# 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

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.

$A=dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }+\sigma _{\lambda }^{\mu }(x^{\alpha })){\dot {\partial }}_{\mu }].\qquad \qquad (3)$ This affine connection defines a unique linear connection

$\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }{\dot {\partial }}_{\mu }]\qquad \qquad (4)$ $A\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\widetilde {x}}^{\nu }+s^{\mu }(x^{\alpha })){\widetilde {\partial }}_{\mu }],\qquad s^{\mu }=-\Gamma _{\lambda }{}^{\mu }{}_{\nu }a^{\nu }+\partial _{\lambda }a^{\mu },$ relative to the affine coordinates (2).

$A=A\Gamma +\sigma \qquad \qquad (5)$ $\sigma =\sigma _{\lambda }^{\mu }(x^{\alpha })dx^{\lambda }\otimes \partial _{\mu }\qquad \qquad (6)$ 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 Lame 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

$s:TX\ni \partial _{\lambda }\to \partial _{\lambda }\rfloor (\theta +\sigma )=(\delta _{\lambda }^{\nu }+\sigma _{\lambda }^{\nu })\partial _{\nu }\in TX,$ $L_{(\sigma )}={\frac {1}{2}}[a_{1}T^{\mu }{}_{\nu \mu }T_{\alpha }{}^{\nu \alpha }+a_{2}T_{\mu \nu \alpha }T^{\mu \nu \alpha }+a_{3}T_{\mu \nu \alpha }T^{\nu \mu \alpha }+a_{4}\epsilon ^{\mu \nu \alpha \beta }T^{\gamma }{}_{\mu \gamma }T_{\beta \nu \alpha }-\mu \sigma ^{\mu }{}_{\nu }\sigma ^{\nu }{}_{\mu }+\lambda \sigma ^{\mu }{}_{\mu }\sigma ^{\nu }{}_{\nu }]{\sqrt {-g}}$ ,
$T^{\alpha }{}_{\nu \mu }=D_{\nu }\sigma ^{\alpha }{}_{\mu }-D_{\mu }\sigma ^{\alpha }{}_{\nu }$ 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.