Gauge group (mathematics)

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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.

It should be emphasized that, 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.


  • 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 Foundarions of Gauge Theory (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

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