Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality

TL;DR

This paper demonstrates that the nonnegative integer coordinates for SL3-webs on punctured surfaces are natural and transform according to cluster algebra rules when changing triangulations, linking webs to tropical points.

Contribution

It proves the naturality of the constructed coordinates under triangulation changes, showing they follow cluster transformation rules, thus connecting webs to tropical cluster varieties.

Findings

Coordinates are compatible with triangulation changes.

Coordinate transformations follow cluster algebra rules.

Webs model tropical points in cluster varieties.

Abstract

In a companion paper (arXiv 2011.01768), we constructed nonnegative integer coordinates $\Phi_\mathscr{T}(\mathscr{W}_{3, \hat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \hat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\hat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\hat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $\Phi_\mathscr{T}(\mathscr{W}_{3, \hat{S}})$ to $\Phi_{\mathscr{T}^\prime}(\mathscr{W}_{3, \hat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \hat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.