Wilson lines and their Laurent positivity

For a marked surface $\Sigma$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line morphism $g_{[c]}:\mathcal{P}_{G,\Sigma} \to G$ associated with the homotopy class of an arc $c$ connecting boundary intervals of $\Sigma$, 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 $\mathcal{O}(\mathcal{P}_{G,\Sigma})$ when $\Sigma$ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov--Shen [GS19], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of $\Sigma$. 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 $\Sigma$, for suitable $f$ and $v$.