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

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.