Liouville sector $(\eff \times \C_{\Re \leq 0}, \fo)$, where $\eff \times \C_{\Re \leq 0} \hookrightarrow M^*_{\C^*}$ is the canonical neighborhood of $\fo \subset \partial_{\infty} M^*_{\C^*}$.
The sector inclusion $(\eff \times \C_{\Re \leq 0}, \fo) \hookrightarrow (M^*_{\C^*}, \fo)$ induces (after appeal to <cit.>) a functor $\W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*}) \to \W(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})$. By composing with stabilization <cit.>, we obtain a functor $\W(\eff) \to \W(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})$, which we call in this paper the cup functor $\cup$ (compare <cit.>, <cit.>, <cit.>).

There is an equivalent realization of the $\cup$ functor with target the quasi-equivalent $\cW(\Sprime)$, where $\Sprime$ is a small deformation of $\ess$ (deformed along the lines of <cit.>) which has exact boundary in the sense of <cit.>. From the fact that $\Sprime$ has exact boundary, one obtains an embedding of Liouville sectors $\eff \times T^*[0,1] \hookrightarrow \Sprime$ covering $\Nbd^Z\partial \Sprime$ — called here a boundary sector embedding — whose induced pushforward can then be composed with stabilization to define $\cup: \cW(\eff) \to \cW(\Sprime)$.

In <ref> below, we recall a particular method of constructing the deformation $\Sprime$, close to that appearing in the proof of <cit.>, as a subsector of $(M^*_{\C^*}, \fo)$ contained in an arbitrarily small cylindrical neighborhood of $\ess$ with boundary outside $\ess$. We also review how the deformation argument provides us with an extended defining function $\Iprime$ on $\Sprime$ which has controlled deviation from, and some compatibilities with, our previously fixed extended 1-defining function $\aye$ for $\partial \ess$. We define $(\Sprime,\Iprime)$ to be the outcome of applying the following lemma to our $(B,\fo) \subset M^*_{\C^*}$ with associated sector and 1-defining function $(\ess,\aye)$ constructed in <ref> and <ref>. We will use the third property of $(\Sprime,\Iprime)$ below in <ref>.

Let $(B,\fo)$ be a sutured Liouville domain with $B$ completing to a Liouville manifold $M$, with $\ess \subset M$ the associated Liouville sector.
Let $\aye$ be any extended defining function defined in a neighbourhood of $\partial \ess$ within $M$ (for instance, the one from <ref> and <ref>).
In that setting, there exists a small deformation $\ess \subset \Sprime \subset (M,\fo)$, along with an extended 1-defining function $\Iprime$ defined near $\partial \Sprime$ satisfying the following properties:

*  $\Sprime$ is contained in $\ess^+:=$ an arbitrarily small outward $X_{\aye}$ pushoff of $\ess$.

There is a boundary sector embedding $\eff \times T^*[0,1] \hookrightarrow \Sprime$ (and in particular $\Sprime$ has exact boundary) living in the complement of $\ess$ and fitting into a commutative diagram:

\begin{equation}\label{cupsquare}
\begin{tikzcd}
\eff \times T^*[0,1]\arrow[hookrightarrow]{d}{}  \arrow[hookrightarrow]{r}{} & (M \backslash \ess^{\circ}, \fo) \arrow[hookrightarrow]{d}{}   \\
\Sprime  \arrow[hookrightarrow]{r}{} & (M,\fo) \end{tikzcd}
\end{equation}

where the horizontal inclusions induce equivalences on wrapped Fukaya categories.

*  There is a Hamiltonian vector field $V$ on $\Sprime \backslash \ess^{\circ}$ which (while not necessarily linear near infinity on the whole region) agrees with $X_{\Iprime}$ near $\partial \Sprime$ and $X_{\aye}$ near $\partial \ess$ and carries $\partial \ess \stackrel{\sim}{\to} \partial \Sprime$.

We begin with $\ess \hookrightarrow (M, \fo)$ and $\aye$ as previously constructed. Let $\ess^+$ be the outward flow of $\ess$ by $X_{\aye}$ for some (unspecified)
small time $a > 0$.
On the region
$\ess^+ \backslash \ess^{\circ} = \Nbd^{Z}(\partial \ess^+)$, $\aye$ determines
an identification with $(\eff \times T^*[0,a], \lambda_{\eff} + \lambda_{T^*[0,a]} +
df)$ for some function $f$ satisfying properties as in <cit.> and where $\eff = \widehat{\fo}$; importantly, $f$ may not be compactly supported
(otherwise we'd be done). After rescaling $\aye$, we identify $T^*[0,a] =
T^*[0,1]$ for simplicity.

Next, as in the proofs of <cit.> and
<cit.> (and closer in construction to
the former), we deform the Liouville one-form $\lambda
\stackrel{\lambda_t}{\leadsto} \lambda'$ within the $\eff \times T^*[0,1]$ region above to turn off the $df$ term in a
subshell $\eff \times T^*[\frac{2}{5},\frac{3}{5}]$ without changing $\lambda$
outside of $\eff \times T^*[\frac{1}{5},\frac{4}{5}]$. Concretely we take the straight line homotopy between $\lambda_0:=\lambda$ and $\lambda_1:= \lambda'= \lambda_{\eff} + \lambda_{T^*[0,1]} + d(\varphi(s) f)$ where $\varphi(s) = 1$ for $s$ outside $[\frac{1}{5},\frac{4}{5}]$ and $\varphi(s) = 0$ for $s$ inside $[\frac{2}{5},\frac{3}{5}]$, and extend this to a homotopy $\lambda_t$ on $M$ which is constant outside this region.
The family $\lambda_t$ is a convex-at-infinity deformation of Liouville structures as in loc. cit. It therefore induces a Liouville isomorphism $\Phi: (M, \lambda_1) \to (M,\lambda_0)$, which is identity inside $\ess$ and outside $\ess^+$ by <cit.>.
Now with respect to the new Liouville form $\lambda_1$ consider  $S_1:= \ess \cup (\eff \times T^*[0,\frac{3}{5}])$ which sits between $\ess$ and $\ess^+$ and is a Liouville subsector of $(M,\lambda_1)$. We observe the following:

* $S_1$ is contained in $\ess^+$;

* $S_1$ has a boundary sector embedding $(\eff \times T^*[\frac{2}{5},\frac{3}{5}],\lambda_{\eff} + \lambda_{T^*[\frac{2}{5},\frac{3}{5}]}) \hookrightarrow S_1$ which lives in $S_1 \backslash \ess^{\circ}$.  Now identify $T^*[\frac{2}{5},\frac{3}{5}]$ (by scaling up in the base and shrinking in the fiber) with $T^*[0,1]$.

* The inclusion of $\eff \times T^*[\frac{2}{5},\frac{3}{5}]$ into $\ess^+ \backslash \ess^{\circ}$ is deformation equivalent to a trivial inclusion, and hence by <cit.> induces an equivalence on wrapped Fukaya categories;

* Similarly, the inclusion of $S_1$ into $\ess^+$ is deformation equivalent to a trivial inclusion, and hence induces an equivalence on wrapped Fukaya categories.

* The vector field $X_{\aye}$ is a Hamiltonian vector field (indeed we haven't changed $\omega$!) taking $\partial \ess$ to $\partial S_1$. The associated Hamiltonian function $\aye$ is an extended 1-defining function near $\partial \ess$ and $\partial S_1$, but may fail to be linear on all of $S_1 \backslash \ess^{\circ}$ (given our modifications to $\lambda$).

Now push forward by $\Phi$: let $\Sprime:= \Phi(S_1)$ with extended 1-defining function $\Iprime := \Phi_*(\aye)$. Since $\Phi|_{\ess} = id$ and $\Phi|_{M \backslash (\ess^+)^\circ} = id$, we see that $\Sprime$ satisfies <ref>.
The remaining conditions other than the top horizontal arrow of <ref> inducing an equivalence follow immediately from taking $\Phi$ of the above bullet points.
To study the effect of the top horizontal arrow of <ref> on wrapped Fukaya categories, first use $\Phi$ of the third bullet point to obtain an equivalence $\W(\eff \times T^*[0,1]) \stackrel{\sim}{\to} \W(\ess^+ \backslash \ess^{\circ})$.

The desired equivalence therefore follows from (by composing with) the claim that the sector embedding $(\ess^+ \backslash \ess^{\circ}) \hookrightarrow (M \backslash \ess^{\circ}, \fo)$ induces an isomorphism on wrapped Fukaya categories.
To establish the claim, it suffices to replace $\ess^+$ by the sufficiently small time outward flow $(\ess^+)'$ of any other extended 1-defining function $I'$ by the following sandwiching argument, compare <cit.>: Denote by $\Nbd^Z(\partial \ess)_{I'}^{+a}$ the complement of $\ess^{\circ}$ in the outward time-$a$ flow of $\ess$ by $X_{I'}$. Then observe that for any $I_1$, $I_2$ there exist sufficiently small $a <c$, $b<d$ and a sequence of inclusions $\Nbd^Z(\partial \ess)_{I_1}^{+a} \hookrightarrow \Nbd^Z(\partial \ess)_{I_2}^{+b} \hookrightarrow \Nbd^Z(\partial \ess)_{I_1}^{+c} \hookrightarrow \Nbd^Z(\partial \ess)_{I_2}^{+d}$. Since any composition of adjacent arrows $\Nbd^Z(\partial \ess)_{I_j}^{+i_1} \hookrightarrow \Nbd^Z(\partial \ess)_{I_j}^{+i_2}$ is a trivial inclusion, inducing an equivalence on wrapped Fukaya categories <cit.>, it follows that $\cW(\Nbd^Z(\partial \ess)_{I_1}^{+a}) \stackrel{\sim}{\to} \cW(\Nbd^Z(\partial \ess)_{I_2}^{+b})$.

It is now enough to show the claim holds after replacing $\aye$ with the extended 1-defining function  $\aye'= -e^s t$, with associated
vector field $X_{\aye'} = -\frac{\partial}{\partial s} + t
\frac{\partial}{\partial t} + Z_{\lambda_{F_0}}$.
Here recall that in local coordinates $\ess$ is the complement
of the set $(\R_{>0})_s \times \R_{|t| < \eps} \times \fo$ in $M$ (with smoothed boundary) where $s$ is the symplectization and $t$ is a Reeb coordinate as in
the previous section, and the Liouville one-form is $\lambda = e^s (dt - \lambda_{\fo})$.
The outward flow $\ess^+$ of $\ess$ by $X_{\aye}$ for a small
time $a > 0$
has $\partial \ess^+$ equal to (a smoothing of) the boundary of
the set $(\R_{>a})_s \times \R_{|t| < e^{-a}\eps} \times {\fo}^{-a}$, where
$\fo^{-a}$ denotes the result of inward flow by $Z_{\lambda_{\fo}}$.
In particular, $S^+$ is the sector constructed from the sutured pair $(B^{+a}, \fo^{-a}) \subset M$ using the neighborhood $\R_{|t| < e^{-a}\eps} \times {\fo}^{-a}$. Hence $(M,\fo)$ is up to deformation the stopped convex completion of $S^+$ by <cit.>, and therefore also the convex completion along the boundary component $\partial S^+$ of $S^+ \backslash \ess^{\circ}$. In particular, we may apply <cit.> to the inclusion $(\ess^+ \backslash \ess^{\circ}) \hookrightarrow (M \backslash \ess^{\circ},
\fo)$ (note while loc. cit. is stated for the “full” stopped completion of a sector, it also applies immediately to the stopped convex completion along a single boundary component of a sector with disconnected boundary) to obtain
an equivalence $\W(\ess^+ \backslash \ess^{\circ}) \stackrel{\sim}{\to} \W(M \backslash \ess^{\circ}, \fo^{-a})$ which factors through the equivalence (by loc. cit. again) $\W(M \backslash \ess^{\circ}, \fo) \to \W(M \backslash \ess^{\circ}, \fo^{-a})$. Hence $\W(\ess^+ \backslash \ess^{\circ}) \stackrel{\sim}{\to} \W(M \backslash \ess^{\circ}, \fo)$ is an equivalence.

Using $\Sprime$ and $\eff \times T^*[0,1] \hookrightarrow \Sprime$ given to us in <ref>, we immediately obtain a boundary inclusion functor

\begin{equation}\label{boundaryinclusionfunctor}
\W(\eff \times T^*[0,1]) \to \W(\Sprime)
\end{equation}

fitting into a commutative diagram

\begin{equation}\label{boundaryinclusionsquare}
\begin{tikzcd}
\W(\eff \times T^*[0,1])\arrow{d}{}  \arrow{r}{} & \W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*}) \arrow{d}{}   \\
\W(\Sprime)  \arrow{r}{} & \W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*}) \end{tikzcd}
\end{equation}

where the horizontal arrows are equivalences. Here we have implicitly applied <cit.> and also used the fact that $(M \backslash \ess^{\circ}, \fo)$
up to deformation is a model for the canonical $\eff \times \C_{\Re \geq 0}$ neighborhood of $\fo$ in $M$ at infinity <cit.>.

Next, there is a Künneth/stabilization embedding $stab: \cW(\eff) \stackrel{\sim}{\to}\cW(\eff \times T^*[0,1])$  <cit.>, defined by multiplying with a cotangent fiber using the Künneth embedding

stab: \cW(\eff) \xrightarrow{- \otimes \R} \cW(\eff) \otimes \cW(T^*[0,1]) \xrightarrow{\text{K\"{u}nneth}} \cW(\eff \times T^*[0,1]).

Here we are considering the object $\R$ of $\cW(T^*[0,1])$ given by the cotangent fibre $\{x=\frac{1}{2}\}$, equipped with the standard spin structure (which in particular induces the standard orientation on $\R$).
The squared phase function is then constant equal to $2\mathrm{arg}(\Omega_{T^*[0,1]}(\partial_y)) = \pi \in S^1$.
We choose the grading to be equal to $\pi \in \R$.
Note that these choices agree with the grading/orientation data on $T^*[0,1]$ and $\R$ induced by the tautological polarization of $T^*[0,1]$.

Composing with the inclusion/equivalence $\W(\eff \times T^*[0,1]) \stackrel{\sim}{\to} \W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*})$ gives a stabilization functor $stab: \cW(\eff) \stackrel{\sim}{\to} \W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*})$.[It is tempting to pick the stabilization functors $stab: \cW(\eff) \stackrel{\sim}{\to}\cW(\eff \times T^*[0,1])$ and $stab: \cW(\eff) \stackrel{\sim}{\to} \W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*})$ completely independently and then argue that they are the “same” after including the first codomain into the second. While this is expected to hold, it is not shown in  <cit.> that Künneth is functorial under arbitrary sector inclusions or independent of all choices made. However, in order to apply the results of <cit.> later, we are free to pick one such stabilization functor once and for all.
The needed structural results from loc. cit. — articulated in <ref> below — hold for any such choice.]
As in <cit.>, we then define the cup functor

\begin{equation}\label{cupeqn}
\cup: \cW(\eff) \to \cW(\Sprime)
\end{equation}

by composing $stab$ with (<ref>). At the start of this subsection, we defined $\cup$ as the composition $\cW(\eff) \stackrel{stab}{\to} \W(\eff \times \C_{\Re \leq 0}, \mathfrak{f}_{\Sigma^*}) \to \W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$. The commutative diagram (<ref>) immediately implies:

For $\Sprime$ as in the previous Lemma, there is a commutative diagram:

\begin{equation}\label{cupfunctorsquare}
\begin{tikzcd}
\W(\eff)\arrow{d}{\cup}  \arrow{r}{=} & \W(\eff) \arrow{d}{\cup}   \\
\W(\Sprime)  \arrow{r}{\sim} & \W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*}) \end{tikzcd}
\end{equation}

Note that $\cup$ is a $\Z$-graded functor with respect to the natural
grading/orientation data on $F$ described in <Ref>. As
described in <ref> for $\cW(\ess)$, there is also a $\Z
\oplus M^*$ graded version $\cW(\Sprime)^{M^*}$ of the wrapped Fukaya category of
$\Sprime$ (with $\cW(\ess)^{M^*} \to \cW(\Sprime)^{M^*}$ a quasi-equivalence), and it is straightforward
to enhance (<ref>) to a $\Z \oplus M^*$-graded functor

\begin{equation}
\cup^{M^*}: \cW(\eff)^{M^*} \to \cW(\Sprime)^{M^*}
\label{eq:gradedcup}
\end{equation}

§.§ Cap functor

We can associate to the cup functor (<ref>) a formal adjoint functor $\cup^*:\mathcal W(\Sprime)\to \operatorname{Mod} \mathcal W(\eff)$ by composing the pullback of modules with Yoneda. In good situations, this functor is representable (i.e., there is an adjoint to the $\cup$-functor).
For example, in the setting of symplectic Landau-Ginzburg models, it is known that the cup functor is spherical <cit.>, and therefore has both left and right adjoints. The right adjoint is usually called the cap functor as for a class of Lagrangian submanifolds $L\subset \ess$ it can be geometrically described as “intersection of $L$ with the fiber $\eff$”.
Several incarnations of this functor have been studied by various authors such as <cit.> with different geometric hypotheses.
In the remainder of this section, we define the version of the cap functor that we will use on a particular class of Lagrangians that we now define.

Let $L\subset \Sprime$ be a compact Lagrangian submanifold with $\partial L\subset \partial \Sprime$. We say that $L$ is $\Iprime$-arclike if $L$ is transverse to $\partial \Sprime$, and $I$ is constant along some neighbourhood of $\partial L$ in $L$. Here $I$ is the same extended $1$-defining function which was used in the definition of the cup functor.

In <Ref>, we introduce a subcategory  $\cP(\Sprime) \subset \cW(\Sprime)$ of negative pushoffs of ($I$-)arclike Lagrangians, and show that intersection with the fiber $\eff$ is adjoint to the cup functor. More precisely, in <Ref>, we show there exists an $A_\infty$ functor

$$\cap: \cP(\Sprime) \to \cF(\eff)$$

which sends a negative-pushoff $L^-$ of an arclike Lagrangian $L$ to the boundary $\partial L$ (which is compact, and in particular defines an object of the compact Fukaya category $\cF(\eff) \subset \cW(\eff)$).
Furthermore, there exist functorial isomorphisms

\[\Hom_{H\cW(\Sprime)}(\cup K,L) \cong \Hom_{H\cW(\eff)}(K,\cap L)\]

for all objects $K$ of $\cW(\eff)$, $L$ of $\cP(\Sprime)$ which lift to an equivalence of suitable $\cW(\eff)\!-\!\cP(\Sprime)$ bimodules on the chain level.
Thus, $\cap$ behaves like a right adjoint to $\cup$ restricted to $\cP(\Sprime)$. We say that $\cap$ is a partially-defined right adjoint to $\cup$ defined along $\cP(\Sprime)$, in the terminology of <ref>. The construction of cap is agnostic as to the grading structures imposed upon our categories; in particular, it also gives us a $\Z \oplus M^*$ graded version of $\cap$:

\begin{equation}\label{gradedcap}
\cap^{M^*}: \cP(\Sprime)^{M^*} \to \cF(\eff)^{M^*}.
\end{equation}

§ HOMOLOGICAL MIRROR SYMMETRY AT LARGE RADIUS

In this section, we recall and analyze homological mirror symmetry relating the sector $\ess$ and its boundary fiber $\eff$ on the A-side to $Y^*$ and its toric boundary $\partial Y^*$ on the B-side.

The main result of this section is <ref>, which (in part) constructs a quasi-equivalence $\Pic_{dg}^{M^*}(\partial Y^*) \simeq \partial \mafuk^{M^*}$ fitting into a diagram of $A_\infty$ categories and functors:

\[\begin{tikzcd}
D^b_{dg} Coh(\partial Y^*) \arrow[leftrightarrow]{r}{GS} & \Perf \cW(\eff) \\
\Pic_{dg}^{M^*}(\partial Y^*)  \arrow[leftrightarrow]{r} \arrow[hookrightarrow]{u} & \partial \mafuk^{M^*}  \arrow[hookrightarrow]{u} \\
\Pic_{dg}^{M^*}(Y^*) \arrow{u}{i^*} \arrow[leftrightarrow]{r}{A} & \mafuk^{M^*} \arrow{u}{\cap_{mon}},\end{tikzcd}.
\]

The top horizontal arrow is a quasi-equivalence constructed by Gammage–Shende <cit.> and the bottom horizontal arrow is a quasi-equivalence constructed by Abouzaid <cit.>.
On the lower right-hand side, we have the “monomially admissible Fukaya categories”:  $\mafuk$ is the category of tropical Lagrangian sections of $S$; $\partial \mafuk$ is the subcategory of the compact Fukaya category $\cF(\eff)$ consisting of boundaries of tropical Lagrangian sections. On the left-hand side, $\Pic_{dg}(Y^*)$ is the category of line bundles on $Y^*$, and $\Pic_{dg}(\partial Y^*)$ is the category of line bundles on $\partial Y^*$ that are restrictions of line bundles on $Y^*$. On both sides, the $M^*$ superscript indicates that these categories are $M^*$-graded. The functors $i^*$, $A$, and $\cap_{mon}$ are $M^*$-graded; we show that the middle horizontal arrow can also be made $M^*$-graded, on the subcategory we are interested in, in <ref>.

§.§ Homological mirror symmetry for Fano toric varieties

We begin with homological mirror symmetry where the toric variety $Y^*$ is on the $B$-side. The relevant dg category associated with $Y^*$ is $D^b_{dg}Coh(Y^*)$.
On the $A$-side, the relevant $A_\infty$ category is the wrapped Fukaya category $\mathcal{W}(\Sprime)$ of $\Sprime$ from <Ref>. We equip cotangent fibers (and Lagrangians Hamiltonian isotopic to them) and the zero section $M^*_{S^1}$ with canonical grading/orientation data coming from the tautological polarization of $T^*M^*_{S^1} \simeq M^*_{\C^*}$ (see <Ref>).

We will recall two independently established statements of homological mirror symmetry for toric varieties from the literature relating $\mathcal{W}(\Sprime)$ and $D^b_{dg}Coh(Y^*)$. We will then prove a few properties of these quasi-equivalences and use a theorem of Bondal and Orlov <cit.> to show that they coincide for smooth Fano toric varieties.

We will use a slight enlargement of the category $\mathcal{W}(\Sprime)$.
Namely, the objects will be pairs $(L,\zeta)$ where $L$ is an exact and cylindrical Lagrangian brane and $\zeta$ is a $\Bbbk$-local system on $L$; counts of holomorphic discs are weighted by the holonomy of the local systems around the boundary. In fact, we will only need to consider local systems on the exact Lagrangian torus $M^*_{S^1}$. Note that the $\Bbbk$-local systems on $\Sprime$
are in natural correspondence with $M_{\Bbbk^*}$ from the isomorphisms $\pi_1(\Sprime) \simeq \pi_1(M^*_{S^1}) \simeq M^*$.

Allowing Lagrangian objects to be equipped with local systems does not change any of the properties that we will use about $\mathcal{W}(\Sprime)$ essentially due to the fact that $\mathcal{W}(\Sprime)$ is generated by simply connected Lagrangians. In particular, the main tools of <cit.> such as the Künneth embedding, wrapping exact triangle, and stop removal are still valid.

One powerful approach to proving homological mirror symmetry for toric varieties is to pass through a microlocal sheaf category.
This method has given the most general proofs, and for us, has the important benefit of having been proven to be compatible with restriction to the toric boundary (see <Ref>).
There is a long history of computing microlocal sheaf categories corresponding to derived categories of toric varieties beginning with the announcement <cit.> and results of <cit.>, and culminating in the general proof of <cit.>.
In these works, the category of interest is the dg category $\mathrm{Sh}_{-\mathbb{L}_{\Sigma^*}}(M^*_{S^1})^c$ of compact objects in the category $\mathrm{Sh}_{-\mathbb{L}_{\Sigma^*}}(M^*_{S^1})$ of sheaves of dg modules on $M^*_{S^1}$ with microsupport in $-\mathbb{L}_{\Sigma^*}$, where the negation denotes applying multiplication by $-1$ only on the $M_\R$ factor of $M^*_{S^1} \times M_\R = T^*M^*_{S^1} \simeq M^*_{\C^*}$. When combined with the general correspondence <cit.> between microlocal sheaf categories and partially wrapped Fukaya categories, we collectively obtain the following (after restricting to our setting):

There are quasi-equivalences of $A_\infty$ categories

\[
\begin{tikzcd} D^b_{dg}Coh(Y^*) \arrow{d}{K }\arrow{dr}{KGPS}\\
\mathrm{Sh}_{-\mathbb{L}_{\Sigma^*}}(M^*_{S^1})^c \arrow{r}{GPS} &  \Perf \mathcal{W}(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*}) \simeq \Perf \mathcal{W}(\Sprime).
\end{tikzcd}
\]

By <cit.>, the functor $K$ agrees with the non-equivariant coherent-constructible correspondence in <cit.>.

The equivalence $GPS$ holds more generally for cotangent bundles of analytic manifolds. We will use the following property of this equivalence.

Let $Q$ be an oriented real analytic manifold and let $\Lambda \subset S^*Q$ be a closed subanalytic isotropic subset. The image of a local system $\zeta$ on $Q$ under the quasi-equivalence

$$ \mathrm{Sh}_{-\Lambda}(Q)^c \xrightarrow{GPS} \Perf \mathcal{W} (T^*Q, \Lambda) $$

from <cit.> is quasi-isomorphic to $(Q, \zeta)$.

The quasi-equivalence $GPS$ is constructed by refining $\Lambda$ to be the conormal to a Whitney triangulation $\mathcal{S}$. We can assume we are in that setting as all linking disks are orthogonal to $(Q, \xi)$ and the stop removal functor is the quotient by linking disks by <cit.>.

In that case, $GPS$ passes through an intermediate quasi-equivalence with perfect complexes over $\mathcal{S}$. The proof of <cit.> shows that $(Q, \Bbbk_Q)$ is identified with the indicator module $1_Q$ in $\text{Perf }\mathcal{S}$ where $\Bbbk_Q$ is the trivial local system. The sheaf $\Bbbk_Q$ is also identified with $1_Q$ by <cit.>. For general local systems, the same calculations apply to show that the local system $HF^*((Q, \zeta), L_q)$ for $q \in Q$ defined using continuation elements as in <cit.> coincides with $\zeta$.

Now, we prove two properties of $KGPS$. We let $[0] \in M_{S^1}^*$ be the additive identity and set $L_{[0]}$ to be the inward conormal to a small ball around $[0]$.[Similar Lagrangians are considered in <cit.>. However, we take the opposite convention as $L_s$ in <cit.> denotes the outward conormal to a small ball around $s$.]

The image of the structure sheaf $\mathcal O_{Y^*}$ under $KGPS$ is quasi-isomorphic to $L_{[0]}$, and for all $\xi \in M_{\Bbbk^*}$, the image of the shifted skyscraper sheaf $\mathcal{O}_{\xi}[-n]$ is quasi-isomorphic to $(M^*_{S^1}, \xi)$.

For the structure sheaf, <cit.> shows that $K$ sends $\mathcal{O}_{Y^*}$ to the sheaf $\Bbbk_{[0]}$ where $[0] \in M_{S^1}^*$ is the unit. In particular, this object co-represents the stalk functor at $[0]$. On the other hand, the calculations in <cit.> imply that the image of $L_{[0]}$ under $GPS^{-1}$ co-represents the stalk functor at $[0]$ when $\mathbb{L}_{\Sigma^*}$ is refined to be the conormal to a Whitney triangulation. However, this property persists under stop removal as $L_{[0]}$ is left-orthogonal to the linking disks of the removed strata.

For $\xi \in M_{\Bbbk^*}$, <cit.> shows that $K$ takes $\mathcal{O}_\xi[-n]$ to $\xi$. We can then apply <Ref> to deduce the result.

A second approach to homological mirror symmetry for toric varieties is to stay within the realm of Floer theory. This route will be beneficial to us for its direct access to Lagrangian submanifolds as objects. Following <cit.>, we consider $A$- and $B$-side categories whose objects are integral piecewise-linear functions $\sF:M_\R \to \R$ which are linear on each cone of $\Sigma^*$.

On the $A$-side, one constructs a Lagrangian submanifold $\Lmon(\sF) \subset T^*M^*_{S^1} \simeq M^*_{\C^*}$ corresponding to each such $\sF$, which comes equipped with a canonical grading as well as an `anchoring' in the sense of <cit.>, i.e., a lift to the universal cover $T^*M^*_\R \to T^*M^*_{S^1}$.
The anchoring of $\Lmon(\sF)$ is the graph of the differential of a smoothing of $\sF$, under the identification $T^*M^*_\R = M^*_\R \times M_\R = T^*M_\R$ (see <Ref> for more details).
As a result, the Lagrangians $\Lmon(\sF)$ are sections of $\pi$ (and hence also $\Log$) and are referred to as tropical Lagrangian sections.
The category of such tropical Lagrangians sections is denoted $\mafuk^{M^*}$, and comes equipped with a $\Z \oplus M^*$-grading (the $\Z$-grading is the cohomological grading, and the $M^*$-grading comes from the anchorings).

On the $B$-side, one considers the equivariant line bundles $\cO(\sF)$ corresponding to support functions.
The category of such equivariant line bundles is denoted $\Pic_{dg}^{M^*}(Y^*)$, and comes equipped with a $\Z \oplus M^*$-grading (the $\Z$-grading is the cohomological grading, and the $M^*$-grading is given by characters of the algebraic torus whose action on the morphism spaces is determined by the equivariant structures).
In <cit.>, Abouzaid constructs a $\Z \oplus M^*$-graded $A_\infty$ quasi-isomorphism

\begin{align*}
\mafuk^{M^*} &\xrightarrow{A} \Pic_{dg}^{M^*}(Y^*),\qquad \text{which identifies} \\
\Lmon(\sF) & \leftrightarrow \cO_{Y^*}(\sF).
\end{align*}

In <cit.>, a $\Z$-graded fully faithful embedding

\begin{align*}
\mafuk^{M^*} & \xhookrightarrow{HH^{pre}} \Perf \cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*}) \qquad \text{sending}\\
\Lmon(\sF) & \mapsto \Lconic(\sF)
\end{align*}

(which forgets the $M^*$-grading) is constructed, and its image is shown to generate in <cit.>.
Denote the composition with the quasi-inverse of the quasi-equivalence $\Perf \cW(\Sprime) \to \Perf \cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ by  $HH:\mafuk^{M^*}\to \Perf\cW(\Sprime)$.
Combining that result with the fact that $D^b_{dg}Coh(Y^*)$ is generated by line bundles <cit.>, one obtains the following when restricting to our setting:

There is a diagram

\[
\begin{tikzcd}
\Pic_{dg}^{M^*}(Y^*) \arrow{r}{A} \arrow[d,hook] & \mafuk^{M^*} \arrow[hook]{d}{HH} \\
D^b_{dg} Coh(Y^*)  \arrow{r}{AHH} &  \Perf\mathcal{W}(\Sprime)
\end{tikzcd}
\]

which commutes up to quasi-isomorphism of $A_\infty$ functors.
The top functor is a $\Z \oplus M^*$-graded quasi-isomorphism, the vertical functors are $\Z$-graded and cohomologically fully faithful, and the bottom functor is a $\Z$-graded quasi-equivalence.

We now check that $AHH$ satisfies the analogue of <Ref>.

The image of the structure sheaf $\mathcal{O}_{Y^*}$ under $AHH$ is quasi-isomorphic to $L_{[0]}$, and for all $\xi \in M_{\Bbbk^*}$, the image of the shifted skyscraper sheaf $\mathcal{O}_\xi[-n]$ under $AHH$ is quasi-isomorphic to $(M^*_{S^1}, \xi)$.

By definition, $AHH$ takes $\mathcal{O}_{Y^*}$ to $\Lconic(0)$. However, $\Lconic(0)$ is Hamiltonian isotopic to $L_{[0]}$ in the complement of $\mathfrak{f}_{\Sigma^*}$ via a linear isotopy of their primitives using the argument in <cit.>.

We now consider the stop removal functor $\mathcal{W}(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*}) \to \mathcal{W}(M^*_{\C^*})$ which is intertwined with the restriction $D^b_{dg}Coh(Y^*) \to D^b_{dg}Coh(M_{\mathbb{G}_m})$ under $AHH$ by <cit.>. Moreover, the stop removal functor is the quotient by linking disks by <cit.>, and the linking disks are orthogonal to $(M^*_{S^1}, \xi)$. In addition, $AHH$ is compatible with taking products. Thus, we are reduced to checking the claim for the quasi-equivalence between $D^b_{dg} Coh(\mathbb{G}_m)$ and $\Perf \mathcal{W}(T^*S^1)$ which identifies $\mathcal{O}_{\mathbb{G}_m}$ with the cotangent fiber $L_{[0]}$.

In that case, we have $HW(L_{[0]}, L_{[0]}) \simeq \Bbbk[z^{\pm}]$ respecting product structures and concentrated in degree zero. In fact, we can choose a wrapping of $L_{[0]}$ so that the powers of $z$ are in bijection with intersection points and this isomorphism holds on the chain level. We therefore need to calculate $HW(L_{[0]}, (T, \xi)) \simeq \Bbbk \langle x \rangle$ as a left $\Bbbk[z^{\pm}]$-module where $x$ is a formal basis element of degree $n$. That is, we need to compute $\mu_2(z,x) $ as a multiple of $x$. There is exactly one disc on the cylinder contributing to this product and the portion of its boundary on $S^1$ sweeps out the circle once positively. Moreover, with the chosen orientation data on $S^1$, this disk contributes positively, and we obtain $\mu_2(z,x) = \xi x$ as desired.

The proof of Proposition <ref> will make use of some facts concerning autoequivalences of $D^b_{dg}Coh(Y^*).$ We first record the following preliminary technical result concerning Fourier-Mukai functors.

<cit.> Let $Y_1$ and $Y_2$ be two smooth projective varieties and let
\begin{align*} G: H^0(D^b_{dg}Coh(Y_1)) \to H^0(D^b_{dg}Coh(Y_2))\end{align*}
be an exact functor between triangulated categories.  If $G$ is a Fourier-Mukai functor,  meaning $G \cong \phi_\mathcal{E}$ for some $\mathcal{E} \in D^b_{dg}Coh(Y_1 \times Y_2)$,  then the cohomology sheaves of $\mathcal{E}$ are uniquely determined (up to isomorphism) by $G$.

We apply this result to obtain the following dg-lift of a celebrated result of Bondal and Orlov:

Let $Y^*$ be any smooth Fano variety.  The autoequivalences of $D^b_{dg} Coh(Y^*)$ are generated by automorphisms of $Y^*$, twists by line bundles, and the shift functor.  In other words,  we have

\begin{align*} \label{eq:BO}
Aut(D^b_{dg} Coh(Y^*)) \cong Aut(Y^*) \ltimes (Pic(Y^*) \oplus \Z)
\end{align*}

where $\Z$ is generated by the shift functor.

In <cit.>,  Bondal and Orlov prove this result at the level of triangulated categories.  To enhance this to the dg-level,  we note that the Fourier-Mukai kernels $\mathcal{E}$ corresponding to each of these autoequivalences are cohomologically concentrated in a single degree.  They thus lift uniquely (up to quasi-isomorphism) to $D^b_{dg} Coh(Y^* \times Y^*)$.

The quasi-equivalences $KGPS$ and $AHH$ are quasi-isomorphic. In particular, $KGPS$ sends $\mathcal {O}_{Y^*}(\sF)\mapsto \Lconic(\sF)$ for every toric divisor on $Y^*$.

By Theorem <ref>,  $G = KGPS^{-1} \circ AHH$ determines an element of $Aut(Y^*) \ltimes (Pic(Y^*) \oplus \Z).$  By the first parts of <Ref> and <Ref>,  $G$ preserves the isomorphism class of the structure sheaf.  Thus, $G$ is induced by an automorphism $f$ of $Y^*$.

Begin by assuming that $\Bbbk$ is algebraically closed. Then, we can identify this automorphism by its action on closed points. The second parts of <Ref> and <Ref> show that $f$ preserves every point in the dense torus orbit of $Y^*$. Thus, $f=id$ and hence  $G$ is quasi-isomorphic to the identity.

For general fields, we base change to the algebraic closure $\overline{\Bbbk}$. Then because both $AHH$ and $KGPS$ are compatible with base-change, the autoequivalence $G$ is quasi-isomorphic to the identity and the automorphism $f$ becomes the identity after base change to $\overline{\Bbbk}$. To conclude, note that $f$ is induced by a map on homogeneous coordinate rings which becomes the identity after tensoring with $\overline{\Bbbk}$. This implies that $f$ is already the identity.

A careful reader will note that <cit.> are all written with $\Bbbk =\C$ for simplicity. However, their arguments work with no modifications over any other coefficient field. For <cit.>, this has been previously claimed in <cit.>.

§.§ Homological mirror symmetry for the affine hypersurface

We now discuss homological mirror symmetry with the toric boundary $\partial Y^*$ on the $B$-side and $\eff$ on the $A$-side, where $\eff$ is the completion of the Liouvile domain $\fo \subset \partial B$ from <Ref>. The relevant categories are $D^b_{dg}Coh(\partial Y^*)$ and $\mathcal{W}(\eff)$ (see <Ref>).

Homological mirror symmetry in this setting was proved by Gammage and Shende <cit.> using <cit.> to convert sheaf-theoretic calculations to Fukaya categories and <cit.> to calculate skeleta.
In line with (<ref>), we let $-\mathfrak{f}_{\Sigma^*}$ be the boundary at infinity of $-\mathbb{L}_{\Sigma^*}$.
The microlocal sheaf analog of $\mathcal{W}(\eff)$ is the dg category $\mu\mathrm{sh}_{-\mathfrak{f}_{\Sigma^*}}(-\mathfrak{f}_{\Sigma^*})^c$ of compact objects in the dg category of microlocal sheaves on $-\mathfrak{f}_{\Sigma^*}$ (see <cit.> for a complete definition).
We let $\mu^* : \mu\mathrm{sh}_{-\mathfrak{f}_{\Sigma^*}}(-\mathfrak{f}_{\Sigma^*})^c \to \mathrm{Sh}_{- \mathbb{L}_{\Sigma^*}}(M^*_{S^1})^c$ be the left adjoint to microlocalization.
Then, 's computation of the skeleton of $F$ suffices to compute the cup functor in terms of these microlocal sheaf categories using the following result of :

There is a diagram of $A_\infty$ functors

\Perf \cW(\eff) \ar{d}{\cup} \ar[leftrightarrow]{r} & \mu\mathrm{sh}_{-\mathfrak{f}_{\Sigma^*}}(-\mathfrak{f}_{\Sigma^*})^c \ar{d}{\mu^*}\\
\Perf \W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})  \ar[leftrightarrow]{r}{GPS} & \mathrm{Sh}_{- \mathbb{L}_{\Sigma^*}}(M^*_{S^1})^c
\end{tikzcd}$$

in which the horizontal functors are quasi-equivalences, and the diagram commutes up to natural quasi-isomorphism.

Then,  compare these microlocal sheaf categories with coherent sheaves:

There is a diagram of $A_\infty$ functors

\[\begin{tikzcd}
\mu\mathrm{sh}_{-\mathfrak{f}_{\Sigma^*}}(-\mathfrak{f}_{\Sigma^*})^c \ar{d}{\mu^*} \ar[leftrightarrow]{r} & D^b_{dg}Coh(\partial Y^*) \arrow{d}{i_!} \\
\mathrm{Sh}_{- \mathbb{L}_{\Sigma^*}}(M^*_{S^1})^c \ar[leftrightarrow]{r}{K} & D^b_{dg}Coh(Y^*) \end{tikzcd}\]

in which the horizontal functors are quasi-equivalences, and the diagram commutes up to natural quasi-isomorphism.

Putting <Ref> together and using <ref> to trade $\cW(\eff) \stackrel{\cup}{\to} \W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ for $\cW(\eff) \stackrel{\cup}{\to} \cW(\Sprime)$, we obtain homological mirror symmetry for the affine hypersurface intertwining $\cup$ and $i_!$ :

There is a diagram of $A_\infty$ functors

\[\begin{tikzcd}
D^b_{dg}Coh(\partial Y^*) \arrow{d}{i_!} \arrow{r}{GS} & \Perf\mathcal W(\eff) \arrow{d}{\cup} \\
D^b_{dg}Coh(Y^*) \arrow{r}{KGPS} & \Perf \mathcal W(\Sprime)\end{tikzcd}\]

in which the horizontal functors are quasi-equivalences, and the diagram commutes up to natural quasi-isomorphism.

Before proceeding further, we need to discuss the pull-back functor $i^*$. Note that there is a formal dg-adjoint (see Appendix <ref>) to $i_!$:

\begin{align*}
(i_!)^*: \Perf D^b_{dg}Coh(Y^*)  \to \operatorname{Mod} D^b_{dg}Coh(\partial Y^*).
\end{align*}

Because $D^b_{dg} Coh(Y^*)$ is already triangulated and split-closed, we may remove `$\operatorname{Perf}$' from the source.
Observe that by the adjunction between $H(i_!)$ and (derived) pull-back, we have that this adjoint functor agrees with the pull-back functor on the level of cohomology, and in particular lands in $\Perf D^b_{dg} Coh(\partial Y^*) = D^b_{dg} Coh(\partial Y^*)$. We take $i^*$ to be the resulting functor:

\begin{align*}
i^*: D^b_{dg}Coh(Y^*)  \to D^b_{dg}Coh(\partial Y^*).
\end{align*}

Because dg-adjoints are unique up to quasi-isomorphism of functors (see Appendix <ref>), the above definition of $i^*$ will agree (up to quasi-isomorphism) with any other definition of pull-back that satisfies a dg-adjunction with $i_!$.

There is a diagram of $A_\infty$ functors

\[\begin{tikzcd}
D^b_{dg}Coh(\partial Y^*)  \arrow{r}{GS} & \Perf \mathcal W(\eff)   \\
D^b_{dg}Coh(Y^*) \arrow{u}{i^*} \arrow{r}{KGPS} & \Perf \mathcal W(\Sprime) \arrow{u}{\cup^*}\end{tikzcd}\]

which commutes up to natural quasi-isomorphism, and where $\cup^*$ is a right-adjoint to $\cup$.

By Theorem <ref> and the naturality of taking the pullback of modules, we have a commutative diagram

\begin{equation}\label{adjointsquare}
\begin{tikzcd}
D^b_{dg}Coh(\partial Y^*)  \arrow{r}{GS} & \operatorname{Mod} \Perf \mathcal W(\eff)   \\
D^b_{dg}Coh(Y^*) \arrow{u}{i^*} \arrow{r}{KGPS} & \Perf \Perf \mathcal W(\Sprime). \arrow{u}{\cup^*}\end{tikzcd}
\end{equation}

Because $\Perf \mathcal W(\Sprime)$ is triangulated and split-closed, we may remove the outer `$\Perf$' from the bottom row. It follows by commutativity of the diagram that $\cup^*$ also lands in $\Perf \Perf \mathcal W(\Sprime) = \Perf \mathcal W(\Sprime)$, which yields the claimed commutative diagram.

§.§ The cap functor

We need a geometric understanding of the equivalence $GS$.
In this subsection, we will show that the restriction of a line bundle to $\partial Y^*$ is mirror to a compact Lagrangian.
We do this by calculating the adjoint $\cap$ to $\cup$ on the subcategory of Lagrangians mirror to line bundles. This $\cap$ functor will be $\ZZ \oplus M^*$ graded.
The results of this subsection invoke one technical result, <ref>, which we temporarily defer to <ref>.

There exists a diagram of $A_\infty$ categories and functors

\begin{equation}
\begin{tikzcd}
\partial \mafuk^{M^*} \arrow{r}{j_A}& \Perf \mathcal W(\eff)\\
\mafuk^{M^*} \arrow{r}{HH} \arrow{u}{\cap_{mon}} & \Perf \mathcal W(\Sprime) \arrow{u}{\cup^*}
\end{tikzcd}
\end{equation}

which commutes up to natural quasi-isomorphism. Furthermore:

* The top and bottom arrows are injective on objects and homotopy equivalences on morphisms (in particular, they are cohomologically fully faithful);

* $\cap_{mon}$ is $\ZZ\oplus M^*$ graded.

The desired diagram follows from commutativity of the outer face of the following  diagram:

\[
\begin{tikzpicture}
\node (v4) at (-7,1) {$\partial \mathcal F_{mon}^{M^*}$};
\node (v3) at (-7,-5) { $\mathcal F_{mon}^{M^*}$};
\node (v5) at (1,1) {$\Perf \mathcal W(\eff)$};
\node (v10) at (1,-5) {$\Perf \mathcal W(\Sprime)$};

\node (v7) at (-3,-2) {$\mathcal P(\Sprime)^{M^*}$};
\draw  (v3) edge[->,dashed] node[midway, fill=white] {$\cap_{mon}$}(v4);
\draw  (v3) edge[->]  node[midway, fill=white] {$HH$} (v10);
\node (v1) at (-3,1) {$\mathcal W(\eff)^{M^*}$};
\draw  (v1) edge[->] (v5);
\draw  (v7) edge[->] node[midway, fill=white]{$\cap^{M^*}$} (v1);

\node at (-1,-1) {\Cref{lem:def_capl}};

\draw  (v4) edge[->,  dashed] (v1);
\draw  (v7) edge[ ->] (v10);
\draw (v10) edge[->] node[midway, fill=white]{$\cup^*$} (v5);
\node at (-3,-4) {\Cref{cor:constructionOfP}};
\draw  (v3) edge[->] node[fill=white, midway]{$HH_P$} (v7);
\end{tikzpicture}
\]

We define $\partial \mafuk^{M^*} \subset \mathcal \cF(F)^{M^*}$ to be the full subcategory whose objects lie in the image of  $ \cap^{M^*}\circ CF^*(\Lmon^+, -)$ where $\cap^{M^*}$ is as in <ref>.   We thus obtain a $\Z \oplus M^*$-graded functor  $\cap_{mon}:\mafuk^{M^*} \to \partial \mafuk^{M^*}$ which makes the upper left square of the following diagram commute.

Let $\Pic_{dg}^{M^*}(\partial Y^*)$ be the full subcategory on objects in the image of $\Pic_{dg}^{M^*}(Y^*)\to D^b_{dg}Coh(Y^*)\to D^b_{dg}Coh(\partial Y^*)$.

There is a diagram of $A_\infty$ functors

\[\begin{tikzcd}
\Pic_{dg}^{M^*}(\partial Y^*)  \arrow[leftrightarrow]{r} & \partial \mafuk^{M^*}  \\
\Pic_{dg}^{M^*}(Y^*) \arrow{u}{i^*} \arrow[leftrightarrow]{r}{A} & \mafuk^{M^*} \arrow{u}{\cap_{mon}},\end{tikzcd}
\]

in which

* the bottom and vertical functors are $\Z \oplus M^*$-graded;

* the top and bottom functors are quasi-isomorphisms;

* the diagram commutes up to natural quasi-isomorphism;

*  the top arrow of the diagram fits into a diagram

\[\begin{tikzcd}
D^b_{dg}Coh(\partial Y^*) \arrow[leftrightarrow]{r}{GS} & \Perf \mathcal W(\eff) \\
\Pic_{dg}^{M^*}(\partial Y^*)  \arrow[leftrightarrow]{r}  \arrow[hookrightarrow]{u} & \partial \mafuk^{M^*}  \arrow[hookrightarrow]{u}
\end{tikzcd}
\]

which commutes up to natural quasi-isomorphism.

The bottom functor is constructed and shown to be a $\Z \oplus M^*$-graded quasi-isomorphism in <ref>.
The left vertical functor is manifestly $\Z \oplus M^*$-graded; the right vertical functor is constructed, and shown to be $\Z \oplus M^*$-graded, in <ref>.

In order to construct the top functor, we observe that both desired commutative diagrams fit in as the left and top faces of the following cube, where every map apart from the dashed line has been defined, and $j_A, j_B, \tilde j_B$ are inclusions of subcategories.

\begin{equation}
\begin{tikzpicture}    \node (v2) at (-7.5,3) {$Pic_{dg}^{M^*}(\partial Y^*)$};
\node (v1) at (-7.5,-2) {$Pic_{dg}^{M^*}(Y^*)$};
\node (v4) at (-5,1.5) {$\partial \mathcal F_{mon}^{M^*}$};
\node (v3) at (-5,-0.5) { $\mathcal F_{mon}^{M^*}$};
\node (v5) at (-1,-0.5) {$\Perf \mathcal W(S)$};
\node (v6) at (-1,1.5) {$\Perf \mathcal W(\eff)$};
\node (v7) at (2.5,-2) {$D^b_{dg}Coh(Y^*)$};
\node (v8) at (2.5,3) {$D^b_{dg}Coh(\partial Y^*)$};
\draw  (v1) edge[->] node[midway, fill=white] {$i^*$} (v2);
\draw  (v3) edge[->] node[midway, fill=white] {$\cap_{mon}$}(v4);
\draw  (v5) edge[->] node[midway, fill=white] {$\cup^*$} (v6);
\draw  (v7) edge[->] node[midway, fill=white] {$i^*$} (v8);
\draw  (v3) edge[->] node[midway, fill=white] {$HH$} (v5);
\draw  (v4) edge[->] node[midway, fill=white]{$j_A$}(v6);
\draw  (v2) edge[->] node[midway, fill=white]{$j_B$} (v8);
\draw  (v1) edge[->] node[midway, fill=white]{$\tilde j_B$}(v7);
\draw  (v1) edge[->] node[midway, fill=white,] {A} (v3);
\draw  (v2) edge [dashed, ->] (v4);
\draw  (v6) edge[<-] node[midway, fill=white] {$GS$} (v8);
\draw  (v5) edge[<-] node[midway, fill=white] {$KGPS$}(v7);
\node at (-6.5,0.5) {};
\node at (-3,-1.5) {\Cref{thm:ahh} and \Cref{prop:sameHMS}};
\node at (0.5,0.5) {\Cref{cor:adjoint_HMS}};
\node at (-3,0.5) {\Cref{lem:innerSquare}};
\end{tikzpicture}
\label{eq:bigDiagram}
\end{equation}

The outer face commutes by definition of the category $\Pic_{dg}^{M^*}(\partial Y^*)$.
The bottom, inner, and right faces commute up to natural quasi-isomorphism by <Ref>, <Ref>, and <Ref> respectively.
We construct the dashed arrow so that the top face of the commutative diagram commutes up to natural quasi-isomorphism, by applying <Ref> to the functors $\cF = GS \circ j_B$ and $\cG = j_A$.
To check the hypotheses of the Lemma, we first observe that both $j_A$ and $GS \circ j_B$ are cohomologically fully faithful, as $j_A$ and $j_B$ are inclusions of subcategories and $GS$ is a quasi-equivalence.
To check that $GS \circ j_B$ and $j_A$ have the same essential image, we use the commutativity of the bottom, inner, right, and outer faces, to show that $j_A \circ \cap_{mon} \circ A$ is naturally isomorphic to $GS \circ j_B \circ i^*$; together with the fact that $A$, $\cap_{mon}$, and the left-hand version of $i^*$ are essentially surjective by definition.
Thus we may construct the dashed arrow, and conclude that it is a quasi-equivalence, by <Ref>.
The commutativity of the left-hand and top square then follows from the commutativity of the remaining squares.

§.§ Arclike Lagrangians mirror to line bundles on toric varieties

In <ref>, we recall the construction of the tropical Lagrangian section $\tilde L_{mon}(\sF)$ from <cit.>. We prove it is arclike, and define $\tilde {L}(\sF)$ to be its negative pushoff in $\Sprime$ in the sense of Appendix <ref>. In <ref>, we recall the construction of the cylindrical (but not arclike) tropical Lagrangian section $\Lconic(\sF)$ from <cit.>. Then, in <ref>, we prove <ref>, which says that the Lagrangians $\Lconic(\sF)$ and $\tilde L(\sF)$ (together with their canonical anchorings) define quasi-isomorphic objects of $\mathcal W(M^*_{\C^*} , \mathfrak f_{\Sigma^*})^{M^*}$. Using this, we easily deduce the following result, which is the main result of this section, and a necessary input for the last subsection's <ref>.

There exists a $\Z \oplus M^*$-graded, cohomologically fully faithful $A_\infty$ functor $HH_P$ which makes the following diagram commute up to natural quasi-isomorphism:

\[\begin{tikzpicture}
\node (v1) at (-1.5,0.5) {$\mafuk^{M^*}$};
\node (v3) at (5.5,0.5) {$\Perf \mathcal W(\Sprime)^{M^*}$};
\node (v4) at (5.5,3) {$\mathcal P(\Sprime)$};
\draw  (v1) edge[->] node[midway, below]{$HH$} (v3);
\draw  (v4) edge[->] node[midway, right]{$\mathcal Y_{(-)}$} (v3);
\draw  (v1) edge[dotted, ->] node[midway, left]{$HH_P$} (v4);
\end{tikzpicture}.
\]

We apply <Ref> to the $\Z \oplus M^*$-graded versions of the functors $\cF = HH$ and $\cG =  \mathcal Y_{(-)}$.
These are both cohomologically fully faithful.

By <Ref>, there exists a quasi-isomorphism in $\cW(M^*_{\C^*}, \mathfrak f_{\Sigma^*})^{M^*}$ between the Lagrangian submanifolds $\Lconic(\sF)$ and $\tilde L(\sF)$. By pulling back this isomorphism along the inclusion (i.e., by applying the quasi-inverse of the inclusion quasi-equivalence) $(i')^*:\Perf\cW(M^*_{\C^*}, \mathfrak f _{\Sigma^*})^{M^*}\to \Perf \cW(\Sprime)^{M^*}$, we obtain quasi-isomorphisms $HH(\Lmon(\sF))= (i')^*\Lconic(\sF) \simeq \mathcal Y_{\tilde L(\sF)}$.
Thus the essential image of $HH$ is contained in that of $\mathcal Y_{(-)}$, so the result follows by <Ref>.

§.§.§ Arclike tropical Lagrangian sections and their negative pushoffs

In this section, we give an explicit construction of Lagrangian sections $\Lmon(\sF)$ which are arclike in $\ess$.
We construct these sections following <cit.>. Let $\sF: M_\R \to \R$ be an integral piecewise linear function on $\Sigma^*$, that is, $\sF$ is the support function of a divisor on $Y^*$. Let $\eta: M_\R \to \R$ be a smooth symmetric mollifier function supported on the ball of radius $1$ around the origin with respect to an arbitrarily chosen Euclidean metric. Then, for small $\delta >0$, we set $\eta_\delta(u) := \delta^{-n}\eta(u/\delta)$ and

\[ H_{mon}^{\sF,\delta}(u) := \int_{M_\R} \sF(u-y) \eta_\delta(y) \, dy .\]

The tropical Lagrangian section associated with $\sF$ is

\[  \Lmon^\delta(\sF) := \text{graph}\left(dH_{mon}^{\sF,\delta}\right) = \left\{ \left(u, dH_{mon}^{\sF,\delta}(u)\right) : u \in M_\R \right\} \subset M_\R \times M^*_{S^1} = T^*M^*_{S^1} \simeq M^*_{\C^*}. \]

Sometimes we will drop $\delta$ from the notation.

In <cit.>, these Lagrangian sections are shown to be mirror objects to line bundles in a modified version of $\mafuk^{M^*}$.
However, it is observed in <cit.> that in the Fano setting, $\Lmon(\sF)$ is also a tropical Lagrangian section in the sense of <cit.>, i.e., is an object of $\mafuk^{M^*}$.

Let $\eps > 0$. We may choose $\delta$ sufficiently small so that:

*  We have

\begin{equation} \label{eq:Lmonisreal} \left( (1 - \psi_\alpha (t,x)) \cdot z^\alpha\right)|_{\Lmon^\delta(\sF)} \in \R_{\geq 0}
\end{equation}

for all $\alpha \in A$ and all $x$. In particular, $ \left( \Im \, W_{t,1} \right)|_{\Lmon^\delta(\sF)} = 0$.

*  $\Lmon^\delta(\sF) $ is contained in the $\eps$-neighbourhood of ${\mathbb L}_{\Sigma^*}$.

*  The tangent space to $\Lmon^\delta(\sF)$ is contained in the $\eps$-neighbourhood of the set

\begin{align}\label{eq:where_TL_lies} Q:=\bigcup_{\sigma \in \Sigma^*} \sigma \times \sigma^\perp \times M_\R \times \sigma^\perp \subset M_\R \times M^*_{S^1} \times M_\R \times M^*_\R = T(M_\R \times M^*_{S^1}).
\end{align}

For each nonzero $\alpha\in A$, we have that

\begin{equation} \label{eq:monomialsupport}
dH_{mon}^{\sF,\delta}(\alpha) = \sF(\alpha)
\end{equation}

in the star of $\alpha$ away from a $\delta$-neighbourhood of the boundary. Since $\sF(\alpha)\in \Z$, this implies that $ \arg(z^\alpha)|_{\Lmon^\delta(\sF)}=0 $ in this region. On the other hand, by <cit.>, we can choose $\varphi$ so that the image under $\Phi$ of the region on which $\psi_\alpha(t,x) \neq 1$ is contained in the interior of the star of $\alpha$.[Note that, from <cit.> and the discussion following Proposition 2.36 in loc. cit., the condition we impose here on $\Phi$ implies that $\varphi$ is adapted to $\Delta^*$.]    In particular, (<ref>) holds over all $x$ satisfying $\psi_\alpha(t,x) \neq 1$ if $\delta$ is sufficiently small. Thus, we see that (<ref>) holds for all nonzero $\alpha$. However, (<ref>) holds trivially for $\alpha = 0$, completing the proof of (<ref>).

Suppose $\left(u,dH_{mon}^{\sF,\delta}(u)\right) \in \Lmon^\delta(\sF)$. In the complement of a neighbourhood $U_\delta$ of the origin in $M_\R^*$ whose size is governed by $\delta$, there is a minimal cone $\tau \in \Sigma^*$ such that $d(u, \tau) <\delta$. Let $\alpha_1, \hdots, \alpha_k \in A$ be the primitive generators of $\tau$. We then have that $dH_{mon}^{\sF,\delta}(u)(\alpha_j) = \sF(\alpha_j)$ for $j = 1,\hdots,k$. In particular, if $v \in \tau$ is a point such that $d(u, v) < \delta$, we have

$$ \left(v, dH_{mon}^{\sF,\delta}(u)\right) \in \tau \times \tau^\perp \subset {\mathbb L}_{\Sigma^*} $$

and the distance between $\left(v,dH_{mon}^{\sF,\delta}(u)\right)$ and $\left(u, dH_{mon}^{\sF,\delta}(u) \right)$ is less than $\delta$.
This completes the proof of (<ref>).

Finally, observe that in the interior of the set of all $u$ with minimal cone $\tau$, we have $dH_{mon}^{\sF,\delta}(u) \in \tau^\perp$, which implies that the tangent space to the graph of $dH_{mon}^{\sF,\delta}$ is contained in $M_\R \times \tau^\perp$.
Therefore, the point $(v,dH_{mon}^{\sF,\delta}(u),T_{(u,dH_{mon}^{\sF,\delta}(u))} \Lmon^\delta(\sF))$ is contained in $Q$, and is distance $\delta$ from $(u,dH_{mon}^{\sF,\delta}(u),T_{(u,dH_{mon}^{\sF,\delta}(u))}\Lmon^\delta(\sF))$ as required. The union of the interiors of these sets, over all cones $\tau$, cover the complement of $U$. Thus, we complete the argument by choosing $\delta < \eps$ sufficiently small that $U_\delta \subset B_\eps(0)$.
This completes the proof of (<ref>).

We obtain as a corollary:

For $\delta$ sufficiently small and the $\aye$ constructed in <ref>, the Lagrangians $\Lmon^\delta(\sF)$ are $\aye$-arclike in $\ess$.

We first check that $\Lmon^\delta(\sF)$ is transverse to the boundary of $\ess$. Recall from <Ref> and the discussion following that $\ess$ is constructed from the data of a sutured Liouville domain $(B, F)$, where $B$ is given by modifying $\Log_t^{-1}(\mathcal C_0)$.

The Liouville vector field $(u\partial_u,0)$ on $M_\R \times M^*_{S^1}$ is transverse to the boundary of $B$ by construction.
Therefore, we have $d(u\partial_u,T\partial B) > \eps$ for some $\eps>0$, where distances are measured with respect to an arbitrary Riemannian metric.
Now let $p \in \Lmon^\delta(\sF) \cap \partial B$.
As $\Lmon^\delta(\sF)$ is a Lagrangian section, the tangent vector $u\partial_u$ lifts to a vector $(u\partial_u,v) \in T_p\Lmon^\delta(\sF)$.
By <Ref>, there exists a tangent vector $(\tilde p,(\tilde u, \tilde v)) \in Q$ which is close to $(p,(u\partial_u,v))$.
We may arrange that this tangent vector lies over $\partial B$, by flowing along the Liouville flow.
By choosing $\delta$ sufficiently small, we may guarantee that $(p,(u\partial_u,v))$ is arbitrarily close to $(\tilde p,(\tilde u,\tilde v))$; and as a consequence that $T_p\partial B$ is arbitrarily close to $T_{\tilde p}\partial B$.

Now assume that $(\tilde p,(\tilde u,\tilde v)) \in (\sigma \times \sigma^\perp) \times (M_\R \times \sigma^\perp)$.
Recall that by construction of $B$, $\mathbb{L}_{\Sigma^*} \cap \partial \Log_t^{-1}(\cC_0) \subset \partial B$.
In particular, $\Log_t(\tilde p) \times \sigma^\perp \subset \partial B$, and hence $\{0\} \times \sigma^\perp \subset T_{\tilde p} \partial B$.
We now have:

\begin{align*}
d\left((p,(u\partial_u,v)),T_p\partial B\right) & > d\left((\tilde p,(\tilde u,\tilde v)),T_{\tilde p}\partial B\right) - \eps/2 \quad\text{for $\delta$ sufficiently small}\\
&= d\left((\tilde p,(\tilde u,0)),T_{\tilde p}\partial B\right) - \eps/2 \quad \text{as $\tilde v \in \sigma^\perp$ so $(0,\tilde{v})\in T_{\tilde p} \partial B$} \\
& \ge d\left((\tilde p,(u\partial_u,0)),T_{\tilde p} \partial B\right) - d\left((\tilde p,(u\partial_u,0)),(\tilde p,(\tilde u,0))\right) - \eps/2 \\
& > \eps - \eps/2 - \eps/2 \quad \text{for $\delta$ sufficiently small} \\
& = 0.
\end{align*}

Therefore, $(u\partial_u,v) \in T_p\Lmon^\delta(\sF)$ is transverse to the hypersurface $T\partial B$, so $\Lmon^\delta(\sF)$ is transverse to $\partial B$ at $p$, and we may choose $\delta$ sufficiently small that this holds for all $p \in \mathbb{L}_{\Sigma^*} \cap \partial B$, by compactness.
We now observe that by construction of $\ess$, $\partial \ess$ agrees with $\partial B$ along a neighbourhood of $\mathbb{L}_{\Sigma^*}$.
By choosing $\delta$ sufficiently small, we may arrange that $\partial \Lmon^\delta(\sF)$ is contained in this neighbourhood by <Ref>; thus $\Lmon^\delta(\sF)$ intersects $\partial \ess$ transversely as required.

By <ref>, the zero sets of $\aye$ and $\Im(W_{t,1})$ agree when we are close to both $\partial \ess$ and $\mathbb L_{\Sigma^*}$. By <Ref>: <ref>, $\Lmon^\delta(\sF)$ is contained in the intersection of the zero set of $\Im(W_{t,1})$ and a small neighbourhood of $\mathbb L_{\Sigma^*}$. Therefore $\Lmon^\delta(\sF)\cap \Nbd(\partial \ess) \subset  \aye^{-1}(0)\cap \Nbd(\partial \ess)$.
Thus, $\Lmon^\delta(\sF)$ is $\aye$-arclike in $\ess$.

By <ref>(<ref>), we may extend (by the flow of $V$ applied to $\partial \Lmon^\delta(\sF)$) $\Lmon^\delta(\sF)$ to an $\Iprime$-arclike Lagrangian

\begin{equation}\label{lmonprime}
(\Lmon^\delta(\sF))' \subset \Sprime.
\end{equation}

whose intersection with $\ess$ is the $\aye$-arclike $\Lmon^\delta(\sF)$ (to see $\Iprime$-arclikeness of the extension, note that tangency of $(\Lmon^\delta(\sF))'$ to $V = X_{\Iprime}$ near $\partial \Sprime$ implies both transversality to $\partial \Sprime$ and containment in an $\Iprime$-level set by <ref>).

Here is an equivalent way to define the extension $(\Lmon^\delta(\sF))'$ of $(\Lmon^\delta(\sF))$.
Observe that $(\Lmon^\delta(\sF))$ itself is tangent to $X_{\aye}$ near and in particular just outside $\partial \ess$ by <ref> and <ref>, and hence determines an $\aye$-arclike Lagrangian in the enlarged sector $S_1 \supset \ess$ constructed in the proof of <ref> with deformed Liouville form $\lambda'$ (in fact $\Lmon^\delta(\sF)$ is tangent to $X_{\aye}$ over all of $S_1 \backslash \ess^{\circ}$). Now apply $\Phi$, the symplectomorphism from $\lambda'$ to $\lambda$ constructed in the same proof, to obtain $(\Lmon^\delta(\sF))'$.

If in the construction of the deformation $\lambda'$, we modified $\lambda'$ by a compactly supported exact 1-form to equal  $\lambda$ on a compact set containing $(\Lmon^\delta(\sF)) \cap (S_1 \backslash \ess^{\circ})$, then we could further arrange that $\Phi = id$ near $\Lmon^\delta(\sF)$ and hence that $(\Lmon^\delta(\sF))'=\Lmon^\delta(\sF)$ at the expense of slightly modifying $\Sprime:= \Phi(S_1)$.

Since $\Sprime$ has exact boundary, we can now take the negative pushoff of $(\Lmon^\delta(\sF))'$ in the sense of <ref>, using the boundary sector $\eff \times T^*[0,1] \hookrightarrow \Sprime$ induced by $\Iprime$.
The outcome is a cylindrical exact Lagrangian whose boundary avoids $\Sprime$

\begin{equation}\label{lmonnegativepushoff}
\tilde{L}(\sF) := ((\Lmon^\delta(\sF))')^- \subset \Sprime;
\end{equation}

this Lagrangian, like $(\Lmon^\delta(\sF))'$ and $\Lmon^\delta(\sF)$, has an anchoring determined by $\sF$ and can be thought of as an object of $\cW(\Sprime)^{M^*}$.
The modifications involved in taking cylindrical pushoff are only made in $\eff \times T^*[0,1] \subset (\Sprime \backslash \ess)$ so in particular

\begin{equation}\label{restrictionofLmonminus}
\tilde{L}(\sF) \cap \ess = \Lmon^\delta(\sF).
\end{equation}

We will drop $\delta$ from the notation when appropriate.

§.§.§ Cylindrized tropical Lagrangian section

By instead using a “cylindrical smoothing” of $\sF$, we can also construct, as in <cit.>, a cylindrical Lagrangian section of $M^*_{\C^*}$ whose Legendrian boundary lies in the complement of $\mathfrak{f}_{\Sigma^*}$, that is, an object of $\mathcal{W}(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$.
The function $\sF$ determines an anchoring of this object, i.e., a lift to $\mathcal{W}(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})^{M^*}$.

We define ${G}_{cyl}^{\sF,\delta}:M_\R \setminus \{0\} \to \mathbb{R}$ by

\begin{equation}
{G}_{cyl}^{\sF,\delta}(u) = \int_{M_\R} \sF(u-y) \eta_{\delta |u|} (y) \, dy =  \int_{M_\R} \sF(u - |u|y)  \eta_\delta(y) \, dy
\label{eq:conicalizedTropicalPrimitive}
\end{equation}

where $|u|$ is measured with respect to the Euclidean metric.
It is then shown in <cit.> that for any $\sF$ there is a constant $C_\sF$  such that

\[ \left| d{G}_{cyl}^{\sF,\delta}(\alpha) -\sF (\alpha) \right| \leq \delta C_\sF \]
on the open cones

\begin{equation}
U^\delta_\alpha = \text{cone} \{u \in M_\R \; : \; | u | = 1,  u \in \st(\alpha), \text{ and } d(u, \sigma) > \delta \text{ for all } \sigma \in \Sigma \text{ such that } \alpha \not \in \sigma \}
\label{eq:conesUAlpha}
\end{equation}

for all nonzero $\alpha \in A$.
In the language of <cit.>, $U^\delta_\alpha$ is the $\delta$-star of $\alpha$.

Given a support function $\sF$ for $\Sigma^*$, we can set

\begin{equation} \label{eq:perturbedPL} \sF_a = \sF - a \sF_K \end{equation}

where $\sF_K$ is the support function of the canonical divisor and $0< a \ll 1$.
We then define a smooth function $H_{cyl}^{\sF,a,\delta}$ by smoothing ${G}_{cyl}^{\sF_a,\delta}$ arbitrarily on a compact neighbourhood of the origin, which may be taken to be arbitrarily small. We denote such a neighbourhood by $\Nbd(0)$ in what follows.  The resulting $H_{cyl}^{\sF,a,\delta}$ will satisfy

\begin{equation}
\left| dH_{cyl}^{\sF,a,\delta}(\alpha) - \sF(\alpha) + a \right| \leq \delta C_{\sF_a}
\label{eq:argumentBound}
\end{equation}

on $U^\delta_\alpha \setminus \Nbd(0)$. Since $\sF(\alpha)$ is an integer and $a$ is not, we may choose $\delta$ small enough that $dH_{cyl}^{\sF,a,\delta}(\alpha) \notin \Z$ on $U^\delta_\alpha \setminus \Nbd(0)$.
On the other hand, if $u \in U^\delta_\alpha$ and $(u,\theta) \in \mathbb{L}_{\Sigma^*}$, then $\theta(\alpha) \in \Z$.
It follows that the Lagrangian

$$\Lconic^{a,\delta}(\sF) := \text{graph}\left(dH_{cyl}^{\sF,a,\delta}\right)$$

avoids ${\mathbb L}_{\Sigma^*}$ outside of the compact set $\pi^{-1}(\Nbd(0))$, for $\delta$ sufficiently small.
Thus, $\Lconic^{a,\delta}(\sF)$ determines an object of $\mathcal{W}(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$. Observe that all choices of $a,\delta$ with $0<a\ll 1$ and $\delta$ sufficiently small yield admissibly Hamiltonian isotopic Lagrangians in $M^*_{\C^*}$; we will therefore drop $a$ and $\delta$ from the notation when appropriate.

§.§.§ Comparing constructions of Lagrangian sections

In the previous two subsections we have constructed two a priori different cylindrical Lagrangians in $\W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ associated to $\sF$:  the cylindrized tropical section $\Lconic^{a,\delta}(\sF)$
and the negative pushoff of the tropical arclike Lagrangian $\tilde{L}(\sF)$
(strictly speaking, each of these depend on small parameters, but only up to
cylindrical isotopy hence isomorphism in the Fukaya category).
<ref> below asserts that they are isomorphic. The obvious
isotopy between these two objects is non-cylindrical (passing through the
non-pushed off $\Lmon^\delta(\sF)$), so doesn't immediately induce an isomorphism.
Instead, we show two objects can be made (after a compactly supported isotopy)
to agree with the arclike Lagrangian $\Lmon^\delta(\sF)$ on a subsector $\ess$, and
also that both objects cofinally wrap away from the subsector.  From here,
locality theory for arclike Lagrangians in Liouville sectors,
developed in <ref>, shows that when tested against objects coming
from the Fukaya category of the (quasi-equivalent) subsector $\ess$, all disks
appearing for either Lagrangian also remain in the subsector, and therefore

Conceptually, we may think of $\Lconic^{a,\delta}(\sF)$ and
$\tilde{L}(\sF)$ as “generalized negative pushoffs” of the
same arclike $\Lmon^\delta(\sF)$, in the sense that they agree with $\Lmon^\delta(\sF)$
within an equivalent subsector which they cofinally wrap away from.
The technology of <ref> can be used to show that any two
generalized negative pushoffs of an arclike Lagrangian are isomorphic objects;
compare <ref>.

$\partial_{\infty}\Lconic^{a,\delta}(\sF)$ admits a cofinal wrapping in $(M^*_{\C^*},\mathfrak{f}_{\Sigma^*})$ converging to $\mathfrak{f}_{\Sigma^*}$ which does not cross $\partial_{\infty} \ess$.

The cofinal wrapping is exhibited in loc. cit.. As also shown there, the boundary at infinity of $\Lconic^{a,\delta}(\sF)$ and its wrappings can be made to take place in an arbitrarily small neighborhood of $\mathfrak{f}_{\Sigma^*}$, which is, in particular, disjoint from $\partial_{\infty} \ess$.

There is a Lagrangian $\Lmc^{a,\delta}(\sF)$ in $M^*_{\C^*}$ which differs from $\Lconic^{a,\delta}(\sF)$ by a compactly supported Hamiltonian isotopy and satisfies

\[ \Lmc^{a,\delta}(\sF) \cap \ess = \Lmon^\delta(\sF) \]

for sufficiently small $\delta$ and $a$.

Pick $\eps>0$ and $R > 0$ so that  $\pi(\ess\cap B_\eps(\mathbb L_{\Sigma^*})) \subset B_R(0) \subset M_\R$.
Let $\rho_1: M_\R\to \R_{\ge 0}$ be smooth, equal to $|u|$ outside $B_{R + b}(0)$ for some $b > 0$, and equal to $1$ on $B_R(0)$, and let $\rho_2: M_\R\to [0,1]$ be smooth, equal to $1$ outside $B_{R + b}(0)$, and equal to $0$ on $B_R(0)$.

We now perform a similar construction to (<ref>) by defining

\begin{equation}
{G}_{mc}^{\sF,\delta} (u) = \int_{M_\R} \sF(u-y) \eta_{\delta \rho_1 (u)} (y) \, dy
\end{equation}

for $u \in M_\R$.
We note that ${G}_{mc}^{\sF,\delta}$ agrees with ${G}_{cyl}^{\sF,\delta}$ on the complement of $B_{R+b}(0)$, and with $H_{mon}^{\sF,\delta}$ on $B_R(0)$. We now set

\[ H_{mc}^{\sF,a,\delta}(u) = G^{\sF,\delta}_{mc}(u) - \rho_2(u)aG^{\sF_K,\delta}_{mc}(u) \]

for all $u \in M_\R$ and define $\Lmc^{a,\delta}(\sF)$ to be the graph of $dH_{mc}^{\sF,a,\delta}$. As before, we note that $H^{\sF, a, \delta}_{mc}$ agrees with $H^{\sF, a, \delta}_{cyl}$ in the complement of $B_{R+b}(0)$ and with $H^{\sF,\delta}_{mon}$ on $B_{R}(0)$.

To show that $\Lmc^{a, \delta}(\sF)$ agrees with $\Lmon^\delta(\sF)$ on $\ess$, it suffices to show that it lies in a small neighbourhood of $\mathbb L_{\Sigma^*}$ as we have chosen $\eps$ and $R$ so that $\ess \cap B_\eps(\mathbb{L}_{\Sigma^*} )\subset \pi^{-1}(B_R(0))$.
By applying the same argument as in <ref>, we can deduce that there exists a choice of $\delta$ and $a$ so that $\Lmc^{a,\delta}(\sF)$ is contained in $B_\eps(\mathbb L_{\Sigma^*})$ by obtaining a uniform bound on $|dH_{mc}^{\sF, a, \delta}(\alpha) - \sF(\alpha)|$ on $U^\delta_\alpha \setminus \Nbd(0)$ for all $\alpha \in A$ that goes to zero as $\delta$ and $a$ go to zero.

We will now produce such a bound. Let $\sF_\sigma$ be the linear function given by restricting $\sF$ to any $\sigma \in \Sigma^*(\dim Y^*)$. Then, for all $u \in U^\delta_\alpha \setminus \Nbd(0)$, we have

\begin{align*}
\left| (dG_{mc}^{\sF, \delta})_u(\alpha) - \sF(\alpha) \right| &= \left| d\rho_1(\alpha) \right| \left| \int_{M_\R} d\sF_{u - \rho_1(u)y}(y) \eta_\delta(y) \, dy \right| \\
&\leq \left| d\rho_1(\alpha) \right| \int_{M_\R} \left| d\sF_{u - \rho_1(u)y} \right| \left| y \right| \eta_\delta(y) \, dy \\
&\leq \delta | d\rho_1(\alpha)| \max_{\sigma \in \Sigma^*(\dim Y^*)} \left| \sF_\sigma \right|
\end{align*}

which uniformly tends to $0$ as $\delta$ goes to zero as $|d\rho_1|$ is bounded. The same calculation shows that for all $u \in U^\delta_\alpha$ with $|u| \geq R + b$ (so that $\rho_2 \equiv 1$) we have

\[ \left| d\left(\rho_2aG^{\sF_K,\delta}_{mc}\right)(\alpha)- a \right| \leq \delta C \]

for some constant $C$. Finally, inside of $B_{R+b}(0)$, the norm of  $d\left(\rho_2 G^{\sF_K,\delta}_{mc} \right)$ is bounded so

\[ \left|  d\left(\rho_2 aG^{\sF_K,\delta}_{mc} \right) \right| \leq a C' \]

for some constant $C'$. Putting all these estimates together, we obtain the desired bound on $|dH_{mc}^{\sF, a, \delta}(\alpha) - \sF(\alpha)|$.

Therefore, we have shown that $\Lmc^{a,\delta}(\sF) \subset B_\eps(\mathbb{L}_{\Sigma^*})$ for any small $\eps > 0$.
In particular, it follows that

\[\Lmc^{a,\delta}(\sF) \cap \ess =
\Lmc^{a, \delta}(\sF) \cap \ess\cap \pi^{-1}(B_R(0))
= \Lmon^\delta(\sF) \]

as required. We also have, by construction, that the flow of $H_{mc}^{\sF,a,\delta} - H_{cyl}^{\sF, a, \delta}$ is a Hamiltonian isotopy, supported in the compact set $\pi^{-1}(B_{R+b}(0))$, from $\Lconic^{a,\delta}(\sF)$ to $\Lmc^{a,\delta}(\sF)$.

There is an isomorphism of objects of $\W(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})^{M^*}$
between  $\Lconic^{a,\delta}(\sF)$ and $\tilde{L}(\sF)$.

We will use the following notation for the tautological sector embeddings: $i: \ess \to \Sprime$, $i': \Sprime \to (M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ and $i' \circ i: \ess \to (M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$. The induced pushforward functors on wrapped Fukaya categories are all equivalences indicated by the left and right vertical arrows in <Ref>.

\[
\begin{tikzpicture}

\fill[rounded corners,fill=gray!20]   (-9.5,-4) rectangle (-5.5,-6.5);

\fill[rounded corners,fill=gray!20]  (2.5,-4) rectangle (5.5,-6.5);

\fill[rounded corners,fill=gray!20]  (-9.5,-0.5) rectangle (5.5,-2.5);

\node (v4) at (-9,-5) {$\Lconic$};

\node (v3) at (3.5,-5) {$\tilde{L}$ };

\node (v6) at (-6.5,-5) {$\Lmc^{a,\delta} $};

\node (v2) at (-6.5,-1.5) {$(i' \circ i)^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}( \Lmc^{a,\delta}, -)$};

\node(v1)  at (-1,-5) {$\Lmc^{a,\delta} \cap \ess= \Lmon^\delta= \tilde{L}\cap \ess$};

\node (v7) at (-1,-1.5) {$CF^*(\Lmon^\delta,-)$};
\node (v5) at (3.5,-1.5) {$i^* \hom_{\cW(\Sprime)}( \tilde{L}, -)$};
\draw  (v6) edge[black, ->, dotted] node[midway, above] {$\cap \ess$} (v1);
\draw  (v3) edge[black, ->,dotted] node[midway, above] {$\cap \ess$} (v1);
\draw  (v1) edge[black, ->,dotted] node[midway, fill=white] {\ref{arclikemodule}} (v7);
\draw  (v4) edge node[midway, fill=gray!20] {\ref{interpolatecyltomon}} (v6);
\draw  (v2) edge[black, ->] node[midway,below ]{\ref{arclikesubsector}} (v7);
\draw  (v5) edge[black, ->] node[midway, below]{\ref{arclikesubsector}} (v7);
\node at (-7.5,-6) {$\mathcal W(M^*_{\C}, \mathfrak f_{\Sigma^*})^{M^*}$};
\node at (4,-6) {$\mathcal W(\Sprime)$};
\node at (-1,-1) {$\operatorname{Mod} \mathcal W(\ess)$};
\draw  (v6) edge[->] node[midway, fill=white]{$(i'\circ i)^* \circ \mathcal Y_{(-)}$} (v2);
\draw  (v3) edge[->] node[midway, fill=white]{$i^* \circ \mathcal Y_{(-)}$}  (v5);
\end{tikzpicture}\]

Schematic diagram of the proof of <ref>.

We prove the isomorphism by exhibiting isomorphisms indicated in <Ref> by horizontal arrows.
First, $\Lconic^{a,\delta}(\sF)$ is isomorphic to $\Lmc^{a,\delta}(\sF)$ (which satisfies all of the hypotheses of <ref>) because they differ by a compactly-supported Hamiltonian isotopy. Now, since $\partial_{\infty} \Lmc^{a,\delta}(\sF) = \partial_{\infty}\Lconic^{a,\delta}(\sF)$ possesses a cofinal wrapping which does not cross $\partial \ess$ by <ref>, and since $\Lmc^{a,\delta}(\sF) \cap \ess$ is the $\aye$-arclike Lagrangian $\Lmon^\delta(\sF)$, <ref> applied to $i' \circ i: \ess \to (M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ implies that there is an isomorphism of $\cW(\ess)$ modules

\begin{equation}\label{arclikecutiso1}
(i' \circ i)^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}( \Lmc^{a,\delta}(\sF), -) \cong CF^*(\Lmon^\delta(\sF),-),
\end{equation}

where $CF^*(\Lmon^\delta(\sF),-)$ is the “arclike” $\cW(\ess)$-module constructed by applying <ref> to $\Lmon^\delta(\sF)$.

Similarly, the negative pushoff $\tilde{L}(\sF)$ also satisfies $\tilde{L}(\sF)\cap \ess = \Lmon^\delta(\sF)$ (see (<ref>)), and $\partial_{\infty} \tilde{L}(\sF)$ admits a cofinal wrapping contained entirely in $\partial_{\infty} (F \times T^*[0,1]) \subset \partial_{\infty} \Sprime$ which in particular avoids $\partial_{\infty} \ess$ (see <ref>). Hence, <ref>, this time applied to $i: \ess \to \Sprime$ again implies that

\begin{equation}\label{arclikecutiso2}
i^* \hom_{\cW(\Sprime)}( \tilde{L}(\sF), -) \cong CF^*(\Lmon^\delta(\sF),-).
\end{equation}

Since we can view $\tilde{L}(\sF)$ as living in $\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})$ via the pushforward $i'$, and $i'$ is a quasi-equivalence, we also know  $\hom_{\cW(\Sprime)}( \tilde{L}(\sF), -) \cong (i')^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}(\tilde{L}(\sF), -)$, from which we collectively deduce that

\begin{equation}\label{arclikecutiso3}
(i' \circ i)^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}( \tilde{L}(\sF), -) \cong CF^*(\Lmon^\delta(\sF),-);
\end{equation}

in other words, (<ref>) and (<ref>) imply that

$$(i' \circ i)^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}( \tilde{L}(\sF), -) \cong (i' \circ i)^* \hom_{\cW(M^*_{\C^*}, \mathfrak{f}_{\Sigma^*})}( \Lmc^{a,\delta}(\sF), -).$$

Now, $i' \circ i$ is a quasi-equivalence, so the same isomorphism holds before applying $(i' \circ i)^*$. Therefore Yoneda implies $\tilde{L}(\sF) \cong \Lmc^{a,\delta}(\sF)$.

§ POWERS OF AN AMPLE LINE BUNDLE

From here on, we drop decorations and use $\cap : \mafuk^{M^*}\to \partial \mafuk^{M^*}$ to denote the cap functor which makes the  diagram from  <ref> commute:

\begin{equation}\label{eq:Pic_sec_cd}\begin{tikzcd}
\Pic_{dg}^{M^*}(\partial Y^*)  \arrow[leftrightarrow]{r} & \partial \mafuk^{M^*}  \\
\Pic_{dg}^{M^*}(Y^*) \arrow{u}{i^*} \arrow[leftrightarrow]{r}{A} & \mafuk^{M^*} \arrow{u}{\cap}.\end{tikzcd}\end{equation}

Recall that all categories have objects indexed by the integral piecewise-linear functions $\sF$.

Let $\sF_\Delta$ denote the function corresponding to the toric boundary divisor; that is, $\sF_\Delta$ is equal to $-1$ on the generator of each ray of $\Sigma^*$.
We now restrict the diagram (<ref>) to the respective subcategories with objects indexed by $k\sF_\Delta$.
We denote the corresponding subcategories by

\begin{equation}
\label{eq:Pic_sec_cd2}\begin{tikzcd}
\cB_0  \arrow[leftrightarrow]{r}{\mir0} & \cA_0  \\
\tilde\cB_0 \arrow{u}{i^*} \arrow[leftrightarrow]{r} & \tilde \cA_0 \arrow{u}{\cap}.\end{tikzcd}
\end{equation}

The objects of $\tilde \cB_0$ are denoted $\tilde\cO(k)$ (powers of the anticanonical bundle on $Y^*$), those of $\cB_0$ are $\cO(k)$ (their restrictions to $\partial Y^*$), those of $\tilde \cA_0$ are $\tilde{L}_k$ (the mirror tropical Lagrangian sections), those of $\cA_0$ are $L_k$ (their boundaries).
In an $A_\infty$ category $\cC$, we use the convention that the cochain-level morphism spaces are denoted $hom^*_\cC(-,-)$, and their cohomologies by $Hom^*_\cC(-,-)$.