Communications in Mathematical Physics

, Volume 247, Issue 3, pp 527–551 | Cite as

A -Structures on an Elliptic Curve

  • A. Polishchuk


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, 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 equivalence by certain triple products.


Vector Bundle Mirror Symmetry Line Bundle Elliptic Curve Natural Restriction 
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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • A. Polishchuk
    • 1
  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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