The Chen-Ruan Cohomology Ring of Mirror Quintic

TL;DR

This paper computes the Chen-Ruan orbifold cohomology ring of the mirror quintic, providing a method for Calabi-Yau hypersurfaces in toric varieties and identifying the obstruction bundle using Riemann bilinear relations.

Contribution

It introduces a general method for computing Chen-Ruan cohomology rings of Calabi-Yau hypersurfaces in toric varieties, including the identification of the obstruction bundle.

Findings

Computed the Chen-Ruan cohomology ring for the mirror quintic.

Outlined a general computational method for Calabi-Yau hypersurfaces.

Proposed a conjecture relating Riemann bilinear relations to the obstruction bundle.

Abstract

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Calabi-Yau hypersurfaces in projective simplicial toric varieties, modulo a conjecture that the Riemann bilinear relations are adequate for identifying the obstruction bundle for any complex orbifold.