Massey and Fukaya Products on Elliptic Curves

This note is a continuation of [6]. Its goal is to show that some higher Massey products on elliptic curve can be computed as higher compositions in Fukaya category of the dual symplectic torus in accordance with the homological mirror conjecture of M. Kontsevich [4]. Namely, we consider triple Massey products of very simple type which are uniquely defined, compute them in terms of theta-functions and compare the result with the series one obtains in Fukaya picture. The identity we get in this way was first discovered by Kronecker. The interesting phenomenon is that although Massey products on elliptic curve are partially defined and multivaled, one always has the corresponding univalued Fukaya product. The conjecture seems to predict that one can choose canonical representatives for all (not necessary well-defined) Massey products such that the A∞-laws are satisfied. In order to understand what kind of functions appear as such representatives we compute higher compositions m3 in Fukaya category of a torus corresponding to four lines forming a trapezoid. The function appearing in this way can be expressed in terms of theta-functions and the basic hypergeometric series 2φ2. We also compute an example of A∞-constraint between m2 and m3 in Fukaya category of a torus. Our computation of univalued triple Massey products on elliptic curve can be easily generalized to the case of higher genus curves. The representation of these products as values of a rational section of some universal bundle is closely related to the cube structure of the theta line bundle on the Jacobian of a curve. We expect also that they can be compared with the Fukaya compositions on the symplectic torus, which is mirror dual to the Jacobian of a curve. These matters will be considered in another paper. Throughout this paper we use the notation e(z) = exp(2πiz).