For instance, these are gauge theory of dislocations in continuous media when
, the generalization of metric-affine gravitation theory when
is a world manifold and, in particular, gauge theory of the fifth force.
admits a natural structure of an affine bundle
, called the affine tangent bundle, possessing bundle atlases with affine transition functions.
onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle
, the affine tangent bundle can be provided with affine bundle coordinates and, in particular, with the linear coordinates (1).
which is associated to a principal connection on an affine frame bundle
, but it takes a form relative to the affine coordinates (2).
is represented by a sum of the extended linear connection
(6) often is treated as a translation gauge field, though it is not a connection.
Let us note that a true translation gauge field (i.e., an affine connection which yields a flat linear connection on
At the same time, one observes such a field in gauge theory of dislocations in continuous media because, in the presence of dislocations, displacement vectors
, of small deformations are determined only with accuracy to gauge translations
describe plastic distortion, covariant derivatives
coincide with elastic distortion, and a strength
are the Lamé parameters of isotropic media.
can be removed by gauge translations and, thereby, it fails to be a dynamic variable.
In gauge gravitation theory on a world manifold
, one can consider an affine, but not linear connection on the tangent bundle
(6) often is treated as a translation gauge field, though it is not a connection.
However, these are different mathematical object because a soldering form is a section of the tensor bundle
, whereas a tetrad field is a local section of a Lorentz reduced subbundle of a frame bundle
In the spirit of the above-mentioned gauge theory of dislocations, it has been suggested that a soldering field
can describe sui generi deformations of a world manifold
Then one considers metric-affine gravitation theory
μ ν α β
is the Levi-Civita symbol, and is the torsion of a linear connection
In particular, let us consider this gauge model in the case of small gravitational and soldering fields whose matter source is a point mass.
Then one comes to a modified Newtonian potential of the fifth force type.