Lagrangian Floer theory on compact toric manifolds, I

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We introduced the notion of weakly unobstructed Lagrangian submanifolds and constructedtheir potential function (PO) purely in terms of A-model data in [FOOO3].In this article, we carry out explicit calculations involving PO on toric manifoldsand study the relationship between this class of Lagrangian submanifolds with theearlier work of Givental [G1], which advocates that the quantum cohomology ringis isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburgsuperpotential. Combining this study with the results from [FOOO3], we also applythe study to various examples to illustrate its implications to symplectic topology ofLagrangian fibers of toric manifolds. In particular, we relate it to the Hamiltoniandisplacement property of Lagrangian fibers and to Entov-Polterovich’s symplecticquasi-states.