Bounded ${\mathfrak{sp}}_4$-laminations and their intersection coordinates

Abstract

We introduce rational bounded ${\mathfrak{sp}}_4$-laminations on a marked surface $\hat{\mathbf{I}}\pounds$ as a proposed topological model for the rational tropical points ${\mathcal{A}}_{S{p}_4,\hat{\mathbf{I}}\pounds }({\mathbb{Q}}^{\mathsf{T}})$ of the Fock--Goncharov moduli space [FG06]. Our space consists of certain equivalence classes of ${\mathfrak{sp}}_4$-webs introduced by Kuperberg [Kup96], together with rational measures. We define tropical coordinate systems using the ${\mathfrak{sp}}_4$-case of the intersection number of Shen--Sun--Weng [SSW25], and establish a bijection using the framework of the graded ${\mathfrak{sp}}_4$-skein algebra. This provides a topological perspective for Fock--Goncharov duality for ${\mathfrak{sp}}_4$.

56 pages, 19 figures