Calibrated geometries and non-perturbative superpotentials in M-theory

Abstract. We consider non-perturbative effects in M-theory compactifications on a seven-manifold of $G_2$ holonomy arising from membranes wrapped on supersymmetric three-cycles. When membranes are wrapped on associative submanifolds they induce a superpotential that can be calculated using calibrated geometry. This superpotential is also derived from compactification on a seven-manifold, to four dimensional anti-de Sitter spacetime, of eleven dimensional supergravity with non-vanishing expectation value of the four-form field strength.