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 ${\mathcal{X}}_{{\mathfrak{sl}}_3,\hat{I}\pounds }$, 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 ${\mathcal{P}}_{PG{L}_3,\hat{I}\pounds }$ 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.

70 pages, 43 figures. Published version (To appear in Algebr. Geom. Topol.), except for an additional correction on a sign in the definition of the Langlands dual coordinates