Gauge group (mathematics)

Agauge groupis a group ofgauge symmetriesof theYang–Mills gauge theoryofprincipal connectionson aprincipal bundle.Given a principal bundle$P\to X$with a structureLie group$G$,a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group$G(X)$of global sections of the associated group bundle${\widetilde {P}}\to X$whose typical fiber is a group$G$which acts on itself by theadjoint representation.The unit element of$G(X)$is a constant unit-valued section$g(x)=1$of${\widetilde {P}}\to X$.

At the same time,gauge gravitation theoryexemplifiesfield theoryon a principalframe bundlewhose gauge symmetries aregeneral covariant transformationswhich 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$G^{0}(X)$of a gauge group$G(X)$which is the stabilizer

$G^{0}(X)=\{g(x)\in G(X)\quad:\quad g(x_{0})=1\in {\widetilde {P}}_{x_{0}}\}$

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

If a structure group$G$is a complex semisimplematrix group,theSobolev completion${\overline {G}}_{k}(X)$of a gauge group$G(X)$can be introduced. It is a Lie group. A key point is that the action of${\overline {G}}_{k}(X)$on a Sobolev completion$A_{k}$of a space of principal connections is smooth, and that an orbit space$A_{k}/{\overline {G}}_{k}(X)$is aHilbert space.It is aconfiguration spaceof 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)ISBN0-444-89708-9.
• Mangiarotti, L.,Sardanashvily, G.,Connections in Classical and Quantum Field Theory(World Scientific, 2000)ISBN981-02-2013-8