On Monday, a bunch of people convened in my office to try and extract some GMN out of Harold. It turned out to be the most insanely productive day since the SQUARE.

Harold had the following perspective on the GMN procedure. It has two steps:

- Choose a point in the Hitchin base. Solve certain flow equations; this gives rise to an essentially combinatorial object called a spectral network on the base curve.
*You can now forget the embedding of the spectral curve into the cotangent bundle of the base and only remember the spectral network and the map from the spectral curve to the base curve.* - From this data, there is a procedure which transfers local systems from the spectral curve to the base curve. This can be expressed as a map between (twisted) path groupoid algebras from the base curve to the spectral curve. This is a local procedure, and is determined by what happens when you cross walls. (The intersections of walls happen at two kinds of vertices, the critical point vertex out of which three flow lines come, and the $13 + 12 –> 12 + 23 + 13$ wall crossing vertex.)

This description, especially the italicized part, helped me a lot to think about “what is a concrete, precise formulation of the philosophy GMN = $\mu^{-1}$”

The answer is *not* to take a spectral curve and hope it’s an exact Lagrangian and then hope that the cluster chart determined by inverse microlocalizing from this curve is equal to the cluster chart determined by GMN from the spectral curve. It should instead be the following.

Fix a base curve $C$ and a spectral curve $\Sigma \subset T^* C$. Then there should be *some mysterious procedure* which produces out of $C$ a (not necessarily complex) Lagrangian $L$. One may either hope this Lagrangian is exact, or one may hope that $\mu^{-1}$ can be extended to non-exact Lagrangians. (I record here David’s belief that mysterious procedure is HK rotation, and if I write $(HK, NS)$ for “Hyperkahler rotate the spectral curve and Narasimhan-Seshadri the line bundle”, and $NH$ for the nonabelian Hodge isomorphism, then $\mu^{-1} \circ (HK, NS) = NH$. I record also both my skepticism and my belief that if this is true it should be (1) essentially tautological and (2) would mean that in the cotangent to a Riemann surface case, all the analysis needed to define $\mu^{-1}$ for non-exact Lagrangians has already been done by the nonabelian Hodge theorists.)

In any case *we are not obliged to think about what this procedure is* in order to make a concrete prediction. Instead, we should believe the following:

- Suppose given a (not necessarily complex) exact Lagrangian $L \subset T^*C$. Then by solving some flow equations, we should produce a spectral network on $C$.
- Inverse microlocalization $\mu^{-1}: Loc(L) \to Loc(C)$ is computed by this spectral network. (
*This is a well defined statement.*) - And maybe someday we will show that the GMN procedure is the same as, first produce $L$ out of the spectral curve $\Sigma$, and then do (1) and (2).

Then David observed that we even basically know how to do (1) and (2). Indeed, determining what happens in the microlocalization means following certain paths and roughly speaking making some adjustment when they cross a disk which meets some sheets of the exact Lagrangian. (Q: how can there be such a disk if the Lagrangian is exact? A: This is why we have a punctured base curve, the compactness assumptions of NZ probably mean that we have to specify asymptotic data of the Lagrangian at infinity and some part of the boundary of the disk goes to infinity?). According to Fukaya-Oh, such disks correspond to Morse flow trees for the (differences between the) functions whose derivatives are the one-forms parameterizing the sheets. So that’s the equation we solve to give us the spectral network, and then we have to make two local calculations (around the spectral network vertices) to convince us that microlocalization is doing the same thing as the spectral network construction. We didn’t succeed in making this calculation, but at least saw some relevant disks. It is possible that Eric has already made at least one of these calculations (the critical point one) in the note which started this whole discussion.

Anyway I think it is now a task for people who can calculate (not me) to prove:

**Theorem.** Let $L \subset T^* C$ be an exact lagrangian. Then spectral network determined by Morse flow trees generated by the differences between the primitives of the one forms parameterizing the branches of $L$ computes the inverse microlocalization map $\mu^{-1}: Loc(L) \to Loc(C)$.

**Remark. **This line of argumentation is not limited to two dimensions and is going to tell us what spectral networks should be in higher dimensions.

Some other observations:

Harold points out: maybe microlocalization works with coefficients in any semiring, which should give a positive structure to the cluster-charts-from-microlocalization construction.

David had previously made what looks like an explicit cluster seed by drawing some circles on the surface from a ruling. The fact that this is a cluster seed is compatible with (implied by if we can find the right Lagrangian) the statement that we get cluster charts because microlocalization carries the pairing on $H^1(L)$ to the pairing on $H^1(C)$. (Philosophy: microlocalization makes clusters because it respects string topology?) Anyway, what this explicitly is in the case that the braid closure comes from a singularity is the classical D-diagram of the singularity (see eg Arnold/Gusein-Zade/Varchenko book). The cluster mutations should correspond to known transformations of these diagrams, and it should be known — from Norbert A’Campo’s theory of planar divides — how these correspond to knot moves. Actually now that I think of it, maybe Norbert knows the name of the Lagrangian we’re looking for.

Alex had the interesting idea that maybe one can cut out ramification points and glue in trefoil knots instead to remember them. I don’t completely understand this idea but there is something compelling about it.

This is indeed the calculation that I worked out in those notes, only I prefer to advocate the philosophy that we *do* think of the Lagrangian as a spectral curve. Why not?

Suppose you wanted a trefoil-type example, so the spectral curve L is y^2 = cubic polynomial in x. Then there would be three branch points, and near these is where the interesting stuff happens. So lets consider a local situation where y^2 = x. This is an exact Lagrangian (since x = d(y^3/3)) and we can ask what algebraically-constructible sheaf F it should correspond to. (Subtlety: the behavior at infinity is weird since it hits horizontal and vertical infinity at the same time), but let’s just look near the branch point x = 0.

Inverse microlocalization says that the fiber at p of the sheaf F (or complex of sheaves) is the Fukaya hom of L with the cotangent fiber T*_p. This is two-dimensional (both intersection points have the same degree by holomorphicity, so disks don’t contribute). Following the intersection points gives a *naive* local system — this is the uncorrected system. At critical junctures (walls), there is a correction to this naive local system. The actual sheaf restriction maps are obtained by studying disks* (*I can explain which ones later) — but as Vivek writes a Morse-theoretic shortcut exists due to Fukaya-Oh. Yet since one of the Lagrangians (a cotangent fiber) is not a graph, we cannot apply Fukaya-Oh until we rotate the whole picture so the cotangent fiber x = const becomes the graph of the one-form const.dy. Conveniently, the Lag becomes a graph, too. x = d(y^3/3). The whole thing is computable, and the Morse continuation maps (up to a crucial sign that Michael Hutchings did not clarify for me — go bug him) compute the correction to the actual monodromy and give a trivial local system. (This would *not* have been the case if you just following critical points/intersection points around, since they do have nontrivial monodromy. I don’t know what the role of Andy’s twisted local systems is.)

*Don’t worry. There can be disks with exact branes as long as there is more than one brane involved. Draw a triangle in the plane. That’s a holomorphic disk connecting three straight exact Lagrangian lines.

There is not a correspondence between branch points and trefoils, as is clear from the above.

However, I have recently been discussing with some of you the possibility of putting two (or more) wild punctures on a curve and realizing that the corresponding moduli space by SFT should be a space of morphisms from one dga to another (or maps from products of dga’s to one). What is the sheaf-theoretic model of such as space? (The Deligne-Malgrange Lags live on different cylinders! Vivek is not totally put off by this, and holds out some hope of the construction/guess where you flip a front diagram.)

How does this jive with Boalch’s work in which the global moduli space is constructed from local fission varieties?

The fission construction is something different — it’s analogous to how we build in STZ the moduli space associated to a braid closure out of the small $G/P$s. Gluing together the local patches containing knots into a global curve is what’s called fusion in the related literature.

About viewing or not viewing the exact lagrangian as a spectral curve: I’m happy to say it’s “like” a spectral curve. I don’t want to say that it is a spectral curve, for various reasons:

(1) the discussion above works in over a base of any dimension, including odd ones, although we will have to make more local calculations in higher dimensions to figure out what the correct notion of spectral network is.

(2) I don’t think it’s literally true that the spectral curve is the thing we microlocalise from, or anyway don’t want to try and literally prove this

(3) or maybe another way to put the above, I think the true story is microlocalization, and in dimension 2 you can become confused and accidentally translate this story through some procedure (possibly nonabelian Hodge theory) and then wind up with GMN. I’d rather not be confused in this way.

I’m comfortable with “like” a spectral curve, too, since as you say we can go through our machinations no matter where the exact Lagrangian came from. But I do want to record Andy’s comment, cited in my lecture note, that the spectral curve is very close to the “naive spectral curve” (characteristic polynomial of the meromorphic connection one-form).

Hi Vivek,

What is the most significant consequence of GMN? I’m not familiar with that and wanted to see why it is so important.

Thanks!