Stability Conditions on Derived Fukaya Categories and Wall-Crossing Structures in Homological Mirror Symmetry

This monograph investigates the geometric and algebraic landscape of Bridgeland stability conditions on the derived Fukaya category DbF(M) of a Calabi-Yau manifold M, within the framework of Homological Mirror Symmetry (HMS). We rigorously analyze the manifold of stability conditions, Stab(DbF(M)), establishing a correspondence between the central charges of stability conditions and the period integrals of the mirror complex variety. A central focus is the phenomenon of wall-crossing, where the set of stable objects undergoes discontinuous changes across real codimension-one walls in the stability manifold. We explicate the Kontsevich-Soibelman wall-crossing formula, interpreting it through the lens of counting invariants for special Lagrangian submanifolds and their holomorphic disc corrections. Furthermore, we discuss the relationship between these algebraic stability structures and the physical BPS state counting in N=2 supersymmetric field theories, elucidating the role of spectral networks and quadratic differentials in low-dimensional examples.