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

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

2000-01-08

arxiv.org/abs/math/0001048v2

Commun.Math.Phys. 247 (2004) 527-551

Mathematics

Algebraic Geometry

19 pages, AMSLatex, references added

Scientific paper

10.1007/s00220-004-1078-7

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.

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