View source for Gauge group (mathematics)

A '''gauge group''' is a group of [[gauge symmetry (mathematics)|gauge symmetries]] of the [[Yang–Mills theory|Yang – Mills gauge theory]] of [[principal connection]]s 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 [[covariant classical field theory|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 <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 space|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 [[path integral formulation|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., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000) {{ISBN|981-02-2013-8}}

== See also ==
*[[Gauge theory]]
*[[Principal bundle]]
*[[Gauge symmetry (mathematics)]]

[[Category:Gauge theories]]
[[Category:Theoretical physics]]