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 equi ...
. Given a principal bundle
with a structure Lie group
, a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group
of global sections of the associated group bundle
whose typical fiber is a group
which acts on itself by the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
. The unit element of
is a constant unit-valued section
of
.
At the same time,
gauge gravitation theory exemplifies
field theory on a principal
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to 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 nat ...
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
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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
, one considers a normal subgroup
of a gauge group
which is the stabilizer
:
of some point
of a group bundle
. It is called the ''pointed gauge group''. This group acts freely on a space of principal connections. Obviously,
. One also introduces the ''effective gauge group''
where
is the center of a gauge group
. This group
acts freely on a space of irreducible principal connections.
If a structure group
is a complex semisimple
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
, the
Sobolev completion of a gauge group
can be introduced. It is a Lie group. A key point is that the action of
on a Sobolev completion
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.
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)
See also
*
Gauge symmetry (mathematics)
*
Gauge theory
*
Gauge theory (mathematics)
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the ...
*
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 equi ...
Differential geometry
Gauge theories
Theoretical physics
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