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


 * $$ A=dx^i\otimes(\partial_i + A^j_i(x^k)\widetilde\partial_j)$$

with respect to the affine bundle coordinates (2). This is a translation gauge field whose coefficients $$ A^j_l$$ describe plastic distortion, covariant derivatives $$D_j u^i =\partial_ju^i- A^i_j$$ coincide with elastic distortion, and a strength $$ F^k_{ji}=\partial_j A^k_i - \partial_i A^k_j$$ is a dislocation density.

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


 * $$ L_{(\sigma)} = \mu D_iu^kD^iu_k + \frac{\lambda}{2}(D_iu^i)^2 - \epsilon F^k{}_{ij}F_k{}^{ij}, $$

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 $$\widetilde g^{\mu\nu}=s^\mu_\alpha s^\nu_\beta g^{\alpha\beta}$$ when a Lagrangian of a soldering field $$\sigma$$ takes a form


 * $$ L_{(\sigma)}=\frac12[a_1T^\mu{}_{\nu\mu} T_\alpha{}^{\nu\alpha}+

a_2T_{\mu\nu\alpha}T^{\mu\nu\alpha}+a_3T_{\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.