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

TL;DR

This paper applies a recent mirror map construction to Calabi-Yau hypersurfaces in weighted projective spaces, computing Yukawa couplings and Chern numbers, confirming integrality constraints and matching predictions in special cases.

Contribution

It extends mirror symmetry computations to weighted projective spaces, providing new calculations of Yukawa couplings and Chern numbers where standard methods are insufficient.

Findings

Computed Chern numbers of holomorphic curves on weighted hypersurfaces.

Confirmed integrality constraints predicted by mirror symmetry.

Achieved agreement with algebraic geometry predictions in special cases.

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.