• More on quiver gauge theory from toric CY3
• Editing the paper

Let's read what did Treumann-Williams-Zaslow do. Input data is a bipartite graph $\Gamma$ on a torus, Output data is a spectral curve.

• One way to do this, is to view the bipartite graph on a torus give rise to Legendrian data, and these Legendrian data gives way to do mirror symmetry to get cohernet sheaves on a toric surface. And the object of such a Lagrangian filling is the spectral curve.
• Another way to do this, is to use the Kasteleyn operator's spectral curve. Consider the second approach, we get not just a specral curve, but a defining function $P(z,w)$, which is a sum of dimer tilting pattern, and also as the zero locus of a determinant of a matrix. The Ronkin function of the Laurent polynomial reflects the Newton polytope's data, also the amoeba data.

Question: how is everything related to toric mirror symmetry for this toric CY?

Let $N_X, M_X$ be the toric CY $X$; and $N_Y, M_Y$ refers to the T-dual side. We are given a collection of points, $A_X \In N_X$, such that, there is a primitive divisor $\xi_X \in M_X$, that pairs $\langle \xi_X, A_X \rangle = 1$. Such a data can be used to construct a toric map $$ W_X = W_{\xi_X}: X \to \C $$ such that $W_X^{-1}(0) = \d X$,and $W_X^{-1}(1) = (\C^*)^2$.

On the other hand, $A_X \In N_X = M_Y$ defines a Laurent polynomial (type) on $Y = (\C^*)^3 = (N_Y)_{\C^*}$. It is homogeneous of degree $1$ with respect to a that distinguished element $\xi_X = \rho_Y \in M_X = N_Y$. (If we choose another degree $1$ polynomial on $Y$, we can divide by it, and obtain a Laurent polynomial on $Y / \C^*$, but let's not do that reduction yet)

To describe the A-model on $Y$, we consider the torus of $(N_Y)_T = (M_X)_T$. So the collection of vectors $A_X \In N_X$ defines a collection of codimension 1 torus with co-orientation in $(N_Y)_T$. All the B-side objects on $X$ can be translated to $A$-side objects on $Y$.

The skeleton is living in $(N_X)_\R \times (M_X)_T$, so you can see the projection of $\Lambda_Y$ to $(N_X)_\R$, is the fan $\Sigma_X$

Where is the dimer, and where is the quiver, and where is the 2-d torus, and where is the spectral curve? The spectral curve of the toric CY, seems to be an B-side object. It is living on $Y$. Indeed, it is the vanishing loci of that Laurent polynomials $W_Y$, then quotient by $\C^*$.

Suppose we want to remember this Laurent polynomial on the quotient of $Y$, what should I do? I can take the monomial embedding, this embeds $Y / \C^*$ to a toric surface called, $\wb Y'$. Or we can even take the original $\Sigma_X$, truncate the tip, use it as the moment polytope of the compactified $Y$, as $\wb Y$. Anyway, I only know that $\wb Y'$ is used in TWZ.

Clue 1 : Spectral Curve $C_Y$ is on the Y-side, living in $(N_Y)'_{\C^*}$. So, TWZ says, the bipartite graph, dimer, is living on $(N_X)'_T = (M_Y)'_T$. Bipartite graph lives on the dual 2-torus as the spectral curve. We have $$ 0 \to \langle \rho_Y \rangle \to N_Y \to N_Y' \to 0 $$ then, if we dualize, we get $$ 0 \to (N_X)' \to N_X \xto{\cdot \xi_X} \Z \to 0 $$ this is the same as $$ 0 \to (M_Y)' \to M_Y \to \Z \to 0 $$ so we have $ (M_Y)'_T \In (M_Y)_T. $ $$(N_X)'_{\C^*} = (N_X)'_{\R} \times (N_X)'_{T} = (M_Y)'_{\R} \times (M_Y)'_{T}\xto{Legendre} (N_Y)'_{\R} \times (M_Y)'_{T} = T^*(M_Y)'_T $$

Now, how to get Legendrian over $(N_X)'_T$? We have a natural candidate, just the $P_\Sigma$ polygon's tangent space to the side.

My starting point is that, toric CY3 $X$ is the B-side, and $Y$ is the $A$-side. Lagrangian in $Y$ corresponds to coherent sheaves on $X$.

Now, things is a bit different. We want to do $X$ as the $A$-side (it is not so perfect, but we will do). It is a bit complicated, since it has Kahler moduli space. And $Y$ is the B-side, with the spectral curve.

There is a bit a mixture of GW-DT on the X-side.

Now, what about dimer? The geometric dimer-torus is on $(N_X)'_T$. One can realize the dimer torus geometrically as $$ (N_X)'_{\C^*} \cong (W_X)^{-1}(1). $$ OK, not too bizarre.

Now, I need the dual cone $$ |\Sigma_X| = \sigma_X \In N_X \Rightarrow \sigma_X^\vee \In M_X $$ So, I do have things living on $(N_X)_T$ and $(N_X)'_T$.

What kind of superpotential do I want to have on $X$? I want sum of monomials, that pairs-positively with the cone $\sigma_X$, that is, maybe the ray generators of $\sigma_X^\vee$.

Ex: $X = \C^3$ , we want $$ \wt W_X = xyz + \epsilon(x+y+z) $$

Ex: $X$ is conifold , say the ray generators of $\sigma_X$ is $$ (0,0,1), (1,0,1), (0,1,1), (1,1,1). $$ with divisors called $D_1, \cdots, D_4$. Then, the dual cone has generators $$ (1,0,0), (0,1,0), (0,-1,1), (-1, 0, 1). $$ Let's call these generators $W_1, \cdots, W_4$, with $W_1 W_4 = W_2 W_3$. We want $$ W = W_{(0,0,1)} + \epsilon(W_1 + W_2 + W_3 + W_4) $$ Suppose we have a corner vertex given by intersection of $D_1, D_2, D_3$, then the function looks like $$ W = xyz + \epsilon(x+y+xz + yz) $$

OK, fine.

$\gdef\La{\Lambda}$

X is the A-side. Except $X$ is the compactification of $(\C^*)^3$.

The skeleton $\Lambda_X \In (N_X)_T \times (M_X)_\R$, where its projection $\La_X$ on $(M_X)_\R$ is the dual cone subdivided by a ray $\xi_X$. And its projection to $(N_X)_T$ is a periodization of $\Sigma_X$.

I am not sure what am I supposed to do with this setup. This should be called: mirror symmetry with toric CY3 as the A-side, and with superpotential of hierarchy.

The B-side, $Y$, will have compactification. The moment polytope of $Y$ is the simplest way to describe it, it is just the truncated $\Sigma_X$, with flat bottom. Then, the superpotential is just these monomials, labelled by $A_X$, and the critical loci is the compactified spectral curve.

We are interested in counting disks ending on a Lagrangian. many ways to cook up Lagrangians here. don't worry.

What do you want? Restrict to one monomial, and look at all the nearby monomials. That's restrict to some open part of the skeleton.

OK, in terms of skeleton and 2d torus on the $X$-side, or rather $X' = W_X^{-1}(1)$ side, I am completely happy. But, where is the dimer? Dimer data is the same as the zigzag path (at least in some nice cases)

What is a dimer? Dimer is a particular Lagrangian filling of a Legendrian boundary condition.

What is dimer mutation? Well, for a given alternativing Lagrangian, the local system on it only support some $(\C^*)^k$ torus open chart worth of objects among all the allowed guys.

We are looking at the moduli space of rank $1$ sheaves on the toric stack $Y'$. Basically, we can change the parameters of the coefficients. What's the big deal then?

What is your starting point? For TWZ or STWZ or STW, their starting point is a singular 2-dimensional skeleton, obtained by gluing disks to a surface. The moduli space of microlocal rank 1 sheaves on it, forms a cluster variety.