SUPERSYMMETRIC CYCLES IN EXCEPTIONAL HOLONOMY MANIFOLDS AND CALABI-YAU 4-FOLDS

We derive in the SCFT and low energy effective action frameworks the n ecessary and sufficient conditions for supersymmetric cycles in except ional holonomy manifolds and Calabi-Yau four-folds. We show that the C ayley cycles in Spirt(7) holonomy eight-manifolds and the associative and coassociative cycles in G(2) holonomy seven-manifolds preserve hal f of the space-time supersymmetry. We find that while the holomorphic and special Lagrangian cycles in Calabi-Yau four-folds preserve half o f the space-time supersymmetry, the Cayley submanifolds are novel as t hey preserve only one quarter of it. We present some simple examples. Finally, we discuss the implications of these supersymmetric cycles on mirror symmetry in higher dimensions.