Hyphs and the Ashtekar-Lewandowski Measure

TL;DR

This paper explores the structure of the space of generalized connections in the Ashtekar framework, introducing new methods for constructing connections and a generalized notion of path independence, which impacts the properties of measures and projections.

Contribution

It introduces a novel construction method for connections based on hyphs, generalizing directedness and enabling Haar measures for various smoothness categories.

Findings

Projections from the space to lattice gauge theory are surjective and open.

A Haar measure can be defined for any compact structure group.

The new notion of path independence generalizes previous concepts.

Abstract

Properties of the space $\Ab$ of generalized connections in the Ashtekar framework are investigated. First a construction method for new connections is given. The new parallel transports differ from the original ones only along paths that pass an initial segment of a fixed path. This is closely related to a new notion of path independence. Although we do not restrict ourselves to the immersive smooth or analytical case, any finite set of paths depends on a finite set of independent paths, a so-called hyph. This generalizes the well-known directedness of the set of smooth webs and that of analytical graphs, respectively. Due to these propositions, on the one hand, the projections from $\Ab$ to the lattice gauge theory are surjective and open. On the other hand, an induced Haar measure can be defined for every compact structure group irrespective of the used smoothness category for the paths.