Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror Symmetry

TL;DR

This paper verifies numerous predictions of mirror symmetry by calculating the number of lines and conics on hypersurfaces, confirming the accuracy of mirror symmetry predictions in algebraic geometry.

Contribution

It extends the verification of mirror symmetry predictions from 3 to 65 cases by explicit calculations of algebraic curves on hypersurfaces.

Findings

Number of verified mirror symmetry predictions increased to 65

Calculations confirm predictions for lines and conics on hypersurfaces

Use of Maple package { extsc{schubert}} for computations

Abstract

Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy certain incidence properties are calculated, and shown to agree with the numbers predicted by Greene, Morrison, and Plesser using mirror symmetry in every instance. This increases the number of verified predictions from 3 to 65. Calculations are performed using the Maple package {\sc schubert} written by Katz and Str{\o}mme.