A mirror pair of Calabi-Yau four-folds in type II string theory

We give some simple examples of mirror Calabi-Yau four-folds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we explicitly give the mirror transformation which maps Type IIA/IIB on such a four-fold 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 four-folds, 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. (C) 1998 Elsevier Science B.V.