Lagrangian boundary is close enough to $D_{\Sigma,j}\cap D_{\Sigma,j+1}$, then
the only Maslov $2$ disks in its small neighborhood are basic broken disks.

Thus $W$ can be decomposed as $W=\overline{W}_{j}+W_{+}$ with

$\begin{array}[]{lcl}\displaystyle\overline{W}_{j}&=&T^{\lambda_{j}}z^{\partial\beta_{\nu_{j}}}+T^{\lambda_{j+1}}z^{\partial\beta_{\nu_{j+1}}}+\displaystyle\sum_{\beta\in\mathcal{A}_{j}}N_{\beta}T^{\delta(\beta)}z^{\partial\beta}\\\
&=&T^{\lambda_{j}}z^{\partial\beta_{\nu_{j}}}+T^{\lambda_{j+1}}z^{\partial\beta_{\nu_{j+1}}}+{\displaystyle\sum_{k=1}^{l}}{l\choose
k}T^{\lambda_{j}+k\lambda_{j+1}}z^{\nu_{j}+k\nu_{j+1}}\end{array}$ (4.21)

when we set the sizes of exceptional divisors to be zero. Here, $l$ is the
number of blowup points in the interior of $D_{\Sigma,j+1}$, so
$\mathcal{A}_{j}=\bigl{\\{}\,\beta_{\nu_{j}}+k\beta_{\nu_{j+1}}:1\leq k\leq
l\,\bigr{\\}}$. From the above discussion, the valuation of each monomial in
$W_{+}$ is bigger than any of those in $\overline{W}_{j}$.

In what follows, we will work with $j=1$ for notational simplicity, where we
further assume $\nu_{1}=(1,0)$, $\nu_{2}=(0,1)$, and
$\lambda_{1}=\lambda_{2}=0$ by some affine coordinate change as before.
However we set the sizes of the exceptional divisors to be generic from now
on. Thus, using the energy formula (4.3), we have

$\displaystyle\begin{split}\overline{W}_{1}=z_{1}+z_{2}&+(T^{-\epsilon_{1}}+T^{-\epsilon_{2}}+\cdots+T^{-\epsilon_{l}})z_{1}z_{2}\\\
&+(T^{-\epsilon_{1}-\epsilon_{2}}+T^{-\epsilon_{1}-\epsilon_{3}}+\cdots+T^{-\epsilon_{l-1}-\epsilon_{l}})z_{1}z_{2}^{2}\\\
&+\cdots+T^{-\epsilon_{1}-\epsilon_{2}-\cdots-\epsilon_{l}}z_{1}z_{2}^{l}\end{split}$
(4.22)

which goes back to (4.21) when $\epsilon_{j}=0$ for all $j$.

Observe that

$\displaystyle\frac{\partial}{\partial z_{1}}\overline{W}_{1}$
$\displaystyle=1+(T^{\epsilon_{1}}+\cdots+T^{\epsilon_{l}})z_{2}+(T^{-\epsilon_{1}}T^{-\epsilon_{2}}+\cdots+T^{-\epsilon_{l-1}}T^{-\epsilon_{-l}})z_{2}^{2}+\cdots+(T^{-\epsilon_{1}}\cdots
T^{\epsilon_{l}})z_{2}^{l}$
$\displaystyle=(1+T^{-\epsilon_{1}}z_{2})(1+T^{-\epsilon_{2}}z_{2})\cdots(1+T^{-\epsilon_{l}}z_{2}).$

Solving $\frac{\partial}{\partial z_{1}}\overline{W}_{1}=0$, we obtain

$z_{2}=-T^{\epsilon_{i}}\quad\text{for }\,\,1\leq i\leq l.$

Therefore, as long as $\epsilon_{j}$’s are generic, $\overline{W}_{1}$ has
$l$-many distinct critical points. In this case, one can check that
$z_{1}=(-1)^{l}T^{\epsilon_{i}+\sum_{j<i}\epsilon_{j}-\epsilon_{i}}$ modulo
higher order terms. See 5.2 for the detailed calculation and some related
discussion. Note that these points are all geometric. For example, when $l=3$,
we have three solutions with valuations

$\mathit{val}(\alpha_{1})=(\epsilon_{1},\epsilon_{1}),\quad\mathit{val}(\alpha_{2})=(\epsilon_{1},\epsilon_{2}),\quad\mathit{val}(\alpha_{3})=(\epsilon_{1}+\epsilon_{2}-\epsilon_{3},\epsilon_{3}),$

where $\epsilon_{1}>\epsilon_{2}>\epsilon_{3}>0$. In fact, the valuation of
the critical point $(z_{1},z_{2})$ lies inside $R_{0}$ (Lemma 2.5) if every
blowup points in $D_{\Sigma,i}$ locate close enough to the corner
$D_{\Sigma,i}\cap D_{\Sigma,i-1}$ for each $i$ (observe that the valuation
does not depend on the location of the blowup center).

On the other hand, applying Kushnirenko’s theorem (Theorem 3.2), we find that
$\rvert\mathrm{Crit}(\overline{W}_{1})\lvert=2V(\Delta_{\overline{W}_{1}})=l$.
This implies that each of the above $l$ distinct points has multiplicity $1$,
i.e., they are non-degenerate. Every such critical point $\alpha_{j}$ of
$\overline{W}_{1}$ can be successfully extended to a geometric critical point
of $W$ using Lemma 3.8 thanks to non-degeneracy of its associated minimal
energy terms.

Hence for each boundary divisor of $X$ containing $l$ points in the blowup
center, we have successfully identified $l$ distinct Morse (non-toric)
critical points for $W$, all located on the left-hand corner of the respective
divisors. Since we have worked with all possible minimal energy terms near
each corner of the moment polytope, Lemma 3.8 guarantees that no new geometric
(non-toric) critical points arise when patching up local models. This proves
(a).

We next show that the non-toric critical points arising from newly added first
order terms do not interfere with any of the previous toric critical points
from the Hori-Vafa part, essential reason being that non-toric blowup are done
in relatively small scale. To see this, observe that geometric critical points
of $W$ other than the newly added ones after blowup are all toric critical
points by Theorem 4.2, i.e., their energy minimizers are basic disks (we have
exhausted all possible non-toric critical points of $W$ above).

We need to show that these toric critical points are in one-to-one
correspondence with the (geometric) critical points of $W_{\Sigma}$. The
argument is completely parallel to the part (i) in the proof of Proposition
4.12. Namely,

  * (a)

when $W_{\Sigma}^{HV}$ is locally convenient, the valuation of a toric
critical point of $W$ sits at a vertex of $\mathrm{Trop}(W_{\Sigma}^{HV})$
which establishes one-to-one correspondence between geometric critical points
of $W_{\Sigma}$ and toric critical points of $W$ (since they are both
determined by inductively solving the same leading order terms at each vertex
of $\mathrm{Trop}(W_{\Sigma}^{HV})$).

  * (b)

If $W_{\Sigma}^{HV}$ is not locally convenient, it fails to be so on the
unique edge $e$ (4.20) over which we pick up four nongenerate critical points
of $W$ by Proposition 4.14. The other critical points can be handled in the
same way as in (a).

So far, we have worked with the fixed expression of $W$ that is valid on
$R_{0}$ (given in Lemma 2.5). As the analytic continuation does not completely
cover the complement of $R_{0}$ (it misses some codimension $1$ region, see
Remark 2.3), we should additionally prove that no critical point occur
outsider $R_{0}$. This will be done in Lemma 4.17 below, which finishes the
proof. ∎

######  Lemma 4.17.

If all the blowup points on $D_{\Sigma,i}$ are close enough to the corner
$D_{\Sigma,i}\cap D_{\Sigma,i-1}$ for every $i$, then the glued LG potential
(regardless of the choice of a chamber) does not have geometric critical
points outside $R_{0}$.

###### Proof.

From the discussion in 4.3, we see that the valuations of toric critical
points of $W$ only depend on $W_{\Sigma}^{HV}$, and in particular there are
only finitely many. Since $R_{0}$ exhausts the whole moment polytope as points
in the blowup centers approach the nearby corners, it contains all toric
critical point at some point.

On the other hand, Theorem 4.5 justifies that the local calculation in the
proof of Theorem 4.16 (see (4.22)) is still valid on any chamber, except that
we need to remove terms containing $\epsilon_{i}$ as we pass to the left side
of the corresponding wall (speaking in the setting of the proof of Theorem
4.16). In particular, if we approach close enough to the boundary of the
moment polytope, we are only left with $z_{1}+z_{2}$ in the local expression
of $W$ which does not admit any critical point. ∎

The following corollary is straightforward.

######  Corollary 4.18.

In the situation of (b) Theorem 4.16, the number of critical of $W$ agrees
with the rank of the cohomology of $X$.

###### Proof.

Corollary 4.13 tells us that $W_{\Sigma}$ has as many geometric critical
points as the rank of $H^{\ast}(X_{\Sigma})$. This is based on the induction
on the number of (toric) blowups, where the base case of the induction is
covered by Lemma 4.9 and Proposition 4.14. Then Theorem 4.16 implies that the
number of geometric critical points of $W$ equals the rank of cohomology of
$X$ since they are both increased by the number of (non-toric) blowups from
$X_{\Sigma}$ to $X$. ∎

We conclude that performing non-toric blowups (with generic $\epsilon_{i}$)
results in the attachment of unimodal 2-cells to the Newton polytope of
$W_{\Sigma}$, which translates to a “splitting” of an unbounded branch of the
tropicalization $\mathrm{Trop}(W_{\Sigma})$.

$\mathrm{Trop}(W_{\Sigma})$ Figure 12. Branching of
$\mathrm{Trop}(W_{\Sigma})$ locally.

In particular, performing non-toric blowups leaves $\mathrm{Trop}(W_{\Sigma})$
unchanged but one branch, specifically the branch representing the corner near
the blowup center. The unique new geometric critical point is supported at the
point where the branch splits, whereas the non-geometric critical points that
has been removed precisely corresponds to the vertices of
$\mathrm{Trop}(W_{\mathfrak{min}})$ arising from the intersection between the
resulting split branches and other branches.

######  Example 4.19.

Let $(X_{\Sigma},D_{\Sigma}=\Sigma_{i=1}^{5}D_{\Sigma,i})$ be the semi-Fano
surface obtained by blowing up $\mathbb{P}^{2}$ twice as illustrated with the
dotted lines in Figure 13(a). The Hori-Vafa part of the potential is given by

$W^{HV}_{\Sigma}=z_{1}+z_{2}+T^{a}\frac{1}{z_{1}z_{2}}+T^{b}\frac{1}{z_{2}}+T^{c}\frac{z_{1}}{z_{2}}.$

The gray region in Figure 13(b) represents the Newton subdivision
$\mathcal{S}_{W^{HV}_{\Sigma}}$ of $X_{\Sigma}$, which is locally convenient.
Now consider the log Calabi-Yau surface $(X,D)$ obtained by taking three non-
toric blowups on $D_{\Sigma,2}$ and one non-toric blowup on $D_{\Sigma,4}$
(indicated by crosses). Then four unimodal 2-cells, namely $\sigma_{1}$,
$\sigma_{2}$, $\sigma_{3}\in\mathrm{Cone}(\nu_{1},\nu_{2})$ and
$\sigma_{4}\in\mathrm{Cone}(\nu_{3},\nu_{4})$ are attached to
$\mathcal{S}_{W^{HV}_{\Sigma}}$. Note that the resulting surface $X$ is no
longer Fano. The blue line in Figure 13(a) depicts the tropicalization
$\mathrm{Trop}(W_{\mathfrak{min}})$ of the minimal energy terms, and the blue
dotted lines indicate branches of $\mathrm{Trop}(W^{HV}_{\Sigma})$ before
splitting. The critical point information of $W$ can be read off of the Newton
subdivision $\mathcal{S}_{W_{\mathfrak{min}}}$ and the tropicalization
$\mathrm{Trop}(W_{\mathfrak{min}})$ as follows:

1. The three vertices of $\mathrm{Trop}(W_{\mathfrak{min}})$ lying away from the boundary divisors corresponds to the three 2-cells of $\mathcal{S}_{W^{HV}_{\Sigma}}$, each of which supports $1$, $1$, and $3$ toric critical points.

2. The three vertices of $\mathrm{Trop}(W_{\mathfrak{min}})$ near the origin, as well as the vertex near the corner $D_{\Sigma,3}\cap D_{\Sigma,4}$, are dual to the unimodal 2-cells $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$, and $\sigma_{4}$, respectively. Each of these vertices supports a single non-toric geometric critical point.

3. On the other hand, the red vertex above $D_{\Sigma,4}$ lying outside the moment polytope is dual to the 2-cell excluded by Proposition 4.6, running across different cones. The excluded 2-cell is represented with dashed line in Figure 13(b).

$b$$c$$a$$D_{\Sigma,2}$$D_{\Sigma,3}$$D_{\Sigma,4}$

(a) The moment polytope of $X_{\Sigma}$ and the tropicalization of
$W_{\mathfrak{min}}$.

$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\sigma_{4}$$\nu_{1}$$\nu_{2}$$\nu_{3}$$\nu_{4}$$\nu_{5}$

(b) The Newton subdivision $\mathcal{S}_{W_{\mathfrak{min}}}$ of the minimal
terms.

Figure 13. Example 4.19.

###  4.5. Bulk-deformed potential (with toric bulk insertions only in non-
Fano)

We finally return to potential functions with bulk, in particular the bulk-
deformation $W^{\mathfrak{b}}$ of $W$ associated with the bulk parameter
$\mathfrak{b}\in H^{even}(X;\Lambda_{+})$ which are pulled back from some
torus invariant cycles in $X_{\Sigma}$. This restriction is to guarantee the
weakly-unobstructedness (Lemma 2.9), and we can consider more general bulk-
insertions in the semi-Fano case where the weakly-unobstructedness is
automatic by the obvious degree reason.

Following [FOOO09, FOOO11], $W^{\mathfrak{b}}$ is defined as

$\displaystyle
W^{\mathfrak{b}}_{u}(\underline{z})=\sum_{k=0}^{\infty}\,\sum_{\beta}\,\frac{1}{k!}\,\mathfrak{q}_{k}(\beta;\mathfrak{b},\dots,\mathfrak{b})\,T^{\omega(\beta)}\,\underline{z}^{\partial\beta}$

where the sum is taken over $\beta\in\pi_{2}(X,L_{u})$. Here,
$\mathfrak{q}_{k}(\beta;\mathfrak{b},\dots,\mathfrak{b})$ is roughly the count
of holomorphic disks with $k$ interior marked points incident to the ambient
cycle $\mathfrak{b}$ and passing through a generic point of the fiber $L_{u}$.
We refer readers to [FOOO09] or [FOOO11] for more details.

Note that the Maslov index of the class $\beta$ can be strictly bigger than
$2$ when it admits the insertion from the point class ($\in H^{4}(X)$), and
hence its contribution to $W^{\mathfrak{b}}$ has relatively high energy in
general. This is also the case for degree-2 bulk insertion $\mathfrak{b}$, as
$\mathfrak{b}$ itself carries a positive energy in its
$\Lambda_{+}$-coefficient. In fact, one can prove that such extra terms of
$W^{\mathfrak{b}}$ arising from nontrivial $\mathfrak{b}$ do not affect the
leading term potential at any geometric critical points. To see this, let us
denote by $W^{\mathfrak{b}}_{\mathfrak{min}}$ the collection of terms in
$W^{\mathfrak{b}}$ induced from the disks that can possibly have minimal
energy at some geometric critical point of $W^{\mathfrak{b}}$.

######  Lemma 4.20.

For generic parameters,
$W^{\mathfrak{b}}_{\mathfrak{min}}=W_{\mathfrak{min}}$.

###### Proof.

Let $\beta$ be any holomorphic disk with
$\mathfrak{q}_{k}(\beta;\mathfrak{b},\dots,\mathfrak{b})\neq 0$, and suppose
that $\partial\beta$ is contained in $\mathrm{Cone}(\nu_{1},\nu_{2})$, where
we assume $\nu_{1}=(1,0)$, $\nu_{2}=(0,1)$ and $\,\lambda_{1}=\lambda_{2}=0$
without loss of generality. Our argument in the proof of Theorem 4.2 remains
valid, showing that disks in the class $\beta$ can be energy minimizing at a
geometric critical point only if $\delta(\beta)=0$ (see (4.5)). As before,
this forces $\pi_{\ast}(\beta)$ to be a linear combination of
$\beta_{\nu_{1}}$ and $\beta_{\nu_{2}}$. Previously, we concluded at this
point that $\beta$ are either s basic or a basic broken disk due to Maslov 2
constraint. Since we now allow bulk-deformations via the point cycle
$D_{\Sigma,1}\cap D_{\Sigma,2}$, there is the third possibility of $\beta$
being the proper transform of a higher Maslov disk in $X_{\Sigma}$, hitting
the corner $D_{\Sigma,1}\cap D_{\Sigma,2}$ (included in $\mathfrak{b}$)
multiple times. However these disks should have strictly bigger energies than
the proper transforms of $\beta_{\nu_{1}}$ and $\beta_{\nu_{2}}$ at any $u\in
B$, and hence cannot serve as a energy minimizing disk anywhere. ∎

Thus we have an analogue of Corollary 4.18 for bulk-deformed potentials.

######  Corollary 4.21.

Let $(X_{\Sigma},D_{\Sigma})$ be a toric surface, and let $(X,D)$ be the
surface obtained after a sequence of non-toric blowups. Generically, the
number of critical of $W^{\mathfrak{b}}$ agrees with the rank of the
cohomology of $X$. Here, $\mathfrak{b}\in H^{even}(X;\Lambda)_{+}$ is a bulk
parameter (which requires to be the lift of a torus-invariant cycle in
$X_{\Sigma}$ when $D$ contains a sphere with a negative Chern-number).

We summarize our discussion so far in the frame of closed-string mirror
symmetry for algebraic surfaces $(X,D)$ obtained by taking (a sequence of)
non-toric blowup on a toric surface $(X_{\Sigma},D_{\Sigma})$. We consider the
quantum cohomology ring $QH^{\ast}_{\mathfrak{b}}(X)=H^{\ast}(X;\Lambda)$
equipped with deformed cup product $\star_{\mathfrak{b}}$. The coefficient of
$z$ in $x\star_{\mathfrak{b}}y$ is given by the Gromov-Witten invariants
$\sum_{n}\sum_{A\in
H_{2}(X)}GW_{0,n+3}^{A}(x,y,z,\mathfrak{b},\mathfrak{b},\cdots,\mathfrak{b})$
which roughly counts the number of holomorphic spheres passing through $x$,
$y$, $z$ and arbitrary many $\mathfrak{b}$’s. For more details, see, for e.g.,
[MS12, Chapter 11].

On the mirror side, we take the Jacobian ideal ring of the potential function
$W$. Lemma 4.17 allows us to regard $W^{\mathfrak{b}}$ as a function defined
on $\mathit{val}^{-1}(R_{0})$, where $R_{0}$ is sufficiently close to the
entire base $B$ (which approximates the moment polytope of $X_{\Sigma}$). Then
the Jacobian ideal ring $\mathrm{Jac}(W^{\mathfrak{b}})$ of $W^{\mathfrak{b}}$
can be defined as in [FOOO16, Definition 1.3.10], but with the moment polytope
replaced by $R_{0}$. Roughly speaking, the Jacobian ideal ring is the quotient
of the convergent power series ring $\Lambda\ll z_{1}^{\pm 1},z_{2}^{\pm
1}\gg$, consisting of power series that converge with respect to the valuation
$z^{\partial\beta_{\nu_{j}}}\mapsto\lambda_{j}+\langle
u,\partial\beta_{\nu_{j}}\rangle$ for all $j$ and $u\in R_{0}$ or its
associated norm.555Our $\lambda_{j}$ corresponds to $-\lambda_{j}$ in
[FOOO16]. (Recall $z_{1}=z^{\partial\beta_{\nu_{1}}}$ and
$z_{2}=z^{\partial\beta_{\nu_{2}}}$.) Notice that the valuation is nothing but
the areas of basic disks. The ideal in the quotient process is generated by
partial derivatives of $W^{\mathfrak{b}}$, but we may need to take its closure
with respect to $T$-adic topology. We refer readers to [FOOO16] for more
details.

We are now ready to state our main theorem:

######  Theorem 4.22 (Theorem I).

In the situation of Corollary 4.21, we have

$\mathrm{Jac}(W^{\mathfrak{b}})=QH^{*}_{\mathfrak{b}}(X).$

###### Proof.

Recall from Lemma 4.9 that for generic parameters, the Newton subdivision is a
locally convenient triangulation, apart from the cell(s) containing the origin
(in their closure). For each convenient triangle $\sigma$, direct computation
on the leading terms yields the desired number ($=\mathrm{Vol}(\sigma)$) of
non-degenerate critical points (Remark 4.10), whereas Proposition 4.14 covers
the case when there exists a non-convenient edge. That is, for generic
parameters, the bulk-deformed potential $W^{\mathfrak{b}}$ is Morse. Hence, by
[FOOO16, Proposition 1.3.16], the left hand side is isomorphic to the semi-
simple ring $\prod_{\alpha\in crit(W_{R_{0}})}\Lambda$ where $\alpha$ runs
over the set of critical points of $W$ over $R_{0}$ which equals the set of
all geometric critical points of $W$ due to Lemma 4.17. On the other hand, the
right hand side is also known to be semi-simple by [Bay04]. We see that both
sides are semi-simple and have the same rank by Corollary 4.21, which
completes the proof. ∎

###  4.6. Continuum of critical points

Even though we can pin-down the precise location of all (geometric) critical
points of $W^{\mathfrak{b}}$ for generic $\omega$, for some special $\omega$,
the bulk deformed potential might have a continuum of critical points, as
observed in [FOOO11]. Interestingly, this phenomena occurs when $X$ with a
non-convenient Newton polytope is equipped with a “locally monotone”
symplectic form. The following example analyzes the critical behavior of
$\mathbb{F}_{k}$ blown up once at the origin.

$a$$b$$c$$\frac{b}{2}$$c$$\frac{b}{2}$$P$$\frac{1}{2}a-\frac{k}{4}b$ (a)
Blowup of $\mathbb{F}_{k}$ when $c<\frac{b}{2}$. The red point
$(\frac{1}{2}a-\frac{k}{4}b,\frac{b}{2})$ supports four toric critical points,
and the green point $(c,c)$ supports a single toric critical point.
$a$$b$$c=\frac{b}{2}$$c=\frac{b}{2}$$\frac{1}{3}(a-\frac{k}{2}b+\frac{b}{2})$$P$$Q$
(b) Blowup of $\mathbb{F}_{k}$ when $c=\frac{b}{2}$. The green point at
$(\frac{1}{3}(a-\frac{k}{2}b+\frac{b}{2}),\frac{b}{2})$ supports three
critical points, and the red point at
$(\frac{1}{2}a-\frac{k}{4}b,\frac{b}{2})$ supports two.
$a$$b$$c=\frac{b}{2}$$P$$Q$ (c) The critical behavior of the bulk-deformed
potential $W^{\mathfrak{b}}$ when $c=\frac{b}{2}$.

Figure 14.

Let $X$ be the blowup of $\mathbb{F}_{k}$ at the origin (Figure 14(a)). The
Hori-Vafa part of the potential is given by

$W^{HV}_{\Sigma}=z_{1}+z_{2}+\frac{T^{b}}{z_{2}}+\frac{T^{a}}{z_{1}z_{2}^{k}}+T^{-c}z_{1}z_{2}.$

First note that if we keep our assumption $c<\frac{b}{2}$ from (4.12), then
the three terms $z_{1}$, $z_{2}$, and $T^{-c}z_{1}z_{2}$ gives rise to a new
toric critical point $\alpha$ with $\mathit{val}(\alpha)=(c,c)$. We observe
the change in the critical behavior of $W_{\Sigma}$ and that of the bulk
deformed potential $W^{\mathfrak{b}}$ when $c=\frac{b}{2}$.

Let $c=\frac{b}{2}$. We follow the steps of the proof of Proposition 4.14
analogously. Normalizing with respect to $z_{2}=T^{b/2}\underline{z_{2}}$, we
have

$W^{HV}_{\Sigma}=z_{1}+T^{b/2}\Big{(}\underline{z_{2}}+\frac{1}{\underline{z_{2}}}\Big{)}+T^{a-\frac{k}{2}b}\frac{1}{z_{1}\underline{z_{2}}^{k}}+T^{-c+\frac{b}{2}}z_{1}\underline{z_{2}},$

and substituting $\underline{z_{2}}=-1+z_{2}^{+}$ gives

$\displaystyle W^{HV}_{\Sigma}$
$\displaystyle=z_{1}+T^{\frac{b}{2}}\Big{(}-({z_{2}}^{+})^{2}-{(z_{2}^{+})}^{3}-\cdots\Big{)}\pm\frac{T^{a-\frac{k}{2}b}}{z_{1}}\Big{(}1+z_{2}^{+}+{(z_{2}^{+})}^{2}+\cdots\Big{)}^{k}+z_{1}(-1+z_{2}^{+})$
$\displaystyle=-T^{\frac{b}{2}}{(z_{2}^{+})}^{2}\pm\frac{T^{a-\frac{k}{2}b}}{z_{1}}+z_{1}z_{2}^{+}+\cdots.$

where the two terms $z_{1}$ and $-T^{-c+\frac{b}{2}}z_{1}$ have canceled out
one another. The leading terms become convenient, hence by Proposition 3.9
(after normalizing), we conclude that there are three critical points located
at $Q=(\frac{1}{3}(a-\frac{k}{2}b+\frac{b}{2}),\frac{b}{2})\in e$, and two
critical points at $P=(\frac{1}{2}a-\frac{k}{4}b,\frac{1}{2}b)$. See Figure
14(b).

Now consider (the $z_{2}^{+}$ expansion of) the bulk-deformed potential:

$W^{\mathfrak{b}}=-T^{\frac{b}{2}}{(z_{2}^{+})}^{2}\pm\frac{T^{a-\frac{k}{2}b}}{z_{1}}-z_{1}z_{2}^{+}+T^{\epsilon}z_{1}+\cdots$

Notice that $T^{\epsilon}z_{1}$ can now become an energy minimizer. If
$0<\epsilon<\frac{1}{3}(a-\frac{k}{2}b-b)$, one can show by following the
steps of the proof of Proposition 4.14, that $T^{\epsilon}z_{1}$ indeed serves
as an energy minimizer666To be precise, one needs to expand the potential to
the third order so that the coefficients satisfies the conditions in Lemma
A.1.. We obtain two critical points that are located at

$\Big{(}\frac{1}{2}(a-\frac{k}{2}b-\epsilon),\,\frac{1}{4}(a-\frac{k}{2}b-b+\epsilon)\Big{)},$

and a single critical point at

$(\epsilon+\frac{b}{2},\epsilon)$

as shown in Figure 14(c). The former two critical points’ change in position
(as $\epsilon$ varies) is depicted in a blue arrow, whereas the latter is
depicted in a red arrow. We omit the details.

##  5\. Homological Mirror Symmetry

We now discuss open-string (homological) mirror symmetry within our geometric
context. For a surface $X$ obtained by taking a non-toric blowup of a toric
surface $X_{\Sigma}$, let us consider the Fukaya category $\mathcal{F}_{0}(X)$
of $X$ consisting the fibers of the SYZ fibration on $X$. More precisely,
these Lagrangian torus fibers are equipped with $\Lambda_{U}$ flat
connections, and their Floer cohomologies are well-defined as long as their
curvature $m_{0}(1)$’s coincide (recall they are all unobstructed by Lemma
2.9). We allow $\mathcal{F}_{0}$ to contain nodal fibers with weak-bounding
cochains from immersed generators if $X$ is semi-Fano, in which case we can
guarantee their weakly-unobstructedness (see Remark 2.10).

Nontrivial objects in $\mathcal{F}_{0}$ can be detected from singularity
information of $W$. Recall from 2.1 that each point in the domain $\check{Y}$
of the mirror LG model $W:\check{Y}\to\Lambda$ represents a Lagrangian brane
supported over a SYZ fiber. An object of $\mathcal{F}_{0}(X)$ corresponding to
a point $p\in\check{Y}$ is nontrivial if and only if $p$ is a critical point
of $W$ (see 5.1 below). On the other hand, we consider the category of
singularities for $W$ on the mirror side. Assuming generic parameters, all of
its (geometric) critical points are non-degenerate, and hence the category
decomposes into skyscraper sheaves supported at critical points of $W$, where
the morphism space between any two different skyscraper sheaves is trivial.

######  Remark 5.1.

Although we focus on $W$ to keep the exposition simple, the entire argument in
this section has a straightforward generalization to the bulk-deformed
potential $W^{\mathfrak{b}}$.

We begin by finding nontrivial objects (generators) of $\mathcal{F}_{0}(X)$
associated with critical points of $W$.

###  5.1. Critical points of $W$ and the Floer cohomology of the associated
Lagrangian

Suppose $p=(z_{1},z_{2})$ is a critical point of $W$. By taking the (unique)
factorization
$p=(T^{\mathit{val}(z_{1})}\underline{z}_{1},T^{\mathit{val}(z_{2})}\underline{z}_{2})$
for $(\underline{z}_{1},\underline{z}_{2})\in\Lambda_{U}^{2}$, we can assign a
Lagrangian torus $L_{u}$ fiber sitting over
$u:=(\mathit{val}(z_{1}),\mathit{val}(z_{2}))$ equipped with the bounding
cochain $b=x_{1}d\theta_{1}+x_{2}d\theta_{2}\in
H^{1}(L_{u};\Lambda_{+})\cong(\Lambda_{+})^{2}$ such that
$e^{b}=(\underline{z}_{1},\underline{z}_{2})$. In view of (2.5), $p$ being a
critical point is equivalent to $b$ being a critical point of $W_{u}(b)$,
where

$W_{u}(b)\cdot[L_{u}]=m_{0}(1)+m_{1}(b)+m_{2}(b,b)+\cdots.$ (5.1)

Differentiating (5.1), we see that the differential $m_{1}^{b,b}$ on
$\hom((L_{u},b),(L_{u},b))\cong H^{\ast}(L_{u},\Lambda)$ vanishes on
$H^{1}(L_{u},\Lambda)$ as it acts trivially on the generators
$d\theta_{1},d\theta_{2}$. Moreover, differentiating (5.1) once more, we see
that $m_{2}^{b,b,b}$ on these generators satisfies

$m_{2}^{b,b,b}(d\theta_{1},d\theta_{2})+m_{2}^{b,b,b}(d\theta_{2},d\theta_{1})=\partial_{12}W\cdot[L_{u}].$

Notice that $d\theta_{1}\wedge d\theta_{2}$ cannot be an image of
$m_{1}^{b,b}$ which vanishes on $d\theta_{1}$ and $d\theta_{2}$, and
$m_{2}^{b,b,b}(d\theta_{1},d\theta_{2})=d\theta_{1}\wedge
d\theta_{2}+c\cdot[L_{u}]$ for some $c\in\Lambda$ where $[L_{u}]$ serves as
the unit. Consequently, the Lagrangian brane $(L_{u},b)$, considered as an
object of $\mathcal{F}_{0}(X)$, has its endomorphism algebra quasi-isomorphic
to the Clifford algebra with respect to the quadratic form $Hess_{b}(W_{u})$
(the Hessian of $W$ at $b$), which equals $Hess_{p}(W)$. Observe that this
calculation is still valid even if there exist nontrivial contributions of
stable disks with negative Maslov indices, and we use crucially the fact that
$\dim L_{u}$ is $2$. Thus we have the following:

######  Proposition 5.2.

For an object
$L(p):=(L_{(\underline{z}_{1},\underline{z}_{2})},b=(x_{1},x_{2}))$
corresponding to the critical point
$p=(\underline{z}_{1}e^{x_{1}},\underline{z}_{2}e^{x_{2}})$ in $X$, its
endomorphism space in $\mathcal{F}_{0}(X)$ is quasi-isomorphic to the Clifford
algebra with respect to $Hess_{p}(W)$.

Moreover, for two distinct critical points $p_{1}$ and $p_{2}$ of $W$,
$\hom_{\mathcal{F}_{0}(X)}(L(p_{1}),L(p_{2}))$ is trivial.

###### Proof.

It only remains to prove the last assertion, so consider $p_{1}\neq p_{2}$
with $\mathit{val}(p_{1})=\mathit{val}(p_{2})$ and $W(p_{1})=W(p_{2})$ (as
otherwise, $\hom_{\mathcal{F}_{0}(X)}(L(p_{1}),L(p_{2}))$ would be
automatically trivial, or not even defined). The situation boils down to
computing the cohomology of

$m_{1}^{b_{1},b_{2}}:CF((L_{u},b_{1}),(L_{u},b_{2}))\to
CF((L_{u},b_{1}),(L_{u},b_{2}))$

for two different weak bounding cochains $b_{1}$ and $b_{2}$ in
$H^{1}(L_{u};\Lambda_{+})$ such that $W_{u}(b_{1})=W_{u}(b_{2})$. We use the
Morse-Bott model for $CF((L_{u},b_{1}),(L_{u},b_{2})$, in that it is still
generated by $[L_{u}]$ and $d\theta_{1}\wedge d\theta_{2}$ in even degree, and
$d\theta_{1},d\theta_{2}$ in odd degree.

Observe first that $m_{1}^{b_{1},b_{2}}([L_{u}])=b_{1}-b_{2}\neq 0$.
Therefore, modulo the image, the odd degree part of
$HF((L_{u},b_{1}),(L_{u},b_{2})$ is generated by one of $d\theta_{i}$, say
$d\theta_{1}$, or it vanishes. However, due to the classical part (the de Rham
differential) of $m_{1}^{b_{1},b_{2}}$,

$m_{1}^{b_{1},b_{2}}(d\theta_{1})=Cd\theta_{1}\wedge d\theta_{2}+E[L_{u}]\neq
0$ (5.2)

for some constants $C$ and $E$ with $C\neq 0$ ($C$ would cancel out if
$b_{1}=b_{2}$), which gives a contradiction.

On the other hand, (5.2) implies that the even degree part of
$HF((L_{u},b_{1}),(L_{u},b_{2})$ should be generated by $[L_{u}]$ unless it is
trivial. Since $[L_{u}]$ is not a cycle, the even degree component should also
vanish. ∎

###  5.2. Nontrivial objects in $\mathcal{F}_{0}(X)$ from non-toric critical
points

We look into the nontrivial object of $\mathcal{F}_{0}(X)$ arising from the
blowup process to see how the category changes under the blowup. Recall that
each blowup (at a non-torus-fixed point) introduces a critical point to $W$
which we call a non-toric critical point. We are interested in the associated
Lagrangian torus fiber (coupled with a suitable bounding cochain), especially
in its behavior when the position of a blowup point changes.

We first find the precise location of such a critical point on the SYZ base
(i.e., the valuation of a non-toric critical point). We also show that
critical values for non-toric critical points of $W$ are mutually distinct.
Clearly, the leading order terms $\overline{W}$ of $W$ only matter for our
purpose.

As before, we look at the local expression of $W$ around exceptional divisors
arising from blowing up at points in $D_{\Sigma,2}$ (and hence, near
$D_{\Sigma,1}\cap D_{\Sigma,2}$ by our earlier choice) in $X_{\Sigma}$.
Without loss of generality, we assume $\nu_{1}=(1,0)$, $\nu_{2}=(0,1)$, and
$\lambda_{1}=\lambda_{2}=0$. In the moment polytope $\Delta_{\Sigma}$,
critical points of our interest lie on the $x$-axis, close to the origin. We
stick to this setting throughout 5.2 and 5.3.

Suppose that there are $k$ distinct such blowup points in $D_{\Sigma,2}$, and
that the sizes of exceptional divisors are $\epsilon_{1},\cdots,\epsilon_{k}$
in increasing order. The leading terms $\overline{W}$ of $W$ locally near the
origin of $\Delta_{\Sigma}$ are given as in (4.22). We have seen in 4.4 that
there are $k$-many critical points of $\overline{W}$ all of which extend to
critical points of $W$ via energy induction. Let
$\alpha_{1},\cdots,\alpha_{k}$ denote these critical points (of $W$) where
$\alpha_{i}$ corresponds to the exceptional divisor of size $\epsilon_{i}$.

Alternatively, we can express $\overline{W}$ as

$\overline{W}=z_{2}+z_{1}\prod_{i=1}^{k}{\left(1+T^{-\epsilon_{i}}z_{2}\right)}.$

Taking the partial derivative with respect to $z_{2}$, we obtain

$\frac{\partial}{\partial
z_{2}}\overline{W}=1+z_{1}\sum_{i=1}^{k}T^{-\epsilon_{i}}\prod_{j\neq
i}{\left(1+T^{-\epsilon_{j}}z_{2}\right)}.$

Substituting $z_{2}=-T^{\epsilon_{i}}$ and solving for $z_{1}$, we have

$z_{1}=-T^{\epsilon_{i}}\prod_{j\neq
i}\frac{1}{1-T^{\epsilon_{i}-\epsilon_{j}}}.$

We can expand each term using Taylor series. If $j<i$, then
$\epsilon_{i}-\epsilon_{j}<0$, and we have

$\frac{1}{1-T^{\epsilon_{i}-\epsilon_{j}}}=-\frac{T^{\epsilon_{j}-\epsilon_{i}}}{1-T^{\epsilon_{j}-\epsilon_{i}}}=-T^{\epsilon_{j}-\epsilon_{i}}\left(1+T^{\epsilon_{j}-\epsilon_{i}}+T^{2(\epsilon_{j}-\epsilon_{i})}+\cdots\right).$

Otherwise, $\frac{1}{1-T^{\epsilon_{i}-\epsilon_{j}}}$ is of valuation $0$.
Therefore,

$z_{1}=(-1)^{k}T^{\epsilon_{i}}T^{\sum_{j<i}\epsilon_{j}-\epsilon_{i}}+T^{\text{h.o.t.}}$

and we thus have

$\mathit{val}(\alpha_{i})=\left(\epsilon_{i}+\sum_{j=1}^{i-1}(\epsilon_{j}-\epsilon_{i}),\,\,\epsilon_{i}\right).$

Due to our choice of $\lambda_{1}=\lambda_{2}=0$, the moment polytope
$\Delta_{\Sigma}$ is located in the 1st quadrant of $\mathbb{R}^{2}$, and our
local calculation is performed near the origin. In particular, $\alpha_{i}$
sits in the interior of $\Delta_{\Sigma}$. Moreover, by choosing the locations
of the blowup points sufficiently close to $D_{\Sigma,1}\cap D_{\Sigma,2}$
(which are irrelevant to $\epsilon_{i}$), we can ensure that all
$\alpha_{1},\cdots,\alpha_{k}$ lie within $R_{0}$ where the local expansion
$W=\overline{W}+T^{h.o.t.}$ is valid (see Figure 15). Therefore we obtain
nontrivial objects $L(\alpha_{1}),\cdots,L(\alpha_{k})$ of
$\mathcal{F}_{0}(X)$ supported over SYZ fibers at
$\mathit{val}(\alpha_{1}),\cdots,\mathit{val}(\alpha_{k})$.

Now, let’s proceed to calculate the valuation of the critical value of $W$ at
$\alpha_{i}$:

$\displaystyle\mathit{val}(W\rvert_{\alpha_{i}})$
$\displaystyle=\mathit{val}\left(T^{\epsilon_{i}}+\left(T^{\epsilon_{i}+\sum_{j=1}^{i-1}(\epsilon_{j}-\epsilon_{i})}\right)\prod_{i=1}^{k}{\left(1+\frac{1}{T^{\epsilon_{i}}}T^{\epsilon_{i}}\right)}\right)$
$\displaystyle=\mathrm{min}\\{\epsilon_{i},\,\epsilon_{i}+\sum_{j=1}^{i-1}(\epsilon_{j}-\epsilon_{i})\\}=\epsilon_{i}.$

Therefore, as long as $\epsilon_{i}$’s are generic, the critical values (of
non-toric critical points) are all distinct.

Moreover, Lemma 4.9 tells us that generically, toric critical points also have
distinct eigenvalues. For the case where the central cell is non-convenient,
one can check by direct computation that the critical values are indeed
distinct for each critical point (4.19) found in the proof of 4.14. The
following is a direct consequence of the discussion so far combined with
Proposition 5.2.

######  Proposition 5.3.

Let $X$ be obtained after a sequence of toric/non-toric blowups on
$X_{\Sigma}$, where we assume that the non-toric blowup centers are close
enough to the corner of the moment polytope.777i.e. blowup points in the
interior of the toric divisor $D_{j}$ are close enough to $D_{\Sigma,j}\cap
D_{\Sigma,j-1}$ for each $j$ according to our earlier convention. Then each
toric/non-toric blowup gives rise to a Floer-nontrivial SYZ fiber in $X$, and
hence a nontrivial object in $\mathcal{F}_{0}(X)$. Its endomorphism algebra is
isomorphic to the Clifford algebra with respect to the Hessian of $W$ at its
corresponding critical point.

For generic parameter, all such Floer-nontrivial SYZ fibers live in mutually
distinct component of $\mathcal{F}_{0}(X)$ when decomposed into potential
values, and hence they do not interact with each other.

When $k=1$, then the unique critical point created by this blowup is located
at $(\epsilon,\epsilon)$ (for $\epsilon=\epsilon_{1}$ in the notation above).
This precisely coincides with the effect of the toric blowup at
$D_{\Sigma,1}\cap D_{\Sigma,2}$ with the size of the exceptional divisor
$\epsilon$. Indeed, the two (toric and non-toric ones) are equivalent in view
of symplectomorphism constructed in the proof of Lemma 2.9. Apart from
technical details (in achieving transversality for Floer theory), the
associated Lagrangian should coincide with what is called the exceptional
brane in [VWX20] (which was constructed in the local model of the toric
blowup).

$R_{0}$$(\epsilon,\epsilon)$

(a) The non-toric critical point $\alpha\in R_{0}$

$R_{0}$$(\epsilon,\epsilon)$

(b) $\alpha\notin R_{0}$

Figure 15. Local picture of the relative position of the blowup center with
respect to $\epsilon$ and corresponding locations of the non-toric critical
point.

###  5.3. Geometric meaning of critical points lying outside $R_{0}$

Finally, we can remove the assumption on the location of the blowup center in
Proposition 5.3 by including nodal fibers under the semi-Fano assumption on
$X$. It is natural to expect that the singularity information of its LG mirror
is independent of positions of the blowup center in any case, but the only
reason we impose this assumption is to ensure the weakly unobstructedness of
nodal fibers (the argument in the proof of Lemma 2.9 is obviously not valid
for these fibers). We speculate that nodal fibers are weakly unobstrcuted for
all bounding cochains formed by immersed generators beyond the semi-Fano case.

For the sake of simplicity, we assume $k=1$, and the argument easily
generalizes for $k>1$. We work with the same setting as above, but now we
allow the blowup points to be projected onto $(\delta,0)$ in the moment
polytope, where $\delta$ is greater than the blowup size $\epsilon$. The
situation is described in Figure 15(b), whereas previously we worked with the
setting in Figure 15(a) (cf. Proposition 5.3).

Notice that for this choice of the blowup point, the critical point
$(\epsilon,\epsilon)$ falls in the chamber $R_{1}$ on the left side of $R_{0}$
sharing a single vertical wall (with respect to our choice of coordinates). On
the other hand, the leading terms for $W_{R_{1}}$ near the origin is given as

$\overline{W}=z_{1}+z_{2},$

which does not admit any critical points (while being convenient). This
seemingly contradictory phenomenon occurs because analytic-continuation of
$W_{R_{0}}$ over $R_{1}$ cannot cover a certain region in $R_{1}$ due to
singularity of the coordinate change (a cluster transformation) between
$R_{0}$ and $R_{1}$ (see Remark 2.3).

More concretely, let $(z_{1}^{\prime},z_{2}^{\prime})$ and $(z_{1},z_{2})$
denote mirror affine coordinates over $R_{1}$ and $R_{0}$, respectively,
induced by the disk classes $\nu_{1}$ and $\nu_{2}$. Recall from (2.12) that
they are related by

$\begin{array}[]{l}z_{1}=z_{1}^{\prime}(1+T^{-\epsilon}z_{2}^{\prime})\\\
z_{2}=z_{2}^{\prime}\end{array}$

in an infinitesimal neighborhood of the wall between $R_{0}$ and $R_{1}$. The
map gives a diffeomorphism away from $\\{z_{2}^{\prime}=-T^{\epsilon}\\}$.
Besides, if $z_{2}^{\prime}=-T^{\epsilon}+T^{h.o.t.}$, then the transition can
change the valuations of points dramatically, and hence the continuation of
$W_{R_{0}}$ in 2.2.1 cannot cover the region over $R_{1}$ given by
$\\{z_{2}^{\prime}\in T^{\epsilon}(-1+\Lambda_{+})\\}$. Notice that the
critical point $\alpha_{0}=(-T^{\epsilon},-T^{\epsilon})$ of $\overline{W}$
and hence $\alpha=(-T^{\epsilon}+T^{h.o.t.},-T^{\epsilon}+T^{h.o.t.})$ of $W$
precisely lie on this region. For this reason, it is impossible to directly
identify the geometric object associated with $\alpha$.

To remedy, we bring in an additional mirror chart induced by the (Floer)
deformation space of the nodal fiber in $X$ responsible for the wall between
$R_{0}$ and $R_{1}$. Such a technique of extending of the mirror space has
first appeared in [HKL18]. See [BECHL21, 7.1] for its usage in a similar
context. Recall that near the exceptional divisor, the SYZ fibration on $X$ is
locally modeled on the one given in [Aur09]. The nodal fiber $\mathbb{L}$ in
our situation is an immersion of a sphere that has a single transversal self-
intersection. It produces two degree 1 immersed generators, say $U$ and $V$.
One can deform the Fukaya algebra $CF(\mathbb{L},\mathbb{L})$ by the weak
bounding cochain $b=uU+vV$ for $(u,v)\in(\Lambda_{+})^{2}$.

The coordinate transitions between this new $(u,v)$-chart and $R_{i}$ are
given as follows. Consider regular fibers $L_{0}$ and $L_{1}$ over $R_{0}$ and
$R_{1}$, respectively and assume that they are close enough to $\mathbb{L}$.
Their projections to the divisor $D_{2}$ is depicted in Figure 16, where the
origin represents $D_{1}\cap D_{2}$. It will be convenient to introduce a
local model for a neighborhood of the fiber $\mathbb{L}$, which is
$\mathbb{C}^{2}\setminus\\{xy=1\\}$ equipped with a Lagrangian torus fibration
given by $(|x|^{2}-|y|^{2},|xy-1|)$. It models a neighborhood of
$D_{2}\setminus D_{1}\cap D_{2}$ where (a part of) $D_{1}$ is given as
$\\{xy=1\\}$. Notice that the conic fibration $(x,y)\mapsto
xy-1=z\in\mathbb{C}^{\times}$ can be interpreted as the reduction of
$\mathbb{C}^{2}\setminus\\{xy=1\\}$ by the circle action $\rho\mapsto(\rho
x,\rho^{-1}y)$. It is nothing but the pull-back of the toric action generated
by $\nu_{2}\in\Sigma$.

The projections of $L_{i}$ and $\mathbb{L}$ to the divisor $D_{2}$ are
depicted in Figure 16. See [BECHL21, 4.2] for more details (Figure 7 therein
will be helpful to understand the picture). In fact, all fibers are sitting
over the concentric circles in the $z:=xy-1$-plane centered at $0$, and
$\mathbb{L}$ corresponds to the fiber $\\{|x|^{2}-|y|^{2}=0,|xy-1|=1\\}$
projecting to $|z|=1$. We may choose $L_{i}$ to be
$L_{0}:=\\{|x|^{2}-|y|^{2}=0,|z|=1+\epsilon\\}$ and
$L_{1}:=\\{|x|^{2}-|y|^{2}=0,|z|=1-\epsilon\\}$ for some small $\epsilon$.

Figure 16. The projections of SYZ fibers near a nodal fiber

Let $\mathbb{L}^{s}$ be a perturbation of $\mathbb{L}$ obtained by slightly
modifying the base circle $|z|=1$ as in Figure 16. We want to compare two
Lagrangians $(\mathbb{L},uU+vV)$ and
$(L_{0},\nabla^{\underline{z}_{1},\underline{z}_{2}})$ each coupled with a
suitable bounding cochain. More precisely, the latter is equipped with the
flat connection $\nabla^{(\underline{z}_{1},\underline{z}_{2})}$ for
$(\underline{z}_{1},\underline{z}_{2})\in\Lambda_{U}^{2}$. In our picture, the
holonomy along the direction of the circle action is set to be
$\underline{z}_{2}$ and that around a fixed choice of a horizontal lift of
$\\{z=1-\epsilon\\}$ is $\underline{z}_{1}$. It was shown in [HKL18] that
$(\mathbb{L},uU+vV)$ and
$(L_{1},\nabla^{(\underline{z}_{1},\underline{z}_{2})})$ are isomorphic to
each other (as objects of the Fukaya category) if and only if

$T^{\delta(s)}\underline{z}_{1}=v,\quad\underline{z}_{2}=uv-1$ (5.3)

where $\delta(s)>0$ is the area difference between two shaded strips in Figure
16. It is positive due to our particular choice of perturbation
$\mathbb{L}^{s}$. We have some flexibility on $\delta$ by adjusting the size
of the base circle for $\mathbb{L}^{s}$.

For the critical point $\alpha=(z_{1},z_{2})$, we have
$\underline{z}_{2}=-1+T^{h.o.t.}$, and (5.3) can be solved to get
$(u,v)\in\Lambda_{+}^{2}$. Since $\alpha_{0}$ is certainly not a critical
point of the global potential of $W$, $\alpha$ carries nontrivial higher order
terms and, $u,v\neq 0$ for such $\alpha$. Thus $(u,v)$ lies in the region
where the coordinate change (5.3) is valid. The Floer potential for
$\mathbb{L}$ can be obtained by applying this coordinate change to $W$ over
$R_{0}$, and from the discussion so far, it admits a critical point $(u,v)$
which transfers to $\alpha$ under (5.3). This gives us a nontrivial object in
$\mathcal{F}_{0}(X)$ whose underlying Lagrangian in $\mathbb{L}$, although
$\alpha$ itself does not directly represent a geometric object.

Finally we remark that in the earlier situation where $R_{0}$ contains all the
critical points (Figure 15(a)), the above process does not create any new
critical points. In fact, one can check by direct order-by-order calculation
similar to the proof of Proposition 4.14 that the Floer potential (for
$\mathbb{L}$) does not admit a critical point over its associated chart
$(\Lambda_{+})^{2}$ if the location of $\mathbb{L}$ is too close to the
corner.

###  5.4. Equivalence of categories

From the discussion so far, $\mathcal{F}_{0}(X)$ is generated by SYZ fibers
$L(p)$ equipped with suitable bounding cochains corresponding to the critical
points $p$ of $W$. For a regular point $p$ of $W$, $L(p)$ gives a trivial
object since the unit class in $\hom(L(p),L(p))$ is an image of the
differential, and hence is zero in the cohomology. Thus the triangulated
category $D^{b}\mathcal{F}_{0}(X)$ is essentially the derived category of
modules over the direct sum of as many Clifford algebras as the number of
critical points of $W$ (associated with the Hessian of $W$ at those critical
points).

######  Remark 5.4.

By [Aur07], the computation of critical values of $W$ (for non-toric critical
points) tells us that the $\epsilon_{i}$-eigenspace $QH(X)_{\epsilon_{i}}$ of
$QH(X)$ with respect $c_{1}(X)\star-$ is $1$-dimensional. Assuming the
extension of [She16, Corollary 1.12] to non-monotone setting, this would imply
that the the nontrivial object $L(\alpha_{i})$ generates the corresponding
component of the genuine Fukaya category.

On the mirror side, we have a decomposition of the singularity category of $W$
into critical values of $W$, that is
$\oplus_{\lambda}D^{b}_{sing}(W^{-1}(\lambda))$. Equivalently, one may
consider the matrix factorizations of $W-\lambda$. Clearly each factor
$D^{b}_{sing}(W^{-1}(\lambda))$ is generated by skyscraper sheaves at critical
points of $W$ whose corresponding critical values are $\lambda$. Let us
consider one of such skyscraper sheaves $\Bbbk(p)$ in
$\mathcal{D}^{b}_{sing}(W^{-1}(\lambda))$. (More precisely, it is the image of
$\mathcal{O}_{p}$ in the quotient category
$\mathcal{D}^{b}_{sing}(W^{-1}(\lambda_{p}))$ by perfect complexes.) It is
well-known that the endomorphism of $\Bbbk(p)$ is quasi-isomorphic to the
Clifford algebra associated with the Hessian of $W$ at the corresponding
point. See for e.g., [Dyc11] or [Tel20]. Thus we proved:

######  Theorem 5.5.

There is an equivalence between $D^{b}\mathcal{F}_{0}(X)$ and
$\oplus_{\lambda}D^{b}_{sing}(W^{-1}(\lambda))(\cong MF(W))$. Both categories
admit orthogonal decomposition with respect to critical (potential) values
$\lambda$, and each summand in the decomposition is generated by the
skyscraper sheaf at the unique critical point whose value is $\lambda$ or its
corresponding SYZ fiber.

##  Appendix A Estimates for the second-order expansion of W

We prove that $(\partial_{x_{+}}W,\partial_{y_{+}}W)=0$ has a unique solution
by considering in a more abstract setup as follows. $F$ below generalizes
$(\partial_{x_{+}}W,\partial_{y_{+}}W)$ minus the constant terms. Readers are
warned that $a,b,c,d$ in the statement represent some general coefficients,
and are irrelevant to the symplectic sizes of toric divisors in
$\mathbb{F}_{k}$ in Proposition 4.14.

######  Lemma A.1.

Let $F:(\Lambda_{+})^{2}\to(\Lambda_{+})^{2}$ is given in the form of

$F(z_{1},z_{2})=\left(az_{1}+bz_{2}+\sum_{v\neq
e_{1},e_{2}}\lambda_{v}z^{v},cz_{1}+dz_{2}+\sum_{v\neq
e_{1},e_{2}}\eta_{v}z^{v}\right)$

for generic coefficients $(a,b,c,d)\in\Lambda_{+}^{4}$, where the terms
$\lambda_{v}z^{v}$, $\eta_{v}z^{v}$ are higher order terms. We assume the
following :

  1. (1)

$ad-bc\neq 0$,

  2. (2)

$\mathit{val}(a)\leq\mathit{val}(b),\mathit{val}(c)$ and
$\mathit{val}(d)\leq\mathit{val}(b),\mathit{val}(c)$ ($a$ and $d$ are not
assumed to be comparable),

  3. (3)

$\mathit{val}(a)+\mathit{val}(d)<\mathit{val}(b)+\mathit{val}(c)$,

  4. (4)

$\mathit{val}(\lambda_{(i,j)})\geq\mathit{val}(d)\,\,\mbox{for}\,\,j\geq
1,\quad\mbox{and}\,\,\mathit{val}(\eta_{(i,j)})\geq\mathit{val}(a)\,\,\mbox{for}\,\,i\geq
1.$ (A.1)

Then $F=(C_{1},C_{2})$ has a unique solution for any $C_{1}$ and $C_{2}$
satisfying $\mathit{val}(C_{1})>\mathit{val}(a)+\epsilon$ and
$\mathit{val}(C_{2})>\mathit{val}(d)+\epsilon$.

###### Proof.

Since $\lambda_{v}z^{v}$, $\eta_{v}z^{v}$ are higher order terms, we have

$\mathit{val}(\lambda_{v}z^{v})>\min\\{\mathit{val}(az_{1}),\mathit{val}(bz_{2})\\},\quad\mathit{val}(\eta_{v}z^{v})>\min\\{\mathit{val}(cz_{1}),\mathit{val}(dz_{2})\\},$

which implies

$\mathit{val}(\lambda_{v})\geq\min\\{\mathit{val}(a),\mathit{val}(b)\\}=\mathit{val}(a)>0,\quad\mathit{val}(\eta_{v})\geq\min\\{\mathit{val}(c),\mathit{val}(d)\\}=\mathit{val}(d)>0.$
(A.2)

If we consider a restriction of $F$ to $(T^{\epsilon}\Lambda_{0})^{2}$, then
under the given conditions, this restriction maps to
$T^{a+\epsilon}\Lambda_{0}\times T^{d+\epsilon}\Lambda_{0}$.

We claim that $F(z_{1},z_{2})=(C_{1},C_{2})$ has a unique solution for any
$C_{1}$ and $C_{2}$ satisfying $\mathit{val}(C_{1})>\mathit{val}(a)+\epsilon$
and $\mathit{val}(C_{2})>\mathit{val}(d)+\epsilon$. To see this, let us first
make the following linear coordinate change

$w_{1}:=az_{1}+bz_{2},\quad w_{2}:=cz_{1}+dz_{2}$

so that

$\mathit{val}(w_{1})\geq\min\\{\mathit{val}(z_{1})+\mathit{val}(a),\mathit{val}(z_{2})+\mathit{val}(b)\\}\geq\mathit{val}(a)+\epsilon$

and likewise,

$\mathit{val}(w_{2})\geq\mathit{val}(d)+\epsilon.$

Therefore $(w_{1}(z),w_{2}(z))$ defines an isomorphism between
$(T^{\epsilon}\Lambda_{0})^{2}$ and $T^{a+\epsilon}\Lambda_{0}\times
T^{d+\epsilon}\Lambda_{0}$. In fact, the inverse coordinate change is
obviously given as

$z_{1}=\frac{dw_{1}-bw_{2}}{ad-bc},\quad z_{2}=\frac{-cw_{1}+aw_{2}}{ad-bc}$

and one can easily check that

$\mathit{val}\left(\frac{dw_{1}}{ad-
bc}\right),\,\,\mathit{val}\left(\frac{bw_{2}}{ad-
bc}\right),\,\,\mathit{val}\left(\frac{cw_{1}}{ad-
bc}\right),\,\,\mathit{val}\left(\frac{aw_{2}}{ad-bc}\right)\geq\epsilon,$
(A.3)

and hence $F(w_{1},w_{2})$ (in new coordinates) gives a map from
$T^{a+\epsilon}\Lambda_{0}\times T^{d+\epsilon}\Lambda_{0}$ to itself. Notice
that non-degeneracy of the leading order of $F$ is crucially used here in this
coordinate change.

We want to find a solution for $F(w_{1},w_{2})=(C_{1},C_{2})\in
T^{a+\epsilon}\Lambda_{0}\times T^{d+\epsilon}\Lambda_{0}$. To this end, we
introduce another function $G:T^{a+\epsilon}\Lambda_{0}\times
T^{d+\epsilon}\Lambda_{0}\to T^{a+\epsilon}\Lambda_{0}\times
T^{d+\epsilon}\Lambda_{0}$ defined by

$\displaystyle G(w_{1},w_{2})$
$\displaystyle:=F(-w_{1},-w_{2})-(C_{1},C_{2})+(w_{1},w_{2})$
$\displaystyle=-(C_{1},C_{2})+\left(\sum_{v\neq(1,0),(0,1)}\tilde{\lambda}_{v}w^{v},\sum_{v\neq(1,0),(0,1)}\tilde{\eta}_{v}w^{v}\right).$

We show that $G$ is a contraction. Since the constant terms play no role, we
will assume $(C_{1},C_{2})=(0,0)$ from now on. We will only deal with the
first component; the second component can be dealt with similarly. A typical
term appearing in the expansion $\lambda_{v}z^{v}$ in $(w_{1},w_{2})$ is of
the form

$m(w_{1},w_{2}):=\lambda_{v}\left(\frac{dw_{1}}{ad-
bc}\right)^{i}\left(\frac{bw_{2}}{ad-bc}\right)^{j}\left(\frac{cw_{1}}{ad-
bc}\right)^{k}\left(\frac{aw_{2}}{ad-bc}\right)^{l}$

with $i+j+k+l\geq 2$, omitting the binomial coefficients in $\mathbb{Z}$.

For simplicity, we put $\tilde{a}:=\frac{a}{ad-bc}$, $\tilde{b}:=\frac{b}{ad-
bc}$, $\tilde{c}:=\frac{c}{ad-bc}$ and $\tilde{d}:=\frac{d}{ad-bc}$. Then
(A.3) translates to

$\mathit{val}(\tilde{c}w_{1}),\,\,\mathit{val}(\tilde{d}w_{1}),\,\,\mathit{val}(\tilde{a}w_{2}),\,\,\mathit{val}(\tilde{b}w_{2})\geq\epsilon>0.$
(A.4)

Now, for two elements $(w_{1},w_{2})$ and $(w_{1}^{\prime},w_{2}^{\prime})$ in
$T^{a+\epsilon}\Lambda_{0}\times T^{d+\epsilon}\Lambda_{0}$, we have

$\displaystyle m(w_{1},w_{2})-m(w_{1}^{\prime},w_{2}^{\prime})=$
$\displaystyle\lambda_{v}(\tilde{d}w_{1})^{i}(\tilde{c}^{k}w_{1})^{k}(\tilde{b}w_{2})^{j}(\tilde{a}w_{2})^{l}-\lambda_{v}(\tilde{d}w_{1}^{\prime})^{i}(\tilde{c}^{k}w_{1}^{\prime})^{k}(\tilde{b}w_{2}^{\prime})^{j}(\tilde{a}w_{2}^{\prime})^{l}$
$\displaystyle=$
$\displaystyle(w_{2}-w_{2}^{\prime})\sum_{\beta}\lambda_{v}(\tilde{d}w_{1})^{i}(\tilde{c}w_{1})^{k}\cdot\tilde{b}^{j}\tilde{a}^{l}w_{2}^{\beta}(w_{2}^{\prime})^{j+l-1-\beta}$
$\displaystyle+(w_{1}-w_{1}^{\prime})\sum_{\beta}\lambda_{v}\tilde{d}^{i}\tilde{c}^{k}w_{1}^{\alpha}(w_{1}^{\prime})^{i+k-1-\alpha}\cdot(\tilde{b}w_{2}^{\prime})^{j}(\tilde{a}w_{2}^{\prime})^{l}$

Let us estimate the valuation of each of summands in the last equation. We
first find a lower bound for the valuation $\nu_{1}$ of the first summand. If
$j=l=0$, then it is obvious from (A.4) that

$\nu_{1}\geq\mathit{val}(w_{2}-w_{2}^{\prime})+\epsilon.$ (A.5)

Suppose now that $j\geq 1$, while $l$ can be possibly zero. Observe that
$\mathit{val}(\tilde{b}^{i_{1}}\tilde{a}^{i_{2}}w_{2}^{j_{1}}(w_{2}^{\prime})^{j_{2}})\geq\epsilon$
as long as $i_{1}+i_{2}\leq j_{1}+j_{2}$ due to (A.4). On the other hand,
using the given condition on $b$ an (A.2), we have

$\mathit{val}(\lambda_{v}\tilde{b})\geq\mathit{val}(a)+\mathit{val}(\tilde{b})\geq\mathit{val}(a)+\mathit{val}(b)-\mathit{val}(a)-\mathit{val}(d)\geq
0.$

Hence rewriting the first summand as

$(w_{2}-w_{2}^{\prime})\left(\lambda_{v}\tilde{b}\right)(\tilde{d}w_{1})^{i}(\tilde{c}w_{1})^{k}\cdot\tilde{b}^{j-1}\tilde{a}^{l}w_{2}^{\beta}(w_{2}^{\prime})^{j+l-1-\beta},$

we see that the estimate (A.5) holds for $j\geq 1$.

Finally, if $j=0$ but $l\geq 1$, then $m(w_{1},w_{2})$ should come from
$\lambda_{v}z^{v}$ with $z^{v}$ divisible by $z_{2}$. Thus (A.1) tells us that
$\mathit{val}(\lambda_{v})\geq\mathit{val}(d)$ in this case. A similar
argument as in the previous paragraph using
$\mathit{val}(\lambda_{v}\tilde{a})\geq 0$ leads to the same inequality (A.5)
for $\nu_{2}$. The valuation $\nu_{2}$ of the second summand can be estimated
in the same way (using $\mathit{val}(\lambda_{v}\tilde{d})\geq 0$ in this
case), resulting in:

$\nu_{2}\geq\mathit{val}(w_{1}-w_{1}^{\prime})+\epsilon.$

In summary, we have

$\mathit{val}(m(w_{1},w_{2})-m(w_{1}^{\prime},w_{2}^{\prime}))\geq\min(\nu_{1},\nu_{2})\geq\min(\mathit{val}(w_{1}-w_{1}^{\prime}),\mathit{val}(w_{2}-w_{2}^{\prime}))+\epsilon,$

and we conclude that

$\displaystyle||G(w_{1},w_{2})-G(w_{1}^{\prime},w_{2}^{\prime})||$
$\displaystyle=e^{-\mathit{val}(G(w_{1},w_{2})-G(w_{1}^{\prime},w_{2}^{\prime}))}$
$\displaystyle\leq
e^{-\min_{m}\\{\mathit{val}(m(w_{1},w_{2})-m(w_{1}^{\prime},w_{2}^{\prime}))\\}}$
$\displaystyle\leq
e^{-\min(\mathit{val}(w_{1}-w_{1}^{\prime}),\mathit{val}(w_{2}-w_{2}^{\prime}))-\epsilon}$
$\displaystyle=e^{-\epsilon}||(w_{1},w_{2})-(w_{1}^{\prime},w_{2}^{\prime})||$

which is as desired.

∎

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