aainstitutetext: DESY Hamburg, Theory Group, Notkestraße 85, D-22607 Hamburg,
Germanybbinstitutetext: Institute of Physics, École Polytechnique Fédérale de
Lausanne (EPFL), Rte de la Sorge, BSP 728, CH-1015 Lausanne,
Switzerlandccinstitutetext: Institut für Physik und Institut für Mathematik,
Humboldt-Universität zu Berlin, IRIS Adlershof, Zum Großen Windkanal 6, 12489
Berlin, Germanyddinstitutetext: Max-Planck-Institut für Gravitationsphysik,
Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany

# Universality at large transverse spin in defect CFT

Madalena Lemos a Pedro Liendo b Marco Meineri c,d Sourav Sarkar
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the spectrum of local operators living on a defect in a generic
conformal field theory, and their coupling to the local bulk operators. We
establish the existence of universal accumulation points in the spectrum at
large $s$, $s$ being the charge of the operators under rotations in the space
transverse to the defect. Our tools include a formula that inverts the bulk to
defect OPE, analogous to the Caron-Huot formula for the four-point function
Caron-Huot:2017vep . Analyticity of the formula in $s$ implies that the
scaling dimensions of the defect operators are aligned in Regge trajectories
${\widehat{\Delta}}(s)$. These results require the correlator of two local
operators and the defect to be bounded in a certain region, a condition that
we do not prove in general. We check our conclusions against examples in
perturbation theory and holography, and we make specific predictions
concerning the spectrum of defect operators on Wilson lines. We also give an
interpretation of the large $s$ spectrum in the spirit of the work of Alday
and Maldacena Alday:2007mf .

###### Keywords:

conformal field theory, defects, conformal bootstrap

††preprint:  DESY 17-239 HU-EP-17/31

##  1 Introduction and summary

The space of conformal field theories (CFTs) is large and varied. Despite
this, remarkable features are common to all members of this special family of
quantum field theories. The most obvious ones are simple consequences of
symmetry, unitarity and locality, which respectively impose that operators
form representations of the conformal group, obey unitarity bounds, and that a
stress tensor exists. In recent years, it has become clear that universality
also emerges at a more detailed level. It was first discovered that the
spectrum and OPE coefficients of operators with asymptotically large spin
($\ell$) is tied to the low-lying (low-twist $\tau=\Delta-\ell$) CFT data
Alday:2007mf ; Komargodski:2012ek ; Fitzpatrick:2012yx . Recently Caron-
Huot:2017vep , it was shown that the relation is analytic rather than
asymptotic, and that operators in every CFT are organized in Regge
trajectories. These facts are best understood as consequences of crossing
symmetry, and as such are still rooted in the presence of conformal invariance
and unitarity constraints, albeit in a far less obvious fashion. For instance,
operators of spin as low as two lie in Regge trajectories, and this is a
consequence of boundedness of the four-point function in the so called Regge
limit, which in turn follows from reality of the OPE coefficients, _i.e._ from
unitarity. The contribution of individual Regge trajectories to the crossing
equation is enhanced and isolated when pairs of operators become light-like
separated. Hence, the lightcone bootstrap is the main source of analytic
information on the _double-twist operators_ that populate the trajectories.
This line of development nicely complements the study of crossing in Euclidean
configuration, which is most sensitive to low-dimensional data, rather than to
operators with low twist. Interestingly, the numerical results obtained by
studying the crossing equation around a Euclidean point arXiv:0807.0004  (see
Rychkov:2016iqz ; Simmons-Duffin:2016gjk  for pedagogical reviews and further
references) are also sensitive to the Lorentzian physics of the lightcone
limit Simmons-Duffin:2016wlq .111In Mazac:2016qev  a new analytic approach has
been put forward, which provides the exact spectrum exchanged in a four-point
function which is extremal in the standard bootstrap sense ElShowk:2012hu .
The combination of the two approaches yields unprecedented amount of
information on a non-supersymmetric, higher dimensional CFT such as the
critical 3d Ising model Simmons-Duffin:2016wlq .

The spectrum of local operators does not exhaust the set of observables in a
CFT. Extended probes arise naturally both from an experimental and from a
theoretical point of view. Boundaries and interfaces, surface operators,
Wilson and ’t Hooft lines are instances of what we refer to collectively as
defects. If the dynamics on a defect preserves the spacetime symmetries that
do not deform it, we call it a _conformal defect_. Conformal defects support a
spectrum of local excitations which are organized according to representations
of the preserved symmetries. Locality implies that the spectrum can be studied
in the highly symmetric case of a flat or spherical defect, in which the
algebra of preserved spacetime transformations is promoted to a full symmetry
group. For reasons to become clear shortly, we are interested in a space-like
$p$-dimensional defect in $d$ dimensions, so that the group is
$SO(p+1,1)\times SO(d-p-1,1)$. We shall also often denote the codimension as
$q$, that is, $p+q=d$. The defect CFT data is constrained by crossing symmetry
of the four-point function of defect primaries, and is tied to the bulk via
crossing symmetry of correlators involving at least two bulk primaries
McAvity:1995zd ; Billo:2016cpy . There is a growing effort in refining our
understanding of the constraints Gadde:2016fbj ; Hogervorst:2017kbj ;
Rastelli:2017ecj ; Fukuda:2017cup ; Lauria:2017 , extracting numerical and
analytic information from them Liendo:2012hy ; Gaiotto:2013nva ;
Gliozzi:2015qsa ; Liendo:2016ymz ; Gliozzi:2016cmg , and performing direct
computations of correlators in specific models, see _e.g._ , Billo:2013jda ;
Cosme:2015cxa ; Bianchi:2016xvf ; Chiodaroli:2016jod ; deLeeuw:2017dkd ;
Giombi:2017cqn ; Soderberg:2017oaa ; Melby-Thompson:2017aip ; deLeeuw:2017cop
; Kim:2017sju .

In this paper, we address the question whether the defect spectra exhibit
universal features akin those briefly discussed above. Of course, crossing
symmetry of the defect four-point function implies the existence of double
twist defect operators organized in Regge trajectories. Furthermore, the
theory on the defect comes naturally endowed with a global symmetry, the
$SO(q-1,1)$ group of boosts and rotations around a defect of codimension $q$.
We call _transverse spin_ , and we denote as $s$, the associated charge. This
global symmetry is the main ingredient of our analysis. In what follows, we
show that the spectrum of any defect CFT includes universal accumulation
points at large $s$. Given any scalar bulk primary operator of dimension
$\Delta_{\phi}$, the defect spectrum contains primaries of dimension

$\widehat{\Delta}\simeq s+\Delta_{\phi}+2m\,,\qquad s\to\infty\,,$ (1)

for asymptotically large $s$ and (non-negative) integer $m$. In fact, the
entire $1/s$ expansion of the $\widehat{\Delta}$ is computable once the bulk
CFT data is known. In eq. (1), and in the rest of the paper, quantum numbers
with a hat refer to the defect spectrum, except for the charge $s$.

It is not difficult to build some intuition for the large $s$ limit. Imagine
to couple a $p$-dimensional CFT to a $d$-dimensional one via a weakly relevant
operator for the lower dimensional theory. Further, assume that a short flow
lands the system on a defect CFT. The $s=0$ sector of the defect spectrum is
perturbatively close to the scaling dimensions of the original $p$-dimensional
theory. On the other hand, operators which transform under transverse
rotations are essentially bulk primaries evaluated at the location of the
defect. The defect primaries in eq. (1) are obtained by decomposing the
conformal family of a scalar bulk primary in representations of
$SO(p+1,1)\times SO(q-1,1)$, _i.e._ , they are of the schematic form
$\partial_{i}^{s}(\partial^{j}\partial_{j})^{m}\phi$, where $i,\,j$ denote
directions orthogonal to the defect. Therefore we shall call _transverse
derivative operators_ the primaries whose scaling dimensions obey eq. (1).
Operators of this kind also appear in the defect spectrum of large $N$
theories. The non-trivial statement, analogous to the case of double-twist
operators in an ordinary CFT, is that their anomalous dimensions222We call
“anomalous dimension” the deviation from eq. (1). This is perhaps non-
standard, but hopefully not confusing. are not only suppressed at weak
coupling or large $N$, but also at large $s$. This expansion parameter is lost
in the special case of a codimension one defect, _i.e._ a boundary or an
interface, transverse derivative operators still exist at small coupling or
large $N$ in this case Rastelli:2017ecj , however, we cannot constrain here
their anomalous dimension in a generic CFT.333This does not exclude that a
certain degree of universality appears in the large ${\widehat{\Delta}}$ limit
also for the spectrum of a boundary CFT Lauria:2017 .

Equation (1) can be obtained by lightcone bootstrap techniques, which we
review and apply to this case in section 2. In section 3, we derive a
Lorentzian inversion formula for the defect OPE (67), analogous to the one
obtained in Caron-Huot:2017vep  for the four-point function of local operators
– see also the recent derivation in Simmons-Duffin:2017nub . The formula
yields scaling dimensions of defect operators and bulk-to-defect OPE
coefficients as analytic functions of the transverse spin, and thus implies
the existence of trajectories in the $({\widehat{\Delta}},s)$ space. It also
resums the results that could be obtained by a systematic analysis of
lightcone expansion, as done in Alday:2015ewa  for the case without defects.
However, (67), and thus the analytic constraint, is valid only for $s$ larger
than a certain minimum $s_{\star}$. Unfortunately, contrary to Caron-
Huot:2017vep , we are not able to prove a theory independent upper bound to
$s_{\star}$. We therefore ignore if there is a universal value of the
transverse spin beyond which the spectrum of any defect CFT is constrained by
analyticity.

In section 4, we adopt a more physical point of view, and discuss the
suppression of anomalous dimensions at large $s$ in a two-dimensional
effective field theory, in a strict analogy with the work of Alday and
Maldacena on large-spin operators Alday:2007mf . This picture gives further
intuition on additional accumulation points which are not of the form (1). In
particular, we discuss adjoint insertions on Wilson lines, and we point out
the connection between their scaling dimension and the inter-quark potential
of a meson made of a heavy and a light (adjoint) quark. We predict the
following asymptotic behavior:

$\widehat{\Delta}\simeq s+\frac{f(\lambda)}{2}\log s\,,$ (2)

up to $1/N$ and $1/s$ corrections. Here we used the standard $\mathcal{N}=4$
notation for the cusp anomalous dimension $f(\lambda)$, even if the result
applies to a generic large-$N$ conformal gauge theory. The important point is
that the coefficient of the logarithm is half of the one appearing in the
twist of single-trace operators.

Section 5 is devoted to the study of specific examples. We confirm the
predictions of the lightcone bootstrap and of the inversion formula in various
weakly-coupled scenarios, as well as in holography. In section 6 we summarize
the main results and conclude with an outlook.

##  2 Lightcone bootstrap with a defect

$z=0$$z=1$$\bar{z}=1$$\bar{z}=0$$z,\bar{z}=0$$\phi(1,1)$$\phi(z,\bar{z})$defect

Figure 1: The configuration of the insertions in
$\braket{\phi(x_{1})\phi(x_{2})}$. We show a two-dimensional plane, transverse
to the defect, where the two operators lie. The defect is space-like, it
intersects the plane at the origin, and the two operators are placed at
$(1,1)$ and $(z,\bar{z})$ respectively.

Let us recall the notation and fix the setup. We consider a flat defect of
codimension $q$ in $d$ spacetime dimensions. We shall also denote the
dimension of the defect by $p$, _i.e._ , $p+q=d$. The defect will always be
space-like, except in section 4. Accordingly, we separate the spacetime
indices ($\mu=0,\dots,d-1$) in two subsets: orthogonal ($i=0,\dots,q-1$) and
parallel ($a=q,\dots,d-1$) to the defect. Our main focus is the two-point
function of identical scalar primaries which belong to the ambient CFT – _bulk
primaries_ for short. The correlator is a function of two cross-ratios, we
refer to appendix A for some technical details and to ref. Billo:2016cpy  for
a general introduction to the topic of defect CFTs. Let us choose
$x_{12}^{a}=0$, and focus on the two-dimensional plane in the transverse space
which contains the origin and the two insertions. The geometry is shown in
fig. 1, and the cross-ratios can be traded for the lightcone coordinates
$x_{2}=(z,\bar{z})$:

$\braket{\phi(x_{1})\phi(x_{2})}=\frac{g(z,\bar{z})}{(\lvert
x_{1}^{i}\rvert\lvert
x_{2}^{i}\rvert)^{\Delta_{\phi}}}=\frac{g(z,\bar{z})}{(z\bar{z})^{\Delta_{\phi}/2}}\,.$
(3)

Some convenient features of the $(z,\bar{z})$ coordinates in the defect CFT
context are discussed in Lauria:2017 . The two-point function can be expanded
either in the bulk or in the defect channel. The former involves the fusion of
the two external operators, while the latter is obtained by separately fusing
each of them with the defect. Agreement of the two expansions imposes the
following crossing constraint:

$g(z,\bar{z})=\left(\frac{(1-z)(1-\bar{z})}{(z\bar{z})^{1/2}}\right)^{-\Delta_{\phi}}\sum_{O}c_{\phi\phi
O}\,a_{O}\,f_{\Delta,J}(z,\bar{z})=\sum_{\widehat{O}}(b_{\phi\widehat{O}})^{2}\,\widehat{f}_{\widehat{\tau},s}(z,\bar{z})\,.$
(4)

The first sum runs over the bulk spectrum, and $f_{\Delta,J}(z,\bar{z})$ is
the bulk conformal block for the exchange of a primary of quantum numbers
$(\Delta,J)$. The prefactor is chosen so that $f_{0,0}(z,\bar{z})=1$. The OPE
data entering the bulk channel is the product of a three-point function
coefficient ($c_{\phi\phi O}$) and the coefficient of the one-point function
of the exchanged bulk operator ($a_{O}$). The second sum in eq. (4) runs over
defect primaries. The latter do not carry $so(p)$ spin when the external
operators are scalars, while a charge $s$ under the transverse $so(q-1,1)$ is
allowed. The conformal blocks $\widehat{f}_{\widehat{\tau},s}(z,\bar{z})$ are
then labeled by $s$ and $\widehat{\tau}=\widehat{\Delta}-s$ which we call the
_transverse twist_. The real numbers $b_{\phi\widehat{O}}$ determine the two-
point function $\braket{\phi\widehat{O}}$.444We use a few slightly different
conventions for the subscripts labeling $b_{\phi\widehat{O}}$ throughout the
text depending on the context and expect that the notation is self-
explanatory. The block in the defect channel is known exactly Billo:2016cpy
(see appendix A for the dictionary between our conventions and those of
Billo:2016cpy ):

$\widehat{f}_{\widehat{\tau},s}(z,\bar{z})=z^{\widehat{\tau}/2}\bar{z}^{\widehat{\tau}/2+s}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,\frac{z}{\bar{z}}\right){}_{2}F_{1}\left(\widehat{\tau}+s,\frac{p}{2},\widehat{\tau}+s+1-\frac{p}{2},z\bar{z}\right)\,.$
(5)

When $q$ is even, an order of limits ambiguity arises in the definition of the
hypergeometric function, one must first take $s$ to be integer, and then $q$
to be even. This prescription is henceforth assumed. The bulk-channel
conformal blocks are not known in closed-form for generic dimension and
codimension. However, in $q=2$ and any $d$ Billo:2016cpy , and $q=3$ and $d=4$
Liendo:2016ymz , the blocks satisfy the same Casimir equation as the blocks of
a four-point function of local operators, so any result in those cases carries
over to the present situation. In this work we are mostly interested in the
lightcone limit (to be defined more precisely below), and in this limit the
bulk blocks are given, for any $d$ and $q$, by555Notice that these are just
the familiar (t-channel) $SL(2,\mathbb{R})$ blocks which appear in the
lightcone decomposition of a four-point function of local operators. This is
not hard to understand: the $SL(2,\mathbb{R})$ transformations act on the $z$
cross-ratios in the same way as they do in the case of a four-point function.
They also act on the defect by simply displacing its intersection with the $z$
plane, as they would on a pair of local operators with
$\Delta_{1}=\Delta_{4}=0$. Finally, the OPE limit itself is the usual
lightcone OPE.

$f_{\Delta,J}(z,\bar{z})=(1-\bar{z})^{\frac{\Delta-J}{2}}\left(2^{-J}\left(1-z\right)^{\frac{\Delta+J}{2}}{}_{2}F_{1}\left(\frac{\Delta+J}{2},\frac{\Delta+J}{2},\Delta+J,1-z\right)+O((1-\bar{z}))\right)\,.$
(6)

One can ask if there can be a solution to the crossing equation (4) with a
finite number of blocks in either the bulk or defect decompositions. On the
bulk side the answer is clearly yes: the trivial defect, _i.e._ , the two-
point function without a defect has a single bulk block, that of the identity,
which is crossing symmetric. Whether it is possible to have a solution with
finitely many non-trivial bulk primaries, on top of the identity, is a
question that we do not address here. On the defect side, it is not hard to
prove that eq. (4) cannot be satisfied by finitely many defect primary
operators if $q>1$. Indeed, the defect block in eq. (5) has an unphysical
singularity when $z\bar{z}=1$, for any value of $\tfrac{z}{\bar{z}}$, which is
not consistent with the singularity structure of the two-point function in a
Euclidean configuration. For $p>1$ the behavior for $z\bar{z}\to 1$ is

$\begin{split}&\widehat{f}_{\widehat{\tau},s}(z,\bar{z})\overset{z\bar{z}\to
1}{\sim}\\\
&\frac{\Gamma\left(\frac{p-1}{2}\right)\Gamma\left(\widehat{\Delta}-\frac{p-2}{2}\right)}{2^{2-p}\sqrt{\pi}\Gamma(\widehat{\Delta})}\left(\frac{\bar{z}}{z}\right)^{s/2}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,\frac{z}{\bar{z}}\right)(1-z\bar{z})^{-p+1}\,,\quad
p>1\,,\end{split}$ (7)

while for $p=1$ the singularity is a logarithm, and the following argument is
unchanged. The case where ${\widehat{\Delta}}=\tfrac{p-2}{2}$ looks different
in (7), but in this case too the argument goes through. It can be easily
checked that all the coefficients of the hypergeometric function in an
expansion around $z=0$, which is a polynomial of degree $s$, are positive.
Through the scalar unitarity bound, the sign of the prefactor in (7) is also
fixed independently of the spectrum. Finally, positivity of the
$(b_{\phi\widehat{O}})^{2}$ in eq. (4) implies that the singularity cannot be
canceled by a finite number of blocks. In the case of a defect of codimension
one the same singularity is instead potentially physical, and the exponent
matches the exchange of the identity in the bulk channel when the theory is
free. This allows for the existence of solutions to the boundary crossing
equation with finitely many blocks McAvity:1995zd ; Liendo:2012hy .

It was shown in Pappadopulo:2012jk ; Fitzpatrick:2012yx ; Komargodski:2012ek
that analytic information can still be extracted from a crossing equation that
contains infinitely many terms, by focusing on a limit that drastically
simplifies one of the channels. Both in the Euclidean and in the lightcone OPE
limits the identity dominates, say, the $t$-channel. This statement is theory
independent, and is therefore the source of the universality of the spectrum
needed in the $s$-channel to ensure crossing symmetry. In the case of the two-
point function with a defect, there are two such simplifying limits. In the
$z\to 0$ limit $\phi(x_{2})$ is light-like separated from the defect, and the
defect OPE is dominated by the operators with the smallest
$\widehat{\tau}=\widehat{\Delta}-s$, as it can be seen in eq. (5). Conversely,
when $(1-\bar{z})\to 0$ $\phi(x_{2})$ is light-like separated from
$\phi(x_{1})$. This is the limit we will study in this section. When
$(1-\bar{z})\to 0$ the identity in the bulk channel dominates the OPE, as it
is clear from (6), so one may hope that some analytic universal statement can
be made about the defect spectrum. It remains to be proven that a specific
sector of the defect spectrum is mostly sensitive to the identity in the
crossed channel, so that a perturbation theory can be set up. We argue in this
direction in subsection 2.1, and also in subsection 3.3 making use of the
inversion formula.

For the moment, we point out that such a sector of the defect spectrum needs
to be perturbatively close to the spectrum of the trivial defect. Indeed, when
there is no defect, translational invariance is preserved, and the identity is
the only exchanged bulk primary

$g(z,\bar{z})=\langle\phi(x_{1})\phi(x_{2})\rangle\left(z\bar{z}\right)^{\Delta_{\phi}/2}=\left(\frac{\left(z\bar{z}\right)^{1/2}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\,.$
(8)

That the identity in the bulk channel is crossing symmetric on its own is a
difference with respect to the four-point function: there the identity is
never crossing symmetric. As anticipated in the introduction, the defect OPE
of a real primary $\phi$ of dimension $\Delta_{\phi}$ is regular and only
contains primaries of the kind

$(\partial_{i})^{n}(\partial_{j}\partial^{j})^{m}\phi\,,\qquad\begin{cases}\widehat{\Delta}=\Delta_{\phi}+n+2m\,,\\\
s=n\,,\end{cases}$ (9)

that is, derivatives of $\phi$ transverse to the defect. For later
convenience, let us also report the defect OPE coefficients:

$b^{2}_{s,m}=\frac{\Gamma\left(\frac{q}{2}+s\right)\Gamma(2m+s+\Delta_{\phi})\Gamma\left(m-\frac{d}{2}+\Delta_{\phi}+1\right)\Gamma\left(m-\frac{p}{2}+s+\Delta_{\phi}\right)}{\Gamma(\Delta_{\phi})\Gamma(m+1)\Gamma(s+1)\Gamma\left(m+\frac{q}{2}+s\right)\Gamma\left(-\frac{d}{2}+\Delta_{\phi}+1\right)\Gamma\left(2m-\frac{p}{2}+s+\Delta_{\phi}\right)}\,.$
(10)

We shall now argue that, at large $s$, the spectrum of any defect does contain
a sector close to the trivial defect, in the same sense as ordinary CFT
spectra are close to generalized free theory in the large spin limit. Our
strategy in the rest of the section is analogous to the one presented in
Komargodski:2012ek ; Fitzpatrick:2012yx . Now, certain weak points in the
argument have been pointed out in Qiao:2017xif . It may be expected that these
issues are easier to tackle here with respect to the analogous ones in the
lightcone bootstrap of the four-point function Komargodski:2012ek ;
Fitzpatrick:2012yx , due to the simplicity of the defect channel blocks.
However, in this work we do not try to solve them. While this section is not
completely rigorous, it does define a calculable perturbative series, whose
predictions we test in some examples in section 5. Furthermore, in subsection
3.3 we shall come back to the question with a more rigorous tool in our hands,
which will allow to rigorously prove part of the results that follow.

###  2.1 The defect spectrum at large transverse spin

In what follows, we would like to analyze the crossing equation (4) in the
bulk light-cone limit, and more specifically in the following region:

$\qquad 1-\bar{z}\ll z<1\,.$ (11)

In this regime, the contribution of the higher-twist bulk operators is
suppressed with respect to the identity (6), while the defect OPE still
converges. We can rewrite the crossing equation as follows:

$1=\lim_{\bar{z}\to
1}\left(\frac{(1-z)(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{\Delta_{\phi}}\sum_{\widehat{\tau},s}(b_{\phi\widehat{O}})^{2}\,\widehat{f}_{\widehat{\tau},s}(z,\bar{z})\,.$
(12)

The conformal blocks of (a subsector of) the defect primaries need to match
the $(1-\bar{z})$ dependence in the prefactor. However, each conformal block
is analytic at $\bar{z}=1$, as long as $z<1$, therefore $\lim_{\bar{z}\to
1}(1-\bar{z})^{\Delta_{\phi}}\widehat{f}_{\widehat{\tau},s}=0$ for every
operator in the spectrum. We are led to conclude that the sum in eq. (12) does
not converge uniformly at $\bar{z}=1$. Let us look for the region which is
responsible for the singularity. At large and positive $\widehat{\tau}$, for
fixed $s$, the blocks are suppressed by $z^{\widehat{\tau}/2}$ for every $z<1$
– see eq. (5). Since $\widehat{\tau}$ is not subject to a unitarity bound, one
should worry about the $\widehat{\tau}\to-\infty$ as well. In perturbation
theory, we expect the transverse twist to be bounded from below. Furthermore,
$\widehat{\tau}$ generates time translations in AdS in the Alday-Maldacena
picture explored in section 4, and this supports the more general assumption
that $\widehat{\tau}$ is bounded from below in a healthy theory. In the
heuristic spirit of the discussion, we will not comment further on this limit.
Finally, precisely in the limit $\bar{z}\to 1$ the sum over transverse spins,
at fixed $\widehat{\tau}$, ceases to be suppressed. In studying this last
region, we replace the blocks with their large $s$ asymptotics:

$\widehat{f}_{\widehat{\tau},s}(z,\bar{z})\underset{\widehat{\tau}\textup{
fixed}}{\overset{s\to\infty}{\sim}}(z\bar{z})^{\widehat{\tau}/2}\bar{z}^{s}\left(\frac{\bar{z}}{(\bar{z}-z)}\right)^{\frac{q-2}{2}}(1-z\bar{z})^{-\frac{p}{2}}\left(1+\mathcal{O}\left(\frac{1}{s}\right)\right)\,.$
(13)

This approximation is obtained in the region $0<z<\bar{z}<1$. Notice that the
blocks are nicely factorized in the pairs $(z\bar{z},\widehat{\tau})$ and
$(\bar{z},s)$ in this limit. If we now plug the asymptotics (13) in eq. (12),
and we analyze the equation order by order in $z$, we see that eq. (12) is
satisfied if the following accumulation points exist in the spectrum:

$\widehat{\tau}=\Delta_{\phi}+2m+\mathcal{O}(s^{-\alpha})\leavevmode\nobreak\
,\quad s\to\infty\,,$ (14)

for non-negative integer $m$, and a real positive $\alpha$ to be determined.
The presence of a finite $s$ correction must be allowed, because we only
treated the crossing equation (12) in the strict infinite $s$ limit. The
operators (14) can obviously be thought of as the transverse derivatives
described in eq. (9). Once we plug the spectrum (14) back in eq. (12), we also
deduce that the OPE coefficients themselves should asymptote the ones in eq.
(10):

$b^{2}_{s,m}=s^{\Delta_{\phi}-1}\left(\frac{1}{\Gamma(\Delta_{\phi})}\binom{m-\frac{d}{2}+\Delta_{\phi}}{m}+\mathcal{O}(s^{-\beta})\right)\leavevmode\nobreak\
,\quad s\to\infty\leavevmode\nobreak\ ,$ (15)

for some positive $\beta$. Let us emphasize what would be needed to make this
argument rigorous – see appendix F in Qiao:2017xif  for a more detailed
discussion. One should prove that the following limit exists at fixed
$\widehat{\tau}$:

$\rho(\widehat{\tau})=\lim_{\bar{z}\to
1}\sum_{s=0}^{\infty}(b_{\phi\widehat{O}})^{2}\,(1-\bar{z})^{\Delta_{\phi}}\bar{z}^{s}\leavevmode\nobreak\
,$ (16)

and that plugging $\rho(\widehat{\tau})$ in eq. (12) one obtains a convergent
sum over $\widehat{\tau}$,666Up to the issue of bounding the spectrum at
negative $\widehat{\tau}$, this step can be done precisely as in
Fitzpatrick:2012yx . so that eq. (12) can be analyzed order by order in $z$.
The rest is then equivalent to the previous discussion: since the trivial
defect in particular solves eq. (12), we obtain $\rho(\widehat{\tau})$ by
plugging the OPE coefficient of the trivial defect in eq. (16). At this point,
in turn, the Hardy-Littlewood tauberian theorem Qiao:2017xif  can be used to
deduce from eq. (16) the asymptotics (15). Finally, we stress that eq. (15)
establishes an averaged property of the spectrum at large spin, while we have
no control on the OPE coefficient of single defect primaries.777Let us also
mention that the power-law form of the subleading corrections to eqs. (14-15)
is assumed, but not implied even by the more rigorous approach.

It is interesting to look in detail at the way the identity is reproduced at
leading order in $1-\bar{z}$. This highlights the relation between the large
$s$ and small $1-\bar{z}$ limits Komargodski:2012ek ; Fitzpatrick:2012yx . Let
us write the r.h.s. of the crossing equation (4) only including the transverse
derivatives and using the large $s$ asymptotics of the blocks eq. (13) and of
the OPE coefficients eq. (15). We also replace the sum over spins by an
integral, that we cutoff at some minimum spin $\Lambda$:

$\left(\frac{\bar{z}}{(\bar{z}-z)}\right)^{\frac{q-2}{2}}(1-z\bar{z})^{-\frac{p}{2}}\sum_{m=0}^{\infty}\binom{m-\frac{d}{2}+\Delta_{\phi}}{m}(z\bar{z})^{\Delta_{\phi}/2+m}\left(\frac{1}{\Gamma(\Delta_{\phi})}\int_{\Lambda}^{\infty}dss^{\Delta_{\phi}-1}\bar{z}^{s}\right)\\\
=\left(\frac{\sqrt{z\bar{z}}}{1-z}\right)^{\Delta_{\phi}}\frac{\Gamma(\Delta_{\phi},-\Lambda\log\bar{z})}{\Gamma(\Delta_{\phi})}\frac{1}{(-\log\bar{z})^{\Delta_{\phi}}}\,.$
(17)

In the $\bar{z}\to 1$ limit, the result matches the bulk identity, for any
finite $\Lambda$, however large. This confirms that only asymptotically large
values of $s$ matter. In fact, $\Lambda$ could even be sent to infinity, as
long as the growth is slower than $1/(1-\bar{z})$. This signals which range of
spin is important in reproducing the bulk OPE limit, and cannot be excluded
from the integral:

$s\sim\frac{1}{1-\bar{z}}\,.$ (18)

An alternative way to understand this fact is through a saddle-point
approximation of the simple integral in eq. (17), as in Komargodski:2012ek ,
which is accurate for large $\Delta_{\phi}$. The relation (18) should be
contrasted with the one relevant to double twist operators, that is $\ell\sim
1/(1-\bar{z})^{1/2}$ Komargodski:2012ek ; Fitzpatrick:2012yx . The different
behavior here is responsible for the different finite spin exponent $\alpha$ –
see eq. (14) below – of the transverse derivatives with respect to the one of
double twists.

Finite spin corrections to eqs. (14) and (15) can be computed by taking into
account subleading contributions to the bulk OPE in the $\bar{z}\to 1$ limit.
A trivial series of corrections is required to match higher orders in
$1-\bar{z}$ coming from the bulk identity block. Only the OPE coefficients are
affected, and all of the corrections are trivially obtained by expanding eq.
(10) at large $s$. More interestingly, new bulk primaries start contributing
at some order in $1-\bar{z}$, according to their twist. For the sake of
simplicity, we restrict the analysis to the correction due to a single bulk
block. The case in which infinitely many primaries with (nearly) degenerate
twist exist can also be dealt with, as done in Alday:2015ota ; Alday:2016jfr
for the four-point function without defects, and is relevant to weakly coupled
CFTs. We leave this analysis for future work. We assume that the leading
contribution after the identity comes from a single bulk block with minimal
twist $\tau_{\textup{min}}$ and dimension $\Delta_{\textup{min}}$:

$\left(\frac{(z\bar{z})^{1/2}}{(1-z)(1-\bar{z})}\right)^{\Delta_{\phi}}\bigr{(}1+c_{\phi\phi
O_{\textup{min}}}\,a_{\textup{min}}\,f_{\Delta_{\textup{min}},J_{\textup{min}}}(z,\bar{z})\bigr{)}\sim\sum_{\begin{subarray}{c}s\
\textup{large}\\\
\widehat{\Delta}-s\,=\,\widehat{\tau}(s)\end{subarray}}\\!\,b_{s,\widehat{\tau}(s)}^{2}\,\widehat{f}_{\widehat{\tau}(s),s}(z,\bar{z})\,.$
(19)

We already assumed that in the regime $\eqref{looseregime}$ we can account for
the leading twist operator $O_{\textup{min}}$ by modifying the trajectory
$\widehat{\tau}(s)$ of the transverse derivative operators and their OPE
coefficient. Let us check that this is sufficient to reproduce the l.h.s., and
let us only consider the leading transverse twist trajectory. This amounts to
taking the small $z$ limit in the bulk collinear block (6):

$f_{\Delta,J}(z,\bar{z})\ \overset{z\to 0}{\sim}\
-2^{\Delta-1}\frac{\Gamma\left(\frac{1}{2}+\frac{\Delta+J}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\Delta+J}{2}\right)}(1-\bar{z})^{\frac{\Delta-J}{2}}\left(2\left(\gamma_{E}+\psi\left(\frac{\Delta+J}{2}\right)\right)+\log
z\right)\,.$ (20)

Note that this contribution is only singular as $\bar{z}\to 1$ if
$\Delta_{\phi}-\frac{\Delta-J}{2}<0$, in which case the rest of the discussion
follows directly. We shall comment below on the opposite scenario. Let us
assume the following parametrization for the leading transverse twist
trajectory:

$\widehat{\tau}(s)=\Delta_{\phi}+\frac{c_{\textup{min}}}{s^{\alpha}}\,.$ (21)

At large $s$, the anomalous dimension produces a logarithm of $z$ on the
r.h.s. of eq. (19), which can be matched to the small $z$ behavior of the
block eq. (20). A short computation yields:

$\alpha=\frac{\tau_{\textup{min}}}{2}=\frac{\Delta_{\textup{min}}-J_{\textup{min}}}{2}\,,$
(22)

and

$c_{\textup{min}}=-c_{\phi\phi
O_{\textup{min}}}\,a_{\textup{min}}2^{\Delta_{\textup{min}}}\frac{\Gamma(\Delta_{\phi})}{\Gamma\left(\Delta_{\phi}-\frac{\tau_{\textup{min}}}{2}\right)}\frac{\Gamma\left(\frac{1}{2}+\frac{\Delta_{\textup{min}}+J_{\textup{min}}}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{\textup{min}}+J_{\textup{min}}}{2}\right)}\,.$
(23)

Similarly, a correction to the OPE coefficient (15) is required to match the
$\log$-independent part of eq. (20):

$b_{s}^{2}=\frac{\Gamma(\Delta_{\phi}+s)}{\Gamma(s+1)\,\Gamma(\Delta_{\phi})}\left(1+\frac{b_{\textup{min}}}{s^{\tau_{\textup{min}}/2}}\right),\\\
b_{\textup{min}}=-c_{\phi\phi
O_{\textup{min}}}\,a_{\textup{min}}\,2^{\Delta_{\textup{min}}}\frac{\Gamma(\Delta_{\phi})}{\Gamma\left(\Delta_{\phi}-\frac{\tau_{\textup{min}}}{2}\right)}\frac{\Gamma\left(\frac{1}{2}+\frac{\Delta_{\textup{min}}+J_{\textup{min}}}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{\textup{min}}+J_{\textup{min}}}{2}\right)}\left(\gamma_{E}+\psi\left(\frac{\Delta_{\textup{min}}+J_{\textup{min}}}{2}\right)\right)\,.$
(24)

Let us pause to comment on the non-singular case
$\Delta_{\phi}-\frac{\Delta-J}{2}>0$. Following Alday:2015eya , one can act
with the defect Casimir operator $\mathcal{C}_{\mathrm{def}}$, written down in
Billo:2016cpy , on both sides of the crossing equation. On the bulk side we
find

$\mathcal{C}_{\mathrm{def}}\left[(1-\bar{z})^{\delta}f(z)\right]=-2\delta(\delta-1)(1-\bar{z})^{\delta-2}f(z)+\mathcal{O}(1-\bar{z})^{\delta-1}\,,$
(25)

and so the leading behavior of (20) can be made singular by repeatedly acting
with the defect Casimir, provided $\tfrac{\Delta-J}{2}-\Delta_{\phi}$ is not a
positive integer. For generic $\Delta$, $\Delta_{\phi}$ the contribution of a
bulk primary is thus Casimir-singular in the sense of Simmons-Duffin:2016wlq .
On the defect side, acting with $\mathcal{C}_{\mathrm{def}}$ introduces the
eigenvalue for the corresponding defect block, which grows as $s^{2}$ for
large $s$ and thus enhances the large $s$ behavior. Therefore, the results
(23-24) are valid also if $\Delta_{\phi}-\frac{\Delta-J}{2}>0$ and non
integer.

It is interesting to notice that unitarity does not fix the sign of
$c_{\textup{min}}$. In other words, the spectrum does not need to be convex.
However, it is intriguing to notice that in all the examples in section 5
$c_{\textup{min}}<0$. In most of the cases the leading correction comes from
the exchange of the stress-tensor block, which always turn out to have a
negative coefficient of the one-point function: $a_{T}<0$. We shall comment
more on this in subsection 5.6.

Here we presented the result for the leading transverse twist correction, but
similar corrections to (14) and (15) for $m\neq 0$ are straightforward to
obtain. The large $s$ expansion of anomalous dimensions and OPE coefficients
can be set up systematically to obtain the contribution of a collinear primary
and all its descendants, as done in Alday:2015ewa  for the four-point function
case. The only requirement is the knowledge of the subleading contributions to
(6). However, we shall pursue a different direction. In section 3, we will
obtain an inversion formula for the defect OPE, analogous to the one found in
Caron-Huot:2017vep  for the four-point function, which allows to resum the
lightcone expansion.

##  3 Inversion of the defect OPE

In this section we describe a general way to extract the defect spectrum given
a two-point function of bulk primaries. The quantum numbers
$(\widehat{\tau},\,s)$ and the defect OPE coefficient
$(b_{\phi\widehat{O}})^{2}$ are extracted by an integral transform of the two-
point function, which is analytic in the transverse spin $s$. This is the
defect analog of the inversion formula found in Caron-Huot:2017vep , which
applies to four-point functions in theories without defects, and most of the
features of the present integral transform, and its derivation, are similar to
Caron-Huot:2017vep . The inversion formula obtained in this section allows to
resum the large $s$ results of section 2, and extract the scaling dimension of
defect operators with finite transverse spin. It also bypasses the need for
some of the assumptions required by the lightcone analysis and discussed in
section 2. The validity of the integral transform, similarly to that of Caron-
Huot:2017vep , depends on the growth of the correlator in a certain region.
Contrary to Caron-Huot:2017vep , though, the behavior of the correlator in
this region is not controlled by an OPE limit, and we cannot place general
bounds on its growth. We shall further comment on this issue in subsection
3.2.

In the rest of this section we derive the inversion formula for the defect OPE
following in the footsteps of Caron-Huot:2017vep . For this purpose we start
by obtaining a Euclidean inversion formula, which simply follows from
orthogonality of partial waves, see for example Hogervorst:2017sfd  for a
detailed derivation in the case of a four-point function in one-dimensional
theories, or Hogervorst:2017kbj  for boundary CFTs. While this Euclidean
formula is not analytic in the transverse spin $s$, it can be manipulated into
a Lorentzian formula that is. A different derivation of the Lorentzian
inversion formula without defects was presented recently in Simmons-
Duffin:2017nub : we leave to future work the extension of that physically more
transparent approach to the present case.

###  3.1 The Euclidean formula

We start by obtaining an Euclidean inversion formula for the defect OPE.
Recall that in our configuration the two operators lie on a plane orthogonal
to the defect. The defect intersects the plane at the origin, with one
external operator placed at $x_{1}=(1,1)$ and the other at
$x_{2}=(z,\bar{z})$, see fig. 1. We introduce the following radial coordinates
for the position of the second operator

$z=rw\,,\qquad\bar{z}=\frac{r}{w}\,,\qquad\eta=\frac{1}{2}\left(w+\frac{1}{w}\right)\,,$
(26)

where in Euclidean signature $\bar{z}=z^{*}$ and so $w$ is a phase. Since
defect blocks (5) factorize,

$\displaystyle\widehat{f}_{\widehat{\tau},s}(z,\bar{z})=\widehat{g}_{s}(\eta)\widehat{f}_{\widehat{\Delta}}(r)\,,$
(27)

where $\widehat{\tau}={\widehat{\Delta}}-s$, we shall treat the parallel
$\widehat{f}_{\widehat{\Delta}}(r)$ and angular $\widehat{g}_{s}(\eta)$ parts
separately.

#### Parallel factor of defect blocks

Let us start by considering the parallel factor of the block. Similarly to
Hogervorst:2017sfd , we re-write the Casimir equation that
$\widehat{f}_{\widehat{\Delta}}(r)$ satisfies Billo:2016cpy  in the form of a
Sturm-Liouville problem

$\mathcal{D}_{\parallel}\widehat{f}(r)={\widehat{\Delta}}({\widehat{\Delta}}-p)\widehat{f}(r)\,,\qquad\mathrm{with}\qquad\mathcal{D}_{\parallel}\widehat{f}(r)=\frac{r^{p+1}}{(1-r^{2})^{p}}\frac{d}{dr}\left(r^{1-p}(1-r^{2})^{p}\;\frac{d\widehat{f}(r)}{dr}\right)\,.$
(28)

The operator $\mathcal{D}_{\parallel}$ defined in (28) is self-adjoint with
respect to the measure

$\mu_{p}(r)=\frac{(1-r^{2})^{p}}{r^{p+1}}\,,$ (29)

in the interval $r\in[0,1]$, provided the functions are well-behaved near
$r=0$ and $r=1$. Concretely, self-adjointness requires that the following
boundary term vanishes

$\begin{split}&\int\limits_{0}^{1}dr\,\mu_{p}(r)\,\mathcal{D}_{\parallel}(\Psi(r))\tilde{\Psi}(r)-\int\limits_{0}^{1}dr\,\mu_{p}(r)\,\Psi(r)\mathcal{D}_{\parallel}(\tilde{\Psi}(r))\\\
&\qquad\qquad\qquad=\int\limits_{0}^{1}dr\,\frac{d}{dr}\left[\mu_{p}(r)r^{2}\left(\frac{d\Psi(r)}{dr}\tilde{\Psi}(r)-\Psi(r)\frac{d\tilde{\Psi}(r)}{dr}\right)\right]\,.\end{split}$
(30)

While for the functions to be square-integrable with respect to the measure
(29) their behavior near $r=0$ and $r=1$ must be such that

$\Psi(r)\underset{r\to
0}{\sim}r^{\frac{p}{2}+\epsilon}\,,\qquad\Psi(r)\underset{r\to
1}{\sim}(1-r)^{-\frac{p+1}{2}+\epsilon^{\prime}}\,,$ (31)

with $\epsilon$, $\epsilon^{\prime}$ positive numbers. However, the parallel
factor in the defect conformal blocks,

$\widehat{f}_{\widehat{\Delta}}(r)=r^{{\widehat{\Delta}}}\;{}_{2}F_{1}\left({\widehat{\Delta}},\frac{p}{2},{\widehat{\Delta}}+1-\frac{p}{2},r^{2}\right)\,,$
(32)

which is an eigenfunction of $\mathcal{D}_{\parallel}$, grows as $(1-r)^{1-p}$
for $r\to 1$ (this growth is logarithmic in the $p=1$ case). Therefore, unless
$p=1,2$ their square is not integrable against the measure (29), and for no
value of $p$ does the boundary term in (30) vanish.

Following Hogervorst:2017sfd ; Caron-Huot:2017vep  we consider a linear
combination of $\widehat{f}_{\widehat{\Delta}}$ that is still an eigenfunction
of $\mathcal{D}_{\parallel}$, with eigenvalue
${\widehat{\Delta}}({\widehat{\Delta}}-p)$, but is regular at $r=1$

$\Psi_{\widehat{\Delta}}(r)=\frac{1}{2}\left(\widehat{f}_{{\widehat{\Delta}}}(r)+\frac{K_{p-{\widehat{\Delta}}}}{K_{\widehat{\Delta}}}\widehat{f}_{p-{\widehat{\Delta}}}(r)\right)=\frac{K_{p-{\widehat{\Delta}}}}{2K_{p}}r^{p-{\widehat{\Delta}}}\;{}_{2}F_{1}\left(\frac{p}{2},p-{\widehat{\Delta}},p,1-r^{2}\right)\,,$
(33)

where we defined

$K_{\widehat{\Delta}}=\frac{\Gamma({\widehat{\Delta}})}{\Gamma\left({\widehat{\Delta}}-\frac{p}{2}\right)}\,.$
(34)

Also, the behavior of $\Psi_{\widehat{\Delta}}(r)$ near $r=1$ is such that the
corresponding boundary term in (30) vanishes. However, near $r=0$ the
functions grow as

$\Psi_{\widehat{\Delta}}(r)\underset{r\to
0}{\sim}\frac{\Gamma(p-{\widehat{\Delta}})\Gamma\left({\widehat{\Delta}}-\frac{p}{2}\right)r^{p-{\widehat{\Delta}}}}{2\Gamma({\widehat{\Delta}})\Gamma\left(\frac{p}{2}-{\widehat{\Delta}}\right)}+\frac{r^{{\widehat{\Delta}}}}{2}\,,$
(35)

and so at best they can be delta-function normalizable, provided we take
$\Re({\widehat{\Delta}})=\tfrac{p}{2}$. Were we to consider normalizable
eigenfunctions of the self-adjoint operator $\mathcal{D}_{\parallel}$,
standard arguments would imply that they are orthogonal. To show orthogonality
in this case, we will work instead with the following regularized functions

$\Psi_{\widehat{\Delta}}^{\mathrm{reg.}}(r)=\frac{K_{p-{\widehat{\Delta}}}}{2K_{p}}r^{p-{\widehat{\Delta}}+\epsilon}\;{}_{2}F_{1}\left(\frac{p}{2},p-{\widehat{\Delta}},p,1-r^{2}\right)\,,\qquad\mathrm{with}\;\;{\widehat{\Delta}}=\frac{p}{2}+\mathrm{i}\,\nu\,,\quad\nu\in\mathbb{R}\,,$
(36)

with $\epsilon>0$ a small number, such that the functions
$\Psi_{\widehat{\Delta}}^{\mathrm{reg.}}(r)$ are normalizable. The operator
$\mathcal{D}_{\parallel}$ is self-adjoint on these functions, since the chosen
regularization makes the boundary term at $r=0$ vanish, while it preserves the
vanishing of the boundary term at $r=1$. Due to the regularization
$\Psi_{\widehat{\Delta}}^{\mathrm{reg.}}(r)$ are not eigenfunctions of
$\mathcal{D}_{\parallel}$ and so orthogonality is not yet immediate.
Nevertheless, starting from self-adjointness (the first line in (37)), we can
evaluate the action of $\mathcal{D}_{\parallel}$ on the regularized functions
to obtain

$\begin{split}0&=\int\limits_{0}^{1}dr\,\mu_{p}(r)\,\left(\mathcal{D}_{\parallel}(\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{1}}(r))\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{2}}(r)-\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{1}}(r)\mathcal{D}_{\parallel}(\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{2}}(r))\right)\\\
&=\left({\widehat{\Delta}}_{1}({\widehat{\Delta}}_{1}-p)-{\widehat{\Delta}}_{2}({\widehat{\Delta}}_{2}-p)\right)\;\int\limits_{0}^{1}dr\,\mu_{p}(r)\,\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{1}}(r)\Psi^{\mathrm{reg.}}_{{\widehat{\Delta}}_{2}}(r)+\mathcal{O}(\epsilon)\,.\end{split}$
(37)

Taking $\epsilon\to 0$ this implies that if
${\widehat{\Delta}}_{1}({\widehat{\Delta}}_{1}-p)\neq{\widehat{\Delta}}_{2}({\widehat{\Delta}}_{2}-p)$
the functions are orthogonal. Finally, all we have to show now is what happens
when the eigenvalues coincide, and for that we need only examine the behavior
of the functions near $r=0$ where the integral develops a singularity. In this
case, taking ${\widehat{\Delta}}_{i}=p/2+\mathrm{i}\,\nu_{i}$, we end up with
integrals of the type

$\int\limits_{0}dr\,r^{-1\pm\mathrm{i}(\nu_{1}\pm\nu_{2})}=\pi\delta(\nu_{1}\pm\nu_{2})+\text{non-
singular}\,,$ (38)

following from the behavior of the measure (29) and (35).888One could
equivalently have shown that the integral of the regularized functions (36)
provides a representation of the delta function as $\epsilon\to 0$, this is
obvious for $p=2$ when the resulting expressions are very simple.

All in all, the functions (33) are orthogonal when
${\widehat{\Delta}}_{i}=p/2+\mathrm{i}\,\nu_{i}$, satisfying999The functions
$\Psi_{\widehat{\Delta}}(r)$ could be made real for $\nu\in\mathbb{R}$ by an
appropriate choice of normalization, but we have not done so. Also, the
orthogonality of (39) is enough for our purposes and thus we do not define a
positive inner product.

$\int\limits_{0}^{1}dr\,\mu_{p}(r)\,\Psi_{{\widehat{\Delta}}_{1}}(r)\Psi_{{\widehat{\Delta}}_{2}}(r)=\frac{\pi}{2}\frac{K_{p-{\widehat{\Delta}}_{2}}}{K_{{\widehat{\Delta}}_{1}}}\left[\delta(\nu_{1}-\nu_{2})+\delta(\nu_{1}+\nu_{2})\right]\,.$
(39)

#### Angular factor of defect blocks

We now turn to the angular factor in the conformal block (27). It is useful to
go back to the representation of the angular factor in (5) as a Gegenbauer
polynomial for integer $s$ via

$w^{-s}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,w^{2}\right)=\binom{s+\frac{q}{2}-2}{\frac{q}{2}-2}^{-1}C_{s}^{q/2-1}\left(\frac{w}{2}+\frac{1}{2w}\right)\,,$
(40)

such that it becomes

$\widehat{g}_{s}(\eta)=\binom{s+\frac{q}{2}-2}{\frac{q}{2}-2}^{-1}\,C_{s}^{(q/2-1)}(\eta)\,.$
(41)

Gegenbauer polynomials are orthogonal with respect to the following measure

$\int\limits_{-1}^{1}d\eta\,\mu_{q}(\eta)C_{s}^{(\frac{q}{2}-1)}(\eta)C_{s^{\prime}}^{(\frac{q}{2}-1)}(\eta)=\frac{2^{3-q}\pi\Gamma(s+q-2)}{\left(s+\frac{q}{2}-1\right)\Gamma(s+1)\Gamma\left(\frac{q}{2}-1\right)^{2}}\delta_{ss^{\prime}}\,,\quad\mu_{q}(\eta)=(1-\eta^{2})^{\frac{q-3}{2}}\,,$
(42)

which we rewrite using the normalization of the conformal block themselves

$\int\limits_{-1}^{1}d\eta\,\mu_{q}(\eta)\widehat{g}_{s}(\eta)\widehat{g}_{s^{\prime}}(\eta)=N_{q,s}\delta_{ss^{\prime}}\,,\qquad
N_{q,s}=2^{3-q}\pi\frac{\Gamma(s+1)\Gamma(s+q-2)}{\Gamma(s+\frac{q}{2})\Gamma\left(s+\frac{q}{2}-1\right)}\,.$
(43)

#### Euclidean inversion formula

Finally we can write the following orthogonal decomposition for the two-point
function, similarly to what has been done for the case of the four-point
function of local operators Dobrev:1975ru  (see also Costa:2012cb ),

$g(r,\eta)=\sum\limits_{s=0}^{\infty}\int_{\gamma}\frac{d{\widehat{\Delta}}}{2\pi\mathrm{i}}b({\widehat{\Delta}},s)\widehat{g}_{s}(\eta)\Psi_{\widehat{\Delta}}(r),\qquad\gamma=\\{{\widehat{\Delta}}:{\widehat{\Delta}}\in\left(p/2-\mathrm{i}\infty,p/2+\mathrm{i}\infty\right)\,\\}\,.$
(44)

Since
$\Psi_{p-{\widehat{\Delta}}}(r)=\frac{K_{\widehat{\Delta}}}{K_{p-{\widehat{\Delta}}}}\Psi_{\widehat{\Delta}}(r)$,
we can assume that

$b(p-{\widehat{\Delta}},s)=\frac{K_{p-{\widehat{\Delta}}}}{K_{\widehat{\Delta}}}b({\widehat{\Delta}},s)\,.$
(45)

The position of the poles and residues of $b({\widehat{\Delta}},s)$ is
revealed by closing the contour $\gamma$. At large ${\widehat{\Delta}}$

$\widehat{f}_{\widehat{\Delta}}(r)\sim
r^{\widehat{\Delta}}(1-r^{2})^{-p/2}\,,$ (46)

so the contour must be closed to the right on the first addend in
$\Psi_{\widehat{\Delta}}$, and to the left on the second - see eq. (33). In
order for the result to agree with the usual conformal block decomposition,
$b({\widehat{\Delta}},s)$ must have single poles in correspondence of the
spectrum, and the residue must coincide, up to a sign, with the OPE
coefficient:101010For defect operators of dimension less than $p/2$ we must
deform the contour such that it picks up the pole on the left and does not
pick up the reflection according to (45) on the right. Similarly if the
operator has dimension exactly $p/2$ we must take the principle-value of the
integral to pick up half of the residue.

$g(r,\eta)=\sum\limits_{s=0}^{\infty}\widehat{g}_{s}(\eta)\sum_{{\widehat{\Delta}}^{*}\in\textup{spectrum}}b_{s,{\widehat{\Delta}}^{*}}^{2}\widehat{f}_{{\widehat{\Delta}}^{*}}(r)\,,\qquad
b_{s,{\widehat{\Delta}}^{*}}^{2}=-\textup{Res}_{{\widehat{\Delta}}={\widehat{\Delta}}^{*}}b({\widehat{\Delta}},s)\,.$
(47)

Not all poles in (44) arise from poles of $b({\widehat{\Delta}},s)$, as the
defect blocks themselves have poles for special values of ${\widehat{\Delta}}$
and $s$. However, since the defect blocks
$\widehat{f}_{{\widehat{\Delta}}}(r)$ have poles for
${\widehat{\Delta}}=\tfrac{p}{2}-n$ Lauria:2017  they are always to the left
of $\tfrac{p}{2}$, and thus are not picked up when we close the contour to the
right. Similarly, for the second addend in $\Psi_{\widehat{\Delta}}$, we close
the contour to the left while $\widehat{f}_{p-{\widehat{\Delta}}}(r)$ only has
poles to the right of $\tfrac{p}{2}$.

Eq. (44) can be easily inverted using the orthogonality relations (39) and
(43), yielding the following Euclidean inversion formula111111We thank D.
Mazáč for collaboration in obtaining this formula.

$b({\widehat{\Delta}},s)=\frac{2}{N_{q,s}}\frac{K_{\widehat{\Delta}}}{K_{p-{\widehat{\Delta}}}}\;\int\limits_{-1}^{1}d\eta\int\limits_{0}^{1}dr\,\mu_{p}(r)\mu_{q}(\eta)\,\widehat{g}_{s}(\eta)\Psi_{\widehat{\Delta}}(r)g(r,\eta)\,.$
(48)

Since in Euclidean signature $\eta$ is nothing more than the cosine of an
angle (see (26)), we change variables in the above integral to obtain

$\begin{split}b({\widehat{\Delta}},s)&=\frac{1}{N_{q,s}}\frac{K_{\widehat{\Delta}}}{K_{p-{\widehat{\Delta}}}}\oint\limits_{|w|=1}\frac{dw}{\mathrm{i}w}\int\limits_{0}^{1}dr\,\mu(r,w)\,\widehat{g}_{s}\left(\frac{1}{2w}+\frac{w}{2}\right)\Psi_{\widehat{\Delta}}(r)\;g\left(r,\frac{1}{2w}+\frac{w}{2}\right)\,,\\\
\mu(r,w)&=\mu_{p}(r)\left|\frac{w}{2\mathrm{i}}-\frac{1}{2\mathrm{i}w}\right|^{q-2}\,.\end{split}$
(49)

The above integral might not converge. Let us first consider the region $r\sim
0$. Here, convergence is controlled by the scaling dimension
${\widehat{\Delta}}_{\textup{min}}$ of the lightest defect primary exchanged
in $g(r,\eta)$. Specifically, the integral converges in the strip

$\left|\Re{\widehat{\Delta}}-\frac{p}{2}\right|<{\widehat{\Delta}}_{\textup{min}}-\frac{p}{2}\leavevmode\nobreak\
.$ (50)

To obtain an analytic function in ${\widehat{\Delta}}$, it is therefore
necessary to subtract from $g(r,\eta)$ the blocks121212Actually, in order to
preserve the analytic structure of the two-point function at $r=1$ for generic
$\eta$, one should subtract the corresponding $\Psi$ rather than the block.
See appendix B.2 of Simmons-Duffin:2017nub  for more details. corresponding to
operators lighter than $p/2$, and invert the subtracted two-point function. If
necessary, the poles corresponding to these operators can be added by hand to
$b({\widehat{\Delta}},s)$. Once the strip (50) exists, the function
$b({\widehat{\Delta}},s)$ can be analytically continued past its edges as
follows. Suppose that we want to extend towards the right – going towards the
left is similar. For ${\widehat{\Delta}}>p/2$, the kernel of eq. (49) has an
expansion in growing powers of $r$ that starts with
$r^{-1-{\widehat{\Delta}}}$. Then we subtract the leading term $g(r,\eta)\sim
r^{{\widehat{\Delta}}_{\textup{min}}}$, and integrate it separately, defining
the result by analytic continuation in ${\widehat{\Delta}}$:

$\int_{0}^{1}\\!dr\,r^{-1-{\widehat{\Delta}}+{\widehat{\Delta}}_{\textup{min}}}=\frac{1}{{\widehat{\Delta}}_{\textup{min}}-{\widehat{\Delta}}}\leavevmode\nobreak\
.$ (51)

This defines the sought analytic continuation of $b({\widehat{\Delta}},s)$ up
to the next exchanged operator. Proceeding order by order in $r$, one
constructs the full function $b({\widehat{\Delta}},s)$.131313In fact, at
${\widehat{\Delta}}=p/2+n$, for integer $n>0$ the combination
$K_{\widehat{\Delta}}/K_{p-{\widehat{\Delta}}}\Psi_{\widehat{\Delta}}$, which
appears in the inversion formula (49), develops a $\log r$ at small $r$. This
is due to the cancellation of a pole in the difference between the block and
its shadow. This is harmless in general, but extra care is needed if an
operator with dimension $p/2+n$ exists in the spectrum.

Divergences at $r=1$, on the other hand, are controlled by the bulk channel
OPE – in particular, they are absent if $\Delta_{\phi}<p/2+1$. They may be
regulated by cutting off the integral at $r<1-\epsilon$. The kernel
$\mu(r,w)\Psi_{\widehat{\Delta}}(r)$ has a regular Taylor expansion close to
$r=1$, therefore the divergent part of the inversion formula as $\epsilon\to
0$ does not contain poles in ${\widehat{\Delta}}$. Hence, it can be safely
dropped without altering the spectrum and the residues of
$b({\widehat{\Delta}},s)$.

###  3.2 The Lorentzian formula

In the Euclidean inversion formula (49) the contour of integration in the
complex $w$ plane is the unit circle, as $w$ is a phase. We now want to deform
the contour in order to integrate over real values of $w$, which correspond to
a Lorentzian configuration. The range of $r$ in the Euclidean formula is
confined between $0$ and $1$.

$z=0$$\bar{z}=0$$z,\bar{z}=0$defect$w\to\infty$$w\to 0$$w=r$$w=\frac{1}{r}$

Figure 2: The positive real axis on the $w$ complex plane, at fixed $r<1$,
maps to the black solid line in the $(z,\bar{z})$ plane. The bulk OPE
singularities correspond to the intersection of the line with the past and
future lightcones of the operator $\phi(1,1)$.

All other points in the Euclidean plane are related to this fundamental region
by inversion. As it is clear from figure 2, at fixed $r<1$ the function
$g(r,w)$ has two copies of the bulk OPE singularity at $w=r$ and $w=1/r$.
Contrary to the case of the four-point function, there is no singularity at
negative values of $w$. Further singularities may lie in the limits $w=0$ and
$w=\infty$, which are double lightcone limits – see the explanation in
appendix B and fig. 6 there. The OPE singularities are generically branch
points, while we know that the correlator is single valued on the circle
$|w|=1$. Therefore the cuts run from 0 to $r$ and from $1/r$ to $\infty$. The
remaining $w$ dependence in the inversion formula comes from $\widehat{g}_{s}$
and the measure. It is useful to consider a larger set of solutions to the
Casimir equation associated to the orthogonal rotations. We make use of (40)
again to go back to a representation of $\widehat{g}_{s}(\eta)$ (41) as a
hypergeometric function

$\widehat{h}_{1}(s,w)=\binom{s+\frac{q}{2}-2}{\frac{q}{2}-2}^{-1}C_{s}^{q/2-1}\left(\frac{w}{2}+\frac{1}{2w}\right)=w^{-s}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,w^{2}\right)\,.$
(52)

Recall that when $q$ is even, an order of limits ambiguity arises in the
definition of the hypergeometric function. The equality (40) holds if we first
take $s$ to be integer, and then $q$ to be even. As before, this prescription
is assumed every time it is necessary.

Other solutions to the Casimir equation for the angular part of the defect
blocks can be obtained by combining the transformations $w\to 1/w$ – which
leaves $\widehat{h}_{1}$ invariant when $s$ is integer – and $s\to 2-q-s$,
both of which are symmetries of the Casimir equation. We will use the two
following solutions

$\displaystyle\widehat{h}_{2}(s,w)$
$\displaystyle\colonequals\widehat{h}_{1}(2-q-s,w)=w^{s+q-2}{}_{2}F_{1}\left(s+q-2,\frac{q}{2}-1,\frac{q}{2}+s,w^{2}\right)\,,$
(53) $\displaystyle\widehat{h}_{3}(s,w)$
$\displaystyle\colonequals\widehat{h}_{2}(s,1/w)=w^{2-q-s}{}_{2}F_{1}\left(s+q-2,\frac{q}{2}-1,\frac{q}{2}+s,\frac{1}{w^{2}}\right)\,,$
(54)

where $\widehat{h}_{2}$ is regular at the origin while $\widehat{h}_{3}$ is
regular at infinity. Ideally, one would like to express $\widehat{h}_{1}$ as a
linear combination of $\widehat{h}_{2}$ and $\widehat{h}_{3}$, but this is
globally possible only for defects of even codimension. Indeed, when $q$ is
even, the discontinuities of $\widehat{h}_{2}$ and $\widehat{h}_{3}$ vanish.
Let us first consider this simpler case.

#### Even $q$

In this case,

$\begin{split}\widehat{h}_{1}(s,w)=(-1)^{\frac{q}{2}-1}\frac{\Gamma(s+1)\Gamma\left(s+q-2\right)}{\Gamma\left(s+\frac{q}{2}-1\right)\Gamma\left(s+\frac{q}{2}\right)}\left(\widehat{h}_{2}(s,w)+\widehat{h}_{3}(s,w)\right)\,,\\\
s=0,1,2\ldots\,,\quad q=2,4,6,\ldots\,.\end{split}$ (55)

After plugging eq. (55) in the Euclidean inversion formula (49), we can deform
the contour towards the interior on $\widehat{h}_{2}$ and towards the exterior
on $\widehat{h}_{3}$ (see fig. 3).

$w$$1$$-1$$\tfrac{1}{r}$$r$$w$$1$$\tfrac{1}{r}$$r$$w$$1$$\tfrac{1}{r}$$C_{+}$$C_{-}$

Figure 3: Deformation of the $|w|=1$ contour of (49) used to define
$b_{0}({\widehat{\Delta}},s)$ and $b_{\infty}({\widehat{\Delta}},s)$ in (58).
For the case of odd codimension it was necessary to add zero in the form of
two contours $C_{+}$ and $C_{-}$ (60).

When deforming the contour towards the interior, divergences may arise when
shrinking the circles around $w=0$ and $w=r$. The former is not an OPE limit.
It lies at the boundary of the region of convergence of the bulk OPE, which is
not positive. In the case of the Caron-Huot formula Caron-Huot:2017vep ,
positivity was used to place a bound on the growth of the correlator in the
Regge limit, which has the same role there as the small $w$ limit here.
Deprived of this tool, we currently have no way to constrain the growth of the
correlator in general. If for small $w$ the two-point function is bounded by a
power, then for $s$ large enough the circle around $w=0$ can be shrunk.
Concretely,

$\mathrm{if}\;\;g(r,w)\lesssim w^{-s_{\star}}\,,\quad\text{as }w\to
0\qquad\text{then the formula is valid for}\;s>s_{\star}\,,$ (56)

since the integrand in (49) is then bounded by $w^{s-1-s_{\star}}$ for $w\to
0$. In all the perturbative examples of section 5, $s_{\star}=0$. We should
also keep in mind that the inversion formula only converges after the light
defect primaries have been subtracted – see the discussion below (50). A
defect block of transverse spin $s$ has $s_{\star}=s$: hence, unless special
cancellations happen, the presence of primaries with ${\widehat{\Delta}}<p/2$
and large $s$ worsens the convergence of the Lorentzian formula.

The point $w=r$ is a bulk OPE singularity, that includes power laws of the
kind $(r-w)^{-\Delta_{\phi}+\tau/2}$. The integral converges for negative
enough $\Delta_{\phi}$, and can then be analytically continued. The procedure
is allowed because the angular integral in the original Euclidean formula is
convergent for all values of $\Delta_{\phi}$. Then we find that the OPE
coefficient can be expressed in terms of the discontinuity of the two-point
function across the branch cut running from $w=0$ to $w=r$:

$\textup{Disc}\,g(r,w)=g(r,w+\mathrm{i}0)-g(r,w-\mathrm{i}0)\,.$ (57)

Note that for even $q$ the measure does not have any branch cuts. Proceeding
in the same way for $\widehat{h}_{3}$, now deforming the contour towards the
exterior, we obtain the following Lorentzian formula:

$b({\widehat{\Delta}},s)=\frac{1}{2}\frac{K_{\widehat{\Delta}}}{K_{p-{\widehat{\Delta}}}}(b_{0}({\widehat{\Delta}},s)+b_{\infty}({\widehat{\Delta}},s))\,,\\\
\begin{aligned}
b_{0}({\widehat{\Delta}},s)&=-\int\limits_{0}^{1}dr\int\limits_{0}^{r}\frac{dw}{\mathrm{i}\pi
w}w^{2-q}(1-w^{2})^{q-2}(1-r^{2})^{p}r^{-p-1}\widehat{h}_{2}(s,w)\Psi_{\widehat{\Delta}}(r)\textup{Disc}\,g(r,w)\,,\\\
b_{\infty}({\widehat{\Delta}},s)&=\int\limits_{0}^{1}dr\int\limits_{1/r}^{\infty}\frac{dw}{\mathrm{i}\pi
w}w^{2-q}(w^{2}-1)^{q-2}(1-r^{2})^{p}r^{-p-1}\widehat{h}_{3}(s,w)\Psi_{\widehat{\Delta}}(r)\textup{Disc}\,g(r,w)\,.\end{aligned}$
(58)

Note that in any theory with $q>2$ the two-point function obeys
$g(r,w)=g(r,\tfrac{1}{w})$, and thus it follows that
$\textup{Disc}\,g(r,w)=-\textup{Disc}\,g(r,1/w)$. In the case $q=2$ the
symmetry of the two-point function is only present in a parity invariant
theory, which is assumed here since we have taken the blocks in eq. (41) to be
symmetric under $w\to\tfrac{1}{w}$. Therefore,

$b_{0}({\widehat{\Delta}},s)=b_{\infty}({\widehat{\Delta}},s)\leavevmode\nobreak\
.$ (59)

#### Odd $q$

When the codimension is odd, the relation (55) cannot be globally valid,
because of the cuts in $\widehat{h}_{2}$ and $\widehat{h}_{3}$. However, we
only need a relation which is valid upon integration. In other words, we can
integrate zero in the form of a combination of $\widehat{h}_{2}$ and
$\widehat{h}_{3}$ along the contours $C_{+}$ and $C_{-}$ in fig. 3. In fact,
we only integrate $\widehat{h}_{3}$ which does not grow at infinity

$b({\widehat{\Delta}},s)=b({\widehat{\Delta}},s)+c_{+}\oint_{C_{+}}dw\,[\dots]\widehat{h}_{3}(s,w)+c_{-}\oint_{C_{-}}dw\,[\dots]\widehat{h}_{3}(s,w)\,.$
(60)

where the dots stand for everything in eq. (49) except
$\widehat{g}_{s}(\eta)$. We need linear combinations of the conformal block
and the $\widehat{h}$’s which does not blow up in $w=0$. This fixes

$c_{\pm}=-e^{\pm\mathrm{i}\frac{\pi}{2}q}\frac{\Gamma(s+1)\Gamma\left(s+q-2\right)}{\Gamma\left(s+\frac{q}{2}-1\right)\Gamma\left(s+\frac{q}{2}\right)}\,.$
(61)

Notice that the combination is different in the upper and lower plane, due to
the cut in $\widehat{h}_{3}$, which extends from $w=-1$ to $w=1$. The
structure of cuts in the integrand is complicated by the contribution of
$\mu(r,w)$ – see eq. (49). As an analytic function of $w$, $\mu(r,w)$ for odd
$q$ has a cut running over the whole real axis. Indeed, let us start from $w$
purely imaginary. In this case

$\displaystyle\left|\frac{w}{2\mathrm{i}}-\frac{1}{2\mathrm{i}w}\right|^{q}$
$\displaystyle=\frac{e^{-\mathrm{i}\frac{\pi}{2}q}}{2^{q}}\left(\frac{(w+1)(w-1)}{w}\right)^{q},\qquad-\mathrm{i}w>0\,,$
(62)
$\displaystyle\left|\frac{w}{2\mathrm{i}}-\frac{1}{2\mathrm{i}w}\right|^{q}$
$\displaystyle=(-1)^{q}\frac{e^{-\mathrm{i}\frac{\pi}{2}q}}{2^{q}}\left(\frac{(w+1)(w-1)}{w}\right)^{q},\qquad-\mathrm{i}w<0\,.$
(63)

So the discontinuity of the measure on the real axis is

$\textup{Disc}\,\mu(u,w)=\mu(u,w+\mathrm{i}0)-\mu(u,w-\mathrm{i}0)=\left(1-(-1)^{q}\right)\mu(u,w+\mathrm{i}0),\qquad
w\in\mathbb{R}\,.$ (64)

We can now deform the contour of integration in $b({\widehat{\Delta}},s)$ to
the real axis, and drop the arcs at infinity in $C_{\pm}$. We first notice
that the discontinuity in $\mu(r,w)$ offsets the difference between $c_{+}$
and $c_{-}$. Let us consider first the region $\lvert w\rvert>1$. There the
only further discontinuity comes from the correlator $g(r,w)$. In the
complementary region $\lvert w\rvert<1$, the integral in
$b({\widehat{\Delta}},s)$ on the r.h.s. of eq. (60) must be taken into
account. This combines with the contributions from $C_{+}$ and $C_{-}$ by
virtue of the following relations:

$\displaystyle\widehat{h}_{1}(s,w)-c_{+}\widehat{h}_{3}(s,w)$
$\displaystyle=-e^{-\mathrm{i}\frac{\pi}{2}q}\frac{\Gamma(s+1)\Gamma\left(s+q-2\right)}{\Gamma\left(s+\frac{q}{2}-1\right)\Gamma\left(s+\frac{q}{2}\right)}\widehat{h}_{2}(s,w)\,,\qquad{\mathcal{I}}w>0\,,\
\lvert\Re w\rvert<1\,,$ (65)
$\displaystyle\widehat{h}_{1}(s,w)-c_{-}\widehat{h}_{3}(s,w)$
$\displaystyle=-e^{\mathrm{i}\frac{\pi}{2}q}\frac{\Gamma(s+1)\Gamma\left(s+q-2\right)}{\Gamma\left(s+\frac{q}{2}-1\right)\Gamma\left(s+\frac{q}{2}\right)}\widehat{h}_{2}(s,w)\,,\qquad{\mathcal{I}}w<0\,,\
\lvert\Re w\rvert<1\,.$ (66)

Putting all together, we recover a result identical to the case of even
codimension, namely eq. (58).

#### Lorentzian inversion formula for codimension $q$

All in all, we obtain the following Lorentzian formula, valid for both even
and odd $q$

$b({\widehat{\Delta}},s)=-\frac{K_{\widehat{\Delta}}}{K_{p-{\widehat{\Delta}}}}\int\limits_{0}^{1}dr\int\limits_{0}^{r}\frac{dw}{\mathrm{i}\pi
w}w^{2-q}(1-w^{2})^{q-2}(1-r^{2})^{p}r^{-p-1}\widehat{h}_{2}(s,w)\Psi_{\widehat{\Delta}}(r)\textup{Disc}g(r,w)\,.$
(67)

Finally, we can change coordinates to $z$ and $\bar{z}$. If we are only
interested in the poles of $b({\widehat{\Delta}},s)$ in ${\widehat{\Delta}}$
corresponding to the exchanged operators, and not their shadows, we need only
keep the second addend in $\Psi_{\widehat{\Delta}}(r)$ – see eq. (33). We find

$\begin{split}\left.b({\widehat{\Delta}},s)\right|_{\textup{poles}}=\int\limits_{0}^{1}\frac{dz}{2z}z^{-\frac{\widehat{\tau}}{2}}\int\limits_{1}^{\frac{1}{z}}\\!\frac{d\bar{z}}{2\pi\mathrm{i}}\,&(1-z\bar{z})(\bar{z}-z)\bar{z}^{-\frac{{\widehat{\Delta}}+s}{2}-2}\;{}_{2}F_{1}\left(s+1,2-\frac{q}{2},\frac{q}{2}+s,\frac{z}{\bar{z}}\right)\\\
&\times{}_{2}F_{1}\left(1-{\widehat{\Delta}},1-\frac{p}{2},1+\frac{p}{2}-{\widehat{\Delta}},z\bar{z}\right)\textup{Disc}\,g(z,\bar{z})\,,\end{split}$
(68)

where we have used hypergeometric identities to simplify the equation. The cut
between $w=0$ and $w=r$ has been mapped to the line $\bar{z}\in[1,1/z]$, and
can be computed by going around the branch point at $\bar{z}=1$. Notice that,
due to the inverse proportionality relation between $w$ and $\bar{z}$ in eq.
(26), $\textup{Disc}\,g(r,w)=-\textup{Disc}\,g(z,\bar{z})$. Eqs. (67) and (68)
are analytic in $s$. However, we stress again that their validity cannot be
established without knowledge of the behavior of $g(r,w)$ for $w\to 0$, or
equivalently $w\to\infty$. If $s_{\star}<\infty$, $s_{\star}$ being defined in
eq. (56), the function $b({\widehat{\Delta}},s)$ defined by eq. (67) is
identical to the function obtained via the Euclidean inversion formula eq.
(49) for all integer values of $s>s_{\star}$. But now, analyticity in $s$
implies that the defect operators organize in analytic trajectories for
$s>s_{\star}$.

Let us also note that, similarly to the formula obtained in Caron-Huot:2017vep
, the discontinuity in eq. (68) vanishes for a single defect block, and thus
its validity cannot be verified term by term in a defect block decomposition.
This is to be contrasted with the Euclidean formula (49), where the poles
precisely arise order by order in the defect OPE expansion of the correlator,
as we discussed around eq. (51).

Poles of $b(s,{\widehat{\Delta}})$ in $\widehat{\tau}$ arise from the lower
bound of integration in $z$, and we can study eq. (68) in an expansion for
small $z$,

$\displaystyle b({\widehat{\Delta}},s)\big{|}_{\mathrm{poles}}$
$\displaystyle=\int\limits_{0}^{1}\frac{dz}{2z}z^{-\frac{\widehat{\tau}}{2}}\sum\limits_{m=0}z^{m}\sum\limits_{k=-m}^{m}c_{m,k}({\widehat{\Delta}},s)B(z,\beta+2k)\,,\quad
B(z,\beta)$
$\displaystyle\colonequals\int\limits_{1}^{\infty}\frac{d\bar{z}}{2\pi\mathrm{i}}\bar{z}^{-\frac{\beta}{2}-1}\textup{Disc}\,g(z,\bar{z})\,,$
(69)

where $\beta={\widehat{\Delta}}+s$, and where $c_{m,k}({\widehat{\Delta}},s)$
are trivially obtained from the $z$ expansion of the integrand in (68), with
$c_{0,0}({\widehat{\Delta}},s)=1$. Note that in eq. (69) we pushed the upper
bound of the $\bar{z}$ integration to infinity, which will not modify the
poles of $b(s,{\widehat{\Delta}})$ in ${\widehat{\Delta}}$, _provided_
$g(z,\bar{z})$ behaves as (56). This follows from the behavior of (69) for
small $z$ and with $\bar{z}\sim\tfrac{1}{z}$.141414The upper bound of the
$\bar{z}$ integration can only produce poles in $s$, and provided
$g(z,\bar{z})$ grows as given in eq. (56) for $w\to 0$, then these poles will
appear only for $s\leqslant s_{\star}$, that is for $s$ outside the range of
applicability of the formula.

In a series expansion for small $z$, the functions $B(\beta,z)$ will give the
following contributions to (69):

$\sum\limits_{m=0}z^{m}\sum\limits_{k=-m}^{m}c_{m,k}({\widehat{\Delta}},s)B(z,\beta+2k)=\sum\limits_{m}b_{m}({\widehat{\Delta}},s)\;z^{\tfrac{\widehat{\tau}_{m}(\beta,s)}{2}}\,,$
(70)

with each term producing a pole for
$\widehat{\tau}=\widehat{\tau}_{m}(\beta,s)$ in $b({\widehat{\Delta}},s)$,
signaling a defect operator with that transverse twist. The OPE coefficients
are obtained from the ${\widehat{\Delta}}$-residue of
$b(s,{\widehat{\Delta}})$, at fixed $s$, according to (47), and so they are
obtained from the coefficients in (70) after correcting by a Jacobian factor
as

$b^{2}_{s,{\widehat{\Delta}}}=\left(1-\frac{d\widehat{\tau}_{m}(\beta,s)}{d\beta}\right)^{-1}b_{m}({\widehat{\Delta}},s)\Big{|}_{\beta=\widehat{\tau}_{m}(\beta,s)+2s}\,.$
(71)

###  3.3 Contributions from a single bulk block

Of course in general one does not have access to the full two-point function.
Representing the two-point function by its bulk OPE, we now discuss what can
be inferred from knowledge of individual exchanged bulk blocks, making contact
with the considerations of section 2.

As discussed in section 2, knowledge of the low twist operators appearing in
the bulk OPE translates into statements about the large transverse spin defect
spectrum. The analysis of section 2 is not free from assumptions, similarly to
the usual lightcone story applied to the four-point function of local
operators. In the latter case, only recently have some of the assumptions
started to be put on a firmer footing Qiao:2017xif . We can now recover the
results of section 2 making use of the inversion formula. _Assuming_ that the
correlation function behaves as in eq. (56), the inversion formula shows that
operators organize in analytic families for $s>s_{\star}$. Now we can prove
that these trajectories have accumulation points for $s\to\infty$ at
$\widehat{\tau}\to\Delta_{\phi}+2m$. Furthermore, unlike in section 2, where
we obtained the contribution of an exchanged bulk block to the defect spectrum
in a $1/s$ expansion, the results obtained through (67) amount to the full
contribution of the block at finite $s>s_{\star}$. These results therefore
resum the $1/s$ expansion that could be obtained from carrying out the
procedure of section 2 to all orders in $1/s$, as done in Alday:2015ewa ;
Alday:2015ota ; Alday:2016jfr ; Simmons-Duffin:2016wlq  for the case without
defects.

Let us see how the transverse derivatives come about in this context. For
large transverse spin $s$, we see from (69) that the integral is dominated by
$\bar{z}\to 1$. From the behavior of the bulk blocks in this limit (6), we
find that the leading contributions come from operators with lowest twist,
$\tau=\Delta-J$, which contribute as

$g(z,\bar{z})\sim(1-\bar{z})^{-\Delta_{\phi}+\frac{\tau}{2}}\left(\textup{a
function of }z\right)+\ldots\,,\qquad\text{for }\bar{z}\to 1\,.$ (72)

The leading contribution is always the identity. As it is expected and we
confirm below, the inversion of the identity yields the spectrum of the
trivial defect, _i.e._ , the transverse derivatives and their OPE
coefficients.

Note that we can perform the $\bar{z}$ integral that defines $B(z,\beta)$ in
eq. (69), _at fixed $z$_, block by block. Indeed, the bulk channel OPE still
converges in the whole region $0<z<1/\bar{z}<1$. This is most easily seen in
the coordinates defined in appendix B, to which we refer. However, it will be
convenient to work in a small $z$ expansion, which is where the poles of
$b({\widehat{\Delta}},s)$ arise from, and so we now discuss the circumstances
under which this is allowed.

#### Small $z$ expansion

The small $z$ expansion does not commute with the infinite sum over bulk
blocks, as it is clear from the fact that while (69) should behave like (70),
all the blocks except the identity contain a single logarithm of $z$ as $z\to
0$, see eq. (20).151515One might complain that using eq. (20) of the block is
not allowed here because the lightcone expansion of the block does not
converge in the whole range $1<\bar{z}<\infty$. One can instead use the
expansion presented in section 4.2.1 of Billo:2016cpy , where the expansion
parameter is $\frac{1-\bar{z}}{\bar{z}}$, which can be integrated in the
desired range. The result still contains a single $\log z$. This is exactly
the same problem discussed in Caron-Huot:2017vep  for the four-point function
without defects. There, in section 4.3.2, a way out was found – see also
Simmons-Duffin:2016wlq : after subtracting a known sum from the inversion
formula, one can commute the small $z$ limit with the block expansion. In this
section, we will content ourselves of computing the contributions of
individual bulk blocks after taking their small $z$ limit, without any
subtraction. We expect that the error we make becomes small when the defect
spectrum differs from the one of the trivial defect by small anomalous
dimensions: indeed, in this case also the r.h.s. of eq. (70) is well
approximated by an expansion up to a single $\log z$. This happens for
instance at large spin, where the analytic functions in $s$ that we find below
resum part of the lightcone expansion. In some specific situations, the result
is actually exact down to $s=s_{\star}$. These are the cases considered in
section 5: defects whose deviation from the trivial one is controlled by a
perturbative parameter. At leading order in this coupling, a single logarithm
of $z$ is all what there is on the r.h.s. of eq. (70). Furthermore, in these
examples a finite number of bulk channel blocks have a non-zero discontinuity,
therefore we are free to take small $z$ block by block.161616A similar
situation was also very recently discussed for the four-point function case in
Alday:2017zzv .

It should be borne in mind that, to go beyond these results, a procedure
similar to Caron-Huot:2017vep  is needed.

#### Identity exchange and the transverse derivative operators

As discussed above, the leading contributions to the large transverse spin
spectrum come from leading twist bulk operators, and thus the identity
operator, which has twist zero, dominates.171717All remaining operators are
constrained by unitarity bounds to have $\tau=\Delta-J>0$, provided $d>2$
which is assumed throughout this work since we consider $q>1$. If this is the
only exchanged bulk operator then from (4) we find $s_{\star}=-\Delta_{\phi}$,
and the inversion formula (67) is valid for all spins starting at $s=0$. This
happens for the trivial defect (8), and thus we recover the full spectrum.
However, if the identity is just part of a more complicated two-point function
the $w\to 0$ behavior, and thus $s_{\star}$ can be modified (see, _e.g._ , the
simple example in subsection 5.1).

Taking the leading small $z$ term in the identity contribution to
$g(z,\bar{z})$ we find

$B(z,\beta)=\int\limits_{1}^{\infty}d\bar{z}\bar{z}^{-\frac{\beta}{2}-1}\frac{1}{2\pi\mathrm{i}}\textup{Disc}\left(\frac{(1-\bar{z})}{\sqrt{z\bar{z}}}\right)^{-\Delta_{\phi}}=\frac{z^{\frac{\Delta\phi}{2}}\Gamma\left(\frac{{\widehat{\Delta}}+s+\Delta_{\phi}}{2}\right)}{\Gamma\left(\frac{{\widehat{\Delta}}+s-\Delta_{\phi}}{2}+1\right)\Gamma\left(\Delta_{\phi}\right)}\,,$
(73)

where we see that in (69) this produces a pole in
$\widehat{\tau}=\Delta_{\phi}$, corresponding to the leading twist defect
primary operator. The residue of $B(z,\beta)$ matches precisely with the OPE
coefficient of the trivial defect (10), for $m=0$. Subleading powers in the
small $z$ expansion of the identity block and of (68) produce poles
corresponding to the rest of the trivial spectrum, (9) with $m>0$.

Since the bulk identity exchange corresponds to the leading contribution to
the spectrum at large $s$, we thus recover the existence of transverse
derivative operators with $\widehat{\tau}\to\Delta_{\phi}+2m$ as $s\to\infty$.
A main difference with respect to section 2 is that now we obtain the full OPE
coefficient (10), instead of an asymptotic series in $1/s$.

Note that the integral in (73) naively diverges for large $\Delta_{\phi}$, but
the result can be defined by analytic continuation and is finite, similarly to
what was observed in Caron-Huot:2017vep .181818The result of the Euclidean
inversion formula gives a finite answer that is analytic in $\Delta_{\phi}$.
For $\Delta_{\phi}<1$ eq. (73) converges and thus it matches the result of the
Euclidean inversion. The integral in (73) can then be analytically continued
from there to $\Delta_{\phi}\geqslant 1$. This will also happen for the
exchange of low dimensional bulk primaries, as the behavior for $\bar{z}\to 1$
of $g(z,\bar{z})$ is controlled by the bulk channel OPE – see (72). For low
dimensional bulk blocks then the result should also be obtained by analytic
continuation.

Finally we note that $\mathrm{i}\,\textup{Disc}g(z,\bar{z})$ in (68) does not
have a definite sign, in contrast to the case of the double-discontinuity in
Caron-Huot:2017vep . This is clear from the identity contribution in (73)
where

$\left(2\mathrm{i}\right)^{-1}\textup{Disc}\left((1-\bar{z})^{-\Delta_{\phi}}\right)=(\bar{z}-1)^{-\Delta_{\phi}}\sin(\pi\Delta_{\phi})\,.$
(74)

Even though positivity of the defect OPE coefficients requires the residues of
$b({\widehat{\Delta}},s)$ in (68) to have definite sign, as is the case for
(73) above, this does not follow form the sign of the discontinuity.

#### Leading bulk twist contribution

The defect operator dimensions and OPE coefficients obtained from the
inversion of the identity block will then be corrected for finite spin by the
presence of all the remaining bulk blocks. We define the anomalous dimension
of the transverse derivative operators whose dimensions approach
$\Delta_{\phi}+2m$ as

$\gamma_{s,m}\colonequals\widehat{\tau}_{m}-(\Delta_{\phi}+2m)\,.$ (75)

As discussed above, if the $\gamma_{s,m}$ are small then we can consider the
the small $z$ limit of the bulk block decomposition.

We can draw from (72) a first general observation: if the exchanged operator
has twist $\tau=2\Delta_{\phi}+2n$, with $n\geqslant 0$ an integer, the
contribution of the relative block to the discontinuity of $g(z,\bar{z})$
vanishes. In other words, exact double twists of the external operator have
zero discontinuity and do not contribute under the inversion formula. Note
that while the discontinuity naively vanishes also for negative integer $n$,
the integral is divergent for $\bar{z}\to 1$ in this case. One must then first
compute the discontinuity for arbitrary $n$ and perform the integration. In
the end, when $n$ is taken to be a negative integer, the zero of the
discontinuity cancels the divergence in the integral, and the final result of
the inversion formula is finite. This is in precise agreement with the results
of section 2.1: the bulk blocks with non vanishing discontinuity either give
singular contributions to $g(z,\bar{z})$ as $\bar{z}\to 1$, or contributions
that can be made singular by acting with the Casimir.191919The same behavior
is observed for the inversion formula of the four-point function with no
defects as pointed out in Caron-Huot:2017vep  – see also Alday:2017vkk .

Then, the contribution of the identity and a bulk primary $O$ of twist $\tau$
and spin $J$ to (70) has the following form in a small $z$ expansion:

$\sum\limits_{m=0}I_{m}({\widehat{\Delta}},s)z^{\frac{\Delta_{\phi}}{2}+m}+c_{\phi\phi
O}a_{O}\sum\limits_{m=0}\left(C_{1}^{m}({\widehat{\Delta}},s)+C_{2}^{m}({\widehat{\Delta}},s)\log{z}\right)z^{\frac{\Delta_{\phi}}{2}+m}\,.$
(76)

The leading contributions to the anomalous dimension, and the correction to
the OPE coefficients (10) are obtained from the above as202020In case $I_{m}$
is zero the denominator should be the first non-zero order, this happens for
instance if the external operator $\phi$ is perturbatively close to the
unitarity bound, since in this case $b_{s,m}^{2}=0$ for $m\neq 0$ and
$\Delta_{\phi}=\tfrac{d}{2}-1$.

$\gamma_{s,m}=\frac{2c_{\phi\phi
O}a_{O}C_{2}^{m}}{I_{m}}\Bigg{|}_{{\widehat{\Delta}}=\Delta_{\phi}+s+2m}\,,\quad\delta
b^{2}_{s,m}=\frac{d\gamma_{s,m}}{d{\widehat{\Delta}}}I_{m}+\gamma_{s,m}\frac{dI_{m}}{d{\widehat{\Delta}}}+c_{\phi\phi
O}a_{O}C_{1}^{m}\Big{|}_{{\widehat{\Delta}}=\Delta_{\phi}+s+2m}\,.$ (77)

Applying these results to the bulk collinear block given in eq. (6) and
expanding the answer for large $s$, we have recovered the results obtained
with the lightcone approach of section 2, for different values of $m$ and to
second order in $\tfrac{1}{s}$. To this order, the Jacobian factor
contribution in (77) is crucial.

This proves the existence of the individual transverse derivative operators,
instead of the averaged statement obtained with the lightcone
analysis.212121To be precise, for a given finite spin, it may happen that the
contributions from the various bulk primaries to the residue of a certain pole
sum up to zero. However, this cannot happen for sufficiently large spin, where
the corrections from different exchanged operators are of different size. In
this sense our results, similarly to those of Caron-Huot:2017vep , establish
the existence of each individual transverse derivative operator for sufficient
large $s$.

While the methods of section 2 only provide an asymptotic series in
$\tfrac{1}{s}$, the inversion formula yields the contribution of a given bulk
block to the anomalous dimension and OPE coefficient of a defect operator of
any transverse spin $s>s_{\star}$. As an example, we compute the full
correction arising from the exchange of a bulk scalar, and from the bulk
stress tensor, to the leading twist defect operator $\gamma_{s,0}$.

#### Scalar operator exchange

Let us first obtain how a scalar operator $O$ of dimension $\Delta$
contributes to $\gamma_{s,0}$. Apart from the scalar operator $O$, here we
only take into account the contribution of the identity. While the bulk blocks
are not known in closed form, we can make use of the representation of the
scalar block as an infinite sum of hypergeometric functions as given in
appendix B of Billo:2016cpy .222222Alternatively we could have used the
recursion relation for the bulk blocks obtained in Billo:2016cpy . Taking the
leading $z\to 0$ term of the block we apply (69) term by term in the block
representation as an infinite sum. This amounts to a representation of the
block as an infinite sum in powers of $\tfrac{\bar{z}-1}{\bar{z}}$ that
converges for all of $\bar{z}$ in the integration region of (69). We then
commute the integral over $\bar{z}$ with the infinite sum, and are able to
resum the result to find

$\gamma_{s,0}\big{|}_{\Delta,J=0}=-c_{\phi\phi
O}a_{O}\frac{2^{\Delta}\Gamma\left(\frac{\Delta+1}{2}\right)\Gamma(\Delta_{\phi})\Gamma(s+1)\,_{3}F_{2}\left(\frac{\Delta-q+2}{2},\frac{\Delta}{2},\frac{\Delta-2\Delta_{\phi}+2}{2};\frac{\Delta}{2}+s+1,\Delta-\frac{d-2}{2};1\right)}{\sqrt{\pi}\Gamma\left(\frac{\Delta}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{\Delta}{2}\right)\Gamma\left(\frac{\Delta}{2}+s+1\right)}\,.$
(78)

We can proceed similarly for subleading transverse twists by keeping more
terms in the small $z$ expansion, but since the resulting anomalous dimension
have longer expressions we do not display them here. Note that by taking the
$z\to 0$ limit of the scalar block we are assuming small anomalous dimensions,
and the result we present here should be seen as the leading contribution in
the small parameter that controls the anomalous dimension.

#### Stress tensor exchange

We now turn to the stress tensor exchange, whose blocks are easily computed by
solving the recursion relation obtained in Billo:2016cpy  for an operator with
$J=2$ and $\Delta=d$. In this case it is easy to obtain a closed form answer
for the $z\to 0$ limit of the block, and again we compute its contribution to
$b(s,\Delta)$ via (69). Again considering only the stress tensor and the
identity, we obtain the following anomalous dimension:

$\gamma_{s,0}\big{|}_{\Delta=d,J=2}=-c_{\phi\phi
T}a_{T}\frac{2^{d}\Gamma\left(\frac{d+3}{2}\right)\Gamma(\Delta_{\phi})\Gamma(s+1)\Gamma\left(\Delta_{\phi}+s-\frac{p}{2}\right)}{\sqrt{\pi}\Gamma\left(1+\frac{d}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{d-2}{2}\right)\Gamma\left(\frac{q}{2}+s\right)\Gamma(\Delta_{\phi}+s)}\,.$
(79)

In section 5 we will only need the contributions (78) and (79), but of course
one can repeat this procedure for subleading twists. Similarly the corrections
to the OPE coefficients of the trivial defect (10) arising from either of
these two exchanges can be computed. Since their expressions are not
particularly illuminating, and in the case of the scalar exchange the infinite
sums were not performed to get a closed form result, we do not present them
here. Note that in computing the OPE coefficient one must include the Jacobian
factor of eq. (71).

#### Behavior of a single bulk block as $w\to 0$

While we cannot bound the growth of $g(z,\bar{z})$ as $w\to 0$, we can check
the behavior of a single bulk block. This is trivial for the cases in which
the blocks are known in closed form, and one finds that for codimension two
and $d=4,6$, the behavior is power-law. _Assuming_ that the behavior of the
bulk block is power-law for all values of $p$ and $q$ we can solve both the
quadratic and quartic Casimir equations in the small $w$ limit to
find232323There is another nontrivial solution allowed by the Casimir that
does not match the behaviors obtained in codimension two.

$f_{\Delta,J}\sim w^{-\frac{p}{2}}f(r)\,,\qquad\text{as}\qquad w\to 0\,,$ (80)

where $f(r)$ is a function of $r$ that is fixed up to two constants by the
Casimir equations. Putting in the behavior of the prefactor in (4) we find
that for a single bulk block $s_{\star}=\tfrac{p}{2}-\Delta_{\phi}$. Indeed,
the anomalous dimensions in eqs. (78) and (79) become singular precisely at
$s=s_{\star}$. Of course, the small $w$ behavior can be different from eq.
(80), for a theory where an infinite number of bulk blocks is exchanged.

##  4 Defect operators and motions in AdS

[linewidth=0.04,linecolor=color830b,dimen=outer,fillstyle=solid,fillcolor=color830b](10.267479,1.4567261)(0.22,0.2)$0$$\pi$$\tau$$\psi=\pi$$\psi=0$$\varphi$$\varphi$$\theta=\frac{\pi}{2}$$\theta=0$$\theta=\frac{\pi}{2}$$\theta=\frac{\pi}{2}$$\theta=0$

Figure 4: A three dimensional version of the configuration discussed in the
text. In red, the line defect marks two lines along $\tau$ at
$\theta=\frac{\pi}{2}$ and $\psi=0,\ \pi$. The particle spins fast along the
$\varphi$ direction.

In sections 2 and 3 we discovered that transverse derivative operators are a
necessary ingredient to ensure crossing symmetry of a correlator. In this
section we give them a more physical characterization, which exploits the
presence of a semi-classical limit of the states in radial quantization at
large $s$ Alday:2007mf . One of the main advantages of a picture directly
based on the Hilbert space is that it provides intuition on the presence of
other accumulation points in the spectrum. In particular, this analysis will
allow us to identify another class of defect operators, which are present on
the spectrum of Wilson lines in large $N$ gauge theories, similar in nature to
the single-trace operators of the ambient CFT.

Historically, the existence of double-twist primaries in the spectrum of an
arbitrary CFT was first established precisely by looking at semi-classical
states on a sphere. We give a brief review of the argument in Alday:2007mf ,
before explaining its consequences for a defect CFT. Consider first the state
created by a descendant of a low-lying scalar primary, schematically
$\partial^{\ell}\phi$. At large spin, the state becomes classical, and can be
approximated by a single particle which rotates close to the speed of light on
the equator. Following Alday:2007mf , we consider the four-dimensional case –
the generalization being straightforward. We parametrize the conformal time
with $\tau$ and the 3-sphere with Hopf angles $(\theta,\varphi,\psi)$, with
$\theta\in[0,\pi/2]$ and $\varphi$ and $\psi$ describing circles:

$ds^{2}=-d\tau^{2}+d\theta^{2}+\cos^{2}\theta\,d\varphi^{2}+\sin^{2}\theta\,d\psi^{2}\leavevmode\nobreak\
.$ (81)

The trajectory of the particle can be chosen to be $\tau=\varphi$, with
$\theta=0$. For reasons to become clear shortly, it is convenient to apply a
Weyl transformation to the cylinder, and turn it into the manifold
$AdS_{3}\times S^{1}$. The authors of Alday:2007mf  chose the following
parametrization in terms of coordinates $(u,\chi,\sigma,\psi)$:

$ds^{2}=-du^{2}+d\chi^{2}-2\sinh 2\sigma\,dud\chi+d\sigma^{2}+d\psi^{2},$ (82)

where

$\displaystyle\sinh\rho$
$\displaystyle=\frac{1}{\tan\theta}\leavevmode\nobreak\ ,$ (83a)
$\displaystyle\sinh 2\sigma$ $\displaystyle=-\sin(\tau-\varphi)\sinh
2\rho\leavevmode\nobreak\ ,$ (83b) $\displaystyle e^{4\mathrm{i}u}$
$\displaystyle=e^{2\mathrm{i}(\tau+\varphi)}\frac{\cos(\tau-\varphi)+\mathrm{i}\cosh
2\rho\sin(\tau-\varphi)}{\cos(\tau-\varphi)-\mathrm{i}\cosh
2\rho\sin(\tau-\varphi)}\leavevmode\nobreak\ ,$ (83c) $\displaystyle\sinh
2\chi$ $\displaystyle=\frac{\cos(\tau-\varphi)\sinh
2\rho}{\sqrt{1+\sin^{2}(\tau-\varphi)\sinh^{2}2\rho}}\,.$ (83d)

This coordinate system has three key features. Firstly, the Hamiltonian equals
the twist, _i.e._
$\mathrm{i}\partial_{u}=\mathrm{i}\partial_{\tau}-(-\mathrm{i}\partial_{\varphi})$.
Secondly, the fast particle on the sphere now sits at $\chi=+\infty$ and
$\sigma=0$. Finally, the killing vector $S=-\mathrm{i}\partial_{\varphi}\sim
e^{2|\chi|}$ for large $\chi$ – the explicit expression can be found in
Alday:2007mf . This construction does not tell us anything new about the
single particle state: of course, the energy of the particle at rest in this
two-dimensional quantum system is finite, which we knew from the start, since
$\Delta-S=\Delta_{\phi}$ for this state. However, consider now the state on
the sphere in which two fast rotating particles are present, the second
trajectory being $\tau=\varphi+\pi$. This state certainly exists, but now its
scaling dimension is not obvious, since the theory is in general strongly
coupled. However, in $AdS$ the particles sit infinitely far apart at
$\chi=\pm\infty$, and the system is gapped due to the unitarity bound, so that
no interaction is left in the infinite spin limit. The authors of Alday:2007mf
concluded that the the twist of such state is
$\tau_{\textup{DT}}=\tau_{1}+\tau_{2}$, where $\tau_{i}$ is the individual
twist of the operators responsible for the single particle states. This
establishes the existence of double-twist operators.

Let us now consider a defect CFT. Since we aim at studying the scaling
dimension of defect operators, we center the radial quantization in a point on
the defect. On the cylinder, then, the defect extends along the time
direction. We still consider a defect of codimension at least two, so we can
localize it at $\theta=\pi/2$. Depending on the dimension, it will cross the
$\psi$ circle in two points, or it will fill it. In this setup, we reconsider
the state of a particle rotating fast along the $\varphi$ circle. This is now
a state with large transverse spin $s$. When we move on to $AdS_{3}\times
S^{1}$, we find the defect at $\chi=\sigma=0$. The same argument as before
allows us to conclude that, in the limit $s\to\infty$, the transverse twist of
this state equals the twist of the bulk single particle state:

$\widehat{\tau}_{\textup{TD}}=\tau_{\phi},\qquad s=\infty\,.$ (84)

We rediscover in this way the transverse derivative operators. This picture
allows us to discuss the finite spin corrections as well, again following
Alday:2007mf . When the transverse spin is large but not infinite, we can use
the mentioned behavior of the spin generator $S\sim e^{2|\chi|}$ to argue that
the wave function of the particle is peaked at a distance from the defect of
order $\frac{1}{2}\log s$. The particle is still localized also in $\sigma$,
because the warp factor in the metric (82) favors the position $\sigma=0$.
Since the system is gapped, the leading correction to the twist is a Yukawa
potential, due to the exchange between the defect and the light particle of
the leading twist state. We find therefore the correction

$\widehat{\tau}_{\textup{TD}}=\tau_{\phi}+\textup{const}\,e^{-\tau_{\textup{min}}\Delta\chi}=\tau_{\phi}+\textup{const}\,s^{-\tau_{\textup{min}}/2}\,.$
(85)

Of course, we also know the precise coefficient from eq. (23). Recall that, in
the case of the double-twist operators, the exponent of the correction is
twice as large in absolute value: in AdS, this is simply a consequence of the
different distance along $\chi$ of the interacting particles in the two cases.

In a gauge theory, one can also consider states in which the rotating
particles are charged, and the authors of Alday:2007mf  consider this
situation as well. In the case of a two-particle state, color flux extends
between the two particles. The warp factor confines the flux close to
$\sigma=0$, so that the twist of the state now equals the mass of a meson
whose constituents are two light quarks. The flux-tube contributes to the mass
with a constant energy per unit length. In the case of adjoint quarks, in
$\mathcal{N}=4$ SYM the energy density is traditionally denoted $f(\lambda)$,
and we keep this notation, even if the argument is valid in general. Since
$\Delta\chi\simeq\log\ell$, the twist of the state is dominated by this
contribution at large spin. Single trace operators precisely exhibit this
behavior:

$\tau_{\textup{S.T.}}\simeq f(\lambda)\log\ell\,.$ (86)

In advocating this picture, we disregarded the possibility that the flux tube
breaks: string-breaking effects are suppressed at large $N$, or in
perturbation theory even at finite $N$, and we restrict our considerations to
those cases.

If we now consider the defect spectrum on a Wilson line, a very similar class
of states emerges. At large $N$, these are created by the insertion of an
adjoint operator $D^{s}\phi$ in the trace, and correspond to a color particle
rotating around the defect. In AdS, we now see that this state is a meson
whose constituents are a heavy quark sitting at the origin in the
$(\chi,\sigma)$ plane, and a light quark placed far away along the $\chi$
direction, as before. Again, the distance between the two sources is half of
what it was in the case of a single trace operator. We predict therefore the
existence of defect operators of transverse twist

$\widehat{\tau}\simeq f(\lambda)\Delta\chi=\frac{f(\lambda)}{2}\log s\,,$ (87)

where $f(\lambda)$ is the same function that appears in the anomalous
dimension of single trace operators (86). It would be interesting to check
this prediction, perhaps along the lines of Giombi:2017cqn .

##  5 Examples

In this section we present a few illustrative examples. We start from a defect
in free theory, which provides a simple instance where the inversion formula
(68) does not converge down to zero transverse spin. We then point out a
consequence of our results for the spectrum of certain Wilson lines in
supersymmetric gauge theories. In subsection 5.3, we define a holographic
defect in a three dimensional CFT and re-derive the large $s$ spectrum in eqs.
(21-23) from a computation in the spirit of section 4. In subsections 5.4 and
5.5, we turn to two examples in which the full defect OPE can be obtained by
applying the inversion formula on a single block.

Finally, in subsection 5.6 we point out an interesting common feature of all
the examples.

###  5.1 Defects in free theory

As a first, simple, example we consider the defect spectra that can appear in
the bulk-to-defect OPE of a free scalar ($\Delta_{\phi}=\tfrac{d-2}{2}$). It
was shown in section B.1.1 of Billo:2016cpy  that only two towers of defect
operators are allowed by the equations of motion:

$\displaystyle\widehat{\tau}$ $\displaystyle=\Delta_{\phi}\,,$
$\displaystyle\widehat{\tau}$ $\displaystyle=\Delta_{\phi}+2-q-2s\,,\quad
s\leqslant\frac{4-q}{2}\,.$ (88)

The first set is the tower of transverse derivatives, which are not allowed to
acquire anomalous dimension. This agrees with the lightcone analysis, and with
the inversion formula (68), since all the operators in the bulk OPE of $\phi$
with itself have zero discontinuity, except the identity. In turn, as remarked
in subsection 3.3, the inversion of the identity precisely yields the spectrum
of the trivial defect, with the OPE coefficients (10). Those vanish for $m>0$
when $\Delta_{\phi}$ is at the unitarity bound, and indeed only the leading
transverse twist trajectory appears in eq. (88). What about the second tower
in eq. (88)? These are isolated operators at low spin, as enforced by the
unitarity bound in eq. (88). The lightcone expansion is blind to this kind of
solutions. As we shall see now in a specific example, these operators also lie
below the radius of convergence $s_{\star}$ of the inversion formula.

The simplest example of a non-trivial defect in free theory is obtained by
integrating a free field on a dimension $p=\tfrac{d}{2}-1$ surface, which
requires even $d\geqslant 4$ – see _e.g._ Billo:2016cpy  for more details. It
follows from (88) that the tower with bounded spin is only present if
$d\leqslant 6$. In this case, a single defect operator with $s=0$ and
${\widehat{\Delta}}=0$ is allowed – the identity operator. The two-point
function of the free field is given by

$\langle\phi(1,1)\phi(z,\bar{z})\rangle=\frac{1}{((1-z)(1-\bar{z}))^{\Delta_{\phi}}}+\frac{a_{\phi}^{2}}{(z\bar{z})^{\Delta_{\phi}/2}}\,,$
(89)

which indeed differs from that of a trivial defect (first addend in (89)) by
the appearance of the defect identity (second addend in (89)). We now want to
use the Lorentzian inversion formula (68) to recover the spectrum. We should
check the behavior of $g(r,w)$ for $w\to 0$ (or similarly $w\to\infty$) before
dropping the arcs near $w=0$ and $w=\infty$ when going from (49) to (67). The
two-point function has the following asymptotics:

$g(r,w)=r^{\Delta_{\phi}}\left\langle\phi(1,1)\phi\left(rw,\frac{r}{w}\right)\right\rangle\sim
a_{\phi}^{2}\,w^{0}+\mathcal{O}(w^{\Delta_{\phi}})\,,\quad\mathrm{for}\;w\to
0\,,$ (90)

and so from (56) we find that the inversion formula (68) is valid for only for
$s>s_{\star}=0$. Indeed, while the inversion of the first addend in (89)
reproduces the spectrum of the trivial defect, the second addend has zero
discontinuity and does not contribute. Since the formula is not valid for
$s=0$, this is not at odds with the presence of the identity in the defect OPE
of $\phi$.

###  5.2 Wilson lines in supersymmetric gauge theories

It was conjectured in Lewkowycz:2013laa  that the one-point function of the
stress tensor in the presence of certain BPS Wilson lines is related to the
so-called Bremsstrahlung function as follows:242424In Lewkowycz:2013laa , the
coefficient of the one-point function of $T_{\mu\nu}$ is called $h$, with
$h=-a_{T}/d$.

$a_{T}=-\frac{\Gamma\left(\frac{d-1}{2}\right)}{\pi^{\frac{d-3}{2}}}\frac{d(d-2)}{d-1}B\leavevmode\nobreak\
.$ (91)

The Bremsstrahlung function measures the energy emitted by an accelerated
charged particle at small velocities. The class of superconformal gauge
theories in which the conjecture holds has not been completely explored yet.
The relation (91) has been checked for $1/2$ BPS Wilson lines in
$\mathcal{N}=4$ SYM by direct computation of the two sides Okuyama:2006jc ;
Gomis:2008qa ; Correa:2012at . Evidence has been put forward also for $1/2$
and $1/6$ BPS Wilson lines in ABJM Lewkowycz:2013laa ; Bianchi:2017svd ;
Bianchi:2017ozk , and for $1/2$ BPS Wilson lines in four dimensional
$\mathcal{N}=2$ theories Fiol:2015spa .

Our results add an entry to the list of observables related to the
Bremsstrahlung function. Indeed, via eqs. (23-24) we see that $B$ controls one
of the leading contributions to the anomalous dimensions and the OPE
coefficients of the transverse derivative operators at large spin. More
precisely, recall that the Ward identities fix the three-point function
$\braket{\phi\phi T_{\mu\nu}}$:

$c_{\phi\phi T}=-\frac{d\,\Delta_{\phi}}{(d-1)S_{d}},\qquad
S_{d}=\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}\,,$ (92)

so that eqs. (23-24) become:252525Our convention for the central charge is the
same as eq. (4.2) in Dolan:2000ut , _i.e._ , the two-point function of
$T_{\mu\nu}$ has coefficient $C_{T}/S_{d}^{2}$. Of course, the combination
$c_{\phi\phi}{}^{T}a_{T}$ is independent of this choice, but notice that
$c_{\phi\phi}{}^{T}=\frac{S_{d}^{2}}{C_{T}}c_{\phi\phi T}$.

$\displaystyle c_{\textup{min,T}}$
$\displaystyle=-\frac{32\,\pi^{\frac{3}{2}}}{(d-1)^{3}}\frac{\Gamma(d+1)\Gamma\left(\frac{d+3}{2}\right)\Gamma(\Delta_{\phi}+1)}{\Gamma\left(\frac{d}{2}\right)^{2}\Gamma\left(\frac{d-2}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{d-2}{2}\right)}\frac{B}{C_{T}}\leavevmode\nobreak\
,$ (93) $\displaystyle b_{\textup{min,T}}$
$\displaystyle=c_{\textup{min}}\big{(}\gamma_{E}+\psi\left(d/2+1\right)\big{)}\leavevmode\nobreak\
.$ (94)

Theories obeying the conjecture (91) also have a protected scalar of dimension
$d-2$ in their spectrum, belonging to the same superconformal multiplet as the
stress tensor. The contribution of this scalar to the anomalous dimensions,
and OPE coefficients, of transverse derivative operators is then of the same
order as that of the stress tensor at large $s$, and we must take it into
account. The coupling of this scalar and of the stress tensor, both to the
Wilson line and to local operators that belong to certain short
representations of the superconformal algebra, are related by the
superconformal algebra. In these cases one can compute the correction to eqs.
(93-94) from the scalar operator. Depending on the theory one might need to
consider other scalars of dimension $d-2$ (or lower).

Let us consider first the case of four-dimensional $\mathcal{N}=4$
superconformal field theories in more detail. From the classification of
irreducible highest-weight representations carried out in detail in
Dolan:2002zh  (see also the summary in Beem:2016wfs ), we see that the only
multiplet present in interacting theories that can contain twist-two conformal
primaries in symmetric traceless representations is the stress tensor
superconformal multiplet.262626Concretely, apart from the stress tensor
multiplet ($\mathcal{B}^{\frac{1}{2},\frac{1}{2}}_{[0,2,0]}$ in the notation
of Dolan:2002zh ), the only multiplets that accommodate twist-two operators
are the free vector multiplet
($\mathcal{B}^{\frac{1}{2},\frac{1}{2}}_{[0,1,0]}$), and multiplets containing
conserved currents of spin greater than two ($\mathcal{C}^{1,1}_{[0,0,0]}$ and
$\mathcal{B}^{\frac{1}{4},\frac{1}{4}}_{[1,0,1],(0,0)}$), which should be
absent in interacting theories Maldacena:2011jn ; Alba:2013yda . The twist-two
symmetric traceless operators present in this superconformal multiplet are the
stress tensor itself, the $SU(4)_{R}$ current – which does not acquire a one-
point function Billo:2016cpy , and the superprimary – a scalar of dimension
two in the ${\bf 20}^{\prime}$ representation of the R-symmetry group
$SU(4)_{R}$. From the two-point function of bulk operators neutral under
$SU(2)_{R}$, we deduce that their defect OPE must contain transverse
derivative operators obeying exactly (93-94), since the scalar ${\bf
20}^{\prime}$ operators cannot be exchanged in the bulk OPE. Considering the
two-point functions of operators which transform non-trivially under
$SU(4)_{R}$, we need to take into account the contribution of the dimension
two scalar. In general, the relation between the coupling to the bulk
operators of the scalar and the stress tensor is not known. However, if we
consider the two point function of half-BPS bulk operators
($\mathcal{B}^{\frac{1}{2},\frac{1}{2}}_{[0,p,0]}$), the superconformal blocks
obtained in Liendo:2016ymz  should allow for this relation to be determined
and for a prediction to be made. This would amount to the anomalous dimension
of long defect operators ($L^{\star}$ in the notation of Liendo:2016ymz )
whose transverse twists approach $p+2m$ as $s\to\infty$, and transforming in
all representations of the $SO(5)_{R}$ R-symmetry, preserved by the line
defect, that appear in the decomposition of the bulk operator’s $[0,p,0]$
irrep of $SU(4)_{R}$.

For four-dimensional $\mathcal{N}=2$ superconformal field theories there are
various multiplets that can accommodate twist two operators. In particular,
the stress-tensor supermultiplet once again contains a dimension two scalar as
its superconformal primary, now neutral under the R-symmetry, this means that
its contribution to (93-94) should generically be included. The relation
between the one-point function of the scalar and the stress tensor is obtained
in Fiol:2015spa , however, the relation between their couplings to bulk
operators is only known when considering half-BPS bulk operators, and only up
to a sign. It would be interesting to work this out and add this correction to
(93-94).

Finally, for three-dimensional $\mathcal{N}=6$ or $\mathcal{N}=8$
superconformal theories, once again there is a scalar in the stress tensor
multiplet of dimension $d-2$, which transforms non-trivially under the
R-symmetry.272727Other scalars of dimension less or equal to one appear in
multiplets that are either free, contain conserved currents of spin greater
than one, or enhance the supersymmetry in the $\mathcal{N}=6$ case, and so we
do not need to consider them. This means that for R-symmetry singlet bulk
operators this primary cannot be exchanged and the result in (93-94) holds.

###  5.3 A holographic line defect in pure gravity

In this subsection we present a holographic example. We find a solution of
Einstein’s equations with cosmological constant that we conjecture dual to a
line defect in three dimensions, similar to the defect constructed in
Horowitz:2014gva  for Einstein-Maxwell theory. We then compute anomalous
dimensions of defect operators due to the exchange of the stress tensor. The
computation is a holographic realization of the argument in section 4, and the
result is in agreement with the lightcone bootstrap. This confirms the
consistency of the whole picture.

#### The metric

$\xi=\infty$$\partial AdS_{4}$$\eta$$\xi$$\xi=0$$\eta=\infty$

(a) $(\xi,\eta)$

$\theta$$r$

(b) $(r,\theta)$

Figure 5: (a) Depiction of the geometry in the coordinates of eqs. (95) and
(96), at fixed $(\tau,\varphi)$. The metric has a conical singularity at
$\xi=0$ (dashed line), and is asymptotically $AdS_{4}$ at large $\xi$. (b)
Depiction of the geometry in global coordinates at fixed $(t,\varphi)$, see
eqs. (103) and (104). The conical singularity now extends along the diameter
of the sphere (dashed line).

In view of the spirit of the argument given in section 4, in this subsection
we work in Lorentzian signature with a time-like defect. It will be convenient
to write the metric using an $AdS_{2}-$slicing – see for example
Horowitz:2014gva :

$ds^{2}=g(\xi)ds^{2}_{AdS_{2}}+\frac{d\xi^{2}}{f(\xi)}+\xi^{2}d\varphi^{2}\,,$
(95)

where the $AdS_{2}$ factor is given by

$ds^{2}_{AdS_{2}}=-\eta^{2}d\tau^{2}+\frac{d\eta^{2}}{\eta^{2}}\,,$ (96)

and $\xi$ and $\varphi$ are radial and angular coordinates respectively. This
metric has a manifest $SO(2,1)\times SO(2)$ symmetry, which is the expected
symmetry group for a line defect in three dimensions. Notice that we used
reparametrization invariance to fix the coefficient of $d\varphi^{2}$.

We would like this space to be asymptotically $AdS_{4}$, so we impose
appropriate boundary conditions at large $\xi$:

$f(\xi)=\xi^{2}+1+\mathcal{O}\left(\frac{1}{\xi}\right)\,,\qquad
g(\xi)=\xi^{2}+1+\mathcal{O}\left(\frac{1}{\xi}\right)\,.$ (97)

The boundary can also be reached sending $\eta\to\infty$, which lands us on a
one-dimensional subspace parametrized by $\tau$, which is the defect locus in
the CFT – see fig. 5. The boundary conditions in eq. (97) are enough to
single-out a one-parameter solution to Einstein’s equations. The function $f$
is fixed in terms of $g$:

$f(\xi)=\frac{4\xi g(\xi)(-1+3g(\xi))}{g^{\prime}(\xi)(4g(\xi)+\xi
g^{\prime}(\xi))}\,.$ (98)

The function $g$ reads

$\displaystyle g(\xi)$
$\displaystyle=\xi^{2}+1+\frac{1}{3}(\xi^{2}+1)\left(-1+\cos{\Theta(\xi)}-\sqrt{3}\sin{\Theta(\xi)}\right)\,,$
(99)

where

$\Theta(\xi)=\frac{2}{3}\arcsin{\left(\frac{{\mathcal{C}}}{(1+\xi^{2})^{\tfrac{3}{2}}}\right)}\,.$
(100)

The parameter ${\mathcal{C}}$ controls the metric close to $\xi=0$. In
particular, turning on a small ${\mathcal{C}}$ one gets

$f(\xi)=1-\frac{4\,{\mathcal{C}}}{3\sqrt{3}}+\mathcal{O}\left(\xi^{2},{\mathcal{C}}^{2}\right)\,.$
(101)

This means that the metric develops a conical singularity at $\xi=0$, which
breaks the isometries of $AdS_{4}$ down to the defect conformal group. The
singularity is a conical defect for ${\mathcal{C}}>0$ and a conical excess in
the opposite case.

#### Single particle states and anomalous dimensions

A basic entry in the $AdS$/CFT dictionary states that the energy of a state in
$AdS$ with respect to global time equals the scaling dimension of the
corresponding CFT operator. The discussion in section 4 suggests that a large
transverse spin operator is dual to a state describing a single particle with
a large momentum in the $\varphi$ direction. Starting with ${\mathcal{C}}=0$,
one obtains free propagation in empty $AdS$, and the spectrum of a trivial
defect in the dual CFT. The conical singularity perturbs the gravitational
potential, therefore the corresponding energy shift is dual to the
contribution of the stress tensor to the scaling dimension of the CFT
operator. In this subsection we demonstrate this matching. The computation is
analogous to the determination of the anomalous dimension of large spin
double-trace operators, as performed in section 2 of Fitzpatrick:2014vua ,
which we follow closely – see also Kaviraj:2015xsa .

The first step is to switch to spherical coordinates:

$\begin{split}\tau&=\frac{1}{\cos{\theta}\sin{\rho(r)}\sec{t}-\tan{t}}\,,\\\
\eta&=\frac{2(\cos{\theta}\sin{\rho(r)}-\sin{\tau})}{\sqrt{-2\cos{2\theta}\sin^{2}{\rho(r)}+\cos{2\rho(r)}+3}}\,,\\\
\xi&=\sin{\theta}\tan{\rho(r)}\,,\end{split}$ (102)

where $\rho(r)=\arctan{r}$. In the new coordinates, the $AdS_{2}$ locus of the
conical singularity extends along $t$ and along a diameter of the sphere at
fixed $t$, see fig. 5. It pierces the cylindrical boundary at the location of
the defect – at $\theta=0$ and $\theta=\pi$ – as depicted in fig. 4.282828But
notice that the labeling of coordinates is different from section 4, where
holography was not involved. Also, the $AdS_{2}\times S^{1}$ frame induced on
the boundary from eq. (95) is not the same as the $AdS_{3}\times S^{1}$ frame
employed in section 4. The wave function of a particle that rotates fast in
the $\varphi$ direction is peaked close to the boundary, therefore we only
need to consider the large $r$ limit to obtain the leading order correction in
$s$. As it can be seen from eq. (100), this is equivalent to keeping only the
leading order in ${\mathcal{C}}$ in all formulae. In this limit the metric is
as follows:

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\varphi\varphi}d\varphi^{2}\,,$
(103)

where

$\displaystyle\begin{split}g_{tt}&=-r^{2}-1-\mathcal{C}\frac{4\left(r^{2}+1\right)}{3\left(r^{2}\cos{2\theta}-r^{2}-2\right)\sqrt{3r^{2}\sin^{2}{\theta}+3}}\,,\\\
g_{rr}&=\frac{1}{(1+r^{2})^{2}}\left(r^{2}+1-\mathcal{C}\sqrt{\frac{2}{3}}\frac{4\left(r^{2}+1\right)\left(2r^{2}\cos{2\theta}+3\cos{2\theta}-2r^{2}-1\right)}{3\left(-r^{2}\cos{2\theta}+r^{2}+2\right)^{5/2}}\right)\,,\\\
g_{\theta\theta}&=r^{2}+\mathcal{C}\sqrt{\frac{2}{3}}\frac{4r^{2}\left(r^{2}\cos{2\theta}+3\cos{2\theta}-r^{2}+1\right)}{3\left(-r^{2}\cos{2\theta}+r^{2}+2\right)^{5/2}}\,,\\\
g_{\varphi\varphi}&=r^{2}\sin{\theta}\,.\end{split}$ (104)

The rest of the exercise is first-order perturbation theory applied to the
Hamiltonian of a free scalar coupled to gravity. The unperturbed wave function
is the following (see _e.g._ , Penedones:2016voo )

$\psi_{m,\ell,s}(t,\rho,\theta,\varphi)=\frac{1}{N_{\Delta_{\phi}m\ell}}e^{-iE_{m,\ell}\,t}Y^{s}_{\ell}(\theta,\varphi)\sin^{\ell}{\rho}\,\cos^{\Delta_{\phi}}\\!{\rho}\,{{}_{2}}F_{1}(-m,\Delta_{\phi}+m+\ell,\ell+\frac{3}{2};\sin^{2}{\rho})\,,$
(105)

where

$E_{m,\ell}=\Delta_{\phi}+2m+\ell\,,$ (106)

$Y^{s}_{\ell}$ is a spherical harmonic and the normalization is

$N_{\Delta_{\phi}m\ell}=(-1)^{m}\sqrt{\frac{m!\Gamma\left(\frac{d}{2}+\ell\right)^{2}\Gamma\left(\frac{2-d}{2}+m+\Delta_{\phi}\right)}{\Gamma\left(\frac{d}{2}+m+\ell\right)\Gamma(m+\ell+\Delta_{\phi})}}\,.$
(107)

In empty $AdS_{4}$, $\psi$ is a descendant of a scalar primary of dimension
$\Delta_{\phi}$. The dual interpretation of the labels is as follows: $m$
counts the number of Laplacians, while $\ell$ and $s$ are the angular momentum
and its $\varphi$-component respectively. A feature of the spectrum of a
trivial defect – see eq. (9) – is that primary states of the defect correspond
to bulk descendants with $s=\ell$, _i.e._ the transverse spin equals the total
spin of the bulk descendant. The quantity $\ell-|s|$ counts the number of
derivatives along the defect. To isolate a defect primary, therefore, we
choose $\ell=s$. Furthermore, we only consider the $m=0$ case for simplicity,
that is, we focus on the leading transverse twist trajectory. It is then easy
to see that $\psi_{0,s,s}(t,\rho,\theta,\varphi)$ in eq. (105) is the wave
function of a defect primary. Indeed,

$K_{a}(P_{z})^{s}\ket{\phi}=0,$ (108)

where $z$ is a complex coordinate parameterizing a plane and $a$ is a
direction orthogonal to the same plane. Alternatively, one can check that the
special conformal generator along the defect annihilates
$\psi_{0,s,s}(t,\rho,\theta,\varphi)$.

The first order correction to the energies (106) is then

$\delta E=\langle m,s|\delta{\mathcal{H}}|m,s\rangle\,,$ (109)

where $\delta{\mathcal{H}}$ is the $\mathcal{O}({\mathcal{C}})$ term in the
Hamiltonian of the scalar field. The computation is simplified by the fact
that $\delta\sqrt{-g}=\mathcal{O}({\mathcal{C}}^{2})$. From the metric in
(103) the kinetic energy of all the coordinates except $\varphi$ receive
corrections. We end up with three integrals

$\delta E=2{\mathcal{C}}\int
drd\Omega_{2}r^{2}\left({\mathcal{I}}_{1}+{\mathcal{I}}_{2}+{\mathcal{I}}_{3}\right)\,,$
(110)

where the integrands ${\mathcal{I}}_{i}$ are given by

$\displaystyle{\mathcal{I}}_{1}$
$\displaystyle=\frac{2r^{2s}(\Delta+s)^{2}\left(r^{2}+1\right)^{-\Delta-s-1}|Y_{s}^{s}(\theta,\varphi)|^{2}}{3\sqrt{3r^{2}\sin^{2}{\theta}+3}\left(r^{2}\cos{2\theta}-r^{2}-2\right)}\,,$
(111) $\displaystyle{\mathcal{I}}_{2}$
$\displaystyle=\frac{2\sqrt{\frac{2}{3}}r^{2s-2}\left(\left(2r^{2}+3\right)\cos{2\theta}-2r^{2}-1\right)\left(r^{2}+1\right)^{-\Delta-s-1}\left(s-\Delta
r^{2}\right)^{2}|Y_{s}^{s}(\theta,\varphi)|^{2}}{3\left(-r^{2}\cos{2\theta}+r^{2}+2\right)^{5/2}}\,,$
(112) $\displaystyle{\mathcal{I}}_{3}$
$\displaystyle=-\frac{2\sqrt{\frac{2}{3}}r^{2s-2}\left(\left(r^{2}+3\right)\cos{2\theta}-r^{2}+1\right)\left(r^{2}+1\right)^{-\Delta-s}|\partial_{\theta}Y^{s}_{s}(\theta,\varphi)|^{2}}{3\left(-r^{2}\cos{2\theta}+r^{2}+2\right)^{5/2}}\,.$
(113)

For $\ell=m=s$ the spherical harmonics simplify significantly:

$\displaystyle|Y_{s}^{s}(\theta,\varphi)|^{2}$
$\displaystyle=\frac{1}{2\pi^{\frac{3}{2}}}\frac{\Gamma(\frac{3}{2}+s)}{\Gamma(1+s)}(\sin{\theta})^{2s}\,,$
(114) $\displaystyle|\partial_{\theta}Y^{s}_{s}(\theta,\varphi)|^{2}$
$\displaystyle=\frac{1}{2\pi^{\frac{3}{2}}}\frac{\Gamma(\frac{3}{2}+s)}{\Gamma(1+s)}s(\cos{\theta})^{2}(\sin{\theta})^{2s-2}\,.$
(115)

The leading large $s$ asymptotics can be easily obtained using a saddle-point
approximation. It turns out that of the three integrals in (110) the first is
the dominant one. After the dust has settled we find

$\delta
E=-{\mathcal{C}}\frac{\Gamma(\Delta_{\phi}+1)}{3\sqrt{3}\Gamma\left(\Delta_{\phi}-\frac{1}{2}\right)}\,s^{-\frac{1}{2}}\,.$
(116)

The exponent of the correction matches the lightcone prediction eq. (22). In
order to compare the prefactor with eq. (23), we need to compute the one-point
function of the stress tensor in this geometry. This is an exercise in
holographic renormalization deHaro:2000vlm , which we perform in Euclidean
signature. One first writes the metric in Fefferman-Graham coordinates
Fefferman:2007rka :

$ds^{2}=\frac{1}{z^{2}}\left(dz^{2}+h_{\mu\nu}(x,z)dx^{\mu}dx^{\nu}\right).$
(117)

We get

$\displaystyle h_{tt}$
$\displaystyle=1+\frac{z^{2}}{2}-\frac{2}{9\sqrt{3}}{\mathcal{C}}\frac{1}{\sin^{3}\theta}z^{3}+\mathcal{O}(z^{4})\,,$
(118) $\displaystyle h_{\theta\theta}$
$\displaystyle=1-\frac{z^{2}}{2}-\frac{2}{9\sqrt{3}}{\mathcal{C}}\frac{1}{\sin^{3}\theta}z^{3}+\mathcal{O}(z^{4})\,,$
(119) $\displaystyle h_{\varphi\varphi}$
$\displaystyle=\sin^{2}\theta\left(1-\frac{1}{2}z^{2}+\frac{4}{9\sqrt{3}}{\mathcal{C}}\frac{1}{\sin^{3}\theta}z^{3}+\mathcal{O}(z^{4})\right).$
(120)

The boundary now sits at $z=0$, and the expectation value of the stress-tensor
is encoded in the coefficient of $z^{d}$ in the boundary expansion of
$h_{\mu\nu}$:

$\braket{T_{\mu\nu}}=\frac{d}{16\pi G_{N}}h^{(d)}_{\mu\nu}.$ (121)

This immediately yields the one-point function on the cylinder. One can then
apply a Weyl transformation to rewrite it on the plane. We report the result
in Cartesian coordinates, which allows for an easy comparison with the form
predicted by conformal invariance eq. (154). Recall that the defect extends
along $t$, while we call $(x,y)$ the orthogonal coordinates:

$\displaystyle\braket{T_{tt}}=-\frac{{\mathcal{C}}}{24\sqrt{3}\pi
G_{N}}\frac{1}{\lvert
x_{\perp}\rvert^{3}}\,,\quad\braket{T_{xx}}=\frac{{\mathcal{C}}}{24\sqrt{3}\pi
G_{N}}\frac{2y^{2}-x^{2}}{\lvert x_{\perp}\rvert^{5}}\,,$ (122)
$\displaystyle\braket{T_{yy}}=\frac{{\mathcal{C}}}{24\sqrt{3}\pi
G_{N}}\frac{2x^{2}-y^{2}}{\lvert
x_{\perp}\rvert^{5}}\,,\quad\braket{T_{xy}}=-\frac{{\mathcal{C}}}{8\sqrt{3}\pi
G_{N}}\frac{xy}{\lvert x_{\perp}\rvert^{5}}\,.$ (123)

This allows to extract the coefficient of the one-point function:

$a_{T}=-\frac{{\mathcal{C}}}{8\sqrt{3}\pi
G_{N}}=-\frac{C_{T}}{384\sqrt{3}}\,{\mathcal{C}}\,.$ (124)

The Newton constant is related to the central charge as follows:292929Our
conventions are summarized in footnote 25.

$G_{N}=\frac{\pi^{\frac{d}{2}-1}\Gamma(d+2)}{2(d-1)\Gamma\left(\frac{d}{2}\right)^{3}C_{T}}\,.$
(125)

The last ingredient is the three-point function coefficient $c_{\phi\phi T}$,
which was given in eq. (92). By plugging eqs. (124) and (92) in the lightcone
bootstrap formula eq. (23), we precisely recover the anomalous dimension eq.
(116).

###  5.4 The replica twist defect

The replica defect arises in the computation of the geometric Rényi entropies
in QFT Calabrese:2004eu ; Hung:2014npa ; Bianchi:2015liz . It is built as a
boundary condition for the tensor product of $n$ copies of the QFT of interest
– $(\textup{QFT})^{n}$. In particular, the fields of consecutive copies are
identified on a codimension one surface ending on the location of the defect,
which has therefore codimension two. Given the replica defect supported on a
surface $\Sigma$ lying on a constant time slice, consider the path-integral
$Z_{n}(\Sigma)$ of the theory in the presence of the defect. Then the
following holds:

$\displaystyle
S_{n}(\Sigma)=\frac{1}{1-n}\log\frac{Z_{n}(\Sigma)}{Z^{n}},\qquad
S_{\textup{EE}}(\Sigma)=\lim_{n\to 1}S_{n}(\Sigma)\leavevmode\nobreak\ ,$
(126)

where $Z$ is the path-integral over a single copy of the QFT, and
$S_{n}(\Sigma)$ and $S_{\textup{EE}}(\Sigma)$ are respectively the Rényi and
entanglement entropies of the density matrix obtained by tracing over the
degrees of freedom enclosed in the region $\Sigma$. The second equation in
(126), in particular, requires a definition of the theory that holds for
complex values of $n$. It was argued in Faulkner:2014jva  – see also
Balakrishnan:2017bjg  – that an analytic continuation exists such that, when
the original theory is a CFT, correlation functions preserve the features of a
defect CFT, in particular the existence of a well defined defect OPE limit.
The crucial step is to consider the orbifolded theory
$(\textup{CFT})^{n}/\mathbb{Z}_{n}$, where only the local operators invariant
under cyclic permutations of the copies are retained. In particular, the
orbifold contains a unique stress tensor out of the $n$ coming from each copy.
This property is crucial in the proof of the quantum null energy condition
Balakrishnan:2017bjg . Notice also that the replica defect is trivial at
$n=1$. Hence, it provides the example of a defect CFT with a perturbative
parameter, such that the leading order deviation from the trivial defect is
physically interesting in its own right.

In this section, we apply the inversion formula to a scalar two-point function
and obtain the anomalous dimensions of a class of defect operators at leading
order in $(n-1)$. The result matches the computation in Balakrishnan:2017bjg ,
performed by direct OPE decomposition of the correlator. The external operator
is the $\mathbb{Z}_{n}$-symmetrization of a scalar primary $\phi$ belonging to
a single copy:

$[\phi]=\sum_{k=1}^{n}\phi^{(k)}\leavevmode\nobreak\ ,$ (127)

where the bracket denotes orbifold operators, and $\phi^{(k)}$ stands for the
operator $\phi$ inserted in the $k$-th copy. We call $[\phi]$ a single-copy
orbifold operator, while multiple-copy operators are the
$\mathbb{Z}_{n}$-symmetrization of products of operators on different replicas
(_e.g._ $\phi_{1}^{(i)}\phi_{2}^{(j)}$ with $i\neq j$). The bulk OPE
$[\phi]\times[\phi]$ contains single-copy and double-copy operators. The
former correspond to the primaries in the fusion $\phi\times\phi$, while the
latter arise from the non-singular fusion $\phi^{(i)}\times\phi^{(j)}$, $i\neq
j$, and have twist $\tau=2\Delta_{\phi}+2m$ with integer $m$. This fusion rule
is independent of the number of copies, and survives in the analytically
continued theory. From the explicit analytic continuation of single-copy
operators Balakrishnan:2017bjg , one sees that the only single-copy operator
with a vev of order $(n-1)$ is the stress-tensor. However, a single block is
not crossing symmetric, and the double-copy operators come to the rescue and
also acquire a one-point function. Since the twists of the latter differ from
$2\Delta_{\phi}$ by an even integer, they do not contribute a discontinuity to
the two-point function $\braket{[\phi][\phi]}$, and so do not affect the OPE
coefficients nor the anomalous dimensions of defect operators. Hence, the
defect spectrum is simply found by inverting the stress tensor block. The one-
point function of the stress-tensor at order $(n-1)$ is known exactly
Hung:2011nu ; Perlmutter:2013gua ; Hung:2014npa :303030Our convention is
related to the one in Hung:2014npa  as follows: $a_{T}=-dh_{n}/(2\pi)$. See
also footnote 25.

$\left.\partial_{n}a_{T}\right|_{n=1}=-d\,\pi^{d/2}\frac{\Gamma\left(d/2\right)}{\Gamma\left(d+2\right)}\frac{C_{T}}{S_{d}^{2}}\leavevmode\nobreak\
,\qquad S_{d}=\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}\,,$ (128)

and the same is true for the OPE coefficient, see eq. (92). To obtain the
leading transverse-twist primaries, we only need to evaluate eq. (79) for
$q=2$, and the result is

$\widehat{\tau}=\Delta_{\phi}-(n-1)\frac{\Delta_{\phi}}{d-1}\frac{\left(\Delta_{\phi}-\frac{d}{2}+1\right)_{s}}{\left(\Delta_{\phi}\right)_{s}}+\mathcal{O}\big{(}(n-1)^{2}\big{)}\leavevmode\nobreak\
,$ (129)

where $(a)_{s}=\Gamma(a+s)/\Gamma(s)$. This is the correct result
Balakrishnan:2017bjg .313131In fact, our anomalous dimension is twice the one
in eq. (3.25) of Balakrishnan:2017bjg . However it matches their eq. (B4), so
we attribute the discrepancy to a typo. However, to confirm its validity down
to finite transverse spin we need to know the behavior of the two-point
function $\braket{[\phi][\phi]}$ as $w\to 0$. In the absence of a general
bound, we resort to the explicit form computed in Balakrishnan:2017bjg :

$g(z,\bar{z})=\left(\frac{(1-z)(1-\bar{z})}{(z\bar{z})^{1/2}}\right)^{-\Delta_{\phi}}\left(1+c_{\phi\phi
T}\,a_{T}\,f_{d,2}(z,\bar{z})+I+\mathcal{O}(n-1)^{2}\right)\,,\\\
I=-(n-1)\int_{0}^{-\infty}\frac{d\lambda}{(\lambda-1)^{2}}\left(1+\frac{1}{w\xi}(1-\lambda)+\frac{w}{\xi}\left(1-\frac{1}{\lambda}\right)\right)^{-\Delta_{\phi}}\,,$
(130)

where $f_{d,2}(z,\bar{z})$ is the bulk stress-tensor block and $\xi$ is
defined in eq. (148). The integral $I$ can be seen in a bulk block
decomposition to encode the contributions of the aforementioned exact double-
twist operators. Expanding $I$ at leading order in $w$ one finds

$\left(\frac{(1-z)(1-\bar{z})}{(z\bar{z})^{1/2}}\right)^{-\Delta_{\phi}}I=(n-1)w^{\Delta}\,\pi\Delta\csc\pi\Delta\,\big{(}1+O(w)\big{)}\\\
+(n-1)w\,\Delta
r^{1-\Delta}\frac{\Gamma(\Delta)\Gamma(\Delta-1)}{\Gamma(2\Delta)}{}_{2}F_{1}\left(\Delta-1,\Delta,2\Delta,1-\frac{1}{r^{2}}\right)\,\big{(}1+O(w)\big{)}\,.$
(131)

This asymptotics, together with the single block asymptotics eq. (80), ensure
that $s_{\star}<0$ – see eq. (56) – and so eq. (129) holds for all $s$.

###  5.5 The Ising twist defect

There exists a conformal defect with codimension two in the 3d Ising model, as
supported by numerical evidence Billo:2013jda  and also by results from the
epsilon expansion and the conformal bootstrap Gaiotto:2013nva . Local
operators odd (even) under the $\mathbb{Z}_{2}$ flavor symmetry of the 3d
Ising CFT are multi-valued (single-valued) around the twist defect. As a
consequence, the $\mathbb{Z}_{2}$ odd (even) defect spectrum take half-integer
(integer) values of the transverse spin $s$.

Following the literature, let us dub $\psi_{s}$ the leading transverse twist
primaries in the defect OPE of the spin field $\sigma$:

$\sigma\sim\sum_{s}\psi_{s}+\textup{higher }\widehat{\tau}\leavevmode\nobreak\
,\qquad s\in\mathbb{N}+\frac{1}{2}\leavevmode\nobreak\ .$ (132)

The dimensions and OPE coefficients of the $\psi_{s}$ have been calculated in
the epsilon expansion in Gaiotto:2013nva , _i.e._ by setting $d=4-\epsilon$
and keeping $q=2$. To leading order in $\epsilon$,

$\displaystyle\widehat{\tau}_{\psi_{s}}$ $\displaystyle=$ $\displaystyle
1-\left(\frac{1}{2}+\frac{1}{24s}\right)\epsilon+O(\epsilon^{2})\leavevmode\nobreak\
,$ (133) $\displaystyle|b_{\sigma\psi_{s}}|$ $\displaystyle=$ $\displaystyle
1+\frac{\psi(1)-\psi(s+1)}{4}\epsilon+O(\epsilon^{2})\leavevmode\nobreak\
,\qquad\psi(z)=\frac{d}{dz}\ln{\Gamma(z)}\leavevmode\nobreak\ .$ (134)

Let us interpret these values from the lightcone bootstrap point of view. The
scaling dimension of $\sigma$ is

$\Delta_{\sigma}=1-\frac{\epsilon}{2}+O(\epsilon^{2})\leavevmode\nobreak\ ,$
(135)

so the $\psi_{s}$ are easily identified as the leading trajectory of
transverse derivative operators. The fusion $\sigma\times\sigma$, at the
leading non-trivial order in $\epsilon$, can be written as follows:

$\sigma\times\sigma\sim
1+\varepsilon+\\{\tau=2\Delta_{\sigma}\\}+\textup{higher
twists}+O(\epsilon^{2})\leavevmode\nobreak\ ,$ (136)

where $\varepsilon$ is the energy operator and $\\{\tau=2\Delta_{\sigma}\\}$
denotes the conserved currents of the free theory, which do not acquire
anomalous dimension at this order Wilson:1973jj  – see also Alday:2015ota  and
Gliozzi:2017gzh  for a more general understanding of this fact. The higher-
twist primaries are decoupled in the free theory, and so their OPE coefficient
is $O(\epsilon)$. Hence, they also appear as operators with
$\tau=2\Delta_{\sigma}+2m$ in this OPE. All together, the only primary
contributing a discontinuity is $\varepsilon$, which is therefore fully