On mirror symmetry for manifolds of exceptional holonomy

We consider Type II string theories on T-n/Z(2)(m) Joyce orbifolds. This class contains orbifolds which can be desingularised to give manifolds of G(2) (n = 7) and Spin(7) holonomy (n = 8). In the G(2) holonomy case we present two types of T-duality transformations which are clearly generalisations of the T-duality/mirror transformation in Calabi-Yau spaces. The first maps Type IIA theory on one such space from this class to Type IIB theory on another such space. The second maps Type IIA (IIB) to Type IIA (IIB). In the case of Spin(7) holonomy we present a T-duality transformation which maps Type IIA (IIB) theory on one such space to Type IIA (IIB) on another such space. As orbifold conformal field theories these T-dual target spaces are related via the inclusion/exclusion of discrete torsion and the T-duality is proven to genus g in string perturbation theory.We then apply a Strominger-Yau-Zaslow-type argument which suggests that manifolds of G2 holonomy which have a "mirror" of the first (second) type admit supersymmetric T-3 (T-4) fibrations and that manifolds of Spin(7) holonomy for which a mirror exists admit fibration by supersymmetric 4-tori. Further evidence for this suggestion is given by examining the moduli space structure of wrapped D-branes. (C) 1998 Elsevier Science B.V.