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.
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