Let me do some concrete stuff tonight.

• Write up and finish the complete intersection project.
• write up the legendrian thickening stuff
• write up the matrix factorization for conifold case. who goes to what.
• dream about, how to do A-side GIT quotient.

So far, what do I know? If I choose a character on the B-side, that means I choose a cocharacter on the A-side. It can mean a weight shifting functor $S$.

We first need to look at the equivariant object $F$, then among those, we need to look at how $F$ maps to $SF$.

We can say $Fuk(Y, W_Y)$ is mirror to $Coh(X, W_X)$. We have $T$ acts on $X$ preserving $W_Y$, and $Y$ maps to $T^\vee$ (how does this map interact with $W_Y$ ?).

In the following, what I wanted should be true for general $X, Y$, not necessarily toric (they can come from toric guys as subvariety), only need some torusaction, and torus invariant.

What's the trouble? Well, we know very well, in the GIT quotient, we can choose GIT quotient parameter. In addition to choosing some ample line bundle