Intersections of Dual $S{L}_3$-Webs

Abstract

We introduce a topological intersection number for an ordered pair of ${\mathrm{SL}}_3$-webs on a decorated surface. Using this intersection pairing between reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs and a collection of $({\mathrm{SL}}_3,\mathcal{X})$-webs associated with the Fock--Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs to the tropical set ${\mathcal{A}}_{{\mathrm{PGL}}_3,\hat{S}}^{+}({\mathbb{Z}}^t)$, as established by Douglas and Sun in \cite{DS20a, DS20b}. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock--Goncharov duality conjecture of higher Teichmüller spaces for ${\mathrm{SL}}_3$.

31 pages, 31 figures