Making enumerative predictions by means of mirror symmetry

TL;DR

This paper reviews how mirror symmetry allows precise enumerative predictions of rational curves on Calabi-Yau threefolds, using calculations on the mirror, and discusses conjectures for fixing constants in the mirror map.

Contribution

It provides a detailed review of the mechanics behind mirror symmetry predictions and discusses conjectures for determining the mirror map constants.

Findings

Predictions of rational curve counts via mirror symmetry.

Discussion of conjectures fixing mirror map constants.

Usefulness of predictions for verifying mirror manifold constructions.

Abstract

Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on the other. We review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ``constants of integration'' in the mirror map should be fixed. Such predictions can be useful for checking whether or not various conjectural constructions of mirror manifolds are producing reasonable answers.