Topological membranes

It is suggested that topological membranes play a fundamental role

in the recently proposed topological M-theory. We formulate a topological theory

of membranes wrapping associative three-cycles in a seven-dimensional target

space with G_2 holonomy. The topological BRST rules and BRST invariant action

are constructed via the Mathai-Quillen formalism. We construct a set of local

and non-local observables for the topological membrane theory. As the BRST

cohomology of local operators turns out to be isomorphic to the de Rham

cohomology of the G_2 manifold, our observables agree with the spectrum of

d=4, N=1 G_2 compactifications of M-theory.