Categorical Mirror Symmetry and Lagrangian Floer Theory for Calabi-Yau Hypersurfaces

This monograph establishes a rigorous categorical framework for Homological Mirror Symmetry (HMS) in the context of Calabi-Yau hypersurfaces within projective space. We construct the derived Fukaya category DF(M) associated with the symplectic A-model, detailing the A-algebraic structures governing Lagrangian Floer cohomology over the Novikov field. Concurrently, we analyze the B-model via the derived category of coherent sheaves Db(Coh(W)) on the mirror manifold. Central to our exposition is the construction of an A-functor inducing an equivalence of triangulated categories, as conjectured by Kontsevich. We address the transversality issues inherent in Lagrangian intersection theory through the method of Kuranishi structures and virtual fundamental chains. Furthermore, we explore the deformation theory of the Fukaya category via Maurer-Cartan elements in the Hochschild cohomology, providing a comprehensive proof of HMS for the quintic threefold case.