# Marginal deformations and defect anomalies

Lorenzo Bianchi  Center for Research in String Theory - School of Physics and
Astronomy Queen Mary University of London, Mile End Road, London E1 4NS, UK

###### Abstract

We deform a defect conformal field theory by an exactly marginal bulk operator
and we consider the dependence on the marginal coupling of flat and spherical
defect expectation values. For odd-dimensional defects, we find a different
qualitative behavior for the flat and spherical case, generalizing to
arbitrary dimensions the line-circle anomaly of superconformal Wilson loops.
In the even-dimensional case, on the other hand, we find a logarithmic
divergence which can be related to a $a$-type anomaly coefficient. This
coefficient, for defect theories, is not invariant on the conformal manifold
and its dependence on the bulk coupling is controlled to all orders by the
one-point function of the associated exactly marginal operator. In particular,
our results imply a non-trivial dependence on the bulk coupling for the
recently proposed defect $C$-function. We finally apply our general result to
a few specific examples, including superconformal Wilson loops and Rényi
entropy.

##  I Introduction and results

Extended probes play a distinguished role in a wide range of physical
phenomena. Wilson and ’t Hooft lines, boundaries, interfaces and twist
operators provide physically interesting examples of a broad class of
observables denoted as defects. In the hope of identifying universal
properties, it is convenient to restrict our attention to non-local operators
preserving conformal symmetry along their profile. The study of conformal
defects started a long time ago in two dimensions Cardy (1984); McAvity and
Osborn (1993), but only recently the constraints of conformal symmetry have
been systematically imposed in higher dimensions Billò  _et al._ (2016).

Among all the examples of conformal defects, superconformal Wilson lines
provide an extremely useful laboratory, both because we have lot of data at
our disposal and because of the variety of techniques that can be used to
access their non-perturbative regime. The most famous example is certainly the
Maldacena Wilson loop in $\mathcal{N}=4$ Super Yang Mills theory Maldacena
(1998). In that case, the expectation value for the circular Wilson loop is
known to all orders in the coupling Erickson  _et al._ (2000); Drukker and
Gross (2001); Pestun (2012) and it is different from the straight line
expectation value, despite the two configurations are mapped to each other by
a conformal transformation. This fact can be attributed to a conformal anomaly
in the transformation relating the straight line and the circle Drukker and
Gross (2001). Similar phenomena have been observed in all those cases where
the expectation value of the circular Wilson loop could be computed exactly,
such as $\mathcal{N}=2$ theories in four dimensions Pestun (2012); Passerini
and Zarembo (2011) or $\mathcal{N}\geq 2$ theories in three dimensions
Kapustin  _et al._ (2010); Griguolo  _et al._ (2013a); Bianchi  _et al._
(2013); Mauri  _et al._ (2018). It is therefore a natural question whether
such an anomaly is a more general feature of conformal defects, i.e. whether
it is always true that the flat defect expectation value is different from the
spherical one. More generally, one may wonder whether it makes sense to
compute the defect expectation value in the flat case where the only available
scales are the IR and UV cut-off (we will say more on this point in the
following). In this letter, we argue that flat and spherical defects exhibit
indeed different qualitative features and that, even assuming one could make
sense of the flat defect expectation value, the spherical one, in general, is
different and it is a non-trivial function of the marginal coupling. Let us
stress, however, that the plane-sphere anomaly is not related to any geometric
invariant and it is qualitatively different from the more familiar case of the
defect Weyl anomaly.

The Weyl anomaly is an important feature of homogeneous conformal field
theories (CFT) in even dimensions. A way to describe such anomaly is through
the expectation value of the trace of the stress tensor $T^{\mu}{}_{\mu}$.
When the theory is embedded in an arbitrary curved manifold $T^{\mu}{}_{\mu}$
acquires a non-vanishing expectation value proportional to a linear
combination of Weyl invariants. The coefficients of this linear combination
are called anomaly coefficients and their number grows with the spacetime
dimension. It is a well-known fact that for $2$ and $4$ dimensions all the
Weyl anomaly coefficients can be related to pieces of conformal data, in
particular to the two- and three-point functions of the stress tensor Osborn
and Petkou (1994).

In the presence of defects, the number of geometric invariants grows
significantly, given the presence of additional ingredients like the extrinsic
curvature of the defect profile. The simplest possible case is that of a two-
dimensional surface, for which the relevant invariants have been classified
Graham and Witten (1999). In that case there is one $a$-type and two $b$-type
anomaly coefficients. Interestingly, the two $b$-type coefficient could be
mapped to the one-point function of the stress tensor operator and the two-
point function of the displacement operator Bianchi  _et al._ (2016a). On the
contrary, the $a$-type coefficient, which is particularly interesting for its
expected monotonicity property under RG flow Jensen and O’Bannon (2016);
Kobayashi  _et al._ (2019), has not been related to any piece of defect
conformal data yet.

In this letter, we study a particular class of conformal field theories
characterized by the presence of a scalar operator of protected dimension $d$,
i.e. an exactly marginal operator. In that case the action can be deformed by

$S\to S+\lambda\int d^{d}x\ \mathcal{O}(x)$ (1)

where we assumed the CFT lives in flat space and for simplicity we restrict to
a single marginal direction. The argument easily generalizes to the case of
several marginal operators. Examples of such theories are very common in the
presence of supersymmetry in four dimensions and less common in three
dimensions (see Cordova  _et al._ (2016) for a full classification), but the
existence of non-supersymmetric conformal manifolds in $d>2$ is an interesting
open question.

We consider the dependence of the defect expectation value on the marginal
coupling $\lambda$ associated to the exactly marginal operator. For the case
of a flat defect, the result is expected to be both IR and UV divergent and
the two cut-offs are the only available scales. It is therefore not clear
whether one could identify a part of the result which is independent of the
regularization scheme in order to make sense of it for general defects. For
the Wilson line, there is a well-defined renormalization procedure that goes
back a long time Dorn and Wieczorek (1981); Aoyama (1982) (a more recent
discussion can be found in Griguolo  _et al._ (2013b)). Our result shows that,
even if one could identify a universal part for the flat defect, it will
always be independent of the marginal coupling.

For a spherical defect the situation is more interesting. The dependence on
the marginal coupling is controlled by the one-point function of the exactly
marginal operator

$\braket{\mathcal{O}(x)}_{\text{plane}}=\frac{C_{O}}{|x_{\perp}|^{d}}$ (2)

and, as expected, it behaves differently for even and odd dimensional defects.
In particular, for a spherical conformal defect $\Sigma$ of dimension $p$ and
codimension $q$ such that $p+q=d$, we find that

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}=\left\\{\begin{array}[]{ll}C_{O}\frac{(-1)^{\frac{p}{2}+1}4\pi^{\frac{p+q}{2}}}{\Gamma(\frac{p}{2}+1)\Gamma(\frac{q}{2})}\log\epsilon&\text{even
}p\\\
C_{O}\frac{(-1)^{\frac{p+1}{2}}2\pi^{\frac{p+q}{2}+1}}{\Gamma(\frac{p}{2}+1)\Gamma(\frac{q}{2})}&\text{odd
}p\end{array}\right.$ (5)

where $\epsilon$ is a UV cut-off and the sphere radius has been set to one.
This is in agreement with general expectations, since we know that for even
$p$ the universal part of the defect expectation value is given by the
coefficient of the logarithmic divergence. In particular, this coefficient can
be expressed as a linear combination of Weyl invariants and the expectation
value of the spherical defect is related to the $a$-type anomaly. More
generally, the spherical defect expectation value has been identified as the
best candidate for a monotonically decreasing function under defect RG flow
both in even and odd dimensions, i.e. a $C$-function Kobayashi  _et al._
(2019) (see also Nozaki  _et al._ (2012); Gaiotto (2014); Estes  _et al._
(2014); Yamaguchi (2002) for previous theorems and conjectures encoded by the
proposal in Kobayashi  _et al._ (2019)). Here we find that this proposed
defect $C$-function depends non-trivially on the bulk marginal couplings. This
is no contradiction with usual results since under a defect RG-flow a
candidate defect $C$-function should not depend on defect marginal couplings,
but could depend in general on bulk marginal couplings.

Note added: While this paper was in the final stage of its preparation, the
preprint Herzog and Shamir (2019) appeared on the arXiv, which overlaps with
this work on several aspects. The authors of Herzog and Shamir (2019)
considered the partition function of a spherical defect in a spherical
background and derived the Wess-Zumino consistency conditions for the
conformal anomaly. Here we consider the expectation value of a spherical
defect in flat space and we include some explicit examples.

##  II Conformal defects and marginal deformations

We start by considering the derivative of the logarithm of the defect
expectation value with respect to the exactly marginal coupling. It is
immediate to see that this derivative is related to the one-point function of
the exactly marginal operator $\braket{\mathcal{O}(x)}_{\Sigma}$ 111One may be
worried of a possible explicit dependence on $\lambda$ in the definition of
the defect. If this was the case, the first-order derivative with respect to
the marginal coupling would give the one-point function of a defect operator,
which vanishes by conformal invariance

$\partial_{\lambda}\log\braket{\Sigma}=\int
d^{d}x\braket{\mathcal{O}(x)}_{\Sigma}$ (6)

where the defect one-point function is normalized by the defect vacuum

$\braket{\mathcal{O}(x)}_{\Sigma}=\frac{\braket{\mathcal{O}(x)\Sigma}}{\braket{\Sigma}}$
(7)

Importantly, the kinematics of this one-point function is completely fixed by
conformal invariance and the dynamical content is encoded in a single constant
$C_{O}$, which would depend, however, on all the free parameters in the bulk
and defect theory and, in particular, on the marginal parameter $\lambda$. The
explicit expression for the one-point function in flat space is trivial and
was given in (2). Also the extension to the spherical case is not particularly
difficult. Actually, it is particularly simple using the embedding formalism
developed in Billò  _et al._ (2016), where one just needs to pick two
different sections of the projective cone. The result is that, for the case of
a spherical defect of unit radius we can split the Euclidean coordinates into
$p+1$ “parallel” and $q-1$ “orthogonal” coordinates

$x^{\mu}=(x_{\parallel}^{\hat{i}},x_{\perp}^{\hat{a}})$ (8)

where we used quotation marks to highlight that the parallel coordinates,
labeled by an index $\hat{i}$, are the $p+1$ directions in which the sphere is
embedded. Consistently the index $\hat{a}$ runs over the $q-1$ orthogonal
directions. Specifically, the sphere is defined by

$x_{\parallel}^{\hat{i}}x_{\parallel\hat{i}}=1$ (9)

In this coordinate system, the one-point function in presence of a spherical
defect reads

$\braket{\mathcal{O}(x)}_{\text{sphere}}=\frac{C_{O}}{\left(x_{\perp}^{2}+\frac{(1-x_{\perp}^{2}-x_{\parallel}^{2})^{2}}{4}\right)^{d/2}}$
(10)

Despite quite unusual, this system of coordinates is particularly useful to
perform the integration in (6).

###  II.1 A trivial case: the flat defect

The first question we would like to address is what happens if one performs
the integral (6) in the case of a flat defect. The answer is quite trivial.
The flat defect case is IR and UV divergent and its expectation value, in
general, is bound to suffer from these divergences. If one tries to regulate
the integral (6) with a IR cutoff $L$ and a UV cutoff $\epsilon$, the result
is

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}$
$\displaystyle=C_{O}S_{q-1}S_{p-1}\\!\\!\int_{\epsilon}^{\infty}\\!\\!\\!\\!dr_{\perp}\frac{1}{r_{\perp}^{d-q+1}}\int_{0}^{L}\\!\\!\\!dr_{\parallel}r_{\parallel}^{p-1}$
$\displaystyle=C_{O}S_{q-1}S_{p-1}\left(\frac{L}{\epsilon}\right)^{p}$ (11)

where $S_{n}=2\frac{\pi^{\frac{n+1}{2}}}{\Gamma(\frac{n+1}{2})}$ is the volume
of the $n$-sphere. The result (II.1) for integer values of $p$ is a power-like
divergence and therefore it is an artifact or the regularization procedure. In
particular, one can consider performing the integral in dimensional
regularization by allowing $p$ to take non-integer values. In that case, the
integrals appearing in (II.1) are the typical examples of scaleless integrals
and therefore must be set to zero in dimensional regularization. The correct
way to interpret this result is to affirm that there is no universal part of
the flat defect expectation value which depends on the marginal coupling. In
other words, even if some symmetry protects the flat expectation value from
the aforementioned divergences or if one could identify a universal part after
renormalization, the result would not depend on $\lambda$. In the literature,
similar issues are discussed in the context of supersymmetric Wilson lines
Griguolo  _et al._ (2013b).

###  II.2 The spherical defect

We will see that the story is quite different for the case of a spherical
defect. Using the coordinates introduced in (9) and trivially performing the
angular integrations we obtain

$\partial_{\lambda}\log\braket{\Sigma}=C_{O}S_{q-2}S_{p}\int\frac{dr_{\perp}dr_{\parallel}r_{\perp}^{q-2}r_{\parallel}^{p}}{\left(r_{\perp}^{2}+\frac{(1-r_{\perp}^{2}-r_{\parallel}^{2})^{2}}{4}\right)^{d/2}}$
(12)

with $r_{\parallel}=|x_{\parallel}|$ and $r_{\perp}=|x_{\perp}|$. The location
of the divergence is geometrically very clear, since the defect is positioned
at $r_{\perp}=0$ and $r_{\parallel}=1$. As a double check of our result, we
perform the integrals in two different ways.

####  II.2.1 Dimensional regularization

In the first example, we will use a defect version of dimensional
regularization such that $d=p+q$ is fixed, but $p$ and $q$ are kept generic.
That means we are not changing the total dimension of the space, but we are
changing defect dimension and codimension. Introducing polar coordinates
$(r_{\parallel}=r\sin\theta,r_{\perp}=r\cos\theta)$ we get

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}=$ (13) $\displaystyle
C_{O}S_{q-2}S_{p}\int_{0}^{\infty}dr\int_{0}^{\frac{\pi}{2}}d\theta\frac{(\cos\theta)^{q-2}(\sin\theta)^{p}}{r\left(\cos\theta^{2}+\frac{(\frac{1}{r}-r)^{2}}{4}\right)^{p+q/2}}$

and, after performing the integrals

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}=C_{O}\frac{2\pi^{\frac{d}{2}}\Gamma(-\frac{p}{2})}{\Gamma(\frac{q}{2})}$
(14)

Clearly this expression has single poles for even values of $p$. This is
expected since for even dimensional defects the universal part of the free
energy is proportional to a logarithmic UV singularity. To extract the
coefficient of the logarithm we can simply take the residue

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}_{\text{univ}}^{\text{even
}p}=C_{O}\frac{(-1)^{\frac{p}{2}+1}4\pi^{\frac{d}{2}}}{\Gamma(\frac{p}{2}+1)\Gamma(\frac{q}{2})}$
(15)

On the other hand, for odd $p$ the result is finite and it is simply

$\displaystyle\partial_{\lambda}\log\braket{\Sigma}^{\text{odd
}p}=C_{O}\frac{2\pi^{\frac{d}{2}}\Gamma(-\frac{p}{2})}{\Gamma(\frac{q}{2})}$
(16)

We will see that these results are perfectly reproduced by a cut-off analysis,
where the physical interpretation of the divergence is more transparent.

####  II.2.2 Cut-off regularization

We compute the integral (12) for integer values of $p$ and $q$ regularizing by
a UV cutoff $\epsilon$ around $r_{\parallel}=1$, where the defect is located.
We then isolate the universal term in the expansion (i.e. the term that is not
affected by a rescaling of the cutoff). For even $p$ it appears as the
coefficient of $\log\epsilon$, while for odd $p$ it is a finite part.

Let us start by even $p$ and $q>1$. One can perform the integral for several
integer values of $p$ and $q$ and then extract the coefficient of the
logarithmic singularity. Doing so, one finds they fit in the pattern

$\partial_{\lambda}\log\braket{\Sigma}^{\text{even
}p}_{\text{univ}}=C_{O}\frac{(-1)^{\frac{p}{2}+1}4\pi^{\frac{p+q}{2}}}{\Gamma(\frac{p}{2}+1)\Gamma(\frac{q}{2})}\log\epsilon$
(17)

in perfect agreement with (15) (in this case however we left the logarithm
explicit since it is related to the actual value of the cutoff).

For odd $p$ and $q>1$ a similar analysis gives

$\partial_{\lambda}\log\braket{\Sigma}^{\text{odd
}p}_{\text{sphere}}=C_{O}\frac{(-1)^{\frac{p+1}{2}}2\pi^{\frac{p+q}{2}+1}}{\Gamma(\frac{p}{2}+1)\Gamma(\frac{q}{2})}$
(18)

which agrees with (16) for odd $p$.

When $q=1$ we do not have orthogonal coordinates in the spherical case (let us
stress again that we denote as parallel all the directions where the sphere is
embedded) and one may be worried that the previous results do not apply to
this specific case. Actually, in this case the integral is simply

$\partial_{\lambda}\log\braket{\Sigma}^{q=1}_{\text{sphere}}=C_{O}S_{p}2^{d}\int
dr_{\parallel}\frac{r_{\parallel}^{p}}{\left|1-r_{\parallel}^{2}\right|^{p+1}}$
(19)

and we find that its universal part agrees perfectly with (16) and (15)
evaluated at $q=1$. Notice however that this is true only if the integral
extends from zero to infinity, so in the case of a codimension one defect (not
a boundary, and not an interface with two different theories on the two
sides).

##  III Examples

In this section, we apply our general result to a few interesting examples.
First, we specify our formula to line defects and we extract predictions for
the Lagrangian expectation value of three- and four-dimensional theories where
the Wilson loop is known exactly. For the case of $\mathcal{N}=4$ SYM we are
particularly lucky because the Lagrangian expectation value can be computed
independently thus confirming the validity of our formula. Then we consider
the case of a two-dimensional defect, where the set of independent Weyl
invariants is well known. This allows us to write down an equation for the
$a$-type anomaly coefficient and to show that it is related to the scalar one-
point function $C_{O}$. Finally, we consider the general case of codimension 2
that is relevant for the twist operator, i.e. the operator whose expectation
value computes the Rényi entropy. For the four-dimensional case, using a
specific feature of the twist operator, we can write down a general formula
relating the one-point function of the stress-tensor to $C_{O}$.

###  III.1 Line defects

If we specify our formula (16) to the case $p=1$ we find

$\partial_{\lambda}\braket{\mathcal{W}}=-2\pi S_{q-1}C_{O}$ (20)

where we used $\mathcal{W}$ to denote the line defect. We will consider the
four-dimensional case, where we can perform explicit checks for the validity
of this formula. Let us point out, however, that marginal deformations exist
also for three-dimensional $\mathcal{N}=2$ SCFT, where an exact expression for
the Wilson loop is available Kapustin  _et al._ (2010). We leave further
analysis in this direction for future work.

####  III.1.1 Wilson loops in four-dimensional SCFT

Exactly marginal deformations exist for four-dimensional theories with
$\mathcal{N}=1$, $\mathcal{N}=2$ and $\mathcal{N}=4$ supersymmetry. For all
these cases we have

$\displaystyle C_{O}$
$\displaystyle=-\frac{1}{8\pi^{2}}\partial_{\lambda}\braket{\mathcal{W}}$ (21)

Focusing on the case of Super Yang-Mills theories, where the marginal coupling
is precisely the Yang-Mills coupling, it is convenient to perform a change of
variables. In (6) we assumed that the coupling $\lambda$ multiplies the
marginal operator, while in the ordinary Yang-Mills lagrangians the inverse of
the ’t Hooft coupling appears in front of the action. Let us therefore
introduce the coupling $\hat{\lambda}=\frac{1}{\lambda}$ and reabsorb a factor
of $\hat{\lambda}$ in the definition of $C_{O}$ such that
$\hat{\lambda}\hat{C}_{O}=C_{O}$. This gives

$\displaystyle\hat{C}_{O}$
$\displaystyle=\frac{1}{8\pi^{2}}\hat{\lambda}\partial_{\hat{\lambda}}\braket{\mathcal{W}}$
(22)

which can be used to extract predictions for the one-point function of the
Lagrangian for those theories where the exact expression of the Wilson loop is
known. This is perhaps not very surprising, but for $\mathcal{N}=4$ SYM it
constitutes a strong consistency check since the one-point function of the
Lagrangian can be computed independently. In the following we will review the
argument which already appeared in Fiol  _et al._ (2016a).

The expectation value of the circular Wilson loop in $\mathcal{N}=4$ SYM is
given by Erickson  _et al._ (2000); Drukker and Gross (2001); Pestun (2012)

$\braket{\mathcal{W}}=\frac{1}{N}L^{1}_{N-1}\left(-\frac{\hat{\lambda}}{4N}\right)e^{\frac{\hat{\lambda}}{8N}}$
(23)

where $L$ is the modified Laguerre polynomial. To obtain an independent
prediction on the Lagrangian one-point function (without taking an explicit
derivative with respect to $\hat{\lambda}$) we can do the following. First of
all, we consider the Bremsstrahlung function, defined as the coefficient of
the second order term in the small angle expansion of the cusp anomalous
dimension Correa  _et al._ (2012)

$\displaystyle\Gamma_{\text{cusp}}(\phi)$
$\displaystyle=-B(\hat{\lambda},N)\phi^{2}+\mathcal{O}(\phi^{4})$ (24)

This function can be computed exactly combining defect techniques with
supersymmetric localization Correa  _et al._ (2012)

$\displaystyle
B(\hat{\lambda},N)=\frac{1}{2\pi^{2}}{\hat{\lambda}}\partial_{\hat{\lambda}}\braket{\mathcal{W}}$
(25)

Then, we introduce the one-point function of the stress tensor (for a straight
Wilson line in direction 4 and $n_{a}=\frac{x_{a}}{|x_{\perp}|}$ with
$a=1,2,3$)

$\displaystyle\braket{T_{ab}}$
$\displaystyle=-\frac{h({\hat{\lambda}},N)}{x_{\perp}^{4}}(\delta_{ab}-2n_{a}n_{b})\,,$
(26) $\displaystyle\braket{T_{a4}}$
$\displaystyle=0\,,\qquad\braket{T_{44}}_{W}=\frac{h({\hat{\lambda}},N)}{x_{\perp}^{4}}\,,$
(27)

and its relation to the Bremsstrahlung function Lewkowycz and Maldacena
(2014); Bianchi  _et al._ (2018)

$h({\hat{\lambda}},N)=\frac{B({\hat{\lambda}},N)}{3}$ (28)

Finally, we use the fact that in $\mathcal{N}=4$ SYM the Lagrangian and the
stress tensor operator belong to the same supermultiplet. Using supersymmetric
Ward identities along the lines of Gomis  _et al._ (2008); Fiol  _et al._
(2016b); Bianchi  _et al._ (2018) one can prove that

$h({\hat{\lambda}},N)=\frac{4}{3}\hat{C}_{O}({\hat{\lambda}},N)$ (29)

The same relation, in slightly different conventions, was found in Fiol  _et
al._ (2012). Therefore, combining all these results we get

${\hat{\lambda}}\partial_{\hat{\lambda}}\log\braket{\mathcal{W}}=8\pi^{2}\hat{C}_{O}(\hat{\lambda},N)$
(30)

in perfect agreement with formula (22). Using the same reasoning backwards,
i.e. starting from the validity of (22), we obtain an alternative derivation
of the exact formula for the Bremsstrahlung function (25) without using the
localization argument of Correa  _et al._ (2012).

###  III.2 Surface defects

As we mentioned, for even dimensional defects, our result relates the one-
point function of the exactly marginal operator to the derivative with respect
to the coupling of the defect $a$-anomaly. For the case $p=2$, the general
form of the universal part of the defect free energy is given by

$\left.\log\braket{\Sigma}\right|_{\text{univ}}=\left(\frac{f_{a}}{2\pi}I_{a}+\frac{f_{b}}{2\pi}I_{b}-\frac{f_{c}}{2\pi}I_{c}\right)\log(\epsilon)$
(31)

where $I_{a}=\int_{\Sigma}R_{\Sigma}$,
$I_{b}=\int_{\Sigma}\tilde{K}_{ij}^{a}\tilde{K}_{ij}^{a}$ and
$I_{c}=\int_{\Sigma}\gamma^{ij}\gamma^{kl}C_{ikjl}$ are three defect Weyl
invariants built out of the 2d Ricci scalar $R_{\Sigma}$, the traceless part
of the extrinsic curvature $\tilde{K}_{ij}^{a}$ and the embedded Weyl tensor
$C_{ikjl}$ contracted with the embedded metric $\gamma^{ij}$. All the
integrals are performed over the defect with the appropriate invariant
measure. Here we are interested in the $I_{a}$ invariant, a.k.a. Euler
characteristics. For a spherical defect that is the only non-vanishing
contribution and we have

$\int_{S^{2}}R_{S^{2}}=8\pi$ (32)

Combining this with (15) we get

$\partial_{\lambda}f_{a}=\frac{\pi}{2}S_{q-1}C_{O}$ (33)

which is one of the main result of the paper as it predicts that the $a$-type
Weyl anomaly coefficient $f_{a}$ is not invariant on the conformal manifold
and its derivative is controlled by the one-point function of the exactly
marginal operator $\mathcal{O}$.

Looking at specific examples, the most studied one is surely the Gukov Witten
defect in $\mathcal{N}=4$ SYM theory Gukov and Witten (2006). In that case,
there is strong evidence that the $a$-anomaly coefficient does not depend on
the marginal coupling Drukker  _et al._ (2008); Jensen  _et al._ (2019) 222The
statement that this coefficient vanishes, made in Drukker  _et al._ (2008),
was corrected in Jensen  _et al._ (2019) finding agreement with the expression
of the entanglement entropy.. Using equation (15) we can actually prove this
statement and make a more general one. We will show that for every surface
defect in four dimensions preserving two supercharges of opposite chirality
$Q_{+}$ and $\tilde{Q}_{\dot{-}}$, the r.h.s. of (33) vanishes. The least
supersymmetric surface in this class is a $\mathcal{N}=(2,0)$ surface with
superconformal algebra $su(1,1|1)\oplus sl(2)\oplus u(1)$ (see Bianchi and
Lemos (2019) for a recent discussion). Without loss of generality we can
restrict to a $\mathcal{N}=1$ bulk four-dimensional theory. The result will
obviously apply to higher supersymmetric cases. The conformal manifold of a
supersymmetric theory is always complex valued and the Lagrangian can be split
in a chiral part belonging to a chiral multiplet and an anti-chiral part
belonging to the conjugate one Green  _et al._ (2010). This supermultiplet
contains a scalar operator $\mathcal{E}$ of dimension $3$ and $r$-charge $2$,
a fermionic operator $\Lambda_{\alpha}$ of dimension $7/2$ and $r$-charge $1$
and the exactly marginal operator $\mathcal{O}$. It is immediate to write down
the supersymmetry transformation of the fermionic operator

$\displaystyle\delta\Lambda_{\alpha}=\xi_{\alpha}\mathcal{O}+\sigma^{\mu}_{\alpha\dot{\alpha}}\partial_{\mu}\mathcal{E}\tilde{\xi}^{\dot{\alpha}}$
(34)

where $\xi_{\alpha}(\tilde{\xi}^{\dot{\alpha}})$ is the Grassmann odd
parameter associated to the supercharge
$Q_{\alpha}(\tilde{Q}_{\dot{\alpha}})$. Clearly, if the defect preserves
$Q_{+}$ the condition $\braket{\delta\Lambda_{+}}_{\Sigma}=0$ readily gives
$\braket{\mathcal{O}}_{\Sigma}=0$. At the same time, if the defect preserves
$U(1)$ R-symmetry $\braket{\mathcal{E}}_{\Sigma}=0$ preventing the marginal
operator from acquiring a non-vanishing one-point function anyway. This
argument does not apply to superconformal defects preserving a combination of
$Q$s and $\tilde{Q}$s (notice also that this is the only way for a surface
defect to preserve a single supercharge). This breaks $U(1)$ R-symmetry as
well as orthogonal rotations, although it may preserve a combination of them.
In this case there is no symmetry setting to zero the one-point function of
$\mathcal{O}$ and one should legitimately expect that $f_{a}$ depends on the
coupling. Gukov Witten defects, despite breaking orthogonal rotations in
general, do not belong to this class as they preserve 4 $Q$s and 4
$\tilde{Q}$s separately. Therefore, we find that the r.h.s. of (33)
consistently vanishes for Gukov Witten defects. Nevertheless, there is still a
large class of defects for which this is not the case, including all non-
supersymmetric surfaces (also when inserted in a bulk supersymmetric theory
where we have examples of conformal manifolds).

###  III.3 Rényi entropies

In Bianchi  _et al._ (2016a), it was pointed out that twist operators in
dimension higher than two can be treated as a conformal defect. This has led
to several important developments Balakrishnan  _et al._ (2017a); Bianchi  _et
al._ (2016b), especially in connection with the proof of the Quantum Null
Energy Condition Balakrishnan  _et al._ (2017b, 2019). The Rényi entropy
$S_{n}$ is related to the twist operator expectation value by Bianchi  _et
al._ (2016a)

$\braket{\tau_{n}}=e^{(1-n)S_{n}}$ (35)

and its universal (regulator independent) part is given by Liu and Mezei
(2013)

$S^{\text{univ}}_{n}=\left\\{\begin{array}[]{ll}(-1)^{\frac{d}{2}}s_{d}\log\epsilon&\text{even
}d\\\ (-1)^{\frac{d-1}{2}}s_{d}&\text{odd }d\end{array}\right.$ (36)

This structure is clearly in agreement with our findings and we can relate the
derivative of $s_{d}$ to the one-point function of $\mathcal{O}$

$(1-n)\partial_{\lambda}s_{d}=\left\\{\begin{array}[]{ll}S_{d-1}2\,C_{O}&\text{even
}d\\\ S_{d-1}\pi\,C_{O}&\text{odd }d\end{array}\right.$ (37)

For $d=4$, we can do something more. In that case the Rényi entropy is a
surface defect and its universal part is described by (31) with the only
difference that all the anomaly coefficients also depends on the replica
number $n$. For this example, however, one additional relation between the
coefficients is available Lewkowycz and Perlmutter (2015)

$f_{c}(n)=\frac{n}{n-1}(a-f_{a}(n)-(n-1)\partial_{n}f_{a}(n))$ (38)

where $a$ is the $a$-anomaly coefficient of the bulk theory. Taking a
derivative with respect to $\lambda$ and using the result and the conventions
of Bianchi  _et al._ (2016a), relating $f_{c}(n)$ to the stress tensor one-
point function we get

$\partial_{\lambda}h(n,\lambda)=\frac{2n}{3}\left(\frac{a}{\pi}-\pi(1-(n-1)\partial_{n})C_{O}(n,\lambda)\right)$
(39)

which provides an intriguing relation between the stress-tensor one-point
function and the marginal operator one-point function.

##  IV Discussion

In this letter we considered the effect of an exactly marginal deformation on
a defect field theory. Thanks to the presence of a non-vanishing one-point
function, the first order derivative of the defect expectation value with
respect to the coupling is non-vanishing and it is determined by a single
piece of defect CFT data. This implies a non-trivial coupling dependence for
the $a$-type anomaly coefficient and for the proposed defect $C$-function. For
the case of the four-dimensional Rényi entropy we also derived an equation
relating the coupling derivative of the stress tensor to the one-point
function of the stress tensor operator.

It would be interesting to analyze further examples, especially in the context
of boundaries. It would also be important to better understand $b$-type
anomalies for higher even dimensional defects and how they could be mapped to
defect conformal data. Finally, one would like to find bounds on the allowed
values of the $a$\- and $b$-type defect anomaly coefficients (or their ratios)
with and without supersymmetry.

###### Acknowledgements.

It is a pleasure to thank Nadav Drukker, Madalena Lemos, Marco Meineri and
Michelangelo Preti for very useful discussions. We would also like to thank
Christopher Herzog and Itamar Shamir for important comments on the manuscript.
This work is supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
749909.

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