Let $X$ be a manifold. Let $L\subset T^{\ast }X$ 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

$${\mu }^{-1}:Lo{c}_r(L,A)\to Lo{c}_{rn}(X,A)$$

(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 $\mathcal{L},{\mathcal{L}}^{\prime }\in Lo{c}_r(L,A)$, then presumably $$Hom({\mu }^{-1}(\mathcal{L}),{\mu }^{-1}({\mathcal{L}}^{\prime }))=Hom(\mathcal{L},{\mathcal{L}}^{\prime })\otimes HF(L,L)$$

So in particular when $r=1$, the RHS vanishes when $\mathcal{L}\ne {\mathcal{L}}^{\prime }$

Henceforth take $r=1$; note the connected component of the identity in $Lo{c}_1(X)$ 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 $\mathrm{\Lambda }$ 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?