aainstitutetext: Departamento de Ciencias Físicas, Facultad de Ciencias
Exactas, Universidad Andrés Bello,
Sazié 2212, Piso 7, Santiago, Chile.bbinstitutetext: Instituto de Física La
Plata - CONICET &
Departamento de Física, Universidad Nacional de La Plata,
C.C. 67, 1900, La Plata, Argentina

# Interpolating Boundary Conditions on $AdS_{2}$

Anthonny F. Canazas Garay b Diego H. Correa a Alberto Faraggi b Guillermo A.
Silva<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider boundary conditions for massless scalars on $AdS_{2}$ that
interpolate between the Dirichlet and Neumann cases while preserving scale
invariance. Assessing invariance under the full $SL(2;\mathds{R})$ conformal
group is not immediate given the non-local nature of the boundary conditions.
To further understand this issue, we compute the holographic 2- and 4-point
correlation functions using the interpolating boundary conditions and study
their transformation properties. Concretely, we look at the fluctuations of an
open string around an $AdS_{2}\subset AdS_{4}\times\mathbb{CP}^{3}$ worldsheet
and identify the scalar field with one of the fluctuations in
$\mathbb{CP}^{3}$, thus obtaining correlators of excitations inserted along a
$1$-dimensional defect in ${\cal N}=6$ super Chern-Simons-matter theory at
strong coupling. Of the two types of boundary conditions discussed, only one
leads to a functional structure expected for correlators of conformal
primaries; the other exhibits covariance under translations and rescalings but
not special conformal transformations.

##  1 Introduction

In the AdS/CFT correspondence Malda  a change in the boundary conditions for
the fields in AdS entails deformations of the dual CFT theory. Turning on the
value of a field at the boundary corresponds to deforming the CFT by a single-
trace operator. This procedure allows us to compute CFT correlation functions
in a straightforward way GKP ; Witten:1998qj . Within a certain range of
parameters, one can freely choose which of the two admissible fall-offs for
the bulk fields is going to be fixed and therefore source the corresponding
dual operator BREITENLOHNER1982197 ; BREITENLOHNER1982249 . As an example,
consider a massive scalar on $AdS_{d+1}$, whose asymptotic behavior is

$\displaystyle\phi(z,x)$ $\displaystyle\underset{z\to
0}{\longrightarrow}z^{\Delta_{-}}\left(\alpha(x)+\cdots\right)+z^{\Delta_{+}}\left(\beta(x)+\cdots\right)\,,$
$\displaystyle\qquad\Delta_{\pm}$
$\displaystyle=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}R^{2}}\,.$ (1)

For $-\frac{d^{2}}{4}<m^{2}R^{2}<-\frac{d^{2}}{4}+1$, one can impose either
Dirichlet or Neumann boundary conditions, i.e. fixing $\alpha(x)$ or
$\beta(x)$, respectively, giving rise to CFT operators with different scale
dimensions, namely, $\Delta_{+}$ or $\Delta_{-}$. This range of masses is
known as the Breitenlohner-Freedman (BF) window BREITENLOHNER1982197 ;
BREITENLOHNER1982249 . Arbitrary combinations of $\alpha(x)$ and $\beta(x)$
which interpolate between Dirichlet and Neumann boundary conditions are also
permitted, and they can be seen to correspond to multi-trace deformations of
the dual CFT. For instance, setting $J(x)=\alpha(x)+\chi\beta(x)$ as the
source describes an interpolation between two fixed points. Since $\alpha(x)$
and $\beta(x)$ have different mass dimensions, this choice introduces a
dimensionful parameter into the problem. As a result, scale invariance is
broken by the boundary condition and a renormalization group flow in the dual
field theory is triggered Witten:2001ua ; Hartman:2006dy ; GM . It is of
particular interest to explore instances of deformations by marginal operators
that preserve the full conformal symmetry within the context of the AdS/CFT
correspondence. Supersymmetric multi-trace boundary conditions for scalar
supermultiplets on $AdS_{d+1}$ were considered in Amsel_2009  for dimensions
$d=2\,,3\,,4$.

In this paper we continue the exploration started in Correa_2020  and
construct boundary conditions for massless scalars on $AdS_{2}$ that
interpolate between the Dirichlet and Neumann cases while preserving scale
invariance. To compensate for the difference in scale dimensions between
$\alpha(x)$ and $\beta(x)$ one can act on $\alpha(x)$ with a differential
operator, thus allowing the introduction of a dimensionless interpolating
parameter. However, having a dimensionless parameter is not sufficient to
guarantee full conformal invariance at the quantum level. One way of testing
if a set of boundary conditions is compatible with a given symmetry is to
compute correlation functions holographically. In particular, for a CFT
defined on an infinite line, a primary operator ${\cal O}(\tau)$ of scale
dimension $\Delta$ has a $4$-point correlator of the form

$\displaystyle\langle{\cal O}(\tau_{1}){\cal O}(\tau_{2}){\cal
O}(\tau_{3}){\cal O}(\tau_{4})\rangle$
$\displaystyle=\frac{G(u)}{(\tau_{1}-\tau_{2})^{2\Delta}(\tau_{3}-\tau_{4})^{2\Delta}}\,,$
$\displaystyle\qquad u$
$\displaystyle=\frac{(\tau_{1}-\tau_{2})(\tau_{3}-\tau_{4})}{(\tau_{1}-\tau_{3})(\tau_{2}-\tau_{4})}\,,$
(2)

where $G(u)$ is an arbitrary function of the unique independent cross-ratio,
$u$, that exits in 1d. Full conformal symmetry, and not just scale invariance,
is needed to conclude this. Our main goal is to compute this $4$-point
function using a set of interpolating boundary conditions and check whether it
respects this structure or not.

The specific setup we consider is that of the fluctuations of an open string
in the $AdS_{4}\times\mathbb{CP}^{3}$ background with induced $AdS_{2}$
geometry. The bosonic spectrum includes two transverse fluctuations in
$AdS_{4}$ and six fluctuations along $\mathbb{CP}^{3}$. They lead to scalar
fields with masses $m^{2}=2$ and $m^{2}=0$, respectively (in the following we
set the AdS radius $R^{2}=1$). The spectrum of fermionic fluctuations includes
two massless spinors and six fields with $|m|=1$. Here we will focus a single
complex massless scalar coming from the fluctiuations along $\mathbb{CP}^{3}$.
In order to check consistency with conformal invariance we will compute
4-point correlations functions using the expansion of the Nambu-Goto action to
quartic order.

The study of this particular worldsheet embedding in
$AdS_{4}\times\mathbb{CP}^{3}$ is motivated by its correspondence to BPS
Wilson loop operators in the ${\cal N}=6$ super Chern-Simons-matter theory. We
remind the reader that $1/6$-BPS bosonic Wilson loops were originally found
and put in correspondence with open strings satisfying Neumann boundary
conditions on two angular coordinates in Drukker_2008 ; CHEN201038 ; Rey_2009
, these were referred as strings ‘smeared’ along a
$\mathbb{CP}^{1}\subset\mathbb{CP}^{3}$. Later, fermionic BPS Wilson loops
involving couplings to fermions were discovered, and mapped to strings
satisfying Dirichlet boundary conditions in all their angular coordinates
Drukker_2010 . On general grounds worldsheet fluctuations for either Neumann
or Dirichlet boundary conditions should correspond, in the dual description,
to 1d operators preserving conformal symmetry. Indeed, correlation functions
for operator insertions have been recently computed in Bianchi_2020 , using
Dirichlet boundary conditions and finding full agreement with expectations.
Since BPS Wilson loops in ${\cal N}=6$ super Chern-Simons-matter theory have a
rich moduli space, the study of supersymmetric boundary conditions for
$AdS_{2}$ fluctuations, interpolating between Dirichlet and Neumann, is then
crucial to understand the Wilson Loops holographic description OUYANG2016215 ;
OUYANG2016496  (see Drukker_2020  for a review of ABJM Wilson loops).

##  2 Interpolating Boundary Conditions

We will work in Euclidean $AdS_{2}$ space with unit radius. In Poincaré
coordinates $(z,\tau)$ the metric takes the form

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{z^{2}}\left(d\tau^{2}+dz^{2}\right)\,,$
$\displaystyle\qquad z$ $\displaystyle>0\,,$ $\displaystyle\quad-\infty$
$\displaystyle<\tau<\infty\,.$ (3)

To regulate possible divergences the boundary is located at $z=\epsilon$, with
induced metric $h=\epsilon^{-2}$ and outer normal vector
$\partial_{n}=-z\partial_{z}$.

###  2.1 Variational Principle and AdS/CFT

Let us start by reviewing the role that boundary terms and the variational
principle play in the calculation of correlation functions in AdS/CFT min1 ;
min2 ; KW ; SK02 . Consider a massless complex scalar field in $AdS_{2}$ with
action

$\displaystyle S_{0}$ $\displaystyle=\int
d^{2}x\sqrt{g}\,\partial_{\mu}\bar{\phi}\partial^{\mu}\phi\,.$ (4)

Setting $d=1$ and $m=0$ in (1) we find that the possible conformal dimensions
are $\Delta_{-}=0$ and $\Delta_{+}=1$. The asymptotic expansion then reads

$\displaystyle\phi(z,\tau)$
$\displaystyle=\alpha(\tau)+z\beta(\tau)+\mathcal{O}(z^{2})\,.$ (5)

Assuming regularity in the bulk ($z\to\infty$), the on-shell variation of the
action takes the form

$\displaystyle\delta S_{0}$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\sqrt{h}\left(\partial_{n}\bar{\phi}\delta\phi+\partial_{n}\phi\delta\bar{\phi}\right)\Big{|}_{z=\epsilon}$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\left(\bar{\beta}(\tau)\delta\alpha(\tau)+\beta(\tau)\delta\bar{\alpha}(\tau)\right)\,.$
(6)

For massless fields we can safely take the regulating parameter $\epsilon\to
0$ since no divergences arise. We learn from (6) that $S_{0}$ is appropriate
for a variational problem in which the function $\alpha(\tau)$ is fixed; only
then does (4) have an actual extremum when $\Box\phi=0$. This corresponds to
the usual Dirichlet boundary conditions in AdS. Moreover, according to the
AdS/CFT dictionary, the $1$-point function (or vev) of the dual operator
${\cal O}_{D}$ in the presence of the source $J_{D}\equiv\alpha$ is

$\displaystyle\langle\bar{\mathcal{O}}_{D}(\tau)\rangle$
$\displaystyle\equiv-\frac{\delta S_{D}[J]}{\delta J_{D}(\tau)}$
$\displaystyle=\bar{\beta}(\tau)\,,$ $\displaystyle\qquad S_{D}$
$\displaystyle\equiv S_{0}\,.$ (7)

Alternatively, in order to impose Neumann boundary conditions we need to add
the boundary term

$\displaystyle S^{N}_{\textrm{bdry}}$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\sqrt{h}\left(\phi\partial_{n}\bar{\phi}+\bar{\phi}\partial_{n}\phi\right)\Big{|}_{z=\epsilon}$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\left(\alpha(\tau)\bar{\beta}(\tau)+\bar{\alpha}(\tau)\beta(\tau)\right)\,.$
(8)

Evaluating the action on-shell one finds

$\displaystyle\delta\left(S_{0}+S^{N}_{\textrm{bdry}}\right)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\left(\bar{\alpha}(\tau)\delta\beta(\tau)+\alpha(\tau)\delta\bar{\beta}(\tau)\right)\,.$
(9)

Now the $1$-point function of the dual operator ${\cal O}_{N}$ sourced by
$J_{N}\equiv\beta$ is

$\displaystyle\langle\bar{\mathcal{O}}_{N}(\tau)\rangle$
$\displaystyle\equiv-\frac{\delta S_{N}[J]}{\delta J_{N}(\tau)}$
$\displaystyle=-\bar{\alpha}(\tau)\,,$ $\displaystyle\qquad S_{N}$
$\displaystyle\equiv S_{0}+S^{N}_{\textrm{bdry}}\,.$ (10)

The possibility of imposing different boundary conditions is due to the fact
that a massless scalar field in $AdS_{2}$ lies inside the BF window.

###  2.2 Local and Non-local Boundary Conditions

Generically, Dirichlet and Neumann boundary conditions are compatible with the
isometries of $AdS_{2}$, leading to correlation functions that exhibit 1d
conformal symmetry. Moreover, when appropriately combined with fermionic
fields, they also respect supersymmetry Sakai:1984vm . In this paper we
explore boundary conditions that interpolate between the Dirichlet and Neumann
cases while preserving scale invariance, and wonder whether they are
consistent with the full conformal group.

To achieve our goal we need a well-posed variational problem in which the
boundary source $J(\tau)$ combines the two fall-off functions $\alpha(\tau)$
and $\beta(\tau)$ in a suitable way. Since these fields have different mass
dimensions, directly adding them necessarily breaks scale invariance
Witten:2001ua ; Hartman_2008 . A specific $AdS_{2}$ interpolation of this kind
was considered in Polchinski:2011im  and interpreted as a Renormalization
Group flow in the dual CFT. As shown in Correa_2020 , however, the difference
in scale dimensions can be compensated by combining $\beta(\tau)$ with the
derivative of $\alpha(\tau)$ or, more conveniently (for reasons that will
become clear below), $\alpha(\tau)$ with an integral of $\beta(\tau)$.
Following these insights, we propose new boundary conditions defined by the
source111Cauchy principal value is implicit in all integrals in order to deal
with divergences at $\tau=\tau^{\prime}$.

$\displaystyle J(\tau)$
$\displaystyle\equiv\cos\chi\,\alpha(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\beta(\tau^{\prime})g(\tau-\tau^{\prime})\,,$
(11)

where $\chi\in\left[0,\frac{\pi}{2}\right]$ is an interpolating parameter and
$g(\tau)$ is a real dimensionless function we will promptly identify. The main
requisite is that it satisfy the closure relation

$\displaystyle\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{\tau}g(\tau-\tau^{\prime\prime})\partial_{\tau^{\prime}}g(\tau^{\prime}-\tau^{\prime\prime})=\delta(\tau-\tau^{\prime})\,,$
(12)

which allows us to implement the boundary condition in the variational problem
via222Recall that in Euclidean signature hermitian conjugation acts as
$\bar{f}(\tau)\equiv f(-\tau)^{*}$. It is easy to see that (13) is real.

$\displaystyle S_{\textrm{bdry}}$
$\displaystyle\equiv-\cos\chi\sin\chi\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau^{\prime}\left(\bar{\beta}(\tau)\beta(\tau^{\prime})-\partial_{\tau^{\prime}}\bar{\alpha}(\tau^{\prime})\partial_{\tau}\alpha(\tau)\right)g(\tau-\tau^{\prime})$
(13) $\displaystyle\hskip
11.0pt+\sin^{2}\chi\int_{-\infty}^{\infty}d\tau\left(\bar{\alpha}(\tau)\beta(\tau)+\alpha(\tau)\bar{\beta}(\tau)\right)\,.$
(14)

Neglecting a total $\tau$-derivative the on-shell variation of the action then
reads

$\displaystyle\delta\left(S_{0}+S_{\textrm{bdry}}\right)$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\left(\langle\bar{\mathcal{O}}(\tau)\rangle\delta
J(\tau)+\langle\mathcal{O}(\tau)\rangle\delta\bar{J}(\tau)\right)\,,$ (15)

where

$\displaystyle\langle\bar{\mathcal{O}}(\tau)\rangle$
$\displaystyle\equiv\cos\chi\,\bar{\beta}(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\partial_{\tau^{\prime}}\bar{\alpha}(\tau^{\prime})\partial_{\tau}g(\tau-\tau^{\prime})\,.$
(16)

We see that $S_{0}+S_{\textrm{bdry}}$ is the appropriate action for the
variational problem at hand, namely, that in which the combination (11) is
fixed. Moreover, according to the AdS/CFT dictionary, the $1$-point function
of the dual operator sourced by $J(\tau)$ is given by expression (16). As
$\chi$ varies, the source and vev interpolate between the Dirichlet case and a
$g$-transformed version of the Neumann case (see discussion below). We remark
that in AdS/CFT boundary actions are defined only up to arbitrary functionals
of the sources. In writing (13) this ambiguity was fixed by demanding that not
only $J(\tau)$ has the correct Dirichlet and Neumann limits, but also
$\langle\bar{\mathcal{O}}(\tau)\rangle$ does. As a consistency check, notice
that $S_{\textrm{bdry}}$ vanishes for $\chi=0$ and reduces to
$S^{N}_{\textrm{bdry}}$ for $\chi=\frac{\pi}{2}$.

Regarding the function $g(\tau)$, some comments are in order. We consider two
alternatives, dubbed _local_ (L) and _non-local_ (NL):

$\displaystyle\begin{split}g(\tau)&=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}\frac{1}{2}\textrm{sign}(\tau)\,,&\quad\textrm{L}\\\
\hfil\\\
\frac{1}{\pi}\ln\left(|\tau|\right)\,,&\quad\textrm{NL}\end{array}\right.\end{split}\qquad\Rightarrow\qquad\begin{split}\partial_{\tau}g(\tau)&=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}\delta(\tau)\,,&\quad\textrm{L}\\\
\hfil\\\
\frac{1}{\pi\tau}\,,&\quad\textrm{NL}\end{array}\right.\,.\end{split}$ (17)

This terminology stems from the fact that

$\displaystyle\partial_{\tau}J(\tau)$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}c_{\chi}\partial_{\tau}\alpha(\tau)+s_{\chi}\beta(\tau)\,,&\quad\textrm{L}\\\
\hfil\\\
c_{\chi}\partial_{\tau}\alpha(\tau)+s_{\chi}\hat{\beta}(\tau)\,,&\quad\textrm{NL}\end{array}\right.\,,$
$\displaystyle\qquad\langle\bar{\mathcal{O}}(\tau)\rangle$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}c_{\chi}\bar{\beta}(\tau)+s_{\chi}\partial_{\tau}\bar{\alpha}(\tau)\,,&\quad\textrm{L}\\\
\hfil\\\
c_{\chi}\bar{\beta}(\tau)+s_{\chi}\partial_{\tau}\hat{\bar{\alpha}}(\tau)\,,&\quad\textrm{NL}\end{array}\right.\,,$
(24)

where we have abbreviated $c_{\chi}=\cos\chi$ and $s_{\chi}=\sin\chi$, and
$\hat{\beta}(\tau)$ denotes the Hilbert transform of ${\beta}(\tau)$, defined
as (see appendix A)

$\displaystyle\hat{\beta}(\tau)$
$\displaystyle\equiv\frac{1}{\pi}\textrm{p.v.}\int_{-\infty}^{\infty}dt\frac{\beta(t)}{\tau-t}\,.$
(25)

So, even though the current (11) sources a non-local combination of
$\alpha(\tau)$ and $\beta(\tau)$, its derivative is local in these fields for
the L choice of boundary conditions. The same is true for the vev in (24).
Actually, modulo an ambiguous term of the kind discussed above, the L boundary
condition is equivalent to one of the cases considered in Correa_2020 . The NL
alternative is a new (non-local) boundary condition for massless fields on
$AdS_{2}$.

The Hilbert transform is a mathematical tool widely used in signal processing
and other areas of physics King_2009 , and its appearance in the present
setting is quite natural. Recall that the fall-off fields $\alpha(\tau)$ and
$\beta(\tau)$ become linked by regularity of $\phi(z,\tau)$ in the interior of
AdS ($z\to\infty$). Indeed, the most general regular solution to the equation
of motion $\Box\phi=0$ can be written as

$\displaystyle\phi(z,\tau)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dw\,e^{-|w|z+iw\tau}\tilde{\phi}(w)\,,$
(26)

with $\tilde{\phi}(w)$ an arbitrary function. Using the fact that the Hilbert
transform takes $e^{iw\tau}\mapsto-i\,\textrm{sign}(w)e^{iw\tau}$, it is
straightforward to see that

$\displaystyle\partial_{z}\hat{\phi}(z,\tau)$
$\displaystyle=\partial_{\tau}\phi(z,\tau)$
$\displaystyle\qquad\Rightarrow\qquad\hat{\beta}(\tau)$
$\displaystyle=\partial_{\tau}{\alpha}(\tau)\,.$ (27)

Since $\hat{\beta}(\tau)$ and $\partial_{\tau}{\alpha}(\tau)$ are locally
related for regular solutions, it is reasonable to combine them as in (24). In
fact, a similar relation holds for massless spinors on $AdS_{2}$, whose
asymptotic expansion reads

$\displaystyle\psi(z,\tau)$ $\displaystyle\underset{z\to
0}{\longrightarrow}z^{\frac{1}{2}}\alpha_{\psi}(\tau)+z^{\frac{1}{2}}\beta_{\psi}(\tau)\,,$
$\displaystyle\qquad P_{+}\alpha_{\psi}$ $\displaystyle=\alpha_{\psi}\,,$
$\displaystyle\quad P_{-}\beta_{\psi}$ $\displaystyle=\beta_{\psi}\,,$
$\displaystyle\quad P_{\pm}$
$\displaystyle=\frac{1}{2}\left(1\pm\gamma_{1}\right)\,.$ (28)

The regular solution to $\not{\nabla}\psi=0$ (in Euclidean signature) is

$\displaystyle\psi(z,\tau)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dw\,\sqrt{z}e^{-wz\gamma+iw\tau}\tilde{\psi}(w)\,,$
$\displaystyle\qquad\gamma\tilde{\psi}(w)$
$\displaystyle=\textrm{sign}(w)\tilde{\psi}(w)\,,$ $\displaystyle\quad\gamma$
$\displaystyle=-i\gamma_{0}\gamma_{1}\,,$ (29)

and one readily finds

$\displaystyle\hat{\psi}(z,\tau)$ $\displaystyle=-i\gamma\psi(z,\tau)$
$\displaystyle\qquad\Rightarrow\qquad\hat{\beta}_{\psi}(\tau)$
$\displaystyle=-i\gamma\alpha_{\psi}(\tau)\,.$ (30)

It is straightforward to write non-local boundary conditions for fermions
analogous to (11). We will not discuss them in this paper.

To close this section let us explain how the Dirichlet and Neumann results can
be recovered as limiting cases of our proposed interpolating boundary
condition. From (11) and (16) we immediately see that

$\displaystyle J_{D}(\tau)$ $\displaystyle=J(\tau)\Big{|}_{\chi=0}\,,$
$\displaystyle\qquad\langle\bar{\mathcal{O}}_{D}(\tau)\rangle$
$\displaystyle=\langle\bar{\mathcal{O}}(\tau)\rangle\Big{|}_{\chi=0}\,,$ (31)

so the $\chi=0$ limit will yield correlation functions corresponding to
Dirichlet boundary conditions, for both the L and NL choices. On the other end
of the interpolation, however, neither the source nor the vev directly reduce
to their Neumann counterparts. Instead, the relation involves a
derivative/Hilbert transform, namely,

$\displaystyle J_{N}(\tau)$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}r>{\displaystyle}l}\partial_{\tau}J(\tau)\Big{|}_{\chi=\frac{\pi}{2}}&\quad\textrm{L}\\\
\hfil\\\
-\partial_{\tau}\hat{J}(\tau)\Big{|}_{\chi=\frac{\pi}{2}}&\quad\textrm{NL}\end{array}\right.\,,$
$\displaystyle\qquad\partial_{\tau}\bar{\mathcal{O}}_{N}(\tau)$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}r>{\displaystyle}l}-\bar{\mathcal{O}}(\tau)\Big{|}_{\chi=\frac{\pi}{2}}&\quad\textrm{L}\\\
\hfil\\\
\hat{\bar{\mathcal{O}}}(\tau)\Big{|}_{\chi=\frac{\pi}{2}}&\quad\textrm{NL}\end{array}\right.\,.$
(38)

The connection between our proposal and the Neumann case is found by
formulating the variational problem in terms of the alternative source
$J^{\prime}(\tau)\equiv\partial_{\tau}J(\tau)$ and using this to compute
correlation functions. After all, we can always integrate by parts and write

$\displaystyle\delta S$
$\displaystyle\sim\int_{-\infty}^{\infty}d\tau\langle\bar{\mathcal{O}}(\tau)\rangle\delta
J(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\langle\bar{\mathcal{O}}^{\prime}(\tau)\rangle\delta
J^{\prime}(\tau)\,,$ (39)

with
$\langle\bar{\mathcal{O}}(\tau)\rangle=-\langle\partial_{\tau}\bar{\mathcal{O}}^{\prime}(\tau)\rangle$.
The problem is that this approach would lead to correlation functions of dual
operators $\bar{\mathcal{O}}^{\prime}(\tau)$ with
“$\Delta_{\mathcal{O}^{\prime}}=0$”, which are not well-defined primaries (cf.
(57)). Using $J(\tau)$ as the source, on the other hand, yields correlators
for the derivatives of such operators,
$\mathcal{O}(\tau)=-\partial_{\tau}\bar{\mathcal{O}}^{\prime}(\tau)$, which
are primaries with $\Delta_{\mathcal{O}}=1$. In general, the relation between
the two sets of correlation functions follows from noticing that,
formally,333In Euclidean signature
$\overline{\partial_{\tau}f(\tau)}=-\partial_{\tau}\bar{f}(\tau)$ and
$\bar{\hat{f}}(\tau)=-\hat{\bar{f}}(\tau)$, hence the factors of $(-1)^{n}$.

$\displaystyle\frac{\delta^{2n}S[J]}{\delta\bar{J}(\tau_{1})\cdots\delta
J(\tau_{2n})}$
$\displaystyle=(-1)^{n}\frac{\partial^{2n}}{\partial\tau_{1}\cdots\partial\tau_{2n}}\left(\frac{\delta^{2n}S^{\prime}[J^{\prime}]}{\delta\bar{J}^{\prime}(\tau_{1})\cdots\delta
J^{\prime}(\tau_{2n})}\right)\,,$ (40)

where $S^{\prime}[J^{\prime}]\equiv S[J]$, up to total boundary derivatives.
Therefore,

$\displaystyle\langle\mathcal{O}(\tau_{1})\cdots\bar{\mathcal{O}}(\tau_{2n})\rangle$
$\displaystyle=(-1)^{n}\langle\partial_{\tau_{1}}\mathcal{O}^{\prime}(\tau_{1})\cdots\partial_{\tau_{2n}}\bar{\mathcal{O}}^{\prime}(\tau_{2n})\rangle$
$\displaystyle=\langle\partial_{\tau_{1}}\mathcal{O}^{\prime}(\tau_{1})\cdots\overline{\partial_{\tau_{2n}}\mathcal{O}^{\prime}(\tau_{2n})}\rangle\,.$
(41)

The NL choice in (38) also suggests considering the operators obtained by
sourcing the Hilbert transform
$J^{\prime\prime}(\tau)\equiv-\partial_{\tau}\hat{J}(\tau)$, in which case one
has

$\displaystyle\delta S$
$\displaystyle\sim\int_{-\infty}^{\infty}d\tau\langle\bar{\mathcal{O}}(\tau)\rangle\delta
J(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\langle\bar{\mathcal{O}}^{\prime\prime}(\tau)\rangle\delta
J^{\prime\prime}(\tau)\,,$ (42)

with
$\langle\bar{\mathcal{O}}(\tau)\rangle=-\langle\partial_{\tau}\hat{\bar{\mathcal{O}}}^{\prime\prime}(\tau)\rangle$.
An analogous application of the chain rule as in (40) gives

$\displaystyle\langle\mathcal{O}(\tau_{1})\cdots\bar{\mathcal{O}}(\tau_{2n})\rangle$
$\displaystyle=\langle\partial_{\tau_{1}}\hat{\mathcal{O}}^{\prime\prime}(\tau_{1})\cdots\partial_{\tau_{2n}}\hat{\bar{\mathcal{O}}}^{\prime\prime}(\tau_{2n})\rangle$
$\displaystyle=\langle\partial_{\tau_{1}}\hat{\mathcal{O}}^{\prime\prime}(\tau_{1})\cdots\overline{\partial_{\tau_{2n}}\hat{\mathcal{O}}^{\prime\prime}(\tau_{2n})}\rangle\,.$
(43)

These relations are crucial when interpreting the $\chi=\frac{\pi}{2}$ limit
of the interpolating boundary conditions: it will yield correlation functions
that are related to those in the Neumann case by taking the Hilbert transform
and/or (NL/L) a derivative with respect to each of the insertion points. In
particular, we expect that

$\displaystyle\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\rangle\Big{|}_{\chi=\frac{\pi}{2}}$
$\displaystyle=\partial_{\tau_{1}}\partial_{\tau_{2}}\langle\mathcal{O}_{N}(\tau_{1})\bar{\mathcal{O}}_{N}(\tau_{2})\rangle\left\\{\begin{array}[]{>{\displaystyle}r>{\displaystyle}l}-1&\quad\textrm{L}\\\
\hfil\\\ 1&\quad\textrm{NL}\end{array}\right.\,,$ (47)

where we have used that successive Hilbert transforms of
$f(\tau_{1}-\tau_{2})$ over $\tau_{1}$ and $\tau_{2}$ give back
$f(\tau_{1}-\tau_{2})$. We will confirm this expectation below.

##  3 Correlation Functions at strong coupling

We now move on to study the tree-level correlation functions that result from
applying the AdS/CFT dictionary to the interpolating boundary conditions
introduced in the previous section. For the purpose of computing $2$-point
functions the free theory discussed so far is sufficient. To compute $4$-point
correlators we consider the interaction vertices arising from the fluctuations
of a type IIA open string around an $AdS_{2}\subset
AdS_{4}\times\mathds{CP}^{3}$ classical worldsheet. The expansion of the
Nambu-Goto action to fourth order in the effective string tension was
performed in Bianchi_2020 . We identify the scalar field discussed in previous
sections with one of the (complex) massless fluctuations in $\mathds{CP}^{3}$.
Interestingly, the quartic interactions involve derivatives of the field. This
forces us to revisit the variational problem and modify the definition of the
source and vev. Performing a first principles derivation of the $4$-point
function we verify that the standard prescription in terms of Witten diagrams
remains valid once the appropriate boundary terms are added to the action.

###  3.1 $2$-point function

The bulk-to-boundary propagator $K(z,\tau;\tau^{\prime})$ for a massless
scalar field is defined such that the regular solution to the equation of
motion is expressed as

$\displaystyle\phi(z,\tau)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau^{\prime}K(z,\tau;\tau^{\prime})J(\tau^{\prime})\,,$
$\displaystyle\qquad\Box K(z,\tau;\tau^{\prime})$ $\displaystyle=0\,,$ (48)

where $J(\tau)$ is the boundary data that is fixed in the variational problem.
In the case of Dirichlet and Neumann boundary conditions they read

$\displaystyle K_{D}(z,\tau;\tau^{\prime})$
$\displaystyle=\frac{1}{\pi}\frac{z}{z^{2}+(\tau-\tau^{\prime})^{2}}\,,$
$\displaystyle\qquad K_{N}(z,\tau;\tau^{\prime})$
$\displaystyle=\frac{1}{2\pi}\ln\left(z^{2}+(\tau-\tau^{\prime})^{2}\right)\,,$
(49)

and satisfy

$\displaystyle K_{D}(z,\tau;\tau^{\prime})$ $\displaystyle\underset{z\to
0}{\longrightarrow}\delta(\tau-\tau^{\prime})\,,$
$\displaystyle\qquad\partial_{z}K_{N}(z,\tau;\tau^{\prime})$
$\displaystyle\underset{z\to 0}{\longrightarrow}\delta(\tau-\tau^{\prime})\,,$
(50)

as appropriate for $J_{D}=\alpha$ and $J_{N}=\beta$, respectively. For the
case at hand, in order to comply with the boundary condition (11), the
propagator must behave as

$\displaystyle
c_{\chi}K(z,\tau;\tau^{\prime})+s_{\chi}\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{z}K(z,\tau^{\prime\prime};\tau^{\prime})g(\tau-\tau^{\prime\prime})$
$\displaystyle\underset{z\to 0}{\longrightarrow}\delta(\tau-\tau^{\prime})\,.$
(51)

We readily find

$\displaystyle K(z,\tau;\tau^{\prime})$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}c_{\chi}K_{D}(z,\tau;\tau^{\prime})+s_{\chi}\partial_{\tau}K_{N}(z,\tau;\tau^{\prime})&\quad\textrm{L}\\\
\hfil\\\
\frac{K_{D}(z,\tau;\tau^{\prime})}{c_{\chi}+s_{\chi}}&\quad\textrm{NL}\end{array}\right.\,.$
(55)

It is easy to check that

$\displaystyle
c_{\chi}K(z,\tau;\tau^{\prime})+s_{\chi}\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{z}K(z,\tau^{\prime\prime};\tau^{\prime})g(\tau-\tau^{\prime\prime})$
$\displaystyle=K_{D}(z,\tau;\tau^{\prime})\,,$ (56)

thus verifying (51) for both L and NL boundary conditions.

From (7) and (10) we find that the $2$-point functions for Dirichlet and
Neumann boundary conditions are given by

$\displaystyle\langle\mathcal{O}_{D}(\tau_{1})\bar{\mathcal{O}}_{D}(\tau_{2})\rangle$
$\displaystyle=\frac{1}{\pi}\frac{1}{(\tau_{1}-\tau_{2})^{2}}\,,\qquad\langle\mathcal{O}_{N}(\tau_{1})\bar{\mathcal{O}}_{N}(\tau_{2})\rangle$
$\displaystyle=-\frac{1}{\pi}\ln\left(|\tau_{2}-\tau_{1}|\right)\,.$ (57)

In the Neumann case, the dual ($\Delta=0$) operator does not transform as a
proper primary. As explained above, it is its derivative that is well-behaved,
that is,

$\displaystyle\langle\partial_{\tau_{1}}\mathcal{O}_{N}(\tau_{1})\overline{\partial_{\tau_{2}}\mathcal{O}_{N}(\tau_{2})}\rangle$
$\displaystyle=-\partial_{\tau_{1}}\partial_{\tau_{2}}\langle\mathcal{O}_{N}(\tau_{1})\bar{\mathcal{O}}_{N}(\tau_{2})\rangle$
$\displaystyle=\frac{1}{\pi}\frac{1}{(\tau_{1}-\tau_{2})^{2}}\,,$ (58)

where we have used that
$\overline{\partial_{\tau}\mathcal{O}_{N}(\tau)}=-\partial_{\tau}\bar{\mathcal{O}}_{N}(\tau)$.
Similarly, according to (16), the $2$-point functions for the interpolating
boundary conditions are

$\displaystyle\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\rangle$
$\displaystyle=\frac{1}{\pi}\frac{1}{(\tau_{1}-\tau_{2})^{2}}\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}1&\qquad\textrm{L}\\\
\hfil\\\
\frac{c_{\chi}-s_{\chi}}{c_{\chi}+s_{\chi}}&\qquad\textrm{NL}\end{array}\right.\,.$
(62)

As advertised, these expressions correctly reproduce the Dirichlet and Neumann
limits (c.f. (47)). We will comment on this result below.

###  3.2 $4$-point function

The computation of 4-point correlation functions depends on the precise form
of the quartic interactions. In the following we consider the Nambu-Goto
action expanded around an open $AdS_{2}$ worldsheet ending on a straight line
at the $AdS_{4}\times\mathbb{CP}^{3}$ boundary. The fluctuations along the
$\mathbb{CP}^{3}$ directions are massless, and we focus on a single complex
combination of these. To fourth order in $\delta
X\sim\lambda^{-\frac{1}{4}}\phi$, the Nambu-Goto action in static gauge
becomes Bianchi_2020

$\displaystyle S_{\lambda}$ $\displaystyle=\int
d^{2}x\sqrt{g}\left[|\partial\phi|^{2}-\sqrt{\frac{2}{\lambda}}\left(|\partial\phi|^{2}|\phi|^{2}+\frac{1}{2}|\partial\phi|^{4}\right)\right]\,,$
(63)

where $g_{\mu\nu}$ is the $AdS_{2}$ worldsheet metric (3), $\lambda=2T^{2}$ is
the ABJM ’t Hooft coupling and $T$ is the effective string tension. The
corresponding Euler-Lagrange equation reads

$\displaystyle\Box\phi$
$\displaystyle=-\sqrt{\frac{2}{\lambda}}\left(|\partial\phi|^{2}\phi-\nabla_{\mu}\left(\partial^{\mu}\phi|\phi|^{2}\right)-\nabla_{\mu}\left(\partial^{\mu}{\phi}|\partial\phi|^{2}\right)\right)\,.$
(64)

A crucial aspect of the action (63) is that the quartic interactions involve
derivatives of the field. This forces us to revisit the variational problem.
Indeed, we now have

$\displaystyle\delta S_{\lambda}$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\left(\bar{\eta}(\tau)\delta\alpha(\tau)+\eta(\tau)\delta\bar{\alpha}(\tau)\right)\,,$
(65)

with

$\displaystyle\eta(\tau)$
$\displaystyle\equiv\beta(\tau)\left(1-\sqrt{\frac{2}{\lambda}}|\alpha(\tau)|^{2}\right)\,.$
(66)

As with $S_{0}$, this action is appropriate for the Dirichlet problem in which
$\alpha(\tau)$ is fixed. The natural analogue of Neumann boundary conditions
corresponds to fixing $\eta(\tau)$ via

$\displaystyle S^{N}_{\textrm{bdry}}$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\left(\alpha(\tau)\bar{\eta}(\tau)+\bar{\alpha}(\tau)\eta(\tau)\right)\,.$
(67)

In order to impose our proposal of interpolating boundary conditions we add to
$S_{\lambda}$ the term

$\displaystyle S_{\textrm{bdry}}$
$\displaystyle=-\cos\chi\sin\chi\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau^{\prime}\left(\bar{\eta}(\tau)\eta(\tau^{\prime})-\partial_{\tau^{\prime}}\bar{\alpha}(\tau^{\prime})\partial_{\tau}\alpha(\tau)\right)g(\tau-\tau^{\prime})$
(68) $\displaystyle\hskip
11.0pt+\sin^{2}\chi\int_{-\infty}^{\infty}d\tau\left(\bar{\alpha}(\tau)\eta(\tau)+\alpha(\tau)\bar{\eta}(\tau)\right)$
(69)
$\displaystyle=-\tan\chi\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau^{\prime}\left(\bar{\eta}(\tau)\eta(\tau^{\prime})-\partial_{\tau^{\prime}}\bar{J}(\tau^{\prime})\partial_{\tau}J(\tau)\right)g(\tau-\tau^{\prime})\,,$
(70)

with $g(\tau)$ as before and

$\displaystyle J(\tau)$
$\displaystyle\equiv\cos\chi\,\alpha(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\eta(\tau^{\prime})g(\tau-\tau^{\prime})\,.$
(71)

The $1$-point function then reads

$\displaystyle\langle\bar{\mathcal{O}}(\tau)\rangle$
$\displaystyle\equiv\cos\chi\,\bar{\eta}(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\partial_{\tau^{\prime}}\bar{\alpha}(\tau^{\prime})\partial_{\tau}g(\tau-\tau^{\prime})$
(72)
$\displaystyle=\frac{1}{\cos\chi}\left(\bar{\eta}(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\partial_{\tau^{\prime}}\bar{J}(\tau^{\prime})\partial_{\tau}g(\tau-\tau^{\prime})\right)\,.$
(73)

We see that the derivative nature of the quartic potential not only alters the
expression for the vev but also requires us to modify the definition of the
source (c.f. (11), (16), (13)). Fortunately, the modification is quite simple,
we just need to replace $\beta(\tau)\to\eta(\tau)$ in every expression. It is
important to mention that the boundary condition (71) is now non-linear in the
field and depends on the coupling $\lambda^{-\frac{1}{2}}$. As we will see
below, this dependence can be treated perturbatively.

####  3.2.1 Bulk-to-bulk propagators

The computation of tree-level correlation functions involves solving the non-
linear equation of motion (64), which in turn requires knowing both the bulk-
to-boundary and bulk-to-bulk propagators. In the massless case, the latter is
defined by

$\displaystyle\Box_{x}G(x;x^{\prime})$
$\displaystyle=\frac{1}{\sqrt{g}}\delta(x;x^{\prime})\,.$ (74)

For Dirichlet and Neumann boundary conditions they coincide with those in flat
space,

$\displaystyle G_{D/N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=\frac{1}{4\pi}\left(\ln\left((z-z^{\prime})^{2}+(\tau-\tau^{\prime})^{2}\right)\mp\ln\left((z+z^{\prime})^{2}+(\tau-\tau^{\prime})^{2}\right)\right)\,,$
(75)

and satisfy

$\displaystyle G_{D}(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=0}$
$\displaystyle=0\,,$
$\displaystyle\qquad\partial_{z}G_{N}(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=0}$
$\displaystyle=0\,.$ (76)

Moreover, they are related to the corresponding AdS2 bulk-to-boundary
propagators (49) by

$\displaystyle K_{D}(z^{\prime},\tau^{\prime};\tau)$
$\displaystyle=-\partial_{z}G_{D}(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=0}\,,$
$\displaystyle\qquad K_{N}(z^{\prime},\tau^{\prime};\tau)$
$\displaystyle=G_{N}(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=0}\,.$ (77)

This follows directly from Green’s third identity

$\displaystyle\phi(z^{\prime},\tau^{\prime})$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\left(\phi(z,\tau)\partial_{z}G(z,\tau;z^{\prime},\tau^{\prime})-\partial_{z}\phi(z,\tau)G(z,\tau;z^{\prime},\tau^{\prime})\right)\Big{|}_{z=0}\,,$
(78)

valid for any harmonic function $\phi$ on the upper half-plane.

In order to identify the correct boundary conditions for $G(x;x^{\prime})$ in
the interpolating case we re-write Green’s identity as

$\displaystyle\phi(z^{\prime},\tau^{\prime})$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau\left[\left(\phi(z,\tau)+\tan\chi\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{z}\phi(z,\tau^{\prime\prime})g(\tau-\tau^{\prime\prime})\right)\partial_{z}G(z,\tau;z^{\prime},\tau^{\prime})\right.$
(79) $\displaystyle\hskip
11.0pt\left.-\partial_{z}\phi(z,\tau)\left(G(z,\tau;z^{\prime},\tau^{\prime})+\tan\chi\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{z}G(z,\tau^{\prime\prime};z^{\prime},\tau^{\prime})g(-\tau+\tau^{\prime\prime})\right)\right]\Bigg{|}_{z=0}\,,$
(80)

where we have swapped the order of integration in the last term. Recalling
that

$\displaystyle\Box\phi$ $\displaystyle=0$
$\displaystyle\qquad\Rightarrow\qquad\int_{-\infty}^{\infty}d\tau\,\partial_{z}\phi(z,\tau)\Big{|}_{z=0}$
$\displaystyle=0\,,$ (81)

we see that a sufficient condition for the second line in (79) to vanish is

$\displaystyle\partial_{\tau}\left(\cos\chi\,G(z,\tau;z^{\prime},\tau^{\prime})+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime\prime}\partial_{z}G(z,\tau^{\prime\prime};z^{\prime},\tau^{\prime})g(-\tau+\tau^{\prime\prime})\right)\Bigg{|}_{z=0}$
$\displaystyle=0\,.$ (82)

The field can then be reconstructed from the boundary data using the bulk-to-
boundary propagator

$\displaystyle K(z^{\prime},\tau^{\prime};\tau)$
$\displaystyle=-\frac{1}{\cos\chi}\partial_{z}G(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=0}\,.$
(83)

Bulk-to-bulk Green functions with boundary conditions (82) can be found with
the help of the relations

$\displaystyle\partial_{\tau^{\prime}}G_{D/N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=\partial_{z^{\prime}}\hat{G}_{N/D}(z,\tau;z^{\prime},\tau^{\prime})\,,\hskip
15.0pt\partial_{z^{\prime}}G_{D/N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=-\partial_{\tau^{\prime}}\hat{G}_{N/D}(z,\tau;z^{\prime},\tau^{\prime})\,,$
(84)

valid for $z>z^{\prime}$, where the Hilbert transforms of the Dirichlet and
Neumann propagators read

$\displaystyle\hat{G}_{D/N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=-\frac{1}{2\pi}\left(\tan^{-1}\left(\frac{\tau-\tau^{\prime}}{|z-z^{\prime}|}\right)\mp\tan^{-1}\left(\frac{\tau-\tau^{\prime}}{z+z^{\prime}}\right)\right)\,.$
(85)

Notice that

$\displaystyle\hat{G}_{D}(z,\tau;z^{\prime},\tau^{\prime})-\hat{G}_{N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=\frac{1}{\pi}\tan^{-1}\left(\frac{\tau-\tau^{\prime}}{z+z^{\prime}}\right)$
(86)

is a harmonic function but

$\displaystyle\hat{G}_{D}(z,\tau;z^{\prime},\tau^{\prime})+\hat{G}_{N}(z,\tau;z^{\prime},\tau^{\prime})$
$\displaystyle=-\frac{1}{\pi}\tan^{-1}\left(\frac{\tau-\tau^{\prime}}{|z-z^{\prime}|}\right)$
(87)

is not. We find

$\displaystyle G(x;x^{\prime})$
$\displaystyle=\left\\{\begin{array}[]{>{\displaystyle}c>{\displaystyle}l}\cos^{2}\chi\,G_{D}(x;x^{\prime})+\sin^{2}\chi\,G_{N}(x;x^{\prime})-\cos\chi\sin\chi\big{(}\hat{G}_{D}(x;x^{\prime})-\hat{G}_{N}(x;x^{\prime})\big{)}&\quad\textrm{L}\\\
\hfil\\\
\frac{\cos\chi\,G_{D}(x;x^{\prime})+\sin\chi\,G_{N}(x;x^{\prime})}{\cos\chi+\sin\chi}&\quad\textrm{NL}\end{array}\right.\,.$
(91)

Satisfyingly, we find from (83) that (91) correctly reproduces (55).

####  3.2.2 First Principles Derivation and Witten Diagrams

The equation of motion (64) can be solved perturbatively in the coupling
$\lambda^{-\frac{1}{2}}$ by expanding the field as:

$\displaystyle\phi$
$\displaystyle=\phi_{0}+\sqrt{\frac{2}{\lambda}}\phi_{1}+\mathcal{O}(\lambda^{-1})$
$\displaystyle\quad\Rightarrow\quad\alpha$
$\displaystyle=\alpha_{0}+\sqrt{\frac{2}{\lambda}}\alpha_{1}+\mathcal{O}(\lambda^{-1})\,,$
$\displaystyle\quad\beta$
$\displaystyle=\beta_{0}+\sqrt{\frac{2}{\lambda}}\beta_{1}+\mathcal{O}(\lambda^{-1})\,.$
(92)

We find that

$\displaystyle\Box\phi_{0}$ $\displaystyle=0\,,$
$\displaystyle\qquad\Box\phi_{1}$ $\displaystyle=J_{\textrm{bulk}}\,,$ (93)

where

$\displaystyle J_{\textrm{bulk}}$
$\displaystyle=-\phi_{0}|\partial\phi_{0}|^{2}+\nabla_{\mu}\left(\partial^{\mu}\phi_{0}|\phi_{0}|^{2}\right)+\nabla_{\mu}\left(\partial^{\mu}\phi_{0}|\partial\phi_{0}|^{2}\right)$
$\displaystyle=\bar{\phi}_{0}(\partial\phi_{0})^{2}+\nabla_{\mu}\left(\partial^{\mu}\phi_{0}|\partial\phi_{0}|^{2}\right)\,.$
(94)

In addition, substituting (92) into (71) we get

$\displaystyle\cos\chi\,\alpha_{0}(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\beta_{0}(\tau^{\prime})g(\tau-\tau^{\prime})$
$\displaystyle=J(\tau)\,,$ (95)
$\displaystyle\cos\chi\,\alpha_{1}(\tau)+\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\beta_{1}(\tau^{\prime})g(\tau-\tau^{\prime})$
$\displaystyle=F(\tau)\,,$ (96)

where

$\displaystyle F(\tau)$
$\displaystyle=\sin\chi\int_{-\infty}^{\infty}d\tau^{\prime}\beta_{0}(\tau^{\prime})\bar{\alpha}_{0}(\tau^{\prime})\alpha_{0}(\tau^{\prime})g(\tau-\tau^{\prime})\,.$
(97)

As usual, the source $J(\tau)$ enters linearly in the boundary condition of
$\phi_{0}$. The novel feature, brought in by the derivative nature of the
vertex, is the non-homogeneous boundary condition for the fluctuation
$\phi_{1}$. The classical solution satisfying (95) and (96) is then given by

$\displaystyle\phi_{0}(z,\tau)$
$\displaystyle=\int_{-\infty}^{\infty}d\tau^{\prime}K(z,\tau;\tau^{\prime})J(\tau^{\prime})\,,$
(98) $\displaystyle\phi_{1}(z,\tau)$
$\displaystyle=\int\frac{dz^{\prime}d\tau^{\prime}}{z^{\prime
2}}\bar{G}(z,\tau;z^{\prime},\tau^{\prime})J_{\textrm{bulk}}(z^{\prime},\tau^{\prime})+\int_{-\infty}^{\infty}d\tau^{\prime}K(z,\tau;\tau^{\prime})F(\tau^{\prime})\,,$
(99)

where $K(z,\tau;\tau^{\prime})$ and $G(z,\tau;z^{\prime},\tau^{\prime})$ are
the bulk-to-boundary and bulk-to-bulk propagators444The conjugation in
$G(z,\tau;z^{\prime},\tau^{\prime})$ is due to the interchange
$\tau\leftrightarrow\tau^{\prime}$. A similar property holds for
$K(z,\tau;\tau^{\prime})$.  introduced in sections 3.1 and 3.2.1. Notice that
$\phi_{0}$ and $\phi_{1}$ have the correct asymptotics as a consequence of
(51) and (82).

According to the AdS/CFT dictionary the $4$-point function in the dual field
theory is given by

$\displaystyle\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle$
$\displaystyle=-\frac{\delta^{4}S[J]}{\delta\bar{J}(\tau_{1})\delta
J(\tau_{2})\delta\bar{J}(\tau_{3})\delta J(\tau_{4})}$
$\displaystyle=\frac{1}{\cos\chi}\frac{\delta^{3}\bar{\eta}(\tau_{4})}{\delta\bar{J}(\tau_{1})\delta
J(\tau_{2})\delta\bar{J}(\tau_{3})}\,,$ (100)

where we have used (72). As done with fields above we expand

$\displaystyle\eta$
$\displaystyle=\eta_{0}+\sqrt{\frac{2}{\lambda}}\eta_{1}+\mathcal{O}(\lambda^{-1})\,,$
$\displaystyle\qquad\eta_{0}$ $\displaystyle=\beta_{0}\,,$
$\displaystyle\qquad\eta_{1}$
$\displaystyle=\beta_{1}-\beta_{0}\bar{\alpha}_{0}\alpha_{0}\,.$ (101)

The term $\eta_{0}$ does not contribute to the $4$-point function since it is
linear in $J$. To compute $\eta_{1}$ we need

$\displaystyle\beta_{1}(\tau)$
$\displaystyle=\int\frac{dz^{\prime}d\tau^{\prime}}{z^{\prime
2}}\partial_{z}\bar{G}(z,\tau;z^{\prime},\tau^{\prime})\Big{|}_{z=\epsilon}J_{\textrm{bulk}}(z^{\prime},\tau^{\prime})+\int_{-\infty}^{\infty}d\tau^{\prime}\partial_{z}K(z,\tau;\tau^{\prime})\Big{|}_{z=\epsilon}F(\tau^{\prime})\,.$
(102)

The first term requires no further manipulation once we recall the relation
(83) between the bulk-to-bulk and bulk-to-boundary propagators. The second
term can be simplified by replacing the definition of $F(\tau)$, swapping the
order of integration, and using the boundary condition for
$K(z,\tau;\tau^{\prime})$. In this process, the $\delta$-function in (51) will
cancel against the term $\beta_{0}\bar{\alpha}_{0}\alpha_{0}$ in $\eta_{1}$.
Putting these ingredients together yields

$\displaystyle\frac{\eta_{1}(\tau)}{\cos\chi}$
$\displaystyle=-\int\frac{dz^{\prime}d\tau^{\prime}}{z^{\prime
2}}J_{\textrm{bulk}}(z^{\prime},\tau^{\prime})\bar{K}(z^{\prime},\tau^{\prime};\tau)-\int_{-\infty}^{\infty}d\tau^{\prime}\beta_{0}(\tau^{\prime})\bar{\alpha}_{0}(\tau^{\prime})\alpha_{0}(\tau^{\prime})K(z,\tau;\tau^{\prime})\Big{|}_{z=\epsilon}\,.$
(103)

Finally, using the expression (98) for $\phi_{0}$ we arrive at

$\displaystyle\frac{\bar{\eta}_{1}(\tau_{4})}{\cos\chi}$
$\displaystyle=-\int_{-\infty}^{\infty}d\tau_{1}d\tau_{2}d\tau_{3}\,\bar{J}(\tau_{1})J(\tau_{2})\bar{J}(\tau_{3})T(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\,,$
(104)

where

$\displaystyle T(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=U(\tau_{1},\tau_{2},\tau_{3},\tau_{4})+V(\tau_{1},\tau_{2},\tau_{3},\tau_{4})+W(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\,,$
(105)

and

$\displaystyle U(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=\int\frac{d\tau
dz}{z^{2}}\partial_{\mu}\bar{K}(z,\tau;\tau_{1})K(z,\tau;\tau_{2})\partial^{\mu}\bar{K}(z,\tau;\tau_{3})K(z,\tau;\tau_{4})\,,$
(106) $\displaystyle V(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=\int\frac{d\tau
dz}{z^{2}}\nabla_{\mu}\left(\partial_{\nu}\bar{K}(z,\tau;\tau_{1})\partial^{\nu}K(z,\tau;\tau_{2})\partial^{\mu}\bar{K}(z,\tau;\tau_{3})\right)K(z,\tau;\tau_{4})\,,$
(107) $\displaystyle W(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=\int_{-\infty}^{\infty}d\tau\,\partial_{z}\bar{K}(z,\tau;\tau_{1})K(z,\tau;\tau_{2})\bar{K}(z,\tau;\tau_{3})K(z,\tau;\tau_{4})\Big{|}_{z=\epsilon}\,.$
(108)

The integrals $U$ and $V$ clearly originate from the bulk current
$J_{\textrm{bulk}}$, while $W$ arises from the boundary term in (103). With
this the $4$-point function becomes

$\displaystyle\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle$
$\displaystyle=-\sqrt{\frac{2}{\lambda}}\Big{[}T(\tau_{1},\tau_{2},\tau_{3},\tau_{4})+(\tau_{1}\leftrightarrow\tau_{3})\Big{]}\,.$
(109)

Before writing an explicit expression for
$\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle$,
let us show the equivalence between the above derivation and the standard
calculation using Witten diagrams. This can be done by a simple manipulation
of the integrals (106)-(108). Indeed, one verifies by integration by parts
that

$\displaystyle U(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=-U^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})-U^{\prime}(\tau_{1},\tau_{4},\tau_{3},\tau_{2})-W(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\,,$
(110) $\displaystyle V(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=-V^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\,,$ (111)

where

$\displaystyle U^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=\int\frac{d\tau
dz}{z^{2}}\partial_{\mu}\bar{K}(z,\tau;\tau_{1})\partial^{\mu}K(z,\tau;\tau_{2})\bar{K}(z,\tau;\tau_{3})K(z,\tau;\tau_{4})\,,$
(112) $\displaystyle V^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=\int\frac{d\tau
dz}{z^{2}}\partial_{\mu}\bar{K}(z,\tau;\tau_{1})\partial^{\mu}K(z,\tau;\tau_{2})\partial_{\nu}\bar{K}(z,\tau;\tau_{3})\partial^{\nu}K(z,\tau;\tau_{4})\,.$
(113)

Then, using the manifest symmetry
$V^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})=V^{\prime}(\tau_{3},\tau_{4},\tau_{1},\tau_{2})$,
the $4$-point correlator becomes

$\displaystyle\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle$
$\displaystyle=\sqrt{\frac{2}{\lambda}}\Big{[}T^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})+(\tau_{1}\leftrightarrow\tau_{3})+(\tau_{2}\leftrightarrow\tau_{4})+(\tau_{1},\tau_{2}\leftrightarrow\tau_{3},\tau_{4})\Big{]}\,,$
(114)

with

$\displaystyle T^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=U^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})+\frac{1}{2}V^{\prime}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\,.$
(115)

This is precisely the expression corresponding to the Witten diagram in figure
1.

$\sqrt{\frac{2}{\lambda}}\left(|\partial\phi|^{2}|\phi|^{2}+\frac{1}{2}|\partial\phi|^{4}\right)$$\mathcal{O}(\tau_{1})$$\bar{\mathcal{O}}(\tau_{2})$$\mathcal{O}(\tau_{3})$$\bar{\mathcal{O}}(\tau_{4})$
Figure 1: Tree-level diagram coming from the expansion of the Nambu-Goto
action.

Finally, the integrals (106)-(108) or (112)-(113) can be computed using the
method of residues. In terms of the invariant cross-ratio

$\displaystyle u$
$\displaystyle=\frac{\tau_{12}\tau_{34}}{\tau_{13}\tau_{24}}\,,$
$\displaystyle\qquad\tau_{ij}$ $\displaystyle=\tau_{i}-\tau_{j}\,,$ (116)

we find

$\displaystyle\sqrt{\frac{\lambda}{2}}\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle_{\textrm{L}}$
$\displaystyle=\frac{A(u)+\cos(2\chi)B(u)+\cos(4\chi)C(u)+\sin(4\chi)S(\tau_{i})}{2\pi^{3}\tau_{13}^{2}\tau_{24}^{2}}$
(117) $\displaystyle\hskip
11.0pt+\frac{\sin^{2}\chi\cos(2\chi)}{\pi^{3}}\left(\frac{1}{\tau_{12}\tau_{14}}-\frac{1}{\tau_{23}\tau_{34}}\right)\left(\frac{1}{\tau_{12}\tau_{23}}-\frac{1}{\tau_{14}\tau_{34}}\right)\,,$
(118)

and

$\displaystyle\sqrt{\frac{\lambda}{2}}\langle\mathcal{O}(\tau_{1})\bar{\mathcal{O}}(\tau_{2})\mathcal{O}(\tau_{3})\bar{\mathcal{O}}(\tau_{4})\rangle_{\textrm{NL}}$
$\displaystyle=\frac{1}{\left(\cos\chi+\sin\chi\right)^{4}}\frac{A(u)+B(u)+C(u)}{2\pi^{3}\tau_{13}^{2}\tau_{24}^{2}}\,,$
(119)

where

$\displaystyle A(u)$
$\displaystyle=-\frac{1}{u^{2}(1-u)^{2}}+\frac{u-3}{4(1-u)^{3}}\ln(u^{2})-\frac{2+u}{4u^{3}}\ln((1-u)^{2})\,,$
(120) $\displaystyle B(u)$ $\displaystyle=\frac{2}{u(1-u)}\,,\qquad
C(u)=-2-\frac{1}{u(1-u)}+\frac{1}{2}\left(2u-1\right)\ln\left(\frac{u^{2}}{(1-u)^{2}}\right)\,,$
(121) $\displaystyle S(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$
$\displaystyle=(1-2u)\Big{[}\textrm{sign}(\tau_{12})+\textrm{sign}(\tau_{23})+\textrm{sign}(\tau_{34})+\textrm{sign}(\tau_{41})\Big{]}\frac{\pi}{2}\,.$
(122)

It is possible to write $A(u)$, $B(u)$ and $C(u)$ in terms of conformal
invariant $D$-functions Beccaria:2019dws . Also, one can check that the
function $S(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$ is invariant under the full
conformal group, including the inversion $\tau\to-1/\tau$. Unlike $u$, it is
not invariant under $\tau\to-\tau$ and $\tau\to 1/\tau$ separately. Finally,
the terms appearing in the second line of (118) cannot be written in terms of
the cross ratio $u$ and spoil the conformal covariance of the 4-point function
in the L case.

##  4 Discussion

In this paper we studied interpolating non-local boundary conditions for
interacting massless scalar fields in $AdS_{2}$ and used them to compute
4-point correlation functions holographically. We considered quartic
interaction terms derived from the expansion of the Nambu-Goto action around
an open string worldsheet in $AdS_{4}\times\mathbb{CP}^{3}$. Since the quartic
terms involve derivatives, they induce additional boundary terms in the
action. For the two types of boundary conditions we introduced, referred to as
L and NL, we computed the tree-level Witten diagrams in $AdS_{2}$ from which
correlation functions of the dual operators on a line were obtained.

In view of the results (62) and (119), we conclude that massless scalars
satisfying NL boundary conditions are dual representations of $\Delta=1$
primaries in the defect CFT1. The values $\chi=\pm\frac{\pi}{4}$ of the
interpolating parameter are special: for $\chi=-\frac{\pi}{4}$ the combination
we want to source in (24) is identically zero, whereas the vev of the dual
operator vanishes (in the presence of sources) for $\chi=\frac{\pi}{4}$.
Notice that these two statements are valid at leading order in
$\lambda^{-1/2}$ as a consequence of (27). Thus, for $\chi=\frac{\pi}{4}$ the
2-point correlator vanishes and becomes negative beyond this point, which
seems problematic as it would violate unitarity. We claim, however, that in
the range $\frac{\pi}{4}<\chi\leq\frac{\pi}{2}$ the lhs of (11) should be
interpreted as the Hilbert transform of the source and not as the source
itself. The physical result is then obtained by transforming back with respect
to $\tau_{1}$ and $\tau_{2}$, a procedure that yields a positive 2-point
function. This interpretation becomes evident at the endpoint $\chi=\pi/2$, as
argued at the end of section 2.2.

The L boundary condition is essentially one of the cases discussed in
Correa_2020 . The expression for the 2-point correlation function (62) is
conformally covariant and does not contain the contact term observed in
Correa_2020 . This slight difference can be explained by rewriting the
boundary term (13) as (cf. (68))

$\displaystyle S_{\textrm{bdry}}$
$\displaystyle=-\tan\chi\int_{-\infty}^{\infty}d\tau\int_{-\infty}^{\infty}d\tau^{\prime}\left(\bar{\beta}(\tau)\beta(\tau^{\prime})-\partial_{\tau^{\prime}}\bar{J}(\tau^{\prime})\partial_{\tau}J(\tau)\right)g(\tau-\tau^{\prime})\,.$
(123)

The term quadratic in the sources is responsible for the cancellation of the
aforementioned contact term and has the effect of giving a vev that
appropriately interpolates between $\beta$ (D) and $\alpha$ (N). On the other
hand, the holographic 4-point correlation function (118) cannot be interpreted
as that of primary operators in a CFT1. This failure of conformal covariance
was also found in the pure Neumann case studied in Beccaria:2019dws  in the
context of ordinary Wilson Loops. In that reference the anomaly disappeared
after integrating over the position in $S^{5}$ around which the Nambu-Goto
action was expanded. No such integration appears to be justified in our case.

The failure of conformal covariance is surprising, as the interpolating
parameter $\chi$ is dimensionless and the correlation functions are consistent
with scale invariance. However, scale invariance does not imply full conformal
symmetry. Given the non-local nature of the boundary conditions we are dealing
with, assessing conformal invariance on the defect appears to be a subtle
question. In principle, conformal covariance of correlation functions is
preserved if sources transform as primary fields. In our case, from

$\alpha^{\prime}(\tau^{\prime})=\alpha(\tau)\,,\qquad\beta^{\prime}(\tau^{\prime})=\frac{\partial\tau}{\partial\tau^{\prime}}\beta(\tau)\,,\qquad\text{under}\
\ \tau^{\prime}=\frac{a\tau+b}{c\tau+d}\,,\qquad\text{with}\ \ ad-bc=1\,,$
(124)

we find the transformation of the source to be

$J^{\prime}(\tau^{\prime})=J(\tau)-J(-d/c)+\cos\chi\alpha(-d/c)\,,$ (125)

where we have used

$\displaystyle\int_{-\infty}^{\infty}d\tau\,\beta(\tau)$ $\displaystyle=0\,.$
(126)

It is worth noticing that, although (125) is not exactly the transformation
rule of a primary field of weight $0$, the derivative $\partial_{\tau}J(\tau)$
does transform as a primary with $\Delta=1$, namely,

$\displaystyle\partial_{\tau^{\prime}}J^{\prime}(\tau^{\prime})$
$\displaystyle=\frac{\partial\tau}{\partial\tau^{\prime}}\partial_{\tau}J(\tau)\,.$
(127)

This disquisition on transformation properties of the sources still does no
explain why that 4-point correlation function (119) is consistent with
conformal covariance while (118) is not. Another difference between L and NL
boundary conditions, perhaps relevant to understand this issue, is that the
former also breaks the invariance under parity transformation $\tau\to-\tau$.

It might be worth mentioning as well that the anomalous terms in (118),
although inappropriate for a 4-point correlation function of primary operators
of scale dimension $\Delta=1$, could appear in 4-point functions among
operators of scale dimensions 3/2 and 1/2. The appearance of fractional scale
dimension for bosonic excitations could point towards derivatives of
fractional order, as the ones defined in Khodaee:2017tbk  to describe fields
in CFT1 that transform under non-local representations of $sl(2)$. Therefore,
it could be interesting to explore the possibility of relating our L boundary
condition to excitations in those more generic representations introduced in
Khodaee:2017tbk .

Let us recall that, as suggested in Correa_2020 , the L boundary condition
could represent the dual description of certain supersymmetric family of
interpolating Wilson Loops in ABJM theory Ouyang:2015bmy ; Ouyang:2015iza .
Thus, it would be interesting to study perturbatively 4-point correlations
functions of insertions along those Wilson loops and appraise their conformal
covariance in the field theory side. In the article Correa_2020  it is also
shown that the L boundary condition describes configurations preserving 4 real
supersymmetries. The consistency of supersymmetry transformations with the NL
boundary condition is another interesting problem to consider in the future.

Finally, one could wonder whether the interpolating boundary conditions
presented here can be generalized for scalar fields in $AdS_{d+1}$. To
guarantee a dimensionless interpolating parameter, one can consider a massive
field with $m^{2}R^{2}=-\frac{d^{2}}{4}+\frac{1}{4}$, such that
$\Delta_{+}-\Delta_{-}=1$. Other masses, such that $\Delta_{+}-\Delta_{-}$
equals a larger integer, lie outside the BF window. For the L choice, the
$\tau$-derivative, could be simply generalized to a directional derivative at
the $d$-dimensional boundary. This, of course, would break the invariance
under rotations as well as the $d$-dimensional conformal invariance. In order
to generalize the NL boundary condition, we could use the following non-local
first order differential operator555This definition follows from the Fourier
transform of the Laplace operator.

$\sqrt{\nabla^{2}}f(x)\equiv-\frac{\Gamma(\tfrac{d+1}{2})}{\pi^{\tfrac{d+1}{2}}}\int
d^{d}x^{\prime}\frac{f(x^{\prime})}{|x-x^{\prime}|^{d+1}}\,.$ (128)

Since $\partial_{\tau}\hat{f}(\tau)=\sqrt{\partial_{\tau}^{2}}f(\tau)$ for
$d=1$, a natural generalization of the NL boundary condition could be

$\partial_{\tau}\hat{J}(\tau)=c_{\chi}\partial_{\tau}\hat{\alpha}(\tau)-s_{\chi}\beta(\tau)\qquad\mapsto\qquad\sqrt{\nabla^{2}}J(x)=c_{\chi}\sqrt{\nabla^{2}}\alpha(x)-s_{\chi}\beta(x)\,.$
(129)

It would be interesting to further analyze these higher-dimensional
generalizations and the correlation functions that can be obtained from them.

## Acknowledgments

We would like to thank Fernando Alday, Max Bañados, Víctor Giraldo Rivera and
Martín Lagares for useful discussions on this problem. This work was supported
by PICT 2020-03749, PICT 2020-03826, PIP 02229, UNLP X791, UNLP X910 and
PUE084 “Búsqueda de nueva física”. DHC and AF would like to acknowledge
support from the ICTP through the Associates Programme (2020-2025 and
2022-2027) and GAS would like to acknowledge support from the ICTP visiting
programme. The work of AFCG and AF is supported by CONICYT FONDECYT Regular
#1201145 and ANID/ACT210100 Anillo Grant “Holography and its applications to
High Energy Physics, Quantum Gravity and Condensed Matter Systems.”

##  Appendix A Hilbert Transform

For any function $f(\tau)$ the Hilbert transform $\hat{f}(\tau)$ is defined as
King_2009

$\displaystyle\hat{f}(\tau)$
$\displaystyle\equiv\frac{1}{\pi}\textrm{p.v.}\int_{-\infty}^{\infty}dt\frac{f(t)}{\tau-t}$
$\displaystyle=\frac{1}{\pi}\lim_{\epsilon\to
0}\int_{-\infty}^{\infty}dt\,f(t)\frac{(\tau-t)}{\epsilon^{2}+(\tau-t)^{2}}\,,$
(130)

where p.v. stands for Cauchy principal value. This can also be written as

$\displaystyle\hat{f}(\tau)$
$\displaystyle=\frac{d}{d\tau}\left(\frac{1}{\pi}\textrm{p.v.}\int_{-\infty}^{\infty}dt\,f(t)\ln\left(|\tau-t|\right)\right)$
$\displaystyle=\frac{d}{d\tau}\left(\frac{1}{2\pi}\lim_{\epsilon\to
0}\int_{-\infty}^{\infty}dt\,f(t)\ln\left(\epsilon^{2}+(\tau-t)^{2}\right)\right)\,.$
(131)

These definitions hold provided that the integrals exists. More generally, one
can consider $f(\tau)$ to be a distribution. In Fourier space the Hilbert
transform is a multiplicative operator, namely,

$\displaystyle\mathcal{F}(\hat{f}\hskip 1.0pt)(w)$
$\displaystyle=-i\,\textrm{sign}(w)\mathcal{F}(f\hskip 0.5pt)(w)\,.$ (132)

A few transform pairs are listed in table 1. Some useful properties are shown
in table 2.

$\begin{array}[]{*2{|>{\displaystyle}c}|}\hline\cr\vphantom{\Big{|}}\textrm{Function}&\textrm{Hilbert
transform}\\\ \hline\cr\hline\cr\vphantom{\Bigg{|}}\textrm{constant}&0\\\
\hline\cr\vphantom{\Bigg{|}}\delta(\tau)&\frac{1}{\pi\tau}\\\
\hline\cr\vphantom{\Bigg{|}}\partial_{\tau}\delta(\tau)&-\frac{1}{\pi\tau^{2}}\\\
\hline\cr\vphantom{\Bigg{|}}\ln(|\tau|)&-\frac{\pi}{2}\textrm{sign}(\tau)\\\
\hline\cr\vphantom{\Bigg{|}}e^{iw\tau}&-i\,\textrm{sign}(w)e^{iw\tau}\\\
\hline\cr\vphantom{\Bigg{|}}\frac{1}{2}\ln\left(z^{2}+\tau^{2}\right)&-\tan^{-1}\left(\frac{\tau}{|z|}\right)\\\
\hline\cr\vphantom{\Bigg{|}}\frac{z}{z^{2}+\tau^{2}}&\textrm{sign}(z)\frac{\tau}{z^{2}+\tau^{2}}\\\
\hline\cr\vphantom{\Bigg{|}}-\frac{2z\tau}{(z^{2}+\tau^{2})^{2}}&\textrm{sign}(z)\frac{z^{2}-\tau^{2}}{(z^{2}+\tau^{2})^{2}}\\\
\hline\cr\end{array}$ Table 1: Hilbert transform pairs.

$\begin{array}[]{*3{|>{\displaystyle}c}|}\hline\cr\vrule\lx@intercol\hfil\textrm{Property}\hfil\lx@intercol\vrule\lx@intercol&\vphantom{\Big{|}}\textrm{Comments}\\\
\hline\cr\hline\cr\vrule\lx@intercol\hfil\displaystyle\int_{-\infty}^{\infty}d\tau\,\hat{f}(\tau)g(\tau)=-\int_{-\infty}^{\infty}d\tau\,f(\tau)\hat{g}(\tau)\hfil\lx@intercol\vrule\lx@intercol&\vphantom{\Bigg{|}}\textrm{anti-
self adjoint}\\\
\hline\cr\hline\cr\vphantom{\Big{|}}\textrm{Function}&\textrm{Hilbert
transform}&\hfil\\\
\hline\cr\hline\cr\vphantom{\Bigg{|}}\hat{f}(\tau)&-f(\tau)&\textrm{inverse}\\\
\hline\cr\vphantom{\Bigg{|}}\partial_{\tau}f(\tau)&\partial_{\tau}\hat{f}(\tau)&\textrm{derivative}\\\
\hline\cr\vphantom{\Bigg{|}}(f*g)(\tau)&(\hat{f}*g)(\tau)=(f*\hat{g})(\tau)&\textrm{convolution}\\\
\hline\cr\vphantom{\Bigg{|}}\bar{f}(\tau)&-\bar{\hat{f}}(\tau)&\bar{f}(\tau)\equiv
f(-\tau)^{*}\\\
\hline\cr\vphantom{\Bigg{|}}f(\tau-a)&\hat{f}(\tau-a)&\textrm{translations}\\\
\hline\cr\vphantom{\Bigg{|}}f(a\tau)&\textrm{sign}(a)\hat{f}(a\tau)&\textrm{rescalings/parity}\\\
\hline\cr\vphantom{\Bigg{|}}f(-1/\tau)&\hat{f}(-1/\tau)-\hat{f}(0)&\textrm{inversion}\\\
\hline\cr\vphantom{\Bigg{|}}\tau
f(\tau)&\tau\hat{f}(\tau)-\frac{1}{\pi}\int_{-\infty}^{\infty}d\tau^{\prime}f(\tau^{\prime})&\hfil\\\
\hline\cr\vphantom{\Bigg{|}}\tau^{2}f(\tau)&\tau^{2}\hat{f}(\tau)-\frac{1}{\pi}\int_{-\infty}^{\infty}d\tau^{\prime}\left(\tau+\tau^{\prime}\right)f(\tau^{\prime})&\hfil\\\
\hline\cr\vphantom{\Bigg{|}}\frac{f(\tau)}{\tau}&\frac{\hat{f}(\tau)-\hat{f}(0)}{\tau}&\hfil\\\
\hline\cr\vphantom{\Bigg{|}}f\left(\frac{a\tau+b}{c\tau+d}\right)&\hat{f}\left(\frac{a\tau+b}{c\tau+d}\right)-\hat{f}\left(\frac{a}{c}\right)&ad-
bc=1\\\
\hline\cr\vphantom{\Bigg{|}}\frac{1}{c\tau+d}f\left(\frac{a\tau+b}{c\tau+d}\right)&\frac{1}{c\tau+d}\hat{f}\left(\frac{a\tau+b}{c\tau+d}\right)&ad-
bc=1\\\
\hline\cr\vphantom{\Bigg{|}}\frac{1}{(c\tau+d)^{2}}f\left(\frac{a\tau+b}{c\tau+d}\right)&\frac{1}{(c\tau+d)^{2}}\hat{f}\left(\frac{a\tau+b}{c\tau+d}\right)+\frac{c}{c\tau+d}\frac{1}{\pi}\int_{-\infty}^{\infty}d\tau^{\prime}f(\tau^{\prime})&ad-
bc=1\\\ \hline\cr\end{array}$ Table 2: Hilbert transform properties.

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