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 ${\displaystyle P\to X}$ with a structure Lie group ${\displaystyle G}$, a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group ${\displaystyle G(X)}$ of global sections of the associated group bundle ${\displaystyle {\widetilde {P}}\to X}$ whose typical fiber is a group ${\displaystyle G}$ which acts on itself by the adjoint representation. The unit element of ${\displaystyle G(X)}$ is a constant unit-valued section ${\displaystyle g(x)=1}$ of ${\displaystyle {\widetilde {P}}\to X}$.

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 ${\displaystyle G^{0}(X)}$ of a gauge group ${\displaystyle G(X)}$ which is the stabilizer

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

of some point ${\displaystyle 1\in {\widetilde {P}}_{x_{0}}}$ of a group bundle ${\displaystyle {\widetilde {P}}\to X}$. It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, ${\displaystyle G(X)/G^{0}(X)=G}$. One also introduces the effective 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 semisimple matrix group, the Sobolev 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 a Hilbert space. It is a configuration space of quantum gauge theory.