Computation of Yukawa Couplings for Calabi-Yau Hypersurfaces in Weighted Projective Spaces

Abstract

Greene, Morrison and Plesser \cite{GMP} have recently suggested a general method for constructing a mirror map between a $d$-dimensional Calabi-Yau hypersurface and its mirror partner for $d > 3$. We apply their method to smooth hypersurfaces in weighted projective spaces and compute the Chern numbers of holomorphic curves on these hypersurfaces. As anticipated, the results satisfy nontrivial integrality constraints. These examples differ from those studied previously in that standard methods of algebraic geometry which work in the ordinary projective space case for low degree curves are not generally applicable. In the limited special cases in which they do work we can get independent predictions, and we find agreement with our results.