Computing amplitudes in topological M-theory

We define a topological quantum membrane theory on a seven dimensional manifold of G(2) holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is CY3 x S-1 quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. In particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the CY3, we find that the amplitudes of our model are related to Joyce invariants.