Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality

TL;DR

The paper constructs a concrete topological model for the positive tropical points of the SL3 moduli framework by encoding reduced SL3-webs on marked surfaces into a KTGS cone via tropical coordinates. It proves naturality: coordinate changes under triangulation flips are tropical A-coordinate cluster transformations, yielding a mapping-class-group-equivariant model for $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$. The authors establish the KTGS cone’s structure for the square, giving an explicit Hilbert basis of 22 webs, a system of 10 tropical skein relations, and a 42-sector decomposition aligned with 42 web families. They further develop two linear isomorphisms connecting the web coordinates to rhombus numbers and to tropical X-coordinates, enabling a detailed sector decomposition of the cone and a surjective projection to a 4-dimensional tropical X-space. Together, these results provide a concrete, natural, topological realization of higher-rank Fock–Goncharov duality in rank 3, with potential implications for higher Teichmüller theory, skein algebras, and explicit bases for SL3 character varieties and their quantizations.

Abstract

In a companion paper (arXiv 2011.01768), we constructed nonnegative integer coordinates $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \hat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\hat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\hat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $Φ_\mathscr{T}(\mathscr{W}_{3, \hat{S}})$ to $Φ_{\mathscr{T}^\prime}(\mathscr{W}_{3, \hat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \hat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.

Figures (28)

• Figure 1: Positive tropical integer $\mathcal{A}$-coordinates for a reduced $\mathrm{SL}_3$-web on the once punctured torus, with respect to an ideal triangulation $\mathcal{T}$.
• Figure 2: Local $\mathrm{SL}_3$ tropical $\mathcal{A}$-coordinate cluster transformation, corresponding to a diagonal flip $\mathcal{T} \to \mathcal{T}^\prime$ in the square. See \ref{['eq:boundarycoords']}-\ref{['equation:mu4']}.
• Figure 3: Sectors and walls in the Knutson--Tao--Goncharov--Shen (KTGS) cone $\Phi_\mathcal{T}(\mathcal{W}_{3, \Box}) \subset \mathbb{Z}_{\geq 0}^{12}$ for a triangulated ideal square $(\Box, \mathcal{T})$. More precisely, displayed is a corresponding sector decomposition $\{ D_i \}_{i=1,2,\dots,42}$ of (a projection to $\mathbb{R}^4$ of a real version of) an isomorphic cone in $\mathbb{Z}_+^8 \times \mathbb{Z}^4$, obtained from the KTGS cone via a transformation defined using the 4 tropical integer $\mathcal{X}$-coordinates. The sectors $D_i$ are grouped depending on which orthant of $\mathbb{R}^4$ they belong to. These sectors are the vertices of a 4-valent graph, where two sectors are connected by an edge if and only if they share a wall; equivalently, their topological types differ by a single web. See Theorem \ref{['thm:second-theorem-intro']}.
• Figure 4: Boundary parallel move in the ideal square.
• Figure 5: Split ideal triangulation.

Theorems & Definitions (126)

• Theorem 1
• Theorem 2
• Definition 1.1
• Definition 1.3
• Definition 1.4
• Proposition 1.5
• proof
• Definition 1.7
• Remark 1.8
• Definition 1.9