Membranes for Topological M-Theory

TL;DR

This paper develops a topological membrane theory on G_2 holonomy manifolds, linking it to topological string theories and exploring its implications for topological M-theory.

Contribution

It formulates a novel topological membrane theory with G_2 holonomy, connecting it to topological strings and M-theory, and introduces a new BRST framework.

Findings

Reduction yields topological A-model strings.

Action is BRST-exact up to topological terms.

Links topological membranes to M-theory and string theories.

Abstract

We formulate a theory of topological membranes on manifolds with G_2 holonomy. The BRST charges of the theories are the superspace Killing vectors (the generators of global supersymmetry) on the background with reduced holonomy G_2. In the absence of spinning formulations of supermembranes, the starting point is an N=2 target space supersymmetric membrane in seven euclidean dimensions. The reduction of the holonomy group implies a twisting of the rotations in the tangent bundle of the branes with ``R-symmetry'' rotations in the normal bundle, in contrast to the ordinary spinning formulation of topological strings, where twisting is performed with internal U(1) currents of the N=(2,2) superconformal algebra. The double dimensional reduction on a circle of the topological membrane gives the strings of the topological A-model (a by-product of this reduction is a Green-Schwarz formulation of topological strings). We conclude that the action is BRST-exact modulo topological terms and fermionic equations of motion. We discuss the role of topological membranes in topological M-theory and the relation of our work to recent work by Hitchin and by Dijkgraaf et al.