In some recent papers, we considered moduli spaces $M(\Sigma, \Lambda)$ (I) or more generally $M(\mathbb{L})$ (II) associated to certain configurations in 4d symplectic geometry. Basically the point was that in this setting, finding an exact Lagrangian in the geometry gives rise to an algebraic torus chart, and disk surgery gives rise to cluster transformations of charts.
Here I want to explain some consequences of these papers which we missed at the time, but have become clear since then. Specifically, the above results come close to showing that the corresponding spaces have a complete cluster structure, i.e., contain all cluster charts, and I will outline now what else is required to finish the argument.
First observe that the spaces constructed in (I) (i.e., those described by legendrian knots which arise as legendrian braid satellites of lifts of simple closed curves) are all quotients of affine varieties by affine algebraic groups. This is basically because such spaces can be glued together from charater spaces of punctured surfaces and the moduli spaces of Stokes data, along the moduli space of local systems on $S^1$, all of the above of which are affine schemes after choosing sufficient framing data. In particular, when all objects in the relevant category $Sh_{\Lambda,1}(\Sigma)$ are simple, the moduli space is a (trivial $\mathbb{G}_m$-torsor over) a smooth affine scheme (!!) (The smoothness because of an $Ext^2$ vanishing I do not describe here.)
On the other hand, a consequence of part (II) is that any moduli space constructed in this manner contains at least one cluster chart, and all of its mutations (!!). In particular this is true for the spaces in (I), including positroid varieties, character varieties, wild character varieties, etc.
Finally, recall that, subject to certain finiteness conditions (I think this subject is discussed in some detail in the Gross-Hacking-Keel papers), one cluster chart plus all its mutations already is, up to codimension 2, the entire cluster variety. In particular, when the cluster variety is affine,
These facts together have a very strong consequence. Below I elide all $X$ versus $A$ versus $A_{prin}$ subtleties.
Corollary. Suppose that (1) all objects in $C_1(\Sigma, \Lambda)$ are simple and (2) the corresponding cluster variety is affine. Then our moduli space $M_1(\Sigma, \Lambda)$ admits a map from the cluster variety.
Proof. Let $Z$ be the union of a single cluster chart and all its mutations. Then there is an inclusion $Z \to M_1(\Sigma, \Lambda)$. Since $M_1(\Sigma, \Lambda)$ is affine, this morphism factors through a morphism from the affinization of $Z$, i.e., from the cluster variety itself.
Remark. Note this requires the result of paper (II) that the mutation at any face cycle of the bipartide graph is contained in the moduli space.
Question. I explained above when the moduli spaces in (I) are certain to be affine. In terms of the curve configurations of (II), when is the moduli space affine?
Expected corollary. There is a complete cluster structure on the positroid strata.
Proof. The positroid strata are affine, and Leclerc has shown that ring of functions on this stratum contains the corresponding cluster algebra as a subalgebra. It follows from this that there’s a morphism from the positroid variety to the cluster variety. On the other hand, we, or anyone else who shows that the positroid variety contains one cluster chart and all its mutations, construct a morphism in the other direction. Presumably one can check that these morphisms are inverse (although I am told that understanding what Leclerc does is nontrivial).
Remark. This result was at least until recently not known, although it follows from a result that has been claimed by Yakimov. It has for a long time been known that the big stratum has a complete cluster structure.
For the other $M_1(\Lambda)$, we do not have the analogue of Leclerc’s result, and even for the positroid strata, I would like to have a purely geometric argument. One way to do this would be to show that $M_1(\Lambda)$ itself is the affinization of the union of one cluster chart and all its mutations, i.e., the complement of these have complex codimension 2 inside $M_1(\Lambda)$. Note that we have an approach towards this in terms of the theory of ruling filtrations. In particular, from Propositions 6.16 and 6.19 of [STZ], plus the translation of these to the circular braid setting via Prop 6.3 of (I), one can deduce:
Corollary. Let $\beta$ be a positive braid. Consider the braid $\Delta^2 \beta$. Consider the moduli space $M_1(S^2, \Delta^2 \beta)$ wherein $\Delta^2 \beta$ has been drawn around a circle inside a sphere. Assume that the braid forms a knot, rather than a link. Then this space is the affinization of the cluster variety.
Proof. The knot-rather-than-link condition ensures that no objects have extra automorphisms; the moduli space is a scheme. In addition to the statements above, it remains only to show that the complement of (a single cluster chart plus all its mutations) is codimension 2. But by Proposition 6.16 of [STZ], we have a stratification of the space enumerated by “rulings”; by Proposition 6.19 of [STZ], all things coming from rulings in which more than two crossings remain unswitched fill up a codimension 2 subset. Finally, a stratum in which a single pair of crossings remain unswitched correponds exactly to the new things in the cluster chart obtained by mutating at the face cycle connecting this pair of unswitched crossings.
Remark. The need for the full twist above may be purely an artifact of not having thought about the correct notion of satellite when we were writing [STZ]. To remove it, one should try and reprove 6.16 and 6.19 of [STZ] using the Legendrian satellite of a braid into the unknot, rather than the “rainbow closure” that we used there, or alternatively, directly in the cuspless circular context above. On the other hand, I do not know for certain that braids not of the form $\Delta^2 \beta$ ever appear as positroid braids. There is a natural significance to $\Delta^2$ — in terms of juggling patterns, it guarantees that every ball which is tossed up falls down within the length of the pattern, which is a requirement for positroid braids.
Remark. We have long believed that the ruling filtration stratification of [STZ] is related to Deodhar stratifications, but have never made this into precise mathematics.
This note records thoughts from during and after a discussion with Thomas Lam and David Speyer.