On topological $M$-theory

Abstract

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 $M\times S^1$, 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.