Let $X$ be a toric CY 3-fold, and this will be our A-side.

Let $N_X$ and $M_X$ be 1PS and character lattice. $\Sigma_X$ lives in $N_X$. Let $P_\Sigma$ denote the compact polytope, being the convex hull of $0$ and the ray generators of $\Sigma$.

Let $Y$ be the toric variety with moment polytope $P_\Sigma$. Let $Y'$ be the toric boundary divisor of $Y$ corresponding to the top face of $P_\Sigma$.

I am reading this paper by Zaslow-Treumann-Williams, https://arxiv.org/pdf/1810.05985.pdf But it does not discuss the relation with toric CY 3fold. Then, let me read this physics paper: FHKV, Dimer models from mirror symmetry and quivering amoebæ.

A toric CY $X$, with fan $\Sigma$; is mirror to another CY (ha? not a LG model?) $$ Y = \{uv = P(z,w) = \sum_{\alpha \in Q_\Sigma} c_\alpha z^\alpha \} $$ So, $Y$ is not toric. Why we want to consider the conic fibration over this 'spectral curve'?

Question: how does this compare with the usual toric mirror symmetry? where the other side is $(\C^*)^3$ with a superpotential? Well the superpotential can be written as $W = u P(z,w)$, and it is not a usual Fukaya-Seidel category, since there is no critical point.

For example, consider one-dimensional lower case, where $W_Y = u(1+z)$ and $X=\C^2$. If we follow the weird mirror construction, we would say: $uv = (1+z)$. Wait, this is one of the cluster mirror symmetry, we do have $X \RM \{x_1 x_2=1\}$ is like self-mirror, with superpotential like $W = u = (1+z)/v$.

Right, so the relation with the usual superpotential is that, we take $uv = 1+z$, the space as is, but we fiber it to $W = u = (1+z)/v$. So, $z$ can be whatever $\C^*$ it wants, that's fine. $v$ can be also whatever $\C^*$-it wants, that's fine. Then, $u$ can be solve. So the Hori-Vafa-Givental mirror can be embedded to this space. For each value of non-zero $u$, the open part does not capture the divisor that $v=0$ and $z=-1$. For the part that $u=0$, the HVG space is $v \in \C^*, z=-1$, but there is more, which is $v=0$ and $z=-1$.

So, how to make it work? Consider the original HVG space, $\C^*_z \times \C^*_v$,then we consider $W=(1+z)/v$, then consider the compactification of $v=0$, and the blow-up so that $W$ makes sense. That is precisely introducing the $u$ variable. OK, but why doens't it affect the Fukaya category? Previously, the fiber over $1$ is: whatever $z$ as long as $z \neq -1$. So a pair-of-pants, and the fiber over $0$ is a cylinder. Now, the fiber over everywhere else is $\C^*$, either parametrized by $z$ or $v$. And the fiber over $0$ is for $z=-1$ dead, and $v \in \C$. OK fine.