Laminations of punctured surfaces as $\tau$-regular irreducible components

Let $\mathbf{\Sigma }:=(\mathrm{\Sigma },\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset \mathrm{\partial }\mathrm{\Sigma }\mathrm{\ne }\mathrm{\varnothing }$ on the boundary, and punctures $\mathbb{P}\subset \mathrm{\Sigma }\setminus \mathrm{\partial }\mathrm{\Sigma }$, and $T$ an arbitrary tagged triangulation of $\mathbf{\Sigma }$ in the sense of Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T),W(T))$ corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelli and the second author is tame, as shown by Schröer and the first two authors. In this paper, we show that there is a natural isomorphism ${\pi }_T:\mathrm{Lam}(\mathbf{\Sigma })\to {\mathrm{DecIrr}}^{\tau }(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\mathrm{Lam}(\mathbf{\Sigma })$ is the set of laminations of $\mathbf{\Sigma }$ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, ${\mathrm{DecIrr}}^{\tau }(A(T))$ denotes the set of generically $\tau$-regular irreducible components of the decorated representation varieties of $A(T)$, with the direct sum of generically $E$-orthogonal irreducible components as partial monoid operation, where $E$ is the symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky, $E(-,\bullet )=\dim {\mathrm{Hom}}_{A(T)}(-,\tau (\bullet ))+\dim {\mathrm{Hom}}_{A(T)}(\bullet ,\tau (-))$.