Orbifold and orientifold compactifications of F-theory and M-theory to six and four dimensions

We study orbifold compactifications of F-theory which lead to N = 1 supersymmetry in six and four space-time dimensions. These are dual to specific orientifolds of M-theory, and in many cases to orientifolds of type IIB string theory. The equivalences are demonstrated by mapping the orbifolding transformations in the F-, M- and string theories to each other using dualities, We observe that M- and F-theory appear to possess a property similar to discrete torsion in string theory. This is related to an ambiguity recently noted by Polchinski in the orientifold projection for six-dimensional models, The four-dimensional compactifications exhibit similar features, from which we predict the existence of certain new orientifolds of type IIB, Some orbifolds with higher supersymmetry are also examined.