Lagrangian Floer theory on compact toric manifolds I

TL;DR

This paper explicitly calculates the potential function for weakly unobstructed Lagrangian submanifolds in toric manifolds, linking Floer theory, quantum cohomology, and Landau-Ginzburg models to deepen understanding of symplectic topology.

Contribution

It provides explicit computations of the potential function on toric manifolds and explores its connection to quantum cohomology and symplectic invariants, advancing Lagrangian Floer theory.

Findings

Explicit formulas for potential functions on toric manifolds

Connection between quantum cohomology and Jacobian rings

Implications for Lagrangian fiber displacements and symplectic quasi-states

Abstract

The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO2], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states.