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}= 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}= 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 give ChernWeil definitions of the Maslov indices of bundle pairs over a Riemann surface \Sigma with boundary, which consists of symplectic vector bundle on \Sigma and a Lagrangian subbundle on \Sigma as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularities, via curvature integral, and find also an analogous topological definition in these cases.

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface with boundary, which consists of symplectic vector bundle on and a Lagrangian subbundle on\partial as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.