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

Abstract

For a finite-dimensional simple Lie algebra g admitting a non-trivial minuscule representation and a connected marked surface Σ with at least two marked points and no punctures, we prove that the cluster algebra Ag,Σ associated with the pair (g, Σ) coincides with the upper cluster algebra Ug,Σ. The proof is based on the fact that the function ring O(A × G,Σ ) of the moduli space of decorated twisted G-local systems on Σ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras Sg,Σ[∂ −1 ] [Mul16, IY21, IY] for g = sl2, sl3, or sp 4 are isomorphic to the cluster algebras Ag,Σ.• Muller [Mul13] proved A = U for locally acyclic cluster algebras. When g = sl 2 and Σ is unpunctured and contains at least two marked points, the cluster algebra A sl 2 ,Σ is locally acyclic, and hence A sl 2 ,Σ = U sl 2 ,Σ .1 In [BMS19], Bucher, Machacek, and Shapiro show that the equality A = U depends on the choice of ground ring. In this paper, we always choose the ground field C. By an easy exercise of linear algebra, all the results of the paper can be generalized to Q.