Gauge group (mathematics) explained
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
G0(X)=\{g(x)\inG(X) : g(x0)=1\in\widetilde
\}
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.
See also
References
- Mitter, P., Viallet, C., On the bundle of connections and the gauge orbit manifold in Yang – Mills theory, Commun. Math. Phys. 79 (1981) 457.
- Marathe, K., Martucci, G., The Mathematical Foundations of Gauge Theories (North Holland, 1992) .
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000)
