Cluster varieties and coordinate systems

$A_2$-laminations as basis for ${\rm PGL}_3$ cluster variety for surface

In this paper we settle Fock-Goncharov's duality conjecture for cluster
varieties associated to their moduli spaces of ${\rm G}$-local systems on a
punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of
type $A_2$, namely ${\rm SL}_3$ and ${\rm PGL}_3$. Based on Kuperberg's
$A_2$-web, we introduce the notion of $A_2$-laminations on $\frak{S}$ defined
as certain $A_2$-webs with integer weights. We introduce coordinate systems for
$A_2$-laminations, and show that $A_2$-laminations satisfying a congruence
property are geometric realizations of the tropical integer points of the
cluster $\mathscr{A}$-moduli space $\mathscr{A}_{{\rm SL}_3,\frak{S}}$. Per
each such $A_2$-lamination, we construct a regular function on the cluster
$\mathscr{X}$-moduli space $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. We show that
these functions form a basis of the ring of all regular functions. For a proof,
we develop ${\rm SL}_3$ classical trace map for any triangulated bordered
surface with marked points, and a state-sum formula for it. Moreover, we
propose generalization to higher ranks.