\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (38)

Although in general the sum contains infinitely many terms, for many cases of
interest it truncates. More specifically, whenever
$\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}$ the exchange diagram reads

$\displaystyle
E_{\Delta_{1}\Delta_{2}}^{\Delta,0}(\xi,\eta)=\sum_{n=1}^{\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}}\frac{\left(-\frac{\Delta_{1}+\Delta_{2}-\Delta-2}{2}\right)_{n-1}\left(-\frac{\Delta_{1}+\Delta_{2}+\Delta-d-2}{2}\right)_{n-1}}{4(1-\Delta_{1})_{n}(1-\Delta_{2})_{n}}\frac{C_{\Delta_{1}-n,\Delta_{2}-n}(\xi,\eta)}{\xi^{n}}\,.$
(39)

We do not repeat the derivation of this formula here, because it follows
easily from [62], and it was reviewed for Witten diagrams with branes in [55].
For general scaling dimensions, we calculate the bulk-exchange diagram in
Mellin space in section 4.3.

Besides scalars, there can also be higher-spin fields in the
$\mathrm{AdS}_{d+1}$ effective action. For example, a spin-two field
$\phi_{\mu\nu}$ couples to the brane as
$\int_{\mathrm{AdS}_{p+1}}g^{ij}\phi_{ij}$, where we contract orthogonal
indices $i,j=p+1,\ldots,d$. As a result, the CFT operator dual to
$\phi_{\mu\nu}$ acquires a one-point function. Furthermore, if the spin-two
field couples to the scalars
$\sim\int_{\mathrm{AdS}_{d+1}}\phi_{\mu\nu}\nabla^{\mu}\phi_{1}\nabla^{\nu}\phi_{2}$,
then it can be exchanged in a two-point function

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(40)

In many situations of interest, the spin-two diagram reduces to a sum of
contact diagrams. This has been discussed for equal external fields in [68,
17], and in appendix A we generalize the discussion to unequal fields.444Spin-
two diagrams for unequal fields were also discussed in [69], but because our
formulation is more streamlined than the reference, we believe it is valuable
to include a detailed discussion in the appendix. Furthermore, we also explain
how to apply the truncation in the presence of branes, which is the focus of
this work. Because the appendix is somewhat technical, we attach to this
publication a Mathematica notebook that computes spin-two diagrams
$E_{\Delta_{1}\Delta_{2}}^{\Delta,2}$.

As a final comment, note that the diagram $E_{\Delta_{1}\Delta_{1}}^{d,2}$,
which corresponds to the exchange of a bulk graviton, contains spurious
divergences. However, we found that the limit
$E_{\Delta_{1}\Delta_{1}}^{d,2}\equiv\lim_{{\varepsilon}\to
0}E_{\Delta_{1}+{\varepsilon},\Delta_{1}+{\varepsilon}}^{d+2{\varepsilon},2}$
leads to sensible results. To motivate this choice of limit, note that it
keeps $\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}$, which is required
for the formulas in appendix A to make sense. As illustration of graviton- and
massive-exchange diagrams, for $d=4$ and $p=1$ we have

$\displaystyle E^{42}_{22}=\frac{4}{3\xi}\big{(}\eta
C_{22}-C_{13}\big{)}\,,\qquad E^{52}_{32}=\frac{3}{5\xi}\big{(}2\eta
C_{23}-C_{23}-C_{14}\big{)}\,.$ (41)

###  3.4 Defect-exchange diagram

####  3.4.1 Scalar diagram

Lastly, we consider a diagram where a brane field is excited by couplings
$\int_{\mathrm{AdS}_{p+1}}\phi_{i}\,\widehat{\phi}$. More specifically, we
introduce the defect exchange diagram

$x_{1}$$x_{2}$$\widehat{z}_{1}$$\widehat{z}_{2}$${\widehat{\Delta}}$
$\displaystyle=\;\int_{\widehat{z}_{1},\widehat{z}_{2}}K_{\Delta_{1}}(x_{1},\widehat{z}_{1})K_{\Delta_{2}}(x_{2},\widehat{z}_{2})\widehat{G}_{\widehat{\Delta}}(\widehat{z}_{1},\widehat{z}_{2})\;\equiv\;\frac{\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}\Delta_{2}}(\xi,\eta)}{|x^{i}_{1}|^{\Delta_{1}}|x_{2}^{i}|^{\Delta_{2}}}\,,$
(42)

where for now we assume the defect field is not charged under transverse
$SO(d-p)$ rotations. The result depends only on $\xi+2\eta$, or equivalently,
it depends only on the cross-ratio $r$ defined in (16). This follows by
observing that the integrand (42) contains no terms of the form
$x_{1}^{i}x_{2}^{i}$.

Since the exchange diagram depends on a single cross-ratio $r$, it is very
efficient to compute it by solving a differential equation. To obtain the
differential equation in question, we employ the method of [70, 56]. Start
giving a name to the $\widehat{z}_{1}$ integral in (42)

$\displaystyle
I(x_{1},\widehat{z}_{2})=\int_{\widehat{z}_{1}}\widehat{G}_{\widehat{\Delta}}(\widehat{z}_{1},\widehat{z}_{2})K_{\Delta_{1}}(x_{1},\widehat{z}_{1})\,.$
(43)

Because this integral preserves the $p$-dimensional conformal group, we have
the relation

$\displaystyle\big{(}\widehat{\mathbf{L}}_{1}+\widehat{\mathcal{L}}_{\widehat{z}_{2}}\big{)}_{AB}\,I(x_{1},\widehat{z}_{2})=0\,,$
(44)

where $\widehat{\mathbf{L}}_{1}$ is a generator of $SO(p+1,1)$ acting on
operator 1, and $\widehat{\mathcal{L}}_{\widehat{z}}$ is the
$\mathrm{AdS}_{p+1}$ symmetry generator. It then follows that

$\displaystyle\left(\frac{1}{2}\,\widehat{\mathbf{L}}_{1}^{2}+{\widehat{\Delta}}({\widehat{\Delta}}-p)\right)I(x_{1},\widehat{z}_{2})=\left(-\nabla^{2}_{\widehat{z}_{2}}+{\widehat{\Delta}}({\widehat{\Delta}}-p)\right)I(x_{1},\widehat{z}_{2})=K_{\Delta_{1}}(x_{1},\widehat{z}_{2})\,,$
(45)

where the last equality is a consequence of the equations of motion for the
brane-to-brane propagator, see (7) with $d\to p$. The final step is to
multiply (45) by $K_{\Delta_{2}}(x_{2},\widehat{z}_{2})$ and integrate over
$\widehat{z}_{2}$. The left-hand side gives a differential operator acting on
$\widehat{E}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}$, and the right-
hand side gives a contact diagram

$\displaystyle\left[-r\partial_{r}r\partial_{r}+\frac{pr(r^{2}+1)}{(1-r)(1+r)}\,\partial_{r}+{\widehat{\Delta}}({\widehat{\Delta}}-p)\right]\widehat{E}_{\Delta_{1},\Delta_{2}}^{\widehat{\Delta},0}(r)=C_{\Delta_{1},\Delta_{2}}(r)\,.$
(46)

In deriving the differential operator, we used that the diagram depends only
on $r$ to simplify expressions. The most general solution of (46) depends on
two free parameters, but they can always be fixed demanding the following
behavior

$\displaystyle\widehat{E}_{\Delta_{1},\Delta_{2}}^{{\widehat{\Delta}},0}(r)=\frac{\pi^{p/2}\Gamma({\widehat{\Delta}})\Gamma\\!\left(\frac{\Delta_{1}-{\widehat{\Delta}}}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}-{\widehat{\Delta}}}{2}\right)\Gamma\\!\left(\frac{\Delta_{1}+{\widehat{\Delta}}-p}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}+{\widehat{\Delta}}-p}{2}\right)}{8\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma({\widehat{\Delta}}+1-\frac{p}{2})}\,r^{{\widehat{\Delta}}}+O(r^{{\widehat{\Delta}}+1})\,.$
(47)

To derive this equation we used the split representation, as in section 4.4.
First one computes (90) exactly in $r$, and then closes the $\nu$ contour to
either left or right. Picking the leading residue gives the contribution (47).

In practice, we have found that solving the ODE provides a very efficient
computational method. Recall that for integer $\Delta_{1}$ and $\Delta_{2}$
the contact diagram is a rational function of $r$ and $\log r$, see (36). In
the examples relevant to section 5, the solution to the ODE also turns out to
be a rational function, but now including $\log r$ and $\log(r+1)$. For
example, in $d=4$ and $p=1$ we obtain

$\displaystyle\widehat{E}_{2,2}^{1,0}\,\propto\,\log(r+1)-\frac{r^{2}}{r^{2}-1}\log
r\,,$ (48)
$\displaystyle\widehat{E}_{3,3}^{2,0}\,\propto\,\frac{-2r^{4}+r^{3}+4r^{2}+r-2}{2\left(r^{2}-1\right)^{2}}-\frac{\left(r^{4}-2r^{2}+3\right)r^{3}}{\left(r^{2}-1\right)^{3}}\,\log
r+\frac{1+r^{2}}{r}\log(r+1)\,.$ (49)

Although the ODE works perfectly for the calculations in section 5, let us
mention that when $\Delta_{1}-\widehat{\Delta}\in 2\mathbb{Z}_{>0}$, we can
instead use a truncation method:

$\displaystyle\widehat{E}_{\Delta_{1}\Delta_{2}}^{\widehat{\Delta},0}(\xi,\eta)=\sum_{n=1}^{\frac{\Delta_{1}-\widehat{\Delta}}{2}}\frac{\left(\\!-\frac{\Delta_{1}-\widehat{\Delta}-2}{2}\right)_{n-1}\left(\\!-\frac{\Delta_{1}+\widehat{\Delta}-p-2}{2}\right)_{n-1}}{4(1-\Delta_{1})_{2n}}C_{\Delta_{1}-2n,\Delta_{2}}(\xi,\eta)\,.$
(50)

(If $\Delta_{2}-\widehat{\Delta}\in 2\mathbb{Z}_{>0}$, one uses the same
formula with $\Delta_{1}\leftrightarrow\Delta_{2}$). Let us stress that the
truncation condition on $\Delta_{1}$, $\Delta_{2}$ and ${\widehat{\Delta}}$
does not cover all cases of interest. As a result, formula (50) is of little
use, and we do not derive it here (in any case, the derivation is identical to
[55] with $d-1\to p$).

####  3.4.2 Transverse-spin diagrams

The previous discussion has a simple generalization to operators that are
charged under transverse spin, i.e. charged under $SO(d-p)$ rotations. As a
simple example, taking the bulk-brane coupling in (29) we find that the spin-
one exchange diagram is given by

$\displaystyle\int_{\widehat{z}_{1},\widehat{z}_{2}}\\!\widehat{z}_{1}^{\,0}\widehat{z}_{2}^{\,0}\widehat{G}_{\widehat{\Delta}}(\widehat{z}_{1},\widehat{z}_{2})\partial_{i}K_{\Delta_{1}}(x_{1},\widehat{z}_{1})\partial_{i}K_{\Delta_{2}}(x_{2},\widehat{z}_{2})\;\equiv\;\frac{\widehat{E}^{\widehat{\Delta},1}_{\Delta_{1}\Delta_{2}}(\xi,\eta)}{|x^{i}_{1}|^{\Delta_{1}}|x_{2}^{i}|^{\Delta_{2}}}\,.$
(51)

A simple calculation shows that the result factorizes as

$\displaystyle\widehat{E}^{\widehat{\Delta},1}_{\Delta_{1}\Delta_{2}}(\xi,\eta)\equiv
4\Delta_{1}\Delta_{2}\eta\,\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}+1,\Delta_{2}+1}(\xi,\eta)\,.$
(52)

It is important to notice the shift in the arguments
$\Delta_{1},\Delta_{2}\to\Delta_{1}+1,\Delta_{2}+1$. More generally, it is not
hard to check that a defect-exchange diagram with transverse spin reduces to a
product of a scalar diagram with shifted arguments, and a factor with simple
$\eta$ dependence:

$\displaystyle\widehat{E}^{\widehat{\Delta},s}_{\Delta_{1}\Delta_{2}}(\xi,\eta)\propto\eta^{s}\,\widehat{E}^{\widehat{\Delta},0}_{\Delta_{1}+s,\Delta_{2}+s}(\xi,\eta)\,.$
(53)

However, we only need the $s=0,1$ cases below.

###  3.5 A detour: connection to analytic functionals555This section lies
somewhat outside the main scope of this work, and can be safely skipped by
readers not interested in analytic functionals.

It is well known that, in CFT without defects, different approaches to
analytic bootstrap are closely related to each other. Indeed, the Lorentzian
inversion formula [71, 72] was resummed into a dispersion relation [73], which
in turn generates the analytic functionals of [74]. Furthermore, the
dispersion relations in position and Mellin space [75] are equivalent [76].
For boundary CFT, some of these connections have been explored in [55, 56,
57].

Motivated by these developments, we show here that in defect CFT there is also
a connection between the dispersion relation, a basis of analytic functionals,
and exchange Witten diagrams. Unlike the rest of this work, this section
relies heavily on analytic bootstrap methods for conformal defects, and the
required background can be found in [65, 77, 54, 78]. Recall that in defect
CFT there are two dispersion relations [54, 78] (or equivalently two inversion
formulas [65, 77]). Here we focus on the dispersion relation that reconstructs
the correlator starting from a single discontinuity
$\operatorname{Disc}F(r,w^{\prime})=F(r,w^{\prime}+i0)-F(r,w^{\prime}-i0)$. We
define the Polyakov-Regge block $P_{\Delta_{1}\Delta_{2}}^{\Delta,J}$ as the
dispersion relation applied to a bulk-channel conformal block
$f_{\Delta,J}$.666In this section, $J$ is the spin of an operator of dimension
$\Delta$, and $\ell$ is the spin of a double-twist operator. Similarly, we
also distinguish transverse-spin $S$ and $s$ below. The expression looks more
elegant in the $r$, $w$ cross ratios of equation (16):

$\displaystyle
P_{\Delta_{1}\Delta_{2}}^{\Delta,J}(r,w)\equiv\int_{0}^{r}\frac{dw^{\prime}}{2\pi
i}\frac{w(1-w^{\prime})(1+w^{\prime})}{w^{\prime}(w^{\prime}-w)(1-ww^{\prime})}\operatorname{Disc}\\!\Big{[}\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{\Delta,J}(r,w^{\prime})\Big{]}\,.$
(54)

The discontinuity vanishes on double-twist operators
$\operatorname{Disc}\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell}=0$,
see e.g. [54]. As a result, the bulk-channel block expansion of the Polyakov-
Regge block is of the form

$\displaystyle
P_{\Delta_{1}\Delta_{2}}^{\Delta,J}(r,w)=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}\bigg{[}f_{\Delta,J}(r,w)+\sum_{n,\ell=0}^{\infty}a_{\ell,n}f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell}(r,w)\bigg{]}\,.$
(55)

To obtain the defect-channel block expansion, one can apply the inversion
formula of [65] to the block $f_{\Delta,J}$. It is not difficult to see that
the expansion has the following structure

$\displaystyle
P_{\Delta_{1}\Delta_{2}}^{\Delta,J}(r,w)=\sum_{n,\ell=0}^{\infty}b_{s,n}\widehat{f}_{\Delta_{1}+s+2n,s}(r,w)+\sum_{n,\ell=0}^{\infty}c_{s,n}\widehat{f}_{\Delta_{2}+s+2n,s}(r,w)\,,$
(56)

although extracting $b_{n,\ell}$ and $c_{n,\ell}$ in closed form is more
challenging. Finally, note that the Polyakov-Regge block decays in the Regge
limit

$\displaystyle\big{|}P_{\Delta_{1}\Delta_{2}}^{\Delta,J}(r,w)\big{|}<|w|^{-{\varepsilon}}\quad\text{for}\quad
w\to\infty\,,\;{\varepsilon}>0\,,$ (57)

a property that follows from definition (54). In the terminology of [56, 74],
we say that the Polyakov-Regge block is superbounded.

At this point, we claim that the functions
$\\{f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell},\widehat{f}_{\Delta_{1}+s+2n,s},\widehat{f}_{\Delta_{2}+s+2n,s}\\}$
form a basis in a suitable space of functions, and restrict our attention to
CFT correlators in such a space.777We expect functions in this space to be
Regge superbounded as in (57). However, one might need further conditions that
ensure that the defect-to-bulk inversion formula [77] converges. We do not
attempt to prove this claim, although there is evidence in the literature for
similar statements being true [56, 74, 79]. We can then introduce a dual basis
$\\{{\alpha}_{\ell,n},{\beta}_{s,n},\gamma_{s,n}\\}$ in the natural way,
called the basis of analytic functionals. The analytic functionals are useful
in the bootstrap program, because when acting on the crossing equation, they
generate sum rules obeyed by the CFT data. For example, we have

$\displaystyle\sum_{{\mathcal{O}}}\lambda_{{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}}a_{\mathcal{O}}\,{\alpha}_{\ell,n}[f_{\Delta,J}]-\sum_{\widehat{\mathcal{O}}}b_{{\mathcal{O}}_{1}\widehat{\mathcal{O}}}b_{{\mathcal{O}}_{2}\widehat{\mathcal{O}}}\,{\alpha}_{\ell,n}[\widehat{f}_{{\widehat{\Delta}},s}]=0\,,$
(58)

and two more expressions with ${\beta}_{s,n}$ and $\gamma_{s,n}$. These sum
rules are valid in any CFT, although they are particularly useful to bootstrap
theories perturbatively around mean-field theory. To compute the action of the
functionals on a generic bulk block, note that acting with the functionals on
(55) and (56) gives

$\displaystyle{\alpha}_{\ell,n}[f_{\Delta,\ell}]=-a_{\ell,n}\,,\qquad{\beta}_{s,n}[f_{\Delta,\ell}]=b_{s,n}\,,\qquad\gamma_{s,n}[f_{\Delta,\ell}]=c_{s,n}\,.$
(59)

Therefore, the expansion coefficients of the Polyakov-Regge block give the
action of analytic functionals on bulk-channel conformal blocks. A similar
story holds for defect-channel Polyakov-Regge blocks
$\widehat{P}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},S}$. These are defined
using the “defect-to-bulk” dispersion relation, namely
$\widehat{P}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},S}=\int
K\operatorname{dDisc}\widehat{f}_{{\widehat{\Delta}},S}$ with a kernel $K$
that can be found in [78]. Their expansion is of the form

$\displaystyle\widehat{P}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},S}(r,w)$
$\displaystyle=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{n,\ell=0}^{\infty}\widehat{a}_{\ell,n}f_{\Delta_{1}+\Delta_{2}+\ell+2n,\ell}(r,w)$
(60)
$\displaystyle=\widehat{f}_{{\widehat{\Delta}},S}+\sum_{n,\ell=0}^{\infty}\widehat{b}_{s,n}\widehat{f}_{\Delta_{1}+s+2n,s}(r,w)+\sum_{n,\ell=0}^{\infty}\widehat{c}_{s,n}\widehat{f}_{\Delta_{2}+s+2n,s}(r,w)\,,$
(61)

which implies the action of analytic functionals

$\displaystyle{\alpha}_{\ell,n}[\widehat{f}_{{\widehat{\Delta}},S}]=\widehat{a}_{\ell,n}\,,\qquad{\beta}_{s,n}[\widehat{f}_{{\widehat{\Delta}},S}]=-\widehat{b}_{s,n}\,,\qquad\gamma_{s,n}[\widehat{f}_{{\widehat{\Delta}},S}]=-\widehat{c}_{s,n}\,.$
(62)

Besides the definition in terms of Polyakov-Regge blocks, the functionals
${\alpha}_{\ell,n}[f]$, ${\beta}_{s,n}[f]$ and $\gamma_{s,n}[f]$ might also
admit a representation as integrals of $\operatorname{Disc}f$ or
$\operatorname{dDisc}f$ against certain kernels. As in [56, 74, 79], one
should obtain the integral form of the functionals by expanding in conformal
blocks the dispersion relation kernel. We do not perform this calculation
here.

Having shown the utility of Polyakov-Regge blocks, we are faced with the
question of how to compute them. One method uses that Polyakov-Regge blocks
are exchange Witten diagrams, with appropriate addition of contact terms to
make them Regge superbounded [56, 74]. For spin $\ell=0,2$, the relations
should be of the form

$\displaystyle
P^{\Delta,0}_{\Delta_{1}\Delta_{2}}=k_{1}E^{\Delta,0}_{\Delta_{1}\Delta_{2}}\,,\qquad
P^{\Delta,2}_{\Delta_{1}\Delta_{2}}=k_{1}k_{2}\left(E^{\Delta,2}_{\Delta_{1}\Delta_{2}}+k_{3}C_{\Delta_{1}\Delta_{2}}\right)\,.$
(63)

To extract the precise relation, we use the decomposition of exchange diagrams
in terms of contact diagrams, see (39) and appendix A. We fix the overall
normalization comparing to the block expansion (55),888This equation depends
on the choice of normalization for the conformal blocks. Here we normalize
blocks in the lightcone limit $|1-z|\ll|1-{\bar{z}}|\ll 1$ as
$f_{\Delta,\ell}(z,{\bar{z}})=(1-z)^{\frac{\Delta-\ell}{2}}(1-{\bar{z}})^{\frac{\Delta+\ell}{2}}+\ldots\,$.
and we fix the contact terms taking the Regge limit (57). In the end, we find

$\displaystyle k_{1}$
$\displaystyle=\frac{2^{\Delta+2}(-1)^{\frac{\Delta_{1}+\Delta_{2}-\Delta+2}{2}}\Gamma\left(\frac{\Delta+1}{2}\right)(1-\Delta_{1})_{\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}}(1-\Delta_{2})_{\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}}}{\pi^{\frac{p+1}{2}}\Gamma\\!\left(\frac{\Delta-p}{2}\right)\left(\frac{\Delta_{1}+\Delta_{2}-\Delta-2}{2}\right)!\left(\frac{d-\Delta-\Delta_{1}-\Delta_{2}+2}{2}\right)_{\frac{\Delta_{1}+\Delta_{2}-\Delta-2}{2}}}\,,$
(64) $\displaystyle k_{2}$
$\displaystyle=\frac{8\left(\Delta^{2}-1\right)}{(\Delta_{1}+\Delta_{2}-\Delta)(\Delta_{1}+\Delta_{2}+\Delta-d)(\Delta^{2}-\Delta^{2}_{12})}\,,$
(65) $\displaystyle k_{3}$
$\displaystyle=\frac{d+1}{2d}\\!-\\!\frac{\Delta_{12}^{2}(\Delta_{1}+\Delta_{2}-d)^{2}}{2d\Delta(\Delta-d)}+\frac{\left(\Delta_{12}^{2}-1\right)\left((\Delta_{1}+\Delta_{2}-d)^{2}-1\right)}{2d(\Delta-1)(\Delta-d+1)}\,.$
(66)

These formulas hold for $\ell=0$, $\Delta_{1}+\Delta_{2}-\Delta\in
2\mathbb{Z}_{>0}$, and for $\ell=2$, $\Delta_{1}+\Delta_{2}-\Delta\in
2\mathbb{Z}_{\geq 0}$. For general dimensions, we expect the formulas to be
more complicated, and perhaps they can be obtained in Mellin space in analogy
to [56, 74].

Although the presentation in section 3.5 has been somewhat schematic, we hope
it will encourage further exploration of these interesting issues.

##  4 Witten diagrams in Mellin space

This section studies the Mellin representation of the two-point function of
local operators in the presence of a defect, initially introduced in [31]. We
show how the Witten diagrams of section 3 translate to Mellin space, and in
particular, we find closed formulas for exchange Mellin amplitudes with
arbitrary scaling dimensions.

###  4.1 Mellin space

The Mellin amplitude is only valid for the connected correlator $F_{c}$, which
we introduced in equation (19). Keeping this in mind, the Mellin amplitude
${\mathcal{M}}(\delta,\rho)$ is

$\displaystyle F_{c}(\xi,\eta)=\int\frac{d\delta\,d\rho}{(2\pi
i)^{2}}\,\frac{\Gamma(\delta)\Gamma(\rho)\Gamma\\!\left(\frac{\Delta_{1}-\delta-\rho}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}-\delta-\rho}{2}\right)}{\xi^{\delta}\,(2\eta)^{\rho}}\,{\mathcal{M}}(\delta,\rho)\,.$
(67)

This construction can be generalized to any correlator in defect CFT [31], but
we do not need it here. Although we are somewhat loose on what the integration
contours are, suffice it to mention that they run parallel to the imaginary
axis. Actually, it is not obvious whether non-perturbative correlators admit a
Mellin representation, and if they do, what is the correct integration
contour. This is not a problem to us, because we work in perturbation theory
and apply the Mellin transform diagram by diagram, so it is possible to find a
contour such that all manipulations are well defined. However, it would be
very interesting to analyze non-perturbative Mellin amplitudes along the lines
of [75, 80].

Leaving these considerations aside, one motivation to work in Mellin space is
that Mellin amplitudes are meromorphic in $\delta$ and $\rho$, at least in
perturbation theory. Furthermore, the location of poles corresponds to
dimensions of operators exchanged in the OPEs. More precisely, assume the OPE
is of the form
${\mathcal{O}}_{1}\times{\mathcal{O}}_{2}\sim{\mathcal{O}}+\ldots\,$, where
operator ${\mathcal{O}}$ has scaling dimension $\Delta$ and spin $\ell$. Then
the integrand of (67) has poles at

$\displaystyle\delta=\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell-2n}{2}\,,\qquad
n=0,1,2,\ldots\,.$ (68)

Similarly, if the defect expansion is of the form
${\mathcal{O}}_{1}\times{\mathcal{D}}\sim\widehat{\mathcal{O}}{\mathcal{D}}+\ldots\,$,
with $\widehat{\mathcal{O}}$ a local operator on the defect of dimension
${\widehat{\Delta}}$ and transverse-spin $s$, then the integrand in (67) has
poles at

$\displaystyle\delta+\rho={\widehat{\Delta}}-s+2n\,,\qquad n=0,1,2,\ldots\,.$
(69)

Let us stress that these are poles of the integrand in (67). In holographic
theories, the spectrum is approximately that of MFT

$\displaystyle\text{Bulk:}\quad{\mathcal{O}}_{1}\Box^{n}\partial_{\mu_{1}}\ldots\partial_{\mu_{\ell}}{\mathcal{O}}_{2}\,,\qquad\text{Defect:}\quad\Box^{n}\partial_{i_{1}}\ldots\partial_{i_{s}}{\mathcal{O}}_{1}\,,\quad\Box^{n}\partial_{i_{1}}\ldots\partial_{i_{s}}{\mathcal{O}}_{2}\,.$
(70)

The integrand in (67) is such that poles by the MFT operators are taken care
of by the gamma functions. As a result, in holographic theories the Mellin
amplitude ${\mathcal{M}}(\delta,\rho)$ has a simpler analytic structure than
the full integrand, as we show in examples below.

###  4.2 Contact diagram

As a first application of Mellin space, let us obtain the Mellin transform of
the contact diagram $C_{\Delta_{1}\Delta_{2}}$. We start from (32) and use
identity (33) twice, so the contact diagram has the form

$\displaystyle
C_{\Delta_{1}\Delta_{2}}(\xi,\eta)=\frac{\pi^{p/2}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-p}{2}\right)}{2\Gamma(\Delta_{1})\Gamma(\Delta_{2})}$
$\displaystyle\int\frac{d\delta\,d\rho}{(2\pi
i)^{2}}\frac{\Gamma(\delta)\Gamma(\rho)\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}}{2}-\delta-\rho\right)}{\xi^{\delta}(2\eta)^{\rho}}$
$\displaystyle\times\int_{0}^{\infty}\frac{ds}{s}\,s^{\Delta_{1}-\delta-\rho}\frac{|x_{1}^{i}|^{\Delta_{1}-\delta-\rho}|x_{2}^{i}|^{\Delta_{2}-\delta-\rho}}{\left(s^{2}|x_{1}^{i}|^{2}+|x_{2}^{i}|^{2}\right)^{\frac{\Delta_{1}+\Delta_{2}}{2}-\delta-\rho}}\,.$
(71)

The $s$ integral is elementary, and after factoring out the prefactor (67) we
get a Mellin amplitude that is constant:

$\displaystyle{\mathcal{M}}[C_{\Delta_{1},\Delta_{2}}]=\frac{\pi^{\frac{p}{2}}\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}-p}{2}\right)}{4\Gamma(\Delta_{1})\Gamma(\Delta_{2})}\,.$
(72)

This is completely analogous to four-point contact Mellin amplitudes, which
are also constant. Regarding the integration contour, observe that the
manipulations that led to (72) are valid provided
$\operatorname{Re}\delta,\operatorname{Re}\rho>0$ and
$0<\operatorname{Re}\delta+\operatorname{Re}\rho<\min(\Delta_{1},\Delta_{2})$.
As a result, we have to obey these constraints when choosing the integration
contour in (67), or equivalently, the contour has to separate the “left” and
“right” families of poles in the integrand.

The fact that contact amplitudes are constant is very desirable, and actually
it motivates the choice of prefactors in the Mellin amplitude (67). One reason
is that large classes of exchange diagrams are sums of contact diagrams, and
in this case, the Mellin amplitudes become rational functions of $\delta$ and
$\rho$. To combine contact amplitudes correctly, there are two observations to
keep in mind. The first observation is that contact diagrams are often
multiplied in position space by powers of the cross-ratios $\xi^{n}$ and
$\eta^{m}$, which in Mellin space are shifts of the variables
$\delta\to\delta+n$ and $\rho\to\rho+m$. However, because of our definition of
the Mellin amplitude (67), the shift is accompanied by extra Pochhammer
symbols:

$\displaystyle\xi^{n}\eta^{m}F(\xi,\eta)$
$\displaystyle\;\leftrightarrow\;\frac{(\rho)_{m}(\delta)_{n}}{2^{m}}\left(\frac{\Delta_{1}-\delta-\rho}{2}\right)_{\frac{-m-n}{2}}\left(\frac{\Delta_{2}-\delta-\rho}{2}\right)_{\frac{-m-n}{2}}\mathcal{M}(\delta+n,\rho+m)\,.$
(73)

The second observation is that the integrand in (67) depends on $\Delta_{1}$
and $\Delta_{2}$, which gives an extra contribution when combining contact
diagrams of different dimensions. These two observations are best illustrated
in practice. For example, the Mellin transform of (41) reads

$\displaystyle{\mathcal{M}}\big{[}E_{2,2}^{4,2}\big{]}=\frac{\pi}{24}\left(1+\frac{3\rho-1}{\delta-1}\right)\,,\qquad{\mathcal{M}}\big{[}E_{3,2}^{5,2}\big{]}=\frac{\sqrt{\pi}}{16}\left(\frac{4}{5}+\frac{2\rho-1}{\delta-1}\right)\,.$
(74)

As anticipated, the exchange diagrams are remarkably simple rational functions
of the Mellin variables.

###  4.3 Bulk-exchange diagram

We already mentioned in section 3.3 that exchange Witten diagrams are hard to
calculate due to the complicated formula for the bulk-to-bulk propagator (8).
Fortunately, there exists a “split representation” of the propagator as a
double integral of bulk-to-boundary propagators

$\displaystyle
G_{\Delta}(z_{1},z_{2})=\int_{-i\infty}^{i\infty}\frac{d\nu}{2\pi
i}\,\frac{2\nu^{2}\,{\mathcal{C}}_{d/2+\nu}\,{\mathcal{C}}_{d/2-\nu}}{\nu^{2}-(\Delta-\frac{d}{2})^{2}}\int_{\partial\mathrm{AdS}_{d+1}}d^{d}x\,K_{\frac{d}{2}+\nu}(z_{1},x)K_{\frac{d}{2}-\nu}(z_{2},x)\,.$
(75)

Here ${\mathcal{C}}_{\Delta}$ is the normalization of the bulk-bulk propagator
(9), the $\nu$ integration goes along the contour $\operatorname{Re}\nu=0$,
and the $x$ integral is over the boundary of $\mathrm{AdS}_{d+1}$, namely
$\mathbb{R}^{d}$. This formula is motivated and proven in the appendix of [7].
For our purposes, it suffices to note that thanks to this formula, exchange
diagram factorize as products of lower-point diagrams, at the expense of
adding extra integrations.

For example, the bulk-exchange diagram defined in (37) is schematically

$\displaystyle E_{\Delta_{1}\Delta_{2}}^{\Delta,0}\;\;=\;\;\leavevmode\hbox
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.82054pt}{-2.5482pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\small{$x_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;.$ (76)

The first factor is a three-point function in the absence of branes

$\displaystyle\int_{z}K_{\Delta_{1}}(x_{1},z)K_{\Delta_{2}}(x_{2},z)K_{\Delta_{3}}(x_{3},z)=\frac{\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}}{|x_{12}|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}|x_{13}|^{\Delta_{1}+\Delta_{3}-\Delta_{2}}|x_{23}|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}\,.$
(77)

Conformal symmetry fixes the form of the right-hand side, and the
normalization follows from explicit calculation [4, 67]

$\displaystyle\lambda_{\Delta_{1}\Delta_{2}\Delta_{3}}=\pi^{d/2}\frac{\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\\!\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}\right)\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}-d}{2}\right)}{2\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta_{3})}\,.$
(78)

For the bulk-exchange diagram, we need this formula with
$\Delta_{3}=\frac{d}{2}+\nu$.

The second factor in (76) is a one-point function of a scalar in the presence
of the brane, which we calculated in (24). Summarizing, the split
representation (75) allows to compute the $\mathrm{AdS}$ integrals, but we are
left with one integral over $x\in\mathbb{R}^{d}$ and a Mellin-Barnes integral
over $\nu$. The $x$ integral reads

$\displaystyle I$
$\displaystyle=\int\frac{d^{d}x_{3}}{|x_{13}|^{\frac{d}{2}+\nu+\Delta_{12}}|x_{23}|^{\frac{d}{2}+\nu-\Delta_{12}}|x_{3}^{i}|^{\frac{d}{2}-\nu}}\,.$
(79)

Now introduce three Schwinger parameters $s,t,u$, integrate over $x_{3}$,
insert an identity $\int_{\lambda}\delta(\lambda-u)$, rescale $s,t,u$, and
integrate over $\lambda$ and $u$ (see the paragraph before (27)). In the end,
one finds the parametric integral

$\displaystyle I$
$\displaystyle=\frac{\pi^{d/2}\Gamma\\!\left(\frac{d+2\nu}{4}\right)}{\Gamma\\!\left(\frac{d-2\nu}{4}\right)\Gamma\\!\left(\frac{d+2\nu-2\Delta_{12}}{4}\right)\Gamma\\!\left(\frac{d+2\Delta_{12}+2\nu}{4}\right)}$
$\displaystyle\quad\times\int_{0}^{\infty}\\!\frac{ds\,dt}{s\,t}\frac{s^{\frac{d-2\nu}{4}}t^{\frac{d+2\Delta_{12}+2\nu}{4}}(t+1)^{\frac{d+2\nu-2p}{4}}(s+t+1)^{\frac{2\nu-d+2p}{4}}}{\left(s(t^{2}|x_{1}^{i}|^{2}+|x_{2}^{i}|^{2})+2\eta
st|x_{1}^{i}||x_{2}^{i}|+\xi
t(s+t+1)|x_{1}^{i}||x_{2}^{i}|\right)^{\frac{d+2\nu}{4}}}\,,$ (80)

where we used the cross-ratios $\xi$ and $\eta$ in (15) to simplify the
denominator. Next, we apply identity (33) twice to split the denominator, and
then integrate over $s$ and $t$

$\displaystyle I$
$\displaystyle=\frac{\pi^{d/2}\Gamma\\!\left(\frac{d+2\nu-2p}{4}\right)|x_{1}^{i}|^{-\frac{d}{4}-\frac{\Delta_{12}}{2}-\frac{\nu}{2}}|x_{2}^{i}|^{-\frac{d}{4}+\frac{\Delta_{12}}{2}-\frac{\nu}{2}}}{2\Gamma\\!\left(\frac{d-2\nu}{4}\right)\Gamma\\!\left(\frac{d+2\nu-2\Delta_{12}}{4}\right)\Gamma\\!\left(\frac{d+2\Delta_{12}+2\nu}{4}\right)}$
$\displaystyle\quad\times\int_{-i\infty}^{i\infty}\frac{d\bar{\delta}\,d\rho}{(2\pi
i)^{2}}\,\frac{\Gamma(\bar{\delta})\Gamma(\rho)\Gamma(\bar{\delta}-\nu)\Gamma\\!\left(\frac{d+2\nu-2\Delta_{12}}{8}-\frac{\bar{\delta}+\rho}{2}\right)\Gamma\\!\left(\frac{d+2\nu+2\Delta_{12}}{8}-\frac{\bar{\delta}+\rho}{2}\right)}{\xi^{\bar{\delta}}(2\eta)^{\rho}\Gamma\\!\left(\bar{\delta}+\frac{d-2p-2\nu}{4}\right)}\,.$
(81)

We finally have all the ingredients for the bulk-exchange diagram (76). The
combination of the split representation (75), the one-point function (21), the
three-point function (78) and the position integral (81) has to be compared to
the Mellin integral (67). For the comparison to work, we shift the Mellin
variable as $\bar{\delta}=\delta-\frac{\Delta_{1}+\Delta_{2}-\nu-d/2}{2}$. All
in all, the bulk-exchange amplitude
${\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\Delta,0}\equiv{\mathcal{M}}\big{[}E_{\Delta_{1}\Delta_{2}}^{\Delta,0}\big{]}$
is given by

$\displaystyle{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\Delta,0}(\delta,\rho)=\frac{\pi^{p/2}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\delta)\Gamma\\!\left(\delta-\frac{\Delta_{1}+\Delta_{2}-d+p}{2}\right)}\int_{-i\infty}^{i\infty}\frac{d\nu}{2\pi
i}\frac{l_{b}(\nu)l_{b}(-\nu)}{(\Delta-\frac{d}{2})^{2}-\nu^{2}}\,,$ (82)

where

$\displaystyle
l_{b}(\nu)=\frac{\Gamma\\!\left(\frac{2\nu+d-2p}{4}\right)\Gamma\\!\left(\frac{2\nu+2\Delta_{1}+2\Delta_{2}-d}{4}\right)\Gamma\\!\left(\delta+\frac{2\nu-2\Delta_{1}-2\Delta_{2}+d}{4}\right)}{4\Gamma(\nu)}\,.$
(83)

Note that the scalar Mellin amplitude only depends on $\delta$. This is a
consequence of our choice of prefactor in (67), because the prefactor captures
the poles in $\delta+\rho$ corresponding to defect operators with MFT
dimensions. On the other hand, (82) captures new poles in $\delta$ due to the
exchange of a bulk operator of dimension $\Delta$, recall (68).

Note also the similarity between our formula and the Mellin amplitude of a
four-point scalar Witten diagram, see equation (38) in [7]. In fact, we can
follow verbatim the analysis of reference [7], which observed that poles in
$\delta$ arise when two poles in the integrand pinch the $\nu$ contour. A
simple application of the residue theorem then gives

$\displaystyle{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\Delta,0}(\delta,\rho)=\sum_{n=0}^{\infty}\frac{R_{n}}{\delta-\frac{\Delta_{1}+\Delta_{2}-\Delta-2n}{2}}\,,$
(84)

where the residues are999To obtain this formula, we use that for a meromorphic
function
$g(z)=\sum_{i}\frac{f_{i}(z)}{z-a_{i}}=\sum_{i}\frac{f_{i}(a_{i})}{z-a_{i}}$,
provided $g(z)$ decays at infinity. We have checked numerically in examples
that (82) and (84) agree, which justifies using this identity.

$\displaystyle
R_{n}=\frac{\pi^{p/2}}{16n!}\frac{\Gamma\\!\left(\frac{\Delta-p}{2}\right)\Gamma\\!\left(\frac{\Delta+\Delta_{1}+\Delta_{2}-d}{2}\right)\left(1+\frac{\Delta-d+p}{2}\right)_{n}\left(1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2}\right)_{n}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma\\!\left(\Delta+n+1-\frac{d}{2}\right)}\,.$
(85)

Here we see that when $\Delta_{1}+\Delta_{2}-\Delta\in 2\mathbb{Z}_{>0}$, the
infinite sum truncates to a finite sum. This finite sum agrees with the sum
over contact diagrams (39) that we found in position space. Although we shall
not need it, the sum can be computed in terms of a generalized hypergeometric
function

$\displaystyle{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{\Delta,0}(\delta,\rho)$
$\displaystyle=\frac{\pi^{p/2}\Gamma\\!\left(\frac{\Delta-p}{2}\right)\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}+\Delta-d}{2}\right)}{16\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma\\!\left(\frac{2\Delta-d+2}{2}\right)\\!\left(\delta-\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}\right)}$
$\displaystyle\qquad\qquad\times\,_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{*{20}{c}}{1+\frac{\Delta-d+p}{2},1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2},\delta+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2}}\\\
{\Delta+1-\frac{d}{2},\delta+1+\frac{\Delta-\Delta_{1}-\Delta_{2}}{2}}\end{array};1\right)\,.$
(88)

The $\ell>0$ blocks for arbitrary dimensions can also be computed using the
split representation, which was developed in [81]. However, that technology is
somewhat involved and we choose not to explore it in the present work.
Actually, for applications to section 5, all $\ell=2$ exchanges truncate to a
sum of contact diagrams, which can be easily Mellin transformed.

###  4.4 Defect-exchange diagram

A similar calculation also applies to the defect-exchange diagram (42). The
idea is to use the split representation for the brane-to-brane propagator,
equation (75) with $d\to p$. The exchange diagram factorizes as a product of
bulk-brane diagrams

$\displaystyle\widehat{E}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}\;\;=\;\leavevmode\hbox
to63.05pt{\vbox to63.38pt{\pgfpicture\makeatletter\hbox{\hskip
34.19492pt\lower-33.11252pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{\small{{\footnotesize$\frac{p}{2}-\nu$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\small{$\widehat{x}$}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;,$ (89)

at the expense of adding integrations over $\widehat{x}$ and $\nu$. The bulk-
to-brane Witten diagrams are kinematically fixed, and their normalization is
computed in section 3.1.2. As a result, we need to compute the integral

$\displaystyle
H=\int\frac{d^{p}\widehat{x}_{3}}{\big{(}(x_{1}^{i})^{2}+(x_{1\hat{3}}^{a})^{2}\big{)}^{\frac{p}{2}+\nu}\big{(}(x_{2}^{i})^{2}+(x_{2\hat{3}}^{a})^{2}\big{)}^{\frac{p}{2}-\nu}}\,.$
(90)

The technique is by now standard. Introduce two Schwinger parameters $s$ and
$t$, integrate over $\widehat{x}_{3}$, and integrate over $t$ with the help of
an auxiliary parameter $\lambda$. The result takes the form

$\displaystyle H$
$\displaystyle=\frac{\pi^{\frac{p}{2}}\Gamma(\frac{p}{2})}{\Gamma(\frac{p}{2}-\nu)\Gamma(\frac{p}{2}+\nu)}\int\frac{ds}{s}\frac{s^{\frac{p}{2}+\nu}}{\big{(}s^{2}|x_{1}^{i}|^{2}+|x_{2}^{i}|^{2}+2\eta
s|x_{1}^{i}||x_{2}^{i}|+\xi s|x_{1}^{i}||x_{2}^{i}|\big{)}^{p/2}}$
$\displaystyle=\frac{\pi^{\frac{p}{2}}}{|x_{1}^{i}|^{\frac{p}{2}+\nu}|x_{2}^{i}|^{\frac{p}{2}-\nu}}\int\frac{d\delta\,d\rho}{(2\pi
i)^{2}}\frac{\Gamma(\delta)\Gamma(\rho)}{\xi^{\delta}(2\eta)^{\rho}}\frac{\Gamma\\!\left(\frac{p/2-\nu-\delta-\rho}{2}\right)\Gamma\\!\left(\frac{p/2+\nu-\delta-\rho}{2}\right)}{2\Gamma(\frac{p}{2}-\nu)\Gamma(\frac{p}{2}+\nu)}\,,$
(91)

where we used (33) twice to factorize the denominator, and then computed an
elementary integral over $s$. Again, the combination of the split
representation (75), the bulk-brane diagram (28) and the boundary integral
(91) is compared with the Mellin amplitude (67). Using notation
$\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}\equiv{\mathcal{M}}\big{[}\widehat{E}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}\big{]}$,
the defect-exchange amplitude reads

$\displaystyle\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}(\delta,\rho)=\frac{\pi^{\frac{p}{2}}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma\\!\left(\frac{\Delta_{1}-\delta-\rho}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}-\delta-\rho}{2}\right)}\int_{-i\infty}^{i\infty}\frac{d\nu}{2\pi
i}\frac{l_{d}(\nu)l_{d}(-\nu)}{({\widehat{\Delta}}-\frac{p}{2})^{2}-\nu^{2}}\,,$
(92)

where

$\displaystyle
l_{d}(\nu)=\frac{\Gamma\\!\left(\frac{2\nu+2\Delta_{1}-p}{4}\right)\Gamma\\!\left(\frac{2\nu+2\Delta_{2}-p}{4}\right)\Gamma\\!\left(\frac{2\nu+p-2\delta-2\rho}{4}\right)}{4\Gamma(\nu)}\,.$
(93)

In this case the amplitude depends only on $\delta+\rho$, and the $\delta$
dependence is accounted for by the prefactor in (67). The interpretation is
similar as before, namely the bulk-channel block expansion contains operators
with MFT dimension $\Delta=\Delta_{1}+\Delta_{2}+\ell+2n$ which are completely
captured by the prefactor. Instead, in the defect-channel expansion there is a
new operator with ${\widehat{\Delta}}$, whose poles are captured by (92).

Once again, we can obtain the poles in $\delta+\rho$ by picking the residues
when two poles pinch the $\nu$ integration contour. We find the following sum

$\displaystyle\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}(\delta,\rho)=\sum_{n=0}^{\infty}\frac{\widehat{R}_{n}}{\delta+\rho-{\widehat{\Delta}}-2n}\,,$
(94)

with residues

$\displaystyle\widehat{R}_{n}=-\frac{\pi^{\frac{p}{2}}}{8n!}\frac{\Gamma\\!\left(\frac{{\widehat{\Delta}}+\Delta_{1}-p}{2}\right)\Gamma\\!\left(\frac{{\widehat{\Delta}}+\Delta_{2}-p}{2}\right)\left(1+\frac{{\widehat{\Delta}}-\Delta_{1}}{2}\right)_{n}\left(1+\frac{{\widehat{\Delta}}-\Delta_{2}}{2}\right)_{n}}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma({\widehat{\Delta}}+n+1-\frac{p}{2})}\,.$
(95)

As expected, the sum truncates when
$\Delta_{1}-{\widehat{\Delta}},\Delta_{2}-{\widehat{\Delta}}\in\mathbb{Z}_{>0}$,
analogously to equation (50). More importantly, even when the sum does not
truncate, our calculation goes through and we obtain

$\displaystyle\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},0}$
$\displaystyle=-\frac{\pi^{p/2}\Gamma\\!\left(\frac{\Delta_{1}+{\widehat{\Delta}}-p}{2}\right)\Gamma\\!\left(\frac{\Delta_{2}+{\widehat{\Delta}}-p}{2}\right)}{8\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma({\widehat{\Delta}}+1-\frac{p}{2})(\delta+\rho-{\widehat{\Delta}})}{}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{*{20}{c}}{1-\frac{\Delta_{1}-{\widehat{\Delta}}}{2},1-\frac{\Delta_{2}-{\widehat{\Delta}}}{2},\frac{{\widehat{\Delta}}-\delta-\rho}{2}}\\\
{{\widehat{\Delta}}+1-\frac{p}{2},1+\frac{{\widehat{\Delta}}-\delta-\rho}{2}}\\\
\end{array};1\right)\,.$ (98)

Although the discussion so far is restricted to scalar exchanges, we showed in
section 3.4.2 that generalization to fields with transverse spin $s>0$ is
straightforward. Indeed, a generic exchange diagrams factorizes as
$\widehat{E}_{\Delta_{1},\Delta_{2}}^{{\widehat{\Delta}},0}\sim\eta^{s}\widehat{E}_{\Delta_{1}+s,\Delta_{2}+s}^{{\widehat{\Delta}},0}$,
where the scalar exchange has shifted arguments. For example, the Mellin
transform of the spin-one exchange (52) reads

$\displaystyle\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},1}(\delta,\rho)=2\Delta_{1}\Delta_{2}\rho{\mathcal{M}}_{\Delta_{1}+1,\Delta_{2}+1}^{{\widehat{\Delta}},0}(\delta,\rho+1)\,,$
(99)

where the factor $\rho$ is explained by (73). More generally, the spin-$s$
exchange diagram is of the form

$\displaystyle\widehat{\mathcal{M}}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},s}(\delta,\rho)\propto(\rho)_{s}\,{\mathcal{M}}_{\Delta_{1}+s,\Delta_{2}+s}^{{\widehat{\Delta}},0}(\delta,\rho+s)\,.$
(100)

##  5 Half-BPS Wilson line in ${\mathcal{N}}=4$ SYM

In this section, we apply the position-space bootstrap to the half-BPS Wilson
line in ${\mathcal{N}}=4$ SYM. More precisely, we bootstrap the correlator of
two chiral-primary operators and a half-BPS Wilson line at leading order in
the supergravity approximation. We start reviewing the setup from the CFT
perspective, and then move on to the holographic description. We derive the
$\mathrm{AdS}$ effective actions for bulk and brane, and input this
information to the position-space bootstrap method. This allows us to
determine the correlator in closed form. The result agrees with a first-
principles Witten diagram calculation, up to contact diagrams that we do not
know how to fix in the first-principles method. We conclude expressing our
result in Mellin space, where it takes a remarkably simple form.

###  5.1 Setup

Before presenting the holographic calculation, let us review the system using
field-theoretic language. The same system was studied in [53, 54], where the
interested reader can find further details.

We work with planar ${\mathcal{N}}=4$ SYM theory in the strong ’t Hooft
coupling limit $\lambda=g^{2}N\to\infty$. The local operators are chiral
primaries in the $[0,k,0]$ representation of $SU(4)_{R}$, that in field theory
language read101010 Here $\ldots$ contains multi-trace operators that ensure
that $S_{k}$ are operators dual to single-particle supergravity modes. The
most important property of this choice of operators is that extremal three-
point functions vanish, namely the OPE coefficient
$\lambda_{S_{k_{1}}S_{k_{2}}S_{k_{1}+k_{2}}}=0$. The difference between
single-trace and $S_{k}$ operators plays only a tangential role in our story,
and we refer to section 2.1 of [18] for a more detailed exposition.

$S_{k}(x,u)\propto\operatorname{tr}\big{(}u\cdot\phi(x)\big{)}^{k}+\ldots\,.$
(101)

As usual, we work with index-free notation, where $u$ is a null six-component
vector $u^{2}=0$ that implements the tracelessness of $S_{k}$. The chiral-
primary operators are scalars with protected scaling dimension $\Delta=k$.

Besides local operators, we also consider a straight half-BPS Wilson line

$W(\theta)=\frac{1}{N}\,\operatorname{tr}\,\operatorname{Pexp}\int_{-\infty}^{\infty}d\tau\,\big{(}i\dot{x}^{\mu}A_{\mu}+|\dot{x}|\,\theta\cdot\phi\big{)}\,.$
(102)

This is a one-dimensional defect that lives in four-dimensional space, or in
other words, the dual setup is $\mathrm{AdS}_{5}$ with a brane extending on
$\mathrm{AdS}_{2}$. Therefore, from now on we set $d=4$ and $p=1$. In (102)
there is a choice of $R$-symmetry direction which we parametrize with the unit
six-component vector $\theta^{2}=1$. For a straight contour, the expectation
value of the Wilson operator is $\langle W(\theta)\rangle=1$, so unlike
section 2.3, we do not need to normalize correlators by $\langle
W(\theta)\rangle$.

Thanks to the index-free notation, we can write the two-point function of
chiral-primary operators as

$\displaystyle\langle
S_{p_{1}}(x_{1},u_{1})S_{p_{2}}(x_{2},u_{2})W(\theta)\rangle=\frac{(u_{1}\cdot\theta)^{p_{1}}(u_{2}\cdot\theta)^{p_{2}}}{|x_{1}^{i}|^{p_{1}}|x_{2}^{i}|^{p_{2}}}\,{\mathcal{F}}(\xi,\eta,\sigma)\,.$
(103)

The function ${\mathcal{F}}(\xi,\eta,\sigma)$ captures the non-trivial
information of this correlator. The first two variables $\xi$, $\eta$ are the
spacetime cross-ratios (15), while $\sigma$ is an $R$-symmetry cross-ratio.
The dependence on $\sigma$ is polynomial, namely

${\mathcal{F}}(\xi,\eta,\sigma)=\sum_{n=0}^{{p_{\mathrm{min}}}}\sigma^{n}\,{\mathcal{F}}_{n}(\xi,\eta)\quad\;\text{with}\quad\;\sigma=\frac{u_{1}\cdot
u_{2}}{(u_{1}\cdot\theta)(u_{2}\cdot\theta)}\,.$ (104)

Here and below we define ${p_{\mathrm{min}}}=\min(p_{1},p_{2})$. We will be
mostly concerned with the connected correlator ${\mathcal{F}}_{c}$, which is
defined analogously to (18). After extracting the overall prefactor in (103),
the connected correlator is

$\displaystyle{\mathcal{F}}(\xi,\eta,\sigma)=\Big{(}\frac{\sigma}{\xi}\Big{)}^{\frac{p_{1}+p_{2}}{2}}+a_{p_{1}}a_{p_{2}}+{\mathcal{F}}_{c}(\xi,\eta,\sigma)\,,$
(105)

where $a_{p}$ is the one-point function $\langle
S_{p}\,W\rangle=a_{p}\big{(}\frac{u_{1}\cdot\theta}{|x^{i}|}\big{)}^{p}$. For
reference, observe that the one-point function is known explicitly

$\displaystyle
a_{p}=\frac{\sqrt{p\lambda}I_{p}(\sqrt{\lambda})}{2^{\frac{p}{2}+1}NI_{1}(\sqrt{\lambda})}+O\big{(}\tfrac{1}{N^{2}}\big{)}\,.$
(106)

Also note that our definition of connected correlator is different than the
one used e.g. in [82].

The Wilson line is a half-BPS object that breaks supersymmetry according to

$\displaystyle
PSU(2,2|4)\;\supset\;OSp(4^{*}|4)\;\supset\;SL(2,\mathbb{R})\oplus
SO(3)_{\text{trans.}}\oplus SO(5)_{R}\,.$ (107)

The $SL(2,\mathbb{R})$ is the group of global conformal transformation along
the line, $SO(3)_{\text{trans.}}$ are rotation in the transverse spacetime
directions, while $SO(5)_{R}$ is the leftover $R$-symmetry. The preserved
$OSp(4^{*}|4)$ symmetry has important consequences for the correlation
function under study, namely the superconformal Ward identity

$\left.\left(\partial_{z}+\frac{1}{2}\partial_{\alpha}\right){\mathcal{F}}(z,{\bar{z}},\alpha)\right|_{z=\alpha}=0\,$
(108)

must be obeyed [58]. This has to be supplemented by another equation with
$z\leftrightarrow{\bar{z}}$. The Ward identity is expressed in terms of
variables $z,{\bar{z}},{\alpha}$ defined as

$\displaystyle\xi=\frac{(1-z)(1-{\bar{z}})}{\sqrt{z{\bar{z}}}}\,,\qquad\eta=\frac{z+{\bar{z}}}{2\sqrt{z{\bar{z}}}}\,,\qquad\sigma=-\frac{(1-{\alpha})^{2}}{2{\alpha}}\,.$
(109)

The superconformal Ward identity is key for the position-space bootstrap of
section 5.4.

###  5.2 Effective action at strong coupling

With this preliminary information in mind, we can now start the holographic
calculation. The chiral-primary operator $S_{k}$ is dual to a supergravity
field $s_{k}$, which arises from the KK reduction of IIB supergravity on
$\mathrm{AdS}_{5}\times S^{5}$. As in the simple example of section 2, the
first step is to obtain the effective action for $s_{k}$. We start in section
5.2.1 reviewing the $\mathrm{AdS}_{5}$ effective action, which is already
known in the literature [11, 12, 13]. We then derive the $\mathrm{AdS}_{2}$
effective action. Besides rederiving vertices that were previously known [34,
37], we also obtain new bulk-brane vertices that, to the best of our
knowledge, have not been previously computed.

####  5.2.1 Effective action in $\mathrm{AdS}_{5}$

We use the ten-dimensional metric $G_{MN}=g_{MN}+h_{MN}$, where $g_{MN}$ is
the background $AdS_{5}\times S_{5}$ metric, and $h_{MN}$ are fluctuations
around it. We use Poincare coordinates $z^{\mu}$ on $\mathrm{AdS}_{5}$, and
stereographic coordinates $y^{A}$ on the $S^{5}$, so the background metric
reads

$\displaystyle ds^{2}_{\mathrm{AdS}_{5}\times
S^{5}}=\frac{dz^{\mu}dz^{\mu}}{(z^{0})^{2}}+\frac{dy^{A}dy^{A}}{(1+\frac{1}{4}y^{B}y^{B})^{2}}\,,\qquad\mu,\nu=0,1,\ldots,4\,,\quad
A,B=5,\ldots,9\,.$ (110)

Let us consider the fluctuations $h_{\mu\nu}$ of the $\mathrm{AdS}_{5}$
metric, which are expanded in $S^{5}$ spherical harmonics to perform the KK
reduction

$\displaystyle
h_{\mu\nu}(z,y)=\sum_{k,I}(h^{I}_{k})_{\mu\nu}(z)Y_{k}^{I}(y)\,.$ (111)

The spherical harmonic $Y_{k}^{I}$ for $k=0,1,2,\ldots$ transforms in the
rank-$k$ symmetric traceless representation of $SO(6)$, with the index
$I=1,\ldots,d_{k}$ labeling the elements in the representation. Appendix C
gives an explicit definition of spherical harmonics, and provides simple rules
to map them to the index-free notation in equations (101)-(104).

Besides fluctuations of the $\mathrm{AdS}_{5}$ metric in (111), there are
fluctuations for the metric on the sphere $h_{AB}$, and for the Ramond-Ramond
four form $C_{4}=\bar{C}_{4}+\delta C_{4}$, which are also expanded in
spherical harmonics. Upon inserting the spherical harmonic expansions in the
IIB supergravity action, one obtains mixing in the kinetic terms. The mixing
problem was first solved in [11], where the reader can find a table of all
fields, and their properties after unmixing. Of all the supergravity fields,
only the ones with even spin and in symmetric-traceless irreps of $SO(6)_{R}$
couple to the fundamental string. This leaves us with the $s$,
$\varphi_{\mu\nu}$ and $t$ fields, whose properties are summarized in table 1.
The standard AdS/CFT dictionary shows that $s_{k}$ is the supergravity field
dual to the CFT operator $S_{k}$, while $\varphi_{\mu\nu}$ and $t$ are
superconformal descendants.

For future reference, let us mention that the fields $s$, $\varphi_{\mu\nu}$
and $t$ are related to $\mathrm{AdS}_{5}$ metric fluctuations as [13]

$\displaystyle(h^{I}_{k})_{\mu\nu}$
$\displaystyle=(\varphi^{I}_{k})_{\mu\nu}+\frac{4}{k+1}\left(\nabla_{\mu}\nabla_{\nu}-\frac{1}{2}k(k-1)g_{\mu\nu}\right)s_{k}^{I}$
$\displaystyle\quad+\frac{4}{k+3}\left(\nabla_{\mu}\nabla_{\nu}-\frac{1}{2}(k+4)(k+5)g_{\mu\nu}\right)t_{k}^{I}\,.$
(112)

There exist similar formulas for the fluctuations $h_{AB}$, $\delta C$, but
they will not play a role in our calculation.

Field | $m^{2}$ | $\Delta$ | $\;\;\ell\;\;$ |  $SU(4)$-irrep
---|---|---|---|---
$s$ | $k(k-4)$ | $k$ | 0 | $[0,k,0]$
$\varphi_{\mu\nu}$ | $(k+2)(k-2)$ | $k+2$ | 2 | $[0,k-2,0]$
$t$ | $k(k+4)$ | $k+4$ | 0 | $[0,k-4,0]$
Table 1: Kaluza-Klein modes that couple to the string worldsheet

Inserting formula (112) and its analogs for $h_{AB}$, $\delta C$ in the IIB
supergravity action, one extracts kinetic terms, cubic couplings, and so on:

$\displaystyle S_{\text{IIB}}$
$\displaystyle=\frac{4N^{2}}{(2\pi)^{5}}\int\\!\frac{d^{5}z}{(z_{0})^{5}}\,\Big{(}L^{(2)}_{\text{IIB}}+L^{(3)}_{\text{IIB}}+\ldots\Big{)}\,.$
(113)

Recall that we are interested in the scattering of $s_{k}$ with a fundamental
string, which is dual to the CFT correlator $\langle
S_{p_{1}}S_{p_{2}}W\rangle$. A simple power-counting argument shows that only
cubic couplings contribute at leading order in the supergravity approximation.
Since we shall not attempt to compute corrections beyond leading order, the
quadratic and cubic terms in (113) are all we need.111111This is a fortunate
state of affairs, since quartic and higher couplings are remarkably
complicated [15]. The kinetic terms for $s$, $\varphi$ and $t$ are [11, 12,
13]

$\displaystyle L^{(2)}_{\text{IIB}}$
$\displaystyle=\sum_{k,I}\Big{(}\zeta^{s}_{k}\,L^{(0)}_{2}(s^{I}_{k})+\zeta^{t}_{k}\,L^{(0)}_{2}(t^{I}_{k})+\zeta^{\varphi}_{k}\,L^{(2)}_{2}(\varphi^{I}_{k})+\ldots\Big{)}\,,$
(114)

where $\zeta^{X}_{k}$ are normalizations presented in appendix B. Besides the
overall normalization, the kinetic terms are standard

$\displaystyle L^{(0)}_{2}(s)$
$\displaystyle=\frac{1}{2}\nabla_{\mu}s\nabla^{\mu}s+\frac{1}{2}m^{2}s^{2}\,,$
(115) $\displaystyle L^{(2)}_{2}(\varphi)$
$\displaystyle=\frac{1}{4}\nabla_{\mu}\varphi_{\nu\rho}\nabla^{\mu}\varphi^{\nu\rho}-\frac{1}{2}\nabla_{\mu}\varphi^{\mu\nu}\nabla^{\rho}\varphi_{\nu\rho}+\frac{1}{2}\nabla_{\mu}\varphi\nabla_{\rho}\varphi^{\mu\rho}$
$\displaystyle\quad-\frac{1}{4}\nabla_{\mu}\varphi\nabla^{\mu}\varphi-\frac{1}{4}(2-m^{2})\varphi_{\mu\nu}\varphi^{\mu\nu}-\frac{1}{4}(2+m^{2})\varphi^{2}\,,$
(116)

where the trace of the spin-two field is $\varphi=g^{\mu\nu}\varphi_{\mu\nu}$,
and the masses appear in table 1. In a similar way, one can write down the
cubic couplings. For our purposes, we only need vertices that contain two $s$
fields, and one $X=s,\varphi,t$ field:

$\displaystyle L_{\text{IIB}}^{(3)}$
$\displaystyle=-\sum_{k,p,q}\Big{(}S^{IJK}_{kpq}s^{I}_{k}\,s^{J}_{p}\,s^{K}_{q}+T^{IJK}_{kpq}s^{I}_{k}\,s^{J}_{p}\,t^{K}_{q}+G^{IJK}_{kpq}T_{\mu\nu}(s_{k}^{I},s_{p}^{J})(\varphi_{q}^{K})^{\mu\nu}+\ldots\Big{)}\,.$
(117)

The precise form of the vertices was obtained in [14] and is reviewed in
appendix B.

####  5.2.2 Effective action in $\mathrm{AdS}_{2}$

Let us now derive the effective action of fluctuations on top of the
fundamental string F1, which lives in the background $\mathrm{AdS}_{5}\times
S^{5}$ just described. Because in this background the NS-NS forms are turned-
off, the string action reads

$\displaystyle S_{\text{F1}}$ $\displaystyle=\frac{\sqrt{\lambda}}{2\pi}\int
d^{2}\widehat{z}\,\sqrt{\det\big{(}G_{MN}(X)\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\big{)}}\,,$
(118)

where we also ignore fermions. The worldsheet coordinates are
$\widehat{z}=(\widehat{z}^{\,0},\widehat{z}^{\,1})$, and $X^{M}(\widehat{z})$
describes the embedding of the string in ten dimensions. The string intersects
the boundary of $\mathrm{AdS}_{5}$ in a contour $\gamma$, and sits at a
particular point in the $S^{5}$. From the CFT perspective, $\gamma$ is the
location of the Wilson loop, while the point in $S^{5}$ is related to the
polarization $\theta$ in (102). The expectation value of the Wilson loop is
computed by the string path integral, which in the limit $\lambda\gg 1$ is
dominated by the saddle point where the string has minimal area.121212 See
section 2 of [38] for a recent pedagogical review of a similar setup. Here we
consider small fluctuations around the minimal-area configuration.

From now on, we restrict our attention to the case when $\gamma$ is a straight
line, a configuration that preserves the $OSp(4^{*}|4)$ group, recall equation
(107). More precisely, the string intersects the boundary of
$\mathrm{AdS}_{5}$ along the $x^{1}$ direction, and is located at
$x^{i=2,3,4}=0$. Furthermore, we choose the string to be located at point
$y^{A}=0$ on $S^{5}$. It is convenient to work in static gauge, in which case
the embedding reads

$\displaystyle X^{0}=\widehat{z}^{\,0}\,,\quad X^{1}=\widehat{z}^{\,1}\,,\quad
X^{i=2,3,4}=X^{A=5,\ldots,9}=0\,.$ (119)

This embedding represent the static configuration of the string. However, we
are interested in allowing small fluctuations around this configuration. In
particular, we allow the orthogonal directions $X^{i}$ and $X^{A}$ to
fluctuate, which can be implemented in a change of the boundary conditions
(119). Motivated by [37],131313To be precise, the authors of [37] split the
coordinates as $z^{\mu}=(z^{\alpha},z^{i})$ for ${\alpha}=0,1$ and $i=2,3,4$.
They used the metric  $\displaystyle ds^{2}_{\mathrm{AdS}_{5}\times
S^{5}}=\left(\frac{1+\frac{1}{4}(z^{i})^{2}}{1-\frac{1}{4}(z^{i})^{2}}\right)^{2}\,\frac{dz^{\alpha}dz^{\alpha}}{(z^{0})^{2}}+\frac{dz^{i}dz^{i}}{(1-\frac{1}{4}(z^{i})^{2})^{2}}+\frac{dy^{A}dy^{A}}{(1+\frac{1}{4}y^{B}y^{B})^{2}}\,,$
(120) and chose static gauge with $X^{\alpha}=\widehat{z}^{\alpha}$,
$X^{i}=x^{i}$ and $X^{A}=y^{A}$. This is equivalent to our choice (121)
because we work in Poincare coordinates, see (110). we require the string to
be embedded as

$\displaystyle
X^{0}=\widehat{z}^{\,0}\,\frac{1-\frac{1}{4}(x^{i})^{2}}{1+\frac{1}{4}(x^{i})^{2}}\,,\quad
X^{1}=\widehat{z}^{\,1}\,,\quad
X^{i}=\frac{x^{i}\widehat{z}^{\,0}}{1+\frac{1}{4}(x^{i})^{2}}\,,\quad
X^{A}=y^{A}\,.$ (121)

For $x^{i}=y^{A}=0$ this naturally reduces to the static configuration (119).
However, we instead treat $x^{i}(\widehat{z})$ and $y^{A}(\widehat{z})$ as
fields that live on the worldsheet, and expand (118) around the background
configuration to find their action. The fields $x^{i}$ and $y^{A}$ have
standard kinetic terms, as well as self-interactions and interactions with
bulk fields. The general structure of the action is

$\displaystyle S_{\text{F1}}$
$\displaystyle=\frac{\sqrt{\lambda}}{2\pi}\int\frac{d^{2}\widehat{z}}{(\widehat{z}^{\,0})^{2}}\,\Big{(}1+L_{\text{F1}}^{(2,0)}+L_{\text{F1}}^{(0,1)}+L_{\text{F1}}^{(1,1)}+L_{\text{F1}}^{(0,2)}+\ldots\Big{)}\,,$
(122)

where $L_{\text{F1}}^{(m,n)}$ corresponds to terms with $m$ worldsheet fields
and $n$ bulk fields. For the kinetic part, we find

$\displaystyle
L_{\text{F1}}^{(2,0)}=\frac{1}{2}g_{2}^{{\alpha}{\beta}}\partial_{\alpha}x^{i}\,\partial_{\beta}x^{i}+x^{i}x^{i}+\frac{1}{2}g_{2}^{{\alpha}{\beta}}\partial_{\alpha}y^{A}\,\partial_{\beta}y^{A}\,,\qquad
g_{2}^{{\alpha}{\beta}}=\delta^{{\alpha}{\beta}}(\widehat{z}^{\,0})^{2}\,.$
(123)

This result is in perfect agreement with [37]. Similarly to the bulk case,
these fields are part of a short multiplet of $OSp(4^{*}|4)$, with quantum
numbers reported in table 2. In defect CFT terminology, the field $y^{A}$ is
dual to the tilt operator, which appears in the Ward identity for the broken
$R$-symmetry $SO(6)_{R}\to SO(5)_{R}$ [83, 84]. Similarly, the field $x^{i}$
is dual to the displacement operator, which appears in the Ward identity for
the the broken translation symmetry [61].

Field | $\;\;m^{2}\;\;$ | $\;\;\Delta\;\;$ | $\,SO(3)_{\text{trans.}}\,$ | $\,\;SO(5)_{R}\;\,$
---|---|---|---|---
$y^{A}$ | 0 | 1 | singlet | vector
$x^{i}$ | 2 | 2 | vector | singlet
Table 2: Bosonic open-string modes on the fundamental string worldsheet.

One can proceed to extract the rest of couplings in a similar manner. For
example, reference [37] computed all quartic vertices of worldsheet fields, in
our notation $L_{\text{F1}}^{(4,0)}$. These terms do not contribute to the
two-point function of chiral-primary operators at leading order, so we do not
present the formulas here. The first vertex of interest couples a bulk field
directly to the string worldsheet

$\displaystyle L_{\text{F1}}^{(0,1)}$
$\displaystyle=\sum_{k,I}\left(\frac{1}{2}(\varphi_{k}^{I})_{\alpha}^{\alpha}-\frac{2k(k-1)}{k+1}s_{k}^{I}-\frac{2(k+4)(k+5)}{k+3}t_{k}^{I}+\ldots\right)Y^{I}_{k}\,.$
(124)

These vertices allow bulk fields to be “absorbed” by the string, recall (11).
The vertex for $s_{k}^{I}$ has appeared multiple times in the literature, see
e.g. [34, 35, 36], but to the best of our knowledge, the vertices for
$\varphi$ and $t$ are new. In (124) we neglect total derivatives
$\partial_{\alpha}V^{\alpha}$, since they drop from Witten diagrams. It is
also implicit in the notation that the spherical harmonic is evaluated at zero
$Y_{k}^{I}=Y_{k}^{I}(y=0)$, and the bulk fields live on the worldsheet, for
example $s_{k}^{I}=s_{k}^{I}(\widehat{z}^{\,0},\widehat{z}^{\,1},0,0,0)$.

Finally, we need vertices where the bulk field $s$ excites a worldsheet
fluctuation. These vertices have not appeared before in the literature, and a
calculation exactly as above gives

$\displaystyle
L_{\text{F1}}^{(1,1)}=-\sum_{k,I}\frac{2k(k-1)}{k+1}\Big{(}\,y^{A}\partial_{A}+\widehat{z}_{0}\,x^{i}\partial_{i}\,+\ldots\Big{)}s^{I}_{k}Y^{I}_{k}\,.$
(125)

To obtain this formula we used multiple integration by parts, and dropped
total derivatives and terms proportional to the equations of motion (EOM).
When computing the correlator $\langle S_{p_{1}}S_{p_{2}}W\rangle$, the EOM
terms will contribute contact diagrams like the one in (31). However, in
section 5.3 we neglect all contact diagrams, and only fix them later with the
bootstrap approach of section 5.4. A more careful first-principle calculation,
that aims at computing the correlator including contact terms, should keep
track of the EOM terms in (125).

###  5.3 Explicit calculation (up to contact terms)

$\displaystyle\langle
S_{p_{1}}S_{p_{2}}W\rangle_{\text{c}}\;=\;\sum_{{\mathcal{O}}=\\{s_{k},\varphi_{k},t_{k}\\}}\;\;\leavevmode\hbox
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Figure 4: Diagrams that contribute to the connected correlator $\langle
S_{p_{1}}S_{p_{2}}W\rangle_{\text{c}}$ at leading order in the supergravity
limit. The quantum numbers of exchanged fields appear in tables 1 and 2.

Having derived the $\mathrm{AdS}_{5}$ and $\mathrm{AdS}_{2}$ effective
actions, we are ready for a first attempt to calculate $\langle
S_{p_{1}}S_{p_{2}}W\rangle_{c}$. Recall that we are interested in the
connected correlator, so we neglect disconnected diagrams as in (20). Figure 4
shows the connected diagrams in the leading supergravity approximation, which
consist of exchange diagrams with $s$, $\varphi_{\mu\nu}$ and $t$ in
$\mathrm{AdS}_{5}$, exchange diagrams with $x$, $y$ in $\mathrm{AdS}_{2}$, and
contact diagrams. The quantum numbers of the exchanged fields are summarized
in tables 1 and 2, while contact interactions are ignored in this section for
reasons to be explained later.

To calculate $\langle S_{p_{1}}S_{p_{2}}W\rangle$, one should view the on-
shell supergravity action with certain boundary conditions as the generating
functional of CFT correlators [2, 3], and then take functional derivatives. In
practice, however, this amounts to computing the correlator with standard
position-space Feynman rules, using propagators (8)-(10) and the effective
actions of section 5.2. For the bulk-to-boundary propagator, we choose the
normalization [34]

$\displaystyle\Pi_{p}(x,z)={\mathcal{N}}_{p}K_{p}(x,z)\,,\qquad{\mathcal{N}}_{p}=2^{p/2-2}\,\frac{p+1}{N\sqrt{p}}\,,$
(126)

which ensures that in the absence of the defect, the two-point function
$\langle S_{p}S_{p}\rangle$ is unit normalized. Besides this, there are
several numerical factors that need to be accounted for. For example, because
of (113), bulk-to-bulk propagators come with a factor
$\frac{(2\pi)^{5}}{4N^{2}\zeta}$, where the normalizations $\zeta$ are shown
in (172). Similarly, brane-to-brane propagators come with a factor
$\frac{2\pi}{\sqrt{\lambda}}$, see (122). Regarding bulk three-point vertices,
they contain an overall factor $\frac{4N^{2}}{(2\pi)^{5}}$. There are three
bulk vertices $X=S,\,T,\,G$, which come with extra factors $6$, $2$ and $2$
respectively, that count the number of different Wick contractions. Each
three-point vertex is of the form

$\displaystyle
X_{p_{1}p_{2}k}^{IJK}Y_{k}^{I}(0)\;\to\;X_{p_{1}p_{2}k}h_{p_{1}p_{2}}^{k}(\sigma)\,,$
(127)

where $X_{p_{1}p_{2}k}$ are found in (175). In equation (127) we have mapped
the spherical harmonic indices $I,J,K$ to the index-free notation of section
5.1. The derivation of this map is presented in appendix C, but here it
suffices to note that the result is exactly the $R$-symmetry block

$\displaystyle
h^{k}_{p_{1}p_{2}}(\sigma)=\sigma^{\frac{p_{1}+p_{2}-k}{2}}\,_{2}F_{1}\\!\left(-\frac{p_{12}+k}{2},\frac{p_{12}-k}{2};-k-1;\frac{\sigma}{2}\right)\,.$
(128)

For integer values of $p_{1}$, $p_{2}$ and $k$ the $R$-symmetry block is a
polynomial in $\sigma$. Finally, the bulk-boundary vertices are proportional
to $\frac{\sqrt{\lambda}}{2\pi}$, with the precise values in (124). As before,
appendix C translates the $R$-symmetry contractions to index-free notation

$\displaystyle
Y_{p_{1}}^{I}(0)Y_{p_{2}}^{J}(0)\;\to\;1\,,\quad\partial_{A}Y_{p_{1}}^{I}(0)\partial_{A}Y_{p_{2}}^{J}(0)\;\to\;p_{1}p_{2}(\sigma-1)\,.$
(129)

All in all, we have the following contributions to the connected correlator

$\displaystyle{\mathcal{F}}_{c}(\xi,\eta,\sigma)$
$\displaystyle\;\sim\;{\mathcal{N}}_{p_{1}}{\mathcal{N}}_{p_{2}}\frac{\sqrt{\lambda}}{2\pi}\Bigg{[}\sum_{k}\frac{6S_{p_{1}p_{2}k}}{\zeta^{s}_{k}}\frac{2k(k-1)}{k+1}E_{p_{1}p_{2}}^{k,0}h_{p_{1}p_{2}}^{k}$
$\displaystyle\quad+\sum_{k}\frac{2T_{p_{1}p_{2}k}}{\zeta^{t}_{k}}\frac{2(k+4)(k+5)}{k+3}E_{p_{1}p_{2}}^{k+8,0}h^{k}_{p_{1}p_{2}}+\sum_{k}\frac{2G_{p_{1}p_{2}k}}{\zeta^{\varphi}_{k}}\frac{1}{2}E_{p_{1}p_{2}}^{k+4,2}h^{k}_{p_{1}p_{2}}$
$\displaystyle\quad+\frac{4p_{1}p_{2}(p_{1}-1)(p_{2}-1)}{(p_{1}+1)(p_{2}+1)}\left(p_{1}p_{2}(\sigma-1)\widehat{E}_{p_{1}p_{2}}^{1,0}+\widehat{E}_{p_{1}p_{2}}^{2,1}\right)\Bigg{]}\,,$
(130)

where the sum runs over $k$’s allowed by $SU(4)_{R}$ symmetry. The couplings
$S$, $T$, $G$ and $\zeta$ are found in appendix B, while
$E_{\Delta_{1}\Delta_{2}}^{\Delta,\ell}$ and
$\widehat{E}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},s}$ are bulk-exchange
and brane-exchange Witten diagrams respectively, which we computed in section
3.

Combining all these ingredients, one can check that (130) cannot be the
correct result, because it does not obey the superconformal Ward identity
(108). The reason (130) is incorrect is that we have completely ignored
contact diagrams. Instead, the correct result takes the form

$\displaystyle{\mathcal{F}}_{c}(\xi,\eta,\sigma)\,=\,\eqref{eq:tentative-
corr}\,+\;C_{p_{1}p_{2}}(\xi,\eta)\sum_{n=0}^{{p_{\mathrm{min}}}}c_{n}\,\sigma^{n}\,.$
(131)

Unfortunately, starting from the IIB and fundamental string actions, we have
been unable to determine coefficients $c_{n}$ that generate correlators
consistent with the Ward identities. The naive way to obtain $c_{n}$ is to
expand the string action up to terms like
$\sim\int_{\mathrm{AdS}_{2}}s_{p_{1}}s_{p_{2}}$, and keep track of the EOM
terms neglected in (125). Because of their $R$-symmetry structure, these terms
can contribute to $c_{0}$ and $c_{1}$, but as we show below, all $c_{n}$ are
non-vanishing in the supersymmetric correlator. This suggests there are other
sources of contact terms, and raises the question of how to do a first-
principles calculation that correctly accounts for them. One possibility is
that in the derivation of the IIB effective action, one generates the missing
contact terms when integrating by parts or in doing certain field
redefinitions. We hope future work will clarify this question, while in the
rest of the paper we determine $c_{n}$ with a bootstrap method.

###  5.4 Position space bootstrap

Although we got far using traditional methods, in the end we failed to find a
correlator consistent with the supersymmetry of the system. To solve this
problem, in this section we determine the correlator with a bootstrap method.
The main idea was put forth in [17]: write an ansatz with all Witten diagrams
allowed by the effective action, and fix the relative coefficients with the
superconformal Ward identity.

Let us spell out the method in detail. First, recall there are bulk-exchange
diagrams with the fields $s$, $\varphi$ and $t$ in table 1. These fields
belong to a superconformal multiplet, where the superprimary $s$ transforms in
the representation $[0,k,0]$ of $SU(4)_{R}$. The value of $k$ is restricted by
$SU(4)_{R}$ symmetry to be $|p_{12}|\leq k\leq p_{1}+p_{2}$ and to run in
steps of two. Furthermore, when the superprimary satisfies $k=p_{1}+p_{2}$ or
$k=|p_{12}|$, the three-point vertices $S$, $T$, $G$ in (175) vanish, so we
take $|p_{12}|<k<p_{1}+p_{2}$. As a result, each multiplet $(s,\varphi,t)$ in
table 1 contributes for $k\in{\mathcal{S}}$, where

$\displaystyle{\mathcal{S}}=\Big{\\{}|p_{12}|+2,|p_{12}|+4,\ldots,p_{1}+p_{2}-4,p_{1}+p_{2}-2\Big{\\}}\,.$
(132)

This concludes the discussion of bulk-exchange diagrams.

Regarding defect-exchange diagrams, the fields $x$, $y$ in table 2 always
contribute, because they are always allowed by $R$-symmetry. Finally, there
can be contact diagrams, so the full ansatz for the connected correlator
is141414One issue we glossed over is what contact diagrams to include in the
ansatz. In fact, (133) ignores contact diagrams with derivatives, which would
come from $\int_{\mathrm{AdS}_{2}}\nabla^{n}s_{1}\nabla^{m}s_{2}$. One
justification is that spin-two diagrams $E^{\Delta,2}_{\Delta_{1}\Delta_{2}}$
and contact diagrams $C_{\Delta_{1}\Delta_{2}}$ have the same Regge behavior.
One the other hand, derivative contact diagrams grow faster in the Regge
limit, and because we do not expect the Regge limit to be controlled only by
contact diagrams, they should be absent. We thank Xinan Zhou for this
argument.

$\displaystyle{\mathcal{F}}_{c}(\xi,\eta,\sigma)$
$\displaystyle=\sum_{k\in{\mathcal{S}}}\Big{(}A^{s}_{k}\,E_{p_{1}p_{2}}^{k,0}(\xi,\eta)\,h_{p_{1}p_{2}}^{k}(\sigma)+A^{\varphi}_{k}\,E_{p_{1}p_{2}}^{k+2,2}(\xi,\eta)\,h_{p_{1}p_{2}}^{k-2}(\sigma)$
$\displaystyle\qquad\qquad+A^{t}_{k}\,E_{p_{1}p_{2}}^{k+4,0}(\xi,\eta)\,h_{p_{1}p_{2}}^{k-4}(\sigma)\Big{)}+C_{p_{1}p_{2}}(\xi,\eta)\sum_{n=0}^{{p_{\mathrm{min}}}}A_{n}^{ss}\,\sigma^{n}$
$\displaystyle\quad+A^{y}\,\widehat{E}_{p_{1}p_{2}}^{1,0}(\xi,\eta)(\sigma-1)+A^{x}\,\widehat{E}_{p_{1}p_{2}}^{2,1}(\xi,\eta)\,,$
(133)

with the sum ranges in (132) and $A$’s numerical coefficients to be
determined. The bulk-exchange diagrams
$E_{\Delta_{1}\Delta_{2}}^{\Delta,\ell}$ can be evaluated as sums of contact
diagrams, see (39) and appendix A. Furthermore, contact diagrams are known in
closed form (35). Finally, the defect-exchange diagrams
$\widehat{E}_{\Delta_{1}\Delta_{2}}^{{\widehat{\Delta}},s}$ can be computed
from the differential equation (46).

For a given choice of $p_{1}$ and $p_{2}$, we build the ansatz as just
explained and impose the superconformal Ward identity (108) order by order in
an expansion around $r=0$. The resulting equations can be solved, and they
fully determine the correlator (133) up to an overall normalization, which of
course cannot be fixed by the Ward identity only. In order to fix the overall
normalization, note that in the strong ’t Hooft coupling limit we have [53]

$\displaystyle\lim_{\lambda\to\infty}\lim_{N\to\infty}\lambda_{S_{p_{1}}S_{p_{2}}S_{k}}a_{S_{k}}=\frac{\sqrt{\lambda}}{N^{2}}\frac{\sqrt{p_{1}p_{2}}\,k}{2^{k/2+1}}+\ldots\,,$
(134)

where $a_{\mathcal{O}}\propto\langle{\mathcal{O}}W\rangle$ and
$\lambda_{{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}}\propto\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}\rangle$.
The Euclidean OPE limit $\xi\to 0$, $\eta\to 1$ is dominated by the lowest-
dimension chiral primary operator $S_{k_{\text{min}}}$, and we find

$\displaystyle\lim_{\xi\to 0}\lim_{\eta\to
1}{\mathcal{F}}_{c}(\xi,\eta,\sigma)=\frac{\sqrt{\lambda}}{N^{2}}\frac{\sqrt{p_{1}p_{2}}\,k_{\text{min}}}{2^{k_{\text{min}}/2+1}}\,\xi^{\frac{k_{\text{min}}-p_{1}-p_{2}}{2}}\,h_{p_{1}p_{2}}^{k_{\text{min}}}(\sigma)+\ldots\,.$
(135)

Comparing this to ansatz (133) fixes the overall normalization of
${\mathcal{F}}_{c}$.

We bootstrapped many correlators with $p_{1},p_{2}\lesssim 30$, and in every
case all coefficients were fixed. All cases with $p_{1},p_{2}\leq 4$ were
computed previously in [53, 54], and we found perfect agreement.151515The
careful reader will note that our correlator for $(p_{1},p_{2})=(2,4)$ differs
with the one in [54]. The difference is only the contribution
$\lambda_{242}a_{2}f_{2,0}^{-2}$ in the bulk OPE. The reason is that [54]
considered the two-point function of single-trace operators
${\mathcal{O}}_{p}$, for which
${\mathcal{O}}_{2}\times{\mathcal{O}}_{4}\sim{\mathcal{O}}_{2}$. However, we
work in the basis natural for supergravity, such that
$\lambda_{S_{2}S_{4}S_{2}}=0$, recall footnote 10. At this point, it is
interesting to compare our method to the analytic bootstrap in [53, 54]. The
analytic bootstrap calculation reconstructed the correlator
${\mathcal{F}}_{c}$ using a dispersion relation or inversion formula. These
formulas suffer from low-spin ambiguities, which in practice means that only
part of the correlator is reconstructed. In the present language, the
dispersion relation reconstructs only the sum over bulk-exchange diagrams,
because both defect-exchange and contact diagrams have vanishing
discontinuity. In [53, 54], the missing part of the correlator was determined
making a very general ansatz and fixing it from superconformal Ward
identities. Remarkably, this ad-hoc procedure managed to reconstruct precisely
the defect-exchange diagrams for the $x^{i}$ and $y^{A}$ fluctuations.
Needless to say, the present method is superior to the one employed in [53,
54]. On one hand, here we have a more clear physical picture of all the terms
that form the correlator. On the other, the present method is easier and
faster, thanks to the efficient methods to compute Witten diagrams.

Because of the efficiency of our method, it is possible to generate many
correlation functions, and by looking at all the examples, we could guess a
closed formula

$\displaystyle{\mathcal{F}}_{c}(\xi,\eta,\sigma)$
$\displaystyle=\frac{\sqrt{\lambda}}{N^{2}}\sum_{k\in{\mathcal{S}}}\frac{\sqrt{p_{1}p_{2}}\,k}{2^{\frac{k}{2}+1}}\Bigg{(}P^{k,0}_{p_{1}p_{2}}h^{k}_{p_{1}p_{2}}+\frac{(p^{2}_{12}-k^{2})\big{(}p^{2}_{12}-(k+2)^{2}\big{)}}{128k(k+1)^{2}(k+3)}P^{k+2,2}_{p_{1}p_{2}}h^{k-2}_{p_{1}p_{2}}$
$\displaystyle\qquad\qquad\quad+\frac{(p_{12}^{2}-(k+2)^{2})(p_{12}^{2}-(k-2)^{2})(p_{12}^{2}-k^{2})^{2}}{16384(k-2)(k-1)^{2}k^{2}(k+1)(k+2)(k+3)}P^{k+4,0}_{p_{1}p_{2}}h^{k-4}_{p_{1}p_{2}}\Bigg{)}$
$\displaystyle\quad+\frac{\sqrt{\lambda}}{N^{2}}\frac{(p_{1}p_{2})^{\frac{1}{2}}(p_{1}-1)(p_{2}-1)}{2^{3-\frac{p_{1}+p_{2}}{2}}\pi}\bigg{(}p_{1}p_{2}(\sigma-1)\widehat{E}_{p_{1}p_{2}}^{1,0}+\widehat{E}_{p_{1},p_{2}}^{2,1}+(1-2\sigma)C_{p_{1}p_{2}}\bigg{)}\,.$
(136)

The sum over $k$ is described in (132), and
$P_{\Delta_{1}\Delta_{2}}^{\Delta\ell}$ are Polyakov-Regge blocks. Recall
that, up to contact terms, Polyakov-Regge blocks are bulk-exchange Witten
diagrams with a different normalization, see (63). The bootstrap result (136)
agrees with the explicit calculation (130), up to the contact terms that we
could not determine in the first-principles calculation. This provides a
strong consistency check for our results. Let us stress that to obtain (136),
the only input from section 5.2 is the field content in the effective actions,
but not the precise value of vertices.

The position-space correlator (136) has a simple interpretation in terms of
conformal blocks. Because Polyakov-Regge blocks are normalized as
$P_{\Delta_{1}\Delta_{2}}^{\Delta\ell}=\xi^{-\frac{\Delta_{1}+\Delta_{2}}{2}}f_{\Delta,\ell}+\ldots$,
the coefficient of $P_{p_{1}p_{2}}^{k,0}h^{k}_{p_{1}p_{2}}$ in (136) is the
OPE coefficient $\lambda_{S_{p_{1}}S_{p_{2}}S_{k}}a_{S_{k}}$, in perfect
agreement with the known value (134). Also note that $\varphi_{\mu\nu}$ and
$t$ are superdescendants of $s$, so their OPE coefficients are fixed in terms
of the superprimary. This is also explicit in the correlator, because the
relative coefficients of the first two lines of (136) precisely form a
superblock (compare with equation (5.12) of [54]). Finally, the last line of
(136) contains bulk-defect OPE coefficients of chiral-primaries $S_{k}$ with
tilt $\widehat{t}$ and displacement $\widehat{D}^{i}$ operators. To present
the OPE coefficients, we use index-free notation
$t(0,\widehat{v})=t^{A}(0)\widehat{v}_{A}$, with the polarization vector
satisfying $\widehat{v}^{\,2}=\widehat{v}\cdot\theta=0$, and we find161616We
use the tilt and displacement operators with unit-normalized two-point
functions. As a result, the Ward identities that define these operators have
extra normalization factors  $\displaystyle\partial_{\mu}T^{\mu
i}=C_{D}^{1/2}\,\widehat{D}^{i}\,\delta_{W}\,,\qquad\partial_{\mu}J^{\mu}_{A}=C_{t}^{1/2}\,\widehat{t}_{A}\,\delta_{W}\,.$
(137)

$\displaystyle\langle
S_{p}(x,u)\,\widehat{t}(0,\widehat{v})\,W(\theta)\rangle$
$\displaystyle=\frac{(u\cdot\theta)^{p-1}(u\cdot\widehat{v})}{x^{2}|x^{i}|^{p-1}}\left(\frac{\lambda^{1/4}}{N}\frac{p^{3/2}}{2^{\frac{p+1}{2}}}+\ldots\right)\,,$
(138) $\displaystyle\langle S_{p}(x,u)\,\widehat{D}^{i}(0)\,W(\theta)\rangle$
$\displaystyle=\frac{x^{i}(u\cdot\theta)^{p}}{x^{4}|x^{j}|^{p}}\left(\frac{\lambda^{1/4}}{N}\frac{p^{3/2}}{2^{\frac{p}{2}}\sqrt{3}}+\ldots\right)\,.$
(139)

To obtain this formula, we used the normalization of defect-exchange Witten
diagrams in (47). The correlator $\langle S_{p}\,\widehat{t}\,W\rangle$ is
captured by the topological sector of [41, 42], and we find perfect agreement.
Summarizing, the correlator (136) is a sum over protected operators, each of
them contributing an exchange Witten diagram, supplemented by extra contact
diagrams fixed by superconformal Ward identities. Besides these highly non-
trivial sanity checks, in appendix D we successfully compare to the
localization calculation of [82], which captures only the topological part of
the correlator.

###  5.5 Mellin space

There are at least three good reasons to map the above results to Mellin
space. First, it would be good to have a bootstrap method directly in Mellin
space, since this could be necessary in setups where the position-space
bootstrap does not work. For example, bulk-exchange diagrams do not truncate
in theories with an $\mathrm{AdS}_{4}$ bulk, making the position space
bootstrap unfeasible. Second, the Mellin amplitude ${\mathcal{M}}$ should be
closely related to a flat-space scattering amplitudes of supergravitons off a
string or brane, see [40] for a recent discussion. By making this link more
precise, one could input information of flat-space amplitudes to bootstrap
subleading corrections in $\frac{1}{\sqrt{\lambda}}$, see similar examples
[85, 22, 25]. Third, thanks to the properties of Mellin amplitudes, it is
plausible the result might be simpler in Mellin space, a hypothesis that we
confirm below.

Keeping this motivation in mind, we Mellin transform the position-space
correlator (136). The first two lines in (136) contain Polyakov-Regge blocks,
which are combinations of bulk-exchange and contact diagrams, see (63). The
Mellin transforms of bulk-exchange and contact diagrams are presented in (88)
and (72) respectively, and they always reduce to rational functions of the
Mellin variables $\delta$ and $\rho$. The last line in (136) contains defect-
exchange diagrams, that in Mellin space do not always reduce to rational
functions, recall (98). For example, for $p_{1}=p_{2}=2$, the Mellin amplitude
reads

$\displaystyle{\mathcal{M}}(\delta,\rho,\sigma)=\frac{\sqrt{\lambda}}{8N^{2}}\Bigg{(}$
$\displaystyle\frac{(\rho\sigma-\sigma+4)\sigma}{\delta-1}-\frac{2\rho+4\sigma-4}{\delta+\rho}+\frac{\rho}{\delta+\rho+2}+(\sigma-1)^{2}$
$\displaystyle+\frac{2^{\delta+\rho+2}\,\Gamma(-\delta-\rho)}{\Gamma\\!\left(\frac{2-\delta-\rho}{2}\right)^{2}}\left(\frac{\delta+2}{\delta+\rho+2}-\sigma\right)\Bigg{)}\,.$
(140)

In this case, the defect-exchange amplitudes
$\widehat{\mathcal{M}}_{2,2}^{1,0}$ and $\widehat{\mathcal{M}}_{2,2}^{2,1}$
simplify in terms of gamma functions, but we have not checked if this holds
for arbitrary $p_{1},p_{2}>2$.

In a similar way, we can Mellin transform the correlator for many values
$p_{1}$, $p_{2}$, and guess a closed formula for ${\mathcal{M}}$. We observe
that the Mellin transform of the first two lines in (136) simplifies
drastically, but there are no important simplifications in the last line:

$\displaystyle{\mathcal{M}}(\delta,\rho,\sigma)$
$\displaystyle=\sum_{n=1}^{{p_{\mathrm{min}}}}\sum_{i=1}^{n}\frac{b_{n,i}\,\rho+c_{n,i}}{\delta-i}\,\sigma^{n}+e\,p_{1}p_{2}(\sigma-1)\widehat{\mathcal{M}}_{p_{1}p_{2}}^{1,0}+e\widehat{\mathcal{M}}_{p_{1}p_{2}}^{2,1}+a\,(\sigma-1)^{2}\,.$
(141)

The numerical constants $a$, $b_{n,i}$, $c_{n,i}$ and $e$ were guessed in
closed form, and take a relatively simple form:

$\displaystyle a$
$\displaystyle=-\frac{\sqrt{\lambda}}{N^{2}}\,\frac{2^{\frac{p_{1}+p_{2}}{2}}\sqrt{p_{1}p_{2}}\left(-\frac{1}{2}\right)_{\frac{p_{1}+p_{2}}{2}}}{16(p_{1}-2)!(p_{2}-2)!}\,,$
$\displaystyle b_{n,i}$
$\displaystyle=a\,\frac{(2-n)_{i-1}(2-p_{1})_{n-2}(2-p_{2})_{n-2}}{2^{n-2}(i-1)!(n-2)!\left(\frac{3-p_{1}-p_{2}}{2}\right)_{n-2}}\,,$
$\displaystyle c_{n,i}$
$\displaystyle=\frac{b_{n,i}}{4}\left(\frac{p_{12}^{2}-1-8(n-p_{1})(n-p_{2})(2n-p_{1}-p_{2})/(i-n)}{p_{1}+p_{2}-2n+1}+4i+6n-5p_{1}-5p_{2}+1\right)\,,$
$\displaystyle e$
$\displaystyle=\frac{\sqrt{\lambda}}{N^{2}}\,\frac{2^{\frac{p_{1}+p_{2}}{2}}}{8\pi}\,(p_{1}p_{2})^{1/2}(p_{1}-1)(p_{2}-1)\,.$
(142)

The formula for $c_{n,i}$ is singular for $n=i$, but the value of $c_{n,n}$ in
(141) is finite and given by

$\displaystyle c_{n,n}$
$\displaystyle=a\,\frac{\left(\frac{p_{1}+p_{2}}{2}-n\right)(2-p_{1})_{n-1}(2-p_{2})_{n-1}}{(-2)^{n-3}(n-1)!\left(\frac{3-p_{1}-p_{2}}{2}\right)_{n-1}}\,.$
(143)

The Mellin amplitude (141) has a simple interpretation in terms of the OPE.
The finite sum captures contributions of single-trace operators in the OPE
$S_{p_{1}}\times S_{p_{2}}\sim S_{k}+\ldots$, compare with the general formula
(68). Similarly, the last two terms correspond to the exchange of tilt and
displacement in the defect expansion $S_{p}\times
W\sim\widehat{t}\,W+\widehat{D}^{i}W+\ldots$. On the other hand, we do not
have a clear interpretation for the term $a(\sigma-1)^{2}$, which should
correspond to a contact interaction, because it is constant.

The result (141) deserves several comments. First, it would be interesting to
find the prescription for taking the flat-space limit and comparing to flat-
space string amplitudes. Second, the sum in (141) is remarkably simple, and is
reminiscent of the elegant formula found in [16, 17] for the four-point
function $\langle S_{p_{1}}S_{p_{2}}S_{p_{3}}S_{p_{4}}\rangle$. In that case,
the simplicity is related to the hidden ten-dimensional conformal symmetry
[86], and it would be fascinating if a similar structure can be found here.
Finally, it is unclear to us whether (141) could have been bootstrapped
directly in Mellin space. In any case, if such bootstrap approach is possible,
it certainly requires superconformal Ward identities in Mellin space, a
subject that we now discuss.

####  5.5.1 Ward identity

Superconformal Ward identities were first implemented in Mellin space in [87],
see also [88, 29]. We adapt these methods to the present setup in appendix E.
The result is that given the decomposition (104) of the correlation function,
the $R$-symmetry channels satisfy

$\displaystyle\sum_{j=0}^{{p_{\mathrm{min}}}}\left[\zeta_{\pm}^{(j)}\big{(}4\xi\partial_{\xi}-\xi\partial_{\eta}+4j\big{)}+\zeta_{\pm}^{(j+1)}(\xi\partial_{\xi}+\eta\partial_{\eta}+j)\right]\frac{{\mathcal{F}}_{j}(\xi,\eta)}{(-2)^{j}}=0\,,$
(144)

where $\zeta_{+}^{(i)}$ are polynomials of the cross-ratios

$\displaystyle\zeta_{+}^{(i)}$ $\displaystyle=\sum_{j=0}^{\left\lfloor
i/2\right\rfloor}\binom{i}{2j}\left(\eta^{2}-1\right)^{j}\left((2\eta+\xi)^{2}-4\right)^{j}(2\eta^{2}+\xi\eta-2)^{i-2j}\,,$
(145a) $\displaystyle\zeta_{-}^{(i)}$ $\displaystyle=\sum_{j=0}^{\left\lfloor
i/2\right\rfloor}\binom{i}{2j+1}\left(\eta^{2}-1\right)^{j}\left((2\eta+\xi)^{2}-4\right)^{j}(2\eta^{2}+\xi\eta-2)^{i-2j-1}\,.$
(145b)

This form of the superconformal Ward identity has a direct translation in
Mellin space, because derivatives map to

$\displaystyle\xi\partial_{\xi}$
$\displaystyle\;\leftrightarrow\;-\delta\,,\qquad\eta\partial_{\eta}\;\leftrightarrow\;-\rho\,,$
(146)

while products by $\xi^{m}\eta^{n}$ act as shifts, see (73). As a result, the
superconformal Ward identity maps into equations that involve shifts of the
Mellin amplitudes and products by rational functions of the Mellin variables.
For example, when $p_{1}=p_{2}=2$ the Ward identity for $\zeta_{-}^{(j)}$
reads

$\displaystyle 0=$
$\displaystyle\,(\delta+\rho){\mathcal{M}}_{0}(\delta,\rho)+2\rho{\mathcal{M}}_{1}(\delta,\rho)+4\rho{\mathcal{M}}_{2}(\delta,\rho)-\frac{2\rho(\rho+1)(\delta+\rho+1)}{(\delta+\rho)^{2}}{\mathcal{M}}_{1}(\delta,\rho+2)$
$\displaystyle+\delta{\mathcal{M}}_{1}(\delta+1,\rho-1)+2\delta{\mathcal{M}}_{2}(\delta+1,\rho-1)-\frac{2\delta\rho(\delta+\rho+1)}{(\delta+\rho)^{2}}{\mathcal{M}}_{1}(\delta+1,\rho+1)$
$\displaystyle-\frac{4\rho(\rho+1)(\delta+2\rho+2)}{(\delta+\rho)^{2}}{\mathcal{M}}_{2}(\delta,\rho+2)-\frac{2\delta\rho(2\delta+5\rho+3)}{(\delta+\rho)^{2}}{\mathcal{M}}_{2}(\delta+1,\rho+1)$
$\displaystyle-\frac{\delta(\delta+1)(\delta+3\rho)}{(\delta+\rho)^{2}}{\mathcal{M}}_{2}(\delta+2,\rho)+\frac{4\rho(\rho+1)(\rho+2)(\rho+3)}{(\delta+\rho)^{2}(\delta+\rho+2)}{\mathcal{M}}_{2}(\delta,\rho+4)$
$\displaystyle+\frac{8\delta\rho(\rho+1)(\rho+2)}{(\delta+\rho)^{2}(\delta+\rho+2)}{\mathcal{M}}_{2}(\delta+1,\rho+3)+\frac{4\delta(\delta+1)\rho(\rho+1)}{(\delta+\rho)^{2}(\delta+\rho+2)}{\mathcal{M}}_{2}(\delta+2,\rho+2)\,.$
(147)

It is not hard to check that the Mellin amplitude (140) indeed satisfies the
Ward identity. Note that ${\mathcal{M}}_{0}$ can be solved directly as a
function of ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$. This seems to be a
general feature, although we leave a more detailed analysis of the Ward
identities for future work.

##  6 Conclusions

In this work we proposed a bootstrap method for holographic defect
correlators, inspired by the position-space bootstrap of [16, 17]. Our focus
was on two-point functions of local operators in the presence of the
holographic defect, a setup that is relatively unexplored in the literature.
The most important ingredient for the bootstrap is knowledge of certain Witten
diagrams, that were partially studied in [55, 56, 57, 31]. The present paper
extends these results, by analyzing systematically all Witten diagrams that
should contribute in the leading supergravity approximation to setups of
arbitrary dimension and codimension. In this respect, the spin-two bulk-
exchange diagram is the most challenging diagram, and its computation in
appendix A constitutes an important new technical result.

To illustrate the position-space bootstrap, we revisited the half-BPS Wilson
line in ${\mathcal{N}}=4$ SYM. References [53, 54] initiated the study of bulk
two-point functions with the Wilson line using analytic bootstrap. Our
calculation clarifies the results in [53, 54], because we show that their
“low-spin ambiguities” correspond to defect-exchange and contact Witten
diagrams. Furthermore, because the present method is more streamlined, we have
succeeded in obtaining closed formulas for correlator of chiral-primary
operators of arbitrary length. These formulas involve finite sums with known
coefficients, see the position space result (136) and its Mellin counterpart
(141). The position-space formula agrees with the explicit calculation (130),
up to contact terms that we were unable to fix without the bootstrap. Although
this provides a strong sanity check for our method, it still begs the question
of how to obtain the correlators from first principles. We hope this will be
clarified in future work.

Our results open a number of new research directions. One possibility is to
extend the calculation in ${\mathcal{N}}=4$ SYM with the half-BPS Wilson line
to subleading orders in the large-$N$ or large-$\lambda$ expansion. Presumably
this requires the use of integrated correlators and a better understanding of
the flat-space limit, as [40] proposes inspired by the success of similar
work, e.g. [22, 25]. Another option is to study other holographic defects at
leading order in the supergravity limit. Some examples in ${\mathcal{N}}=4$
SYM are Wilson lines in symmetric or antisymmetric representations [89, 90,
35, 91], ’t Hooft lines [92, 93], surface operators [94, 95, 96], or
interfaces [97, 98, 99]. The present methods are not restricted to four
dimensions, and they should also apply to the $3d$ ABJM or $6d$ $(2,0)$
theories. In this respect, recently [59] studied the M2-brane surface defect
of $6d$ $(2,0)$ with the analytic bootstrap of [53, 54], and it seems this
setup is also amenable to the Witten diagram bootstrap [100].

Finally, we have outlined in section 3.5 how some of our results are
interconnected with other analytic bootstrap methods, such as analytic
functionals. However, these links could be extended and made more precise. For
example, it would be good to understand under what circumstances the Mellin
representation is valid for non-perturbative defect correlators, in the spirit
of [75, 80], and also to derive dispersion relations directly in Mellin space.
Alternatively, one could explore the crossing-symmetric Mellin bootstrap of