The cluster algebras A are a class of commutative algebras equipped with a distinguished family of generators called cluster variables. The upper cluster algebras U is the intersection of Laurent polynomial rings associated with all clusters. By Laurent phenomenon, A⊂U as a subalgebra, but in general they are not equal. For a finite-dimensional simply-connected connected simple Lie group G over C and a connected marked surface Σ, we can associate a cluster algebra AG,Σ.

In this seminar, we introduce a recent work by Ishibashi–Oya–Shen that the cluster algebra AG,Σ coincides with its upper cluster algebra UG,Σ. The main tool is AG,Σ×, the moduli space of decorated twisted G-local systems on Σ, introduced by Fock–Goncharov, and Wilson lines introduced by Ishibashi– Oya. The proof is based on the fact that the function ring O(AG,Σ×) is generated by matrix coefficients of Wilson lines.