Gauge group

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.