Membranes for topological M-theory

We formulate a theory of topological membranes on manifolds with G_2holonomy. The BRST charges of the theories are the superspace Killing vectors(the generators of global supersymmetry) on the background with reducedholonomy G_2. In the absence of spinning formulations of supermembranes, thestarting point is an N=2 target space supersymmetric membrane in seveneuclidean dimensions. The reduction of the holonomy group implies a twisting ofthe rotations in the tangent bundle of the branes with ``R-symmetry'' rotationsin the normal bundle, in contrast to the ordinary spinning formulation oftopological strings, where twisting is performed with internal U(1) currents ofthe N=(2,2) superconformal algebra. The double dimensional reduction on acircle of the topological membrane gives the strings of the topological A-model(a by-product of this reduction is a Green-Schwarz formulation of topologicalstrings). We conclude that the action is BRST-exact modulo topological termsand fermionic equations of motion. We discuss the role of topological membranesin topological M-theory and the relation of our work to recent work by Hitchinand by Dijkgraaf et al.Comment: 22 pp, plain tex. v2: refs. adde