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 that in the B-side, we need to choose some point in the G-equivairaint ample cone, and assuming affine, some character of $T$. I just need some simple enough reason to say that, this choice, a cocharacter of $T^\vee$, means taking certain tropical limit in that fibration. Why is that?

If B-side has inequivalent quotient, how does A-side say?

Ex 1 Say B side, we have weight $(1,1)$ for $\C^*$ acting on $\C^2$. On the A-side, we have $W = y + t/y$ over space $\C^*_t$.

• On the B-side, the quotient is $\P^1$ or empty set
• On the A-side, the mutual critical fiber for $(y_1+y_2, y_1y_2): (\C^*)^2 \to \C \times \C^*$ has critical point at the diagonal $y_1=y_2=y$, then the image of the critical loci is $W^2 = 4t$.

Ex 2 Another example, where we have $\C^*$ acts on $\C^3$ by weight $(1,1,-1)$, the mirror superpotential have $t = y_1 y_2 / y_3$ with $W = y_1 + y_2 + y_3$, and we have $W = y_1 + y_2 + y_1 y_2 / t$. So, we have critical loci, which is at $$ 1 + y_2/t = 0, \quad 1 + y_1/t = 0 $$ so $y_1 = y_2 = -t$ is the critical loci. Pick any $t$, say $t=1$. What is $$ Fuk((\C^*)^2, y_1 + y_2 + y_1 y_2) $$ It is definitely not generated by the critical point. What is the fiber at $y_1 + y_2 + y_1 y_2 = 0$? Well, the fiber is then $\C^* \RM \{-1\}$. Generic fiber is a $4$ punctured $\P^1$, and special fiber is a $3$ punctured $\P^1$. This is also visible from the change of the Newton polytope for the defining hypersurface, over the value 0, one lose the constant term.