††institutetext:  a Mathematics Department, King’s College London,
The Strand, London, WC2R 2LS, UK
b SISSA and INFN,
Via Bonomea 265, 34136, Trieste, Italy
c CMSA, Harvard University,
20 Garden St, Cambridge, MA 02138

# Anomalies from correlation functions in defect conformal field theory

Christopher P. Herzoga and Itamar Shamirb,c
christopher.herzog<EMAIL_ADDRESS>
###### Abstract

In previous work, we showed that an anomaly in the one point function of
marginal operators is related by the Wess-Zumino condition to the Euler
density anomaly on a two dimensional defect or boundary. Here we analyze in
detail the two point functions of marginal operators with the stress tensor
and with the displacement operator in three dimensions. We show how to get the
boundary anomaly from these bulk two point functions and find perfect
agreement with our anomaly effective action. For a higher dimensional
conformal field theory with a four dimensional defect, we describe for the
first time the anomaly effective action that relates the Euler density term to
the one point function anomaly, generalizing our result for two dimensional
defects.

##  1 Introduction

Our goal is to explore in more depth the connection exposed in refs.
Herzog:2019rke ; Bianchi:2019umv  between bulk marginal operators and Euler
density terms in the trace anomaly for boundary and defect conformal field
theories. These theories present an important frontier in the study of quantum
field theory more generally. Conformal field theories provide a good
description of a number of real world experimental systems at the point of a
second order phase transition, and moreover these experimental systems almost
always involve boundaries, interfaces, and defects which can have important
physical effects. From a more abstract point of view, conformal field theories
describe the fixed points of the renormalization group flow of relativistic
quantum field theories. Thus, they are good starting points, with enhanced
symmetry, from which to explore the landscape of quantum field theory. Defect
conformal field theory provides an opportunity for a better treatment of
extended objects in quantum field theory, such as Wilson lines. There is also
the possibility of realizing the renormalization group spatially, through
domain walls that separate the high and low energy fixed points Gaiotto:2012np
.

Against this backdrop, there has been intense interest in the trace anomaly of
the stress tensor. While the stress tensor is classically traceless in
conformal field theories, through quantum effects the trace may fail to vanish
in even dimensions when a background metric is introduced. The trace is then
equal to a sum of curvature invariants with special properties. We focus here
on the Euler density, i.e. the curvature invariant that integrates to the
Euler characteristic on a compact manifold without boundary. Writing the
expectation value of the trace as $\langle
T^{\mu}{}_{\mu}\rangle=aE_{d}+\ldots$, where $E_{d}$ is the Euler density, the
coefficient $a$ has a number of remarkable properties. Chief among them is
that in two and four dimensional relativistic QFT, the coefficient $a$ must
decrease under renormalization group flow Zamolodchikov:1986gt ;
Komargodski:2011vj , providing a primitive ordering on the space of quantum
field theories.

Interestingly, the introduction of defects and boundaries provides an
opportunity for new types of Euler density terms associated with even
dimensional sub-manifolds, $\langle
T^{\mu}{}_{\mu}\rangle=fE_{p}\delta^{(q)}(z)+\ldots$, where $p$ is the
dimensionality of the submanifold, $p+q=d$ is the space-time dimension, and
the coordinates $z$ parametrize the normal directions to the defect. We use
the coefficient $f$ to denote the “charge” associated with the boundary or
defect anomaly associated with the Euler density. This contribution persists
even when the ambient space-time is odd dimensional and there would not
normally be a trace anomaly. In this work, we are most interested in the cases
of two and four dimensional boundaries and defects.

The possibility of marginal operators, i.e. operators of dimension exactly
equal to the space-time dimension, means that fixed points in the
renormalization group may actually be fixed manifolds, where one can move
around in theory space by tuning the value of the associated marginal
couplings for the operators. In the case of conformal field theory without
boundary or defect, the coefficent $a$ is known to be independent of the value
of these marginal couplings (due to Wess-Zumino consistency Osborn:1991gm ;
Wess:1971yu ). It is natural then to wonder what the story is for boundary and
surface Euler density contributions. It is not true in general that the
boundary and defect coefficients $f$ are independent of all marginal couplings
Herzog:2019rke ; Bianchi:2019umv . The natural generalization of the statement
in the bulk is that $f$ is independent only of marginal couplings associated
to operators that live purely on the boundary or defect. This restriction is
consistent with a similar restriction on the monotonicity of $f$ for surface
defects under renormalization group flow. This monotonicity is proven for
boundary and defect renormalization group flows Jensen:2015swa , not for bulk
flows.

A bulk marginal operator ${\mathcal{O}}_{I}$, on the other hand, will in
general have a nonzero one point function in boundary and defect CFT whose
form is fixed by conformal invariance up to a single constant $f_{I}$. Our
work Herzog:2019rke  along with Bianchi:2019umv  suggests that the derivative
of $f$ with respect to a bulk marginal coupling $\lambda^{I}$ is proportional
to the coefficient $f_{I}$ of the one-point functions of the associated
marginal operator.

In more detail, in ref. Herzog:2019rke  we began an exploration of the
connection between bulk marginal operators and $f$ for two dimensional
boundaries and defects. There is an anomaly in the one point function
$\langle\mathcal{O}_{I}\rangle$ of these marginal operators. Promoting the
theory to curved space time with a position dependent $\lambda^{I}(x)$ allows
us to construct an anomaly effective action (14). For the effective action to
be Wess-Zumino consistent, the coefficients $f_{I}$ must be proportional to
$\partial f/\partial\lambda^{I}$.

In this work, we explore the connection between $f$ and $\lambda^{I}$ in two
directions. In the first instance, we investigate more carefully the emergence
of the full (consistent) anomaly in (14) from the correlation functions
$\langle T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ and $\langle D({\bf
x_{1}})\mathcal{O}_{I}(x_{2})\rangle$ where $D({\bf x})$ is the displacement
operator, i.e. the operator dual to the position of the boundary. This anomaly
effective action was obtained in Herzog:2019rke  by invoking the Wess-Zumino
consistency condition, and is here verified by explicitly computing the scale
dependence of the aforementioned two point functions.

Specifically the two point function $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ gives rise to the Euler
density part of the anomaly effective action and is reminiscent in this regard
of the well known computation of the trace anomaly of $\langle
T^{\mu\nu}(x_{1})T^{\lambda\rho}(x_{2})\rangle$ in two dimensional CFTs. While
the two dimensional anomaly appears in the coincident limit $x_{1}\to x_{2}$,
in our case the anomaly appears in the double limit where $x_{1}\to x_{2}$ and
both points are taken to the boundary. To see this boundary contribution, we
rewrite $\langle T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ in terms of a
$d+1$ dimensional variable $w$, making manifest the coincident boundary
singularities which are otherwise hidden from sight. The resolution of these
singularities via the method of differential regularization Freedman:1991tk
leads in $d=3$ to the trace anomaly localized on the boundary. As the boundary
is even dimensional, this picture is consistent with the absence of intrinsic
scale anomalies in odd dimensions, mentioned above. There is another source of
the anomaly which must be taken into consideration if we are to account for
the full anomaly. The resolution of singularities following differential
regularization works by the extraction of derivatives which reduces the degree
of divergence. When there is a boundary, this procedure invariably leads to
boundary terms which also contribute to the anomaly, if the boundary is even
dimensional.111In higher codimension, this two point function involves an
undetermined function of two cross ratios which makes isolating the anomaly a
more challenging task that we leave for the future.

In the second instance, we generalize the anomaly effective action (14) for
four dimensional defects and boundaries in higher dimensional conformal field
theory. In other words we find a solution for the Wess-Zumino completion of
the one point anomaly, and show that this solution includes the four
dimensional Euler density $E_{4}$. The effective action we need in this case
turns out to be significantly more complicated than the surface case,
involving several dozen invariants built out of intrinsic and extrinsic
curvatures. Nevertheless, we are able to show that this structure produces the
correct anomaly for the $\langle D^{i}({\bf
x}_{1})\mathcal{O}_{I}(x_{2})\rangle$ two point function. (Note that in defect
conformal field theory, the displacement operator becomes a vector quantity,
with indices that run over the directions normal to the defect.) We also
verify that the derivative with respect to $\lambda^{I}$ of the effective
action on the sphere matches the integrated one point function of
$\mathcal{O}_{I}$, providing another cross check of our lengthy result.

In section 2 we begin by reviewing the construction in ref. Herzog:2019rke  of
the anomaly effective action for a three dimensional conformal field theory
with a two dimensional boundary. We then compute from this action the
anomalous contribution to the two point functions $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ and $\langle D({\bf
x}_{1})\mathcal{O}_{I}(x_{2})\rangle$. Finally, we review from ref.
McAvity:1995zd  the conservation and trace Ward identities for $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$. We show that these Ward
identities are consistent with the anomaly in the two-point functions. In
section 3 we rewrite the correlation function $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ in a way adapted to reveal its
anomaly. Then in section 4, we match the anomalies in $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ and $\langle D({\bf
x}_{1})\mathcal{O}_{I}(x_{2})\rangle$ to the result from the effective action.
Section 5 contains the computation of the anomaly effective action for four
dimensional defects along with the match of the anomaly in $\langle D^{i}({\bf
x}_{1})\mathcal{O}_{I}(x_{2})\rangle$. A final discussion section concludes
with some remarks about the bootstrap and conformal geometry. Appendix A
contains some identities for the $J_{(\alpha,\beta,\gamma)}$ functions we use
to rewrite $\langle T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ in order
to better reveal its anomalous structure, and a computation of the Ward
identity.

##  2 Anomalous correlation functions in three dimensions

###  2.1 The one point anomaly

Let us begin by a review of the result of our previous work Herzog:2019rke .
Consider a boundary conformal field theory (BCFT) in $d$ dimensions. Let us
also assume that this theory admits a set of exactly marginal bulk operators
$\mathcal{O}_{I}$, with a conformal manifold spanned by the associated
couplings $\lambda^{I}$. We use coordinates $x^{\mu}=(x^{a},z)$ where $x^{a}$
(or $\mathbf{x}$ which we use interchangeably) are coordinates along the
boundary and $z\leq 0$ is normal to the boundary, such that the boundary is
specified by $z=0$. The reason for working with negative values of $z$ is so
that the unit normal to the boundary $n^{\mu}$ is directed along the positive
$z$ direction. In other words $\partial_{\mu}z=n_{\mu}$. For the time being we
are considering flat space.

The one point function of exactly marginal operators is fixed by conformal
symmetry up to a coefficient which can depend on the couplings $\lambda^{I}$.
It is given by

$\displaystyle\langle O_{I}(x)\rangle=\frac{f_{I}(\lambda)}{|z|^{d}}.$ (1)

The basic claim is that this one point function has a scale anomaly, a
consequence of the nonintegrability of the correlation function near the
boundary for $d\geq 1$. For the effective action $W[\lambda]$ to be well-
defined, we must resolve the boundary singularity; for $d$ integer, resolution
requires violating the scaling symmetry. There are many techniques for
revealing the violation of scaling symmetry. Laplace transform, Fourier
transform, and differential regularization are all commonly used. In
Herzog:2019rke  we used differential regularization Freedman:1991tk , but will
use the Laplace transform here instead. The Fourier transform method is most
useful in analyzing scale anomalies associated with coincidence singularities
in correlation functions. In converting these functions from real space to
momentum space, the anomalies show up as short distance divergences in the
Fourier transform where the insertion points of the correlation function
coincide. Our systems have scale anomalies associated with taking a boundary
or defect limit of the insertion points, and a Laplace transform feels more
appropriate:

$\displaystyle\int_{-\infty}^{0}{\rm
d}z\,e^{sz}|z|^{-\Delta}=\Gamma(1-\Delta)s^{\Delta-1},$ (2)

where $\Delta$ is the dimension of the operator we consider. Clearly the
problem appears for integer $\Delta$ when the Gamma function has a pole. Other
values of $\Delta>1$ are well-defined by analytic continuation. For positive
integer values $d$ we expand with $\Delta=d+\epsilon$ for small $\epsilon$. We
find

$\displaystyle\frac{(-1)^{d}}{(d-1)!}\left(\frac{s^{d-1}}{\epsilon}+s^{d-1}\left(\log(s/\Lambda)-\psi^{(0)}(d)\right)+O(\epsilon)\right),$
(3)

where $\psi^{(0)}(d)=-\gamma+\sum_{k=1}^{d-1}\frac{1}{k}$ is the polygamma
function, and $\gamma$ the Euler-Mascheroni constant. The divergent term, of
order $\epsilon^{-1}$, is a polynomial in $s$ and thus can be removed by a
local counter-term on the boundary. For the second term we introduce a scale
$\Lambda$ into the logarithm. Thus, the one point function acquires an
anomalous scale dependence, leading to

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
O_{I}\rangle=f_{I}(\lambda)\frac{(-1)^{d-1}}{(d-1)!}s^{d-1}.$ (4)

Clearly the anomaly is a local term as it should be. In position space, this
monomial in $s$ can be produced by derivatives acting on a delta function:

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
O_{I}\rangle=f_{I}(\lambda)\frac{1}{(d-1)!}\partial_{z}^{d-1}\delta(z).$ (5)

Let us now discuss the one point anomaly from the point of view of the
effective action $W=-\log Z$, where $Z$ is the Euclidean path integral. We use
the effective action as a way of summarizing the anomalous scale dependence of
correlation functions. We start by recalling some basic facts about this
construction. We define the effective action by coupling to an arbitrary
background metric $g_{\mu\nu}$ and taking the couplings to be space dependent
$\lambda^{I}(x)$. The effective action also depends on the geometry of the
boundary. This is described by an embedding $x^{\mu}=X^{\mu}(u^{a})$ where
$u^{a}$ are coordinates on the boundary. Since we start from a BCFT in flat
space, the effective action should satisfy certain conditions, at least
classically. First, the effective action should be diffeomorphism invariant.
More precisely, we restrict to the diffeomorphisms that keep the boundary
unchanged up to a reparametrization of the boundary coordinates $u^{a}$.
Second, the effective action should be invariant under a Weyl transformation
$\delta_{\sigma}g_{\mu\nu}=e^{2\sigma}g_{\mu\nu}$.

When either of these conditions fails to be satisfied by the effective action,
we have an anomaly. Sometimes we can satisfy either condition but not at the
same time, in which case we always choose to uphold diff invariance, whenever
possible. In this paper we consider Weyl anomalies, i.e. $\delta_{\sigma}W\neq
0$. It is known from general principles that $\delta_{\sigma}W$ must satisfy
certain conditions. It must be diff invariant and a local function of the
background fields as well as the variation parameter $\delta\sigma$.
Furthermore, the effective action can be modified by adding local diff
invariant counter terms, and hence the anomaly is well-defined only up to Weyl
variation of such terms. Lastly, the anomaly should satisfy the Wess-Zumino
consistency condition Wess:1971yu ; Osborn:1991gm : because the Weyl action is
Abelian, the effective action must satisfy

$\displaystyle[\delta_{\sigma_{1}},\delta_{\sigma_{2}}]W=0.$ (6)

Returning to the one point anomaly, the scale transformation of the one point
function that we found above can be related to a Weyl transformation with a
constant parameter $\delta\sigma=-\delta\log\Lambda$, evaluated in flat
space.222The relative sign is because positive $\delta\sigma$ increases length
scale and decreases energy scales. Thus, the dependence of the one point
function on $\Lambda$ implies we must include in our anomalous effective
action the term

$\displaystyle\delta_{\sigma}W=-\frac{1}{(d-1)!}\int{\rm
d}^{d-1}x\,\delta\sigma f_{I}\partial_{n}^{d-1}\lambda^{I},$ (7)

written in flat space. The idea here is that taking the scale variation of

$\displaystyle\frac{\delta}{\delta\lambda^{I}(x)}W=\langle O_{I}(x)\rangle,$
(8)

we can reproduce (5).

Promoting the result (7) to a curved space and boundary poses certain
challenges. It is not a priori clear what $\partial_{n}^{d-1}$ should mean in
a curved background, and the most obvious choices, e.g. promoting
$\partial_{n}$ to a covariant derivative normal to the boundary, will not be
Wess-Zumino consistent on their own. To address, these issues, first we must
introduce some notation. We use $\nabla$ for the bulk connection and
$\bar{\nabla}$ for the induced boundary connection. We often use the short
hand notation

$\displaystyle\nabla_{\mu_{1}\ldots\mu_{k}}=\nabla_{\mu_{1}}\ldots\nabla_{\mu_{k}}.$
(9)

Similarly barred curvatures such as the Ricci scalar $\bar{R}$ correspond to
the boundary. The bulk metric is decomposed on the boundary as
$g_{\mu\nu}=n_{\mu}n_{\nu}+h_{\mu\nu}$ where $h_{\mu\nu}$ is the induced
metric on the boundary and $n_{\mu}$ is the normal. We also need the extrinsic
curvature, which can be defined by

$\displaystyle
K_{\mu\nu}=h_{\mu}{}^{\rho}h_{\nu}{}^{\sigma}\nabla_{\rho}n_{\sigma},$ (10)

and $K=g^{\mu\nu}K_{\mu\nu}$. We have the following Weyl transformations

$\displaystyle\delta_{\sigma}\Gamma_{\mu\nu}^{\rho}=\partial_{\mu}\delta\sigma\delta_{\nu}{}^{\rho}+\partial_{\nu}\delta\sigma\delta_{\mu}{}^{\rho}-g_{\mu\nu}\partial^{\rho}\delta\sigma,$
(11)
$\displaystyle\delta_{\sigma}\bar{R}=-2\delta\sigma\bar{R}-2(d-2)\bar{\nabla}\delta\sigma,$
(12) $\displaystyle\delta_{\sigma}K=-\delta\sigma
K+(d-1)\partial_{n}\delta\sigma.$ (13)

Note that $\Gamma_{\mu\nu}^{\rho}$ is the bulk connection while $\bar{R}$ is
the $(d-1)$ dimensional boundary curvature.

For the rest of this section we mainly focus on $d=3$ with a two dimensional
boundary. (We will discuss the case of a four dimension boundary or defect in
the next section.) In this case it is easy to verify that the following is a
solution of the Wess-Zumino condition Herzog:2019rke

$\displaystyle\delta_{\sigma}W=-\frac{1}{4}\int{\rm
d}^{2}u\sqrt{h}\,\delta\sigma\left(2f_{I}\nabla_{nn}\lambda^{I}+f_{I}K\nabla_{n}\lambda^{I}-f\bar{R}\right),$
(14)

where the condition fixes $\partial_{I}f=f_{I}$. We use $\nabla_{nn}\equiv
n^{\mu}n^{\nu}\nabla_{\mu}\nabla_{\nu}$.

Next we examine in detail what (14) teaches us about the anomalies in the flat
space two points functions $\langle
D(\mathbf{x}_{1})\mathcal{O}_{I}(x_{2})\rangle$ and $\langle
T^{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$.

###  2.2 The effective action point of view

After presenting the one point anomaly and its WZ completion in three
dimensions, our goal in the rest of the section is to show how this anomaly
emerges from higher point correlation functions. We can discover which
correlation functions are related to the anomaly by taking derivatives with
respect to the background bulk metric $g_{\mu\nu}$, the embedding
$X^{\mu}(u)$, and the coupling constant $\lambda^{I}$. Here $u^{a}$ represent
coordinates on the defect. We will keep the dimension unspecified when
possible. A variation of the effective action with respect to one of the
backgrounds is related to an insertion of the conjugate operator as follows

$\displaystyle\delta W=\int{\rm d}^{d}x\sqrt{g}\left(\frac{1}{2}\delta
g_{\mu\nu}\langle T^{\mu\nu}\rangle+\delta\lambda^{I}\langle
O_{I}\rangle\right)+\int{\rm d}^{d-1}u\sqrt{h}\,\delta X^{\mu}\langle
D_{\mu}\rangle.$ (15)

Here $O^{I}$ are exactly marginal operators as above, $T_{\mu\nu}$ is the
stress-energy tensor and $D_{\mu}$ is a boundary operator that we shortly
relate to the displacement operator $D$.

The symmetries of the effective action imply the following (classical) Ward
identities. Bulk diffeomorphism invariance, which acts infinitesimally by
$x^{\mu}\to x^{\mu}+\epsilon^{\mu}(x)$, leads to the following small shifts:

$\displaystyle\delta
g_{\mu\nu}=\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu},\qquad\delta\lambda^{I}=\epsilon^{\mu}\partial_{\mu}\lambda^{I},\qquad\delta
X^{\mu}=\epsilon^{\mu}.$ (16)

Enforcing diffeomorphism invariance on the path integral gives the Ward
identity

$\displaystyle\partial_{\mu}T^{\mu\nu}=\mathcal{O}_{I}\partial^{\nu}\lambda^{I}+\delta(x^{n})D^{\nu},$
(17)

written in flat space.333As discussed before, only diff transformations for
which $\epsilon^{n}|_{\rm bdy}=0$ are symmetries. In practice we allow for any
$\epsilon^{\mu}$, even those which displace the boundary. As a result, the
displacement operator appears in the Ward identity. Diffeomorphism invariance
on the defect, $u^{a}\to u^{a}+\kappa^{a}(u)$, changes the embedding
infinitesimally

$\displaystyle\delta X^{\mu}=\kappa^{a}\partial_{a}X^{\mu},$ (18)

implying

$\displaystyle\kappa^{a}\partial_{a}X^{\mu}D_{\mu}=0.$ (19)

The vector $\kappa^{a}\partial_{a}X^{\mu}$ is tangent to the boundary,
implying that the only non-vanishing component of $D^{\mu}$ is the normal
component $D^{n}\equiv D$. Using the standard pillbox argument, i.e.
integrating (17) on a small box on the boundary, we get the relation
$T_{nn}=-D$ in our conventions (taking constant $\lambda^{I}$).

Now, let us find a Ward identity for $\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ by taking a $\lambda^{I}$
derivative of (17). In doing so, note that the stress-energy tensor itself has
an implicit $\lambda$ dependence,
$\partial_{I}T_{\mu\nu}=\delta_{\mu\nu}\mathcal{O}_{I}$. We get

$\displaystyle\partial^{\mu}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=-\delta^{(d)}(x_{12})\langle\partial_{\nu}\mathcal{O}_{I}(x_{2})\rangle+n_{\nu}\delta(z_{1})\langle
D(\vec{x}_{1})\mathcal{O}_{I}(x_{2})\rangle.$ (20)

Lastly, Weyl invariance implies the (classical) Ward identity
$T^{\mu}{}_{\mu}=0$. Taking a derivative with respect to $\lambda$ leads
to444At this point we might worry about possible anomalies here. The one point
anomaly in (7) would suggest a contribution of the form
$\delta(z_{1})\partial_{z_{1}}^{d-1}\delta^{(d)}(x_{12})$ on the right hand
side. Note however that this contribution can be subtracted by adding a local
boundary term to $\langle T_{\mu\nu}\mathcal{O}\rangle$.

$\displaystyle\langle
T^{\mu}{}_{\mu}(x_{1})\mathcal{O}(x_{2})\rangle=-d\,\delta^{(3)}(x_{12})\langle
O(x_{2})\rangle.$ (21)

There is an ambiguity in the two point function that we need to take into
account because it affects the scale dependence. It comes about as the freedom
to add a quasi-local term with some coefficient McAvity:1995zd ; Osborn:1993cr

$\displaystyle\langle T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle\to\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle-c\,\delta_{\mu\nu}\delta^{(d)}(x_{12})\langle\mathcal{O}_{I}(x_{2})\rangle.$
(22)

This arbitrariness is a consequence of the singularity when the two operators
approach each other, namely $\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle\sim\delta_{\mu\nu}z_{2}^{-d}x_{12}^{-d}$
in the limit $x_{1}\to x_{2}$ (see (2.3) below), and reflects the
arbitrariness of regularizing the singularity at $x_{1}=x_{2}$. As a
consequence, the Ward identities (20) and (21) should be altered to allow for
the ambiguity in the definition of the two point function:

$\displaystyle\partial^{\mu}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-\delta^{(d)}(x_{12})\langle\partial_{\nu}\mathcal{O}_{I}(x_{2})\rangle+c\partial_{\nu
1}\delta^{(d)}(x_{12})\langle\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle\quad+n_{\nu}\delta(z_{1})\langle
D(\mathbf{x}_{1})\mathcal{O}_{I}(x_{2})\rangle,$ (23)

where we note that the modification is by a total derivative in $x_{1}$ and
hence does not affect the integrated form of the Ward identity, and555The
scaling dimension of $\mathcal{O}$ often appears on the right hand side of the
trace Ward identity. To see the dimension more reliably, one has to consider
the dilatation current $x^{\nu}T_{\mu\nu}$ for which it is easy to verify that
$\displaystyle\partial^{\mu}\langle
x_{1}^{\nu}T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle=-\delta^{(3)}(x_{12})\left(x_{2}^{\mu}\partial_{\mu
2}+d\right)\langle\mathcal{O}(x_{2})\rangle+\partial_{{\mu
1}}\left(c\,x^{\mu}_{1}\delta^{(d)}(x_{12})\right)\langle\mathcal{O}(x_{2})\rangle.$
(24) The ambiguity appears only in a total derivative term.

$\displaystyle\langle
T^{\mu}{}_{\mu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=-d(1-c)\,\delta^{(3)}(x_{12})\langle\mathcal{O}_{I}(x_{2})\rangle.$
(25)

Next, we consider the functional derivative of the anomalous effective action.
We are only interested in the derivative evaluated in flat space and with a
planar defect, which we call the trivial background. We may always choose
coordinates such that the embedding is given by $X^{\mu}=(u^{a},0)$. Note that
$\partial_{a}X^{\mu}$ gives a vector tangent to the boundary. For a planar
defect in flat space we simply write $\partial_{a}X^{\mu}=\delta_{a}^{\mu}$. A
small variation can be absorbed by a boundary change of coordinates unless it
is directed along the normal. Hence, we take $\delta X^{\mu}=n^{\mu}\delta
Z(u^{a})$. We then have $\delta Z\,D$ instead of (15). The induced metric is
given by $h_{ab}=(\partial_{a}X^{\mu})(\partial_{b}X^{\nu})g_{\mu\nu}$ and we
note that it is unchanged by a variation in $X^{\mu}$ to linear order in
$\delta Z(u^{a})$ (for a trivial background). The change in the normal vector
is $\delta_{X}n_{a}=-\partial_{a}\delta Z$. Using the definition of the
extrinsic curvature (10), we get the variation

$\displaystyle\delta_{X}K=\nabla^{\rho}\delta_{X}n_{\rho}=-\partial^{c}\partial_{c}\delta
Z,$ (26)

evaluated on the trivial background. As a function in the bulk,
$\lambda(x^{\mu})$ doesn’t change when we vary the embedding $X^{\mu}$. But in
the boundary anomaly it appears as a pullback to the boundary
$\lambda(X^{\mu}(u))$. We thus get the variation

$\displaystyle\delta_{X}\lambda=(\partial_{\mu}\lambda)\delta
X^{\mu}=(\partial_{n}\lambda)\delta Z.$ (27)

Next, consider a variation of the bulk metric $\delta g_{\mu\nu}$. To leading
order, the change in the induced metric is simply $\delta_{g}h_{ab}=\delta
g_{ab}$. Thus the variation of the boundary Ricci scalar is given as usual by

$\displaystyle\delta_{g}\bar{R}=(\partial^{a}\partial^{b}-\delta^{ab}\partial^{c}\partial_{c})\delta
g_{ab}.$ (28)

Regardless of the metric, $\partial_{a}X^{\mu}$ are tangents to the boundary.
But as the metric changes the normal must change to remain orthogonal to these
vectors and to remain of unit length:

$\displaystyle\delta_{g}n^{\mu}=-\frac{1}{2}n^{\mu}\delta
g_{nn}-h^{\mu\nu}\delta g_{\nu n}\
,\;\;\;\delta_{g}n_{\mu}=\frac{1}{2}n_{\mu}\delta g_{nn}.$ (29)

Lastly we also need

$\displaystyle\delta_{g}K=-\partial^{a}\delta
g_{an}+\frac{1}{2}h^{ab}\partial_{n}\delta g_{ab},$ (30)

and

$\displaystyle\delta_{g}\Gamma^{n}_{nn}=\frac{1}{2}\partial_{n}\delta
g_{nn},\qquad\delta_{g}\Gamma^{a}_{nn}=h^{ab}\partial_{n}\delta
g_{nb}-\frac{1}{2}\partial^{a}\delta g_{nn}.$ (31)

We now use these derivatives to obtain the anomalous scale variation of the
two point functions of the marginal operators with the the stress tensor and
the displacement operator: $\langle\mathcal{O}_{I}T_{\mu\nu}\rangle$ and
$\langle\mathcal{O}_{I}D\rangle$. We take two derivatives of (14), one with
respect to $\lambda^{I}$ and one with respect to either the metric or the
location of the boundary. For a constant Weyl transformation
$\delta\sigma\equiv-\delta\log\Lambda$, what we should get is the scale
dependence.

We find for the displacement operator

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
D(\mathbf{x}_{1})\mathcal{O}_{I}(x_{2})\rangle=\frac{3}{4}f_{I}\partial^{c}\partial_{c}\delta^{(2)}(\mathbf{x}_{12})\delta^{\prime}(z_{2})+\frac{1}{2}f_{I}\delta^{(2)}(\mathbf{x}_{12})\delta^{\prime\prime\prime}(z_{2}),$
(32)

and for the stress-energy tensor we write it in the following way

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=-\frac{1}{2}f_{I}\Delta_{\mu\nu}\delta^{(2)}(\mathbf{x}_{12})\delta(z_{1})\delta(z_{2}),$
(33)

where $\Delta_{\mu\nu}$ is a two derivative operator. We find

$\displaystyle\Delta_{ab}=(\partial_{a}\partial_{b}-\delta_{ab}\Box_{2})-\frac{1}{2}\delta_{ab}\partial_{z_{1}}\partial_{z_{2}},$
(34)
$\displaystyle\Delta_{an}=-\frac{3}{2}\partial_{a1}\partial_{z_{2}}-\partial_{a1}\partial_{z_{1}},$
(35)
$\displaystyle\Delta_{nn}=-\partial^{c}\partial_{c}+2\Box_{2}+\partial_{z_{1}}\partial_{z_{2}}.$
(36)

where $\Box_{2}$ is the laplacian with respect to $x^{\mu}_{2}$. It is easy to
verify that this form of the anomaly satisfies the Ward identities (2.2) and
(25) provided we choose $c=1$. For example, in the parallel direction $\nu=a$
the only contribution from the right hand side of (2.2) comes from the second
term. Using the one point anomaly (5), the scale variation of this term is

$\displaystyle\frac{cf_{I}}{2}\partial_{a1}\delta^{(3)}(x_{12})\delta^{\prime\prime}(z_{2})=2cf_{I}\partial_{a1}\partial_{+}^{2}\delta^{(2)}(\mathbf{x}_{12})\delta(z_{1})\delta(z_{2}),$
(37)

which matches with the result from $\partial^{\mu 1}\Delta_{\mu a}$. Here
$\partial_{+}=\frac{1}{2}(\partial_{z_{1}}+\partial_{z_{2}})$. The other
component $\nu=n$ is verified similarly. Changing $c$ has the effect of
shifting the anomaly by

$\displaystyle\delta_{\mu\nu}\delta^{(3)}(x_{12})\Lambda\partial_{\Lambda}\langle\mathcal{O}_{I}(x_{2})\rangle=2f_{I}\delta_{\mu\nu}\partial_{+}^{2}\delta^{(2)}(\mathbf{x}_{12})\delta(z_{1})\delta(z_{2})$
(38)

with an arbitrary coefficient.

###  2.3 Construction of the two point functions

Next we review for completeness the construction of the relevant boundary
correlation functions. The correlation functions we consider are completely
fixed by conformal symmetry up to an overall constant. We follow McAvity and
Osborn McAvity:1995zd  with minor alteration of notation.

We already discussed how conformal symmetry fixes the one point function (1)
of a scalar operator up to an undetermined constant,
$\langle\mathcal{O}\rangle=f|z|^{-\Delta}$, where our interest in this work is
primarily in marginal operators $\mathcal{O}_{I}$ of dimension $\Delta=d$. We
furthermore are interested in the two point functions of $\mathcal{O}_{I}$
with the displacement operator and the stress tensor. Conformal symmetry fixes
these two point functions as well, up to constants which we call $c_{DI}$ and
$c_{TI}$. The result for $\langle D(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ is
simplest to state. We can use translation symmetry to place $D$ at the origin
of the planar boundary:

$\displaystyle\langle D(0)\mathcal{O}_{I}(x)\rangle$ $\displaystyle=$
$\displaystyle\frac{c_{DI}}{x^{2d}}\ .$ (39)

The result for $\langle T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
requires some unpacking:

$\displaystyle\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=\frac{c_{TI}v^{d}}{(x_{1}-x_{2})^{2d}}\left(X_{\mu}X_{\nu}-\frac{1}{d}\delta_{\mu\nu}\right)\
,$ (40)

where $v$ is an invariant cross ratio constructed from $x_{1}$ and $x_{2}$
while $X_{\mu}$ is a weight zero vector that transforms covariantly under the
residual conformal group left unbroken by the boundary. This two point
function has the right transformation properties with respect to the conformal
group. Also it is traceless, and it is conserved $\partial^{\mu}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=0$.

To unpack $v$ and $X_{\mu}$, recall that unlike the situation without a
boundary, which requires four points to construct a conformally invariant
cross ratio, in the presence of a boundary only two points are needed:

$\displaystyle v^{2}=\frac{(x_{1}-x_{2})^{2}}{(x_{1}-x_{2})^{2}+4z_{1}z_{2}}.$
(41)

Sometimes formulae are more elegant with an alternate form,
$\xi=v^{2}/(1-v^{2})$. With these cross ratios in hand, one then defines

$\displaystyle X_{\mu}\equiv z_{1}\frac{v}{\xi}\frac{\partial}{\partial
x^{\mu}_{1}}\xi=v\left(\frac{2z_{1}}{x_{12}^{2}}(x_{12})_{\mu}-n_{\mu}\right),$
(42)

Note $X^{\mu}$ is normalized to satisfy the identity $X^{\mu}X_{\mu}=1$.
Moreover it has the boundary limit $X_{\mu}\to-n_{\mu}$ for $z_{1}\to 0$.

Using the Ward identities shows that the constants $f_{I}$, $c_{DI}$, and
$c_{TI}$ are not independent. First recall that the displacement operator is
obtained from the stress-energy tensor by the boundary limit of the normal-
normal component (see below (19))

$\displaystyle T_{nn}(\mathbf{x}_{1},z_{1})\xrightarrow[z_{1}\to
0]{}-D(\mathbf{x}_{1}).$ (43)

Taking also $\mathbf{x}_{1}\to 0$, we get

$\displaystyle\langle
T_{nn}(x_{1})\mathcal{O}_{I}(x_{2})\rangle\xrightarrow[x_{1}\to
0]{}\frac{c_{TI}}{(x_{2})^{2d}}\left(\frac{d-1}{d}\right),$ (44)

which fixes

$\displaystyle c_{DI}=-c_{TI}\left(\frac{d-1}{d}\right).$ (45)

To relate $f_{I}$ to $c_{TI}$, we expand the two point function in the
coincident limit $x_{1}=x_{2}$ (away from the boundary). Keeping both the
leading and sub-leading orders, we obtain

$\displaystyle\langle T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle\xrightarrow[x_{2}\to
x_{1}]{}\frac{c_{TI}}{(2z_{2})^{d}}\frac{1}{d(d-2)}\left\\{\partial_{\mu}\partial_{\nu}\left(\frac{1}{x^{d-2}}\right)+\frac{1}{2z_{2}}\frac{(d-2)}{(d-4)}\partial_{n}\partial_{\mu}\partial_{\nu}\left(\frac{1}{x^{d-4}}\right)\right.$
$\displaystyle\quad\qquad+\left.\frac{1}{z_{2}}(d-1)\left(2n_{(\nu}\partial_{\mu)}-\frac{1}{(d-1)}\delta_{\mu\nu}\partial_{n}\right)\left(\frac{1}{x^{d-2}}\right)\right\\}+\ldots$
(46)

where we set $x\equiv x_{12}$ to simplify the notation. Using now

$\displaystyle\partial^{2}\left(\frac{1}{x^{d-2}}\right)=-(d-2)S_{d}\delta^{(d)}(x),$
(47)

where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the volume of the $(d-1)$-dimensional
sphere, we find

$\displaystyle\partial^{\mu}_{1}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$ $\displaystyle=-\partial_{\nu
2}\left(\frac{f_{I}}{z_{2}^{d}}\right)\delta^{(d)}(x_{12})+\frac{d}{(d-1)}\left(\frac{f_{I}}{z_{2}^{d}}\right)\partial_{\nu
1}\delta^{(d)}(x_{12}),$ (48) $\displaystyle\langle
T^{\mu}{}_{\mu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=\frac{d}{(d-1)}\left(\frac{f_{I}}{z_{2}^{d}}\right)\delta^{(d)}(x_{12}).$
(49)

with the identification

$\displaystyle
f_{I}=-\frac{c_{TI}(d-1)S_{d}}{2^{d}d^{2}}=\frac{c_{DI}S_{d}}{2^{d}d}.$ (50)

The coincident limit relations (48) and (49) match the Ward identities (2.2)
and (25) in the bulk provided we choose $c=d/(d-1)$.666We can restore $c=1$
that we got for the anomaly in the previous section by the modification in
(22). We deal with this issue in the next section. For reference, let us give
the explicit expression for these coefficients in $d=3$. We get

$\displaystyle c_{DI}=\frac{6f_{I}}{\pi},\qquad c_{TI}=-\frac{9f_{I}}{\pi}.$
(51)

Finally, to match the boundary contribution of the displacement operator to
the Ward identity (2.2), we must be more specific about the domain of validity
of the result for $\langle T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$.
The expression we considered is valid for $z_{1,2}\leq 0$ and we take it to
vanish otherwise. More explicitly the expression for the two point function
should contain the Heaviside step function $\Theta(-z_{1})\Theta(-z_{2})$.
Clearly taking the derivative $\partial/\partial x_{1}^{\mu}$ now leads to an
additional term

$\displaystyle-\delta(z_{1})\langle
T_{n\nu}(x_{1})\mathcal{O}(x_{2})\rangle=\delta(z_{1})n_{\nu}\langle
D(\mathbf{x}_{1})\mathcal{O}(x_{2})\rangle,$ (52)

since the boundary limit of $T_{na}$ vanishes as can be easily checked.

##  3 Rewriting the two point function

Consider the two point function of $\mathcal{O}_{I}$ with the stress-energy
tensor. The challenge here is that the anomaly doesn’t simply show up in the
coincidence limit of the two operators. It appears when the two operators
collide on a point on the boundary. In practice, this means that the effect we
are looking for is localised in a four dimensional space, i.e. we should find
a delta function of the form $\delta^{(3)}(\mathbf{x}_{12})\delta(z_{2})$.

It is convenient to introduce $z_{\pm}=z_{1}\pm z_{2}$ with
$z_{-}\in\mathbb{R}$ and $z_{+}\leq 0$. We have the standard relations

$\displaystyle\partial_{\pm}=\frac{1}{2}(\partial_{z_{1}}\pm\partial_{z_{2}}),\qquad\delta(z_{+})\delta(z_{-})=\frac{1}{2}\delta(z_{1})\delta(z_{2}).$
(53)

For simplicity of notation we shall use $x\equiv x_{1}-x_{2}=x_{12}$ and
$\mathbf{x}\equiv\mathbf{x}_{1}-\mathbf{x}_{2}$ and

$\displaystyle
x^{2}=\mathbf{x}^{2}+z_{-}^{2},\qquad\bar{x}^{2}=\mathbf{x}^{2}+z_{+}^{2}.$
(54)

With this notation in place we have

$\displaystyle v^{2}=\frac{x^{2}}{\bar{x}^{2}},$ $\displaystyle
X_{a}=\frac{(z_{-}+z_{+})x_{a}}{x\bar{x}},$ $\displaystyle
X_{n}=\frac{\left(z_{-}z_{+}-\vec{x}^{\,2}\right)}{x\bar{x}},$ (55)

and from (40),

$\displaystyle\langle
T_{ab}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=\frac{c_{TI}}{x^{d+2}\bar{x}^{d+2}}\left((z_{-}+z_{+})^{2}x_{a}x_{b}-\frac{1}{d}\delta_{ab}\,x^{2}\bar{x}^{2}\right),$
(56) $\displaystyle\langle
T_{an}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=\frac{c_{TI}(z_{-}+z_{+})x_{a}}{x^{d+2}\bar{x}^{d+2}}\left(z_{-}z_{+}-\vec{x}^{\,2}\right),$
(57) $\displaystyle\langle
T_{nn}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=\frac{c_{TI}}{x^{d+2}\bar{x}^{d+2}}\left(-(z_{-}+z_{+})^{2}\vec{x}^{\,2}+\frac{d-1}{d}x^{2}\bar{x}^{2}\right).$
(58)

Now we use the Feynman parameter identity

$\displaystyle\frac{1}{(x^{2})^{m_{1}}(\bar{x}^{2})^{m_{2}}}=\frac{\Gamma(m_{1}+m_{2})}{\Gamma(m_{1})\Gamma(m_{2})}\int_{0}^{1}{\rm
d}t\frac{t^{m_{1}-1}(1-t)^{m_{2}-1}}{(tx^{2}+(1-t)\bar{x}^{2})^{m_{1}+m_{2}}}.$
(59)

This formula actually gives us the $(d+1)$ perspective we were looking for
since we can now define a new $(d+1)$ dimensional variable $w$ by

$\displaystyle w=\left(x^{a},\sqrt{t}z_{-},\sqrt{(1-t)}z_{+}\right),$ (60)
$\displaystyle w^{2}=tx^{2}+(1-t)\bar{x}^{2}.$ (61)

Using this trick we write the two point function in terms of two differential
operators, $D^{(2)}_{\mu\nu}$ and $D^{(4)}_{\mu\nu}$ which are quadratic and
quartic in derivatives respectively, as follows

$\displaystyle\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=c^{\prime}_{TI}\int_{0}^{1}{\rm
d}t\,t^{d/2-1}(1-t)^{d/2-1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left(D_{\mu\nu}^{(2)}\frac{1}{(w^{2})^{d-1}}+\frac{1}{2(d-2)}D^{\prime(4)}_{\mu\nu}\frac{1}{(w^{2})^{d-2}}\right),$
(62)

where

$\displaystyle
c^{\prime}_{TI}=c_{TI}\frac{\Gamma(d-1)}{8\Gamma(\frac{d}{2}+1)^{2}}\xrightarrow[d\to
3]{}-\frac{2f_{I}}{\pi^{2}}.$ (63)

We added a prime to $D^{\prime(4)}_{\mu\nu}$ above because
$D^{\prime(4)}_{an}$ has a $t$-dependence that we will shortly get rid of to
define the unprimed operators. This dependence also prevents us from taking
the derivatives out of the integral.

It will prove useful to slightly generalize our $t$-integral in the following
way. We define the functions

$\displaystyle
J_{(\alpha,\beta,\gamma)}(x^{2},\bar{x}^{2})\equiv\int_{0}^{1}dt\,\frac{t^{\alpha/2-1}(1-t)^{\beta/2-1}}{(tx^{2}+(1-t)\bar{x}^{2})^{\gamma/2}},$
(64)

which for $\gamma=\alpha+\beta$ collapses back to to the Feynman parameter
identity. The full integral can be expressed in terms of the Hypergeometric
function777In fact the integral in (64) is tightly related to the Euler
representation of the Hypergeometric function  $\displaystyle
B(b,c-b){}_{2}F_{1}(a,b,c,z)=\int_{0}^{1}dt\,t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}.$
(65) as

$\displaystyle
J_{(\alpha,\beta,\gamma)}(x^{2},\bar{x}^{2})=\frac{B\left(\tfrac{\alpha}{2},\tfrac{\beta}{2}\right)}{\bar{x}^{\gamma}}{}_{2}F_{1}\left(\frac{\alpha}{2},\frac{\gamma}{2},\frac{\alpha+\beta}{2},1-\frac{x^{2}}{\bar{x}^{2}}\right)$
(66)

where $B\left(\tfrac{\alpha}{2},\tfrac{\beta}{2}\right)$ is the Beta function.
The $J$ functions satisfy various algebraic and differential identities which
can be related to analogous identities of the Hypergeometric functions. We
explore some of them in appendix A. It is convenient for notational purposes
to define a map $P_{(m,n)}$ by

$\displaystyle
P_{(m,n)}J_{(\alpha,\beta,\gamma)}=J_{(\alpha+m,\beta+n,\gamma)}.$ (67)

With this notation in place we use the $J$ functions to write the two point
correlation function in the form

$\displaystyle\langle
T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle=c^{\prime}_{TI}\left(D^{(2)}_{\mu\nu}J_{(d,d,2d-2)}+\frac{1}{2(d-2)}D^{(4)}_{\mu\nu}J_{(d,d,2d-4)}\right)$
(68)

where we find the following

$\displaystyle
D^{(2)}_{ab}=(d-2)\partial_{a}\partial_{b}-\delta_{ab}\Box_{2},$
$\displaystyle D^{(4)}_{ab}=-\Box_{2}\partial_{a}\partial_{b},$ $\displaystyle
D_{an}^{(2)}=-(d-2)\partial_{a}\partial_{z_{2}},$ $\displaystyle
D_{an}^{(4)}=-P_{(-2,0)}\Box_{2}\partial_{a}\partial_{-},$ $\displaystyle
D^{(2)}_{nn}=-(d-2)\partial^{c}\partial_{c}+(d-1)\Box_{2},$ $\displaystyle
D^{(4)}_{nn}=\Box_{2}\partial^{a}\partial_{a}.$ (69)

Here $P_{(-2,0)}$ reflects the $t$ dependence of $D^{\prime(4)}_{an}$
mentioned before. Recall that $\Box_{2}$ is the laplacian with respect to
$x_{2}$, that is
$\Box_{2}=\vec{\partial}^{\,2}+(\partial_{+}-\partial_{-})^{2}.$

It is immediate to see that
$\delta^{\mu\nu}D^{(2)}_{\mu\nu}=\delta^{\mu\nu}D^{(4)}_{\mu\nu}=0$ and hence
the trace Ward identity (25) is satisfied with $c=1$. For completeness we show
in appendix A that also the conservation Ward identity (2.2) is likewise
satisfied with $c=1$.

##  4 Computation of the anomaly

We began section 2 by describing the anomaly in the one point function of a
marginal operator $\mathcal{O}_{I}$. We now proceed to give an independent
derivation on the anomalies in the two point functions of $\mathcal{O}_{I}$
with the displacement operator $D$ and the stress-energy tensor $T_{\mu\nu}$
in the special case of a 2d surface and 3d bulk.

Let us begin with the former which is the simpler one. We may take, as above,
$D$ to be at the origin such that

$\displaystyle\langle
D(0)\mathcal{O}_{I}(x)\rangle=\frac{c_{DI}}{(\mathbf{x}^{2}+z^{2})^{3}}.$ (70)

We can obtain the scale dependence by first performing a Fourier transform
with respect to $\mathbf{x}$. Expanding near $z\to 0^{-}$, we find the
singular contributions are

$\displaystyle\int{\rm
d}^{2}\mathbf{x}\,e^{i\mathbf{x}\cdot\mathbf{k}}\frac{(2\pi)^{-1}}{(\mathbf{x}^{2}+z^{2})^{3}}\to\frac{1}{4z^{4}}-\frac{\mathbf{k}^{2}}{16z^{2}}-\frac{\mathbf{k}^{4}}{256}\left(2\log\left(\frac{z^{2}\mathbf{k}^{2}}{4}\right)-3+4\gamma\right).$
(71)

Using (5) to find the scale dependence of the $1/z^{4}$ and $1/z^{2}$ terms,
we get

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
D(0)\mathcal{O}_{I}(x)\rangle=\frac{\pi
c_{DI}}{12}\delta^{(2)}(\mathbf{x})\delta^{\prime\prime\prime}(z)+\frac{\pi
c_{DI}}{8}\vec{\partial}^{\,2}\delta^{(2)}(\mathbf{x})\delta^{\prime}(z)\ .$
(72)

Comparing this with (32), we see that the two expressions match when using the
relation $c_{DI}=6f_{I}/\pi$ found in (51).

Let us examine the anomaly in (70) using another method, differential
regularization Freedman:1991tk . We find differential regularization to be
preferable in analyzing the anomaly in $\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$, and we can learn some
valuable lessons here in the displacement operator case about how to apply it
properly. At first glance this two point function looks precisely like a bulk
two point function in three dimension (without boundary), and such odd
dimensional correlation functions are well known not to exhibit scale
anomalies. The anomaly in our case must depend on the restriction to half
space $z\leq 0$, which we make explicit by introducing a step function
$\Theta(-z)$. The key to the anomaly, i.e. the scale dependence, is the
failure of (70) to furnish a well defined distribution. In other words,
integrating (70) against a well behaved function does not lead to a convergent
result. The trick of differential regularization is to extract derivatives so
as to reduce the degree of divergence. We thus write

$\displaystyle\frac{\Theta(-z)}{(\mathbf{x}^{2}+z^{2})^{3}}=\frac{1}{24}\Theta(-z)(\vec{\partial}^{\,2}+\partial_{z}^{2})^{2}\frac{1}{(\mathbf{x}^{2}+z^{2})}.$
(73)

The distribution is then defined by integration by parts, but doing that we
acquire some boundary terms. Pulling the step function inside the derivatives
we find terms with the structure
$\partial_{z}^{k}(\delta(z)\partial_{z}^{n}\frac{1}{(\mathbf{x}^{2}+z^{2})})$.
When $n$ is odd, this leads to $z\delta(z)$ inside the brackets which is a
local term (i.e. possibly non vanishing only when $\mathbf{x}^{2}=0$) that we
can ignore. We end up with

$\displaystyle\frac{1}{24}(\vec{\partial}^{\,2}+\partial_{z}^{2})^{2}\frac{\Theta(-z)}{(\mathbf{x}^{2}+z^{2})}+\frac{1}{24}(2\vec{\partial}^{\,2}\partial_{z}+\partial_{z}^{3})\frac{\delta(z)}{\mathbf{x}^{2}}+\frac{1}{24}\partial_{z}\left(\delta(z)\partial_{z}^{2}\frac{1}{(\mathbf{x}^{2}+z^{2})}\right).$
(74)

The terms we keep are localized on the boundary but are not fully local and
cannot be ignored. By explicitly evaluating the normal derivative in the
brackets of the last term and using the delta function to set $z=0$, we arrive
at

$\displaystyle\frac{1}{24}(\vec{\partial}^{\,2}+\partial_{z}^{2})^{2}\frac{\Theta(-z)}{(\mathbf{x}^{2}+z^{2})}+\frac{1}{24}\left(\frac{3}{2}\vec{\partial}^{\,2}\partial_{z}+\partial_{z}^{3}\right)\frac{\delta(z)}{\mathbf{x}^{2}}.$
(75)

Lastly, $\mathbf{x}^{-2}$ is logarithmically divergent as a two dimensional
distribution on the boundary. Using again differential regularization in the
form
$\mathbf{x}^{-2}=\frac{1}{8}\Box_{\mathbf{x}}\log^{2}\left(\mathbf{x}^{2}\Lambda^{2}\right)$
leads to the anticipated scale dependence

$\displaystyle\frac{\partial}{\partial\log\Lambda}\frac{1}{\mathbf{x}^{2}}=2\pi\delta^{(2)}(\mathbf{x}),$
(76)

which clearly reproduces the anomaly in (72). The lesson from this computation
is that the process of rendering the two point function well defined as a
distribution leads to boundary terms. The naive form of (70) (naive since it’s
secretly ill-defined) does not make these boundary terms obvious. The new
boundary terms may then lead to anomalies. In fact in the case of (70), the
full anomaly comes from the boundary, which is consistent with the absence of
“intrinsic” anomalies in odd dimensions.

Now let us consider the $\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$. There are several sources of
scale dependence hidden here, which we will consider one by one. As noted
above, due to translation invariance in the directions parallel to the
boundary, the effective configuration space of the two point function is $d+1$
dimensional $(\vec{x},z_{1},z_{2})$. The most obvious source for the anomaly
comes from the $d+1$ dimensional perspective, which we made manifest by using
the $w$ variable in (3). Beside that, in a similar fashion to the computation
of the $\langle OD\rangle$ anomaly above, there will be contributions to the
anomaly localized at $z_{1}=z_{2}=0$.

In $d=3$ the $w$ variable is four dimensional and the integrand in (3)
contains the factors $w^{-4}$ and $w^{-2}$. Viewed as a four dimensional
distribution, $w^{-2}$ is completely well defined as it is clear that the
negative powers are over-powered by the measure ${\rm d}^{4}w$.

In contrast, $w^{-4}$ suffers from a logarithmic divergence and is thus not
well-defined. Proceeding as above using differential regularization, we write

$\displaystyle\frac{1}{w^{4}}=\Box_{w}\left(-\frac{1}{4}\frac{\log(w^{2}\Lambda^{2})}{w^{2}}+{\rm
const}\right),$ (77)

which is valid anywhere away from $w=0$, to give a definition of $w^{-4}$ that
works also at the singularity. The point is that the right hand side is well-
defined after integration by parts which reduces the degree of divergence at
$w=0$. The price for this resolution of the singularity is the acquired scale
dependence,

$\displaystyle\frac{\partial}{\partial\log\Lambda}\left(\frac{1}{w^{4}}\right)=-\frac{1}{2}\Box_{w}\left(\frac{1}{w^{2}}\right)=2\pi^{2}\delta^{(4)}(w).$
(78)

However, we must remember that $w$ takes values only in a quarter of four
dimensional space (since $z_{1,2}\leq 0$) and therefore we get only a quarter
of the naive result. Taking this into account and combining with the
$t$-integral, we get

$\displaystyle
c^{\prime}_{TI}\int_{0}^{1}t^{3/2-1}(1-t)^{3/2-1}\frac{\partial}{\partial\log\Lambda}\left(\frac{1}{w^{4}}\right)=-\frac{f_{I}}{2}\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2}).$
(79)

The first contribution that we find to the anomaly is hence

$\displaystyle\frac{\partial}{\partial\log\Lambda}$ $\displaystyle\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=-\frac{f_{I}}{2}D_{\mu\nu}^{(2)}\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2})+\ldots$
(80)

Comparing this with (34), we see that we are already quite close to the result
but there are some pieces that are missing.

We turn now to contributions to the anomaly that come from boundary terms in
the correlation function. Recall from the example of $\langle DO\rangle$
worked out above, that such boundary terms arise when we integrate by parts
the differential form (68). As a practical matter these contributions are
straightforward to implement by introduction of step functions as in $\langle
DO\rangle$. We thus include $\Theta(-z_{1})\Theta(-z_{2})$ explicitly in the
definition of the correlation function (68).

We consider first the boundary terms coming from $D^{(2)}_{\mu\nu}$. Commuting
the step function with the normal derivatives in $D^{(2)}_{\mu\nu}$ gives

$\displaystyle\Theta(-z_{2})\partial_{z_{2}}J_{(d,d,2d-2)}=\partial_{z_{2}}\left(\Theta(-z_{2})J_{(d,d,2d-2)}\right)+\delta(z_{2})\frac{B(\frac{d}{2},\frac{d}{2})}{(\mathbf{x}^{2}+z_{1}^{2})^{d-1}},$
$\displaystyle\Theta(-z_{2})\partial_{z_{2}}^{2}J_{(d,d,2d-2)}=\partial_{z_{2}}^{2}\left(\Theta(-z_{2})J_{(d,d,2d-2)}\right)+\delta^{\prime}(z_{2})\frac{B(\frac{d}{2},\frac{d}{2})}{(\mathbf{x}^{2}+z_{1}^{2})^{d-1}}+\delta(z_{2})\partial_{z_{2}}J_{(d,d,2d-2)}.$

The last term on the second line vanishes since

$\displaystyle\delta(z_{2})\partial_{z_{2}}\frac{1}{w^{2d-2}}\sim\delta(z_{2})\frac{z_{2}+(2t-1)z_{1}}{w^{2d}}=\delta(z_{2})\frac{(2t-1)z_{1}}{(\mathbf{x}^{2}+z_{1}^{2})^{d}}$
(81)

and the $t$-integral over $2t-1$ vanishes. Specializing to three dimensions we
find

$\displaystyle\Theta(-z_{2})D^{(2)}_{\mu\nu}J_{(3,3,4)}=D^{(2)}_{\mu\nu}\left(\Theta(-z_{2})J_{(3,3,4)}\right)+\begin{Bmatrix}-\delta_{ab}\delta^{\prime}(z_{2})\\\
-\delta(z_{2})\partial_{a}\\\
2\delta^{\prime}(z_{2})\end{Bmatrix}\frac{B(\tfrac{3}{2},\tfrac{3}{2})}{(\mathbf{x}^{2}+z_{1}^{2})^{2}}.$
(82)

We can now evaluate the anomaly on the boundary term similarly to $\langle
DO\rangle$. The quickest way is to proceed as in (71), by taking the Fourier
transform

$\displaystyle\int{\rm
d}^{2}\mathbf{x}\,e^{i\mathbf{x}\cdot\mathbf{k}}\frac{1}{(\mathbf{x}^{2}+z_{1}^{2})^{2}}=\frac{\pi}{z_{1}^{2}}+\ldots$
(83)

where the ellipses represent non-singular contributions. Regularizing
$1/z^{2}$ à la (5) yields

$\displaystyle\frac{\partial}{\partial\log\Lambda}\frac{1}{(\mathbf{x}^{2}+z_{1}^{2})^{2}}=\pi\delta^{(2)}(\mathbf{x})\delta^{\prime}(z_{1}).$
(84)

Adding the overall coefficient
$c^{\prime}_{TI}B(\tfrac{3}{2},\tfrac{3}{2})=-\frac{1}{4\pi}f_{I}$, this leads
to an update of (80) in the form

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-\frac{f_{I}}{2}\left(D_{\mu\nu}^{(2)}+\begin{Bmatrix}-\frac{1}{2}\delta_{ab}\partial_{z_{1}}\partial_{z_{2}}\\\
-\frac{1}{2}\partial_{z_{1}}\partial_{a}\\\
\partial_{z_{1}}\partial_{z_{2}}\end{Bmatrix}\right)\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2})+\ldots$
$\displaystyle=-\frac{f_{I}}{2}\left(\Delta_{\mu\nu}+\begin{Bmatrix}0\\\
\partial_{+}\partial_{a}\\\
0\end{Bmatrix}\right)\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2})+\ldots$
(85)

The ellipses represent the contribution we have not considered yet. In the
second line we use $\Delta_{\mu\nu}$ from (34) which is the result for the
anomaly found from the effective action. We thus arrive to the desired result
in the $(ab)$ and $(nn)$ components.

We now turn to consider the boundary terms coming from $D^{(4)}_{\mu\nu}$. The
first observation to make is that since $D^{(4)}_{\mu\nu}$ act on $w^{-2}$ in
(3), this term can have an anomaly only if it is localized to two dimensions.
This means that commuting the step function through the normal derivatives
needs to produce two delta functions $\delta(z_{1})\delta(z_{2})$ to get a
contribution. It is easy to check that this kind of boundary term cannot come
from the $(ab)$ or $(nn)$ components.

Let us then consider $D^{(4)}_{an}$. Working directly in three dimensions, we
start from (keeping only normal derivatives in $\Box_{2}$)

$\displaystyle\Theta\partial_{z_{2}}^{2}\partial_{-}J_{(1,3,2)}$
$\displaystyle=\frac{1}{2}\partial_{z_{2}}\left(\partial_{z_{1}}\partial_{z_{2}}\Theta
J_{(1,3,2)}\right)-\partial_{-}\left(\partial_{z_{2}}\Theta\partial_{z_{2}}J_{(1,3,2)}\right)+\ldots$
$\displaystyle=\frac{\pi}{4}\delta(z_{1})\delta^{\prime}(z_{2})\frac{1}{\mathbf{x}^{2}}+\partial_{-}\left(\Theta(-z_{1})\delta(z_{2})\partial_{z_{2}}J_{(1,3,2)}\right)+\ldots$
(86)

up to terms we can ignore. We use $\Theta=\Theta(-z_{1})\Theta(-z_{2})$ for
brevity. Considering the second term we simplify by

$\displaystyle\delta(z_{2})\partial_{z_{2}}\int
dt\frac{t^{-1/2}(1-t)^{1/2}}{(\mathbf{x}^{2}+tz_{-}^{2}+(1-t)z_{+}^{2})}=\frac{\pi}{4}\delta(z_{2})\partial_{z_{1}}\frac{1}{(\mathbf{x}^{2}+z_{1}^{2})}.$
(87)

Now commuting $\Theta(-z_{1})$ through the derivative produces the additional
delta function $\delta(z_{1})$. Putting (87) back in (4) together with an
overall factor of $-c^{\prime}_{TI}/2$ we get

$\displaystyle-\frac{\pi
c^{\prime}_{TI}}{8}\partial_{a}\partial_{+}\left(\delta(z_{1})\delta(z_{2})\frac{1}{\mathbf{x}^{2}}\right)+\ldots$
(88)

and using (76), this procedure leads to the anomaly

$\displaystyle-\frac{f_{I}}{2}\left(-\partial_{a}\partial_{+}\right)\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2}).$
(89)

We can now see that adding this last correction (89) to (4) reproduces
precisely the result from the effective action in (34)

$\displaystyle\frac{\partial}{\partial\log\Lambda}\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-\frac{f_{I}}{2}\Delta_{\mu\nu}\delta^{(2)}(\mathbf{x})\delta(z_{1})\delta(z_{2}).$
(90)

##  5 Anomaly for four dimensional defects

Up to this point we have considered the anomaly for a two dimensional
boundary. Recall, that the main result from Herzog:2019rke  is that the one
point anomaly of exactly marginal bulk operators requires the introduction of
additional curvature boundary terms to satisfy the Wess-Zumino consistency
conditions. Specifically, the completion of the anomaly in two dimensions
includes the curvature $R$, which is also the two dimensional Euler density
$E_{2}$. It is natural to propose that for any even dimensional defect the WZ
completion of the one point anomaly includes the Euler density. The goal of
this section is to verify this proposition explicitly for a four dimensional
defect where888This Euler density is normalized such that the Euler
characteristic of the four manifold $M$ is given by
$\chi=\frac{1}{32\pi^{2}}\int_{M}E_{4}\,\sqrt{g}d^{4}x$.

$\displaystyle
E_{4}=\bar{R}^{\mu\nu\lambda\rho}\bar{R}_{\mu\nu\lambda\rho}-4\bar{R}^{\mu\nu}\bar{R}_{\mu\nu}+\bar{R}^{2}\
.$ (91)

(Recall barred objects are associated with the induced metric on the defect.)

Let us recall the standard treatment of conformal anomalies in four
dimensions. There are two kinds of anomalies, usually dubbed $a$ and $c$, that
can appear. They correspond to the following Weyl transformation of the
effective action

$\displaystyle\delta_{\sigma}W=\frac{1}{16\pi^{2}}\int
d^{4}x\sqrt{g}\,\delta\sigma(cW^{2}_{\mu\nu\rho\sigma}-aE_{4}).$ (92)

Here $W_{\mu\nu\rho\sigma}$ is the Weyl tensor. The consistency of these
anomalies means that they both satisfy the Wess-Zumino condition, but they do
so in different ways. The Weyl tensor is of course Weyl invariant and
therefore the $c$ anomaly is immediately Wess-Zumino consistent. Contrary to
that the transformation of the Euler density contain a non-homogeneous part

$\displaystyle\delta_{\sigma}E_{4}=-4\delta\sigma
E_{4}+8G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\delta\sigma.$ (93)

Here $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is as usual the Einstein
tensor. Nonetheless a quick computation reveals that the Wess-Zumino condition
still holds provided one is allowed to integrate by parts. This extra
complication has an interesting consequence; it is not possible for $a$ to be
a non-trivial function on the conformal manifold. No such conclusion can be
arrived at for the $c$ anomaly by these means. The reason is that if $a$
depends on a marginal coupling $\lambda$, the Wess-Zumino condition must hold
even if we let the coupling be an arbitrary function of position $\lambda(x)$.
Indeed, it was shown by Osborn in Osborn:1991gm  that the additional term

$\displaystyle[\delta_{\sigma_{1}},\delta_{\sigma_{2}}]W=\frac{1}{2\pi^{2}}\int
d^{4}x\sqrt{g}\,\delta\sigma_{[2}\nabla_{\nu}\delta_{\sigma_{1}]}G^{\mu\nu}\nabla_{\mu}a$
(94)

cannot be cancelled by adding terms constructed from the metric and $\lambda$
intrinsic to four dimensions. The exact same considerations hold when
considering the Euler density in any even dimensional space. As we have shown
in the two dimensional case, we claim that the situation changes when the
Euler density anomaly is realized on a four dimensional defect in a larger
ambient space.

We work in a general setting where the four dimensional space supporting the
anomaly can be either a boundary or a submanifold of arbitrary codimension.
While we focused in the earlier sections on the two dimensional boundary case,
in ref. Herzog:2019rke  we have shown how to solve the WZ condition for a two
dimensional defect in an ambient space with arbitrary codimension. In such a
setting we use the coordinates $x^{\mu}=(x^{a},z^{i})$ where $x^{a}$ is two
dimensional and the world volume of the defect is specified by the equations
$z^{i}=0$. The bulk metric now decomposes as
$g_{\mu\nu}=h_{\mu\nu}+\delta_{ij}n^{i}_{\mu}n^{j}_{\nu}$. The one point
function of a bulk marginal operator is given by $|z|^{-d}$. Its corresponding
anomaly is

$\displaystyle\frac{\partial}{\partial\log\Lambda}\frac{1}{|z|^{d}}=c_{2,q}\Box_{q}\delta^{(q)}(z),$
(95)

where $q$ is the codimension (the dimension of $z^{i}$) and
$c_{2,q}=\operatorname{Vol}(S^{q-1})/2q$.999Note that for $q=1$ the zero-
dimensional sphere has two points, but to compare to the first part of this
paper one should take the volume to be one instead of two. The boundary has
one side, while the defect has two. With that the anomalous effective action
in (14) generalizes to

$\displaystyle\delta_{\sigma}W=-c_{2,q}\int{\rm
d}^{2}u\sqrt{h}\,\delta\sigma\left(f_{I}\delta^{ij}\nabla_{ij}\lambda^{I}+\frac{(2-q)}{2}f_{I}K^{i}\nabla_{i}\lambda^{I}-\frac{q}{2}f\bar{R}\right),$
(96)

where $\nabla_{ij}=n^{\mu}_{i}n^{\nu}_{j}\nabla_{\mu}\nabla_{\nu}$ and the
extrinsic curvature now has a normal index

$\displaystyle K^{i}_{\mu\nu}\equiv
h_{\mu}{}^{\rho}h_{\nu}{}^{\sigma}\nabla_{\rho}n^{i}_{\sigma},$ (97)

and $K^{i}=g^{\mu\nu}K^{i}_{\mu\nu}$. The Ricci scalar has a bar on it to
indicate it is a two dimensional quantity. It will be advantageous to rewrite
this formula in a way that will facilitate the analysis in the four
dimensional case. Recall that the WZ condition also fixed
$f_{I}=\partial_{I}f$. We can thus substitute the term in the brackets by

$\displaystyle\left(\delta^{ij}\nabla_{ij}+\frac{(2-q)}{2}K^{i}\nabla_{i}-\frac{q}{2}\bar{R}\right)f-f_{IJ}\nabla_{i}\lambda^{I}\nabla^{i}\lambda^{J}.$
(98)

However, the last term is independently WZ consistent and can thus be dropped.
Note that the remaining term $\nabla^{i}{}_{i}f$ reproduces the one point
anomaly and the discarded term only affects two point functions in the rigid
limit. Therefore, to obtain the simplest possible solution we may ignore
higher order terms in $\lambda$.

Consider now a four dimensional defect. For the one point anomaly we find

$\displaystyle\frac{\partial}{\partial\log\Lambda}\frac{1}{|z|^{d}}=c_{4,q}\Box_{q}^{2}\delta^{(q)}(z),$
(99)

where

$\displaystyle c_{4,q}=\frac{\operatorname{Vol}(S^{q-1})}{8q(q+2)}\ .$ (100)

This matches our general formula (5) for $q=1$ with arbitrary $d$ provided the
$\operatorname{Vol}$ is treated appropriately as noted in footnote 9. It is
the claim of this section that the Wess-Zumino completion of the one point
anomaly in (99) includes the four dimensional Euler density $E_{4}$. To show
that, we need to start by matching the anomaly with a term in the variation of
the effective action. We may lift the Laplacian in the normal directions
$\Box_{q}$ to $\nabla_{i}\nabla^{i}$ as in (96), which leads to

$\displaystyle\delta_{\sigma}W=c_{4,q}\int
d^{4}x\sqrt{h}\delta\sigma\nabla^{i}{}_{i}\nabla^{j}{}_{j}f.$ (101)

Note however that $\nabla^{i}\nabla^{j}\nabla_{j}\nabla_{i}$ also leads to
$\Box_{q}^{2}$ in flat space, in addition to $\nabla_{i}\nabla^{i}\nabla^{2}$
which incorporates the same normal derivatives. Indeed we see below that the
Wess-Zumino condition relates $E_{4}$ to a particular linear combination of
them.

To generate the additional terms needed to obtain a WZ consistent anomaly, we
used the computer algebra package xAct xAct  to search through a large
“dictionary” of possibilities. The rules are as follows. Our “alphabet” was
the bulk $d$-dimensional curvature $R_{\mu\nu\rho\sigma}$, connection
$\nabla_{\mu}$ and the extrinsic curvature $K^{i}_{\mu\nu}$ contracted with
either the metric $g_{\mu\nu}$ or the normal vectors to the 4d surface $n^{\mu
i}$. The connection can act on the Weyl variation parameter $\delta\sigma$,
and the anomaly coefficient $f$. “Words” fourth order in derivatives were
constructed with this alphabet. We identify ten classes of words:

$\begin{array}[]{lll}R^{2}&RK^{2}&K^{4}\\\ &RK\nabla&K^{3}\nabla\\\
&R\nabla^{2}&K^{2}\nabla^{2}\\\ &&K\nabla^{3}\\\ &&\nabla^{4}\end{array}$

We look for linear combinations of such words which satisfy the WZ condition.
We searched for two kinds of solutions. First we consider a list which starts
with the Euler density $E_{4}$. Let us denote the list of words by $X_{A}$
with $A=1,2,\ldots$ We look for a linear combination such that

$\displaystyle\delta_{\sigma^{\prime}}\left(\sum a_{A}X_{A}+\delta\sigma
fE_{4}\right)=0$ (102)

up to total derivatives and terms symmetric in exchanging $\delta\sigma$ with
$\delta\sigma^{\prime}$. Our solution requires only words of these types:
$\nabla^{4}$, $K\nabla^{3}$, $K^{2}\nabla^{2}$, $RK\nabla$, and $K^{3}\nabla$.
In fact we further restrict in most cases to terms with no derivatives acting
on $\delta\sigma$. We weaken this constraint for the $K\nabla^{3}$ terms, in
which case we allow for a single derivative to act on $\delta\sigma$, and in
the $\nabla^{4}$ case, where we allow single derivatives and also some
$\nabla_{\mu}\nabla_{\nu}\delta\sigma$ terms. The second kind of solution we
find, so called kernel solutions, do not include the Euler density, i.e. $\sum
b_{A}\delta_{\sigma^{\prime}}X_{A}\sim 0$. These represent the non-uniqueness
of the completion of the anomaly and can be used to simplify its form. It is
an important consistency condition that these kind of solutions do not
contribute to the one point anomaly.

A list of words which was sufficient to solve the problem at hand, along with
the coefficients $a_{A}$ solving (102) and three of nine kernel solutions, is

$\begin{array}[]{c l c | c c c}&&a_{A}&\lx@intercol\hfil{\rm kernel}\hfil\lx@intercol\\\ \nabla^{4}:&X_{1}=\delta\sigma{\nabla_{i}}^{i}\nabla^{2}f&-\frac{6q^{2}+12q+8}{q(q+2)}&0&\frac{1}{3}&0\\\ &X_{2}=\delta\sigma\nabla^{i\;j}_{\;i\;j}f&2&0&\frac{1}{3}&0\\\ &X_{3}=\delta\sigma\nabla^{i\;j}_{\;j\;i}f&4&0&-\frac{2}{3}&0\\\ &X_{4}=(\nabla^{i}\delta\sigma)\nabla_{i}\nabla^{2}f&\frac{(q-2)(3q^{2}-20q-4)}{q^{2}(q+2)}&0&\frac{(16-q)(q-2)}{6q(q+2)}&\frac{q-2}{q}\\\ &X_{5}=(\nabla^{i}\delta\sigma)\nabla^{j}_{\;\;ij}f&\frac{6q^{2}-40q-8}{q^{2}}&0&\frac{16-q}{3q}&\frac{2(q+2)}{q}\\\ &X_{6}=(\nabla^{i}\delta\sigma)\nabla_{i\;\;j}^{\;\;j}f&\frac{3q^{2}-20q-4}{q^{2}}&0&\frac{16-q}{6q}&-\frac{q+2}{q}\\\ &X_{7}=(\nabla_{\;\;i}^{i}\delta\sigma)\nabla_{\;\;j}^{j}f&-\frac{4q^{2}+22q+4}{q^{2}}&-q-2&\frac{q-16}{3q(q-2)}&\frac{q+2}{q}\\\ &X_{8}=(\nabla_{\;\;i}^{i}\delta\sigma)\nabla^{2}f&4&q-2&0&-1\end{array}$ (103) $\begin{array}[]{c l c | c c c }K\nabla^{3}:&X_{9}=\delta\sigma K^{i}\nabla^{\;\;j}_{i\;j}f&1&0&1&\frac{q+2}{3}\\\ &X_{10}=\delta\sigma K^{i}\nabla^{\;\;j}_{j\;i}f&0&0&0&\frac{q+2}{3}\\\ &X_{11}=\delta\sigma K^{i}\nabla_{i}\nabla^{2}f\ ,&\frac{5q^{2}+14q-8}{q(q+2)}&0&-\frac{4}{3}&-\frac{2}{3}(q+2)\\\ &X_{12}=\delta\sigma K^{i}_{\mu\nu}\nabla_{i}\nabla^{\mu\nu}f&-16&0&2&\frac{2}{3}(q+2)\\\ &X_{13}=(\nabla_{i}\delta\sigma)K^{i}_{\mu\nu}\nabla^{\mu\nu}f&0&0&0&1\\\ \end{array}$ (104) $\begin{array}[]{c l c | c c c }R\nabla^{2}:&X_{14}=\delta\sigma R^{\mu\nu}\nabla_{\mu\nu}f&\frac{(q+2)(3q-2)}{q}&0&\frac{4-q}{6}&\frac{q+2}{3}\\\ &X_{15}=\delta\sigma R^{i}_{\;i}\nabla^{j}_{\;j}f&-\frac{4q^{2}-18q-4}{q(q+2)}&2&\frac{20-3q}{3(q^{2}-4)}&-\frac{2}{3}\\\ &X_{16}=\delta\sigma R^{i}_{\;j}\nabla^{j}_{\;i}f&\frac{4q}{(q+2)}&0&-\frac{8}{3(q+2)}&-\frac{4}{3}\\\ &X_{17}=\delta\sigma R\nabla^{2}f&-\frac{5q^{4}+2q^{3}+6q^{2}-24q-24}{q(q+2)^{2}(q+3)}&\frac{q}{q+3}&\frac{q^{3}-12q^{2}+22q+24}{6(q^{2}-4)(q+3)}&-\frac{q+1}{q+3}\\\ &X_{18}=\delta\sigma R\nabla^{i}_{\;i}f\ ,&\frac{2q^{3}-9q^{2}+14q+8}{q(q+2)(q+3)}&-\frac{q}{q+3}&\frac{5q^{2}-8q-32}{6(q^{2}-4)(q+3)}&\frac{2(q+2)}{3(q+3)}\\\ &X_{19}=\delta\sigma R_{\mu}^{i}\nabla_{i}\nabla^{\mu}f&-\frac{16q^{2}+56q+16}{q(q+2)}&0&\frac{4(q+1)}{3(q+2)}&0\\\ &X_{20}=\delta\sigma{R^{i}}_{\mu i\nu}\nabla^{\mu\nu}f&-\frac{3q^{2}+4q+4}{q}&0&\frac{1}{6}(q-4)&-\frac{1}{3}(q+2)\\\ &X_{21}=\delta\sigma R^{i}_{\;i}\nabla^{2}f&\frac{7q^{2}-12q+4}{(q+2)^{2}}&-2&-\frac{q^{2}-12q+48}{6(q^{2}-4)}&1\\\ &X_{22}=\delta\sigma n^{\lambda i}n^{\rho}_{i}\nabla^{\mu}f\nabla_{\mu}S_{\lambda\rho}&-\frac{4q^{2}+22q+12}{q}&-2(q+2)&\frac{2q^{2}-3q-16}{3(q-2)}&0\\\ &X_{23}=\delta\sigma n^{\mu j}n^{\nu}_{j}\nabla^{i}f\nabla_{i}S_{\mu\nu}&\frac{2(q+2)}{q}&2(q+2)&\frac{q+12}{3(q-2)}&0\\\ &X_{24}=\delta\sigma R^{ij}_{\;\;\;\;ij}\nabla^{2}f&-\frac{2q-8}{(q+2)}&0&0&0\\\ &X_{25}=\delta\sigma R^{ij}_{\;\;\;\;ij}\nabla^{k}_{\;k}f&2&0&0&0\\\ &X_{26}=\delta\sigma R^{ik}_{\;\;\;\;jk}\nabla^{j}_{\;i}f&-4&0&0&0\end{array}$ (105) $\begin{array}[]{c l c | c c c }K^{2}\nabla^{2}:&X_{27}=\delta\sigma(\operatorname{tr}K^{2})^{i}_{\;i}\nabla^{2}f&\frac{3(q+6)}{(q+2)}&0&-2&-(q+2)\\\ &X_{28}=\delta\sigma(\operatorname{tr}K^{2})^{i}_{\;i}\nabla^{j}_{\;j}f&-3&0&2&q+2\\\ &X_{29}=\delta\sigma(\operatorname{tr}K^{2})^{i}_{\;j}\nabla^{j}_{\;i}f&-18&0&4&2(q+2)\\\ &X_{30}=\delta\sigma K^{i}K_{i}^{\mu\nu}\nabla_{\mu\nu}f&4&0&0&0\\\ &X_{31}=\delta\sigma{K^{i\mu}}_{\lambda}K^{\lambda\nu}_{i}\nabla_{\mu\nu}f&-4&0&0&0\end{array}$ (106) $\begin{array}[]{c l c | c c c }RK\nabla:&X_{32}=\delta\sigma R^{i}_{\;i}K^{j}\nabla_{j}f&\frac{14q+4}{q(q+2)}&2&\frac{16-q}{3(q^{2}-4)}&-\frac{1}{3}\\\ &X_{33}=\delta\sigma R^{i}_{\;j}K^{j}\nabla_{i}f&-\frac{2}{(q+2)}&0&-\frac{2}{q+2}&-1\\\ &X_{34}=\delta\sigma RK^{i}\nabla_{i}f&-\frac{10q^{2}-4}{q(q+2)(q+3)}&-\frac{q}{q+3}&\frac{q^{2}-5q-8}{3(q^{2}-4)(q+3)}&\frac{q+2}{3(q+3)}\\\ &X_{35}=\delta\sigma R^{i}_{\mu}K_{i}^{\mu\nu}\nabla_{\nu}f&\frac{4q-8}{q(q+2)}&-4&-\frac{2(5q+4)}{3(q^{2}-4)}&-\frac{4}{3}\\\ &X_{36}=\delta\sigma R^{i}_{\mu}K_{i}\nabla^{\mu}f&-\frac{2(q+1)(4-3q)}{q(q+2)}&0&-\frac{1}{3}&-\frac{1}{3}(q+1)\end{array}$ (107) $\begin{array}[]{c l c | c c c }K^{3}\nabla:&X_{37}=\delta\sigma K^{i}(\operatorname{tr}K^{2})^{j}_{\;j}\nabla_{i}f\ ,&2&0&0&0\\\ &X_{38}=\delta\sigma K^{i}(\operatorname{tr}K^{2})^{j}_{\;i}\nabla_{j}f\ ,&-4&0&0&0\\\ &X_{39}=\delta\sigma(\operatorname{tr}K^{3})^{i\;j}_{\;i}\nabla_{j}f&-10&0&4&2(q+2)\end{array}$ (108)

The notation $\nabla_{ij\cdots k}$ means $n^{\mu}_{i}n^{\nu}_{j}\cdots
n^{\lambda}_{k}\nabla_{\mu}\nabla_{\nu}\cdots\nabla_{\lambda}$, with the
normal vectors outside the derivatives. $S_{\mu\nu}$ is the Schouten
tensor.101010
$S_{\mu\nu}\equiv\frac{1}{d-2}\left(R_{\mu\nu}-\frac{1}{2(d-1)}g_{\mu\nu}R\right)$
We can add multiples of the three kernel solutions to produce other consistent
forms of the anomaly. In fact we found nine kernel solutions in total, the
remaining six of which we describe in the Technical Details section below.

### Consistency Checks

Our first consistency check involves computing the effective action for the
defect conformal field theory on a sphere. The second will compute the
anomalies in the two point function of the displacement operator with a
marginal operator, $\langle D^{i}({\bf x})\mathcal{O}_{I}(x^{\prime})\rangle$.

By looking at the theory on the sphere, we can single out the Euler density
contribution to the effective action. All the other terms will vanish. On the
other hand, we can also compute the integrated one point function
$\langle\mathcal{O}_{I}(x)\rangle$ on the sphere, which has a natural
intepretation as a derivative of the effective action with respect to the
marginal coupling $\lambda_{I}$. As our (102) also relates the anomaly in the
one-point function to the Euler density term, we can check for consistency.

To be concrete, we consider the line element

$\displaystyle
ds^{2}=d\theta^{2}+\sin^{2}\theta\,d\Omega_{4}^{2}+\cos^{2}\theta\,d\Omega_{q-1}^{2}\
,$ (109)

where $d\Omega_{p}^{2}$ is the line element on a sphere of unit radius. The
defect lies along $\theta=\frac{\pi}{2}$. By conformal invariance, a marginal
operator must have the one point function

$\displaystyle\langle\mathcal{O}_{I}\rangle_{S^{d}}=\frac{f_{I}}{\cos^{d}\theta}\
.$ (110)

The derivative of the partition function $W$ with respect to $\lambda_{I}$
contains a log divergence that we can obtain by integrating the
$\langle\mathcal{O}_{I}\rangle_{S^{d}}$ over the sphere with a cut-off,
$0<\theta<\frac{\pi}{2}-\delta$. We find that

$\displaystyle\frac{\partial_{I}W}{f_{I}\operatorname{Vol}(S^{q-1})}\sim-\pi^{2}\log\delta\sim\pi^{2}\log\Lambda\
.$ (111)

where in the last similarity relation, we replaced a short distance cut-off
$\delta$ with a large energy cut-off $\Lambda$.

We should now compare this result with the Euler density term in the anomaly
effective action, evaluated in the same geometry. On the sphere $E_{4}$
evaluates to 24 while $\operatorname{Vol}(S^{4})=8\pi^{2}/3$. Supposing that
the Euler density contribution to the Weyl variation of the effective action
has the form $C\int d^{4}x\sqrt{h}fE_{4}$, where $C$ is a constant we need to
determine, we find that

$\displaystyle\delta_{\sigma}W=64\pi^{2}Cf\ .$ (112)

Comparing the scale variation of (111) against the $\lambda_{I}$ derivative of
(112), we learn that

$\displaystyle C=-\frac{\operatorname{Vol}(S^{q-1})}{64}\ .$ (113)

We now need to check whether this choice of constant $C$ is consistent with
the solution (102). If the coefficient of the Euler density term is $Cf$, as
above, then the coefficient of the $(\Box_{q})^{2}f$ is $-\frac{8C}{q(q+2)}$
for WZ consistency. (This result comes from looking at just the first few
words in the $\nabla^{4}$ category, in particular $a_{1}+a_{2}+a_{3}$.)
Furthermore, from the flat space limit, we anticipate that the coefficient of
this term should be related to the one point function via

$\displaystyle-\frac{8}{q(q+2)}C=c_{4,q}\ ,$ (114)

as we discussed above. Using (100), we find agreement with (113).

Our second consistency check is to look at the $\langle D^{i}({\bf
x})\mathcal{O}_{I}(x^{\prime})\rangle$ two point function, where $D^{i}({\bf
x})$ is the displacement operator. Conformal invariance and a Ward identity
Billo:2016cpy  fixes the form of this two point function

$\displaystyle\langle\mathcal{O}_{I}({\bf
x},z)D^{i}(0)\rangle=-\frac{12df_{I}}{\pi^{2}}\frac{z^{i}}{|z|^{q}({\bf
x}^{2}+z^{2})^{5}}\ .$ (115)

To quickly identify the anomalous contributions to this correlation function,
we compute the Fourier transform

$\displaystyle\int d^{4}x\,e^{ik\cdot{\bf x}}\langle\mathcal{O}_{I}({\bf
x},z)D^{i}(0)\rangle=$
$\displaystyle\frac{df_{I}}{q(q^{2}-4)}\partial_{i}\left(-\frac{1}{8d}(\Box_{q})^{3}+\frac{k^{2}}{16}(\Box_{q})^{2}-\frac{k^{4}}{64}(q+2)\Box_{q}+\ldots\right)|z|^{2-q}\log(|z|\Lambda)\
.$

Now we take a scale variation and obtain

$\displaystyle\Lambda\partial_{\Lambda}\int d^{4}x\,e^{ik\cdot{\bf
x}}\langle\mathcal{O}_{I}({\bf x},z)D^{i}(0)\rangle=$
$\displaystyle-\frac{f_{I}\operatorname{Vol}(S^{q-1})}{8q(q+2)}\partial_{i}\left((\Box_{q})^{2}-\frac{k^{2}d}{2}\Box_{q}+\frac{k^{4}}{8}d(q+2)+\ldots\right)\delta^{(q)}(z)\
.$

The coefficients of these three terms match the appropriate terms in our
(102):

$\displaystyle a_{1}+a_{2}+a_{3}$ $\displaystyle=$
$\displaystyle-\frac{8}{q(q+2)}\ ,$ (118) $\displaystyle
3a_{1}+4a_{2}+4a_{3}-a_{9}-a_{10}-a_{11}$ $\displaystyle=$
$\displaystyle-\frac{4d}{q(q+2)}\ ,$ (119) $\displaystyle
2a_{1}-a_{11}-a_{12}$ $\displaystyle=$ $\displaystyle-\frac{d}{q}\ .$ (120)

The first line matches the leading term in the Fourier transform. In this
case, we are looking for an anomalous contribution with five normal
derivatives. Such a contribution comes from varying the words $X_{1}$,
$X_{2}$, and $X_{3}$ with respect to the embedding function dependence in the
coupling, $\delta_{X}\lambda=(\partial_{n}^{i}\lambda)\delta Z^{i}$.

The second line matches the second term in the Fourier transform. Now we are
looking for terms with three normal and two parallel derivatives. The word
$X_{1}$ contributes with weight three because we can vary either the two unit
normal vectors $n^{i}$ (using $\delta_{X}n_{a}^{i}=-\partial_{a}\delta Z^{i}$)
or $\lambda$ with respect to the embedding $Z^{i}$. The words $X_{2}$ and
$X_{3}$ come with weight four because we can vary any of the four unit normals
$n^{i}$. Finally $X_{9}$ and $X_{11}$ come with weight -1 through a variation
with respect to the extrinsic curvature,
$\delta_{X}K^{i}_{ab}=-\partial_{a}\partial_{b}\delta Z^{i}$.

The third line matches the third term in the Fourier transform. The word
$X_{1}$ contributes with weight two because of the two $n^{i}$, while $X_{11}$
and $X_{12}$ contribute with weight minus one through the extrinsic curvature
dependence.

Note that these sums (118)-(120) provide a further opportunity for a
consistency check. We can check that adding the kernel solutions to (102) does
not change the value of these linear combinations. Thus the kernel solutions
do not contribute to the anomaly in $\langle\mathcal{O}_{I}({\bf
x},z)D^{i}(0)\rangle$.

### Technical Details

The word list in the table above is actually a subset of the total set of
words we considered. The eight words

$\displaystyle(\nabla_{\;\;i}^{j}\delta\sigma)\nabla_{\;\;j}^{i}f\
,\;\;(\nabla_{i}\delta\sigma)K^{i}\nabla^{j}_{\;j}f\
,\;\;(\nabla_{i}\delta\sigma)K^{j}\nabla^{i}_{\;j}f\
,\;\;(\nabla^{\mu}\delta\sigma)K^{i}_{\mu\nu}\nabla^{\nu}\nabla_{i}f\ ,$
$\displaystyle(\nabla_{\mu}\delta\sigma)K^{i}\nabla^{\mu}\nabla_{i}f\
,\;\;\delta\sigma K^{i}K_{i}\nabla^{j}_{\;j}f\ ,\;\;\delta\sigma
K^{i}K_{j}\nabla^{j}_{\;i}f\ ,\;\;\delta\sigma K^{i}K_{i}K^{j}\nabla_{j}f\ ,$
(121)

show up in the six additional kernel solutions we found below. We also allowed
for the following four words for which we found no use at all

$\displaystyle\delta\sigma\nabla^{4}f\
,\;\;(\nabla^{\mu}\delta\sigma)\nabla_{\mu}\nabla_{\;\;i}^{i}f\
,\;\;\delta\sigma K^{i}K_{i}\nabla^{2}f\ ,\;\;\delta\sigma
R^{\mu\nu}K^{i}_{\mu\nu}\nabla_{i}f\ .$ (122)

To verify the relation (102), we needed to subtract some total derivatives. As
the code works implicitly with bulk quantities, the following pair of
identities for converting bulk derivatives to boundary derivatives is useful:

$\displaystyle\bar{\nabla}_{\mu}(h^{\mu}{}_{\nu}V^{\nu})$ $\displaystyle=$
$\displaystyle{h^{\nu}}_{\mu}\nabla_{\nu}V^{\mu}-K\cdot n_{\mu}V^{\mu}\ ,$
(123)
$\displaystyle\bar{\nabla}_{\mu}(h^{\mu}{}_{\nu}V^{\nu\rho\sigma}n_{\rho}\cdot
n_{\sigma})$ $\displaystyle=$
$\displaystyle{h_{\mu}}^{\nu}(\nabla_{\nu}V^{\mu\rho\sigma})n_{\rho}\cdot
n_{\sigma}+V^{\mu\rho\sigma}(K_{\mu\rho}\cdot n_{\sigma}+K_{\mu\sigma}\cdot
n_{\rho})$ (124) $\displaystyle-K\cdot n_{\mu}V^{\mu\rho\sigma}n_{\rho}\cdot
n_{\sigma}\ .$

For the second identity, we need to take advantage of the standard fact that
$(\nabla_{\nu}n^{\mu i})n_{\mu j}+(\nabla_{\nu}n_{\mu j})n^{\mu i}=0$. We
considered the divergence of the following five tangent vectors:

$\begin{array}[]{c|c}&b_{A}\\\
V_{1}^{\mu}=\delta\sigma(\nabla^{\mu}\delta\sigma^{\prime})\nabla^{2}f&-\frac{8}{q}-\frac{8}{q+2}\\\
V_{2}^{\mu}=\delta\sigma(\nabla^{\nu}\delta\sigma^{\prime})\nabla_{\mu\nu}f&\frac{8}{q}-6q\\\
V_{3}^{\mu}=\delta\sigma(\nabla^{\mu}\delta\sigma^{\prime})\nabla_{nn}f&12\\\
V_{4}^{\mu}=\delta\sigma(\nabla_{n}\delta\sigma^{\prime})\nabla^{\mu}\nabla_{n}f&40\\\
V_{5}^{\mu}=\delta\sigma(\nabla_{nn}\delta\sigma^{\prime})\nabla^{\mu}f&\frac{14q+4}{q}\end{array}$
(125)

The defect derivative we take is written $\bar{\nabla}_{\mu}\equiv
h_{\lambda\rho}\nabla^{\rho}h^{\lambda}_{\mu}$ where
$h_{\mu\nu}=g_{\mu\nu}-n^{\mu}n^{\nu}$ is the projector. The numbers in the
right hand column give the coefficients necessary to verify the relation
(102). In other words, we are really verifying that

$\displaystyle\delta_{\sigma^{\prime}}\left(\sum a_{A}X_{A}+\delta\sigma
fE_{4}\right)=\sum b_{A}\bar{\nabla}_{\mu}V_{A}^{\mu}\ ,$ (126)

where equality is up to terms symmetric under
$\delta\sigma^{\prime}\leftrightarrow\delta\sigma$.

There are in fact nine trivial solutions that can be generated from our
$X_{A}$ along with (121). Three are Weyl covariant. We construct the traceless
part of the extrinsic curvature,

$\displaystyle\hat{K}^{i}_{\mu\nu}=K^{i}_{\mu\nu}-\frac{1}{d-q}h_{\mu\nu}K^{i}\
,$ (127)

and a Laplacian type operator

$\displaystyle\Box_{ij}\equiv\delta_{ij}\,\nabla^{2}-(d-2)\nabla_{ij}-\frac{2(d-2)}{d-q}K_{(i}\nabla_{i)}\
.$ (128)

Both these transform covariantly under a Weyl transformation, i.e.
$\delta_{\sigma}\hat{K}^{i}_{\mu\nu}=\delta\sigma\hat{K}^{i}_{\mu\nu}$ and
$\delta_{\sigma}\Box_{ij}=-2\delta\sigma\Box_{ij}$. With these we can form
three Weyl covariant terms
$(\operatorname{tr}\hat{K}^{3})^{i\;j}_{\;i}\nabla_{j}f$,
$(\operatorname{tr}\hat{K}^{2})^{i}_{\;i}\Box^{j}_{\;j}f$ and
$(\operatorname{tr}\hat{K}^{2})^{i}_{\;j}\Box^{j}_{\;i}f$ of dimension $4$
that can be added to the anomalous effective action without affecting the WZ
consistency.

There are three more “trivial” solutions that vary to produce terms symmetric
under the switch $\delta\sigma^{\prime}\leftrightarrow\delta\sigma$:

$\displaystyle(\nabla_{i}\delta\sigma)\left((1-q)\nabla^{i}\nabla^{2}-2\nabla^{ji}_{\;\;j}+(1+q)\nabla^{ij}_{\;\;j}\right)f-(\nabla^{i}_{\;i}\delta\sigma)\nabla^{j}_{\;j}f+q(\nabla^{i}_{\;j}\delta\sigma)\nabla^{j}_{\;i}f\
,$ (129)
$\displaystyle\hat{K}_{\mu\nu}^{i}(\nabla^{\mu}\delta\sigma)\nabla^{\nu}\nabla_{i}f+\frac{1}{2}K_{\mu\nu}^{i}(\nabla_{i}\delta\sigma)\nabla^{\mu\nu}f\
,$ (130)
$\displaystyle(\nabla^{i}_{\;i}\delta\sigma)\nabla^{2}f-\frac{q+2}{2}K^{i}(\nabla^{\mu}\delta\sigma)\nabla_{\mu}\nabla_{i}f+2K^{i}(\nabla^{j}\delta\sigma)\nabla_{ij}f+K^{i}(\nabla_{i}\delta\sigma)\nabla^{j}_{\;j}f$
$\displaystyle+q(\nabla^{i}\delta\sigma)\nabla_{i}\nabla^{2}f-(q+2)(\nabla^{i}_{\;j}\delta\sigma)\nabla^{j}_{\;i}f-(q+2)(\nabla^{i}\delta\sigma)\nabla_{i\;j}^{\;\;j}f\
.$ (131)

These three terms vanish for constant $\delta\sigma$. That makes six. The
final three, mentioned already briefly above in describing the word list
table, are trivial up to terms symmetric in $\delta\sigma^{\prime}$ and
$\delta\sigma$ and up to a total derivative. We wrote them as the last three
columns in the word list.

All of these nine trivial solutions could in principle be added to the anomaly
effective action, and encode anomalous contributions to the correlation
functions that are independent of the Euler density term we are interested in.
The three
$\delta\sigma(\operatorname{tr}\hat{K}^{3})^{i\;j}_{\;i}\nabla_{j}f$,
$\delta\sigma(\operatorname{tr}\hat{K}^{2})^{i}_{\;i}\Box^{j}_{\;j}f$ and
$\delta\sigma(\operatorname{tr}\hat{K}^{2})^{i}_{\;j}\Box^{j}_{\;i}f$ for
example encode anomalies in the correlation functions $\langle
D^{i}(x_{1})D^{j}(x_{2})\mathcal{O}_{I}(x_{3})\rangle$ and $\langle
D^{i}(x_{1})D^{j}(x_{2})D^{k}(x_{3})\mathcal{O}_{I}(x_{4})\rangle$.

##  6 Discussion

We had two goals in exploring the relationship between Euler density type
conformal anomalies and anomalies in one point functions. First, we wanted to
obtain the full structure of the anomaly (14), realized on a two dimensional
boundary, from two point correlations functions. More specifically the bulk of
our attention was dedicated to analyzing $\langle
T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle$. While the anomaly in a two point
function is usually found in a coincident limit, the story here is richer. The
relevant singularity appears in the double limit where both $x_{1}\to x_{2}$
and $z_{1}+z_{2}\to 0$.

To access this singularity, we wrote the two point function in terms of a
$d+1$ dimensional variable $w$ in (3). This presentation of the two point
function in terms of the $t$-integral has a close relation to the conformal
block expansion that will be discussed elsewhere Shamirtoappear . This method
of analyzing $\langle T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle$ naturally
accommodates more complicated two point functions, such as two currents
$\langle j_{\mu}(x_{1})j_{\nu}(x_{2})\rangle$ and two stress tensors $\langle
T_{\mu\nu}(x_{1})T_{\rho\sigma}(x_{2})\rangle$. Unlike the two point functions
and three point functions that one studies in the context of anomalies in two
and four dimensions respectively, the two point function in the boundary case
has a more intricate structure as it can depend on a non-trivial cross ratio.
While in the case we considered – $\langle
T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle$ – this dependence is fixed by the
Ward identity, $\langle j_{\mu}(x_{1})j_{\nu}(x_{2})\rangle$ and $\langle
T_{\mu\nu}(x_{1})T_{\rho\sigma}(x_{2})\rangle$ depend on a general function of
the cross ratio. Through an operator product expansion, either bringing the
operators close together or close to the boundary, these two point functions
can be expressed as a sum over conformal blocks, a sum whose form is fixed by
conformal invariance up to the choice of constant coefficients multiplying the
blocks. We hope that by looking at the anomaly structure of these two point
functions using our $w$ variable, we can further constrain these conformal
block expansions. This type of bootstrap approach might help to pinpoint
interesting interacting boundary conformal field theories and calculate their
higher point correlation functions. In higher codimension, $\langle
T_{\mu\nu}(x_{1})\mathcal{O}(x_{2})\rangle$ also depends on cross ratios and
can be expanded in conformal blocks. There too, it will be interesting to look
at the interplay between anomalies and the conformal block expansion.

The second goal we pursued here was to find the Wess-Zumino completion of the
one point anomaly in the case of a four dimensional defect and show that it
contains the Euler density. The solution we found passes several consistency
checks, including matching the hemisphere partition function with the one
point anomaly, but its full form contains more than fifty terms. We do not
make any claim this is the simplest solution. It is quite possible that a
clever reorganization of the terms will collapse this expression to something
that fits in two or three lines.

We have conjectured that the same structure persists for any even dimensional
defect. It will be nice to find a general proof of this claim although so far
as we can see it is not of immediate interest to physics. The anomaly
effective actions we found, here for a four dimensional defect and in
Herzog:2019rke  for a two dimensional defect, strongly suggest some underlying
geometrical structure. Such a geometrical picture can render these anomalies
more transparent and natural and perhaps even provide a constructive way to
build them efficiently. It might also provide an interesting connection to
discussions of conformal geometry in the mathematics literature. For example,
is there a connection between our anomaly actions and the notion of
Q-curvature (see for instance Qcurvature )?

## Acknowledgments

We would like to thank F. Benini, M. Gillioz, A. Schwimmer and M. Serone for
discussion. C. H. was supported in part by a Wolfson Fellowship from the Royal
Society and by the U.K. Science & Technology Facilities Council Grant
ST/P000258/1. I. S. is supported in part by the MIUR-SIR grant RBSI1471GJ
“Quantum Field Theories at Strong Coupling: Exact Computations and
Applications” and by INFN Iniziativa Specifica ST&FI.

##  Appendix A Some properties of $J_{(\alpha,\beta,\gamma)}$ and the Ward
identity

Let us elaborate of some basic properties of the $J$ functions, defined in
(64), and use them to verify the Ward identity with $c=1$ directly in the form
(68). The first relation we mention is

$\displaystyle
J_{(\alpha,\beta,\gamma)}=J_{(\alpha+2,\beta,\gamma)}+J_{(\alpha,\beta+2,\gamma)}.$
(132)

This is obtained by inserting $1=t+(1-t)$ in the definition (64) of
$J_{(\alpha,\beta,\gamma)}$. It is useful to define derivatives as follows

$\displaystyle\widetilde{\partial}_{-}=P_{(-2,0)}\partial_{-},\qquad\widetilde{\partial}_{+}=P_{(0,-2)}\partial_{+},\qquad\widetilde{\partial}_{1,2}=\widetilde{\partial}_{+}\pm\widetilde{\partial}_{-}.$
(133)

The $P$’s are defined in (67) by
$P_{(m,n)}J_{(\alpha,\beta,\gamma)}=J_{(\alpha+m,\beta+n,\gamma)}$. The idea
here is that $\widetilde{\partial}_{\pm}J_{(\alpha,\beta,\gamma)}=-\gamma
z_{\pm}J_{(\alpha,\beta,\gamma+2)}$, without changing $\alpha$ or $\beta$.

For our purposes the most important property of the $J$ functions comes from
the $(d+1)$ dimensional point of view. The variable
$w=(\mathbf{x},\sqrt{t}z_{-},\sqrt{1-t}z_{+})$ has a corresponding Laplacian
given by

$\displaystyle\Box_{w}=\partial^{a}\partial_{a}+\frac{1}{t}\partial_{-}^{2}+\frac{1}{(1-t)}\partial_{+}^{2}.$
(134)

This operator acts on functions of $w$ which always appear as integrands of
$t$-integrals, as in (64). It is useful to pull the differential operators out
of the integral such that we get an operator acting on the $J$ functions. This
is achieved by shifting $\alpha$ and $\beta$ to account for the explicit
dependence on $t$. To be economical (though abusive) with notation we will use
the same notation $\Box_{w}$ to refer also to the action on the $J$ functions,
namely

$\displaystyle\Box_{w}J_{(\alpha,\beta,\gamma)}\equiv(\partial^{a}\partial_{a}+\partial_{-}\widetilde{\partial}_{-}+\partial_{+}\widetilde{\partial}_{+})J_{(\alpha,\beta,\gamma)}.$
(135)

This Laplacian is related to the bulk Laplacian
$\Box_{2}=\vec{\partial}^{\,2}+\partial_{z_{2}}^{2}$ by

$\displaystyle\Box_{w}=\Box_{2}+P_{(2,2)}\widetilde{\partial}_{1}^{2}.$ (136)

We then get from $\Box_{w}w^{-\gamma}=\gamma(\gamma+1-d)w^{-\gamma/2-1}$ the
relation111111When $\gamma=d-1$ we get instead a delta function in $w$ space
leading to
$\displaystyle\Box_{w}J_{(\alpha,\beta,d-1)}=-(d-1)S_{d+1}B\left(\tfrac{\alpha-1}{2},\tfrac{\beta-1}{2}\right)\delta^{(d-1)}(\mathbf{x})\delta(z_{-})\delta(z_{+}).$
(137)

$\displaystyle\Box_{w}J_{(\alpha,\beta,\gamma)}=\gamma(\gamma+1-d)J_{(\alpha,\beta,\gamma+2)}.$
(138)

Another fundamental relation that we will need for the Ward identity is given
by

$\displaystyle-\widetilde{\partial}_{1}\widetilde{\partial}_{2}J_{(\alpha+2,\beta+2,\gamma)}=\gamma(\alpha-1)J_{(\alpha,\beta+2,\gamma+2)}-\gamma(\beta-1)J_{(\alpha+2,\beta,\gamma+2)}.$
(139)

This can be proven by considering the expression

$\displaystyle-2\gamma\int_{0}^{1}dt\,t^{\alpha/2}(1-t)^{\beta/2}\partial_{t}w^{-\gamma-2}$
(140)

and evaluating it once by applying the derivative forward and once by
integrating by parts. In the first instance, evaluating the $t$-derivative
gives us

$\displaystyle\gamma(\gamma+2)\left(z_{-}^{2}-z_{+}^{2}\right)J_{(\alpha+2,\beta+2,\gamma+4)}$
$\displaystyle=-\widetilde{\partial}_{1}\widetilde{\partial}_{2}J_{(\alpha+2,\beta+2,\gamma)}+\gamma\left(J_{(\alpha,\beta+2,\gamma+2)}-J_{(\alpha+2,\beta,\gamma+2)}\right)$

and in the second integrating by parts gives $\gamma(\alpha
J_{(\alpha,\beta+2,\gamma+2)}-\beta J_{(\alpha+2,\beta,\gamma+2)})$. A
generalization of (139) that we will use below is

$\displaystyle-\widetilde{\partial}_{1}^{2}\widetilde{\partial}_{2}J_{(\alpha+2,\beta+2,\gamma)}=\gamma\widetilde{\partial}_{1}\left((\alpha-2)J_{(\alpha,\beta+2,\gamma+2)}-(\beta-2)J_{(\alpha+2,\beta,\gamma+2)}\right)+\gamma\widetilde{\partial}_{2}J_{(\alpha,\beta,\gamma+2)}.$
(141)

To obtain this from (139) the important thing to note here is that we can’t
use (139) straightforwardly because its right hand side depends explicitly on
$\alpha$ and $\beta$, and $\widetilde{\partial}_{1}$ includes $P$’s which
modify them. Therefore, to use (139) one has to first break
$\widetilde{\partial}_{1}$ along (133).

Let us now consider the Ward identity, starting with the parallel components.
Using the form in (68) we get

$\displaystyle c^{\prime-1}_{TI}\partial^{\mu 1}\langle T_{\mu
a}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=\left((d-2)\vec{\partial}^{\,2}-\Box_{2}-(d-2)\partial_{z_{1}}\partial_{z_{2}}\right)\partial_{a}J_{(d,d,2d-2)}$
$\displaystyle\quad-\frac{1}{2(d-2)}\left(\vec{\partial}^{\,2}+\widetilde{\partial}_{-}\partial_{z_{1}}\right)\partial_{a}\Box_{2}J_{(d,d,2d-4)}.$
(142)

Focusing on the second line we proceed as follows

$\displaystyle\left(\vec{\partial}^{\,2}+\widetilde{\partial}_{-}\partial_{z_{1}}\right)J_{(d,d,2d-4)}$
$\displaystyle=\left(\Box_{w}-\widetilde{\partial}_{-}\partial_{-}-\widetilde{\partial}_{+}\partial_{+}+\widetilde{\partial}_{-}\partial_{z_{1}}\right)J_{(d,d,2d-4)}$
$\displaystyle=2(d-2)(d-3)J_{(d,d,2d-2)}-\partial_{+}\widetilde{\partial}_{2}J_{(d,d,2d-4)}.$
(143)

In the first line we use the definition of $\Box_{w}$ in (135) and in the
second (138). Putting this back in (A) we get after some shuffling

$\displaystyle c^{\prime-1}_{TI}\partial^{\mu 1}\langle T_{\mu
a}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-2(d-2)\partial_{a}\partial_{z_{2}}\partial_{+}J_{(d,d,2d-2)}+\frac{1}{2(d-2)}\partial_{a}\partial_{+}\widetilde{\partial}_{2}\Box_{2}J_{(d,d,2d-4)}.$
(144)

Consider the second term. First use (136) and then evaluate $\Box_{w}$ and use
(141) to get

$\displaystyle\frac{1}{2(d-2)}\widetilde{\partial}_{2}\Box_{2}J_{(d,d,2d-4)}$
$\displaystyle=\frac{1}{2(d-2)}\widetilde{\partial}_{2}\left(\Box_{w}-P_{(2,2)}\widetilde{\partial}_{1}^{2}\right)J_{(d,d,2d-4)}$
$\displaystyle=(d-2)\widetilde{\partial}_{2}J_{(d,d,2d-2)}+(d-2)\widetilde{\partial}_{1}\left(J_{(d,d+2,2d-4)}-J_{(d+2,d,2d-4)}\right)$
$\displaystyle=2(d-2)\partial_{z_{2}}J_{(d,d,2d-2)}$ (145)

which shows that (A) vanishes away from coincident points.

Now consider the behavior of $J$ in the limit $x^{2}\to 0$ with $z_{+}$ fixed.
The leading order singularity is

$\displaystyle
J_{(\alpha,\beta,\gamma)}\to\frac{B\left(\tfrac{\beta}{2},\tfrac{\gamma-\beta}{2}\right)}{\bar{x}^{\beta}x^{\gamma-\beta}}.$
(146)

Hence when $\gamma-\beta=d-2$ the bulk Laplacian $\Box_{2}$ gives a delta
function

$\displaystyle\Box_{2}J_{(\alpha,\beta,\beta+d-2)}\supset-(d-2)S_{d}\frac{B\left(\tfrac{\beta}{2},\tfrac{\gamma-\beta}{2}\right)}{(2|z_{2}|)^{\beta}}\delta^{(d)}(x).$
(147)

With this we get a contribution from the last terms of (144) since
$\widetilde{\partial}_{2}$ includes $P_{(0,-2)}\partial_{+}$ giving us
$\gamma-\beta=d-2$. We thus find

$\displaystyle\partial^{\mu 1}\langle T_{\mu
a}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=-\frac{1}{2}c^{\prime}_{TI}\partial_{a}\partial_{+}^{2}S_{d}\frac{B\left(\tfrac{d-2}{2},\tfrac{d-2}{2}\right)}{(2|z_{2}|)^{d-2}}\delta^{(d)}(x)=\partial_{a}\delta^{(d)}(x)\langle\mathcal{O}(x_{2})\rangle,$
(148)

matching the Ward identity in (2.2) with $c=1$.

Now for the normal component. For the first step we get

$\displaystyle c^{\prime-1}_{TI}\partial^{\mu 1}\langle T_{\mu
n}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-2(d-2)\partial_{+}\vec{\partial}^{\,2}J_{(d,d,2d-2)}+(d-1)\partial_{z_{1}}\Box_{2}J_{(d,d,2d-2)}$
$\displaystyle\quad+\frac{1}{2(d-2)}(\partial_{z_{1}}-\widetilde{\partial}_{-})\vec{\partial}^{\,2}\Box_{2}J_{(d,d,2d-4)}.$
(149)

We develop the second line as follows. First simplify the brackets
$\partial_{z_{1}}-\widetilde{\partial}_{-}=P_{(0,2)}\widetilde{\partial}_{2}$
and then using (136) to get

$\displaystyle\frac{1}{2(d-2)}\widetilde{\partial}_{2}\vec{\partial}^{\,2}\left(\Box_{w}-P_{(2,2)}\widetilde{\partial}_{1}^{2}\right)J_{(d,d+2,2d-4)}$
$\displaystyle=2(d-2)\partial_{+}\vec{\partial}^{\,2}J_{(d,d,2d-2)}$
$\displaystyle\quad-2(d-1)\widetilde{\partial}_{1}\vec{\partial}^{\,2}J_{(d+2,d+2,2d-2)}.$
(150)

Thus

$\displaystyle c^{\prime-1}_{TI}\partial^{\mu 1}\langle T_{\mu
n}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=(d-1)\left(\partial_{z_{1}}\Box_{2}-2P_{(2,2)}\widetilde{\partial}_{1}\vec{\partial}^{\,2}\right)J_{(d,d,2d-2)}$
$\displaystyle=(d-1)\left((\partial_{z_{1}}-2P_{(2,2)}\widetilde{\partial}_{1})\Box_{2}+2P_{(2,2)}\widetilde{\partial}_{1}\partial_{z_{2}}^{2}\right)J_{(d,d,2d-2)}$
$\displaystyle=(d-1)\left(-\Pi\partial_{z_{2}}\Box_{2}+2\widetilde{\partial}_{1}\partial_{z_{2}}^{2}\right)J_{(d+2,d+2,2d-2)}$
(151)

To get to the second line $\vec{\partial}^{\,2}=\Box_{2}-\partial_{z_{2}}^{2}$
was used. $\Pi\equiv P_{(0,-2)}-P_{(-2,0)}$ is ubiquitous in such
computations. Next we use again (136) and evaluate $\Box_{w}$ to get

$\displaystyle=-2(d-1)^{3}\Pi\partial_{z_{2}}J_{(d+2,d+2,2d)}+(d-1)\left(P_{(2,2)}\Pi\widetilde{\partial}_{1}+2\partial_{z_{2}}\right)\partial_{z_{2}}\widetilde{\partial}_{1}J_{(d+2,d+2,2d-2)}$
$\displaystyle=-2(d-1)^{3}\Pi\partial_{z_{2}}J_{(d+2,d+2,2d)}+(d-1)\partial_{z_{2}}\widetilde{\partial}_{1}\widetilde{\partial}_{2}J_{(d+2,d+2,2d-2)},$
(152)

and finally using (139) one last time we get that the entire expression
vanishes. To get the contribution at $x_{1}=x_{2}$ we use again (147). Since
we need $\gamma-\beta=d-2$ the only contribution comes from the last line of
(A), in the form

$\displaystyle-c^{\prime}_{TI}(d-1)\partial_{z_{2}}\Box_{2}J_{(d+2,d,2d-2)}$
$\displaystyle\supset
c^{\prime}_{TI}(d-1)(d-2)S_{d}\partial_{z_{2}}\left(\frac{B\left(\tfrac{d}{2},\tfrac{d-2}{2}\right)}{(2|z_{2}|)^{d}}\delta^{(d)}(x)\right)$
$\displaystyle=-f_{I}\partial_{z_{2}}\left(\frac{1}{|z_{2}|^{d}}\delta^{(d)}(x)\right),$
(153)

and therefore

$\displaystyle\partial^{\mu 1}\langle T_{\mu
n}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle=-\delta^{(d)}(x)\partial_{z_{2}}\langle\mathcal{O}_{I}(x_{2})\rangle+\partial_{z_{1}}\delta^{(d)}(x)\langle\mathcal{O}_{I}(x_{2})\rangle$
$\displaystyle\quad+(\text{displacement op.})$ (154)

To get the contribution of the displacement operator in the second line we
simply repeat the argument in (52); namely there is an implicit
$\Theta(-z_{1})\Theta(-z_{2})$ in the definition of $\langle
T_{\mu\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle$. When acting with
$\partial^{\mu 1}$ we get the additional contribution

$\displaystyle-\delta(z_{1})\Theta(-z_{2})\langle
T_{n\nu}(x_{1})\mathcal{O}_{I}(x_{2})\rangle=\delta(z_{1})\Theta(-z_{2})n_{\nu}\langle
D(\mathbf{x}_{1})\mathcal{O}_{I}(x_{2})\rangle.$ (155)

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