Skein and cluster algebras of punctured surfaces

We prove the full Fock–Goncharov conjecture for ${\mathcal{A}}_{S{L}_2,{\mathrm{\Sigma }}_{g,p}}$, the $\mathcal{A}$-cluster variety of the moduli of decorated twisted $S{L}_2$-local systems on triangulable surfaces ${\mathrm{\Sigma }}_{g,p}$ with at least 2 punctures. Equivalently, we show that the tagged skein algebra $S{k}^{ta}(\mathrm{\Sigma })$, or the middle cluster algebra $\mathrm{m}\mathrm{i}\mathrm{d}(\mathcal{A})$, coincides with the upper cluster algebra $U(\mathrm{\Sigma })$. Inspired by the work of Shen–Sun–Weng, we introduce the localized cluster variety $\mathring{\mathcal{A}}$ as the algebraic version of the decorated Teichmüller space ${\mathcal{T}}^d(\mathrm{\Sigma })$. We show its global section $\mathrm{\Gamma }(\mathring{\mathcal{A}},{\mathcal{O}}_{\mathring{\mathcal{A}}})$ equals the classical Roger–Yang skein algebra $S{k}_{q\to 1}^{RY}(\mathrm{\Sigma })$, thereby providing a quantization of ${\mathcal{T}}^d(\mathrm{\Sigma })$ in terms of the Roger–Yang skein algebra $S{k}_q^{RY}(\mathrm{\Sigma })$. As a consequence of our geometric characterizations, we deduce normality and the Gorenstein property of the tagged skein algebra $S{k}^{ta}(\mathrm{\Sigma })$ and the classical Roger–Yang skein algebra $S{k}_{q\to 1}^{RY}(\mathrm{\Sigma })$, as well as finite generation of upper cluster algebra $U(\mathrm{\Sigma })$.