论文标题
威尔逊线及其劳伦（Laurent）的积极性
Wilson lines and their Laurent positivity
论文作者
论文摘要
For a marked surface $Σ$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line morphism $g_{[c]}:\mathcal{P}_{G,Σ} \to G$ associated with the homotopy class of an arc $c$ connecting boundary intervals of $Σ$, which is the comparison element of pinnings via parallel-transport. Wilson线的矩阵系数给出了一组生成的函数代数$ \ MATHCAL {O}（\ MATHCAL {p} _ {g，σ}）$当$σ$没有刺穿时。威尔逊线具有相对于Goncharov-Shen [GS19]引入的胶合形态的乘法性质，因此，对于给定的$σ$的理想三角剖分，可以将其分解为三角形。我们表明，矩阵系数$ c_ {f，v}^v（g _ {[c]）$在goncharov-shen坐标系统中给出带有积极积分系数的laurent多项式 - 与任何装饰的三角形相关的$σ$，用于合适的$ f $和$ v $。
For a marked surface $Σ$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line morphism $g_{[c]}:\mathcal{P}_{G,Σ} \to G$ 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 $\mathcal{O}(\mathcal{P}_{G,Σ})$ when $Σ$ has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov--Shen [GS19], 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 $v$.
