A quasi-summary (to be taken with a grain of salt) of my understanding of the paper Eric alluded to as well as the paper of Smith on the $A_1$ Hitchin 3-folds, mostly to draw out some subtle differences between the two settings that might reward reflection.

3-folds from Toric Spectral Curves
Start with a complete fan $\Sigma^\vee$ in $\mathbb{Z}^2$, and let $\Delta$ be the convex hull of the generators of its 1-dimensional cones. I write $\Sigma^\vee$ to emphasize that this is not the dual fan of $\Delta$. The following is the construction of the mirror family to the affine cone over the toric surface $X_{\Sigma^\vee}$.

Let $P(z,w)$ be a Laurent polynomial with Newton polygon $\Delta$, $\Sigma_P \subset \mathbb{C}^*_z \times \mathbb{C}^*_w$ its vanishing locus/spectral curve, and $X_P \subset \mathbb{C}^*_z \times \mathbb{C}^*_w \times \mathbb{C}_u \times \mathbb{C}_v$ be the vanishing locus of $P(z,w) – uv$. We think of $X_P$ as fibered over a plane $\mathbb{C}_W$ via $(z,w,u,v) \mapsto P(z,w) = uv$, so the spectral curve sits inside $X_P$ over $W=0$. More generally, for each value of $W$ we have a curve $\Sigma_W$ defined by $P(z,w) = W$, $u=v=0$ inside $X_P$.

We build a basis of $H_3(X_P,\mathbb{Z})$ out of the following $S^3$’s. Choose a collection of nonintersecting vanishing paths between $0$ and the critical values $\{W_i\}$ of $P(z,w)$ in the $W$-plane. Each path gives you a vanishing thimble in $\mathbb{C}^*_z \times \mathbb{C}^*_w$ where a cycle in $\Sigma_P$ pinches off as you go to the critical value $W_i$, and a vanishing thimble in $\mathbb{C}_u \times \mathbb{C}_v$ where the cycle in the generic fiber $\mathbb{C}^*$ of $(u,v) \mapsto W$ pinches off as you go to $0$. The fiberwise product of these thimbles is a Lagrangian 3-sphere $S^3_i$ in $X_P$. By construction, these $S^3$’s only intersect inside of $\Sigma_P$, so their intersection numbers are controlled by those of the 1-cycles $W_i \cap \Sigma_P$ in $\Sigma_P$. Note that these intersections are vanishing cycles of $\Sigma_P$ that come with disks naturally attached to them inside $X_P$, since they bound vanishing thimbles in the family $\{\Sigma_W\}$ over $\mathbb{C}_W$.

Now the Newton polynomial of $P$ also encodes a bipartite graph $\Gamma$ on $T^2$, well-defined up to square moves, by the construction we’re used to (I’m not totally clear on how they think about reading off $\Gamma$ a priori, though part of their claim is how to read it off directly from the choices of vanishing paths made above). We are also used to (it seems they were the first to compute this? I don’t know the earlier dimer literature.) the fact that the genus of the conjugate surface $C_\Gamma$ of $\Gamma \subset T^2$ has the same genus and number of punctures as $\Sigma_P$. Recall that $C_\Gamma$ inherits a homology basis from the face cycles of $\Gamma \subset T^2$. Their claim seems to be that for each choice of $\Gamma$ in its square-move equivalence class one can identify $C_\Gamma$ and $\Sigma_P$ so that this basis is identified with the basis of vanishing cycles for some choice of vanishing paths in the $W$-plane. I cannot find if and where this is actually proved, though they do at least make some effort to relate how the face-cycle bases transform under square-moves to Picard-Lefschetz transformations (i.e. they compute one example). Moreover, they do not bother formulating a precise statement about what is being claimed mathematically, so it’s rather hard to read around all the physical smoke and mirrors. In any case, the quiver dual to the bipartite graph exactly encodes the intersection numbers of the above basis of $H_3(X_P,\mathbb{Z})$. Moreover, given the claim we can (topologically? smoothly?) identify the $T^2$ containing $\Gamma$ with a torus in $X_P$ obtained by gluing together the actual vanishing thimbles as the faces of $\Gamma \subset T^2$.

I have never totally understood what’s going on with the amoebae sections of this paper or related ones — I don’t know if there are claims made about how $\Sigma_P$ intersects $U(1)^2$ inside $\mathbb{C}^2$.

3-folds from Hitchin Spectral Curves
Let $\phi$ be a quadratic differential on a curve $C$, $\Sigma_\phi = \{\lambda| \lambda^2 – \phi =0\} \subset T^*C$ its spectral curve, and $X_\phi = \{(\lambda_1,\lambda_2,\lambda_3)| \sum \lambda_i – \phi = 0$ the associated 3-fold in the total space of $K_C^{\oplus 3}$. The most manageable case is when $\phi$ has at least one pole and all its poles haver order at least 3. In this case the projection $X_\phi \to C$ has generic fiber a smooth affine quadric, has empty fiber above the poles of $\phi$, and has a fiber with an $A_1$-singularity above the zeroes of $\phi$.

We build a basis of $H_3(X_\phi,\mathbb{Z})$ out of the following $S^3$’s. Choose a collection of nonintersecting vanishing paths between the zeros of $\phi$ which is Poincare dual to an ideal triangulation (i.e. whose vertices are at the poles of $\phi$). The matching cycle above each path is a Lagrangian 3-sphere $S^3_i$ in $X_\phi$. By construction, these $S^3$’s only intersect above the zeros of $\phi$, so their intersection numbers are controlled by the combinatorics of the chosen triangulation. The standard association of a quiver to a triangulation exactly encodes these intersection numbers. Note that it also encodes the intersection numbers of the associated matching 1-cycles in $\Sigma_\phi$, which are vanishing cycles of the entire family of spectral curves over the Hitchin base.

Questions
In each of the above cases, we build a basis of 3-spheres by playing two slightly different fibration games. But it seems reasonable to ask if the two games are secretly the same. For example, if I take $X_P$ and project it to, say, $\mathbb{C}^*_z$, are the 3-spheres constructed above matching cycles with respect to the singular fibers of this projection? The spirit of the question is to try to make as precise as possible the distance between the two classes of example.

Smith’s main result is that the subcategory of the Fukaya category of $X_\phi$ generated by these 3-spheres is equivalent to the finite-dimensional derived category of the 3CY completion of the quiver with potential of the triangulation. He (and others) speculate that the full derived category of perfect complexes is essentially the wrapped Fukaya category of $X_\phi$. My question to those more attuned to the symplectic world: have either of these questions been addressed in the toric case? Those spaces seem to have been studied much more extensively, and I know there are at least some precise statements relating those quivers to exceptional collections on the mirror to $X_P$ (i.e. the affine cone over $X_{\Sigma^\vee}$), but I dont really know where these results are or exactly what they say. In particular, what they say regarding the geometric realization of the relationship between the perfect and finite-dimensional categories of the quiver with potential. This is in a sense a key relationship for us, as there is a precise sense in which the triangulated quotient of these is the correct categorical object to regard as controlling the coordinate ring of the cluster variety (hence, knotty character variety).