For a marked surface Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line function g[c] : PG,Σ → G associated with the homotopy class of an arc c connecting boundary intervals of Σ. We show that g[c] defines a morphism of algebraic stacks with respect to the algebraic structure on PG,Σ investigated by Shen [She20]. Combining with Shen’s result, we show that the cluster Poisson algebra with respect to the natural cluster Poisson structure on PG,Σ coincides with the ring of global functions with respect to the Betti structure. Moreover we show that the matrix coefficients cVf,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.
|Publication status||Published - 2020 Nov 28|
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