On topological M-theory

TL;DR

This paper develops a gauge-fixed topological membrane theory on G2-manifolds, showing its localization on associative submanifolds and its reduction to known models like the A-model on Calabi-Yau spaces, with generalizations to other p-branes.

Contribution

It constructs a gauge-fixed action for topological membranes on G2-manifolds and demonstrates its localization properties and reductions to established topological theories.

Findings

Path integral localizes on associative submanifolds.

Reduces to A-model on Calabi-Yau when compactified on S^1.

Generalizes to topological p-branes using vector cross product structures.

Abstract

We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically 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.