affil0affil0affiliationtext: Department of Physics, University of Illinois,
1110 West Green St., Urbana IL 61801, U.S.A.

# Obstruction Tensors in Weyl Geometry
and Holographic Weyl Anomaly

Weizhen<EMAIL_ADDRESS>and Manthos
<EMAIL_ADDRESS>
###### Abstract

Recently a generalization of the Fefferman-Graham gauge for asymptotically
locally AdS spacetimes, called the Weyl-Fefferman-Graham (WFG) gauge, has been
proposed. It was shown that the WFG gauge induces a Weyl geometry on the
conformal boundary. The Weyl geometry consists of a metric and a Weyl
connection. Thus, this is a useful setting for studying dual field theories
with background Weyl symmetry. Working in the WFG formalism, we find the
generalization of obstruction tensors, which are Weyl-covariant tensors that
appear as poles in the Fefferman-Graham expansion of the bulk metric for even
boundary dimensions. We see that these Weyl-obstruction tensors can be used as
building blocks for the Weyl anomaly of the dual field theory. We then compute
the Weyl anomaly for $6d$ and $8d$ field theories in the Weyl-Fefferman-Graham
formalism, and find that the contribution from the Weyl structure in the bulk
appears as cohomologically trivial modifications. Expressed in terms of the
Weyl-Schouten tensor and extended Weyl-obstruction tensors, the results of the
holographic Weyl anomaly up to $8d$ also reveal hints on its expression in any
dimension.

###### Contents

  1. 1 Introduction
  2. 2 Obstruction Tensors
  3. 3 Weyl-Fefferman-Graham Gauge
  4. 4 Weyl-Obstruction Tensors
  5. 5 Holographic Weyl Anomaly
    1. 5.1 Weyl-Ward Identity
    2. 5.2 Weyl Anomaly in $2d$ and $4d$
    3. 5.3 Weyl Anomaly in $6d$
    4. 5.4 Weyl Anomaly in $8d$
    5. 5.5 Building Blocks of the Weyl Anomaly
  6. 6 The Role of Weyl Structure
  7. 7 Conclusions
  8. A Weyl Geometry
  9. B Solving the Bulk Einstein Equations
  10. C Expansions of the Raychaudhuri Equation and $\sqrt{-\det h}$
  11. D Dilatation Operator Method
    1. D.1 Review of the Algorithm
    2. D.2 Results and Anomaly in $8d$

##  1 Introduction

There is an important fact about the asymptotic AdS geometry: the conformal
boundary of a $(d+1)$-dimensional asymptotically locally AdS (AlAdS) spacetime
carries not a metric but a conformal class of metrics, i.e. the boundary
enjoys Weyl symmetry. This is due to the fact that the asymptotic boundary is
formally located at conformal infinity[1]. In holographic theories[2], the
(background) Weyl symmetry is implied by diffeomorphism invariance in the bulk
spacetime (called _Weyl diffeomorphism_). Usually when discussing AdS/CFT, one
picks a specific representative of the conformal class. For example, the most
commonly used choice for studying the conformal boundary of an AlAdS spacetime
is the Fefferman-Graham (FG) gauge [3, 4]. However, the FG gauge explicitly
breaks the Weyl symmetry by fixing a specific boundary metric.

In a suitable coordinate system $\\{z,x^{\mu}\\}$ ($\mu=0,\cdots,d-1$), the
metric of any $(d+1)$-dimensional AlAdS spacetime can be expanded with respect
to the bulk coordinate $z$ into two series, called the Fefferman-Graham
expansion[5, 6]. The first series has the boundary metric in the leading
order, while the subleading terms are determined by the bulk equations of
motion; the leading order of the second series represents the vacuum
expectation value of the energy-momentum tensor operator of the boundary field
theory, which cannot be determined in the absence of an interior boundary
condition[6].

When the spacetime dimension is odd, both series in the FG expansion are power
series to infinite order; however, in an even-dimensional spacetime, a
logarithmic term will occur at order $O(z^{d-2})$, causing an obstruction to
the power series expansion[7]. This logarithmic term in $d=2k$ (with $k$ an
integer) gives rise to the _obstruction tensor_ ${\cal O}^{(2k)}_{\mu\nu}$.
The obstruction tensor was first proposed in [3] as a symmetric traceless
tensor of type $(0,2)$, which is Weyl-covariant with Weyl weight $2k-2$
($k\geqslant 2$), and was precisely defined using the ambient metric in [7]
(see also [4]). It is also convenient to define the _extended obstruction
tensor_[8] which has a pole at $d=2k$, and whose residue gives rise to the
obstruction tensor. The obstruction tensor for $d=4$ is also known as the Bach
tensor[9], which is the only Weyl-covariant tensor in $4d$ that is
algebraically independent of the Weyl tensor. It has been shown in [7] that
the only irreducible Weyl-covariant tensors in $2k$-dimension with $k\geqslant
2$ are the obstruction tensor ${\cal O}^{(2k)}_{\mu\nu}$ and the Weyl tensor
(which has weight $0$), while in any odd dimension $d=2k+1$ with $k\geqslant
2$ the Weyl tensor is the only one (in $3d$ where the Weyl tensor becomes
trivial, it is the Cotton tensor).

The origin of the obstruction tensor in the FG expansion is that the two
series will mix if the spacetime dimension $d$ is even, and the solution to
the equations of motion encounters a pole. Hence, another way to formulate the
FG expansion is to use the technique of dimensional regularization, i.e. to
regard $d$ as a variable (formally complex)[5, 6]. Using this formulation, in
this paper we will describe a practical way of reading off the obstruction
tensor from the pole of the FG expansion in an even dimension.

Even though the FG gauge is quite convenient to use, the Weyl symmetry in the
boundary will be broken when the boundary metric is fixed. More specifically,
one can introduce a Penrose-Brown-Henneaux (PBH) transformation [10, 11, 12]
in the bulk and induce a Weyl transformation on the boundary, but the
subleading terms in the $z$-expansion will not transform in a Weyl-covariant
way if the form of the FG ansatz is to be preserved. In order to resolve this
issue, one can relax the FG ansatz of the bulk metric to the Weyl-Fefferman-
Graham (WFG) ansatz [5]. In the WFG gauge, the form of the bulk metric is
preserved under a Weyl diffeomorphism, and all the terms in the $z$-expansion
transform in a Weyl-covariant way, which brings a powerful reorganization of
the holographic dictionary. Unlike the FG gauge, where the bulk Levi-Civita
(LC) connection induces on the conformal boundary also a LC connection (of the
boundary metric), in the WFG gauge, the bulk LC connection gives a Weyl
connection on the boundary[5]. Having the induced metric together with the
Weyl connection, the bulk geometry induces on the boundary a Weyl-covariant
geometry[13, 14, 15].

On the boundary, the induced metric and the Weyl connection act as non-
dynamical backgrounds of the dual quantum field theory. Similar to the FG
case, the metric is the source of the energy-momentum tensor operator on the
boundary. However, the Weyl connection does not source any current since it
comes from a pure gauge mode of the bulk metric. Despite being pure gauge, the
appearance of the Weyl connection on the boundary is far from innocuous since
it makes the geometric quantities on the boundary Weyl-covariant.
Specifically, we will show that the obstruction tensors in the WFG gauge are
promoted to Weyl-obstruction tensors, which will play an important role in the
construction of the Weyl anomaly in this paper.

The Weyl anomaly is reflected by the nonvanishing trace of the energy-momentum
tensor in even dimensions, which has been computed for various conformal field
theories[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The results in $2d$ and
$4d$ are well-known:

$\displaystyle 2d:\langle T^{\mu}{}_{\mu}\rangle=-\frac{c}{24\pi}R\,,\qquad
4d:\langle T^{\mu}{}_{\mu}\rangle=cW^{2}-aE^{(4)}\,,$ (1)

where $W^{2}$ is the contraction of two Weyl tensors, and $E^{(4)}$ is the
Euler density in $4d$. In the context of holography, the Weyl anomaly was
first suggested in [27], and was then calculated from the bulk in [28] and
[24]. For a holographic theory where we have the vacuum Einstein theory in the
bulk, one gets $a=c$ in the 4-dimensional boundary theory as a constraint on
the central charges. In the FG gauge, after going through the holographic
renormalization procedure by adding counterterms to cancel the divergence
extracted by the regulator, one finds that the holographic Weyl anomaly in an
even dimension corresponds to the logarithmic term in the bulk volume
expansion. In mathematical literature this is also referred to as the
Q-curvature [29, 30, 31, 32] (see [33] for a short review), which has been
studied by means of obstruction tensors and extended obstruction tensors in
[7] and [8]. Going into the WFG gauge, it was shown in [5] using dimensional
regularization that the Weyl anomaly in $2k$-dimension can be extracted
directly from the variation of the pole term at the $O(z^{2k-d})$-order of the
“bare” on-shell action under the $d\to 2k^{-}$ limit. This is the method we
will use for computing the Weyl anomaly in this work.

Our goal in this paper is to find the holographic Weyl anomaly in higher
dimensions using the advantages of the WFG gauge, and organize the results in
a form that manifests its general structure. It has been shown in [5] that, up
to total derivatives, the Weyl anomaly in $2d$ and $4d$ in the WFG gauge has
the same form of that in the FG gauge, but now become Weyl-covariant. We
generalize these results to $6d$ and $8d$ by calculating the Weyl anomaly
explicitly, and we find that the same statement still holds. Furthermore, we
discover that by promoting the obstruction tensors in the FG gauge to the
Weyl-obstruction tensors in the WFG gauge, one can use them as natural
building blocks for the Weyl anomaly. In this way, we will see clearly how the
WFG gauge Weyl-covariantizes the Weyl anomaly without introducing additional
nontrivial cocycles. Our results also reveal some interesting clues about the
general form of the holographic Weyl anomaly in any dimension.

This paper will be organized as follows. In Section 2 we briefly introduce the
obstruction tensors and extended obstruction tensors in the FG gauge and their
properties. In Section 3 we review the WFG gauge as the Weyl-covariant
modification of the FG gauge, and how the bulk LC connection induces a Weyl
connection on the conformal boundary. More details about the Weyl connection
and Weyl geometry are given in Appendix A. In Section 4 we generalize the
results of Section 2 to Weyl-obstruction tensors and extended Weyl-obstruction
tensors by solving the Einstein equations in the WFG gauge. The expansions of
the Einstein equations can be found in Appendix B. Using the Weyl-Schouten
tensor and extended Weyl-obstruction tensors as building blocks, in Section 5
we will derive the holographic Weyl anomaly in the WFG gauge in $6d$ and $8d$
after we review the results in $2d$ and $4d$. More details of the calculation
are provided in Appendix C. As a consistency check, we also compute the $8d$
holographic Weyl anomaly in the FG gauge using a completely different
approach—the dilatation operator method[34, 35]—which will be presented in
Appendix D. The result agrees with what we get in Section 5. The expressions
for the holographic Weyl anomaly up to $8d$ also suggest the pattern in any
dimension, which will be discussed in the end of Section 5. In Section 6 we
discuss some aspects of the Weyl structure observed from the role it plays in
the formulas for the Weyl-obstruction tensors and Weyl anomaly that we
derived. Finally, in Section 7 we summarize our results and point out possible
directions for future studies.

##  2 Obstruction Tensors

The obstruction tensor is known as the only irreducible conformal covariant
tensor besides the Weyl tensor in an even-dimensional spacetime. The general
references for the obstruction tensor are [7, 4], where it was defined
precisely in terms of the ambient metric. Instead of providing the formal
definition, we will derive the obstruction tensors explicitly in the FG gauge
for up to $d=6$ by solving the bulk equations of motion order by order. The
same method will also be used in Section 4 for the Weyl-obstruction tensor.

According to the Fefferman-Graham theorem [3] the metric of a
$(d+1)$-dimensional asymptotically locally AdS (AlAdS) spacetime can always be
expressed in the following form

$\text{d}s^{2}=L^{2}\frac{\text{d}z^{2}}{z^{2}}+h_{\mu\nu}(z;x)\text{d}x^{\mu}\text{d}x^{\nu}\,,\qquad\mu,\nu=0,\cdots,d-1\,,$
(2)

where the coordinate $z$ can be considered as a “radial” coordinate, and $z=0$
is the “location” of the conformal boundary. When
$h_{\mu\nu}=L^{2}\eta_{\mu\nu}/z^{2}$ with $\eta_{\mu\nu}$ the flat metric,
this represents the Poincaré metric for $AdS_{d+1}$ spacetime. Near the
conformal boundary, $h_{\mu\nu}$ can be expanded with respect to $z$ as
follows[5]:

$h_{\mu\nu}(z;x)=\frac{L^{2}}{z^{2}}\left[\gamma^{(0)}_{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\gamma^{(2)}_{\mu\nu}(x)+\cdots\right]+\frac{z^{d-2}}{L^{d-2}}\left[\pi^{(0)}_{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\pi^{(2)}_{\mu\nu}(x)+\cdots\right]\,.$
(3)

As we mentioned in Section 1, the conformal boundary carries a conformal class
of metrics. In the FG expansion $\gamma^{(0)}_{\mu\nu}$ serves as the
“canonical” representative of the conformal class sourcing the energy-momentum
tensor of the dual field theory on the boundary, while $\pi^{(0)}_{\mu\nu}$
corresponds to the expectation value of the energy-momentum tensor[6]. Once
$\gamma^{(0)}_{\mu\nu}$ is given, each term in the first series can be
determined by solving the vacuum Einstein equations with negative cosmological
constant in the bulk. Similarly, once $\pi^{(0)}_{\mu\nu}$ is given, the
second series will be determined. However, $\pi^{(0)}_{\mu\nu}$ is not
completely arbitrary but is actually constrained by the Einstein equations. To
be more specific, the $zz$-component of the Einstein equations tells us that
$\pi^{(0)}_{\mu\nu}$ is traceless while the $z\mu$-components indicate that it
is also divergence-free.

Nevertheless, subtleties will arise when the boundary dimension $d$ is an even
integer, since the two series in (3) mix into one. To resolve this issue for
an even $d=2k$, we treat $d$ formally as a variable $d\in\mathbb{C}$ in the
expansion (3) and let $d$ approach $2k$ from below. As we will see explicitly,
when the Einstein equations are satisfied, $\gamma^{(2k)}_{\mu\nu}$ has a
first order pole at $d=2k$. For any integer $k\geqslant 2$, up to some factor,
the coefficient of the pole term (which is actually a meromorphic function of
the boundary dimension) is what we define as the _obstruction tensor_ ,
denoted by $\mathcal{O}^{(2k)}_{\mu\nu}$:

$\gamma^{(2k)}_{\mu\nu}=\frac{c_{(2k)}}{d-2k}\mathcal{O}^{(2k)}_{\mu\nu}+\tilde{\gamma}_{\mu\nu}^{(2k)}\,,\qquad
c_{(2k)}=-\frac{L^{2k}}{2^{2k-3}k!}\frac{\Gamma(d/2-k+1)}{\Gamma(d/2-1)}\,,$
(4)

where the normalization factor $c_{(2k)}$ has been chosen so that the
obstruction tensor agrees with the convention of [4], and the tensor
$\tilde{\gamma}^{(2k)}_{\mu\nu}$ is analytic at $d=2k$.

Besides holographic dimensional regularization [6], another common approach is
to introduce a logarithmic term for $d=2k$[24], which turns out to be
proportional to the obstruction tensor. This is also the origin of the name
obstruction tensor, as it obstructs the existence of a formal power series
expansion. Note that the tensor $\mathcal{O}^{(2k)}_{\mu\nu}$ is well-defined
in any $d\geqslant 2k$, but only behaves as an “obstruction” when $d=2k$. The
relation between the two approaches will be cleared up at the end of this
section once we show how to correctly take the limit for an even $d$ in
holographic dimensional regularization.

Now we present the obstruction tensors explicitly. First, by solving the bulk
Einstein equations to the $O(z^{2})$-order one finds that

$\displaystyle\frac{\gamma^{(2)}_{\mu\nu}}{L^{2}}=-\frac{1}{d-2}\left(R^{(0)}_{\mu\nu}-\frac{R^{(0)}}{2(d-1)}\gamma_{\mu\nu}^{(0)}\right)\,,$
(5)

where $R^{(0)}_{\mu\nu}$ and $R^{(0)}$ represent the Ricci tensor and Ricci
scalar of $\gamma_{\mu\nu}^{(0)}$ at the boundary, respectively. One can
recognize $\gamma^{(2)}_{\mu\nu}/L^{2}$ as the Schouten tensor $P_{\mu\nu}$ at
the boundary (with a minus sign):

$\displaystyle P_{\mu\nu}$
$\displaystyle=\frac{1}{d-2}\left(R^{(0)}_{\mu\nu}-\frac{R^{(0)}}{2(d-1)}\gamma_{\mu\nu}^{(0)}\right)\,.$
(6)

Indeed we notice that there is a first order pole when $d=2$ as expected.
However, it is easy to see that the residue of the pole vanishes identically
for $d=2$. This is the reason why it is often stated that there is no
obstruction tensor for $d=2$.

At the $O(z^{4})$-order, the Einstein equations give us

$\displaystyle\frac{\gamma^{(4)}_{\mu\nu}}{L^{4}}=-\frac{1}{4(d-4)}B_{\mu\nu}+\frac{1}{4}P_{\rho\mu}P^{\rho}{}_{\nu}\,.$
(7)

Note that on the boundary, the tensor indices are lowered and raised using
$\gamma^{(0)}_{\mu\nu}$ and its inverse $\gamma_{(0)}^{\mu\nu}$. The tensor
$B_{\mu\nu}$ is the Bach tensor, which is defined as

$\displaystyle B_{\mu\nu}$
$\displaystyle=\nabla_{(0)}^{\lambda}\nabla^{(0)}_{\lambda}P_{\mu\nu}-\nabla_{(0)}^{\lambda}\nabla^{(0)}_{\nu}P_{\mu\lambda}-W^{(0)}_{\rho\nu\mu\lambda}P^{\lambda\rho}\,,$
(8)

where $\nabla^{(0)}_{\mu}$ is the derivative operator on the boundary
associated with $\gamma^{(0)}_{\mu\nu}$, and $W^{(0)}_{\rho\mu\nu\lambda}$ is
the Weyl tensor of $\gamma_{\mu\nu}^{(0)}$. We notice that the first term has
a pole at $d=4$ and it follows from (4) that the obstruction tensor for $d=4$
is just the Bach tensor, i.e. ${\cal O}^{(4)}_{\mu\nu}=B_{\mu\nu}$.

Similarly, if we move on to the $O(z^{6})$-order of the Einstein equations, we
find that $\gamma^{(6)}_{\mu\nu}$ has a pole at $d=6$ and can be written as

$\displaystyle\frac{\gamma^{(6)}_{\mu\nu}}{L^{6}}$
$\displaystyle=-\frac{1}{24(d-6)(d-4)}{\cal
O}^{(6)}_{\mu\nu}-\frac{1}{6(d-4)}B_{\rho\mu}P^{\rho}{}_{\nu}\,.$ (9)

From (4) one can see that ${\cal O}^{(6)}_{\mu\nu}$ is the obstruction tensor
for $d=6$, now given by

$\displaystyle{\cal O}^{(6)}_{\mu\nu}={}$
$\displaystyle\nabla_{(0)}^{\lambda}\nabla^{(0)}_{\lambda}B_{\mu\nu}-2W^{(0)}_{\rho\nu\mu\lambda}B^{\lambda\rho}-4B_{\mu\nu}P+2(d-4)\big{(}2P^{\rho\lambda}\nabla^{(0)}_{\lambda}C_{(\mu\nu)\rho}+\nabla^{(0)}_{\lambda}PC_{(\mu\nu)}{}^{\lambda}$
$\displaystyle\qquad\qquad-C^{\rho}{}_{\mu}{}^{\lambda}C_{\lambda\nu\rho}+\hat{\nabla}_{(0)}^{\lambda}P^{\rho}{}_{(\mu}C_{\nu)\rho\lambda}-W^{(0)}_{\rho\mu\nu\lambda}P^{\lambda}{}_{\sigma}P^{\sigma\rho}\big{)}\,,$
(10)

where $P\equiv P_{\mu\nu}\gamma_{(0)}^{\mu\nu}$, and $C_{\mu\nu\rho}$ is the
Cotton tensor on the boundary defined as

$\displaystyle
C_{\mu\nu\rho}=\nabla^{(0)}_{\rho}P_{\mu\nu}-\nabla^{(0)}_{\nu}P_{\mu\rho}\,.$
(11)

Let us make a few remarks on some important properties of the obstruction
tensors. First, they are symmetric traceless tensors for any boundary
dimension $d$. The traceless condition can be derived from the $zz$-component
of the Einstein equations at the $O(z^{2k})$-order. Also, the obstruction
tensor ${\cal O}^{(2k)}_{\mu\nu}$ is divergence-free when $d=2k$. For
instance, divergence of the Bach tensor gives

$\displaystyle\nabla_{(0)}^{\nu}B_{\nu\mu}=(d-4)P^{\nu\rho}C_{\rho\nu\mu}\,.$
(12)

The divergence of the Bach tensor can be read from the $O(z^{4})$-order of the
$z\mu$-component of Einstein equations. In general, at any $O(z^{2k})$-order
one finds that the divergence of ${\cal O}^{(2k)}_{\mu\nu}$ is proportional to
$d-2k$ and thus vanishes when $d=2k$. The divergence of ${\cal
O}^{(2k)}_{\mu\nu}$ can also be obtained by using the following identity

$\displaystyle\nabla_{(0)}^{\nu}P_{\nu\mu}=\nabla^{(0)}_{\mu}P\,.$ (13)

This is equivalent to the contracted Bianchi identity at the boundary (see
Appendix A), which can also be read from the leading order of the
$z\mu$-component of Einstein equations. Finally, a notable feature of ${\cal
O}^{(2k)}_{\mu\nu}$ is that it is Weyl-covariant when $d=2k$ with Weyl weight
$2k-2$ (for a proof see [7]).

For convenience, we can also absorb the $d$-dependent factors in
$\gamma^{(2k)}_{\mu\nu}$ by introducing Graham’s extended obstruction tensor
$\Omega^{(k-1)}_{\mu\nu}$ ($k\geqslant 2$) in $d>2k$:

$\displaystyle\Omega^{(1)}_{\mu\nu}=-\frac{1}{d-4}B_{\mu\nu}\,,\qquad\Omega^{(2)}_{\mu\nu}=\frac{1}{(d-6)(d-4)}\mathcal{O}^{(6)}_{\mu\nu}\,,\qquad\cdots$
(14)

The extended obstruction tensor $\Omega^{(k)}_{\mu\nu}$ was precisely defined
in [8] in the context of the ambient metric. The general relation between the
obstruction tensor and extended obstruction tensor is

$\displaystyle\Omega^{(k)}_{\mu\nu}=\frac{(-1)^{k}}{2^{k}}\frac{\Gamma(d/2-k-1)}{\Gamma(d/2-1)}{\cal
O}^{(2k+2)}_{\mu\nu}\qquad(k\geqslant 1)\,.$ (15)

We finish this section by describing how to get the $d\to 2k^{-}$ limit of the
two series in (3) properly. By taking the limit carefully we will recover a
logarithmic term in the expansion whose coefficient is exactly the obstruction
tensor for $d=2k$, which also justifies the name “obstruction” as we mentioned
before. There are two issues one has to deal with while taking the $d\to
2k^{-}$ limit. First, as we already noted, $\gamma^{(2k)}_{\mu\nu}$ has a pole
at $d-2k$, so it diverges in this limit. Second, the two series mix since both
$\gamma^{(2k)}_{\mu\nu}$ and $\pi^{(0)}_{\mu\nu}$ appear at the same order
$O(z^{2(k-1)})$ in (3), for $d=2k$. To keep the $O(z^{2k})$-order finite we
pose that $\pi_{\mu\nu}^{(0)}$ should also have a pole for $d=2k$ proportional
to ${\cal O}^{(2k)}_{\mu\nu}$ so that the divergence in
$\gamma^{(2k)}_{\mu\nu}$ gets canceled, i.e. we claim that
$\pi^{(0)}_{\mu\nu}$ has the following form:

$\pi^{(0)}_{\mu\nu}=-\frac{c_{(2k)}}{d-2k}{\cal
O}^{(2k)}_{\mu\nu}+\tilde{\pi}^{(0)}_{\mu\nu}\,,$ (16)

where $\tilde{\pi}^{(0)}_{\mu\nu}$ is finite at $d=2k$. Substituting back (16)
and (4) to (3) we get

$h_{\mu\nu}(z;x)=\sum_{n=0}^{k-1}\gamma_{\mu\nu}^{(2n)}\left(\frac{z}{L}\right)^{2n-2}+\big{(}\tilde{\gamma}_{\mu\nu}^{(2k)}+\tilde{\pi}^{(0)}_{\mu\nu}\big{)}\left(\frac{z}{L}\right)^{2k-2}-c_{(2k)}\left(\frac{z}{L}\right)^{2k-2}\text{ln}\left(\frac{z}{L}\right){\cal
O}^{(2k)}_{\mu\nu}+o\big{(}(z/L)^{d}\big{)}\,.$ (17)

This makes contact with the expansion with a logarithmic term (for an even
$d$) presented in the literature, e.g. [24, 37, 36].

##  3 Weyl-Fefferman-Graham Gauge

This section is a brief review of the Weyl-Fefferman-Graham (WFG) formalism
established in [5]. At the end of this section we introduce the “Weyl
quantities” that will appear in later sections.

The Fefferman-Graham ansatz (2) is quite convenient for calculations,
especially in the context of holographic renormalization. In this setup, one
can induce a Weyl transformation of the boundary metric by a bulk
diffeomorphism, namely the PBH transformation[10],

$z\to z^{\prime}=z/{\cal{B}}(x)\,,\qquad x^{\mu}\to
x^{\prime\mu}=x^{\mu}+\xi^{\mu}(z;x)\,,$ (18)

where $\xi^{\mu}(z;x)$ vanish at the boundary $z=0$. The functions
$\xi^{\mu}(z;x)$ can be found (infinitesimally) in terms of ${\cal{B}}(x)$ by
the constraint that the form of the FG ansatz is preserved under the
transformation. However, under the PBH transformation, the subleading terms in
the FG expansion (3) do not transform in a Weyl-covariant way. The source of
this complication is the compensating diffeomorphisms $\xi^{\mu}(z;x)$
introduced for preserving the FG ansatz.

This above-mentioned issue motivated the authors of [5] to replace the FG
ansatz with

$\displaystyle\text{d}s^{2}=L^{2}\left(\frac{\text{d}z}{z}-a_{\mu}(z;x)\text{d}x^{\mu}\right)^{2}+h_{\mu\nu}(z;x)\text{d}x^{\mu}\text{d}x^{\nu}\,,$
(19)

which was named the Weyl-Fefferman-Graham ansatz. With the additional Weyl
structure $a_{\mu}$ added, the form of the WFG ansatz is now preserved under
the Weyl diffeomorphism

$\displaystyle z\to z^{\prime}=z/{\cal B}(x)\,,\qquad x^{\mu}\to
x^{\prime\mu}=x^{\mu}\,.$ (20)

It is not hard to see that the Weyl diffeomorphism (20) induces the following
transformation of the fields $a_{\mu}$ and $h_{\mu\nu}$:

$\displaystyle a_{\mu}(z;x)\to a^{\prime}_{\mu}(z^{\prime};x)=a_{\mu}({\cal
B}(x)z^{\prime};x)-\partial_{\mu}\ln{\cal B}(x)\,,\quad h_{\mu\nu}\to
h^{\prime}_{\mu\nu}(z^{\prime};x)=h_{\mu\nu}({\cal B}(x)z^{\prime};x)\,.$ (21)

Thus, we can now induce a Weyl transformation on the boundary and preserve the
form of the metric without introducing the irritating $\xi^{\mu}(z;x)$. Note
that according to the FG theorem, any AlAdS spacetime can always be expressed
in the FG form, and so (19) can be transformed into (2) under a suitable
diffeomorphism. This indicates that $a_{\mu}$ is actually pure gauge in the
bulk. Another way of going back to the FG gauge is to simply set $a_{\mu}$ to
zero; in this perspective, the FG gauge is nothing but a special case of the
WFG gauge with a fixed gauge.

The main utility of the WFG gauge is that all the terms (except one) in the
$z$-expansions of $h_{\mu\nu}(z;x)$ and $a_{\mu}(z;x)$ transform as Weyl
tensors under Weyl diffeomorphisms. To see this, let us expand $h_{\mu\nu}$
and $a_{\mu}$ near $z=0$:

$\displaystyle h_{\mu\nu}(z;x)$
$\displaystyle=\frac{L^{2}}{z^{2}}\left[\gamma^{(0)}_{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\gamma^{(2)}_{\mu\nu}(x)+\cdots\right]+\frac{z^{d-2}}{L^{d-2}}\left[\pi^{(0)}_{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\pi^{(2)}_{\mu\nu}(x)+\cdots\right]\,,$
(22) $\displaystyle a_{\mu}(z;x)$
$\displaystyle=\left[a^{(0)}_{\mu}(x)+\frac{z^{2}}{L^{2}}a^{(2)}_{\mu}(x)+\cdots\right]+\frac{z^{d-2}}{L^{d-2}}\left[p^{(0)}_{\mu}(x)+\frac{z^{2}}{L^{2}}p^{(2)}_{\mu}(x)+\cdots\right]\,.$
(23)

In the FG gauge where $a_{\mu}$ is turned off, the FG expansion only includes
(22), and the subleading terms $\gamma^{(2k)}_{\mu\nu}$ in the first series
are determined solely by the boundary induced metric $\gamma^{(0)}_{\mu\nu}$
and its derivatives. Now with the extra series (23), $\gamma^{(2k)}_{\mu\nu}$
will also depend on $a^{(0)}_{\mu}$, $a^{(2)}_{\mu}$, $a^{(4)}_{\mu}$, etc.
Moving on, from the transformations (21) under a Weyl diffeomorphism, one
finds the transformation of each term in the expansions (22) and (23) as
follows[5]:

$\displaystyle\gamma^{(2k)}_{\mu\nu}(x)$
$\displaystyle\to\gamma^{(2k)}_{\mu\nu}(x){\cal
B}(x)^{2k-2}\,,\qquad\pi^{(k)}_{\mu\nu}(x)\to\pi^{(2k)}_{\mu\nu}(x){\cal
B}(x)^{d-2+2k}\,,$ (24) $\displaystyle a^{(2k)}_{\mu}(x)$ $\displaystyle\to
a^{(2k)}_{\mu}(x){\cal B}(x)^{2k}-\delta_{k,0}\partial_{\mu}\ln{\cal
B}(x)\,,\qquad p^{(2k)}_{\mu}(x)\to p^{(2k)}_{\mu}(x){\cal B}(x)^{d-2+2k}\,.$
(25)

Indeed, we see that almost all the terms in the expansions transform Weyl-
covariantly. The only exception is $a^{(0)}_{\mu}$, which transforms
inhomogeneously under Weyl transformation, and thus does not have a definite
Weyl weight. All the other terms in the expansions (22) and (23) can be viewed
as tensor fields on the boundary and we can easily read off their Weyl weights
from the power of ${\cal B}(x)$ appearing in (24) and (25).

For a metric in the form of (19) defined on the bulk manifold $M$, one can
choose a dual form basis and its corresponding vector basis as follows:

$\displaystyle\bm{e}^{z}$
$\displaystyle=L\frac{\text{d}z}{z}-La_{\mu}(z;x)\text{d}x^{\mu}\,,\qquad\bm{e}^{\mu}=\text{d}x^{\mu}\,,$
(26) $\displaystyle\underline{e}_{z}$
$\displaystyle=\frac{z}{L}\underline{\partial}_{z}\equiv\underline{D}_{z}\,,\qquad\underline{e}_{\mu}=\underline{\partial}_{\mu}+za_{\mu}(z;x)\underline{\partial}_{z}\equiv\underline{D}_{\mu}\,.$
(27)

Then the tangent space at any point $(z,x^{\mu})\in M$ can be spanned by the
basis $\\{\underline{D}_{z},\underline{D}_{\mu}\\}$, and the basis vectors
$\\{\underline{D}_{\mu}\\}$ form a $d$-dimensional distribution on $M$ which
belongs to the kernel of $\bm{e}^{z}$. The Lie brackets of these basis vectors
are

$\displaystyle[\underline{D}_{\mu},\underline{D}_{\nu}]=Lf_{\mu\nu}\underline{D}_{z}\,,\qquad[\underline{D}_{z},\underline{D}_{\mu}]=L\varphi_{\mu}\underline{D}_{z}\,,$
(28)

where $\varphi_{\mu}\equiv D_{z}a_{\mu}$ and $f_{\mu\nu}\equiv
D_{\mu}a_{\nu}-D_{\nu}a_{\mu}$ ($D_{z}$ and $D_{\mu}$ represent taking the
derivatives along $\underline{e}_{z}$ and $\underline{e}_{\mu}$). According to
the Frobenius theorem, the condition for the distribution spanned by
$\\{\underline{D}_{\mu}\\}$ to be integrable is that
$[\underline{D}_{\mu},\underline{D}_{\nu}]=0$, i.e. $f_{\mu\nu}=0$. In this
case, this distribution defines a hypersurface. For instance, in the FG gauge
where $a_{\mu}$ is turned off, the distribution $\\{\underline{D}_{\mu}\\}$
becomes $\\{\underline{\partial}_{\mu}\\}$, which generates a foliation of
constant-$z$ surfaces. However, $\\{\underline{D}_{\mu}\\}$ in the WFG gauge
is not necessarily an integrable distribution, and thus one needs to keep in
mind that the boundary hypersurface $z=0$ is in general not part of a
foliation.

Suppose $\nabla$ is the Levi-Civita (LC) connection on $M$. One can find the
connection coefficients of $\nabla$ in the frame
$\\{\underline{D}_{z},\underline{D}_{\mu}\\}$ from its definition (83):

$\displaystyle\nabla_{\underline{D}_{\mu}}\underline{D}_{\nu}=\Gamma^{\lambda}{}_{\mu\nu}\underline{D}_{\lambda}+\Gamma^{z}{}_{\mu\nu}\underline{D}_{z}\,.$
(29)

The coefficients $\Gamma^{\lambda}{}_{\mu\nu}$ in the above equation define
the induced connection coefficients on the distribution over $M$ spanned by
$\\{\underline{D}_{\mu}\\}$ (see [38]). Expanding
$\Gamma^{\lambda}{}_{\mu\nu}$ with respect to $z$, at the leading order one
finds that

$\displaystyle\Gamma^{\lambda}_{(0)}{}_{\mu\nu}$
$\displaystyle=\frac{1}{2}\gamma_{(0)}^{\lambda\rho}\big{(}\partial_{\mu}\gamma^{(0)}_{\nu\rho}+\partial_{\nu}\gamma^{(0)}_{\mu\rho}-\partial_{\rho}\gamma^{(0)}_{\mu\nu}\big{)}-\big{(}a^{(0)}_{\mu}\delta^{\lambda}{}_{\nu}+a^{(0)}_{\nu}\delta^{\lambda}{}_{\mu}+a^{(0)}_{\rho}\gamma_{(0)}^{\lambda\rho}\gamma^{(0)}_{\mu\nu}\big{)}\,.$
(30)

We can see that (30) gives exactly the connection coefficients of a torsion-
free connection with Weyl metricity [see (94) in Appendix A, where $A_{\mu}$
and $g_{\mu\nu}$ correspond to $a^{(0)}_{\mu}$ and $\gamma^{(0)}_{\mu\nu}$].
That is, on the boundary with $z\to 0$ we have a connection $\nabla^{(0)}$
satisfying

$\displaystyle\nabla^{(0)}_{\mu}\gamma^{(0)}_{\nu\rho}=2a^{(0)}_{\mu}\gamma^{(0)}_{\nu\rho}\,.$
(31)

This indicates that although $a_{\mu}$ is pure gauge in the bulk, its leading
order $a_{\mu}^{(0)}$ serves as a Weyl connection at the conformal boundary.
Together with the induced metric $\gamma^{(0)}_{\mu\nu}$, they provide a Weyl
geometry at the boundary [13]. Under a boundary Weyl transformation

$\displaystyle\gamma_{\mu\nu}^{(0)}\to{\cal
B}(x)^{-2}\gamma^{(0)}_{\mu\nu}\,,\qquad a_{\mu}^{(0)}\to
a_{\mu}^{(0)}-\partial_{\mu}\text{ln}{\cal}B(x)\,,$ (32)

for any tensor $T$ (with indices suppressed) with Weyl weight $w_{T}$ on the
boundary, we have

$\displaystyle T\to
B^{w_{T}}T\,,\qquad(\nabla^{(0)}_{\mu}T+w_{T}a^{(0)}_{\mu}T)\to
B^{w_{T}}(\nabla^{(0)}_{\mu}T+w_{T}a^{(0)}_{\mu}T)\,.$ (33)

One can also absorb the Weyl connection and define $\hat{\nabla}^{(0)}$ such
that

$\displaystyle\hat{\nabla}^{(0)}_{\mu}T\equiv\nabla^{(0)}_{\mu}T+w_{T}a^{(0)}_{\mu}T\,,$
(34)

which renders $\hat{\nabla}^{(0)}_{\mu}T$ Weyl-covariant. Particularly, Eq.
(31) indicates that $\hat{\nabla}^{(0)}$ is metricity-free, which makes it
convenient for boundary calculations.

Now that we have the Weyl geometry on the boundary, the geometric quantities
there are promoted to the “Weyl quantities”. More precisely, for any geometric
quantity constructed by the boundary metric $\gamma^{(0)}_{\mu\nu}$ and the LC
connection in the FG case, we now have a Weyl-covariant counterpart of it
constructed by $\gamma^{(0)}_{\mu\nu}$, $a^{(0)}_{\mu}$ and
$\hat{\nabla}^{(0)}$ in the WFG case. For instance, we have the Weyl-Riemann
tensor $\hat{R}^{\mu}{}^{(0)}_{\nu\rho\sigma}$, Weyl-Ricci tensor
$\hat{R}^{(0)}_{\mu\nu}$ and Weyl-Ricci scalar $\hat{R}^{(0)}$. In addition,
$f_{\mu\nu}$ induces on the boundary a tensor
$f^{(0)}_{\mu\nu}=\partial_{\mu}a^{(0)}_{\nu}-\partial_{\nu}a^{(0)}_{\mu}$,
namely the curvature of the Weyl connection $a^{(0)}$, which is obviously
Weyl-invariant. We can also define the Weyl-Schouten tensor $\hat{P}_{\mu\nu}$
and Weyl-Cotton tensor $\hat{C}_{\mu\nu\rho}$ on the boundary as follows:

$\displaystyle\hat{P}_{\mu\nu}$
$\displaystyle=\frac{1}{d-2}\bigg{(}\hat{R}^{(0)}_{\mu\nu}-\frac{1}{2(d-1)}\hat{R}^{(0)}\gamma^{(0)}_{\mu\nu}\bigg{)}\,,$
(35) $\displaystyle\hat{C}_{\mu\nu\rho}$
$\displaystyle=\hat{\nabla}^{(0)}_{\rho}\hat{P}_{\mu\nu}-\hat{\nabla}^{(0)}_{\nu}\hat{P}_{\mu\rho}\,.$
(36)

One should notice that the symmetry of the indices of a “Weyl quantity” is not
necessarily the same as the corresponding quantity defined with the LC
connection. For instance, the Weyl-Ricci tensor is not symmetric, with its
antisymmetric part $\hat{R}^{(0)}_{[\mu\nu]}=-(d-2)f^{(0)}_{\mu\nu}/2$, and
hence the Weyl-Schouten tensor $\hat{P}_{\mu\nu}$ also contains an
antisymmetric part $\hat{P}_{[\mu\nu]}=-f^{(0)}_{\mu\nu}/2$. In the next
section, we will see that the obstruction tensors also have their Weyl-
covariant counterparts. More details of the above Weyl quantities are
exhibited in Appendix A.

##  4 Weyl-Obstruction Tensors

In the previous section we saw that the WFG gauge in the bulk induces a Weyl
geometry on the boundary. Now we would like to determine the higher order
terms in the expansion (22) and find the obstruction tensors with the Weyl
connection turned on. The method is exactly analogous to that in Section 2 for
the FG gauge. By solving the bulk Einstein equations order by order in the WFG
gauge, we find that $\gamma^{(2k)}_{\mu\nu}$ still has the same form as (4),
except that the obstruction tensor ${\cal O}^{(2k)}_{\mu\nu}$ is now promoted
to the _Weyl-obstruction tensor_ $\hat{{\cal O}}^{(2k)}_{\mu\nu}$. Unlike
${\cal O}^{(2k)}_{\mu\nu}$, which is only Weyl-covariant in $2k$-dimension,
the Weyl-obstruction tensors $\hat{{\cal O}}^{(2k)}_{\mu\nu}$ are Weyl-
covariant with a weight $2k-2$ in any dimension; that is, under a Weyl
transformation (32) it transforms in any $d$ as $\hat{{\cal
O}}^{(2k)}_{\mu\nu}\to{\cal B}(x)^{2k-2}\hat{{\cal O}}^{(2k)}_{\mu\nu}$.

In principle, $\gamma^{(2k)}_{\mu\nu}$ at any order can be obtained from the
Einstein equations by iteration. In this section, we will show solutions of
$\gamma^{(2k)}_{\mu\nu}$ obtained from Einstein equations up to $k=3$, and
read off the corresponding Weyl-obstruction tensors from them. Some detailed
expansions of Einstein equations can be found in Appendix B.

First, the leading order of the $\mu\nu$-components of the Einstein equations
gives

$\displaystyle\frac{\gamma^{(2)}_{\mu\nu}}{L^{2}}$
$\displaystyle=-\frac{1}{d-2}\bigg{(}\hat{R}^{(0)}_{(\mu\nu)}-\frac{1}{2(d-1)}\hat{R}^{(0)}\gamma^{(0)}_{\mu\nu}\bigg{)}\,.$
(37)

We notice that this is the symmetric part of the Weyl-Schouten tensor defined
in (35) with a minus sign, i.e.

$\displaystyle\frac{\gamma^{(2)}_{\mu\nu}}{L^{2}}$
$\displaystyle=-\hat{P}_{(\mu\nu)}=-\hat{P}_{\mu\nu}-\frac{1}{2}f^{(0)}_{\mu\nu}\,.$
(38)

Similar to the FG gauge, one can check that the residue of the pole in (37)
vanishes identically when $d=2$. Hence, there is no Weyl-obstruction tensor
for $d=2$ and so no logarithmic term will appear in the metric expansion in
the $d\to 2^{-}$ limit.

Then, solving the $O(z^{2})$-order of the $\mu\nu$-components of the Einstein
equations yields

$\displaystyle\frac{\gamma^{(4)}_{\mu\nu}}{L^{4}}$
$\displaystyle=-\frac{1}{4(d-4)}\hat{\cal
O}^{(4)}_{\mu\nu}+\frac{1}{4}\hat{P}^{\rho}{}_{\mu}\hat{P}_{\rho\nu}-\frac{1}{2L^{2}}\hat{\nabla}^{(0)}_{(\mu}a_{\nu)}^{(2)}\,,$
(39)

where $\hat{\cal O}^{(4)}_{\mu\nu}$ is the Weyl-obstruction tensor for $d=4$,
namely the Weyl-Bach tensor $\hat{B}_{\mu\nu}$, given by

$\displaystyle\hat{\cal
O}^{(4)}_{\mu\nu}=\hat{B}_{\mu\nu}=\hat{\nabla}^{(0)}_{\lambda}\hat{\nabla}_{(0)}^{\lambda}\hat{P}_{\mu\nu}-\hat{\nabla}^{(0)}_{\lambda}\hat{\nabla}^{(0)}_{\nu}\hat{P}_{\mu}{}^{\lambda}-\hat{W}^{(0)}_{\rho\nu\mu\lambda}\hat{P}^{\lambda\rho}\,.$
(40)

If we compare (39) with the corresponding result (7) in the FG case, we see
that the form of the expression stays almost the same, with all the LC
quantities now being promoted to the corresponding Weyl quantities. Besides,
in the WFG gauge $\gamma^{(4)}_{\mu\nu}$ also has an additional term involving
$a^{(2)}_{\mu}$, which does not contribute to the pole at $d=4$.

Moving on to the $O(z^{4})$-order of the Einstein equations we get

$\begin{split}\frac{\gamma^{(6)}_{\mu\nu}}{L^{6}}=&-\frac{1}{24(d-6)(d-4)}\hat{\cal
O}^{(6)}_{\mu\nu}+\frac{1}{6(d-4)}\hat{B}_{\rho(\mu}\hat{P}^{\rho}{}_{\nu)}-\frac{1}{3L^{4}}\hat{\nabla}^{(0)}_{(\mu}a^{(4)}_{\nu)}\\\
&-\frac{1}{L^{4}}a^{(2)}_{\mu}a^{(2)}_{\nu}+\frac{1}{6L^{2}}a^{(2)}\cdot
a^{(2)}\gamma^{(0)}_{\mu\nu}+\frac{1}{6L^{2}}\hat{\nabla}^{(0)}_{(\mu}(\hat{P}^{\rho}{}_{\nu)}a^{(2)}_{\rho})+\frac{1}{2L^{4}}\hat{\gamma}_{(2)}^{\sigma}{}_{\mu\nu}a^{(2)}_{\sigma}\,,\end{split}$
(41)

where
$\hat{\gamma}_{(2)}^{\sigma}{}_{\mu\nu}\equiv-\frac{L^{2}}{2}(\hat{\nabla}^{(0)}_{\mu}\hat{P}^{\lambda}{}_{\nu}+\hat{\nabla}^{(0)}_{\nu}\hat{P}_{\mu}{}^{\lambda}-\hat{\nabla}_{(0)}^{\lambda}\hat{P}_{\mu\nu})$,
and $\hat{\cal O}^{(6)}_{\mu\nu}$ is the Weyl-obstruction tensor for $d=6$:

$\begin{split}\hat{\cal
O}^{(6)}_{\mu\nu}={}&\hat{\nabla}_{(0)}^{\lambda}\hat{\nabla}^{(0)}_{\lambda}\hat{B}_{\mu\nu}-2\hat{W}^{(0)}_{\rho\nu\mu\lambda}\hat{B}^{\lambda\rho}-4\hat{P}\hat{B}_{\mu\nu}+2\hat{P}_{\rho(\nu}\hat{B}^{\rho}{}_{\mu)}-2\hat{B}^{\rho}{}_{(\mu}\hat{P}_{\nu)\rho}\\\
&+2(d-4)\bigg{(}\hat{\nabla}_{(0)}^{\lambda}\hat{C}_{\lambda\rho(\mu}\hat{P}^{\rho}{}_{\nu)}-\hat{P}^{\lambda\rho}\hat{\nabla}^{(0)}_{(\mu}\hat{C}_{\nu)\rho\lambda}+2\hat{P}^{(\rho\lambda)}\hat{\nabla}^{(0)}_{\lambda}\hat{C}_{(\mu\nu)\rho}+\hat{\nabla}^{(0)}_{\lambda}\hat{P}^{\rho\lambda}\hat{C}_{(\mu\nu)\rho}\\\
&\qquad\qquad\qquad-\hat{C}^{\rho}{}_{\mu}{}^{\lambda}\hat{C}_{\lambda\nu\rho}+\hat{\nabla}_{(0)}^{\lambda}\hat{P}^{\rho}{}_{(\mu}\hat{C}_{\nu)\rho\lambda}-\hat{W}^{(0)}_{\rho(\nu\mu)\lambda}\hat{P}^{\lambda}{}_{\sigma}\hat{P}^{\sigma\rho}\bigg{)}\,.\end{split}$
(42)

It is easy to verify that (41) and (42) go back to the FG expressions (9) and
(2) when we turn off the Weyl structure $a_{\mu}$. Note that when the Weyl
connection is turned off, the first term inside the parentheses of (42)
vanishes due to (115), and the second term there vanishes since the LC
Schouten tensor $\mathring{P}_{\mu\nu}$ is symmetric. Once again, we observe
that all the $a^{(2)}_{\mu}$ and $a^{(4)}_{\mu}$ terms that appear in
$\gamma^{(6)}_{\mu\nu}$ do not contribute to the pole at $d=6$ and thus are
not part of the obstruction tensor $\hat{\cal O}_{\mu\nu}^{(6)}$. We will
discuss this more in Section 6.

Just as ${\cal O}_{\mu\nu}^{(2k)}$ derived in the FG gauge, all the $\hat{\cal
O}_{\mu\nu}^{(2k)}$ are also symmetric traceless tensors, and they are
divergence-free when $d=2k$. These properties can either be verified by using
the result from the $\mu\nu$-components of the Einstein equations (“evolution
equations”), or read off from the $zz$\- and $z\mu$-components of the Einstein
equations (“constraint equations”). More specifically, plugging
$\gamma^{(2k)}_{\mu\nu}$ into the $zz$-component of the Einstein equations we
can see that $\mathcal{\hat{O}}^{(2k)}_{\mu\nu}$ is traceless in any
dimension, and the same result can also be obtained by taking the trace of the
$\mu\nu$-components of the Einstein equations. To see that $\hat{\cal
O}_{\mu\nu}^{(2k)}$ is divergence-free when $d=2k$, we can plug
$\gamma^{(2k)}_{\mu\nu}$ into the $z\mu$-components of the Einstein equations.
For instance, the $O(z^{4})$-order of the $z\mu$-equations gives

$\displaystyle\hat{\nabla}_{(0)}^{\nu}\hat{B}_{\nu\mu}=(d-4)\hat{P}^{\nu\rho}(\hat{C}_{\rho\nu\mu}+\hat{C}_{\mu\nu\rho})\,,$
(43)

and so the divergence of $\hat{B}_{\mu\nu}$ vanishes when $d=4$. In the FG
gauge where the Schouten tensor is symmetric, the second term in the bracket
vanishes and so (43) goes back to (12). On the other hand, the divergence of
$\hat{\cal O}_{\mu\nu}^{(2k)}$ can also be derived from a direct calculation
by using repeatedly the Weyl-Bianchi identity

$\displaystyle\hat{\nabla}_{(0)}^{\nu}\hat{P}_{\nu\mu}=\hat{\nabla}^{(0)}_{\mu}\hat{P}\,,$
(44)

which can be read off from the $O(z^{2})$-order of the $z\mu$-equation. The
above discussion indicates that the $zz$\- and $z\mu$-components of the
Einstein equations do not contain more information about
$\gamma^{(2k)}_{\mu\nu}$ than the $\mu\nu$-components of Einstein equations.
Note that here we only talk about the equations of motion for
$\gamma^{(2k)}_{\mu\nu}$. At $O(z^{d})$-order the $zz$\- and $z\mu$-equations
do provide new constraints on $\pi^{(0)}_{\mu\nu}$, while the
$\mu\nu$-equations on $\pi^{(0)}_{\mu\nu}$ become trivial.

It is also convenient to define the _extended Weyl-obstruction tensor_
$\hat{\Omega}^{(k)}_{\mu\nu}$ as the Weyl-covariant version of the extended
obstruction tensor defined in (14). For example, for $k=1$ and $k=2$ we have

$\displaystyle\hat{\Omega}^{(1)}_{\mu\nu}=-\frac{1}{d-4}\hat{B}_{\mu\nu}\,,\qquad\hat{\Omega}^{(2)}_{\mu\nu}=\frac{1}{(d-6)(d-4)}\hat{\mathcal{O}}^{(6)}_{\mu\nu}\,.$
(45)

Similar to the FG case, the Weyl-obstruction tensor $\hat{\cal
O}_{\mu\nu}^{(2k+2)}$ is also proportional to the residue of the extended
Weyl-obstruction tensor $\hat{\Omega}^{(k)}_{\mu\nu}$. Both the Weyl-
obstruction tensors and the extended Weyl-obstruction tensors can be defined
following [7, 8] by promoting the ambient metric to the “Weyl-ambient metric”.
We will discuss this in detail in a separate publication.

##  5 Holographic Weyl Anomaly

###  5.1 Weyl-Ward Identity

In this section, we first discuss the anomalous Weyl-Ward identity for a
general field theory on a background Weyl geometry following [5], and then we
focus on holographic theories in the WFG gauge. Later, we will compute the
Weyl anomaly for a holographic theory in the WFG gauge up to $d=8$.

Essentially, for a $d$-dimensional field theory333From now on, we will work in
the Euclidean signature. We also adopt natural units where $c=\hbar=1$.
coupled to a background metric $\gamma^{(0)}_{\mu\nu}$ and a Weyl connection
$a^{(0)}_{\mu}$, the Weyl anomaly comes from an additional exponential factor
arising in the path integral after applying a Weyl transformation:

$\displaystyle Z[\gamma^{(0)},a^{(0)}]=\text{e}^{-{\cal A}[{\cal
B}(x);\gamma^{(0)},a^{(0)}]}Z[\gamma^{(0)}/{\cal
B}(x)^{2},a^{(0)}-\text{d}\ln{\cal B}(x)]\,.$ (46)

The anomaly ${\cal A}[{\cal B}(x);g,a]$ should satisfy the 1-cocycle condition
[39, 40]

$\displaystyle{\cal A}[{\cal B^{\prime\prime}}{\cal
B^{\prime}};\gamma^{(0)},a^{(0)}]={\cal A}[{\cal
B^{\prime}};\gamma^{(0)},a^{(0)}]+{\cal A}[{\cal
B^{\prime\prime}};\gamma^{(0)}/({\cal
B^{\prime}})^{2},a^{(0)}-\text{d}\ln{\cal B^{\prime}}]\,.$ (47)

For any non-exact Weyl-invariant $d$-form $\bm{A}[\gamma_{(0)},a_{(0)}]$, one
can check that ${\cal A}[{\cal B}(x);\gamma^{(0)},a^{(0)}]=\int(\ln{\cal
B})\bm{A}$ satisfies the cocycle condition, and thus it is a possible
candidate for the Weyl anomaly. However, if $\bm{A}$ is exact, ${\cal A}$
would be cohomologically trivial since it can be written as the difference of
a Weyl-transformed local functional. The linearly independent choices of
$\bm{A}$ in non-trivial cocycles correspond to different central charges.

In general, the background fields $\gamma^{(0)}_{\mu\nu}$ and $a^{(0)}_{\mu}$
are the sources of the energy-momentum tensor operator $T^{\mu\nu}$ and the
Weyl current operator $J^{\mu}$, respectively:

$\displaystyle\langle
T^{\mu\nu}(x)\rangle=\frac{2}{\sqrt{-\det\gamma^{(0)}}}\frac{\delta
S}{\delta\gamma^{(0)}_{\mu\nu}(x)}\,,\qquad\langle
J^{\mu}(x)\rangle=-\frac{1}{\sqrt{-\det\gamma^{(0)}}}\frac{\delta S}{\delta
a^{(0)}_{\mu}(x)}\,.$ (48)

Expanding the quantum effective action $S\equiv-\ln Z$ to the first order
under an infinitesimal Weyl transformation and integrating by parts, for a
theory with a Weyl anomaly we obtain

$\displaystyle\frac{1}{\sqrt{-\det\gamma^{(0)}}}\frac{\delta\cal
A}{\delta\ln{\cal
B}(x)}=\big{\langle}T^{\mu\nu}(x)\gamma^{(0)}_{\mu\nu}(x)+\hat{\nabla}^{(0)}_{\mu}J^{\mu}(x)\big{\rangle}\,.$
(49)

This is the (anomalous) Weyl-Ward identity. As we can see, besides the trace
of the energy-momentum tensor that appears in the usual case, the divergence
of the Weyl current also contributes to the Ward identity when the Weyl
connection is turned on.

Let us now focus on a holographic field theory dual to the vacuum Einstein
theory in the $(d+1)$-dimensional bulk. The holographic dictionary provides
the relation between the on-shell classical bulk action $S_{bulk}$ and quantum
effective action $S_{bdr}$ of the field theory on the boundary [27]:

$\displaystyle\exp\left(-S_{bulk}[g;\gamma_{(0)},a_{(0)}]\right)=\exp\left(-S_{bdr}[\gamma_{(0)},a_{(0)}]\right)\,,$
(50)

where $\gamma_{(0)}$ and $a_{(0)}$ are the boundary values of $h$ and $a$ as
shown in (22) and (23). Since $a_{\mu}$ is pure gauge in the bulk,
$a^{(0)}_{\mu}$ could be gauged away and hence it is not expected to source
any current on the boundary. The role of the $a^{(0)}_{\mu}$, however, is
important since it makes the energy-momentum tensor along with all the
geometric quantities on the boundary Weyl-covariant. On the other hand, the
$p^{(0)}_{\mu}$ also plays a role in the Weyl-Ward identity. In the FG gauge,
$\pi^{(0)}_{\mu\nu}$ corresponds to the expectation value of $T_{\mu\nu}$; the
Ward identity for the Weyl symmetry shows that the trace of
$\pi^{(0)}_{\mu\nu}$ vanishes, which can be read off from the $O(z^{d})$-order
of the $zz$-component of the Einstein equations[6]. In the WFG gauge, this
equation now gives

$\displaystyle
0=\frac{d}{2L^{2}}\gamma_{(0)}^{\mu\nu}\pi^{(0)}_{\mu\nu}+\hat{\nabla}^{(0)}\cdot
p_{(0)}\,.$ (51)

Besides $\pi^{(0)}_{\mu\nu}$, there is an additional term
$\hat{\nabla}^{(0)}\cdot p_{(0)}$ which represents a gauge ambiguity of
$a_{\mu}$. This suggests that the energy-momentum tensor in the WFG gauge
acquires an extra piece, which now can be considered as an “improved” energy-
momentum tensor $\tilde{T}_{\mu\nu}$ (à la[41, 42]):

$\langle\kappa^{2}\tilde{T}_{\mu\nu}\rangle=\frac{d}{2L^{2}}\pi^{(0)}_{\mu\nu}+\hat{\nabla}^{(0)}_{(\mu}p^{(0)}_{\nu)}\,,$
(52)

where $\kappa^{2}=8\pi G$.444The energy-momentum tensor (52) in the WFG gauge
can be verified using the prescription introduced in[37]. It is easy to see
that the trace of this energy-momentum tensor gives the right-hand side of
(51). One can also find that the $z\mu$-components of the Einstein equations
at the $O(z^{d})$-order give exactly the conservation law
$\langle\hat{\nabla}_{(0)}^{\mu}\tilde{T}_{\mu\nu}\rangle=0$ [see (132)],
which is the Ward identity corresponding to the boundary diffeomorphisms.
Therefore, in the holographic case we can write the anomalous Weyl-Ward
identity (49) as

$\frac{1}{\sqrt{-\det\gamma^{(0)}}}\frac{\delta\cal A}{\delta\ln{\cal
B}(x)}=\big{\langle}\tilde{T}^{\mu\nu}(x)\gamma^{(0)}_{\mu\nu}(x)\big{\rangle}\,.$
(53)

Notice that one should distinguish $p^{(0)}_{\mu}$ and the Weyl current
$J_{\mu}$. Unlike $\pi_{\mu\nu}^{(0)}$ which is sourced by
$\gamma^{(0)}_{\mu\nu}$, $p^{(0)}_{\mu}$ is not sourced by $a^{(0)}_{\mu}$
since $a_{\mu}$ is pure gauge in the bulk. In the boundary field theory, the
Weyl current $J_{\mu}$ vanishes identically, while $p^{(0)}_{\mu}$ contributes
to the expectation value of $\tilde{T}_{\mu\nu}$ as an “improvement”. In a
generic non-holographic field theory defined on the background with Weyl
geometry, there may exist a nonvanishing $J_{\mu}$ sourced by the Weyl
connection $a^{(0)}_{\mu}$ (see [5] for an example).

Using the basis $\\{\bm{e}^{z},\bm{e}^{\mu}=\text{d}x^{\mu}\\}$ in (26), the
bulk on-shell Einstein-Hilbert action with negative cosmological constant can
be written as

$\displaystyle S_{bulk}$
$\displaystyle=\frac{1}{2\kappa^{2}}\int_{M}\sqrt{-\det
g}\,(R-2\Lambda)\bm{e}^{z}\wedge\text{d}x^{1}\wedge\cdots\wedge\text{d}x^{d}\,.$
(54)

Note that $\sqrt{-\det g}=\sqrt{-\det h}$. Considering the vacuum Einstein
equation in the bulk and the expansion

$\displaystyle\sqrt{-\det h}$
$\displaystyle=\left(\frac{L}{z}\right)^{d}\sqrt{-\det\gamma^{(0)}}\left(1+\frac{1}{2}\left(\frac{z}{L}\right)^{2}X^{(1)}+\frac{1}{2}\left(\frac{z}{L}\right)^{4}X^{(2)}+\cdots+\frac{1}{2}\left(\frac{z}{L}\right)^{d}Y^{(1)}+\cdots\right)\,,$
(55)

one can expand (54) as

$\displaystyle S_{bulk}$
$\displaystyle=-\frac{L^{-2}}{\kappa^{2}}\int_{M}\left(\frac{L}{z}\right)^{d}\left(d+\frac{d}{2}\left(\frac{z}{L}\right)^{2}X^{(1)}+\frac{d}{2}\left(\frac{z}{L}\right)^{4}X^{(2)}+\cdots+\frac{d}{2}\left(\frac{z}{L}\right)^{d}Y^{(1)}+\cdots\right)\bm{e}^{z}\wedge
vol_{\Sigma}\,,$ (56)

where
$vol_{\Sigma}\equiv\sqrt{-\det\gamma^{(0)}}\text{d}x^{1}\wedge\cdots\wedge\text{d}x^{d}$.

When the bulk action transforms under a Weyl diffeomorphism, the corresponding
boundary theory undergoes a Weyl transformation. However, the diffeomorphism
invariance of the bulk Einstein theory does not imply the Weyl invariance on
the boundary when there is an anomaly[43], since it follows from (46) that

$\displaystyle
0=S_{bulk}[g|z^{\prime},x^{\prime}]-S_{bulk}[g|z,x]=S_{bdr}[\gamma^{\prime}_{(0)},a^{\prime}_{(0)}|x]-S_{bdr}[\gamma_{(0)},a_{(0)}|x]+{\cal
A}[{\cal B}]\,,$ (57)

where $(z^{\prime},x^{\prime})=(z/{\cal B},x)$ for the bulk and
$\gamma^{\prime}_{(0)}=\gamma_{(0)}/{\cal B}^{2}$,
$a^{\prime}_{(0)}=a_{(0)}-\text{d}\ln{\cal B}$ for the boundary.

Normally, to compute the Weyl anomaly first one needs to regularize the bulk
on-shell action (56) by introducing a cutoff surface at some small value of
$z=\epsilon$, and then add counterterms to cancel the divergences when
$\epsilon\to 0$[24]. This is essentially how the Weyl anomaly arises since the
regulator breaks the Weyl symmetry and causes the appearance of a
logarithmically divergent term. However, since we do not assume that we have
an integrable distribution when the Weyl structure is turned on, the cutoff
regularization scheme is inconvenient for the WFG gauge. It has been
elucidated in [5] using dimensional regularization that the Weyl anomaly can
be extracted from the pole of $S_{bulk}$ that arises in an even dimension. By
evaluating the difference of the pole term in $S_{bulk}$ under a Weyl
diffeomorphism, one finds that the Weyl anomaly ${\cal A}_{k}$ of the
$2k$-dimensional boundary theory is

$\displaystyle{\cal A}_{k}=\frac{k}{\kappa^{2}L}\int\ln{\cal
B}X^{(k)}_{d=2k}vol_{\Sigma}\,.$ (58)

Therefore, to find the Weyl anomaly in $2k$-dimension, we only have to compute
$X^{(k)}$ coming from the expansion of $\sqrt{-\det h}$.

###  5.2 Weyl Anomaly in $2d$ and $4d$

Now let us apply (58) to $2d$ and $4d$. Here we first go over the WFG results
presented in [5], and then make a few important remarks. To find the
holographic Weyl anomaly in $2d$ and $4d$ all we have to do is plug in the
expressions of $X^{(1)}$ and $X^{(2)}$ obtained from the $zz$-components of
the Einstein equations (see Appendix B), that is,

$\displaystyle X^{(1)}=-\frac{L^{2}}{2(d-1)}\hat{R}\,,\qquad
X^{(2)}=-\frac{L^{4}}{4(d-2)^{2}}\bigg{(}\hat{R}_{\mu\nu}\hat{R}^{\nu\mu}-\frac{d}{4(d-1)}\hat{R}^{2}\bigg{)}-\frac{L^{2}}{2}\hat{\nabla}\cdot
a^{(2)}\,.$ (59)

[From now on we will drop the label “(0)” for the boundary curvature
quantities and derivative operator when there is no confusion.] First we look
at the Weyl anomaly in $d=2$:

$\displaystyle{\cal A}_{1}$ $\displaystyle=\frac{1}{\kappa^{2}L}\int\ln{\cal
B}X^{(1)}_{d=2}vol_{\Sigma}=-\frac{L}{16\pi G}\int\ln{\cal
B}\hat{R}\sqrt{-\det\gamma^{(0)}}\text{d}^{2}x\,,$ (60)

where in the second equality we used (59). Then, it follows from (53) that the
Weyl-Ward identity now reads

$\displaystyle\langle{\tilde{T}}^{\mu}{}_{\mu}\rangle=-\frac{L}{16\pi
G}\hat{R}\,.$ (61)

We can see that the right-hand side of this result has exactly the same form
as what we get from the standard calculation in the FG gauge, except that the
curvature scalar now is Weyl-covariant. Similarly, plugging (59) into (58), we
find that the Weyl anomaly in $d=4$ can be written as

$\displaystyle{\cal A}_{2}$ $\displaystyle=\frac{2}{\kappa^{2}L}\int\ln{\cal
B}X^{(2)}_{d=4}vol_{\Sigma}=-\frac{L}{8\pi
G}\int\bigg{[}\frac{L^{2}}{8}\Big{(}\hat{R}_{\mu\nu}\hat{R}^{\nu\mu}-\frac{1}{3}\hat{R}^{2}\Big{)}+\hat{\nabla}\cdot
a^{(2)}\bigg{]}\ln{\cal B}\sqrt{-\det\gamma^{(0)}}\text{d}^{4}x\,.$ (62)

Again, one can immediately tell that the right-hand side of this result
matches the standard FG result (e.g. [24]) if we turn off the Weyl structure.

There are a few things worth paying attention to: first, in the $2d$ Weyl
anomaly (60), the Weyl-Ricci scalar is also the Weyl-Euler density $E^{(2)}$
in $2d$, i.e. the Euler density Weyl-covariantized by the Weyl connection.
Furthermore, we can rewrite the $4d$ Weyl anomaly (62) as

$\displaystyle{\cal A}_{2}$ $\displaystyle=-\frac{L}{8\pi
G}\int\bigg{[}\frac{L^{2}}{16}\Big{(}\hat{W}_{\mu\nu\rho\sigma}\hat{W}^{\rho\sigma\mu\nu}-\hat{E}^{(4)}\Big{)}+\hat{\nabla}\cdot
a^{(2)}\bigg{]}\ln{\cal B}\sqrt{-\det\gamma^{(0)}}\text{d}^{4}x\,,$ (63)

where $\hat{E}^{(4)}$ is the Weyl-Euler density in $4d$:

$\displaystyle\hat{E}^{(4)}=\hat{R}_{\mu\nu\rho\sigma}\hat{R}^{\rho\sigma\mu\nu}-4\hat{R}_{\mu\nu}\hat{R}^{\nu\mu}+\hat{R}^{2}\,.$
(64)

Traditionally, the Euler density $E^{(2k)}$ without the Weyl connection is
called the type A Weyl anomaly, which is topological in $2k$-dimension and not
Weyl-invariant, while the type B Weyl anomaly is the Weyl-invariant part of
the anomaly [23]. Here we find that in the WFG gauge, this classification of
the Weyl anomaly is still available, with the Weyl-Euler density now Weyl-
invariant since the curvature quantities in this setup are endowed with Weyl
covariance.

Also, notice that the subleading term $a^{(2)}_{\mu}$ of $a_{\mu}$ only makes
an appearance in the anomaly through a cohomologically trivial term, i.e. we
can express it as a Weyl-transformed local functional as follows:

$\displaystyle\int\text{d}^{4}x\sqrt{-\det\gamma_{(0)}}\ln{\cal
B}\,\hat{\nabla}_{\mu}a_{(2)}^{\mu}=\int\text{d}^{4}x\sqrt{-\det\gamma^{\prime}_{(0)}}\,a^{\prime(0)}_{\mu}a^{\prime\mu}_{(2)}-\int\text{d}^{4}x\sqrt{-\det\gamma_{(0)}}\,a^{(0)}_{\mu}a_{(2)}^{\mu}\,,$
(65)

where $a^{\prime\mu}_{(2)}={\cal B}^{4}a_{(2)}^{\mu}$, and the boundary term
due to integrating by parts is ignored. We will see that this is a generic
feature of the Weyl anomaly in the WFG gauge for any dimension.

Although in (60) and (62) we expressed the holographic Weyl anomaly in $2d$
and $4d$ in terms of curvature to match the corresponding familiar results in
the FG gauge, we can also express them alternatively in terms of the Weyl-
Schouten tensor:

$\displaystyle\frac{X^{(1)}}{L^{2}}=-\hat{P}\,,\qquad\frac{X^{(2)}}{L^{4}}=-\frac{1}{4}\text{tr}(\hat{P}^{2})+\frac{1}{4}\hat{P}^{2}-\frac{1}{2L^{2}}\hat{\nabla}\cdot
a^{(2)}\,.$ (66)

Then (60) and (62) can be written as

$\displaystyle{\cal A}_{1}$
$\displaystyle=-\frac{L}{\kappa^{2}}\int\text{d}^{2}x\sqrt{-\det\gamma^{(0)}}\ln{\cal
B}\hat{P}\,,$ (67) $\displaystyle{\cal A}_{2}$
$\displaystyle=-\frac{L^{3}}{\kappa^{2}}\int\text{d}^{4}x\sqrt{-\det\gamma^{(0)}}\ln{\cal
B}\bigg{(}\frac{1}{2}\text{tr}(\hat{P}^{2})-\frac{1}{2}\hat{P}^{2}+\frac{1}{L^{2}}\hat{\nabla}\cdot
a^{(2)}\bigg{)}\,.$ (68)

In higher dimensions, $X^{(k)}$ can be expressed in terms of
$\gamma_{\mu\nu}^{(0\leqslant j\leqslant 2k)}$ (see Appendix C). By solving
the Einstein equations we have seen that these terms can all be expressed in
terms of $\hat{P}_{\mu\nu}$ and $\hat{\cal O}^{(2<j<2k)}_{\mu\nu}$. Therefore,
we will use the Weyl-Schouten tensor and Weyl-obstruction tensors as the
building blocks for the Weyl anomaly in even dimensions.

###  5.3 Weyl Anomaly in $6d$

After revisiting the results in $2d$ and $4d$, we will now present our
computations for $6d$ and $8d$. In principle, $X^{(k)}$ can be obtained by
solving Einstein equations as we have done for $2d$ and $4d$. However, as the
dimension goes higher, computing the curvature will become extremely tedious.
To facilitate the computation in higher dimensions, we can use a more
efficient way of organizing the Einstein equations which helps us avoid the
curvature tensors, namely to use the Raychaudhuri equation of the congruence
generated by $\underline{D}_{z}$. The details of the Raychaudhuri equation and
its expansions are given in Appendix C.

To solve for $X^{(3)}$, we need to expand $\sqrt{-\det h}$ to the order
$O(z^{6-d})$. Using (144) and plugging the results we have got for
$\gamma^{(2)}_{\mu\nu},\gamma^{(4)}_{\mu\nu}$ and $X^{(1)},X^{(2)}$ into
(147), we obtain

$\displaystyle\frac{X^{(3)}}{L^{6}}=$
$\displaystyle-\frac{1}{12}\text{tr}(\hat{P}^{3})+\frac{1}{8}\text{tr}(\hat{P}^{2})\hat{P}-\frac{1}{24}\hat{P}^{3}+\frac{1}{12}\text{tr}(\hat{\Omega}^{(1)}\hat{P})$
$\displaystyle+\frac{1}{6L^{4}}(d-6)a^{2}_{(2)}-\frac{1}{3L^{4}}\hat{\nabla}\cdot
a^{(4)}-\frac{1}{12L^{2}}\hat{\nabla}_{\mu}\big{[}a^{(2)}_{\nu}(3\hat{P}^{\mu\nu}+\hat{P}^{\nu\mu}-3\hat{P}\gamma_{(0)}^{\mu\nu})\big{]}\,,$
(69)

where we used the extended Weyl-obstruction tensor
$\hat{\Omega}^{(1)}_{\mu\nu}$ defined in (45). Notice first that the
$a^{(2)}_{\mu}$ quadratic term in $X^{(3)}$ vanishes in $6d$, and thus does
not contribute to the Weyl anomaly. Then, it follows from (58) that the Weyl
anomaly in $6d$ is

$\displaystyle{\cal A}_{3}={}$ $\displaystyle\frac{3}{\kappa^{2}L}\int\ln{\cal
B}X^{(3)}_{d=6}vol_{\Sigma}$ $\displaystyle={}$
$\displaystyle-\frac{L^{5}}{\kappa^{2}}\int\text{d}^{6}x\sqrt{-\det\gamma^{(0)}}\ln{\cal
B}\bigg{(}\frac{1}{4}\text{tr}(\hat{P}^{3})-\frac{3}{8}\text{tr}(\hat{P}^{2})\hat{P}+\frac{1}{8}\hat{P}^{3}-\frac{1}{4}\text{tr}(\hat{\Omega}^{(1)}\hat{P})$
$\displaystyle+\frac{1}{L^{4}}\hat{\nabla}\cdot
a^{(4)}+\frac{1}{4L^{2}}\hat{\nabla}_{\mu}\big{[}a^{(2)}_{\nu}(3\hat{P}^{\mu\nu}+\hat{P}^{\nu\mu}-3\hat{P}\gamma_{(0)}^{\mu\nu})\big{]}\bigg{)}\,.$
(70)

Just as what we have shown for the $4d$ case, the subleading terms in the
expansion of $a_{\mu}$ appear only in total derivatives and thus only
contribute to cohomologically trivial terms in the $6d$ Weyl anomaly. When we
turn off $a_{\mu}^{(0)}$ and $a_{\mu}^{(2)}$, this result agrees with the
holographic Weyl anomaly in the FG gauge computed in [24].

Usually, the Weyl anomaly in $6d$ is written as a linear combination of the
$6d$ Euler density and three conformal invariants in $6d$ (see [21, 23, 24]),
which represents the four central charges in $6d$. The result we obtained can
also be written in this way, which means the classification of type A and type
B anomalies still holds for the WFG gauge in $6d$. However, as we will discuss
shortly, the expression we have in (69) in terms of $\hat{P}_{\mu\nu}$ and
$\hat{\Omega}^{(1)}_{\mu\nu}$ reveals some interesting aspects of the Weyl
anomaly.

###  5.4 Weyl Anomaly in $8d$

Expanding $\sqrt{-\det h}$ to the order $O(z^{8-d})$, we have $X^{(4)}$ in
(148). Using (145) and plugging the results up to $\gamma^{(6)}_{\mu\nu}$ and
$X^{(3)}$ into (148), we have

$\displaystyle\frac{X^{(4)}}{L^{8}}=$
$\displaystyle-\frac{1}{32}\text{tr}(\hat{P}^{4})+\frac{1}{24}\text{tr}(\hat{P}^{3})\hat{P}+\frac{1}{64}(\text{tr}(\hat{P}^{2}))^{2}-\frac{1}{32}\text{tr}(\hat{P}^{2})\hat{P}^{2}+\frac{1}{192}\hat{P}^{4}$
$\displaystyle-\frac{1}{24}\text{tr}(\hat{\Omega}^{(1)}\hat{P})\hat{P}+\frac{1}{24}\text{tr}(\hat{\Omega}^{(1)}\hat{P}^{2})-\frac{1}{96}\text{tr}(\hat{\Omega}^{(1)}\hat{\Omega}^{(1)})-\frac{1}{96}\text{tr}(\hat{\Omega}^{(2)}\hat{P})$
$\displaystyle+\frac{d-8}{4L^{6}}a^{(4)}\cdot
a^{(2)}+\frac{d-8}{12L^{4}}a^{(2)}_{\mu}a^{(2)}_{\nu}(\hat{P}^{\mu\nu}-\hat{P}\gamma_{(0)}^{\mu\nu})+\text{total
derivatives}\,.$ (71)

As expected, all the terms in (5.4) that involve $a^{(2)}_{\mu}$,
$a^{(4)}_{\mu}$, $a^{(6)}_{\mu}$ either vanish when $d=8$ or contribute only
to the total derivatives. The details of the total derivatives are given in
(149). Plugging (5.4) into (58), we obtain the holographic Weyl anomaly in
$8d$:

$\displaystyle{\cal A}_{4}={}$ $\displaystyle\frac{4}{\kappa^{2}L}\int\ln{\cal
B}X^{(4)}_{d=8}vol_{\Sigma}$ $\displaystyle={}$
$\displaystyle-\frac{L^{7}}{\kappa^{2}}\int\text{d}^{8}x\sqrt{-\det\gamma^{(0)}}\ln{\cal
B}\bigg{(}\frac{1}{8}\text{tr}(\hat{P}^{4})-\frac{1}{6}\text{tr}(\hat{P}^{3})\hat{P}-\frac{1}{16}(\text{tr}(\hat{P}^{2}))^{2}+\frac{1}{8}\text{tr}(\hat{P}^{2})\hat{P}^{2}-\frac{1}{48}\hat{P}^{4}$
$\displaystyle+\frac{1}{6}\text{tr}(\hat{\Omega}^{(1)}\hat{P})\hat{P}-\frac{1}{6}\text{tr}(\hat{\Omega}^{(1)}\hat{P}^{2})+\frac{1}{24}\text{tr}(\hat{\Omega}^{(1)}\hat{\Omega}^{(1)})+\frac{1}{24}\text{tr}(\hat{\Omega}^{(2)}\hat{P})+\text{total
derivatives}\bigg{)}\,.$ (72)

Once again, we can see that the subleading terms in $a_{\mu}$ only have
cohomologically trivial contributions. If we go back to the FG gauge, then
this result agrees with the renormalized volume coefficient for $k=4$ shown in
[8]. One can also write the FG version of the above result in the traditional
way as a linear combination of the type A and type B anomalies, i.e. the Euler
density and Weyl invariants (the list of Weyl invariants in $8d$ can be found
in [44]). We naturally expect that this classification can also be applied to
the holographic Weyl anomaly in the WFG gauge for higher dimensions.

###  5.5 Building Blocks of the Weyl Anomaly

As we have seen, if we ignore the total derivatives that depend on the
subleading terms of the $a_{\mu}$ expansion, $X^{(1)}$ corresponds to the
Weyl-Ricci scalar (i.e. the $2d$ Weyl-Euler density) and $X^{(2)}$ corresponds
to the classic “$a=c$” result. For the Weyl anomaly in $6d$ and $8d$ both
$X^{(3)}$ and $X^{(4)}$ can also be written as linear combinations of the
Weyl-Euler density and type B anomalies. This is true for both the FG and WFG
cases, just the quantities in the latter are Weyl-covariant. One just needs to
substitute the Weyl quantities with their LC counterparts (i.e. set $a_{\mu}$
to zero) to get the Weyl anomaly in the FG case. However, when expressing them
in terms of the Weyl-Schouten tensor and extended Weyl-obstruction tensors (or
Schouten tensor and extended obstruction tensors in the FG case), we observe
that the polynomial terms of $X^{(k)}/L^{2k}$ (without the total derivative
terms) in $2k$-dimensions, denoted by $\bar{X}^{(k)}$, have the following
structures:

$\displaystyle\bar{X}^{(1)}$
$\displaystyle=-\delta^{\mu}_{\nu}\hat{P}^{\nu}{}_{\mu}\,,$ (73)
$\displaystyle 2\bar{X}^{(2)}$
$\displaystyle=\frac{1}{2}\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{P}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\,,$
(74) $\displaystyle 6\bar{X}^{(3)}$
$\displaystyle=-\frac{1}{4}\delta^{\mu_{1}\mu_{2}\mu_{3}}_{\nu_{1}\nu_{2}\nu_{3}}\hat{P}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}-\frac{1}{2}\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{\Omega}_{(1)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\,,$
(75) $\displaystyle 24\bar{X}^{(4)}$
$\displaystyle=\frac{1}{8}\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\hat{P}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}\hat{P}^{\nu_{4}}{}_{\mu_{4}}+\frac{1}{2}\delta^{\mu_{1}\mu_{2}\mu_{3}}_{\nu_{1}\nu_{2}\nu_{3}}\hat{\Omega}_{(1)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}$
$\displaystyle\quad+\frac{1}{4}\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{\Omega}_{(1)}^{\nu_{1}}{}_{\mu_{1}}\hat{\Omega}_{(1)}^{\nu_{2}}{}_{\mu_{2}}+\frac{1}{4}\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{\Omega}_{(2)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\,,$
(76)

where the Kronecker $\delta$ symbol is defined as

$\displaystyle\delta^{\mu_{1}\cdots\mu_{s}}_{\nu_{1}\cdots\nu_{s}}=s!\delta^{\mu_{1}}{}_{[\nu_{1}}\cdots\delta^{\mu_{s}}{}_{\nu_{s}]}\,.$
(77)

From (73)–(5.5) we can see that $\bar{X}^{(k)}$ contains all kinds of possible
combinations of $\hat{P}_{\mu\nu}$ and $\hat{\Omega}^{(2<j<2k)}_{\mu\nu}$
whose Weyl weights add up to be $2k$, i.e. the Weyl weight of $X^{(k)}$. Using
this pattern, one can directly write down the terms in the holographic Weyl
anomaly in any dimension. For instance, we can easily predict without explicit
calculation that $\bar{X}^{(5)}$ is the linear combination of the following
terms:

$\displaystyle\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}}\hat{P}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}\hat{P}^{\nu_{4}}{}_{\mu_{4}}\hat{P}^{\nu_{5}}{}_{\mu_{5}}\,,\quad\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\hat{\Omega}_{(1)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}\hat{P}^{\nu_{4}}{}_{\mu_{4}}\,,$
$\displaystyle\delta^{\mu_{1}\mu_{2}\mu_{3}}_{\nu_{1}\nu_{2}\nu_{3}}\hat{\Omega}_{(2)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}\,,\quad\delta^{\mu_{1}\mu_{2}\mu_{3}}_{\nu_{1}\nu_{2}\nu_{3}}\hat{\Omega}_{(1)}^{\nu_{1}}{}_{\mu_{1}}\hat{\Omega}_{(1)}^{\nu_{2}}{}_{\mu_{2}}\hat{P}^{\nu_{3}}{}_{\mu_{3}}\,,\quad\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{\Omega}_{(2)}^{\nu_{1}}{}_{\mu_{1}}\hat{\Omega}_{(1)}^{\nu_{2}}{}_{\mu_{2}}\,,\quad\delta^{\mu_{1}\mu_{2}}_{\nu_{1}\nu_{2}}\hat{\Omega}_{(3)}^{\nu_{1}}{}_{\mu_{1}}\hat{P}^{\nu_{2}}{}_{\mu_{2}}\,.$

These terms represent the independent central charges that appear in the
holographic Weyl anomaly in $d=10$.

Based on the above pattern, it is natural to expect a general expression that
can generate the holographic Weyl anomaly in any dimension, which is an analog
of the exponential structure given by the Chern class that generates the
chiral anomaly in any dimension (see, e.g. [45, 46, 47]). It has been
suggested in [23] that the type A Weyl anomaly can be generated by a mechanism
similar to that for the chiral anomaly. The expressions for the Weyl anomaly
in terms of the (Weyl-) Schouten tensor and the extended (Weyl-) obstruction
tensors suggest a similar mechanism for the holographic Weyl anomaly.

##  6 The Role of Weyl Structure

Now that we have obtained the Weyl-obstruction tensors and Weyl anomaly, let
us provide some observations on how the $a_{\mu}$ mode (23) is involved. We
have already mentioned that according to the FG theorem, this mode is pure
gauge in the bulk. Now we have a few clear manifestations of this from our
calculations.

The first one is that the subleading terms $a^{(2k)}_{\mu}$ with $k>0$ in the
expansion of $a_{\mu}$ cannot be determined from the Einstein equations when
$a^{(0)}_{\mu}$ is given. This is different from the expansion of $h_{\mu\nu}$
where the subleading terms $\gamma_{\mu\nu}^{(2k)}$ can be solved (on-shell)
in terms of $\gamma_{\mu\nu}^{(0)}$.

The second one is that $a_{\mu}$ appears only inside total derivatives in
$X^{(k)}$, and thus represents cohomologically trivial modifications of the
boundary Weyl anomaly. For $a_{\mu}^{(2k)}$ with $k\geqslant 2$, this can
easily be seen from the expressions (68), (5.3) and (5.4). What is not
explicit in these formulas is that $a^{(0)}_{\mu}$ also appears inside a total
derivative. This can be verified by separating the LC quantities out of the
Weyl quantities in $X^{(k)}$. For instance, denote the LC Schouten tensor as
$\mathring{P}_{\mu\nu}$ and the LC connection as $\mathring{\nabla}$, and then
$X^{(1)}$ in $2d$ and $X^{(2)}$ in $4d$ can be written as

$\displaystyle L^{-2}X^{(1)}_{d=2}={}$ $\displaystyle
L^{-2}\mathring{X}^{(1)}_{d=2}+\mathring{\nabla}\cdot a^{(0)}\,,$ (78)
$\displaystyle L^{-4}X^{(2)}_{d=4}={}$ $\displaystyle
L^{-4}\mathring{X}^{(2)}_{d=4}-\frac{1}{2}\mathring{\nabla}_{\mu}(\mathring{P}^{\mu\nu}a^{(0)}_{\nu}-\mathring{P}a_{(0)}^{\mu})$
$\displaystyle-\frac{1}{4}\mathring{\nabla}_{\mu}(a^{(0)}_{\nu}\mathring{\nabla}^{\nu}a_{(0)}^{\mu}-a_{(0)}^{\mu}\mathring{\nabla}\cdot
a_{(0)})-\frac{1}{4}\mathring{\nabla}_{\mu}(a_{(0)}^{\mu}a_{(0)}^{2})-\frac{1}{2L^{2}}\mathring{\nabla}\cdot
a^{(2)}\,,$ (79)

where $L^{-2}\mathring{X}^{(1)}=-\mathring{P}$ and
$L^{-4}\mathring{X}^{(2)}=\frac{1}{4}\mathring{P}^{2}-\frac{1}{4}\text{tr}(\mathring{P}^{2})$.555Note
that $\mathring{\nabla}\cdot a^{(2)}$ is equivalent to $\hat{\nabla}\cdot
a^{(2)}$, since in $2k$-dimension $\hat{\nabla}$ and $\mathring{\nabla}$ give
the same result when acting on a vector with Weyl weight $+2k$ (see Appendix
A). Notice that although the terms involving $a^{(0)}_{\mu}$ are total
derivatives, they are not Weyl-covariant and so one cannot naively assume that
they are trivial cocycles. However, by finding suitable local counterterms, we
have checked that all the terms involving $a_{\mu}^{(0)}$ are indeed part of a
trivial cocycle for $2d$ and $4d$. As $a_{\mu}$ is pure gauge, we expect this
to be generally true.

In principle, the Weyl connection $a^{(0)}_{\mu}$ on the boundary brings new
Weyl-invariant objects, such as $\text{tr}(f_{(0)}^{2})$, which could lead to
new central charges in the Weyl anomaly. However, up to $d=8$ we find the
classification of type A and type B anomalies is still available, and in such
a basis the nonvanishing central charges are still the same as those in the FG
case. Once this can be carried over to higher dimensions, then $a^{(0)}_{\mu}$
appearing in total derivatives in $X^{(k)}$ can also be deduced by considering
the Weyl anomaly as the sum of the type A and type B anomalies. In the FG
gauge, under a Weyl transformation the type B anomaly is invariant while the
type A anomaly, i.e. the Euler density, gets an extra total derivative
involving $\ln{\cal B}$. Since the Weyl connection makes the Weyl anomaly in
the WFG gauge Weyl-invariant, the terms with $a^{(0)}_{\mu}$ in the Weyl-Euler
density should exactly compensate the extra total derivative, and hence they
must form a total derivative.

Another observation we have mentioned is that although the subleading terms in
the expansion of $a_{\mu}$ make an appearance in $\gamma^{(2k)}_{\mu\nu}$,
they do not appear in the Weyl-obstruction tensors. Up to $k=3$, we have seen
explicitly in (37), (39) and (41) that the terms with $a_{\mu}^{(2)}$ and
$a_{\mu}^{(4)}$ do not contribute to the pole at $d=2k$ in
$\gamma^{(2k)}_{\mu\nu}$. What is also true but not as obvious, is that the
terms with $a^{(0)}_{\mu}$ do not contribute to the pole at $d-2$ in the Weyl-
Schouten tensor and are proportional to $d-2k$ in Weyl-obstruction tensors.
For instance, one can separate the $a^{(0)}_{\mu}$ from $\hat{P}_{\mu\nu}$ and
get

$\displaystyle\hat{P}_{\mu\nu}=\mathring{P}_{\mu\nu}+\mathring{\nabla}_{\nu}a^{(0)}_{\mu}+a^{(0)}_{\mu}a^{(0)}_{\nu}-\frac{1}{2}a_{(0)}^{2}\gamma^{(0)}_{\mu\nu}\,,$
(80)

while the only pole on the right-hand side is in the LC Schouten tensor
$\mathring{P}_{\mu\nu}$. Similarly, expressing the Weyl-Bach tensor in terms
of LC quantities we have

$\displaystyle\hat{B}_{\mu\nu}=\mathring{B}_{\mu\nu}+(d-4)(a_{(0)}^{\lambda}\mathring{C}_{\lambda\nu\mu}-2a_{(0)}^{\lambda}\mathring{C}_{\mu\nu\lambda}+a_{(0)}^{\lambda}a_{(0)}^{\rho}\mathring{W}_{\rho\mu\lambda\nu})\,.$
(81)

Thus, when $d=4$, $a^{(0)}_{\mu}$ does not contribute to the pole in
$\gamma^{(4)}_{\mu\nu}$, and the Weyl-Bach tensor $\hat{B}_{\mu\nu}$ is
equivalent to the LC Bach tensor $\mathring{B}_{\mu\nu}$. One should naturally
expect that this is also true for any Weyl-obstruction tensors, i.e.
$\hat{\cal O}^{(2k)}_{\mu\nu}$ is equivalent to the LC obstruction tensor
$\mathring{\cal O}^{(2k)}_{\mu\nu}$ when $d=2k$. Note that when $d>2k$, the
$a^{(0)}_{\mu}$ terms are included in the Weyl-obstruction tensor so that
$\hat{\cal O}^{(2k)}_{\mu\nu}$ is always Weyl-covariant.

The statement that any term in the expansion of $a_{\mu}$ does not appear in
the pole of $\hat{\gamma}^{(2k)}_{\mu\nu}$ is consistent with the following
claim: when $d=2k$, the Weyl-obstruction tensor $\hat{\cal O}_{(2k)}^{\mu\nu}$
satisfies

$\displaystyle\hat{\cal
O}_{(2k)}^{\mu\nu}=\frac{1}{\sqrt{-\det\gamma^{(0)}}}\frac{\delta}{\delta\gamma^{(0)}_{\mu\nu}}\int\text{d}^{d}x\sqrt{-\det\gamma^{(0)}}X^{(k)}\,.$
(82)

The FG version of this relation for $\mathring{\cal O}_{(2k)}^{\mu\nu}$ was
proved in [7] (see also [37]). If the claim above can be proved for the WFG
gauge, then the reason that none of the terms in the expansion of $a_{\mu}$
contributes to $\hat{\cal O}^{(2k)}_{\mu\nu}$ at $d=2k$ will be
straightforward: as they only appear in total derivative terms in $X^{(k)}$,
they will be dropped in the variation above. Hence, this can be viewed as
another manifestation of $a_{\mu}$ being pure gauge in the bulk. We have
verified by brute force that for $k=2$ the variation in (82) indeed gives the
Weyl-Bach tensor when $d=4$, and a rigorous proof for any $k$ is worth further
study.

Based on the FG version of relation (82), there is another approach of finding
the (LC) obstruction tensors and Weyl anomaly in even dimensions called the
dilatation operator method [35]. As a consistency check, we also computed the
$8d$ Weyl anomaly in the FG gauge using this method. We will briefly introduce
this method in Appendix D and show there that the result in $8d$ agrees with
what we have in (5.4) when the Weyl structure is turned off.

##  7 Conclusions

In this work, we first derived the obstruction tensor from the pole at $d=2k$
of the (on-shell) $\gamma^{(2k)}_{\mu\nu}$ in the FG expansion of an AlAdS
spacetime using dimensional regularization. Under an appropriate analytical
continuation when $d$ approaches an even integer, this approach is equivalent
to the one with a logarithmic term in [4, 7]. We defined the pole term in the
expansion to be Graham’s extended obstruction tensor, whose residue is the
obstruction tensor (up to a constant factor). Then, after introducing the WFG
ansatz, we generalized the Schouten tensor and obstruction tensors in the FG
gauge to the Weyl-Schouten tensor and Weyl-obstruction tensors in the WFG
gauge, which are now Weyl-covariant in any dimension. By solving the bulk
Einstein equations, we computed the Weyl-obstruction tensors in $4d$ (i.e. the
Weyl-Bach tensor) and $6d$ explicitly, and found that they have almost the
same form as the corresponding obstruction tensors, with everything Weyl-
covariantized and some extra terms due to the Weyl-Schouten tensor being not
symmetric. This is a natural manifestation of the fact that the WFG gauge
Weyl-covariantizes the boundary geometry. We observed that all the subleading
terms in the expansion of $a_{\mu}$ do not contribute to the Weyl-obstruction
tensor. We also found that when $d=2k$, the Weyl-obstruction tensor ${\cal
O}^{(2k)}_{\mu\nu}$ is equivalent to its LC counterpart, and so
$a^{(0)}_{\mu}$ does not contribute to the obstruction either. When $d>2k$,
the $a^{(0)}_{\mu}$ terms are included in the ${\cal O}^{(2k)}_{\mu\nu}$ to
make it Weyl-covariant.

As the main result of this paper, we computed the Weyl anomaly in $6d$ and
$8d$ in the WFG gauge by using the Weyl-Schouten tensor and extended Weyl-
obstruction tensors as the building blocks. The Weyl anomaly shown in (5.3)
and (5.4) indeed go back to the corresponding FG results when the Weyl
structure $a_{\mu}$ is turned off, but now they become Weyl-covariant. In
addition, we also re-expressed the Weyl anomaly in $2d$ and $4d$ in terms of
the Weyl-Schouten tensor. By observing the pattern of the Weyl anomaly in
different dimensions, we suspect there exists a general formulation that can
generate the holographic Weyl anomaly in any dimension, which will be explored
in future work.

In the boundary field theory, both the induced metric $\gamma^{(0)}_{\mu\nu}$
and the Weyl connection $a^{(0)}_{\mu}$ are non-dynamical background fields.
However, only $\gamma^{(0)}_{\mu\nu}$ is sourcing a current operator, namely
the energy-momentum tensor, while $a^{(0)}_{\mu}$ does not source any current
since $a_{\mu}$ is pure gauge in the bulk. From the Weyl-Ward identity (53),
we can see that the trace of the energy-momentum tensor obtains a contribution
from $p^{(0)}_{\mu}$ due to the gauge freedom of WFG. Together we can regard
it as an improved energy-momentum tensor $\tilde{T}_{\mu\nu}$. For non-
holographic field theories with background Weyl geometry the corresponding
Weyl current $J^{\mu}$ of the Weyl connection does not need to vanish. The
Weyl current in the general case deserves further investigation.

An important corollary in our analysis is that the Weyl structure $a_{\mu}$
only appears as a trivial cocycle in the Weyl anomaly, and thus only
contributes cohomologically trivial modifications. From the Weyl anomaly up to
$8d$ we can directly see this for the subleading terms of $a_{\mu}$ as they
appear only in total derivative terms in $X^{(k)}$. For the leading term
$a_{\mu}^{(0)}$ this is less obvious since it plays the role of the boundary
Weyl connection, but one can verify that by writing the anomaly in terms of
the boundary LC connection, the terms involving $a_{\mu}^{(0)}$ also represent
trivial cocycles. This indicates a striking feature of the WFG gauge, namely
$a^{(0)}_{\mu}$ manages to make the expressions Weyl-covariant without
introducing new central charges, which, once again, is consistent with the
fact that $a_{\mu}$ is pure gauge in the bulk. Nonetheless, these
cohomologically trivial terms might have significant effects in the presence
of corners, i.e. spacelike codimension-2 surfaces.666We thank Rob Leigh and
Luca Ciambelli for pointing this out in conversations. The recent construction
proposed in [48, 49] may be useful for the analysis of these effects.

In this paper we introduced the obstruction tensor and extended obstruction
tensor as the pole of $\gamma^{(2k)}_{\mu\nu}$. However, as we have mentioned,
they can also be defined using the ambient construction. What we have found
but not demonstrated in this paper is that the Weyl-obstruction tensors and
extended Weyl-obstruction tensors can be defined in a similar way by promoting
the ambient metric to the Weyl-ambient metric. We expect to discuss the Weyl-
ambient construction in detail in a future publication.

Finally, although this paper focuses on the holographic Weyl anomaly, we
believe that the (Weyl-) Schouten tensor and extended (Weyl-) obstruction
tensors can also be used as the building blocks for the Weyl anomaly of other
theories in general. How can these building blocks arise in a non-holographic
context requires a deep understanding of the Lorentz-Weyl structure of a frame
bundle, which encodes all the local Lorentz and Weyl transformations. To
achieve this, the picture of Atiyah Lie algebroids introduced in [50] for
gauge theories can be used to organize the Weyl and Lorentz anomalies in a
geometric fashion. By means of this geometric picture, we look forward to
carrying over the holographic results obtained in this paper to the
construction of Weyl anomaly in the general case.

## Acknowledgements

We would like to thank Rob Leigh for suggesting the problem and providing
constant support to us. We are also grateful to Luca Ciambelli for many
valuable discussions and carefully going through our manuscript. This work was
partially supported by the U.S. Department of Energy under contract DE-
SC0015655.

##  Appendix A Weyl Geometry

This appendix provides a brief review of Weyl geometry[13, 14]. We will mainly
introduce the geometric quantities equipped with Weyl connection as well as
some useful relations we used in the previous sections. We use $a,b,\cdots$ to
label the internal Lorentz indices and $\mu,\nu,\cdots$ to label the spacetime
indices. For clarity, we also put $\circ$ on the top of LC quantities, e.g.
$\mathring{R}^{a}{}_{bcd}$, $\mathring{P}_{ab}$, etc.

Given a generalized Riemannian manifold $(M,g)$ with a connection $\nabla$, in
an arbitrary basis $\\{\underline{e}_{a}\\}$, the connection coefficients
$\Gamma^{c}{}_{ab}$ are defined as

$\displaystyle\nabla_{\underline{e}_{a}}\underline{e}_{b}=\Gamma^{c}{}_{ab}\underline{e}_{c}\,.$
(83)

The torsion tensor and Riemann curvature tensor of $\nabla$ in this basis are
given by

$\displaystyle T^{c}{}_{ab}\underline{e}_{c}$
$\displaystyle\equiv\nabla_{\underline{e}_{a}}\underline{e}_{b}-\nabla_{\underline{e}_{b}}\underline{e}_{a}-[\underline{e}_{a},\underline{e}_{b}]\,,$
(84) $\displaystyle R^{a}{}_{bcd}\underline{e}_{a}$
$\displaystyle\equiv\nabla_{\underline{e}_{c}}\nabla_{\underline{e}_{d}}\underline{e}_{b}-\nabla_{\underline{e}_{d}}\nabla_{\underline{e}_{c}}\underline{e}_{b}-\nabla_{[\underline{e}_{c},\underline{e}_{d}]}\underline{e}_{b}\,.$
(85)

When $\nabla$ is associated with $g$ and is torsion-free, it is called a Levi-
Civita (LC) connection, denoted by $\mathring{\nabla}$. Using
$\mathring{\Gamma}$ to denote the LC connection coefficients, we have
$\mathring{\nabla}_{\underline{e}_{a}}\underline{e}_{b}=\mathring{\Gamma}^{c}{}_{ab}\underline{e}_{c}$.
By definition, the conditions satisfied by the LC connection coefficients
$\mathring{\Gamma}^{c}{}_{ab}$ are

$\displaystyle 0$
$\displaystyle=(\mathring{\nabla}g)(\underline{e}_{a},\underline{e}_{b},\underline{e}_{c})=\mathring{\nabla}_{\underline{e}_{c}}g(\underline{e}_{a},\underline{e}_{b})-\mathring{\Gamma}^{d}{}_{ca}g(\underline{e}_{d},\underline{e}_{b})-\mathring{\Gamma}^{d}{}_{cb}g(\underline{e}_{d},\underline{e}_{a})\,,$
(86) $\displaystyle 0$
$\displaystyle=T^{a}{}_{bc}=\mathring{\Gamma}^{c}{}_{ab}-\mathring{\Gamma}^{c}{}_{ba}-C_{ab}{}^{c}\,,$
(87)

where $C_{\mu\nu}{}^{\rho}$ are the commutation coefficients defined by
$[\underline{e}_{a},\underline{e}_{b}]=C_{ab}{}^{c}\underline{e}_{c}$. Denote
$g_{ab}\equiv g(\underline{e}_{a},\underline{e}_{b})$ as the component of the
metric in the frame $\\{\underline{e}_{a}\\}$. From these conditions
$\mathring{\Gamma}^{c}{}_{ab}$ can be derived as

$\displaystyle\mathring{\Gamma}^{c}{}_{ab}=$
$\displaystyle\,\frac{1}{2}g^{cd}\big{(}\underline{e}_{a}(g_{db})+\underline{e}_{b}(g_{ad})-\underline{e}_{d}(g_{ab})\big{)}-\frac{1}{2}g^{cd}(C_{ad}{}^{e}g_{eb}+C_{bd}{}^{e}g_{ae}-C_{ab}{}^{e}g_{ed})\,.$
(88)

Now we will work in a coordinate basis
$\\{\underline{\partial}_{\mu}\\}$.777Note that $\underline{e}_{a}\equiv
e_{a}^{\mu}\underline{\partial}_{\mu}$ and $\bm{e}^{a}\equiv
e^{a}_{\mu}\text{d}x^{\mu}$ have Weyl weights $+1$ and $-1$ respectively,
while $\underline{\partial}_{\mu}$ and $\text{d}x^{\mu}$ have no Weyl weights.
This is because the Weyl transformation of the frame only comes from the
soldering of the vector bundle associated with the frame bundle to the tangent
space of $M$. Consider a Weyl transformation

$\displaystyle g\to{\cal B}^{-2}g\,.$ (89)

The metricity tensor $\nabla g$ will not transform covariantly under (89). To
restore the Weyl covariance, one can introduce a Weyl connection
$A=A_{\mu}\text{d}x^{\mu}$ which transforms under a Weyl transformation as

$\displaystyle A_{\mu}\to A_{\mu}-\nabla_{\mu}\ln{\cal B}\,.$ (90)

Then, we obtain an object that is Weyl-covariant:

$\displaystyle(\nabla_{\mu}g_{\nu\rho}-2A_{\mu}g_{\nu\rho})\to{\cal
B}^{-2}(\nabla_{\mu}g_{\nu\rho}-2A_{\mu}g_{\nu\rho})\,.$ (91)

More generally, for a tensor $T$ of an arbitrary type (with indices
suppressed) that transforms under a Weyl transformation with a specific Weyl
weight $\omega_{T}$, i.e. $T\to B^{\omega_{T}}T$, we can define

$\displaystyle\hat{\nabla}_{\mu}T\equiv\nabla_{\mu}T+w_{T}A_{\mu}T\,.$ (92)

In this way, $\hat{\nabla}$ acting on $T$ will also transform Weyl-covariantly
as $\hat{\nabla}_{\mu}T\to B^{\omega_{T}}\hat{\nabla}_{\mu}T$.

Now we choose the connection $\nabla$ by setting the Weyl metricity as follows

$\displaystyle 0$
$\displaystyle=\nabla_{\mu}g_{\nu\rho}-2A_{\mu}g_{\nu\rho}=\hat{\nabla}_{\mu}g_{\nu\rho}\,.$
(93)

We will also require $\nabla$ defined in the above equation to be torsion-
free. With the existence of the Weyl metricity, the connection coefficients of
$\nabla$ in the coordinate basis become

$\displaystyle\Gamma^{\rho}{}_{\mu\nu}={}$
$\displaystyle\frac{1}{2}g^{\rho\sigma}(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\nu\sigma}-\partial_{\sigma}g_{\mu\nu})-(A_{\mu}\delta^{\rho}{}_{\nu}+A_{\nu}\delta^{\rho}{}_{\mu}-g^{\rho\sigma}A_{\sigma}g_{\mu\nu})\,.$
(94)

We can see that this is different from the familiar Christoffel symbols due to
the extra terms involving the Weyl connection. When $\nabla$ and
$\mathring{\nabla}$ act on a vector, their difference can be reflected by

$\displaystyle\nabla_{\mu}v^{\nu}=\mathring{\nabla}_{\mu}v^{\nu}-(A_{\mu}\delta^{\nu}{}_{\rho}+A_{\rho}\delta^{\nu}{}_{\mu}-g^{\nu\sigma}A_{\sigma}g_{\mu\rho})v^{\rho}\,.$
(95)

It is worthwhile to notice that if $v^{\nu}$ has Weyl weight $d=\dim M$, then
it follows from (92) and (95) that
$\hat{\nabla}_{\mu}v^{\mu}=\mathring{\nabla}_{\mu}v^{\mu}$.

Now one can compute the Riemann tensor of $\nabla$ and its contractions.
Denoting the coordinate components of the Riemann tensor of
$\mathring{\nabla}$ as $\mathring{R}^{\mu}{}_{\nu\rho\sigma}$, one finds from
(85) that

$\displaystyle R^{\mu}{}_{\nu\rho\sigma}={}$
$\displaystyle\mathring{R}^{\mu}{}_{\nu\rho\sigma}+\mathring{\nabla}_{\sigma}A_{\nu}\delta^{\mu}{}_{\rho}-\mathring{\nabla}_{\rho}A_{\nu}\delta^{\mu}{}_{\sigma}+(\mathring{\nabla}_{\sigma}A_{\rho}-\mathring{\nabla}_{\rho}A_{\sigma})\delta^{\mu}{}_{\nu}+\mathring{\nabla}_{\rho}A^{\mu}g_{\nu\sigma}-\mathring{\nabla}_{\sigma}A^{\mu}g_{\nu\rho}$
$\displaystyle+A_{\nu}(A_{\sigma}\delta^{\mu}{}_{\rho}-A_{\rho}\delta^{\mu}{}_{\sigma})+A^{\mu}(g_{\nu\sigma}A_{\rho}-g_{\nu\rho}A_{\sigma})+A^{2}(g_{\nu\rho}\delta^{\mu}{}_{\sigma}-g_{\nu\sigma}\delta^{\mu}{}_{\rho})\,,$
(96) $\displaystyle R_{\mu\nu}={}$
$\displaystyle\mathring{R}_{\mu\nu}-\frac{d}{2}F_{\mu\nu}+(d-2)(\mathring{\nabla}_{(\mu}A_{\nu)}+A_{\mu}A_{\nu})+(\mathring{\nabla}\cdot
A-(d-2)A^{2})g_{\mu\nu}\,,$ (97) $\displaystyle R={}$
$\displaystyle\mathring{R}+2(d-1)\mathring{\nabla}\cdot A-(d-1)(d-2)A^{2}\,,$
(98)

where $R_{\mu\nu}\equiv R^{\rho}{}_{\mu\rho\nu}$, $R\equiv
R_{\mu\nu}g^{\mu\nu}$, and we defined the curvature of $A_{\mu}$ as
$F_{\mu\nu}=\mathring{\nabla}_{\mu}A_{\nu}-\mathring{\nabla}_{\nu}A_{\mu}$. It
is easy to see from (A) that, unlike $\mathring{R}^{\mu}{}_{\nu\rho\sigma}$,
the $R^{\mu}{}_{\nu\rho\sigma}$ of $\nabla$ now is not antisymmetric in the
first two indices, and it does not have the interchange symmetry for the two
index pairs. Also, the $R_{\mu\nu}$ of $\nabla$ is not symmetric due to the
appearance of the $F_{\mu\nu}$ term.

On the other hand, from (83) we have the connection coefficients
$\hat{\Gamma}^{c}{}_{ab}$ for $\hat{\nabla}$:

$\displaystyle\hat{\Gamma}^{c}{}_{ab}\underline{e}_{c}=\hat{\nabla}_{\underline{e}_{a}}\underline{e}_{b}=\nabla_{\underline{e}_{a}}\underline{e}_{b}+A(\underline{e}_{a})\underline{e}_{b}=\Gamma^{c}{}_{ab}\underline{e}_{c}+A(\underline{e}_{a})\underline{e}_{b}\,,$
(99)

where we used the fact that the basis vector $\underline{e}_{a}$ has Weyl
weight $+1$. Plugging this into (85), we find that the Riemann tensor of
$\hat{\nabla}$ and its contractions satisfy

$\displaystyle\hat{R}^{\mu}{}_{\nu\rho\sigma}=$ $\displaystyle
R^{\mu}{}_{\nu\rho\sigma}+\delta^{\mu}{}_{\nu}F_{\rho\sigma}\,,\qquad\hat{R}_{\mu\nu}=R_{\mu\nu}+F_{\mu\nu}\,,\qquad\hat{R}=R\,.$
(100)

We refer to $\hat{R}^{\mu}{}_{\nu\rho\sigma}$, $\hat{R}_{\mu\nu}$ and
$\hat{R}$ as the Weyl-Riemann tensor, Weyl-Ricci tensor, and Weyl-Ricci
scalar, respectively.888Note that this is different from [5], in which the
quantities defined using $\nabla$ instead of $\hat{\nabla}$ are called Weyl
quantities. Similar to the curvature tensors for $\nabla$, the Weyl-Riemann
tensor is not antisymmetric in the first two indices and does not have the
interchange symmetry for the two index pairs, and the Weyl-Ricci tensor is not
symmetric. Also notice that the Weyl-Weyl tensor, namely the traceless part of
the Weyl-Riemann tensor, is equal to the LC Weyl tensor, i.e.

$\displaystyle\hat{W}^{\mu}{}_{\nu\rho\sigma}=\mathring{W}^{\mu}{}_{\nu\rho\sigma}\,.$
(101)

Unlike the LC curvature quantities, which transform in a non-covariant way
under the Weyl transformation, the Weyl-Riemann tensor, Weyl-Ricci tensor, and
Weyl-Ricci scalar transform under the Weyl transformation as

$\displaystyle\hat{R}^{\mu}{}_{\nu\rho\sigma}\to\hat{R}^{\mu}{}_{\nu\rho\sigma}\,,\qquad\hat{R}_{\mu\nu}\to\hat{R}_{\mu\nu}\,,\qquad\hat{R}\to{\cal
B}^{2}\hat{R}\,.$ (102)

Furthermore, we can define the Weyl-Schouten tensor $\hat{P}_{\mu\nu}$ and
Weyl-Cotten tensor $\hat{C}_{\mu\nu\rho}$ as

$\displaystyle\hat{P}_{\mu\nu}$
$\displaystyle=\frac{1}{d-2}\bigg{(}\hat{R}_{\mu\nu}-\frac{1}{2(d-1)}\hat{R}g_{\mu\nu}\bigg{)}\,,$
(103) $\displaystyle\hat{C}_{\mu\nu\rho}$
$\displaystyle=\hat{\nabla}_{\rho}\hat{P}_{\mu\nu}-\hat{\nabla}_{\nu}\hat{P}_{\mu\rho}\,.$
(104)

Although the LC Schouten tensor $\mathring{P}_{\mu\nu}$ defined by
substituting $\hat{R}_{\mu\nu}$ and $\hat{R}$ in (103) with $R_{\mu\nu}$ and
$R$ is a symmetric tensor, $\hat{P}_{\mu\nu}$ has an antisymmetric part
$\hat{P}_{[\mu\nu]}=-F_{\mu\nu}/2$. In terms of the LC connection, the Bach
tensor is defined by (the indices of the components are raised and lowered by
$g$)

$\displaystyle\mathring{B}_{\mu\nu}=\mathring{\nabla}^{\rho}\mathring{\nabla}_{\rho}\mathring{P}_{\mu\nu}-\mathring{\nabla}^{\rho}\mathring{\nabla}_{\nu}\mathring{P}_{\mu\rho}-\mathring{W}_{\sigma\nu\mu\rho}\mathring{P}^{\rho\sigma}\,,$
(105)

which satisfies $\mathring{B}_{\mu\nu}\to{\cal B}^{2}\mathring{B}_{\mu\nu}$ in
$4d$. Now we can define the Weyl-Bach tensor

$\displaystyle\hat{B}_{\mu\nu}=\hat{\nabla}^{\rho}\hat{\nabla}_{\rho}\hat{P}_{\mu\nu}-\hat{\nabla}^{\rho}\hat{\nabla}_{\nu}\hat{P}_{\mu\rho}-\hat{W}_{\sigma\nu\mu\rho}\hat{P}^{\rho\sigma}\,.$
(106)

Similar to the LC Bach tensor, the Weyl-Bach tensor is also symmetric and
traceless; however, it is Weyl-covariant in any dimension. Following (A)–(98),
here we list the above-mentioned Weyl quantities in terms of their
corresponding LC quantities:

$\displaystyle\hat{P}_{\mu\nu}$
$\displaystyle=\mathring{P}_{\mu\nu}+\mathring{\nabla}_{\nu}A_{\mu}+A_{\mu}A_{\nu}-\frac{1}{2}A^{2}g_{\mu\nu}\,,$
(107) $\displaystyle\hat{C}_{\mu\nu\rho}$
$\displaystyle=\mathring{C}_{\mu\nu\rho}-A_{\sigma}\mathring{W}^{\sigma}{}_{\mu\rho\nu}\,,$
(108) $\displaystyle\hat{B}_{\mu\nu}$
$\displaystyle=\mathring{B}_{\mu\nu}+(d-4)(A^{\rho}\mathring{C}_{\rho\nu\mu}-2A^{\rho}\mathring{C}_{\mu\nu\rho}+A^{\rho}A^{\sigma}\mathring{W}_{\sigma\mu\rho\nu})\,.$
(109)

The Bianchi identity for $\hat{\nabla}$ reads

$\displaystyle\hat{\nabla}_{\mu}\hat{R}^{\lambda}{}_{\nu\rho\sigma}+\hat{\nabla}_{\rho}\hat{R}^{\lambda}{}_{\nu\sigma\mu}+\hat{\nabla}_{\sigma}\hat{R}^{\lambda}{}_{\nu\mu\rho}=0\,.$
(110)

Noticing that $\hat{\nabla}_{\mu}g_{\nu\rho}=0$, the contraction of the above
equation gives

$\displaystyle\hat{\nabla}^{\mu}\hat{G}_{\mu\nu}=0\,,$ (111)

where we defined the Weyl-Einstein tensor
$\hat{G}_{\mu\nu}\equiv\hat{R}_{\mu\nu}-\frac{1}{2}\hat{R}g_{\mu\nu}$. Using
(103), this identity can also be expressed using the Weyl-Schouten tensor as

$\displaystyle\hat{\nabla}^{\mu}\hat{P}_{\mu\nu}=\hat{\nabla}_{\nu}\hat{P}\,.$
(112)

where $\hat{P}$ is the trace of $\hat{P}_{\mu\nu}$. Starting from (106) and
using (112) repeatedly, one obtains

$\displaystyle\hat{\nabla}^{\mu}\hat{B}_{\mu\nu}=(d-4)\hat{P}^{\mu\rho}(\hat{C}_{\rho\mu\nu}+\hat{C}_{\nu\mu\rho})\,.$
(113)

Note that since $\mathring{P}$ is symmetric, the above equation in the LC case
becomes

$\displaystyle\mathring{\nabla}^{\mu}\mathring{B}_{\mu\nu}=(d-4)\mathring{P}^{\mu\rho}\mathring{C}_{\rho\mu\nu}\,.$
(114)

It is also useful to notice that in the LC case, the divergence of the Cotton
tensor vanishes

$\displaystyle\mathring{\nabla}^{\mu}\mathring{C}_{\mu\nu\rho}=0\,,$ (115)

while for the Weyl-Cotton tensor we have instead

$\displaystyle\hat{\nabla}^{\mu}\hat{C}_{\mu\nu\rho}=\hat{W}_{\sigma\rho\lambda\nu}F^{\sigma\lambda}\,.$
(116)

In the end of this appendix, we list the Weyl weights of the above-mentioned
Weyl quantities:

Table 1: Weyl weights of Weyl-covariant quantities $\underline{e}_{a}$ | $\bm{e}^{a}$ | $g_{\mu\nu}$ | $g^{\mu\nu}$ | $\hat{R}^{\mu}{}_{\nu\rho\sigma}$ | $\hat{R}_{\mu\nu}$ | $\hat{R}$ | $F_{\mu\nu}$ | $\hat{P}_{\mu\nu}$ | $\hat{C}_{\mu\nu\rho}$ | $\hat{B}_{\mu\nu}$
---|---|---|---|---|---|---|---|---|---|---
$+1$ | $-1$ | $-2$ | $+2$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $+2$

##  Appendix B Solving the Bulk Einstein Equations

To solve for $\gamma^{(2k)}_{\mu\nu}$ in the WFG gauge from the Einstein
equations, we first introduced the following notations:

$\displaystyle\varphi_{\mu}$ $\displaystyle\equiv D_{z}a_{\mu}\,,\qquad
f_{\mu\nu}\equiv
D_{\mu}a_{\nu}-D_{\nu}a_{\mu}\,,\qquad\rho_{\mu\nu}\equiv\frac{1}{2}D_{z}h_{\mu\nu}\,,\qquad\theta\equiv\text{tr}\rho\,,$
$\displaystyle\psi_{\mu\nu}$
$\displaystyle\equiv\rho_{\mu\nu}+\frac{L}{2}f_{\mu\nu}\,,\qquad\gamma^{\lambda}{}_{\mu\nu}\equiv\Gamma^{\lambda}{}_{\mu\nu}=\frac{1}{2}h^{\lambda\rho}(D_{\mu}h_{\rho\nu}+D_{\nu}h_{\mu\rho}-D_{\rho}h_{\nu\mu})\,.$
(117)

Since the integral curves of $\underline{D}_{z}$ form a congruence, some of
these quantities can be interpreted as the properties of this congruence:
$\varphi^{\mu}$ is the acceleration, $f_{\mu\nu}$ is the twist, $\theta$ is
the expansion and $\sigma_{\mu\nu}\equiv\rho_{\mu\nu}-\frac{1}{d}\theta
h_{\mu\nu}$ is the shear. By plugging in the expansions (22) and (23), one can
obtain the expansions of the quantities above. A list of these expansions
enough for capturing the first two leading orders of the Einstein equations
can be found in the Appendix of [5].

Using the connection coefficients $\Gamma^{\lambda}{}_{\mu\nu}$ in the bulk,
one can compute the curvature tensors and the Einstein tensor. Then, the
vacuum Einstein equations can be written as

$\displaystyle 0$
$\displaystyle=G_{zz}+g_{zz}\Lambda=-\frac{1}{2}\text{tr}(\rho\rho)-\frac{3L^{2}}{8}\text{tr}(ff)-\frac{1}{2}\bar{R}+\frac{1}{2}\theta^{2}+\Lambda\,$
(118) $\displaystyle 0$
$\displaystyle=G_{z\mu}+g_{z\mu}\Lambda=\nabla_{\nu}\psi^{\nu}{}_{\mu}-D_{\mu}\theta+L^{2}f_{\nu\mu}\varphi^{\nu}\,,$
(119) $\displaystyle 0$
$\displaystyle=G_{\mu\nu}+g_{\mu\nu}\Lambda=\bar{G}_{\mu\nu}-(D_{z}+\theta)\psi_{\mu\nu}-L\nabla_{\nu}\varphi_{\mu}+2\rho_{\nu\rho}\rho^{\rho}{}_{\mu}+\frac{L^{2}}{2}f_{\nu\rho}f^{\rho}{}_{\mu}-L^{2}\varphi_{\mu}\varphi_{\nu}$
$\displaystyle\qquad\qquad\qquad\qquad+h_{\mu\nu}\left(L\nabla_{\mu}\varphi^{\mu}+D_{z}\theta+\frac{1}{2}\text{tr}(\rho\rho)-\frac{L^{2}}{8}\text{tr}(ff)+L^{2}\varphi^{2}+\frac{1}{2}\theta^{2}+\Lambda\right)\,.$
(120)

where $\Lambda=-\frac{d(d-1)}{2L^{2}}$ is the cosmological constant, and
$\bar{R}=h^{\mu\nu}\bar{R}_{\mu\nu}$ with

$\displaystyle\bar{R}_{\mu\nu}=D_{\rho}\gamma^{\rho}{}_{\nu\mu}-D_{\nu}\gamma^{\rho}{}_{\rho\mu}+\gamma^{\rho}{}_{\rho\sigma}\gamma^{\sigma}{}_{\nu\mu}-\gamma^{\rho}{}_{\nu\sigma}\gamma^{\sigma}{}_{\rho\mu}\,.$
(121)

Denote $m_{(2k)\nu}^{\mu}\equiv\gamma_{(0)}^{\mu\rho}\gamma^{(2k)}_{\rho\nu}$
and $n_{(2k)\nu}^{\mu}\equiv\gamma_{(0)}^{\mu\rho}\pi^{(2k)}_{\rho\nu}$.
Expanding (118)–(B) using (22) and (23), one can solve the Einstein equations
order by order. First, the $zz$-component of the Einstein equations gives

$\displaystyle 0={}$
$\displaystyle\bigg{[}\frac{d(d-1)}{2L^{2}}+\Lambda\bigg{]}-\frac{z^{2}}{L^{2}}\bigg{[}\frac{R^{(0)}}{2}+\frac{d-1}{L^{2}}X^{(1)}\bigg{]}+\frac{z^{4}}{L^{4}}\bigg{[}\frac{d}{2L^{2}}(X^{(1)})^{2}-\frac{2(d-1)}{L^{2}}X^{(2)}-\frac{1}{2L^{2}}\text{tr}(m_{(2)}^{2})$
$\displaystyle-\frac{3L^{2}}{8}\text{tr}(f_{(0)}^{2})-\frac{1}{2}\Big{(}\gamma_{(0)}^{\lambda\nu}\hat{\nabla}^{(0)}_{\lambda}\hat{\nabla}_{\mu}\big{(}m_{(2)}{}^{\mu}{}_{\nu}-\text{tr}(m_{(2)})\delta^{\mu}{}_{\nu}\big{)}+2(d-1)\hat{\nabla}\cdot
a^{(2)}-\text{tr}\big{(}m_{(2)}\gamma_{(0)}^{-1}R^{(0)}\big{)}\Big{)}\bigg{]}$
$\displaystyle+\cdots-\frac{z^{d}}{L^{d}}(d-1)\bigg{[}\frac{d}{2L^{2}}Y^{(1)}+\hat{\nabla}\cdot
p_{(0)}\bigg{]}+\cdots\,,$ (122)

where $X^{(1)}$, $X^{(2)}$ and $Y^{(1)}$ are given in expansion (55), which
can be expressed in terms of the expansion of $h_{\mu\nu}$ as

$\displaystyle X^{(1)}$ $\displaystyle=\text{tr}(m_{(2)})\,,\quad
X^{(2)}=\text{tr}(m_{(4)})-\frac{1}{2}\text{tr}(m_{(2)}^{2})+\frac{1}{4}\left(\text{tr}(m_{(2)})\right)^{2}\,,\,\cdots\,,Y^{(1)}=\text{tr}(n_{(0)})\,,\,\cdots\,.$
(123)

At the $O(1)$-order, the $zz$-equation is trivially satisfied, and at the
$O(z^{2})$-order, we can find that

$\displaystyle X^{(1)}=-\frac{L^{2}}{2(d-1)}R^{(0)}=-L^{2}\hat{P}\,.$ (124)

Then, using the above result we can obtain from the $O(z^{4})$-order that

$\displaystyle X^{(2)}$
$\displaystyle=-\frac{1}{4}\text{tr}(m_{(2)}^{2})+\frac{1}{4}(X^{(1)})^{2}-\frac{L^{2}}{2}\hat{\nabla}\cdot
a^{(2)}-\frac{L^{4}}{16}\text{tr}(f^{(0)}f^{(0)})$
$\displaystyle=-\frac{L^{4}}{4}\text{tr}(\hat{P}^{2})+\frac{L^{4}}{4}\hat{P}^{2}-\frac{L^{2}}{2}\hat{\nabla}\cdot
a^{(2)}\,,$ (125)

where we used (38). Also notice that the $O(z^{d})$-order gives the Weyl-Ward
identity

$\displaystyle 0=\frac{d}{2L^{2}}Y^{(1)}+\hat{\nabla}\cdot p_{(0)}\,.$ (126)

Now we look at the $\mu\nu$-components of the Einstein equations:

$\displaystyle 0={}$
$\displaystyle\bigg{[}G^{(0)}_{\mu\nu}+\frac{d}{2}f^{(0)}_{\mu\nu}-\frac{d-2}{L^{2}}X^{(1)}\gamma^{(0)}_{\mu\nu}+\frac{d-2}{L^{2}}\gamma^{(2)}_{\mu\nu}\bigg{]}+\frac{z^{2}}{L^{2}}\bigg{[}\frac{1}{2}\hat{\nabla}_{\lambda}\Big{(}\gamma_{(0)}^{\lambda\xi}\Big{(}\hat{\nabla}_{\nu}\gamma^{(2)}_{\xi\mu}+\hat{\nabla}_{\mu}\gamma^{(2)}_{\xi\nu}-\hat{\nabla}_{\xi}\gamma^{(2)}_{\mu\nu}\Big{)}\Big{)}$
$\displaystyle-\frac{1}{2}\gamma^{(0)}_{\mu\nu}\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}\Big{(}\gamma_{(2)}^{\mu\nu}-X^{(1)}\gamma_{(0)}^{\mu\nu}\Big{)}-\frac{1}{2}\hat{\nabla}_{(\mu}\hat{\nabla}_{\nu)}X^{(1)}+(d-4)\big{(}\hat{\nabla}^{(0)}_{(\mu}a_{\nu)}^{(2)}-\gamma_{\mu\nu}^{(0)}\hat{\nabla}\cdot
a^{(2)})$
$\displaystyle+\frac{2(d-4)}{L^{2}}\gamma^{(4)}_{\mu\nu}+\frac{2}{L^{2}}m_{(2)}^{\rho}{}_{\mu}\gamma^{(2)}_{\rho\nu}+\frac{L^{2}}{2}f^{(0)}_{\nu\rho}f^{(0)}_{\sigma\mu}\gamma_{(0)}^{\sigma\rho}+\Big{(}\frac{1}{2}\text{tr}(m_{(2)}\gamma_{(0)}^{-1}{R}^{(0)})-\frac{L^{2}}{8}\text{tr}(f^{(0)}f^{(0)})$
$\displaystyle-\frac{2(d-4)}{L^{2}}X^{(2)}+\frac{d-3}{2L^{2}}(X^{(1)})^{2}+\frac{1}{2L^{2}}\text{tr}(m_{(2)}^{2})\Big{)}\gamma^{(0)}_{\mu\nu}\bigg{]}+\cdots\,.$
(127)

Note that
$\gamma_{(2)}^{\mu\nu}\equiv(\gamma_{(0)}^{-1}\gamma^{(2)}\gamma_{(0)}^{-1})^{\mu\nu}$
is not the inverse of $\gamma^{(2)}_{\mu\nu}$. Plugging in the results we got
from the $zz$-equation, we obtain from the first two leading orders of (B)
that

$\displaystyle\gamma^{(2)}_{\mu\nu}={}$
$\displaystyle-\frac{L^{2}}{d-2}\bigg{(}R^{(0)}_{(\mu\nu)}-\frac{1}{2(d-1)}R^{(0)}\gamma^{(0)}_{\mu\nu}\bigg{)}\,,$
(128) $\displaystyle\gamma^{(4)}_{\mu\nu}={}$
$\displaystyle-\frac{L^{2}}{4(d-4)}\bigg{(}2\hat{\nabla}_{\lambda}\hat{\nabla}_{(\mu}m_{(2)}{}^{\lambda}{}_{\nu)}-\hat{\nabla}\cdot\hat{\nabla}\gamma^{(2)}_{\mu\nu}-\hat{\nabla}_{(\mu}\hat{\nabla}_{\nu)}X^{(1)}-\frac{1}{L^{2}}\gamma^{(0)}_{\mu\nu}\text{tr}(m_{(2)}^{2})+\frac{4}{L^{2}}m_{(2)}^{\rho}{}_{\mu}\gamma^{(2)}_{\rho\nu}$
$\displaystyle\qquad\qquad\qquad+L^{2}f^{(0)}_{\nu\rho}f^{(0)}_{\sigma\mu}\gamma_{(0)}^{\sigma\rho}-\frac{L^{2}}{4}\text{tr}(f^{(0)}f^{(0)})\gamma^{(0)}_{\mu\nu}\bigg{)}-\frac{L^{2}}{2}\hat{\nabla}^{(0)}_{(\mu}a_{\nu)}^{(2)}\,.$
(129)

Furthermore, expanding (B) to the $O(z^{4})$-order one obtains

$\displaystyle\gamma^{(6)}_{\mu\nu}=$
$\displaystyle-\frac{L^{2}}{3(d-6)}\bigg{[}\hat{\nabla}_{\lambda}\hat{\gamma}^{\lambda}_{(4)\mu\nu}-\frac{1}{2}\hat{\nabla}_{(\mu}\hat{\nabla}_{\nu)}\text{tr}(m_{(4)})-\hat{\nabla}_{\lambda}(\hat{\gamma}^{\sigma}_{(2)\mu\nu}{m}_{(2)}^{\lambda}{}_{\sigma})+\hat{\nabla}_{(\nu}(\hat{\gamma}^{\sigma}_{(2)\mu)\lambda}{m}_{(2)}^{\lambda}{}_{\sigma})$
(130)
$\displaystyle+\frac{1}{2}\hat{\nabla}_{\sigma}X^{(1)}\hat{\gamma}^{\sigma}_{(2)\mu\nu}-\hat{\gamma}^{\sigma}_{(2)\mu\lambda}\hat{\gamma}^{\lambda}_{(2)\sigma\nu}-\frac{2}{L^{2}}(m_{(2)}^{3})^{\rho}{}_{\nu}\gamma^{(0)}_{\mu\rho}+\frac{8}{L^{2}}\gamma^{(4)}_{\rho(\mu}m_{(2)}^{\rho}{}_{\nu)}-\frac{1}{L^{2}}\gamma^{(4)}_{\mu\nu}X^{(1)}$
$\displaystyle-\frac{L^{2}}{2}f^{(0)}_{\sigma\mu}f^{(0)}_{\nu\rho}\gamma_{(2)}^{\rho\sigma}+L^{2}f^{(2)}_{\sigma(\mu}f^{(0)}_{\nu)\rho}\gamma_{(0)}^{\rho\sigma}-\frac{1}{L^{2}}\gamma^{(0)}_{\mu\nu}\Big{(}\text{tr}(m_{(4)}m_{(2)})-\frac{1}{2}\text{tr}(m_{(2)}^{3})-\frac{L^{4}}{8}\text{tr}(m_{(2)}f_{(0)}^{2})$
$\displaystyle-\frac{L^{4}}{4}\hat{\nabla}_{\rho}a^{(2)}_{\sigma}f_{(0)}^{\rho\sigma}-\frac{L^{2}}{4}\hat{\nabla}_{\sigma}X^{(1)}a_{(2)}^{\sigma}+\frac{L^{2}}{2}\hat{\nabla}_{\lambda}(\gamma_{(2)}^{\lambda\rho}a^{(2)}_{\rho})\Big{)}+2\hat{\nabla}_{\lambda}({m}_{(2)}^{\lambda}{}_{(\nu}a^{(2)}_{\mu)})-2\gamma^{(0)}_{\sigma(\nu}\hat{\gamma}^{\sigma}_{(2)\mu)\lambda}a_{(2)}^{\lambda}$
$\displaystyle-a^{(2)}_{(\nu}\hat{\nabla}_{\mu)}X^{(1)}-\hat{\nabla}_{(\nu}(X^{(1)}a^{(2)}_{\mu)})\bigg{]}-\frac{L^{2}}{3}\hat{\nabla}_{(\mu}a^{(4)}_{\nu)}-L^{2}a^{(2)}_{\mu}a^{(2)}_{\nu}+\frac{L^{2}}{6}a^{(2)}\cdot
a^{(2)}\gamma^{(0)}_{\mu\nu}+\frac{L^{2}}{3}\hat{\gamma}^{\lambda}_{(2)\mu\nu}a^{(2)}_{\lambda}\,,$

where
$f^{(2)}_{\sigma\mu}\equiv\hat{\nabla}_{\sigma}a^{(2)}_{\mu}-\hat{\nabla}_{\mu}a^{(2)}_{\sigma}$,
and

$\displaystyle\hat{\gamma}^{\lambda}_{(2)\mu\nu}$
$\displaystyle=\frac{1}{2}\gamma_{(0)}^{\lambda\rho}(\hat{\nabla}^{(0)}_{\mu}\gamma^{(2)}_{\nu\rho}+\hat{\nabla}^{(0)}_{\nu}\gamma^{(2)}_{\mu\rho}-\hat{\nabla}^{(0)}_{\rho}\gamma^{(2)}_{\mu\nu})=-\frac{L^{2}}{2}(\hat{\nabla}^{(0)}_{\mu}\hat{P}^{\lambda}{}_{\nu}+\hat{\nabla}^{(0)}_{\nu}\hat{P}_{\mu}{}^{\lambda}-\hat{\nabla}_{(0)}^{\lambda}\hat{P}_{\mu\nu})\,.$
(131)

(In the second step we used
$\hat{\nabla}^{(0)}_{\mu}f^{(0)}_{\nu\rho}+\hat{\nabla}^{(0)}_{\nu}f^{(0)}_{\rho\mu}+\hat{\nabla}^{(0)}_{\rho}f^{(0)}_{\mu\nu}=0$.)
The $\gamma^{(4)}_{\mu\nu}$ and $\gamma^{(6)}_{\mu\nu}$ above can be organized
in to (39) and (41), respectively.

Finally, the $z\mu$-component of the Einstein equations gives

$\displaystyle 0=$
$\displaystyle-\frac{L}{d-2}\frac{z^{2}}{L^{2}}\gamma_{(0)}^{\alpha\beta}\hat{\nabla}_{\alpha}^{(0)}\hat{G}^{(0)}_{\beta\mu}+L^{-1}\frac{z^{4}}{L^{4}}\bigg{[}\hat{\nabla}_{\alpha}\big{(}2m_{(4)\mu}^{\alpha}-(m_{(2)}^{2})^{\alpha}{}_{\mu}\big{)}+\frac{1}{2}m_{(2)\mu}^{\alpha}\hat{\nabla}_{\alpha}X^{(1)}$
$\displaystyle+\frac{L^{2}}{2}\bigg{(}\hat{\nabla}\cdot\hat{\nabla}a_{\mu}^{(2)}-\hat{\nabla}_{\mu}\hat{\nabla}\cdot
a^{(2)}+(R^{(0)}_{\beta\mu}+4f^{(0)}_{\beta\mu})\gamma_{(0)}^{\alpha\beta}a^{(2)}_{\alpha}-\hat{\nabla}_{\alpha}\big{(}f^{(0)}_{\beta\mu}m_{(2)\rho}^{\alpha}\gamma_{(0)}^{\rho\beta}\big{)}$
$\displaystyle-f^{(0)}_{\nu\rho}\gamma_{(0)}^{\alpha\nu}\hat{\nabla}_{\alpha}m_{(2)}^{\rho}{}_{\mu}+\frac{1}{2}f^{(0)}_{\beta\mu}\gamma_{(0)}^{\alpha\beta}\hat{\nabla}_{\alpha}X^{(1)}\bigg{)}-2\hat{\nabla}_{\mu}X^{(2)}+\frac{1}{2}\hat{\nabla}_{\mu}(X^{(1)})^{2}-\frac{1}{4}\hat{\nabla}_{\mu}\text{tr}(m_{(2)}^{2})\bigg{]}+\cdots$
$\displaystyle+\frac{z^{d}}{L^{d}}\bigg{[}\frac{d}{2L}\hat{\nabla}_{\alpha}n_{(0)\mu}^{\alpha}+\frac{L}{2}(\hat{\nabla}\cdot\hat{\nabla}p_{\mu}^{(0)}+\hat{\nabla}_{\alpha}\hat{\nabla}_{\mu}p_{(0)}^{\alpha})\bigg{]}+\cdots\,.$
(132)

One can observe that the $O(z^{2})$-order of the above equation is exactly the
contraction of the Weyl-Bianchi identity as shown in (111). By plugging in the
results we got from the $zz$-equation, the $O(z^{4})$-order can be organized
into the identity (43), which demonstrates the divergence of the Bach tensor.
Also, the $O(z^{d})$-order gives the conservation law of the improved energy-
momentum tensor defined in (52).

##  Appendix C Expansions of the Raychaudhuri Equation and $\sqrt{-\det h}$

Using the components of the Einstein equations (118)–(B), one can construct
the following equation [5]:

$\displaystyle 0$ $\displaystyle=\frac{g^{MN}(G_{MN}+\Lambda
g_{MN})}{d-1}+(G_{zz}+\Lambda g_{zz})$
$\displaystyle=D_{z}\theta+L\nabla_{\nu}\varphi^{\nu}+L^{2}\varphi^{2}+\text{tr}(\rho\rho)+\frac{L^{2}}{4}\text{tr}(ff)-\frac{d}{L^{2}}\,,$
(133)

where the indices $M,N$ represent the bulk components as $M=(z,\mu)$. This
equation can be recognized as the Raychaudhuri equation of the congruence
generated by $\underline{D}_{z}$. Expanding each term in the above equation,
we can write down a general expansion of this equation to any order. This
combination of the components of the Einstein equations contains all the
information we need for deriving $X^{(k)}$. We here provide some details of
deriving $X^{(3)}$ and $X^{(4)}$ by means of the Raychaudhuri equation.

First, it is useful to expand the inverse of $h_{\mu\nu}$:

$\displaystyle h^{\mu\nu}(z;x)$
$\displaystyle=\frac{z^{2}}{L^{2}}\left[\gamma_{(0)}^{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\gamma_{(2)}^{\mu\nu}(x)+...\right]+\frac{z^{d+2}}{L^{d+2}}\left[\pi_{(0)}^{\mu\nu}(x)+\frac{z^{2}}{L^{2}}\pi_{(2)}^{\mu\nu}(x)+...\right]$
(134)
$\displaystyle=\frac{z^{2}}{L^{2}}\left[\gamma_{(0)}^{\mu\nu}(x)-\frac{z^{2}}{L^{2}}\tilde{m}_{(2)\rho}^{\mu}\gamma_{(0)}^{\rho\nu}(x)-\frac{z^{4}}{L^{4}}\tilde{m}_{(4)\rho}^{\mu}\gamma_{(0)}^{\rho\nu}(x)+\cdots\right]+\frac{z^{d+2}}{L^{d+2}}\left[\tilde{n}_{(2)\rho}^{\mu}\gamma_{(0)}^{\rho\nu}(x)+\cdots\right]\,,$

where
$\tilde{m}_{(2k)\nu}^{\mu}\equiv-\gamma_{(2k)}^{\mu\rho}\gamma^{(0)}_{\rho\nu}$,
$\tilde{n}_{(2k)\nu}^{\mu}\equiv-\pi_{(2k)}^{\mu\rho}\gamma^{(0)}_{\rho\nu}$.
The above expansion can be solved order by order in terms of
$m_{(2k)\nu}^{\mu}$ and $n_{(2k)\nu}^{\mu}$:

$\displaystyle\gamma_{(0)}^{\mu\nu}$
$\displaystyle=(\gamma^{(0)}_{\mu\nu})^{-1}\,,\qquad\tilde{m}_{(2)\nu}^{\mu}=m_{(2)\nu}^{\mu}\,,\qquad\tilde{m}_{(4)\nu}^{\mu}=m_{(4)\nu}^{\mu}-m_{(2)\rho}^{\mu}m_{(2)\nu}^{\rho}\,,\qquad\cdots$
(135) $\displaystyle\tilde{n}_{(0)\nu}^{\mu}$
$\displaystyle=n_{(0)\nu}^{\mu}\,,\qquad\tilde{n}_{(2)\nu}^{\mu}=n_{(2)\nu}^{\mu}-m_{(2)\rho}^{\mu}n_{(0)\nu}^{\rho}-n_{(0)\rho}^{\mu}m_{(2)\nu}^{\rho}\,,\qquad\cdots.$

Also by taking the inverse of the metric, one finds the following relation:

$\displaystyle
m_{(2p)}-\tilde{m}_{(2p)}=\sum^{p-1}_{k=1}\tilde{m}_{(2k)}m_{(2p-2k)}\,.$
(136)

Specifically, we have

$\displaystyle m_{(2)}-\tilde{m}_{(2)}=0\,,\qquad
m_{(4)}-\tilde{m}_{(4)}=m^{2}_{(2)}\,,\qquad
m_{(6)}-\tilde{m}_{(6)}=m_{(2)}m_{(4)}+\tilde{m}_{(4)}m_{(2)}\,.$ (137)

Now we expand the quantities defined in (117) to an arbitrary order by
plugging the expansions (22), (23) and (134) into their definitions. For the
purpose of finding the Weyl anomaly, here we only keep the $m_{(2p)}$ and
$a_{(2p)}$ terms in the first series of $h_{\mu\nu}$ and $a_{\mu}$ and neglect
the $n_{(2p)}$ and $p_{(2p)}$ terms. The expansions of these quantities are

$\displaystyle\rho^{\mu}{}_{\nu}=$
$\displaystyle-\delta^{\mu}{}_{\nu}+\frac{1}{2}\sum_{p=1}^{\infty}\left(\frac{z}{L}\right)^{2p}\Big{[}p(m_{(2p)}+\tilde{m}_{(2p)})+\sum_{k=1}^{p-1}(2k-p)\tilde{m}_{(2k)}m_{(2p-2k)}\Big{]}^{\mu}{}_{\nu}+O(z^{d})\,,$
(138) $\displaystyle\theta=$
$\displaystyle-\frac{d}{L}+\frac{1}{2L}\sum^{\infty}_{p=1}\bigg{(}\frac{z}{L}\bigg{)}^{2p}\bigg{[}p\text{tr}(m_{(2p)}+\tilde{m}_{(2p)})+\sum^{p-1}_{k=1}(2k-p)\text{tr}\tilde{m}_{(2k)}m_{(2p-2k)}\bigg{]}+O(z^{d})\,,$
(139) $\displaystyle\varphi_{\mu}={}$
$\displaystyle\frac{1}{L}\sum^{\infty}_{p=0}\left(\frac{z}{L}\right)^{2p}2pa_{\mu}^{(2p)}+O(z^{d-2})\,,$
(140) $\displaystyle f_{\mu\nu}=$
$\displaystyle\sum_{p=0}^{\infty}\left(\frac{z}{L}\right)^{2p}\big{[}f^{(2p)}_{\mu\nu}+\sum_{q=1}^{p-1}2q(a_{\mu}^{(2p-2q)}p_{\nu}^{(2q)}-a_{\nu}^{(2p-2q)}p_{\mu}^{(2q)})\big{]}+O(z^{d-2})\,,$
(141) $\displaystyle\gamma^{\lambda}{}_{\mu\nu}={}$
$\displaystyle\gamma^{\lambda}_{(0)\mu\nu}-\sum_{p=1}^{\infty}\left(\frac{z}{L}\right)^{2p}\bigg{(}\sum_{q=0}^{p-1}\tilde{m}_{(2q)}^{\lambda}{}_{\rho}\hat{\gamma}^{\rho}_{(2p-2q)\mu\nu}+\frac{1}{2}\sum_{q=0}^{p-1}[\tilde{m}_{(2q)}\gamma_{(0)}^{-1}]^{\lambda\rho}\sum_{k=0}^{p-q-1}(2k-2)\qquad$
$\displaystyle\times(a^{(2p-2q-2k)}_{\mu}\gamma^{(2k)}_{\nu\rho}+a^{(2p-2q-2k)}_{\nu}\gamma^{(2k)}_{\mu\rho}-a^{(2p-2q-2k)}_{\rho}\gamma^{(2k)}_{\mu\nu})\bigg{)}+O(z^{d-2})\,,$
(142)

where

$\displaystyle f^{(0)}_{\mu\nu}$
$\displaystyle=\partial_{\mu}a_{\nu}^{(0)}-\partial_{\nu}a_{\mu}^{(0)}\,,\qquad
f^{(2k)}_{\mu\nu}=\hat{\nabla}^{(0)}_{\mu}a_{\nu}^{(2k)}-\hat{\nabla}^{(0)}_{\nu}a_{\mu}^{(2k)}\quad(k>0)\,,$
$\displaystyle\gamma^{\lambda}_{(0)\mu\nu}$
$\displaystyle=\frac{1}{2}\gamma_{(0)}^{\lambda\rho}\big{(}\partial_{\mu}\gamma^{(0)}_{\nu\rho}+\partial_{\nu}\gamma^{(0)}_{\mu\rho}-\partial_{\rho}\gamma^{(0)}_{\mu\nu}\big{)}-\big{(}a^{(0)}_{\mu}\delta^{\lambda}{}_{\nu}+a^{(0)}_{\nu}\delta^{\lambda}{}_{\mu}-a^{(0)}_{\rho}\gamma_{(0)}^{\lambda\rho}\gamma^{(0)}_{\mu\nu}\big{)}\,,$
$\displaystyle\hat{\gamma}^{\lambda}_{(2k)\mu\nu}$
$\displaystyle=\frac{1}{2}\gamma_{(0)}^{\lambda\rho}(\hat{\nabla}^{(0)}_{\mu}\gamma^{(2k)}_{\nu\rho}+\hat{\nabla}^{(0)}_{\nu}\gamma^{(2k)}_{\mu\rho}-\hat{\nabla}^{(0)}_{\rho}\gamma^{(2k)}_{\mu\nu})\quad(k>0)\,.$

Expanding everything in (C) using (138)–(142), we obtain the following
equation:

$\displaystyle 0={}$
$\displaystyle\frac{1}{L^{2}}p(p-1)\text{tr}(m_{(2p)}+\tilde{m}_{(2p)})+\frac{1}{L^{2}}\sum^{p-1}_{q=1}(p-1)(2q-p)\text{tr}\tilde{m}_{(2q)}m_{(2p-2q)}$
$\displaystyle-\sum^{p-1}_{q=1}2q\hat{\nabla}_{\mu}a_{\nu}^{(2q)}\big{[}\tilde{m}_{(2p-2q-2)}\gamma_{(0)}^{-1}\big{]}^{\mu\nu}-\sum^{p-1}_{q=1}\sum^{q-1}_{k=0}(2p-2q+2k)2ka_{\mu}^{(2p-2q)}a_{\nu}^{(2k)}\big{[}\tilde{m}_{(2q-2k-2))}\gamma_{(0)}^{-1}\big{]}^{\mu\nu}$
$\displaystyle-\sum^{p-1}_{q=1}\sum_{k=0}^{q-1}\sum_{n=0}^{p-q-1}na_{\lambda}^{(2n)}[\tilde{m}_{(2p-2q-2n-2)}\gamma^{-1}_{(0)}]^{\mu\nu}\bigg{(}\tilde{m}_{(2k)}^{\lambda}{}_{\rho}\hat{\gamma}^{\rho}_{(2q-2k)\mu\nu}$
$\displaystyle\quad-[\tilde{m}_{(2k)}\gamma_{(0)}^{-1}]^{\lambda\rho}\sum_{m=0}^{q-k-1}(2-2m)(a^{(2q-2k-2m)}_{\mu}\gamma^{(2m)}_{\nu\rho}+a^{(2q-2k-2m)}_{\nu}\gamma^{(2m)}_{\mu\rho}-a^{(2q-2k-2m)}_{\rho}\gamma^{(2m)}_{\mu\nu})\bigg{)}$
$\displaystyle+\frac{1}{4L^{2}}\sum_{q=1}^{p-1}(p-q)\text{tr}\bigg{[}(m_{(2p-2q)}+\tilde{m}_{(2p-2q)})\Big{[}q(m_{(2q)}+\tilde{m}_{(2q)})+\sum_{k=1}^{q-1}2(2k-q)\tilde{m}_{(2k)}m_{(2q-2k)}\Big{]}\bigg{]}$
$\displaystyle+\frac{1}{4L^{2}}\sum_{q=1}^{p-1}\sum_{k=1}^{q-1}\sum_{m=1}^{p-q-1}(2k-q)(2m-p+q)\text{tr}\big{[}\tilde{m}_{(2k)}m_{(2q-2k)}\tilde{m}_{(2m)}m_{(2p-2q-2m)}\big{]}$
$\displaystyle+\frac{L^{2}}{4}\sum_{q=1}^{p-1}\sum_{k=0}^{q-1}\big{[}f^{(2k)}_{\mu\rho}+\sum_{m=1}^{k-1}2m(a_{\mu}^{(2k-2m)}a_{\rho}^{(2m)}-a_{\rho}^{(2k-2m)}a_{\mu}^{(2m)})\big{]}[\tilde{m}_{(2q-2k-2)}\gamma^{-1}_{(0)}]^{\rho\nu}$
$\displaystyle\quad\times\sum_{n=0}^{p-q-1}\big{[}f^{(2n)}_{\nu\sigma}+\sum_{s=1}^{n-1}2s(a_{\nu}^{(2n-2s)}a_{\sigma}^{(2s)}-a_{\sigma}^{(2n-2s)}a_{\nu}^{(2s)})\big{]}[\tilde{m}_{(2p-2q-2n-2)}\gamma^{-1}_{(0)}]^{\sigma\mu}\,.$
(143)

From this equation, one can find $\text{tr}(m_{(2p)}+\tilde{m}_{(2p)})$ in
terms of $m_{(2q)}$ and $\tilde{m}_{(2q)}$ for all $q<p$.

Taking $p=3$ we get the Raychaudhuri equation at the $O(z^{6})$-order:

$\displaystyle 0={}$
$\displaystyle\frac{6}{L^{2}}\text{tr}(m_{(6)}+\tilde{m}_{(6)})+\frac{4}{L^{2}}\text{tr}(m_{(4)}m_{(2)})-\frac{4}{L^{2}}\text{tr}(m_{(2)}^{3})-\frac{L^{2}}{2}m_{(2)}^{\mu}{}_{\alpha}f^{\alpha}_{(0)\beta}f^{\beta}_{(0)\mu}$
$\displaystyle+4\hat{\nabla}\cdot
a^{(4)}-2m_{(2)}^{\mu}{}_{\rho}\gamma_{(0)}^{\rho\nu}\hat{\nabla}_{\nu}a_{\mu}^{(2)}-2\gamma_{(0)}^{\mu\nu}\hat{\gamma}^{\lambda}_{(2)\mu\nu}a^{(2)}_{\lambda}-2(d-6)a^{2}_{(2)}+\frac{L^{2}}{2}f^{(2)}_{\mu\nu}f_{(0)}^{\nu\mu}\,.$
(144)

And for $p=4$, we have the Raychaudhuri equation at the $O(z^{8})$-order:

$\displaystyle 0={}$
$\displaystyle\frac{12}{L^{2}}\text{tr}(m_{(8)}+\tilde{m}_{(8)})+\frac{9}{L^{2}}\text{tr}(m_{(6)}m_{(2)})-\frac{22}{L^{2}}\text{tr}(m_{(4)}m_{(2)}^{2})+\frac{6}{L^{2}}\text{tr}(m_{(2)}^{4})+\frac{4}{L^{2}}\text{tr}(m_{(4)}^{2})$
$\displaystyle+\frac{L^{2}}{4}f^{(0)}_{\mu\rho}f_{(0)}^{\nu\sigma}m_{(2)}^{\rho}{}_{\nu}m_{(2)}^{\mu}{}_{\sigma}+\frac{L^{2}}{2}f^{(0)}_{\mu\rho}f_{(0)}^{\rho\sigma}(m_{(2)}^{2})^{\mu}{}_{\sigma}-\frac{L^{2}}{2}f^{(0)}_{\mu\rho}f_{(0)}^{\rho\sigma}(m_{(4)})^{\mu}{}_{\sigma}+6\hat{\nabla}\cdot
a^{(6)}$
$\displaystyle-4\hat{\nabla}_{\mu}a_{\nu}^{(4)}\gamma_{(2)}^{\mu\nu}+L^{2}\hat{\nabla}_{[\mu}a^{(4)}_{\rho]}f_{(0)}^{\rho\mu}-4a_{\sigma}^{(4)}\gamma_{(0)}^{\mu\nu}\hat{\gamma}_{(2)}^{\sigma}{}_{\mu\nu}-6(d-8)a^{(4)}\cdot
a^{(2)}-2\hat{\nabla}_{\mu}a_{\nu}^{(2)}\gamma_{(4)}^{\mu\nu}$
$\displaystyle-2a_{\sigma}^{(2)}\gamma_{(0)}^{\mu\nu}\hat{\gamma}_{(4)}^{\sigma}{}_{\mu\nu}+2\hat{\nabla}_{\mu}a_{\nu}^{(2)}(m_{(2)}^{2})^{\mu}{}_{\rho}\gamma_{(0)}^{\rho\nu}+L^{2}\hat{\nabla}_{[\mu}a^{(2)}_{\rho]}\hat{\nabla}^{[\rho}a_{(2)}^{\mu]}-2L^{2}\hat{\nabla}_{[\mu}a^{(2)}_{\rho]}f_{(0)}^{\rho\sigma}m_{(2)}^{\mu}{}_{\sigma}$
$\displaystyle+2a_{\sigma}^{(2)}\gamma_{(2)}^{\mu\nu}\hat{\gamma}_{(2)}^{\sigma}{}_{\mu\nu}+2a_{\lambda}^{(2)}\gamma_{(0)}^{\mu\nu}m_{(2)}^{\lambda}{}_{\sigma}\hat{\gamma}_{(2)}^{\sigma}{}_{\mu\nu}+2(d-8)a^{(2)}_{\mu}a^{(2)}_{\nu}\gamma_{(2)}^{\mu\nu}+2X^{(1)}a^{(2)}\cdot
a^{(2)}\,.$ (145)

Now let us look at the expansion of $\sqrt{-\det h}$. Using the fact that
$\theta=D_{z}(\ln\sqrt{-\det h})$, we can write down the expansion of
$\sqrt{-\det h}$ to any order as

$\displaystyle\sqrt{-\det h}={}$
$\displaystyle\sqrt{-\det\gamma_{(0)}}\bigg{(}\frac{z}{L}\bigg{)}^{-d}\sum_{0}^{\infty}\frac{1}{n!}$
(146)
$\displaystyle\times\left[\frac{1}{2}\sum^{\infty}_{m=1}\bigg{(}\frac{z}{L}\bigg{)}^{2m}\bigg{[}\frac{1}{2}\text{tr}(m_{(2m)}+\tilde{m}_{(2m)})+\sum^{m-1}_{k=1}\bigg{(}\frac{k}{m}-\frac{1}{2}\bigg{)}\text{tr}(\tilde{m}_{(2k)}m_{(2m-2k)})\bigg{]}\right]^{n}\,.$

Comparing with (55), at the $O(z^{6})$-order and the $O(z^{8})$-order, the
above equation gives respectively

$\displaystyle X^{(3)}={}$
$\displaystyle\frac{1}{2}\text{tr}(m_{(6)}+\tilde{m}_{(6)})-\frac{1}{6}\text{tr}(m_{(2)}^{3})+\frac{1}{2}X^{(1)}X^{(2)}-\frac{1}{12}(X^{(1)})^{3}\,,$
(147) $\displaystyle X^{(4)}={}$
$\displaystyle\frac{1}{2}\text{tr}(m_{(8)}+\tilde{m}_{(8)})-\frac{1}{2}\text{tr}(m_{(4)}m_{(2)}^{2})+\frac{1}{4}\text{tr}(m_{(2)}^{4})$
$\displaystyle+\frac{1}{2}X^{(3)}X^{(1)}-\frac{1}{4}X^{(2)}(X^{(1)})^{2}+\frac{1}{4}(X^{(2)})^{2}+\frac{1}{32}(X^{(1)})^{4}\,.$
(148)

Now solving for $\text{tr}(m_{(6)}+\tilde{m}_{(6)})$ from (144) and plugging
(37), (39) and (66) into (147), we can organize all the $m_{(2)}$ and
$f_{(0)}$ terms in $X^{(3)}$ and get (69). Similarly, plugging
$\text{tr}(m_{(8)}+\tilde{m}_{(8)})$ obtained from (145) into (148), the
expression for $X^{(4)}$ can be organized in terms of the Weyl-Schouten tensor
and extended Weyl-obstruction tensors as

$\displaystyle\frac{24}{L^{2}}X^{(4)}={}$ $\displaystyle
L^{6}\bigg{(}\frac{1}{8}\hat{P}^{4}-\frac{3}{4}\text{tr}(\hat{P}^{2})\hat{P}^{2}+\frac{3}{8}[\text{tr}(\hat{P}^{2})]^{2}+\text{tr}(\hat{P}^{3})\hat{P}-\frac{3}{4}\text{tr}(\hat{P}^{4})-\text{tr}(\hat{\Omega}_{(1)}\hat{P})\hat{P}+\text{tr}(\hat{\Omega}_{(1)}\hat{P}^{2})$
$\displaystyle-\frac{1}{4}\text{tr}(\hat{\Omega}_{(1)}^{2})-\frac{1}{4}\text{tr}(\hat{\Omega}_{(2)}\hat{P})\bigg{)}+2(d-8)\big{[}3a^{(4)}\cdot
a^{(2)}+a^{(2)}_{\mu}a^{(2)}_{\nu}(\hat{P}^{\mu\nu}-\hat{P}\gamma_{(0)}^{\mu\nu})\big{]}-6\hat{\nabla}\cdot
a^{(6)}$
$\displaystyle-L^{2}\hat{\nabla}_{\mu}\big{[}a_{\nu}^{(4)}(4\hat{P}^{\mu\nu}+2\hat{P}^{\nu\mu}-4\hat{P}\gamma_{(0)}^{\mu\nu})\big{]}-\frac{L^{2}}{2}\hat{\nabla}_{\mu}\big{[}a^{(2)}_{\nu}(3\hat{\nabla}^{\nu}a_{(2)}^{\mu}+\hat{\nabla}^{\mu}a^{\nu}_{(2)}-3\hat{\nabla}\cdot
a_{(2)}\gamma_{(0)}^{\mu\nu})\big{]}$
$\displaystyle+L^{4}\hat{\nabla}_{\mu}\big{[}a^{(2)}_{\nu}(3\hat{P}^{\mu\nu}\hat{P}+\hat{P}^{\nu\mu}\hat{P})\big{]}+\frac{3L^{4}}{2}\hat{\nabla}^{\mu}\big{[}a^{(2)}_{\mu}(\text{tr}(\hat{P}^{2})-\hat{P}^{2})\big{]}-\frac{3L^{4}}{2}\hat{\nabla}_{\mu}(a_{\nu}^{(2)}\hat{\Omega}_{(1)}^{\mu\nu})$
$\displaystyle-\frac{L^{4}}{4}\hat{\nabla}_{\mu}\big{[}a_{\nu}^{(2)}(3\hat{P}^{\rho\mu}\hat{P}^{\nu}{}_{\rho}-5\hat{P}^{\rho\mu}\hat{P}_{\rho}{}^{\nu}+7\hat{P}^{\mu\rho}\hat{P}_{\rho}{}^{\nu}-9\hat{P}^{\mu\rho}\hat{P}^{\nu}{}_{\rho})\big{]}\,,$
(149)

which leads to (5.4).

##  Appendix D Dilatation Operator Method

In this appendix we will derive the holographic Weyl anomaly in $8d$ using the
recursive algorithm of [35] which we will refer to as the dilatation operator
method. We point out that this method uses the usual FG gauge where the Weyl
connection is turned off. This alternative way of finding the Weyl anomaly
will provide a nontrivial consistency check of the results presented in
Section 5. For completeness we start with a brief review of the algorithm of
the dilatation operator method. We then apply the algorithm one step further
than [35] and compute the holographic Weyl anomaly in $8d$.

###  D.1 Review of the Algorithm

We start by using the metric in the FG gauge (2)

$\text{d}s^{2}=\text{d}r^{2}+h_{\mu\nu}(r;x)\text{d}x^{\mu}\text{d}x^{\nu}\,,\qquad\mu,\nu=1,\cdots,d\,,$
(150)

where we changed the coordinates by setting $r=-L\text{ln}\left(z/L\right)$.
The Einstein-Hilbert action in the bulk manifold $M$ with a Gibbons-Hawking
boundary term is

$S_{\text{EH-
GH}}=\frac{1}{2\kappa^{2}}\int_{M}\text{d}r\text{d}^{d}x\sqrt{-\text{det}g}(R-2\Lambda)+\frac{1}{\kappa^{2}}\int_{\partial
M_{r_{c}}}\text{d}^{d}x\sqrt{-\det h}K\,,$ (151)

where $\kappa^{2}=8\pi G$, $\Lambda=-\frac{d(d-1)}{2L^{2}}$, and $\partial
M_{r_{c}}$ is a cutoff surface at some large value of $r_{c}$.999We abuse
notation and call the cuttof surface $\partial M_{r_{c}}$ even though it is
not necessarily the boundary of $M$.  Taking a metric variation and evaluating
the result on-shell one gets

$\delta S_{EHGH}^{\text{ o.s}}=\int_{\partial
M_{r_{c}}}\text{d}^{d}x\pi^{\mu\nu}\delta
h_{\mu\nu}\,,\qquad\pi^{\mu\nu}=\frac{1}{2\kappa^{2}}\sqrt{-\det
h}\left(Kh^{\mu\nu}-K^{\mu\nu}\right)\,,$ (152)

where $K_{\mu\nu}=\frac{1}{2}\partial_{r}h_{\mu\nu}$ is the extrinsic
curvature tensor in the FG gauge, and $h_{\mu\nu}$ is the induced metric on
$\partial M_{r_{c}}$. The boundary tensor density $\pi^{\mu\nu}$ appears in
many different contexts; it was first defined in [51] and was later used in
[52] to define a boundary stress tensor in an asymptotically AdS spacetime
(with the inclusion of necessary counterterms to cancel divergences). It also
appears as the conjugate momenta in the ADM formalism [53].101010The sign
difference in the definition of $\pi^{\mu\nu}$ in [53] arises because here we
consider the radial evolution which is in a spacelike direction.

The $rr$\- and $r\mu$-components of the Einstein equations, i.e.
$G^{rr}+\Lambda g^{rr}=0$ and $G^{r\mu}=0$, can be written in terms of the
conjugate momenta as follows:

$\displaystyle\frac{2\kappa^{2}}{\sqrt{-\det
h}}\left(\pi\indices{{}^{\mu}_{\nu}}\pi^{\nu}_{\mu}-\frac{1}{d-1}\pi^{2}\right)+\frac{1}{2\kappa^{2}}\sqrt{-\det
h}\left(\mathring{R}-2\Lambda$ $\displaystyle=0\,,$ (153)
$\displaystyle\mathring{\nabla}_{\rho}\pi^{\rho\mu}$ $\displaystyle=0\,,$
(154)

where $\pi\equiv h_{\mu\nu}\pi^{\mu\nu}$ ,$\mathring{\nabla}$ is the LC
connection of the induced metric $h_{\mu\nu}$ on $\partial M_{r_{c}}$, and
$\mathring{R}$ is the LC Ricci scalar of $h_{\mu\nu}$. Note that the indices
are raised and lowered using the induced metric $h_{\mu\nu}$. Eq.
$\eqref{EE_p1}$ and (154) are the well-known Hamiltonian and momentum
constraints in the ADM language [53].

The dilatation operator method of solving the constraint equations uses an
asymptotic expansion of the conjugate momenta in terms of the induced metric.
One assumes a Hamilton-Jacobi functional $S[h]$ such that

$\pi^{\mu\nu}=\frac{\delta S[h]}{\delta h_{\mu\nu}}\,,\qquad
S[h]\equiv\int_{\partial M_{r_{c}}}\text{d}^{d}x\mathcal{L}[h]\,,$ (155)

where $S[h]$ is a local diffeomorphic invariant functional of the induced
metric.111111In [35] the functional $S[h]$ is used to derive the boundary
terms for a well-defined bulk variational problem. Since these are tangential
remarks for the calculation of the Weyl anomaly, we simply neglect them and
refer the reader there for more details.

The momentum constraint (154) is now trivially satisfied. The Hamiltonian
constraint (153) can be solved asymptotically by writing

$\mathcal{L}=\sum_{k=0}\mathcal{L}_{(2k)}[h]\,,\qquad\delta_{D}\mathcal{L}_{(2k)}[h]=(d-2k)\mathcal{L}_{(2k)}[h]\,,$
(156)

where $\delta_{D}$ is the dilatation operator [34] (acting on metric
functionals), defined as

$\displaystyle\delta_{D}\equiv\int\text{d}^{d}x\left[2h_{\mu\nu}\frac{\delta}{\delta
h_{\mu\nu}}\right]\,.$ (157)

It is useful to keep in mind that for an AlAdS spacetime the radial derivative
$\partial_{r}$ asymptotes the dilatation operator, i.e.
$\delta_{D}\sim\frac{\partial}{\partial r}$ (see [35]). We can view the
dilatation operator expansion as another asymptotic expansion near the
conformal boundary since $h_{\mu\nu}\sim
e^{2r/L}\gamma_{\mu\nu}^{(0)}+\cdots$.

The expansion (156) together with (155) implies an expansion of $\pi^{\mu\nu}$
in terms of the dilatation weight:

$\pi^{\mu\nu}_{(2k)}=\frac{\delta}{\delta h_{\mu\nu}}\int_{\partial
M_{r_{c}}}\text{d}^{d}x\mathcal{L}_{(2k)}\,,\qquad\pi^{\mu\nu}=\sum_{k=0}\pi^{\mu\nu}_{(2k)}\,.$
(158)

The $\mathcal{L}_{(2k)}[h]$ are defined only up to total derivative terms in
$\partial M_{r_{c}}$. To set up the recursive algorithm we need the following
relation:

$\pi_{(2k)}=\frac{L}{2}\mathcal{Q}_{(2k)}\,,\qquad\pi_{(2k)}:=h_{\mu\nu}\pi^{\mu\nu}_{(2k)}\,,$
(159)

where we defined the _Q-curvature_

$\displaystyle\mathcal{Q}_{(2k)}:=\frac{d-2k}{L}\mathcal{L}_{(2k)}\,.$ (160)

The Q-curvature is the main quantity we are interested in, since it is
proportional to the Weyl anomaly in a specific even dimension [see (163)].
Plugging $\eqref{pi_2n- definition}$ into $\eqref{EE_p1}$ and making use of
$\eqref{trace_of_pi}$ we find

$\mathcal{Q}_{(2k)}=\frac{2\kappa^{2}}{\sqrt{-\det
h}}\sum_{m=1}^{k-1}\left(\pi\indices{{}_{(2m)}^{\mu}{}_{\nu}}\pi\indices{{}_{(2k-2m)}^{\nu}{}_{\mu}}-\frac{1}{d-1}\pi_{(2m)}\pi_{(2k-2m)}\right)\,,$
(161)

where we used the initial values

$\mathcal{Q}_{(2)}=\frac{\sqrt{-\det
h}}{2\kappa^{2}}\mathring{R}\,,\qquad\pi_{(0)}^{\mu\nu}=\frac{(d-1)}{2\kappa^{2}L}\sqrt{-\det
h}h^{\mu\nu}\,.$ (162)

Equations $\eqref{Q-Algorithm}$, $\eqref{pi_2n- definition}$ and the initial
values (162), are enough to fix the iterative algorithm of the dilatation
operator method. Expanding on this a little more, given the value of
$\mathcal{Q}_{(2)}$ we can use (158) to find $\pi_{(2)}^{\mu\nu}$. Since we
now have $\pi_{(2)}^{\mu\nu}$ and $\pi_{(0)}^{\mu\nu}$, we can find
$\mathcal{Q}_{(4)}$ from (161). The process can then be iterated to compute
$\pi_{(2k)}^{\mu\nu}$ and $\mathcal{Q}_{(2k)}$ to any order. The recursive
algorithm has been solved up to $\pi_{(4)}^{\mu\nu}$ and $\mathcal{Q}_{(6)}$
in [35]. In the next section we will push the calculation one step further for
finding $\pi^{\mu\nu}_{(6)}$ and $\mathcal{Q}_{(8)}$. This will enable us to
find the Weyl anomaly in $8d$.

###  D.2 Results and Anomaly in $8d$

We have explained the algorithm for solving the Hamiltonian and momentum
constraints. We now focus on the Q-curvature, which is expressed in terms of
the conjugate momenta in (161). The Weyl anomaly in $d$-dimension corresponds
to the Q-curvature for $d=2k$[35]:

${\cal A}_{k}=-L\int\text{d}^{d}x\ln{\cal B}{\cal Q}_{(2k)}^{d=2k}\,.$ (163)

We now present the results of the algorithm presented in D.1. First, we review
the results for $\pi_{(2k)}^{\mu\nu}$ and $\mathcal{Q}_{(2k)}$ up to $k=3$:

$\begin{split}\pi_{(0)}^{\mu\nu}&=\frac{(d-1)}{2\kappa^{2}L}\sqrt{-\det
h}h^{\mu\nu}\,,\\\ \pi_{(2)}^{\mu\nu}&=-\frac{\sqrt{-\det
h}L}{2\kappa^{2}}\left(\mathring{P}^{\mu\nu}-\mathring{P}h^{\mu\nu}\right)\,,\\\
\pi_{(4)}^{\mu\nu}&=-\frac{\sqrt{-\det
h}L^{3}}{2\kappa^{2}(d-4)(d-2)}\left[\mathring{B}^{ij}+(d-4)\left(\mathring{P}^{\mu}_{~{}\lambda}\mathring{P}^{\lambda\nu}-\mathring{P}\mathring{P}^{\mu\nu}-\frac{1}{2}h^{\mu\nu}(\text{tr}(\mathring{P}^{2})-\mathring{P}^{2})\right)\right]\,,\end{split}$
(164)

and

$\begin{split}\mathcal{Q}_{(2)}&=\frac{\sqrt{-\det
h}}{2\kappa^{2}}\mathring{R}\,,\\\ \mathcal{Q}_{(4)}&=\frac{\sqrt{-\det
h}L^{2}}{2\kappa^{2}}\left[\text{tr}(\mathring{P}^{2})-\mathring{P}^{2}\right]\,,\\\
\mathcal{Q}_{(6)}&=\frac{\sqrt{-\det
h}L^{4}}{\kappa^{2}(d-4)(d-2)}\left[\text{tr}(\mathring{P}\mathring{B})+(d-4)\bigg{(}\text{tr}(\mathring{P}^{3})-\frac{3}{2}\mathring{P}\text{tr}(\mathring{P}^{2})+\frac{1}{2}\mathring{P}^{3}\bigg{)}\right]\,.\end{split}$
(165)

We can see that $\mathcal{Q}_{(2)}$, $\mathcal{Q}_{(4)}$ and
$\mathcal{Q}_{(6)}$ correspond to the LC counterparts of the Weyl anomaly
shown in (67), (68) and (5.3), respectively. Using $\mathcal{Q}_{(6)}$ in
(165) we can calculate $\pi_{(6)}^{\mu\nu}$ from (158) by taking a metric
variation of $\mathcal{Q}_{(6)}$. The result is as follows:

$\displaystyle\pi_{(6)}^{\mu\nu}={}$
$\displaystyle-\frac{L^{5}}{\kappa^{2}(d-6)(d-4)(d-2)^{2}}\bigg{[}\mathcal{\mathring{O}}_{(6)}^{\mu\nu}+(d-6)\bigg{(}\mathring{P}\indices{{}^{(\mu}_{\lambda}}\mathring{B}^{\nu)\lambda}-\mathring{P}\mathring{B}^{\mu\nu}-2\mathring{P}_{\rho\lambda}\mathring{\nabla}^{\lambda}\mathring{C}^{(\mu\nu)\rho}$
$\displaystyle-(\mathring{\nabla}_{\lambda}P)\mathring{C}^{(\mu\nu)\lambda}+\mathring{C}^{\rho\mu\lambda}\mathring{C}\indices{{}_{\lambda}^{\nu}{}_{\rho}}-\frac{1}{2}\mathring{C}\indices{{}^{\mu}{}^{\rho\lambda}}\mathring{C}\indices{{}^{\nu}_{\rho\lambda}}+\mathring{P}\mathring{W}\indices{{}^{\mu}_{\alpha\beta}{}^{\nu}}\mathring{P}^{\alpha\beta}+\frac{1}{2}\mathring{\nabla}^{\lambda}\mathring{\nabla}_{\lambda}(\mathring{P}^{\mu\beta}\mathring{P}\indices{{}^{\nu}_{\beta}}-\mathring{P}\mathring{P}^{\mu\nu})$
$\displaystyle-\frac{(d-2)}{4(d-1)}\mathring{\nabla}^{\mu}\mathring{\nabla}^{\nu}(\text{tr}(\mathring{P}^{2})-\mathring{P}^{2})-\frac{1}{4(d-1)}h^{\mu\nu}\mathring{\nabla}^{\lambda}\mathring{\nabla}_{\lambda}(\text{tr}(\mathring{P}^{2})-\mathring{P}^{2})+(d-4)\mathring{P}^{\mu}_{~{}~{}\alpha}\mathring{P}^{\alpha\beta}\mathring{P}\indices{{}_{\beta}^{\nu}}$
$\displaystyle-\frac{(3d^{2}-12d+8)}{4(d-1)}\mathring{P}^{\mu\nu}(\text{tr}(\mathring{P}^{2})-\mathring{P}^{2})-(d-4)\mathring{P}\mathring{P}^{\mu}{}_{\lambda}\mathring{P}^{\lambda\nu}-\frac{1}{2}h^{\mu\nu}\Big{(}\text{tr}(\mathring{P}\mathring{B})+(d-4)\text{tr}(\mathring{P}^{3})$
$\displaystyle-\frac{(3d^{2}-14d+10)}{2(d-1)}\mathring{P}\text{tr}(\mathring{P}^{2})+\frac{(d^{2}-4d+2)}{2(d-1)}\mathring{P}^{3}\Big{)}\bigg{)}\bigg{]}\,,$
(166)

where ${\cal\mathring{O}}^{\mu\nu}_{(6)}$ is the LC obstruction tensor defined
in (2). We have also checked that $\pi_{(6)}^{\mu\nu}$ is divergence-free in
any dimension, as is required by (154).121212This calculation was done thanks
to the Mathematica package diffgeo.m by Matthew Headrick.

By plugging (D.2) and (164) into (161) we find after some reorganization the
expression of $\mathcal{Q}_{(8)}$ as follows:

$\displaystyle\mathcal{Q}_{(8)}={}$ $\displaystyle\frac{2L^{6}\sqrt{-\det
h}}{\kappa^{2}(d-6)(d-4)(d-2)^{2}}\bigg{[}\frac{(d-2)}{2(d-4)}\text{tr}(\mathring{P}\mathring{\mathcal{O}}_{(6)})+(d-6)\bigg{(}\frac{3}{4(d-4)}\text{tr}(\mathring{B}^{2})+\frac{5d-16}{2(d-4)}\text{tr}(\mathring{P}^{2}\mathring{B})$
$\displaystyle-\frac{5d-16}{2(d-4)}\mathring{P}\text{tr}(\mathring{P}\mathring{B})-\frac{5d^{2}-20d+8}{16(d-1)}\mathring{P}^{4}+\frac{16-5d}{2}\mathring{P}\text{tr}(\mathring{P}^{3})+\frac{15d^{2}-62d+40}{8(d-1)}\mathring{P}^{2}\text{tr}(\mathring{P}^{2})$
$\displaystyle-\frac{13d^{2}-44d+24}{16(d-1)}(\text{tr}(\mathring{P}^{2}))^{2}+\frac{7d-20}{4}\text{tr}(\mathring{P}^{4})\bigg{)}+\mathring{\nabla}_{\mu}K^{\mu}\bigg{]}\,,$
(167)

where
$K^{\mu}=\frac{(d-6)}{2(d-4)}K_{0}^{\mu}+\frac{(d-6)}{2}K_{1}^{\mu}+\frac{(d-6)}{2}K_{2}^{\mu}+\frac{(d-6)(d-2)}{4(d-1)}K_{3}^{\mu}$,
with