Английская Википедия:Gauge group (mathematics)
Шаблон:Short description 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 <math>P\to X </math> with a structure Lie group <math>G</math>, a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group <math>G(X) </math> of global sections of the associated group bundle <math> \widetilde P\to X</math> whose typical fiber is a group <math>G</math> which acts on itself by the adjoint representation. The unit element of <math>G(X) </math> is a constant unit-valued section <math>g(x)=1</math> of <math> \widetilde P\to X</math>.
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 <math>G^0(X) </math> of a gauge group <math>G(X) </math> which is the stabilizer
- <math>G^0(X)=\{g(x)\in G(X)\quad : \quad g(x_0)=1\in \widetilde P_{x_0}\} </math>
of some point <math>1\in \widetilde P_{x_0} </math> of a group bundle <math> \widetilde P\to X</math>. It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, <math> G(X)/G^0(X)=G</math>. One also introduces the effective gauge group <math> \overline G(X)=G(X)/Z</math> where <math>Z</math> is the center of a gauge group <math>G(X) </math>. This group <math> \overline G(X)</math> acts freely on a space of irreducible principal connections.
If a structure group <math> G</math> is a complex semisimple matrix group, the Sobolev completion <math>\overline G_k(X)</math> of a gauge group <math> G(X)</math> can be introduced. It is a Lie group. A key point is that the action of <math>\overline G_k(X)</math> on a Sobolev completion <math>A_k</math> of a space of principal connections is smooth, and that an orbit space <math>A_k/\overline G_k(X)</math> is a Hilbert space. It is a configuration space of quantum gauge theory.
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) Шаблон:ISBN.
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) Шаблон:ISBN
See also
Шаблон:Theoretical-physics-stub
Шаблон:Geometry-stub
