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.

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) ISBN 0-444-89708-9.
• Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8

See also

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