Joyce's stability and special Lagrangian conjecture

Let $(M,\omega,\Omega)$ be a compact Calabi–Yau $n$-fold, where $\omega$ is a Kähler form and $\Omega$ is a holomorphic volume form. Let $\operatorname{DFuk}(M)$ denote its derived Fukaya category, whose objects are Lagrangian branes $\mathbb{L}$.

Joyce's conjecture. There exists a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on $\operatorname{DFuk}(M)$ such that the central charge is induced by

$$K_0\left(\operatorname{DFuk}(M)\right)\xrightarrow{\mathbb{L}\mapsto[L]}H_n(M,\mathbb{Z})\xrightarrow{[L]\mapsto\int_{[L]}\Omega}\mathbb{C},$$

where $L\subseteq M$ is the underlying Lagrangian of $\mathbb{L}$, and an object $\mathbb{L}$ is $\sigma$-semistable if and only if its underlying Lagrangian admits a possibly singular special Lagrangian representative.

This is a simplified version of Joyce's conjecture relating Bridgeland stability in derived Fukaya categories to special Lagrangians. The conjecture remains largely open; the paper constructs higher-dimensional examples in a non-geometric setting.