Gauge group (mathematics)

A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle with a structure Lie group , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group of global sections of the associated group bundle whose typical fiber is a group which acts on itself by the adjoint representation. The unit element of is a constant unit-valued section of . At the same time, gauge gravitation theory exemplifies field theory on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group. In the physical literature on gauge theory, a structure group of a principal bundle often is called the gauge group. In quantum gauge theory, one considers a normal subgroup of a gauge group which is the stabilizer of some point of a group bundle . It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, . One also introduces the effective gauge group where is the center of a gauge group . This group acts freely on a space of irreducible principal connections. If a structure group is a complex semisimple matrix group, the Sobolev completion of a gauge group can be introduced. It is a Lie group. A key point is that the action of on a Sobolev completion of a space of principal connections is smooth, and that an orbit space is a Hilbert space. It is a configuration space of quantum gauge theory.