Regular Connections among Generalized Connections

TL;DR

This paper investigates the properties of regular connections within the space of generalized connections in the Ashtekar framework, focusing on their density and measure-zero characteristics across various groups and smoothness categories.

Contribution

It establishes conditions under which the space of regular connections is dense or measure-zero in the space of generalized connections, extending to gauge orbits.

Findings

Regular connections are not dense in the space of generalized connections for nontrivial groups.

The space of regular connections has measure zero in the Ashtekar-Lewandowski measure.

Results apply across different structure groups and smoothness categories.

Abstract

The properties of the space $\A$ of regular connections as a subset of the space $\Ab$ of generalized connections in the Ashtekar framework are studied. For every choice of compact structure group and smoothness category for the paths it is determined whether $\A$ is dense in $\Ab$ or not. Moreover, it is proven that $\A$ has Ashtekar-Lewandowski measure zero for every nontrivial structure group and every smoothness category. The analogous results hold for gauge orbits instead of connections.