Lagrangian Floer theory on compact toric manifolds I

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.