Naturality of $\mathrm{SL}_{3}$ quantum trace maps for surfaces

Abstract

Fock–Goncharov’s moduli spaces \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} of framed \mathrm{PGL}_{3} -local systems on punctured surfaces \mathfrak{S} provide prominent examples of cluster \mathscr{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 \mathscr{X} -chart for \mathscr{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.