The Extended Loop Group: An Infinite Dimensional Manifold Associated with the Loop Space

TL;DR

This paper introduces an extended infinite-dimensional manifold structure related to loop space, providing new coordinates and group extensions with applications in field theory, quantum gravity, and knot theory.

Contribution

It proposes a novel extension of the loop group with a new coordinate system, forming an infinite-dimensional Lie group with potential applications in physics and topology.

Findings

Coordinates transform under infinite-dimensional linear representations.

The extended group contains ordinary loops as a subgroup.

The algebraic structure is analyzed in detail.

Abstract

A set of coordinates in the non parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analized in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.