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.

In these cases, it makes sense to ask, on the B-model side, what is the endomorphism ring of the structure sheaf. That is the case if we have Coh as the B-side. Not so much when we have MF. So that we know, what is the weight of each endomorphism.

Suppose we have some basic MF on $\C^N$, like $W = x_1\cdots x_N$. And suppose we have some $\C^*$-action, say, given by weights $a_1, \cdots, a_N$, such that $\sum_i a_i = 0$, so $W$ is invariant. We can ask: what's going on here?

Say, we have $(1,1,-2)$ weights. What is MF? It is some two-periodic chain complex, living on $W=0$.

Is there a know-it-all sheaf, like structure sheaf on MF category? No, unfortunately no. Instead, we have three basic MFs.

For the non-equivariant MF, there are three basic ones. And they have $(\C^*)^3$ equivariant lifts. Such lifts collapse to $\C^*$-lift for each $\C^* \to (\C^*)^3$. We have various weights. If $x,y$ has weight $1$ and $z$ has weight $-2$, then the three basic ones has different weights $(x,yz), (z,xy), (y, zx)$. The $(z,xy)$ one is between $O(k)$ and $O(k+2)$. Somehow, they don't fit from a window, and cannot be used.

What if we choose some different weights? Like $(2,3,-5)$? Then, the window size is $5$. We cannot use $(z,xy)$, since it has size $5$.

I think we should take MF first, then take GIT quotient.

Consider the example of $(1,1,-1,-1)$, say with variable $x,y,z,w$. Window size is 2. Allowed factorization $(x, ..), (y, ...), \cdots$ 4 of them, and $(xz, yw), (xw, yz)$. two of them. All six makes geometric sense on the A-side.

No, we should not use window so early. We are talking about GIT quotient. We should talk about polarization, unstable orbits. So what are those? Is polarization still about a line bundle?

Following Segal.

What's hom between MF? It is the dg hom between curved 2-periodic chain complexes, the beautiful thing is that, the curvature cancels out, and the result is an ordinary 2-periodic chain complex.

Then, you take global section to get usual hom. Of course, sheaf hom is better.

But, how to deal with removing unstable loci? What's the approach of Segal? Well, he took GIT quotient first, without worrying about $W$. Then, the usual window works. Lemma 3.5 there.