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The non-Abelian Weyl\(-\)Yang\(-\)Kaluza\(-\)Klein gravity model

Abstract

The Weyl\(-\)Yang gravitational gauge theory is investigated in the structure of a pure higher-dimensional non-Abelian Kaluza\(-\)Klein background. We construct the dimensionally reduced field equations and stress-energy-momentum tensors as well as the four dimensional modified Weyl\(-\)Yang\(+\)Yang\(-\)Mills theory from an arbitrary curved \(internal\) space in the anholonomic frame which are the extensions of our previous model for the non-Abelian case. In particular, the coset space reduction is considered to explicitly obtain the interactions between the gravitational and the gauge fields. The resulting equations not only appear to be generalization of the well-established equations of non-Abelian theory but also contain intrinsically the generalized gravitational source term which does not exist in the literature so far and the Yang\(-\)Mills force density which is exactly equivalent to the negative gradient of a Yang\(-\)Mills quadratic invariant.

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Acknowledgments

I would like to thank S. Başkal for bringing references [89] and [104] into my attention.

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Correspondence to Halil Kuyrukcu.

Appendix: The non-Abelian Kaluza\(-\)Klein theory in the anholonomic frame

Appendix: The non-Abelian Kaluza\(-\)Klein theory in the anholonomic frame

A brief review to the major steps of non-Abelian KK framework unifying gravitation and Yang\(-\)Mills theories in more than five dimensions is appropriate. First, let us remember some basic notions of that theory in the usual way for convenient reading. In what follows, the Greek indices \(\mu ,\nu ,\ldots =0,\ldots ,3\) refer to the \(external\) 4D flat Minkowski (Ricci flat) spacetime \(M_{4}\), admitting the metric \({g}_{\mu \nu }(x)\) with usual signature and collectively coordinates \(x\in M_{4}\). The Latin indices \(i,j,\ldots =5,\ldots ,(4+N)\) refer to the curved \(internal\) \(N\)-dimensional compact sub-space \(M_{N}\) (such as simply the hypersphere or hypertorus), admitting the metric \({g}_{ij}(y)\) with Euclidean signature and collectively coordinates \(y\in M_{N}\), whereas the Latin capital indices \(A,B,\ldots =0,\ldots ,3,5,\ldots ,(4+N)\) refer to the whole \((4+N)\)-dimensional Minkowski space \(M_{4+N}\), with the metric \(\hat{g}_{AB}(x,y)\) and associated with the collectively event \(z\in M_{4+N}\), \(z=(x,y)\). The quantities with/without the hat symbol demonstrate the ones in the \((4+N)\)-dimensional entire space/in the usual 4D \(external\) space. The stable ground state of the generalized 5D KK theory is assumed to be, at least locally, a direct space product of the form \(M_{4+N}=M_{4}\times M_{N}\) which is a trivial principal bundle over a \(M_{4}\) with fibres \(M_{N}\), instead of assuming to be only \(M_{4+N}\). Finally, the Greek indices \(\alpha ,\beta ,\ldots \) refer to any compact isometry (Lie) group \(G\) of \(M_{N}\), running over the rank of \(G\), i.e. \(\alpha ,\beta ,\ldots =1,\ldots ,dim(G)\). It does not matter whether the indices of internal space are in upper or lower form. The isometries of the \(internal\) manifold produce linearly independent space-like Killing vector fields \(\xi ^{i}_{\alpha }(y)\) each corresponding to a metric symmetry in an elegant way. The symmetries of \(M_{N}\) appear to be gauge group in the real 4D world for the effective observer as to be the 5D KK approach. The Killing vector fields \(\xi ^{i}_{\alpha }(y)\) respectively satisfy the Lie’s equation

$$\begin{aligned}{}[\xi _{\alpha },\xi _{\beta }]^{i}\equiv \xi ^{j}_{\alpha }\partial _{j}\xi ^{i}_{\beta } -\xi ^{j}_{\beta }\partial _{j}\xi ^{i}_{\alpha }&= -f_{\alpha \beta } \,^{\gamma }\xi ^{i}_{\gamma }, \end{aligned}$$
(70)

corresponding to the Lie algebra by the Lie bracket and the following isometry condition

$$\begin{aligned} \mathcal {L}_{\xi }g_{ij}\equiv \xi ^{\alpha k}\partial _{k}g_{ij}+g_{ik}\partial _{j}\xi ^{\alpha k}+g_{jk}\partial _{i}\xi ^{\alpha k}=0, \end{aligned}$$
(71)

which gives Killing’s equation \(D_{(i}\xi ^{\alpha }_{j)}=0\). Here, \(f_{\alpha \beta } \,^{\gamma }\) are the real antisymmetric \(f_{\alpha \beta } \,^{\gamma }=-f_{\beta \alpha } \,^{\gamma }\) structure constants of \(G\). The \((4+N)\)-dimensional metric \(\hat{g}_{AB}(x,y)\) can be written in terms of 1-forms massless gauge fields (Yang\(-\)Mills vector bosons) \(A^{\alpha }_{\mu }(x)\) of the arbitrary gauge group \(G\) and the Killing vector fields \(\xi ^{i}_{\alpha }(y)\) in the higher-dimensional spacetime \(M_{4}\times M_{N}\) as follows

$$\begin{aligned} \hat{g}_{AB}(x,y)=\left( \begin{array}{c|c} {g}_{\mu \nu }(x)+ {g}_{ij}(y)\xi ^{\alpha i}(y)\xi ^{\beta j}(y)A^{\alpha }_{\mu }(x)A^{\beta }_{\nu }(x) &{} {g}_{ij}(y)\xi ^{\alpha i}(y)A^{\alpha }_{\mu }(x) \\ \hline {g}_{ij}(y)\xi ^{\beta j}(y)A^{\beta }_{\nu }(x) &{} {g}_{ij}(y) \\ \end{array} \right) \end{aligned}$$
(72)

It is very useful and convenient to make the metric \(\hat{g}_{AB}(x,y)\) block diagonal for calculations. It can be achieved by choosing the basis in the so-called anholonomic (noncoordinate, horizontal lift) basis [12] with

$$\begin{aligned} \hat{E}^{\mu }(x)=dx^{\mu },\quad \hat{E}^{i}(x,y)=dy^{i}+\xi ^{\alpha i}(y)A^{\alpha }_{\mu }(x)dx^{\mu }. \end{aligned}$$
(73)

The dual basis can be found by the help of the identity \(\hat{E}^{A}\hat{\iota }_{B}=\hat{\delta }^{A}\,_{B}\) in the following forms

$$\begin{aligned} \hat{\iota }_{\mu }(x,y)=\partial _{\mu }-\xi ^{\alpha i}(y)A^{\alpha }_{\mu }(x)\partial _{i}, \quad \hat{\iota }_{i}(y)=\partial _{i}, \end{aligned}$$
(74)

where we use \(\partial /\partial x^{\mu }=\partial _{\mu }\) and \(\partial /\partial y^{i}=\partial _{i}\) shortly. Hence, the metric components in Eq. (72) reduce to a simple form in which both \({g}_{\mu \nu }\) and \({g}_{ij}\) are now diagonal so that the \((4+N)\)-dimensional metric \(\hat{g}_{AB}(x,y)\) becomes

$$\begin{aligned} \hat{g}_{AB}(x,y)=\left( \begin{array}{c|c} {g}_{\mu \nu }(x) &{} 0 \\ \hline 0 &{} {g}_{ij}(y) \\ \end{array} \right) . \end{aligned}$$
(75)

Now, it is very easy to raise and lower indices by taking into account the form of \(\hat{g}_{AB}(x,y)\) in Eq. (75). By employing necessary relations that can be found in [111] for this basis, the nonvanishing components of the \((4+N)\)-dimensional curvature tensor \(\hat{R}^{A}\,_{BCD}\) decompose into

$$\begin{aligned}&\hat{R}^{\mu }\,_{\nu \lambda \sigma }={R}^{\mu }\,_{\nu \lambda \sigma } -\frac{1}{4}{g}_{ij}\xi ^{\alpha i}\xi ^{\beta j} \left( 2F^{\alpha \mu }\,_{\nu }F^{\beta }_{\lambda \sigma } +F^{\alpha \mu }\,_{\lambda }F^{\beta }_{\nu \sigma } -F^{\alpha \mu }\,_{\sigma }F^{\beta }_{\nu \lambda }\right) , \nonumber \\&\hat{R}^{i}\,_{\nu \lambda \sigma }=\frac{1}{2}\xi ^{\alpha i} \mathcal {D}_{\nu }F^{\alpha }_{\lambda \sigma },\nonumber \\&\hat{R}^{i}\,_{\nu j\sigma }=\frac{1}{2}D_{j}\xi ^{\alpha i} F^{\alpha }_{\nu \sigma } -\frac{1}{4}{g}_{ij}\xi ^{\alpha i}\xi ^{\beta j}F^{\alpha }_{\sigma \tau }F^{\beta \tau }\,_{\nu },\nonumber \\&\hat{R}^{\mu }\,_{\nu ij}=D_{j}\xi ^{\alpha }_{i} F^{\alpha \mu }\,_{\nu } +\frac{1}{4}\xi ^{\alpha }_{i}\xi ^{\beta }_{j} \left( F^{\alpha \mu \tau }F^{\beta }_{\tau \nu } -F^{\beta \mu \tau }F^{\alpha }_{\tau \nu }\right) ,\nonumber \\&\hat{R}^{i}\,_{jkl}={R}^{i}\,_{jkl}. \end{aligned}$$
(76)

We present these expressions, since they are the basis of what follows. For completeness, the electromagnetic field strength tensor \(F^{\alpha }_{\mu \nu }(x)\) and the Yang-Mills total covariant derivative in Eq. (76) are explicitly given as

$$\begin{aligned} F^{\alpha }_{\mu \nu }=\partial _{\mu }A^{\alpha }_{\nu }-\partial _{\nu }A^{\alpha }_{\nu } +f_{\beta \gamma }\,^{\alpha }A^{\beta }_{\mu }A^{\gamma }_{\nu },\qquad \mathcal {D}_{\mu } F^{\alpha }_{\nu \lambda }=D_{\mu } F^{\alpha }_{\nu \lambda } +f_{\beta \gamma }\,^{\alpha }A^{\beta }_{\mu }F^{\gamma }_{\nu \lambda }. \end{aligned}$$
(77)

The reduced forms of Ricci tensor \(\hat{R}_{AB}\) are, on the other hand, expressed from the Eq. (76) as

$$\begin{aligned} \hat{R}_{\mu \nu }&\equiv \mathcal {P}_{\mu \nu }= R_{\mu \nu }-\frac{1}{2}{g}_{ij}\xi ^{\alpha i}\xi ^{\beta j}F^{\alpha }_{\mu \lambda }F^{\beta }_{\nu }\,^{\lambda },\end{aligned}$$
(78)
$$\begin{aligned} \hat{R}_{i\nu }&\equiv \mathcal {Q}_{i\nu }=\xi ^{\alpha }_{i} \mathcal {D}_{\mu }F^{\alpha \mu }\,_{\nu },\end{aligned}$$
(79)
$$\begin{aligned} \hat{R}_{ij}&\equiv \mathcal {U}_{ij}= R_{ij}+\frac{1}{4}\xi ^{\alpha }_{i}\xi ^{\beta }_{j} F^{\alpha }_{\lambda \tau }F^{\beta \lambda \tau }. \end{aligned}$$
(80)

Here, \(R_{\mu \nu }\) and \(R_{ij}\) are the 4D Ricci tensors of \(external\) and \(internal\) spaces, respectively. The Ricci tensors contracting to get Ricci scalar, we find the curvature invariant corresponding to the non-Abelian ansatz (72) is given as

$$\begin{aligned} \hat{R}(x,y) = R(x)+R(y)-\frac{1}{4}{g}_{ij}(y)\xi ^{\alpha i}(y)\xi ^{\beta j}(y)F^{\alpha }_{\lambda \tau }(x)F^{\beta \lambda \tau }(x), \end{aligned}$$
(81)

where \(R(x)\) and \(R(y)\) are scalar curvatures in four and \(N\) dimensions, respectively [112]. As it is known, to obtain conventional form of the 4D gauge fields in Eq. (81), i.e. to construct a desired 4D field theory which only includes the graviton and massless Yang-Mills fields, we must select and normalize the Killing vector fields such as

$$\begin{aligned} {g}_{ij}(y)\xi ^{\alpha i}(y)\xi ^{\beta j}(y)=c\delta ^{\alpha \beta }, \end{aligned}$$
(82)

for some constant \(c\). We also add an appropriate cosmological constant term \(\hat{\varLambda }\) to the action of the theory to avoid contributions from non-zero \(R(y)\) term. Then the Eq. (81) is completely independent of the \(internal\) coordinates. Unfortunately, the Eq. (82) is generally not true for any choice of \(internal\) space, and we need to write a matrix term \(\varPsi ^{\alpha \beta }(y)\) to the right-hand side of the Eq. (82) rather than \(\delta ^{\alpha \beta }\)

$$\begin{aligned} \xi ^{\alpha i}(y)\xi ^{\beta }_{i}(y)=\varPsi ^{\alpha \beta }(y), \end{aligned}$$
(83)

which however leads to a well-known consistency problem in the non-Abelian KK framework. Thanks to supergravity theories, one can prove that \(\varPsi ^{\alpha \beta }(y)=\delta ^{\alpha \beta }\) by making use of supergravities [113, 114].

Now, we can obtain an appropriate vacuum solution looking for the equations of motions \(\hat{R}_{AB}=0\) just as classical 5D KK theory. The last Eq. (80) does not vanish \(\hat{R}_{ij}\ne 0\) (not Ricci flat) because the \(internal\) space has to be curved for any non-Abelian group. However, by adding suitable matter fields to the Hilbert\(-\)Einstein Lagrangian, we can obtain an acceptable vacuum solution of the form \(M_{4}\times M_{N}\) which is called \(spontaneous~compactification\) by Cremmer and Scherk [115117].

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Kuyrukcu, H. The non-Abelian Weyl\(-\)Yang\(-\)Kaluza\(-\)Klein gravity model. Gen Relativ Gravit 46, 1751 (2014). https://doi.org/10.1007/s10714-014-1751-x

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Keywords

  • Kaluza\(-\)Klein theories
  • Modified theories of gravity
  • Non-Abelian