Tropical Fock–Goncharov coordinates for SL 3 -webs on surfaces II: naturality

Source abstract

In a companion article, we constructed nonnegative integer coordinates Φ 𝒯 ( 𝒲 3 , S ^ ) ⊂ ℤ ≥ 0 N for the collection 𝒲 3 , S ^ of reduced SL 3 -webs on a finite-type punctured surface S ^ , depending on an ideal triangulation 𝒯 of S ^ . We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation 𝒯 ′ is chosen, then the coordinate change map relating Φ 𝒯 ( 𝒲 3 , S ^ ) to Φ 𝒯 ′ ( 𝒲 3 , S ^ ) is a tropical 𝒜 -coordinate cluster transformation. We can therefore view the webs 𝒲 3 , S ^ as a concrete topological model for the Fock–Goncharov–Shen positive integer tropical points 𝒜 PGL 3 , S ^ + ( ℤ t ) .