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

TL;DR

This paper studies polynomial solutions to $rak{g}_2$ affine Toda equations, analyzing the associated almost-complex curves in a pseudo-sphere, their asymptotic polygons, and the related moduli spaces, revealing symmetry properties and geometric structures.

Contribution

It introduces a new geometric correspondence between polynomial sextic differentials and asymptotic polygons, and establishes the existence and uniqueness of solutions to the affine Toda equations.

Findings

Asymptotic boundary of curves forms polygons with degree-related vertices.

The polygons satisfy an annihilator property linked to a $rak{g}_2'$-invariant metric.

The boundary map between moduli spaces is conjectured to be a homeomorphism.

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 $\nu_q: \mathbb{C} \rightarrow \hat{\mathbb{S}}^{2,4}$. 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 $q \in H^0(\mathcal{K}_\mathbb{C}^6)$.