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

TL;DR

This paper proves a key part of the homological mirror symmetry conjecture for elliptic curves by classifying $A_{}$-structures on vector bundles, showing they are uniquely determined by triple products.

Contribution

It establishes the equivalence of two $A_{}$-structures on elliptic curve vector bundles, advancing the understanding of mirror symmetry for elliptic curves.

Findings

$A_{}$-structures are uniquely determined by triple products.

The paper confirms the homological mirror symmetry conjecture for the elliptic curve's transversal part.

It characterizes $A_{}$-structures with specific restrictions on the differentials and compositions.

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.