Special Lagrangian sections and stability conditions on threefolds

We study a class of Lagrangian submanifolds, given by sections of a special Lagrangian fibration, contained in certain almost Calabi-Yau threefolds (mirrors of polarised toric threefolds satisfying suitable assumptions). We show that, for a Lagrangian section $L$ in this class, the shift $L[2]$ defines an object in the heart of a natural Bridgeland stability condition on the relevant Fukaya-Seidel category, and that if $L[2]$ is semistable with respect to this stability condition, then it is isomorphic to a special Lagrangian. For mirrors of weak Fanos, the central charge of the stability condition is very close to periods of the holomorphic volume form. These results are consistent with Joyce's interpretation of the Thomas-Yau conjecture. As part of the proof we describe a set of line bundles and polarisations on suitable toric threefolds for which semistability with respect to the Bridgeland stability conditions constructed by Bernardara-Macrì-Schmidt-Zhao implies the existence of a deformed Hermitian Yang-Mills connection.