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

Abstract

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

69 pages, 28 figures. Version 3: Final version after publication

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