A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle.
with a structure Lie group
, a gauge group is defined to be a group of its vertical automorphisms, that is, its group of bundle automorphisms.
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
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
It is called the pointed gauge group.
This group acts freely on a space of principal connections.
One also introduces the effective gauge group
is the center of a gauge group
acts freely on a space of irreducible principal connections.
is a complex semisimple matrix group, the Sobolev completion
A key point is that the action of
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.
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