Gauge Orbit Types for Generalized Connections

Abstract

Different versions for defining Ashtekar's generalized connections are investigated depending on the chosen smoothness category for the paths and graphs -- the label set for the projective limit. Our definition covers the analytic case as well as the case of webs. Then the orbit types of the generalized connections are determined for compact structure groups. The stabilizer of a connection is homeomorphic to the holonomy centralizer, i.e. the centralizer of its holonomy group, and the homeomorphism class of the gauge orbit is completely determined by the holonomy centralizer. Furthermore, the stabilizers of two connections are conjugate in the gauge group if and only if their holonomy centralizers are conjugate in the structure group. Finally, the gauge orbit type of a connection is defined to be the conjugacy class of its holonomy centralizer equivalently to the standard definition via stabilizers.