a contradiction to the preservation of the valuation. Furthermore, using Lemma
7.2, we obtain that the constant term of $h_{2}$ has to be $1$.

For the lower wall crossing map the same reasoning applies by first modifying
the locality embedding with a symplectomorphism as in the end of the previous
proof. ∎

Finally, we come to the step which relates the upper and lower wall-crossing
isomorphisms.

######  Proposition 8.4.

We have the equality

(62)
$1+a_{1}\eta+a_{2}\eta^{2}+\ldots=\eta^{k}(1+b_{1}\eta^{-1}+b_{2}\eta^{-2}+\ldots).$

The proof of this proposition relies on a _monodromy argument_ which formulate
now. Consider an admissible convex polygon $S\subset B_{k}$ as shown in Figure
3 for definiteness.

Let us call $X$ (resp. $Y$) the union of the upper, lower and left (resp.
right) edges of $S$. Note that $X$ (resp. $Y$) lies in the image of $E_{+}$
(resp. $E_{-}$). Therefore we can use the locality isomorphisms with respect
to the $+$ (resp. $-$) embedding for $X$ (resp. $Y$) as in Diagram (48). We
use this along with Proposition 3.2 to identify

$\mathcal{F}_{k}(X)\simeq\mathcal{F}_{0}(E_{+}^{-1}(X))\simeq\mathcal{O}_{0}(E_{+}^{-1}(X))\simeq\mathcal{O}_{0}(hull(E_{+}^{-1}(X))),$

where $hull(E_{+}^{-1}(X))$ is the convex hull of $E_{+}^{-1}(X)$ \- an
admissible convex polygon. Of course the same analysis can be made for $Y$ as
well.

Note that unlike what happened in Diagram (48) for admissible polygons
contained in the upper or lower half of the eigenline, we have exactly one
identification for $X$ and $Y$ depending on which eigenray they intersect.

Figure 3. Drawn inside $B_{k}$ with the dashed ray representing the defining
monodromy invariant ray

######  Proposition 8.5.

The restriction map

$\mathcal{F}_{k}(S)\to\mathcal{F}_{k}(X)=\mathcal{O}_{0}(hull(E_{+}^{-1}(X)))$

has in its image the element with formal expression $\xi.$

We first finish the proof of Proposition 8.4 using this statement.

###### Proof of Proposition 8.4.

Take an element $A\in\mathcal{F}_{k}(S)$ which has image $A_{X}=\xi$ under the
identification $\mathcal{F}_{k}(X)=\mathcal{O}_{0}(hull(E_{+}^{-1}(X)))$ as in
the statement of Proposition 8.5.

Restrict $A$ to either the upper edge $U$ or the lower edge $L$ of $P$. Using
the left side of Diagram (48), we obtain elements
$A_{U}\in\mathcal{O}_{0}(E_{+}^{-1}(U))$ and
$A_{L}\in\mathcal{O}_{0}(E_{+}^{-1}(L))$. The formal expression for both of
these elements is $\xi$.

Now use the upper and the lower wall-crossing isomorphisms to get elements in
$A_{U}^{\prime}\in\mathcal{O}_{0}(hull(E_{-}^{-1}(U)))$ and
$A_{L}^{\prime}\in\mathcal{O}_{0}(hull(E_{-}^{-1}(L)))$. We a priori know that
both of these elements have to be restrictions of
$A_{Y}\in\mathcal{O}_{0}(hull(E_{-}^{-1}(Y)))$ and in particular they have to
have the same formal expression.

Writing this down using Proposition 8.3 we obtain

$\xi\cdot(1+a_{1}\eta+a_{2}\eta^{2}+\ldots)=\xi\cdot\eta^{k}(1+b_{1}\eta^{-1}+b_{2}\eta^{-2}+\ldots).$

It is easy to see that one can cancel $\xi$ from this relation. Indeed, we can
restrict to a polygon where $\xi$ is invertible. The claim follows.

∎

######  Corollary 8.6.

The wall-crossing isomorphisms (both!) are given by

$\eta\mapsto\eta,\xi\mapsto\xi(1+\eta)^{k}.$

###### Proof.

For $k=1$, we uniquely determine the wall-crossing transformations from
equation (62). For $k>1$, (62) does not have a unique solution, but notice
that we can resolve the the $k$-fold singularity into $k$ simple nodal
singularities lying on one monodromy invariant line by nodal sliding. This can
be achieved by modifying $\pi_{k}$ in a small neighborhood of
$\pi_{k}^{-1}(0)$, and in particular does not modify $\pi_{k}^{-1}(S)\subset
M_{k}$ as a set or its relative $SH$. For this resolved singularity the wall
crossing around all $k$ singularities is the same as the wall crossing around
the unresolved one, and it factors as a $k$-fold iteration of the wall
crossing isomorphism for the case $k=1$.

∎

Finally, we can prove Theorem 6.1.

###### Proof of Theorem 6.1.

As commented in Section 6.2 this amounts to matching the A- and B- side wall
crossing. The formula for the B-side wall crossing is given in Proposition
3.10, which matches with Corollary 8.6. ∎

###  8.1. Proof of Proposition 8.5

Let us now prove the intermediary Proposition 8.5

###### Proof of Proposition 8.5.

Let $L$ be the edge of $X$ on the left. By functoriality with respect to
inclusions and our knowledge of the restriction map
$\mathcal{O}_{0}(hull(E_{+}^{-1}(X)))\to\mathcal{O}_{0}(E_{+}^{-1}(L))$ it
suffices to prove that the restriction map to $L$ contains $\xi$, where we
identified $\mathcal{F}_{k}(L)$ with $\mathcal{O}_{0}(E_{+}^{-1}(L))$ via the
locality isomorphism as well.

A Hamiltonian $H$ is called _$Q$ -admissible_ for a subset $Q\subset B_{k}$ if
$H<0$ on $\pi_{k}^{-1}(Q)$. A dissipative $H$ is called _$\xi$ -full_ if $\xi$
is in the image of the natural map $HF^{0}(H;\Lambda)\to
SH^{0}_{M_{k}}(\pi_{k}^{-1}(L),\Lambda)=\mathcal{F}_{k}(L)$.

It suffices to show that there is an $S$-admissible Hamiltonian $H$ which is
$\xi$-full. Indeed, the map $HF^{0}(H;\Lambda)\to\mathcal{F}_{k}(L)$ factors
through the restriction map $\mathcal{F}_{k}(S)\to\mathcal{F}_{k}(L)$.

Let $B_{+}$ be the image of $E_{+}$ and $M_{+}:=\pi_{k}^{-1}(B_{+})$. A
function $h:B_{k}\to{\mathbb{R}}$ is said to be _convex on the left_ if
$dh(e_{1})\geq 0$ on the eigenray $B_{k}\setminus B_{+}$ and on $B_{+}$ any
co-vector $\nu$ with $\nu(e_{1})<0$ is obtained at most once as $dh_{b}$ for
some $b\in B_{+}$, using the canonical identification of the cotangent spaces
of points in $B_{+}$

For a function $h$ which is convex on the left we show now that if the level
sets are sufficiently close to being rectangular in a neighbourhood of $L$ we
have the following property. Equip $M$ with a $S$-compatible primitive (given
by Euler primitive outside an open subset whose closure is contained in the
interior of $S$). Consider the homology class $a$ in $H_{1}(M_{+},\mathbb{Z})$
given by the primitive covector (called $-e_{1}^{\vee}$) of the left edge
pointing out of $S$. Let $\tilde{a}$ be its image in $H_{1}(M,\mathbb{Z})$.
Then if $h$ has a $1$-periodic orbit $\gamma$ corresponding to
$-e_{1}^{\vee}$, this orbit maximizes action among all orbits in the class
$\tilde{a}$. To see this note the action of an orbit representing
$e_{1}+ne_{2}^{\vee}$ is arbitrarily close to $-\lambda_{1}-|n|\lambda_{2}$
where $\lambda_{1}$ is the distance from the origin to $L$ and $\lambda_{2}$
is half the length of $L$ by Proposition 5.3 (using the flat metric in the
domain of $E^{+}$).

Call a convex from the left $h$ which has sufficiently large slope to the left
of $L$ and with level sets sufficiently close to rectangular for the property
of the previous paragaraph to hold _well-behaved_. It is straightforward that
we can construct acceleration data for $L$ which consist of well behaved
Hamiltonians.

Given $F:M_{k}\times S^{1}\to\mathbb{R}$ that is an appropriate small time
dependent perturbation of $f\circ\pi_{k}$ for a well behaved $f$, let
$\gamma^{f}$ be the $1$-periodic orbit corresponding to the negative of
$e_{1}^{\vee}$ in degree $0$. By standard Morse-Bott considerations, the only
possible contributions to the differential of $\gamma^{f}$ are those
connecting it to degree $1$ periodic orbits which don’t arise from the
perturbation of $-e_{1}^{\vee}$. By action considerations we conclude that
$\gamma^{f}$ is a cycle. Fix such $H,G$ so that $H$ is $S$-admissible and $G$
is $L$-admissible. Assume that a neighborhood of $\gamma^{H}$ coincides with a
neighborhood of $\gamma^{G}$ and that $H$ and $G$ coincide there. Then
$\gamma^{H}$ maps precisely to $\gamma^{G}$. Indeed, we have the constant
solution, and we can have no other by the action minimization.

To conclude, we turn to show that at least for carefully chosen $G$ and almost
complex structures, the generator $\gamma^{G}$ maps to a non-zero scalar
multiple of $\xi$. By nodal sliding to the right (similar to the argument in
locality for complete embeddings) we can assume that any relevant Floer
trajectory with $\gamma^{G}$ as input with energy $\leq 1$ is contained in a
fixed compact neighbourhood $K$ of $\pi^{-1}_{k}(L)$ inside $M_{+}$. Thus, up
to $O(1)$ we can guarantee that $\gamma^{G}$ maps to $T^{\epsilon}\xi$ where
$\epsilon<1$. To see the last point let $G_{loc}$ be an extension of $G|_{K}$
to $M_{0}$ which is still convex from the left. Then at least for $G_{loc}$
large enough we know that $\xi$ is in the image of the map $HF^{*}(G_{loc})\to
SH^{*}_{M_{0}}(\pi^{-1}(L))$. But the only thing that can map to $\xi$ is
$\gamma^{G}$. We now rule out the possibility of an $O(1)$ correction. Such a
correction would amount to a continuation trajectory from $\gamma^{G}$ to some
orbit $\delta$ of some well behaved Hamiltonian $G^{\prime}>G$ so that
$\delta$ is in the class $\tilde{a}$ in $M$ but not in $M^{+}$. By definition
of well behavedness such a $\delta$ has lower action and such a trajectory is
impossible.

∎

######  Remark 8.7.

Let us explain what happened a posteriori. The $A$-side wall crossing maps as
defined by Diagram (48) gives us an isomorphism of the rigid analytic spaces

$(\Lambda^{*})^{2}\setminus\\{val(\eta^{-})=0\\}\to(\Lambda^{*})^{2}\setminus\\{val(\eta^{+})=0\\}.$

This is a morphism between two disconnected spaces and it appears that we have
two entirely independent maps. But it turns out that in fact this map extends
to an isomorphism

$(\Lambda^{*})^{2}\setminus\\{\eta^{-}=-1\\}\to(\Lambda^{*})^{2}\setminus\\{\eta^{+}=-1\\}.$

If we knew this fact a priori, then Proposition 8.4 would follow as both maps
(and in particular the pullback of the regular function $\xi^{+}$) are
determined by this one map. In fact our results prove that it is the
restriction of the transition map from Proposition 3.10.

Our monodromy argument is a replacement of this point. Its intuition is the
knowledge that the function $\xi^{+}$ extends to a function $Y_{1}^{an}$ and
hence maybe we should use the mirror of this global function to obtain the
desired relationship between how the mirror of $\xi^{+}$ transforms under the
two wall crossing transformations.

##  9\. Hartogs’s property

Our goal in this section is to prove the following statement, which will allow
us to extend the isomorphism of Theorem 6.1 to all of $B_{k}$.

######  Proposition 9.1.

Let $P\subset B_{k}$ be an admissible convex polygon containing a node in its
interior. Then, the restriction map

$\mathcal{F}_{k}(P)\to\mathcal{F}_{k}(\partial P)$

is an isomorphism.

Recall that the same statement holds if we replace $\mathcal{F}_{k}$ with
$\mathcal{O}_{k}$, see Proposition 3.17. The rough idea of the proof is that
one can choose the acceleration data for $\partial P$ and $P$ so that in a
large region where the degree $0$ orbits of the Hamiltonians are contained the
acceleration data are exactly the same. We start with an abstract lemma.

######  Lemma 9.2.

Let $C\to D$ be a chain map of non-negatively graded chain complexes. Assume
that $C^{0}\to D^{0}$ is an isomorphism and $C^{1}\to D^{1}$ is injective.
Then, $H^{0}(C)\to H^{0}(D)$ is an isomorphism.

###### Proof.

For a formal proof, truncate the complexes at degree $2$ and consider the long
exact sequence. ∎

######  Lemma 9.3.

Let us take a map of $1$-rays over the Novikov ring $\Lambda_{\geq 0}$

(67)

Assume that each of these chain complexes are non-negatively graded, in
degrees $0$ and $1$ they are free of finite rank,

$f_{i}^{0}:\mathcal{C}_{i}^{0}\otimes\Lambda_{\geq
0}/\Lambda_{>0}\to\mathcal{C}_{0}^{{}^{\prime}1}\otimes\Lambda_{\geq
0}/\Lambda_{>0}$

is an isomorphism,

$f_{i}^{1}:\mathcal{C}_{i}^{1}\otimes\Lambda_{\geq
0}/\Lambda_{>0}\to\mathcal{C}_{i}^{{}^{\prime}1}\otimes\Lambda_{\geq
0}/\Lambda_{>0}$

is injective and both $H^{0}(\widehat{tel}(\mathcal{C}))$ and
$H^{0}(\widehat{tel}(\mathcal{C}^{\prime}))$ have finite torsion.

Then, the induced map

$H^{0}(\widehat{tel}(\mathcal{C}))\to
H^{0}(\widehat{tel}(\mathcal{C}^{\prime}))$

is an isomorphism.

###### Proof.

We have the commutative diagram:

(72)

and, under the finite torsion assumption, the vertical maps are isomorphisms
by [9, Proposition 1.2].

Therefore, it suffices to show that

$H^{0}(\mathcal{C}_{i}/T^{r}\mathcal{C}_{i})\to
H^{0}(\mathcal{C}_{i}^{\prime}/T^{r}\mathcal{C}_{i}^{\prime})$

is an isomorphism for every $r\geq 0$ and $i=1,2,\ldots$. Fix an $i$ and $r$
for the rest of the argument.

Now, by Nakayama’s lemma, we see that

$f_{i}^{0}:\mathcal{C}_{i}^{0}\otimes\Lambda_{\geq
0}/\Lambda_{>r}\to\mathcal{C}_{i}^{{}^{\prime}0}\otimes\Lambda_{\geq
0}/\Lambda_{>r}$

is an isomorphism.

Note that, moreover,

$f_{i}^{1}:\mathcal{C}_{i}^{1}\otimes\Lambda_{\geq
0}/\Lambda_{>r}\to\mathcal{C}_{i}^{{}^{\prime}1}\otimes\Lambda_{\geq
0}/\Lambda_{>r}$

preserves the module norms and hence is injective for all $r\geq 0$. Here we
are also using that $\mathcal{C}_{i}^{1}$ is free to conclude that the module
semi-norms are in fact norms.

Finally, we use Lemma 9.2.

∎

###### Proof of Proposition 9.1.

We construct acceleration data for $P$ and $\partial P$ and homotopy data for
the restriction map so that the hypotheses of the previous lemma hold.

Let us call a smooth function $f:B_{1}\to\mathbb{R}$ strongly non-degenerate
if

  * •

In a neighborhood of the node $f$ only depends on the monodromy invariant
integral affine coordinate $u$ and $0<\frac{\partial f}{\partial u}(0)<1$.

  * •

The Hessian

(73) $\left(\frac{\partial^{2}f}{\partial q_{i}\partial q_{j}}(b)\right)$

with respect to an (and hence any) integral affine coordinate system
$q_{1},q_{2}$ is non-degenerate for every $b\in B^{reg}_{1}$ such that
$df_{b}\in T_{\mathbb{Z}}^{*}B$.

  * •

It is dissipative for some geometrically bounded almost complex structure.

Note that the first bullet point implies that the only $1$-periodic orbit of
the Hamiltonian $f\circ\pi_{1}$ which lies on the nodal fiber
$\pi_{1}^{-1}(0)$ is the constant orbit at the focus-focus singularity.
Moreover, the Morse index of this critical point of $f\circ\pi_{1}$ is $2$. To
see this note that in a neighborhood of the singular point there are complex
valued coordinates $z_{1},z_{2}$ such that $u\circ\pi_{1}$ is the map
$(z_{1},z_{2})\mapsto|z_{1}|^{2}-|z_{2}|^{2}$.

Using the strategy described at the end of Section 6.1 (label!), we choose

  * •

a sequence of strongly non-degenerate smooth functions $f_{1}<f_{2}<\ldots$
and $g_{1}<g_{2}<\ldots$

  * •

a sequence of compact domains $D_{1}\subset D_{2}\subset\ldots$ inside $B_{1}$

such that

  * •

$f_{1}<f_{2}<\ldots$’s form a cofinal sequence among functions that are
negative on $P$

  * •

$g_{1}<g_{2}<\ldots$’s form a cofinal sequence among functions that are
negative on $\partial P$

  * •

$f_{i}=g_{i}$ outside of $D_{i}$ for every $i\geq 0$

  * •

$f_{i}$ is $C^{2}$-small inside $D_{i}$.

  * •

$g_{i}$ is so that if $(dg_{i})_{b}\in T_{\mathbb{Z}}^{*}B$ for $b\in D_{i}$,
then the Hessian from (73) is not positive definite.

We carefully perturb the Morse-Bott families of orbits with the condition that
outside of $D_{i}$ we use the same perturbations for both $f_{i}$ and $g_{i}$.
After choosing the relevant monotone interpolations, the conditions of Lemma
9.3 are easily checked. ∎

We finally prove Theorem 1.8

###### Proof of Theorem 1.8.

We want to show that we can extend the isomorphism of Theorem 6.1. By another
application of Lemma 2.1, it suffices to extend it to admissible convex
polygons containing the node. We define these isomorphisms by the diagram

(78)

where the vertical maps are given by Propositions 9.1 and 3.17, for any such
$P\subset B_{k}.$

We need to check that, for an admissible convex polygon $Q\subset P\subset
B_{k}$,

(83)

commutes.

First, note that when $Q\subset\partial P$, this is immediate as the
restriction maps factor through $\partial P$. Now assume that $Q\subset P$
intersects $\partial P$ and let $q\in Q\cap\partial P$. Then $q\in\partial Q$.
Consider the diagram

(88)

We already know the outer rectangle and the square on the right commute.
Noting that $\mathcal{O}_{k}(Q)\to\mathcal{O}_{k}(\\{q\\})$ is injective
finishes the proof. This injectivity is immediate (see [3, Lemma 10]) from the
fact that $p_{k}^{-1}(\\{q\\}\subset p_{k}^{-1}(Q)$ is an affinoid (in fact
rational) subdomain, which was shown in the proof of Proposition 3.14.

Finally, assume that $Q$ is arbitrary. We can find an admissible convex
polygon $Q^{\prime}\subset P$ which intersects both $Q$ and $\partial P$. Let
$q^{\prime}\in\partial Q^{\prime}\cap\partial Q$. Both squares in the diagram

(93)

commute so the composition square also commutes. We finish this time using the
injectivity of $\mathcal{O}_{k}(Q)\to\mathcal{O}_{k}(\\{q^{\prime}\\}).$

∎

##  10\. Construction of mirrors

###  10.1. Symplectic cluster manifolds

Let $\mathcal{R}$ be an eigenray diagram. As explained in Section 7 of [9],
one can associate to $\mathcal{R}$ a symplectic manifold $M_{\mathcal{R}}$
together with a nodal Lagrangian fibration
$\pi_{\mathcal{R}}:M_{\mathcal{R}}\to B_{\mathcal{R}}$. The pre-image of an
admissible polygon containing a singular point with monodromy $A_{k}$ is
modeled on $\pi_{k}:M_{k}\to B_{k}$. There are some choices involved in the
construction near each node but by abuse of notation we will shall not
distinguish them.

A _symplectic cluster manifold_ is symplectic manifold which is
symplectomorphic to $M_{\mathcal{R}}$ for some eigenray diagram $\mathcal{R}$.
A given symplectic cluster manifold may have multiple eigenray diagram
representations as a result of nodal slide and branch move operations on
eigenray diagrams. See Section 7.2 of [9].

The following is Proposition 7.17 of [9].

######  Proposition 10.1.

Symplectic cluster manifolds are geometrically of finite type.

If $M$ is a symplectic cluster manifold, then $c_{1}(M)=0$ and there is a
preferred trivialization up to homotopy of $\Lambda^{n}T_{\mathbb{C}}M,$ which
makes the regular fibers of any associated
$\pi_{\mathcal{R}}:M_{\mathcal{R}}\to B_{\mathcal{R}}$ have Maslov
homomorphism $\pi_{1}(T^{2},\star)\to\mathbb{R}$ zero. We use this grading
datum in defining relative $SH$ without further mention.

For a small admissible polygon we say it has multiplicity $k>0$ if it contains
a singular value of multiplicity $k$. If it contains no singular value we say
it has multiplicity $0$.

######  Proposition 10.2.

Let $P\subset B_{\mathcal{R}}$ be a small admissible polygon of multiplicity
$k$. Then there exists a complete embedding $\iota:M_{k}\to M_{\mathcal{R}}$,
a convex admissible polygon $P_{0}\subset B_{k}$, and an integral affine
embedding $f:U\to B_{\mathcal{R}}$ from neighborhood of $P_{0}$ so that

  * •

$f(P_{0})=P$

  * •

$\pi_{\mathcal{R}}^{-1}(P)\subset\iota(M_{k})$

  * •

$\pi_{\mathcal{R}}\circ\iota|_{\pi_{k}^{-1}(P_{0})}=f\circ\pi_{k}$.

###### Proof.

If the multiplicity is $k>0$ we can choose Lagrangian tails $L_{i}$ lying over
each ray not emanating from the singular value inside $P$. The complement of
these tails is symplectomorphic to $M_{k}$ according to [9, Theorem 1.8].
Moreover, this symplectomorphism can be taken to intertwine the Lagrangian
fibrations away from arbitrarily small neighborhoods of the removed rays. If
the multiplicity is $0$ and $P$ intersects no rays we can again do the same
thing. If $P$ meets some ray $l$ then by removing tails over rays which don’t
meet $P$ the complement is symplectomorphic to $M_{k}$ in a fibration
preserving manner near $P$. We can then perform a branch move on $l$, which
has no effect on the fibration, and reduce to the previous case. ∎

We can now prove the five properties listed in Theorem 1.5.

###### Proof of Theorem 1.5.

The locality isomorphism of Theorem 6.3 and Proposition 10.2 reduce the result
to the same statements for $\mathcal{F}_{k}$ on $B_{k}$. Theorem 1.8 reduces
it to $\mathcal{O}_{k}$ on $B_{k},$ which is covered by Corollary 3.16. ∎

###  10.2. Gluing

For $P$ a small admissible polygon in $B_{\mathcal{R}}$, we consider the
affinoid domains

$M(P):=M(\mathcal{F}_{\mathcal{R}}(P)).$

Let us denote by

$\iota_{Q\subset P}:M(Q)\to M(P)$

the maps induced by the restriction maps.

We denote the collection of all small admissible polygons in $B_{\mathcal{R}}$
by $\\{P_{i}\\}_{i\in I}$. Let us define, for each pair $i,j\in I,$ the
affinoid subdomains

$U_{ij}:=\iota_{P_{i}\cap P_{j}\subset P_{i}}(M(P_{i}\cap P_{j})).$

Then, for all $i,j\in I$, we automatically obtain the isomorphism

$\psi_{ij}:=\iota_{P_{i}\cap P_{j}\subset P_{j}}\circ\iota_{P_{i}\cap
P_{j}\subset P_{i}}^{-1}:U_{ij}\to U_{ji}.$

######  Lemma 10.3.

For $i,j,k\in I,$ we have

$\psi_{ij}(U_{ij}\cap U_{ik})\subset U_{ji}\cap U_{jk}.$

###### Proof.

Note that we have

$U_{ij}\cap U_{ik}=im(\mathcal{M}(P_{i}\cap P_{j}\cap P_{k}))$

inside $\mathcal{M}(P_{i})$ by the strong cocycle condition, and the map
$\iota_{P_{i}\cap P_{j}\cap P_{k}\subset P_{i}}$ is the same as

$\iota_{P_{i}\cap P_{j}\subset P_{i}}\circ\iota_{P_{i}\cap P_{j}\cap
P_{k}\subset P_{i}\cap P_{j}}$

by the presheaf property of restriction maps.

Therefore, we have a canonical commutative diagram

(98)

where the image of $M(P_{i}\cap P_{j}\cap P_{k})$ in $U_{ij}$ and $U_{ji}$ is
equal to $U_{ij}\cap U_{ik}$ and $U_{ji}\cap U_{jk},$ respectively. The proof
is immediate. ∎

For $i,j,k\in I,$ let us give the name

$\psi_{ijk}:U_{ij}\cap U_{ik}\to U_{ji}\cap U_{jk}$

to the map obtained by restricting $\psi_{ij}$.

######  Lemma 10.4.

The collection of maps $\psi_{ij}$ satisfies $\psi_{ij}\circ\psi_{ji}=id$,
$\psi_{ii}=id$ and the cocycle condition

$\psi_{ijk}=\psi_{kji}\circ\psi_{ikj}.$

###### Proof.

The first two are obvious. The cocycle condition follows from the
commutativity of the diagram of isomorphisms:

(103)

∎

As in [3, Proposition 5] we now define the rigid analytic space by gluing
$M(P_{i})$ along $U_{ij}$’s using $\psi_{ij}$’s. As a set

(104) $\mathcal{Y}_{\mathcal{R}}=\coprod_{i\in I}M(P_{i})/\sim$

where $\sim$ is the relation $x\in U_{ij}\sim\psi_{ij}(x)\in U_{ji}$.

Note that by construction there are canonical embeddings of each
$\mathcal{M}(P_{i})$ into $\mathcal{Y}_{\mathcal{R}}$ and we call their images
(which are admissible opens) $X_{i}\subset\mathcal{Y}_{\mathcal{R}}.$ If
$P_{i}\subset P_{j}$, then $X_{i}\subset X_{j}$.

We now want to construct a map $p_{\mathcal{R}}:\mathcal{Y}_{\mathcal{R}}\to
B_{\mathcal{R}}$ using the properties listed in Theorem 1.5.

######  Proposition 10.5.

Let $y\in\mathcal{Y}_{\mathcal{R}}$. Denote by $I_{y}\subset I$ the set of
indices for which $P_{i}$ is a small admissible polygon such that $y\in
X_{i}$. Then the set

$\bigcap_{i\in I_{y}}P_{i}\subset B_{\mathcal{R}}$

is a singleton $\\{b_{y}\\}$.

###### Proof.

We first claim that there is a sequence of small admissible polytopes

$P_{i(1)}\supset P_{i(2)}\supset\ldots$

such that

  * •

$i(k)\in I_{y}$ for all $k\geq 1$.

  * •

$\bigcap_{k\geq 1}P_{i(k)}$ is a singleton.

Note that we have an explicit PL isomorphism $\mathbb{R}^{2}\simeq
B_{\mathcal{R}}$ by construction. Let us use the flat metric on
$\mathbb{R}^{2}$ to equip $B_{\mathcal{R}}$ with a metric space structure.

By construction $I_{y}$ is non-empty. Let $i(1)\in I_{y}$. Partition
$P_{i(1)}=\bigcup_{k=1}^{n}P_{j(k)}$ into admissible convex polygons (which
are automatically small) with half the diameter. Using the independence
property we know that $y\in X_{j(k)}$ for some $k$. We let $i(2)=j(k)$ and
thus continue inductively. The polygons $P_{i(k)}$ are compact in the standard
topology on $B_{\mathcal{R}}$ so we deduce that the intersection
$\bigcap_{k\geq 1}P_{i(k)}$ is non-empty and of $0$ diameter. That is, a
singleton.

It follows, first, that the intersection $\bigcap_{i\in I_{y}}P_{i}$ has at
most one element. It remains to show non-emptyness. Let $b_{y}$ be the unique
element in $\bigcap_{k\geq 1}P_{i(k)}$. Let $i\in I_{y}$ and assume by
contradiction that $b_{y}\not\in P_{i}$. Then for $k$ large enough, $P_{i}\cap
P_{i(k)}=\emptyset$. But then we cannot have $y\in X_{i}\cap X_{i(k)}$ by the
gluing construction. This contradiction completes the proof. ∎

We thus define a canonical map

$p_{\mathcal{R}}:\mathcal{Y}_{\mathcal{R}}\to B_{\mathcal{R}}$

by $y\to b_{y}$.

######  Lemma 10.6.

For any small admissible polygon $P_{i}$ we have
$p_{\mathcal{R}}^{-1}(P_{i})=X_{i}$ .

###### Proof.

For $y\in X_{i}$ we have $b_{y}\in P_{i}$ by construction. So
$p_{\mathcal{R}}(X_{i})\subset P_{i}$. Conversely, suppose $b\in P_{i}$. We
want to show that $p_{\mathcal{R}}^{-1}(b)\subset X_{i}$. Let $y\in
p_{\mathcal{R}}^{-1}(b),$ that is $b_{y}=b.$

Let $P_{j}$ be any small admissible polygon such that $y\in X_{j}$. Let
$P_{k}=P_{i}\cap P_{j}$. Note that $b$ is contained in $P_{k}$. Then we claim
that $y\in X_{k}$. Assuming $y\notin X_{k}$, by the separation property, there
is a $P_{l}\subset P_{j}$ which does not contain $b$ so that $y\in X_{l}$.
This contradicts the construction of $b=b_{y}$. Therefore, we have that $y\in
X_{k}\subset X_{i}$.

∎

######  Proposition 10.7.

If $P\subset B_{\mathcal{R}}$ is a small admissible polygon of multiplicity
$k$ there is a commutative diagram of sets

(109)

where the upper horizontal arrow is an analytic embedding and the lower arrow
is a nodal integral affine embedding.

###### Proof.

By Lemma 10.6 we have
$p_{\mathcal{R}}^{-1}(P_{i})=X_{i}\simeq\mathcal{M}(P_{i})$. The upper
horizontal arrow is the one induced by the locality isomorphism and the lower
arrow is the inverse of the map $f$ from Lemma 10.2 restricted to $f^{-1}(P)$.
It remains to show commutativity.

For this note, that for $P_{i}\subset P_{j}$ admissible polygons, locality and
the isomorphism of Theorem 1.8 intertwine the inclusion
$\mathcal{M}(P_{i})\to\mathcal{M}(P_{j})$ with the inclusion
$p_{k}^{-1}(P_{i})\to p_{k}^{-1}(P_{j})$. Second, for any $y\in Y_{k}$ we have
tautologically that $p_{k}(y)=\cap_{i:y\in p_{k}^{-1}(P_{i})}P_{i}$ where $i$
runs over all small admissible polygons in $B_{k}$. The commutativity thus
follows by construction of $p_{\mathcal{R}}$. ∎

###### Proof of Theorem 1.1.

We have already constructed $\mathcal{Y}_{\mathcal{R}}$ with its map
$p_{\mathcal{R}}\to B_{\mathcal{R}}$. That $p_{\mathcal{R}}$ is a Stein
continuous map follows from Lemma 10.6 and the properties of the gluing
construction of analytic spaces. Let us denote the structure sheaf of
$\mathcal{Y}_{\mathcal{R}}$ by $\mathcal{O}_{\mathcal{R}}.$

By construction and Lemma 10.6, for a small admissible polygon $P$,

$(p_{\mathcal{R}})_{*}\mathcal{O}_{\mathcal{R}}(P)=\mathcal{F}_{\mathcal{R}}(P).$

Invoking Lemma 2.1 we obtain the desired isomorphism of sheaves of algebras
$\mathcal{F}_{\mathcal{R}}\simeq
p_{\mathcal{R}}^{*}\mathcal{O}_{\mathcal{R}}$.

Let us now start proving the numbered assertions. The items (1) and (2) follow
immediately from Proposition 10.7 and the discussion in Section 3.2 for
$p_{k}:Y_{k}\to B_{k}$. For item (3), we in addition need Proposition 3.18 and
the fact that $\Omega_{0}$ is invariant under the action of the integral
affine transformations $SL(2,\mathbb{Z})\ltimes\mathbb{R}^{2}$ on
$Y_{0}^{an}=(\Lambda^{*})^{2}$.

Note that if we take a cover of $B_{\mathcal{R}}$ by small admissible
polygons, we can construct $\mathcal{Y}_{\mathcal{R}}$ by gluing only the
corresponding affinoid domains because of the invariance property. If the
cover is sufficiently fine, namely if for every point of $B_{\mathcal{R}}$
there is a descending chain of members of the cover with intersection
precisely equalling that point, then we can construct $p_{\mathcal{R}}$ only
using this cover. Items (4) and (5) follow from this point.

For (4), we note that $B_{\mathcal{R}_{1}}$ and $B_{\mathcal{R}_{2}}$ are
canonically identified as nodal integral affine manifolds and we use the cover
by the admissible polygons which are small with respect to both eigenray
representations.

For (5), first assume that we can find a small admissible polygon in
$B_{\mathcal{R}_{1}}$ which contains the nodal slide locus in its interior. We
consider the covers of $B_{\mathcal{R}_{1}}$ and $B_{\mathcal{R}_{2}}$
(canonically identified by a PL isomorphism) by this small admissible polygon
and all the small admissible polygons which do not intersect the nodal slide
locus. Clearly, the resulting rigid analytic spaces are the same. Moreover, we
see that the isomorphism respects the fibrations outside of the nodal slide
locus. The general case follows by iterating this step.

Finally, for (6), it suffices to note that both sides can be defined as an
inverse limit over the preimage of an exhaustion of $B_{\mathcal{R}}$ by
admissible polygons.

∎

######  Remark 10.8.

Our proof of the first three bullet points rely on a computation (not just
properties) and is ad-hoc. At least when all the multiplicities are $1$, we
expect to be able to give a much more conceptual proof based on the fact that
there is a Lagrangian section $L$ satisfying the local generation property
after recasting the construction of $\mathcal{Y}_{\mathcal{R}}$ using the
relative Lagrangian Floer homology of $L$.

The intrinsic meanings of (2) and (3) are clear. One might consider replacing
(1) with the statement that $p_{\mathcal{R}}$ is the projection to the
essential skeleton defined using the canonical volume form on
$\mathcal{Y}_{\mathcal{R}}$.

##  11\. Future work

###  11.1. Higher dimensions

In this section we want to briefly explain how our methods would extend to
higher dimensions. Let us start with a generalization that is essentially
straightforward.

Any triple $(\mu,f,Z)$, where

  * •

$\mu:\mathbb{R}^{n}\to\mathbb{R}$ an integral affine function

  * •

$f\in\mathbb{Z}^{n}$ such that $d\mu(f)=0$

  * •

a properly embedded connected codimension $1$ smooth submanifold $Z$ of

$H:=\\{x\in\mathbb{R}^{n}\mid\mu(x)=0\\}$

which is transverse to $f$ everywhere

defines a subset $S(\mu,f,Z)$ of $\mathbb{R}^{n}$ by considering the closure
of the $f$-positive side of $Z$ inside $H$.

Given $(\mu,f,Z)$ inside $\mathbb{R}^{n}$ we can define a nodal integral
affine manifold with the singular locus $Z$ by regluing a connected
neighborhood of $S:=S(\mu,f,Z)$ by the identity on the $\mu(x)>0$ side and by
the transvection

(110) $v\mapsto v-\mu(v)\cdot f.$

on the other.

We define a shear-cut diagram $\mathcal{R}$ in $\mathbb{R}^{n}$ as a finite
collection of such triples $(\mu_{i},f_{i},Z_{i}),i\in I$ such that
$S_{i}:=S(\mu_{i},f_{i},Z_{i})$ are pairwise disjoint. Given a shear cut locus
$\mathcal{R}$, we can define a Lagrangian fibration
$\pi_{\mathcal{R}}:M_{\mathcal{R}}\to B_{\mathcal{R}}$ with only focus-focus
singularities such that the induced nodal integral affine structure is
isomorphic to the one induced on $\mathbb{R}^{n}$ by $\mathcal{R}$ [2].

We have analogues of the nodal slide and branch cut operations. For nodal
slides, we can isotope $Z_{i}$ inside $H_{i}=\\{\mu_{i}(x)=0\\}$ as a properly
embedded smooth submanifold keeping it transverse to $f_{i}$. This does not
change $M_{\mathcal{R}}$ but modifies the fibration $\pi_{\mathcal{R}}$.
Assuming that $H_{i}$ is disjoint from all the other $S_{j}$’s, we can apply a
branch move by replacing $f_{i}$ with $-f_{i}$. This only changes how we
represent our Lagrangian fibration using a shear-cut diagram.

Let us identify $B_{\mathcal{R}}$ with $\mathbb{R}^{n}$ using the canonical PL
homeomorphism and denote the singularity locus $\bigcup_{i\in I}Z_{i}$ by $Z$.
We now start assuming that $Z_{i}$ are in fact affine. This means that there
are neighborhoods of $Z_{i}$ which are for some $k_{i}$ isomorphic to a
neighborhood of $\mathbb{R}^{n-2}\times\\{\text{node}\\}$ inside
$\mathbb{R}^{n-2}\times B_{k_{i}}.$ Let us fix such isomorphisms and call this
data a framing.

We define an admissible convex polygon in $B_{\mathcal{R}}$ to be a convex
rational polygon if it does not intersect $Z$, and if it does intersect, as a
product of admissible convex rational polygons inside $\mathbb{R}^{n-2}$ and
$B_{k_{i}}$ with respect to a fixed framing.

Using locality for complete embeddings, Kunneth theorem e.g. [8, Theorem 10.6]
and some torsion analysis it should be easy to prove Theorem 1.5 and then
construct a mirror $p_{\mathcal{R}}:\mathcal{Y}_{\mathcal{R}}\to
B_{\mathcal{R}}$.

We can go further and consider more complicated constructions. To illustrate
the idea let us focus on dimension $3$. In that case we consider
$(\mu,\Delta,Z)$ where $\mu:{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ is an
integral affine map, $\Delta\subset{\mathbb{R}}^{2}$ is a tropical curve, and
$Z\subset\mu^{-1}(\Delta)$ is a codimension $1$-graph projecting bijectively
onto $\Delta$ under $\mu$. We label the two components of
$\mu^{-1}(\Delta)\setminus Z$ by $j\in\\{\pm\\}$, and denote the closure by
$S_{j}(\mu,\Delta,Z)$. Assume that we are given $\mathcal{R}$, a collection of
$(\mu_{i},\Delta_{i},Z_{i})$ and a choice of signs $j_{i}$ such that
$S_{j_{i}}(\mu_{i},\Delta_{i},Z_{i})$ are pairwise disjoint. Then, we can form
by the same method as above an integral affine manifold with nodal
singularities on the edges. At each vertex we get the singular integral affine
structure of the _positive vertex_. Accordingly we can form a symplectic
manifold $M_{\mathcal{R}}$ by gluing a neighborhood of the singular fibers of
the Gross-fibration corresponding to $\Delta$. See [9] for a more detailed
discussion of the local models. We expect the methods of this paper to extend
to allow the proof of Theorem 1.5 and moreover to compute the local models.

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