• More on quiver gauge theory from toric CY3

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 is on the Y-side, living in $(N_Y)'$. So, TWZ says, the Legendrian curve, 2d torus, dimer, is living on $(N_X)'_T = (M_Y)'_T$, 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. $ $$(N_X)'_{\C^*} = (N_X)'_{\R} \times (N_X)'_{T} = (M_Y)'_{\R} \times (M_Y)'_{T}\xto{Legendre} (N_Y)'_{\R} \times (M_Y)'_{T} = T^*(M_Y)'_T $$