A quantization of topological M theory

TL;DR

This paper proposes a method to quantize topological M theory using Hamiltonian decomposition and loop quantum gravity techniques, resulting in a diffeomorphism-invariant framework with a new basis of states.

Contribution

It introduces a novel quantization approach for topological M theory, connecting it to LQG methods and providing a basis for diffeomorphism-invariant states.

Findings

The theory has 2 degrees of freedom per point.

The Hamiltonian constraint is polynomial and regulatable.

Complex and symplectic structures are non-commuting.

Abstract

A conjecture is made as to how to quantize topological M theory. We study a Hamiltonian decomposition of Hitchin's 7-dimensional action and propose a formulation for it in terms of 13 first class constraints. The theory has 2 degrees of freedom per point, and hence is diffeomorphism invariant, but not strictly speaking topological. The result is argued to be equivalent to Hitchin's formulation. The theory is quantized using loop quantum gravity methods. An orthonormal basis for the diffeomorphism invariant states is given by diffeomorphism classes of networks of two dimensional surfaces in the six dimensional manifold. The hamiltonian constraint is polynomial and can be regulated by methods similar to those used in LQG. To connect topological M theory to full M theory, a reduction from 11 dimensional supergravity to Hitchin's 7 dimensional theory is proposed. One important conclusion is that the complex and symplectic structures represent non-commuting degrees of freedom. This may have implications for attempts to construct phenomenologies on Calabi-Yau compactifications.