Wilson lines and their Laurent positivity

TL;DR

This paper demonstrates that Wilson line functions on moduli spaces of local systems for marked surfaces can be expressed as Laurent polynomials with positive coefficients in certain coordinate systems, revealing their positivity properties.

Contribution

It establishes the Laurent positivity of Wilson line matrix coefficients in Goncharov--Shen coordinates for surfaces without punctures, connecting geometric and algebraic structures.

Findings

Wilson lines generate the function algebra of the moduli space.

Wilson lines decompose into triangular pieces via ideal triangulations.

Matrix coefficients are Laurent polynomials with positive coefficients.

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 $v$.