Stratification of the Generalized Gauge Orbit Space

TL;DR

This paper analyzes the structure of the space of generalized connections under gauge transformations, establishing stratification, openness, and denseness of certain orbit types, and linking gauge orbit types to Howe subgroups.

Contribution

It introduces a topological stratification of the generalized gauge orbit space and proves key properties like openness and denseness of strata, extending previous Sobolev connection results.

Findings

The gauge orbit space is topologically regularly stratified.

Strata are open and dense within the space of generalized connections.

The set of all gauge orbit types corresponds to conjugacy classes of Howe subgroups.

Abstract

The action of Ashtekar's generalized gauge group $\Gb$ on the space $\Ab$ of generalized connections is investigated for compact structure groups $\LG$. First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\Ab$. This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\Ab$ is topologically regularly stratified by $\Gb$. These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\LG$. Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.