# Affine gauge theory

Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold $X$. For instance, these are gauge theory of dislocations in continuous media when $X=\mathbb {R} ^{3}$, the generalization of metric-affine gravitation theory when $X$ is a world manifold and, in particular, gauge theory of the fifth force.

## Affine tangent bundle

Being a vector bundle, the tangent bundle $TX$ of an $n$-dimensional manifold $X$ admits a natural structure of an affine bundle $ATX$, called the affine tangent bundle, possessing bundle atlases with affine transition functions. It is associated to a principal bundle $AFX$ of affine frames in tangent space over $X$, whose structure group is a general affine group $GA(n,\mathbb {R} )$.
The tangent bundle $TX$ is associated to a principal linear frame bundle $FX$, whose structure group is a general linear group $GL(n,\mathbb {R} )$. This is a subgroup of $GA(n,\mathbb {R} )$ so that the latter is a semidirect product of $GL(n,\mathbb {R} )$ and a group $T^{n}$ of translations.
There is the canonical imbedding of $FX$ to $AFX$ onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle $TX$ as the affine one.
Given linear bundle coordinates
    $(x^{\mu },{\dot {x}}^{\mu }),\qquad {\dot {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\dot {x}}^{\nu },\qquad \qquad (1)$
on the tangent bundle $TX$, the affine tangent bundle can be provided with affine bundle coordinates
    $(x^{\mu },{\widetilde {x}}^{\mu }={\dot {x}}^{\mu }+a^{\mu }(x^{\alpha })),\qquad {\widetilde {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\widetilde {x}}^{\nu }+b^{\mu }(x^{\alpha }).\qquad \qquad (2)$
and, in particular, with the linear coordinates (1).

## Affine gauge fields

The affine tangent bundle $ATX$ admits an affine connection $A$ which is associated to a principal connection on an affine frame bundle $AFX$. In affine gauge theory, it is treated as an affine gauge field.
Given the linear bundle coordinates (1) on $ATX=TX$, an affine connection $A$ is represented by a connection tangent-valued form
    $A=dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }^{\mu }_{\nu }(x^{\alpha }){\dot {x}}^{\nu }+\sigma _{\lambda }^{\mu }(x^{\alpha })){\dot {\partial }}_{\mu }].\qquad \qquad (3)$
This affine connection defines a unique linear connection
    $\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+\Gamma _{\lambda }^{\mu }_{\nu }(x^{\alpha }){\dot {x}}^{\nu }{\dot {\partial }}_{\mu }]\qquad \qquad (4)$
on $TX$, which is associated to a principal connection on $FX$.
Conversely, every linear connection $\Gamma$ (4) on $TX\to X$ is extended to the affine one $A\Gamma$ on $ATX$ which is given by the same expression (4) as $\Gamma$ with respect to the bundle coordinates (1) on $ATX=TX$, but it takes a form
    $A\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }^{\mu }_{\nu }(x^{\alpha }){\widetilde {x}}^{\nu }+s_{\lambda }^{\mu }(x^{\alpha })){\widetilde {\partial }}_{\mu }],\qquad s_{\lambda }^{\mu }=-\Gamma _{\lambda }^{\mu }_{\nu }a^{\nu }+\partial _{\lambda }a^{\mu },$
relative to the affine coordinates (2).
Then any affine connection $A$ (3) on $ATX\to X$ is represented by a sum
    $A=A\Gamma +\sigma \qquad \qquad (5)$
of the extended linear connection $A\Gamma$ and a basic soldering form
    $\sigma =\sigma _{\lambda }^{\mu }(x^{\alpha })dx^{\lambda }\otimes \partial _{\mu }\qquad \qquad (6)$
on $TX$, where ${\dot {\partial }}_{\mu }=\partial _{\mu }$ due to the canonical isomorphism $VATX=ATX\times _{X}TX$ of the vertical tangent bundle $VATX$ of $ATX$.
Relative to the linear coordinates (1), the sum (5) is brought into a sum $A=\Gamma +\sigma$ of a linear connection $\Gamma$ and the soldering form $\sigma$ (6). In this case, the soldering form $\sigma$ (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 $TX$) is well defined only on a parallelizable manifold $X$.

## Gauge theory of dislocations

In field theory, one meets a problem of physical interpretation of translation gauge fields because there are no fields subject to gauge translations $u(x)\to u(x)+a(x)$. 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 $u^{k}$, $k=1,2,3$, of small deformations are determined only with accuracy to gauge translations $u^{k}\to u^{k}+a^{k}(x)$.
In this case, let $X=\mathbb {R} ^{3}$, and let an affine connection take a form
    $A=dx^{i}\otimes (\partial _{i}+A_{i}^{j}(x^{k}){\widetilde {\partial }}_{j})$
with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients $A_{l}^{j}$ describe plastic distortion, covariant derivatives $D_{j}u^{i}=\partial _{j}u^{i}-A_{j}^{i}$ coincide with elastic distortion, and a strength $F_{ji}^{k}=\partial _{j}A_{i}^{k}-\partial _{i}A_{j}^{k}$ is a dislocation density.
Equations of gauge theory of dislocations are derived from a gauge invariant Lagrangian density
    $L_{(\sigma )}=\mu D_{i}u^{k}D^{i}u_{k}+{\frac {\lambda }{2}}(D_{i}u^{i})^{2}-\epsilon F^{k}_{ij}F_{k}^{ij},$
where $\mu$ and $\lambda$ are the Lamé parameters of isotropic media. These equations however are not independent since a displacement field $u^{k}(x)$ can be removed by gauge translations and, thereby, it fails to be a dynamic variable.

## Gauge theory of the fifth force

In gauge gravitation theory on a world manifold $X$, one can consider an affine, but not linear connection on the tangent bundle $TX$ of $X$. Given bundle coordinates (1) on $TX$, it takes the form (3) where the linear connection $\Gamma$ (4) and the basic soldering form $\sigma$ (6) are considered as independent variables.
As was mentioned above, the soldering form $\sigma$ (6) often is treated as a translation gauge field, though it is not a connection. On another side, one mistakenly identifies $\sigma$ with a tetrad field. However, these are different mathematical object because a soldering form is a section of the tensor bundle $TX\otimes T^{*}X$, whereas a tetrad field is a local section of a Lorentz reduced subbundle of a frame bundle $FX$.
In the spirit of the above-mentioned gauge theory of dislocations, it has been suggested that a soldering field $\sigma$ can describe sui generi deformations of a world manifold $X$ which are given by a bundle morphism
    $s:TX\ni \partial _{\lambda }\to \partial _{\lambda }\rfloor (\theta +\sigma )=(\delta _{\lambda }^{\nu }+\sigma _{\lambda }^{\nu })\partial _{\nu }\in TX,$
where $\theta =dx^{\mu }\otimes \partial _{\mu }$ is a tautological one-form.
Then one considers metric-affine gravitation theory $(g,\Gamma )$ on a deformed world manifold as that with a deformed pseudo-Riemannian metric ${\widetilde {g}}^{\mu \nu }=s_{\alpha }^{\mu }s_{\beta }^{\nu }g^{\alpha \beta }$ when a Lagrangian of a soldering field $\sigma$ takes a form
    $L_{(\sigma )}={\frac {1}{2}}[a_{1}T^{\mu }_{\nu \mu }T_{\alpha }^{\nu \alpha }+a_{2}T_{\mu \nu \alpha }T^{\mu \nu \alpha }+a_{3}T_{\mu \nu \alpha }T^{\nu \mu \alpha }+a_{4}\epsilon ^{\mu \nu \alpha \beta }T^{\gamma }_{\mu \gamma }T_{\beta \nu \alpha }-\mu \sigma ^{\mu }_{\nu }\sigma ^{\nu }_{\mu }+\lambda \sigma ^{\mu }_{\mu }\sigma ^{\nu }_{\nu }]{\sqrt {-g}}$,
where $\epsilon ^{\mu \nu \alpha \beta }$ is the Levi-Civita symbol, and
    $T^{\alpha }_{\nu \mu }=D_{\nu }\sigma ^{\alpha }_{\mu }-D_{\mu }\sigma ^{\alpha }_{\nu }$
is the torsion of a linear connection $\Gamma$ with respect to a soldering form $\sigma$.
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.