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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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=\delta^{IJ}\,I_{12}\,,\\\ \text{Fermions:}\qquad&\leavevmode\hbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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=i\delta^{ab}\,\not{\partial}_{1}I_{12}\,,\end{split}$ (2.85)

where here we keep the propagator function in $d=4-\varepsilon$ dimensions:

$I_{12}:=\frac{\Gamma\left(1-\frac{\varepsilon}{2}\right)}{4\pi^{2-\varepsilon/2}_{\phantom{12}}x_{12}^{2\left(1-\varepsilon/2\right)}}\,,$
(2.86)

with $x_{12}:=x_{1}-x_{2}$. At $d=4$, (2.86) reduces to (2.47). The scalar
propagator satisfies Green’s equation

$\partial_{1}^{2}I_{12}=-\delta^{(d)}(x_{12})\,,$ (2.87)

where $\delta^{(d)}(x)$ refers to the $d$-dimensional Dirac delta function.

The interaction terms yield the following insertion rules:

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&=-\frac{\lambda}{2}(\delta^{IJ}\delta^{KL}+\delta^{IK}\delta^{JL}+\delta^{IL}\delta^{JK})X_{1234}\,,\\\
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&=g\,\not{\partial}_{1}(\Sigma^{I})^{ab}\not{\partial}_{3}Y_{123}\,,\end{split}$
(2.88)

with the $X$\- and $Y$-integrals given in Appendix C.2.1. Note that one has to
add a factor $1/n!$ when $n$ vertices of the same kind are being inserted,
while symmetry factors have been taken into account.

####  2.3.2 The Wilson–Fisher–Yukawa fixed point

The action (2.81) accepts three fixed points in three dimensions: the trivial
one $g=\lambda=0$, the Wilson–Fisher (WF) fixed point
$g=0,\lambda=\lambda_{\star}$ corresponding to the (conformal) $O(N)$ models,
and the Wilson–Fisher–Yukawa (WFY) fixed point. Here we present the
$\beta$-functions of the couplings and the value they take at the WFY fixed
point. The lowest-lying scalar and fermionic operators are also discussed.

##### The critical point

The $\beta-$functions of the couplings in (2.81) are known up to several loop
orders for each model [313, 314, 315, 316].171717We restrict ourselves from
now on to the models that preserve the $O(N)$ symmetry. For general Yukawa and
four-scalar couplings, they can be found up to two loops for instance in the
appendix of [308]. Although we keep $N$ arbitrary, the expressions given here
should be considered with care and not extended beyond $N=3$.

The $\beta-$functions are given by

$\displaystyle\begin{split}\beta_{\lambda}&=-\varepsilon\lambda+\frac{1}{(4\pi)^{2}}\left(8g^{2}\lambda{N_{f}}-48g^{4}{N_{f}}+\frac{N+8}{3}\lambda^{2}\right)+\ldots\,,\\\
\beta_{g}&=-\varepsilon\frac{g}{2}+\frac{\kappa_{1}g^{3}}{(4\pi)^{2}}+\ldots\,,\end{split}$
(2.89)

with $\kappa_{1}$ defined in (2.91). The WFY fixed point can be reached for
the following values of the couplings at one loop in $\varepsilon:=4-d$:

$\displaystyle\frac{\lambda_{\star}}{(4\pi)^{2}}=\frac{3\kappa_{2}}{2\kappa_{1}(N+8)}\varepsilon+{\mathcal{O}}(\varepsilon^{2})\,,\qquad\frac{g^{2}_{\star}}{(4\pi)^{2}}=\frac{\varepsilon}{2\kappa_{1}}+{\mathcal{O}}(\varepsilon^{2})\,,$
(2.90)

where we see that $g\sim\mathcal{O}(\sqrt{\varepsilon})$, while
$\lambda\sim\mathcal{O}(\varepsilon)$. Furthermore, we have defined

$\displaystyle\begin{split}\kappa_{1}&:=2{N_{f}}-N+4\,,\\\
\kappa_{2}&:=2(4-N)-\kappa_{1}+\sqrt{12\left(N^{2}+16\right)+\kappa_{1}(\kappa_{1}+12(N+4))}\,.\end{split}$
(2.91)

Note that all the dependence on ${N_{f}}$ is contained in $\kappa_{1}$. At
$\varepsilon=0$, the WFY fixed point corresponds to the free theory, which
flows at $\varepsilon=1$ to an interacting model.

##### Bulk operators

We now turn our attention to the lowest-lying bulk operators associated with
the action (2.81) for arbitrary values of $N$ and ${N_{f}}$. In Section 5.3,
we calculate correlation functions involving the fundamental scalars and
fermions in the presence of a line defect. As we expand these correlators
using conformal blocks, operators with higher scaling dimensions naturally
emerge. To ensure the consistency of our analysis, we provide here a selection
of relevant operators that are useful for conducting sanity checks.

The lowest-lying scalar operators are the fundamental scalars, which we refer
to as

$\mathcal{O}^{I}(x):=\frac{1}{\sqrt{n_{I}}}\phi^{I}(x)\,,$ (2.92)

with $I=1,\ldots,N$ the $O(N)$ index. The normalization constant $n_{I}$ is
chosen as usual such that the two-point function is unit-normalized. The
scaling dimension $\Delta_{I}$ of this operator was computed in [301] at the
WFY fixed point (2.90) and found to be

$\Delta_{I}=1+\frac{N-4}{2\kappa_{1}}\varepsilon+\mathcal{O}(\varepsilon^{2})\,.$
(2.93)

The explicit expression for the normalization constant is given by

$n_{I}=\frac{1}{4\pi^{2}}\left(1-\frac{2\kappa_{1}+(N-4)(1+\aleph)}{2\kappa_{1}}\varepsilon+{\mathcal{O}}(\varepsilon^{2})\right)\,.$
(2.94)

In the above, $\kappa_{1}$ is defined in (2.91) while $\aleph$ is a constant
arising from dimensional regularization and given in (C.2).

The next set of operators that we consider are the fermions. We denote
fermionic operators as $\mathcal{O}_{\bar{\ell},\ell}$, with $\ell$
corresponding to the number of fermions $\psi^{a}$ and $\bar{\ell}$ to the
number of antifermions ${\bar{\psi}}^{a}$. The variables $s$ and $\bar{s}$ are
polarization spinors, introduced to avoid the cluttering of indices and
defined via (A.8). Using this notation, we denote the elementary fermions as

$\mathcal{O}_{1,0}^{a}(\bar{s},x):=\frac{1}{\sqrt{n_{a}}}{\bar{\psi}}^{a}(\bar{s},x)\,,\quad\mathcal{O}_{0,1}^{a}(s,x):=\frac{1}{\sqrt{n_{a}}}\psi^{a}(s,x)\,.$
(2.95)

The scaling dimension of these operators has also been computed in [301] at
the WFY fixed point, yielding

$\Delta_{a}=\frac{3}{2}-\frac{2\kappa_{1}-N}{4\kappa_{1}}\varepsilon+{\mathcal{O}}(\varepsilon^{2})\,.$
(2.96)

As mentioned earlier, the block expansions of the operators defined in (2.92)
and (2.95) include operators with higher leading-order scaling dimensions in
$\varepsilon$. Particularly, for $\Delta\sim 2$, we have the following
operators:

$\mathcal{O}^{2}:=\frac{1}{\sqrt{n_{\mathcal{O}^{2}}}}\phi^{2}\,,\qquad
T^{IJ}:=\frac{1}{\sqrt{n_{T}}}\left(\phi^{I}\phi^{J}-\frac{\delta^{IJ}}{N}\phi^{2}\right)\,,$
(2.97)

where, as usual, $n_{\mathcal{O}^{2}}$ and $n_{T}$ are the respective
normalization constants.

The number of operators grows rapidly as we consider higher combinations of
fundamental fields. For future purposes, we conclude this section by
introducing the fermion bilinear, defined as

$\mathcal{O}_{1,1}^{aa}(x):=\frac{1}{\sqrt{n_{\mathcal{O}_{1,1}^{aa}}}}{\bar{\psi}}^{a}(x)\psi^{a}(x)\,,$
(2.98)

with its corresponding scaling dimension given by

$\Delta_{\mathcal{O}_{1,1}^{aa}}=3+{\mathcal{O}}(\varepsilon)\,.$ (2.99)

The couplings and operators get renormalized. We define the bare couplings and
(elementary) fields as

$\displaystyle\lambda_{0}$ $\displaystyle:=\mu^{\varepsilon}\lambda
Z_{\lambda}\,,\qquad$
$\displaystyle\mathcal{O}^{I}_{0}:=Z_{I}\mathcal{O}^{I}\,,$ (2.100)
$\displaystyle g_{0}$ $\displaystyle:=\mu^{\varepsilon/2}gZ_{g}\,,\qquad$
$\displaystyle(\mathcal{O}_{\bar{\ell},\ell}^{a})_{0}:=Z_{a}\mathcal{O}_{\bar{\ell},\ell}^{a}\,,$

where we introduced rescaled couplings
$g\to\mu^{\varepsilon/2}g,\lambda\to\mu^{\varepsilon}\lambda$ to ensure that
the couplings in the renormalized Lagrangian are dimensionless, and with
$\ell+\bar{\ell}=1$. The expressions for the renormalization factors $Z$ up to
${\mathcal{O}}(\varepsilon^{2})$ can be found in Appendix B of [5].

The renormalization factors allow us to obtain the anomalous dimensions
$\gamma_{I}$ and $\gamma_{a}$ for the scalar and fermionic fields, which are
given here to first order in the couplings:

$\displaystyle\begin{split}\gamma_{I}&=\frac{d\log
Z_{I}}{d\log\mu}=\frac{2g^{2}{N_{f}}}{(4\pi)^{2}}+\ldots\,,\\\
\gamma_{a}&=\frac{d\log
Z_{a}}{d\log\mu}=\frac{g^{2}N}{2(4\pi)^{2}}+\ldots\,.\end{split}$ (2.101)

This leads to the following values for the conformal dimensions evaluated at
the WFY fixed point defined in (2.90):

$\displaystyle\begin{split}\Delta_{I}&=1-\frac{\varepsilon}{2}+\gamma_{I}=1-\varepsilon\frac{4-N}{2\kappa_{1}}+{\mathcal{O}}(\varepsilon^{2})\,,\\\
\Delta_{a}&=\frac{3}{2}-\frac{\varepsilon}{2}+\gamma_{a}=\frac{3}{2}-\frac{\varepsilon}{4}\left(2-\frac{N}{\kappa_{1}}\right)+{\mathcal{O}}(\varepsilon^{2})\,.\end{split}$
(2.102)

####  2.3.3 The magnetic line

Magnetic lines are natural defects to consider in the $O(N)$ model, which
corresponds to (2.81) setting $g=0$. It was shown in [208] that a defect CFT
continues to exist for this type of defect at non-zero values of the Yukawa
coupling. In the following, we present the necessary toolkit for describing
such a model. We start with the defect action and Feynman rules, then present
the critical point for the defect coupling. We then discuss the defect
operators, as well as the renormalization of the defect coupling.

##### Defect Feynman rules

Adding a magnetic line defect to the action results in a breaking of the
conformal group $SO(d+1,1)\to SO(2,1)\times SO(d-1)$, following the concepts
of Section 2.1.2. The action reads

$S_{\text{defect}}:=S_{\text{Y}}+h\int_{-\infty}^{\infty}d\tau\,\theta\cdot\phi(\tau)\,,$
(2.103)

where the scalar fields $\phi^{I}$ must have a scaling dimension below $1$ for
the deformation to be relevant. Here $h$ is the coupling of the defect, which
extends in the Euclidean time direction $\tau$, while $\theta^{I}$ is an
$O(N)$ vector that defines the polarization of the defect. We choose here
$\phi^{1}$ to be the scalar coupling to the defect, i.e.,

$\theta=(1,0,\ldots,0)\,.$ (2.104)

Note that one could also choose the line defect to couple to a pseudoscalar,
and this would not introduce additional technical difficulties.

The defect introduces a new vertex

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:=-h\,\delta^{I1}\int_{-\infty}^{\infty}d\tau_{2}\,I_{12}\,,$ (2.105)

with $x_{2}:=(\tau_{2},0,0,0)$ the point on the line that has to be integrated
over. As for the bulk Feynman rules, one should add a factor $1/n!$ when $n$
vertices are inserted, while symmetry factors have to be accounted for if more
couplings to the line are being considered.181818These rules differ from
(2.66) due to the absence of gauge symmetry. In other words, there is no path
ordering on the defect.

##### The critical point

In the following, we show that there exists a non-trivial critical value for
the coupling $h$ such that the defect CFT exists and can be taken across
dimensions by varying $\varepsilon$, just as in the bulk. This has been done
for the $O(N)$ model in [317, 205], while correlation functions were
considered in [182]. With fermions, this was considered in [208] and [318] for
the cases $N=1,2,3$.

We can compute the $\beta$-function $\beta_{h}$ by calculating the divergences
of the one-point function of the (renormalized) scalar $\mathcal{O}^{I}$. We
obtain [208, 318]

$\beta_{h}=-\frac{\varepsilon h}{2}+\frac{1}{(4\pi)^{2}}\frac{\lambda
h^{3}}{6}+\ldots\,.$ (2.106)

Using the values for $\lambda$ and $g$ at the WFY fixed point given in (2.90),
we find the corresponding defect fixed point

$h^{2}_{\star}=-\frac{2(N-4)(N+8)}{\kappa_{2}}+{\mathcal{O}}(\varepsilon)\,,$
(2.107)

where the ${\mathcal{O}}(\varepsilon)$ term is given in Appendix B of [5]. It
is interesting to note that there exists a non-trivial critical point even at
$\varepsilon=0$. This can be understood by the fact that free theories can
also accommodate conformal defects.

If we include the finite part of the one-point function, we can extract the
one-point coefficient $a_{I}$ :

$\langle\\!\langle\,\mathcal{O}^{I}(x)\,\rangle\\!\rangle=\frac{a_{I}}{|x^{\bot}|^{\Delta_{I}}}\>,\quad
a_{I}^{2}=-\frac{(N-4)(N+8)}{2\kappa_{2}}+{\mathcal{O}}(\varepsilon)\>.$
(2.108)

The ${\mathcal{O}}(\varepsilon)$ term is lengthy and given in the Mathematica
notebook attached to [5].

##### Defect operators

The fundamental fields discussed in 2.3.2 give rise to a rich set of defect
operators on the magnetic line. In the defect CFT, the $O(N)$ symmetry breaks
down to $O(N-1)$, resulting in the emergence of two fundamental scalars:

$\mathcal{O}^{I=1,\ldots,N}\longrightarrow\\{\hat{\mathcal{O}}^{1}\,,\hat{\mathcal{O}}^{i}\\}\,.$
(2.109)

Here the indices split as $I=\\{1,i\\}$, with $i=2\,,\ldots\,,N$ introduced in
Section 2.3.3. Specifically, the fundamental defect scalars are defined as

$\hat{\mathcal{O}}^{1}(\tau):=\frac{1}{\sqrt{\hat{n}_{1}}}\phi^{1}(\tau)\,,\quad\hat{\mathcal{O}}^{i}(\tau):=\frac{1}{\sqrt{\hat{n}_{i}}}\phi^{i}(\tau)\,.$
(2.110)

Recall that, according to (2.104), $\phi^{1}$ is the operator that couples to
the line defect, and it plays a role similar to $\phi^{6}$ in the case of the
Wilson line. The conformal dimension of $\hat{\mathcal{O}}^{1}$ can be
obtained from the $\beta$-function of the defect coupling:

$\hat{\Delta}_{1}=\left.1+\frac{\partial\beta_{h}}{\partial
h}\right|_{h=h_{\star}}=1-\frac{N-4}{\kappa_{1}}\varepsilon+{\mathcal{O}}(\varepsilon^{2})\,.$
(2.111)

On the other hand, $\hat{\mathcal{O}}^{i}$ plays the same role as
$\hat{\mathcal{O}}_{\hat{\Delta}=1}$ (see (2.74)) in the case of the
${\mathcal{N}}=4$ SYM theory. And similarly, the scaling dimension of
$\hat{\mathcal{O}}^{i}$ is protected:

$\hat{\Delta}_{i}=1\,.$ (2.112)

This operator is commonly referred to as the tilt operator. Note that for
$N=1$, there will be no tilt operator, but only
$\hat{\mathcal{O}}^{1}=\hat{\mathcal{O}}$ on the defect.

The next scalar operator of interest is the displacement operator, denoted by
$\mathbb{D}_{\mu}$. It is related to the bulk stress-energy tensor through the
Ward identity

$\partial_{\mu}T_{\mu\nu}=\delta^{d-1}(x^{\bot})\mathbb{D}_{\nu}\,,$ (2.113)

where the index $\nu$ on the right-hand side represents the transverse
components, i.e., $\nu=1,\ldots,d$. The displacement has a transverse spin
$s=1$ and a protected conformal dimension

$\hat{\Delta}_{\mathbb{D}}=2\,.$ (2.114)

It can be constructed by taking a transverse derivative of the field
$\hat{\mathcal{O}}^{1}$:

$\mathbb{D}_{\mu}:=\frac{1}{\sqrt{\hat{n}_{\mathbb{D}}}}\partial^{\bot}_{\mu}\mathcal{O}^{1}\>.$
(2.115)

While there exist additional operators $\partial^{\bot}_{\mu}\mathcal{O}^{i}$
corresponding to taking the transverse derivative of the tilt operator, we
will not be considering them here. Their correlators can however be computed
similarly to those involving the displacement operator.

At $\varepsilon\sim 0$, we encounter non-scalar operators that lie between the
fundamental scalars and the displacement. These are the elementary defect
Dirac fermions $\hat{\mathcal{O}}_{1,0}$ and $\hat{\mathcal{O}}_{0,1}$,
defined similarly to (2.95) as

$\hat{\mathcal{O}}_{1,0}^{a}(\bar{s},\tau):=\frac{1}{\sqrt{\hat{n}_{a}}}\hat{{\bar{\psi}}}^{a}(\bar{s},\tau)\,,\quad\hat{\mathcal{O}}_{0,1}^{a}(s,\tau):=\frac{1}{\sqrt{\hat{n}_{a}}}\hat{\psi}^{a}(s,\tau)\,.$
(2.116)

The conformal dimensions of these operators are given by

$\hat{\Delta}_{a}=\frac{3-\varepsilon}{2}+\hat{\gamma}_{a}\>,$ (2.117)

where the anomalous dimension $\hat{\gamma}_{a}$ can be extracted from the
two-point function. This will be discussed in Section 5.2.1.

We renormalize the defect coupling in a similar way to the bulk couplings. We
define the bare coupling $h_{0}$ in terms of the renormalized coupling $h$ as

$h_{0}=\mu^{\varepsilon/2}\,h\,Z_{h}\,.$ (2.118)

We require that (2.106) vanishes, or equivalently that

$\langle\\!\langle\,\mathcal{O}^{I}(x)\,\rangle\\!\rangle=\text{finite}\,.$
(2.119)

$Z_{h}$ is given in Appendix B of [5] up to $\mathcal{O}(\varepsilon^{-2})$.

Note that the one-point function of a single fermion $\mathcal{O}_{1,0}^{a}$
is zero. In Section 5.3.1, we show that the one-point function of the fermion
bilinear is also finite at ${\mathcal{O}}(\varepsilon)$, and therefore the
perturbative defect CFT generated by the magnetic line forms a consistent
model.

## CHAPTER 3 Bootstrapping holographic defect correlators

One of our general goals is to understand how the dynamics of a bulk theory
are affected by the presence of a defect. When the symmetry is broken,
correlation functions of local operators do not obey the rules of conformal
field theory anymore. New degrees of freedom arise: the operators acquire a
vacuum expectation value, while two-point functions are not fixed anymore
kinematically. Such systems have received a lot of attention in recent years
[112, 285, 319, 320, 321, 168, 175, 182, 184, 185], as they are relevant in
different fields of theoretical physics ranging from holography to condensed
matter.

In this chapter, we study two-point functions of scalar operators in the
presence of a line defect, using modern analytic bootstrap methods. In this
context, the Lorentzian inversion formula presented in (2.38) has led to
spectacular results in recent years. If one is interested in the correlator
itself, it is desirable to bypass the inversion formula and replace it with a
dispersion relation. The result is the position-space formula (2.39)
reminiscent of the dispersion relation (2.40) familiar from scattering
amplitudes. Here we wish to develop a similar formula for defect CFT
correlators. We show in particular that two-point functions of scalar
operators can be reconstructed from a single discontinuity. This is in
contrast to the bulk case, where the analogous relation contains a double
discontinuity.

We apply the dispersion relation to the special case where the bulk theory is
four-dimensional ${\mathcal{N}}=4$ Super Yang–Mills, with the defect being the
supersymmetric Wilson line introduced in Section 2.2. The reasons for this
choice are simple: ${\mathcal{N}}=4$ SYM is rich in symmetries and offers
powerful handles to conduct an analytical study. Moreover, the intricate web
of dualities surrounding it has provided insights into various regimes that
may be inaccessible in other models. There exists in particular a plethora of
exact results for half-BPS operators, while Wilson lines play a special role
in the context of the AdS/CFT correspondence. It is therefore natural to study
the dynamics of half-BPS operators and Wilson lines together in the
${\mathcal{N}}=4$ SYM theory.

The content of this chapter is mostly based on the publications [1, 3].
Section 3.4 presents the results of the unpublished notes [9].

###  3.1 An invitation

The defect CFT generated by the supersymmetric Wilson line in
${\mathcal{N}}=4$ SYM has emerged as a subject of considerable attention in
recent years. To situate the computations performed here within the ongoing
research, we begin our study by providing a concise overview of the relevant
literature. We collect useful results exact in the coupling constant, and
present our setup of interest: the two-point functions of half-BPS single-
trace operators. As explained in Section 2.1.2, this is the first correlator
to have non-trivial kinematics when considering bulk operators only in the
presence of the defect. This correlator has been considered perturbatively in
the weak- and strong-coupling regimes, and here we summarize the known results
to prepare the ground for the rest of this chapter.

####  3.1.1 Exact results

Numerous results have been obtained in ${\mathcal{N}}=4$ SYM, thanks to
conformal symmetry, supersymmetry, and integrability. In particular, the half-
BPS operators defined in (2.52) have been the objects of intense study since
the early days of the AdS/CFT correspondence. Their scaling dimensions are
protected and they do not need to be renormalized. This makes it possible to
obtain exact results, even for finite parameters $g$ and $N$.

From the conformal block point of view, we have seen in (2.30) that the OPE
coefficients $a_{\Delta}$ and $\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}$ play
a prominent role in the defect two-point functions. In the following, we list
the exact results for these coefficients in the coupling $\lambda$, while
taking the limit $N\to\infty$. Moreover, we present the strong-coupling limits
of bulk-defect two-point functions of half-BPS operators. The fact that such
expressions are reachable makes ${\mathcal{N}}=4$ SYM a natural playground for
bootstrap techniques, due to the abundance of symmetry and the amount of
information already present in the literature.

##### Three-point functions

Three-point functions of half-BPS operators are the first correlators to
consider. Like the two-point functions without defects, they do not receive
quantum corrections. At large $N$, their values were determined in [322]:

$\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}=\frac{\sqrt{\Delta_{1}\Delta_{2}\Delta_{3}}}{N}\,,$
(3.1)

following the definitions of Section 2.1.1 adapted for supersymmetric
theories. To the best of our knowledge, there does not exist a closed-form
expression for three-point functions at finite $N$. However, since they are
protected, it is easy to compute them case by case using the trace algebra of
$SU(N)$ as well as elementary Wick combinatorics.

In Section 3.3, we compare the outputs of our computations with three-point
functions of single- and double-trace operators. For this purpose, we note the
special case

$\lambda_{\Delta_{1}\Delta_{2}(\Delta_{1},\Delta_{2})}=1\,,$ (3.2)

where the third operator in the subscript should be understood as the double-
trace operator $\mathcal{O}_{\Delta_{1},\Delta_{2}}$ defined in (2.55). Note
that this formula is only valid at large $N$.

##### One-point functions

As we have seen in Section 2.1.2, operators acquire an expectation value in
the presence of a defect. For the case of a supersymmetric Wilson-line defect,
one-point function coefficients have been computed with localization
techniques [209, 323, 101], and at large $N$ they read

$a_{\Delta}=\frac{\sqrt{\lambda\Delta}}{\vphantom{I_{1}(\sqrt{\lambda})}2^{\Delta/2+1}N}\frac{I_{\Delta}(\sqrt{\lambda})}{I_{1}(\sqrt{\lambda})}\,,$
(3.3)

with $I_{\alpha}(z)$ the Bessel function defined in (2.63). We follow the
conventions defined in (2.67). (3.3) can be expanded in the weak- and strong-
coupling regimes, yielding

$\begin{split}a_{\Delta}&\overset{\lambda\sim
0}{=}\frac{\sqrt{\Delta\lambda^{\Delta}}}{2^{3\Delta/2}N\Gamma(\Delta+1)}\left(1-\lambda\frac{\Delta-1}{8(\Delta+1)}+\mathcal{O}(\lambda^{2})\right)\,,\\\
a_{\Delta}&\overset{\lambda\gg
1}{=}\frac{\sqrt{\lambda\Delta}}{2^{\Delta/2+1}N}\left(1-\frac{\Delta^{2}-1}{2\sqrt{\lambda}}+\mathcal{O}(1/\lambda)\right)\,.\end{split}$
(3.4)

Note that analytical expressions were also derived at finite $N$ [324], where
the Bessel functions are replaced by Laguerre polynomials, in a way similar to
(2.61).

The expression (3.3) generalizes straightforwardly to multi-trace operators.
In particular, we need for later use the case where the operator is a double-
trace:

$a_{(\Delta_{1},\Delta_{2})}=\frac{\sqrt{\Delta_{1}\Delta_{2}}\lambda}{2^{\frac{\Delta_{1}+\Delta_{2}+4}{2}}N^{2}}\frac{I_{\Delta_{1}+\Delta_{2}-1}(\sqrt{\lambda})}{\vphantom{2^{\frac{\Delta_{1}+\Delta_{2}+4}{2}}}I_{1}(\sqrt{\lambda})}\,.$
(3.5)

##### Bulk-defect two-point functions

We conclude this section by giving the strong-coupling limit of bulk-defect
two-point functions of half-BPS operators. These correlators are defined in
(2.78), and the OPE coefficients at strong coupling and in the planar limit
read [105]

$\begin{split}b_{\Delta_{1}\hat{\Delta}_{2}}=\
&\frac{\lambda^{\frac{2-\hat{\Delta}_{2}}{4}}}{2^{\frac{\Delta_{1}+2-3\hat{\Delta}_{2}}{2}}N}\frac{\sqrt{\Delta_{1}}}{\sqrt{\hat{\Delta}_{2}!}}\frac{\Gamma\left(\frac{\Delta_{1}+\hat{\Delta}_{2}+1}{2}\right)}{\Gamma\left(\frac{\Delta_{1}-\hat{\Delta}_{2}+1}{2}\right)}\\\
&\times\left(1+\frac{4(1-\Delta_{1}^{2})+\hat{\Delta}_{2}(5+\hat{\Delta}_{2})}{8\sqrt{\lambda}}+{\mathcal{O}}(\lambda^{-1})\right)\,.\end{split}$
(3.6)

In the course of our computation, we encounter the OPE coefficients
corresponding to bulk-defect two-point functions with double-trace defect
operators, defined as

$\hat{\mathcal{O}}_{0,\hat{\Delta}}(\hat{u},\tau):=\mathcal{W}_{\ell}[\
]\,\operatorname{tr}(\hat{u}\cdot\phi(x))^{\hat{\Delta}}\,,$ (3.7)

with $\mathcal{W}_{\ell}[\ ]$ meaning that no field is inserted on the Wilson
line. To the best of our knowledge, the coefficients
$b_{\Delta_{1}(0,\hat{\Delta}_{2})}$ have not appeared in the literature yet.
An interesting outcome of our analysis is a prediction for these values. We
should stress however that our correlators contain information about
infinitely many unprotected operators as well.

####  3.1.2 Two-point functions of half-BPS operators

We are interested here in the connected two-point function, defined as

$\begin{split}\langle\\!\langle\,\mathcal{O}_{\Delta_{1}}(u_{1},x_{1})\mathcal{O}_{\Delta_{2}}(u_{2},x_{2})\,\rangle\\!\rangle:=\
&\langle\,\mathcal{O}_{\Delta_{1}}(u_{1},x_{1})\mathcal{O}_{\Delta_{2}}(u_{2},x_{2})\mathcal{W}_{\ell}\,\rangle\\\
&-\langle\,\mathcal{O}_{\Delta_{1}}(u_{1},x_{1})\mathcal{O}_{\Delta_{2}}(u_{2},x_{2})\,\rangle\langle\,\mathcal{W}_{\ell}\,\rangle\,.\end{split}$
(3.8)

The explicit definition of the Wilson line is given in (2.59), while the half-
BPS operators are given in (2.52).

The structure of the correlator can be found in (2.68). Without loss of
generality, we consider as before $\Delta_{1}\leq\Delta_{2}$ to streamline the
notation. The reduced correlator ${\mathcal{F}}(z,{\bar{z}};\sigma)$ depends
on two spacetime cross-ratios and one $R$-symmetry variable $\sigma$.
Determining this function at strong coupling is the goal of this chapter.

Note that the second term on the right-hand side of (3.8) (the disconnected
part) is only non-zero for the case $\Delta_{1}=\Delta_{2}$, as implied by the
form of conformal two-point functions given in (2.8). We should stress that
the reduced correlator ${\mathcal{F}}(z,{\bar{z}};\sigma)$ depends on the
external dimensions $\Delta_{1},\Delta_{2}$, although not explicitly
indicated.

##### Superconformal Ward identities

We have seen in (2.70) that the reduced correlator
${\mathcal{F}}(z,{\bar{z}};\sigma)$ is a polynomial of degree $\Delta_{1}$
(for $\Delta_{1}\leq\Delta_{2}$) in $\sigma$, with each power corresponding to
a different $R$-symmetry channel. For example, in the simplest case
$\Delta_{1}=\Delta_{2}=2$, we have

${\mathcal{F}}(z,{\bar{z}};\sigma)=\Omega^{2}\,F_{0}(z,{\bar{z}})+\Omega\,F_{1}(z,{\bar{z}})+F_{2}(z,{\bar{z}})\,,$
(3.9)

where the definition of $\Omega\sim\sigma$ can be found in (2.71). As a
consequence, the full correlator is known if the individual $R$-symmetry
channels $F_{j}(z,{\bar{z}})$ are known. We often use the case
$\Delta_{1}=\Delta_{2}=2$ as an example to illustrate our methods in a simple
setting.

The $R$-symmetry channels are related to each other by symmetry. This can be
described concretely with the help of the superconformal Ward identities [285]

$\begin{split}\left.\left(\partial_{z}+\frac{1}{2}\partial_{\alpha}\right){\mathcal{F}}(z,{\bar{z}};\sigma)\right|_{z=\alpha}&=0\,,\\\
\left.\left(\partial_{{\bar{z}}}+\frac{1}{2}\partial_{\alpha}\right){\mathcal{F}}(z,{\bar{z}};\sigma)\right|_{{\bar{z}}=\alpha}&=0\,,\end{split}$
(3.10)

where we have defined for convenience the help $R$-symmetry variable $\alpha$,
which relates to $\sigma$ in the following way:

$\sigma:=-\frac{(1-\alpha)^{2}}{2\alpha}\,.$ (3.11)

The identities (3.10) impose powerful constraints since the differential
operators are of first order in the variables. They will be particularly
helpful in Section 3.3 to fix the remaining open coefficients of our
correlators.

##### Crossing equation

As introduced in Section 2.2.2, the two-point functions of half-BPS operators
can be expanded in superblocks. Superblocks are useful, as they encode the
contributions of superdescendants in the block expansion. The expansions in
the bulk and defect channels (2.72) and (2.73) lead to a supersymmetric
crossing equation, which takes the form

$\Omega^{\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{\vphantom{\hat{\mathcal{O}}}\mathcal{O}}\lambda_{\Delta_{1}\Delta_{2}\mathcal{O}}a_{\mathcal{O}}\,{\mathcal{G}}_{\mathcal{O}}(z,{\bar{z}},\sigma)=\sum_{\hat{\mathcal{O}}}b_{\Delta_{1}\hat{\mathcal{O}}}b_{\Delta_{2}\hat{\mathcal{O}}}\,\hat{\mathcal{G}}_{\hat{\mathcal{O}}}(z,{\bar{z}},\sigma)\,.$
(3.12)

This equation relates the bulk OPE coefficients
$\lambda_{\Delta_{1}\Delta_{2}\mathcal{O}}$ to the defect CFT data composed of
one-point functions $a_{\mathcal{O}}$ and bulk-defect correlators
$b_{\Delta\hat{\mathcal{O}}}$. Here we adjusted the notation of (2.72) and
(2.73) such that the sum is over the operators $\mathcal{O}$ and
$\hat{\mathcal{O}}$, instead of the quantum numbers that describe these
operators, namely $\Delta$, $\ell$ and $k$ on the left-hand side and
$\hat{\Delta}$, $s$ and $\hat{k}$ on the right-hand side.

##### The topological subsector

The superconformal Ward identities (3.10) admit an interesting special limit
of the kinematical variables. If one sets $z={\bar{z}}=\alpha$ in (3.9), we
find that the sum of the $R$-symmetry channels is equal to

${\mathcal{F}}(x,x,x)=\sum_{j=0}^{\Delta_{1}}\Omega_{\text{topo}}^{j}\,F_{\Delta_{1}-j}(x,x)=c_{-}\,\Theta(-x)+c_{+}\,\Theta(x)\,,$
(3.13)

where we have defined the topological variable

$\Omega_{\text{topo}}:=\left.\Omega\right|_{z={\bar{z}}=\alpha=x}=\frac{1}{2}\operatorname{sgn}(x)\,.$
(3.14)

This is called the topological sector since the correlator is constant on both
sides of the point $x=0$. The $c_{\pm}$ are functions of the coupling constant
$\lambda$ only, and they have been determined to all orders using localization
techniques (see (A.2) in [105]).

The study of the topological sector in the block expansion leads to an
interesting truncation of the bulk and defect OPEs. The resulting crossing
relations are called microboostrap equations. We take a closer look at the
topological sector in Section 3.4.

####  3.1.3 A first glimpse at weak coupling

To familiarize ourselves with the setup, we start by considering the two-point
functions of half-BPS operators in the weak-coupling regime. This was
considered in [10] for the simplest case $\Delta_{1}=\Delta_{2}=2$, and we
review the results at the lower orders for gaining some physical intuition.

##### Leading order

The first connected contribution was found to be

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{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{41.25641pt}{0.0pt}\pgfsys@lineto{41.25641pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{15.6491pt}{28.45276pt}\pgfsys@lineto{41.25641pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{41.25641pt}{28.45276pt}\pgfsys@moveto{46.37769pt}{28.45276pt}\pgfsys@curveto{46.37769pt}{31.28119pt}{44.08484pt}{33.57404pt}{41.25641pt}{33.57404pt}\pgfsys@curveto{38.42798pt}{33.57404pt}{36.13513pt}{31.28119pt}{36.13513pt}{28.45276pt}\pgfsys@curveto{36.13513pt}{25.62433pt}{38.42798pt}{23.33148pt}{41.25641pt}{23.33148pt}\pgfsys@curveto{44.08484pt}{23.33148pt}{46.37769pt}{25.62433pt}{46.37769pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{41.25641pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{15.6491pt}{23.33147pt}\pgfsys@moveto{17.78296pt}{23.33147pt}\pgfsys@curveto{17.78296pt}{24.50998pt}{16.8276pt}{25.46533pt}{15.6491pt}{25.46533pt}\pgfsys@curveto{14.47058pt}{25.46533pt}{13.51523pt}{24.50998pt}{13.51523pt}{23.33147pt}\pgfsys@curveto{13.51523pt}{22.15295pt}{14.47058pt}{21.1976pt}{15.6491pt}{21.1976pt}\pgfsys@curveto{16.8276pt}{21.1976pt}{17.78296pt}{22.15295pt}{17.78296pt}{23.33147pt}\pgfsys@closepath\pgfsys@moveto{15.6491pt}{23.33147pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{20.77039pt}{28.45276pt}\pgfsys@moveto{22.90425pt}{28.45276pt}\pgfsys@curveto{22.90425pt}{29.63127pt}{21.9489pt}{30.58662pt}{20.77039pt}{30.58662pt}\pgfsys@curveto{19.59187pt}{30.58662pt}{18.63652pt}{29.63127pt}{18.63652pt}{28.45276pt}\pgfsys@curveto{18.63652pt}{27.27425pt}{19.59187pt}{26.3189pt}{20.77039pt}{26.3189pt}\pgfsys@curveto{21.9489pt}{26.3189pt}{22.90425pt}{27.27425pt}{22.90425pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{20.77039pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{41.25641pt}{23.33147pt}\pgfsys@moveto{43.39027pt}{23.33147pt}\pgfsys@curveto{43.39027pt}{24.50998pt}{42.43492pt}{25.46533pt}{41.25641pt}{25.46533pt}\pgfsys@curveto{40.0779pt}{25.46533pt}{39.12254pt}{24.50998pt}{39.12254pt}{23.33147pt}\pgfsys@curveto{39.12254pt}{22.15295pt}{40.0779pt}{21.1976pt}{41.25641pt}{21.1976pt}\pgfsys@curveto{42.43492pt}{21.1976pt}{43.39027pt}{22.15295pt}{43.39027pt}{23.33147pt}\pgfsys@closepath\pgfsys@moveto{41.25641pt}{23.33147pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{36.13512pt}{28.45276pt}\pgfsys@moveto{38.26898pt}{28.45276pt}\pgfsys@curveto{38.26898pt}{29.63127pt}{37.31363pt}{30.58662pt}{36.13512pt}{30.58662pt}\pgfsys@curveto{34.9566pt}{30.58662pt}{34.00125pt}{29.63127pt}{34.00125pt}{28.45276pt}\pgfsys@curveto{34.00125pt}{27.27425pt}{34.9566pt}{26.3189pt}{36.13512pt}{26.3189pt}\pgfsys@curveto{37.31363pt}{26.3189pt}{38.26898pt}{27.27425pt}{38.26898pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{36.13512pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\color[rgb]{0.94921875,0.57421875,0.11328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@stroke{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}{}\pgfsys@moveto{15.6491pt}{0.0pt}\pgfsys@moveto{17.64099pt}{0.0pt}\pgfsys@curveto{17.64099pt}{1.1001pt}{16.74919pt}{1.9919pt}{15.6491pt}{1.9919pt}\pgfsys@curveto{14.549pt}{1.9919pt}{13.6572pt}{1.1001pt}{13.6572pt}{0.0pt}\pgfsys@curveto{13.6572pt}{-1.1001pt}{14.549pt}{-1.9919pt}{15.6491pt}{-1.9919pt}\pgfsys@curveto{16.74919pt}{-1.9919pt}{17.64099pt}{-1.1001pt}{17.64099pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{15.6491pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\color[rgb]{0.94921875,0.57421875,0.11328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@stroke{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}{}\pgfsys@moveto{41.25641pt}{0.0pt}\pgfsys@moveto{43.2483pt}{0.0pt}\pgfsys@curveto{43.2483pt}{1.1001pt}{42.3565pt}{1.9919pt}{41.25641pt}{1.9919pt}\pgfsys@curveto{40.15631pt}{1.9919pt}{39.26451pt}{1.1001pt}{39.26451pt}{0.0pt}\pgfsys@curveto{39.26451pt}{-1.1001pt}{40.15631pt}{-1.9919pt}{41.25641pt}{-1.9919pt}\pgfsys@curveto{42.3565pt}{-1.9919pt}{43.2483pt}{-1.1001pt}{43.2483pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{41.25641pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{\lambda}{4N^{2}}\Omega\,.$
(3.15)

The half-BPS operators are represented by  with the smaller circles indicating
the number of scalars inside the trace (here two per operator).

In terms of the $R$-symmetry channels defined in (3.9), this is equivalent to

$F_{0}(z,{\bar{z}})=F_{2}(z,{\bar{z}})=0\,,\quad
F_{1}(z,{\bar{z}})=\frac{\lambda}{4N^{2}}\,.$ (3.16)

In the topological limit, we observe as anticipated in (3.13) that all the
dependence on the kinematic variables drops apart from the sign function:

${\mathcal{F}}(x,x,x)=-\frac{\lambda}{4N^{2}}\operatorname{sgn}(x)\,.$ (3.17)

##### Next-to-leading order

At next-to-leading order (NLO), only the channel without bulk vertices could
be determined in a closed form:

$\displaystyle F_{2}(z,{\bar{z}})$ $\displaystyle=\leavevmode\hbox
to58.91pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
1.0pt\lower-2.1919pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{11.9501pt}{0.0pt}\pgfsys@lineto{11.9501pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{19.34766pt}{0.0pt}\pgfsys@lineto{19.34766pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{37.55785pt}{0.0pt}\pgfsys@lineto{37.55785pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{44.9554pt}{0.0pt}\pgfsys@lineto{44.9554pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{15.6491pt}{28.45276pt}\pgfsys@moveto{20.77037pt}{28.45276pt}\pgfsys@curveto{20.77037pt}{31.28119pt}{18.47752pt}{33.57404pt}{15.6491pt}{33.57404pt}\pgfsys@curveto{12.82066pt}{33.57404pt}{10.52782pt}{31.28119pt}{10.52782pt}{28.45276pt}\pgfsys@curveto{10.52782pt}{25.62433pt}{12.82066pt}{23.33148pt}{15.6491pt}{23.33148pt}\pgfsys@curveto{18.47752pt}{23.33148pt}{20.77037pt}{25.62433pt}{20.77037pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{15.6491pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{41.25641pt}{28.45276pt}\pgfsys@moveto{46.37769pt}{28.45276pt}\pgfsys@curveto{46.37769pt}{31.28119pt}{44.08484pt}{33.57404pt}{41.25641pt}{33.57404pt}\pgfsys@curveto{38.42798pt}{33.57404pt}{36.13513pt}{31.28119pt}{36.13513pt}{28.45276pt}\pgfsys@curveto{36.13513pt}{25.62433pt}{38.42798pt}{23.33148pt}{41.25641pt}{23.33148pt}\pgfsys@curveto{44.08484pt}{23.33148pt}{46.37769pt}{25.62433pt}{46.37769pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{41.25641pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{11.9501pt}{25.03856pt}\pgfsys@moveto{14.08397pt}{25.03856pt}\pgfsys@curveto{14.08397pt}{26.21707pt}{13.12862pt}{27.17242pt}{11.9501pt}{27.17242pt}\pgfsys@curveto{10.77159pt}{27.17242pt}{9.81624pt}{26.21707pt}{9.81624pt}{25.03856pt}\pgfsys@curveto{9.81624pt}{23.86005pt}{10.77159pt}{22.9047pt}{11.9501pt}{22.9047pt}\pgfsys@curveto{13.12862pt}{22.9047pt}{14.08397pt}{23.86005pt}{14.08397pt}{25.03856pt}\pgfsys@closepath\pgfsys@moveto{11.9501pt}{25.03856pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{19.34766pt}{25.03856pt}\pgfsys@moveto{21.48152pt}{25.03856pt}\pgfsys@curveto{21.48152pt}{26.21707pt}{20.52617pt}{27.17242pt}{19.34766pt}{27.17242pt}\pgfsys@curveto{18.16914pt}{27.17242pt}{17.21379pt}{26.21707pt}{17.21379pt}{25.03856pt}\pgfsys@curveto{17.21379pt}{23.86005pt}{18.16914pt}{22.9047pt}{19.34766pt}{22.9047pt}\pgfsys@curveto{20.52617pt}{22.9047pt}{21.48152pt}{23.86005pt}{21.48152pt}{25.03856pt}\pgfsys@closepath\pgfsys@moveto{19.34766pt}{25.03856pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}{}\pgfsys@moveto{37.55785pt}{25.03856pt}\pgfsys@moveto{39.69171pt}{25.03856pt}\pgfsys@curveto{39.69171pt}{26.21707pt}{38.73636pt}{27.17242pt}{37.55785pt}{27.17242pt}\pgfsys@curveto{36.37933pt}{27.17242pt}{35.42398pt}{26.21707pt}{35.42398pt}{25.03856pt}\pgfsys@curveto{35.42398pt}{23.86005pt}{36.37933pt}{22.9047pt}{37.55785pt}{22.9047pt}\pgfsys@curveto{38.73636pt}{22.9047pt}{39.69171pt}{23.86005pt}{39.69171pt}{25.03856pt}\pgfsys@closepath\pgfsys@moveto{37.55785pt}{25.03856pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{44.9554pt}{25.03856pt}\pgfsys@moveto{47.08926pt}{25.03856pt}\pgfsys@curveto{47.08926pt}{26.21707pt}{46.13391pt}{27.17242pt}{44.9554pt}{27.17242pt}\pgfsys@curveto{43.77689pt}{27.17242pt}{42.82153pt}{26.21707pt}{42.82153pt}{25.03856pt}\pgfsys@curveto{42.82153pt}{23.86005pt}{43.77689pt}{22.9047pt}{44.9554pt}{22.9047pt}\pgfsys@curveto{46.13391pt}{22.9047pt}{47.08926pt}{23.86005pt}{47.08926pt}{25.03856pt}\pgfsys@closepath\pgfsys@moveto{44.9554pt}{25.03856pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{0.94921875,0.57421875,0.11328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@stroke{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\color[rgb]{0.94921875,0.57421875,0.11328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@stroke{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}{}\pgfsys@moveto{37.55785pt}{0.0pt}\pgfsys@moveto{39.54974pt}{0.0pt}\pgfsys@curveto{39.54974pt}{1.1001pt}{38.65794pt}{1.9919pt}{37.55785pt}{1.9919pt}\pgfsys@curveto{36.45775pt}{1.9919pt}{35.56595pt}{1.1001pt}{35.56595pt}{0.0pt}\pgfsys@curveto{35.56595pt}{-1.1001pt}{36.45775pt}{-1.9919pt}{37.55785pt}{-1.9919pt}\pgfsys@curveto{38.65794pt}{-1.9919pt}{39.54974pt}{-1.1001pt}{39.54974pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{37.55785pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\color[rgb]{0.94921875,0.57421875,0.11328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.94921875,0.57421875,0.11328125}\pgfsys@color@rgb@stroke{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.94921875}{0.57421875}{0.11328125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.94921875,0.57421875,0.11328125}{}\pgfsys@moveto{44.9554pt}{0.0pt}\pgfsys@moveto{46.9473pt}{0.0pt}\pgfsys@curveto{46.9473pt}{1.1001pt}{46.0555pt}{1.9919pt}{44.9554pt}{1.9919pt}\pgfsys@curveto{43.8553pt}{1.9919pt}{42.9635pt}{1.1001pt}{42.9635pt}{0.0pt}\pgfsys@curveto{42.9635pt}{-1.1001pt}{43.8553pt}{-1.9919pt}{44.9554pt}{-1.9919pt}\pgfsys@curveto{46.0555pt}{-1.9919pt}{46.9473pt}{-1.1001pt}{46.9473pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{44.9554pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=\frac{\lambda}{32\pi^{2}N^{2}}\biggl{\\{}\frac{\pi^{2}}{4}+2\log
2\left(H_{-1}(r)+H_{1}(r)\right)$ $\displaystyle\phantom{=\
}+\left(H_{-1,0}(r)+H_{1,0}(r)\right)-2\left(H_{-1,-1}(r)+H_{1,-1}(r)\right)\biggr{\\}}\,,$
(3.18)

where the functions $H_{\vec{a}}(x)$ are harmonic polylogarithm (HPL)
functions, defined in (C.3) and for which important properties are listed in
Appendix C.1.2. It was possible to compute this diagram, because in the
conformal frame (2.27) it depends only on the distance $r=z{\bar{z}}$ between
the defect and one of the operators. It is interesting to note that this
expression contains only transcendental functions of weight $w=2$, without
rational functions in front.

It has not been possible to determine the other two $R$-symmetry channels
$F_{0}$ and $F_{1}$ in a closed form. Instead, the OPE coefficients of the
block expansions were fixed by using the constraints imposed by the
superconformal Ward identities (3.10). The remaining degrees of freedom were
fixed by high-precision numerical integration. In this sense, a full solution
of the correlator was found, although it would be desirable to have an
analytical expression in terms of transcendental functions like (3.18). In
fact, little is known in general about such correlators at weak coupling. In
the $O(N)$ model, it was shown recently that the two-point function of
elementary scalars is equal to the derivative of a bulk block at NLO [184,
185]. These blocks are not known in closed form, and it might be the case that
the two problems are intimately connected.111This point is discussed in more
detail in Section 5.3.2.

To summarize, the two-point functions in the presence of the line defect at
weak coupling turned out to be a significantly more complicated object to
study than the four-point function of bulk operators. We now consider the
strong-coupling end of the correlator, with the hope that the situation is
more favorable in this regime.

####  3.1.4 The supergravity perspective

The AdS/CFT correspondence is introduced in Section 2.2.1. There we have seen
that the strong-coupling limit of ${\mathcal{N}}=4$ SYM admits a description
in terms of classical IIB supergravity. Although this formulation will not be
used to carry out our calculations, it is useful for understanding the
perturbative structure of our correlator in the strong-coupling regime.

The dual of a supersymmetric Wilson loop is a string worldsheet extending
inside AdS5, whose boundary corresponds to the path of the loop [88, 325]. It
can be represented graphically as the Witten diagram

$\langle\,\mathcal{W}_{\ell}\,\rangle=\ \leavevmode\hbox to46.32pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =1\,.$ (3.19)

Here the red curved line corresponds to the string worldsheet, while the black
circle represents the boundary of the AdS spacetime. In Section 2.2.2, we
discussed the fact that $\langle\,\mathcal{W}_{\ell}\,\rangle$ does not
receive corrections and thus (3.19) remains exact, even at higher-loop orders.

##### Leading order

We are interested in the interplay between the supersymmetric Wilson line and
local operators. In the holographic description, the single-trace half-BPS
operators $\mathcal{O}_{\Delta}$ are dual to certain Kaluza-Klein modes
arising from the compactification of the IIB action on $S^{5}$. The two-point
function consists of two of these modes, sourced at the boundary of AdS5 and
propagating through the bulk before being absorbed by the string worldsheet.
Since we subtract in (3.8) the disconnected contributions, the first diagram
at strong coupling corresponding to this process is

$\displaystyle=\langle\\!\langle\,\mathcal{O}_{\Delta_{1}}(u_{1},x_{1})\,\rangle\\!\rangle\langle\\!\langle\,\mathcal{O}_{\Delta_{2}}(u_{2},x_{2})\,\rangle\\!\rangle$
$\displaystyle=\frac{\lambda}{\vphantom{2^{\frac{\Delta_{1}+\Delta_{2}+4}{2}}}N^{2}}\frac{\sqrt{\Delta_{1}\Delta_{2}}}{2^{\frac{\Delta_{1}+\Delta_{2}+4}{2}}}\frac{(u_{1}\cdot\theta)^{\Delta_{1}}(u_{2}\cdot\theta)^{\Delta_{2}}}{|x_{1}^{\perp}|^{\Delta_{2}}|x_{2}^{\perp}|^{\Delta_{1}}}\,,$
(3.20)

where the lines propagating in the bulk space correspond to $\Delta_{j}$
propagators. This is a factorized diagram of one-point functions since the
operators do not interact in the bulk. The one-point functions of
$\mathcal{O}_{\Delta}$ in the presence of the Wilson line are

$\leavevmode\hbox to48.26pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@moveto{-9.38902pt}{0.0pt}\pgfsys@curveto{-9.38902pt}{1.1001pt}{-10.28082pt}{1.9919pt}{-11.38092pt}{1.9919pt}\pgfsys@curveto{-12.48102pt}{1.9919pt}{-13.37282pt}{1.1001pt}{-13.37282pt}{0.0pt}\pgfsys@curveto{-13.37282pt}{-1.1001pt}{-12.48102pt}{-1.9919pt}{-11.38092pt}{-1.9919pt}\pgfsys@curveto{-10.28082pt}{-1.9919pt}{-9.38902pt}{-1.1001pt}{-9.38902pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@moveto{24.89615pt}{0.0pt}\pgfsys@curveto{24.89615pt}{1.17851pt}{23.9408pt}{2.13387pt}{22.76228pt}{2.13387pt}\pgfsys@curveto{21.58377pt}{2.13387pt}{20.62842pt}{1.17851pt}{20.62842pt}{0.0pt}\pgfsys@curveto{20.62842pt}{-1.17851pt}{21.58377pt}{-2.13387pt}{22.76228pt}{-2.13387pt}\pgfsys@curveto{23.9408pt}{-2.13387pt}{24.89615pt}{-1.17851pt}{24.89615pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
=\langle\\!\langle\,\mathcal{O}_{\Delta}(u,x)\,\rangle\\!\rangle=\frac{\sqrt{\lambda\Delta}}{2^{\frac{\Delta+2}{2}}N}\frac{(u\cdot\theta)^{\Delta}}{\vphantom{2^{\frac{\Delta+2}{2}}}|x^{\perp}|^{\Delta}}\,.$
(3.21)

The diagrams (3.20) and (3.21) have been computed in [326, 101, 102, 327],
where they were shown to match the strong-coupling limit obtained through
localization.

##### Next-to-leading order

The first non-trivial correction to the two-point function contains an
interaction vertex in the bulk. In the exchanged line, one sums over all
Kaluza-Klein modes that can couple to two $\mathcal{O}_{\Delta}$’s.
Schematically, we have the bulk-exchange diagram

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23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@moveto{20.62798pt}{-12.80365pt}\pgfsys@curveto{20.62798pt}{-11.62514pt}{19.67262pt}{-10.66978pt}{18.49411pt}{-10.66978pt}\pgfsys@curveto{17.3156pt}{-10.66978pt}{16.36024pt}{-11.62514pt}{16.36024pt}{-12.80365pt}\pgfsys@curveto{16.36024pt}{-13.98216pt}{17.3156pt}{-14.93752pt}{18.49411pt}{-14.93752pt}\pgfsys@curveto{19.67262pt}{-14.93752pt}{20.62798pt}{-13.98216pt}{20.62798pt}{-12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@moveto{20.62798pt}{12.80365pt}\pgfsys@curveto{20.62798pt}{13.98216pt}{19.67262pt}{14.93752pt}{18.49411pt}{14.93752pt}\pgfsys@curveto{17.3156pt}{14.93752pt}{16.36024pt}{13.98216pt}{16.36024pt}{12.80365pt}\pgfsys@curveto{16.36024pt}{11.62514pt}{17.3156pt}{10.66978pt}{18.49411pt}{10.66978pt}\pgfsys@curveto{19.67262pt}{10.66978pt}{20.62798pt}{11.62514pt}{20.62798pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (3.22)

There are two additional diagrams

$\leavevmode\hbox to46.32pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@lineto{-12.51927pt}{8.5359pt}\pgfsys@lineto{-6.82881pt}{8.5359pt}\pgfsys@lineto{-6.82881pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{-6.82881pt}{8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@lineto{-12.51927pt}{8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@lineto{18.49411pt}{-12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.51927pt}{8.5359pt}\pgfsys@lineto{18.49411pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@moveto{-10.52737pt}{-8.5359pt}\pgfsys@curveto{-10.52737pt}{-7.4358pt}{-11.41917pt}{-6.544pt}{-12.51927pt}{-6.544pt}\pgfsys@curveto{-13.61937pt}{-6.544pt}{-14.51117pt}{-7.4358pt}{-14.51117pt}{-8.5359pt}\pgfsys@curveto{-14.51117pt}{-9.636pt}{-13.61937pt}{-10.5278pt}{-12.51927pt}{-10.5278pt}\pgfsys@curveto{-11.41917pt}{-10.5278pt}{-10.52737pt}{-9.636pt}{-10.52737pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{-12.51927pt}{-8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{-12.51927pt}{8.5359pt}\pgfsys@moveto{-10.52737pt}{8.5359pt}\pgfsys@curveto{-10.52737pt}{9.636pt}{-11.41917pt}{10.5278pt}{-12.51927pt}{10.5278pt}\pgfsys@curveto{-13.61937pt}{10.5278pt}{-14.51117pt}{9.636pt}{-14.51117pt}{8.5359pt}\pgfsys@curveto{-14.51117pt}{7.4358pt}{-13.61937pt}{6.544pt}{-12.51927pt}{6.544pt}\pgfsys@curveto{-11.41917pt}{6.544pt}{-10.52737pt}{7.4358pt}{-10.52737pt}{8.5359pt}\pgfsys@closepath\pgfsys@moveto{-12.51927pt}{8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@moveto{20.62798pt}{-12.80365pt}\pgfsys@curveto{20.62798pt}{-11.62514pt}{19.67262pt}{-10.66978pt}{18.49411pt}{-10.66978pt}\pgfsys@curveto{17.3156pt}{-10.66978pt}{16.36024pt}{-11.62514pt}{16.36024pt}{-12.80365pt}\pgfsys@curveto{16.36024pt}{-13.98216pt}{17.3156pt}{-14.93752pt}{18.49411pt}{-14.93752pt}\pgfsys@curveto{19.67262pt}{-14.93752pt}{20.62798pt}{-13.98216pt}{20.62798pt}{-12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@moveto{20.62798pt}{12.80365pt}\pgfsys@curveto{20.62798pt}{13.98216pt}{19.67262pt}{14.93752pt}{18.49411pt}{14.93752pt}\pgfsys@curveto{17.3156pt}{14.93752pt}{16.36024pt}{13.98216pt}{16.36024pt}{12.80365pt}\pgfsys@curveto{16.36024pt}{11.62514pt}{17.3156pt}{10.66978pt}{18.49411pt}{10.66978pt}\pgfsys@curveto{19.67262pt}{10.66978pt}{20.62798pt}{11.62514pt}{20.62798pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
,\qquad\leavevmode\hbox to46.32pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{18.49411pt}{-12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{18.49411pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\color[rgb]{0.703125,0.1875,0.1875}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@stroke{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@moveto{-9.38902pt}{0.0pt}\pgfsys@curveto{-9.38902pt}{1.1001pt}{-10.28082pt}{1.9919pt}{-11.38092pt}{1.9919pt}\pgfsys@curveto{-12.48102pt}{1.9919pt}{-13.37282pt}{1.1001pt}{-13.37282pt}{0.0pt}\pgfsys@curveto{-13.37282pt}{-1.1001pt}{-12.48102pt}{-1.9919pt}{-11.38092pt}{-1.9919pt}\pgfsys@curveto{-10.28082pt}{-1.9919pt}{-9.38902pt}{-1.1001pt}{-9.38902pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@moveto{20.62798pt}{-12.80365pt}\pgfsys@curveto{20.62798pt}{-11.62514pt}{19.67262pt}{-10.66978pt}{18.49411pt}{-10.66978pt}\pgfsys@curveto{17.3156pt}{-10.66978pt}{16.36024pt}{-11.62514pt}{16.36024pt}{-12.80365pt}\pgfsys@curveto{16.36024pt}{-13.98216pt}{17.3156pt}{-14.93752pt}{18.49411pt}{-14.93752pt}\pgfsys@curveto{19.67262pt}{-14.93752pt}{20.62798pt}{-13.98216pt}{20.62798pt}{-12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@moveto{20.62798pt}{12.80365pt}\pgfsys@curveto{20.62798pt}{13.98216pt}{19.67262pt}{14.93752pt}{18.49411pt}{14.93752pt}\pgfsys@curveto{17.3156pt}{14.93752pt}{16.36024pt}{13.98216pt}{16.36024pt}{12.80365pt}\pgfsys@curveto{16.36024pt}{11.62514pt}{17.3156pt}{10.66978pt}{18.49411pt}{10.66978pt}\pgfsys@curveto{19.67262pt}{10.66978pt}{20.62798pt}{11.62514pt}{20.62798pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ ,$ (3.23)

which are respectively defect-exchange and contact diagrams.

A quantitative description of these diagrams can be found in [187]. Instead of
an explicit calculation using the effective action in AdS${}_{5}\times S^{5}$,
here we bootstrap the result using the dispersion relation derived in Section
3.2. This formula reconstructs a correlator from its singular part, which is
mathematically captured by a discontinuity:

$\leavevmode\hbox to46.32pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\color[rgb]{1,0.67578125,0.67578125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.67578125,0.67578125}\pgfsys@color@rgb@stroke{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0.67578125,0.67578125}{}\pgfsys@moveto{-16.09541pt}{16.09541pt}\pgfsys@curveto{-9.8187pt}{5.2238pt}{-9.8187pt}{-5.2238pt}{-16.09541pt}{-16.09541pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@moveto{20.62798pt}{12.80365pt}\pgfsys@curveto{20.62798pt}{13.98216pt}{19.67262pt}{14.93752pt}{18.49411pt}{14.93752pt}\pgfsys@curveto{17.3156pt}{14.93752pt}{16.36024pt}{13.98216pt}{16.36024pt}{12.80365pt}\pgfsys@curveto{16.36024pt}{11.62514pt}{17.3156pt}{10.66978pt}{18.49411pt}{10.66978pt}\pgfsys@curveto{19.67262pt}{10.66978pt}{20.62798pt}{11.62514pt}{20.62798pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
\sim\int\operatorname{Disc}\ \leavevmode\hbox to46.32pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip
23.16228pt\lower-23.16228pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{1}{0.67578125}{0.67578125}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{12.57141pt}{12.57141pt}{22.76228pt}{0.0pt}{22.76228pt}\pgfsys@curveto{-12.57141pt}{22.76228pt}{-22.76228pt}{12.57141pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-12.57141pt}{-12.57141pt}{-22.76228pt}{0.0pt}{-22.76228pt}\pgfsys@curveto{12.57141pt}{-22.76228pt}{22.76228pt}{-12.57141pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-11.38092pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{18.49411pt}{-12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{18.49411pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.703125,0.1875,0.1875}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.703125}{0.1875}{0.1875}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@moveto{20.62798pt}{-12.80365pt}\pgfsys@curveto{20.62798pt}{-11.62514pt}{19.67262pt}{-10.66978pt}{18.49411pt}{-10.66978pt}\pgfsys@curveto{17.3156pt}{-10.66978pt}{16.36024pt}{-11.62514pt}{16.36024pt}{-12.80365pt}\pgfsys@curveto{16.36024pt}{-13.98216pt}{17.3156pt}{-14.93752pt}{18.49411pt}{-14.93752pt}\pgfsys@curveto{19.67262pt}{-14.93752pt}{20.62798pt}{-13.98216pt}{20.62798pt}{-12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{-12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@moveto{20.62798pt}{12.80365pt}\pgfsys@curveto{20.62798pt}{13.98216pt}{19.67262pt}{14.93752pt}{18.49411pt}{14.93752pt}\pgfsys@curveto{17.3156pt}{14.93752pt}{16.36024pt}{13.98216pt}{16.36024pt}{12.80365pt}\pgfsys@curveto{16.36024pt}{11.62514pt}{17.3156pt}{10.66978pt}{18.49411pt}{10.66978pt}\pgfsys@curveto{19.67262pt}{10.66978pt}{20.62798pt}{11.62514pt}{20.62798pt}{12.80365pt}\pgfsys@closepath\pgfsys@moveto{18.49411pt}{12.80365pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (3.24)

A crucial property of holographic CFTs is that the discontinuity is
dramatically simpler than the correlator. In particular, we show in the course
of the computation that the discontinuity receives corrections only from a
finite number of single-trace operators exchanged in the bulk. Each of these
contributions is schematically a product of a (tree-level) one-point function
and a three-point function:

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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