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.
