Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Abstract

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,\Sigma}$, which we call unbounded $\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs. We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $\mathcal{P}_{PGL_3,\Sigma}$ of Goncharov--Shen, we describe a geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with pinnings via ``shearings''. We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads to a quantum duality map.