Thomas–Yau–Joyce stability conjecture for special Lagrangians and dHYM solutions

Let $\check{\mathcal{X}}$ be the mirror geometry, let $L$ be a Lagrangian in $\check{\mathcal{X}}$, and let $\mathcal{E}$ be a holomorphic vector bundle on $\mathcal{X}$. A Bridgeland stability condition is a stability condition on the relevant derived category. Thomas–Yau–Joyce conjecture. There is a Bridgeland stability condition on $D^b\operatorname{Fuk}(\check{\mathcal{X}})$, respectively on $D^b\operatorname{Coh}(\mathcal{X})$, such that $L$, respectively $\mathcal{E}$, is stable if and only if $L$ contains a special Lagrangian, respectively $\mathcal{E}$ admits a metric solving the deformed Hermitian–Yang–Mills equation. This conjectural correspondence is motivated by mirror symmetry and the Donaldson–Uhlenbeck–Yau picture; the source presents it as a broad proposal, with substantial analytic and algebro-geometric cases still unresolved.