3-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 ca ses of one-parameter families of d-folds realized as Fermat type hyper surfaces embedded in weighted projective spaces and a two-parameter fa mily 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 coefficient s count the number of maps with a definite degree which map each of th e three-points 0, 1 and infinity On the world sheet on some homology c ycle of M associated with a cohomology element. From these analyses, w e can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology rin g of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associa ted with Kahler forms of M. For that reason, the charge conservation o f operators turns out to be a classical one. Furthermore, because thei r first Chern classes c(1) vanish, their topological selection rules d o not depend on the degree of maps (in particular, a nilpotent propert y of operators O-(1)O-(d) = 0 is satisfied). Then these fusion couplin gs {kappa(l)} are represented as some series adding up all degrees of maps.