A Mirror Pair of Calabi-Yau Fourfolds in Type II String Theory

TL;DR

This paper constructs explicit examples of mirror Calabi-Yau fourfolds in Type II string theory using toroidal orbifolds, confirming mirror symmetry and exploring its implications for M-theory and gauge theory dualities.

Contribution

It provides explicit mirror transformations for Calabi-Yau fourfolds realized as toroidal orbifolds, including the role of discrete torsion and implications for M-theory dualities.

Findings

Mirror symmetry confirmed for genus g in string path integral.

Explicit mirror transformations involving discrete torsion.

Mirror symmetry exchanges Coulomb and Higgs branches in M-theory.

Abstract

We give some simple examples of mirror Calabi-Yau fourfolds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we give explicitly the mirror transformation which maps Type IIA/IIB on such a fourfold to Type IIA/IIB on the mirror. The mirrors are related by the inclusion/exclusion of discrete torsion. Implicit in the result is a confirmation of mirror symmetry to genus $g$ in the string path integral. Finally, by considering the relationship between M-theory and Type IIA theory, we show how in M-theory on mirror Calabi-Yau fourfolds, mirror symmetry exchanges the Coulomb branch with (one of the) Higgs branches of the theory. This result is relevant to duality in N=2 supersymmetric gauge theories in three dimensions.