$A_{\infty }$-structures on an elliptic curve

Abstract

The main result of this paper is the proof of the "transversal part" of the homological mirror symmetry conjecture for an elliptic curve which states an equivalence of two $A_{\infty }$-structures on the category of vector bundles on an elliptic curves. The proof is based on the study of $A_{\infty }$-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, $m_1$ should be zero, $m_2$ should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to homotopy by certain triple products.

19 pages, AMSLatex, references added

References in corpus (3)

• Introduction to A-infinity algebras and modules
• Homological mirror symmetry with higher products
• Homotopy Algebras are Homotopy Algebras

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