Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

Abstract

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2\pi} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2\pi} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.