Three-point functions on the sphere of Calabi-Yau d-folds

Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space P-d+1[2, 2, 2,..,2, 2,1,1](2(d + 1)). These three-point functions (O-a(1))O-b((l-1))O-c((d-l)) are expanded by indeterminates ql = e(2 pi itl) associated with a set of Kahler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and infinity On the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kahler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c(1) vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators O-(1)O-(d) = 0 is satisfied). Then these fusion couplings {kappa(l)} are represented as some series adding up all degrees of maps.