Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$

For solutions to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ with respect to \emph{polynomial} holomorphic sextic differential $q$, we study the associated almost-complex curves $\nu_q: \mathbb{C} \rightarrow \hat{\mathbb{S}}^{2,4}$. The asymptotic boundary $\Delta := \partial_{\infty}(\nu_q)$ of $\nu_q$ is found to be a polygon in $\mathsf{Ein}^{2,3}$ with $\mathsf{deg} q + 6$ vertices. The polygon $\Delta$ satisfies an \emph{annihilator property}, which is related to a $\mathsf{G}_2'$-invariant discrete metric $d_3: \mathsf{Ein}^{2,3} \times \mathsf{Ein}^{2,3} \rightarrow \{0,1,2,3\}$ on $\mathsf{Ein}^{2,3}$. In fact, we show $\mathsf{G}_2' = \mathsf{Isom}(d_3) \cap \mathsf{Diff}(\mathsf{Ein}^{2,3})$. The asymptotic boundary defines a map $\alpha: \mathsf{MS}_{k} \rightarrow \mathsf{MP}_{k+6}$ between the equidimensional moduli spaces of holomorphic polynomial sextic differentials of degree $k$ and of annihilator polygons with $k+6$ vertices and is conjectured to be a homeomorphism onto its image. We also discuss the relationship between $\nu_q$ and a related minimal surface $f_q: \mathbb{C} \rightarrow \mathsf{G}_2'/K$ in the symmetric space $\mathsf{G}_2'/K$, showing how to realize their mutual harmonic lift to $\mathsf{G}_2'/T$ geometrically. Before beginning the geometry, we prove the existence and uniqueness of a complete (real) solution to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ associated to polynomial $q \in H^0(\mathcal{K}_\mathbb{C}^6)$.