A∞-structures on an Elliptic Curve

Let E be an elliptic curve over a field k. Let us denote by Vect(E) the category of algebraic vector bundles on E, where as space of morphisms from V1 to V2 we take the graded space Hom(V1, V2)⊕Ext (V1, V2) with the natural composition law. In this paper we study extensions of this (strictly associative) composition to A∞-structures on Vect(E) (see section 1 for the definition). The motivation comes from the homological mirror symmetry for elliptic curves formulated by Kontsevich (see [7]) which provides two such extensions in the case k = C and states that they should be equivalent. We recall the definitions of these A∞-structures in section 1.5. One of them is an A∞-version of the derived category of vector bundles while another comes from a general construction in symplectic geometry due to Fukaya. Roughly speaking, one can associate to an indecomposable vector bundle on E a geodesic circle on the torus R/Z with a local system on it. Then the second A∞-structure is defined using generating series counting holomorphic maps from the disk bounding given geodesic circles. In [13] we checked that the double products defined in this way coincide with the standard composition law on Vect(E). In this paper we use this together with some calculations of triple products (see [11]) to prove the essential part of the homological mirror conjecture for E. Namely, we construct a homotopy between transversal products given by these two A∞-structures. This means that we are looking only at the products such that the corresponding geodesic circles form a transversal configuration. The advantage is that in this case the homotopy can be constructed in a canonical way. We leave to a future investigation more subtle points of defining non-transversal products in the Fukaya category and of extending the above homotopy to the entire derived categories. Although we are mainly interested in the case of a complex elliptic curve we formulate a result on A∞-structures on the category L of line bundles on E which is valid over any field k. More precisely, we axiomatize the notion of transversality and prove that if one imposes some natural restrictions on a transversal A∞-structure on L (in particular m1 = 0, m2 is equal to the standard composition) then such a structure is uniquely determined (up to homotopy) by certain triple products. Namely, these are triple products which are invariant under any homotopy. We apply this result to the two A∞-structures arising in the homological mirror symmetry and then use the isogenies between elliptic curves (as in [13]) to construct the required homotopy on the category of vector bundles on E. The natural framework for the generalization of our result which is valid over any field k should involve the notion of a triangulated A∞-category (as sketched in [8]). Our result seems to imply that there exists a unique up to homotopy triangulated A∞-structure on the derived category of an elliptic curve which is compatible with the standard products and with Serre duality (see section 1.3 for the definition of the latter compatibility). Indeed, the triple products appearing in our statement are univalued Massey products which are uniquely determined by the double products in the case when A∞-structure is triangulated. One may hope that such a uniqueness of A∞-structure on the derived category holds for other varieties (e.g. for abelian varieties of arbitrary dimension). The main reason why in the case of A∞-structures