Computing Amplitudes in topological M-theory

TL;DR

This paper introduces a topological quantum membrane theory on G2 manifolds, demonstrating its relation to known topological string models and invariants, and providing a new framework for computing Gopakumar-Vafa and Joyce invariants.

Contribution

It defines a new topological membrane model on G2 manifolds and connects its amplitudes to established topological string invariants and geometrical invariants.

Findings

Reduces to A-model amplitudes for membranes wrapping S^1 in CY_3×S^1

Computes Gopakumar-Vafa invariants at genus zero

Relates membrane amplitudes to Joyce invariants for homology spheres

Abstract

We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\times S^1$ quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. In particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the $CY_3$, we find that the amplitudes of our model are related to Joyce invariants.