The gravitating coupled vortex conjecture for triples on the projective line

Let $T=(E_1,E_2,\phi)$ be a triple over $\mathbb{P}^1$, and let $\mathcal{M}_\sigma=\mathcal{M}_\sigma(n_1,n_2,d_1,d_2)$ denote the moduli space of $\sigma$-polystable triples. The group $\operatorname{SL}(2,\mathbb{C})$ acts naturally on $\mathcal{M}_\sigma$ through its action on $\mathbb{P}^1$.

Gravitating coupled vortex conjecture. The pair $(\mathbb{P}^1,T)$ admits a solution to the gravitating coupled vortex equations if and only if $T$ is $\sigma$-polystable and the point $T\in\mathcal{M}_\sigma$ is GIT polystable for the natural action of $\operatorname{SL}(2,\mathbb{C})$ on $\mathcal{M}_\sigma$ induced by the action of $\operatorname{SL}(2,\mathbb{C})$ on $\mathbb{P}^1$.

The higher-rank existence problem for gravitating coupled vortex equations is described as entirely open in the source. The conjecture is intended as an analogue of the abelian case and connects analytic existence with stability of triples and GIT polystability.