A∞-Structures on an Elliptic Curve

The main result of this paper is the proof of the ‘‘transversal part’’ of the homological mirror symmetry conjecture for an elliptic curve that states an equivalence of two A∞-categories: one is built using holomorphic vector bundles on an elliptic curve and another is a subcategory in the Fukaya A∞-category of a torus. The proof is based on the study of A∞-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, m1 should be zero, m2 should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to equivalence by certain triple products.