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

DOI10.1007/S00220-004-1078-7zbMATH Open1075.14036arXivmath/0001048OpenAlexW1573579263MaRDI QIDQ1885638

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Publication date: 11 November 2004

Published in: (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/math/0001048

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