††institutetext: School of Mathematical Sciences, Highfield, University of
Southampton, SO17 1BJ Southampton, UK

# Real time holography for higher spin theories

Zezhuang Hao<EMAIL_ADDRESS>
###### Abstract

Real time holography is studied in the context of the embedding space
formalism. In the first part of this paper, we present matching conditions for
on-shell integer spin fields when going from Euclidean to Lorentzian signature
on AdS background. Using the BTZ black hole as an example, we discuss various
ways of lifting the physical solution from the AdS surface to the whole
embedding space. The BTZ propagator for higher spin field is expressed
elegantly in terms of the embedding coordinates. In the second part of the
paper, we develop the proposed duality between higher spin theory and vector
models. We obtain a specific map between the field configurations of these two
theories in real time, so called Lorentzian AdS/CFT map. We conclude by
exploring the matching conditions for higher spin fields satisfying the
proposed bulk quadratic action. The physical and ghost modes can be treated
independently during the Wick rotation; only physical modes are considered to
be external modes.

##  1 Introduction

AdS/CFT correspondence Maldacena:1997re ; gubser1998gauge ; witten1998anti ;
Aharony:1999ti  has been brought up over two decades and most of the checks
are carried out in the large ’t Hooft limit. At such limit the bulk theory
tends to stay in the low energy region thus they are described by the well
studied model, for example, semiclassical field theory or supergravity. When
the ’t Hooft constant goes to the small limit the bulk theory will enter the
high energy region thus the higher spin excitations and gravitational
corrections should not be ignored while it was proposed that the spectra of
string theory could be approximately described by the higher spin field theory
and the boundary theory will be free, leading to the proposal that Vasiliev’s
higher spin Vasiliev:1990en ; Vasiliev:1999ba ; Vasiliev:2003ev  theory in the
AdS bulk is dual to the free $O(N)$ vector model on the boundary Sezgin:2002rt
; 2002 ; Sezgin:2003pt . The Higher spin/ Vector model duality has been
verified by studying the correlation function specifically giombi2010higher ;
Giombi:2012ms  and recently a derivation of the duality is also given by
constructing a map between the boundary fields in the bi-local form and higher
spin fields on the bulk, so called AdS/CFT map deMelloKoch:2018ivk ;
Aharony:2020omh ; Aharony:2021ovo ; Aharony:2022feg . In their work, the
AdS/CFT map is constructed between the Euclidean AdS and CFT while here in
this article we will first obtain the Lorentzian version of AdS/CFT map based
on the study of real time holography then further study its various
implications on the understanding of holography principle.

In the first part, we developed the approach to real time holography
Skenderis:2008dg ; Skenderis:2008dh  from scalar case to the higher spin case
by checking the match of higher spin bulk boundary propagators at the joint
surface between the Lorentzian and Euclidean AdS space then obtain the match
conditions for the higher spin fields. In fact, the idea of gluing the
Euclidean gravitational action to the Lorentzian part has been discussed dated
back to the study of wavefunction of the universe hartle1983wave  in which the
Euclidean part is used to prepare the initial state of the universe so called
Hartle-Hawking state. After that, the complex version of the metric has also
been studied for long Gibbons:1976ue ; louko1997complex ; kontsevich2021wick ;
witten2021note  while till now the physical meaning of Euclidean saddle and
its connection to the Lorentzian part are still not fully understood. However
the relation between Euclidean and Lorentzian signature in quantum field
theory has been well studied and it turns out that there is a one-to-one
correspondence between the Euclidean and Lorentzian QFT governed by the
analytic continuation osterwalder1973axioms ; osterwalder1975axioms . Now
based on the holography principle, one should also expect the existence of
similar match conditions that will govern the quantum gravity theory when
going between different signatures. Such conditions at low energy could be
understood as the Wick rotation of the QFT on the curved background.

Starting from the simplest condition, by making both of the Euclidean and
Lorentzian action dynamical and assuming that they are smoothly connected at
the joint surface, one could obtain the matching conditions for the scalar
fields on the AdS background. It turns out that bulk-boundary propagator
should also be joint smoothly at the surface where we perform the match. While
for the higher spin theory, we choose to follow the opposite approach since
the higher spin action is absent and it is easier to study the bulk-boundary
propagator for higher spin particles in the embedding space Costa:2011mg ;
Costa:2014kfa , which means we will obtain the match conditions by studying
the bulk-boundary propagator. In section 2, we will first study the real time
holography in embedding space then further present the matching condition for
the higher spin fields. Moreover, if the fields are expanded in terms of the
conformal basis, then we will see that the match between Euclidean and
Lorentzian fields can be written as the match of the coefficients.

More precisely, the matching condition are directly imposed on the AdS
spacetime since in our case the physical space is AdS even though it is easier
to the study the higher spin fields especially the higher spin bulk boundary
propagator in embedding space treating the AdS as the hyperboloid embedded in
a one dimension higher flat spacetime. By considering the field configuration,
we can see that the physical fields on the AdS hyperboloid is obtained by the
pushing back of the fields in embedding space thus there is a redundancy for
the fields at the direction which is norm to the hyperboloid. In terms of
coefficients we can see that there also exists extra degrees of freedom to
choose the coefficients in embedding space which will give us the same higher
spin fields, so called the pure gauge of coefficients. By checking the match
of higher spin propagator explicitly, it turns that the pure gauge should also
be fixed during the match therefore such redundancy will be eliminated and the
fields will be uniquely fixed by the matching the physical fields in the
chosen gauge. Moreover, we find that for the matching condition for the higher
spin field, one needs to distinguish the spatial and time direction because
the extra factor $i$ should be introduced along the time direction.

Before going into the study of Lorentzian AdS/CFT map, we analysis the
difference between AdS space and embedding space. In this article, we insist
that the AdS spacetime is physical and the onshell fields are solutions of
corresponding equation of motion on the AdS background. For scalar fields, the
equation of motion is KG equation. For the higher spin fields, they are
governed by the nonlinear Vasilive’s equation while here all we need to known
is that the equation of motion could be reduced to the second order at linear
level Bouatta:2004kk ; Costa:2014kfa  similar to the KG equation. Therefore,
due to the absence of dynamics for the embedding space, there are various ways
for us to extend the onshell fields from the AdS hyperboloid to the embedding
space. Here we will use the BTZ solution as an example to illustrate two
extension methods. One is the direct extension of the field according to the
AdS radius R while the other also requires the modification of the scale
dimension $\Delta\rightarrow\Delta_{R}$. Both methods will give us smooth
functions in embedding space while the second approach will automatically
introduce the dynamical structure to the embedding space, which could possibly
make it physical.

In the second part, we establish the general principle for Lorentzian
continuation for field theories then generalise the Euclidean AdS/CFT map to
the Lorentzian version and then apply such map to the vector models. In the
work Aharony:2020omh ; Aharony:2021ovo , the higher spin fields in the bulk
are expanded by the bulk-boundary propagator as the basis while the bi-local
field on the boundary are expanded by the three point correlation functions
chosen as the conformal basis. The completeness of the conformal basis are
provided by summing over the principal series and the map between field
configuration is then manifested by establishing the map between the
coefficients. After analysing the real time holography for higher spin fields
on the bulk side, in order to obtain the Lorentzian AdS/CFT map, one also
needs to figure out the Lorentzian conformal basis and the corresponding
completeness relations so that one can decompose the boundary fields into
coefficients. The Lorentzian version of the completeness relation is obtained
following the principle developed in section 3. Moreover, by comparing the
degrees of freedom from both sides, one can conclude that the AdS/CFT map is
invertible at large N limit.

At the end, we study the match conditions for the quadratic action which is
treated as the effective action for higher spin fields. By mapping the action
for the vector model on the boundary to the AdS bulk, it is proposed that the
higher spin field theory is governed by the quadratic action provided the
existence of proper gauge fixing procedure and the corresponding ghost modes
contribute the unphysical modes coming from the higher order term. In section
4, we will just consider the scalar components of the quadratic action and
present the match conditions for the quadratic action and it turns out that
physical modes and ghost modes behave independently during the Wick rotation
of the bulk theory.

##  2 Real-Time Holography in Embedding Space

In this section, we will discuss real-time holographic fields within the
embedding space formalism. Beginning with a brief summary of the approach to
real-time holography developed in Skenderis:2008dg ; Skenderis:2008dh , which
illustrates the discussions with the case of a free scalar field, we then lift
the results to the embedding space and present the matching condition for
higher spin fields.

###  2.1 Real-Time Holography: Review

In holography, when studying the field configuration or constructing the
dictionary between two theories, people often choose to first specify the
behaviour of the field at the boundary of AdS and then use them to write down
the bulk-boundary propagator or the source of the operator which belongs to
the boundary CFT witten1998anti ; Freedman:1998tz ; Costa:2014kfa , for
further development one can see the application of Fefferman-Graham expansion
AST_1985__S131__95_0 ; graham1999volume ; skenderis2002lecture .

This is enough for us to deal with Euclidean AdS/CFT since in Euclidean
signature, spatial and time directions are indistinguishable and the data on
the boundary will in principle encode the whole information of the field. As
for the Lorentzian signature, specifying the behaviour of the field at the
spatial boundary will no longer enable us to uniquely determine the bulk field
due to the lack of information about the field at the far past or the far
future, i.e., the boundary of the time direction. Moreover it is interesting
to note that, if there is a black hole in the bulk, one can show that it is
possible to determine such information concerning the initial and final states
by specifying the behavior of the modes at the horizon son2002minkowski ;
herzog2003schwinger  .

In fact, the problem of specifying the initial data for a gravitational theory
has been discussed by Hartle and Hawking hartle1983wave  when studying the
quantum gravity wavefunction of the universe. In order to specify the initial
condition of the Lorentzian evolution they choose to glue part of Euclidean
path integral to the initial codimension one surface of the Lorentzian
spacetime. If we denote the action of the Euclidean and Lorentzian spacetime
as $S_{E}$ and $S_{L}$, the weight of the quantum gravity path integral can be
represented as

$e^{-S_{E}}\;e^{iS_{L}},$ (1)

in which the term $e^{-S_{E}}$ can be treated as a norm factor resulting from
the preparation of initial state. At quantum level, especially for the quantum
field on the curved space time, this enables us to pick out a preferred vacuum
and Hilbert space wald1994quantum ; Witten:2021jzq .

At the same time, on the field theory side, one can also construct various
contours in complex time plane to calculate the corresponding vacuum-vacuum,
thermal or out of time order correlation function landsman1987real ;
Maldacena:2013xja . The field theory along these contours are continuous, for
example, one can see the study of analyticity of the Wightman functions
osterwalder1973axioms ; osterwalder1975axioms . This implies that on the bulk
side the Euclidean action and Lorentzian action should also match smoothly.
Rather than regarding the Euclidean action as a norm factor, one should
instead also treat the Euclidean action as the dynamical part, i.e., consider
the total action $-S_{E}+iS_{L}$ , filling the contour with the bulk geometry,
and impose the matching condition at the joint surface.

Before presenting the matching conditions, here we first clarify the concept
that what we mean by matching two theories together. Consider two physical
systems that live in two regions labelled by I and II, the dynamics of two
systems are governed by the action $S_{I}$, $S_{II}$, respectively. Moreover,
we denote the intersection of these two regions as $\Sigma$. It could be a
purely mathematical surface or a physically measurable junction, like the
domain wall and we propose the condition that two theories joint smoothly at
the surface $\Sigma$ to be

$\delta S_{I}|_{\Sigma}=\delta S_{II}|_{\Sigma},$ (2)

in which $\delta S|_{\Sigma}$ represents the boundary term of the variation.
It is obvious if $S_{II}=S_{I}$ then $\Sigma$ will not exist physically. In
our case, the region $\rm I$ and $\rm II$ now becomes spacetime with different
signatures, the dynamics are governed by the action $S_{L}$, $S_{E}$.

$S_{I}$$S_{II}$$\Sigma$III Figure 1: Two systems distribute in region $I$ and
$II$, joint at the surface $\Sigma$.

As for the full quantum gravity theory, the matching condition following from
(2) is still unknown but, if we just consider the scalar field, and in the
context of saddle point approximation, we have

$\phi_{E}=\phi_{L},\qquad i\partial_{t}\phi_{L}+\partial_{\tau}\phi_{E}=0,$
(3)

in which subscripts $\rm E,L$ are used to represent that the field lie along
the imaginary and real part of the contour and we will stick with such
convention throughout the article.

Thus this motivates us, taking the vacuum-vacuum contour for example, to write
down the duality relation Skenderis:2008dg ; Skenderis:2008dh

$\langle 0|T\;{\rm exp}\left(-i\int
d^{d}x\sqrt{-g}\phi_{0}\mathcal{O}\right)|0\rangle=\int_{\phi\sim\phi_{0}}DgD\phi\;{\rm
exp}\left(-S_{E}+iS_{L}\right),$ (4)

in which on the left hand side $g$ represents the bulk metric while on the
right hand it represents the induced conformal structure on the boundary. As
for the scalar field lives in the bulk with boundary value $\phi_{0}$, on the
boundary CFT theory, it can be regarded as the source of the operator
$\mathcal{O}$ while such correspondence are called real time gauge/gravity
duality. In practice, we evaluate the right hand side at the AdS saddle point
and the formula has been used to calculate the Wightman functions and produced
right results. Finite temperature correlation functions are also studied
provided the matching condition for thermal contour is applied on the bulk
side. Moreover, it is interesting to note that, by identifying pair of sources
along the thermal contour vanRees:2009rw , one can recover the ingoing waves
and retarded correlations functions when there is a black hole in the bulk.

As for the higher spin field, the matching condition will be quite
complicated. Classically, the equation of motion for higher spin fields will
be non-linear Vasiliev:1990en ; Vasiliev:1999ba ; Vasiliev:2003ev  while one
needs to impose more physical restrictions when dealing with the matching in
the context of quantum theory. One can see the discussion of massless spin two
field in louko1997complex ; kontsevich2021wick ; witten2021note . Here, we
will only discuss the matching condition at the classic and linear level,
i.e., the equation of motion are linear equations. In this case, we can write
down the matching condition for higher spin fields by the investigation of the
higher spin bulk-boundary propagator in embedding space.

###  2.2 Holographic Scalar Fields in Embedding Space Formalism

We start from the study of scalar fields in embedding space and one can find a
brief introduction of embedding space together with solutions of the KG
equation in Appendix A. For the embedding scalar bulk-boundary propagators,
i.e., the Green function for the particle moving on the $AdS$ background with
given asymptotic limit, they are deduced to be

$\displaystyle G^{E}_{\Delta}(X_{1},X_{2})=\frac{1}{(-2X_{1}\cdot
X_{2}+i\epsilon)^{\Delta}_{E}}+Y^{E}_{\Delta}(X_{1},X_{2}),$ (5)
$\displaystyle G^{L}_{\Delta}(X_{1},X_{2})=\frac{1}{(-2X_{1}\cdot
X_{2}+i\epsilon)^{\Delta}_{L}}+Y^{L}_{\Delta}(X_{1},X_{2}),$ (6)

in which we use $E$ and $L$ to represent different values in the two
signatures while $X_{1}$, $X_{2}$ are two points in embedding space. In our
expression, propagators are separated into two terms, the first is the
regularised sources with proper $i\epsilon$ prescription and the second is the
contribution from normalizable modes denoted as $Y_{\Delta}$. By solving the
Klein-Gordon equation on the surface $X^{2}=-1$, we obtain 111In fact, it is
subtle to lift a function from the AdS spacetime to the embedding space and
there are various ways to do this. In the section 2.4, we will use BTZ
solution as an example to discuss this in detail.

$Y_{\Delta}^{L}(X)=\frac{1}{(2\pi)^{d}}\int
dK\;e^{iK^{\mu}X_{\mu}/X^{+}}\;\theta(-K^{2})\;B_{L}(K)\;(X^{+})^{-d/2}\;J_{\Delta}(|K|X^{+})$
(7)

in which $K=(w,k^{i})$ for $1\leq i\leq d$ is the momentum space coordinate
and $J_{\Delta}$ is the Bessel function written in terms of the scale
dimension $\Delta$. $J_{\Delta}$ can be regarded as orthogonal basis of the
normalizable modes and the coefficients $B(K)$ are determined by the boundary
condition of the propagator. We can check that these two terms behave like
$z^{\Delta}$ and $z^{d-\Delta}$ respectively when $z\rightarrow 0$ and they
are two independent solutions of the asymptotic Klein-Gordon equation.

To obtain the Euclidean version of the normalizable modes $Y^{E}_{\Delta}$, we
first do the Wick rotation on $Y^{L}_{\Delta}$, i.e., taking
$X^{0}\rightarrow-iX^{0}$, which will result in blow up modes when
$X^{0}\rightarrow\pm\infty$. To get rid of this, we use the absolute value of
$|X^{0}K_{0}|$ in the exponential term, which leads to

$Y^{E}_{\Delta}(X)=\frac{1}{(2\pi)^{d}}\int
dK\;e^{(-|X^{0}K_{0}|+iX^{i}K_{i})/X^{+}}\;\theta(-K^{2})\;B_{E}(K)\;(X^{+})^{-d/2}\;J_{\Delta}(|K|X^{+}).$
(8)

Given the bulk-boundary propagator, we treat them as a set of complete basis
so that in general we can expand an arbitrary scalar field $\Phi_{\Delta}(X)$
as

$\displaystyle\Phi_{\Delta}(X)=\int dP\;C_{\Delta}(P)\;G_{\Delta}(X,P),$ (9)

in which the integral means that we are suming over the null rays and
$C_{\Delta}(P)$ can be treated as the coefficients of the basis while $\Delta$
is the scale dimension of the dual boundary fields, related to the mass of
$\Phi_{\Delta}(X)$. Noting that there are two sets of basis thus we have two
possible ways of expansion

$\displaystyle\Phi_{\Delta}^{E}(X)=\int
dP\;C^{E}_{\Delta}(P)\;G^{E}_{\Delta}(X,P),$ (10)
$\displaystyle\Phi_{\Delta}^{L}(X)=\int
dP\;C^{L}_{\Delta}(P)\;G^{L}_{\Delta}(X,P),$ (11)

o$t$$\tau$$\tau$$T$$-T$(a)

$t$$t=T$(b)

Figure 2: (a) A toy Wick rotation contour, vertical lines represent the
imaginary time $\tau$ while horizontal lines represent the real time $t$. The
red point at $(T,0)$ is the matching surface $\Sigma$. The dashed curve means
that we can deform the contour apart from the axes in principle. (b) The
global $AdS_{2}$ is illustrated by the strip while our embedding coordinate
covers the region between two blue lines, which is a hyperboloid. Moreover,
the Poincar$\acute{{\rm e}}$ coordinates covers half of the hyperboloid showed
in the triangle. The red line represents the matching surface $\Sigma$ at
$t=T$.

where one is for imaginary time while the other is for real time. We should
keep in mind that these two kinds of expansions only work in their own proper
region. Taking the contour in Fig.2 for example, which starts from
$(-T,+\infty)$ and ends at $(T,-\infty)$ with two corners at $(\pm T,0)$, we
can interpret that two vertical lines are used to prepare the initial and
final quantum state and the horizontal line represents the evolution of time
from $-T$ to $T$.

To study the behavior of the fields along this contour, we should apply the
Euclidean expansion $\Phi_{E}$ on $\tau$ axes and the Lorentzian expansion
$\Phi_{L}$ on $t$ axes. For the contour going through the plane, like the
dashed line around $-T$, these two kinds of basis will mix and it will go
beyond our discussion. Since the contour with corners are not smooth, to make
the fields $\Phi^{E}_{\Delta}$ and $\Phi^{L}_{\Delta}$ consistent along the
whole contour, matching conditions should be imposed at the singular surface.
Consider the surface at $t=T$ and we denote it as $\Sigma$ , illustrated in
Fig.2, the matching conditions Skenderis:2008dh ; Skenderis:2008dg  are

$\displaystyle\Phi^{E}_{\Delta}(X)\arrowvert_{\Sigma}=\Phi^{L}_{\Delta}(X)\arrowvert_{\Sigma},$
(12)
$\displaystyle\partial_{\tau}\Phi^{E}_{\Delta}(X)\arrowvert_{\Sigma}+i\partial_{t}\Phi^{L}_{\Delta}(X)\arrowvert_{\Sigma}=0,$
(13)

and these two conditions enable us to solve $C^{L}_{\Delta}$, $C^{E}_{\Delta}$
and $B_{L}$, $B_{E}$. Since the basis of the source and the normalizable modes
are independent, we will deal with $B$ and $C$ separately. First for the
source terms, with the help of (A.159) and (A.158), we choose to push forward
the derivative of the time to the embedding space

$\displaystyle\partial_{\tau}\frac{1}{(-2X\cdot
P)^{\Delta}_{E}}=\frac{2\Delta}{(-2X\cdot
P)^{\Delta+1}_{E}}(-X^{0}P^{+}+X^{+}P^{0}),$ (14)
$\displaystyle\partial_{t}\frac{1}{(-2X\cdot
P)^{\Delta}_{L}}=\frac{2\Delta}{(-2X\cdot
P)^{\Delta+1}_{L}}(X^{0}P^{+}-X^{+}P^{0}).$ (15)

Moreover, in order to use the equation (12) and (13) to solve the
coefficients, we need to figure out the form of a function when it is
restricted on the surface $\Sigma$ from the Euclidean and Lorentzian point of
view. For Lorentzian signature, since the contour lies exactly along the real
$t$ axes, the restricted function on $\Sigma$ means that we take $t=T$. For
the Euclidean signature, since the contour shifts away from the pure imaginary
axes $\tau$ and in order to reflect such shift in the theory, we need to make
the variable in the function shift as $X^{0}\rightarrow X^{0}+iT$ and then set
$\tau=0$. Moreover, after taking the coordinate rotation
$P^{E}_{0}=iP^{L}_{0}$ into consideration and substituting (14), (15) into the
matching condition equations, we find the solution should be

$C^{E}_{\Delta}=C^{L}_{\Delta}.$ (16)

As for the normalizable term, by directly comparing the integrands, from (12)
we have

$B_{L}(K)+B^{-}_{L}(K)=B_{E}(K)+B_{E}^{-}(K),$ (17)

while from (13) we obtain222In fact here we take $T=0$ otherwise there will be
a phase factor $e^{\pm iwT}$ in front of the coefficients. Since $B$ should
only depend on the boundary conditions rather than the choice of contour, it
will not change the result.

$B_{L}(K)-B^{-}_{L}(K)=B_{E}(K)+B_{E}^{-}(K),$ (18)

in which we define $B^{-}(w,k^{i})=B(-w,k^{i})$ and these two equations will
lead to

$B_{E}=B_{L}=0,$ (19)

which tells us normalizable modes do not contribute to the bulk-boundary
propagator.

###  2.3 Matching Conditions for Higher Spin Fields

Now, we are going to discuss the matching condition of higher spin fields and
lift them to the embedding space as briefly introduced in Appendix A and
extensively studied in weinberg2010six ; Costa:2011mg ; Costa:2014kfa ;
Parisini:2022wkb . The $\rm AdS_{d+1}$ spacetime is regarded as a hyperboloid
$X^{2}=-1$ in the embedding space, as we have introduced before. Here we use
the embedding coordinates in the light cone gauge
$X^{A}=(X^{+},X^{-},X^{\mu})$ and the $\rm AdS$ Poincar$\acute{{\rm e}}$
coordinates $y^{a}=(y^{\mu},z)$. Moreover, the Jacobian matrix between these
two coordinates is given by

$\frac{\partial X}{\partial
y^{\mu}}=\frac{1}{z}(\;0,\;2y_{\mu},\;\delta^{\nu}_{\mu}\;)\qquad{\rm
and}\qquad\frac{\partial X}{\partial z}=(-\frac{1}{z^{2}},\;1,\;0,\dots,0),$
(20)

which tell us the rule to push back a vector $H_{A}$ in embedding space to a
vector $H_{a}$ in $AdS$ space as a submanifold thus we have

$H_{a}(X)=\frac{\partial X^{A}}{\partial y^{a}}H_{A}(X).$ (21)

Noting that the dimension of $\rm AdS$ is lower than the embedding space,
there is a redundancy when we transform form $H_{A}$ to $H_{a}$. To construct
a one-to-one correspondence between symmetric tensors in these two spaces, we
should impose the transverse condition $X^{A}H_{A}(X)=0$ to eliminate the
extra degree of freedom while we can understand it as restricting the vector
$H_{A}$ tangent to the $\rm AdS$ submanifold.

Before writing down the matching conditions in embedding space, based on the
study of scalar field matching, we first propose the two conditions in the
$\rm AdS$ space

$\displaystyle H_{a}^{E}(X)|_{\Sigma}=H_{a}^{L}(X)|_{\Sigma},$ (22)
$\displaystyle\partial_{\tau}H^{E}_{a}(X)|_{\Sigma}+i\partial_{t}H^{L}_{a}(X)|_{\Sigma}=0.$
(23)

These two equations specify conditions at the joint surface $\Sigma$ up to the
first order. Since perturbatively physical equations with integer spin, like
the curved background Maxwell equation, are second order differential
equations333For the fields with half integer spin, the equation of motion will
become first order while one needs to consider the match of spin structure
between two different signatures. ; the above conditions at the boundary
enable us to determine the tensor field in a unique way. As for the the first
matching condition, using the Jacobian matrix, we can write it in terms of the
embedding coordinate as

$\left(\frac{\partial X^{A}}{\partial
y^{a}}\right)^{E}H^{E}_{A}(X)=\left(\frac{\partial X^{A}}{\partial
y^{a}}\right)^{L}H^{L}_{A}(X),$ (24)

in which we omit the joint surface restriction and again we use $E,L$ to
distinguish Jacobian matrices in Euclidean and Lorentzian signature,
respectively. As for the second matching condition, since the derivative with
respect to time $\partial_{t}=\partial_{\tau}=\partial_{0}$ is involved, we
need to study the matching conditions in different spacetime directions
separately. Taking the Euclidean signature for example, for $a\neq 0$, we have

$\partial_{\tau}\left(\;\frac{\partial X^{A}}{\partial
y^{a}}H_{A}(X)\;\right)=\frac{\partial X^{A}}{\partial
y^{a}}\;\partial_{\tau}H_{A}(X),$ (25)

while for $a=0$

$\partial_{\tau}\left(\;\frac{\partial X^{A}}{\partial
y^{0}}H_{A}(X)\;\right)=\frac{\partial X^{A}}{\partial
y^{0}}\;\partial_{\tau}H_{A}(X)+2X^{+}H_{+},$ (26)

which tells us that $\partial_{\tau}$ will commute with the Jacobian matrix
except for the $y^{0}=\tau$ direction in which we get the extra $2X^{+}H_{-}$
contribution. This leads us to define an operator valued matrix $M$ as

$M:=\frac{\partial X}{\partial
y}\partial_{\tau}+\left(\begin{array}[]{ccc}2X^{+}&0\\\ \\\
0&0\end{array}\right)=\frac{\partial X}{\partial
y}\partial_{\tau}+\mathcal{T},$ (27)

in which the second term $\mathcal{T}$ is a $(d+1)\times(d+2)$ matrix which
has $(0,\dots,d-1,z)$ as rows and $(-,+,0,\dots,d-1)$ as columns while $(0,-)$
is the only nonzero element. With the help of matrix $M$, we can write (25)
and (26) in a compact form as

$\partial_{\tau}\left(\;\frac{\partial X^{A}}{\partial
y^{a}}H_{A}(X)\;\right)=M_{a}^{A}\;H_{A},$ (28)

and the second matching condition can be written as

$M^{E}\cdot H^{E}(X)+iM^{L}\cdot H^{L}(X)=0,$ (29)

in which $M^{E}$ is the matrix we have already discussed in (27) acting on the
column vector $H_{A}$ and $M^{L}$ is

$M^{L}=\left(\frac{\partial X}{\partial
y}\right)^{L}\partial_{t}-\left(\begin{array}[]{ccc}2X^{+}&0\\\ \\\
0&0\end{array}\right).$ (30)

Equation (24) together with (29) give us matching conditions for the vector
fields while it is straight forward to generalise them to the high spin
fields. Before doing that, we first hide the spin index into the polynomial
via the operator $K_{A}$ Costa:2014kfa

$H_{A_{1}\dots A_{J}}(X)=\frac{1}{J!(\frac{d-1}{2})_{J}}K_{A_{1}}\dots
K_{A_{J}}H(X,W),$ (31)

in which $H(X,W)=W^{A_{1}}\dots W^{A_{J}}H_{A_{1}\dots A_{J}}(X)$ is a
polynomial in terms of $W$. Moreover, we should note that $K_{A}$ only
involves the variable $W$ provided $H_{A_{1}\dots A_{j}}$ is a symmetric
traceless tensor. Thus $K_{A}$ acts on the polynomial $H(X,W)$ independently
and we can write higher spin field matching conditions in terms of the
embedding polynomials as

$\displaystyle\left(\frac{\partial X^{A_{1}}}{\partial
y^{a_{1}}}K_{A_{1}}\right)^{E}\dots\left(\frac{\partial X^{A_{J}}}{\partial
y^{a_{J}}}K_{A_{J}}\right)^{E}H^{E}(X,W)$ (32)
$\displaystyle=\left(\frac{\partial X^{A_{1}}}{\partial
y^{a_{1}}}K_{A_{1}}\right)^{L}\dots\left(\frac{\partial X^{A_{J}}}{\partial
y^{a_{J}}}K_{A_{J}}\right)^{L}H^{L}(X,W)$

and

$\displaystyle\sum_{i=1}^{J}\left(\frac{\partial X^{A_{1}}}{\partial
y^{a_{1}}}K_{A_{1}}\right)^{E}\dots\left(T_{a_{i}}^{A_{i}}K_{A_{i}}\right)^{E}\dots\left(\frac{\partial
X^{A_{J}}}{\partial y^{a_{J}}}K_{A_{J}}\right)^{E}H^{E}(X,W)$ (33)
$\displaystyle+i\sum_{i=1}^{J}\left(\frac{\partial X^{A_{1}}}{\partial
y^{a_{1}}}K_{A_{1}}\right)^{L}\dots\left(T_{a_{i}}^{A_{i}}K_{A_{i}}\right)^{L}\dots\left(\frac{\partial
X^{A_{J}}}{\partial y^{a_{J}}}K_{A_{J}}\right)^{L}H^{L}(X,W)=0,$

in which we take $T^{A_{J}}_{a_{J}}=M^{A_{J}}_{a_{J}}$ for short. At first
sight, the matching conditions in embedding space are more complicated than
those in $\rm AdS$ space since more operators and transform matrices are
involved but we should note that the tensor field $H_{a_{1}\dots a_{J}}$ will
be simplified a lot once we write it into the form of polynomial $H(X,W)$.

### Examples

Here we will see how the matching conditions work based on the investigation
of the fields with spin $J=1$ and $J=2$. During the discussion, we use the
spin $J$ bulk-to-boundary propagator of dimension $\Delta$ written as

$G_{\Delta,J}(X,P;W,Z)=\frac{((-2P\cdot X)(W\cdot Z)+2(W\cdot P)(Z\cdot
X))^{J}}{(-2P\cdot X)^{\Delta+J}},$ (34)

in which $X,P$ are points live on the bulk and boundary associated with the
polynomial variables $W,Z$. The above expression is determined by imposing the
boundary condition on bulk Green function

$\lim_{z_{2}\to
0}z_{2}^{-\Delta}G_{\Delta,J}(X_{1},X_{2};Z_{1},Z_{2})=G_{\Delta,J}(X_{1},X_{2}^{\infty};Z_{1},Z_{2}).$
(35)

Here we are abusing the notion $G_{\Delta,J}$. On the left it represents the
bulk Green function while on the right it represents a bulk-boundary operator.
The limit $z\to 0$ means that we are sending a bulk point
$X=\frac{1}{z}(1,z^{2}+y^{2},y^{\mu})$ to the boundary
$X^{\infty}=(1,y^{2},y^{\mu})$ thus the explicit physical meaning of
$G_{\Delta,J}$ depends on the position of its second point. Moreover, after
taking the limit in (35), we can see the formula in $\eqref{bbo}$ only encodes
the information about the sources while the information about the normalizable
modes proportional to $z^{d-\Delta}$ is absent.

As we have mentioned before, to extract the tensor from a given polynomial, we
need to act the operator Costa:2011mg ; Costa:2014kfa

$D^{A}_{Z}=\left(\frac{d}{2}-1+Z\cdot\frac{\partial}{\partial
Z}\right)\frac{\partial}{\partial
Z_{A}}-\frac{1}{2}Z^{A}\frac{\partial^{2}}{\partial Z\cdot\partial Z}$ (36)

and

$K^{W}_{A}=\left(\frac{d-1}{2}+W\cdot\frac{\partial}{\partial
W}\right)\frac{\partial}{\partial W^{A}}$ (37)

on the polynomial corresponding to the bulk and boundary points, respectively.
In the following two examples, we will try to deal with the field of spin
$J=1$ and $J=2$ separately and the above operators will be used extensively
together with the higher spin field expansion

$H(X,W)=\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}C_{\Delta,J}(P,D_{Z})G_{\Delta,J}(X,P;W,Z),$
(38)

in which the higher spin fields and the bulk-to-boundary propagator are both
written in terms of polynomials. As for the $C_{\Delta,J}(P,D_{Z})$, we can
treat it as an operator polynomial $C^{A_{1}\cdots
A_{J}}_{\Delta,J}D^{Z}_{A_{1}\cdots A_{J}}$ thus determining the matching
condition for $H^{A_{1}\cdots A_{J}}$ is equivalent to determining the
condition for $C_{\Delta,J}^{A_{1}\cdots A_{J}}$.

i) Spin $J=1$ Match

For the spin 1 case, we first write down the expansion of fields in terms of
the polynomial

$H(X,W)=\frac{1}{\frac{d}{2}-1}\;\int
dP\;C^{A}(P)\;D^{Z}_{A}\;G_{\Delta,J=1}(X,P;W,Z),$ (39)

in which $C^{A}$ are coefficients carrying the tensor indexes. After
substituting $G_{\Delta,J}$, $D^{Z}_{A}$ into the integrand and applying
$K_{A}^{W}$ on the polynomial, we obtain the tensor in embedding space

$H_{A}(X)=\int dP\;\frac{-2(P\cdot X)C_{A}+2(C\cdot X)P_{A}}{(-2P\cdot
X)^{\Delta+1}}$ (40)

in which we can check that the transverse condition $X\cdot H=0$ is
automatically satisfied for arbitrary $C^{A}(P)$. Moreover, one can also
directly see from the above expression that there is a redundancy of the
coefficients

$C^{A}(P)\longrightarrow C^{A}(P)+\lambda P^{A},$ (41)

which tells us a shift of the coefficient by $\lambda P^{A}$ will give us the
same higher spin field. We call this the pure gauge of coefficients since it
works the same way as the pure gauge for CFT spin fields and later we will see
that the consistence of matching conditions requires us to fix the pure gauge.

Now, if we just consider the integrand, the matching condition (24) tells us
that

$\displaystyle-(P\cdot X)_{E}\;C^{E}_{a}+(C^{E}\cdot X)_{E}\;P_{a}=-(P\cdot
X)_{L}\;C_{a}^{L}+(C^{L}\cdot X)_{L}\;P_{a},$ (42)

in which $C^{E}_{A}$, $C^{L}_{A}$ are the tensor coefficients in Euclidean and
Lorentzian signature and they have already been pulled back to the AdS space
thus labelled by $C^{E}_{a}$, $C^{L}_{a}$. Here we should note that the above
equation is restricted on the surface $\Sigma$ which we did not write down
explicitly for short. Since $P_{a}$ serve as variables in the integrand, to
solve the above matching condition for arbitrary $P$, we should make each term
on both sides fit. Furthermore, noting that $(P\cdot X)_{E}=(P\cdot X)_{L}$ is
guaranteed by the rotation of embedding coordinates, therefore the non trivial
conditions are determined as

$\displaystyle C_{a}^{E}|_{\Sigma}=C_{a}^{L}|_{\Sigma},\qquad(C^{E}\cdot
X)_{E}|_{\Sigma}=(C^{L}\cdot X)_{L}|_{\Sigma}.$ (43)

The first equation is just the statement that the coefficients are continuous
at the joint surface $\Sigma$ in $\rm AdS$ spacetime and the second one can be
simplified to $iC^{E}_{0}=C^{L}_{0}$, which is the feature of Wick rotation on
coefficient tensor fields associated with the rotation of embedding space
coordinates.444Here $C_{A=0}$ represents the tensor in embedding space while
later we will meet the zero component in $AdS$ space and we denote it as
$C_{a=0}$.

Here, we should stop to check that if the two conditions in (43) could be
compatible. Actually, if we consider the coordinate transformation of time
direction under the condition $\tau=iT,t=T$ and $C^{E}_{0}=iC_{0}^{L}$, the
relation we get in $\rm AdS$ spacetime should be $C_{a=0}^{E}=iC_{a=0}^{L}$ at
the joint surface rather than the first one in (43)
$(C^{E}_{a=0}=C^{L}_{a=0})$, which implies in fact the matching condition we
should impose on the tensor field is that

$\displaystyle H_{a}^{E}(X)|_{\Sigma}=H_{a}^{L}(X)|_{\Sigma},\qquad{\rm
for}\qquad a\neq 0$ (44) $\displaystyle
H_{0}^{E}(X)|_{\Sigma}=iH_{0}^{L}(X)|_{\Sigma}.$ (45)

And the modification of the matching condition in time direction results from
the convention when we are doing the calculation while we can treat it as the
feature of the rotation of vector fields, which we do not need to consider in
scalar field matching.

Physically, suppose that we have the higher spin action in the hand and try to
match actions of different signatures together, the special matching condition
(45) for time component will make the invariant term like

$\eta^{ab}H_{a}^{L}H_{b}^{L}=\delta^{ab}H_{a}^{E}H_{b}^{E}$ (46)

joint smoothly at the surface $\Sigma$. Thus now we can see that the condition
(45) becomes obvious from physics point of view even though it was derived by
checking the consistence of the propagator matching. Next, we come to study
the second matching equation. Based on the above discussion, we determine the
modified matching conditions as

$\displaystyle\partial_{\tau}H^{E}_{a}(X)|_{\Sigma}+i\partial_{t}H^{L}_{a}(X)|_{\Sigma}=0,\qquad{\rm
for}\qquad a\neq 0$ (47)
$\displaystyle\partial_{\tau}H^{E}_{0}(X)|_{\Sigma}-\partial_{t}H^{L}_{0}(X)|_{\Sigma}=0,$
(48)

in which we have taken the rotation of $H_{0}$ into consideration. As for the
derivative with respect to time, we need to substitute (A.159) and (A.158)
into (40) and then obtain

$\displaystyle\partial_{\tau}H_{A}(X)$ $\displaystyle=$ $\displaystyle\int
dP\;\frac{-2(-2X^{0}P_{-}+X^{+}P_{0})C_{A}+2(-2X^{0}C_{-}+X^{+}C_{0})\;P_{A}}{(-2P\cdot
X)_{E}^{\Delta+1}}$ (49) $\displaystyle+$
$\displaystyle\frac{2(\Delta+1)(-2X^{0}P_{-}+X^{+}P_{0})(-2(X\cdot
P)_{E}C_{A}+2(C\cdot X)_{E}P_{A})}{(-2X\cdot P)_{E}^{\Delta+2}}$

for Euclidean signature and

$\displaystyle\partial_{t}H_{A}(X)$ $\displaystyle=$ $\displaystyle\int
dP\;\frac{-2(2X^{0}P_{-}+X^{+}P_{0})\;C_{A}+2(-2X^{0}C_{-}+X^{+}C_{0})\;P_{A}}{(-2P\cdot
X)_{L}^{\Delta+1}}$ (50) $\displaystyle+$
$\displaystyle\frac{2(\Delta+1)(2X^{0}P_{-}+X^{+}P_{0})(-2(X\cdot
P)_{L}\;C_{A}+2(C\cdot X)_{L}\;P_{A})}{(-2X\cdot P)_{L}^{\Delta+2}}$

for Lorentzian signature. It will be convenient to note the differences
between them mainly come from the inner product $(\cdot)_{L/E}$ and the sign
in front of $\pm X^{0}P_{-}$. Then, using the matching condition (29), we have

$\displaystyle\Delta(-X\cdot P)_{E}\;(-2X^{0}P_{-}+X^{+}P_{0})C_{a}+((-X\cdot
P)_{E}\;(-2X^{0}C_{-}+X^{+}C_{0})$ $\displaystyle+(\Delta+1)(C\cdot
X)_{E}\;(-2X^{0}P_{-}+X^{+}P_{0}))P_{a}+i(\cdots)_{L}=0,$ (51)

in which $a\neq 0$ and $(\cdots)_{L}$ represents the Lorentzian version of the
formula we wrote down explicitly and one can check that the above equation is
trivial provided the conditions in (43) are satisfied. The new restriction
comes from the study of the zero component for $a=0$, taking the extra
contribution $2X^{+}H_{-}$ into consideration, we have

$\displaystyle\Delta(-X\cdot P)_{E}\;(-2X^{0}P_{-}+X^{+}P_{0})C_{a=0}+(P\cdot
X)_{E}^{2}\;C_{-}X^{+}-(P\cdot X)_{E}(C\cdot X)_{E}P_{-}X^{+}$
$\displaystyle+((-X\cdot P)_{E}\;(-2X^{0}C_{-}+X^{+}C_{0})+(\Delta+1)(C\cdot
X)_{E}\;(-2X^{0}P_{-}+X^{+}P_{0}))P_{0}$ $\displaystyle-(\cdots)_{L}=0.$ (52)

To solve it, one should impose the condition

$C_{-}^{E}|_{\Sigma}=C_{-}^{L}|_{\Sigma},$ (53)

which gives us the proper gauge of the embedding coordinates on the joint
surface $\Sigma$. To understand such an extra condition, first we consider the
degrees of freedom of the matching conditions in terms of the coefficients
$C^{A}$. Since we are dealing with $d+1$ dimensional AdS spacetime, there are
$d+1$ equations for us to solve in (43), together with the gauge condition
(53), we have $d+2$ matching conditions, which uniquely fix the $d+2$
coefficients $C^{A}$ in embedding space. We can also understand this in a way
that is similar to the direct match of $H_{A}$. As we have mentioned before,
after imposing the transverse condition, there will be a one to one
correspondence between the fields in the embedding space and the fields on the
AdS surface. But the transverse conditions will not introduce any restriction
on the coefficients therefore we have the pure gauge redundancy. Now we see
that the pure gauge should be fixed when doing the matching.

ii) Spin $J=2$ Match

For the spin 2 field, we just consider the symmetric and traceless tensor
field $H_{AB}$ for simplicity. In this case, the polynomial can be written as

$H(X,W)=\frac{2}{(d+1)(d-1)}\;\int
dP\;C^{AB}(P)\;D^{Z}_{A}D_{B}^{Z}\;G_{\Delta,J=2}(X,P;W,Z),$ (54)

which corresponds to the tensor

$\displaystyle H_{AB}(X,W)=\int dP\;\frac{4}{(-2X\cdot
P)^{\Delta+2}}\;((X\cdot P)^{2}C_{AB}$ (55)
$\displaystyle+P_{A}P_{B}X^{C}X^{D}C_{CD}-(X\cdot P)P_{A}X^{C}C_{CB}-(X\cdot
P)P_{B}X^{C}C_{CA}).$

After imposing the matching condition, we have

$\displaystyle(X\cdot
P)_{E}^{2}\;C^{E}_{ab}+P_{a}P_{b}X^{c}X^{d}C^{E}_{cd}-(X\cdot
P)_{E}\;P_{a}\;X^{c}C^{E}_{cb}-(X\cdot P)_{E}\;P_{b}\;X^{c}C^{E}_{ca}$
$\displaystyle=(X\cdot
P)_{L}^{2}\;C^{L}_{ab}+P_{a}P_{b}X^{c}X^{d}C^{L}_{cd}-(X\cdot
P)_{L}\;P_{a}\;X^{c}C^{L}_{cb}-(X\cdot P)_{L}\;P_{b}\;X^{c}C_{ca}^{L},$ (56)

in which the tensor $C_{AB}$, $P_{A}$, $X_{A}$ together with the inner product
have been pulled back to the $AdS$ Poincar$\acute{{\rm e}}$ coordinates.
Similar to the study of spin $J=1$ case, we deduce the solution to be

$\displaystyle C_{ab}^{E}|_{\Sigma}=C_{ab}^{L}|_{\Sigma}\quad{\rm for}\quad
a\neq 0,\qquad C^{E}_{0b}|_{\Sigma}=iC^{L}_{0b}|_{\Sigma}\quad{\rm for}\quad
a=0,$ (57) $\displaystyle
C^{E}_{ab}\;X^{a}|_{\Sigma}=C^{L}_{ab}\;X^{a}|_{\Sigma},$ (58)

in which the first implies the continuation of the tensor field, the second
corresponds to the Wick rotation on the time direction and the third implies
the match of inner product, and we note that they are compatible by imposing
$C^{E}_{0B}=iC^{L}_{0B}$.

Now, we come to study the second matching condition and check if the
restrictions showed in (58) are enough. Substituting (A.159) into (55), we see
that the derivative with respect to the Euclidean time is given by

$\displaystyle\partial_{\tau}H_{AB}(X,W)=4\int dP\;\frac{2}{(-X\cdot
P)_{E}^{\Delta+2}}((-2X^{0}P_{-}+X^{+}P_{0})((X\cdot P)_{E}\;C_{AB}$ (59)
$\displaystyle-
P_{(A}X^{C}C_{CB)})+P_{A}P_{B}X^{D}(-2X^{0}C_{-D}+X^{+}C_{0D})-(X\cdot
P)_{E}P_{(A}(-2X^{0}C_{-B)}+X^{+}C_{0B)}))$
$\displaystyle+\frac{2(\Delta+2)(-2X^{0}P_{-}+X^{+}P_{0})}{(-2X\cdot
P)_{E}^{\Delta+3}}\;((X\cdot
P)_{E}\;C_{AB}+P_{A}P_{B}X^{C}X^{D}C_{CD}-2(X\cdot
P)_{E}\;P_{(A}X^{C}C_{CB)}).$

in which we use the convention for symmetrising the free tensor index
$C_{(AB)}=\frac{1}{2}(C_{AB}+C_{BA})$. And, as for the Lorentzian time
derivative, all we need to do is to change the sign in front of $X^{0}P_{-}$
and the notion of inner product while we will not show that explicitly here.
Consider the integrand, we will obtain

$\displaystyle
P_{a}P_{b}((\Delta+2)(-2X^{0}P_{-}+X^{+}P_{0})X^{c}X^{d}C_{cd}+(-2P\cdot
X)_{E}X^{d}(-2X^{0}C_{-d}+X^{+}C_{0d}))$ $\displaystyle-P_{(a}(X\cdot
P)_{E}(\Delta(-2X^{0}P_{-}+X^{+}P_{0})X^{c}C_{cb)}+(-2P\cdot
X)_{E}\;(-2X^{0}C_{-b)}+X^{+}C_{0b)}))$
$\displaystyle+\Delta(-2X^{0}P_{-}+X^{+}P_{0})(X\cdot
P)_{E}^{2}\;C_{ab}+i(\cdots)_{L}=0$ (60)

for $a,b\neq 0$ from the second the matching condition, which will be trivial
provided the restrictions we imposed in (58) are satisfied. After all, to make
the matching condition fits at the $a=0$ and $b\neq 0$ direction, one should
also impose the gauge constraint

$C^{E}_{-b}|_{\Sigma}=C^{L}_{-b}|_{\Sigma},$ (61)

which results from the matching condition

$\partial_{\tau}H^{E}_{0b}-\partial_{t}H^{L}_{0b}=0.$ (62)

###  2.4 Embedding the BTZ

Based on the study of match conditions on the AdS background, we now come to a
more interesting case which is the BTZ black hole. The treatment of BTZ black
hole is similar to the $\rm AdS_{3}$ since they are locally isomorphic
therefore all the results for AdS could be generalised to the BTZ black hole.
Also, as we have mentioned, there is not a unique way to lift a AdS solution
to the entire embedding space. In this section, we will use BTZ solutions as
an example to illustrate more detail on this therefore achieve a bettering
understanding on how to deal with AdS solutions in embedding space or vice
versa.

First, we will introduce BTZ black hole Banados:1992gq ; Banados:1992wn  in
embedding space via identifying points along the trajectory generated by the
chosen Killing vector in ${\rm AdS_{3}}$. We start from the embedding of
Lorentzian $\rm AdS_{3}$ given by

$-(X^{-1})^{2}-(X^{0})^{2}+(X^{1})^{2}+(X^{2})^{2}=-R^{2},$ (63)

in which the $SO(2,2)$ symmetry is manifested and $R$ is related to the
cosmological constant by $-\Lambda=R^{-2}$. The $\rm AdS_{3}$ in embedding
space has the metric

$ds^{2}=-(dX^{-1})^{2}-(dX^{0})^{2}+(dX^{1})^{2}+(dX^{2})^{2},$ (64)

and in order to get the black hole geometry, we introduce the Killing vector

$\xi=\frac{r_{+}}{R}\left(X^{-1}\frac{\partial}{\partial
X^{2}}+X^{2}\frac{\partial}{\partial X^{-1}}\right),$ (65)

where $r_{+}$ is a constant characterising the size of horizon. Given the
initial point $P$, the Killing vector will generate a curve $c(t)=e^{t\xi}P$
in which $\xi$ serves as the tangent vector along $c(t)$. Here, we are going
to identify the points on the curve such that $t\in 2\pi\mathbb{Z}$ and those
points which are invariant under the transformation $e^{2\pi\mathbb{Z}\xi}$
will become singularities of the quotient space ${\rm AdS_{3}}\setminus\sim$.
Although the quotient space satisfies the Einstein equation at the regular
points, we still need to get rid of the closed timelike curves, resulting from
the identification procedure, to obtain a reasonable causal structure.
According to the property of the Killing field, $\xi\cdot\xi$ is preserved
along the curve $c(t)$, thus the necessary condition for the absence of closed
timelike curves is $\xi^{2}>0$ everywhere in the manifold and in terms of the
embedding coordinates, we have

$(X^{-1})^{2}-(X^{2})^{2}=X^{+}X^{-}>0,$ (66)

which gives us the black hole geometry with a proper causal structure. To see
the effect of identification, we will introduce the $(t,r,\phi)$ coordinate
defined as

$\displaystyle X^{-1}$ $\displaystyle=$ $\displaystyle\frac{Rr}{r_{+}}\;{\rm
cosh}(\frac{r_{+}}{R}\phi),$ (67) $\displaystyle X^{2}$ $\displaystyle=$
$\displaystyle\frac{Rr}{r_{+}}\;{\rm sinh}(\frac{r_{+}}{R}\phi),$ (68)
$\displaystyle X^{0}$ $\displaystyle=$
$\displaystyle\frac{R}{r_{+}}\sqrt{r^{2}-r_{+}^{2}}\;{\rm
sinh}(\frac{r_{+}t}{R^{2}}),$ (69) $\displaystyle X^{1}$ $\displaystyle=$
$\displaystyle\frac{R}{r_{+}}\sqrt{r^{2}-r_{+}^{2}}\;{\rm
cosh}(\frac{r_{+}t}{R^{2}}),$ (70)

in which $-\infty<t,\phi<\infty$, $r\geq r_{+}$, and for $0\leq r\leq r_{+}$
the expressions for $X^{-1}$, $X^{2}$ are the same while we have

$\displaystyle X^{0}$ $\displaystyle=$
$\displaystyle\frac{-R}{r_{+}}\sqrt{r_{+}^{2}-r^{2}}\;{\rm
cosh}(\frac{r_{+}t}{R^{2}}),$ (71) $\displaystyle X^{1}$ $\displaystyle=$
$\displaystyle\frac{-R}{r_{+}}\sqrt{r_{+}^{2}-r^{2}}\;{\rm
sinh}(\frac{r_{+}t}{R^{2}}),$ (72)

for $X^{0}$, $X^{1}$. These two patches together will cover the $\rm AdS_{3}$
and Killing vector (65) becomes

$\xi=\frac{\partial}{\partial\phi}$ (73)

once we have pushed it back to the hyperboloid described by $(t,r,\phi)$.
Therefore, in the new coordinates, the identification of points under
$e^{2\pi\mathbb{Z}\xi}$ is equivalent to imposing $\phi\cong\phi+2\pi$ and
then we obtain the black hole metric

$ds^{2}=-\left(\frac{r^{2}-r_{+}^{2}}{R^{2}}\right)dt^{2}+\left(\frac{R^{2}}{r^{2}-r_{+}^{2}}\right)dr^{2}+\frac{r^{2}}{R^{2}}d\phi^{2},$
(74)

from which we can see that the horizon lies at $r=r_{+}$, thus in embedding
coordinates, the horizon is

$X^{+}X^{-}=R^{2}.$ (75)

For simplicity, we usually choose to perform the coordinates transformation

$t\longrightarrow\frac{r_{+}}{R^{2}}t,\qquad
r\longrightarrow\frac{r}{r_{+}},\qquad\phi\longrightarrow\frac{r_{+}}{R}\phi$
(76)

and get

$ds^{2}=-(r^{2}-1)dt^{2}+\frac{dr^{2}}{r^{2}-1}+r^{2}d\phi^{2},$ (77)

with the periodic condition $\phi=\phi+\frac{2\pi r_{+}}{R}$.

Now, we come to introduce the solution of the Kelin-Gordon equation for the
scalar field $\Phi$ on the BTZ black hole background (74), written as

$\nabla^{2}\Phi_{\Delta}-m^{2}\Phi_{\Delta}=0,$ (78)

in which $\nabla$ represents the Laplacian operator on the curved spacetime
$G$. The detail of the solution is shown in the Appendix B and here we just
discuss the results. As for the solution near the horizon $r=r_{+}$, there are
two independent modes

$\psi_{\pm}=e^{\frac{ir_{+}}{R^{2}}(\pm\omega
t-kR\phi)}f_{\Delta}(\pm\omega,k,\frac{r}{r_{+}}),$ (79)

from which we can see that the behaviour of the modes near the horizon depends
on the frequency $\omega$ which characterizes the propagation of the modes
along the circle, called left or right moving modes. If we consider the
solution at the spatial infinity making $r\to\infty$, the modes now become

$\psi_{\pm}=e^{\frac{ir_{+}}{R^{2}}(\omega
t-kR\phi})f_{\Delta^{\pm}}(\omega,k,\frac{r}{r_{+}}),$ (80)

where the scale dimension is given by the dictionary $\Delta(\Delta-d)=m^{2}$
treated as a constant parameter independent of the spacetime coordinates. We
can see that these two modes will behave like $r^{-\Delta}$ and $r^{\Delta-2}$
asymptotically, corresponding to the source and normalizable modes, which have
been studied in the vacuum $\rm AdS_{3}$ case.

It is worthwhile to note that, although we obtain four different kinds of
modes in total, this dose not mean that the scalar field $\Phi$ should be the
linear combination of these four modes. The reason that we obtain four modes
here is that we are expanding the same function around different singular
points thus the basis changes. Near horizon, the basis carries the information
of the direction of the propagation while, at the infinity, the basis carries
the information of asymptotic behavior of the field according to the radius
$r$.

#### Embedding the Solution

In the above section, we have studied the embedding structure of the BTZ black
hole and obtained the solution of Klein-Gordon equation on AdS background
while in this section we are going to lift the solution from the AdS
hyperboloid to the embedding space.

There are various ways of extending a function on a hyperboloid to the
embedding space. Since we can foliate the embedding space with AdS surfaces of
different radii $R$, a natural embedding way is just treating the radii $R$ as
a variable and extending the solution obtained from some standard surface
$R=R_{0}$ to the surfaces with different radii $R$. This is easy to do
mathematically while we should note that the extended function will not be the
solution of Kelin-Gordon equation on other AdS surfaces except for the surface
$R_{0}$. Vice versa, to fully retain the physical meaning of the extended
function, one can solve the KG equation (78) on each surface but it would be
hard to smoothly glue them together.

Here, instead of considering the extension of the solution directly, we
consider the extension of the KG equation. Using the generator of the
$SO(2,2)$ group

$J_{AB}=X_{A}\frac{\partial}{\partial X^{B}}-X_{B}\frac{\partial}{\partial
X^{A}},$ (81)

we can construct the quadratic Casimir directly in embedding space 2016

$\frac{1}{2}J_{AB}J^{BA}\Phi_{\Delta}(X)=R^{2}\nabla^{2}_{AdS}\;\Phi_{\Delta}(X),$
(82)

in which $\nabla_{AdS}=\nabla$ is the Laplacian operator for the BTZ black
geometry which is locally isometric to $\rm AdS_{3}$. The above expression can
be treated as the decomposition of an embedding operator along the AdS
surfaces. Given the quadratic Casimir, the equation of motion in embedding
space is deduced to be

$\frac{1}{2}J_{AB}J^{BA}\Phi_{\Delta}(X)=m^{2}R^{2}\Phi_{\Delta}(X),$ (83)

from which we see that the mass term $m^{2}R^{2}$ now depends on the radii of
the surfaces, and it will be reduced to the KG equation at radii $R=1$. Since
$[J_{AB},\;X^{2}+R^{2}]=0$, we can solve the Equation (83) on each surface,
following the same method of solving KG equation, provided we make

$m^{2}R^{2}=\Delta_{R}(\Delta_{R}-2).$ (84)

Now, if we write solution in terms of the $(r,t,\phi,R)$ coordinates i.e,
$\Phi(X)=\Phi(t,r,\phi,R)$, the modes will become

$\psi_{\pm}=e^{\frac{ir_{+}}{R^{2}}(\pm\omega
t-kR\phi)}f_{\Delta_{R}}(\pm\omega,k,\frac{r}{r_{+}})$ (85)

at the horizon while, at the infinity, the two modes are

$\psi_{\pm}=e^{\frac{ir_{+}}{R^{2}}(\omega
t-kR\phi)}f_{\Delta_{R}^{\pm}}(\omega,k,\frac{r}{r_{+}}),$ (86)

in which we not only make the phase part $R$ dependent but also the scale
dimension $R$ dependent. These solutions are smooth in embedding space and
they are also solutions of the scalar KG equation on each AdS surface with
mass $mR$, i.e, now the equation becomes

$\nabla^{2}\Phi_{\Delta_{R}}-m^{2}R^{2}\Phi_{\Delta_{R}}=0,$ (87)

which can be understood that we are considering excitation spectra or
fluctuation of particle with mass of order $R$ on each AdS surfaces.

### BTZ Propagator

In this part, based on the study of the geometry of BTZ black holes, we will
generalise the vacuum AdS propagators to the black hole case and discuss the
thermal feature of the boundary theories. Using these propagators, following
the same procedure, one can write down on shell higher spin fields propagating
on the black hole background therefore obtain the match conditions. It turns
out that the match condition will have the same form as the ${\rm AdS_{3}}$
case since the local property of the propagator remain the same.

As we have already known, BTZ geometry is obtained from $\rm AdS_{3}$ by the
identification of the points $\phi\cong\phi+2\pi$. In the embedding
coordinates, using (67) and (68), one can deduce that this is equivalent to

$X^{\pm}\cong e^{\pm 2\pi r_{+}}X^{\pm}.$ (88)

Here, to study the identified points more carefully, we introduce the notion

$\widetilde{X}^{n}:=(e^{+2n\pi r_{+}}X^{+},\;e^{-2n\pi
r_{+}}X^{-}\;X^{0},\;X^{1}),$ (89)

in which we use the superscript $n$ to represent the winding number of the
coordinates. Moreover, we should note that the points $\widetilde{X}^{n}$ are
distinguishable in the $\rm AdS_{3}$ geometry while they form a cover of a
single point $X$ of BTZ black hole

$X=\widetilde{X}^{0}\cong\widetilde{X}^{1}\cdots\widetilde{X}^{n}\cong\widetilde{X}^{n+1}\cdots.$
(90)

Following such convention, one can directly write down the the BTZ boundary-
bulk propagator

$G^{BTZ}_{\Delta,J}(X,P;W,Z)=\sum_{n=-\infty}^{\infty}\frac{((-2P\cdot\widetilde{X}^{n})(W\cdot
Z)+2(W\cdot
P)(Z\cdot\widetilde{X}^{n}))^{J}}{(-2P\cdot\widetilde{X}^{n})^{\Delta+J}}$
(91)

with the help of the method of images introduced in 1999 ; Kraus:2002iv .
Basically, the infinite sum over the winding number on the right hand side is
used to construct a function that is invariant at the points
$\widetilde{X}^{n}$ so that the relation (90) is manifested in the context of
the BTZ geometry.

To see how this works in a more specific way, we go back to the $(t,r,\phi)$
coordinates. The bulk points $X$ are well defined in the previous section and
here for the boundary points $P$, we write them as

$P=(P^{+},\;P^{-},\;P^{0},\;P^{1})=(e^{r_{+}\phi^{\prime}},\;e^{-r_{+}\phi^{\prime}},\;{\rm
sinh}(r_{+}t^{\prime}),\;{\rm cosh}(r_{+}t^{\prime})),$ (92)

in which the light ray condition $P^{2}=0$ is satisfied and the
$(t^{\prime},\phi^{\prime})$ are in fact coordinates of a cylinder. With these
coordinates, for the scalar case, one can write the bulk boundary operator as

$G^{+}_{\Delta,0}(X,P)=\sum_{n=-\infty}^{\infty}\frac{1}{\left(-\frac{\sqrt{r^{2}-r_{+}^{2}}}{r_{+}}{\rm
cosh}(r_{+}\delta t)+\frac{r}{r_{+}}{\rm cosh}(r_{+}(\delta\phi+2\pi
n))\right)^{\Delta}}$ (93)

for $r>r_{+}$ and $\delta\phi=\phi-\phi^{\prime}$, $\delta t=t-t^{\prime}$.
This is the bulk-boundary propagator when the bulk point is outside the
horizon. For the inside horizon propagator, we should use the coordinate (71),
(72) and then obtain

$G^{-}_{\Delta,0}(X,P)=\sum_{n=-\infty}^{\infty}\frac{1}{\left(-\frac{\sqrt{r_{+}^{2}-r^{2}}}{r_{+}}{\rm
sinh}(r_{+}\delta t)+\frac{r}{r_{+}}{\rm cosh}(r_{+}(\delta\phi+2\pi
n))\right)^{\Delta}}.$ (94)

For the boundary correlation functions, the polynomial of higher spin two
point function

$\langle\mathcal{O}_{J}(P_{1})\mathcal{O}_{J}(P_{2})\rangle(Z_{1},Z_{2})=\sum_{n=-\infty}^{\infty}\frac{((-2\widetilde{P}^{n}_{1}\cdot
P_{2})(Z_{1}\cdot Z_{2})+2(\widetilde{P}_{1}^{n}\cdot Z_{2})(P_{2}\cdot
Z_{1}))^{J}}{(-2\widetilde{P}_{1}^{n}\cdot P_{2})^{\Delta+J}}$ (95)

is obtained by the projection of the bulk-boundary propagator to the
$r^{\Delta}$ term, in which $\widetilde{P}^{n}$ are defined as

$\widetilde{P}^{n}:=(e^{+2n\pi r_{+}}P^{+},\;e^{-2n\pi
r_{+}}P^{-},\;P^{0},\;P^{1})$ (96)

thus the temperature $T=r_{+}/2\pi$ can be deduced after going to the
Euclidean signature $t=-i\tau$.

##  3 General Principles of Lorentzian Continuation

In this section we will do the Wick rotation of CFT completeness relation by
matching the scale dimension and spin in different signatures properly then
present a Lorentzian AdS/CFT map based the derivation of Euclidean AdS/CFT map
in Aharony:2020omh ; Aharony:2021ovo . Given the Lorentzian AdS/CFT map, we
will present the matching conditions for the quadratic action then obtain the
matching conditions for higher order action.

###  3.1 Scale Dimension and Spin

First we will discuss the behavior of scale dimension $\Delta$ and spin $J$ in
different signatures from both the conformal field theory and gravity theory
point of view, which plays the central role in the context of Wick rotation
and AdS/CFT matching.

| Symmetry Group |  Principal Series | Parameter
---|---|---|---
Euclidean | $SO(d+1,1)\sim SO(1,1)+SO(d)$ |  $\quad\mathcal{E}_{\Delta,\;J}$ |  $\Delta=\frac{d}{2}+i\mathbb{R}$, $J\in\mathbb{Z}$
| $SO(d,2)\sim SO(1,1)+$ |  | $\Delta=\frac{d}{2}+i\mathbb{R}$
Lorentzian |  |  $\;\mathcal{P}_{\Delta,\;J,\;\lambda}$ |
| $SO(1,1)+SO(d-2)$ |  |  $J=\frac{d-2}{2}+i\mathbb{R}$, $\lambda\in\mathbb{Z}$
Table 1: Harmonic analysis of the conformal symmetry for $d$ spacetime
dimensional Euclidean and Lorentzian conformal field theory. The principal
series for Euclidean and Lorentzian signature are labelled by $\mathcal{E}$
and $\mathcal{P}$, respectively.

We start from the study of representation theory of the conformal symmetry
group. As for the scale dimension induced by the dilaton operator $D$, it
generates a noncompact direction, i.e noncompact subgroup $SO(1,1)$, in both
signatures and takes the value $\Delta=\frac{d}{2}+i\mathbb{R}$ on the same
principal series provided the representation is unitary while the story for
the spin is different.

For the spin in Euclidean signature, it represents the compact group $SO(d)$
thus takes the integer value $J\in\mathbb{Z}$ and in Lorentzian signature it
represents the noncompact time direction generated by the operator $M_{01}$
therefore becomes continuous on the principal series
$J=\frac{d-2}{2}+i\mathbb{R}$. The decomposition of the symmetry group and the
range of parameters are summarized in Table.1. We should note that in
Lorentzian signature we usually take the representation of compact group
$SO(d-2)$ to be trivial therefore make $\lambda=1$. In the end, to get the
physical operators, we need to analytically continue the $\Delta,J$ from
principal series to real axes while the rotation of the scale dimension is
shown in LHS of Fig.3.

$\Delta=\frac{d}{2}+is$$\Delta_{0}$$\Delta$

$\Delta=\frac{d}{2}+is$$\Delta_{0}$$\Delta$

Figure 3: The analytic continuation of the scale dimension is shown on the
left hand in which principal series lies on the red line and physical scale
dimension distributes along the blue line on the real axes, with the lower
bound $\Delta_{0}$. In the right figure, the motion of the poles according to
the changing of the lower bound $\Delta_{0}$ is illustrated. In order to
eliminate such effects caused by the motion of poles, an extra part of the
contour winding around the singularity should be added into the original
contour $\gamma$. The new one is denoted as $\gamma_{J}$ and illustrated by
the red curves.

Practically, in order to get an analytic function, the contour along the
principal series is not enough and we need to deform the contour especially
when we meet poles in the complex plane. For example, in the completeness
relation (3.2) or (3), $\Delta_{0}=\frac{d}{2}$ is defined on the principal
series. But physically, as the lower bound of the scale dimension,
$\Delta_{0}$ itself depends on the theory thus it varies along the real axes
according to the interaction picture we have. Note that $\Delta_{0}$ serves as
a parameter in the constant factor thus the position of the pole of
coefficients $N_{\Delta,J}$ 555Given the complete basis which goes through the
principal series of $\Delta,J$, the coefficients $N_{\Delta,J}$ for arbitrary
field configuration are automatically determined after performing the mode
expansion. More careful study will be shown later. will move when we change
the value of $\Delta_{0}$. Taking these effects into consideration, if one
wants to extend the equation apart from $\Delta=\frac{d}{2}$ analytically,
extra contours going around the poles should be added properly, illustrated in
RHS of Fig.3, and the Euclidean case is discussed in Aharony:2020omh . To
distinguish the deformed contours from the original one $\gamma$, we denote
them as $\gamma_{E}$ and $\gamma_{L}$ corresponding to the Euclidean and
Lorentzian CFT 666In fact we should impose that $\gamma_{E}=\gamma_{L}$ to
make fields satisfy the match condition and we label them together as
$\gamma_{J}$, which means that the deformation of the principal series depends
on the spin $J$. .

The above discussion comes from the study of the symmetry group on field
theory side, which must have implications in the bulk theory with gravity due
to the correspondence. The connections come in when we try to construct a map
between the bulk fields and the boundary fields via the decomposition (10) and
(11). In (10) and (11), we just state without explanation that $\Delta$ should
lie on the principal series even though harmonic analysis does not apply to
the gravity theory. Here we point out that if one wants to complete the map
between the bulk and boundary field, one should choose the contour
$\gamma_{E}$ in (10) and the contour $\gamma_{L}$ in (11) and such choice of
contour in the bulk theory can be interpreted as selecting proper modes of the
bulk-boundary propagator, i.e, we have the correspondence

${\rm
Poles\;\;of}\;\;N_{J}(\Delta)\;,\;N_{\Delta,J}\qquad\longleftrightarrow\qquad{\rm
Bulk\;\;Modes},$ (97)

This is similar to the case we have met in the study of quasinormal modes
birmingham2001choptuik ; birmingham2002conformal ; birmingham2003relaxation
of BTZ black holes in which the pole of frequency in the BTZ solutions is
related to the pole of boundary CFT correlation functions. In the study of SYK
model Maldacena:2016upp ; maldacena2016remarks ; Polchinski:2016xgd ;
Kitaev:2017awl , four point functions are also expanded by the conformal basis
consisting of continuous modes on the principal series together with discrete
modes resulting from poles of $N_{J}(\Delta)$ and such discrete mode in the
bulk corresponds Goldstsone boson mode called dilaton that describes the
symmetry broken of $\rm AdS_{2}$. In the the context of HS/CFT duality, we
will use three point correlation functions as the basis of the boundary
bilocal fields and it turns out the both of the physical and ghost modes in
the bulk are generated by the boundary discrete modes.

Having discussed the role of scale dimension, we now come to the study of spin
$J$ while it will become more complicated even though we just stay on the CFT
side. The subtlety firstly comes in when we try to analytically extend the
spin $J$ in Lorentzian signature apart from the principal series since the
physical spin are discrete integer numbers mack1977all  on real axes rather
than a continuous interval and there is no mechanism telling us how the basis
collapse or whether the physical basis is still complete.

$J=\frac{d-2}{2}+is$$J$$\Gamma$

$J=\frac{d-2}{2}+is$$J$$\Gamma$

Figure 4: The contour in the complex $J$ plane we meet during the Wick
rotation is illustrated. The blue contour denoted as $\Gamma$ is the one we
used to take the place of the sum over non-negative spin while the red line is
the spin principal series. Wick rotation can be understood as the deformation
of $\Gamma$ to the principal series. The figure on the left shows the
configuration when the spacetime dimension $d$ is odd so that the principal
series will not pass through any integer point on the real axes. The figure on
the right shows the even dimension case in which we need to deform the
principal series around $(\frac{d-2}{2},\;0)$ in order to make contours not
meet with each other so that the Wick rotation is continuous.

We will also meet similar difficulty when doing rotation between two
signatures. For example, in the study of partial wave decomposition of
conformal four point functions Dobrev:1977qv ; simmons2018spacetime ;
kravchuk2018light , we need to sum over all possible spin $J$ for every
representations. This is represented as a sum over non-negative integers in
Euclidean signature. In Lorentzian signature, in order to make such
completeness still valid, we need to sum over the principal series, which now
becomes a continuous integral. We see that such a gap between integer and
continuous number arises again while this time it can be resolved by applying
the complex analysis technics.

The key idea comes from the study of Sommerfeld-Waston transform
eden2002analytic ; gribov2003theory ; cornalba2008eikonal , which tells us
that it is possible to rewrite the sum of discrete numbers into an integral
along the proper contour in the complex $J$ plane. More precisely, we have

$\sum_{J=0}^{\infty}\longrightarrow\frac{i}{2}\int_{\Gamma}\;\frac{dJ}{{\rm
sin}(\pi J)},$ (98)

in which the contour $\Gamma$ is shown in Fig.4. At this stage, we are still
in Euclidean signature and the next thing we need to do is to deform the
contour from $\Gamma$ to the Lorentzian principal series. During the
deformation, various physical phenomenon will show up and it usually depends
on the pole structure of the integrand. Specifically, we can write the
integrand into the product of the coefficients $N_{J}(\Delta)$ and the basis
$I^{E}_{\Delta,J}$ while the pole structure is hidden in the analytic
extension of $N_{J}(\Delta)$, labelled by $N(\Delta,J)$. After figuring out
the Lorentzian version of the basis $I_{\Delta,J}^{L}$, one may write down the
formula in Lorentzian signature. We summarise the Wick rotation procedure into
the formula

$\sum_{J=0}^{\infty}\int_{\gamma}\;\frac{d\Delta}{2\pi
i}\;N_{J}(\Delta)\;I^{E}_{\Delta,J}\longrightarrow\frac{i}{2}\int_{\gamma_{J}}\;\frac{dJ}{{\rm
sin}(\pi J)}\int_{\gamma}\;\frac{d\Delta}{2\pi
i}\;N(\Delta,J)\;I^{L}_{\Delta,J},$ (99)

which is the combination of following four steps

i) Write the discrete sum $\sum$ in terms of the integral along $\Gamma$.

ii) Do the space time rotation on the basis $I^{E}_{\Delta,J}$ in order to get
the Lorentzian expression $I^{L}_{\Delta,J}$.

iii) Analytically extend the coefficients $N_{J}(\Delta)$ to the complex plane
to get $N(\Delta,J)$.

iv) Rotate the contour $\Gamma$ to Lorentzian principal series $\gamma_{J}$
while the pole structure should be taken into consideration.

The above method has already been applied to study the CFT partial wave
decomposition Hartman:2015lfa ; simmons2018spacetime ; kravchuk2018light  and
the poles are called Regge poles gribov2003theory ; cornalba2008eikonal  in
the context of scattering amplitudes. We will use it to deal with the
conformal completeness relation in the next section.

###  3.2 CFT Completeness Relation

In conformal field theory, three-point correlation functions are elementary
building blocks of the higher point functions and it is believed that, after
imposing proper constraints, they will encode the whole information of a CFT
following the idea of conformal bootstrap. Moreover, according to the harmonic
analysis of the conformal symmetry group $SO(d+1,1)$ for a $d$ dimensional
Euclidean CFT Dobrev:1977qv ; Karateev:2018oml ; Aharony:2020omh , we can
regard three-point functions as the basis of the fields and the orthogonality
of the basis is given by the completeness relation

$\displaystyle\delta(x_{1},x_{3})\delta(x_{2},x_{4})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{J=0}^{\infty}\int_{\gamma}\frac{d\Delta}{2\pi
i}\int d^{d}x_{5}\;{N_{J}}(\Delta)$ $\displaystyle\times$
$\displaystyle\langle
O_{\Delta_{0}}(x_{1})O_{\Delta_{0}}(x_{2})O_{\Delta,J}^{\mu_{1}\cdots\mu_{J}}(x_{5})\rangle\langle
O_{\widetilde{\Delta}_{0}}(x_{3})O_{\widetilde{\Delta}_{0}}(x_{4})O^{\widetilde{\Delta},J}_{\mu_{1}\cdots\mu_{J}}(x_{5})\rangle,$

in which $x_{i}^{\mu}$ for $0\leq\mu\leq d-1$ are spacetime coordinates and
$\Delta$ is the scale dimension. Due to the harmonic analysis, the
representation of $SO(d+1,1)$ will be unitary provided $\Delta$ lies on the
principal series $\gamma=\frac{d}{2}+is$ and we denote the shadow transform of
the operator $O_{\Delta}(x_{i})$ as $O_{\widetilde{\Delta}}(x_{i})$, in which
$\widetilde{\Delta}=d-\Delta$. For simplicity, we will label them as $O_{i}$
and $\widetilde{O}_{i}$ in the later discussion. Moreover, given a CFT, we
should note that the scale dimension usually has a lower bound named
$\Delta_{0}$ and for free theory it takes the value $\frac{d-2}{2}$.

The completeness relation (3.2) can be used to expand local or bi-local fields
in Euclidean signature while we need to do the Wick rotation on it to study
the field theory in Lorentzian signature. In order to apply the Wick rotation
skill, we first choose to integrate over $x_{1},x_{4}$ on the two delta
functions, making them a constant, and then focus on the integrand term on the
RHS which is labelled as 777More precisely, we should use $I^{E}_{\Delta,J}$.

$I_{\Delta,J}:=\int d^{d}x_{1}d^{d}x_{4}d^{d}x_{5}\;\langle
O_{1}O_{2}O_{5}\rangle\langle\widetilde{O}_{3}\widetilde{O}_{4}\widetilde{O}_{5}\rangle.$
(101)

The main task of the rest of this section is to find the expression of
$I_{\Delta,J}$ in Lorentzian signature.

There are five spacetime points in the product of two three-points functions
and we use the conformal symmetry to fix three of them, i.e, we set
$x_{2}=(0,\dots,0)$, $x_{3}=(1,0,\dots,0)$ and
$x_{5}=(\Lambda,0,\dots,0)$888Usually we take $\Lambda\longrightarrow\infty$
but here we make $\Lambda$ a finite number. and then obtain

$I_{\Delta,J}=\int d^{d}x_{1}d^{d}x_{4}\;\langle
O_{1}O_{2}O_{5}\rangle\langle\widetilde{O}_{3}\widetilde{O}_{4}\widetilde{O}_{5}\rangle$
(102)

in which there are only two variables $x_{1}$ and $x_{4}$ left and they are
explicitly shown in the integral. We should note that there will be an overall
constant term arising from the volume of integral over $x_{5}$ when we choose
the gauge but we do not need to consider that since it will go away once we
ungauge the fixed points back to the integral in the end. To perform the Wick
rotation, we introduce the normal Feynman continuation in which we take
$x^{0}=(i+\epsilon)t$, $u=x^{1}-t$ and $v=x^{1}+t$. Therefore we obtain

$I_{\Delta,J}=-\frac{1}{4}\int
dv_{1}du_{1}dv_{4}du_{4}d^{d-2}x_{1}d^{d-2}x_{4}\;\langle
O_{1}O_{2}O_{5}\rangle\langle\widetilde{O}_{3}\widetilde{O}_{4}\widetilde{O}_{5}\rangle,$
(103)

and the singularities at the coincident points $x_{1}\thicksim x_{2},x_{5}$,
$x_{4}\sim x_{3},x_{5}$ are given by

$\displaystyle u_{12}v_{12}+i\epsilon=0\qquad u_{15}v_{15}+i\epsilon=0,$ (104)
$\displaystyle u_{43}v_{43}+i\epsilon=0\qquad u_{45}v_{45}+i\epsilon=0.$ (105)

As for the integral, we can think about fixing $u_{1}$ and $u_{4}$ and then
investigating the behavior of $I_{\Delta,J}$ on the $v_{1}$ and $v_{4}$
complex plane. Due to the introduction of the $i\epsilon$ expression,
singularities will shift apart from the real axes and their positions are
determined by the equation (104) and (105). More explicitly, singularities of
$v$ will shift to the upper half plane if $u$ is negative and vice versa.

Since we are interested in the nontrivial $I_{\Delta,J}$, the singularities
$x_{1}\sim x_{2}$ and $x_{1}\sim x_{5}$ can not lie in the same half plane
otherwise we can deform the integral contour along the real axes to the
infinity and make $I_{\Delta,J}$ vanish. The same argument holds for the
$x_{4}\thicksim x_{3}$ and $x_{4}\thicksim x_{5}$ singularities, which means
that the sign of $u_{12},u_{15}$ and $u_{43},u_{45}$ should be different, i.e,
it requires that

$u_{2}<u_{1}<u_{5},\qquad u_{3}<u_{4}<u_{5}.$ (106)

$v_{1}$$1\thicksim 2$$1\thicksim 5$$1\thicksim 4$

$v_{4}$$4\thicksim 3$$4\thicksim 5$$4\thicksim 1$

Figure 5: One possible choice of integral contour and branch cuts are shown in
the $v_{1}$ and $v_{4}$ plane, in which blue points are coincident points in
the correlation functions and wavy lines are branch cuts in the complex plane.
In the $v_{1}$ plane the contour winding around $1\thicksim 2$ will generate
the commutator $[O_{1},O_{2}]$ while the contour around $4\thicksim 5$ in the
$v_{4}$ plane will generate the commutator
$[\widetilde{O}_{5},\widetilde{O}_{4}]$.

The next step is to determine the deformation of the integral contours of
$v_{1}$ and $v_{4}$ and we illustrate one way of the deformation in Fig.5, in
which we let the $v_{1}$ integral go around $1\thicksim 2$ and the $v_{4}$
integral go around $4\thicksim 5$. This means that we restrict the integral in
the region $v_{1}\leq 0,v_{4}\geq\Lambda$ and, at the same time, it induces
the term

$\langle[O_{1},O_{2}]\;O_{5}\rangle\langle\widetilde{O}_{3}\;[\widetilde{O}_{5},\widetilde{O}_{4}]\rangle,$
(107)

in which the winding of the contour around the branch cuts will produce a
commutator $[,\;]$ and the order of the operator in the commutator is
determined by the direction of the contour. After taking all kinds of
deformation into consideration, we obtain

$(\langle[O_{1},O_{2}]\;O_{5}\rangle+\langle
O_{2}\;[O_{5},O_{1}]\rangle)\times(\langle\widetilde{O}_{3}\;[\widetilde{O}_{5},\widetilde{O}_{4}]\rangle+\langle[\widetilde{O}_{4},\widetilde{O}_{3}]\;\widetilde{O}_{5}\rangle),$
(108)

in which each term corresponds to a region in the $(v_{1},v_{4})$ plane and
they together contribute to $I_{\Delta,J}$. After all, we should sum over all
possible $\Delta$ and $J$ in $I_{\Delta,J}$ to get the completeness relation.
Now we should note that such problem has already been extensively discussed
and resolved in (99) and here we just write down the result in embedding space

$\displaystyle\delta(P_{1},P_{3})\delta(P_{2},P_{4})=\int_{\gamma_{J}}\frac{idJ}{{\rm
2sin}(\pi
J)}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}\int_{\gamma}\frac{d\Delta}{2\pi
i}N(\Delta,J)$ $\displaystyle\times$
$\displaystyle(\langle[O_{\Delta_{0}}(P_{1}),O_{\Delta_{0}}(P_{2})]\;O_{\Delta,J}(P,D_{Z})\rangle+\langle
O_{\Delta_{0}}(P_{2})\;[O_{\Delta,J}(P,D_{Z}),O_{\Delta_{0}}(P_{1})]\rangle)$
$\displaystyle\times$ $\displaystyle\langle
O_{\widetilde{\Delta}_{0}}(P_{3})\;[O_{\widetilde{\Delta},J}(P,Z),O_{\widetilde{\Delta}_{0}}(X_{4})]\rangle+\langle[O_{\widetilde{\Delta}_{0}}(P_{4}),O_{\widetilde{\Delta}_{0}}(P_{3})]\;O_{\widetilde{\Delta},J}(P,Z)\rangle)+\cdots.$

in which $P,Z$ are points in the embedding space corresponding to the boundary
CFT, the overall constant $N(\Delta,J)$ will now become analytic function that
depends on $\Delta,J$ and $\cdots$ represents the contribution from its poles.
$D_{Z}$ is the operator used to contract the spin indexes associated with the
factor $\frac{1}{J!(\frac{d}{2}-1)_{J}}$.

### Physical Basis

We have discussed that it is still unclear how to get physical spin from the
principal series in Lorentzian signature while we found the connection between
the Euclidean physical spin and the Lorentzian spin principal series via the
Wick Rotation of the CFT completeness relation. Here we are going to directly
rotate the physical Euclidean CFT to the physical Lorentzian CFT since spins
are integers on both sides and we summarise all of these ideas in the
following figure.

|  |
---|---|---
|  |
$\textstyle{{\rm
Euclidean\;CFT\;on}\;\mathcal{E}_{\Delta,J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
A.C.}$ $\scriptstyle{\rm A.C.}$5 Wick Rotation Completeness Relation 4
$\textstyle{{\rm
Lorentzian\;CFT\;on\;}\mathcal{P}_{\Delta,J,\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
A.C.}$ $\scriptstyle{\rm A.C.}$6 Physical Euclidean CFT
---
$\Delta\geq\Delta_{0},\;J\in\mathbb{Z}$
7 Bi-local Field Matching $\scriptstyle{\rm AdS/CFT}$ $\scriptstyle{\rm
AdS/CFT}$1 Physical Lorentzian CFT
---
$\Delta\geq\Delta_{0},\;J\in\mathbb{Z}$
$\scriptstyle{\rm AdS/CFT}$ $\scriptstyle{\rm AdS/CFT}$2 Euclidean AdS Higher
Spin Field Matching Higher Spin Field Matching 3$\textstyle{{\rm
Lorentzian\;AdS}\;}$

The work about the analytic continuation of Euclidean CFT labelled by  1 and
Euclidean AdS/CFT map labelled by  5 has already been discussed in
Aharony:2020omh  while we have studied Wick Rotation of the completeness
relation  4 and high spin field matching  3 in previous sections. In order to
verify the AdS/CFT map in Lorentzian signature, denoted as  2, we need to find
results in the physical Lorentzian CFT. Although the detail of analytic
continuation of Lorentzian CFT is unclear, we find that it is possible to
consider the direct correspondence between physical CFTs, labeled by  7.

7 can be regarded as the combination of  4,  5 and  6 and its net effect is
just that we do the Wick rotation of space time while keep the spin physical.
Now, we consider the behavior of spin in  4 and  6, in  4 we extend physical
Euclidean spin to Lorentzian principal series while in  6 we need to extend
spin on Lorentzian series to physical Lorentzian spin and in both cases the
contour will move across poles on complex plane determined by $N(\Delta,J)$
but from opposite directions. Therefore we can see that the net contribution
from the poles will be zero and then we obtain the physical CFT completeness
relation written as

$\displaystyle\delta(P_{1},P_{3})\delta(P_{2},P_{4})=\sum_{J=0}^{\infty}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}\int_{\gamma}\frac{d\Delta}{2\pi
i}N(\Delta,J)$ $\displaystyle\times$
$\displaystyle(\langle[O_{\Delta_{0}}(P_{1}),O_{\Delta_{0}}(P_{2})]\;O_{\Delta,J}(P,D_{Z})\rangle+\langle
O_{\Delta_{0}}(P_{2})\;[O_{\Delta,J}(P,D_{Z}),O_{\Delta_{0}}(P_{1})]\rangle)$
$\displaystyle\times$ $\displaystyle(\langle
O_{\widetilde{\Delta}_{0}}(P_{3})\;[O_{\widetilde{\Delta},J}(P,Z),O_{\widetilde{\Delta}_{0}}(P_{4})]\rangle+\langle[O_{\widetilde{\Delta}_{0}}(P_{4}),O_{\widetilde{\Delta}_{0}}(P_{3})]\;O_{\widetilde{\Delta},J}(P,Z)\rangle).$

###  3.3 AdS/CFT Map

In this part, we will use the CFT completeness relation to expand bi-local
fields $\Phi(P_{1},P_{2})$ on the boundary, together with expansion of higher
spin fields on the bulk, then construct a map between them in Lorentzian
signature so called Lorentzian AdS/CFT map .

#### Bi-local Fields Matching

Before going into the derivation of Lorentzian AdS/CFT map, we first discuss
the match of bi-local fields during the Wick rotation, as the building block
of our construction. First, following the convention from the previous
section, we denote the bi-local fields in Euclidean and Lorentzian signature
as $\Psi_{E}(P_{1},P_{2})$ and $\Psi_{L}(P_{1},P_{2})$, respectively. From the
holography point of view, the match of bi-local fields is conceptually
different from the match of bulk fields. But if we just consider the match of
two theories with actions, the matching rule (2) is universal. Therefore,
consider the continuity of the field at the matching surface $\Sigma$, we
immediately obtain the first matching condition, written as

$\Psi_{E}(P_{1},P_{2})|_{\Sigma}=\Psi_{L}(P_{1},P_{2})|_{\Sigma}.$ (111)

We should note that such matching condition is universal and it does not
depend on how we construct the bi-local fields while, for the second matching
condition, one needs to consider the detail structure of the theory. First, we
consider the $O(N)$ vector model in which the bi-local field is defined as

$\Psi(P_{1},P_{2}):=\frac{1}{N}\sum_{I=1}^{N}\phi_{I}(P_{1})\phi_{I}(P_{2}),$
(112)

where $\phi_{I}$ are real scalar fields. Since for each scalar $\phi_{I}$ one
should impose the condition (13), we then obtain the second matching
condition, written as

$\partial_{\tau_{a}}\Psi(P_{1},P_{2})+i\partial_{t_{a}}\Psi_{L}(P_{1},P_{2})=0\qquad{\rm
for}\qquad a=1,2.$ (113)

Then we consider the $U(N)$ vector model in which the bi-local field is given
by

$\Psi(P_{1},P_{2}):=\frac{1}{N}\sum_{I=1}^{N}\phi^{*}_{I}(P_{1})\phi_{I}(P_{2}),$
(114)

where $\phi_{I}$ become complex with the conjugate $\phi^{*}_{I}$ while the
matching condition of the complex conjugate field $\phi^{*}_{I}$ is the
complex conjugate of (13). Therefore, for the $U(N)$ vector model, the second
matching condition now becomes 999We should note that in the bi-local form
$\phi$ and $\phi^{*}$ are not independent if one requires the bi-local field
is hermitian, i.e., $\Phi^{*}(P_{2},P_{1})=\Phi(P_{1},P_{2})$. If we treat
$\phi$ and $\phi^{*}$ independently, there will be no sign difference in the
matching condition as derived in the Appendix C.

$\displaystyle\partial_{\tau_{1}}\Psi(P_{1},P_{2})-i\partial_{t_{1}}\Psi_{L}(P_{1},P_{2})=0$
(115)
$\displaystyle\partial_{\tau_{2}}\Psi(P_{1},P_{2})+i\partial_{t_{2}}\Psi_{L}(P_{1},P_{2})=0,$
(116)

in which one can check that the matching conditions are hermitian in the sense
that we treat 1,2 as the matrix indices. From the above discussion we can see
that the explicit form of the second matching condition depends on the theory
itself and there also exist physical models which do not require the second
matching condition. For example, in SYK model the bi-local field is defined as

$\Psi(P_{1},P_{2}):=\frac{1}{N}\sum_{I=1}^{N}\langle\chi_{I}(P_{1})\chi_{I}(P_{2})\rangle,$
(117)

in which $\chi_{I}$ are Majorana fermions. In this case, the requirement of
the first matching condition is still necessary while there is no second
matching condition since the on shell equation for $\chi_{I}$ is the first
order differential equation. Moreover, one should impose proper reality and
hermiticity condition on the spinor and action, respectively
nicolai1978possible ; van1996euclidean .

#### Derivation

To make the Lorentzian results compatible with the Euclidean case, also for
simplicity, we first define the Lorentzian three-point function basis as

$\displaystyle\langle
O_{1}O_{2}O_{3}\rangle_{L}:=\langle[O_{1},O_{2}]\;O_{3}\rangle+\langle
O_{2}\;[O_{3},O_{1}]\rangle,$ (118)
$\langle\widetilde{O}_{1}\widetilde{O}_{2}\widetilde{O}_{3}\rangle_{L}:=\langle\widetilde{O}_{1}\;[\widetilde{O}_{3},\widetilde{O}_{2}]\rangle+\langle[\widetilde{O}_{2},\widetilde{O}_{1}],\widetilde{O}_{3}]\rangle,$
(119)

in which we can see that Lorentzian three-point function basis are in fact
combinations of three-point functions and its shadow counterpart is not just
the shadow transform of each operator while we should also take the effect of
commutator into consideration. But $\langle O_{1}O_{2}O_{3}\rangle_{L}$ and
$\langle\widetilde{O}_{1}\widetilde{O}_{2}\widetilde{O}_{3}\rangle_{L}$ are
still orthogonal in the sense of (3). Therefore, from the bi-local field
expansion

$\displaystyle\Psi(P_{1},P_{2})_{E/L}=\sum_{J=0}^{\infty}\int_{\gamma_{L}}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}\;\bar{C}_{\Delta,J}^{E/L}(P,D_{Z})\langle
O_{\Delta_{0}}(P_{1}),O_{\Delta_{0}}(P_{2})O_{\Delta,J}(P,Z)\rangle_{E/L},$

we can determine the coefficients $\bar{C}$ to be

$\displaystyle\bar{C}_{\Delta,J}^{E/L}(P,Z)=\frac{N_{\Delta,J}}{2}\int
dX_{1}dX_{1}\;\Phi(P_{1},P_{2})\langle
O_{\widetilde{\Delta}_{0}}(P_{1}),O_{\widetilde{\Delta}_{0}}(P_{2})O_{\widetilde{\Delta},J}(P,Z)\rangle_{E/L}.$
(120)

As for the higher spin fields, we will focus on the study of transverse tensor
fields and the transverse bulk completeness relation is given by

$\delta^{TT}(X_{1},X_{2})(W_{12})^{J}=\int_{\gamma}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}\frac{N_{\Delta,J}}{\alpha_{J}}G_{\Delta,J}(X_{1},P;W_{1},D_{Z})G_{\widetilde{\Delta},J}(X_{2},P;W_{2},Z),$

which comes from the zero spin term of the full completeness relation and
$\alpha_{J}$ are constants that depend on the spin. Therefore, given the off-
shell tensor field expansion

$H(X,W)=\int_{\gamma}\frac{d\Delta}{2\pi
i}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}C_{\Delta,J}(P,D_{Z})G_{\Delta,J}(X,P;W,Z),$
(121)

we can use the bulk completeness relation to deduce the coefficient
polynomial, which is given by

$C_{\Delta,J}(P,Z)=\frac{N_{\Delta,J}}{\alpha_{J}}\frac{1}{(\frac{d-1}{2})_{J}J!}\int
dX\;H(X,K_{W})\;G_{\widetilde{\Delta},J}(X,P;W,Z).$ (122)

Now, we can construct a map between the bulk coefficients $C_{\Delta,J}$ and
the CFT coefficients $\bar{C}_{\Delta,J}$, given by

$\displaystyle
C_{\Delta,J}^{E}(P,Z)=f^{E}_{\Delta,J}\;\bar{C}_{\Delta,J}^{E}(P,Z),\qquad
C_{\Delta,J}^{L}(P,Z)=f^{L}_{\Delta,J}\;\bar{C}_{\Delta,J}^{L}(P,Z),$ (123)

in which $f^{E}_{\Delta,J}$ and $f^{L}_{\Delta,J}$ are functions on the scale
dimension and spin. Although one can propose the AdS/CFT map in Euclidean and
Lorentzian signature separately, we will see that in fact $f^{E}_{\Delta,J}$
and $f^{L}_{\Delta,J}$ are not independent. According to the matching of bi-
local field and the Wick rotation of the completeness relation, we know
$\bar{C}^{E}_{\Delta,J}=\bar{C}^{L}_{\Delta,J}$ while we have
$C_{-}^{E}=C_{-}^{L}$ from the study of higher spin field matching on the
bulk. Taking these into consideration, we can conclude

$f_{\Delta,J}:=f^{E}_{\Delta,J}=f^{L}_{\Delta,J},$ (124)

which tells us that the AdS/CFT map should be invariant during the Wick
rotation. Moreover, we can transfer the map between coefficients into the map
between fields directly with the help of (120) and (122). We can deduce the
CFT to AdS map to be

$\displaystyle H(X,W)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\gamma}\frac{d\Delta}{2\pi
i}f_{\Delta,J}N_{\Delta,J}\int\frac{dP}{J!(\frac{d}{2}-1)_{J}}\int
dP_{1}dP_{2}\;G^{L}_{\Delta,J}(X,P;W,D_{Z})$ (125) $\displaystyle\times$
$\displaystyle\langle
O_{\widetilde{\Delta}_{0}}(P_{1}),O_{\widetilde{\Delta}_{0}}(P_{2})O_{\widetilde{\Delta},J}(P,Z)\rangle_{L}\;\Psi(P_{1},P_{2}),$

and the AdS to CFT map to be

$\displaystyle\Psi(P_{1},P_{2})$ $\displaystyle=$
$\displaystyle\sum_{J=0}^{\infty}\int_{\gamma}\frac{d\Delta}{2\pi
i}\frac{N_{\Delta,J}}{\alpha_{J}\;f_{\Delta,J}}\int\frac{dP}{(\frac{d-1}{2})_{J}(\frac{d}{2}-1)_{J}J!^{2}}$
(126) $\displaystyle\times$ $\displaystyle\int dX\langle
O_{\Delta_{0}}(P_{1}),O_{\Delta_{0}}(P_{2})O_{\Delta,J}(P,Z)\rangle_{L}G^{L}_{\widetilde{\Delta},J}(X,P;K_{W},Z)H(X,W),$

therefore completes our derivation.

Here, we should note that the AdS/CFT map provides us with a machinery to
build connections between bi-local fields and higher spin fields and the
fields could be off-shell and both take values in embedding space. In order to
emphasis that the higher spin fields live on the bulk AdS and the bi-local
fields live on the boundary CFT, we need to specify their physical regions.
That is to say, the physical region for $H$ is the AdS surface $X^{2}=-1$ and
the physical region for $\Psi$ is often taken to be light rays on the cone
$X^{2}=0$. Moreover, for the on-shell AdS/CFT map, i.e, after solving equation
of motion on the physical region, we will see that the on-shell AdS/CFT map is
invertible in the large $N$ limit.

#### Vector Models

Now, we will try to reconstruct the on-shell AdS/CFT map in a more compact way
based on the study of vector models and BTZ solutions. Since equations of
motion for higher spin fields are also second order differential equations, we
write the on shell, for example spin 1 field $H^{A}$, as

$H^{A}(\omega,k,R)=a^{+}(w,k)\;\psi_{+}^{A}(\omega,k,R)+a^{-}(\omega,k)\;\psi_{-}^{A}(\omega,k,R),$
(127)

in which $a_{\pm}(\omega,k)$ are coefficients written in momentum space and
$\psi^{A}_{\pm}$ are the basis of the solution space, which carries the spin
index $A$. Moreover, we should note that such expression will also hold for
the arbitrary spin $J$ field $H^{A_{1}\dots A_{J}}$. Since the dimension of
the solution space is always 2, after fixing the basis, we choose to write the
higher spin fields as a column vector

$H^{A}=\left(\begin{array}[]{ccc}a^{+}\\\ a^{-}\end{array}\right)^{A}.$ (128)

As for the boundary $O(N)$ vector model, following the same argument, we write
the vector fields $\phi_{i}$ for $i=1,\dots N$ in terms of

$\phi_{i}=\bar{a}^{+}\bar{\psi}^{i}_{+}+\bar{a}^{-}\bar{\psi}^{i}_{-},$ (129)

where $\bar{a}^{\pm}$, $\bar{\psi}_{\pm}$ are the coefficients and the basis
of the boundary solution. Moreover, since the bi-local field is the
combination of the scalar $\Psi=\sum_{i}\frac{1}{N}\phi_{i}\phi_{i}$, we write
it into a $2N$ dimensional column vector

$\Psi=\left(\begin{array}[]{ccc}\bar{a}^{+}_{1}\\\ \bar{a}^{-}_{1}\\\
\vdots\\\ \bar{a}_{N}^{+}\\\ \bar{a}_{N}^{-}\end{array}\right).$ (130)

Now, given the CFT to AdS map, one can construct a $2\times 2N$ matrix $T^{A}$

$H^{A}=T^{A}\cdot\Psi,$ (131)

in which $\cdot$ represents the product of matrix and the elements of $T^{A}$
can be uniquely determined by mapping the basis $\bar{\psi}^{i}_{\pm}$ to the
bulk. Therefore, we can define all the spin fields together as an infinite
dimensional vector $H$

$H:=\bigoplus_{J=0}^{\infty}H^{A_{1}\cdots A_{J}}$ (132)

and encode all the information about the CFT to AdS map into a $\infty\times
2N$ matrix $\mathcal{T}$

$\mathcal{T}=\bigoplus_{J=0}^{\infty}T^{A_{1}\dots A_{J}}:\Psi\longmapsto H,$
(133)

in which $T^{A_{1}\cdots A_{J}}$ are $2\times 2N$ matrices representing the
map $\Psi\to H^{A_{1}\dots A_{J}}$. Following similar procedure, one can also
construct the AdS to CFT map as a $2N\times\infty$ matrix, named
$\bar{\mathcal{T}}$, and the AdS-CFT map is invertible
$\mathcal{T}^{-1}=\bar{\mathcal{T}}$provided $N\to\infty$.

Here, we should emphasise the physical meaning of the parameter $N$. It is
shown in Aharony:2020omh  that both the loop Feynman diagrams of the bulk
theory and bi-local field theory are suppressed in the large $N$ limit, which
is crucial for studying the theory in the bulk thus it implies a semi-
classical higher spin theory on the AdS background. In the $\rm AdS_{3}$ case,
one can further see this explicitly by recasting the higher spin field into
the ${\rm SL}(2,\mathbb{R})\times{\rm SL}(2,\mathbb{R})$ Chern-Simons theory
achucarro1986chern ; witten19882+ , after comparing to the Einstein-Hilbert
action, then obtain the relation $G_{N}\sim 1/N$ henneaux2010nonlinear ;
campoleoni2010asymptotic ; gaberdiel2011ads . Therefore, we see that gravity
effect is indeed weak in the large $N$ limit and the semi-classical
approximation is reliable.

These tell us that the bulk theory is only well defined in the large $N$ limit
even though the AdS/CFT map also works in the finite $N$ region. The AdS/CFT
map of finite $N$ will give us insights to study the quantum gravity
definition of the higher spin theory or the back reaction of the bulk
geometry. Moreover, the lose of energy of the bulk geometry in the large $N$
limit will finally leads to a low energy bulk theory even though more spin
fields are excited. Again as for the $\rm AdS_{3}$, instead of considering the
change of energy resulting form the back reaction, they 101010In fact, the
boundary CFTs they are considering are the WZW coset models and the $O(N)$
vector models are regarded as UV or IR fixed points. In addition to the
parameter $N$, there is also an integer parameter $k$ which describes the
level of WZW model. These two parameters together form the ’t Hooft coupling,
corresponding to the parameter $N$ and $\Delta$ in the bulk.  introduce a pair
of massive complex scalar fields in addition to the higher spin fields and
their asymptotic behavior $\Delta$ plays the role of parameter $\alpha$ in the
standard AdS/CFT literature gaberdiel2011ads .

##  4 Match Conditions for the Quadratic Action

As it was discussed in Aharony:2020omh , the AdS/CFT map mainly works for the
off-shell field while the matching condition introduced in section 2 is
applied to the on-shell field. In fact, on-shell fields belong to the subset
of the off-shell fields. The expression (121) of the off-shell fields is
basically the expansion of an arbitrary function by the given basis
$G_{\Delta,J}$ thus all the information of a physical systems is encoded in
the coefficients $C_{\Delta,J}$. Suppose that the mass of the spin $J$ field
is given by the scale dimension $\Delta_{J}$, although it is hard to solve
$C_{\Delta,J}$ directly, we can deduce the coefficient will take the form

$C_{\Delta,J}(P)\rightarrow\delta(\Delta-\Delta_{J})C_{\Delta_{J}}(P),$ (134)

in which $C_{\Delta_{J}}(P)$ is the source of the spin $J$ field. Therefore,
we can see that the mass spectra of the theory is determined by the pole
structure of these coefficients $C_{\Delta,J}$, which will give us the AdS/CFT
dictionary, together with the CFT coefficients $\bar{C}_{\Delta,J}$. So in
order to complete the construction of Lorentzian AdS/CFT map, one needs to
specify the matching conditions for the off-shell field.

###  4.1 The Quadratic Action

The match condition for a generic off-shell field is hard to study without
specifying the equation of motion or the action for a given theory. In this
article we will focus on the study of a special class of off-shell fields
which govern the higher spin field theory dual to the vector model on the
boundary and they are described by the quadratic action shown in
deMelloKoch:2018ivk ; Aharony:2020omh . For simplicity, we just consider the
scalar action $S_{E}$

$\displaystyle
S_{E}=\frac{1}{2}\int_{M_{E}}\;\sqrt{G}(\partial^{\mu}\partial^{\nu}\Phi\;\partial_{\mu}\partial_{\nu}\Phi+M_{1}^{2}\;\partial_{\mu}\Phi\;\partial^{\mu}\Phi+M_{2}^{2}\;\partial_{\nu}\Phi\;\partial^{\nu}\Phi+M_{1}^{2}M_{2}^{2}\;\Phi^{2})$
(135)

in which $\Phi$ is the off-shell scalar field propagating in the manifold
$M_{E}$ with AdS background described by the metric $G$ while $M_{1}$ and
$M_{2}$ are mass of the particles. Here we should note that they are spin
dependent parameters coming from the pole structure of $C_{\Delta,J}$ and
fixed in our context since we are dealing with scalar fields setting $J=0$.
Those off-shell scalar fields are not totally free while they should obey the
equation of motion given by the action $S_{E}$, the variation is given by

$\displaystyle\delta S_{E}$ $\displaystyle=$
$\displaystyle\int_{M_{E}}\sqrt{G}(\partial^{\mu}\partial^{\nu}\Phi\;\partial_{\mu}\partial_{\nu}\delta\Phi+M_{1}^{2}\;\partial_{\mu}\Phi\;\partial^{\mu}\delta\Phi+M_{2}^{2}\;\partial_{\nu}\Phi\;\partial^{\nu}\delta\Phi+M_{1}^{2}M_{2}^{2}\;\Phi\;\delta\Phi)$
(137) $\displaystyle=$
$\displaystyle\int_{\partial{M_{E}}}\sqrt{G}\left(\left(2\partial_{t}\;\partial^{i}\partial_{i}\Phi+M_{1}^{2}\;\partial_{t}\Phi+M_{2}^{2}\;\partial_{t}\Phi\right)\delta\Phi+\partial^{t}\partial_{t}\Phi\;\delta\partial_{t}\Phi\right)$
$\displaystyle+\int_{M_{E}}\left(\partial^{\nu}\partial^{\mu}\sqrt{G}\;\partial_{\nu}\partial_{\mu}\Phi-
M_{1}^{2}\partial_{\mu}\sqrt{G}\;\partial^{\mu}\Phi-
M_{2}^{2}\partial^{\nu}\sqrt{G}\;\partial_{\nu}\Phi+\sqrt{G}\;M_{1}^{2}M_{2}^{2}\Phi\right)\delta\Phi$

in which the first term results from the boundary $\partial M_{E}$ and the
second term gives rise to the equation of motion written as

$\displaystyle\frac{1}{\sqrt{G}}(\partial^{\mu}\partial^{\nu}\sqrt{G}\;\partial_{\nu}\partial_{\mu}\Phi-
M_{1}^{2}\partial_{\mu}\sqrt{G}\;\partial^{\mu}\Phi-
M_{2}^{2}\partial^{\nu}\sqrt{G}\;\partial_{\nu}\Phi)+M_{1}^{2}M_{2}^{2}\Phi$
$\displaystyle=$ $\displaystyle 0.$ (138)

Given that the AdS/CFT map is also valid in Lorentzian signature, the
Lorentzian version of the action $S_{L}$ should exist and can be deduced to
take the form of

$\displaystyle
S_{L}=-\frac{1}{2}\int_{M_{L}}\;\sqrt{-G}(\partial^{\mu}\partial^{\nu}\Phi\;\partial_{\mu}\partial_{\nu}\Phi+M_{1}^{2}\;\partial_{\mu}\Phi\;\partial^{\mu}\Phi+M_{2}^{2}\;\partial_{\nu}\Phi\;\partial^{\nu}\Phi+M_{1}^{2}M_{2}^{2}\;\Phi^{2}).$
(139)

The matching condition of the total action $iS_{L}-S_{E}$ at the joint surface
$\Sigma=\partial M_{E}=-\partial M_{L}$ is then deduced to be

$\displaystyle
i\left(2\partial_{t}\;\partial^{i}\partial_{i}\Phi_{L}+M_{1}^{2}\;\partial_{t}\Phi_{L}+M_{2}^{2}\;\partial_{t}\Phi_{L}\right)\delta\Phi_{L}+\left(2\partial_{\tau}\;\partial^{i}\partial_{i}\Phi_{E}+M_{1}^{2}\;\partial_{\tau}\Phi_{E}+M_{2}^{2}\;\partial_{\tau}\Phi_{E}\right)\delta\Phi_{E}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
i\partial^{2}_{t}\Phi_{L}\;\delta\partial_{t}\Phi_{L}-\partial^{2}_{\tau}\Phi_{E}\;\delta\partial_{\tau}\Phi_{E}$
$\displaystyle=$ $\displaystyle 0,$

in which we have used the contraction relation
$\partial_{t}^{2}=-\partial^{t}\partial_{t}$ and
$\partial_{\tau}^{2}=\partial^{\tau}\partial_{\tau}$. Moreover, if we impose
the condition that the charge $\partial^{i}\partial_{i}\Phi$ is conserved at
the joint surface

$i\partial_{t}\partial^{i}\partial_{i}\Phi_{L}+\partial_{\tau}\partial^{i}\partial_{i}\Phi_{E}=0.$
(140)

Then the matching conditions can be simplified to

$\displaystyle\Phi_{E}-\Phi_{L}=0,$ (141) $\displaystyle
i\partial_{t}\Phi_{L}+\partial_{\tau}\Phi_{E}=0,$ (142)
$\displaystyle\partial^{2}_{t}\Phi_{L}+\partial^{2}_{\tau}\Phi_{E}=0,$ (143)

so called offshell matching conditions even though $\Phi_{E}$, $\Phi_{L}$ now
satisfy the quadratic equation (138). Moreover, after rewriting the equation
into the form of

$(\nabla^{2}_{AdS}-M^{2}_{1})(\nabla^{2}_{AdS}-M_{2}^{2})\Phi(X)=0$ (144)

with the $\nabla_{AdS}$ on AdS background. Then one can obtain the solution in
terms of the bulk boundary propagator written as

$\Phi(X)=\int dP\;\frac{1}{(X\cdot P)^{\Delta_{1}}}\;C_{\Delta_{1}}(P)+\int
dP\frac{1}{(X\cdot P)^{\Delta_{2}}}\;C_{\Delta_{2}}(P),$ (145)

in which $\Delta_{1}$ and $\Delta_{2}$ are given by the relation

$M_{1}^{2}=\Delta_{1}(\Delta_{1}-d),\qquad
M_{2}^{2}=\Delta_{2}(\Delta_{2}-d).$ (146)

$C_{\Delta_{1}}(P)$ and $C_{\Delta_{2}}(P)$ can be treated as coefficients or
sources associated to the mode of scale dimension $\Delta_{1},\Delta_{2}$.
Therefore one can check the offshell match conditions (141) and (142) are
solved by the matching of coefficients

$C_{\Delta_{1}}^{E}=C_{\Delta_{1}}^{L},\qquad
C_{\Delta_{2}}^{E}=C_{\Delta_{2}}^{L}$ (147)

and the condition (143) is then automatically preserved, which is shown in the
appendix D. In fact, since we have seen that the offshell solution is the
linear combination of two onshell fields, the offshell matching condition
(140) can be rewritten in term of $\partial^{3}_{t},\partial^{3}_{t}$ after
using the onshell equation of motion for each propagator with mass
$M_{1},M_{2}$.

Now we come back to the duality between the higher spin theory and the vector
model. As it is known that the higher spin theory in the bulk is described by
the Vasiliev equations and the corresponding action is still absent. Thus,
with the help of AdS/CFT map, it is possible for us to reconstruct the bulk
action from the action of boundary free vector model deMelloKoch:2018ivk ;
Aharony:2020omh . This is done by first decomposing the boundary action into
the coefficients $\bar{C}_{\Delta,J}$ then mapping it to bulk coefficients
$C_{\Delta,J}$ via the AdS/CFT map (123) therefore rewriting the action in
terms of the bulk fields. The spectra is determined by the pole structure of
$C_{\Delta,J}$ as shown in (134). It turns out that the bulk action is
quadratic consisting of higher spin fields and the scalar part is described by
(135) while the mass $M_{1}$ and $M_{2}$ are given by

$\Delta_{1}=d,\qquad\Delta_{2}=d-2.$ (148)

From the the above procedure, we can see that one of the mode described by
$C_{\Delta_{1}}$ is physical and the mode described by $C_{\Delta_{2}}$ with
negative mass $M^{2}_{2}<0$ is unphysical based on the observation that it
will contribute to the offshell field $\Phi$ in a negative way
$C_{\Delta_{2}}<0$. Such unphysical modes are identified as ghost modes coming
from the gauge fixing of the higher spin fields. Moreover, from the match
condition (147), one can see that the physical and unphysical modes will
behave independently during the Wick rotation.

Furthermore, as pointed out in the work Skenderis:2009nt ; Skenderis:2009kd
during the study of three-dimensional Einstein gravity, the higher-derivative
terms in the action will introduce extra propagating degrees of freedom and
they are identified as ghost modes. Those ghost modes make the theory unstable
and violate the unitary condition. Such problem is rescued by considering the
topologically massive gravity at the chiral point so that the left-moving
sector will be gauge fixed and the extra degrees of freedom are then
eliminated. Here for the higher spin fields, to make the theory physical, one
can also choose to set the negative modes to zero in Euclidean signature by
fixing the gauge of the higher spin fields , i.e $C_{\Delta_{2}}^{E}=0$. This
naturally tells us that the ghost modes will not contribute to the external
legs of the Feynman diagrams since the source is turned off in real time by
checking the match condition (147). However, the ghost modes will contribute
to the Feynman diagram at the loop level while the detail goes beyond the
study of classical match condition in this article.

###  4.2 Analyticity

At first sight, it looks confusing since, starting from different actions, we
eventually arrive at the same matching condition (147) . As we will see, this
becomes nature if one considers the behavior of the field and propagator over
the whole complex time plane. The analytic property of the propagator
$G_{\Delta}(X,P)$ implies that the condition (147) could be universal for
arbitrary higher order matching.

The complex time plane is described by the variable $z=t+i\tau$,
$\bar{z}=t-i\tau$ and the function on the complex plane is denoted as

$\Phi_{C}(z,\bar{z})=\Phi_{C}(t,\tau).$ (149)

o$t$$\tau$$\Phi_{C}(t,\tau$)$\Phi_{E}(-\tau)$$\Phi_{L}(t)$ Figure 6: The
complexification of the field $\Phi_{E}$ and $\Phi_{L}$ into a function
$\Phi_{C}$ with complex variables is depicted.

Although the complex time $t+i\tau$ will not make sense physically and we are
just interested in the Euclidean field $\Phi_{E}(\tau)$ and the Lorentzian
field $\Phi_{L}(t)$, we can still treat these two fields as living on the
boundary of the complex field $\Phi_{C}(t,\tau)$. More precisely, as
illustrated in Fig.6 we can impose the condition

$\Phi_{C}(t,0)=\Phi_{L}(t),\qquad\Phi_{C}(0,\tau)=\Phi_{E}(-\tau),$ (150)

which specify the boundary value of $\Phi_{C}$ while the value in the interior
is still unknown. But after taking the first order matching condition (142)
into consideration

$\partial_{t}\Phi_{L}(t)-i\partial_{\tau}\Phi_{E}(\tau)=\partial_{\bar{z}}\Phi_{C}(t,\tau),$
(151)

we find that a sufficient condition for the complex field is the analyticity
,i.e, an analytic complex function will naturally induced a pair of field
$\Phi_{L}$ and $\Phi_{E}$ which satisfy the first order matching. We can also
see that the second order matching condition (143) can be written as
$\partial_{z}\partial_{\bar{z}}\Phi_{C}=0$ thus now becomes trivial.
Furthermore, using the relation

$\partial_{t}=\frac{\partial_{z}+\partial_{\bar{z}}}{2},\qquad\partial_{t}=i\frac{\partial_{z}-\partial_{\bar{z}}}{2},$
(152)

one can write the $\partial_{t}$, $\partial_{\tau}$ into the derivative
$\partial_{z}$ and $\partial_{\bar{z}}$ and check that the third order
matching condition (140) becomes trivial provided
$\partial_{\bar{z}}\Phi_{C}=0$.

Therefore, we can propose that the higher order matching condition are
generated by the derivatives $\partial_{z}$, $\partial_{\bar{z}}$ acting on
the complex function and the matching condition for arbitrary higher order
action is

$\partial_{\bar{z}}\Phi_{C}(z,\bar{z})=0.$ (153)

This assumption is reasonable since as for the boundary quantum field theory,
we have the analytic Wightman functions so we should expect to get analytic
fields on the bulk via the help of holography principle. Although the Wightman
functions are characterised by a series of axioms osterwalder1973axioms ;
osterwalder1975axioms  and here we start from the study of variation of the
action $\delta S$.

##  5 Discussion

In this paper, we have discussed various aspects of AdS/CFT map between the
higher spin fields in the bulk and the bi-local fields on the boundary. We
obtain the Lorentzian AdS/CFT map by doing the Wick rotation of higher spin
fields, bi-local fields and the CFT completeness relation at the same time.
Moreover, we should emphasize that the matching conditions are imposed at the
linear and classic level and the complete matching for quantum gravity is
still absent. The direct quantization of the two matching conditions we have
determined in this article depends on the matching of spectra and energy
between two signatures. Furthermore, for a complete match of field theories at
quantum level, more sophisticated conditions should be imposed and one can see
the construction in the recent work kontsevich2021wick .

Comparing to the Euclidean AdS/CFT map, though more complicated, the
Lorentzian AdS/CFT map provides us with a powerful tool to study the real time
duality, especially the study of black holes. Via the map, one can obtain the
classic solution of the higher spin field in the bulk by dealing with the
boundary CFT in the bi-local form, which is easier to work with since the
boundary CFT is usually well understood.

As for the boundary CFT, we have discussed the matching conditions for the
$O(N)$ vector model, $U(N)$ vector model, and the SYK model in the context of
vacuum-vacuum contour. It will be interesting to investigate how other
contours, like thermal Christodoulou:2016nej ; Pantelidou:2022ftm ;
Pantelidou:2023pjn  or out of time order contour, work in these theories and
then possibly calculate the corresponding correlation functions.

As we have discussed, both the mathematical structure and physical
interpretation of the matching of the pole structure which show up on two
sides of the AdS/CFT map is still unclear. It will be interesting to
investigate this in some well studied models, for example, the duality between
the JT gravity and the SYK model or the duality between the Higher spin on BTZ
background and coset WZW models on the boundary.

Moreover, the study of AdS/CFT map of finite $N$ may lead to a definition of
quantum higher spin theory in the bulk since the theory of finite $N$ on the
boundary is well defined. After getting the solution of the boundary theory of
finite $N$, one can directly substitute it into AdS/CFT map the obtain a
function on the bulk. This solution will carry the quantum information of the
bulk theory, one can treat it as the solution of the bulk equation will some
proper quantum corrected effective potential or use it to deduce the
background then study the back reaction of the bulk geometry.

As for fermion fields, the matching conditions will become more subtle since
one needs to consider the representation of the symmetry group in different
signatures and fermions may transform in different ways, which makes it hard
to impose the condition at the joint surface. Moreover, given a Riemannian
manifold $X$ with boundary $Y=\partial X$, one can classify the anomaly via
studying the Dirac index by gluing a tube $Y\times\mathbb{R}_{+}$ to the
boundary, which leads to the APS index theorem atiyah1975spectral ;
witten2016fermion , while recently the theorem is formulated in the Lorentzian
manifold bar2019index ; bar2020local . In our case, to prepare a Hartle-
Hawking state, we need to study the Dirac operator on the Euclidean manifold
with boundary but instead we should glue the Lorentzian manifold to the
boundary $Y$. It is worthwhile to explore what the matching condition is in
order to make the Dirac operator well defined.

## Acknowledgments

I would like to thank my father Qinghe Hao and mother Xiulan Xu for providing
funding for the tuition and accommodation fees when studying his PhD at the
University of Southampton. I also wish to thank Marika Taylor for the guidance
throughout this project. I am grateful to Enrico Parisini and Kostas Skenderis
for insightful discussions on ambient space and real time holography.

## Appendix

##  Appendix A Embedding space

###  A.1 Bulk embedding space

In this section we review the embedding of hyperbolic space and AdS space into
flat space in one dimension higher, focussing in particular on the differences
between Euclidean AdS (hyperbolic space) and Lorentzian AdS.

We view $H^{d+1}$ as a spacelike surface in $R^{d+1,1}$. The coordinates of
$R^{d+1,1}$ are denoted as $X=(X^{+},X^{-},X^{\mu})$ with $\mu=0,\cdots(d-1)$,
with the metric being:

$ds^{2}=-dX^{-}dX^{+}+\delta_{\mu\nu}dX^{\mu}dX^{\nu}$ (A.154)

Then $H^{d+1}$ can be embedded as the spacelike surface $X^{2}=-1$, which can
be parameterized in Poincaré coordinates as

$X=\frac{1}{z}\left(1,z^{2}+y^{2},y^{\mu}\right).$ (A.155)

Note that as $z\rightarrow 0$ the embedding approaches $\frac{1}{z}P$ where
$P$ is null.

Lorentzian AdSd+1 is embedded into $R^{d,2}$ as follows. The coordinates of
$R^{d,2}$ are denoted as $X=(X^{+},X^{-},X^{\mu})$ with $\mu=0,\cdots(d-1)$,
with the metric being:

$ds^{2}=-dX^{-}dX^{+}+\eta_{\mu\nu}dX^{\mu}dX^{\nu}$ (A.156)

AdSd+1 can be embedded as the surface $X^{2}=-1$ with signature (d,1), which
can again be parameterized in Poincaré coordinates as (A.155) but with the
induced metric now being

$ds^{2}=\frac{1}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dy^{\mu}dy^{\nu}\right).$
(A.157)

To distinguish different time directions it is convenient to use $t$ and
$\tau$ to label the Lorentzian and Euclidean time directions; they share the
same coordinate expression $t,\tau=y^{0}=X^{0}/X^{+}$. We will find it
convenient to treat $X^{0}$ as a complex number with $t$ and $\tau$ the real
and imaginary part, respectively. In the embedding space the derivatives with
respect to $t$ and $\tau$ can be expressed as

$\displaystyle\frac{\partial}{\partial t}=\frac{\partial X^{\mu}}{\partial
t}\frac{\partial}{\partial X^{\mu}}=-\frac{2t}{z}\frac{\partial}{\partial
X^{-}}+\frac{1}{z}\frac{1}{\partial X^{0}}=-2X^{0}\frac{\partial}{\partial
X^{-}}+X^{+}\frac{\partial}{\partial X^{0}},$ (A.158)
$\displaystyle\frac{\partial}{\partial\tau}=\frac{\partial
X^{\mu}}{\partial\tau}\frac{\partial}{\partial
X^{\mu}}=\frac{2\tau}{z}\frac{\partial}{\partial
X^{-}}+\frac{1}{z}\frac{1}{\partial X^{0}}=2X^{0}\frac{\partial}{\partial
X^{-}}+X^{+}\frac{\partial}{\partial X^{0}}.$ (A.159)

###  A.2 Conformal light cone

In this section we summarise the embedding of flat space into the conformal
light cone of a flat space in two dimensions higher, focussing on the
differences between Euclidean and Lorentzian flat space.

We denote the coordinates of the embedding space as $P=(P^{+},P^{-},P^{\mu})$
with $\mu=0,\cdots(d-1)$. In the case of Euclidean $d$-dimensional flat space
the metric on the embedding space $R^{d+1,1}$ is

$ds^{2}=-dP^{+}dP^{-}+\delta_{\mu\nu}dP^{\mu}dP^{\nu}$ (A.160)

The relation between the embedding coordinates and the flat space coordinates
$x^{\mu}$ is determined by the conformal light cone conditions:

$P^{2}=0\qquad P=\lambda P$ (A.161)

which are solved by $P=(1,x^{2},x^{\mu})$ with $x^{2}=x^{\mu}x_{\mu}$.

For Lorentzian $d$-dimensional flat space the metric on the embedding space
$R^{d,2}$ is

$ds^{2}=-dP^{+}dP^{-}+\eta_{\mu\nu}dP^{\mu}dP^{\nu}$ (A.162)

The relation between the embedding coordinates and the flat space coordinates
$x^{\mu}$ is still determined by the conformal light cone conditions (A.161).

###  A.3 Solutions

Here we briefly present the solutions that are used in section 2.2 and one can
see more detail in Skenderis:2008dg . Given the real time action on
$AdS_{d+1}$ background

$S=-\frac{1}{2}\int
d^{d+1}x\sqrt{-G}(\partial_{\mu}\Phi\partial^{\mu}\Phi+m^{2}\Phi^{2})$ (A.163)

with the metric $G$ in A.157, we have the equation of motion

$z^{d+1}\partial_{z}(z^{-d+1}\partial_{z}\Phi)+z^{2}\square_{0}\Phi-m^{2}\Phi=0$
(A.164)

in which $\square_{0}$ is the Laplacian corresponding to the induced metric at
$z=0$. In momentum space $q=(\omega,\vec{k})$, the solutions take the form

$e^{-i\omega t+\vec{k}\cdot\vec{x}}z^{d/2}K_{l}(qz),\qquad e^{-i\omega
t+\vec{k}\cdot\vec{x}}z^{d/2}I_{l}(qz),$ (A.165)

in which $K_{l}$, $I_{l}$ are two types of Bessel functions.
$\Delta=\frac{d}{2}+l$ is defined as $m^{2}=\Delta(\Delta-d)$. For spacelike
momenta $q^{2}>0$ these modes are well behaved, while for timelike momenta,
one need to specify how the contour of $\omega$ winds around the branch cuts
of $q=\sqrt{q^{2}}$. Usually we use the $i\epsilon$ prescription

$q_{\epsilon}=\sqrt{-\omega^{2}+{\vec{k}}^{2}-i\epsilon}$ (A.166)

to specify that we are applying the Feynman contours $C$ over the complex
$\omega$ plane. Moreover, the solutions behave like

$z^{\frac{d}{2}}K_{l}\sim\frac{z^{d/2-l}}{q^{l}},\qquad
z^{\frac{d}{2}}I_{l}\sim\frac{z^{\frac{d}{2}+l}}{q^{-l}}$ (A.167)

for $z\longrightarrow 0$ and

$z^{\frac{d}{2}}K_{l}\sim\sqrt{\frac{z^{d-1}}{q}}e^{-qz},\qquad
z^{\frac{d}{2}}I_{l}\sim\sqrt{\frac{z^{d-1}}{q}}e^{qz}$ (A.168)

when $z\longrightarrow\infty$. According to their asymptotics behavior,
$K_{l}$ are called source modes and contributes to the bulk-boundary
propagator $X_{L}$ as

$X_{L}(t,\vec{x},z)=\frac{1}{(2\pi)^{d}}\int_{C}d\omega\int
d\vec{k}e^{-i\omega
t+i\vec{k}\cdot\vec{x}}\frac{2^{l+1}q^{l}_{\epsilon}}{\Gamma(l)}z^{\frac{d}{2}}K_{l}(q_{\epsilon}z),$
(A.169)

which leads to the familiar space time expression

$X_{L}=i\Gamma(l)\Gamma(l+\frac{d}{2})\pi^{-\frac{d}{2}}\frac{z^{l+\frac{d}{2}}}{(-t^{2}+\vec{x}^{2}+z^{2}+i\epsilon)^{l+\frac{d}{2}}}.$
(A.170)

As for $I_{l}$, they are called normalizable modes and we should note they
will become infinite at $z=\infty$ when $q$ is space like. Therefore, we only
consider the time like contribution

$Y_{L}(t,\vec{x},z)=\frac{1}{(2\pi)^{d}}\int_{C}d\omega\int
d\vec{k}e^{-i\omega
t+i\vec{k}\cdot\vec{x}}\;\theta(-q^{2})b(\omega,\vec{k})\;z^{\frac{d}{2}}J_{l}(|q|z),$
(A.171)

in which $b(\omega,\vec{k})$ are undetermined coefficients and $J_{l}$ is
defined as $I_{l}(-i|q|z)=e^{-i\pi l/2}J_{l}(|q|z)$. One can obtain the
Euclidean version by doing the Wick rotation $t=-i\tau$. For source
contribution we have

$X_{E}=i\Gamma(l)\Gamma(l+\frac{d}{2})\pi^{-\frac{d}{2}}\frac{z^{l+\frac{d}{2}}}{(\tau^{2}+\vec{x}^{2}+z^{2}+i\epsilon)^{l+\frac{d}{2}}}$
(A.172)

and the normalizable term is

$Y_{E}(t,\vec{x},z)=\frac{1}{(2\pi)^{d}}\int_{C}d\omega\int
d\vec{k}e^{-|\omega\tau|+i\vec{k}\cdot\vec{x}}\;\theta(-q^{2})b(\omega,\vec{k})\;z^{\frac{d}{2}}J_{l}(|q|z),$
(A.173)

in which we have chosen the positive frequency when $\tau>0$. After lifting
these to the embedding space, one should be able to recover the expressions
shown at the beginning of section 2.2 by making

$G_{\Delta}=X+Y.$ (A.174)

##  Appendix B Solving BTZ

We start from the Klein-Gordon equation for the scalar field $\Phi$, written
as

$\nabla^{2}\Phi-m^{2}\Phi=0,$ (B.175)

in which the Laplacian operator $\nabla^{2}$ has the form of

$\nabla^{2}\Phi=\frac{1}{\sqrt{-G}}\partial_{\mu}(\sqrt{-G}G^{\mu\nu}\partial_{\nu}\Phi).$
(B.176)

After using the BTZ black hole metric (77), we obtain the equation of motion
in terms of the $(r,t,\phi)$ coordinates

$-\frac{1}{r^{2}-1}\partial_{t}^{2}\Phi+\frac{1}{r^{2}}\partial_{\phi}^{2}\Phi+(r^{2}-1)\partial_{r}^{2}\Phi+\frac{3r^{2}-1}{r}\partial_{r}\Phi-\Delta(\Delta-2)=0,$
(B.177)

in which we make $m^{2}=\Delta(\Delta-2)$. The modes of the above equation are
given by

$\psi=e^{i\omega t-ik\phi}f_{\Delta}(\omega,k,r),$ (B.178)

in which $\omega,k$ tell us how the modes propagate along the circle and
$f_{\Delta}(\omega,k,r)$ is given by

$f_{\Delta}(\omega,k,r)=C_{\omega
k\Delta}\left(1-\frac{1}{r^{2}}\right)^{-\frac{i\omega}{2}}r^{-\Delta}H(\frac{1}{r^{2}})$
(B.179)

where $C_{\omega k\Delta}$ are normalization constants while the function
$H(\frac{1}{r^{2}})$ is determined by Euler’s hypergeometric differential
equation

$z(1-z)H^{\prime\prime}+(\Delta-(\Delta+1-i\omega)z)H^{\prime}-\frac{1}{4}(\Delta-i(\omega-k))(\Delta-i(\omega+k))H=0,$
(B.180)

where we have $z=\frac{1}{r^{2}}$ and the solutions are built from
hypergeometric functions $F(a,b;c;z)$. Therefore in this case we have

$a=\frac{\Delta}{2}-\frac{i}{2}(\omega+k),\qquad
b=\frac{\Delta}{2}-\frac{i}{2}(\omega-k),\qquad c=\Delta=l+1.$ (B.181)

Note that hypergeometric functions are locally expressed as power series and
it converges when $|z|<1$ while the function over the whole complex $z$ plane
can be obtained by the analytic continuation. Moreover, for physical systems,
$\omega$ and $k$ are usually nonintegral thus the form of the solutions are
mainly determined by the value of $\Delta$. Now, we discuss the solutions in
two cases.

i) None of the numbers $c$, $c-a-b$; $a-b$ is equal to an integer.

In this case, solutions can be expanded as combination of two independent
power series at each singular point $z=0,1,\infty$. Here we will only consider
the behavior of the solution around the horizon $z=1$ and the infinity $z=0$.
At $z=1$, solutions depend on the frequency $\omega$ and two independent
solutions are given by

$H_{-\omega}(z)=F(\;\frac{\Delta}{2}-\frac{i}{2}(\omega+k),\;\frac{\Delta}{2}-\frac{i}{2}(\omega-k);\;1-i\omega;1-z)$
(B.182)

and

$H_{\omega}(z)=(1-z)^{i\omega}F(\;\frac{\Delta}{2}+\frac{i}{2}(\omega-k),\;\frac{\Delta}{2}+\frac{i}{2}(\omega+k);\;1+i\omega;1-z).$
(B.183)

Taking these two into consideration at the same time, we can write the modes
as

$\psi_{\pm}=e^{\pm i\omega t-ik\phi}f_{\Delta}(\pm\omega,k,r),$ (B.184)

in which

$f_{\Delta}(\pm\omega,k,r)=C_{\omega
k\Delta}\left(1-\frac{1}{r^{2}}\right)^{-\frac{i\omega}{2}}r^{-\Delta}H_{\pm\omega}(\frac{1}{r^{2}})$
(B.185)

At $z=0$, we also have two independent solutions while they now depend on the
scale dimension $\Delta$ and we write them as

$H_{\Delta^{+}}(z)=F(\;\frac{\Delta}{2}-\frac{i}{2}(\omega+k),\;\frac{\Delta}{2}-\frac{i}{2}(\omega-k);\;\Delta;z)$
(B.186)

and

$H_{\Delta^{-}}(z)=z^{1-\Delta}F(\;1-\frac{\Delta}{2}-\frac{i}{2}(\omega+k),\;1-\frac{\Delta}{2}-\frac{i}{2}(\omega-k);\;2-\Delta;z).$
(B.187)

Similar to the $z=1$ case, we can write the modes as

$\psi_{\pm}=e^{i\omega t-ik\phi}f_{\Delta^{\pm}}(\omega,k,r),$ (B.188)

in which

$f_{\Delta^{\pm}}(\omega,k,r)=C_{\omega
k\Delta}\left(1-\frac{1}{r^{2}}\right)^{-\frac{i\omega}{2}}r^{-\Delta}H_{\Delta^{\pm}}(\frac{1}{r^{2}}).$
(B.189)

ii) $c=\Delta=l+1$ is an integer for $l=1,2,3,\dots$

In this case, we will try to get the solutions of the equation from two ways.
From one hand, we can apply the formula in abramowitz1988handbook  for integer
$l$ directly and then obtain the fundamental system of the solution, given by

$H_{1(0)}(z)=F(a,b;l+1,z),$ (B.190)

and

$\displaystyle H_{2(0)}(z)=F(a,b;l+1;z)\;{\rm
ln}z+\sum_{n=1}^{m}\frac{(a)_{n}(b)_{n}}{(1+l)_{n}n!}z^{n}(\psi(a+n)-\psi(a)+\psi(b+n)$
(B.191)
$\displaystyle-\psi(b)-\psi(l+1+n)+\psi(l+1)-\psi(n+1)+\psi(1))-\sum_{n=1}^{\infty}\frac{(n-1)!(-m)_{n}}{(1-a)_{n}(1-b)_{n}}z^{-n}.$
(B.192)

These are two independent solutions at $z=0$ and one can obtain the solution
at $z=1,\infty$ through the analytic continuation.

From the other hand, we can treat the integer case $\Delta=l+1$ as the limit
of the general case $\Delta=l+1+\delta$ after taking $\delta\to 0$. This
allows us to use the solutions for non-integer $\Delta$ thus their physical
meanings are retained. To take the $\delta\to 0$ limit, we need to take care
of the coefficients of the power series since they may have poles at integer
$\Delta$. Taking the solution $H_{-\omega}$ for example. If we expand
$H_{-\omega}$ around $z=0$ as power series, the term $\Gamma(k-l)$ will show
up in the denominator of the coefficients of $z^{k}$ which leads to the poles
when $k-l=1$. In fact, we can resolve those poles by choosing the
hypergeometric function transformation $z\to 1-z$ for integer $\Delta$ then
get

$\displaystyle
H_{-\omega}(z)=\frac{\Gamma(l)\Gamma(a+b-l)}{\Gamma(a)\Gamma(b)}z^{-l}\sum_{n=0}^{l-1}\frac{(a-l)_{n}(b-l)_{n}}{n!(1-m)_{n}}z^{n}-\frac{(-1)^{l}\Gamma(a+b-l)}{\Gamma(a-l)\Gamma(b-l)}$
(B.193)
$\displaystyle\times\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{n!(m+n)!}z^{-n}\left({\rm
ln}z-\psi(n+1)-\psi(n+l+1)+\psi(a+n)+\psi(b+n)\right),$ (B.194)

from which we can see that the infinite term in the coefficients are
transformed to the function ${\rm ln}z$. Given the above solution, we choose
the normalization constants to be

$C_{\pm\omega
kl}=\frac{\Gamma(\frac{1}{2}(l+1)+\frac{i}{2}(\pm\omega-k))\Gamma(\frac{1}{2}(l+1)+\frac{i}{2}(\pm\omega+k))}{\Gamma(l)\Gamma(1\pm
i\omega)}$ (B.195)

so that the coefficient of the leading term $r^{l-1}$ in $\psi_{\pm}$ turns to
be 1. Therefore, the solution can be written as

$\psi=e^{\pm i\omega
t-ik\phi}(r^{l-1}+\cdots+\alpha(\pm\omega,k,l)r^{-l-1}[{\rm
ln}(r^{2})+\beta(\pm\omega,k,l)]+\cdots),$ (B.196)

in which we just show the $r^{l-1}$ and $r^{-l-1}$ term and the coefficients
$\alpha$, $\beta$ are given by

$\alpha(\pm\omega,k,l)=(-1)^{l}\frac{(\frac{i}{2}(\pm\omega+k)+\frac{1}{2}(1+l))_{l}(\frac{i}{2}(\pm\omega-k)+\frac{1}{2}(1+l))_{l}}{l!(l-1)!},$
(B.197)
$\beta(\pm\omega,k,l)=-\psi(\frac{i}{2}(\pm\omega+k)+\frac{1}{2}(1+l))-\psi(\frac{i}{2}(\pm\omega-k)+\frac{1}{2}(1+l))+\psi(1)+\psi(l+1).$
(B.198)

##  Appendix C Complex Scalar Matching

In this section, we will derive the matching conditions for the complex scalar
fields. We start from writing down the action for the complex scalar $\phi$.
In Lorentzian signature we have

$S_{L}=\frac{1}{2}\int_{M_{L}}\sqrt{-G}(-\partial_{\mu}\phi_{L}^{*}\;\partial^{\mu}\phi_{L}-m^{2}\phi_{L}^{*}\phi_{L})$
(C.199)

while for Euclidean signature the action is

$S_{E}=\frac{1}{2}\int_{M_{E}}\sqrt{G}(\partial_{\mu}\phi_{E}^{*}\;\partial^{\mu}\phi_{E}+m^{2}\phi_{E}^{*}\phi_{E}),$
(C.200)

in which we denote the Lorentzian spacetime and Euclidean spacetime as $M_{L}$
and $M_{E}$ joint at the codimension one surface $\Sigma$. As we have
discussed, the continuation of the state implies the first matching condition
$\phi_{E}=\phi_{L}$. Moreover, the stationarity of the total on-shell action
with respect to $\phi$ and $\phi^{*}$ tells us

$\frac{\delta}{\delta\phi}\;(\;iS_{L}-S_{E})=\int_{\Sigma}\sqrt{K}(-i\partial_{t}\phi^{*}_{L}-\partial_{t}\phi^{*}_{E})=0,$
(C.201)
$\frac{\delta}{\delta\phi^{*}}\;(\;iS_{L}-S_{E})=\int_{\Sigma}\sqrt{K}(-i\partial_{t}\phi_{L}-\partial_{t}\phi_{E})=0,$
(C.202)

in which $K$ is the intrinsic curvature induced on $\Sigma$. Therefore, we
obtain the second matching condition written as

$i\partial_{t}\phi_{L}+\partial_{t}\phi_{E}=0\qquad{\rm and}\qquad
i\partial_{t}\phi^{*}_{L}+\partial_{t}\phi^{*}_{E}=0.$ (C.203)

##  Appendix D Quadratic Matching

Here we present the detail of the verification of (143). The Euclidean and
Lorentzian propagator are given by

$G_{\Delta}^{E}(X,P)=\frac{1}{(-2X\cdot P)_{E}^{\Delta}}\qquad
G_{\Delta}^{L}(X,P)=\frac{1}{(-2X\cdot P)_{L}^{\Delta}}.$ (D.204)

Acting $\partial^{2}_{t}$ and $\partial_{\tau}^{2}$ on them, we obtain

$\partial_{\tau}^{2}\frac{1}{(-2X\cdot
P)_{E}^{\Delta}}=\frac{4\Delta(\Delta+1)}{(-2X\cdot
P)_{E}^{{\Delta+2}}}(-X^{0}P^{-}+X^{+}P^{0})^{2}-\frac{2\Delta}{(-2X\cdot
P)^{\Delta+1}_{E}}X^{+}P^{-}$ (D.205)

and

$\partial_{t}^{2}\frac{1}{(-2X\cdot
P)_{L}^{\Delta}}=\frac{4\Delta(\Delta+1)}{(-2X\cdot
P)_{L}^{{\Delta+2}}}(X^{0}P^{-}-X^{+}P^{0})^{2}+\frac{2\Delta}{(-2X\cdot
P)^{\Delta+1}_{L}}X^{+}P^{-}.$ (D.206)

Acting $\partial^{3}_{t}$ and $\partial^{3}_{\tau}$ on the propagator we have

$\partial_{\tau}^{3}\frac{1}{(-2X\cdot
P)_{E}^{\Delta}}=\frac{8\Delta(\Delta+1)(\Delta+2)}{(-2X\cdot
P)_{E}^{{\Delta+2}}}(-X^{0}P^{-}+X^{+}P^{0})^{3}-X^{+}P^{-}\frac{8\Delta(\Delta+1)}{(-2X\cdot
P)^{\Delta+1}_{E}}(-X^{0}P^{-}+X^{+}P^{0})$ (D.207)

and

$\partial_{t}^{3}\frac{1}{(-2X\cdot
P)_{L}^{\Delta}}=\frac{8\Delta(\Delta+1)(\Delta+2)}{(-2X\cdot
P)_{L}^{{\Delta+2}}}(X^{0}P^{-}-X^{-}P^{0})^{3}+X^{+}P^{-}\frac{8\Delta(\Delta+1)}{(-2X\cdot
P)^{\Delta+1}_{L}}(X^{0}P^{-}-X^{+}P^{0}).$ (D.208)

One can directly check that (143) is true at the joint and surface $t=T$,
$\tau=iT$, after taking the rotation of coordinates $P_{0}^{E}=iP_{0}^{L}$
into consideration.

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