In this paper we partially 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 ${ \rm SL}_3$-webs, we introduce the notion of ${ \rm SL}_3$-laminations on $\frak{ S}$ defined as certain ${ \rm SL}_3$-webs with integer weights. We introduce coordinate systems for ${ \rm SL}_3$-laminations, and show that ${ \rm SL}_3$-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 ${ \rm SL}_3$-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 ${ \rm SL}_3$ quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on $\mathscr{ X}_{ { \rm PGL}_3,\frak{ S}}$. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher `bangles' bases, and the corresponding `bracelets' versions can also be considered as direct analogs of Fock-Goncharov's and Allegretti-Kim's bases for the ${ \rm SL}_2$-${ \rm PGL}_2$ case.
Categories: math.AG math-ph math.GT math.MP math.QA