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.
