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

TL;DR

This paper establishes the existence of Quantum $A_{ abla}$-relations on the Fukaya category of the elliptic curve, connecting advanced algebraic structures with quantum geometry via operads and master equations.

Contribution

It introduces and proves Quantum $A_{ abla}$-relations on the elliptic curve's Fukaya category using Feynman transforms of modular operads, linking to BV geometry.

Findings

Existence of Quantum $A_{ abla}$-relations proved.

Relations are solutions to the quantum master equation.

Uses Feynman transform of modular operads.

Abstract

We define and prove the existence of the Quantum $A_{\infty}$-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.