MIRROR SYMMETRY AND AN EXACT CALCULATION OF AN (N–2)-POINT CORRELATION FUNCTION ON A CALABI-YAU MANIFOLD EMBEDDED IN CPN−1

Abstract

We consider an (N–2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of the Fermat type) in CPN−1 and its mirror manifold. We introduce an (N–2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of algebraic geometry. In enumerating the holomorphic curves in the general-dimensional Calabi-Yau manifolds, we extend the method of counting rational curves on the Calabi-Yau three-fold using the Shubert calculus on Gr (2, N). The agreement of the two calculations for the (N–2)-point function establishes “the mirror symmetry at the correlation function level” in the general-dimensional case.