Wilson lines and their Laurent positivity

For a marked surface $$\Sigma $$Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism $$g_{[c]}:{\mathcal {P} }_{G,\Sigma } \rightarrow G$$g [ c ] : P G , Σ → G associated with the homotopy class of an arc c connecting boundary intervals of $$\Sigma $$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 $$\mathcal {O}({\mathcal {P} }_{G,\Sigma })$$O ( P G , Σ ) when $$\Sigma $$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 $$\Sigma $$Σ . We show that the matrix coefficients $$c_{f,v}^V(g_{[c]})$$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 $$\Sigma $$Σ , for suitable f and v.