In this talk, no $G_2$ background is assumed and all relevant terminology will be defined.
We
discuss the non-abelian Hodge theory on the punctured sphere for the split real Lie group $G’_2$. We
study almost-complex curves $\nu_q : C \rightarrow {\mathbb S}^{2,4}$ in the pseudosphere ${\mathbb S}^{2,4}$ associated to polynomial sextic
differential $q$.
Focusing on the asymptotic geometry, we detect stable regions and critical lines
where the limits of $\nu$ along rays change. Moreover, we find such polynomial almost-complex curves
have polygonal boundaries in $Ein^{2,3}$ satisfying a condition we call the annihilator property. Time
permitting, we discuss a conjectural homeomorphism from a moduli space of sextic differentials to
a moduli space of annihilator polygons.