Summary of discussion with David the other night – let me know if anything sounds wrong:

Claim: Let $\Sigma \subset N_\mathbb{R} \cong \mathbb{R}^2$ be a complete fan, $\Lambda_\Sigma \subset T^* (M_\mathbb{R}/M)$ the associated Lagrangian, and $n \in N_\mathbb{R}$.

1. Let $\mu_n: Sh_{cc}(M_\mathbb{R}/M, \Lambda_\Sigma) \to dg-Vect$ take a sheaf to its microstalk at $(M,n)$. If $-n$ is in the interior of a maximal cone $\sigma$ and $p_\sigma$ is the associated T-fixed point, then $\mu_n \circ CCC \cong \Gamma \circ (\mathcal{O}_{p_\sigma} \otimes {-})[2]$. If $-n$ is in the interior of a 1-dimensional cone $\sigma$ and $\mathbb{P}^1_\sigma$ is the associated T-divisor, then $\mu_n \circ CCC \cong \Gamma \circ (\mathcal{O}_{\mathbb{P}^1_\sigma} \otimes {-})[2]$.
2. If $-n$ is in the interior of a 1-dimensional cone $\sigma$, let $\mu’_n: Sh_{cc}(M_\mathbb{R}/M, \Lambda_\Sigma) \to dg-Vect$ the functor that takes a sheaf to its microstalk at $(M+m,n)$, where $m$ is any nonlattice vector in $n^\bot$. Then $\mu’_n \circ CCC \cong \Gamma \circ (\mathcal{O}_{\mathbb{P}^1_\sigma}(-1) \otimes {-})[2]$.

   First a sanity check: the statements all check out for $\mathcal{O}_{X_\Sigma} = CCC(\mathbb{C}_M)$.

   The main ideas of the claim are:

   1. Since taking a microstalk is an exact functor, it is determined by its restriction to the full subcategory of constructible theta sheaves. It is basically straightforward (with David’s help :)) to compute the relevant microstalks of theta sheaves, which tells us how the composition of $CCC$ and taking a microstalk acts on coherent theta sheaves.
   2. Recall that the (nonequivariant) $CCC$ is proved by identifying both sides with the triangulated envelope of a combinatorially-defined category $\Sigma^\vee$ (or $C(\Sigma)$ in David’s paper). Its objects are the duals of cones of $\Sigma$, and $Hom(\sigma^\vee,\tau^\vee)$ is $\tau^\vee$ if $\sigma^\vee \subset \tau^\vee$, zero otherwise. Now a functor $\Sigma^\vee \to dg-Vect$ is essentially the same thing as a sheaf on $X_\Sigma$; more precisely, a quasicoherent sheaf $F$ gives such a functor via $\sigma^\vee \mapsto F(X_\sigma)$. Thus we can think of such a functor in two ways: as a quasicoherent sheaf $F_{qc}$ on $X_\Sigma$ and as a covariant functor $F_{fun}: Perf(X_\Sigma) \to dg-Vect$ since $Perf(X_\Sigma) \cong \Sigma^\vee$ (let’s use subscripts to distinguish the two ways of thinking about $F$). The relationship between the two perspectives is that $F_{fun} \cong \Gamma \circ (F_{qc} \otimes {-})$. Of course, this isn’t a “toric” fact, it’s just some general nonsense about Cech covers.
   3. Now using idea #2 we can unwind idea #1 to produce the purely algebro-geometric descriptions of the compositions of $CCC$ and microstalks stated in the claim.

      In particular, since the $MR\Delta$ locus in $Sh_{cc}(M_\mathbb{R}/M, \Lambda_\Sigma)$ is cut out by the microstalk functors in the claim, the $CCC(MR\Delta)$ locus is cut out by their coherent counterparts given in the claim. And it’s manifestly the case that a torsion-free rank one sheaf on a divisor in the linear system $\Delta$ satisfies these conditions. Essentially, they say a sheaf is $CCC(MR\Delta)$ if the intersection of its support with the divisor at infinity is a specific finite number of points away from the $T$-fixed locus, counted by the ranks of their stalks.

      A feature of the answer is that now the hard part isn’t showing that the things we expect to be $CCC(MR\Delta)$ actually are, but figuring the correct way to “get rid of” all the unwanted stuff on either side of $CCC$. In particular, on the coherent side we can build a lot of crap we aren’t really interested in using skyscraper sheaves (Q: is every $CCC(MR\Delta)$ sheaf the direct sum of a $CCC(MR\Delta’)$ for some other $\Delta’$ and a bunch of skyscraper sheaves?). So a first pass at what we want is something like “mod out by skyscraper sheaves.” This is closely related to the idea of “modding out by local systems” on the constructible side, but seems like a hint that that isn’t actually going far enough. For example, if you take $CCC$ of a skyscraper sheaf at a non-$T$-fixed point on the divisor at infinity, you don’t get a local system, but you do get a sheaf whose stalks all have the same rank (right?). So perhaps we really want to say something like “mod out by all MR whatever sheaves whose stalks all have the same rank”.