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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

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

on the tangent bundle $TX$, the affine tangent bundle can be provided with affine bundle coordinates

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

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

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

This affine connection defines a unique linear connection

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

on $TX$, which is associated to a principal connection on $FX$.

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

$A\mathrm{\Gamma }=d{x}^{\lambda }\otimes [{\partial }_{\lambda }+({\mathrm{\Gamma }}_{\lambda }{}^{\mu }{}_{\nu }(x^{\alpha }){\tilde{x}}^{\nu }+s^{\mu }(x^{\alpha })){\tilde{\partial }}_{\mu }],s^{\mu }=-{\mathrm{\Gamma }}_{\lambda }{}^{\mu }{}_{\nu }{a}^{\nu }+{\partial }_{\lambda }{a}^{\mu },$

relative to the affine coordinates (2).

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

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

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

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

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 $A=\mathrm{\Gamma }+\sigma$ of a linear connection $\mathrm{\Gamma }$ 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

$A=d{x}^i\otimes ({\partial }_i+A_i^j(x^k){\tilde{\partial }}_j)$

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

$L_{(\sigma )}=\mu D_i{u}^k{D}^i{u}_k+\frac{\lambda }{2}(D_i{u}^i{)}^2-ϵF^k{}_{ij}{F}_k{}^{ij},$

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 $\mathrm{\Gamma }$ (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 $\theta =d{x}^{\mu }\otimes {\partial }_{\mu }$ is a tautological one-form.

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

$L_{(\sigma )}=\frac{1}{2}[a_1{T}^{\mu }{}_{\nu \mu }{T}_{\alpha }{}^{\nu \alpha }+a_2{T}_{\mu \nu \alpha }{T}^{\mu \nu \alpha }+a_3{T}_{\mu \nu \alpha }{T}^{\nu \mu \alpha }+a_4{ϵ}^{\mu \nu \alpha \beta }{T}^{\gamma }{}_{\mu \gamma }{T}_{\beta \nu \alpha }-\mu {\sigma }^{\mu }{}_{\nu }{\sigma }^{\nu }{}_{\mu }+\lambda {\sigma }^{\mu }{}_{\mu }{\sigma }^{\nu }{}_{\nu }]\sqrt{-g}$,

where ${ϵ}^{\mu \nu \alpha \beta }$ 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 $\mathrm{\Gamma }$ 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.

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.

External links

• G. Sardanashvily, Gravity as a Higgs field. III. Nongravitational deviations of gravitational field, arXiv: gr-qc/9411013.

See also

• Gauge gravitation theory
• Metric-affine gravitation theory
• Connection (affine bundle)
• Dislocations
• Fifth force