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

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 ${\nu }_q:\mathbb{C}\to {\widehat{\mathbb{S}}}^{2,4}$. The asymptotic boundary $\mathrm{\Delta }:={\mathrm{\partial }}_{\mathrm{\infty }}({\nu }_q)$ of ${\nu }_q$ is found to be a polygon in ${\mathsf{E}\mathsf{i}\mathsf{n}}^{2,3}$ with $\mathsf{d}\mathsf{e}\mathsf{g}q+6$ vertices. The polygon $\mathrm{\Delta }$ satisfies an \emph{annihilator property}, which is related to a ${\mathsf{G}}_2^{\mathrm{\prime }}$-invariant discrete metric $d_3:{\mathsf{E}\mathsf{i}\mathsf{n}}^{2,3}\times {\mathsf{E}\mathsf{i}\mathsf{n}}^{2,3}\to \{0,1,2,3\}$ on ${\mathsf{E}\mathsf{i}\mathsf{n}}^{2,3}$. In fact, we show ${\mathsf{G}}_2^{\mathrm{\prime }}=\mathsf{I}\mathsf{s}\mathsf{o}\mathsf{m}(d_3)\cap \mathsf{D}\mathsf{i}\mathsf{f}\mathsf{f}({\mathsf{E}\mathsf{i}\mathsf{n}}^{2,3})$. The asymptotic boundary defines a map $\alpha :{\mathsf{M}\mathsf{S}}_k\to {\mathsf{M}\mathsf{P}}_{k+6}$ 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 $f_q:\mathbb{C}\to {\mathsf{G}}_2^{\mathrm{\prime }}\mathrm{/}K$ in the symmetric space ${\mathsf{G}}_2^{\mathrm{\prime }}\mathrm{/}K$, showing how to realize their mutual harmonic lift to ${\mathsf{G}}_2^{\mathrm{\prime }}\mathrm{/}T$ 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 $q\in H^0({\mathcal{K}}_{\mathbb{C}}^6)$.