$\mathcal{A}=\mathcal{U}$ for cluster algebras from moduli spaces of $G$-local systems

Abstract

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\Sigma$ with at least two marked points and no punctures, we prove that the cluster algebra ${\mathcal{A}}_{\mathfrak{g},\Sigma }$ associated with the pair $(\mathfrak{g},\Sigma )$ coincides with the upper cluster algebra ${\mathcal{U}}_{\mathfrak{g},\Sigma }$. The proof is based on the fact that the function ring $\mathcal{O}({\mathcal{A}}_{G,\Sigma }^{\times })$ of the moduli space of decorated twisted $G$-local systems on $\Sigma$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras ${\mathcal{S}}_{\mathfrak{g},\Sigma }[{\partial }^{-1}]$ [Muller,IY23,IY22] for $\mathfrak{g}={\mathfrak{sl}}_2,{\mathfrak{sl}}_3,$ or ${\mathfrak{sp}}_4$ are isomorphic to the cluster algebras ${\mathcal{A}}_{\mathfrak{g},\Sigma }$.

41 pages, 20 figures. v2: Section 6 is added. v3: Typos are corrected. Journal version

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