Let \( X \) be a manifold. Let \( L \subset T^* X \) be an exact Lagrangian (equipped with a pinning etc.) such that \( L \to X \) is proper map generically of degree \( n \). Since in this case \( L \) is bounded away from infinity, the Nadler-Zaslow construction gives a map
$$\mu^{-1}: Loc_r(L, A) \to Loc_{rn}(X, A)$$
(D.T. seems to believe that there is a direct definition of this map; I do not know this definition.) Here \(A \) is an arbitrary ring of coefficients; since the above construction makes sense functorially in \( A \), it determines an algebraic morphism of varieties.
We have a series of motivating questions.
Question. How injective is this map?
Sketch/hope: very much so — if \( \mathcal{L}, \mathcal{L}’ \in Loc_r(L, A) \), then presumably $$Hom(\mu^{-1}(\mathcal{L}), \mu^{-1}(\mathcal{L’}) ) = Hom(\mathcal{L}, \mathcal{L’}) \otimes HF(L, L)$$
So in particular when \(r = 1\), the RHS vanishes when \( \mathcal{L} \ne \mathcal{L’} \)
Henceforth take \( r = 1 \); note the connected component of the identity in \(Loc_1(X)\) is a torus.
Question. When does \( \mu^{-1} \) give cluster coordinates on \( Loc(X) \)? Or maybe it is better to say: when can we describe a natural subvariety of \( Loc(X) \) on which the image of \( \mu^{-1} \) is dense?
Question. Having fixed the original \( L \), what is the collection of other exact Lagrangians \( M \) such that \( \mu^{-1}(Loc(L)) \cap \mu^{-1}(Loc(M)) \) is nonempty? Such that it is dense in both factors?
Conjecture. (Neitzke-Zaslow) Take \( X \) to be a holomorphic curve with a puncture, and assume \( L \) is holomorphic. Then \( \mu^{-1} \) is the GMN procedure.
Remark. This conjecture is almost certainly not hard to prove if it’s true and would be pretty cool.
(Big) Question. GMN makes sense and is most relevant for non-exact Lagrangians; how does the Nadler-Zaslow correspondence extend?
Conjecture. (Treumann) Take \(X \) to be the complex plane, and \( \Lambda \) to be the Legendrian of an irregular singularity. Then every cluster chart of the cluster structure on the wild character variety comes as above from an exact Lagrangian. In particular there are infinitely many non-isomorphic Lagrangians bounding the (3, whatever) torus knots.
Remark. I think what this conjecture is really saying is the following. Assume GMN = microlocalization. Then GMN gives a wall-and-chamber decomposition of the Hitchin base, and different cluster charts for each chamber; each corresponds to a different cluster chart. The above conjecture probably amounts to the assertion that there is an exact Lagrangian in each. (I don’t particularly believe this conjecture though.)
Question. How are the charts arising from the above construction related to the charts arising from ruling filtrations?
Where does the “wall-and-chamber decomposition of the Hitchin base” occur in GMN?
Here is a comment on the last question. Briefly, though any given front diagram won’t lead to very many cluster charts, you get “a lot” by looking at all possible front projections. But I don’t know whether “a lot” is “as many as there are clusters.”
Let \(\Phi\) be a front diagram, \(\mathcal{M}_1(\Phi)\) its moduli space of sheaves of microlocal rank one. Call a ruling \(R\) of \(\Phi\) “chart-like” if the set of sheaves that admit a ruling filtration inducing \(R\) is open and isomorphic (via STZ Prop. 5.12) to the \(\mathrm{GL}(1)\)-character variety of the filling surface of \(R\).
Stick to rainbow-closures. After all I doubt that the Legendrian figure-eight, or \(m(8_{21})\), have cluster structures.
A rainbow closure has a unique chart-like ruling, the all-crossings ruling. The set of all rainbow projections of a fixed Legendrian knot \(\Lambda\) (assuming there is at least one) therefore leads to a set of torus charts of \(\mathcal{M}_1(\Lambda)\).
For the \((2,n)\) knot it is easy to write down \(n+2\) charts this way. That’s a complete list for \(n = 3\), but only 7 of the expected 21 charts for \(n = 5\). I’d like to see the other ones, or if they don’t come from rainbow projections understand why.