The unmentioned context for David T.’s previous post is the following conjecture of mine.
Construction. Let $S$ be a surface, and $C_i$ be oriented simple curves on it. Lift the $C_i$ to Legendrians on the cocircle bundle by taking the positive conormal. Form a symplectic 4-manifold $M$ by Weinstein handle attachment to the lifts of the $C_i$. Then:
(1). This symplectic manifold has a Lagrangian skeleton given by attaching disks $D_i$ to $S$ along the $C_i$.
(2). Rank one local systems along $S$ give an algebraic torus $H^1(S, \mathbb{G}_m)$ worth of objects inside $Fuk(M)$.
(3). Lagrangian surgery along the disks gives, for each $k$, a new exact Lagrangian surface $\mu_k S$ inside $M$
(4). There are canonical new curves $\mu_k C_i$ on $\mu_k S$, unique up to isotopy preserving all structures and differing from the $C_i$ only in a neighborhood of the surgery, such that the skew symmetric matrix $\langle \mu_k C_i, \mu_k C_j \rangle$ differs from $\langle C_i, C_j \rangle$ by a mutation.
Conjecture.
(5). The tori $H^1(S, \mathbb{G}_m) \subset Fuk(M) \supset H^1(\mu_k S, \mathbb{G}_m)$ are related by the cluster transformation corresponding to mutation at $k$.
(6). There exist canonical new Lagrangian disks $\mu_k D_i$, attached to $\mu_k S$ along the curves $\mu_k C_i$, such that the result $\mu_k S \cup \bigcup \mu_k D_i$ is again a Lagrangian skeleton for $M$.
(7). Therefore, the cluster variety with seed given by the skew symmetric matrix $\langle C_i, C_j \rangle$ can be realized as a moduli space of objects in $Fuk(M)$.
I believe the main thing needed to prove the above is to find some way to describe the disks $\mu_k D_i$.
This conjecture is motivated by a calculation of Paul Seidel (proposition 11.8, page 110), and by the fact that it would have the following desirable consequence.
Let $\Gamma$ be a bicolored graph on a surface $X$, let $\Lambda$ be the Legendrian lift of the alternating strand diagram, and let $M_1(\Lambda)$ be the moduli of rank one sheaves along $\Lambda$. We show in the upcoming [STWZ] that $\Gamma$ determines an exact Lagrangian $S$ ending on $\Lambda$, and inverse microlocalization gives a chart $H^1(S, \mathbb{G}_m) \to M_1(\Lambda)$. Moreover a square move $\Gamma \to \Gamma’$ is covered by a Legendrian isotopy $\Lambda \to \Lambda’$, and the corresponding comparison of charts
$$H^1(S, \mathbb{G}_m) \to M_1(\Lambda) = M_1(\Lambda’) \leftarrow H^1(S’, \mathbb{G}_m)$$
is given by the corresponding cluster transformation. This story recovers exactly the corresponding Postnikov/Fock-Goncharov/etc. story.
This setup has a deficiency inherited from the bicolored graphs: not all cluster charts can be obtained from bicolored graphs, hence not all cluster charts can be obtained from the above exact Lagrangians. There may however be other exact Lagrangians ending on $\Lambda$, and David T. has long held the belief that these account for the remaining cluster charts. And indeed:
Corollary of the above conjecture: all cluster charts come from Lagrangians, and mutation is Lagrangian disk surgery.
Proof. By construction, the Lagrangian $S$ above meets the zero section of $T^*X$ in $\Gamma$. In particular, each face of the graph $\Gamma$ gives a Lagrangian disk $D_i$, which is attached to $S$ along a cycle we call $C_i$. Now we work always in the neighborhood of the union of $S \coprod D_i$ as in the above conjecture; its validity establishes the existence and correct behavior of all desired Lagrangians.