# Affine gauge theory

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

## Affine tangent bundle

Being a vector bundle, the tangent bundle ${\displaystyle TX}$ of an ${\displaystyle n}$-dimensional manifold ${\displaystyle X}$ admits a natural structure of an affine bundle ${\displaystyle ATX}$, called the affine tangent bundle, possessing bundle atlases with affine transition functions. It is associated to a principal bundle ${\displaystyle AFX}$ of affine frames in tangent space over ${\displaystyle X}$, whose structure group is a general affine group ${\displaystyle GA(n,{\mathbb {R} })}$.

The tangent bundle ${\displaystyle TX}$ is associated to a principal linear frame bundle ${\displaystyle FX}$, whose structure group is a general linear group ${\displaystyle GL(n,{\mathbb {R} })}$. This is a subgroup of ${\displaystyle GA(n,{\mathbb {R} })}$ so that the latter is a semidirect product of ${\displaystyle GL(n,{\mathbb {R} })}$ and a group ${\displaystyle T^{n}}$ of translations.

There is the canonical imbedding of ${\displaystyle FX}$ to ${\displaystyle AFX}$ onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle ${\displaystyle TX}$ as the affine one.

Given linear bundle coordinates

${\displaystyle (x^{\mu },{\dot {x}}^{\mu }),\qquad {\dot {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\dot {x}}^{\nu },\qquad \qquad (1)}$

on the tangent bundle ${\displaystyle TX}$, the affine tangent bundle can be provided with affine bundle coordinates

${\displaystyle (x^{\mu },{\widetilde {x}}^{\mu }={\dot {x}}^{\mu }+a^{\mu }(x^{\alpha })),\qquad {\widetilde {x}}'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}{\widetilde {x}}^{\nu }+b^{\mu }(x^{\alpha }).\qquad \qquad (2)}$

and, in particular, with the linear coordinates (1).

## Affine gauge fields

The affine tangent bundle ${\displaystyle ATX}$ admits an affine connection ${\displaystyle A}$ which is associated to a principal connection on an affine frame bundle ${\displaystyle AFX}$. In affine gauge theory, it is treated as an affine gauge field.

Given the linear bundle coordinates (1) on ${\displaystyle ATX=TX}$, an affine connection ${\displaystyle A}$ is represented by a connection tangent-valued form

${\displaystyle A=dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }+\sigma _{\lambda }^{\mu }(x^{\alpha })){\dot {\partial }}_{\mu }].\qquad \qquad (3)}$

This affine connection defines a unique linear connection

${\displaystyle \Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\dot {x}}^{\nu }{\dot {\partial }}_{\mu }]\qquad \qquad (4)}$

on ${\displaystyle TX}$, which is associated to a principal connection on ${\displaystyle FX}$.

Conversely, every linear connection ${\displaystyle \Gamma }$ (4) on ${\displaystyle TX\to X}$ is extended to the affine one ${\displaystyle A\Gamma }$ on ${\displaystyle ATX}$ which is given by the same expression (4) as ${\displaystyle \Gamma }$ with respect to the bundle coordinates (1) on ${\displaystyle ATX=TX}$, but it takes a form

${\displaystyle A\Gamma =dx^{\lambda }\otimes [\partial _{\lambda }+(\Gamma _{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\widetilde {x}}^{\nu }+s^{\mu }(x^{\alpha })){\widetilde {\partial }}_{\mu }],\qquad s^{\mu }=-\Gamma _{\lambda }{}^{\mu }{}_{\nu }a^{\nu }+\partial _{\lambda }a^{\mu },}$

relative to the affine coordinates (2).

Then any affine connection ${\displaystyle A}$ (3) on ${\displaystyle ATX\to X}$ is represented by a sum

${\displaystyle A=A\Gamma +\sigma \qquad \qquad (5)}$

of the extended linear connection ${\displaystyle A\Gamma }$ and a basic soldering form

${\displaystyle \sigma =\sigma _{\lambda }^{\mu }(x^{\alpha })dx^{\lambda }\otimes \partial _{\mu }\qquad \qquad (6)}$

Relative to the linear coordinates (1), the sum (5) is brought into a sum ${\displaystyle A=\Gamma +\sigma }$ of a linear connection ${\displaystyle \Gamma }$ and the soldering form ${\displaystyle \sigma }$ (6). In this case, the soldering form ${\displaystyle \sigma }$ (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 ${\displaystyle TX}$) is well defined only on a parallelizable manifold ${\displaystyle X}$.

## 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 ${\displaystyle u(x)\to u(x)+a(x)}$. 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 ${\displaystyle u^{k}}$, ${\displaystyle k=1,2,3}$, of small deformations are determined only with accuracy to gauge translations ${\displaystyle u^{k}\to u^{k}+a^{k}(x)}$.

In this case, let ${\displaystyle X={\mathbb {R} }^{3}}$, and let an affine connection take a form

${\displaystyle A=dx^{i}\otimes (\partial _{i}+A_{i}^{j}(x^{k}){\widetilde {\partial }}_{j})}$

with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients ${\displaystyle A_{l}^{j}}$ describe plastic distortion, covariant derivatives ${\displaystyle D_{j}u^{i}=\partial _{j}u^{i}-A_{j}^{i}}$ coincide with elastic distortion, and a strength ${\displaystyle F_{ji}^{k}=\partial _{j}A_{i}^{k}-\partial _{i}A_{j}^{k}}$ is a dislocation density.

Equations of gauge theory of dislocations are derived from a gauge invariant Lagrangian density

${\displaystyle L_{(\sigma )}=\mu D_{i}u^{k}D^{i}u_{k}+{\frac {\lambda }{2}}(D_{i}u^{i})^{2}-\epsilon F^{k}{}_{ij}F_{k}{}^{ij},}$

where ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ are the Lame parameters of isotropic media. These equations however are not independent since a displacement field ${\displaystyle u^{k}(x)}$ 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 ${\displaystyle X}$, one can consider an affine, but not linear connection on the tangent bundle ${\displaystyle TX}$ of ${\displaystyle X}$. Given bundle coordinates (1) on ${\displaystyle TX}$, it takes the form (3) where the linear connection ${\displaystyle \Gamma }$ (4) and the basic soldering form ${\displaystyle \sigma }$ (6) are considered as independent variables.

As was mentioned above, the soldering form ${\displaystyle \sigma }$ (6) often is treated as a translation gauge field, though it is not a connection. On another side, one mistakenly identifies ${\displaystyle \sigma }$ with a tetrad field. However, these are different mathematical object because a soldering form is a section of the tensor bundle ${\displaystyle TX\otimes T^{*}X}$, whereas a tetrad field is a local section of a Lorentz reduced subbundle of a frame bundle ${\displaystyle FX}$.

In the spirit of the above mentioned gauge theory of dislocations, it has been suggested that a soldering field ${\displaystyle \sigma }$ can describe sui generi deformations of a world manifold ${\displaystyle X}$ which are given by a bundle morphism

${\displaystyle s:TX\ni \partial _{\lambda }\to \partial _{\lambda }\rfloor (\theta +\sigma )=(\delta _{\lambda }^{\nu }+\sigma _{\lambda }^{\nu })\partial _{\nu }\in TX,}$

Then one considers metric-affine gravitation theory ${\displaystyle (g,\Gamma )}$ on a deformed world manifold as that with a deformed pseudo-Riemannian metric ${\displaystyle {\widetilde {g}}^{\mu \nu }=s_{\alpha }^{\mu }s_{\beta }^{\nu }g^{\alpha \beta }}$ when a Lagrangian of a soldering field ${\displaystyle \sigma }$ takes a form

${\displaystyle L_{(\sigma )}={\frac {1}{2}}[a_{1}T^{\mu }{}_{\nu \mu }T_{\alpha }{}^{\nu \alpha }+a_{2}T_{\mu \nu \alpha }T^{\mu \nu \alpha }+a_{3}T_{\mu \nu \alpha }T^{\nu \mu \alpha }+a_{4}\epsilon ^{\mu \nu \alpha \beta }T^{\gamma }{}_{\mu \gamma }T_{\beta \nu \alpha }-\mu \sigma ^{\mu }{}_{\nu }\sigma ^{\nu }{}_{\mu }+\lambda \sigma ^{\mu }{}_{\mu }\sigma ^{\nu }{}_{\nu }]{\sqrt {-g}}}$,
${\displaystyle T^{\alpha }{}_{\nu \mu }=D_{\nu }\sigma ^{\alpha }{}_{\mu }-D_{\mu }\sigma ^{\alpha }{}_{\nu }}$

is the torsion of a linear connection ${\displaystyle \Gamma }$ with respect to a soldering form ${\displaystyle \sigma }$.

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.

## References

• A. Kadic, D. Edelen, A Gauge Theory of Dislocations and Disclinations, Lecture Notes in Physics 174 (Springer, New York, 1983), ISBN 3-540-11977-9
• G. Sardanashvily, O. Zakharov, Gauge Gravitation Theory (World Scientific, Singapore, 1992), ISBN 981-02-0799-9
• C. Malyshev, The dislocation stress functions from the double curl T(3)-gauge equations: Linearity and look beyond, Annals of Physics 286 (2000) 249.