Homological Perturbation Theory and Homological Mirror Symmetry

Abstract

In this article, we discuss an application of homological perturbation theory (HPT) to homological mirror symmetry (HMS) based on Kontsevich and Soibelman’s proposal [Kontsevich, M., Soibelman, Y. (2001) Homological mirror symmetry and torus fibrations]. After a brief review of Morse theory, Morse homotopy and the corresponding Fukaya categories, we explain the idea of deriving a Fukaya category from a DG category via HPT, which is expected to give a solution to HMS, and apply it to the cases of \({\mathbb{R}}^{2}\) discussed in [Kajiura, H. (2007) An A ∞ -structure for lines in a plane] and then T2. A finite dimensional A ∞ -algebra obtained from the Fukaya category on T2 is also presented.