The Quantum $A_{\mathrm{\infty }}$-Relations on the Elliptic Curve

We define and prove the existence of the Quantum $A_{\mathrm{\infty }}$-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.