Recall that given a collection of curves $C_i$ on a surface $\Sigma$, we construct a 4-manifold $W$ by Weinstein handle attachment of disks to $T^* \Sigma$ along the Legendrian lifts of the $C_i$. It contains a Lagrangian skeleton $\mathbb{L}$, formed as the union of $\Sigma$, the positive conormals to the $C_i$ and the 2-d disks $D_i$ inside the attached handles.
We define $$\mu loc(\mathbb{L}) := \mu loc(\Sigma \cup \bigcup N^+ C_i) \times_{loc(\bigcup C_i)} loc(\bigcup D_i)$$
and can now prove that the moduli of “rank one” objects in this space has a cluster atlas corresponding to the skew symmetric matrix $\langle C_i, C_j\rangle$.
I.e., we construct cluster (X-)varieties as moduli of microlocal sheaves.
Assuming the conjecture of Kontsevich that, whenever you have the skeleton of a Weinstein manifold, you have $\mu loc(\mathbb{L}) = Fuk(W)$, this proves the conjecture stated in a previous post.
I’m going around giving lectures on this topic and will record here some interesting developments.
I’ve been talking a bunch with Sean Keel this week about how this picture is related to the picture of cluster varieties developed in these papers. Their story is that cluster varieties are naturally blowups of toric varieties, and that moreover the functions on these toric varieties have a natural interpretation via mirror symmetry.
The relevance of mirror symmetry is that one can approach algebrao-geometric questions about a cluster variety $Z$ by passing to its SYZ mirror $Z^\vee$ and doing symplectic geometry there. The perspective coming from the discussions with Sean is that what we are doing is close enough to mirror symmetry (SYZ mirror symmetry is about moduli of tori; we’re doing moduli of some other kind of Lagrangians…) that it’s at least reasonable to see how much of the GHKK perspective works with $W$ in place of $Z^\vee$.
(1). Because of the fibre product definition, there is a natural morphism $\mu loc(\mathbb{L}) \to \mu loc(\Sigma \cup \bigcup N^+ C_i)$. In some cases, this matches the GHK blowup. Of course it’s better in the usual way our definition of cluster varieties is better; i.e., it’s defined automatically and everywhere, no “outside of codimension 2” nonsense.
In the context of GHK, it is natural to ask whether $\mu loc_1(\Sigma \cup \bigcup N^+ C_i)$ is a toric variety. The answer seems to be yes: first of all, we are taking moduli of objects in an abelian category so the space is a stack in the usual sense and most likely just a $\mathbb{G}_m$ gerbe; second of all the torus $loc_1(\Sigma)$ acts; third of all, the usual combinatorial description of the moduli space is as an open subset of the everywhere 1d reps of a quiver, subject to some relations of the form $ab = cd$; I am told that such relations amount to saying that this is an open subset of spec of a monoid.
It would be interesting to see what toric variety this is. Probably the combinatorics is related to any previous work which has been done on toric varieties via microlocal sheaves…
(2). Certainly any functions on $\mu loc(\Sigma \cup \bigcup N^+ C_i)$ pull back to functions on $\mu loc(\mathbb{L})$. But, there are some obvious such functions: any oriented curve which meets every one of the $C_i$ positively has a well defined holonomy on $\mu loc(\Sigma \cup \bigcup N^+ C_i)$, which is sometimes zero. In the relevant case of rank 1, any monomial of such functions makes sense, since the holonomies take value in $\mathbb{C}$ and can be multiplied.
I will call such curves “test curves”.
Optimistic conjecture: in the $A = X$ situation (i.e. unimodular matrix) where the $X$ cluster variety $\mu loc_1(\mathbb{L})$ is also an $A$ cluster variety and we have a right to expect affine-ness, there exists, for any skew symmetric matrix, some collection of curves which (1) realizes this matrix and (2) admits enough test curves to give a basis of the functions on $\mu loc(\Sigma \cup \bigcup N^+ C_i)$.
Note this conjecture is in the direction of the Laurent phenomenon, since it is saying that at least some of the monodromy monomials (more precisely a cone of them) on $loc_1(\Sigma)$ extend over all of $\mu loc_1(\mathbb{L})$.
One might ask whether, more generally, the algebra of functions on $\mu loc_1(\Sigma \cup \bigcup N^+ C_i)$ is generated by test curves, I presently have no opinion.
Then there is the very good question of what happens to test curves under mutation. They certainly do not get sent to test curves — the curve which results from the mutation now crosses the mutated curve the wrong way! But, it still must have an interpretation as a function on $\mu loc(\mathbb{L})$. What’s going on?
I think the point is that there are also microlocal test curves. That is, if you come in to $C_i$ going the other way, you can go up the conormal (!) After all, in a different arboreal realization, that would be the flat direction. That is, there’s a sensible path category on an arboreal space (for now let us stay in the A2 or smooth locus) and microlocal sheaves have well defined monodromies around loops in this path algebra.
(3) According to GHKK, the functions on the cluster variety are supposed to count disks on the mirror. If we regard W as being mirror-like, we can take literally the same formula. Let us work again in the Fukaya context and assume that enough SFT is defined. Then for a Reeb orbit $\rho$ in $\partial W$, we can define a function by saying that its value on an object $(L, \mathcal{L})$ is the appropriate sum over holomorphic cylinders with one end on $\rho$ and the other on $L$. Here you weight the cylinders as usual by their symplectic area and the holonomy of $\mathcal{L}$ around the end of the cylinder on $L$.
(4) As in GHK, GHKK, etc., one can has the following relation:
$$SH^*(W) = HH^*(Fuk(W)) = HH^*(\mu loc(\mathbb{L})) = \Omega^*(\mbox{moduli of objects}) \mapsto \Omega^*(\mbox{moduli of rank one objects})$$
In particular, one expects that appropriately degreed symplectic cohomology on $W$ gives functions on the cluster variety (!!). This is related to (3) by the usual relation of contact homology and symplectic homology. One might conjecture that these are all the functions.
The literal correctness of the above statement probably requires some careful meditation on wrapped versus infinitesimal Fukaya categories, A versus X, etc.
One of the motivation questions at the beginning of this blog was trying to understand what, precisely, is the relation of Nadler-Zaslow inverse microlocalization to Gaiotto-Moore-Neitzke nonabelianization.
Talking to Andy has resolved a confusion in my head about this. He will write more soon, but just to remind myself:
The relation is almost certainly a form of hyperkahler rotation. While NZ makes a sheafified Floer complex whose stalk takes intersections between the fibre and an exact Lagrangian, and differential counts holomorphic disks between them, the GMN procedure instead has to do with taking something like a complex with basis the intersections with holomorphic Lagrangian spectral curve and fibre, and differential involving special lagrangian disks between them.
Incidentally, resolving another confusion: the spectral networks are not themselves disks, but rather a combinatorial encoding of all possible paths the disks might travel.
A remark of Paul Seidel about the formula for the quiver potential via cyclic words in the curves: it looks kind of like something which one might encounter in computing contact homology of $\partial W$ in terms of cyclic homology of the DGA for $(T^\infty \Sigma, \bigcup C_i)$.
See e.g. section 4.3 of Effects of Legendrian Surgery.
Emmy Murphy suggests: do Kirby calculus to identify the 4-manifold here with the one implicated in our description of cluster varieties a la Fock and Goncharov.
Tony Pantev asks: is there some sense in which $\mu loc(\mathbb{L})$ is a perverse schober? (Maybe in the disjoint circles setting ?)
If the connected curves are $\mathcal{C} = \{C_1,\ldots,C_n\}$, it looks like the torus-orbits in your toric variety are indexed by subsets of $\mathcal{C}$ that are pairwise disjoint. That’s close to a standard construction in geometric group theory, the curve complex.