On the counting of holomorphic discs in toric Fano manifolds
We first compute three-point open Gromov-Witten numbers of Lagrangian torus fibers in toric Fano manifolds and show that they depend on the choice of three points, hence they are not invariants. Then, we find a sufficient (but restrictive) condition on a family of chains on a Lagrangian submanifold which enables invariant countings of holomorphic discs. This condition is equivalent to being a homology cycle of the coalgebra, which is the anti-symmetrized version of \AI-algebra by Fukaya, Oh, Ohta and Ono. We call this homology the big Floer cohomology, and we show that under some restrictions, the count of holomorphic discs intersecting cycles (in symplectic manifold) at fixed interior marked points, and intersecting a big Floer cohomology cycle at the boundary is invariant under various choices (except for the choice of the complex structure). For the case of the Clifford torus, we obtain a generalized count of holomorphic discs intersecting three cycles at the boundary.