Wilson lines and their Laurent positivity

For a marked surface Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism g [ c ] : P G , Σ → G associated with the homotopy class of an arc c connecting boundary intervals of Σ , which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra O ( P G , Σ ) when Σ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [ 18 ], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of Σ . We show that the matrix coefficients c f , v V ( g [ c ] ) give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of Σ , for suitable f and v .