Joyce's Bridgeland stability conjecture for special Lagrangians

Let $(M^n,\omega,\Omega)$ be a possibly non-compact Calabi–Yau manifold, with Kähler form $\omega$ and holomorphic volume form $\Omega$, suitably convex at infinity. Let $D\!\operatorname{Fuk}(M)$ denote its Fukaya category, whose objects are Lagrangian branes $\mathbb{L}$. Joyce's conjecture. There exists a natural stability condition $(Z,\mathcal{P})$ on $D\!\operatorname{Fuk}(M)$ such that the central charge $Z$ is the composition of natural maps

$$K_0(D\!\operatorname{Fuk}(M)) \xrightarrow{\mathbb{L} \mapsto [\mathcal{L}]} H_n(M,\mathbb{Z}) \xrightarrow{[\mathcal{L}] \mapsto \int_{[\mathcal{L}]}\Omega} \mathbb{C},$$

where $\mathcal{L}\subset M$ is the Lagrangian submanifold underlying $\mathbb{L}$, and each isomorphism class of semistable objects of phase $\pi\phi$—equivalently, each isomorphism class in $\mathcal{P}(\phi)$—contains a unique representative Lagrangian brane $\mathbb{L}$ whose underlying Lagrangian submanifold is a possibly immersed, singular special Lagrangian of phase $e^{\operatorname{i}\pi\phi}$, meaning a weak solution of

$$\iota^*_{\mathcal{L}}\operatorname{Im}(e^{-\operatorname{i}\pi\phi}\Omega)=0.$$

The conjecture proposes a Bridgeland-stability description of special Lagrangians, with the period of the holomorphic volume form providing the central charge. The paper presents its results as consistent with Joyce's interpretation of the Thomas–Yau conjecture, but the general statement recalled here is not established in the source.