\(A_\infty\)-structures on an elliptic curve (Q1885638)

This article studies two \(A_\infty\) structures on the category of the algebraic vector bundles on an elliptic curve \(E\); one is built using holomorphic vector bundles on \(E\) and another is a subcategory in the Fukaya \(A_\infty\)-category of the torus \(E\), which naturally comes from the symplectic geometrical approach. By introducing {\`\`}transversal{\'\'} \(A_\infty\)-structures, the author gives a proof of the {\`\`}transversal part{\'\'} of the homological mirror symmetry conjecture on \(E\), which is based on the study of \(A_\infty\) structures on the category of the line bundles on an elliptic curve \(E\) satisfying natural restrictions.