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 {\displaystyle P\to X} with a structure Lie group G {\displaystyle G} , a gauge group is defined to be a group of its vertical automorphisms, that is, its group of bundle automorphisms. This group is isomorphic to the group G ( X ) {\displaystyle G(X)} of global sections of the associated group bundle P ~ → X {\displaystyle {\widetilde {P}}\to X} whose typical fiber is a group G {\displaystyle G} which acts on itself by the adjoint representation. The unit element of G ( X ) {\displaystyle G(X)} is a constant unit-valued section g ( x ) = 1 {\displaystyle g(x)=1} of P ~ → X {\displaystyle {\widetilde {P}}\to 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. 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 ) {\displaystyle G^{0}(X)} of a gauge group G ( X ) {\displaystyle G(X)} which is the stabilizer
G 0 ( X ) = { g ( x ) ∈ G ( X ) : g ( x 0 ) = 1 ∈ P ~ x 0 } {\displaystyle G^{0}(X)=\{g(x)\in G(X)\quad :\quad g(x_{0})=1\in {\widetilde {P}}_{x_{0}}\}}
of some point 1 ∈ P ~ x 0 {\displaystyle 1\in {\widetilde {P}}_{x_{0}}} of a group bundle P ~ → X {\displaystyle {\widetilde {P}}\to 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 {\displaystyle G(X)/G^{0}(X)=G} . One also introduces the effective gauge group G ¯ ( X ) = G ( X ) / Z {\displaystyle {\overline {G}}(X)=G(X)/Z} where Z {\displaystyle Z} is the center of a gauge group G ( X ) {\displaystyle G(X)} . This group G ¯ ( X ) {\displaystyle {\overline {G}}(X)} acts freely on a space of irreducible principal connections. If a structure group G {\displaystyle G} is a complex semisimple matrix group, the Sobolev completion G ¯ k ( X ) {\displaystyle {\overline {G}}_{k}(X)} of a gauge group G ( X ) {\displaystyle G(X)} can be introduced. It is a Lie group. A key point is that the action of G ¯ k ( X ) {\displaystyle {\overline {G}}_{k}(X)} on a Sobolev completion A k {\displaystyle A_{k}} of a space of principal connections is smooth, and that an orbit space A k / G ¯ k ( X ) {\displaystyle A_{k}/{\overline {G}}_{k}(X)} is a Hilbert space. It is a configuration space of quantum gauge theory.
See also Gauge symmetry (mathematics) Gauge theory Gauge theory (mathematics) Principal bundle
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