An Algebraic Topological Construct of Classical Loop Gravity and the prospect of Higher Dimensions

TL;DR

This paper reformulates classical gravity using algebraic topological loop constructs, introduces a loop calculus, and explores the potential for extending general relativity to higher dimensions through novel loop variables.

Contribution

It develops a new algebraic topological framework for classical gravity based on loop groups and introduces a loop calculus with applications to higher-dimensional theories.

Findings

Loop products define Maurer-Cartan structure in the loop space.

New loop variables 'momenta' and 'velocity' behave as cobordant loops in higher dimensions.

A toy model demonstrates the algebraic topological approach to gravity.

Abstract

In this paper, classical gravity is reformulated in terms of loops, via an algebraic topological approach. The main component is the loop group, whose elements consist of pairs of cobordant loops. A Chas-Sullivan product is described on the cobordism, and three other products, namely the 'vertical', 'horizontal' and 'total' products are re-introduced. (They have already been defined in an earlier paper by the author). A loop calculus is introduced on the space of loops, consisting of the loop variation functional,the loop derivative, Mandelstam derivative, and what the author wishes to call, the Gambini-Pullin contact functional. The loop derivative happens to be a generator of the group of loops, and the Gambini-Pullin functional is an infinitesimal generator of diffeomorphisms. A toy model of gravity is formulated in terms of the above, and it is proven that the total product provides for the Maurer-Cartan structure of the space. Further, new quantities labeled as 'momenta', 'velocity' and 'energy' are introduced in terms of the loop products and loop derivatives. The prospect of re-constructing general relativity in higher dimensions is explored, with the 'momenta' and 'velocity' being the basic loop variables. In the process, it is proven that the 'momenta' and 'velocity' behave as cobordant loops in higher dimensions.