A Spinorial Hamiltonian Approach to Gravity

TL;DR

This paper develops a spinorial Hamiltonian framework for General Relativity applicable in higher dimensions, generalizing Ashtekar variables and enabling new approaches to problems involving spinors and holonomy groups.

Contribution

It introduces a novel spinorial Hamiltonian formulation for gravity in arbitrary dimensions, extending Ashtekar variables and exploring algebraic properties of spinors in higher dimensions.

Findings

Reduces to Ashtekar variables in 4D

Provides a spinorial ADM formalism in higher dimensions

Offers a new perspective for studying manifolds with special holonomy

Abstract

We give a spinorial set of Hamiltonian variables for General Relativity in any dimension greater than 2. This approach involves a study of the algebraic properties of spinors in higher dimension, and of the elimination of second-class constraints from the Hamiltonian theory. In four dimensions, when restricted to the positive spin-bundle, these variables reduce to the standard Ashtekar variables. In higher dimensions, the theory can either be reduced to a spinorial version of the ADM formalism, or can be left in a more general form which seems useful for the investigation of some spinorial problems such as Riemannian manifolds with reduced holonomy group. In dimensions $0 \pmod 4$, the theory may be recast solely in terms of structures on the positive spin-bundle $\mathbb{V}^+$, but such a reduction does not seem possible in dimensions $2 \pmod 4$, due to algebraic properties of spinors in these dimensions.