Strominger-Yau-Zaslow proposed that the mirror of a Kaehler manifold can be constructed from deformations of fibers of a Lagrangian torus fibration.
We consider a non-commutative deformation space of a set of Lagrangian submanifolds which are not necessarily tori. It possesses an important mirror property, namely there exists a natural functor from the Fukaya category to the category of modules over the deformation space.
General construction of non-commutative mirror will be explained in the first part of the talk. In the latter half, we will apply it to examples of orbifold spheres, and see that the resulting non-commutative algebras are related to the deformation quantization of Fano hypersurfaces.
This is a joint work with C.-H. Cho.