VGIT take 2, this time with superpotential on the B-side. On one hand, this story is well understood by other people, not me; on the other hand, the story of window on A-side is not so well understand, and we don't know what the mirror of taking quotient is.

Well, the mirror of LG model is another LG model. The most foundational one is the one proven by Nadler, mirror of pair-of-pants. (indeed, this is the building block of everything)

The next step is to turn on some group action on the B-side, and turn on some fibration on the A-side. (or vice versa)

But why the recipe works? Big question, ask Vivek.

Now, Vivek says: one should take categorical fixed point, namely consider the equivariant category. Say $Coh(X) \cong Fuk(Y)$, and $T$ acts on $X$, with $Y \to T^\vee$ (I don't know why that is the case, and how to generalize this to non-abelian group. Ask Teleman), then we have $$ Coh(X)^T \cong Fuk(Y)^T \cong Fuk(\wt Y) $$ where $\wt Y$ is pullback of $Y$ along the universal cover $\wt T \to T$.

That's a good first step, and necessary. Now, we need to pass to categorical GIT quotient, namely, throw away some 'unstable loci' on the B-side, and correspondingly, do some stop removal on the A-side.

How does that work?