Superpotentials and Membrane Instantons

We investigate nonperturbative effects in M -theory compactifications arising from wrapped membranes. In particular, we show that in d = 4,N = 1 compactifications along manifolds of G2 holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize manyM -theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invariants of the 3cycle. We discuss briefly some applications of these results. For example, using mirror symmetry we derive a counting formula for supersymmetric three-cycles in certain CalabiYau manifolds.