The Lagrangian Cubic Equation

TL;DR

This paper investigates algebraic identities involving the quantum homology class of Lagrangian submanifolds, especially spheres, and explores their connections with Floer theory invariants in symplectic geometry.

Contribution

It introduces new identities for the quantum homology class of Lagrangians and relates these to Floer invariants, focusing on Lagrangian spheres.

Findings

Derived identities involving the quantum homology class [L]

Established connections between quantum homology and Floer invariants

Analyzed special cases for Lagrangian spheres

Abstract

Let $M$ be a closed symplectic manifold and $L \subset M$ a Lagrangian submanifold. Denote by $[L]$ the homology class induced by $L$ viewed as a class in the quantum homology of $M$. The present paper is concerned with properties and identities involving the class $[L]$ in the quantum homology ring. We also study the relations between these identities and invariants of $L$ coming from Lagrangian Floer theory. We pay special attention to the case when $L$ is a Lagrangian sphere.