Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

The Strominger-Yau-Zaslow approach to homological mirror symmetry starts from a Lagrangian torus fibration on the complement of an anticanonical divisor, and constructs the mirror as a moduli space of weakly unobstructed objects of the Fukaya category supported on the fibers. However, in the presence of holomorphic discs of negative Maslov index, the geometry of the mirror may be deformed beyond the familiar world of Landau-Ginzburg models. We propose a Morse-theoretic construction of the Fukaya-Floer algebra of a family of Lagrangian tori, which recovers a (suitably deformed) Cech model for the algebra of polyvector fields on the mirror, as well as a functor from Lagrangian sections of the SYZ fibration to modules over this algebra.