Wilson lines and their Laurent positivity

Abstract

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 $v$.

arXiv comment58 pages, 13 figures. v3: A major revision shortening the length of the paper by removing minor sections; modifying the presentations of Wilson lines as stack morphisms. v4: Minor corrections of typos. Journal version

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