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 $$P\to X $$ with a structure Lie 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 the adjoint 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 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 $$G^0(X) $$ 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 the pointed gauge group. This group acts freely on a space of principal connections. Obviously, $$ G(X)/G^0(X)=G$$. One also introduces the effective 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 semisimple matrix group, the Sobolev 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 a Hilbert space. It is a configuration space of quantum gauge theory.