Floer Cohomology of Torus Fibers and Real Lagrangians in Fano Toric Manifolds

This article deals with the Floer cohomology (with $\mathbb Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. This description is used to show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is nonzero, then the Floer cohomology of the pair is nonzero. Finally, we develop some applications to nondisplaceability and the minimum number of intersection points under Hamiltonian isotopy.