# A conformally invariant Yang-Mills type energy and equation on 6-manifolds

A. Rod Gover, Lawrence J. Peterson, and Callum Sleigh  A.R.G.: Department of
Mathematics
The University of Auckland
Private Bag 92019
Auckland 1142
New Zealand<EMAIL_ADDRESS>L.J.P.: Department of Mathematics
The University of North Dakota
101 Cornell Street Stop 8376
Grand Forks, ND 58202-8376
USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We define a conformally invariant action $\mathcal{S}$ on gauge connections on
a closed pseudo-Riemannian manifold $M$ of dimension 6. At leading order this
is quadratic in the gauge connection. The Euler-Lagrange equations of
$\mathcal{S}$, with respect to variation of the gauge connection, provide a
higher-order conformally invariant analogue of the (source-free) Yang-Mills
equations.

For any gauge connection $A$ on $M$, we define $\mathcal{S}(A)$ by first
defining a Lagrangian density associated to $A$. This is not conformally
invariant but has a conformal transformation analogous to a $Q$-curvature.
Integrating this density provides the conformally invariant action.

In the special case that we apply $\mathcal{S}$ to the conformal Cartan-
tractor connection, the functional gradient recovers the natural conformal
curvature invariant called the Fefferman-Graham obstruction tensor. So in this
case the Euler-Lagrange equations are exactly the “obstruction-flat” condition
for 6-manifolds. This extends known results for 4-dimensional pseudo-
Riemannian manifolds where the Bach tensor is recovered in the Yang-Mills
equations of the Cartan-tractor connection.

2020 Mathematics Subject Classification. primary: 53C18, 53C07; secondary:
58E15, 81T13, 53A55.
A.R.G. gratefully acknowledges support from the Royal Society of New Zealand
via Marsden Grants 16-UOA-051 and 19-UOA-008. C.S. Acknowledges support of an
FRDF Postdoctoral Fellowship at the University of Auckland.

##  1\. Introduction

Central to both the standard model in physics and our understanding of
differentiable 4-manifolds are the extremely well-known gauge field equations
known as the Yang-Mills equations [D, DK, tH, YM]. A feature of these
equations, that is crucially important in both spheres, is their conformal
invariance in dimension 4 [D, FP-1978, FP-1984, JR, Tb]. On the background of
a Riemannian or pseudo-Riemannian 4-manifold $(M,\mbox{{g}})$, consider a
gauge connection $A$ with curvature $F_{ab}$ (where the gauge field indices
are suppressed and the tensor indices are abstract). The Yang-Mills Lagrangian
density is (up to a constant factor we shall ignore)
$\mathcal{L}_{A}:=\operatorname{Trace}(\mbox{{g}}^{ab}\mbox{{g}}^{cd}F_{ac}\circ
F_{bd})$. For any $\Upsilon\in C^{\infty}(M)$, let
$\hat{\mbox{{g}}}_{ab}=e^{2\Upsilon}\mbox{{g}}_{ab}$. Under the conformal
transformation $\mbox{{g}}_{ab}\mapsto\hat{\mbox{{g}}}_{ab}$, the Lagrangian
transforms to $e^{-4\Upsilon}\mathcal{L}_{A}$, and this exactly balances the
corresponding transformation of the metric measure $d\mu^{\mbox{{\tiny
g}}}\mapsto d\mu^{\hat{\mbox{{\tiny g}}}}=e^{4\Upsilon}d\mu^{\mbox{{\tiny
g}}}$. The action $\mathcal{S}$ given by

(1) $\mathcal{S}(A):=\int_{M}\mathcal{L}_{A}\,d\mu^{\mbox{{\tiny g}}},$

for compactly supported $F$, is thus conformally invariant. It follows at once
that the functional gradient, with respect to variation of connection, is also
conformally invariant. The Euler-Lagrange equations that these determine, that
is the (source-free) Yang-Mills equations

(2) $\delta_{A}F=0,$

are thus conformally invariant as well. Here $\delta_{A}$ is the formal
adjoint of the connection-twisted exterior derivative $d_{A}$.

This argument for conformal invariance evidently fails in dimensions other
than 4, and it is well-known, and easily verified, that the equations (2) are
not conformally invariant in other dimensions [Br-1981]. It is thus natural to
ask whether, in dimensions $n$ other than 4, there is a replacement
action/energy and corresponding natural Euler Lagrange PDE system that is
conformally invariant. That is, on a smooth $n$-manifold we seek a map from
pairs $(\mbox{{g}},A)$, where g is a pseudo-Riemannian metric of any given
signature $(p,q)$, and $A$ is a principal connection, to corresponding
Lagrangian densities $\mathcal{L}_{A}$ of conformal weight $-n$ so that the
expression (1) is again conformally invariant. (We will discuss densities and
weights in Section 2.2 below.) There is an important question here of whether
the conformally invariant 4-dimensional Yang-Mills equations are an isolated
phenomenon, or whether rather they are the dimension 4 case of more general
picture. From elementary conformal geometry one does not expect such a
possibility in odd dimensions.

These questions are interesting from the perspective of gauge theory in
general, but they also have an interesting and direct motivation in conformal
geometry linked to the Bach tensor and its generalisations. On pseudo-
Riemannian 4-manifolds, the Bach tensor is an important conformal invariant
given by

(3) $B_{ab}:=\nabla^{c}A_{acb}+P^{cd}C_{cadb}.$

Here and below, $A_{acb}$, $P^{cd}$, and $C_{cadb}$ are the Cotton, Schouten,
and Weyl tensors, respectively, and $\nabla$ is the Levi-Civita connection.
See Section 2.1. In other dimensions $n\geq 3$ we will also say that $B_{ab}$,
as given in (3), is the Bach tensor. Metrics g that are conformal to Einstein
metrics are Bach-flat (meaning $B^{\mbox{{\tiny g}}}_{ab}=0$). So in dimension
4 the equations $B^{\mbox{{\tiny g}}}_{ab}=0$ provide, in a sense, a
conformally invariant weakening of the Einstein equations, an idea that dates
back to the original work of Bach for a “conformal gravity” theory [B-1921]
and that is still exploited [AO, CLW, GV, H, M]. Self-dual 4-manifolds are
also Bach-flat and so there is considerable interest in determining to what
extent there are other examples on closed Riemannian manifolds [CGGGR, LeB].
The tensor (3) is not conformally invariant in dimensions other than 4, but in
higher even dimensions there is a conformally invariant higher-order analogue
of (3) that takes the form

$\mathcal{O}_{ab}=\Delta^{n/2-2}\nabla^{c}\nabla^{d}C_{acbd}+\mbox{lower-order
terms}.$

Fefferman and Graham discovered the tensor $\mathcal{O}_{ab}$ as an
obstruction to their “ambient metric” construction. See [FG-1985]. For this
reason, $\mathcal{O}_{ab}$ is often referred to as the Fefferman-Graham
obstruction tensor. In dimension 4, the tensors $\mathcal{O}_{ab}$ and
$B_{ab}$ are equal. We will often write $\mathcal{O}_{ab}^{n}$ instead of
$\mathcal{O}_{ab}$ to emphasise the dimension.

The Bach tensor also arises as the functional gradient (now with respect to
metric variations and where we are ignoring overall constant factors) of the
integral of the square of the conformal Weyl tensor $|C|^{2}_{\mbox{{\tiny
g}}}=C^{abcd}C_{abcd}$. (Here we use the metric to raise and lower indices.)
As in the case of $\mathcal{L}_{A}$, the weights conspire so that
$\int_{M}|C|^{2}_{\mbox{{\tiny g}}}\,d\mu^{\mbox{{\tiny g}}}$ is conformally
invariant in dimension 4. On closed manifolds, an alternative Lagrangian
density is provided by Branson’s much studied $Q$-curvature $Q^{\mbox{{\tiny
g}}}$ of [BrO, Br-1993, Br-1995]. This alternative density generalises, in
that in all even dimensions there is such a $Q$-curvature, and the Fefferman-
Graham obstruction tensor is the functional gradient of

(4) $\int_{M}Q^{\mbox{{\tiny g}}}\,d\mu^{\mbox{{\tiny g}}}$

with respect to metric variations [GH]. The situation is slightly more subtle
now, however, because the $Q$-curvature is not itself conformally invariant
(whether we treat it as a function or as a conformal density). Under conformal
transformation, $Q$ changes by a term in which a linear operator acts on the
logarithm of the conformal factor. This term is of divergence form, and for
this reason, (4) is conformally invariant [Br-1995].

On a general Riemannian or pseudo-Riemannian manifold there is no natural
conformally invariant connection on the tangent bundle, but in dimensions
$n\geq 3$ there is a natural conformally invariant connection on a related
bundle of rank $n+2$. This is the conformal tractor connection [BEG] due, in
its original form, to Cartan and Thomas [C, T]. See Section 6.1 below. A nice
feature of this connection is that in dimension 4, its associated Yang-Mills
equations recover exactly the Bach-flat condition $B_{ab}=0$ [Mer, GSS]: Lemma
4.2 in [GSS] states that in dimension 4, in an informal tractor notation,

(5)
$\delta_{\nabla^{\mathcal{T}}}F_{\nabla^{\mathcal{T}}}=c\cdot(0,0,0,B_{ab}),$

where $\nabla^{\mathcal{T}}$ is the conformal tractor connection, and $c$ is
an (explicit non-zero) constant. One might hope that a higher-dimensional
conformally invariant Yang-Mills type theory would generalise this picture.

In the current article, we treat these questions in dimension $n=6$. It seems
to us that the results should assist with extending many of the studies and
directions mentioned above from the confines of dimension 4 to higher even
dimensions. For convenience, we will always view the gauge connection $A$ as a
linear connection acting on some vector bundle $V$, and we let $F_{A}$ denote
the curvature of $A$. Our work will rely on the operator $Q_{2}^{A}$ defined
in Section 3.2. This operator belongs to a family of operators defined in
[BG-2005, BG-2007]. In what follows, we will also work with a bilinear mapping
$\left<\cdot,\cdot\right>$ which we define in Section 3.

For closed (i.e. compact without boundary) 6-manifolds, we provide the sought
map from pairs $(\mbox{{g}},A)$, consisting of a metric and a connection, to
Lagrangian densities $\mathcal{L}_{A}$ so that the action $\mathcal{S}$ given
in (1) is conformally invariant. The Lagrangian density we provide, namely
$\left<F_{A},Q_{2}^{A}F_{A}\right>$, is an analogue of the $Q$-curvature. Here
we mean that this density is not conformally invariant but rather transforms,
under conformal rescaling of the metric g on $M$, by a linear divergence-type
operator acting on the logarithm of the conformal factor. This is an immediate
consequence of Proposition 3.3 below, which was first derived in [BG-2007]. If
$M$ is closed, it follows that

(6)
$\mathcal{S}(A)=\int_{M}\left<F_{A},Q_{2}^{A}F_{A}\right>\,d\mu^{\mbox{{\tiny
g}}}$

is invariant under conformal change of the metric on $M$. See Proposition 4.2.

An important feature of the Lagrangian density
$\left<F_{A},Q_{2}^{A}F_{A}\right>$ is that it is quadratic in the jets of the
connection $A$ at leading order. So if we consider the functional gradient of
$\mathcal{S}$, with respect to variation of connection, we find that this
functional gradient is linear at leading order. The functional gradient
therefore provides a conformally invariant map $\mathfrak{D}$ from connections
$A$ to 1-forms taking values in $\operatorname{End}(V)$:

$A\mapsto\mathfrak{D}A.$

We thus obtain a higher-order conformally invariant analogue of the Yang-Mills
equations. See Proposition 4.3 and Definition 5.1. By construction these
equations are conformally invariant, but a direct verification is provided in
Proposition 5.2.

As an application and test of this, in Section 6 we show that the non-linear
operator $\mathfrak{D}$ applied to the conformal tractor connection exactly
recovers the Fefferman-Graham obstruction tensor of conformal 6-manifolds.
Thus we obtain a perfect parallel of the situation in dimension 4, as
discussed above, and at the same time a new perspective on the Fefferman-
Graham obstruction tensor. See Theorem 6.1 which, again in an informal tractor
notation, states that

(7) $\mathfrak{D}(\nabla^{\mathcal{T}})=k\cdot(0,0,0,\mathcal{O}_{ab}),$

where $\nabla^{\mathcal{T}}$ is the conformal tractor connection, and $k$ is
an (explicit non-zero) constant. Compare this to (5) above.

It is not difficult to see that there are other conformally invariant actions
on connections in dimension 6. For example, apart from divergences, we can add
cubic terms
$\operatorname{Trace}(\mbox{{g}}^{af}\mbox{{g}}^{bc}\mbox{{g}}^{de}F_{ab}\circ
F_{cd}\circ F_{ef})$ to the Lagrangian density. However Theorem 6.1 (i.e. (7))
suggests that (6) is distinguished.

Finally, in Section 6.3 we discuss the link between our action (27), when
specialised to the case that $A$ is the conformal tractor connection, and a
conformally invariant action we construct using the Fefferman-Graham invariant
of Proposition 3.4 of [FG-1985].

It seems that recently there has been some interest in “higher order Yang-
Mills” flows [K, Z]. In these works, the generation of the equations
emphasises the use of broadly analogous higher-order actions on Riemannian
manifolds, rather than attention to conformal properties, but it is possible
that some of the results could be applied to the equations we develop here. In
[K] Kelleher asks the question of whether there might be conformally invariant
higher-order Yang-Mills type equations, and suggests that, if so, conformal
invariance might be a good distinguishing property. In [Bz] the author looks
at a notion of Yang-Mills for higher dimensions that is not closely related to
the programme here.

##  2\. Notation and background

Throughout our work, $M$ will be a smooth manifold of dimension $n$. Our main
interest lies in smooth manifolds of dimension $6$, but in this section, we
work with general manifolds in general dimensions $n$. We let $TM$ and
$T^{*}M$ denote the tangent and cotangent bundles, respectively, of $M$. We
will often use Penrose’s abstract index notation. Lowercase indices $a$, $b$,
$c$, and so forth will be associated to $TM$ and $T^{*}M$, and uppercase
indices such as $B$, $C$, and $D$ will be associated to other finite-
dimensional vector bundles. Parentheses will indicate symmetrisation of
indices. For example, if $T_{abc}$ is a rank 3 tensor, then
$T_{a(bc)}=(1/2)(T_{abc}+T_{acb})$. We will always assume that
$\mbox{{g}}_{ab}$ (or just g) denotes a pseudo-Riemannian metric on $M$ and
that $R_{ab}{}^{c}{}_{d}$ is the Riemannian curvature tensor of the Levi-
Civita connection $\nabla$ associated to $\mbox{{g}}_{ab}$. We use the sign
convention for $R_{ab}{}^{c}{}_{d}$ such that

(8) $(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})v^{c}=R_{ab}{}^{c}{}_{d}v^{d}$

for all vector fields $v^{c}$ on $M$. Let $k\in\mathbb{Z}_{>0}$ and a smooth
tensor field $T^{c_{1}\ldots c_{k}}$ on $M$ be given. Then

(9) $\nabla_{a}\nabla_{b}T^{c_{1}\ldots
c_{k}}=\nabla_{b}\nabla_{a}T^{c_{1}\ldots
c_{k}}+R_{ab}{}^{c_{1}}{}_{i}T^{i\ldots
c_{k}}+\cdots+R_{ab}{}^{c_{k}}{}_{i}T^{c_{1}\ldots i},$

by (8). We may use the metric $\mbox{{g}}_{ab}$ to raise or lower any of the
free indices in (9). We let $\Delta:=\nabla_{a}\nabla^{a}$. For any vector
bundle $W$ over $M$, let $\Gamma(W)$ denote the space of all smooth sections
of $W$. The notation $W^{*}$ will always denote the dual of $W$. Our scaling
convention for the wedge product of differential forms will be such that for
all $p,\,q\in\mathbb{Z}_{\geq 0}$, all $x\in M$, and all $p$-forms $\omega$
and $q$-forms $\psi$ at $x$,

(10) $\omega\wedge\psi=\frac{(p+q)!}{p!q!}\,\mbox{Alt}(\omega\otimes\psi).$

###  2.1. Conformal geometry

Let a manifold $M$ and pseudo-Riemannian metrics g and $\hat{\mbox{{g}}}$ on
$M$ be given, and suppose there is an $\Upsilon\in C^{\infty}(M)$ such that
$\hat{\mbox{{g}}}=e^{2\Upsilon}\mbox{{g}}$. Then we say that
$\hat{\mbox{{g}}}$ is conformal to g. Let c denote the set of all pseudo-
Riemannian metrics on $M$ which are conformal to g. We say that c is a
conformal structure on $M$ or a conformal class on $M$, and we say that the
pair $(M,\mbox{c})$ is a conformal manifold. Throughout our work,
$\hat{\mbox{{g}}}$ will always denote the metric $e^{2\Upsilon}\mbox{{g}}$
(for some g and $\Upsilon$). For any object, such as a tensor, that depends on
the choice of the metric in c, a hat $\widehat{}$ will indicate the object as
determined by the metric $\hat{\mbox{{g}}}$.

We will need to introduce several classical tensors that occur naturally in
conformal geometry due to their simple transformation laws under conformal
rescaling. The first tensor we will need is the symmetric tensor
$P_{ab}\in\Gamma(S^{2}T^{*}M)$, called the _Schouten tensor_ , defined in
general dimensions $n\geq 3$ by

$P_{ab}:=\frac{1}{n-2}\left(\mbox{{Ric}}_{ab}-\frac{1}{2(n-1)}\,\mbox{{g}}_{ab}\,\mbox{{Sc}}\right).$

Here $\mbox{{Ric}}_{ab}=R^{c}{}_{acb}$ is the Ricci tensor, and
$\mbox{{Sc}}=\mbox{{Ric}}_{a}{}^{a}$ is the scalar curvature. The tensors $P$,
$R$, Ric, and Sc depend on the choice of metric g, and we may write
$P^{\mbox{{\tiny g}}}$, $R^{\mbox{{\tiny g}}}$, and so forth to indicate this
dependence. We will usually omit g from our notation, however.

By using the metric to raise indices, we may associate $P_{ab}$ to an element
of $\Gamma(\operatorname{End}(T^{*}M))$. We will let the symbol $P\\#$ denote
the corresponding derivation on general tensor fields. For example, the action
of $P\\#$ on a two-form $\omega_{ab}$ is

$(P\\#\omega)_{ab}=\tensor{P}{{}_{a}^{c}}\tensor{\omega}{{}_{c}{}_{b}}+\tensor{P}{{}_{b}^{c}}\tensor{\omega}{{}_{a}{}_{c}}.$

We will use this later. For all $r,s\in\mathbb{Z}_{\geq 0}$, let $T^{(r,s)}$
denote the bundle of tensors of contravariant rank $r$ and covariant rank $s$.
For any vector bundle $W$, we may extend the action of $P\\#$ (trivially) to
sections of $T^{(r,s)}\otimes W$ as follows. First, let

$P\\#(\omega\otimes U)=(P\\#\omega)\otimes U$

for all $\omega\in\Gamma(T^{(r,s)})$ and all $U\in\Gamma(W)$. Then extend the
action $P\\#$ linearly to sums of sections of the form $\omega\otimes U$,
where $\omega\in\Gamma(T^{(r,s)})$ and $U\in\Gamma(W)$.

Our work will also involve the scalar field $J$ given by
$J=P_{a}{}^{a}=P_{ab}\mbox{{g}}^{ab}$. From the second Bianchi identity, it
follows that

(11) $\nabla_{a}P^{a}{}_{b}=\nabla_{b}J.$

The Cotton tensor $A_{abc}$ is defined by

(12) $A_{abc}:=\nabla_{b}P_{ca}-\nabla_{c}P_{ba}.$

The Cotton tensor is trace-free. The Weyl tensor is given by

(13)
$C_{abcd}:=R_{abcd}+\mbox{{g}}_{cb}P_{ad}-\mbox{{g}}_{ca}P_{bd}+\mbox{{g}}_{da}P_{bc}-\mbox{{g}}_{db}P_{ac}.$

In all dimensions $n\geq 3$, the Bach tensor $B_{ab}$ is given by (3). We note
here that $B_{ab}$ is trace-free. A short computation shows that $B_{ab}$ is
symmetric.

###  2.2. Conformal density bundles

One may simplify many of the calculations and fomulae of conformal geometry by
using the language of weighted density bundles. By its definition, a conformal
structure c determines a ray sub-bundle $\mathcal{Q}$ of $S^{2}T^{*}M$. This
sub-bundle is actually a principal $\mathbb{R}^{+}$-bundle
$\pi:\mathcal{Q}\rightarrow M$. For every $\mbox{{g}}\in\mbox{c}$ and every
$x\in M$, the fibre of $\mathcal{Q}$ over $x$ consists of all metrics at $x$
which are conformal to g at $x$. For each $w\in\mathbb{R}$ we will denote by
$\mathcal{E}[w]$ the induced line bundle arising from the representation of
$\mathbb{R}^{+}$ on $\mathbb{R}$ given by $t\mapsto t^{-w/2}$. We note that
each bundle $\mathcal{E}[w]$ is canonically oriented. Sections of
$\mathcal{E}[w]$ are called _conformal densities of weight $w$_ and maybe
identified (via the associated bundle construction) with homogeneous functions
of degree $w$ on $\mathcal{Q}$.

There is a tautological function g on $\mathcal{Q}$ taking values in
$S^{2}T^{*}M$ which simply assigns to a point
$(\mbox{{g}}_{x},x)\in\mathcal{Q}$ the metric $\mbox{{g}}_{x}$ at $x$. This
determines a section g (or $\textbf{\mbox{{g}}}_{ab}$) of
$S^{2}T^{*}M\otimes\mathcal{E}[2]$ called the _conformal metric_. For any
nonvanishing $\sigma\in\Gamma(\mathcal{E}[1])$, it follows
$\sigma^{-2}\textbf{\mbox{{g}}}_{ab}$ is a metric in the conformal class, and
we use the term _conformal scale_ for $\sigma$. In a similar fashion, we may
define a section $\textbf{\mbox{{g}}}^{ab}$ of $S^{2}TM\otimes\mathcal{E}[-2]$
in such a way that $\textbf{\mbox{{g}}}^{ab}=\textbf{\mbox{{g}}}^{-1}$. A
given pseudo-Riemannian metric g determines a positive section
$\xi^{\mbox{{\tiny g}}}$ of $\mathcal{E}[1]$ via the relation
$\mbox{{g}}_{ab}=(\xi^{\mbox{{\tiny g}}})^{-2}\textbf{\mbox{{g}}}_{ab}$. We
say that $\xi^{\mbox{{\tiny g}}}$ is the scale density associated to g. For
any metric g, the Levi-Civita connection acts on sections of $\mathcal{E}[w]$
as follows. Let $f\in C^{\infty}(M)$ be given. Then $\nabla((\xi^{\mbox{{\tiny
g}}})^{w}f)=(\xi^{\mbox{{\tiny g}}})^{w}df$, where $d$ is the exterior
derivative on functions. Note that $\nabla\xi^{\mbox{{\tiny g}}}=0$.

Note that $\textbf{\mbox{{g}}}_{ab}$ and $\textbf{\mbox{{g}}}^{ab}$ are
conformally invariant, so we may use $\textbf{\mbox{{g}}}^{ab}$ and
$\textbf{\mbox{{g}}}_{ab}$ to raise and lower tensor indices in a conformally
invariant way. This fits with the notion of weighted tensors. From this point
forward, all tensors will implicitly be the product of a tensor and a density
of some weight $w$. We say that the tensor is weighted, or that it carries a
weight. The Riemannian curvature tensor $R_{ab}{}^{c}{}_{d}$ will carry a
weight of 0. On the other hand, $\textbf{\mbox{{g}}}_{ab}$ will carry a weight
of 2, and $\textbf{\mbox{{g}}}^{ab}$ will carry a weight of $-2$. If we raise
or lower an index, we will always use the conformal metric to do this. Thus
$R_{abcd}$ will have weight $2$, and $R_{ab}{}^{cd}$ will carry a weight of
$-2$. The tensors $P_{ab}$, $J$, $A_{abc}$, $C_{abcd}$, and $B_{ab}$ will
carry weights $0$, $-2$, $0$, $2$, and $-2$, respectively. Elements and
sections of vector bundles will also carry weights.

To work with weighted tensors and vectors efficiently, we introduce additional
notations. For every $k\in\mathbb{Z}_{\geq 0}$, let $\Lambda^{k}(M)$ denote
the bundle of $k$-forms on $M$, and for every $w\in\mathbb{R}$, let
$\Lambda^{k}(M,w)$ denote $\Lambda^{k}(M)\otimes\mathcal{E}[w]$. Let
$\Omega^{k}(M,w)$ denote $\Gamma(\Lambda^{k}(M,w))$. Similarly, for any vector
bundle $W$ over $M$, let $\Lambda^{k}(W)$ denote the bundle of $k$-forms on
$M$ with values in $W$, and let $\Lambda^{k}(W,w)$ denote
$\Lambda^{k}(W)\otimes\mathcal{E}[w]$. Let $\Omega^{k}(W,w)$ denote
$\Gamma(\Lambda^{k}(W,w))$. Note that $\Lambda^{k}(M)=\Lambda^{k}(M,0)$.
Similar remarks apply to $\Lambda^{k}(W)$, $\Omega^{k}(M)$, and
$\Omega^{k}(W)$.

Remembering that tensors carry weights, we now state the transformation rules
for $P_{ab}$ and $J$ under conformal change of metric. In general dimensions
$n$, the Schouten tensor transforms according to the rule

$\widehat{P}_{ab}=P_{ab}-\nabla_{a}\Upsilon_{b}+\Upsilon_{a}\Upsilon_{b}-\frac{1}{2}\textbf{\mbox{{g}}}_{ab}\Upsilon_{c}\Upsilon^{c}.$

Here $\nabla$ is the Levi-Civita connection associated to the original metric
g, and (for example) $\Upsilon_{a}=\nabla_{a}\Upsilon$. Also,
$\Upsilon^{c}=\textbf{\mbox{{g}}}^{cd}\Upsilon_{d}$. We will use these
notational conventions throughout our work. Note that $\Upsilon^{c}$ carries a
weight of $-2$. In general dimensions $n$, the conformal transformation rule
for $J$ is as follows:

$\widehat{J}=J-\nabla_{a}\Upsilon^{a}+\left(1-\frac{n}{2}\right)\Upsilon_{a}\Upsilon^{a}.$

###  2.3. Integration of densities

We now consider integration of densities. Let a conformal manifold
$(M,\mbox{c})$, a section $\phi$ of $\mathcal{E}[-n]$, which is of compact
support, and $\mbox{{g}}\in\mbox{c}$ be given. Then $\phi=(\xi^{\mbox{{\tiny
g}}})^{-n}f$ for some $f\in C^{\infty}(M)$. Let $d\mu^{\mbox{{\tiny g}}}$
denote the pseudo-Riemannian measure on $M$ associated to g, and define
$\int_{M}\phi\,d\mu$ by

(14) $\int_{M}\phi\,d\mu:=\int_{M}f\,d\mu^{\mbox{{\tiny g}}}.$

This integral is independent of the choice of the $\mbox{{g}}\in\mbox{c}$ that
we use in its definition. To see this, begin by recalling our convention that
$\hat{\mbox{{g}}}=e^{2\Upsilon}\mbox{{g}}$. One can show that
$\xi^{\mbox{{\tiny g}}}=e^{\Upsilon}\xi^{\hat{\mbox{{\tiny g}}}}$. Thus
$(\xi^{\mbox{{\tiny g}}})^{-n}=e^{-n\Upsilon}(\xi^{\hat{\mbox{{\tiny
g}}}})^{-n}$, and hence with $f$ as above,

$\phi=(\xi^{\mbox{{\tiny g}}})^{-n}f=e^{-n\Upsilon}(\xi^{\hat{\mbox{{\tiny
g}}}})^{-n}f=(\xi^{\hat{\mbox{{\tiny g}}}})^{-n}e^{-n\Upsilon}f.$

So if we use $\hat{\mbox{{g}}}$ to define $\int_{M}\phi\,d\mu$, we find that

$\int_{M}\phi\,d\mu=\int_{M}e^{-n\Upsilon}f\,d\mu^{\hat{\mbox{{\tiny
g}}}}=\int_{M}e^{-n\Upsilon}fe^{n\Upsilon}\,d\mu^{\mbox{{\tiny
g}}}=\int_{M}f\,d\mu^{\mbox{{\tiny g}}}.$

This is consistent with (14).

##  3\. Bundles and connections

In our discussions of connections and curvature on general vector bundles, our
conventions and basic definitions will essentially follow those given in
standard references such as [DK]. In this section, we describe our conventions
and define some of the vector bundles and operators that our work will use. We
also state some conformal transformation rules for connections and other
operators. For consistency with work in other contexts, we work in general
dimensions $n$ unless we indicate otherwise.

Let a finite-dimensional vector bundle $W$ over $M$, a connection $A$ on $W$,
and a nonnegative integer $k$ be given. Let $\nabla$ denote the standard
coupling of $A$ and the Levi-Civita connection. When $\nabla$ acts on weighted
tensors associated to $TM$ or $T^{*}M$, it acts as the Levi-Civita connection.
If $\nabla$ acts on a section of $W$, then it acts as $A$. By linearity and
the Leibniz property, $\nabla$ is extended to weighted sections of tensor
products of these bundles. The symbol $\nabla$ will denote a coupled
connection throughout much of our work.

Let $w\in\mathbb{R}$ be given. The connection $A$ induces a twisted exterior
derivative

$d_{A}:\Omega^{k}(W,w)\rightarrow\Omega^{k+1}(W,w).$

By (10),

(15) $(d_{A}\omega)_{a_{1}\ldots a_{k+1}}{}^{B}=\frac{1}{k!}\sum_{\sigma\in
S_{k+1}}(\mbox{sgn\ }\sigma)\nabla_{a_{\sigma(1)}}\omega_{a_{\sigma(2)}\ldots
a_{\sigma(k+1)}}{}^{B}$

for all $\omega\in\Omega^{k}(W,w)$. Here $S_{k+1}$ denotes the set of all
permutations of $\\{1,\ldots,k+1\\}$.

To give a formula for the formal adjoint of $d_{A}$, we begin by letting a
metric $h$ (of some signature) on $W$ be given, and we suppose that $h$ is
preserved by the connection $A$. Also let $k\in\mathbb{Z}_{\geq 0}$ and
$w_{1}$, $w_{2}\in\mathbb{R}$ be given, and suppose that $w_{1}+w_{2}=2k-n$.
We will use the conformal metric g on $M$ together with $h$ to define a
bilinear mapping

$\langle\cdot,\cdot\rangle:\Lambda^{k}(W,w_{1})\times\Lambda^{k}(W,w_{2})\rightarrow\mathcal{E}[-n].$

Specifically, for all $x\in M$ and all $\omega\in\Lambda^{k}(W,w_{1})$ and all
$\eta\in\Lambda^{k}(W,w_{2})$ at $x$, we let

(16) $\langle\omega,\eta\rangle=\frac{1}{k!}\,\omega_{a_{1}\ldots
a_{k}}{}^{B}\eta^{a_{1}\ldots a_{k}}{}_{B}.$

Now again let $k\in\mathbb{Z}_{\geq 0}$ and $w_{1},w_{2}\in\mathbb{R}$ be
given, but now suppose that $w_{1}+w_{2}=2(k+1)-n$. The formal adjoint of
$d_{A}:\Omega^{k}(W,w_{1})\rightarrow\Omega^{k+1}(W,w_{1})$ with respect to
$\langle\cdot,\cdot\rangle$ is easily computed to be the operator
$\delta_{A}:\Omega^{k+1}(W,w_{2})\rightarrow\Omega^{k}(W,w_{2}-2)$ given by

(17) $(\delta_{A}\eta)_{a_{1}\ldots
a_{k}}{}^{B}=-\nabla_{b}\eta^{b}{}_{a_{1}\ldots a_{k}}{}^{B}.$

Once again let $k\in\mathbb{Z}_{\geq 0}$ be given. For any $w\in\mathbb{R}$,
an obvious adaptation of the above discussion defines our conventions for the
exterior derivative $d:\Omega^{k}(M,w)\rightarrow\Omega^{k+1}(M,w)$ and its
formal adjoint $\delta$. By removing the subscript $A$ and the index $B$ from
(15) and (17), we obtain symbolic formulae for $(d\omega)_{a_{1}\ldots
a_{k+1}}$ and $(\delta\eta)_{a_{1}\ldots a_{k}}$.

We will always assume that the connection $A$ is independent of any choice of
metric from the conformal class c on $M$. The coupled connection $\nabla$ will
transform under conformal change of the metric g, however, and we will need
certain conformal transformation rules involving $\nabla$. Specifically, let
$w\in\mathbb{R}$, a one-form $\eta$, and a two-form $\omega$ be given. Suppose
that $\eta$ and $\omega$ have weight $w$ and that both take values in
$\mathbb{R}$ or $W$. Then

$\widehat{\nabla}_{a}\eta_{b}=\nabla_{a}\eta_{b}+(w-1)\Upsilon_{a}\eta_{b}-\Upsilon_{b}\eta_{a}+\textbf{\mbox{{g}}}_{ab}\Upsilon^{c}\eta_{c}\,,$

and

(18)
$\widehat{\nabla}_{a}\omega_{bc}=\nabla_{a}\omega_{bc}+(w-2)\Upsilon_{a}\omega_{bc}-\Upsilon_{b}\omega_{ac}-\Upsilon_{c}\omega_{ba}+\textbf{\mbox{{g}}}_{ab}\Upsilon^{d}\omega_{dc}+\textbf{\mbox{{g}}}_{ac}\Upsilon^{d}\omega_{bd}.$

Both of these transformation rules follow from elementary computations and the
fact that $A$ is conformally invariant. From (17) and (18), it follows that

(19)
$(\widehat{\delta}_{A}\omega)_{c}=(\delta_{A}\omega)_{c}+(4-n-w)\Upsilon^{a}\omega_{ac}.$

Our work will focus on vector bundles $adV$ which we will define as follows.
Let $\pi:P\rightarrow M$ be an arbitrary $G$-bundle, for some Lie group $G$.
Any finite-dimensional $G$-module $\mathbb{V}$ gives rise to an associated
vector bundle $V$ over $M$. Let $A$ denote a $G$-connection on the bundle $P$.
The same notation will be used to denote the corresponding associated linear
connection on the associated vector bundle $V$. The adjoint representation
induces a bundle of Lie-algebras over $M$, and the derivative of the
$G$-representation acting on $\mathbb{V}$ then gives a vector bundle $adV$,
which is a sub-bundle of $\operatorname{End}(V)\cong V\otimes V^{*}$. Let
$x\in M$ and elements $\omega$ and $\eta$ of $adV$ at $x$ be given. We will
let $\omega\eta$ denote the composition $\omega\circ\eta$. We may write
$\omega\eta$ in index notation as $\omega^{B}{}_{D}\eta^{D}{}_{C}$. We will
use this notation when we define a metric on $adV$ below. Our work will not
require the existence of a metric on $V$.

We will work with operators acting on $\Omega^{k}(adV,w)$ for various
$k\in\mathbb{Z}_{\geq 0}$ and $w\in\mathbb{R}$. To define these operators, let
$k\in\mathbb{Z}_{\geq 0}$ be given. Let $\nabla$ denote the connection $A$,
and recall that $\nabla$ determines a dual connection $\nabla^{*}$ on $V^{*}$.
Specifically, define $\nabla^{*}$ by the formula
$(\nabla^{*}_{X}u)(v)=X(u(v))-u(\nabla_{X}v)$. Here $X\in\Gamma(TM)$,
$u\in\Gamma(V^{*})$, and $v\in\Gamma(V)$.

By coupling $\nabla$ and $\nabla^{*}$, we determine a connection on $adV$, and
we in turn couple this connection with the Levi-Civita connection on $M$ to
obtain a connection on $\Omega^{k}(adV,w)$. Then (15) defines an exterior
derivative $d_{A}:\Omega^{k}(adV,w)\rightarrow\Omega^{k+1}(adV,w)$ associated
to $A$. When we work with this $d_{A}$, we will append a lower index $C$ to
each side of (15).

To give a formula for the formal adjoint of $d_{A}$, we will need a metric on
$adV$. For any $x\in M$ and any $\omega$ and $\eta$ in $adV$ at $x$, let
$h(\omega,\eta)=\omega^{B}{}_{C}\eta^{C}{}_{B}$. Then by construction, $h$ is
preserved by the connection $A$. For the general theory, and for the purposes
of deriving equations, we will assume that $G$ is such that $h$ is non-
degenerate, so $h$ is a bundle metric on $adV$. (For example this is always
the case if $G$ is semisimple. Then $h$ is a multiple of the Killing form.)
Now let $k\in\mathbb{Z}_{\geq 0}$ be given. Let $W=adV$, and let
$\langle\cdot,\cdot\rangle$ be the bilinear mapping described in (16) and in
the discussion preceding (16). In this context, we may rewrite (16) as

$\langle\omega,\eta\rangle=\frac{1}{k!}\,\omega_{a_{1}\ldots
a_{k}}{}^{B}{}_{C}\,\eta^{a_{1}\ldots a_{k}C}{}_{B}.$

and (17) as

(20) $(\delta_{A}\eta)_{a_{1}\ldots
a_{k}}{}^{B}{}_{C}=-\nabla_{b}\eta^{b}{}_{a_{1}\ldots a_{k}}{}^{B}{}_{C}.$

######  Remark 3.1.

Note that since the $h$ is defined by the canonical self-duality of
$\operatorname{End}(V)$, it is non-degenerate but with mixed signature. Indeed
if, for example, we use any choice of positive definite metric to identify $V$
with $V^{*}$, and hence to identify $\operatorname{End}(V)$ with
$\otimes^{2}V$, then it is easily seen that $h$ is positive definite on the
symmetric part $S^{2}V\subset\otimes^{2}V$, and negative definite on the skew
part $\Lambda^{2}V\subset\otimes^{2}V$. Of course the signature of $h$ is
independent of the choice made.

Although for the general theory we assume assume that $G$ is such that $h$ is
non-degenerate on $adV$, the equations that we derive can be used more widely.

We will sometimes consider the induced connections or twisted exterior
derivatives on $V$ and on $adV$ at the same time, and we will use the same
notations (such as $d_{A}$) for the versions of these operators on both $V$
and $adV$.

The space of all connections on $V$ will be denoted $\mathcal{A}$. For any
$A_{1},\ A_{2}\in\mathcal{A}$, one can easily show that
$A_{1}-A_{2}\in\Omega^{1}(adV)$. So once a connection $A_{0}\in\mathcal{A}$
has been chosen, any other connection $A\in\mathcal{A}$ can be written as
$A_{0}+a$, for some $a\in\Omega^{1}(adV)$. Thus $\mathcal{A}$ is an affine
space modeled on the vector space $\Omega^{1}(adV)$. Suppose we choose a local
frame for $V$ and let $A_{0}=d$ be the trivial connection for this frame (the
exterior derivative on components). Then for any $A\in\mathcal{A}$, there is
an $a\in\Omega^{1}(adV)$ such that $A=d+a$. This is the usual local expression
for a linear connection.

Now let $A$, $A_{0}\in\mathcal{A}$ be given. Then $A=A_{0}+a$ for some
$a\in\Omega^{1}(adV)$, as we noted above. Next, let $k\in\mathbb{Z}_{\geq 0}$
be given, and note that $A$ and $A_{0}$ induce connections on
$T^{(0,k)}\otimes\operatorname{End}(V)$. Let $A$ and $A_{0}$, respectively,
denote these induced connections. Then for any $\phi_{j_{1}\ldots
j_{k}}{}^{B}{}_{C}\in\Gamma(T^{(0,k)}\otimes\operatorname{End}(V))$,
elementary computations show that

(21) $A_{i}\phi_{j_{1}\ldots j_{k}}{}^{B}{}_{C}=(A_{0})_{i}\phi_{j_{1}\ldots
j_{k}}{}^{B}{}_{C}+a_{i}{}^{B}{}_{E}\phi_{j_{1}\ldots
j_{k}}{}^{E}{}_{C}-\phi_{j_{1}\ldots j_{k}}{}^{B}{}_{E}a_{i}{}^{E}{}_{C}.$

When we regard $A$ and $A_{0}$ as connections on
$T^{(0,k)}\otimes\operatorname{End}(V)$, we will thus say that
$A=A_{0}+[a,\cdot]$. In this context, we will also let $A_{0}+a$ denote
$A_{0}+[a,\cdot]$. We do this, for example, in (29).

###  3.1. Curvature

There is a (gauge) invariant section $F_{A}\in\Omega^{2}(adV)$ such that
$d_{A}^{2}S^{B}=F_{A}S^{B}$ for all $S^{B}\in\Gamma(V)$. Thus

(22)
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})S^{B}=(F_{A})_{ab}{}^{B}{}_{E}S^{E}$

for all $S^{B}\in\Gamma(V)$. Here $\nabla$ is again the coupled connection.
Because the Levi-Civita connection is torsion free, $F_{A}$ is the usual
_curvature_ of the connection $A$. Similarly, there is an
$F_{A}^{*}\in\Omega^{2}(adV^{*})$ such that
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})S_{B}=(F_{A}^{*})_{ab}{}^{C}{}_{B}S_{C}$
for all $S_{B}\in\Gamma(V^{*})$. One can use the duality of $V$ and $V^{*}$ to
show that $(F_{A}^{*})_{ab}{}^{C}{}_{B}=-(F_{A})_{ab}{}^{C}{}_{B}$. Thus

(23)
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})S_{B}=-(F_{A})_{ab}{}^{E}{}_{B}S_{E}$

for all $S_{B}\in\Gamma(V^{*})$. The curvature $F_{A}$ satisfies the Bianchi
identity $d_{A}F_{A}=0$, as noted in [DK].

###  3.2. The coupled $Q_{2}^{A}$-operator

As we noted in Section 1, the $Q$-curvature is defined in [BrO, Br-1993,
Br-1995]. In [BG-2005, BG-2007], the first author and Branson define and
discuss a family of linear operators $Q_{k}$, for $n$ even and
$k=0,1,2,\ldots,n/2+1$, mapping $\Omega^{k}(M)$ to $\Omega^{k}(M,2k-n)$. The
$Q_{k}$-operators generalise Branson’s $Q$-curvature in several ways.
Branson’s original $Q$-curvature is equal to $Q_{0}1$. We now restrict to
manifolds of dimension $n=6$. The case of interest to us then is the case
$k=2$, where the operator $Q_{2}$ is given as follows:

(24) $Q_{2}\omega:=d\delta\omega-4P\\#\omega+2J\omega.$

In [BG-2007], the above-mentioned authors also discuss a generalised family of
operators $Q_{k}$ which act on bundle-valued $k$-forms. We need only the $k=2$
case here, and we construct such generalised operators by replacing the
exterior derivative and its adjoint in (24) with the twisted exterior
derivative $d_{A}$ and its adjoint $\delta_{A}$, respectively. We do this in
the next definition. In this definition and in much of what follows, we
suppress bundle indices. We will do so without further comment, since the
context should make this clear.

######  Definition 3.2.

Let a pseudo-Riemannian manifold $(M,g)$ of dimension $n=6$, a vector bundle
$W$ over $M$, and a bundle metric $h$ on $W$ be given. Also let $A$ be a
conformally invariant connection on $W$ that preserves $h$. Let the operator
$Q_{2}^{A}:\Omega^{2}(W)\rightarrow\Omega^{2}(W,-2)$ be defined as follows:

$Q^{A}_{2}\omega=d_{A}\delta_{A}\omega-4P\\#\omega+2J\omega.$

The following proposition describes the behaviour of $Q_{2}^{A}$ under
conformal rescaling.

######  Proposition 3.3.

Let $n$, $W$, $A$, and $\omega$ be as in Definition 3.2. Suppose also that
$\omega$ is $d_{A}$-closed. Then under a conformal rescaling of the metric g
on $M$, the operator $Q_{2}^{A}$ obeys the following transformation rule:

(25)
$\widehat{Q}^{A}_{2}\omega=Q^{A}_{2}\omega+2\delta_{A}d_{A}(\Upsilon\omega).$

Proposition 3.3 is straightforward to verify, and is a special case of Theorem
5.3 of [BG-2007]. Readers familiar with the Branson’s original $Q$-curvature
will recognise the similarity between (25) and the corresponding
transformation rule for the original $Q$-curvature. We will apply $Q^{A}_{2}$
to the curvature of a connection in Section 4, below. We will apply
Proposition 3.3 in sections 4 and 5. Finally in this Section, we present here
a result that we will need in those Sections.

######  Proposition 3.4.

The operator $Q_{2}^{A}$ is formally self-adjoint.

###### Proof.

For any $\eta_{ab},\,\omega_{ab}\in\Omega^{2}(W)$,

$P_{a}{}^{c}\eta^{ab}\omega_{cb}+P_{b}{}^{c}\eta^{ab}\omega_{ac}=P_{c}{}^{a}\eta^{cb}\omega_{ab}+P_{c}{}^{b}\eta^{ac}\omega_{ab}=P_{a}{}^{c}\eta_{cb}\omega^{ab}+P_{b}{}^{c}\eta_{ac}\omega^{ab},$

so the result follows from Definition 3.2. ∎

##  4\. The conformally invariant action in six dimensions

Let $M$ be a closed 6-manifold equipped with a conformal class c of pseudo-
Riemannian metrics and let notation be as in Section 3 above. So $A$ denotes a
connection on a vector bundle $V$. Note that

(26) $\left<F_{A},Q_{2}^{A}F_{A}\right>\in\Gamma(\mathcal{E}[-6]),$

and define the integration of densities as in Section 2.3. This allows us to
define an action on $A$ as follows.

######  Definition 4.1.

Suppose that $n=6$ and that $M$ is closed. Define an action $\mathcal{S}(A)$
by

(27) $\mathcal{S}(A):=\int_{M}\left<F_{A},Q_{2}^{A}F_{A}\right>d\mu.$

######  Proposition 4.2.

The action $\mathcal{S}(A)$ is invariant under conformal change of the metric
g on $M$.

###### Proof.

Note that $d_{A}F_{A}=0$, by the Bianchi identity. Thus

$\int_{M}\langle F_{A},\widehat{Q}_{2}^{A}F_{A}\rangle\,d\mu=\int_{M}\langle
F_{A},Q_{2}^{A}F_{A}+2\delta_{A}d_{A}(\Upsilon F_{A})\rangle\,d\mu$

by Proposition 3.3. But $M$ has empty boundary and $A$ preserves the pairing.
Thus

$\int_{M}\langle F_{A},2\delta_{A}d_{A}(\Upsilon
F_{A})\rangle\,d\mu=\int_{M}\langle d_{A}F_{A},2d_{A}(\Upsilon
F_{A})\rangle\,d\mu=0,$

and the proposition follows. ∎

###  4.1. Euler-Lagrange equations for $\mathcal{S}$

We will now derive the Euler-Lagrange equations for the conformally invariant
action $\mathcal{S}$. Here and below, $[\delta_{A}F_{A},F_{A}]$ will denote
$(\delta_{A}F_{A})F_{A}-F_{A}\delta_{A}F_{A}$. This notation includes an
implicit contraction of the free lowercase index of $\delta_{A}F_{A}$ with the
first lowercase index of the other $F_{A}$ within the brackets.

######  Proposition 4.3.

The Euler-Lagrange equations for the action $\mathcal{S}$ are

$\delta_{A}Q^{A}_{2}F_{A}-[\delta_{A}F_{A},F_{A}]=0.$

###### Proof.

For $a\in\Omega^{1}(adV)$ and $t\in\mathbb{R}$, let $A+ta$ be a path of
connections on $V$ starting at $A$. Then, since
$(F_{A+ta})_{ij}{}^{B}{}_{C}=(F_{A})_{ij}{}^{B}{}_{C}+t(d_{A}a)_{ij}{}^{B}{}_{C}+t^{2}(a\wedge
a)_{ij}{}^{B}{}_{C}$, it follows that

$\left.\frac{d}{dt}\mathcal{S}(A+ta)\,\right|_{t=0}=\int_{M}\left(\left<d_{A}a,Q^{A}_{2}F_{A}\right>+\left<F_{A},Q^{A}_{2}d_{A}a\right>+\langle
F_{A},\dot{Q}^{A}_{2}F_{A}\rangle\right)d\mu.$

Here $\dot{Q}_{2}^{A}$ denotes the linearisation of $Q_{2}^{A}$. Thus by
Proposition 3.4,

(28)
$\left.\frac{d}{dt}\mathcal{S}(A+ta)\,\right|_{t=0}=\int_{M}\left(2\langle
d_{A}a,Q_{2}^{A}F_{A}\rangle+\langle
F_{A},\dot{Q}_{2}^{A}F_{A}\rangle\right)d\mu.$

Consider the integral of $\langle F_{A},\dot{Q}_{2}^{A}F_{A}\rangle$. Only the
leading term of $Q_{2}^{A}$ will contribute to this integral. We find that

(29) $\frac{d}{dt}\left.\int_{M}\langle
F_{A},d_{A+ta}\delta_{A+ta}F_{A}\rangle\,d\mu\,\right|_{t=0}=\int_{M}2\langle\dot{\delta}_{A}F_{A},\delta_{A}F_{A}\rangle\,d\mu.$

By using (20) and (21), one can easily show that
$\delta_{A+ta}F_{A}=\delta_{A}F_{A}-t[a,F_{A}]$. Here the bracket includes an
implicit contraction between the lowercase index of $a$ and the first
lowercase index of $F_{A}$. Thus $\dot{\delta}_{A}F_{A}=-[a,F_{A}]$. From this
it follows that

(30) $\int_{M}\langle
F_{A},\dot{Q}_{2}^{A}F_{A}\rangle\,d\mu=-\int_{M}2\langle[a,F_{A}],\delta_{A}F_{A}\rangle\,d\mu.$

Now again let $\nabla$ denote the coupling of $A$ and the Levi-Civita
connection, and note that

(31)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&2(a_{c}{}^{G}{}_{H}F^{c}{}_{d}{}^{H}{}_{I}\nabla_{e}F^{edI}{}_{G}-F_{cd}{}^{G}{}_{H}a^{cH}{}_{I}\nabla_{e}F^{edI}{}_{G})=\vspace{0.8ex}\\\
&2(a_{c}{}^{G}{}_{H}F_{d}{}^{cH}{}_{I}(-1)\nabla_{e}F^{edI}{}_{G}-a^{cH}{}_{I}F_{dc}{}^{G}{}_{H}(-1)\nabla_{e}F^{edI}{}_{G})=\vspace{0.8ex}\\\
&-2(-a_{c}{}^{G}{}_{H}F_{d}{}^{cH}{}_{I}(\delta_{A}F)^{dI}{}_{G}+a^{cH}{}_{I}(\delta_{A}F)^{dI}{}_{G}F_{dc}{}^{G}{}_{H})=-2\langle
a,[\delta_{A}F_{A},F_{A}]\rangle.\end{array}$

In most of the expressions in (31), we have omitted the subscript $A$ from
$F$, but we have included the indices of $F_{A}$. From (28), (30), and (31),
it follows that

$\begin{array}[]{ll}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\frac{d}{dt}S(A+ta)\,\bigg{|}_{t=0}=\int_{M}\left(2\langle
d_{A}a,Q^{A}_{2}F_{A}\rangle-2\langle
<EMAIL_ADDRESS>&\displaystyle\int_{M}2\left\langle
a,\left(\delta_{A}Q^{A}_{2}F_{A}-[\delta_{A}F_{A},F_{A}]\right)\right\rangle
d\mu,\end{array}$

and the proposition follows. ∎

##  5\. A higher-order analogue of the Yang-Mills equations

We continue the setting and notation of the previous section.

###  5.1. Definition of $\mathfrak{D}$

The results in the previous section motivate the following definition:

######  Definition 5.1.

Define an operator
$\mathfrak{D}:\mathcal{A}\rightarrow\Omega^{1}(\operatorname{End}(V))$ as
follows:

$\mathfrak{D}A:=\delta_{A}Q_{2}^{A}F_{A}-[\delta_{A}F_{A},F_{A}].$

By comparing Proposition 4.3 to the usual derivation of the Yang-Mills
equations as the Euler-Lagrange equations for the functional $||F_{A}||^{2}$,
one can view the equation $\mathfrak{D}A=0$ as a conformally invariant
analogue, for dimension 6, of the usual source-free Yang-Mills equations in
four dimensions.

###  5.2. Conformal invariance of $\mathfrak{D}$

The fact that the action $\mathcal{S}(A)$ is conformally invariant suggests
that the Euler-Lagrange equations $\mathfrak{D}A=0$ are also conformally
invariant. This is verified explicitly in the following proposition:

######  Proposition 5.2.

On psuedo-Riemannian 6-manifolds, the operator $\mathfrak{D}$ is invariant
under conformal change of the metric g on $M$.

###### Proof.

In this proof, we often omit indices. Note first that $F_{A}$ and
$Q_{2}^{A}F_{A}$ carry weights $0$ and $-2$, respectively. Thus

$\widehat{\delta}_{A}\widehat{Q}_{2}^{A}F_{A}=\widehat{\delta}_{A}(Q_{2}^{A}F_{A}+2\delta_{A}d_{A}(\Upsilon
F_{A}))=\delta_{A}Q_{2}^{A}F_{A}+2\delta_{A}^{2}d_{A}(\Upsilon F_{A}),$

by (19) and (25) since $n=6$. Similarly,

(32)
$-[\widehat{\delta}_{A}F_{A},F_{A}]=-[\delta_{A}F_{A},F_{A}]+2[\iota(d\Upsilon)F_{A},F_{A}],$

where $\iota(d\Upsilon)$ denotes interior multiplication by $\Upsilon^{a}$. In
the second term on the right-hand side of (32), the bracket notation includes
a contraction of the free lowercase index of $\iota(d\Upsilon)F_{A}$ with the
first lowercase index of the second $F_{A}$ within the brackets. To complete
the proof, it suffices to show that

(33) $\delta^{2}_{A}d_{A}(\Upsilon F_{A})+[\iota(d\Upsilon)F_{A},F_{A}]=0.$

To do this, we begin by recalling that $d_{A}F_{A}=0$. From this it follows
that

(34) $(\delta^{2}_{A}d_{A}(\Upsilon
F))_{c}=2\nabla^{[e}\nabla^{d]}(\Upsilon_{d}F_{ec})+\nabla^{[e}\nabla^{d]}(\Upsilon_{c}F_{de}).$

Here and for the rest of this proof, we omit the subscript $A$ from $F$.
Again, $\nabla$ denotes the coupling of $A$ and the Levi-Civita connection. By
(8), (22), and (23), the right-hand side of (34) is equal to

(35)
$\begin{array}[]{l}R_{ed}{}^{d}{}_{j}\Upsilon^{j}F^{e}{}_{c}{}^{G}{}_{H}+R_{ed}{}^{e}{}_{j}\Upsilon^{d}F^{j}{}_{c}{}^{G}{}_{H}-R_{ed}{}^{j}{}_{c}\Upsilon^{d}F^{e}{}_{j}{}^{G}{}_{H}\vspace{0.5ex}\\\
-\frac{1}{2}R_{ed}{}^{j}{}_{c}\Upsilon_{j}F^{deG}{}_{H}+\frac{1}{2}R_{ed}{}^{d}{}_{j}\Upsilon_{c}F^{jeG}{}_{H}+\frac{1}{2}R_{ed}{}^{e}{}_{j}\Upsilon_{c}F^{djG}{}_{H}\vspace{0.5ex}\\\
+F_{ed}{}^{G}{}_{I}\Upsilon^{d}F^{e}{}_{c}{}^{I}{}_{H}-F_{ed}{}^{I}{}_{H}\Upsilon^{d}F^{e}{}_{c}{}^{G}{}_{I}+\frac{1}{2}F_{ed}{}^{G}{}_{I}\Upsilon_{c}F^{deI}{}_{H}-\frac{1}{2}F_{ed}{}^{I}{}_{H}\Upsilon_{c}F^{deG}{}_{I}.\end{array}$

Here we have included the indices associated to $V$ and $V^{*}$. The sum of
the first six terms of (35) is zero, and the sum of the last two terms of (35)
is also zero. Thus

$(\delta_{A}^{2}d_{A}(\Upsilon
F))_{c}{}^{G}{}_{H}=-(\iota(d\Upsilon)F)_{e}{}^{G}{}_{I}F^{e}{}_{c}{}^{I}{}_{H}+F^{e}{}_{c}{}^{G}{}_{I}\,(\iota(d\Upsilon)F)_{e}{}^{I}{}_{H}=-[\iota(d\Upsilon)F,F]_{c}{}^{G}{}_{H}.$

This establishes (33). ∎

##  6\. Applications

In this section we specialise to the case that $A$ is the conformal tractor
connection of [BEG]. As usual we work on a closed conformal 6-manifold
$(M,\mbox{c})$. In this case, in terms of a metric $\mbox{{g}}\in\mbox{c}$,
our Lagrangian density (26) is

(36)
$\left<F_{A},Q_{2}^{A}F_{A}\right>=4A_{abc}\nabla^{b}P^{ac}-JC_{abcd}C^{abcd}+4C_{abcd}\nabla^{d}\nabla^{b}P^{ac}+4P_{ab}C^{a}{}_{cde}C^{bcde}.$

If we work modulo divergences, we may simplify the right-hand side of (36)
further. (By divergences, we mean terms of the form $\nabla_{i}T^{i}$, where
$T^{i}\in\Gamma(TM\otimes\mathcal{E}[-6])$.) We find that

$\mathcal{S}{(A)}=\int_{M}(8A_{abc}\nabla^{c}P^{ab}-JC_{abcd}C^{abcd}+4P_{ab}C^{a}{}_{cde}C^{bcde})\,d\mu.$

For this action we will compute the Euler-Lagrange equations of Proposition
4.3. The main result is Theorem 6.1 below which shows that these equations
recover the condition of $\mathcal{O}_{ab}^{6}=0$, where
$\mathcal{O}_{ab}^{6}$ is the Fefferman-Graham obstruction tensor of [FG-2012,
FG-1985] in dimension $6$. Thus Proposition 4.3, and the operator of
Definition 5.1 so defined, provide a new perspective and way of constructing a
symbolic formula for the Fefferman-Graham obstruction tensor
$\mathcal{O}_{ab}^{6}$. This symbolic formula will express
$\mathcal{O}_{ab}^{6}$ in terms of the Levi-Civita connection and tensors
associated to the Riemannian curvature tensor.

###  6.1. The standard tractor bundle

We begin by reviewing some basic facts concerning the standard tractor bundle
and the conformal tractor connection. We are interested in pseudo-Riemannian
metrics in dimension $n=6$, but our discussion of tractors in this subsection
treats pseudo-Riemannian metrics in general dimensions $n$. We refer the
reader to [BEG] and [GP-2003] for further information.

Let a manifold $M$ of dimension $n\geq 3$ and a conformal class c of pseudo-
Riemannian metrics on $M$ be given. Let $(p,q)$ denote the signature of the
metrics in c. The conformal class c determines a certain $(n+2)$-dimensional
vector bundle over $M$ commonly known as the (standard) tractor bundle. We let
$\mathcal{T}$ (or $\mathcal{T}^{B}$ in abstract index notation) denote this
bundle. The conformal class also determines a connection on $\mathcal{T}$
which we call the tractor connection. Let $\nabla^{\mathcal{T}}$ or simply
$\nabla$ denote this connection. The symbol $\nabla$ will also denote the
coupled Levi-Civita tractor connection. This connection acts on powers of
$\mathcal{T}$ in the obvious way.

A choice of metric $\mbox{{g}}\in\mbox{c}$ determines a vector bundle
isomorphism

$I_{\mbox{{\tiny
g}}}:\mathcal{E}[1]\oplus(T^{(0,1)}\otimes\mathcal{E}[1])\oplus\mathcal{E}[-1]\rightarrow\mathcal{T}^{B}.$

The chosen metric g also determines algebraic splitting operators

$Y^{B}\in\Gamma(\mathcal{T}^{B}\otimes\mathcal{E}[-1]),\hskip
4.30554ptZ^{Bc}\in\Gamma(\mathcal{T}^{B}\otimes
T^{(1,0)}\otimes\mathcal{E}[-1]),\hskip
4.30554ptX^{B}\in\Gamma(\mathcal{T}^{B}\otimes\mathcal{E}[1])$

such that for all
$(\sigma,\mu_{c},\rho)\in\mathcal{E}[1]\oplus(T^{(0,1)}\otimes\mathcal{E}[1])\otimes\mathcal{E}[-1]$,

$I_{\mbox{{\tiny g}}}(\sigma,\mu_{c},\rho)=\sigma Y^{B}+\mu_{c}Z^{Bc}+\rho
X^{B}.$

The splitting operator $X^{B}$ is conformally invariant. When the coupled
Levi-Civita tractor connection acts on the three splitting operators, it obeys
the following rules:

(37) $\nabla_{a}Y^{B}=P_{ac}Z^{Bc},\hskip
12.91663pt\nabla_{a}Z^{B}{}_{c}=-P_{ac}X^{B}-\textbf{\mbox{{g}}}_{ac}Y^{B},\hskip
12.91663pt\nabla_{a}X^{B}=Z^{B}{}_{a}.$

The conformal structure c also determines a metric $h$ on $\mathcal{T}^{B}$
which has signature $(p+1,q+1)$. We say that $h$ is the tractor metric. The
tractor connection $\nabla$ preserves $h$. We may use $h$ to raise and lower
tractor indices, and this operation commutes with the action of $\nabla$. One
can show that $Y_{B}X^{B}=1$ and $Z_{Ba}Z^{B}{}_{c}=\textbf{\mbox{{g}}}_{ac}$.
All other contractions of tractor indices involving pairs of the splitting
operators $Y^{B}$, $Z^{Bc}$, and $X^{B}$ are zero.

Let $\Omega_{ab}{}^{D}{}_{E}$ denote the curvature of $\nabla^{\mathcal{T}}$.
Thus
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})V^{D}=\Omega_{ab}{}^{D}{}_{E}V^{E}$
for all $V^{D}\in\mathcal{T}$. We say that $\Omega_{ab}{}^{D}{}_{E}$ is the
tractor curvature. One can show that

(38)
$\Omega_{ab}{}^{D}{}_{E}=\tensor{Z}{{}^{D}}^{c}\tensor{Z}{{}_{E}}^{e}C_{abce}-X^{D}Z_{E}{}^{e}\tensor{A}{{}_{eab}}+X_{E}Z^{D}{}^{e}\tensor{A}{{}_{eab}}\,,$

and

(39)
$\nabla^{a}\Omega_{ac}{}^{D}{}_{E}=(n-4)Z^{Dd}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{cde}}-\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{B}{{}_{ec}}+\tensor{X}{{}_{E}}Z^{De}\tensor{B}{{}_{ec}}.$

Observe that, according to (19), in dimension $n=4$ the operator
$\delta_{\nabla^{\mathcal{T}}}$ is conformally invariant on weight 0 tractor-
bundle-valued 2-forms. So $\nabla^{a}\Omega_{ac}{}^{D}{}_{E}$ is conformally
invariant in dimension four. On the other hand, in any dimension

(40) $X_{E}Z^{Dd}-X^{D}Z_{E}{}^{d}$

is conformally invariant, by the conformal transformation rules for $Z^{Dd}$
and $Z_{E}{}^{d}$ given in equation (4) of [GP-2003]. Thus (40) provides, up
to a constant factor, the standard and well-known conformally invariant
injection of 1-forms into the adjoint tractor bundle $\Lambda^{2}\mathcal{T}$.
Thus in dimension four, the display (39) is recovering the result (5)
mentioned in the Introduction.

###  6.2. The obstruction tensor in dimension 6

In this subsection, we will describe symbolic computations which establish the
following theorem:

######  Theorem 6.1.

Suppose that $n=6$. Then

$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}=(X_{E}Z^{Dd}-X^{D}Z_{E}{}^{d})16\mathcal{O}_{cd}^{6}\,,$

and hence

(41)
$\mathcal{O}_{ce}^{6}=\frac{1}{32}(Y^{E}Z_{De}-Y_{D}Z^{E}{}_{e})(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}.$

Here and throughout our work, our scaling convention for
$\mathcal{O}_{ab}^{6}$ is the same as in [GP-2006]. One can expand the right-
hand side of (41) and compute a symbolic formula which expresses
$\mathcal{O}_{ce}^{6}$ in terms of the Weyl tensor, the Schouten tensor, and
$J$.

###### Proof of Theorem 6.1.

We begin by applying Definition 5.1. In this definition, we replace $A$ with
$\nabla^{\mathcal{T}}$ and $F$ with the tractor curvature $\Omega$. We work in
dimension 6. We conclude that

(42)
$\begin{array}[]{ll}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}=(\delta
<EMAIL_ADDRESS>&-(\delta\Omega)_{b}{}^{D}{}_{G}\Omega^{b}{}_{c}{}^{G}{}_{E}+\Omega^{b}{}_{c}{}^{D}{}_{G}(\delta\Omega)_{b}{}^{G}{}_{E}.\end{array}$

In (42) and for the rest of this subsection, we omit the subscript
$\nabla^{\mathcal{T}}$ from $\delta$ and $d$. Our plan will be to consider the
various summands on the right-hand side of (42) separately.

Consider the first summand, namely $(\delta d\delta\Omega)_{c}{}^{D}{}_{E}$.
From (15), (20), and (39), it follows that

(43)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&-4\nabla_{a_{1}}(\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{a_{2}de}})+2\nabla_{a_{1}}(\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{B}{{}_{ea_{2}}})-2\nabla_{a_{1}}(\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{e}}\tensor{B}{{}_{ea_{2}}}).\end{array}$

On the right-hand side of this display, we antisymmetrise the indices $a_{1}$
and $a_{2}$. By (37) and (43),

(44)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&4\tensor{P}{{}_{a_{1}}^{d}}\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{a_{2}de}}+4\tensor{Y}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{a_{2}a_{1}e}}+4\tensor{Z}{{}^{Dd}}\tensor{P}{{}_{a_{1}}^{e}}\tensor{X}{{}_{E}}\tensor{A}{{}_{a_{2}de}\vspace{0.5ex}}\\\
&+4\tensor{Z}{{}^{Dd}}\tensor{Y}{{}_{E}}\tensor{A}{{}_{a_{2}da_{1}}}-4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\nabla_{a_{1}}\tensor{A}{{}_{a_{2}de}}+2Z^{D}{}_{a_{1}}\tensor{Z}{{}_{E}^{e}}\tensor{B}{{}_{ea_{2}}\vspace{0.5ex}}\\\
&+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}-2\tensor{Z}{{}_{Ea_{1}}}\tensor{Z}{{}^{De}}\tensor{B}{{}_{ea_{2}}}-2\tensor{X}{{}_{E}}\tensor{Z}{{}^{De}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}.\end{array}$

Here we again antisymmetrise the indices $a_{1}$ and $a_{2}$. From (20), (37),
(44), and the fact that $A_{abc}$ is trace-free, it follows that $(\delta
d\delta\Omega)_{a_{2}}{}^{D}{}_{E}$ is given by the formula in Figure 1.

$\displaystyle
4\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}_{a_{1}}^{d}}\tensor{P}{{}^{a_{1}e}}\tensor{A}{{}_{a_{2}de}}+4\tensor{X}{{}^{D}}\tensor{Y}{{}_{E}}\tensor{P}{{}_{a_{1}}^{d}}\tensor{A}{{}_{a_{2}d}^{a_{1}}}-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{a_{1}}^{d}}\nabla^{a_{1}}\tensor{A}{{}_{a_{2}de}}$
$\displaystyle-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}(\nabla^{a_{1}}\tensor{P}{{}_{a_{1}}^{d}})\tensor{A}{{}_{a_{2}de}}-4\tensor{Z}{{}^{Da_{1}}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{a_{1}}^{d}}\tensor{A}{{}_{a_{2}de}}+4\tensor{Y}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}^{a_{1}e}}\tensor{A}{{}_{a_{2}a_{1}e}}$
$\displaystyle-4\tensor{Y}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\nabla^{a_{1}}\tensor{A}{{}_{a_{2}a_{1}e}}-4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}^{a_{1}}_{d}}\tensor{A}{{}_{a_{2}a_{1}e}}+4\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}^{a_{1}d}}\tensor{P}{{}_{a_{1}}^{e}}\tensor{A}{{}_{a_{2}de}}$
$\displaystyle+4\tensor{Y}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}_{a_{1}}^{e}}A_{a_{2}}{}^{a_{1}}{}_{e}-4\tensor{Z}{{}^{Dd}}\tensor{X}{{}_{E}}\tensor{P}{{}_{a_{1}}^{e}}\nabla^{a_{1}}\tensor{A}{{}_{a_{2}de}}-4\tensor{Z}{{}^{Dd}}\tensor{X}{{}_{E}}(\nabla^{a_{1}}\tensor{P}{{}_{a_{1}}^{e}})\tensor{A}{{}_{a_{2}de}}$
$\displaystyle-4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{a_{1}}}\tensor{P}{{}_{a_{1}}^{e}}\tensor{A}{{}_{a_{2}de}}+4\tensor{X}{{}^{D}}\tensor{Y}{{}_{E}}\tensor{P}{{}^{a_{1}d}}\tensor{A}{{}_{a_{2}da_{1}}}-4\tensor{Z}{{}^{Dd}}\tensor{Y}{{}_{E}}\nabla^{a_{1}}\tensor{A}{{}_{a_{2}da_{1}}}$
$\displaystyle-4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}^{a_{1}}_{e}}\tensor{A}{{}_{a_{2}da_{1}}}-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}^{a_{1}d}}\nabla_{a_{1}}\tensor{A}{{}_{a_{2}de}}-4\tensor{Y}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\nabla_{a_{1}}A_{a_{2}}{}^{a_{1}}{}_{e}$
$\displaystyle-4\tensor{Z}{{}^{Dd}}\tensor{X}{{}_{E}}\tensor{P}{{}^{a_{1}e}}\nabla_{a_{1}}\tensor{A}{{}_{a_{2}de}}-4\tensor{Z}{{}^{Dd}}\tensor{Y}{{}_{E}}\nabla_{a_{1}}\tensor{A}{{}_{a_{2}d}^{a_{1}}}+4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\nabla^{a_{1}}\nabla_{a_{1}}\tensor{A}{{}_{a_{2}de}}$
$\displaystyle+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}^{a_{1}}_{a_{1}}}\tensor{B}{{}_{ea_{2}}}+2\tensor{Y}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{\mbox{{g}}}{{}^{a_{1}}_{a_{1}}}\tensor{B}{{}_{ea_{2}}}+2\tensor{Z}{{}^{D}_{a_{1}}}\tensor{X}{{}_{E}}\tensor{P}{{}^{a_{1}e}}\tensor{B}{{}_{ea_{2}}}+2\tensor{Z}{{}^{D}_{a_{1}}}\tensor{Y}{{}_{E}}\tensor{B}{{}^{a_{1}}_{a_{2}}}$
$\displaystyle-2\tensor{Z}{{}^{D}_{a_{1}}}\tensor{Z}{{}_{E}^{e}}\nabla^{a_{1}}\tensor{B}{{}_{ea_{2}}}+2\tensor{X}{{}^{D}}\tensor{Y}{{}_{E}}\nabla_{a_{1}}B^{a_{1}}{}_{a_{2}}-2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\nabla^{a_{1}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}$
$\displaystyle-2\tensor{Z}{{}^{Da_{1}}}\tensor{Z}{{}_{E}^{e}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}-2\tensor{Z}{{}_{Ea_{1}}}\tensor{X}{{}^{D}}\tensor{P}{{}^{a_{1}e}}\tensor{B}{{}_{ea_{2}}}-2\tensor{Z}{{}_{Ea_{1}}}\tensor{Y}{{}^{D}}\tensor{B}{{}^{a_{1}}_{a_{2}}}-2\tensor{X}{{}_{E}}\tensor{Z}{{}^{De}}\tensor{P}{{}^{a_{1}}_{a_{1}}}\tensor{B}{{}_{ea_{2}}}$
$\displaystyle-2\tensor{Y}{{}_{E}}\tensor{Z}{{}^{De}}\tensor{\mbox{{g}}}{{}^{a_{1}}_{a_{1}}}\tensor{B}{{}_{ea_{2}}}+2\tensor{Z}{{}_{Ea_{1}}}\tensor{Z}{{}^{D}{}^{e}}\nabla^{a_{1}}\tensor{B}{{}_{ea_{2}}}-2\tensor{X}{{}_{E}}\tensor{Y}{{}^{D}}\nabla_{a_{1}}B^{a_{1}}{}_{a_{2}}$
$\displaystyle+2\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{e}}\nabla^{a_{1}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}+2\tensor{Z}{{}_{E}^{a_{1}}}\tensor{Z}{{}^{De}}\nabla_{a_{1}}\tensor{B}{{}_{ea_{2}}}.$
Figure 1. Symbolic formula for $(\delta d\delta\Omega)_{a_{2}}{}^{D}{}_{E}$ in
dimension 6

In this formula, we antisymmetrise the lower indices $a_{1}$ and $a_{2}$. The
upper index $a_{1}$ does not participate in the antisymmetrisation. The index
$a_{2}$ is a free index. One may replace this index with the index $c$ to
conform with (42).

Now consider the second summand on the right-hand side of (42), namely
$-4(\delta(P\\#\Omega))_{c}{}^{D}{}_{E}$. By (38),

$\displaystyle-4(P\\#\Omega)_{bc}{}^{D}{}_{E}=\,$
$\displaystyle-4\tensor{P}{{}_{b}^{k}}\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}_{kcde}}+4\tensor{P}{{}_{b}^{k}}\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{ekc}}-4\tensor{P}{{}_{b}^{k}}\tensor{X}{{}_{E}}\tensor{Z}{{}^{Dd}}\tensor{A}{{}_{dkc}}$
$\displaystyle-4\tensor{P}{{}_{c}^{k}}\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}_{bkde}}+4\tensor{P}{{}_{c}^{k}}\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{ebk}}-4\tensor{P}{{}_{c}^{k}}\tensor{X}{{}_{E}}\tensor{Z}{{}^{Dd}}\tensor{A}{{}_{dbk}}.$

From this and from (20), it follows that

$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&4\nabla^{b}(\tensor{P}{{}_{b}^{k}}\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}_{kcde}})-4\nabla^{b}(\tensor{P}{{}_{b}^{k}}\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{ekc}})+4\nabla^{b}(\tensor{P}{{}_{b}^{k}}\tensor{X}{{}_{E}}\tensor{Z}{{}^{Dd}}\tensor{A}{{}_{dkc}})\vspace{0.5ex}\\\
&+4\nabla^{b}(\tensor{P}{{}_{c}^{k}}\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}_{bkde}})-4\nabla^{b}(\tensor{P}{{}_{c}^{k}}\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{ebk}})+4\nabla^{b}(\tensor{P}{{}_{c}^{k}}\tensor{X}{{}_{E}}\tensor{Z}{{}^{Dd}}\tensor{A}{{}_{dbk}}).\end{array}$

Thus from (37) and the fact that the Cotton tensor and the Weyl tensor are
trace-free, it follows that $-4(\delta(P\\#\Omega))_{c}{}^{D}{}_{E}$ is given
by the formula in Figure 2.

$\displaystyle-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{b}^{k}}\tensor{P}{{}^{bd}}\tensor{C}{{}_{kcde}}-4\tensor{Y}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}^{d}{}^{k}}\tensor{C}{{}_{kcde}}-4\tensor{Z}{{}^{D}{}^{d}}\tensor{X}{{}_{E}}\tensor{P}{{}_{b}^{k}}\tensor{P}{{}^{be}}\tensor{C}{{}_{kcde}}$
$\displaystyle-4\tensor{Z}{{}^{D}{}^{d}}\tensor{Y}{{}_{E}}\tensor{P}{{}^{e}{}^{k}}\tensor{C}{{}_{kcde}}+4\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}(\nabla^{b}\tensor{P}{{}_{b}^{k}})\tensor{C}{{}_{kcde}}+4\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{b}^{k}}\nabla^{b}\tensor{C}{{}_{kcde}}$
$\displaystyle+4\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}_{b}^{k}}\tensor{P}{{}^{be}}\tensor{A}{{}_{ekc}}+4\tensor{X}{{}^{D}}\tensor{Y}{{}_{E}}\tensor{P}{{}^{e}{}^{k}}\tensor{A}{{}_{ekc}}-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{b}^{k}}\nabla^{b}\tensor{A}{{}_{ekc}}$
$\displaystyle-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}(\nabla^{b}\tensor{P}{{}_{b}^{k}})\tensor{A}{{}_{ekc}}-4\tensor{Z}{{}^{D}{}^{b}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{b}^{k}}\tensor{A}{{}_{ekc}}-4\tensor{X}{{}_{E}}\tensor{X}{{}^{D}}\tensor{P}{{}_{b}^{k}}\tensor{P}{{}^{bd}}\tensor{A}{{}_{dkc}}$
$\displaystyle-4\tensor{X}{{}_{E}}\tensor{Y}{{}^{D}}\tensor{P}{{}^{d}{}^{k}}\tensor{A}{{}_{dkc}}+4\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{d}}\tensor{P}{{}_{b}^{k}}\nabla^{b}\tensor{A}{{}_{dkc}}+4\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{d}}(\nabla^{b}\tensor{P}{{}_{b}^{k}})\tensor{A}{{}_{dkc}}$
$\displaystyle+4\tensor{Z}{{}_{E}^{b}}\tensor{Z}{{}^{D}{}^{d}}\tensor{P}{{}_{b}^{k}}\tensor{A}{{}_{dkc}}-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{c}^{k}}\tensor{P}{{}^{bd}}\tensor{C}{{}_{bkde}}-4\tensor{Z}{{}^{D}{}^{d}}\tensor{X}{{}_{E}}\tensor{P}{{}_{c}^{k}}\tensor{P}{{}^{be}}\tensor{C}{{}_{bkde}}$
$\displaystyle+4\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}(\nabla^{b}\tensor{P}{{}_{c}^{k}})\tensor{C}{{}_{bkde}}+4\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{c}^{k}}\nabla^{b}\tensor{C}{{}_{bkde}}+4\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{P}{{}_{c}^{k}}\tensor{P}{{}^{be}}\tensor{A}{{}_{ebk}}$
$\displaystyle-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{c}^{k}}\nabla^{b}\tensor{A}{{}_{ebk}}-4\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}(\nabla^{b}\tensor{P}{{}_{c}^{k}})\tensor{A}{{}_{ebk}}-4\tensor{Z}{{}^{D}{}^{b}}\tensor{Z}{{}_{E}^{e}}\tensor{P}{{}_{c}^{k}}\tensor{A}{{}_{ebk}}$
$\displaystyle-4\tensor{X}{{}_{E}}\tensor{X}{{}^{D}}\tensor{P}{{}_{c}^{k}}\tensor{P}{{}^{bd}}\tensor{A}{{}_{dbk}}+4\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{d}}\tensor{P}{{}_{c}^{k}}\nabla^{b}\tensor{A}{{}_{dbk}}+4\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{d}}(\nabla^{b}\tensor{P}{{}_{c}^{k}})\tensor{A}{{}_{dbk}}$
$\displaystyle+4\tensor{Z}{{}_{E}^{b}}\tensor{Z}{{}^{D}{}^{d}}\tensor{P}{{}_{c}^{k}}\tensor{A}{{}_{dbk}}.$
Figure 2. Symbolic formula for $-4(\delta(P\\#\Omega))_{c}{}^{D}{}_{E}$ in
dimension 6

We now consider the third summand on the right-hand side of (42), namely
$2(\delta(J\Omega))_{c}{}^{D}{}_{E}$. By (20) and (38),

$\begin{array}[]{ll}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
<EMAIL_ADDRESS>&-2\nabla^{a}(J\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}_{acde}}-J\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{eac}}+J\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{e}}\tensor{A}{{}_{eac}}).\end{array}$

From (37) and from the fact that the Cotton and Weyl tensors are trace-free,
it follows that

(45) $\begin{array}[]{ll}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
<EMAIL_ADDRESS>&-2\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}(\nabla^{a}J)\tensor{C}{{}_{acde}}+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}J\tensor{P}{{}^{ad}}\tensor{C}{{}_{acde}}-2\tensor{Z}{{}^{D}{}^{d}}\tensor{Z}{{}_{E}^{e}}J\nabla^{a}\tensor{C}{{}_{acde}\vspace{0.5ex}}\\\
&+2\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{d}}J\tensor{P}{{}^{ae}}\tensor{C}{{}_{acde}}+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}(\nabla^{a}J)\tensor{A}{{}_{eac}}-2\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{e}}(\nabla^{a}J)\tensor{A}{{}_{eac}\vspace{0.5ex}}\\\
&+2\tensor{Z}{{}^{D}{}^{a}}\tensor{Z}{{}_{E}^{e}}J\tensor{A}{{}_{eac}}-2\tensor{Z}{{}_{E}^{a}}\tensor{Z}{{}^{D}{}^{e}}J\tensor{A}{{}_{eac}}+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}J\nabla^{a}\tensor{A}{{}_{eac}\vspace{0.5ex}}\\\
&-2\tensor{X}{{}_{E}}\tensor{Z}{{}^{D}{}^{e}}J\nabla^{a}\tensor{A}{{}_{eac}}.\end{array}$

The fourth summand is
$-(\delta\Omega)_{b}{}^{D}{}_{G}\Omega^{b}{}_{c}{}^{G}{}_{E}$. By (20), (38)
and (39),

$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&(2\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{G}^{k}}\tensor{A}{{}_{bdk}}-\tensor{X}{{}^{D}}\tensor{Z}{{}_{G}^{k}}\tensor{B}{{}_{kb}}+\tensor{X}{{}_{G}}\tensor{Z}{{}^{Dk}}\tensor{B}{{}_{kb}})\times\vspace{0.5ex}\\\
&(\tensor{Z}{{}^{Gi}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}^{b}_{cie}}-\tensor{X}{{}^{G}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{e}^{b}{}_{c}}+\tensor{X}{{}_{E}}\tensor{Z}{{}^{Ge}}\tensor{A}{{}_{e}^{b}{}_{c}}).\end{array}$

From this and from the rules for the tractor metric as applied to the
splitting operators $Y^{B}$, $Z^{Bc}$, and $X^{B}$, it follows that

(46)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&2\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{bd}^{i}}\tensor{C}{{}^{b}_{cie}}+2\tensor{Z}{{}^{Dd}}\tensor{X}{{}_{E}}\tensor{A}{{}_{bd}^{e}}\tensor{A}{{}_{e}^{b}{}_{c}}-\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{B}{{}^{i}_{b}}\tensor{C}{{}^{b}_{cie}}\\\
&-\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{B}{{}^{e}_{b}}\tensor{A}{{}_{e}^{b}{}_{c}}.\end{array}$

The fifth summand is
$\Omega^{b}{}_{c}{}^{D}{}_{G}(\delta\Omega)_{b}{}^{G}{}_{E}$. By (20), (38)
and (39),

$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&(\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{G}^{k}}\tensor{C}{{}^{b}_{cdk}}-\tensor{X}{{}^{D}}\tensor{Z}{{}_{G}^{k}}\tensor{A}{{}_{k}^{b}{}_{c}}+\tensor{X}{{}_{G}}\tensor{Z}{{}^{Dk}}\tensor{A}{{}_{k}^{b}{}_{c}}\vspace{0.5ex})\times\\\
&(-2\tensor{Z}{{}^{Gi}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}_{bie}}+\tensor{X}{{}^{G}}\tensor{Z}{{}_{E}^{e}}\tensor{B}{{}_{eb}}-\tensor{X}{{}_{E}}\tensor{Z}{{}^{Ge}}\tensor{B}{{}_{eb}}).\end{array}$

Thus

(47)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&-2\tensor{Z}{{}^{Dd}}\tensor{Z}{{}_{E}^{e}}\tensor{C}{{}^{b}_{cd}{}^{i}}\tensor{A}{{}_{bie}}-\tensor{Z}{{}^{Dd}}\tensor{X}{{}_{E}}\tensor{C}{{}^{b}_{cd}{}^{e}}\tensor{B}{{}_{eb}}+2\tensor{X}{{}^{D}}\tensor{Z}{{}_{E}^{e}}\tensor{A}{{}^{ib}_{c}}\tensor{A}{{}_{bie}\vspace{0.5ex}}\\\
&+\tensor{X}{{}^{D}}\tensor{X}{{}_{E}}\tensor{A}{{}^{eb}_{c}}\tensor{B}{{}_{eb}}.\end{array}$

Our next step will be to combine (implicitly) several of the symbolic formulae
that we computed above. Note first, however, that in the symbolic formula
appearing in Figure 1, the index $a_{2}$ is a free index. After we perform the
antisymmetrisation of the lower indices $a_{1}$ and $a_{2}$, we may replace
the index $a_{2}$ with the index $c$. We can then combine the symbolic
formulae appearing in figures 1 and 2 together with the symbolic formulae
appearing on the right-hand sides of (45), (46), and (47). The result will be
a symbolic formula for $(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$,
which we will refer to as the “total” symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$. As we will now show, one
can simplify this total symbolic formula and obtain a new total symbolic
formula for $(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
every term contains an occurrence of either $X_{E}Z^{Dd}$ or
$X^{D}Z_{E}{}^{d}$.

Note first that none of the terms in the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ contain an occurrence of
$Y_{E}Y^{D}$.

Now consider the terms in the total symbolic formula that contain $X^{D}Y_{E}$
or $X_{E}Y^{D}$. The sum of these terms is

(48)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&-2P^{a_{1}d}A_{a_{1}dc}+\nabla_{a_{1}}B^{a_{1}}{}_{c}-\nabla_{c}B^{a_{1}}{}_{a_{1}}+4P^{ek}A_{ekc}).\end{array}$

This follows by inspection and the fact that $A_{abc}$ is antisymmetric in $b$
and $c$. But $P_{ab}$ is symmetric, and $A_{abc}$ and $B_{ab}$ are trace-free.
Thus (48) is equal to

$(X^{D}Y_{E}-X_{E}Y^{D})(\nabla_{a}B^{a}{}_{c}+2P^{ek}A_{ekc}).$

A short symbolic computation using [W] and [L], along with (3), (9), (11),
(12), and (13), shows that $\nabla_{a}B^{a}{}_{c}+2P^{ek}A_{ekc}=0$ in
dimension 6. Thus (48) is zero.

Next, consider the terms in the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which $X^{D}X_{E}$
appears. The sum of these terms is

(49) $\begin{array}[]{ll}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle
<EMAIL_ADDRESS>&+4P_{b}{}^{k}P^{be}A_{ekc}-4P_{b}{}^{k}P^{bd}A_{dkc}+4P_{c}{}^{k}P^{be}A_{ebk}-4P_{c}{}^{k}P^{bd}A_{dbk}-B^{e}{}_{b}A_{e}{}^{b}{}_{c}\vspace{0.5ex}\\\
&+A^{eb}{}_{c}B_{eb}).\end{array}$

Note that $P_{a_{1}}{}^{d}P^{a_{1}e}$ is symmetric in $d$ and $e$ and that
$A_{cde}$ is antisymmetric in $d$ and $e$. Thus in the parenthesised
expression in (49), terms 1 and 3 are zero. Since $A_{a_{1}de}$ is
antisymmetric in $d$ and $e$, it follows that terms 2 and 4 cancel. Terms 5
and 6 cancel, terms 7 and 8 cancel, and terms 9 and 10 cancel. Thus (49) is
zero.

We now consider the terms in the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
$Y^{D}Z_{E}{}^{i}$, $Y_{E}Z^{Di}$, $Y^{D}Z_{Ei}$, or $Y_{E}Z^{D}{}_{i}$
appears. Here $i$ is a dummy index. The total symbolic formula may use some
index other than $i$ here. We will also consider the terms of the total
symbolic formula in which $Y^{D}Z_{Ec}$ or $Y_{E}Z^{D}{}_{c}$ appears. The sum
of the terms under consideration is as follows:

(50)
$\begin{array}[]{l}(Y^{D}Z_{E}{}^{i}-Y_{E}Z^{Di})(-2\nabla^{a_{1}}A_{ca_{1}i}+2\nabla^{a_{1}}A_{a_{1}ci}-2\nabla_{a_{1}}A_{c}{}^{a_{1}}{}_{i}+2\nabla_{c}A_{a_{1}}{}^{a_{1}}{}_{i}\vspace{0.5ex}\\\
\hskip
15.0694pt+\mbox{{g}}^{a_{1}}{}_{a_{1}}B_{ic}-\mbox{{g}}^{a_{1}}{}_{c}B_{ia_{1}}-B_{ic}-4P^{dk}C_{kcdi})\vspace{0.5ex}\\\
+(Y^{D}Z_{Ec}-Y_{E}Z^{D}{}_{c})B^{a_{1}}{}_{a_{1}}\,.\end{array}$

This follows from inspection of the total symbolic formula, from the fact that
$A_{abc}$ is antisymmetric in $b$ and $c$, and from the fact that $C_{abcd}$
is antisymmetric in $c$ and $d$. Now recall that the Cotton tensor is trace-
free and the Bach tensor is symmetric and trace free. Since $n=6$, it follows
that (50) is equal to

(51)
$\begin{array}[]{l}(Y^{D}Z_{E}{}^{i}-Y_{E}Z^{Di})(4B_{ci}-4(\nabla^{a_{1}}A_{ca_{1}i}+P^{kd}C_{kcdi})+2\nabla^{a_{1}}A_{a_{1}ci}).\end{array}$

It is well-known that $\nabla_{a}A^{a}{}_{bc}=0$. (See formula (2.7) of [GN],
for example.) Thus (51) is zero, by (3), and hence (50) is zero as well.

Now consider the terms of the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
$Z^{Di}Z_{E}{}^{k}$, $Z^{D}{}_{c}Z_{E}{}^{e}$, or $Z^{De}Z_{Ec}$ appears. Here
$i$, $k$, and $e$ are dummy indices; in the total symbolic formula, any of
these indices may appear as a lower index. The sum of the terms under
consideration is given by the symbolic expression on the left-hand side of the
equation in Figure 3.

$\begin{array}[]{l}Z^{Di}Z_{E}{}^{k}(-2P_{i}{}^{d}A_{cdk}+2P_{c}{}^{d}A_{idk}-2P^{a_{1}}{}_{i}A_{ca_{1}k}+2P^{a_{1}}{}_{i}A_{a_{1}ck}-2P_{k}{}^{e}A_{cie}+2P_{c}{}^{e}A_{kie}\vspace{0.5ex}\\\
\hskip
15.0694pt-2P^{a_{1}}{}_{k}A_{cia_{1}}+2P^{a_{1}}{}_{k}A_{a_{1}ic}+2\nabla^{a_{1}}\nabla_{a_{1}}A_{cik}-2\nabla^{a_{1}}\nabla_{c}A_{a_{1}ik}-\nabla_{i}B_{kc}-\nabla_{i}B_{kc}\vspace{0.5ex}\\\
\hskip
15.0694pt+\nabla_{c}B_{ki}+\nabla_{k}B_{ic}+\nabla_{k}B_{ic}-\nabla_{c}B_{ik}+4(\nabla^{b}P_{b}{}^{a})C_{acik}+4P_{b}{}^{a}\nabla^{b}C_{acik}\vspace{0.5ex}\\\
\hskip
15.0694pt-4P_{i}{}^{a}A_{kac}+4P_{k}{}^{a}A_{iac}+4(\nabla^{b}P_{c}{}^{a})C_{baik}+4P_{c}{}^{a}\nabla^{b}C_{baik}-4P_{c}{}^{a}A_{kia}\vspace{0.5ex}\\\
\hskip
15.0694pt+4P_{c}{}^{a}A_{ika}-2(\nabla^{a}J)C_{acik}-2J\nabla^{a}C_{acik}+2JA_{kic}-2JA_{ikc}+2A_{bi}{}^{a}C^{b}{}_{cak}\vspace{0.5ex}\\\
\hskip 15.0694pt-2C^{b}{}_{ci}{}^{a}A_{bak})\vspace{0.5ex}\\\
+Z^{D}{}_{c}Z_{E}{}^{e}\nabla^{a_{1}}B_{ea_{1}}-Z^{De}Z_{Ec}\nabla^{a_{1}}B_{ea_{1}}\vspace{0.5ex}\\\
\hskip 15.0694pt=\vspace{0.5ex}\\\
Z^{Di}Z_{E}{}^{k}(-4P_{i}{}^{d}A_{cdk}+2P_{c}{}^{a}A_{ika}+2P^{a_{1}}{}_{i}A_{a_{1}ck}-4P_{k}{}^{e}A_{cie}-2P_{c}{}^{a}A_{kia}\vspace{0.5ex}\\\
\hskip
15.0694pt+2P^{a_{1}}{}_{k}A_{a_{1}ic}+2\nabla^{a_{1}}\nabla_{a_{1}}A_{cik}-2\nabla^{a_{1}}\nabla_{c}A_{a_{1}ik}-2\nabla_{i}B_{kc}+2\nabla_{k}B_{ic}\vspace{0.5ex}\\\
\hskip
15.0694pt+2(\nabla^{a}J)C_{acik}+4P_{b}{}^{a}\nabla^{b}C_{acik}-4P_{i}{}^{a}A_{kac}+4P_{k}{}^{a}A_{iac}+4(\nabla^{b}P_{c}{}^{a})C_{baik}\vspace{0.5ex}\\\
\hskip
15.0694pt+4P_{c}{}^{a}\nabla^{b}C_{baik}-2J\nabla^{a}C_{acik}+2JA_{kic}-2JA_{ikc}+2A_{bi}{}^{a}C^{b}{}_{cak}-2C^{b}{}_{ci}{}^{a}A_{bak})\vspace{0.5ex}\\\
+Z^{D}{}_{c}Z_{E}{}^{e}\nabla^{a_{1}}B_{ea_{1}}-Z^{De}Z_{Ec}\nabla^{a_{1}}B_{ea_{1}}\end{array}$
Figure 3. Sum of the terms of the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
$Z^{Di}Z_{E}{}^{k}$, $Z^{D}{}_{c}Z_{E}{}^{e}$, or $Z^{De}Z_{Ec}$ appears.

This follows from direct inspection of the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$. The equation in the
figure follows from simplification of its left-hand side. To perform this
simplification, one may use the symmetry of $B_{ab}$, the antisymmetry of
$A_{abc}$ in $b$ and $c$, and (11). By using [W] and [L], one can show that
the right-hand side of the equation is zero. To do this, one may begin with
the right-hand side of the equation and use (3) and (12) and to express
$A_{abc}$ and $B_{ab}$ in terms of $P_{ab}$ and $C_{abcd}$. If one then
applies (9) in a suitable way, all derivatives of order greater than 1 cancel.
One may then use the Bianchi identities and (13) to show that the remaining
terms cancel.

Finally, consider the terms of the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
$X_{E}Z^{D}{}^{i}$, $X^{D}Z_{E}{}^{i}$, $X_{E}Z^{D}{}_{c}$, or
$X^{D}Z_{E}{}_{c}$ occurs. Here $i$ is a dummy index, as before. In the total
symbolic formula, this $i$ may appear as a lower index. The sum of the terms
under consideration is given by the leftmost member of the equation in Figure
4.

$\begin{array}[]{ll}X_{E}Z^{Di}(-2P_{a_{1}}{}^{e}\nabla^{a_{1}}A_{cie}+2P_{c}{}^{e}\nabla^{a_{1}}A_{a_{1}ie}-2(\nabla^{a_{1}}P_{a_{1}}{}^{e})A_{cie}+2(\nabla^{a_{1}}P_{c}{}^{e})A_{a_{1}ie}\vspace{0.5ex}\\\
\hskip
15.0694pt-2P^{a_{1}e}\nabla_{a_{1}}A_{cie}+2P^{a_{1}e}\nabla_{c}A_{a_{1}ie}+P_{i}{}^{e}B_{ec}-P^{a_{1}}{}_{a_{1}}B_{ic}+P^{a_{1}}{}_{c}B_{ia_{1}}+\nabla^{a_{1}}\nabla_{a_{1}}B_{ic}\vspace{0.5ex}\\\
\hskip
15.0694pt-\nabla^{a_{1}}\nabla_{c}B_{ia_{1}}-4P_{b}{}^{k}P^{be}C_{kcie}+4P_{b}{}^{k}\nabla^{b}A_{ikc}+4(\nabla^{b}P_{b}{}^{k})A_{ikc}-4P_{c}{}^{k}P^{be}C_{bkie}\vspace{0.5ex}\\\
\hskip
15.0694pt+4P_{c}{}^{k}\nabla^{b}A_{ibk}+4(\nabla^{b}P_{c}{}^{k})A_{ibk}+2JP^{ae}C_{acie}-2(\nabla^{a}J)A_{iac}-2J\nabla^{a}A_{iac}\vspace{0.5ex}\\\
\hskip
15.0694pt+2A_{bi}{}^{e}A_{e}{}^{b}{}_{c}-C^{b}{}_{ci}{}^{e}B_{eb})\vspace{0.5ex}\\\
\hskip 30.1388pt-Z^{D}{}_{c}X_{E}P^{a_{1}e}B_{ea_{1}}\vspace{0.5ex}\\\
+X^{D}Z_{E}{}^{i}(-2P_{a_{1}}\\!{}^{d}\nabla^{a_{1}}A_{cdi}+2P_{c}{}^{d}\nabla^{a_{1}}A_{a_{1}di}-2(\nabla^{a_{1}}P_{a_{1}}{}^{d})A_{cdi}+2(\nabla^{a_{1}}P_{c}{}^{d})A_{a_{1}di}\vspace{0.5ex}\\\
\hskip
15.0694pt-2P^{a_{1}d}\nabla_{a_{1}}A_{cdi}+2P^{a_{1}d}\nabla_{c}A_{a_{1}di}+P^{a_{1}}{}_{a_{1}}B_{ic}-P^{a_{1}}{}_{c}B_{ia_{1}}-\nabla^{a_{1}}\nabla_{a_{1}}B_{ic}\vspace{0.5ex}\\\
\hskip
15.0694pt+\nabla^{a_{1}}\nabla_{c}B_{ia_{1}}-P_{i}{}^{e}B_{ec}-4P_{b}{}^{k}P^{bd}C_{kcdi}-4P_{b}{}^{k}\nabla^{b}A_{ikc}-4(\nabla^{b}P_{b}{}^{k})A_{ikc}\vspace{0.5ex}\\\
\hskip
15.0694pt-4P_{c}{}^{k}P^{bd}C_{bkdi}-4P_{c}{}^{k}\nabla^{b}A_{ibk}-4(\nabla^{b}P_{c}{}^{k})A_{ibk}+2JP^{ad}C_{acdi}+2(\nabla^{a}J)A_{iac}\vspace{0.5ex}\\\
\hskip
15.0694pt+2J\nabla^{a}A_{iac}-B^{a}{}_{b}C^{b}{}_{cai}+2A^{ab}{}_{c}A_{bai})\vspace{0.5ex}\\\
\hskip 30.1388pt+Z_{Ec}X^{D}P^{a_{1}e}B_{ea_{1}}\vspace{0.5ex}\\\ \hskip
15.0694pt\vspace{0.5ex}=\\\
(X_{E}Z^{Di}-X^{D}Z_{E}{}^{i})(-2P_{a_{1}}{}^{e}\nabla^{a_{1}}A_{cie}+2P_{c}{}^{e}\nabla^{a_{1}}A_{a_{1}ie}-2(\nabla^{a_{1}}P_{a_{1}}{}^{e})A_{cie}\vspace{0.5ex}\\\
\hskip
15.0694pt+2(\nabla^{a_{1}}P_{c}{}^{e})A_{a_{1}ie}-2P^{a_{1}e}\nabla_{a_{1}}A_{cie}+2P^{a_{1}e}\nabla_{c}A_{a_{1}ie}+P_{i}{}^{e}B_{ec}-P^{a_{1}}{}_{a_{1}}B_{ic}\vspace{0.5ex}\\\
\hskip
15.0694pt+P^{a_{1}}{}_{c}B_{ia_{1}}+\nabla^{a_{1}}\nabla_{a_{1}}B_{ic}-\nabla^{a_{1}}\nabla_{c}B_{ia_{1}}-4P_{b}{}^{k}P^{be}C_{kcie}+4P_{b}{}^{k}\nabla^{b}A_{ikc}\vspace{0.5ex}\\\
\hskip
15.0694pt+4(\nabla^{b}P_{b}{}^{k})A_{ikc}-4P_{c}{}^{k}P^{be}C_{bkie}\vspace{0.5ex}+4P_{c}{}^{k}\nabla^{b}A_{ibk}+4(\nabla^{b}P_{c}{}^{k})A_{ibk}+2JP^{ae}C_{acie}\\\
\hskip
15.0694pt-2(\nabla^{a}J)A_{iac}-2J\nabla^{a}A_{iac}\vspace{0.5ex}+2A_{bi}{}^{e}A_{e}{}^{b}{}_{c}-C^{b}{}_{ci}{}^{e}B_{eb}-\textbf{\mbox{{g}}}_{ic}P^{a_{1}e}B_{ea_{1}})\\\
\hskip 15.0694pt\vspace{0.5ex}=\\\
(X_{E}Z^{Di}-X^{D}Z_{E}{}^{i})(-8P_{ab}\nabla^{a}A_{(ic)}{}^{b}+2P_{c}{}^{e}\nabla^{a_{1}}A_{a_{1}ie}-4A_{(ic)a}\nabla^{a}J\vspace{0.5ex}\\\
\hskip
15.0694pt+2(\nabla^{a_{1}}P_{c}{}^{e})A_{a_{1}ie}+2P^{a_{1}e}\nabla_{c}A_{a_{1}ie}+P_{i}{}^{e}B_{ec}-3JB_{ic}+5P_{c}{}^{k}B_{ik}+\Delta
B_{ic}\vspace{0.5ex}\\\ \hskip
15.0694pt-\nabla^{a_{1}}\nabla_{c}B_{ia_{1}}+4P_{b}{}^{k}P^{be}C_{ieck}+4(\nabla^{b}P_{c}{}^{k})A_{ibk}-2A_{bie}A^{e}{}_{c}{}^{b}+B_{eb}C_{i}{}^{e}{}_{c}{}^{b}\vspace{0.5ex}\\\
\hskip 15.0694pt-\textbf{\mbox{{g}}}_{ic}P^{a_{1}e}B_{ea_{1}})\end{array}$
Figure 4. Sum of the terms in the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$ in which
$X_{E}Z^{D}{}^{i}$, $X^{D}Z_{E}{}^{i}$, $X_{E}Z^{D}{}_{c}$, or
$X^{D}Z_{E}{}_{c}$ occurs.

This follows from direct inspection of the total symbolic formula for
$(\mathfrak{D}\nabla^{\mathcal{T}})_{c}{}^{D}{}_{E}$. The equation in the
figure follows from the symmetries of $A_{abc}$ and $C_{abcd}$ and from (3)
and (11). In dimension $n=6$, the rightmost member of this equation is equal
to $(X_{E}Z^{Dd}-X^{D}Z_{E}{}^{d})16\mathcal{O}_{cd}^{6}$. This follows from a
direct computation using [W] and [L], together with (3), (9), (11), (12), and
(13). The proof of Theorem 6.1 is thus complete. ∎

###  6.3. A comment on the Fefferman-Graham invariant in dimension 6

Fefferman and Graham introduced an interesting natural scalar conformal
invariant FGI in Proposition 3.4 of [FG-1985] (see also [CG]). This invariant
has conformal weight $-6$ and so on closed conformal 6-manifolds,

(52) $\int_{M}\mbox{{FGI}}~{}d\mu$

is a well-defined global conformal invariant. Moreover at leading order FGI is
quadratic in the jets of the metric. Thus (52) is an important action to
consider for metric variations. In fact, it is closely linked to the
specialisation to the tractor connection of our action (27) as follows.

Let $A$ denote the tractor connection, and let $F_{A}$ denote the tractor
curvature as given in (38). Then in dimension $n=6$,

(53)
$\begin{array}[]{ll}\lx@intercol\<EMAIL_ADDRESS>&2\,\langle
F_{A},Q_{2}^{A}F_{A}\rangle+8C_{abcd}C^{a}{}_{e}{}^{c}{}_{i}C^{bedi}-4C_{abcd}C^{a}{}_{e}{}^{c}{}_{i}C^{bide}+\mbox{{(}{divergences}\hskip
1.07639pt{)}}.\end{array}$

Here $\langle\cdot,\cdot\rangle$ and $Q_{2}^{A}$ are as in Definition 4.1. The
equality in (53) follows from a short symbolic computation using the computer
algebra system Mathematica, [W], together with Lee’s tensor calculus software
package Ricci, [L].

In even dimensions, elementary representation theory shows that the Weyl
tensor (and its irreducible parts in dimension 4) and the Fefferman-Graham
obstruction tensor are the only conformal invariants that at leading order are
linear in the jets of the metric [GH]. Using this and Theorem 6.1, it is not
difficult to conclude that with respect to metric variations both (27) (with
$A$ the conformal tractor connection) and (52) must yield
$\mathcal{O}_{ce}^{6}$ as their functional gradient, up to conformally
invariant lower-order terms.

## References

  * [AO] G. Anastasiou, R. Olea: From conformal to Einstein gravity. Phys. Rev. D 94 (2016), no. 8, 086008, 4 pp.
  * [B-1921] R. Bach: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. $\bf{9}$ (1921), no. 1-2, 110-135.
  * [Bz] J.C. Baez: Higher Yang-Mills Theory. arXiv:hep-th/0206130.
  * [BEG] T.N. Bailey, M.G. Eastwood, A.R. Gover: Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. $\bf{24}$ (1994), no. 4, 1191-1217.
  * [BSS] E. Bergshoeff, A. Salam, E. Sezgin: Supersymmetric $R^{2}$ actions, conformal invariance and the Lorentz Chern-Simons term in 6 and 10 dimensions. Nuclear Phys. B 279 (1987), no. 3-4, 659-683.
  * [Br-1993] T.P. Branson: The functional determinant. Global Analysis Research Center Lecture Notes Series, Number 4 (Seoul National University, 1993).
  * [Br-1981] T.P. Branson: Quasi-invariance of the Yang-Mills equations under conformal transformations and conformal vector fields. J. Differential Geometry 16 (1981), no. 2, 195-203.
  * [Br-1995] T.P. Branson: Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc. $\bf{347}$ (1995), no. 10, 3671-3742.
  * [BG-2005] T.P. Branson, A.R. Gover: Conformally invariant operators, differential forms, cohomology and a generalisation of $Q$-curvature. Comm. Partial Differential Equations 30 (2005), no. 10-12, 1611-1669.
  * [BG-2007] T.P. Branson, A.R. Gover: Pontrjagin forms and invariant objects related to the $Q$-curvature. Commun. Contemp. Math. $\bf{9}$ (2007), no. 3, 335-358.
  * [BrO] T.P. Branson, B. Ørsted: Explicit functional determinants in four dimensions. Proc. Amer. Math. Soc. 113 (1991), no. 3, 669-682.
  * [CGGGR] E. Calvi$\tilde{\mbox{n}}$o-Louzao, X. García-Martínez, E. García-Río, I. Gutiérrez-Rodríguez, R. Vázquez-Lorenzo: Conformally Einstein and Bach-flat four-dimensional homogeneous manifolds. J. Math. Pures Appl. (9) 130 (2019), 347-374.
  * [CG] A. Čap, A.R. Gover: Standard tractors and the conformal ambient metric construction. Ann. Global Anal. Geom. 24 (2003), no. 3, 231-259.
  * [C] É. Cartan: Les groupes d’holonomie des espaces généralisés. Acta Math. 48 (1926), no. 1-2, 1-42.
  * [CLW] X. Chen, C. LeBrun, B. Weber: On conformally Kähler, Einstein manifolds. J. Amer. Math. Soc. 21 (2008), no. 4, 1137-1168.
  * [D] S.K. Donaldson: An application of gauge theory to four-dimensional topology. J. Differential Geom. 18 (1983), no. 2, 279-315.
  * [DK] S.K. Donaldson, P.B. Kronheimer: The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990\.
  * [FG-2012] C. Fefferman, C.R. Graham: The ambient metric. Princeton University Press, Princeton, NJ, 2012.
  * [FG-1985] C. Fefferman, C.R. Graham: Conformal invariants. In: Elie Cartan et les mathématiques d’aujourd’hui, 95-116. Astérisque, Numéro Hors Série, Société Mathématique de France, Paris, 1985.
  * [FP-1984] E.S. Fradkin, M.Ya. Palchik: Conformal invariance in quantum Yang-Mills theory. Phys. Lett. B 147 (1984), no. 1-3, 86-90.
  * [FP-1978] E.S. Fradkin, M.Ya. Palchik: Recent developments in conformal invariant quantum field theory. Phys. Rep. 44 (1978), no. 5, 249-349.
  * [GN] A.R. Gover, P. Nurowski: Obstructions to conformally Einstein metrics in $n$ dimensions. J. Geom. Phys. 56 (2006), no. 3, 450-484.
  * [GP-2006] A.R. Gover, L.J. Peterson: The ambient obstruction tensor and the conformal deformation complex. Pacific J. Math. 226 (2006), no. 2, 309-351.
  * [GP-2003] A.R. Gover, L.J. Peterson: Conformally invariant powers of the Laplacian, $Q$-curvature, and tractor calculus. Comm. Math. Phys. 235 (2003), no. 2, 339-378.
  * [GSS] A.R. Gover, P. Somberg, V. Souček: Yang-Mills detour complexes and conformal geometry. Comm. Math. Phys. 278 (2008), no. 2, 307-327.
  * [GH] C.R. Graham, K. Hirachi: The ambient obstruction tensor and $Q$-curvature. In: AdS/CFT correspondence: Einstein metrics and their conformal boundaries, 59-71. IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005.
  * [GV] M.J. Gursky, J.A. Viaclovsky: Rigidity and stability of Einstein metrics for quadratic curvature functionals. J. Reine Angew. Math. 700 (2015), 37-91.
  * [H] G. Huang: Rigidity of Riemannian manifolds with positive scalar curvature. Ann. Global Anal. Geom. 54 (2018), no. 2, 257-272.
  * [JR] R. Jackiw, C. Rebbi: Conformal properties of a Yang-Mills pseudoparticle. Phys. Rev. D (3) 14 (1976), no. 2, 517-523.
  * [K] C. Kelleher: Higher order Yang-Mills Flow. Calc. Var. Partial Differential Equations 58 (2019), no. 3, Paper No. 100, 45 pp.
  * [LeB] C. LeBrun: Bach-flat Kähler surfaces. J. Geom. Anal. 30 (2020), no. 3, 2491-2514.
  * [L] J.M. Lee: “Ricci” software package. sites.math.washington.edu/$\sim$lee/Ricci, 2016, Version 1.61.
  * [M] J. Maldacena: Einstein gravity from conformal gravity. arXiv:1105.5632.
  * [Mer] S.A. Merkulov: A conformally invariant theory of gravitation and electromagnetism. Classical Quantum Gravity 1 (1984), no. 4, 349-354.
  * [Tb] C. Taubes: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differential Geometry 17 (1982), no. 1, 139-170.
  * [T] T.Y. Thomas: On conformal geometry. Proc. Natl. Acad. Sci. USA 12 (1926), no. 5, 352-359.
  * [tH] G. ’t Hooft, ed.: 50 years of Yang-Mills theory. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
  * [W] Wolfram Research, Inc.: “Mathematica” software. Champaign, Illinois, 2016, Version 11.0.
  * [YM] C.N. Yang, R.L. Mills: Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. (2) 96 (1954), 191-195.
  * [Z] P. Zhang: Gradient flows of higher order Yang-Mills-Higgs Functionals. J. Aust. Math. Soc. (2021) 1-31. doi:10.1017/S1446788721000057.