This serves as a reminder that I need to do work, rather than just sitting around.
There are three approaches,
They each need to solve some technical problems.
The equation for $X_1$ is $x x' = 1 + q$, $q \in \C^*, x,x' \in \C$. We have several approaches to build a Liouville structure on this space. They are related here but have different abilities to generalize to higher dimension.
Let $X$ be an complex $n$ dimension smooth toric variety, and let $T = U(1)^n$ the compact torus acting on $X$. Assume $X$ is equipped with a $T$-invariant symplectic form $\omega$, and $\Delta_X$ is the moment polytope of $X$.
Theorem $$ Loc(X)^{Loc(T)} \cong \mu sh(\wt \Delta_X) $$ where $\wt \Delta_X \to \Delta_X$ is certain FLTZ Lagrangian skeleton standing over $\Delta$, with $n-k$ dimension torus fiber over the $k$-dimensional boundary strata.
$\gdef\CS{ {\C^*}}$ $\gdef\v{\vee}$ $\gdef\ycal{\mathcal{Y}}$ $\gdef\zcal{\mathcal{Z}}$
No, I don't want to consider complete intersection, I just want to consider A-model where I compactify $(\C^*)^n$ somehow.
For example, $X_A = (\C^*)^3 \cup D_{1,1,1}$, and $W_A = x+y+z$. And the mirror is $X_B = \C^3$, with $W_B = xyz$.
Suppose I take products of these type. I get HMS still.
Then, I consider some torus action on the B-side, and fibration to dual torus on the A-side. We have $$ Coh([X_B / T], W) = Coh(X_B,W_B)^T \cong Fuk(X_A, W_A)^T \cong Fuk(\wt X_A, \wt W_A) $$
Now, here is the interesting thing: if we take GIT quotient, that corresponds to localization by the unstable loci.