Loop Variable Representation of Classical Higher Dimensional Gravity and the Hilbert Space Grassmannian

TL;DR

This paper develops a loop variable framework for 5+1 dimensional gravity, expressing constraints as polynomials, and explores the associated Hilbert space Grassmannian, advancing the understanding of higher-dimensional quantum gravity.

Contribution

It introduces a novel loop variable representation for 5+1D gravity and analyzes the Grassmannian structure of the Hilbert space, extending loop quantum gravity methods to higher dimensions.

Findings

Constraints expressed as polynomials in loop variables

Master Constraint method applied to resolve constraints

Properties of the Hilbert space Grassmannian analyzed

Abstract

In this paper, an attempt is made to represent 5+1 dimensional gravity (via ADM formalism) in terms of the loop constructions introduced by the author in a companion paper. The "momenta" and "velocity" from the earlier paper, which were proven to be cobordant loops in 6 D; are used as the new loop variables. In the process, the Hamiltonian, Diffeomorphism and Gauss constraints are written in polynomials of these loop variables. Other constraints such as the "Q" constraint and simplicity constraints arise due to greater degrees of freedom. We then undergo the Master Constraint treatment to resolve the constraints. Then, a pre-quantum version of the theory is examined; and the properties of the Grassmannian of the Hilbert Space are explored.