Gauge group (mathematics)
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]\displaystyle{ P\to X }[/math] with a structure Lie group [math]\displaystyle{ G }[/math], a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group [math]\displaystyle{ G(X) }[/math] of global sections of the associated group bundle [math]\displaystyle{ \widetilde P\to X }[/math] whose typical fiber is a group [math]\displaystyle{ G }[/math] which acts on itself by the adjoint representation. The unit element of [math]\displaystyle{ G(X) }[/math] is a constant unit-valued section [math]\displaystyle{ g(x)=1 }[/math] of [math]\displaystyle{ \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]\displaystyle{ G^0(X) }[/math] of a gauge group [math]\displaystyle{ G(X) }[/math] which is the stabilizer
- [math]\displaystyle{ G^0(X)=\{g(x)\in G(X)\quad : \quad g(x_0)=1\in \widetilde P_{x_0}\} }[/math]
of some point [math]\displaystyle{ 1\in \widetilde P_{x_0} }[/math] of a group bundle [math]\displaystyle{ \widetilde P\to X }[/math]. It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, [math]\displaystyle{ G(X)/G^0(X)=G }[/math]. One also introduces the effective gauge group [math]\displaystyle{ \overline G(X)=G(X)/Z }[/math] where [math]\displaystyle{ Z }[/math] is the center of a gauge group [math]\displaystyle{ G(X) }[/math]. This group [math]\displaystyle{ \overline G(X) }[/math] acts freely on a space of irreducible principal connections.
If a structure group [math]\displaystyle{ G }[/math] is a complex semisimple matrix group, the Sobolev completion [math]\displaystyle{ \overline G_k(X) }[/math] of a gauge group [math]\displaystyle{ G(X) }[/math] can be introduced. It is a Lie group. A key point is that the action of [math]\displaystyle{ \overline G_k(X) }[/math] on a Sobolev completion [math]\displaystyle{ A_k }[/math] of a space of principal connections is smooth, and that an orbit space [math]\displaystyle{ 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) ISBN:0-444-89708-9.
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN:981-02-2013-8
See also
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