Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces

Abstract

Fock–Goncharov’s moduli spaces ${\mathcal{X}}_{{\mathrm{PGL}}_3,\mathfrak{S}}$ of framed ${\mathrm{PGL}}_3$-local systems on punctured surfaces $\mathfrak{S}$ provide prominent examples of cluster $\mathcal{X}$-varieties and higher Teichmüller spaces. In a previous paper of the author (2022), building on the works of others, the so-called ${\mathrm{SL}}_3$ quantum trace map is constructed for each triangulable punctured surface $\mathfrak{S}$ and an ideal triangulation $\Delta$ of $\mathfrak{S}$, as a homomorphism from the stated ${\mathrm{SL}}_3$-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster $\mathcal{X}$-chart for ${\mathcal{X}}_{{\mathrm{PGL}}_3,\mathfrak{S}}$ associated to $\Delta$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\mathrm{SL}}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\mathrm{SL}}_3$-${\mathrm{PGL}}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.