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.

with a structure Lie group

, a gauge group is defined to be a group of its vertical automorphisms, that is, its group of bundle automorphisms.

of global sections of the associated group bundle

whose typical fiber is a group

which acts on itself by the adjoint representation.

The unit element of

is a constant unit-valued section

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

It is called the pointed gauge group.

This group acts freely on a space of principal connections.

One also introduces the effective gauge group

is the center of a gauge group

acts freely on a space of irreducible principal connections.

is a complex semisimple matrix group, the Sobolev completion

A key point is that the action of

of a space of principal connections is smooth, and that an orbit space

is a Hilbert space.

It is a configuration space of quantum gauge theory.

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