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--- blog:2023-04-30 [2023/04/30 13:16] – created pzhou
+++ blog:2023-04-30 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ====== 2023-04-30 ======
   * More on quiver gauge theory from toric CY3
+  * Editing the paper
 ===== quiver gauge theory from toric CY3 =====
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 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 $\Sigma_X$
+==== Dimer? Quiver? Spectral Curve? 2d Torus? ====
 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 would take the quotient, then compactify. That amount to a collection of vectors in $N_Y / \rho_Y = M_X / \xi_X$. Or rahter, we can use the monomial embedding of Mikhalkin.
+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.
+==== The tension ====
+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 $(N_X)'_T$.
+==== Toric CY3 X, as the A-side ====
+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.
+==== The 3d and 2d torus on the X-side ====
+$\gdef\La{\Lambda}$
+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. **
+==== Remaining Questions ====
+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.