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 P → X with a structure Lie group G , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group G ( X ) of global sections of the associated group bundle P ~ → X whose typical fiber is a group G which acts on itself by the adjoint representation. The unit element of G ( X ) is a constant unit-valued section g ( x ) = 1 of P ~ → X .
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.
It should be emphasized that, 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 G 0 ( X ) of a gauge group G ( X ) which is the stabilizer
G 0 ( X ) = { g ( x ) ∈ G ( X ) : g ( x 0 ) = 1 ∈ P ~ x 0 } of some point 1 ∈ P ~ x 0 of a group bundle P ~ → X . It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, G ( X ) / G 0 ( X ) = G . One also introduces the effective gauge group G ¯ ( X ) = G ( X ) / Z where Z is the center of a gauge group G ( X ) . This group G ¯ ( X ) acts freely on a space of irreducible principal connections.
If a structure group G is a complex semisimple matrix group, the Sobolev completion G ¯ k ( X ) of a gauge group G ( X ) can be introduced. It is a Lie group. A key point is that the action of G ¯ k ( X ) on a Sobolev completion A k of a space of principal connections is smooth, and that an orbit space A k / G ¯ k ( X ) is a Hilbert space. It is a configuration space of quantum gauge theory.
