A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a

principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...

. Given a principal bundle

P\to X

with a structure

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

G

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

\widetilde P\to X

. At the same time,

gauge gravitation theory In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. ''Gauge gravitation theory'' should not be confused with th ...

exemplifies field theory on a principal

frame bundle In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...

whose gauge symmetries are

general covariant transformations In physics, general covariant transformations are symmetries of gravitation theory on a world manifold X. They are gauge transformations whose parameter functions are vector fields on X. From the physical viewpoint, general covariant transfor ...

which are not elements of a gauge group. In the physical literature on

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

, 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)=\

of some point

1\in \widetilde P_

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 In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

. 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) . * Mangiarotti, L., Sardanashvily, G., ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000) Differential geometry Gauge theories Theoretical physics {{geometry-stub

See also

References