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Agauge group is a group ofgauge symmetries of theYang–Mills gauge theory ofprincipal connections on aprincipal bundle. Given a principal bundle${\displaystyle P\to X}$ with a structureLie group${\displaystyle G}$, 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${\displaystyle G(X)}$ of global sections of the associated group bundle${\displaystyle \tilde{P}\to X}$ whose typical fiber is a group${\displaystyle G}$ which acts on itself by theadjoint representation. The unit element of${\displaystyle G(X)}$ is a constant unit-valued section${\displaystyle g(x)=1}$ of${\displaystyle \tilde{P}\to X}$.

At the same time,gauge gravitation theory exemplifiesfield theory on a principalframe bundle whose gauge symmetries aregeneral covariant transformations which are not elements of a gauge group.

In the physical literature ongauge theory, a structure group of a principal bundle often is called thegauge group.

Inquantum gauge theory, one considers a normal subgroup${\displaystyle G^0(X)}$ of a gauge group${\displaystyle G(X)}$ which is the stabilizer

${\displaystyle G^0(X)=\{g(x)\in G(X):g(x_0)=1\in {\tilde{P}}_{x_0}\}}$

of some point${\displaystyle 1\in {\tilde{P}}_{x_0}}$ of a group bundle${\displaystyle \tilde{P}\to X}$. It is called thepointed gauge group. This group acts freely on a space of principal connections. Obviously,${\displaystyle G(X)/G^0(X)=G}$. One also introduces theeffective gauge group${\displaystyle \overline{G}(X)=G(X)/Z}$ where${\displaystyle Z}$ is the center of a gauge group${\displaystyle G(X)}$. This group${\displaystyle \overline{G}(X)}$ acts freely on a space of irreducible principal connections.

If a structure group${\displaystyle G}$ is a complex semisimplematrix group, theSobolev completion${\displaystyle {\overline{G}}_k(X)}$ of a gauge group${\displaystyle G(X)}$ can be introduced. It is a Lie group. A key point is that the action of${\displaystyle {\overline{G}}_k(X)}$ on a Sobolev completion${\displaystyle A_k}$ of a space of principal connections is smooth, and that an orbit space${\displaystyle A_k/{\overline{G}}_k(X)}$ is aHilbert space. It is aconfiguration space of quantum gauge theory.

• Gauge symmetry (mathematics)
• Gauge theory
• Gauge theory (mathematics)
• Principal bundle

• 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

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