# Yang-Mills sources in biconformal gravity

Davis W. Muhwezi and James T. Wheeler Utah State University Department of
Physics<EMAIL_ADDRESS>State University Department of Physics,
<EMAIL_ADDRESS>
###### Abstract

Biconformal gravity, based on gauging of the conformal group to $2n$
dimensions, reproduces $n$-dim scale-covariant general relativity on the co-
tangent bundle in any dimension. We generalize this result to include Yang-
Mills matter sources formulated as $SU\left(N\right)$ gauge theories with a
twisted action on the full $2n$-dimensional biconformal space. We show that
the coupling of the sources to gravity does not stop the reduction of
effective dimension $2n\rightarrow n$ of the gravity theory, and instead
forces the Yang-Mills source to reduce to ordinary $n$-dimensional Yang-Mills
theory on the gravitating cotangent bundle, with the usual Yang-Mills energy
tensor as gravitational source. The results apply as well to gravity with
sources on Kähler manifolds and in double field theories.

##  1 Introduction

In general relativity the coupling of matter sources to gravity is
accomplished by making the matter action invariant under general coordinate
transformations, then adding it to the Einstein-Hilbert action. In
formulations of general relativity based on the conformal group, there may be
additional conditions. Here we examine $SU\left(N\right)$ gauge theories as
sources for gravity in a large class of spaces of doubled dimension.

Doubled dimension of spacetime arises in various contexts, frequently related
to the idea of a relativistic phase space. Born reciprocity (M. Born, [1, 2])
was one early suggestion aimed at unifying relativity and quantum theory. The
reciprocity involves the scaled symplectic exchange $x^{\alpha}\rightarrow
ap^{\alpha},p^{\beta}\rightarrow-bx^{\beta}$, thereby preserving Hamilton’s
equations. Further developments include the study of Kähler manifolds, with
mutually compatible metric, symplectic, and complex structures such that any
two of the structures yield the third.

In 1982, using a gauge theory approach to gravity, Ivanov and Niederle [3, 4]
showed that general relativity can arise in a space of doubled dimension
called biconformal space. Generalizing the 8-dimensional quotient of the
conformal group of spacetime by its homogeneous Weyl subgroup led the authors
to a class of curved geometries. From an action quadratic in the curvatures
they found that suitable constraints reduced the field equations to the
Einstein equation in 4-dimensions.

The group quotient approach used in [3, 4] generalizes to arbitrary dimension
$n$ and signature $\left(p,q\right)$. The quotient of conformal group of an
$n$ dimensional space by its Weyl subgroup yields a $2n$ dimensional
biconformal manifold. In [5] it was shown that biconformal spaces of any
dimension $2n$ admit an action _linear_ in the curvature with field equations
reducing to the vacuum Einstein equation in $n$ dimensions. Unlike the other
double field theories cited below, this reduction in both field count and the
number of independent variables occurs _by virtue of the field equations_.
Starting from the most general action linear in the curvatures–which takes
essentially the same form in any dimension–it has now been shown that the
field equations generically lead to scale invariant general relativity on the
co-tangent bundle [5, 6]. As in Riemannian geometry, these biconformal spaces
are taken to be torsion free.

Because biconformal spaces have natural symplectic structure [7] they give an
arena appropriate to quantum problems [8], and when they are required to fully
reproduce the properties of phase space, the $\left(3,1\right)$ signature of
spacetime emerges necessarliy from an originally Euclidean space [9, 7, 10].
It is further shown in [6, 7, 10, 11] that these spaces are generically
Kähler, and that they share the properties of double field theories.

Some years after the first biconformal spaces, another form of doubled
dimension called double field theories arose as a means of making the $O(d,d)$
symmetry of $T$-duality manifest. By introducing scalars to produce an
additional $d$ dimensions, Duff [12]] doubled the X(ς,τ) string variables to
make this $O(d,d)$ symmetry manifest. Siegel brought the idea to full fruition
by deriving results from superstring theory [13, 14, 15]. Allowing fields to
depend on all $2d$ coordinates, Siegel introduced generalized Lie brackets,
gauge transformations, covariant derivatives, and a section condition on the
full doubled space, thereby introducing torsions and curvatures in addition to
the manifest $T$-duality. By restricting half the coordinates–called imposing
a _section condition_ –one recovers the $d$-dimensional theory.

In any of these doubled dimension gravity theories it is desirable to
understand matter couplings, and preferable to introduce the matter fields _ab
initio_ in the doubled space. Carrying out such an investigation in
biconformal spaces simultaneously gives results applicable to other doubled
spaces. Because biconformal gravity is a gauge theory, it allows direct
extension of the symmetry to include gravitational sources from
$SU\left(N\right)$ gauge theories. However, such Yang-Mills type sources must
be written in the space of doubled dimension. This means the introduction of
far more potential source fields,
$\frac{2n\left(2n-1\right)}{2}\times\left(N^{2}-1\right)$ instead of only
$\frac{n\left(n-1\right)}{2}\times\left(N^{2}-1\right)$, each depending on
$2n$ independent variables instead of only $n$. There are two principal
questions we address here. First is the question of the correct form of the
Yang-Mills action in the doubled dimension. We find that the usual $\int
tr\left(\boldsymbol{\mathcal{F}}\,^{*}\boldsymbol{\mathcal{F}}\right)$ form
must be augmented with a “twist” to reproduce familiar results, similar to the
twist found in studies of double field theories [16, 17, 18, 19]. The second
question is whether the increase in fields and independent variables spoils
the gravitational reduction, or at the other extreme, shares the reduction to
$n$ dimensions with gravity. We find that both gravity and sources reduce to
$n$-dimensions. The reduction again occurs by virtue of the field equations,
with no need for section conditions.

A further result arises from this investigation. With the exceptions of a
study of biconformal supersymmetry [20], which necessarily includes matter
fields, and of a brief study [21] with scalar field sources, previous
solutions and further properties of biconformal gravity [4, 5, 6, 7, 10, 22,
23] have been based on pure gravity solutions. The field equations for pure
biconformal gravity arise by variation of the gauge fields, and it has not
been necessary to introduce a metric. However, the actions for Yang-Mills
theories involve the Hodge dual and therefore a metric. But biconformal spaces
possess both conformal and Kähler structures, it is not clear which of these
should take precedence in defining the orthonormality of the basis gauge
fields. The ambiguity in identifying the metric is confounded with the
specification of the biconformal matter action. The outcome of the current
investigation is that only the Killing form reproduces the expected coupling
to general relativity.

Concretely, biconformal spaces are spanned by two sets of frame fields,
$\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$, called the solder form and the
co-solder form. In solutions, the solder form, $\mathbf{e}^{a}$, reduces via
the field equations to the usual solder form on spacetime. In order to write
any Yang-Mills action we are compelled to place an orthonormality condition on
these forms,

$\displaystyle\left\langle\mathbf{e}^{a},\mathbf{e}^{b}\right\rangle$
$\displaystyle=$ $\displaystyle M^{ab}$
$\displaystyle\left\langle\mathbf{e}^{a},\mathbf{f}_{b}\right\rangle$
$\displaystyle=$ $\displaystyle M_{\;\;\;b}^{a}$
$\displaystyle\left\langle\mathbf{f}_{a},\mathbf{f}_{b}\right\rangle$
$\displaystyle=$ $\displaystyle M_{ab}$

where some invariant matrix,

$M^{AB}=\left(\begin{array}[]{cc}M^{ab}&M_{\;\;\;b}^{a}\\\
M_{a}^{\;\;\;b}&M_{ab}\end{array}\right)$

must be specified. A central hurdle in the course of our study was that there
are two natural candidates for $M^{AB}$: the Killing form of the conformal
group restricted to the base manifold, and the Kähler metric of the Kähler
structure. We carry out the search for a suitable Yang-Mills action for each
candidate symmetric form. Ultimately, we find that only the Killing form can
give the expected coupling to gravitation. It is therefore the Killing form
that provides the orthonormality of the solder and co-solder forms throughout
the remainder of the paper.

Before proceeding with our investigation of $SU\left(N\right)$ sources in
biconformal/double-field-theory/Kahler gravity, we look briefly at sources in
other gravity theories. We first exhibit general relativity with Yang-Mills
sources as a gauge theory. This displays our general approach to gauging, as
well as the reduced result we hope to achieve. In the remainder of this
Section, we look at the modifications of other conformally based gravity
theories required in order to include sources. In general relativity, we must
extend the symmetry of source actions to general coordinate from global
Lorentz. By analogy, we expect that including matter in conformally based
theories may require some additional conditions. We briefly discuss sources in
scale-invariant gravity and Weyl gravity, before concluding the Section with a
discussion of sources in doubled dimension theories.

###  1.1 Yang-Mills matter in general relativity

Defining a projection on the quotient of the
$\frac{n\left(n+1\right)}{2}$-dimensional Poincarè group $\mathcal{P}$ by its
$\frac{n\left(n-1\right)}{2}$-dimensional Lorentz subgroup $\mathcal{L}$ gives
a principal fiber bundle with Lorentz fibers over an $n$-dimensional Minkowski
spacetime. Generalizing the base space by changing the Maurer-Cartan
connection forms of the Poincarè group and perhaps changing the manifold, the
fiber structure is maintained by demanding horizontality of the curvature and
torsion. The result is a Riemann-Cartan geometry characterized by curvature
and torsion with local Lorentz symmetry.

Concretely, the generalization of the connection
$\left(\tilde{\mathbf{e}}^{b},\tilde{\boldsymbol{\omega}}_{\;\;\;b}^{a}\right)\Rightarrow\left(\mathbf{e}^{b},\boldsymbol{\omega}_{\;\;\;b}^{a}\right)$
takes the Maurer-Cartan equations of the Poincarè group,

$\displaystyle\mathbf{d}\tilde{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\tilde{\boldsymbol{\omega}}_{\;\;\;b}^{c}\land\tilde{\boldsymbol{\omega}}_{\;\;\;c}^{a}$
$\displaystyle\mathbf{d}\tilde{\mathbf{e}}^{a}$ $\displaystyle=$
$\displaystyle\tilde{\mathbf{e}}^{b}\land\tilde{\boldsymbol{\omega}}_{\;\;\;b}^{a}$

to the Cartan equations,

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\land\boldsymbol{\omega}_{\;\;\;c}^{a}+\mathbf{R}_{\;\;\;b}^{a}$
(1) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\land\boldsymbol{\omega}_{\;\;\;b}^{a}+\mathbf{T}^{a}$
(2)

where horizontality of the curvature $\mathbf{R}^{ab}$ and the torsion
$\mathbf{T}^{a}$ is captured by omitting any occurrence of the spin connection
when writing them expressly as 2-forms

$\displaystyle\mathbf{R}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}R_{\;\;\;bcd}^{a}\mathbf{e}^{c}\land\mathbf{e}^{d}$
$\displaystyle\mathbf{T}^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}T_{\;\;\;bc}^{a}\mathbf{e}^{b}\land\mathbf{e}^{c}$

Horizontality insures the survival of the principal fiber bundle by
guaranteeing that integrals of the curvatures over an area, or equivalenty of
the connection forms over closed curves, are independent of lifting.

Completing the description of the Riemann-Cartan geometry is the demand for
integrability of the Cartan equations, which follows by exterior
differentiation of Eqs.(1) and (2):

$\displaystyle\mathbf{D}\mathbf{R}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\mathbf{D}\mathbf{T}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\land\mathbf{R}_{\;\;\;b}^{a}$

When torsion vanishes, this construction describes a general $n$-dimensional
Riemannian spacetime with local Lorentz symmetry.

To include an additional $SU\left(N\right)$ Yang-Mills symmetry in the fiber
bundle we extend the $\mathcal{P}/\mathcal{L}$ quotient to the quotient of the
product $\mathcal{P}\times SU\left(N\right)$ by the product $\mathcal{L}\times
SU\left(N\right)$:

$\mathcal{P}\times SU\left(N\right)/\mathcal{L}\times SU\left(N\right)$

and carry out the same procedure. This still results in an $n$-dimensional
spacetime but now the fibers of the principal bundle are isomorphic to
$\mathcal{L}\times SU\left(N\right)$. The Cartan Eqs.(1) and (2) are augmented
by a third equation,

$\displaystyle\mathbf{d}\mathbf{A}^{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}c_{\;\;\;jk}^{i}\mathbf{A}^{j}\land\mathbf{A}^{k}+\mathbf{F}^{i}$

where indices beginning with $i$ have range $i,j,k,\ldots=1,2,\ldots,N^{2}-1$,
and $\mathbf{F}^{i}$ is horizontal

$\displaystyle\mathbf{F}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F_{\;\;\;ab}^{i}\mathbf{e}^{a}\land\mathbf{e}^{b}$

The integrability condition is $\boldsymbol{\mathcal{D}}\mathbf{F}^{i}=0$,
where $\boldsymbol{\mathcal{D}}$ is the $SU\left(N\right)$ covariant
derivative. Here, indices from the first part of the alphabet have range
$a,b,\ldots=1,\ldots,n$.

To build a physical theory, we write an action functional using any of the
tensor fields arising from the construction,
$\mathbf{R}_{\;\;\;b}^{a},\mathbf{T}^{a},\mathbf{F}^{i},\mathbf{e}^{a},\eta_{ab},e_{ab\cdots
c}$, together with any representations of the original group.

In any dimension of spacetime, the action coupling the $SU\left(N\right)$
Yang-Mills field to general relativity is written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{GR}+S_{\mathrm{YM}}$
$\displaystyle=$
$\displaystyle\intop\mathbf{R}^{ab}\wedge\mathbf{e}^{c}\wedge\ldots\wedge\mathbf{e}^{d}e_{abc\ldots
d}-\frac{\kappa}{2}\intop\mathbf{F}^{i}\wedge^{*}\mathbf{F}_{i}$

where ${}^{*}\mathbf{F}_{i}$ is the Hodge dual of the 2-form $\mathbf{F}_{i}$.
We vary the action with respect to the solder form, $\mathbf{e}^{a}$, the spin
connection, $\boldsymbol{\omega}_{\;\;\;b}^{a}$, and the Yang-Mills connection
$\mathbf{A}^{i}$. Making the usual assumptions for the gravity theory to
reduce to general relativity, this results in

$\displaystyle R_{ab}-\frac{1}{2}\eta_{ab}R$ $\displaystyle=$
$\displaystyle\kappa\left(\eta^{cd}F_{\;\;\;ac}^{i}F_{i\;bd}-\frac{1}{4}\eta_{ab}F^{i\,cd}F_{i\,cd}\right)$
$\displaystyle\tilde{D}^{c}F_{\;\;\;ac}^{i}$ $\displaystyle=$ $\displaystyle
0$ (3)

where $\tilde{D}^{c}$ is covariant with respect to both local Lorentz and
local $SU\left(N\right)$ transformations. These methods generalize immediately
to additional internal symmetries, such as the $SU\left(3\right)\times
SU\left(2\right)\times U\left(1\right)$ of the standard model.

Our main result is to show that Eqs.(3) in $n$-dimensions follow from the
field equations of biconformal gravity coupled to a twisted Yang-Mills matter
action, formulated in $2n$-dimensions.

###  1.2 Sources for scale-invariant gravity

In addition to general covariance, scale-invariant gravity typically requires
tracelessness of the energy tensor. This condition arises when local scale
invariance is a symmetry of the action. Let $S_{g}$ be any locally scale
invariant gravity action, with sources supplied by adding a matter action,

$S_{matter}=\int\mathcal{L}_{matter}\sqrt{-g}d^{4}x$

Then the source for gravity is the energy-momentum tensor given by the metric
variation,

$\displaystyle\delta_{g}S_{matter}$ $\displaystyle=$
$\displaystyle\int\frac{\delta\left(\mathcal{L}_{matter}\sqrt{-g}\right)}{\delta
g^{\alpha\beta}}\delta g^{\alpha\beta}d^{4}x$ $\displaystyle=$
$\displaystyle\int T_{\alpha\beta}\delta g^{\alpha\beta}\sqrt{-g}d^{4}x$

If the energy-momentum tensor is scale _invariant_ , we may apply Noether’s
theorem to the scaling symmetry transformation, $\delta g^{\alpha\beta}=-2\phi
g^{\alpha\beta}$ immediately. This gives

$0\equiv\delta_{symmetry}S_{matter}=-2\int T_{\alpha\beta}\phi
g^{\alpha\beta}\sqrt{-g}d^{4}x=-2\int
T_{\;\;\;\alpha}^{\alpha}\phi\sqrt{-g}d^{4}x$

and since this must hold for any $\phi$, the energy tensor must be traceless,
$T_{\;\;\;\alpha}^{\alpha}=0$.

The tracelessness follows from the assumption that the only conformal
dependence resides in the single metric factor. However, it is more often the
case that the energy tensor $T_{\alpha\beta}$ is _covariant_ under conformal
transformations. For example, consider a simple massless scalar field $\phi$
with action

$S_{scalar}=\frac{1}{2}\int
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\sqrt{-g}d^{4}x$

This action is scale invariant under the simultaneous transformations

$\displaystyle\widetilde{g}_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
e^{2\alpha}g_{\alpha\beta}$ $\displaystyle\widetilde{\phi}$ $\displaystyle=$
$\displaystyle e^{-\alpha}\phi$

since then

$\displaystyle\tilde{S}_{scalar}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\tilde{g}^{\mu\nu}\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}\sqrt{-\tilde{g}}d^{4}x$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int
e^{-2\alpha}g^{\mu\nu}\partial_{\mu}\left(e^{-\alpha}\phi\right)\partial_{\nu}\left(e^{-\alpha}\phi\right)e^{4\alpha}\sqrt{-g}d^{4}x$
$\displaystyle=$ $\displaystyle S_{scalar}$

The energy-momentum tensor for $\phi$ is given by metric variation

$\displaystyle\delta_{g}S_{scalar}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\left(\delta
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\sqrt{-g}-\frac{1}{2}\left(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\right)g_{\mu\nu}\delta
g^{\mu\nu}\sqrt{-g}\right)d^{4}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\delta
g^{\mu\nu}\left(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\right)\right)\sqrt{-g}d^{4}x$

as

$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\right)$

This is clearly not traceless even though the action is scale invariant.
Indeed, under conformal transformation $T_{\mu\nu}$ has conformal weight $-2$,

$\tilde{T}_{\mu\nu}=e^{-2\alpha}T_{\mu\nu}$

and this covariance is all that is required to make the action invariant.

In cases such as this, an energy-momentum tensor with definite, nonzero
conformal weight may serve as the source for the curvature of a Weyl geometry.
By including the Weyl vector in these geometries, even the Weyl-Ricci tensor
takes on definite conformal weight:

$\displaystyle\tilde{C}_{\;\;\;bcd}^{a}$ $\displaystyle=$ $\displaystyle
e^{-2\alpha}C_{\;\;\;bcd}^{a}$ $\displaystyle\tilde{\mathcal{R}}_{ab}$
$\displaystyle=$ $\displaystyle
e^{-2\alpha}\left(\mathcal{R}_{ab}+W_{\left(a;b\right)}-W_{a}W_{b}+\frac{1}{2}W^{2}\eta_{ab}\right)$
$\displaystyle\tilde{\mathcal{R}}$ $\displaystyle=$ $\displaystyle
e^{-2\alpha}\left(\tilde{\mathcal{R}}+W_{\;\;\;;a}^{a}+W^{2}\right)$

where $\mathcal{R}_{ab}$ is the Schouten tensor and $W_{a}$ the Weyl vector. A
scale-invariant action functional in a Weyl geometry differs from the
Einstein-Hilbert action. A full discussion of such theories is given in [24].

###  1.3 Sources for Weyl gravity

We present two possibilities for the inclusion of sources in Weyl gravity.

Weyl, or conformal, gravity begins with the free action,

$\displaystyle S_{Weyl}$ $\displaystyle=$ $\displaystyle\intop
C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}\sqrt{-g}d^{4}x$

which is quadratic in the conformal curvature, with metric variation leading
to the Bach equation [25],

$W^{\mu\nu}\;\;=\;\;D_{\alpha}D_{\beta}C^{\mu\alpha\nu\beta}-\frac{1}{2}C^{\mu\alpha\nu\beta}R_{\alpha\beta}=0$
(4)

The Bach tensor $W^{\mu\nu}$ is automatically traceless by the tracelessness
of the conformal curvature, $g_{\mu\nu}C^{\mu\alpha\nu\beta}=0$. Because
Eq.(4) arises by metric variation it is necessarily symmetric. Since the
metric variation has the form

$\delta_{g}S_{Weyl}=\intop W^{\mu\nu}\delta g_{\mu\nu}\sqrt{-g}d^{4}x$

the change of the metric, $\delta g_{\mu\nu}=h_{\mu;\nu}+h_{\nu;\mu}$ under a
change of coordinates implies

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\intop
W^{\mu\nu}\left(h_{\mu;\nu}+h_{\nu;\mu}\right)\sqrt{-g}d^{4}x$
$\displaystyle=$ $\displaystyle-2\intop
W_{\;\;\quad;\nu}^{\mu\nu}h_{\mu}\sqrt{-g}d^{4}x$

and since $h_{\mu}$ is arbitrary, the Bach tensor is divergence free,

$\displaystyle W_{\;\;\quad;\nu}^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0$

By rewriting the conformal curvature $C^{\alpha\beta\mu\nu}$ in terms of the
Riemann tensor minus its traces, and using the Gauss-Bonnet density for the
Euler character to remove the Riemann squared term, $S_{Weyl}$ becomes
proportional to
$S^{\prime}=\int\left(R_{\alpha\beta}R^{\alpha\beta}-\frac{1}{3}R^{2}\right)\sqrt{-g}d^{4}x$.
Variation of the metric now leads to a field equation written entirely in
terms of the Ricci tensor,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
R_{\quad\;\;\;;\beta}^{\mu\nu;\beta}\text{\textminus}R_{\quad\;\;\;;\beta}^{\mu\beta;\nu}-R_{\quad\;\;\;;\beta}^{\nu\beta;\mu}\text{\textminus}2R^{\mu\beta}R_{\;\;\;\beta}^{\nu}\text{\textminus}\frac{1}{6}g^{\mu\nu}R{}_{\;\;\;;\beta}^{;\beta}$
(5)
$\displaystyle+\frac{1}{2}g^{\mu\nu}R^{\alpha\beta}R_{\alpha\beta}+\frac{2}{3}R{}^{;\mu\nu}+2RR^{\mu\nu}\text{\textminus}\frac{1}{6}g^{\mu\nu}R{}^{2}$

Clearly, vacuum solutions to the Einstein equation are also solutions to
Eq.(5). Thus, it is fair to argue that evidence for vacuum general relativity
is also evidence for vacuum Weyl gravity.

It is well known that the converse is false. There are explicit solutions to
Eq.(5) that are not solutions to vacuum general relativity.

We cannot make a similar claim that solutions to general relativity with
sources gives evidence for Weyl gravity with sources. As we show below, there
are alternative candidates for sources for the Bach equation. Without a clear
specification of the source, the question of the relationship between
solutions in general relativity and solutions in Weyl gravity is ill-posed.

Nonetheless, many argue for or assume [26, 27] direct addition of a matter
action with traceless energy tensor,

$S=\intop
C^{\alpha\beta\mu\nu}C_{\alpha\beta\mu\nu}\sqrt{-g}d^{4}x+S_{matter}$ (6)

The resulting metric variation leads to

$W^{\mu\nu}=\kappa T^{\mu\nu}$ (7)

which is consistent with the properties of the Bach tensor as long as the
energy-momentum tensor is traceless.

However, the resulting solutions may differ substantially from the verified
predictions of general relativity. Specifically, Flanagan has argued that the
fourth order theory fails to agree with solar system values, and Yoon has
criticized the Newtonian limit and the application to galactic rotation curves
[for the critique and response see [28, 29, 30]]. On the other hand, Palatini-
style variation has been shown for the vacuum case to reduce to scale
covariant general relativity [31], and a novel coupling to matter might give
better agreement with known results.

Because of the fourth order character of Eq.(7), the coupling constant must
have units that differ from the Einstein coupling by some inverse length scale
squared, $L^{-2}$. This is expressible as a (very small) mass,
$m\sim\frac{\hbar}{Lc}$. Therefore, the nonvacuum solutions to Eq.(7) and
solutions to general relativity with the same energy-momentum tensor will
differ by terms proportional to $m^{2}$. For sufficiently close agreement with
general relativity, this mass must be extremely small. The length scale $L$
might represent a radius of inversion symmetry. Since inversion is a conformal
symmetry, this is a possible interpretation. Consistency would require this
length scale to be cosmologically large, with the mass $m$ correspondingly
small. The mass-dependent corrections might then be consistent with
experiment.

As a second possible coupling of matter to Weyl gravity, we note that the
introduction of a scale between general relativity and Weyl gravity is
eliminated if the source field equations are also modified to fourth order
equations. For example, we may modify the Klein-Gordon action to

$\displaystyle S_{MKG}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\left(D_{\alpha}D_{\mu}\phi
D^{\alpha}D^{\mu}\phi+m^{2}D_{\mu}\phi D^{\mu}\phi\right)\sqrt{-g}d^{4}x$

so that the field equation becomes

$D_{\mu}\square D^{\mu}\phi-m^{2}\square\phi=0$

and commuting $\square
D^{\mu}\phi=D^{\mu}\square\phi+R^{\mu\alpha}\phi_{;\alpha}$ this becomes

$\square\left(\square\phi-m^{2}\phi\right)+\frac{1}{2}R^{;\alpha}\phi_{;\alpha}+R^{\mu\alpha}\phi_{;\alpha\mu}=0$

Because we have modified the mass term as well as the original kinetic term,
solutions to the original Klein-Gordon equation satisfy this modified field
equation up to terms proportional to the curvature. In regions of weak
gravity, at least some solutions will be close to the corresponding solution
fields in general relativity. Notice that if we identify
$B_{\alpha}=D_{\alpha}\phi$, the action $S_{MKG}$ describes a restriction of a
Proca field.

Varying the metric, the energy tensor for this modified Klein-Gordon field is
found to be

$\mathcal{T}_{\alpha\beta}=g^{\mu\nu}D_{\alpha}D_{\mu}\phi
D_{\beta}D_{\nu}\phi+\frac{1}{2}m^{2}D_{\alpha}\phi
D_{\beta}\phi-\frac{1}{4}\left(D_{\nu}D_{\mu}\phi
D^{\nu}D^{\mu}\phi+m^{2}D_{\mu}\phi D^{\mu}\phi\right)g_{\alpha\beta}$

This is more transparent if we let $B_{\alpha}=D_{\alpha}\phi$,

$\mathcal{T}_{\alpha\beta}=D_{\alpha}B^{\mu}D_{\beta}B_{\mu}+\frac{1}{2}m^{2}B_{\alpha}B_{\beta}-\frac{1}{4}g_{\alpha\beta}\left(D_{\nu}B_{\mu}D^{\nu}B^{\mu}+m^{2}B_{\mu}B^{\mu}\right)$

Notice that $\mathcal{T}_{\alpha\beta}$ is traceless if we set $m^{2}=0$. To
provide a conformal source, we therefore consider a massless modified Klein-
Gordon field. The resulting conserved gravitational equation becomes

$\displaystyle W_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa\mathcal{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa\left(g^{\mu\nu}D_{\alpha}D_{\mu}\phi
D_{\beta}D_{\nu}\phi-\frac{1}{4}g_{\alpha\beta}\left(D_{\nu}D_{\mu}\phi
D^{\nu}D^{\mu}\phi\right)\right)$

Additional curvature terms will occur for higher spin fields, but the basic
conclusion remains the same. Solutions to the usual field equation continue to
solve the field equation, up to terms proportional to the curvature. As a
second example, the action for the general fourth order modification of the
Maxwell field may be written as

$S=\frac{1}{4}\int D_{\alpha}F_{\mu\nu}D^{\alpha}F^{\mu\nu}\sqrt{-g}d^{4}x$

where $F_{\left[\alpha\beta;\mu\right]}=0$ guarantees the existence of a
potential and makes this form unique. This results in the higher order field
equation

$D^{\nu}\square F_{\mu\nu}=0$

which we may once again commute to express in terms of the usual lower order
expression $D_{\nu}F^{\alpha\nu}$ plus curvature terms:

$\square D_{\nu}F^{\alpha\nu}+terms\>linear\>in\>FR=0$

As we found for the scalar field, symmetry and vanishing divergence of the
energy tensor,

$\mathcal{T}_{\rho\sigma}=D_{\rho}F^{\alpha\beta}D_{\sigma}F_{\alpha\beta}+2D^{\beta}F_{\rho}^{\;\;\;\mu}D_{\beta}F_{\sigma\mu}-\frac{1}{2}\left(D_{\alpha}F_{\mu\nu}D^{\alpha}F^{\mu\nu}\right)g_{\rho\sigma}$

are guaranteed by the metric variation and diffeomorphism invariance.

Tracelessness of $\mathcal{T}_{\alpha\beta}$, however, requires an additional
constraint on the derivatives of the field,
$\mathcal{T}_{\;\;\;\alpha}^{\alpha}=D^{\sigma}F^{\alpha\beta}D_{\sigma}F_{\alpha\beta}$.
Using the field equation, we may express this as a vanishing divergence,

$\displaystyle\mathcal{T}_{\;\;\;\alpha}^{\alpha}$ $\displaystyle=$
$\displaystyle
D_{\sigma}\left(F_{\alpha\beta}D^{\sigma}F^{\alpha\beta}-2\square
F^{\alpha\sigma}A_{\alpha}\right)\overset{!}{=}0$

indicating the presence of an additional conserved current.

Other field equations might be modified in a similar way. Of course, though
these modifications do not necessarily introduce a new mass scale, they still
require additional initial conditions.

###  1.4 Sources for biconformal gravity, Kähler gravity, and double field
theory

The situation may not be as complicated for biconformal gravity [4, 23, 5, 6],
double field theory [12, 13, 14, 15, 33, 35], or gravity on a Kähler manifold
[7, 6]. Each of these cases starts as a fully $2n$-dimensional theory but
ultimately describes gravity on an $n$-dimensional submanifold. It is
desirable to have a fully $2n$-dimensional form of the matter action which
nonetheless also reduces to the expected $n$-dimensional source as a
consequence of the field equations. It is this condition we address. We
discuss the issue in biconformal space, since biconformal gravity is already a
gauge theory and it generically includes the structures of both double field
theory and Kähler manifolds [7, 6].

For matter fields we restrict our attention to Yang-Mills type sources. We
find that although the usual form of $2n$-dimensional Yang-Mills action gives
nonstandard coupling to gravity, including a “twist” matrix in the action
corrects the problem.

Biconformal gravity arises as follows. The quotient of the conformal group
$\mathcal{C}_{p,q}=SO\left(p+1,q+1\right)$ of an
$SO\left(p,q\right)$-symmetric space $(p+q=n$, metric $\eta_{ab})$ by its
homogeneous Weyl subgroup, $\mathcal{W}_{p,q}\equiv SO\left(p,q\right)\times
SO\left(1,1\right)$, leads to a $2n$-dimensional homogeneous space with local
$\mathcal{W}_{p,q}$ symmetry. This homogeneous space, discussed in [4] and
[23] and studied extensively in [6, 7, 10], is found to have compatible
symplectic, metric and complex structures, making it Kähler [7]. In addition,
the restriction of the Killing form to the base manifold is nondegenerate and
scale invariant, and the volume form of the base manifold is scale invariant.
The homogeneous space and its curved generalizations are called biconformal
spaces.

Ivanov and Niederle [4], wrote a gravity theory on an 8-dimensional
biconformal space, using the curvature-quadratic action of Weyl gravity. By a
suitable restriction of the coordinate transformations of the extra
$4$-dimensions, they showed that $4$-dimensional general relativity describes
the remaining subspace.__ Subsequently, Wehner and Wheeler [5] introduced a
class of $\mathcal{W}$-invariant actions _linear_ in the curvatures, defining
biconformal gravity. Curvature-linear actions are possible because the
$2n$-dimensional volume element is scale invariant. Unlike the 4-dimensional
theories above with actions quadratic in the curvature, the linear action
functionals take the same form in any dimension. The doubled dimension is
understood in terms of the symplectic structure, leading to a phase space
interpretation for generic solutions. Lagrangian submanifolds represent the
physical spacetime and have the original dimension. The class of torsion-free
biconformal spaces has been shown to reduce to general relativity on the
cotangent bundle of spacetime [6]. These reductions of the model work for any
signature $\left(p,q\right)$.

The most general action linear in the biconformal curvatures is given by

$S=\lambda\int e_{ac\cdots d}^{\quad\quad be\cdots
f}\left(\alpha\boldsymbol{\Omega}_{\;\;b}^{a}+\beta\delta_{\;b}^{a}\boldsymbol{\Omega}+\gamma\mathbf{e}^{a}\wedge\mathbf{f}_{b}\right)\land\mathbf{e}^{c}\land\cdots\land\mathbf{e}^{d}\land\mathbf{f}_{e}\land\cdots\land\mathbf{f}_{f}$
(8)

where $\boldsymbol{\Omega}_{\;\;b}^{a}$ is the curvature of the spin
connection and $\boldsymbol{\Omega}$ is the dilatational curvature. Here
$\lambda=\frac{\left(-1\right)^{n}}{\left(n-1\right)!\left(n-1\right)!}$ is a
convenient constant, chosen to eliminate a combinatoric factor and to make our
sign conventions agree with [6]. The cotangent bundle is spanned by the pair,
$\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$, called the solder form and the
co-solder form, respectively. The variation is taken with respect to all
$\frac{\left(n+1\right)\left(n+2\right)}{2}$ gauge fields.

The reduction of a fully $2n$-dimensional gravity theory to dependence only on
the fields of $n$-dimensional gravity is a remarkable feature of biconformal
gravity. While it has been shown to be a double field theory [12, 13, 14, 15,
34, 35, 6], double field theories require the assumption that fields depend on
only half the coordinates. This artificial constraint is called a section
condition. In sharp contrast, biconformal solutions do _not_ require a section
condition, _reducing as a consequence of the field equations_ of torsion-free
biconformal spaces. Thus, using the torsion-free field equations, the
components of the $\frac{\left(n+1\right)\left(n+2\right)}{2}$
curvatures–initially dependent on $2n$ independent coordinates–reduce to the
usual Riemannian curvature tensor in $n$ dimensions. Correspondingly, the
$n$-dim solder form determines all fields, up to coordinate and gauge
transformations. Generic, torsion-free, vacuum solutions describe
$n$-dimensional scale-covariant general relativity on the co-tangent bundle.

As noted in the introduction, with the exceptions biconformal supergravity
[20] and a scalar field example [32], studies of biconformal spaces [23, 5, 6,
9, 7, 10, 8, 22, 11] have considered pure gravity biconformal spaces, leading
to vacuum general relativity.

Here we consider $SU\left(N\right)$ Yang-Mills fields as gravitational
sources. The central issue is to show that even with a completely general
$SU\left(N\right)$ gauge theory over a $2n$-dimensional biconformal space, a
full $2n\rightarrow n$ reduction occurs, both for gravity and the Yang-Mills
field.

As with the Riemann-Cartan construction of general relativity above, the
development of biconformal spaces from group symmetry makes it straightforward
to include the additional symmetry of sources. By extending the quotient to

$\displaystyle\mathcal{M}^{2n}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{p,q}\times SU\left(N\right)/\mathcal{W}_{p,q}\times
SU\left(N\right)$

the local symmetry is enlarged by $SU\left(N\right)$ and we may add a Yang-
Mills or similar action to Eq.(8). This construction gives the form of the
Yang-Mills field in terms of the potentials, but not the form of the action.

There are then two basic parts to our investigation.

First, we must determine a suitable $2n$-dimensional action functional for the
sources. To accomplish this requires two interdependent specifications:

  * •

Find a form of the Yang-Mills action which gives the usual $n$-dimensional
Yang-Mills source to the Einstein tensor.

  * •

Fix the orthornormality condition for the solder and co-solder form,
$M^{AB}=\left\langle\tilde{\mathbf{e}}^{A},\tilde{\mathbf{e}}^{B}\right\rangle$
where $\tilde{\mathbf{e}}^{A}=\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$.
This determines for form of the Hodge dual and ultimately the metric variation
of the matter action.

In accomplishing these steps we show that the standard Yang-Mills action

$S_{YM}^{0}=\int tr\left(\mathbf{F}\land^{*}\mathbf{F}\right)$

cannot give the right couplings, and find a satisfactory modification

$S_{YM}=\int tr\left(\bar{\mathbf{F}}\land^{*}\mathbf{F}\right)$

where the twisted field, $\bar{\mathbf{F}}$ is defined below. The twisted
action, together with the restricted Killing form as orthonormal metric, give
the desired reduction.

The second challenge is then to use the field equations to show:

  * •

The number of field components in $2n$ dimensions reduces to the expected
number on $n$ dimensional spacetime.

  * •

The functional dependence of the fields reduces from $2n$ to $n$ independent
variables.

  * •

The gravitational source is the usual Yang-Mills stress-energy tensor.

  * •

The $SU\left(N\right)$ field equation is the usual $n$-dim Yang-Mills field
equation.

The remainder of our presentation proceeds as follows. In the next Section, we
introduce our notation and other conventions. In Section 3, we show that the
usual form of Yang-Mills action, $S_{YM}^{0}$, cannot produce the usual
coupling to gravity. The twisted form $\bar{\mathbf{F}}$ is developed in
Section 4, and the variation of $S_{YM}$ carried out. The Yang-Mills
potentials are identified and varied in Section 5\. The next Section contains
the reduction of the gravitational field equations, as far as possible with
the presence of sources. This reduction closely follows reference [6]. We find
that the field equations imply certain restrictions that must be applied to
the matter sources.

The reduction of the number of fields and the number of independent variables
is shown in Section 7 to follow from the full gravitational equations. We find
that the reduction of fields that is necessary in the purely gravitational
sector also forces reduction of the source fields. The Section concludes with
the emergence of both the usual Yang-Mills gravitational source in
$n$-dimensions, and the usual $n$-dimensional Yang-Mills equation. The final
Section includes a brief review of the main results.

##  2 Notation and conventions

###  2.1 Conventions with biconformal tensors

####  2.1.1 Differential forms

The co-tangent space of biconformal manifolds are spanned by two sets of
opposite conformal weight orthonormal frame fields,
$\tilde{\mathbf{e}}^{A}=\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$, with
lowercase Latin indices $a,b,\ldots=1,2,\ldots,n$ indicating the use of these
frames and upper case Latin $A,B,\ldots=1,2,\ldots,2n$ to denote the pair.
Coordinate indices are lower case Greek, $\mu,\nu,\ldots=1,2,\ldots,n$ so that
we have, for example,

$\mathbf{e}^{a}=e_{\mu}^{\;\;\;a}\mathbf{d}x^{\mu}+e^{\mu a}\mathbf{d}y_{\mu}$

A general 2-form may be written in the orthonormal basis as

$\displaystyle\boldsymbol{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F_{ab}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+F_{\;\;\;b}^{a}\mathbf{f}_{a}\wedge\mathbf{e}^{b}+\frac{1}{2}F^{ab}\mathbf{f}_{a}\wedge\mathbf{f}_{b}$

It is important to realize that $F_{ab},F_{\;\;\;b}^{a}$ and $F^{ab}$ are
distinct fields. Therefore, we cannot raise and lower indices in the usual way
unless we choose different names for the separate independent components. As
compensation for this, the raised or lowered position of an index reflects its
conformal weight. Thus, $F_{ab}$ has weight $-2$ while $F^{ab}$ has weight
$+2$. When practical these distinct fields will be given different names,

$\displaystyle\boldsymbol{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F_{ab}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+G_{\;\;\;b}^{a}\mathbf{f}_{a}\wedge\mathbf{e}^{b}+H^{ab}\mathbf{f}_{a}\wedge\mathbf{f}_{b}$

but this can lead to an unnecessary profusion of field names.

When we need to explicitly refer to internal $SU\left(N\right)$ indices, the
field and its components will be given an additional index from the lower case
Latin set $\left\\{i,j,k\right\\}$. Other lower case Latin indices refer to
the null orthonormal frame field,
$\left(\mathbf{e}^{b},\mathbf{f}_{a}\right)$. Thus $F_{\;\;\;ab}^{i}$
represents the components of
$\frac{1}{2}F_{\;\;\;ab}^{i}G_{i}\mathbf{e}^{a}\wedge\mathbf{e}^{b}$ where
$G_{i}$ is a generator of $SU\left(N\right)$ as $i$ runs from 1 to $N^{2}-1$.
In most cases the internal index is suppressed.

Differential forms are written in boldface and always multiplied with the
wedge product. For brevity in some longer expressions we omit the explicit
wedge between forms. Thus, for example,

$\mathbf{f}_{a}\wedge\mathbf{f}_{b}\wedge\mathbf{f}_{c}\wedge\mathbf{e}^{d}\wedge\mathbf{e}^{e}\Longleftrightarrow\mathbf{f}_{abc}\mathbf{e}^{de}$

The bold font shows that these are differential forms, and therefore are to be
wedged together.

As a compromise between keeping track of conformal weights, while being able
to assess the symmetry or antisymmetry of components, we introduce a weight +1
basis
$\mathbf{e}^{A}\equiv\left(\mathbf{e}^{a},\eta^{ab}\mathbf{f}_{b}\right)$,
where $\eta_{ab}$ is the $n$-dimensional metric of the original
$SO\left(p,q\right)$-symmetric space, _not_ the metric of the biconformal
space. Thus, we may write

$\displaystyle\boldsymbol{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F_{AB}\mathbf{e}^{A}\wedge\mathbf{e}^{B}$
$\displaystyle=$
$\displaystyle\frac{1}{2}F_{ab}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+\mathcal{G}_{ab}\eta^{ac}\mathbf{f}_{c}\wedge\mathbf{e}^{b}+\frac{1}{2}\mathcal{H}_{ab}\eta^{ac}\eta^{bd}\mathbf{f}_{c}\wedge\mathbf{f}_{d}$

where we have defined

$\displaystyle\mathcal{G}_{ab}$ $\displaystyle\equiv$
$\displaystyle\eta_{ac}G_{\;\;\;b}^{c}$ $\displaystyle\mathcal{H}_{ab}$
$\displaystyle\equiv$ $\displaystyle\eta_{ac}\eta_{bd}H^{cd}$

The use of a different font is important because we are not using the
biconformal metric, $K_{AB}$, to change index positions. It is the original
index positions and font, $G_{\quad b}^{a}$, $H^{ab}$, that are the proper
field components.

The matrix components $F_{AB}$ are then written as

$\displaystyle F_{AB}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}F_{ab}&F_{a}^{\;\;\;b}\\\
F_{\;\;\;b}^{a}&F^{ab}\end{array}\right)$ (11) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}F_{ab}&-\mathcal{G}_{ba}\\\
\mathcal{G}_{ab}&\mathcal{H}_{ab}\end{array}\right)$ (14)

where the form of the upper right corner follows because

$\displaystyle-\mathcal{G}_{ba}\mathbf{e}^{a}\land\eta^{bc}\mathbf{f}_{c}$
$\displaystyle=$
$\displaystyle\mathcal{G}_{ba}\eta^{bc}\mathbf{f}_{c}\land\mathbf{e}^{a}$
$\displaystyle=$ $\displaystyle
G_{\;\;\;a}^{c}\mathbf{f}_{c}\land\mathbf{e}^{a}$

With this convention, we can meaningfully define the transpose of matrices.
Specifically, while the transpose of

$\left(\begin{array}[]{cc}F_{ab}&F_{a}^{\;\;\;b}\\\
F_{\;\;\;b}^{a}&F^{ab}\end{array}\right)$

is ill-defined because of the mixed indices on the off-diagonal terms, the
transpose of $F_{AB}$ in the weight +1 basis is

$\left(\begin{array}[]{cc}F_{ab}&-\mathcal{G}_{ba}\\\
\mathcal{G}_{ab}&\mathcal{H}_{ab}\end{array}\right)^{t}=\left(\begin{array}[]{cc}F_{ba}&\mathcal{G}_{ba}\\\
-\mathcal{G}_{ab}&\mathcal{H}_{ba}\end{array}\right)$ (15)

For $\mathcal{H}_{ba}=-\mathcal{H}_{ab}$ and $F_{ba}=-F_{ab}$ this is
manifestly antisymmetric, as befits a 2-form. Notice that the effect of two
transposes is the identity, so this operation provides an involutive
automorphism even though $\eta_{ab}$ is not the biconformal metric.

From $\left[\mathcal{G}_{ab}\right]^{t}=\mathcal{G}_{ba}$ we have

$\left[\eta_{ac}G_{\;\;\;b}^{c}\right]^{t}=\left[\eta_{bc}G_{\;\;\;a}^{c}\right]$

and therefore,
$\left[G_{\;\;\;b}^{d}\right]^{t}=\eta^{ad}\eta_{bc}G_{\;\;\;a}^{c}$.

####  2.1.2 Metric

Because all quadrants of the metric are used in the variation, it is desirable
to retain both the name and index positions throughout. Since

$K_{AB}=\left(\begin{array}[]{cc}K_{ab}&K_{a}^{\;\;\;b}\\\
K_{\;\;\;b}^{a}&K^{ab}\end{array}\right)$

contains all index positions, the inverse metric and its components are
written with an overbar,

$\bar{K}^{AB}=\left(\begin{array}[]{cc}\bar{K}^{ab}&\bar{K}_{\;\;\;b}^{a}\\\
\bar{K}_{a}^{\;\;\;b}&\bar{K}_{ab}\end{array}\right)$

Thus $K_{ab}$ is the first quadrant of the metric, while $\bar{K}_{ab}$ is the
final quadrant of the inverse metric. Here any changes of index position must
be indicated with an explicit factor of $\eta_{ab}$ or $\eta^{ab}$

####  2.1.3 Volume form

It is convenient to define a volume form as $\boldsymbol{\Phi}\equiv{}^{*}1$
but in defining the Hodge dual operation a number of ambiguities need to be
clarified. Because up and down indices have distinct conformal weight, we may
partially order the indices on the Levi-Civita tensor. We establish the
following conventions:

  1. 1.

All factors of $\mathbf{f}_{a}$ are written first, followed by all of the
$\mathbf{e}^{b}$.

  2. 2.

The Levi-Civita _tensor_ is written as $e_{\quad\quad de\cdots f}^{ab\cdots
c}$, with all $n$ up indices first. The partially ordered antisymmetric
_symbol_ is written as $\varepsilon_{\quad\quad de\cdots f}^{ab\cdots c}$.

  3. 3.

When taking the dual of a $p$-form, we sum the components on the first $p$
indices of the Levi-Civita tensor, then introduce factors of $-1$ to move
indices to the default positions. For example, the dual of
$\mathcal{\boldsymbol{H}}=H_{\;\;\;b}^{a}\mathbf{f}_{a}\wedge\mathbf{e}^{b}$
is

${}^{*}\mathcal{\boldsymbol{H}}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(n-1\right)!\left(n-1\right)!}H_{\;\;\;b}^{a}e_{a\quad\quad\;\;\;e\cdots
f}^{\;\;\;bc\cdots
d}\mathbf{f}_{c}\wedge\cdots\wedge\mathbf{f}_{d}\wedge\mathbf{e}^{e}\wedge\cdots\wedge\mathbf{e}^{f}$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{n}}{\left(n-1\right)!\left(n-1\right)!}H_{\;\;\;b}^{a}e_{\quad\quad
ae\cdots f}^{bc\cdots
d}\mathbf{f}_{c}\wedge\cdots\wedge\mathbf{f}_{d}\wedge\mathbf{e}^{e}\wedge\cdots\wedge\mathbf{e}^{f}$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{n}}{\left(n-1\right)!\left(n-1\right)!}H_{\;\;\;b}^{a}e_{\quad\quad
ae\cdots f}^{bc\cdots d}\mathbf{f}_{c\cdots d}\mathbf{e}^{e\cdots f}$

Notice that in the final step, the number of wedged forms in
$\mathbf{f}_{c\cdots d}$ may be inferred from the Levi-Civita tensor. Since
the Levi-Civita tensor always has $n$ up and $n$ down indices, the number of
basis forms of each type is unambiguous. For example,

$e_{\qquad e\cdots f}^{abc\cdots d}\mathbf{f}_{c\cdots
d}\wedge\mathbf{e}^{e\cdots f}=e_{\qquad e\cdots f}^{abc\cdots
d}\underbrace{\mathbf{f}_{c}\wedge\ldots\wedge\mathbf{f}_{d}}_{n-2}\wedge\underbrace{\mathbf{e}^{e}\wedge\ldots\wedge\mathbf{e}^{f}}_{n}$

is a $2n-2$ form that includes the wedge product of $n-2$ factors of
$\mathbf{f}_{a}$ and $n$ factors of $\mathbf{e}^{a}$.

  4. 4.

An $m$-form is a polynomial with each term having different numbers of
$\mathbf{e}$’s and $\mathbf{f}$’s, we write the terms in order of increasing
number of $\mathbf{f}$’s.

  5. 5.

The partial ordering of indices on the Levi-Civita tensor reduces the
normalization of the volume element from $\frac{1}{\left(2n\right)!}$ to
$\frac{1}{n!n!}$.

It has been noted elsewhere [36] that there are alternative duals in
biconformal space. For instance, we may use the symplectic form instead of the
metric to connect indices. The difference resides in the relative signs
between the $\mathbf{e}^{a}$, $\mathbf{f}_{a}$, and mixed terms. Here we use
only the Hodge dual, taking care to keep the correct signs.

With these conventions in mind, we define

$\displaystyle\boldsymbol{\Phi}$ $\displaystyle\equiv$ ${}^{*}1$ (16)
$\displaystyle=$ $\displaystyle\frac{1}{n!n!}e_{\qquad e\cdots f}^{c\cdots
d}\mathbf{f}_{c\cdots d}\wedge\mathbf{e}^{e\cdots f}$ $\displaystyle=$
$\displaystyle\frac{1}{n!n!}\sqrt{K}\varepsilon_{\qquad e\cdots f}^{c\cdots
d}\mathbf{f}_{c\cdots d}\wedge\mathbf{e}^{e\cdots f}$

and consequently,

$\displaystyle\mathbf{f}_{c\cdots d}\wedge\mathbf{e}^{e\cdots f}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{K}}\varepsilon_{\qquad c\cdots
d}^{e\cdots f}\boldsymbol{\Phi}$ (17) $\displaystyle=$
$\displaystyle\bar{e}_{c\cdots d}^{\qquad e\cdots f}\boldsymbol{\Phi}$

where the overbar denotes the contravariant form of the Levi-Civita tensor.
The contravariant form satisfies

$\displaystyle e_{\qquad c\cdots d}^{a\cdots b}\bar{e}_{a\cdots b}^{\qquad
c\cdots d}$ $\displaystyle=$ $\displaystyle n!n!$

We also need the reduction formulas,

$\displaystyle e_{\qquad mnc\cdots d}^{e\cdots f}\bar{e}_{\qquad e\cdots
f}^{ghc\cdots d}$ $\displaystyle=$ $\displaystyle
n!\left(n-2\right)!\left(\delta_{m}^{g}\delta_{n}^{h}-\delta_{m}^{h}\delta_{n}^{g}\right)$
$\displaystyle e_{\qquad ne\cdots f}^{mc\cdots d}\bar{e}_{\qquad gc\cdots
d}^{he\cdots f}$ $\displaystyle=$
$\displaystyle\left(n-1\right)!\left(n-1\right)!\delta_{n}^{h}\delta_{g}^{m}$
(18)

####  2.1.4 Further notation

The antisymmetric projection operator mapping for
$\left(\begin{array}[]{c}1\\\ 1\end{array}\right)$ tensors,

$\displaystyle\Delta_{db}^{ac}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(\delta_{d}^{a}\delta_{b}^{c}-\eta^{ac}\eta_{db}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\eta^{ce}\eta_{bf}\left(\delta_{d}^{a}\delta_{e}^{f}-\delta_{e}^{a}\delta_{d}^{f}\right)$

arises frequently.

###  2.2 Conventions for invariant matrices

Possible actions can be constructed using curvatures naturally arising in a
theory, together with any invariant tensors consistent with the gauging. The
biconformal gauging of the conformal group has a surprising number of
invariant objects. These invariant structures arise from internal symmetries
of the conformal group are induced into generic biconformal spaces [7].

The conformally invariant Killing form, restricted to the base manifold:

$K_{AB}=\left(\begin{array}[]{cc}0&\delta_{a}^{\;\;\;b}\\\
\delta_{\;\;\;b}^{a}&0\end{array}\right)$ (19)

The symplectic form, underlying dilatations,

$\Omega_{AB}=\left(\begin{array}[]{cc}0&\delta_{a}^{\;\;\;b}\\\
-\delta_{\;\;\;b}^{a}&0\end{array}\right)$ (20)

Interestingly, this form manifests Born reciprocity [1, 2]. The complex
structure, arising from the symmetry between translations of the origin and
translations of the point at infinity (i.e., special conformal
transformations),

$\displaystyle J_{\;\;\;B}^{A}$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{cc}0&-\eta^{ab}\\\
\eta_{ab}&0\end{array}\right)$ (23)

The symmetric Kähler form, arising from the compatibility,
$g\left(u,v\right)=\Omega\left(u,Jv\right)$, of the symplectic and complex
structures,

$\displaystyle g_{AB}$ $\displaystyle=$
$\displaystyle\Omega_{AC}J_{\;\;\;B}^{C}$ (28) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&\delta_{a}^{\;\;\;c}\\\
-\delta_{\;\;\;c}^{a}&0\end{array}\right)\left(\begin{array}[]{cc}0&-\eta^{cb}\\\
\eta_{cb}&0\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\eta_{ab}&0\\\
0&\eta^{ab}\end{array}\right)$ (31)

These three Kähler structures satisfy

$g\left(u,v\right)=\Omega\left(u,Jv\right)$ (32)

Notice that the symmetric Kähler form is not invariant under the conformal
structure.

###  2.3 Kähler or Killing?

As noted in the introduction, vacuum biconformal gravity depends only on the
variation of the gauge fields and does not require introduction of a metric
while Yang-Mills actions make use of the Hodge dual and a metric is required.
Given the presence of two natural symmetric forms in biconformal spaces, i.e.,
the scale invariant Killing form, Eq.(19), and the Kähler metric, Eq.(31), we
must make a choice of how to specify the orthonormality relation of the solder
and co-solder forms. With
$\tilde{\mathbf{e}}^{A}=\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$, and using
an overbar to denote the inverse, we may specify orthonormality by either

$\left\langle\tilde{\mathbf{e}}^{A},\tilde{\mathbf{e}}^{B}\right\rangle\equiv\bar{K}^{AB}$
(33)

or

$\left\langle\tilde{\mathbf{e}}^{A},\tilde{\mathbf{e}}^{B}\right\rangle\equiv\bar{g}^{AB}$
(34)

but clearly not both. Once we choose either Eq.(33) or Eq.(34), the
corresponding symmetric form becomes the metric, with its variation following
from the variation of the solder and co-solder forms.

This choice between the symmetric Killing and Kähler forms is not arbitrary.
We therefore studied both cases, ultimately showing that the correct matter
couplings arise only if we use the _Killing form_ , $K_{AB}$, to determine
orthonormality of the basis forms. By inverting the components of the basis
forms, we then have the metric in arbitrary coordinates (indicated by
_coordinate_ indices $M,N$, distinct from _orthonormal_ indices $A,B$)

$K_{MN}=\tilde{e}_{M}^{\;\;\;\;A}\tilde{e}_{N}^{\;\;\;\;B}K_{AB}$ (35)

This is our final choice of metric, and is the only choice which yields the
expected Yang-Mills source for general relativity.

The gravitational field equations follow by variation of the connection forms,
including the solder and co-solder forms $\tilde{e}_{M}^{\;\;\;\;A}$. The
variation of the metric then follows from Eq.(35). Until the variation is
complete, we need the general form of the inverse metric,

$\bar{K}^{AB}=\left(\begin{array}[]{cc}\bar{K}^{ab}&\bar{K}_{\;\;\;b}^{a}\\\
\bar{K}_{a}^{\;\;\;b}&\bar{K}_{ab}\end{array}\right)$

Once the variations are expressed in terms of $\delta\bar{K}^{AB}$ or
$\delta\bar{g}^{AB}$ any remaining components may be returned to the
orthonormal form, Eq.(19) or Eq.(31).

Let the solder and co-solder variations be given by

$\displaystyle\delta\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;c}^{a}\mathbf{e}^{c}+B^{ac}\mathbf{f}_{c}$ (36)
$\displaystyle\delta\mathbf{f}_{c}$ $\displaystyle=$ $\displaystyle
C_{cd}\mathbf{e}^{d}+D_{c}^{\;\;\;d}\mathbf{f}_{d}$ (37)

Then for variation of the components of the Killing metric we expand Eq.(33)

$\displaystyle\delta\bar{K}^{ab}$ $\displaystyle=$
$\displaystyle\delta\left\langle\mathbf{e}^{a},\mathbf{e}^{b}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
A_{\;\;\;c}^{a}\mathbf{e}^{c}+B^{ac}\mathbf{f}_{c},\mathbf{e}^{b}\right\rangle+\left\langle\mathbf{e}^{a},A_{\;\;\;c}^{b}\mathbf{e}^{c}+B^{bc}\mathbf{f}_{c}\right\rangle$
$\displaystyle=$ $\displaystyle
A_{\;\;\;c}^{a}\bar{K}^{cb}+B^{ac}\bar{K}_{c}^{\;\;\;b}+A_{\;\;\;c}^{b}\bar{K}^{ac}+B^{bc}\bar{K}_{\;\;\;c}^{a}$

Since the coefficients $A_{\;\;\;c}^{a},B^{ac},C_{cd},D_{c}^{\;\;\;d}$ now
represent the variation, we may return the remaining components of
$\bar{K}^{AB}$ to the null orthonormal form of Eq.(19),

$\delta\bar{K}^{ab}=B^{ab}+B^{ba}$

Computing the remaining components in the same way, we arrive at the full set,

$\displaystyle\delta\bar{K}^{ab}$ $\displaystyle=$ $\displaystyle
B^{ab}+B^{ba}$ $\displaystyle\delta\bar{K}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle A_{\;\;\;b}^{a}+D_{b}^{\;\;\;a}$
$\displaystyle\delta\bar{K}_{a}^{\;\;\;b}$ $\displaystyle=$ $\displaystyle
D_{a}^{\;\;\;b}+A_{\;\;\;a}^{b}$ $\displaystyle\delta\bar{K}_{ab}$
$\displaystyle=$ $\displaystyle C_{ab}+C_{ba}$ (38)

We also considered the analogous calculation if we were to choose the Kähler
inner product, Eq.(34). This variation gives

$\displaystyle\delta\bar{g}^{ab}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;c}^{a}\eta^{cb}+A_{\;\;\;c}^{b}\eta^{ac}$
$\displaystyle\delta\bar{g}_{\;\;\;b}^{a}$ $\displaystyle=$ $\displaystyle
B^{ac}\eta_{bc}+C_{bc}\eta^{ac}$ $\displaystyle\delta\bar{g}_{a}^{\;\;\;b}$
$\displaystyle=$ $\displaystyle C_{ac}\eta^{cb}+B^{bc}\eta_{ac}$
$\displaystyle\delta\bar{g}_{ab}$ $\displaystyle=$ $\displaystyle
D_{a}^{\;\;\;c}\eta_{cb}+D_{b}^{\;\;\;c}\eta_{ac}$ (39)

As we indicate when we perform the gravitational variation, Eq.(89) in Section
6 below, it is the $B^{ab}$ part of the variation which ultimately reduces to
the Einstein equation. From Eqs.(38) and (39) above we see that the different
possible choices for the metric lead to completely different sources for
gravity. Using the Killing form to define orthonormality of the basis, it is
the coefficient of the $\delta\bar{K}^{ab}$ variation that provides the
gravitational source, while the opposite choice of the Kähler form couples the
coefficients of the cross-terms $\delta\bar{g}_{\;\;\;b}^{a}$ and
$\delta\bar{g}_{a}^{\;\;\;b}$ to gravity. Only one of these can give the usual
Yang-Mills energy tensor. To determine which, we checked each of the two inner
products, for each proposed action functional below until it became clear that
_we must use the Killing form_.

Rather than presenting these distinct variations here, we continue with the
Killing metric and its variation, as given by Eqs.(19), (33), and (38).
Details of the failure of the Kähler choice are given in Appendix A.

##  3 Constructing the Yang-Mills action

In spacetime, the action for a Yang-Mills field may be written as

$\displaystyle S_{\mathrm{YM}}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}\intop
tr\boldsymbol{\mathcal{F}}\wedge^{*}\boldsymbol{\mathcal{F}}$ (40)

and it is natural to consider the same form in biconformal space. However, as
we show in this Section, this usual form leads to nonstandard coupling to
gravity. We show in the next Section that a twisted action is required to give
the usual coupling.

Expanded into independent components in the $\tilde{\mathbf{e}}^{A}$ basis,

$\displaystyle\boldsymbol{\mathcal{F}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{F}_{\quad
ab}^{i}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+\mathcal{F}_{\quad
b}^{ia}\mathbf{f}_{a}\wedge\mathbf{e}^{b}+\frac{1}{2}\mathcal{F}^{iab}\mathbf{f}_{a}\wedge\mathbf{f}_{b}$

where $i$ is an index of the internal Lie algebra. This index can be
suppressed without loss of generality in the action and gravitational field
equations. Also, for the Yang-Mills field it proves more transparent to give
the three coefficients distinct names. Finally, since we must insure
antisymmetry of the twisted field, it is most transparent to use the uniform
weight basis,
$\mathbf{e}^{A}=\left(\mathbf{e}^{a},\eta^{bc}\mathbf{f}_{c}\right)=\left(\mathbf{e}^{a},\mathbf{f}^{b}\right)$.
We therefore write

$\displaystyle\boldsymbol{\mathcal{F}}\;\;=\;\;\frac{1}{2}\mathcal{F}_{AB}\mathbf{e}^{A}\wedge\mathbf{e}^{b}$
$\displaystyle=$
$\displaystyle\frac{1}{2}F_{ab}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+\mathcal{G}_{ab}\mathbf{f}^{a}\wedge\mathbf{e}^{b}+\frac{1}{2}\mathcal{H}_{ab}\mathbf{f}^{a}\wedge\mathbf{f}^{b}$
(41)

In discussions of the Yang-Mills field equations (as opposed to the
gravitational equations), the internal index becomes important and will be
shown where necessary.

Note that $F_{ab}$ and $\mathcal{H}_{ab}$ are antisymmetric. The cross-term
may be written as

$\displaystyle G_{\quad b}^{a}\mathbf{f}_{a}\wedge\mathbf{e}^{b}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(G_{\quad
b}^{a}\right)\mathbf{f}_{a}\wedge\mathbf{e}^{b}+\frac{1}{2}\left(-G_{\quad
a}^{b}\right)\mathbf{e}^{a}\wedge\mathbf{f}_{b}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{ab}\mathbf{f}^{a}\wedge\mathbf{e}^{b}$

where $\mathcal{G}_{ab}$ may be asymmetric. As a matrix, $\mathcal{F}_{AB}$
takes the form given in Eq.(14).

###  3.1 The Hodge dual and standard Yang-Mills action

With these factors in mind, we find the Hodge dual of the Yang-Mills field. In
general terms the Hodge dual of a 2-form is given by

${}^{*}\boldsymbol{\mathcal{F}}$ $\displaystyle=$
${}^{*}\left(\frac{1}{2}\mathcal{F}_{AB}\tilde{\mathbf{e}}^{A}\wedge\tilde{\mathbf{e}}^{B}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(2n-2\right)!}\frac{1}{2}\mathcal{F}_{AB}\bar{K}^{AC}\bar{K}^{BD}\varepsilon_{CDE\ldots
F}\tilde{\mathbf{e}}^{E}\wedge\ldots\wedge\tilde{\mathbf{e}}^{F}$

However, we need to separate the distinct quadrants of each inverse metric.
Expanding each upper case index $A,B,\ldots$, as a raised, weight +1 index and
a lowered weight -1 index in turn, then collecting like terms leads to

${}^{*}\boldsymbol{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{n!\left(n-2\right)!}\left(\frac{1}{2}F_{ab}\bar{K}^{am}\bar{K}^{bn}+G_{\;\;\;b}^{a}\bar{K}_{a}^{\;\;\;m}\bar{K}^{bn}+\frac{1}{2}H^{ab}\bar{K}_{a}^{\;\;\;m}\bar{K}_{b}^{\;\;\;n}\right)\varepsilon_{\;\;\;\qquad
mne\cdots f}^{c\cdots d}\mathbf{f}_{c\cdots d}\mathbf{e}^{e\cdots f}$ (42)
$\displaystyle+\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!\left(n-1\right)!}\left(\frac{1}{2}F_{ab}\bar{K}_{\;\;\;m}^{a}\bar{K}^{bn}+G_{\;\;\;b}^{a}\bar{K}_{am}\bar{K}^{bn}+\frac{1}{2}H^{ab}\bar{K}_{am}\bar{K}_{b}^{\;\;\;n}\right)\varepsilon_{\;\;\;\qquad
ne\cdots f}^{mc\cdots d}\mathbf{f}_{c\cdots d}\mathbf{e}^{e\cdots f}$
$\displaystyle+\frac{\left(-1\right)^{n}}{\left(n-1\right)!\left(n-1\right)!}\left(\frac{1}{2}F_{ab}\bar{K}^{am}\bar{K}_{\;\;\;n}^{b}+G_{\;\;\;b}^{a}\bar{K}_{a}^{\;\;\;m}\bar{K}_{\;\;\;n}^{b}+\frac{1}{2}H^{ab}\bar{K}_{a}^{\;\;\;m}\bar{K}_{bn}\right)\varepsilon_{\;\;\;\qquad
me\cdots f}^{nc\cdots d}\mathbf{f}_{c\cdots d}\mathbf{e}^{e\cdots f}$
$\displaystyle+\frac{1}{n!\left(n-2\right)!}\left(\frac{1}{2}F_{ab}\bar{K}_{\;\;\;m}^{a}\bar{K}_{\;\;\;n}^{b}+G_{\;\;\;b}^{a}\bar{K}_{am}\bar{K}_{\;\;\;n}^{b}+\frac{1}{2}H^{ab}\bar{K}_{am}\bar{K}_{bn}\right)\varepsilon_{\;\;\;\qquad
e\cdots f}^{mnc\cdots d}\mathbf{f}_{c\cdots d}\mathbf{e}^{e\cdots f}$

where one readily sees the advantage of omitting the wedge between forms,
$\mathbf{e}^{e}\wedge\ldots\wedge\mathbf{e}^{f}\longleftrightarrow\mathbf{e}^{e\ldots
f}$.

We form the usual Yang-Mills Lagrangian density as the wedge product,
$\boldsymbol{\mathcal{F}}\wedge{}^{*}\boldsymbol{\mathcal{F}}$, eliminating
the basis forms in favor of the volume form $\boldsymbol{\Phi}$. After a bit
of algebra, we find

$\displaystyle\boldsymbol{\mathcal{F}}\wedge{}^{*}\boldsymbol{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}F_{mn}\bar{K}^{am}\bar{K}^{bn}+G_{\;\;\;n}^{m}\bar{K}_{\;\;\;m}^{a}\bar{K}^{bn}+\frac{1}{2}H^{mn}\bar{K}_{\;\;\;m}^{a}\bar{K}_{\;\;\;n}^{b}\right)F_{ab}\boldsymbol{\Phi}$
(43)
$\displaystyle+\left(F_{mn}\bar{K}_{a}^{\;\;\;m}\bar{K}^{bn}+G_{\;\;\;n}^{m}\left(\bar{K}_{am}\bar{K}^{bn}-\bar{K}_{a}^{\;\;\;n}\bar{K}_{\;\;\;m}^{b}\right)+H^{mn}\bar{K}_{am}\bar{K}_{\;\;\;n}^{b}\right)G_{\;\;\;b}^{a}\boldsymbol{\Phi}$
$\displaystyle+\left(\frac{1}{2}F_{mn}\bar{K}_{a}^{\;\;\;m}\bar{K}_{b}^{\;\;\;n}+G_{\;\;\;n}^{m}\bar{K}_{am}\bar{K}_{b}^{\;\;\;n}+\frac{1}{2}H^{mn}\bar{K}_{am}\bar{K}_{bn}\right)H^{ab}\boldsymbol{\Phi}$

and the matter action is given by Eq.(40). The full action is the combination
of Eq.(8) and Eq.(40),

$S=S_{G}+S_{YM}$

Substituting the null orthonormal form of the Killing metric, Eq.(19) into
Eq.(43) the lagrange density reduces to

$\boldsymbol{\mathcal{F}}\>{}^{*}\boldsymbol{\mathcal{F}}=\left(H^{ab}F_{ab}-G_{\;\;\;a}^{b}G_{\;\;\;b}^{a}\right)\boldsymbol{\Phi}$
(44)

We find that this Hodge dual form of the action is identical to the form of
the action given in [20], despite the claim in [20] that the action is
independent of the metric. The presence of the metric is concealed in [20]
because the Killing metric in this basis is comprised of Kronecker deltas.

Although varying the Yang-Mills potentials in Eq.(43), or equivalently in
Eq.(44), yields the usual Yang-Mills field equation, we show below that
varying the metric gives a nonstandard coupling to gravity. In the next
Section we define a twisted action that gives the usual coupling to gravity,
while still leading to the correct Yang-Mills field equations.

###  3.2 The failure of the $\mathcal{F}\wedge^{*}\mathcal{F}$ action

Variation of the usual Lagrangian Eq.(43) gives

$\displaystyle\delta_{g}\int\boldsymbol{\mathcal{F}}\wedge{}^{*}\boldsymbol{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\int\left(F_{mn}F_{ab}\delta\bar{K}^{am}\eta^{bn}\right)\boldsymbol{\Phi}$
$\displaystyle+\left(G_{\;\;\;n}^{m}F_{ab}\eta^{bn}\delta\bar{K}_{\;\;\;m}^{a}+F_{mn}G_{\;\;\;b}^{a}\eta^{bn}\delta\bar{K}_{a}^{\;\;\;m}\right)\boldsymbol{\Phi}$
$\displaystyle+\left(G_{\;\;\;n}^{m}G_{\;\;\;b}^{a}\eta^{bn}\delta\bar{K}_{am}+G_{\;\;\;n}^{m}G_{\;\;\;b}^{a}\eta_{am}\delta\bar{K}^{bn}\right)\boldsymbol{\Phi}$
$\displaystyle+\left(H^{mn}G_{\;\;\;b}^{a}\eta_{am}\delta\bar{K}_{\;\;\;n}^{b}+G_{\;\;\;n}^{m}H^{ab}\eta_{am}\delta\bar{K}_{b}^{\;\;\;n}\right)\boldsymbol{\Phi}$
$\displaystyle+\left(H^{mn}H^{ab}\eta_{bn}\delta\bar{K}_{am}\right)\boldsymbol{\Phi}$

where after variation we returned the remaining inverse metric components to
the orthonormal form. The metric variations are now given by Eq.(38).
Substituting, the variation yields the gravitational field equations,

$\displaystyle\alpha\Omega_{\;\;\;b\quad m}^{n\quad
b}-\alpha\Omega_{\;\;\;b\quad a}^{a\quad
b}\delta_{m}^{n}+\beta\Omega_{\;\;\;m}^{n}-\beta\Omega_{\;\;\;a}^{a}\delta_{m}^{n}+\Lambda\delta_{m}^{n}$
$\displaystyle=$
$\displaystyle-\kappa\left(H^{bn}F_{bm}-G_{\;\;\;m}^{b}G_{\;\;\;b}^{n}\right)$
(45)
$\displaystyle-\kappa\left(\frac{1}{2}\delta_{m}^{n}\left(F_{bc}H^{bc}-G_{\;\;\;c}^{b}G_{\;\;\;b}^{c}\right)\right)$
$\displaystyle\alpha\Omega_{\;\;\;n\quad a}^{a\quad
m}-\alpha\Omega_{\;\;\;b\quad a}^{a\quad
b}\delta_{n}^{m}+\beta\Omega_{\;\;\;n}^{m}-\beta\Omega_{\;\;\;a}^{a}\delta_{n}^{m}+\Lambda\delta_{n}^{m}$
$\displaystyle=$
$\displaystyle-\kappa\left(H^{bm}F_{bn}-G_{\;\;\;n}^{b}G_{\;\;\;b}^{m}\right)$
(46)
$\displaystyle-\kappa\left(\frac{1}{2}\delta_{n}^{m}\left(F_{bc}H^{bc}-G_{\;\;\;c}^{b}G_{\;\;\;b}^{c}\right)\right)$
$\displaystyle\alpha\Omega_{\;\;\;nam}^{a}+\beta\Omega_{nm}$ $\displaystyle=$
$\displaystyle\kappa\left(F_{am}G_{\;\;\;n}^{a}+F_{an}G_{\;\;\;m}^{a}\right)$
(47) $\displaystyle\alpha\Omega_{\;\;\;b}^{n\quad bm}+\beta\Omega^{nm}$
$\displaystyle=$
$\displaystyle-\kappa\left(H^{nb}G_{\;\;\;b}^{m}+H^{mb}G_{\;\;\;b}^{n}\right)$
(48)

The problem with this coupling to gravity becomes evident when the
gravitational equations require vanishing momentum curvatures on the left side
of Eq.(48). With this, Eq.(48) implies

$\displaystyle H^{nb}G_{\;\;\;b}^{m}+H^{mb}G_{\;\;\;b}^{n}$ $\displaystyle=$
$\displaystyle 0$

The failure of this result is not immediate, but the reduction of Eqs.(45) and
(46) gives a second constraint on $H^{ab}$ and $G_{\;\;\;b}^{a}$. The two
constraints lead, at the most general, to vanishing $H^{ab}$ and symmetric
$G_{\;\;\;b}^{a}$, leaving the source for the Einstein equation, Eq.(47)
linear in $F_{ab}$ and therefore incompatible with the usual energy source for
general relativity.

The problem cannot be altered by a different choice of the inner product (see
Appendix A), but must lie in the use of the usual action. In the context of
other double field theories, a twist allows dimensional reduction to preserve
gauging of supersymmetries [16, 18, 19]. Here we find that including a twist
insures consistency under dimensional reduction not only for supersymmetry,
but also gives the correct coupling of Yang-Mills sources to gravity.

We now turn to the definition of the twisted Yang-Mills action, and its metric
variation. Then, in Section 5, we vary the Yang-Mills potentials. The
remainder of our presentation details the reduction of the full set of field
equations to reproduce scale covariant general relativity with Yang-Mills
sources in the usual $n$-dimensional form.

##  4 Metric variation of the twisted Yang-Mills Action

Instead of the usual spacetime action, Eq.(40), we consider a biconformal
Yang-Mills theory with an action functional of the form

$S_{\mathrm{TYM}}=-\frac{\kappa}{2}\intop
tr\bar{\mathcal{F}}\wedge^{*}\mathcal{F}$ (49)

where $*$ is the usual Hodge dual, $\mathcal{F}$ is a curvature $2$-form,
$\bar{\mathcal{F}}$ is a twisted conjugate curvature, and the trace is over
the $SU\left(N\right)$ generators. The twist matrix is formed using both the
Killing metric and the Kähler form, $K_{\;\;\;B}^{A}\equiv\bar{K}^{AC}g_{CB}$.
While the twist matrix is similar to that used to preserve supersymmetry in
other double field theories, [16, 18, 19], we define the twisted Yang-Mills
field by

$\bar{\mathcal{F}}_{AB}=\frac{1}{2}\left(K_{A}^{\;\;\;C}\mathcal{F}_{CB}+\mathcal{F}_{AC}K_{\;\;\;B}^{C}\right)$

We find that this form is necessary to preserve the antisymmetry of the field
while giving the required interchange of source fields. Variation of the
inverse Killing form $\bar{K}^{AB}$ then gives the source for the
gravitational field equations.

###  4.1 Details of the twist

The twist is accomplished using $K_{\;\;\;B}^{A}\equiv\bar{K}^{AC}g_{CB}$
where $K_{\;\;\;B}^{A}=K_{B}^{\;\;\;A}$, since both $g_{AB}$ and $K_{AB}$ are
symmetric. In the null-orthonormal form and the $\mathbf{e}^{A}$ basis, this
matrix is simply

$\displaystyle K_{\;\;\;B}^{A}$ $\displaystyle=$ $\displaystyle
K_{B}^{\;\;\;A}=\left(\begin{array}[]{cc}0&\delta_{b}^{a}\\\
\delta_{b}^{a}&0\end{array}\right)$

and the required form of the field is

$\displaystyle\bar{\mathcal{F}}_{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(K_{A}^{\;\;\;C}\mathcal{F}_{CB}+\mathcal{F}_{AC}K_{\;\;\;B}^{C}\right)$
(53) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\mathcal{G}_{\left[ab\right]}&\frac{1}{2}\left(F_{ab}+\mathcal{H}_{ab}\right)\\\
\frac{1}{2}\left(F_{ab}+\mathcal{H}_{ab}\right)&\mathcal{G}_{\left[ab\right]}\end{array}\right)$

where $\mathcal{F}_{AB}$ is given by Eq.(14). Note that this transformation
maintains the antisymmetry while interchanging the diagonal and anti-diagonal
elements.

However, the reduced form of the twisted field in the orthonormal frame given
in Eq.(53) is insufficient. Until we complete the metric variation we must use
the generic form of the metric in computing the twist matrix,

$\displaystyle K_{\;\;\;B}^{A}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bar{K}^{ac}&\bar{K}_{\;\;\;e}^{a}\eta^{ec}\\\
\eta^{ae}\bar{K}_{e}^{\;\;\;c}&\eta^{ae}\eta^{cf}\bar{K}_{ef}\end{array}\right)\left(\begin{array}[]{cc}\eta_{cb}&0\\\
0&\eta_{bc}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\bar{K}^{ac}\eta_{cb}&\bar{K}_{\;\;\;b}^{a}\\\
\bar{K}_{\;\;\;b}^{a}&\eta^{ae}\bar{K}_{eb}\end{array}\right)$ $\displaystyle
K_{A}^{\;\;\;B}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\eta_{ad}\bar{K}^{db}&\bar{K}_{a}^{\;\;\;b}\\\
\bar{K}_{a}^{\;\;\;b}&\bar{K}_{ae}\eta^{eb}\end{array}\right)$

with symmetry given by
$\bar{K}_{\;\;\;b}^{a}=\eta^{ae}\bar{K}_{e}^{\;\;\;c}\eta_{cb}$ (see Appendix
B for details of the symmetry). Then $\bar{\mathcal{F}}_{AB}$ becomes

$\displaystyle\bar{\mathcal{F}}_{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(K_{A}^{\;\;\;C}\mathcal{F}_{CB}+\mathcal{F}_{AC}K_{\;\;\;B}^{C}\right)$
(62) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}\eta_{ad}\bar{K}^{dc}F_{cb}+\bar{K}_{a}^{\;\;\;c}\mathcal{G}_{cb}&-\eta_{ad}\bar{K}^{dc}\mathcal{G}_{bc}+\bar{K}_{a}^{\;\;\;c}\mathcal{H}_{cb}\\\
\eta_{ad}\bar{K}_{\;\;\;e}^{d}\eta^{ec}F_{cb}+\bar{K}_{ae}\eta^{ec}\mathcal{G}_{cb}&-\eta_{ad}\bar{K}_{\;\;\;e}^{d}\eta^{ec}\mathcal{G}_{bc}+\bar{K}_{ae}\eta^{ec}\mathcal{H}_{cb}\end{array}\right)$
$\displaystyle+\frac{1}{2}\left(\begin{array}[]{cc}F_{ac}\bar{K}^{ce}\eta_{eb}-\mathcal{G}_{ca}\eta^{cd}\bar{K}_{d}^{\;\;\;e}\eta_{eb}&F_{ac}\bar{K}_{\;\;\;b}^{c}-\mathcal{G}_{ca}\eta^{ce}\bar{K}_{eb}\\\
\mathcal{G}_{ac}\bar{K}^{ce}\eta_{eb}+\mathcal{H}_{ac}\eta^{cd}\bar{K}_{d}^{\;\;\;e}\eta_{eb}&\mathcal{G}_{ac}\bar{K}_{\;\;\;b}^{c}+\mathcal{H}_{ac}\eta^{ce}\bar{K}_{eb}\end{array}\right)$

and we check that 62 agrees with Eq.(53) when we restore the null orthonormal
frame for the metric. The twisted field is simpler when written as a 2-form,

$\displaystyle\bar{\boldsymbol{\mathcal{F}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\mathcal{F}}_{AB}\mathbf{e}^{A}\wedge\mathbf{e}^{B}$
(63) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(F_{ac}\bar{K}^{ce}\eta_{eb}+\bar{K}_{a}^{\;\;\;c}\mathcal{G}_{cb}\right)\mathbf{e}^{a}\wedge\mathbf{e}^{b}$
$\displaystyle+\frac{1}{2}\left(F_{ac}\bar{K}_{\;\;\;b}^{c}\eta^{bf}+\bar{K}_{a}^{\;\;\;c}\mathcal{H}_{cb}\eta^{bf}-\mathcal{G}_{ca}\eta^{ce}\bar{K}_{eb}\eta^{bf}-\mathcal{G}_{bc}\bar{K}^{ce}\eta_{ea}\eta^{bf}\right)\mathbf{e}^{a}\wedge\mathbf{f}_{f}$
$\displaystyle+\frac{1}{2}\left(\mathcal{H}_{ac}\eta^{ce}\bar{K}_{eb}+\mathcal{G}_{ac}\bar{K}_{\;\;\;b}^{c}\right)\eta^{af}\mathbf{f}_{f}\wedge\eta^{bg}\mathbf{f}_{g}$

We may interchange $\bar{K}_{a}^{\;\;\;b}$ and $\bar{K}_{\;\;\;a}^{b}$ when
convenient since these have identical variations and both restrict to
$\delta_{b}^{a}$ in the null orthonormal basis.

###  4.2 The action

The dual field is given by Eq.(42). To avoid duplication of indices when we
wedge the dual field together with the twisted field, we rename the indices in
the twisted field

$\displaystyle\bar{\boldsymbol{\mathcal{F}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(F_{rq}\bar{K}^{qt}\eta_{ts}+\bar{K}_{r}^{\;\;\;q}\mathcal{G}_{qs}\right)\mathbf{e}^{r}\wedge\mathbf{e}^{s}$
$\displaystyle+\frac{1}{2}\left(F_{rc}\bar{K}_{\;\;\;s}^{c}+\bar{K}_{r}^{\;\;\;c}\mathcal{H}_{cs}\right)\eta^{sw}\mathbf{e}^{r}\wedge\mathbf{f}_{w}$
$\displaystyle+\frac{1}{2}\left(-\mathcal{G}_{sq}\bar{K}^{qt}\eta_{tr}-\mathcal{G}_{qr}\eta^{qt}\bar{K}_{ts}\right)\eta^{sw}\mathbf{e}^{r}\wedge\mathbf{f}_{w}$
$\displaystyle+\frac{1}{2}\left(\mathcal{G}_{rq}\bar{K}_{\;\;\;s}^{q}+\mathcal{H}_{rq}\eta^{qt}\bar{K}_{ts}\right)\eta^{rw}\eta^{sx}\mathbf{f}_{w}\wedge\mathbf{f}_{x}$

Each term of the wedge product is proportional to the volume form, using
$\mathbf{f}_{c\cdots d}\wedge\mathbf{e}^{e\cdots f}=\bar{e}_{c\cdots
d}^{\qquad e\cdots f}\boldsymbol{\Phi}$. Then, replacing the double Levi-
Civita tensors with Kronecker deltas according to Eqs.(18), we fully
distribute the lengthy expression. We summarize the essential features here,
but details including the full wedge product
$\bar{\boldsymbol{\mathcal{F}}}\wedge{}^{*}\boldsymbol{\mathcal{F}}$ and its
reduction to
$\left.\bar{\boldsymbol{\mathcal{F}}}\wedge{}^{*}\boldsymbol{\mathcal{F}}\right|_{contributing}$
below are given in Appendix C.

Since we are interested only in the variation, and will return the metric to
the null orthonormal form of Eq.(19) after variation, certain terms clearly do
not contribute to the field equations. For example, in the product

$\displaystyle\left(-\mathcal{G}_{sq}\bar{K}^{qt}\eta_{tr}-\mathcal{G}_{qr}\eta^{qt}\bar{K}_{ts}\right)\eta^{sw}\left(\frac{1}{2}F_{ab}\bar{K}_{\;\;\;m}^{a}\bar{K}^{bn}+\frac{1}{2}\mathcal{H}_{gh}\eta^{ga}\eta^{hb}\bar{K}_{am}\bar{K}_{b}^{\;\;\;n}\right)$

none of the four terms after distribution will contribute to the field
equation because the variation of any one factor of metric components always
leaves an unvaried $\bar{K}_{ts}$ or $\bar{K}^{bn}$,

$\delta\left(\bar{K}_{\;\;\;m}^{a}\bar{K}_{ts}\bar{K}^{bn}\right)=\delta\bar{K}_{\;\;\;m}^{a}\,\cancelto{0}{\bar{K}_{ts}\bar{K}^{bn}}+\bar{K}_{\;\;\;m}^{a}\delta\bar{K}_{ts}\,\cancelto{0}{\bar{K}^{bn}}+\bar{K}_{\;\;\;m}^{a}\,\cancelto{0}{\bar{K}_{ts}}\delta\bar{K}^{bn}$

Dropping such terms, we are only required to vary terms linear in
$\bar{K}_{ab}$ or $\bar{K}^{ab}$, or cubic in the off diagonal components
$\bar{K}_{\;\;\;b}^{a}$. Collecting these and using the symmetries of the
fields, finally yields,

$\displaystyle\left.\bar{\boldsymbol{\mathcal{F}}}\wedge{}^{*}\boldsymbol{\mathcal{F}}\right|_{contributing}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}F_{bc}F_{da}\eta^{ac}+F_{bc}\mathcal{H}^{ac}\eta_{ad}+\frac{1}{2}\left(\mathcal{G}_{ad}-2\mathcal{G}_{da}\right)\eta_{bc}\mathcal{G}^{ac}\right)\bar{K}^{bd}\boldsymbol{\Phi}$
(64)
$\displaystyle+\left(\mathcal{G}_{ba}H^{dc}+F_{ab}\mathcal{G}^{cd}\right)\bar{K}_{\;\;\;e}^{b}\bar{K}_{\;\;\;d}^{e}\bar{K}_{\;\;\;c}^{a}\boldsymbol{\Phi}$
$\displaystyle+\left(\frac{1}{2}\mathcal{H}_{ac}\mathcal{H}^{ab}\eta^{cd}+F_{ac}\mathcal{H}^{ab}\eta^{cd}+\frac{1}{2}\left(\mathcal{G}^{ba}-2\mathcal{G}^{ab}\right)\eta^{cd}\mathcal{G}_{ca}\right)\bar{K}_{bd}\boldsymbol{\Phi}$

We may now vary the metric.

To carry out the variation of the Yang-Mills potentials we may write the
action in the null orthonormal frame. This form is still contained in the
expression above, given by the purely off-diagonal terms, cubic in
$\bar{K}_{\;\;\;b}^{a}$. This simpler form of the action follows immediately
as

$S_{TYM}=\kappa\int
tr\left(\bar{\boldsymbol{\mathcal{F}}}\wedge{}^{*}\boldsymbol{\mathcal{F}}\right)=\kappa\int
tr\left(\mathcal{G}^{ab}\left(\mathcal{H}_{ab}+F_{ab}\right)\right)\boldsymbol{\Phi}$
(65)

The variation of the potentials is carried out in Section 5.

###  4.3 Metric variation

Using the variation of the Killing metric given in Eq.(38) and the variation
of the volume form given by

$\displaystyle\delta\boldsymbol{\Phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}K_{AB}\delta\bar{K}^{AB}\boldsymbol{\Phi}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left[K_{ab}\left(B^{ab}+B^{ba}\right)+K_{\;\;\;b}^{a}\left(D_{a}^{\;\;\;b}+A_{\;\;\;a}^{b}\right)+K_{b}^{\;\;\;a}\left(A_{\;\;\;a}^{b}+D_{a}^{\;\;\;b}\right)+K^{ab}\left(C_{ab}+C_{ba}\right)\right]\boldsymbol{\Phi}$
$\displaystyle=$
$\displaystyle-\delta_{\;\;\;b}^{a}\left(A_{\;\;\;a}^{b}+D_{a}^{\;\;\;b}\right)\boldsymbol{\Phi}$

the variation of Eq.(64) yields

$\displaystyle\delta\left(\bar{\boldsymbol{\mathcal{F}}}\wedge{}^{*}\boldsymbol{\mathcal{F}}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}F_{bc}F_{da}\eta^{ac}+F_{bc}\mathcal{H}^{ac}\eta_{ad}+\frac{1}{2}\left(\mathcal{G}_{ad}-2\mathcal{G}_{da}\right)\eta_{bc}\mathcal{G}^{ac}\right)2B^{\left(bd\right)}\boldsymbol{\Phi}$
(66)
$\displaystyle+\left(\left(2\mathcal{G}^{ad}-\mathcal{G}^{da}\right)F_{ab}-\left(2\mathcal{G}^{da}-\mathcal{G}^{ad}\right)\mathcal{H}_{ab}-\left(F_{ac}+\mathcal{H}_{ac}\right)\mathcal{G}^{ac}\delta_{\;\;\;b}^{d}\right)\left(A_{\;\;\;d}^{b}+D_{d}^{\;\;\;b}\right)\boldsymbol{\Phi}$
$\displaystyle+\left(\frac{1}{2}\mathcal{H}_{ac}\mathcal{H}^{ab}\eta^{cd}+F_{ac}\mathcal{H}^{ab}\eta^{cd}+\frac{1}{2}\left(\mathcal{G}^{ba}-2\mathcal{G}^{ab}\right)\eta^{cd}\mathcal{G}_{ca}\right)2C_{\left(bd\right)}\boldsymbol{\Phi}$

This variation couples to the $\left(\mathbf{e}^{a},\mathbf{f}_{a}\right)$
variation of the gravity action, Eq.(8). The resulting field equations and
their reduction to the gravity theory are given in Section (6) below. Before
considering reduction of the field equations, we turn to variation of the
Yang-Mills potentials to find the Yang-Mills equations.

##  5 Yang-Mills potentials and field equations

We also need to vary the potential to find the Yang-Mills field equations. We
start with the action of Eq.(65) in the null orthonormal basis,

$\displaystyle S_{TYM}$ $\displaystyle=$ $\displaystyle\kappa\int
tr\bar{\boldsymbol{\mathcal{F}}^{i}}\wedge{}^{*}\boldsymbol{\mathcal{F}}^{i}=\kappa\int\mathcal{G}^{iab}\left(\mathcal{H}_{i\;ab}+F_{i\;ab}\right)\boldsymbol{\Phi}$
(67)

In this Section, we make the internal symmetry explicit, varying $S_{YM}$ with
respect to the $SU\left(N\right)$ potentials. It is most convenient to work in
the $\bar{\mathbf{e}}^{A}=\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$ basis.
Internal indices are labeled with letters $i,j,k$, while frame indices are
chosen from the beginning of the alphabet, $a,b,c,\ldots$.

###  5.1 The field components and potentials

The $SU\left(N\right)$ field is given in the $\bar{\mathbf{e}}^{A}$ basis by
Eq.(68),

$\displaystyle\boldsymbol{\mathcal{F}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F_{\;\;\;ab}^{i}\mathbf{e}^{a}\wedge\mathbf{e}^{b}+\mathcal{G}_{\;\;\;cb}^{i}\eta^{ac}\mathbf{f}_{a}\wedge\mathbf{e}^{b}+\frac{1}{2}\mathcal{H}_{\;\;\;ab}^{i}\eta^{ac}\eta^{bd}\mathbf{f}_{c}\wedge\mathbf{f}_{c}$
(68)

where as a matrix,

$\displaystyle\mathcal{F}_{\;\;\;AB}^{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}F_{\;\;\;ab}^{i}&G_{\;\;a}^{i\quad\;\;b}\\\
G_{\;\;\quad b}^{i\;a}&\mathcal{H}_{\;\;\;ab}^{i}\end{array}\right)$

with $G_{\;\;a}^{i\quad\;\;b}=-G_{\;\;\quad b}^{i\;a}$.

The field is given in terms of its $U\left(1\right)$ or $SU\left(N\right)$
potential by the Cartan equation,

$\displaystyle\boldsymbol{\mathcal{F}}^{i}$ $\displaystyle=$
$\displaystyle\mathbf{d}\boldsymbol{\mathcal{A}}^{i}-\frac{1}{2}c_{\;\;\;jk}^{i}\boldsymbol{\mathcal{A}}^{j}\land\boldsymbol{\mathcal{A}}^{k}$

where the potentials are biconformal 1-forms,

$\boldsymbol{\mathcal{A}}^{i}=A_{\;\;\;a}^{i}\mathbf{e}^{a}+B^{i\;a}\mathbf{f}_{a}$

In terms of $A_{\;\;\;a}^{i},B^{i\;a}$, the field becomes

$\displaystyle\boldsymbol{\mathcal{F}}^{i}$ $\displaystyle=$
$\displaystyle\mathbf{D}A_{\;\;\;a}^{i}\land\mathbf{e}^{a}-\frac{1}{2}\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}\land
A_{\;\;\;b}^{k}\mathbf{e}^{b}$ (70)
$\displaystyle+\mathbf{D}B^{i\;a}\land\mathbf{f}_{a}+B^{i\;a}\mathbf{S}{}_{a}-\frac{1}{2}\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}\land
B^{k\;b}\mathbf{f}_{b}$

where $\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}$ is the $SU\left(N\right)$
connection in the adjoint representation,

$\displaystyle\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}$ $\displaystyle\equiv$
$\displaystyle c_{\;\;\;jk}^{i}\boldsymbol{\mathcal{A}}^{j}$

and the Weyl-covariant derivatives of the potentials are given by

$\displaystyle\mathbf{D}A_{\;\;\;a}^{i}$ $\displaystyle=$
$\displaystyle\mathbf{d}A_{\;\;\;a}^{i}-\boldsymbol{\omega}_{\;\;\;c}^{a}A_{\;\;\;a}^{i}+A_{\;\;\;a}^{i}\boldsymbol{\omega}$
$\displaystyle\mathbf{D}B^{i\;a}$ $\displaystyle=$
$\displaystyle\mathbf{d}B^{i\;a}-B^{i\;a}\boldsymbol{\omega}_{\;\;\;a}^{c}-B^{i\;a}\boldsymbol{\omega}$

Note that the covariant derivative of $\eta^{ab}$ does not necessarily vanish,
$\mathbf{D}\eta^{ab}=\mathbf{d}\eta^{ab}-2\boldsymbol{\omega}\eta^{ab}$ where
$\mathbf{d}\eta^{ab}$ takes into account the conformal equivalence class,
$\eta_{ab}\in\left\\{\left.e^{2\varphi}\eta_{ab}^{0}\right|\varphi=\varphi\left(x,y\right)\right\\}$.

Now we separate Eq.(70) into its independent
$\mathbf{e}^{a}\land\mathbf{e}^{b}$, $\mathbf{f}_{a}\land\mathbf{e}^{b}$ and
$\mathbf{f}_{a}\land\mathbf{f}_{b}$ parts. We begin by expanding the exterior
derivatives in the orthonormal basis

$\displaystyle\mathbf{d}A_{\;\;\;a}^{i}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;a,b}^{i}\mathbf{e}^{b}+A_{\;\;\;a}^{i\quad\;\;,b}\mathbf{f}_{b}$
$\displaystyle\mathbf{d}B^{i\;a}$ $\displaystyle=$ $\displaystyle
B_{\;\;\;\;\;,b}^{i\;a}\mathbf{e}^{b}+B^{i\;\;a,b}\mathbf{f}_{b}$

and similarly the covariant exterior derivatives

$\displaystyle\mathbf{D}A_{\;\;\;a}^{i}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;a;b}^{i}\mathbf{e}^{b}+A_{\;\;\;a}^{i\quad\;\;;b}\mathbf{f}_{b}$
$\displaystyle\mathbf{D}B^{i\;a}$ $\displaystyle=$ $\displaystyle
B_{\;\;\;\;\;;b}^{i\;a}\mathbf{e}^{b}+B^{i\;a;b}\mathbf{f}_{b}$

We also expand the $SU\left(N\right)$ connection,
$\boldsymbol{\mathcal{A}}^{j}G_{j}$, in the adjoint representation:

$\displaystyle\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}$ $\displaystyle=$
$\displaystyle
c_{\;\;\;jk}^{i}A_{\;\;\;a}^{j}\mathbf{e}^{a}+c_{\;\;\;jk}^{i}B^{j\;a}\mathbf{f}_{a}$
$\displaystyle=$
$\displaystyle\boldsymbol{\alpha}_{\;\;\;k}^{i}+\boldsymbol{\beta}_{\;\;\;k}^{i}$

Finally, writing the general form of the co-torsion

$\displaystyle\mathbf{S}{}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}S_{abc}\mathbf{e}^{b}\wedge\mathbf{e}^{c}+S_{a\quad\;\;c}^{\;\;\;b}\mathbf{f}_{b}\wedge\mathbf{e}^{c}+\frac{1}{2}S_{a}^{\;\;\;bc}\mathbf{f}_{b}\land\mathbf{f}_{c}$

we arrive the component fields in terms of the potentials:

$\displaystyle F_{\;\;\;ab}^{i}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;b;a}^{i}-A_{\;\;\;a;b}^{i}-c_{\;\;\;jk}^{i}A_{\;\;\;a}^{j}A_{\;\;\;b}^{k}+B^{i\;c}S_{cab}$
(71) $\displaystyle G_{\;\;\quad b}^{i\;a}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;b}^{i\quad\;\;;a}-B_{\;\;\;;b}^{i\;a}-c_{\;\;\;jk}^{i}B^{j\;a}A_{\;\;\;b}^{k}+B^{i\;c}S_{c\quad\;\;b}^{\;\;\;a}$
(72) $\displaystyle H^{i\;\;ab}$ $\displaystyle=$ $\displaystyle
B^{i\;\;b;a}-B^{i\;\;a;b}-c_{\;\;\;jk}^{i}B^{j\;a}B^{k\;b}+B^{i\;c}S{}_{c}^{\;\;\;ab}$
(73)

These expressions are covariant with respect to both $SU\left(N\right)$ and
Weyl group transformations.

###  5.2 The field equations for the potentials

Using Eqs.(71)-(73) in the action, Eq.(67), the variation of $A_{\;\;\;a}^{i}$
gives

$\displaystyle\delta_{A}S_{TYM}$ $\displaystyle=$
$\displaystyle\kappa\int\left(\delta_{A}\mathcal{G}^{iab}\left(\mathcal{H}_{i\;ab}+F_{i\;ab}\right)\right)\boldsymbol{\Phi}$
$\displaystyle+\kappa\int\mathcal{G}^{iab}\left(\cancel{\delta_{A}\mathcal{H}_{i\;ab}}+\delta_{A}F_{i\;ab}\right)\boldsymbol{\Phi}$
$\displaystyle=$ $\displaystyle\kappa\int\left(\delta
A_{\;\;\;b}^{k}\eta^{bc}\left(-\left(\mathcal{H}_{k\;ac}+F_{k\;ac}\right)^{;a}-c_{\;\;\;jk}^{i}B^{j\;a}\left(\mathcal{H}_{i\;ac}+F_{i\;ac}\right)\right)\right)\boldsymbol{\Phi}$
$\displaystyle+\kappa\int\delta
A_{\;\;\;b}^{k}\left(-\left(\mathcal{G}_{k}^{\;\;\;ab}-\mathcal{G}_{k}^{\;\;\;ba}\right)_{;a}-c_{\;\;\;kj}^{i}A_{\;\;\;a}^{j}\left(\mathcal{G}_{i}^{\;\;\;ba}-\mathcal{G}_{i}^{\;\;\;ab}\right)\right)\boldsymbol{\Phi}$

Therefore the field equation becomes

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}\left(\mathcal{H}_{k\;ac}+F_{k\;ac}\right)^{;a}+\eta^{bc}\left(\mathcal{H}_{i\;ac}+F_{i\;ac}\right)\beta_{\;\;\;k}^{i\;\;\quad
a}$
$\displaystyle+\left(\mathcal{G}_{k}^{\;\;\;ab}-\mathcal{G}_{k}^{\;\;\;ba}\right)_{;a}+\left(\mathcal{G}_{i}^{\;\;\;ab}-\mathcal{G}_{i}^{\;\;\;ba}\right)\alpha_{\;\;\;ka}^{i}$

For the $B^{i\;a}$ variation, recalling that $i,j,$ are internal indices and
$a,b,c,e$ orthonormal indices, we find

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}\left(\left(\mathcal{H}_{j\;ac}+F_{j\;ac}\right)_{;b}+\alpha_{\;\;\;jb}^{i}\left(\mathcal{H}_{i\;ac}+F_{i\;ac}\right)+S_{a\quad\;\;b}^{\;\;\;e}\left(\mathcal{H}_{j\;ec}+F_{j\;ec}\right)\right)$
$\displaystyle+\left(\mathcal{G}_{j\;ab}-\mathcal{G}_{j\;ba}\right)^{;b}+\left(\mathcal{G}_{i\;ab}-\mathcal{G}_{i\;ba}\right)\beta_{\;\;\;j}^{i\;\;\quad
b}+\frac{1}{2}\left(\mathcal{G}_{j\;bc}-\mathcal{G}_{j\;cb}\right)S{}_{a}^{\;\;\;bc}$
$\displaystyle+\frac{1}{2}\left(\mathcal{G}_{j}^{\;\;\;bc}-\mathcal{G}_{j}^{\;\;\;cb}\right)S_{abc}$

Notice that there is no dynamical equation for the symmetric part of
$\mathcal{G}_{\;\;\;cd}^{k}$. Moreover, the action depends only on the
antisymmetric part of $\mathcal{G}_{\;\;\;cd}^{k}$. Therefore, from here on we
assume that like $F_{\;\;\;ab}^{i}$ and $H^{i\,ab}$,
$\mathcal{G}_{\;\;\;cd}^{k}$ is antisymmetric,
$\mathcal{G}_{\;\;\;ab}^{k}\equiv\mathcal{G}_{\;\;\;\left[ab\right]}^{k}$.
Also notice that there is no separate Yang-Mills field equation for $F_{ab}$
and $H_{ab}$. Both field equations contain only their sum,
$\mathcal{H}_{\;\;\;ab}^{k}+F_{\;\;\;ab}^{k}$, although $F_{ab}$ and
$\mathcal{H}_{ab}$ enter the gravitational equations separately. We therefore
define a new field,

$\displaystyle K_{\;\;\;ab}^{k}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(\mathcal{H}_{\;\;\;ab}^{k}+F_{\;\;\;ab}^{k}\right)$

In terms of these, the field equations become

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}K_{k\;ac}^{\;\;\quad;a}+\eta^{bc}K_{i\;ac}\beta_{\;\;\;k}^{i\;\;\quad
a}+\mathcal{G}_{k\;\;\quad;a}^{\;\;\;ab}+\mathcal{G}_{i}^{\;\;\;ab}\alpha_{\;\;\;ka}^{i}$
(74) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}\left(K_{j\;ac;b}+K_{i\;ac}\alpha_{\;\;\;jb}^{i}+K_{j\;ec}S_{a\quad\;\;b}^{\;\;\;e}\right)$
(75)
$\displaystyle+\mathcal{G}_{j\;ab}^{\;\;\quad;b}+\mathcal{G}_{i\;ab}\beta_{\;\;\;j}^{i\;\;\quad
b}+\frac{1}{2}\mathcal{G}_{j\;bc}S{}_{a}^{\;\;\;bc}+\frac{1}{2}\mathcal{G}_{j}^{\;\;\;bc}S_{abc}$

These have the expected form of a divergence of the field strength.

The field equations of an $SU\left(N\right)$ gauge theory on a
$2n$-dimensional space given by Eqs.(74) and (75) together with the
gravitational sources of Eqs.(66) complete the first stage of our
investigation. These expressions give a satisfactory formulation of
biconformal Yang-Mills theory.

The second stage of this study is to understand how the $SU\left(N\right)$
gravitational sources together with their field equations affect the
gravitational and Yang-Mills solutions. Specifically, we want to know if the
reduction to the co-tangent bundle of $n$-dimensional spacetime still occurs,
and if so, whether the usual Yang-Mills field equations and gravitational
sources result.

This is indeed what happens. Once we have reduced the gravitational field
equations to more simply describe the underlying geometry, we will return to
Eqs.(74) and (75). Our final result is to show that the reduction of the
underlying geometry to the co-tangent bundle of general relativity
simultaneously forces the reduction of the source equations to the usual Yang-
Mills sources for general relativity.

In the next Section we carry out the gravitational reduction. Since the pure
gravitational field case is presented in detail elsewhere [6], we are able to
simply state some of the conclusions, focussing on those features which are
different in the presence of matter. Subsequently, in Sec.(7), we give
particular attention to the resulting twofold reduction of the
$SU\left(N\right)$ fields: (1) from
$\left(N^{2}-1\right)\times\frac{2n\left(2n-1\right)}{2}$ field components to
$\left(N^{2}-1\right)\times\frac{n\left(n-1\right)}{2}$ components, and (2)
the restriction of the number of effective independent variables,
$\mathcal{\boldsymbol{F}}\left(x,y\right)\rightarrow\mathcal{\boldsymbol{F}}\left(x\right)$.

##  6 Reduction of the gravitational field equations

Although the reduction of the gravitational field equations has been presented
in detail elsewhere [6] and we only highlight the features which differ in the
presence of matter, the presentation is still somewhat lengthy. To aid in
following the discussion, we begin with a short description of the steps
before providing details. The basic steps are the following:

  * •

Present the structure equations and Bianchi identities and their immediate
consequences for the form of the torsion-free biconformal curvatures
(Subsections 6.1 \- 6.2.1). By manipulating the field equations, we arrive at
a reduced form for the components of the curvature tensors (Subsections 6.2.1
\- 6.2.4). (These calculations involve the sources in essential ways that
require more detailed presentation here.)

  * •

The Cartan structure equations show that the solder form $\mathbf{e}^{a}$ is
in involution. The Frobenius theorem therefore allows us to set
$\mathbf{e}^{a}=e_{\mu}^{\;\;\;a}\mathbf{d}x^{\mu}$ and study the restricted
solution on the $x^{\mu}=constant$ submanifolds spanned by $n$ additional
coordinates $y_{\mu}$. Because of the reduced form of the curvature
components, this restricted solution completely determines the
$\mathbf{e}^{a}=0$ pieces of the connection forms. We then extend the
connection forms back to the full biconformal space. At this point, the
connection forms have far fewer than their original number of components
(Summarized in Subsection 6.2.5).

  * •

Substitute the reduced connection forms into the structure equations and
impose the field equations. This is done one structure equation at a time,
each time reducing the degrees of freedom of the full system. Ultimately, all
connection and curvature components are determined by the solder form,
$\mathbf{e}^{a}$, and the components of the solder form itself only depend on
half the original coordinates, $e_{\mu}^{\;\;\;a}\left(x,y\right)\Rightarrow
e_{\mu}^{\;\;\;a}\left(x\right)$ (Subsections 6.2.6, 6.2.7).

###  6.1 Field equations for the twisted action

The full action is

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{G}+S_{YM}$ (76)
$\displaystyle=$ $\displaystyle\lambda\int e_{ac\cdots d}^{\quad\quad be\cdots
f}\left(\alpha\boldsymbol{\Omega}_{\;\;\;b}^{a}+\beta\delta_{\;\;\;b}^{a}\boldsymbol{\Omega}+\gamma\mathbf{e}^{a}\wedge\mathbf{f}_{b}\right)\land\mathbf{e}^{c}\land\cdots\land\mathbf{e}^{d}\land\mathbf{f}_{e}\land\cdots\land\mathbf{f}_{f}$
$\displaystyle-\frac{\kappa}{2}\intop
tr\overline{\mathcal{F}}\wedge^{*}\mathcal{F}$

We choose the combinatoric factor $\lambda$ so the final coupling is $\kappa$.

####  6.1.1 Curvatures, Bianchi identities, and gravity variation

The gravitational field equations are given by varying $S_{G}$ in Eq.(76) with
respect to all connection 1-forms, then combining with the Yang-Mills sources
found in Eq.(66). The curvature components are given in terms of the
connection by the Cartan structure equations,

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\boldsymbol{\omega}_{\;\;\;c}^{a}+2\Delta_{db}^{ac}\mathbf{f}_{c}\mathbf{e}^{d}+\boldsymbol{\Omega}_{\;\;\;b}^{a}$
(77) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{c}\boldsymbol{\omega}_{\;\;\;c}^{a}+\boldsymbol{\omega}\mathbf{e}^{a}+\mathbf{T}{}^{a}$
(78) $\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{c}\mathbf{f}_{c}+\mathbf{f}_{a}\boldsymbol{\omega}+\mathbf{S}{}_{a}$
(79) $\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{c}\mathbf{f}_{c}+\boldsymbol{\Omega}$ (80)

We also have the integrability conditions–generalized Bianchi identities–which
follow from the Poincarè lemma, $\mathbf{d}^{2}\equiv 0$. Exterior
differentiation of the Cartan equations, Eqs.(77)-(80) yields conditions which
must be satisfied in order for solutions to exist. These take the form

$\displaystyle\mathbf{D}\boldsymbol{\Omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle
2\Delta_{db}^{ac}\mathbf{f}_{c}\mathbf{T}^{d}-2\Delta_{db}^{ac}\mathbf{S}_{c}\mathbf{e}^{d}$
(81) $\displaystyle\mathbf{D}\mathbf{T}{}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{c}\boldsymbol{\Omega}_{\;\;\;c}^{a}-\boldsymbol{\Omega}\mathbf{e}^{a}$
(82) $\displaystyle\mathbf{D}\mathbf{S}{}_{a}$ $\displaystyle=$
$\displaystyle-\boldsymbol{\Omega}_{\;\;\;a}^{c}\mathbf{f}_{c}+\mathbf{f}_{a}\boldsymbol{\Omega}$
(83) $\displaystyle\mathbf{D}\boldsymbol{\Omega}$ $\displaystyle=$
$\displaystyle-\mathbf{T}^{c}\mathbf{f}_{c}+\mathbf{e}^{c}\mathbf{S}_{c}$ (84)

Conversely, the converse to the Poincarè lemma tells us that in a star-shaped
region the integrability conditions Eqs.(81)-(84) imply the original Cartan
equations, up to boundary terms.

For example, consider a Newtonian force written as minus the gradient of a
potential, $\mathbf{F}=-\mathbf{d}V$. The Poincarè lemma shows that this
equation has no solution unless

$\displaystyle 0$ $\displaystyle\equiv$
$\displaystyle\mathbf{d}^{2}V=-\mathbf{d}\mathbf{F}$

so the force must be curl-free. Conversely, we may solve the this “Bianchi
identity” first. In a star-shaped region any curl-free force must be a
gradient, $\mathbf{F}=\mathbf{d}U$. This reproduces the potential up to a
“boundary” term satisfying

$\displaystyle\mathbf{d}\left(U+V\right)$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle U+V$ $\displaystyle=$ $\displaystyle constant$

This in turn provides the usual additive constant to the potential. These
principles hold for any set of equations where we may apply the Poincarè lemma
and its converse, allowing us use both the original equations and their
integrability conditions throughout the development of a solution. In the end,
one set of equations cannot be satisfied without simultaneously satisfying the
other up to boundary terms.

The variation of the gravitational part of the action, Eq.(76), is discussed
in detail in [6], and involves only variation of the connection forms, so we
simply state the result. The variation of the spin connnection
$\boldsymbol{\omega}_{\;\;\;b}^{a}$ and Weyl vector $\boldsymbol{\omega}$ give

$\displaystyle T_{\quad e}^{ae}-T_{\quad e}^{ea}-S_{e}^{\;\;ae}$
$\displaystyle=$ $\displaystyle 0$ (85) $\displaystyle
T_{\;\;ca}^{a}+S_{c\quad a}^{\;\;a}-S_{a\quad c}^{\;\;a}$ $\displaystyle=$
$\displaystyle 0$ (86) $\displaystyle\alpha\Delta_{sb}^{ar}\left(T_{\quad
a}^{mb}-\delta_{a}^{m}T_{\quad e}^{eb}-\delta_{a}^{m}S_{c}^{\;\;bc}\right)$
$\displaystyle=$ $\displaystyle 0$ (87)
$\displaystyle\alpha\Delta_{sb}^{ar}\left(\delta_{c}^{b}T_{\;\;ad}^{d}+S_{c\quad
a}^{\;\;\;b}-\delta_{c}^{b}S_{d\quad a}^{\;\;\;d}\right)$ $\displaystyle=$
$\displaystyle 0$ (88)

and these acquire no sources since the Yang-Mills action is independent of the
these connection forms, as noted in [5]. The variation of the solder and co-
solder forms lead to

$\displaystyle\left[\alpha\left(\Omega_{\;\;\;b\quad m}^{n\quad
b}-\Omega_{\;\;\;b\quad a}^{a\quad
b}\delta_{m}^{n}\right)+\beta\left(\Omega_{\;\;\;m}^{n}-\Omega_{\;\;\;a}^{a}\delta_{m}^{n}\right)+\Lambda\delta_{m}^{n}\right]A_{\;\;\;n}^{m}$
$\displaystyle\left[\alpha\left(\Omega_{\;\;\;n\quad a}^{a\quad
m}-\Omega_{\;\;\;b\quad a}^{a\quad
b}\delta_{n}^{m}\right)+\beta\left(\Omega_{\;\;\;n}^{m}-\Omega_{\;\;\;a}^{a}\delta_{n}^{m}\right)+\Lambda\delta_{n}^{m}\right]D_{n}^{\;\;\;m}$
$\displaystyle-\left[\alpha\Omega_{\;\;\;nam}^{a}+\beta\Omega_{nm}\right]B^{mn}$
$\displaystyle\left[\alpha\Omega_{\;\;\;b}^{n\quad
bm}+\beta\Omega^{nm}\right]C_{mn}$ (89)

with the arbitrary variations $A_{\;\;\;n}^{m},B^{mn},C_{mn}$ and
$D_{n}^{\;\;\;m}$ defined in Eqs.(36) and (37). In [6] the expressions above
are equated to zero, but they now acquire sources.

####  6.1.2 Combined equations

Equating corresponding parts of Eqs.(66) and Eqs.(89) and symmetrizing
appropriately,

$\displaystyle\left[\alpha\left(\Omega_{\;\;\;c\quad b}^{a\quad
c}-\Omega_{\;\;\;d\quad c}^{c\quad
d}\delta_{b}^{a}\right)+\beta\left(\Omega_{\;\;\;b}^{a}-\Omega_{\;\;\;c}^{c}\delta_{b}^{a}\right)+\Lambda\delta_{b}^{a}\right]$
$\displaystyle=$ $\displaystyle\kappa W_{\;\;\;b}^{a}$ (90)
$\displaystyle\left[\alpha\left(\Omega_{\;\;\;b\quad c}^{c\quad
a}-\Omega_{\;\;\;d\quad c}^{c\quad
d}\delta_{b}^{a}\right)+\beta\left(\Omega_{\;\;\;b}^{a}-\Omega_{\;\;\;c}^{c}\delta_{b}^{a}\right)+\Lambda\delta_{b}^{a}\right]$
$\displaystyle=$ $\displaystyle\kappa W_{\;\;\;b}^{a}$ (91)
$\displaystyle-\left[\alpha\Omega_{\;\;\;bca}^{c}+\beta\Omega_{ba}\right]$
$\displaystyle=$ $\displaystyle\kappa T_{ab}$ (92)
$\displaystyle\left[\alpha\Omega_{\;\;\;c}^{b\quad
ca}+\beta\Omega^{ba}\right]$ $\displaystyle=$ $\displaystyle\kappa S^{ab}$
(93)

where, recalling the antisymmetry of $\mathcal{G}^{ab}$,

$\displaystyle T_{ab}$ $\displaystyle\equiv$ $\displaystyle
tr\left(F_{ac}F_{bd}\eta^{cd}+H_{ac}F_{bd}\eta^{cd}+H_{bc}F_{ad}\eta^{cd}+3\eta^{cd}\mathcal{G}_{ac}\mathcal{G}_{bd}\right)$
(94) $\displaystyle S^{ab}$ $\displaystyle\equiv$
$\displaystyle\eta^{bd}\eta^{ac}tr\left(\eta^{ef}H_{ec}H_{fd}+\eta^{ef}H_{ec}F_{fd}+\eta^{ef}H_{ed}F_{fc}+3\eta^{fe}\mathcal{G}_{ec}\mathcal{G}_{fd}\right)$
(95) $\displaystyle W_{\;\;\;b}^{a}$ $\displaystyle\equiv$ $\displaystyle
tr\left(3\mathcal{G}^{ca}F_{cb}+3\mathcal{G}^{ca}\mathcal{H}_{cb}-\left(F_{cd}+\mathcal{H}_{cd}\right)\mathcal{G}^{cd}\delta_{\;\;\;b}^{a}\right)$
(96)

In addition, we have the field equations for the $U\left(1\right)$ field,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}K_{k\;ac}^{\;\;\quad;a}+\eta^{bc}K_{i\;ac}\beta_{\;\;\;k}^{i\;\;\quad
a}+\mathcal{G}_{k\;\;\quad;a}^{\;\;\;ab}+\mathcal{G}_{i}^{\;\;\;ab}\alpha_{\;\;\;ka}^{i}$
(97) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}\left(K_{j\;ac;b}+K_{i\;ac}\alpha_{\;\;\;jb}^{i}+K_{j\;ec}S_{a\quad\;\;b}^{\;\;\;e}\right)$
(98)
$\displaystyle+\mathcal{G}_{j\;ab}^{\;\;\quad;b}+\mathcal{G}_{i\;ab}\beta_{\;\;\;j}^{i\;\;\quad
b}+\frac{1}{2}\mathcal{G}_{j\;bc}S{}_{a}^{\;\;\;bc}+\frac{1}{2}\mathcal{G}_{j}^{\;\;\;bc}S_{abc}$

Our gravitational solution now follows many of the steps presented in detail
in [6].

###  6.2 Solving the field equations for the twisted action

For the remainder of the gravitational solution, the particular forms of
$T_{ab},S^{ab}$ and $W_{\;\;\;b}^{a}$ make little difference; indeed, the
solution of this Section holds for the metric variation of any sources at all.
While the form of these source tensors varies with the fields and the matter
action, the positions in which they occur in the field equations and their
symmetries follow knowing only the variation of $\bar{K}^{AB}$. For the
present, this is all we need.

We first turn to the consequences of vanishing torsion, $\mathbf{T}^{a}=0$.

####  6.2.1 Vanishing torsion

Similarly to general relativity, with vanishing torsion the torsion Bianchi
identity, Eq.(82), simplifies to an algebraic relation,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathbf{e}^{c}\boldsymbol{\Omega}_{\;\;\;c}^{a}-\boldsymbol{\Omega}\mathbf{e}^{a}$

which must hold independently of any sources. The algebraic condition
$\mathbf{e}^{c}\boldsymbol{\Omega}_{\;\;\;c}^{a}=\boldsymbol{\Omega}\mathbf{e}^{c}$
expands to three independent component equations,

$\displaystyle\Omega_{\;\;\left[bcd\right]}^{a}$ $\displaystyle=$
$\displaystyle\delta_{[b}^{a}\Omega_{cd]}$ (99)
$\displaystyle\Omega_{\;\;b\quad d}^{a\quad c}-\Omega_{\;\;d\quad b}^{a\quad
c}$ $\displaystyle=$ $\displaystyle\delta_{b}^{a}\Omega_{\quad
d}^{c}-\delta_{d}^{a}\Omega_{\quad b}^{c}$ (100)
$\displaystyle\Omega_{\;\;\;b}^{a\quad cd}$ $\displaystyle=$
$\displaystyle\delta_{b}^{a}\Omega^{cd}$ (101)

Since $\eta_{ea}\Omega_{\;\;b}^{a\quad cd}=-\eta_{ba}\Omega_{\;\;e}^{a\quad
cd}$, the $ab$ trace of Eq.(101) leaves $\Omega^{cd}=0$. Therefore each term
vanishes separately,

$\displaystyle\Omega_{\;\;b}^{a\quad cd}$ $\displaystyle=$ $\displaystyle 0$
(102) $\displaystyle\Omega^{cd}$ $\displaystyle=$ $\displaystyle 0$ (103)

The $ad$ contraction of Eq.(100) gives

$\Omega_{\;\;b\quad a}^{a\quad c}=-\left(n-1\right)\Omega_{\quad b}^{c}$ (104)

Combining Eqs.(102) and (103) with Eq.(93) we immediately find that the
gravitational fields force a constraint on the source fields. This is our
_first source constraint_ :

$S^{ab}=0$ (105)

We next look at the field equations for the curvature.

####  6.2.2 Curvature equations

We now combine the vanishing torsion simplifications with the curvature and
dilatation field equations, Eqs.(90) and (91). The reduction of these
equations begins by noting that the difference between Eq.(90) and Eq.(91)
immediately gives equality of the traces,

$\displaystyle\Omega_{\;\;\;b\quad c}^{c\quad a}$ $\displaystyle=$
$\displaystyle\Omega_{\;\;\;c\quad b}^{a\quad c}$ (106)

Next, formally lowering an index in Eq.(100)

$\eta_{ea}\Omega_{\;\;b\quad d}^{a\quad c}-\eta_{ea}\Omega_{\;\;d\quad
b}^{a\quad c}=\eta_{eb}\Omega_{\quad d}^{c}-\eta_{ed}\Omega_{\quad b}^{c}$

we cycle $ebd$, then add the first two and subtract the third. Using the the
antisymmetry of the curvature on the first two indices,
$\eta_{ea}\Omega_{\;\;d\quad b}^{a\quad c}=-\eta_{da}\Omega_{\;\;e\quad
b}^{a\quad c}$ we find

$\displaystyle\Omega_{\;\;e\quad d}^{a\quad c}$ $\displaystyle=$
$\displaystyle-2\Delta_{de}^{ab}\Omega_{\quad b}^{c}$ (107)

Substituting Eq.(107) into the trace symmetry, Eq.(106) to the two
contractions of Eq.(107) constrains the cross-dilatation,

$\displaystyle-\delta_{d}^{a}\Omega_{\quad
c}^{c}+\eta^{ae}\eta_{cd}\Omega_{\quad e}^{c}$ $\displaystyle=$
$\displaystyle-\left(n-1\right)\Omega_{\quad d}^{a}$

Contracting with $\eta_{ba}$, we see that the antisymmetric part vanishes,

$\displaystyle\left(n-2\right)\left(\eta_{bc}\Omega_{\quad
d}^{c}-\eta_{cd}\Omega_{\quad b}^{c}\right)$ $\displaystyle=$ $\displaystyle
0$

in dimensions greater than 2, while an explicit check confirms the vanishing
antisymmetry in 2-dimensions as well. Therefore, the symmetric part,
$\eta_{bc}\Omega_{\quad d}^{c}+\eta_{dc}\Omega_{\quad
b}^{c}=\frac{2}{n}\eta_{bd}\Omega_{\quad c}^{c}$, becomes a solution for the
full cross-dilatation in terms of its trace,

$\Omega_{\quad d}^{c}=\frac{1}{n}\delta_{b}^{a}\Omega_{\quad c}^{c}$ (108)

This, in turn, combines with Eq.(107) to give the cross-curvature in terms of
the trace of the dilatation,

$\displaystyle\Omega_{\;\;b\quad d}^{a\quad c}$ $\displaystyle=$
$\displaystyle-\frac{2}{n}\Delta_{db}^{ac}\Omega_{\quad e}^{e}$ (109)

We have one remaining cross-curvature field equation, Eq.(90), which couples
the cross-dilatation trace, $\Omega_{\;\;\;a}^{a}$, to the Yang-Mills source
fields. Using Eqs.(108) and (109) to replace the cross-curvature and the
cross-dilatation in Eq.(90), and simplifying,

$\displaystyle\frac{1}{n}\left(n-1\right)\left[\left(\left(n-1\right)\alpha-\beta\right)\Omega_{\quad
c}^{c}+n\Lambda\right]\delta_{b}^{a}$ $\displaystyle=$ $\displaystyle
W_{\;\;\;b}^{a}$

so that $W_{\;\;\;b}^{a}=f\delta_{b}^{a}$ for some function $f$. The constant
$\Lambda$ is given by $\Lambda=\left(n-1\right)\alpha-\beta+n^{2}\gamma$.
Contracting then substituting back, the gravitational field equations force a
_second source constraint_ :

$\displaystyle W_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{n}W_{\;\;\;c}^{c}\delta_{b}^{a}$ (110)

where

$\displaystyle W_{\;\;\;c}^{c}$ $\displaystyle=$
$\displaystyle\left(n-1\right)\left[\left(\left(n-1\right)\alpha-\beta\right)\Omega_{\quad
c}^{c}+n\Lambda\right]$

The traced source tensor on the right, $W_{\;\;\;c}^{c}$, therefore drives the
entire cross-curvature and cross-dilatation. It is striking that the only
source dependence for these components is the Yang-Mills Lagrangian density,

$\displaystyle W_{\;\;\;a}^{a}$ $\displaystyle=$ $\displaystyle
3\mathcal{G}^{i\;ac}\left(\mathcal{H}_{i\;ac}+F_{i\;ac}\right)=3\mathcal{L}$

####  6.2.3 Spacetime terms

Finally, we combine the remaining field equation, Eq.(92),

$\alpha\Omega_{\;\;\;bca}^{c}+\beta\Omega_{ba}=-\kappa T_{ab}$

and the corresponding part of the vanishing torsion Bianchi, Eq.(99), which
expanded becomes

$\Omega_{\;\;bcd}^{a}+\Omega_{\;\;cdb}^{a}+\Omega_{\;\;dbc}^{a}=\delta_{b}^{a}\Omega_{cd}+\delta_{c}^{a}\Omega_{db}+\delta_{d}^{a}\Omega_{bc}$

The $ac$ trace reduces this to

$\Omega_{\;\;bcd}^{c}-\Omega_{\;\;dcb}^{c}=-\left(n-2\right)\Omega_{bd}$

Combining this with the antisymmetric part of the field equation,
$\alpha\left(\Omega_{\;\;\;nam}^{a}-\Omega_{\;\;\;man}^{a}\right)=-2\beta\Omega_{nm}$
shows that

$\displaystyle\left(\left(n-2\right)\alpha-2\beta\right)\Omega_{ab}$
$\displaystyle=$ $\displaystyle 0$

so that generically (i.e., unless $\left(n-2\right)\alpha=2\beta$), the
spacetime dilatation vanishes. Note that this is true for any symmetric source
tensor, so spacetime dilatation is never driven by ordinary matter. As a
result,

$\displaystyle\Omega_{\;\;\;acb}^{c}=\Omega_{\;\;\;\left(a\left|c\right|b\right)}^{c}$
$\displaystyle=$ $\displaystyle-\frac{\kappa}{\alpha}T_{ab}$ (111)
$\displaystyle\Omega_{nm}$ $\displaystyle=$ $\displaystyle 0$ (112)

####  6.2.4 Dilatation

Having reduced the dilatational curvature to a single function,

$\displaystyle\boldsymbol{\Omega}$ $\displaystyle=$
$\displaystyle\chi\mathbf{e}^{a}\wedge\mathbf{f}_{a}$ (113)

where $\chi\equiv-\frac{1}{n}\Omega_{\;\;\;a}^{a}$, we can now use the
dilatational integrability condition, Eq.(84), to press further. Substituting
Eq.(113) into the Bianchi identity, Eq.(84),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathbf{d}\boldsymbol{\Omega}-\mathbf{e}^{b}\wedge\mathbf{S}_{b}$
$\displaystyle=$
$\displaystyle\mathbf{d}\chi\wedge\mathbf{e}^{a}\wedge\mathbf{f}_{a}-\left(1+\chi\right)\mathbf{e}^{a}\wedge\mathbf{S}_{a}$

where we have used
$\mathbf{d}\left(\mathbf{e}^{a}\wedge\mathbf{f}_{a}\right)=\mathbf{D}\left(\mathbf{e}^{a}\wedge\mathbf{f}_{a}\right)=\cancel{\mathbf{T}^{a}}\wedge\mathbf{f}_{a}-\mathbf{e}^{a}\wedge\mathbf{S}_{a}$.
Setting $\mathbf{d}\chi=\chi_{c}\mathbf{e}^{c}+\chi^{c}\mathbf{f}_{c}$,
expanding the co-torsion into components, and combining like forms yields
three independent equations,

$\displaystyle\left(1+\chi\right)S_{\left[acd\right]}$ $\displaystyle=$
$\displaystyle 0$ (114) $\displaystyle\left(1+\chi\right)\left(S_{c\;\;\quad
d}^{\quad a}-S_{d\;\;\quad c}^{\quad a}\right)$ $\displaystyle=$
$\displaystyle\chi_{d}\delta_{c}^{a}-\chi_{c}\delta_{d}^{a}$ (115)
$\displaystyle\left(1+\chi\right)S_{a}^{\quad cd}$ $\displaystyle=$
$\displaystyle\chi^{d}\delta_{a}^{c}-\chi^{c}\delta_{a}^{d}$ (116)

We may now use the co-torsion field equations to gain insight into $\chi$.

With vanishing torsion, the co-torsion field equations Eqs.(85)-(88) reduce to

$\displaystyle S_{e}^{\;\;ae}$ $\displaystyle=$ $\displaystyle 0$ (117)
$\displaystyle S_{c\quad a}^{\;\;a}-S_{a\quad c}^{\;\;a}$ $\displaystyle=$
$\displaystyle 0$ (118) $\displaystyle\alpha\Delta_{sb}^{ar}\left(S_{c\quad
a}^{\;\;\;b}-\delta_{c}^{b}S_{d\quad a}^{\;\;\;d}\right)$ $\displaystyle=$
$\displaystyle 0$ (119)

Using the field equation Eq.(118) to replace the co-torsion terms in the trace
of the Bianchi identity, Eq.(115), gives

$\displaystyle\left(1+\chi\right)\left(S_{c\;\;\quad a}^{\quad
a}-S_{a\;\;\quad c}^{\quad a}\right)$ $\displaystyle=$
$\displaystyle-\left(n-1\right)\chi_{c}$ $\displaystyle\chi_{c}$
$\displaystyle=$ $\displaystyle 0$

Then, combining Eq.(117) with the $ad$ trace of Eq.(116),

$\displaystyle\left(1+\chi\right)S_{a}^{\quad ca}$ $\displaystyle=$
$\displaystyle-\left(n-1\right)\chi^{c}$ $\displaystyle\chi^{c}$
$\displaystyle=$ $\displaystyle 0$

and therefore,

$\mathbf{d}\boldsymbol{\chi}=0$ (120)

The dilatation therefore takes the form

$\boldsymbol{\Omega}=\chi\mathbf{e}^{a}\wedge\mathbf{f}_{a}$

with $\chi$ _constant_. The remainder of the development of the solution
continues as in the homogeneous case but with a different constant value,

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{1}{\left(n-1\right)\alpha-\beta}\left(\frac{1}{n-1}\Lambda-\frac{\kappa}{n\left(n-1\right)}W_{\;\;\;c}^{c}\right)$

for the magnitude of the dilatation cross-term.

Importantly, the constancy of $\chi$ implies the constancy of
$W_{\;\;\;c}^{c}$, and via the second source constraint, Eq.(110), the (mostly
zero) constancy of all of $W_{\;\;\;b}^{a}$.

####  6.2.5 The Frobenius theorem and the final reduction

With vanishing torsion, 78 shows that the solder form becomes involute, and we
may write

$\displaystyle\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle
e_{\alpha}^{\;\;\;a}\mathbf{d}x^{\alpha}$

where $x^{\alpha}$ comprise $n$ of the $2n$ coordinates. Holding $x^{\alpha}$
constant, $\mathbf{e}^{a}=0$, and the residual field equations describe a
submanifold. Here the discussion exactly parallels that of [6]:

  1. 1.

Solve for the connection on the $\mathbf{e}^{a}=0$ submanifold. Here, because
the curvature and dilatation vanish (see Eqs.(102) and (103)), we may gauge
the restricted components of the spin connection and Weyl vector to zero. A
careful coordinate choice puts the submanifold basis in the form
$\mathbf{h}_{a}=e_{a}^{\;\;\;\mu}\mathbf{d}y_{\mu}$.

  2. 2.

Now let $x^{\alpha}$ vary, extending the solution back to the full biconformal
space. This allows all connection forms to acquire an additional
$\mathbf{d}x^{\alpha}$ or $\mathbf{e}^{a}$ term,

$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;bc}^{a}\mathbf{e}^{c}$
$\displaystyle\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle
e_{\alpha}^{\;\;\;a}\mathbf{d}x^{\alpha}$ $\displaystyle\mathbf{f}_{a}$
$\displaystyle=$ $\displaystyle
e_{a}^{\;\;\;\mu}\mathbf{d}y_{\mu}+c_{ab}\mathbf{e}^{b}$
$\displaystyle\boldsymbol{\omega}$ $\displaystyle=$ $\displaystyle
W_{a}\mathbf{e}^{a}$ (121)

  3. 3.

Note that Eq.(78) is now purely quadratic in $\mathbf{e}^{a}$, and therefore
requires the coefficients to depend only on the $x$-coordinates,
$e_{\alpha}^{\;\;\;a}=e_{\alpha}^{\;\;\;a}\left(x\right)$. Solving for the
connection separates it into a compatible piece and a Weyl vector piece,

$\boldsymbol{\omega}_{\;\;\;b}^{a}=\boldsymbol{\alpha}_{\;\;\;b}^{a}-2\Delta_{db}^{ac}W_{c}\mathbf{e}^{d}$

where
$\mathbf{d}\mathbf{e}^{a}=\mathbf{e}^{b}\boldsymbol{\alpha}_{\;\;\;b}^{a}$.

  4. 4.

Substitute these reduced forms of $\mathbf{e}^{a},\mathbf{f}_{b}$ into the
dilatation, Eq.(80) and solve for the Weyl vector. This yields

$\boldsymbol{\omega}=-\left(1+\chi\right)y_{a}\mathbf{e}^{a}$

where $y_{a}=e_{a}^{\;\;\;\mu}y_{\mu}$.

These steps give the final expressions for the connection forms, except for
the form of $c_{ab}=c_{ba}$ in the expansion of the co-solder form
$\mathbf{f}_{a}$.

We note that the submanifolds found by setting either $\mathbf{e}^{a}=0$ by
holding $x^{\mu}$ constant, or $\mathbf{h}_{a}=0$ by holding $y_{\mu}$
constant are Lagrangian submanifolds.

####  6.2.6 The curvature

The final steps in the gravitational reduction are to substitute the partial
solution for the connection forms Eq.(121) into Eqs.(77) and (79) to impose
the final field equations.

To express the remaining undetermined component of the curvature,
$\Omega_{\;\;\;bcd}^{a}$, we define the Schouten tensor

$\displaystyle\boldsymbol{\mathcal{R}}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{n-2}\left(R_{ab}-\frac{1}{2\left(n-1\right)}\eta_{ab}R\right)\mathbf{e}^{b}$

where $R_{ab}=\left(n-2\right)\mathcal{R}_{ab}+\eta_{ab}\mathcal{R}$ is the
Ricci tensor. The generalization of the Schouten tensor to an integrable Weyl
geometry is then (see [24])

$\displaystyle\boldsymbol{\mathscr{R}}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\mathcal{R}}_{a}+\mathbf{D}_{\left(\alpha,x\right)}W_{a}+W_{a}\boldsymbol{\omega}-\frac{1}{2}\eta_{ab}W^{2}\mathbf{e}^{b}$

In terms of the Schouten tensor, the decomposition of the Riemann curvature
2-form into the Weyl curvature 2-form and trace parts is

$\mathbf{R}_{\;\;\;b}^{a}=\mathbf{C}_{\;\;\;b}^{a}-2\Delta_{db}^{ac}\mathcal{\boldsymbol{R}}_{c}\wedge\mathbf{e}^{d}$

Because of the manifest involution of
$\mathbf{h}_{a}=e_{a}^{\;\;\;\mu}\left(x\right)\mathbf{d}y_{\mu}$, the
subspace spanned by the solder form, $\mathbf{e}^{a}$, is a submanifold.
Because $\Omega_{ab}=0$ the submanifold geometry is always an _integrable_
Weyl geometry, so the Weyl vector may be removed from the spacetime
submanifold by a gauge transformation. The spacetime submanifold is simply a
Riemannian geometry with local scale invariance.

Now, introducing the reduced form of the connection into Eq.(77) and imposing
the corresponding field equation, Eq,(92) shows that

$\displaystyle\frac{1}{1+\chi}\mathscr{R}_{ab}+c_{ab}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\alpha}T_{ab}$ (122)

with the full spacetime component of the biconformal curvature given by the
Weyl curvature, $\Omega_{\;\;\;bcd}^{a}=C_{\;\;\;bcd}^{a}$.

####  6.2.7 The co-torsion

A similar introduction of the reduced connection into Eq.(79) for the co-
torsion shows that the momentum and cross-terms vanish, while (following the
somewhat intricate calculation of [6]) the remaining component is given by

$\displaystyle\mathbf{S}_{a}$ $\displaystyle=$
$\displaystyle\mathbf{d}_{\left(x\right)}\mathbf{c}_{a}+\mathbf{c}_{b}\wedge\boldsymbol{\omega}_{\;\;a}^{b}+\boldsymbol{\omega}\wedge\mathbf{c}_{a}$
(123)

where $\mathbf{c}_{a}$, in turn, is determined by Eq.(122).

Expanding $\mathbf{c}_{a}$ fully to separate the Weyl vector parts,

$\displaystyle\mathbf{c}_{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{1+\chi}\left(\boldsymbol{\mathcal{R}}_{a}+\mathbf{D}_{\left(\alpha,x\right)}W_{a}+W_{a}\boldsymbol{\omega}-\frac{1}{2}\eta_{ab}W^{2}\mathbf{e}^{b}\right)-\frac{\kappa}{\alpha}\mathbf{T}_{a}$
$\displaystyle=$
$\displaystyle\mathbf{b}_{a}-\frac{1}{1+\chi}\left(\mathbf{D}_{\left(\alpha,x\right)}W_{a}+W_{a}\boldsymbol{\omega}-\frac{1}{2}\eta_{ab}W^{2}\mathbf{e}^{b}\right)$

where $\mathbf{T}_{a}=T_{ab}\mathbf{e}^{b}$ is the remaining source field and
$\mathbf{b}_{a}\equiv-\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{a}-\frac{\kappa}{\alpha}\mathbf{T}_{a}$.
Then substituting into Eq.(123), after multiple cancellations the co-torsion
becomes

$\displaystyle\mathbf{S}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\chi}W_{b}\mathbf{R}{}_{\;\;\;a}^{b}-\mathbf{D}_{\left(\alpha,x\right)}\left(\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{a}+\frac{\kappa}{\alpha}\mathbf{T}_{a}\right)$
(124)
$\displaystyle+2\Delta_{ca}^{db}W_{d}\left(\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{b}+\frac{\kappa}{\alpha}\mathbf{T}_{b}\right)\wedge\mathbf{e}^{c}$

with the cross term and momentum term of the co-torsion vanishing.

This result is quite similar to an integrability condition. It is shown in
[24] that the condition for the existence of a conformal gauge in which the
Einstein equation, $G_{ab}=\kappa T_{ab}$, holds is

$0=\phi_{,b}\mathbf{R}{}_{\;\;\;a}^{b}-\mathbf{D}_{\left(\alpha,x\right)}\left(\boldsymbol{\mathcal{R}}_{a}-\kappa\boldsymbol{\mathcal{T}}_{a}\right)+2\Delta_{ca}^{db}\phi_{,d}\left(\boldsymbol{\mathcal{R}}_{b}-\kappa\boldsymbol{\mathcal{T}}_{b}\right)\wedge\mathbf{e}^{c}$
(125)

where

$\boldsymbol{\mathcal{T}}_{a}=\frac{1}{n-2}\left(T_{ab}-\frac{1}{n-1}T\eta_{ab}\right)$

When $\boldsymbol{\mathcal{T}}_{a}=0$, Eq.(125) reduces to the well-known
condition,
$\mathbf{D}_{\left(\alpha,x\right)}\boldsymbol{\mathcal{R}}_{a}-\varphi_{,b}\mathbf{C}_{\;\;\;a}^{b}=0$,
for the existence of a Ricci flat conformal gauge.

There are two differences between Eq.(124) and Eq.(125). First, the co-torsion
on the left hand side of Eq.(124) obstructs the integrability condition,
Eq.(125), and we cannot set $\mathbf{S}_{a}=0$ because the Triviality Theorem
shows that when both torsion and co-torsion vanish, biconformal space must be
trivial. The second difference is that the Weyl vector on the right is not
integrable on the _full_ biconformal space.

These issues have a common solution. The part of structure equation for the
co-solder form involving $\mathbf{h}_{a}$ is

$\displaystyle\mathbf{d}\mathbf{h}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{c}\land\mathbf{h}_{c}+\mathbf{h}_{a}\land\boldsymbol{\omega}$
(126)

Therefore, as briefly noted above,
$\mathbf{h}_{a}=e_{a}^{\;\;\;\mu}\mathbf{d}y_{\mu}$ is in involution. Holding
$y_{\mu}=y_{\mu}^{0}$ constant shows that $\mathbf{e}^{a}$ spans a
submanifold. On that submanifold, the Weyl vector becomes exact,

$\boldsymbol{\omega}=W_{a}\mathbf{e}^{a}=\mathbf{d}\left(y_{\mu}^{0}x^{\mu}\right)$

This means that on the $y_{\mu}=y_{\mu}^{0}$ spacetime submanifold, the right
side takes the form of the integrability condition.

At the same time, we may use the form of $\mathbf{S}_{a}$ as the covariant
derivative of $\mathbf{c}_{a}$, Eq.(123) with a suitable choice of $c_{ab}$.
In [6] it is shown that $c_{ab}$ is symmetric and divergence free. While the
interpretation given in [6] of a phenomenological energy tensor is consistent,
it is at odds with the more fundamental interpretation of sources given here.
Instead, we identify $c_{ab}$ as proportional to the Minkowski metric–the only
invariant, symmetric tensor available. It is also divergence free with respect
to the compatible connection, since $D_{\left(\alpha,x\right)}\eta_{ab}=0$.
However, as noted in the previous Section the fully biconformal-covariant
derivative of $\eta_{ab}$ does not necessarily vanish. Since by Eq.(123) the
co-torsion is given by the full biconformal derivative of $c_{ab}$, the
identification $c_{ab}=\Lambda_{0}\eta_{ab}$, implies

$\mathbf{S}_{a}=2\left(1+\chi\right)\Delta_{ea}^{bc}y_{b}\Lambda_{0}\eta_{cd}\mathbf{e}^{d}\wedge\mathbf{e}^{e}$

thereby avoiding the Triviality Theorem. This residual form of the co-torsion
may now be combined into the right hand side of Eq.(124).

Combining these observations, on the $\mathbf{h}_{a}=0$ spacetime submanifold
with $c_{ab}=\Lambda_{0}\eta_{ab}$, setting $\phi_{,a}=y_{a}^{0}$, and using
$\mathbf{D}_{\left(\alpha,x\right)}\left(\Lambda_{0}\eta_{ab}\right)=0$, it
follows that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{1+\chi}\phi_{,b}\mathbf{R}{}_{\;\;\;a}^{b}-\mathbf{D}_{\left(\alpha,x\right)}\left(\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{a}+\frac{\kappa}{\alpha}\mathbf{T}_{a}+\Lambda_{0}\eta_{ab}\mathbf{e}^{b}\right)$
(127)
$\displaystyle+2\Delta_{ca}^{db}\phi_{,d}\left(\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{b}+\frac{\kappa}{\alpha}\mathbf{T}_{b}+\Lambda_{0}\eta_{be}\mathbf{e}^{e}\right)\wedge\mathbf{e}^{c}$

This is now the condition for the existence of a conformal transformation such
that

$\displaystyle\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{b}+\Lambda_{0}\eta_{be}\mathbf{e}^{e}$
$\displaystyle=$ $\displaystyle-\frac{\kappa}{\alpha}\mathbf{T}_{b}$

Expressed in terms of the Einstein tensor,

$\displaystyle G_{ab}+\Lambda_{C}\eta_{ab}$ $\displaystyle=$
$\displaystyle-\left(n-2\right)\frac{\kappa\left(1+\chi\right)}{\alpha}\left(T_{ab}-\eta_{ab}T\right)$
(128)

where the net effect of $c_{ab}$ is a cosmological constant,
$\Lambda_{C}=-\left(n-1\right)\left(n-2\right)\left(1+\chi\right)\Lambda_{0}$.

If we make the conformal transformation that produces Eq.(128), the co-torsion
equation (127) reduces to $\phi_{,b}\mathbf{R}{}_{\;\;a}^{b}=0$. In generic
spacetimes this requires $W_{\mu}=y_{\mu}^{0}=0$.

##  7 The source for gravity

###  7.1 The reduction of sources forced by coupling to gravity

The necessary source constraints from the gravitational couplings, Eqs.(105)
and (110), may be written as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\;\;ac}^{i}\mathcal{H}_{i\>bd}\eta^{cd}+\left(\mathcal{H}_{\;\;ac}^{i}F_{i\>bd}+\mathcal{H}_{\;\;bc}^{i}F_{i\>ad}\right)\eta^{cd}+3\mathcal{G}_{\;\;ac}^{i}\mathcal{G}_{i\>bd}\eta^{cd}$
(129) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathcal{G}^{i\;ca}\left(F_{\;\;cb}^{i}+\mathcal{H}_{\;\;cb}^{i}\right)-\frac{1}{n}\delta_{\;\;\;b}^{a}\mathcal{G}^{i\;cd}\left(F_{\;\;cd}^{i}+\mathcal{H}_{\;\;cd}^{i}\right)$
(130)

where the full contraction of the second,
$\mathcal{G}^{i\;cd}\left(F_{\;\;cd}^{i}+\mathcal{H}_{\;\;cd}^{i}\right)$, is
constant.

These conditions must continue to hold for small physical variations of the
independent potentials, $A_{\;\;\;a}^{i}$ and $B^{ia}$. We may imagine two
nearby solutions differing only in one or both of the potentials and look at
their difference. The change in $F_{\;\;\;ab}^{i}$ as we change
$A_{\;\;\;a}^{i}$ is given by

$\displaystyle\delta F_{\;\;\;ab}^{i}$ $\displaystyle=$ $\displaystyle\delta
A_{\;\;\;b;a}^{i}-\delta A_{\;\;\;a;b}^{i}-c_{\;\;\;jk}^{i}\delta
A_{\;\;\;a}^{j}A_{\;\;\;b}^{k}-c_{\;\;\;jk}^{i}A_{\;\;\;a}^{j}\delta
A_{\;\;\;b}^{k}$ $\displaystyle=$ $\displaystyle\left(\delta
A_{\;\;\;b;a}^{i}-\alpha_{\;\;\;ka}^{i}\delta
A_{\;\;\;b}^{k}\right)-\left(\delta
A_{\;\;\;a;b}^{i}-\alpha_{\;\;\;jb}^{i}\delta A_{\;\;\;a}^{j}\right)$
$\displaystyle=$ $\displaystyle\mathcal{D}_{a}\left(\delta
A_{\;\;\;b}^{i}\right)-\mathcal{D}_{b}\left(\delta A_{\;\;\;a}^{i}\right)$

where $\mathcal{D}_{a}$ is covariant with respect to local Lorentz,
dilatational and $SU\left(N\right)$ transformations. Similarly we find for
$G_{\;\;\quad b}^{i\;a}$ and $H^{i\;\;ab}$,

$\displaystyle\delta_{A}G_{\;\;\quad b}^{i\;a}$ $\displaystyle=$
$\displaystyle\mathcal{D}^{a}\left(\delta A_{\;\;\;b}^{i}\right)$
$\displaystyle\delta_{A}H^{i\;\;ab}$ $\displaystyle=$ $\displaystyle 0$

Of course, under changes of gauge, these fields are invariant.

The conditions (129) and (130) must continue to hold throughout such small
changes. Substituting these variations into the first constraint,

$0=\delta
F_{\;\>ed}^{i}\left(\mathcal{H}_{i\;ac}\delta_{b}^{e}+\mathcal{H}_{i\;bc}\delta_{a}^{e}\right)\eta^{cd}+3\left(\eta_{ae}\delta
G_{\;\;\;\;c}^{i\;e}\mathcal{G}_{i\>bd}+\mathcal{G}_{i\;ac}\eta_{be}\delta
G_{\;\;\;\;d}^{i\;e}\right)\eta^{cd}$ (131)

The first term of Eq.(129) has dropped out because $H^{i\;\;ab}$ is
independent of $A_{\;\;\;a}^{i}$.

Now we expand Eq.(131) in terms of the variation $\delta A_{\;\;\;a}^{i}$ and
its derivatives,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\delta
A_{\;\;\;d,e}^{i}\left(\mathcal{H}_{i\;ac}\delta_{b}^{e}\eta^{cd}+\mathcal{H}_{i\;bc}\delta_{a}^{e}\eta^{cd}-\mathcal{H}_{i\;ac}\delta_{b}^{d}\eta^{ce}-\mathcal{H}_{i\;bc}\delta_{a}^{d}\eta^{ce}\right)$
$\displaystyle+\delta
A_{\;\;\;c}^{i\quad\;\;,e}3\left(\eta_{ae}\mathcal{G}_{i\>bd}+\mathcal{G}_{i\;ad}\eta_{be}\right)\eta^{cd}$
$\displaystyle\delta
A_{\;\;\;f}^{k}\left(-\omega_{\;\;\;de}^{f}+W_{e}\delta_{d}^{f}+\alpha_{\;\;\;kd}^{i}\delta_{e}^{f}\right)\left(\mathcal{H}_{k\;ac}\delta_{b}^{e}\eta^{cd}+\mathcal{H}_{k\;bc}\delta_{a}^{e}\eta^{cd}-\mathcal{H}_{k\;ac}\delta_{b}^{d}\eta^{ce}-\mathcal{H}_{k\;bc}\delta_{a}^{d}\eta^{ce}\right)$
$\displaystyle+\delta
A_{\;\;\;f}^{k}\left(3c_{\;\;\;kj}^{i}B^{j\;e}\right)\left(\eta_{ae}\mathcal{G}_{i\>bd}\eta^{fd}+\mathcal{G}_{i\;ad}\eta_{be}\eta^{fd}\right)$

where we collect terms proportional to $\delta A_{\;\;\;f}^{k},\delta
A_{\;\;\;f,e}^{k}$ and $\delta A_{\;\;\;f}^{k\;\;\quad,e}$ separately, noting
that the gravitational solution reduces the $y$-covariant derivative to a
partial, $\delta A_{\;\;\;b}^{i\quad\;\;;a}=\delta
A_{\;\;\;b}^{i\quad\;\;,a}$.

While the field equations determine the second derivatives of the potentials,
the potential itself and its first derivative are arbitrary initial conditions
on any Cauchy surface. Therefore, the three variations $\delta
A_{\;\;\;f}^{k},\delta A_{\;\;\;f,e}^{k}$ and $\delta
A_{\;\;\;f}^{k\;\;\quad,e}$ are independent, and the coefficient of each must
vanish separately:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathcal{H}_{i\;ac}\delta_{b}^{e}\eta^{cd}+\mathcal{H}_{i\;bc}\delta_{a}^{e}\eta^{cd}-\mathcal{H}_{i\;ac}\delta_{b}^{d}\eta^{ce}-\mathcal{H}_{i\;bc}\delta_{a}^{d}\eta^{ce}$
(132) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
3\left(\eta_{ae}\mathcal{G}_{i\>bd}+\mathcal{G}_{i\;ad}\eta_{be}\right)\eta^{cd}$
(133) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(-\omega_{\;\;\;de}^{f}+W_{e}\delta_{d}^{f}+\alpha_{\;\;\;kd}^{i}\delta_{e}^{f}\right)\left(\mathcal{H}_{k\;ac}\delta_{b}^{e}\eta^{cd}+\mathcal{H}_{k\;bc}\delta_{a}^{e}\eta^{cd}-\mathcal{H}_{k\;ac}\delta_{b}^{d}\eta^{ce}-\mathcal{H}_{k\;bc}\delta_{a}^{d}\eta^{ce}\right)$
(134)
$\displaystyle+\left(3c_{\;\;\;kj}^{i}B^{j\;e}\right)\left(\eta_{ae}\mathcal{G}_{i\>bd}\eta^{fd}+\mathcal{G}_{i\;ad}\eta_{be}\eta^{fd}\right)$

For the $x$-derivative part of the constraint, Eq.(132), we contract $eb$ and
lower the $d$ index to show that $\mathcal{H}_{i\;ac}$ must vanish,

$0=n\mathcal{H}_{i\;ac}$

Similarly, contracting $ac$ in Eq.(133) expressing the independence of the
$y$-derivative, shows that $\mathcal{G}_{i\>be}$ must also vanish.

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 3\mathcal{G}_{i\>be}$

With these two conditions, the final equation Eq.(134) is identically
satisfied.

These conditions satisfy both gravitational conditions on the sources,
Eqs.(105) and (110).

###  7.2 The source for gravity

We have shown that

$\displaystyle\frac{1}{1+\chi}\boldsymbol{\mathcal{R}}_{b}+\Lambda_{0}\eta_{be}\mathbf{e}^{e}$
$\displaystyle=$ $\displaystyle-\frac{\kappa}{\alpha}\mathbf{T}_{b}$

Expressed in terms of the Einstein tensor this is

$\displaystyle G_{ab}+\Lambda_{C}\eta_{ab}$ $\displaystyle=$
$\displaystyle-\left(n-2\right)\left(1+\chi\right)\frac{\kappa}{\alpha}\left(T_{ab}-\eta_{ab}T\right)$

where the vanishing of $\mathcal{H}_{i\;ab}$ and $\mathcal{G}_{i\;ab}$ leave
us with

$\displaystyle T_{ab}$ $\displaystyle=$ $\displaystyle
F_{\;\;ca}^{i}F_{i\>db}\eta^{cd}$

The trace of this is well-known to be gauge dependent, and the conformal
symmetry requires the energy tensor to be trace free. Therefore, we are
justified in adjusting the $SU\left(N\right)$ gauge to give

$\displaystyle G_{ab}+\Lambda_{C}\eta_{ab}$ $\displaystyle=$
$\displaystyle-\lambda\left(F_{\;\;ca}^{i}F_{i\>db}\eta^{cd}-\frac{1}{4}\eta_{ab}\left(\eta^{ce}\eta^{df}F_{\;\;cd}^{i}F_{i\>ef}\right)\right)$

where

$\lambda=\left(n-2\right)\left(1+\chi\right)\frac{\kappa}{\alpha}$

###  7.3 The Yang-Mills equation

With $\mathcal{H}_{\;\;\;ab}^{k}=G_{\;\;\quad b}^{i\;a}=0$, Eqs.(74) and (75)
reduce to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}F_{k\;ac}^{\;\;\quad;a}+\eta^{bc}F_{i\;ac}\beta_{\;\;\;k}^{i\;\;\quad
a}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta^{bc}\left(F_{j\;ac;b}+F_{i\;ac}\alpha_{\;\;\;jb}^{i}\right)$

where

$\displaystyle\boldsymbol{\mathcal{A}}_{\;\;\;k}^{i}$ $\displaystyle=$
$\displaystyle
c_{\;\;\;jk}^{i}A_{\;\;\;a}^{j}\mathbf{e}^{a}+c_{\;\;\;jk}^{i}B^{j\;a}\mathbf{f}_{a}$
$\displaystyle=$
$\displaystyle\boldsymbol{\alpha}_{\;\;\;k}^{i}+\boldsymbol{\beta}_{\;\;\;k}^{i}$

We may use these results and the form of the co-torsion,

$\mathbf{S}_{a}=2\left(1+\chi\right)\Delta_{ea}^{bc}y_{b}\Lambda\eta_{cd}\mathbf{e}^{d}\wedge\mathbf{e}^{e}$

to solve for the potentials,

$\displaystyle F_{\;\;\;ab}^{i}$ $\displaystyle=$ $\displaystyle
A_{\;\;\;b;a}^{i}-A_{\;\;\;a;b}^{i}-c_{\;\;\;jk}^{i}A_{\;\;\;a}^{j}A_{\;\;\;b}^{k}+\left(1+\chi\right)\Lambda
B^{i\;c}\left(\Delta_{bc}^{gf}y_{g}\eta_{fa}-\Delta_{ac}^{gf}y_{g}\eta_{fb}\right)$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
A_{\;\;\;b}^{i\quad\;\;,a}-B_{\;\;\;;b}^{i\;a}-c_{\;\;\;jk}^{i}B^{j\;a}A_{\;\;\;b}^{k}$