Homological mirror symmetry on noncommutative two-tori

Abstract

Homological mirror symmetry is a conjecture that a category constructed in the A-model and a category constructed in the B-model are equivalent in some sense. We construct a cyclic differential graded (DG) category of holomorphic vector bundles on noncommutative two-tori as a category in the B-model side. We define the corresponding Fukaya's category in the A-model side, and prove the equivalence of the two categories at the level of cyclic categories. We further write down explicitly Feynman rules for higher Massey products derived from the cyclic DG category. As a background of these arguments, a physical explanation of the mirror symmetry for noncommutative two-tori is also given.