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

Abstract

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 ${\nu }_q:\mathbb{C}\to {\hat{\mathbb{S}}}^{2,4}$. 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 ${\mathsf{G}}_2^{\prime }$-invariant discrete metric $d_3:{\mathsf{Ein}}^{2,3}\times {\mathsf{Ein}}^{2,3}\to \{0,1,2,3\}$ on ${\mathsf{Ein}}^{2,3}$. In fact, we show ${\mathsf{G}}_2^{\prime }=\mathsf{Isom}(d_3)\cap \mathsf{Diff}({\mathsf{Ein}}^{2,3})$. The asymptotic boundary defines a map $\alpha :{\mathsf{MS}}_k\to {\mathsf{MP}}_{k+6}$ 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 $f_q:\mathbb{C}\to {\mathsf{G}}_2^{\prime }/K$ in the symmetric space ${\mathsf{G}}_2^{\prime }/K$, showing how to realize their mutual harmonic lift to ${\mathsf{G}}_2^{\prime }/T$ 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 $q\in H^0({\mathcal{K}}_{\mathbb{C}}^6)$.

arXiv commentThe main results and structure are the same, but thorough edits have been implemented for clarity and precision. Most notably -- the proof of Lemma 4.11 in the first version was incorrect, and has been replaced with a different argument, and Conjecture 8.13 (from the old version) was updated slightly