A moment of clarity, inspired by a talk of Harold Williams and subsequent conversations:
A cluster chart on a Poisson algebraic variety $X$ is just an open torus $T = Spec\, \mathbb{C}[x_1^\pm,\ldots, x_n^\pm]$ so that the Poisson structure on $X$ restricts to an integral coordinate Poisson structure on $T$, i.e., $\omega(x_i, x_j) = n_{ij} \delta_{ij}$ for some $n_{ij} \in \mathbb{Z}$. This gives rise to a quiver where the $n_{ij}$ count arrows; for the variety to be cluster it should be the corresponding cluster variety. The point is that the notion of cluster chart makes sense without first picking a quiver.
The reason an exact Lagrangian surface in the cotangent bundle of a surface, $L \subset T^*\Sigma$, gives rise to cluster coordinates is going to be something like: microlocalization takes the Goldman bracket to some multiple of the Goldman bracket.
Also, since the character variety of a surface is affine, to show that GMN = microlocalization, it suffices to show that functions on the character variety pull back to the same things. I.e., it is enough to show that “holonomy around a loop” in the base is computed in the same way on both sides.
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Suppose we can produce one cluster chart by naming one exact lagrangian submanifold $L \subset T^* \Sigma$. This should correspond to some Fock-Goncharov configuration. Then we should figure out how to lift the corresponding cluster moves to some sort of surgeries on $L$.
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I record an unrelated thought. Harold had some isomorphism of cluster varieties between a double bruhat cell and a wild character variety, which was shown I think roughly speaking by matching cluster charts. I believe there should be a direct argument having to do with a picture Harold drew months ago, which builds a knot out of the brick diagram he was using to get cluster charts.
Suppose we can produce one cluster chart by naming one exact lagrangian submanifold $L \subset T^* \Sigma$. This should correspond to some Fock-Goncharov configuration. Then we should figure out how to lift the corresponding cluster moves to some sort of surgeries on $L$.
Seidel explained something like this to me this Spring. If $L$ is a Lagrangian surface, $\gamma \subset L$ is a simple essential loop in $L$, and $D$ is a Lagrangian disk ending on $\gamma$, then one should be able to do a surgery to produce an $L’$, so that $L \to L’$ was an edge in the mutation graph. I don’t know if it is difficult to produce an exact such $L’$. Perhaps the remaining problem is to classify exact Lagrangians in a regular neighborhood of $L \cup D$, which is homeomorphic to an interesting Weinstein manifold.
You can sort of see the input (not to say the output) of this in Fock-Goncharov, in hK rotated form. Take a spectral curve $Y \subset T^* X$, and two critical values $x_1$ and $x_2$. (Is that the kind of thing Fock and Goncharov start with, when they write down cluster transforms?) I think Fukaya-Oh produces a holomorphic disk with boundary on the loop in $Y$ given by the two branches of a gradient path from $x_1$ to $x_2$. In another symplectic structure, the spectral curve and the FO disk become special Lagrangian.
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Quick comment noted in discussion with Harold in Austin: take a diagram with crossings ready for an R3 move, and consider the \cM_1 space. The left-right flags looks like two different lines with two different points in P^2. The two \cM_1 spaces on either side of the R3 move are obtained by adding to this data either (a) the point at the intersection of the lines, or (b) the line connecting the two points in those planes. Harold says that these descriptions are related by a cluster transformation (in what seed/basis I don’t know).
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This mutation is exactly the move sending >–< to the same picture rotated 90 degrees.
The horizontal line connects two zeros of the quadratic differential (and there is a disk
bounding both branches lying over it) and the moment of mutation comes when this line
shrinks to zero and branches collide. Then the associated triangulation undergoes a
flip and we move to a different cluster chart.
Are you describing something which turns an exact lagrangian into an exact lagrangian?
I think we’re describing a path in the space of exact Lagrangians, in which one circle collapses and another grows. This is a path across a wall, i.e. in the resulting Strebel-type flow, there is a moment where the associated triangulation on a four-gon flips from one diagonal to the other.
can you expand on this? what data is needed to define such a path; what are its equations, what is this flow you are describing…?
I am referring to the WKB triangulation from Section 6 of GMN. The characteristic polynomial of a rank-two Higgs field is a quadratic differential when pushed down to the base curve. The constant phase (e.g., real) flow lines emanating from zeroes of this differential cut up the curve into cells — and there is a dual triangulation (note edges map to edges under duality). As you vary the input data, the flow lines move around and sometimes the triangulation jumps — see Figure 27 of GMN.
WEIRD THING THAT WE NEED EXTENSIVE CONVERSATIONS WITH ANDY PRESENT TO FIGURE OUT: To me, “varying the input data” should mean looking at a family of points in the Hitchin space and hence a family of Higgs bundles. But to Andy and GMN, they vary the phase of the flows and this corresponds to a circle in the twistor family of kahler structures! Andy says these are related by some “WKB philosophy” (quotes mine), but I don’t know what this means. I don’t think Marco Gualtieri does, either. I’ve been asking….
Having said that, though we do have a family of Lagrangians corresponding to the spectral curves, I was wrong about a circle collapsing and growing. The appearance of a BPS state is what happens at the wall, and this corresponds to a holomorphic disk. In the picture on the base, it is just a finite flow line between zeroes, but in the cotangent bundle this lifts to a disk bounding the two sheets. I think the “vary the phase” problem corresponds here to varying the fiber of the corresponding flat bundle, so it is at these phases where one requires wall-crossing input into the nonabelianization map.
Okay, now I’m stuck trying to reconcile with Section 9 of GMN.