# The non-Abelian Weyl-Yang-Kaluza-Klein gravity model

Halil Kuyrukcu<EMAIL_ADDRESS>Physics Department, Bülent Ecevit
University, 67100, Zonguldak, Turkey

###### Abstract

The Weyl-Yang gravitation gauge theory is investigated in the framework of a
pure higher-dimensional non-Abelian Kaluza-Klein background. We construct the
dimensionally reduced field equations and energy-momentum tensors as well as
the four dimensional modified Weyl-Yang+Yang-Mills theory from an arbitrary
curved $internal$ space which is a extension of our previous model. In
particular, the coset space case is considered to obtain explicitly the
interactions between the gravitational and the gauge fields. The results not
only appear to be generalization of the well-established equations of non-
Abelian theory but also contain intrinsically the generalized gravitational
source term and the Lorentz force density.

PACS numbers: 04.50.Cd, 04.50.Kd, 04.50.-h, 11.25.Mj

##  I Introduction

The Kaluza-Klein (KK) theories Kaluza ; Klein ; Mandel ; KleinN  base on the
idea of a unification of well-known gravitation and $U(1)$ Abelian gauge
interactions on a circle bundle with purely geometrical point of view in the
four-dimensional ($4$D) effective theory, when extra space dimension is
compactified (i.e. invisible through Kaluza’s cylinder condition). This idea
can naturally be generalized to the case of a non-Abelian gauge fields
Einstein ; Witt ; Rayski ; Kerner ; Trautman ; Cho ; Choo ; Chooo ; Chang
describe the strong and weak interactions leads to most natural generalization
of Einstein-Maxwell theory on a principal fibre bundle structure in the
$(4+N)$-dimensional spacetime, i.e. the usual $4$D spacetime ($external$) plus
$N$-dimensional compact sub-space ($internal$), usually preferred to be the
coset space Helgason ; Kobayashi ; Gilmore ; witten77 ; forgacs ; witten ;
kapetanakis  or the group manifold scherk79  (and references therein). The
gauge symmetry group of $external$ space comes remarkably from the isometry
group of $internal$ space in the KK-type multi-dimensional unified field
theories, and therefore they have been extensively discussed from many
different angles over the years by other unification theories (for a complete
overview see Appelquist ; Bailin ; Overduin ) such that the supergravity Duff
and superstring Schwarz  theories.

To obtain equations of motions of Abelian or non-Abelian KK theories (most of
gravitational theories) we conventionally prefer to employ well-established
Hilbert-Einstein action as a gravitational Lagrangian. However, there is an
alternatively Yang-Mills-style gravitational formulations (also known as gauge
theories of gravitation) which had already been considered very early by Weyl
weyl1 , Lanczos lanczos , Lichnerowicz lichnerowicz , Stephenson stephenson ,
Higgs higgs , Kilmister and Newman kilmister , later improved by Yang yang
and further investigated by Thompson thompson , Pavelle pavelle  and Fairchild
fairchild .

The Weyl-Yang gravitational Lagrangian is a quadratic in Riemann-Christofell
tensor, and two distinct Einsteinian field equations are obtained by
considering the metric and the affine connection are independent dynamical
field variables without torsion and with torsion Szczyrba ; mashhoon88 ;
mashhoon91 . The theory may be called Yang-Mills approach to gravity because
the equations of motions are similarly achieved as those of the Yang-Mills
yangmills  or the well-known classical electromagnetic theory. Actually, these
Maxwellian field equations may appear a special limit of those are given by
Hehl and $et~{}al$ hehl78  considering nonzero torsion is also independent
variable within the framework of the theory so-called Poincaré Gauge theory of
Gravity hehl76 , and references given there. Although the initial appeal for
quadratic-type of alternative gravitational theory is faded as a gauge
formulation of gravitational physics, it is interesting in its own right, it
is still prevailing as effective theories of modified gravity, specifically in
quantum gravity lausher2002  and loop quantum cosmology cognola2013 .
Recently, that simple quadratic gravitational Lagrangian model is shown to be
very useful to overcome some cosmological problems in the real four space-time
dimensions gerard ; cook ; gonzalez ; chen ; chen1 ; yeung ; wang2013 . The
physical plane-wave solutions of the theory in the four and five dimensions is
also developed and discussed by Başkal baskal  and Kuyrukcu kuyrukcu ,
respectively.

In the present paper, we completely generalize methods and results from our
previous work Başkal and Kuyrukcu sibelhalil  to the non-Abelian case without
including any scalar fields by taking into account spinless and torsionless
Weyl-Yang gravity model in the context of the more than five dimensional pure
KK theories. As is well-known in KK theories, the $4$D matter is induced from
geometry in higher dimensions that the space-time is empty. In our approach,
the $4$D matter-spin source term is induced from those that matter carrying
energy-momentum but not possessing any spin tensor. We also extend the reduced
equations to more physical forms by taking the compact $internal$ space as a
homogeneous coset spaces background needed in KK theories. We can achieve this
construction the following outline of our article. In Sec. II and III, we
begin a brief review of both of the non-Abelian KK theory and Weyl-Yang
gravity model, respectively, to remind the reader some basic elements of those
theories for convenient reading, and to introduce the relations which we
employ all along in the work. In Sec. IV, we perform a popular dimensional KK
reduction procedure to obtain the modified $4$D Weyl-Yang+Yang-Mills action
from a pure $(4+N)$-dimensional Weyl-Yang gravitational Lagrangian without
supplementary matter fields. In that respect, the dimensionally reduced field
equations which contain naturally the generalized gravitational source term
and the Lorentz force density and stress-energy tensors are simultaneously
investigated, and they are compared with the our previous gravity model and
standard higher-dimensional KK theories in Sec. V and Sec. VI, respectively.
As a by-product, the coset space case of the theory is also included which
leads to more physical results and some plain solution of the field equations
in some detail. Finally, the last section demonstrates a brief discussion and
conclusions obtained from our analysis.

##  II The non-Abelian Kaluza-Klein theory

We briefly review here the major steps of non-Abelian KK framework unifying
gravitation and Yang-Mills theories in more than five dimensions. First, let
us remember some basic notions of that theory in the usual way for convenient
reading. In what follows, the Greek indices $\mu,\nu,...=0,...,3$ refer to the
$external$ $4D$ flat Minkowski (Ricci flat) space-time $M_{4}$, admitting the
metric ${g}_{\mu\nu}(x)$ with usual signature and collectively coordinates
$x\in M_{4}$. The Latin indices $i,j,...=5,...,4+N$ refer to the curved
$internal$ $N$-dimensional compact subspace $M_{N}$ 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 $A,B,...=0,...,3,5,...,4+N$ 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 $4$D $external$ space,respectively. 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 $\alpha,\beta,...$
refer to any compact isometry (Lie) group $G$ of $M_{N}$, running over the
rank of $G$, i.e. $\alpha,\beta,...=1,...,dim(G)$. The isometries of the
$internal$ manifold produce linearly independent space-like Killing vectors
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
vectors fields $\xi^{i}_{\alpha}(y)$ satisfy the Lie’s equation

$\displaystyle[\xi_{\alpha},\xi_{\beta}]^{i}\equiv\xi^{j}_{\alpha}\partial_{j}\xi^{i}_{\beta}-\xi^{j}_{\beta}\partial_{j}\xi^{i}_{\alpha}$
$\displaystyle=$ $\displaystyle-f_{\alpha\beta}\,^{\gamma}\xi^{i}_{\gamma},$
(1)

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

$\displaystyle\mathcal{L}_{\xi}g_{ij}\equiv\xi^{\alpha
k}\partial_{k}g_{ij}+g_{ik}\partial_{j}\xi^{\alpha
k}+g_{jk}\partial_{i}\xi^{\alpha k}=0,$ (2)

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

$\displaystyle\hat{g}_{\mu\nu}(x,y)={g}_{\mu\nu}(x)+{g}_{ij}(y)\xi^{\alpha
i}(y)\xi^{\beta j}(y)A^{\alpha}_{\mu}(x)A^{\beta}_{\nu}(x),$
$\displaystyle\hat{g}_{\mu j}(x,y)={g}_{ij}(y)\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x),$ (3) $\displaystyle\hat{g}_{ij}(x,y)={g}_{ij}(y).$

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

$\displaystyle\hat{E}^{\mu}(x,y)=dx^{\mu},$
$\displaystyle\hat{E}^{i}(x,y)=dy^{i}+\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x)dx^{\mu}.$ (4)

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

$\displaystyle\hat{\iota}_{\mu}(x,y)=\partial_{\mu}-\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x)\partial_{i},$ (5)
$\displaystyle\hat{\iota}_{i}(x,y)=\partial_{i}.$

Hence, the metric components in equation (II) reduce a simple forms so that
the $(4+N)$-dimensional metric $\hat{g}_{AB}(x,y)$ becomes

$\displaystyle\hat{g}_{AB}=\left(\begin{array}[]{cc}{g}_{\mu\nu}(x)&0\\\
0&{g}_{ij}(y)\\\ \end{array}\right).$ (8)

It is very easy to raise and lower indices by taking into account the form of
$\hat{g}_{AB}(x,y)$ in equation (8). By employing necessary relations can be
found in misner  for this basis, the non-zero components of the
$(4+N)$-dimensional Riemann tensor $\hat{R}^{A}\,_{BCD}$ decomposes into

$\displaystyle\hat{R}^{\mu}\,_{\nu\lambda\sigma}={R}^{\mu}\,_{\nu\lambda\sigma}-l_{\alpha\beta}(F^{\alpha\mu}\,_{\nu}F^{\beta}_{\lambda\sigma}+F^{\alpha\mu}\,_{[\lambda}F^{\beta}_{|\nu|\sigma]}),$
$\displaystyle\hat{R}^{i}\,_{\nu\lambda\sigma}=\frac{1}{2}\xi^{\alpha
i}\mathcal{D}_{\nu}F^{\alpha}_{\lambda\sigma},$
$\displaystyle\hat{R}^{i}\,_{\nu j\sigma}=\frac{1}{2}D_{j}\xi^{\alpha
i}F^{\alpha}_{\nu\sigma}-\frac{1}{2}l_{\alpha\beta}F^{\alpha}_{\sigma\tau}F^{\beta\tau}\,_{\nu},$
(9) $\displaystyle\hat{R}^{\mu}\,_{\nu
ij}=D_{j}\xi^{\alpha}_{i}F^{\alpha\mu}\,_{\nu}+\frac{1}{4}\xi^{\alpha}_{i}\xi^{\beta}_{j}(F^{\alpha\mu\tau}F^{\beta}_{\tau\nu}-F^{\beta\mu\tau}F^{\alpha}_{\tau\nu}),$
$\displaystyle\hat{R}^{i}\,_{jkl}={R}^{i}\,_{jkl},$

where the following abbreviation are introduced

$\displaystyle l_{\alpha\beta}=\frac{1}{2}{g}_{ij}\xi^{\alpha i}\xi^{\beta
j}.$ (10)

We give 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
equation (II) are explicitly given by

$\displaystyle F^{\alpha}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A^{\alpha}_{\nu}-\partial_{\nu}A^{\alpha}_{\nu}+f_{\beta\gamma}\,^{\alpha}A^{\beta}_{\mu}A^{\gamma}_{\nu},$
(11) $\displaystyle\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}$ $\displaystyle=$
$\displaystyle
D_{\mu}F^{\alpha}_{\nu\lambda}+f_{\beta\gamma}\,^{\alpha}A^{\beta}_{\mu}F^{\gamma}_{\nu\lambda}.$
(12)

The reduced forms of Ricci tensor $\hat{R}_{AB}$ are, on the other hand,
expressed as

$\displaystyle\hat{R}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\mathcal{P}_{\mu\nu}=R_{\mu\nu}-l_{\alpha\beta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda},$
(13) $\displaystyle\hat{R}_{i\nu}$ $\displaystyle\equiv$
$\displaystyle\mathcal{Q}_{i\nu}=\xi^{\alpha}_{i}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu},$
(14) $\displaystyle\hat{R}_{ij}$ $\displaystyle\equiv$
$\displaystyle\mathcal{U}_{ij}=R_{ij}+\frac{1}{4}\xi^{\alpha}_{i}\xi^{\beta}_{j}F^{\alpha}_{\lambda\tau}F^{\beta\lambda\tau}.$
(15)

Here, $R_{\mu\nu}$ and $R_{ij}$ are the $4$D Ricci tensors of $external$ and
$internal$ spaces, respectively. In the usual manner, we assume that a matter-
free Hilbert-Einstein Lagrangian which is linear in curvature components
proportional to the dimensional constant $1/{2\hat{\kappa}^{2}}$ leads to

$\hat{S}_{HE}(x,y)=\frac{1}{2\hat{\kappa}^{2}}\int_{M_{4+N}}\hat{R}\,\sqrt{-\hat{g}}\,d^{4}x\,d^{N}y,$
(16)

where $\hat{R}$ is the $(4+N)$-dimensional curvature scalar of the Riemann
space and the $\hat{\kappa}$ is coupling constant which satisfy
$\hat{\kappa}^{2}\equiv 8\pi\hat{\mathcal{G}}/c^{4}$ together with the
universal gravitational constant $\hat{\mathcal{G}}$. The Ricci tensor
contracting to get Ricci scalar, we find the curvature invariant corresponding
to the non-Abelian ansatz is given by

$\hat{R}(x,y)=R(x)+R(y)-\frac{1}{2}l_{\alpha\beta}(y)F^{\alpha}_{\lambda\tau}(x)F^{\beta\lambda\tau}(x),$
(17)

where $R(x)$ and $R(y)$ are scalar curvatures in four and $N$ dimensions,
respectively. As is well-known, to obtain conventional form of the $4$D gauge
fields in equation (17), i.e. to construct a desired $4$D field theory which
only includes the graviton and massless Yang-Mills fields, we must select and
normalize the Killing vectors fields such that

$2l_{\alpha\beta}\equiv{g}_{ij}(y)\xi^{\alpha i}(y)\xi^{\beta
j}(y)=c\delta^{\alpha\beta},$ (18)

for some constant $c$ so that the right-hand side of equation (17) is
independent of the $internal$ coordinates Duff84 ; Duff85 . We also add an
appropriate cosmological constant term $\hat{\Lambda}$ to the action of the
theory (16) to avoid contributions from non-zero $R(y)$ term.

We can obtain an appropriate vacuum solution looking for the equations of
motion $\hat{R}_{AB}=0$ as classical KK framework. However, the last equation
(15) is not vanish $\hat{R}_{ij}\neq 0$ (not Ricci flat) because of the
$internal$ space has to be curved for any non-Abelian group. This is well-
known consistency problem of the non-Abelian KK theories and the supergravity
theories as well. However, by adding suitable matter fields to the action
(16), we can obtain acceptable vacuum solution of the form $M_{4}\times M_{N}$
which is called $spontaneous~{}compactification$ by Cremmer and Scherk Cremmer
. Finally, one also can consider generalize the KK ansatz (II) including
massless scalar fields $\phi$ Awada ; Appequist  which play an important role
in supergravity but it will not be discussed in this work.

##  III The Weyl-Yang gravity model in a (4+N) dimensions

The dynamics of Weyl-Yang theory of gravity is determined by the curvature-
squared gravitational action on $M_{4+N}$ (or may be alternatively called
Stephenson-Kilmister-Yang action)

$\hat{S}_{WY}(x,y)=\hat{k}\int_{M_{4+N}}\hat{R}_{ABCD}^{2}\,\sqrt{-\hat{g}}\,d^{4}x\,d^{N}y,$
(19)

for matter-free gravity in the $(4+N)$-dimensional space-time with the
shorthand notation $\hat{R}_{ABCD}^{2}\equiv\hat{R}_{ABCD}\hat{R}^{ABCD}$ and
a real coupling constant $\hat{k}$. This Riemannian model is introduced by
Yang yang  as an alternative approach to the Einstein’s theory of General
Relativity by employing perfect analogy with Yang-Mills theories. The Yang’s
theory is developed by Fairchild fairchild  applying à la Palatini variation
(P-variation) principle palatini  to obtain Euler-Lagrange equations of the
theory in a natural manner. Thus, the independent variation of the action (19)
with respect to the $(4+N)$-dimensional connection (gauge potential)
$\hat{\Gamma}^{A}\,_{BC}$ and the metric $\hat{g}_{AB}$ gives the desired
Yang-Mills style field equations. If spin and torsion vanish, from the
connection variation, symbolically as
$\delta\hat{S}_{\hat{\Gamma}}[\hat{g},\hat{\Gamma}]=0$, we obtain third-order
Yang’s pure gravitational field equation

$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}=0,$ (20)

which can also equivalently be written by consecutive contractions and
applying the Bianchi identities as follows

$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}\equiv\hat{D}_{C}\hat{R}_{BD}-\hat{D}_{D}\hat{R}_{BC}=0.$
(21)

Obviously, Yang’s equation (20) generalize Einstein’s field equations and
contains naturally vacuum Einsteinian solutions as well as non-Einsteinian
ones. The matter-source term $\hat{S}_{BCD}$, antisymmetric in $C$ and $D$,
may also be added to the right hand side of the equation (20), namely,

$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}=\hat{\lambda}\hat{S}_{BCD},$ (22)

together with some suitable coupling constant $\hat{\lambda}$ kilmister ;
camenzind75 ; pavelle76 ; fairchild77 ; tseytlin . This gravitational current
term is interpreted as the covariant derivative of the Einstein’s matter
stress-energy tensor by Kilmister kilmister66  later Camenzind camenzind75
without a variational principle or more conveniently the spin tensor of the
matter fields by Fairchild fairchild . We can also rewrite sourceless field
equations (20) in many other ways, see baekler ; garecki .

From the metric variation
$\delta\hat{S}_{\hat{g}}[\hat{g},\hat{\Gamma}]=\hat{T}_{AB}$, we arrive at the
symmetric second-rank gravitational energy-momentum tensor together with
choosing appropriate $\hat{k}$ in the following form

$\hat{T}_{AB}\equiv\hat{R}_{ACDE}\hat{R}_{B}\,^{CDE}-\frac{1}{4}\hat{g}_{AB}\hat{R}^{2}_{CDEF}.$
(23)

It is called Stephenson equation for the case $\hat{T}_{AB}=0$ stephenson
which is divergence-free $\hat{D}_{A}\hat{T}^{A}\,_{B}=0$ but not traceless in
higher dimensions. It is actually $\hat{T}=-(N/4)\hat{R}^{2}_{ABCD}$ and
totally traceless only $N=0$ case, i.e. for the usual $4$D space-time. Let us
finally mention that, the equation (23) is indeed the contraction form of the
well-known Bel-Robinson superenergy tensor bel ; robinson , and it appear
perfectly analogous to stress-energy of classical electromagnetic theories as
well.

##  IV The Reduction of the Quadratic Curvature

The method of dimensional reduction, from higher-dimensional theory to the
actual $4$D space-time, bring into the clear the types of gravitational and
gauge fields together with the forms of interactions between the constituent
fields. The reduced form of the $(4+N)$-dimensional quadratic curvature after
performing dimensional reduction procedure is given by

$\displaystyle\hat{R}_{ABCD}^{2}$ $\displaystyle=$
$\displaystyle\hat{R}_{\mu\nu\lambda\sigma}^{2}+4(\hat{R}_{i\nu\lambda\sigma}^{2}+\hat{R}_{i\nu
k\sigma}^{2}+\hat{R}_{ijk\sigma}^{2})$ (24)
$\displaystyle+2\hat{R}_{ij\lambda\sigma}^{2}+\hat{R}_{ijkl}^{2}.$

The substitutions of Riemann’s components in equation (II) into above with a
straightforward calculation and help of the relation

$\displaystyle
R_{\mu\nu\lambda\sigma}F^{\alpha\mu\lambda}F^{\beta\nu\sigma}=\frac{1}{2}R_{\mu\nu\lambda\sigma}F^{\alpha\mu\nu}F^{\beta\lambda\sigma},$
(25)

gives the dimensionally reduced invariant in the following form

$\displaystyle\hat{R}_{ABCD}^{2}$
$\displaystyle=R_{\mu\nu\lambda\rho}^{2}-3l_{\alpha\beta}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\beta\lambda\rho}$
(26)
$\displaystyle+\frac{1}{2}l_{\alpha\beta}l_{\gamma\eta}\left(\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+3\mathcal{F}^{2}_{\alpha\gamma}\mathcal{F}^{2}_{\beta\eta}+4\mathcal{F}^{4}_{\alpha\gamma\eta\beta}\right)$
$\displaystyle+3(m_{\alpha\beta\gamma}\mathcal{F}^{3}_{\alpha\beta\gamma}+k_{\alpha\beta}\mathcal{F}^{2}_{\alpha\beta})$
$\displaystyle+2l_{\alpha\beta}\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}+R_{ijkl}^{2}.$

Here, we have shortly defined the notations

$\displaystyle\mathcal{F}_{\alpha\beta}^{2}=F^{\alpha}_{\mu\nu}F^{\beta\mu\nu},$
$\displaystyle\mathcal{F}_{\alpha\beta\gamma}^{3}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma}\,{}^{\mu},$
$\displaystyle\mathcal{F}^{3~{}\alpha\beta\gamma}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\nu},$
(27)
$\displaystyle\mathcal{F}_{\alpha\beta\gamma\eta}^{4}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\nu}F^{\eta\nu\mu},$
$\displaystyle\mathcal{F}^{4~{}\alpha\beta\gamma\eta}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\rho}F^{\eta\rho}\,_{\nu},$

and, the invariant coefficients are

$\displaystyle k_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
D_{i}\xi^{\alpha}_{j}D^{i}\xi^{\beta j},$ (28) $\displaystyle
m_{\alpha\beta\gamma}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}D^{i}\xi^{\gamma j}.$ (29)

Furthermore, in equation (26) and what follows, the gauge
$\mathcal{D}_{[\mu}F^{\alpha}_{\nu\lambda]}=0$ and the gravitational
$D_{[\tau}R_{\mu\nu]\lambda\sigma}=0$ Bianchi identities are used, whenever we
need them. We also recognize the fact that, the equation (26) is a common
result for all the basis, because $\hat{R}_{ABCD}^{2}$ is clearly an
invariant. We can overcome the term $D_{i}\xi^{\alpha}_{j}$ by taking into
account Wu and Zee’s assumption Yong-shi , then we get simple formula for
covariant derivative of Killing vector fields as

$D_{i}\xi^{\alpha}_{j}=\frac{1}{2}f^{\alpha\beta\gamma}\xi^{\beta}_{i}\xi^{\gamma}_{j}.$
(30)

However, in this case the $internal$ manifold is reduced to be homogeneous
space which is first discussed by Luciani Luciani  in KK theories. Thus, the
extra space can now be written as the symmetric coset space $M_{N}=G/H$. $H$
is defined the isotropy subgroup of $G$, $H\subset G$ with $N=dim(G)-dim(H)$.
All the Killing vector terms of equation (26) are then disappeared, just as
expected, by employing equations (18) and (30) with $c=1$

$\displaystyle\hat{R}_{ABCD}^{2}$
$\displaystyle=R_{\mu\nu\lambda\rho}^{2}-\frac{3}{2}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\alpha\lambda\rho}+\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\alpha\nu\lambda}$
$\displaystyle+\frac{3}{4}(f^{\alpha\gamma\eta}f^{\beta\gamma\eta}\mathcal{F}^{2}_{\alpha\beta}+2f^{\alpha\beta\gamma}\mathcal{F}^{3}_{\alpha\beta\gamma})$
$\displaystyle+\frac{1}{8}[\mathcal{F}^{4}_{\alpha\beta\alpha\beta}+3(\mathcal{F}^{2}_{\alpha\beta})^{2}+4\mathcal{F}^{4}_{\alpha\beta\beta\alpha}]+R_{ijkl}^{2}.$

The equation (IV) defines a new Lagrangian formalism in a background Yang-
Mills fields. It is easy see that, in addition to the standard $4$D Weyl-Yang
Lagrangian term $R_{\mu\nu\lambda\rho}^{2}$, the effective $4$D action (IV)
also contains the well-known $RF^{2}$-type term ( liu  and references therein)
which describes a non-minimal coupling between curvature and the non-Abelian
gauge field and the self-interacting Yang-Mills field invariants which are the
cubic term $F^{3}$ and the quartic terms $F^{4}$ as well. The invariant form
$F^{3}$ appears only non-Abelian theories, and it is first studied by Alekseev
and Arbuzov Alekseev82 ; Alekseev84 . The ordinary Yang-Mills Lagrangian term
$\mathcal{F}^{2}_{\alpha\alpha}$ can be obtained from the
$f^{\alpha\gamma\eta}f^{\beta\gamma\eta}\mathcal{F}^{2}_{\alpha\beta}$ term,
if we normalize the structure constants such that
$f^{\alpha\gamma\eta}f^{\beta\gamma\eta}=(N-1)\delta^{\alpha\beta}$. By virtue
of Leibniz rule, the term with covariant derivative of gauge field tensor may
be rewritten symbolically as
$(\mathcal{D}F)^{2}=\mathcal{D}(F\mathcal{D}F)-F\mathcal{D}^{2}F$ of which
first term is total derivative so it can be dropped from the action, thus we
just have the $F\mathcal{D}^{2}F$-type interactions. Finally, the last term
$R_{ijkl}^{2}$ can be interpreted as the cosmological constant term as well as
the $4$D Ricci scalar of $internal$ space $R(y)$ or it can be just ignored so
that it does not contain physical fields. As a conclusion, the equation (IV)
leads to a modified $4$D Weyl-Yang+Yang-Mills action, and in fact it is some
part of second-order Euler-Poincaré (Gauss-Bonnet) curvature invariant Huang ;
m ller ; dereli ; kimm . We should also emphasise that, although the
dimensional reduction of quadratic Lagrangian from $4+N$ to $4$ dimensions is
discussed very early by Cho and $et~{}al$ Choo , they not have obtained proper
Yang-Mills term $\mathcal{F}^{2}_{\alpha\alpha}$ but rather discussed only
$(\mathcal{D}F)^{2}$ term.

##  V The Reduction of the Field Equations

Now, we ready to work out the reduced components of $(4+N)$-dimensional
source-free field equations $\hat{D}_{A}\hat{R}^{A}\,_{BCD}=0$ in (20) by
taking into account the KK reduction scheme, as traditional way. Thus, we have
the six different equations of motions (one $4$D part and five $N$-dimensional
parts) are complementary each other

$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{\nu\lambda\sigma}=0,$ (32)
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{i\lambda\sigma}=0,\qquad\qquad\hat{D}_{A}\hat{R}^{A}\,_{\nu
j\sigma}=0,$ (33) $\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{i\lambda
k}=0,\qquad\qquad\hat{D}_{A}\hat{R}^{A}\,_{\nu jk}=0,$ (34)
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{ijk}=0.$ (35)

After some lengthy but careful manipulations, on account of mainly equation
(II) together with the Killing vector relations (equations (1) and (2)) and
the dual basis equation (5), one obtains the $4$D part of the
$(4+N)$-dimensional Yang’s equation in (32) as the $4$D Yang’s equation with
current term

$\displaystyle D_{\mu}R^{\mu}\,_{\nu\lambda\sigma}=S_{\nu\lambda\sigma},$ (36)

where

$\displaystyle
S_{\nu\lambda\sigma}(x,y)=l_{\alpha\beta}[J^{\alpha}_{\nu}F^{\beta}_{\lambda\sigma}+J^{\alpha}_{[\lambda}F^{\beta}_{|\nu|\sigma]}+2\mathcal{D}_{[\sigma}(F^{\alpha}_{\lambda]\mu}F^{\beta\mu}\,_{\nu})].$

Here, $J^{\alpha}_{\nu}$ is usual non-zero four current term of Yang-Mills
theory, i.e. $J^{\alpha}_{\nu}=\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}$. The
term $S_{\nu\lambda\sigma}(x,y)$ may be interpreted as the gravitational
source like-term of the $4$D matter, and it consists of various combinations
of $F$ and $DF$ terms. It is no difficult to conjecture that, the equation (V)
satisfies the cyclic symmetry $S_{[\nu\lambda\sigma]}=0$ and covariant
conservation $\mathcal{D}_{\nu}S^{\nu}\,_{\lambda\sigma}=0$ properties as
expected from a source term. F. Öktem comments that “Those identities then
reveal that matter is endowed with angular momentum” oktem . This result (36)
very valuable from the physical point of view in case the $4$D matter-spin
tensor term is induced from in higher dimensions that matter carrying energy-
momentum but not possessing any spin tensor. In this sense our approach is
formally similar to the KK theories. Besides, we can remark that the first
reduced field equation (36) governs mainly the gravitational fields, since the
$DR$ term have the highest order derivative of the $external$ metric
$g_{\mu\nu}(x)$.

The reduced form of field equations (33) however give more complicated
relations

$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}\mathcal{D}^{\mu}F^{\alpha}_{\lambda\sigma}$
$\displaystyle=$
$\displaystyle-\xi^{\alpha}_{i}R_{\mu\nu\lambda\sigma}F^{\alpha\mu\nu}+4\xi^{\alpha
j}D_{j}\xi^{\beta}_{i}F^{[\alpha}_{\lambda\mu}F^{\beta]\mu}\,_{\sigma}$
$\displaystyle+\xi^{\alpha}_{i}l_{\beta\gamma}(\mathcal{F}^{2}_{\alpha\beta}F^{\gamma}_{\lambda\sigma}+2\mathcal{F}^{3~{}\alpha\beta\gamma}_{[\sigma\lambda]})$
$\displaystyle-2F^{\alpha}_{\lambda\sigma}D_{j}D^{j}\xi^{\alpha}_{i},$
$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}\mathcal{D}_{\sigma}F^{\alpha\mu}\,_{\nu}$
$\displaystyle=$
$\displaystyle-\xi^{\alpha}_{i}R_{\mu\nu\tau\sigma}F^{\alpha\tau\mu}+2\xi^{\alpha
j}D_{j}\xi^{\beta}_{i}F^{\alpha}_{\sigma\mu}F^{\beta\mu}\,_{\nu}$
$\displaystyle-\frac{1}{2}\xi^{\alpha}_{i}l_{\beta\gamma}(\mathcal{F}^{2}_{\alpha\beta}F^{\gamma}_{\nu\sigma}-2\mathcal{F}^{3~{}\alpha\beta\gamma}_{\nu\sigma})$
$\displaystyle+F^{\alpha}_{\nu\sigma}D_{j}D^{j}\xi^{\alpha}_{i}.$

The equations (V) and (V) basically govern the Yang-Mills gauge fields in view
of $\mathcal{D}\mathcal{D}\mathcal{F}$, and they include non-minimal $RF$ type
couplings as well as the cubic terms $F^{3}$. Additionally, we can obtain the
covariant derivative of the non-Abelian current
$\mathcal{D}_{\mu}J^{\alpha}_{\nu}$ by summing above two equations ((V) and
(V)) as well as employing Bianchi identities with following relation

$\displaystyle\xi^{\alpha}_{i}[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]F^{\alpha\lambda}\,_{\sigma}$
$\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}(R^{\lambda}\,_{\tau\mu\nu}F^{\alpha\tau}\,_{\sigma}-R^{\tau}\,_{\sigma\mu\nu}F^{\alpha\lambda}\,_{\tau})$
(40) $\displaystyle+2\xi^{\beta
j}D_{j}\xi^{\alpha}_{i}F^{\alpha}_{\mu\nu}F^{\beta\lambda}\,_{\sigma}.$

Hence, the result is

$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}J^{\alpha}_{\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\xi^{\alpha}_{i}(\mathcal{F}^{2}_{\alpha\beta}\mathcal{F}^{\beta}_{\mu\nu}-\mathcal{F}^{3~{}\alpha\beta\beta}_{\mu\nu}+2\mathcal{F}^{3~{}\alpha\beta\beta}_{\nu\mu})$
(41)
$\displaystyle+\xi^{\alpha}_{i}F^{\alpha}_{\nu\lambda}R^{\lambda}\,_{\mu}+\xi^{\alpha}_{j}F^{\alpha}_{\mu\nu}R_{i}\,^{j}.$

We can prove that, the contraction of free indices gives precisely the
covariant conservation law for external current such that
$\mathcal{D}_{\mu}J^{\alpha\mu}=0$.

In an equivalent way, the field equations with two $internal$ space indices
(34) are respectively reduced as

$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}(\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}+F^{\alpha}_{\lambda\tau}J^{\beta\tau})+2J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}=0,$
(42)
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}F^{[\alpha}_{\lambda\tau}J^{\beta]\tau}+2J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}=0.$
(43)

Another interesting result is obtained, if we eliminate the common term
$J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}$ of equations (42) and (43) by
adding the equations together. Then, we read

$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}[\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}+F^{(\alpha}_{\lambda\tau}J^{\beta)\tau}]=0,$
(44)

or, alternatively

$\displaystyle
f_{\lambda}^{\alpha\beta}+f_{\lambda}^{\beta\alpha}=-2\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}.$
(45)

Here, $f_{\lambda}^{\alpha\alpha}=F^{\alpha}_{\lambda\tau}J^{\alpha\tau}$ is
confidentially interpreted as a Lorentz-like force density of the non-Abelian
theory. Although such a terms generally emerge from the higher-dimensional
geodesic equations Kerner , they naturally appears in the reduced field
equations of our approach. It is difficult to understand physical meaning of
this circumstance (45), however we can, at least theoretically, say that the
Yang-Mills invariant $\mathcal{F}^{2}_{\alpha\beta}$ is defined as a field
whose gradient is equal and opposite to the generalized Lorentz force density.

Finally, from the last field equation (35), we again obtain the Yang’s
equation with source term but in this case for the $internal$ space

$D_{l}R^{l}\,_{ijk}=S_{ijk},$ (46)

where the unphysical source-like term is found to be

$S_{ijk}(x,y)=-\frac{1}{2}\mathcal{F}^{2}_{\alpha\beta}(\xi^{\alpha}_{[k}D_{j]}\xi^{\beta}_{i}-\xi^{\alpha}_{i}D_{k}\xi^{\beta}_{j}).$
(47)

which provides the same conservation and symmetry conditions as the term
$S_{\nu\lambda\sigma}$ in (V). In the absence of Yang-Mills fields we only
have source-free Yang’s equation for $external$
$D_{\mu}R^{\mu}\,_{\nu\lambda\sigma}=0$ and $internal$ $D_{l}R^{l}\,_{ijk}=0$
spaces respectively.

Attention will now be turned to investigate the couplings between the
components of $(4+N)$-dimensional Ricci tensor $\mathcal{P}_{\mu\nu}$,
$\mathcal{Q}_{i\nu}$, $\mathcal{U}_{ij}$ in equations (13)-(15) and the gauge
fields $F$ together with Killing fields $\xi$. By virtue of alternative field
equation (21), one can recognize the fact that
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set are
embedded in the Weyl-Yang field equations of non-Abelian theory (32)-(35).
Hence, from the reduced equations ((36) with (V), (V), (V), (42), (43) and
(46) with (47)) we deduce the following compact embedded equations

$\displaystyle\mathcal{D}_{[\lambda}\mathcal{P}_{|\nu|\sigma]}+\frac{1}{4}\xi^{\alpha
i}(F^{\alpha}_{\nu[\lambda}\mathcal{Q}_{|i|\sigma]}-F^{\alpha}_{\lambda\sigma}\mathcal{Q}_{i\nu})=0,$
(48)
$\displaystyle\mathcal{D}_{[\lambda}\mathcal{Q}_{|i|\sigma]}+\xi^{\alpha}_{i}F^{\alpha\mu}\,_{[\sigma}\mathcal{P}_{|\mu|\lambda]}+\xi^{\alpha
j}F^{\alpha}_{\lambda\sigma}\mathcal{U}_{ij}=0,$ (49)
$\displaystyle\mathcal{D}_{\sigma}\mathcal{Q}_{i\nu}+\xi^{\alpha}_{i}F^{\alpha\mu}\,_{\nu}\mathcal{P}_{\mu\sigma}-\xi^{\alpha
j}F^{\alpha}_{\nu\sigma}\mathcal{U}_{ij}=0,$ (50)
$\displaystyle\mathcal{D}_{\lambda}\mathcal{U}_{ik}+\frac{1}{4}(2D_{k}\mathcal{Q}_{i\lambda}+\xi^{\alpha}_{i}F^{\alpha}_{\lambda}\,{}^{\tau}\mathcal{Q}_{k\tau})=0,$
(51) $\displaystyle
D_{k}\mathcal{Q}_{j\nu}+\frac{1}{2}F^{\alpha}_{\nu}\,{}^{\tau}\xi^{\alpha}_{[j}\mathcal{Q}_{k]\tau}=0,$
(52) $\displaystyle D_{[i}\mathcal{U}_{|k|j]}=0,$ (53)

on account of identities as mentioned before and the relation
$D_{j}D_{k}\xi^{\alpha}_{l}=\xi^{\alpha}_{i}R^{i}\,_{jkl}$ weinberg . The
equation (49) can be obtain from the equation (50), thus it is not essential.
The equations (48)-(53) contain the non-Abelian covariant derivative of
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set, as well
as the ordinary derivative of $\\{\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set
and various $\mathcal{P}F$, $\mathcal{Q}F$-type coupling terms. Above
generalized field equations are welcome from another point of view that any
solution to the
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set solves
the embedded equations (48)-(53) naturally.

There is a mismatch between the left-hand side of the field equation (36)
which depends only on $external$ coordinates $x$ and the right-hand side which
depends both on $external$ and $internal$ coordinates $x,y$. Therefore, to not
only avoid this problem but also to reduce equations (48)-(53) to the more
simpler and physical forms, we restrict our considerations such that the
$internal$ space is taken to be the homogeneous coset space the type $G/H$. It
is easy to see from equation (18) with $c=1$ that the source term (V) yields

$\displaystyle
S_{\nu\lambda\sigma}(x)=\frac{1}{2}[J^{\alpha}_{\nu}F^{\alpha}_{\lambda\sigma}+J^{\alpha}_{[\lambda}F^{\alpha}_{|\nu|\sigma]}+2\mathcal{D}_{[\sigma}(F^{\alpha}_{\lambda]\mu}F^{\alpha\mu}\,_{\nu})],$

which depend only on $external$ coordinates $x$ as it should be. Next by using
equation (30) and reduce form of equation (40)

$\displaystyle[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]F^{\alpha\lambda}\,_{\sigma}$
$\displaystyle=$ $\displaystyle
R^{\lambda}\,_{\tau\mu\nu}F^{\alpha\tau}\,_{\sigma}-R^{\tau}\,_{\sigma\mu\nu}F^{\alpha\lambda}\,_{\tau}$
(55)
$\displaystyle+f^{\beta\gamma\alpha}F^{\beta}_{\mu\nu}F^{\gamma\lambda}\,_{\sigma}.$

the first embedded equation (48) becomes

$\begin{array}[]{l}\mathcal{D}_{\lambda}(R_{\sigma\nu}-\frac{1}{2}F^{\alpha}_{\sigma\mu}F^{\alpha}_{\nu}\,{}^{\mu})-\mathcal{D}_{\sigma}(R_{\lambda\nu}-\frac{1}{2}F^{\alpha}_{\lambda\mu}F^{\alpha}_{\nu}\,{}^{\mu})\\\\[8.61108pt]
+\frac{1}{4}\left[F^{\alpha}_{\nu\lambda}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\sigma}-F^{\alpha}_{\nu\sigma}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\lambda}-2F^{\alpha}_{\lambda\sigma}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}\right]=0.\end{array}$
(56)

Second one (49) simultaneously yields

$\displaystyle\xi^{\alpha}_{i}[\mathcal{D}_{\lambda}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\sigma})-\mathcal{D}_{\sigma}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\lambda})$
$\displaystyle+F^{\alpha\mu}\,_{\sigma}(R_{\mu\lambda}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\lambda}\,{}^{\tau})-F^{\alpha\mu}\,_{\lambda}(R_{\mu\sigma}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\sigma}\,{}^{\tau})$
$\displaystyle+\frac{1}{2}F^{\beta}_{\lambda\sigma}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})]=0.$
(57)

The third equation (50) implies that

$\begin{array}[]{l}\xi^{\alpha}_{i}[\mathcal{D}_{\sigma}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu})+F^{\alpha\mu}\,_{\nu}(R_{\mu\sigma}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\sigma}\,{}^{\tau})\\\\[8.61108pt]
+\frac{1}{4}F^{\beta}_{\nu\sigma}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})]=0.\end{array}$
(58)

and the equations (51), (52) are similarly reorganized as

$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}[\mathcal{D}_{\lambda}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})+F^{\alpha}_{\lambda\tau}\mathcal{D}_{\mu}F^{\beta\mu\tau}$
$\displaystyle+f^{\gamma\beta\alpha}\mathcal{D}_{\mu}F^{\gamma\mu}\,_{\lambda}]=0,$
(59)
$\displaystyle\xi^{\alpha}_{j}\xi^{\beta}_{k}[F^{\alpha}_{\nu\tau}\mathcal{D}_{\mu}F^{\beta\mu\tau}-F^{\beta}_{\nu\tau}\mathcal{D}_{\mu}F^{\alpha\mu\tau}+2f^{\gamma\beta\alpha}\mathcal{D}_{\mu}F^{\gamma\mu}\,_{\nu}]=0,$

respectively. Finally, the last equation (53), on the other hand, can be
expressed in the following form

$\displaystyle\xi_{i}^{\alpha}\xi_{j}^{\beta}\xi_{k}^{\gamma}[f_{\eta\alpha\gamma}(\mathcal{F}^{2}_{\beta\eta}-f^{\zeta\pi\beta}f^{\eta\pi\zeta})-f_{\eta\beta\gamma}(\mathcal{F}^{2}_{\alpha\eta}-f^{\zeta\pi\alpha}f^{\eta\pi\zeta})$
$\displaystyle+2f_{\eta\alpha\beta}(\mathcal{F}^{2}_{\eta\gamma}-f^{\zeta\pi\eta}f^{\gamma\pi\zeta})]=0.$
(61)

If we remove all the Killing vector terms from embedded equations (78)-(V),
then everything would be fine. Furthermore, it is possible to recognize the
following plain solutions in the reduced equations

$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F^{\alpha}_{\mu\tau}F^{\alpha}_{\nu}\,{}^{\tau},$
$\displaystyle\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}$ $\displaystyle=$
$\displaystyle 0,$ (62) $\displaystyle F^{\alpha}_{\mu\nu}F^{\beta\mu\nu}$
$\displaystyle=$ $\displaystyle f^{\eta\gamma\alpha}f^{\beta\gamma\eta}.$

For $N=1$ case i.e. $G=U(1)$ and $G/H=S^{1}$ which gives the usual $5$D KK
ground state $M_{4}\times S^{1}$, we dramatically obtain well-known KK field
equations in the case of scalar field $\phi(x)=1$ where $F^{\alpha}_{\mu\nu}$
reduces to $F_{\mu\nu}$, $\mathcal{D}_{\mu}$ to ${D}_{\mu}$ and
$f^{\alpha\beta\gamma}$ to $0$. That is

$\displaystyle R_{\mu\nu}=\frac{1}{2}F_{\mu\tau}F_{\nu}\,^{\tau},\qquad
D_{\mu}F^{\mu}\,_{\nu}=0,\qquad F_{\mu\nu}F^{\mu\nu}=0.$

In that respect, the equations (V) and (V) vanish identically out, and the
remaining four reduced field equations (56)-(V) (with the current term (V))
precisely gives those of 5D Weyl-Yang-Kaluza-Klein model sibelhalil , as
expected. Besides, the equation (45) is reduced to be more interesting
expression $f_{\lambda}=-D_{\lambda}\mathcal{F}^{2}$ (also see sibelhalil ).

##  VI The Reduction of the Stress-Energy Tensor

In this section, we obtain the components of energy-momentum tensor of our
model $\hat{T}_{AB}$ (23) in the $(4+N)$-dimensional entire space. The
computations are more straightforward by taking into account non-trivial
Riemann tensors (II) and reduced form of the quadratic curvature term
$\hat{R}^{2}_{ABCD}$ (26), however we should recall that the noncoordinate
components of the full metric (8) are also used here. It is useful to
decompose bak (separate) the $\hat{T}_{\mu\nu}$ as well as $\hat{T}_{ij}$
component of the stress-energy tensor into trace-free
$\hat{T}_{\mu\nu}^{(tf)}$ and trace $\hat{T}_{\mu\nu}^{(t)}$ parts

$\displaystyle\hat{T}_{\mu\nu}=\hat{T}_{\mu\nu}^{(tf)}+\hat{T}_{\mu\nu}^{(t)}$
(64)

because not only the former look nice but also the later can be employed to
find the trace of $\hat{T}_{AB}$ more easily. Hence, after performing some
manipulations, the traceless parts are found to be

$\displaystyle\hat{T}^{(tf)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle{T}^{(g)}_{\mu\nu}+3(k_{\alpha\beta}+\frac{1}{2}l_{\alpha\gamma}l_{\beta\eta}\mathcal{F}^{2}_{\gamma\eta})T^{(em)\alpha\beta}_{\mu\nu}$
$\displaystyle+3l_{\alpha\beta}{T}^{(c)\alpha\beta}_{\mu\nu}+3m_{\alpha\beta\gamma}{T}^{(1)\alpha\beta\gamma}_{\mu\nu}$
$\displaystyle+\frac{1}{2}(l_{\alpha\beta}l_{\gamma\eta}+4l_{\alpha\eta}l_{\gamma\beta}){T}^{(2)\alpha\gamma\beta\eta}_{\mu\nu}+2l_{\alpha\beta}{T}^{(3)\alpha\beta}_{\mu\nu}.$

The first term of equation (VI) is labelled as ${T}^{(g)}_{\mu\nu}$ for the
reason that it exactly gives pure gravitational stress-energy tensor of Weyl-
Yang gauge theory in the actual $4$D $external$ space-time

$\displaystyle{T}^{(g)}_{\mu\nu}=R_{\mu\lambda\sigma\rho}R_{\nu}\,^{\lambda\sigma\rho}-\frac{1}{4}g_{\mu\nu}R_{\tau\lambda\sigma\rho}R^{\tau\lambda\sigma\rho}.$
(66)

Another well-known term $T^{(em)\alpha\beta}_{\mu\nu}$ is also welcome from
the equation (VI). That is gauge invariant energy-momentum tensor of the non-
Abelian Yang-Mills fields which can be written

$\displaystyle{T}^{(em)\alpha\beta}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}-\frac{1}{4}g_{\mu\nu}F^{\alpha}_{\tau\lambda}F^{\beta\tau\lambda}.$
(67)

The resting non-trivial terms of equation (VI) have more complicated
structures because they come, as might be expected, from higher-order
quadratic action. We respectively identify these terms as the stress-energy
tensor of the $RF^{2}$-type non-minimal coupling fields

$\displaystyle{T}^{(c)\alpha\beta}_{\mu\nu}=F^{\beta\rho\sigma}F^{\alpha}_{(\mu}\,{}^{\lambda}R_{\nu)\lambda\sigma\rho}-\frac{1}{4}g_{\mu\nu}F^{\alpha\tau\lambda}F^{\beta\rho\sigma}R_{\tau\lambda\sigma\rho},$

that of the cubic fields

$\displaystyle{T}^{(1)\alpha\beta\gamma}_{\mu\nu}=\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}-\frac{1}{4}g_{\mu\nu}\mathcal{F}^{3}_{\alpha\beta\gamma},$
(69)

of the quartic constituent fields

$\displaystyle{T}^{(2)\alpha\gamma\beta\eta}_{\mu\nu}=\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\beta\eta}-\frac{1}{4}g_{\mu\nu}\mathcal{F}^{4}_{\alpha\gamma\beta\eta},$
(70)

and of the $(\mathcal{D}F)^{2}$-type interactions

$\displaystyle{T}^{(3)\alpha\beta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\beta\sigma\rho}-\frac{1}{4}g_{\mu\nu}\mathcal{D}_{\sigma}F^{\alpha}_{\tau\rho}\mathcal{D}^{\sigma}F^{\beta\tau\rho}.$
(71)

The trace part which consists of the remaining terms of $\hat{T}_{\mu\nu}$
(64) is become

$\displaystyle\hat{T}_{\mu\nu}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{\mu\nu}R^{2}_{ijkl}-\frac{3}{2}m_{\alpha\beta\gamma}\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}-\frac{3}{2}k_{\alpha\beta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}$
(72)
$\displaystyle+\frac{1}{2}l_{\alpha\beta}[l_{\gamma\eta}(\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\beta\eta}-2\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\eta\beta})$
$\displaystyle+2\mathcal{D}_{\sigma}F^{\alpha}_{\mu\rho}\mathcal{D}^{\sigma}F^{\beta}_{\nu}\,{}^{\rho}-3\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\beta\sigma\rho}].$

Another reduced component of the energy-momentum tensor is the $\hat{T}_{\mu
i}$ which turns out that

$\displaystyle\hat{T}_{\mu i}$
$\displaystyle=\frac{1}{2}\xi^{\alpha}_{i}R_{\mu\lambda\sigma\rho}\mathcal{D}^{\lambda}F^{\alpha\sigma\rho}+\xi^{\alpha}_{j}D^{j}\xi^{\beta}_{i}F^{\beta\sigma\rho}\mathcal{D}_{(\mu}F^{\alpha}_{\rho)\sigma}$
$\displaystyle+\xi^{\alpha}_{i}l_{\beta\gamma}[\frac{3}{2}F^{\beta}_{\mu}\,{}^{\lambda}F^{\sigma\rho}_{\gamma}\mathcal{D}_{\sigma}F^{\alpha}_{\rho\lambda}-F^{\alpha\sigma\lambda}F^{\beta}_{\lambda}\,{}^{\rho}\mathcal{D}_{(\mu}F^{\gamma}_{\sigma)\rho}].$

Similarly, the $i-j$ component of equation 23) can be split into a trace-free
and a trace part as follows

$\displaystyle\hat{T}_{ij}=\hat{T}_{ij}^{(tf)}+\hat{T}_{ij}^{(t)},$ (74)

where

$\displaystyle\hat{T}_{ij}^{(tf)}$ $\displaystyle=$
$\displaystyle\frac{N}{4}{T}^{(g)}_{ij}+\frac{N}{16}[l_{\gamma\eta}(\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+4\mathcal{F}_{\alpha\gamma\eta\beta}^{4})$
$\displaystyle+4\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}]{T}^{(1)\alpha\beta}_{ij}$
$\displaystyle+\frac{3N}{4}[\mathcal{F}^{2}_{\alpha\beta}{T}^{(2)\alpha\beta}_{ij}+\mathcal{F}^{3}_{\alpha\beta\gamma}{T}^{(3)\alpha\beta\gamma}_{ij}].$

The ${T}^{(g)}_{ij}$ appear formally to be $4$D stress-energy tensor of Weyl-
Yang theory but in this case for the $internal$ space. That is,
${T}^{(g)}_{ij}$ is written in terms of the Riemann tensors of $internal$
space

$\displaystyle{T}^{(g)}_{ij}=R_{inkl}R_{j}\,^{nkl}-\frac{1}{N}g_{ij}R_{mnkl}R^{mnkl}.$
(76)

Other unphysical energy-momentum tensors correspondingly are

$\displaystyle{T}^{(1)\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}\mathcal{\xi}^{\alpha}_{k}\xi^{\beta
k},$ (77) $\displaystyle{T}^{(2)\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle
D_{n}\xi^{\alpha}_{i}D^{n}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}D_{n}\xi^{\alpha}_{m}D^{n}\xi^{\beta
m},$ (78) $\displaystyle{T}^{(3)\alpha\beta\gamma}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{n}\xi^{\beta}_{(i}D^{n}\xi^{\gamma}_{j)}-\frac{1}{N}g_{ij}\xi^{\alpha}_{n}\xi^{\beta}_{m}D^{n}\xi^{\gamma
m},$ (79)

together with more complicated trace part

$\displaystyle\hat{T}_{ij}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{ij}[R^{2}_{\mu\nu\lambda\rho}-\frac{3}{2}l_{\alpha\beta}(2R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\beta\lambda\rho}$
(80) $\displaystyle-
l_{\gamma\eta}\mathcal{F}^{2}_{\alpha\gamma}\mathcal{F}^{2}_{\beta\eta})]-(\frac{N-4}{4})R_{inkl}R_{j}\,^{nkl}$
$\displaystyle-(\frac{3N-6}{4})[D_{n}\xi^{\alpha}_{i}D^{n}\xi^{\beta}_{j}\mathcal{F}^{2}_{\alpha\beta}+\xi^{\alpha}_{n}\xi^{\beta}_{(i}D^{n}\xi^{\gamma}_{j)}\mathcal{F}^{3}_{\alpha\beta\gamma}]$
$\displaystyle-\frac{1}{16}\xi^{\alpha}_{i}\xi^{\beta}_{j}\big{\\{}l_{\gamma\eta}[(N+4)\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+2(N-2)\mathcal{F}^{4}_{\alpha\gamma\eta\beta}]$
$\displaystyle+4(N-1)\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}\big{\\}}.$

In $5$D KK theories, $\hat{T}_{\mu 5}$ may be interpreted as the current
density by help of conservation law $\hat{D}_{\mu}\hat{T}^{\mu}\,_{5}=0$. In
our approach, it is not only easy to find physical roles or consequences of
components $\hat{T}_{\mu i}$ and $\hat{T}_{ij}$ but also to obtain an
analogues of those in the $4$D Einstein’s theory of gravity. Now, we can
calculate the trace of the full $\hat{T}_{AB}$ (i.e. $\hat{T}$) by using

$\displaystyle\hat{T}\equiv\hat{g}^{AB}\hat{T}_{AB}=\hat{g}^{\mu\nu}\hat{T}^{(t)}_{\mu\nu}+\hat{g}^{ij}\hat{T}^{(t)}_{ij}.$
(81)

The trace is easily obtained by employing in the manner of equations (72) and
(80). The result will not here be considered further, however the resulting
form can be checked on due to the fact that $\hat{T}=-(N/4)\hat{R}^{2}_{ABCD}$
(from equation (23)) with equation (26). It is no difficult to conjecture
that, the absence of non-Abelian gauge fields $A_{\mu}^{\alpha}=0$ we only get
pure gravitational stress-energy tensor $\hat{T}_{\mu\nu}={T}^{(g)}_{\mu\nu}$
with $\hat{T}_{\mu i}=\hat{T}_{ij}=0$.

Attention will now be turned to investigate the reduced energy-momentum
tensors which are already obtained below for the case where the $internal$
space is the homogeneous coset space. On account of all necessarily
assumptions which are mentioned previous sections, the traceless part
$\hat{T}_{\mu\nu}^{(tf)}$ in (VI) is become

$\displaystyle\hat{T}_{\mu\nu}^{(tf)}$ $\displaystyle=$
$\displaystyle{T}^{(g)}_{\mu\nu}+\frac{3}{8}(2f^{\alpha\gamma\eta}f^{\beta\gamma\eta}+\mathcal{F}^{2}_{\alpha\beta}){T}^{(em)\alpha\beta}_{\mu\nu}+\frac{3}{2}{T}^{(c)\alpha\alpha}_{\mu\nu}$
(82)
$\displaystyle+\frac{3}{2}f^{\alpha\beta\gamma}{T}^{(1)\alpha\beta\gamma}_{\mu\nu}+\frac{1}{8}{T}^{(2)\alpha\beta\alpha\beta}_{\mu\nu}+\frac{1}{2}{T}^{(2)\alpha\beta\beta\alpha}_{\mu\nu}$
$\displaystyle+{T}^{(3)\alpha\alpha}_{\mu\nu},$

and the trace part in (72) is

$\displaystyle\hat{T}_{\mu\nu}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{\mu\nu}R^{2}_{ijkl}-\frac{3}{8}f^{\alpha\gamma\eta}f^{\beta\gamma\eta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}$
(83)
$\displaystyle-\frac{3}{4}f^{\alpha\beta\gamma}\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}\
+\frac{1}{8}(\mathcal{F}_{\mu\nu}^{4~{}\alpha\beta\alpha\beta}-2\mathcal{F}_{\mu\nu}^{4~{}\alpha\beta\beta\alpha})$
$\displaystyle+\frac{1}{2}\mathcal{D}_{\sigma}F^{\alpha}_{\mu\rho}\mathcal{D}^{\sigma}F^{\beta}_{\nu}\,{}^{\rho}-\frac{3}{4}\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\alpha\sigma\rho}.$

Hence, we recognize the fact that the component $\hat{T}_{\mu\nu}$ (the
summation of equation (82) and equation (83)) is Killing term-free equation,
and the resulting expression is more convenient to bring out type of
interactions between constituent fields.

Next, the $\hat{T}_{\mu i}$ (VI) yields

$\displaystyle\hat{T}_{\mu i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\xi^{\alpha}_{i}[R_{\mu\lambda\sigma\rho}\mathcal{D}^{\lambda}F^{\alpha\sigma\rho}+f^{\alpha\beta\gamma}F^{\beta\sigma\rho}\mathcal{D}_{(\mu}F^{\gamma}_{\rho)\sigma}$
(84)
$\displaystyle+\frac{3}{2}F^{\beta}_{\mu}\,{}^{\lambda}F^{\beta\sigma\rho}\mathcal{D}_{\sigma}F^{\alpha}_{\rho\lambda}-F^{\alpha\sigma\lambda}F^{\beta}_{\lambda}\,{}^{\rho}\mathcal{D}_{(\mu}F^{\beta}_{\sigma)\rho}].$

Let us finally evaluate the last component $\hat{T}_{ij}$ of the stress-energy
tensor $\hat{T}_{AB}$. The equation (VI) is reduced to

$\displaystyle\hat{T}_{ij}^{(tf)}$ $\displaystyle=$
$\displaystyle\frac{N}{4}{T}^{(g)}_{ij}+\frac{N}{16}(\mathcal{F}^{4}_{\alpha\gamma\beta\gamma}+4\mathcal{F}^{4}_{\alpha\gamma\gamma\beta}+4\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}$
(85)
$\displaystyle+3\mathcal{F}^{2}_{\eta\zeta}f^{\eta\gamma\alpha}f^{\zeta\gamma\beta}){T}^{\alpha\beta}_{ij}+\frac{3N}{8}\mathcal{F}^{3}_{\eta\alpha\gamma}f^{\gamma\eta\beta}{\widetilde{T}}^{\alpha\beta}_{ij}.$

Here, we write to show that

$\displaystyle{T}^{\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}\delta^{\alpha\beta},$
(86) $\displaystyle{\widetilde{T}}^{\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{(i}\xi^{\beta}_{j)}-\frac{1}{N}g_{ij}\delta^{\alpha\beta}.$
(87)

It means that, the ${T}^{(1)\alpha\beta}_{ij}$ in (77) and the
${T}^{(2)\alpha\beta}_{ij}$ in (78) reduced to the ${T}^{\alpha\beta}_{ij}$
equation (86) , the ${T}^{(3)\alpha\beta\gamma}_{ij}$ in (79) to the
${\widetilde{T}}^{\alpha\beta}_{ij}$ equation (87), respectively. As another
point of view, the equation (80) which is trace part of $\hat{T}_{ij}$ is
equivalent to

$\displaystyle\hat{T}_{ij}^{(t)}$
$\displaystyle=-\frac{1}{4}g_{ij}[R^{2}_{\mu\nu\lambda\rho}-\frac{3}{2}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\alpha\lambda\rho}+\frac{3}{8}(\mathcal{F}^{2}_{\alpha\beta})^{2}]$
$\displaystyle-(\frac{N-4}{4})R_{inkl}R_{j}\,^{nkl}-(\frac{3N-6}{8})f^{\gamma\eta\beta}\xi^{\alpha}_{(i}\xi^{\beta}_{j)}\mathcal{F}^{3}_{\eta\alpha\gamma}$
$\displaystyle-\frac{1}{32}\xi^{\alpha}_{i}\xi^{\beta}_{j}[(N+4)\mathcal{F}^{4}_{\alpha\gamma\beta\gamma}+4(N-2)\mathcal{F}^{4}_{\alpha\gamma\gamma\beta}$
$\displaystyle+8(N-1)\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}+6(N-2)f^{\eta\gamma\alpha}f^{\zeta\gamma\beta}\mathcal{F}^{2}_{\eta\zeta}].$

The compatibility is also welcome here for the $5$D KK picture where
$\xi^{\alpha}_{i}=1$ and $R_{ijkl}=0$. The equations (82), (83), (84) and (VI)
exactly reduce to the those of our previous model sibelhalil , and the
equation (85) goes explicitly to $0$. It should be emphasised that, we use in
(84) well-known tensorial rule, the inner product of a symmetric and an
antisymmetric tensor vanishes.

##  VII Conclusions

In this paper, we have completely generalized methods and results from our
previous work Başkal and Kuyrukcu sibelhalil  to the non-Abelian case without
including any scalar fields by taking into account spinless and torsionless
Weyl-Yang gravity model in the context of the more than five dimensional pure
KK theories. We have firstly given a brief review of both of the non-Abelian
KK theory and Weyl-Yang gravity model, respectively. Next, we have performed a
popular dimensional KK reduction procedure to obtain the modified $4$D Weyl-
Yang+Yang-Mills action from a pure $(4+N)$-dimensional Weyl-Yang gravitational
Lagrangian without supplementary matter fields. In that respect, the
dimensionally reduced field equations which contain naturally the generalized
gravitational source term and the Lorentz force density and stress-energy
tensors are simultaneously investigated, and they are compared with the our
previous gravity model and standard higher-dimensional KK theories,
respectively. In our approach, the $4$D matter-spin source term is induced
from those that matter carrying energy-momentum but not possessing any spin
tensor. We also extend the reduced equations to more physical forms by taking
the compact $internal$ space as a homogeneous coset spaces background needed
in KK theories.

S. Başkal comments that “The $5$D Weyl-Yang theory appears to be more complete
compared to that of the standard KK model by accommodating terms that can be
identified as the Lorentz force density.” sibelhalil . Our theory which is
extension of $5$D Weyl-Yang model to the case of a higher dimensional non-
Abelian KK theory has effectively completed this remark with the generalized
gravitational source term which is not exist in the literature.

###### Acknowledgements.

I would like to thank S. Başkal for showing me ref. dereli .

## References

  * (1) Th. Kaluza, Sitz. Preuss. Akad. Wiss. Phys. Math. K1., 966 (1921).
  * (2) O. Klein, Z. Phys. 37, 895 (1926).
  * (3) H. Mandel, Z. Phys. 39, 136 (1926).
  * (4) O. Klein, Nature 118, 516 (1926).
  * (5) A. Einstein and P. Bergmann, Ann. Math. 39, 683 (1938).
  * (6) B. S. De Witt, Relativity, Groups and Topology, edited by B. S. De Witt and C. De Witt (Gordon and Breach, New York, 1964), p. 725 ; Dynamical Theory of Groups and Fields (Gordon and Breach, New York, London, Paris, 1965), p. 139
  * (7) J. Rayski, Acta Phys. Polon. 27, 947 (1965) ; 28, 87 (1965).
  * (8) R. Kerner, Ann. Inst. Henri Poincaré 9, 143 (1968).
  * (9) A. Trautman, Rep. Math. Phys. 1, 29 (1970).
  * (10) Y.M. Cho, J. Math. Phys. 16, 2029 (1975).
  * (11) Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12, 1711 (1975).
  * (12) Y.M. Cho and P.S. Jang, Phys. Rev. D 12, 3789 (1975).
  * (13) L.N. Chang, K.I. Macrae and F. Mansouri, Phys. Rev. D 13, 235 (1976).
  * (14) S. Helgason, Differential Geometry and Symmetric Spaces, (Academic, New York, 1962).
  * (15) S. Kobayashi and K. Nomizu, Foundations of Differential Geometry , (Wiley, New York, 1969), Vol. 2.
  * (16) R. Gilmore, Lie Groups, Lie Algebras, and Some Their Applications , (Wiley, New York, 1974).
  * (17) E. Witten, Phys. Rev. Lett. 38, 121 (1977).
  * (18) P. Forgacs and N. S. Manton, Commun. Math. Phys. 72, 15 (1980).
  * (19) E. Witten, Nucl. Phys. B186, 412 (1981).
  * (20) D. Kapetanakis and G. Zoupanos, Phys. Rept. 219, 1 (1992).
  * (21) J. Scherk and J. H. Schwarz, Nucl. Phys. B153, 61 (1979).
  * (22) T. Appelquist, A. Chodos and P.G.O. Freund, Modern Kaluza-Klein theories, (Addison-Wesley, Reading, MA, 1987).
  * (23) D. Bailin and A. Love, Rep. Prog. Phys. 50, 1087 (1987).
  * (24) J. M. Overduin and P. S. Wesson, Phys. Rep. 283, 303 (1997).
  * (25) M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Rep. 130, 1 (1986).
  * (26) J. H. Schwarz, Phys. Rep. 89, 223 (1982).
  * (27) H. Weyl, Ann. Phys. (Leipzig) 59, 103 (1919).
  * (28) C. Lanczos, Rev. Mod. Phys. 21, 497 (1949).
  * (29) A. Lichnerowicz, C. R. Acad. Sci. 247, 433 (1958).
  * (30) G. Stephenson, Nuovo Cimento 9, 263 (1958).
  * (31) P. W. Higgs, Nuovo Cimento 11, 816 (1959).
  * (32) C. W. Kilmister and D. J. Newman, Proc. Cambridge Philos. Soc. 57, 851 (1961).
  * (33) C. N. Yang, Phys. Rev. Lett. 33, 445 (1974).
  * (34) A. H. Thompson, Phys. Rev. Lett. 34, 507 (1975).
  * (35) R. Pavelle, Phys. Rev. Lett. 34, 1114 (1975).
  * (36) E. E. Fairchild Jr., Phys. Rev. D 14, 384 (1976); $ibid$. 14, 2833 (1976).
  * (37) P. Baekler, F. W. Hehl and E. W. Mielke, in Proceedings of the Second Marcel Grossmann Meetings on General Relativity, Amsterdam, 1982, edited by R. Ruffini (North-Holland, Amsterdam, 1982) p. 413.
  * (38) V. Szczyrba, Phys. Rev. D 3, 351 (1987).
  * (39) J. Maluf, Classical Quant. Grav. 5, L81 (1988).
  * (40) J. Maluf, J. Math. Phys. 32, 1556 (1991).
  * (41) C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).
  * (42) F. W. Hehl, Y. Ne’eman, J. Nitsch and P. von der Heyde, Phys. Lett. 78B, 102 (1978)
  * (43) F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976)
  * (44) O. Lauscher and M. Reuter, Phys. Rev. D 66, 025026 (2002).
  * (45) G. Cognola et al., Phys. Rev. D 77, 046009 (2008).
  * (46) J.-M. Gerard, Classical Quant. Grav. 24, 1867 (2007).
  * (47) R. J. Cook, (2008), arXiv:0810.4495 [gr-qc].
  * (48) G. R. Gonzalez-Martin, (2009), arXiv:0910.3380 [physics.gen-ph].
  * (49) P. Chen, Modern Phys. Lett. A 25, 2795 (2010),
  * (50) P. Chen, K. Izumi and N.-E. Tung, (2013), arXiv:1304.6334 [gr-qc].
  * (51) Y. Yang and W. B. Yeung, (2013), arXiv: 1111.7062 [physics.gen-ph]; (2012), arXiv:1205.2690 [physics.gen-ph]; (2012), arXiv:1210.0529 [physics.gen-ph]; (2013), arXiv:1303.3801 [physics.gen-ph].
  * (52) A. Wang, Phys. Rev. Lett. 110, 091101 (2013).
  * (53) S. Başkal, Prog. Theor. Phys. 102, 803 (1999).
  * (54) H. Kuyrukcu, Classical Quant. Grav. 30, 155013 (2013).
  * (55) S. Başkal and H. Kuyrukcu, Gen. Relativ. Gravit. 45, 359 (2013).
  * (56) C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, edited by W. H. Freeman (San Francisco, 1973), Chap. 8.
  * (57) M. J. Duff, B. E. W. Nilsson, C. N. Pope and N. P. Warner, Phys. Lett. 149B, 90 (1984).
  * (58) M. J. Duff and C. N. Pope, Nucl. Phys. B225, 355 (1985).
  * (59) E. Cremmer and J. Scherk, Nucl. Phys. B103, 399 (1976); B108, 409 (1976); B118, 61 (1977).
  * (60) M. A. Awada, Phys. Lett. 127B, 415 (1983).
  * (61) T. Appequist and A. Chodos, Phys. Rev. D 28, 772 (1983).
  * (62) A. Palatini, Rend. Circ. Mat. Palermo 43, 203 (1919).
  * (63) M. Camenzind, J. Math. Phys. 16, 1023 (1975).
  * (64) R. Pavelle, Phys. Rev. Lett. 37, 961 (1976).
  * (65) E. E. Fairchild Jr., Phys. Rev. D 16, 2438 (1977).
  * (66) A. A. Tseytlin, Phys. Rev. D 26, 3327 (1982).
  * (67) C. W. Kilmister, Perspectives in Geometry and Relativity; Essays in Honor of Vaclav Hlavaty, edited by B. Hoffmann (Bloomington, Indiana, 1966), p. 201-2011.
  * (68) P. Baekler and P. B. Yasskin, Gen. Relativ. Gravit. 16, 1135 (1984).
  * (69) J. Garecki, Classical Quant. Grav. 2, 403 (1985).
  * (70) L. Bel, C.R. Acad. Sci. 247, 1094 (1958); $ibid$. 248, 1297 (1959).
  * (71) I. Robinson, Kings College Lectures 1958 (unpublished); Classical Quant. Grav. 14, 4331 (1997).
  * (72) Y.-S. Wu and A. Zee, Nucl. Phys. B237, 586 (1984).
  * (73) J. F. Luciani, Nucl. Phys. 135, 111 (1978).
  * (74) Y. Liu and J. Jing, Gen. Relativ. Gravit. 44, 1739 (2012).
  * (75) A. I. Alekseev and B. A. Arbuzov and V. A. Baikov, Theor. Mat. Phys. 52, 187 (1982).
  * (76) A. I. Alekseev and B. A. Arbuzov, Theor. Mat. Phys. 59, 372 (1984).
  * (77) W. H. Huang, Phys. Lett. B203, 105 (1988).
  * (78) F. Müller-Hoissen, Classical Quant. Grav. 5, L35 (1988).
  * (79) T. Dereli and G. Üçoluk, Classical Quant. Grav. 7, 533 (1990).
  * (80) D. S. Kimm , J. Korean Phys. Soc. 23, 359 (1990).
  * (81) F. Öktem, Doğa Bilim Dergisi A1 9, 3 (1985).
  * (82) S. Weinberg, Gravitation and cosmology, edited by J. Wiley and Sons, Inc. (New York, 1972), Chap. 13.


# BiasKG: Adversarial Knowledge Graphs to Induce Bias in Large Language Models

Chu Fei Luo
Vector Institute
Queen’s University
&Ahmad Ghawanmeh
Ernst & Young
&Xiaodan Zhu
Queen’s University
&Faiza Khan Khattak
Vector Institute

###### Abstract

Modern large language models (LLMs) have a significant amount of world
knowledge, which enables strong performance in commonsense reasoning and
knowledge-intensive tasks when harnessed properly. The language model can also
learn social biases, which has a significant potential for societal harm.
There have been many mitigation strategies proposed for LLM safety, but it is
unclear how effective they are for eliminating social biases. In this
work111Accepted at TrustNLP@NAACL, we propose a new methodology for attacking
language models with knowledge graph-augmented generation. We refactor natural
language stereotypes into a knowledge graph, and use adversarial attacking
strategies to induce biased responses from several open- and closed-source
language models. We find our method increases bias in all models, even those
trained with safety guardrails. This demonstrates the need for further
research in AI safety, and further work in this new adversarial space.222Our
code and data are publicly available at
https://github.com/VectorInstitute/biaskg

##  1 Introduction

Modern large language models (LLMs) learn a significant amount of world
knowledge from their pre-training data, which enables strong performance in
commonsense reasoning and knowledge-intensive tasks Yu et al. (2023). However,
world knowledge also implies that language models learn common social biases
in society such as stereotypes Kotek et al. (2023); Gallegos et al. (2023).
Social biases can affect a variety of language modelling tasks that are not
directly harmful, such as writing recommendation letters Wan et al. (2023). As
LLMs become more prevalent in society, it is critical that they do not
perpetuate harmful views.

Creating language models aligned to human values is a rapidly growing field of
interest known as AI alignment Ji et al. (2023). Conversely, there are many
strategies to adversarially attack alignment strategies, such as red-teaming
Ganguli et al. (2022). There are then works that use the red-teaming
strategies to train guardrails to recognize and refuse biased queries Inan et
al. (2023); Bai et al. (2022). However, bad actors have the potential to
invent new adversarial attacks and strategies. Also, these works mainly focus
on harmful or unethical instructions (eg. how to rob a bank) rather than the
more subtle issue of social bias. This is a critically under-explored area,
and it is important to understand how language models can be exploited beyond
their intended use.

The main contributions of this work include:

  * •

We refactor an existing dataset of harmful stereotypes, the Social Bias
Inference Corpus (SBIC), into a knowledge graph with 51,371 relations. We call
this a bias knowledge graph
(BiasKG).333https://huggingface.co/datasets/chufeiluo/biaskg

  * •

We propose to use this stereotype knowledge graph as a source of superior
knowledge to adversarially induce biased responses in LLMs. To our best
knowledge, this is a recent approach that has not been applied to the issue of
bias.

  * •

We present empirical results on the efficacy of the above methods across
several open- and closed-source models. Our results show that our method does
increase biased responses from a model, even those trained with safety
alignment.

Figure 1: Our proposed pipeline.

##  2 Related Work

#### Adversarial attacking.

Adversarial attacking is a prominent field of research for language models
Zhang et al. (2020). For large language models, the most common method of
attack is adversarial prompting Kumar et al. (2023); Liu et al. (2023). This
is a broad category for an attack which inserts some adversarial tokens in the
input prompt. These tokens can be nonsensical Zhang et al. (2020), but the
specific act of overriding safeguards with natural language instructions is
known as red-teaming Ganguli et al. (2022). Others attempt to borrow concepts
from other fields such as elevating permissions Liu et al. (2023).

#### Knowledge Graphs and Retrieval-Augmented Generation

Knowledge graphs (KGs) are a form of structured data that encodes entities and
their inter-relationships in a (startnode, edge, endnode) format Ji et al.
(2021). There are knowledge graphs with a set of pre-defined possible
relationships, such as ConceptNet Speer et al. (2017), and dynamic knowledge
graphs that allow free-form relationships between entities Andrus et al.
(2022). For LLMs, it is most common to use knowledge graphs Retrieval-
Augmented Generation (RAG) Lewis et al. (2020). Please refer to Appendix A for
more related work.

Knowledge Graph Conversion Example
---
Examples:
Target minority: black folks
Stereotype: black women are basic
Graph:
‘(black women, are, basic)‘
‘(black women, belong to, black people)‘
‘(black women, belong to, women)‘

Table 1: Our hand-crafted demonstration for converting the SBIC dataset into a
dynamic knowledge graph.

##  3 Methodology

We introduce a new approach to adversarially induce bias in LLMs with our
BiasKG pipeline, illustrated in Figure 1. First, we convert free-form
knowledge from the Social Bias Inference Corpus (SBIC) Sap et al. (2020) into
a structured knowledge graph. We store the knowledge graph via semantic
embeddings, and retrieve triplets with a 2-hop retrieval algorithm. Lastly, we
use the retrieved nodes to attack the LLM as Superior Knowledge, following
previous work Liu et al. (2023).

###  3.1 Constructing BiasKG

We construct our bias knowledge graph (BiasKG) from free-text stereotypes
sourced from the SBIC dataset Sap et al. (2020). The knowledge graph is
defined as a directed graph $G=(E,R,S)$, where $E$ stands for the set of
entities (nodes), $R$ the set of relationships (edges), and $S$ is the set of
all entity-relationship-entity triplets in the graph. Each directed triplet
$(e_{s},r,e_{e})\in S$ signifies a relationship $r$ connecting the starting
node entity $e_{s}$ and end node $e_{e}$. Notably, $e_{s}$ is always a
sensitive attribute (eg. black person, black man, etc.). We define a sensitive
attribute as a potential target for social bias and discrimination, such as
race, gender, religion, etc. The start node $e_{s}$ can be a compound of
several attributes (eg. a white man is both caucasian and male), so we allow
$e_{e}$ to decompose sensitive attributes with a “belongs to” edge.

We convert free-form text into triplets through prompting GPT-4 OpenAI (2023)
with the prompt defined in Table 1. We use manually crafted few-shot
demonstrations as well as natural language instructions to enforce the
constraints of the knowledge graph defined above. Statistics on the knowledge
graph can be found in Table 2.

Stat. | Count (%)
---|---
SBIC entries | 25,602
Total KG relations | 51,371
Sensitive Attributes | 3,015
Stereotypes | 10,333
Total Entities | 13,348
Edge Types | 4,806

Table 2: Statistics on our knowledge graph.

Setting | Baseline | BiasKG
---|---|---
Explanation? | Y | N | Y | N
| DR $\uparrow$ | NM $\downarrow$ | DR $\uparrow$ | NM $\downarrow$ | DR $\uparrow$ | NM $\downarrow$ | DR $\uparrow$ | NM $\downarrow$
GPT-3.5-turbo | 14.1 | (0.2) | 7.3 | (4.8) | 26.4 | (1.3) | 13.6 | (4.9)
GPT-4 | 1.5 | (0.1) | 1.6 | (0.1) | 7.0 | (16.6) | 8.9 | (2.9)
Mistral-7b | 36.3 | (0.0) | 48.3 | (0.0) | 59.7 | (0.0) | 46.2 | (0.0)
Llama2-7b | 65.9 | (1.0) | 63.2 | (5.3) | 66.0 | (0.7) | 43.4 | (34.5)
Llama2-13b | 54.6 | (17.1) | 61.8 | (7.1) | 61.6 | (7.4) | 57.9 | (12.6)

Table 3: Summary of Deception Rate (DR %) and No Match rate (NM %) across five
generative LLMs, open- and closed-source. Model temperature: 0.1

###  3.2 Top k Retrieval

While converting the stereotypes knowledge to KG format enforces structure to
the data, it is relatively noisy due to the minimal constraints we place on
its construction. We first encode all graph data and the input query into a
shared embedding representation. Then, we filter the triplets through a 2-hop
retrieval process. Our algorithm is inspired by multi-hop question answering
Yang et al. (2018) that retrieves one set of documents, then recursively
branches from that set to retrieve further related information. We use this
technique to discover stereotypes associated with both compound and decomposed
sensitive attributes, per the structure we defined in Section 3.1.

#### Embedding representations

We define the embedding function $\phi:E\cup S\cup c\rightarrow\mathbb{R}^{d}$
that can map entities, triplets $\in G$, as well as the input context $c$, to
a vectorized embedding space. This encodes our knowledge graph into two sets
of vectors: ${\bf V_{S}}=\\{\phi(e_{s},r,e_{e})\ \forall\ (e_{s},r,e_{e})\in
S\\}$ representing KG triplets , and ${\bf V_{E}}=\\{\phi(e)\ \forall\ e\in
E\\}$ representing all unique entities. Throughout the retrieval process, we
periodically prune the search by taking top $k$, defined as ranking a set of
embeddings by the cosine similarity to a target and retaining $k$ results by
the highest score.

#### BiasKG graph search

After encoding the knowledge graph and input context, we formulate our
retrieval as a cosine similarity ranking. We compute the cosine similarity of
all entities to the input context, $\cos(v_{c},v_{e})\forall v_{e}\in V_{E}$,
and retain the top $k$ ranked entities, $E_{0}$. From the top $k$ nodes
$E_{0}$, we derive the corresponding subgraph which is the set of triplets
$S_{E_{0}}=\\{(e_{s},r,e_{e})\in S|e_{s}\in E_{0}\\}$. Next, we obtain the set
of all end nodes $E_{1}=\\{e_{2}:(e_{s},r,e_{e})\in S_{E_{0}}\ \land\
e_{e}\notin E_{0}\\}$. Finally, for each entity $\in E_{0}$, we derive its top
k most relevant outgoing entities by cosine similarity
$E_{1}=argmax_{k}\cos(v_{c},v_{E_{2}})$.

BBQ
---
Explanation? | Y | N
Temperature | 0.1 | 0.5 | 1.0 | 0.1 | 0.5 | 1.0
GPT-3.5-turbo | 26.4 | 25.0 | 23.5 | 13.6 | 14.4 | 17.0
GPT-4 | 8.1 | 8.2 | 8.2 | 10.8 | 10.8 | 11.2
Mistral-7b | 59.7 | 43.9 | 37.7 | 46.2 | 37.8 | 37.9
Llama2-7b | 66.7 | 67.1 | 68.1 | 78.1 | 78.1 | 79.4
Llama2-13b | 71.7 | 70.1 | 82.5 | 71.1 | 77.7 | 85.1

Table 4: Summary of the Deception Rate (%) in five generative LLMs, varying
temperatures.

#### Finalizing the top k triplets

We take the set of all entities from this retrieval process,
$E^{\prime}=E_{0}\cup E_{1}$ and their relevant triplets as set
$S_{E^{\prime}}$. Formally, $S_{E^{\prime}}=\\{(e_{s},r,e_{e})\in S\ |\
e_{s}\in E^{\prime}\ \lor\ e_{e}\in E^{\prime}\\}$. We re-rank all triplets in
$S_{E^{\prime}}$ by cosine similarity to the original input context $c$ to
obtain the final top k triplets, $T=argmax_{k}\cos(v_{c},V_{E_{2}})$

###  3.3 BiasKG for Augmented Generation

#### Prompting

Based on previous work Liu et al. (2023), we present the set of top $k$
triplets $T$ as a “superior source of knowledge and wisdom.” Please refer to
Appendix B for the full prompt template. We prompt the model for a JSON
output, as shown in Appendix C. To resolve any generation errors, such as
incorrect JSON formatting or API rate limits, we allow a maximum of three
retries. We experimented with a maximum of 10 retries, and empirically found
we reach a valid output on 1.5 retries on average. In addition to our BiasKG,
we test the effect of reasoning by asking for an Explanation field in the
prompt, as previous works found reasoning has a significant impact on a
language model’s bias Dige et al. (2023).

#### Baseline

We compare our BiasKG method to a baseline which simply omits the retrieved
triplets and the portion of the system prompt mentioning superior commonsense
knowledge. Every other component of the prompt remains constant.

##  4 Experiment Settings

We report results on Bias Benchmark for QA (BBQ) Parrish et al. (2022), a
question answering bias dataset with 58492 samples that tests bias for eleven
individual and combined sensitive attributes. The dataset is cloze-form
format, with an input context consisting of a scenario, a question based on
the scenario, and three possible answers. We experiment with GPT-3.5-turbo
Ouyang et al. (2022), GPT-4 OpenAI (2023), Mistral-7b Jiang et al. (2023),
Llama2-7b, and Llama2-13b Touvron et al. (2023). Please refer to Appendix C.2
for further model details.

For embedding representations, we use OpenAI’s text-embedding-ada-002
model444https://platform.openai.com/docs/guides/embeddings/embedding-models.
We report the Deception rate as the rate of biased responses as defined in
BBQ, and No Match rate as the rate of invalid responses. We report the mean
result over three unique runs. For more experimental details, please refer to
Appendix C.1.

##  5 Results & Discussion

Our experiment results are summarized in Table 3. Overall, we observe an
increase in Deception rate for all models after the application of our BiasKG
except for the Llama-2 suite of models. The most significant change was seen
in GPT-4, where the deception rate increased from 1.5% to 7.0%, almost
tripling the deception rate. Mistral-7b model, which increased from 36.3% to
59.7%.

We also find the No Match rate is high for some models (eg. Llama2-13b)
despite incorporating retries and multiple runs. After manual inspection of
the generations, the majority of these errors appear to be _refusals_ such as
“I cannot make any assumptions about the individuals based on their background
or demographic information…” The No Match rate generally decreases for the
Llama 2 models when we ask for an explanation, indicating the BiasKG
effectively bypasses guardrails, but does not necessarily increase the model’s
bias.

Figure 2: Deception rate averaged over all temperatures based on prompt
template. Top figure is gpt-3.5-turbo, bottom figure depicts gpt-4. Figure 3:
The average cosine similarity of the top k entities organized by sensitive
attribute.

#### Most Vulnerable Social Groups

We also show the per-attribute Deception rate for GPT-3.5-turbo and GPT-4 in
Figure 2. While the weighted deception rate is low, there are several
sensitive attributes with deception rates above 20%. We examine the mean
cosine similarity among the top $k$ nodes for every sample in BBQ, averaged
per sensitive attribute, as a proxy for average stereotype data available. As
shown in Figure 3, the cosine similarity varies between 0.74 and 0.78, which
is not a significant variation compared to attack effectiveness. This
indicates there are certain sensitive attributes tested by BBQ that are more
susceptible to our BiasKG attack, independent of data available.

#### Impact of Temperature

We experiment with increasing temperatures across all models, as summarized in
Table 4. The effect of temperature varies by model. For Llama2-7b and
Llama2-13b, the Deception Rate increases as temperature increases. Mistral-7b
has decreasing deception rate as temperature increases, and GPT-3.5-turbo
varies by whether or not we ask for an explanation from the model.

#### Significance of Explanation

The explanation component played a critical role in the deception rates. We
observed when models were explicitly asked for their reasoning, they generally
exhibited higher deception rates compared to where no explanation was
requested. The only exception is GPT-4, where the Deception Rate was 1.9%
lower. GPT-4, which has superior reasoning capabilities due to model scaling
Kaplan et al. (2020), is able to produce non-biased justifications that
provide stronger evidence to resist deception. Asking a smaller model for an
explanation, such as GPT-3.5 or Mistral-7b, appears to generate biased
reasoning, likely because the LLM relies more on the biased knowledge
provided.

##  6 Conclusion

Our results underscore that even models trained with safety guardrails can
have their responses manipulated to exhibit bias, emphasizing the complexity
and gravity of the bias problem in AI and the need for continued research in
AI safety.

## Limitations

We applied the BiasKG method specifically to induce social bias in language
models. Further investigations are necessary to determine its effectiveness
for addressing other types of biases, such as bias in healthcare and finance.
Another limitation is the choice of embedding model, text-embedding-ada-002 is
relatively low performing in semantic similarity benchmarks such as MTEB
Muennighoff et al. (2023). While there were other options for embedding model
choice, our paper is meant to establish a proof of concept for this
methodology, and text-embedding-ada-002 was sufficient for our purposes.

## Intended Use

There are two main intended uses for our work: a method of automatically
benchmarking LLMs for resilience against adversarial attacks, and a case study
in how RAG can be used to adversarially attack a language model. Automatic
benchmarking methods are important for rigorous evaluation of AI safety due to
the large range of possible inputs for a language model.

## Broader Impact Statement

This paper focuses on uncovering the limitations of language models and their
potential for misuse. We introduce a novel technique that leverages knowledge
graphs to identify vulnerabilities in language models, highlighting areas
where improvements are needed.

### Acknowledgements

This work has resulted from a larger collaborative initiative involving the
Vector Institute for AI and its industry partners. The authors extend their
appreciation to Tahniat Khan, the project manager, for her efforts in
coordinating this project. We also express our thanks to Deval Pandya, Vice
President of AI Engineering at the Vector Institute, for his valuable support.

The authors would also like to acknowledge the leaders at Ernst & Young (EY)
for their exceptional support and commitment to advancing artificial
intelligence research. Special thanks to Mario Schlener, Managing Partner for
Risk Consulting Canada, whose strategic vision exemplifies EY’s dedication to
fostering innovation and thought leadership in the industry. We also recognize
the expert oversight of Yara Elias, Kiranjot Dhillon, and Rasoul Shahsavarifar
from AI Risk Canada, whose contributions were integral to the project’s
success. This partnership not only reflects EY’s investment in AI but also
sets a foundation for continued research collaboration and driving progress in
the field.

## References

  * Andrus et al. (2022) Berkeley R Andrus, Yeganeh Nasiri, Shilong Cui, Benjamin Cullen, and Nancy Fulda. 2022.  Enhanced story comprehension for large language models through dynamic document-based knowledge graphs.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 36, pages 10436–10444.
  * Bai et al. (2022) Yuntao Bai, Saurav Kadavath, Sandipan Kundu, Amanda Askell, Jackson Kernion, Andy Jones, Anna Chen, Anna Goldie, Azalia Mirhoseini, Cameron McKinnon, et al. 2022.  Constitutional ai: Harmlessness from ai feedback.  _arXiv preprint arXiv:2212.08073_.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901.
  * Dige et al. (2023) Omkar Dige, Jacob-Junqi Tian, David Emerson, and Faiza Khan Khattak. 2023.  Can instruction fine-tuned language models identify social bias through prompting?  _arXiv preprint arXiv:2307.10472_.
  * Gallegos et al. (2023) Isabel O Gallegos, Ryan A Rossi, Joe Barrow, Md Mehrab Tanjim, Sungchul Kim, Franck Dernoncourt, Tong Yu, Ruiyi Zhang, and Nesreen K Ahmed. 2023.  Bias and fairness in large language models: A survey.  _arXiv preprint arXiv:2309.00770_.
  * Ganguli et al. (2022) Deep Ganguli, Liane Lovitt, Jackson Kernion, Amanda Askell, Yuntao Bai, Saurav Kadavath, Ben Mann, Ethan Perez, Nicholas Schiefer, Kamal Ndousse, et al. 2022.  Red teaming language models to reduce harms: Methods, scaling behaviors, and lessons learned.  _arXiv preprint arXiv:2209.07858_.
  * Inan et al. (2023) Hakan Inan, Kartikeya Upasani, Jianfeng Chi, Rashi Rungta, Krithika Iyer, Yuning Mao, Michael Tontchev, Qing Hu, Brian Fuller, Davide Testuggine, et al. 2023.  Llama guard: Llm-based input-output safeguard for human-ai conversations.  _arXiv preprint arXiv:2312.06674_.
  * Ji et al. (2023) Jiaming Ji, Tianyi Qiu, Boyuan Chen, Borong Zhang, Hantao Lou, Kaile Wang, Yawen Duan, Zhonghao He, Jiayi Zhou, Zhaowei Zhang, et al. 2023.  Ai alignment: A comprehensive survey.  _arXiv preprint arXiv:2310.19852_.
  * Ji et al. (2021) Shaoxiong Ji, Shirui Pan, Erik Cambria, Pekka Marttinen, and S Yu Philip. 2021.  A survey on knowledge graphs: Representation, acquisition, and applications.  _IEEE transactions on neural networks and learning systems_ , 33(2):494–514.
  * Jiang et al. (2023) Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, et al. 2023.  Mistral 7b.  _arXiv preprint arXiv:2310.06825_.
  * Jiang et al. (2022) Liwei Jiang, Jena D. Hwang, Chandra Bhagavatula, Ronan Le Bras, Jenny Liang, Jesse Dodge, Keisuke Sakaguchi, Maxwell Forbes, Jon Borchardt, Saadia Gabriel, Yulia Tsvetkov, Oren Etzioni, Maarten Sap, Regina Rini, and Yejin Choi. 2022.  Can machines learn morality? the delphi experiment.
  * Kaplan et al. (2020) Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. 2020.  Scaling laws for neural language models.  _arXiv preprint arXiv:2001.08361_.
  * Kotek et al. (2023) Hadas Kotek, Rikker Dockum, and David Sun. 2023.  Gender bias and stereotypes in large language models.  In _Proceedings of The ACM Collective Intelligence Conference_ , pages 12–24.
  * Kumar et al. (2023) Aounon Kumar, Chirag Agarwal, Suraj Srinivas, Soheil Feizi, and Hima Lakkaraju. 2023.  Certifying llm safety against adversarial prompting.  _arXiv preprint arXiv:2309.02705_.
  * Lewis et al. (2020) Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, et al. 2020.  Retrieval-augmented generation for knowledge-intensive nlp tasks.  _Advances in Neural Information Processing Systems_ , 33:9459–9474.
  * Liu et al. (2023) Yi Liu, Gelei Deng, Zhengzi Xu, Yuekang Li, Yaowen Zheng, Ying Zhang, Lida Zhao, Tianwei Zhang, and Yang Liu. 2023.  Jailbreaking chatgpt via prompt engineering: An empirical study.  _arXiv preprint arXiv:2305.13860_.
  * Muennighoff et al. (2023) Niklas Muennighoff, Nouamane Tazi, Loic Magne, and Nils Reimers. 2023.  MTEB: Massive text embedding benchmark.  In _Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics_ , pages 2014–2037, Dubrovnik, Croatia. Association for Computational Linguistics.
  * OpenAI (2023) OpenAI. 2023.  Gpt-4 technical report.  _ArXiv_ , abs/2303.08774.
  * Ouyang et al. (2022) Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. 2022.  Training language models to follow instructions with human feedback.  _Advances in neural information processing systems_ , 35:27730–27744.
  * Pan et al. (2024) Shirui Pan, Linhao Luo, Yufei Wang, Chen Chen, Jiapu Wang, and Xindong Wu. 2024.  Unifying large language models and knowledge graphs: A roadmap.  _IEEE Transactions on Knowledge and Data Engineering (TKDE)_.
  * Parrish et al. (2022) Alicia Parrish, Angelica Chen, Nikita Nangia, Vishakh Padmakumar, Jason Phang, Jana Thompson, Phu Mon Htut, and Samuel Bowman. 2022.  BBQ: A hand-built bias benchmark for question answering.  In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 2086–2105, Dublin, Ireland. Association for Computational Linguistics.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Sap et al. (2020) Maarten Sap, Saadia Gabriel, Lianhui Qin, Dan Jurafsky, Noah A. Smith, and Yejin Choi. 2020.  Social bias frames: Reasoning about social and power implications of language.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 5477–5490, Online. Association for Computational Linguistics.
  * Speer et al. (2017) Robyn Speer, Joshua Chin, and Catherine Havasi. 2017.  Conceptnet 5.5: An open multilingual graph of general knowledge.  pages 4444–4451.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. 2023.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_.
  * Wan et al. (2023) Yixin Wan, George Pu, Jiao Sun, Aparna Garimella, Kai-Wei Chang, and Nanyun Peng. 2023.  " kelly is a warm person, joseph is a role model": Gender biases in llm-generated reference letters.  _arXiv preprint arXiv:2310.09219_.
  * Xu et al. (2023) Xilie Xu, Keyi Kong, Ning Liu, Lizhen Cui, Di Wang, Jingfeng Zhang, and Mohan Kankanhalli. 2023.  An llm can fool itself: A prompt-based adversarial attack.  _arXiv preprint arXiv:2310.13345_.
  * Yang et al. (2018) Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William W Cohen, Ruslan Salakhutdinov, and Christopher D Manning. 2018.  Hotpotqa: A dataset for diverse, explainable multi-hop question answering.  _arXiv preprint arXiv:1809.09600_.
  * Yu et al. (2023) Jifan Yu, Xiaozhi Wang, Shangqing Tu, Shulin Cao, Daniel Zhang-Li, Xin Lv, Hao Peng, Zijun Yao, Xiaohan Zhang, Hanming Li, et al. 2023.  Kola: Carefully benchmarking world knowledge of large language models.  _arXiv preprint arXiv:2306.09296_.
  * Zhang et al. (2020) Wei Emma Zhang, Quan Z Sheng, Ahoud Alhazmi, and Chenliang Li. 2020.  Adversarial attacks on deep-learning models in natural language processing: A survey.  _ACM Transactions on Intelligent Systems and Technology (TIST)_ , 11(3):1–41.
  * Zhang et al. (2022) Xikun Zhang, Antoine Bosselut, Michihiro Yasunaga, Hongyu Ren, Percy Liang, Christopher D Manning, and Jure Leskovec. 2022.  Greaselm: Graph reasoning enhanced language models for question answering.  _arXiv preprint arXiv:2201.08860_.
  * Zhao et al. (2023) Wayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, et al. 2023.  A survey of large language models.  _arXiv preprint arXiv:2303.18223_.
  * Zou et al. (2023) Andy Zou, Zifan Wang, J Zico Kolter, and Matt Fredrikson. 2023.  Universal and transferable adversarial attacks on aligned language models.  _arXiv preprint arXiv:2307.15043_.

##  Appendix A Extended Related Work

#### Large language models.

A large language model generally refers to an auto-regressive language model
generally over 5 billion parameters in size Zhao et al. (2023). These larger
models are enabled by self-supervised pre-training, commonly next word
prediction, to increase scale and overall performance Radford et al. (2019);
Kaplan et al. (2020). However, the nature of self-supervised pre-training
means there is less control over the information learned, large language
models have historically demonstrated a propensity for toxic language Brown et
al. (2020). This can have surprising effects when querying large language
models on topics of morality and social bias (Jiang et al., 2022).

#### Adversarial attacking.

Adversarial attacking is a prominent field of research for language models
Zhang et al. (2020). Other than the methods mentioned in the main body, there
are further methods for red-teaming and adversarial attacks. Several recent
works attempt to automatically generate adversarial prompts, Zou et al. (2023)
formulate this as prompt optimization, while Xu et al. (2023) generate
candidate prompt attacks by querying a language model. Others attempt to
engineer human parallels to jailbreak the LLM Liu et al. (2023).

#### Knowledge Graphs and Retrieval-Augmented Generation

Knowledge graphs (KGs) are a form of structured data that encodes entities and
their inter-relationships in a (startnode, edge, endnode) format Ji et al.
(2021). Knowledge graphs can be used to enhance language model outputs at the
input level Pan et al. (2024) as well as the embedding level Zhang et al.
(2022). For LLMs it is most common to use them in Retrieval-Augmented
Generation (RAG) Lewis et al. (2020), where the knowledge graph is added to
the input prompt in text form.

##  Appendix B Prompt Templates

Our unified prompt template is shown in Table 5. Additionally, the full system
prompt for the dynamic knowledge graph generation is shown in Table 6.

Wrapper Template for User Prompt
---
Commonsense information is below. Consider it as your superior source of
knowledge and wisdom:
<top $k$ triplets>
Given the Commonsense information and not prior knowledge answer the query.
Query: <user query>
Output must be in JSON format:
{{
“Response”: “[Only output either “A” or “B” or “C”]”
“Explanation”: “[Provide the expanation justifying your selection]”
}}

Table 5: Wrapper template serving to present triplets as superior wisdom and
common sense.

Knowledge Graph Generation Prompt
---
You are a helpful assistant tasked with converting social stereotypes into a
dynamic knowledge graph. This is for a research project analyzing common
social biases. Instructions: Please present the graph in triplets in the
format ‘(start_node, edge, end_node)‘ where nodes are subjects and an edge is
the relationship between them. Please have the start_node be a target group
where possible. Also, please decompose compound target groups. The input will
be in the format: Target minority: Stereotype:.
Examples:
Target minority: black folks
Stereotype: black women are basic
Graph:
‘(black women, are, basic)‘
‘(black women, belong to, black people)‘
‘(black women, belong to, women)‘

Table 6: Few-shot prompt for converting social stereotypes into a knowledge
graph.

##  Appendix C Additional Experiment Details

###  C.1 Experiment Hyperparameters

For the close-sourced models, we used OpenAI’s Chat Completions API
555https://platform.openai.com/docs/api-reference/chat/create. Experiments
with GPT-3.5-turbo completed in 2 hours for one run of 58K samples, and 6
hours for GPT-4. Other than temperature, we keep the recommended settings from
the OpenAI API (top p = 1). For the open-sourced models, we download the
models from HuggingFace666https://huggingface.co/, and use the vLLM library
for serving the models 777https://github.com/vllm-project/vllm. We run
experiments on a cluster of 12 Nvidia a40 GPUs with 48GB of vRAM. One
experiment with 3 runs and 3 maximum retries ran approximately 10 GPU hours
for Llama2-7b and Mistral-7b, and 20 GPU hours for Llama2-13b.

###  C.2 Additional Model Details

We experiment with the following models: Llama2-(7b, 13b) Touvron et al.
(2023), a suite of open-source models trained using Reinforcement Learning
with Human Feedback (RLHF), as well as safety fine-tuning to enhance
helpfulness.

Mistral-7b Jiang et al. (2023), a model trained with instruction tuning;
rather than reinforcement learning, they fine-tune directly on instruction
data.

GPT-3.5-turbo Ouyang et al. (2022), a closed-source LLM that has been fine-
tuned with RLHF.

GPT-4 OpenAI (2023), a closed-source LLM trained with further human feedback.
The exact number of parameters is unknown, but it is known to be larger than
GPT-3.5.

###  C.3 Influence of Top K

We chose top $k$ empirically, but perform additional experiments with a small
balanced subset of BBQ. The subset was balanced over three factors, the
sensitive attribute (e.g. age, nationality, etc.), ambiguity (e.g. ambiguous
entries and non-ambiguous entries), and finally, polarity (e.g. negative and
non-negative).

The ablation study in Table 7 reveals that the number of retrieved triplets
(k) can impact the deception rate. For instance, in the GPT-3.5-turbo model,
we observed a rise in deception rate from 14.1% to 17.0% as we increased the
value of k from 1 to 10. However, not all models exhibited this trend,
indicating that the impact of the retrieval number on the outcome of an
adversarial attack can vary among different language models. However, there is
a weak correlation between the top $k$ value and the deception rate.

Top k | 0 | 1 | 3 | 5 | 10
---|---|---|---|---|---
GPT-3.5-turbo | 14.6 | 17.5 | 18.1 | 19.3 | 19.7

Table 7: Ablation studies varying the top k choice during retrieval.

Setting | Context Condition | Question Polarity
---|---|---
Type | Ambiguous | Unambiguous | Negative | Non-negative
| Baseline | BiasKG | Baseline | BiasKG | Baseline | BiasKG | Baseline | BiasKG
GPT-3.5-turbo | 20.9 | 20.3 | 14.4 | 15.1 | 14.2 | 13.8 | 21.2 | 21.6
GPT-4 | 21.3 | 24.5 | 3.8 | 4.7 | 2.6 | 16.7 | 3.3 | 12.6

Table 8: Deception Rate (DR %) results for ambiguity and polarity across
GPT-3.5-turbo and GPT-4. Model temperature: 0.1

###  C.4 Polarity and Ambiguity

We further dissect the effect of our BiasKG methodology based on question
ambiguity and polarity. We subset the BBQ dataset Parrish et al. (2022) based
on whether the bias-related context in the question is explicit (unambiguous)
or implicit (ambiguous), and whether the expected response supports (negative)
or refutes (non-negative) the social bias.

The results, presented in Table 8, indicate a complex interplay between
BiasKG’s impact, the prompt’s ambiguity, and the answer’s polarity. For
example, with GPT-3.5-turbo, BiasKG increases the deception rate in
unambiguous contexts, but does not have the same effect on the ambiguous
contexts. A similar effect occurs for the question polarity where the BiasKG
only increases the deception rate in non-negative scenarios. As for GPT-4, the
results are less convoluted. BiasKG increases deception rate regardless of
ambiguity and polarity.

Overall, deception rates are much higher in ambiguous context conditions. This
makes sense as the model will shift to utilize the BiasKG inputs as an attempt
to resolve ambiguity.