Wilson lines and their Laurent positivity · arXivDesk
2011.14260Nov 29, 202058 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
For a marked surface Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism g[c]:PG,Σ→G
Nearby in the stack
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
O(PG,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
cf,vV(g[c])
give Laurent polynomials with positive integral coefficients in the Goncharov--Shen coordinate system associated with any decorated triangulation of