Gauge Orbit Types for Theories with Classical Compact Gauge Group

TL;DR

This paper classifies the orbit types of gauge transformations acting on connections for classical compact gauge groups over 4-dimensional simply connected manifolds, completing a comprehensive classification for these groups.

Contribution

It provides a complete classification of orbit types for all classical compact gauge groups on 4-manifolds, including new results on SO(n) bundles and Howe subgroups.

Findings

Classification of principal SO(n) bundles over 4-manifolds

Complete orbit type classification for all classical groups

Identification of Howe subgroups of SO(n)

Abstract

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or $Sp(n)$ over a closed, simply connected manifold of dimension 4. Complemented with earlier results on U(n) and SU(n) this completes the classification of the orbit types for all classical compact gauge groups over such space-time manifolds. On the way we derive the classification of principal bundles with structure group SO(n) over these manifolds and the Howe subgroups of SO(n).