We construct a gauge-fixed action for topological membranes on G 2-manifolds such that its bosonic part is the standard membrane theory in a particular gauge. We prove that the path integral in this gauge localizes on associative submanifolds. Moreover on M × S 1, the theory naturally reduces to the standard A-model on Calabi-Yau manifold and to a membrane theory localized on special Lagrangian submanifolds. We discuss some properties of topological membrane theory on G 2-manifolds. We also generalize our construction to topological p-branes on special manifolds by exploring a relation between vector cross product structures and TFTs.
