YITP-22-99
---

Late-time correlation functions in dS3/CFT2 correspondence

Heng-Yu Chen,a Shi Chena and Yasuaki Hikidab

aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan

bCenter for Gravitational Physics and Quantum Information, Yukawa Institute
for Theoretical Physics,

Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

We compute the late-time correlation functions on three-dimensional de Sitter
spacetime for a higher-spin gravity theory. For this, we elaborate on the
formulation to obtain the wave functional of universe from a dual conformal
field theory, which is used to compute the late-time correlation functions. We
argue that the relation to direct bulk Feynman diagram computations in the in-
in formulation. We furthermore provide a precise prescription to construct a
higher-spin dS3 holography as an analytic continuation of Gaberdiel-Gopakumar
duality for AdS3. Part of results here were already reported in an earlier
letter. We explain the details of their derivations and extend the analysis to
more generic cases in this paper. Previously, we have examined two- and three-
point functions and a simple four-point correlator at the leading order in
Newton constant. Here we also evaluate more complicated four-point
correlators. Finally, we study late-time correlators in an alternative limit
of dS3/CFT2 with critical level coset, such as, two-point correlator on
conical defect geometry. We also examine one-loop corrections to two-point
correlator on dS3.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Future directions
    2. 1.2 Organization
  2. 2 General strategy
    1. 2.1 Analytic continuation from AdSd+1 to dSd+1
    2. 2.2 Late-time correlators from the wave functional of universe
    3. 2.3 Relation to the in-in formulation
    4. 2.4 Higher order contributions in $G_{N}$
  3. 3 Higher-spin $\text{dS}_{3}/\text{CFT}_{2}$ correspondence
    1. 3.1 Review of Gaberdiel-Gopakumar duality
    2. 3.2 Analytic continuation from AdS3 to dS3
      1. 3.2.1 Two-point correlators
      2. 3.2.2 Three-point correlators
  4. 4 Four-point correlators dual to AdS3 and dS3
    1. 4.1 The case dual to AdS3
      1. 4.1.1 Four-point function $G_{-+}(z)$
      2. 4.1.2 Four-point functions $G_{\pm\pm}(z)$
    2. 4.2 Analytic continuation from AdS3 to dS3
      1. 4.2.1 Four-point correlator from $G_{-+}(z)$
      2. 4.2.2 Four-point correlator from $G_{\pm\pm}(z)$
  5. 5 Example of other late-time correlator
    1. 5.1 Higher-spin dS3/CFT2 with truncated spin
    2. 5.2 Two-point correlator on dS3 conical defect
  6. 6 Comments on the loops in dS spacetime
    1. 6.1 One-loop corrections to two-point correlator
    2. 6.2 The expansion of four-point correlation functions at $\mathcal{O}(c^{-2})$
  7. A Partial wave (conformal block) expansions
    1. A.1 Crossing symmetry under $s$-$t$ channel exchange
    2. A.2 Expansions of $G_{\pm\pm}(z)$
    3. A.3 Shadow blocks for conserved currents
  8. B Bulk interpretations
    1. B.1 Embedding formalism for Euclidean AdSd+1
    2. B.2 Three-point function in the holographic gauge
    3. B.3 Geodesic Witten diagrams and two-dimensional global conformal blocks

##  1 Introduction and summary

Our universe experienced an inflationary era at the early time, during which
its geometry can be approximated by de Sitter (dS) spacetime. In order to
understand the beginning of our universe, it is important to formulate quantum
gravity on dS spacetime, which has been understood very little so far. In the
case of anti-de Sitter (AdS) spacetime, the situation is quite different
mainly due to AdS/CFT correspondence proposed by Maldacena [1, 2, 3]. AdS/CFT
correspondence is supposed to formulate quantum gravity on AdS spacetime by
its boundary conformal field theory (CFT), and there have been a lot of
developments since its discovery. It is thus natural to expect that an
explicit formulation of dS/CFT correspondence [4, 5, 6] would be helpful for
our purpose. Recently, the late-time correlation functions on three-
dimensional dS (dS3) have been analyzed by making use of the holography in
[7]. The late-time correlators on dS are expected to be useful to understand
what happened during the inflation era [6, 8, 9, 10]. In [7], only partial
results were reported and derivations were explained quite briefly. In this
paper, we present our full results on late-time correlation functions on dS
spacetime and explain our methods and calculations in more details.

Currently, dS/CFT correspondence is not as well understood compared with its
AdS/CFT counterpart. One of the main reasons is that only very few concrete
calculable examples have been constructed so far. In [11] it was proposed that
a higher-spin gravity on dS4 is dual to a 3d Sp$(N)$ vector model. The
proposal may be regarded as an “analytic continuation” of Klebanov-Polyakov
duality [12] between higher-spin gravity on AdS4 and 3d O$(N)$ vector model.
For previous works, see, e.g., [13, 14, 15, 16]. Recently, an explicit class
of dS3/CFT2 correspondence was proposed in [17, 18, 7] between a (higher-spin)
gravity on dS3 and a certain 2d CFT with an imaginary central charge. In these
papers, the relation to Gaberdiel-Gopakumar duality [19] involving a higher-
spin gravity on AdS3 was also clarified. The higher-spin theory is given
explicitly by 3d Prokushkin-Vasiliev theory [20], which includes higher-spin
gauge fields with spin $s=2,3,\ldots$ and two complex scalars with mass:

$\displaystyle\ell^{2}_{\text{AdS}}m^{2}=-1+\hat{\lambda}^{2}\,.$ (1.1)

Here $\ell_{\text{AdS}}$ denotes the AdS radius and a dimensionless parameter
$\hat{\lambda}$ is introduced. The dual 2d CFT is supposed to be a coset

$\displaystyle\frac{\text{SU}(N)_{k}\times\text{SU}(N)_{1}}{\text{SU}(N)_{k+1}}\,$
(1.2)

and the classical limit of gravity theory corresponds to taking the ’t Hooft
limit of the coset:

$\displaystyle N,k\to\infty,\quad\lambda=\frac{N}{N+k}\to\text{fixed}\,.$
(1.3)

The ’t Hooft parameter $\lambda$ is identified with $\hat{\lambda}$ appeared
in the dual gravity theory. In the bulk, we map from AdS3 to dS3 mainly by
replacing $\ell_{\text{AdS}}$ by $-i\ell$ with $\ell$ being the dS radius. It
was shown that the asymptotic symmetry near the future infinity is given by a
W-algebra with an imaginary central charge [21] (see also [4, 6]). We thus
identify the dual CFT with fixed $\lambda$ but taking an analytic continuation
such that the central charge becomes purely imaginary as $-ic^{(g)}$ with real
$c^{(g)}$. Making use of the simplicity and explicitness of the lower-
dimensional theories, we could investigate various properties of dS/CFT
correspondence.

In AdS/CFT correspondence, the scattering amplitudes or more strictly
transition amplitudes of gravity theory between the boundaries of AdSd+1 can
be obtained from the correlation functions of its dual CFT [2, 3]. It was
similarly proposed that the correlation functions of fields in a gravity
theory located at the points on its late time boundary of dSd+1 can be
computed from the correlation functions of the dual CFT [6]. For this, we
prepare the wave functional of the universe by integrating over bulk fields
$\psi_{j}$ (including a metric $g$) subject to boundary conditions
$\psi_{j}=\psi_{j}^{0}$ (in particular, $g=h$) at late time $t=t_{\infty}$ as

$\displaystyle\Psi[\psi_{j}^{0}]=\int\mathcal{D}\psi_{j}\exp\left(iS[\psi_{j}]\right)$
(1.4)

where $S[\psi_{j}]$ is the action of the gravity theory. The proposal by [6]
is that the wave functional can be computed from certain dual CFT as

$\displaystyle\Psi[\psi_{j}^{0}]=\left\langle\exp\left(\ell^{d}\int
d^{d}\vec{x}\psi_{j}^{0}\mathcal{O}^{j}\right)\right\rangle\,.$ (1.5)

Here $\mathcal{O}^{j}$ are CFT operators dual to $\psi_{j}$. The correlation
functions of $\psi_{j}^{0}$ are then computed as expectation values as

$\displaystyle\langle\,\psi_{1}^{0}(\vec{x}_{1})\cdots\psi_{m}^{0}(\vec{x}_{m})\,\rangle=\int[{\cal
D}\psi_{j}^{0}]\,\left|\Psi\left[\psi_{l}^{0}\right]\right|^{2}\,\psi_{1}^{0}(\vec{x}_{1})\cdots\psi_{m}^{0}(\vec{x}_{m})\,.$
(1.6)

In our previous work [7], we briefly explained how to compute late-time
correlation functions of bulk theory from an analytic continuation of the
coset CFT (1.2). We also provided final results on two- and three-point
correlators. Furthermore, we examined a simple four-point correlator. In this
paper, we explain in more details our methods and how to derive these results.
We consider more complicated four-point correlators as well.

Some earlier works on the bulk late-time correlators have been made in the in-
in formalism and relation to bulk correlators on AdS backgrounds have been
revealed [22, 23] (see also [24]). In particular, they found certain phase
factors should be associated when corresponding Feynman diagrams for an
identical theory on AdS and dS backgrounds are compared. We provide arguments
on the relation to our results. For instance, we obtain the phase factors
associated with the correlators, which reproduce the same phase factors up to
a subtlety associate with massless limit of higher-spin fields. We see that
the two formulations are consistent with each other. As an advantage, our
formalism would allow us to compute non-trivial correlators, which are
difficult to compute directly from the bulk theory due to, e.g., the lack of
knowledge of quantum aspects on the (higher-spin) gravity theory.

###  1.1 Future directions

Our formalism is generic enough to apply to other cases. In order to
illustrate this, we compute some interesting examples, which could be useful
to reveal important properties of dS/CFT correspondence. There are several
open questions we would like to address in near future. In this paper, we only
explain how to apply our method to compute bulk correlators related to these
questions, and we postpone detailed analysis based on the correlators to
separate publications.

It is important to consider bulk late-time correlations on more generic
asymptotic dS3 geometry. In [17, 18], dS3 conical geometry was constructed
based on the analysis for AdS3 case in [25]. In this paper, we compute bulk
late-time correlations of scalar fields on dS3 conical geometry, though its
bulk interpretation will be examined elsewhere. Generically, it is important
problem to read off the properties of the asymptotic dS3 geometry from late-
time two-point functions of bulk scalar field. In particular, we are
interested in the properties of higher-spin black hole with asymptotic dS3
condition, see, e.g., [26]. For asymptotic AdS3 case, refer, e.g., [27, 28].

Throughout this paper, we mainly consider the leading order effects in the
Newton constant $G_{N}$, and it is important to systematically analyze higher
order gravitational effects. It is exciting if we can deal with even finite
$G_{N}$ effects, see, e.g., [24, 29]. In order to show that our method can be
applied to higher orders in $G_{N}$, we examine a type of one-loop corrections
to two-point correlator on dS3 at late time, though the full detailed analysis
would be presented elsewhere. We analyze the expansions of four-point
correlators up to the next non-trivial order as well.

There are other examples, which will not be addressed within this paper. For
instance, the original Gaberdiel-Gopakumar duality has been extended to
include supersymmetry or matrix-valued fields in order to see the relation to
superstring theory. Concretely, $\mathcal{N}=3$ supersymmetry with matrix-
valued extension was considered in [30, 31, 32]. The $\mathcal{N}=4$
supersymmetry and its relation to strings on
$\text{AdS}_{3}\times\text{S}^{3}\times\text{T}^{4}$ was analyzed in [33, 34].
It is interesting to consider an analytic continuation of these extended
cases. It is also worthwhile extending the current analysis to the higher
dimensional example of [11].

###  1.2 Organization

This paper is organized as follows. In the next section, we explain the
generic method to compute late-time correlators on dSd+1 by using the wave
functional of universe as in (1.6). In particular, we examine what kind of
phase factors arise from the map from AdSd+1 to dSd+1 for operators dual to
bulk fields. We also compare our analysis to generic arguments with bulk
Feynman diagrams and the in-in formulation [22, 23]. In section 3, we
introduce our version of dS3/CFT2 correspondence by providing a prescription
to perform an analytic continuation of Gaberdiel-Gopakumar duality in [19]. We
also compute two- and three-point correlators on dS3 at late time. In section
4, which contains some of the main results in this paper, we present the
computations of four-point correlators on dS3 at late time. In a previous
paper [7], a simple four-point correlator has been analyzed. Here we explain
the computation in more details and generalize the analysis to more
complicated four-point correlators. In section 5, we consider alternative
limit of dS3/CFT2 correspondence, which was introduced in [17, 18]. With the
setup, we deal with two-point correlators on dS3 conical defect. In section 6,
we examine the higher order corrections to bulk correlators in $G_{N}$, such
as, one-loop corrections on two-point function on dS3. We also analyze the
structure of $1/c^{2}$-expansions of four-point functions of scalar operators
in the dual CFT. In appendix A, we collect detailed analysis on conformal
block and partial wave expansions. In appendix B, we review the embedding
formalism for AdSd+1 and summarize some bulk analyses specific to three
dimensions, including holographic dual configurations for three-point
functions and conformal blocks involving holomorphic conserved tensors.

##  2 General strategy

The main purpose of this work is to compute correlation functions of higher-
spin gravity on dS3 at late time from dual CFT2. For this, we utilize the wave
functional prescription in [6], which will be explained to some details in
this section. Before focusing on an explicit setting, we consider a bulk
theory defined on $(d+1)$-dimensional dS spacetime and including massive
scalar fields and (massive) symmetric tensor fields in order to make argument
as generic as possible. We assume that dual CFT exists as well.

###  2.1 Analytic continuation from AdSd+1 to dSd+1

We start from the physical quantities on Euclidean AdS space and then discuss
the map to those on Lorentzian dS spacetime. In the bulk theory on Euclidean
AdSd+1, we include at least one scalar field $\phi^{\text{AdS}}$ and symmetric
tensor fields $\sigma_{i_{1}\cdots i_{s}}^{\text{AdS}}$. We denote the dual
scalar operator by $\mathcal{O}_{\text{AdS}}$ and the dual symmetric traceless
tensor by $J^{i_{1}\cdots i_{s}}_{\text{AdS}}$. We denote the mass of a bulk
field as $m$, then the conformal dimension of dual operator is
$\Delta=\Delta_{\pm}$ depending on the boundary condition assigned to the
given bulk field. Here $\Delta_{\pm}$ satisfy

$\displaystyle\ell_{\text{AdS}}^{2}\,m^{2}=-(\Delta_{+}\Delta_{-}+s)\,,\qquad\Delta_{-}=d-\Delta_{+}$
(2.1)

with the AdS radius $\ell_{\text{AdS}}$. We represent the spin of the field by
$s$ and set $s=0$ for the scalar field. We will later consider the case where
the dual symmetric tensor currents are conserved, and in that case
$\Delta_{+}=s+d-2$. For the metric of Euclidean AdSd+1, we adopt the Poincaré
coordinates as

$\displaystyle
ds^{2}=\ell_{\text{AdS}}^{2}\,\frac{dy^{2}+d\vec{x}^{2}}{y^{2}}$ (2.2)

with $y\geq 0$. The boundary of AdS space is located at $y=0$. The bulk fields
on AdSd+1 behave near the boundary $y\to+0$ as

$\displaystyle\begin{aligned}
\phi^{\text{AdS}}(y,\vec{x})&\sim\phi_{+}^{\text{AdS}}(\vec{x})\,y^{\Delta_{+}}+\phi_{-}^{\text{AdS}}(\vec{x})\,y^{\Delta_{-}}\,,\\\
\sigma_{i_{1}\cdots i_{s}}^{\text{AdS}}(y,\vec{x})&\sim\sigma_{i_{1}\cdots
i_{s}}^{+,\text{AdS}}(\vec{x})\,y^{\Delta_{+}-s}+\sigma_{i_{1}\cdots
i_{s}}^{-,\text{AdS}}(\vec{x})\,y^{\Delta_{-}-s}\ .\end{aligned}$ (2.3)

It is convenient to define the $d$-dimensional boundary metric as

$\displaystyle ds^{2}=d\vec{x}^{2}\,,$ (2.4)

such as to be independent of the AdS radius $\ell_{\text{AdS}}$. With the
boundary metric above, the boundary couplings between the bulk fields and the
CFT operators are given by

$\displaystyle\ell^{d}_{\text{AdS}}\int
d^{d}\vec{x}\,\phi^{\text{AdS}}_{\pm}\,\mathcal{O}^{\pm}_{\text{AdS}}\,,\quad\ell^{d}_{\text{AdS}}\int
d^{d}\vec{x}\,\sigma^{\pm,\text{AdS}}_{i_{1}\cdots i_{s}}\,J^{i_{1}\cdots
i_{s}}_{\pm,\text{AdS}}\,.$ (2.5)

We then consider the bulk theory on Lorentzian dS spacetime including at least
a scalar field $\phi$ and symmetric tensor fields $\sigma_{i_{1}\cdots i_{s}}$
as in the case of Euclidean AdS space. We also denote the dual scalar operator
by $\mathcal{O}$ and the dual symmetric tensor operators by $J^{i_{1}\cdots
i_{s}}$. The conformal dimension of dual operator is given by:

$\displaystyle\ell^{2}\,m^{2}=\Delta_{+}\Delta_{-}+s\,,\qquad\Delta_{-}=d-\Delta_{+}\,,$
(2.6)

where $m$ corresponds to the mass of bulk field. We express the Poincaré patch
of Lorentzian dSd+1 with the metric

$\displaystyle ds^{2}=\ell^{2}\,\frac{-d\eta^{2}+d\vec{x}^{2}}{\eta^{2}}\,,$
(2.7)

where we consider the region with $-\infty<\eta\leq 0$. The future infinity is
located at $\eta\to-0$. We may relate the dS metric to the AdS one given in
(2.2) by:

$\displaystyle y=-i\eta\ ,\qquad\ell_{\text{AdS}}=-i\ell\,.$ (2.8)

In the following, we will see how such an analytic continuation acts on bulk
fields and CFT operators.

We begin with the simpler case with scalar field and then move to more
complicated case with symmetric tensor fields. Near the future infinity
$\eta\to-0$, the bulk scalar field behaves as

$\displaystyle\phi(\eta,\vec{x})\sim\phi_{+}(\vec{x})\,(-\eta)^{\Delta_{+}}+\phi_{-}(\vec{x})\,(-\eta)^{\Delta_{-}}\,.$
(2.9)

As mentioned in [6], $\phi^{\text{AdS}}$ may be identified with $\phi$ under
the map (2.8), which leads to

$\displaystyle\phi_{\pm}=e^{-i\frac{\pi}{2}\Delta_{\pm}}\,\phi_{\pm}^{\text{AdS}}\,.$
(2.10)

In AdS/CFT correspondence, the boundary couplings between bulk fields and CFT
operators are given by (2.5). On the other hand, in dS/CFT correspondence, we
have the boundary coupling as in (1.5) instead. The two types of boundary
couplings can be identified if we relate the corresponding dual CFT operators
as:

$\displaystyle\mathcal{O}_{\pm}=e^{i\frac{\pi}{2}(\Delta_{\pm}-d)}\,\mathcal{O}_{\pm}^{\text{AdS}}\,.$
(2.11)

In the case of spin-$s$ tensor fields, their behaviors near the future
infinity are given by

$\displaystyle\sigma_{i_{1}\cdots
i_{s}}(\eta,\vec{x})\sim\sigma^{+}_{i_{1}\cdots
i_{s}}(\vec{x})\,(-\eta)^{\Delta_{+}-s}+\sigma^{-}_{i_{1}\cdots
i_{s}}(\vec{x})\,(-\eta)^{\Delta_{-}-s}\,.$ (2.12)

Here notice that the indices of bulk fields are lowered and raised by the bulk
metric $g_{\mu\nu}\sim\ell^{2}$ and $g^{\mu\nu}\sim\ell^{-2}$. On the other
hand, the boundary metric is defined as in (2.4), which is independent of
$\ell$. It is thus convenient to define boundary operators, whose indices are
raised and lowered by the boundary metric independent of $\ell$, see, e.g.,
[11] for a similar argument. Therefore, we assign

$\displaystyle\ell^{-s}\,\sigma^{\pm}_{i_{1}\cdots
i_{s}}(\vec{x})=i^{\Delta_{\pm}-s}\,\ell_{\text{AdS}}^{-s}\,\sigma^{\pm,\text{AdS}}_{i_{1}\cdots
i_{s}}(\vec{x})\,,$ (2.13)

which leads to

$\displaystyle\sigma^{\pm}_{i_{1}\cdots
i_{s}}(\vec{x})=e^{-i\frac{\pi}{2}\Delta_{\pm}}\,\sigma^{\pm,\text{AdS}}_{i_{1}\cdots
i_{s}}(\vec{x})\,,\qquad J^{\pm}_{i_{1}\cdots
i_{s}}(\vec{x})=e^{i\frac{\pi}{2}(\Delta_{\pm}-d)}\,J^{\pm,\text{AdS}}_{i_{1}\cdots
i_{s}}(\vec{x})\,.$ (2.14)

For instance, a spin-$s$ conserved current have conformal dimension
$\Delta_{+}=s+d-2$, and hence symmetric tensor currents are related as

$\displaystyle J^{+}_{i_{1}\cdots
i_{s}}(\vec{x})=e^{i\frac{\pi}{2}(s-2)}\,J^{+,\text{AdS}}_{i_{1}\cdots
i_{s}}(\vec{x})\,.$ (2.15)

In particular, the energy momentum tensor with $s=2$ does not receive any
phase factor and the standard convention of energy momentum tensor is
preserved by this prescription.

###  2.2 Late-time correlators from the wave functional of universe

In the previous subsection, we obtained the phase factors for operators dual
to bulk fields by making use of the map from AdS to dS as in (2.8). In this
subsection, we explain how to compute correlation functions of the bulk fields
at late time from these dual operators, see, e.g., appendix A of [10] as well.
As explained in the introduction, we utilize the wave functional of universe
to compute the correlation functions on dS at late time. The proposal of [6]
is that the wave functional can be evaluated from a certain dual CFT as in
(1.5). In AdS/CFT correspondence, the boundary values of bulk fields are
fixed, and they can be regarded as the sources in dual CFT [2, 3]. However, in
dS/CFT correspondence, boundary values $\psi_{j}^{0}$ at late time
$t=t_{\infty}$ are not constants but path-integral variables. Therefore, it is
natural to expand the wave functional by the boundary fields $\psi^{0}_{j}$
as:

$\displaystyle\begin{aligned} \Psi[\psi_{l}^{0}]&=\exp\left[\sum_{m\geq 2}\int
d^{d}\vec{x}_{1}\cdots
d^{d}\vec{x}_{m}C_{m}(\\{\vec{x}_{l}\\})\prod_{j=1}^{m}\psi_{j}^{0}(\vec{x}_{j})\right]\,.\end{aligned}$
(2.16)

Thus, it is better to state that the coefficient functions
$C_{m}(\\{\vec{x}_{l}\\})$ could be evaluated by the dual CFT living on the
late boundary of dS. We will specify the precise relation between
$C_{m}(\\{\vec{x}_{l}\\})$ and the $m$-point CFT correlation function in the
concrete example considered later. The correlation functions of $\psi_{j}^{0}$
are computed as expectation values as in (1.6).

There are a lot of computations of late-time correlation functions of bulk
theory such as in [6, 8, 9, 10], see also [35]. They are however analyzed
usually in momentum basis. This is because their prime interest is to deduce
the information after the inflation ends, where scale invariance is broken and
only translation invariance remains, thus the momentum representation is
useful. Generically, CFT correlation functions in momentum basis are much more
difficult to treat than those in coordinate basis. In order to avoid
unnecessary complication, we mainly work in coordinate basis in this paper. If
one wishes, the expressions obtained here could be mapped to those in momentum
basis by performing Fourier transformations in principle.

In the following sections, we explicitly compute bulk correlation functions on
dS3 at late time by applying the method explained above. We will
systematically consider the order-by-order expansion of these correlation
functions in the Newton constant $G_{N}$ (see (3.3) below for the relation
with other parameters). In this subsection, we concentrate on the leading
order contributions. We will discuss the higher order effects in subsection
2.4. At the leading non-trivial order in $G_{N}$, bulk correlators can be
expressed by CFT correlators as follows. The two-point function of scalar
field is obtained through functional inverse as:111The overall minus sign in
front of the two point function comes from the convergence of momentum
Gaussian path integral, while the sign of the ${\rm Re}\langle\dots\rangle$
come from the branch choice of the analytic continuation in (2.9).

$\displaystyle\langle\,\phi(\vec{x}_{1})\,\phi(\vec{x}_{2})\,\rangle=-\frac{1}{2\,\text{Re}\,\langle\,\mathcal{O}_{i}(\vec{x}_{1})\,\mathcal{O}_{i}(\vec{x}_{2})\,\rangle}\,.$
(2.17)

For symmetric tensor fields, they are written as

$\displaystyle\langle\,\sigma_{i_{1}\cdots
i_{s}}(\vec{x}_{1})\,\sigma^{j_{1}\cdots
j_{s}}(\vec{x}_{2})\,\rangle=-\frac{\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\hat{x}_{12})}{2\,\text{Re}\,\langle\,J_{s}(\vec{x}_{1})\,J_{s}(\vec{x}_{2})\,\rangle}\,.$
(2.18)

Here we have used the expression

$\displaystyle\langle\,J_{i_{1}\cdots i_{s}}(\vec{x}_{1})\,J^{j_{1}\cdots
j_{s}}(\vec{x}_{2})\,\rangle=\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\hat{x}_{12})\,\langle\,J_{s}(\vec{x}_{1})\,J_{s}(\vec{x}_{2})\,\rangle$
(2.19)

with the unit vector $\hat{x}=\vec{x}/|\vec{x}|$ and the bi-tensorial
projector into symmetric traceless tensor $\Pi^{j_{1}\cdots
j_{s}}_{i_{1}\cdots i_{s}}(\hat{x})$, which captures the index structure. Here
we have also set $\vec{x}_{12}=\vec{x}_{1}-\vec{x}_{2}$. We should take care
of defining the inverse of two-point functions in coordinate basis, which will
be explicitly given later. Similarly, the three-point function we will use is

$\displaystyle\begin{aligned}
&\langle\,\phi_{1}(\vec{x}_{1})\,\phi_{2}(\vec{x}_{2})\,\sigma_{i_{1}\cdots
i_{s}}(\vec{x}_{3})\,\rangle\\\
&\quad=\int\prod_{l=1}^{3}\mathrm{d}^{d}\vec{x}_{l}^{\prime}\frac{2\,\text{Re}\,\langle\,\mathcal{O}_{1}(\vec{x}_{1}^{\prime})\,\mathcal{O}_{2}(\vec{x}_{2}^{\prime})\,J_{j_{1}\cdots
j_{s}}(\vec{x}_{3}^{\prime})\,\rangle\,\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\vec{x}_{33^{\prime}})}{\left[\prod_{i=1}^{2}(-2\,\text{Re}\,\langle\,\mathcal{O}_{i}(\vec{x}_{i})\,\mathcal{O}_{i}(\vec{x}_{i}^{\prime})\,\rangle)\right](-2\,\text{Re}\,\langle\,J_{s}(\vec{x}_{3})\,J_{s}(\vec{x}_{3}^{\prime})\,\rangle)}\
.\end{aligned}$ (2.20)

We also consider scalar four-point function, which can be decomposed into:

$\displaystyle\langle\,\phi_{1}(\vec{x}_{1})\,\phi_{2}(\vec{x}_{2})\,\phi_{3}(\vec{x}_{3})\,\phi_{4}(\vec{x}_{4})\,\rangle=\int\prod_{l=1}^{4}d^{d}\vec{x}_{l}^{\prime}\frac{\langle\mathcal{O}^{4}\rangle_{c}+\langle\mathcal{O}^{4}\rangle_{d}}{\prod_{i=1}^{4}(-2\,\text{Re}\,\langle\,\mathcal{O}_{i}(\vec{x}_{i})\,\mathcal{O}_{i}(\vec{x}_{i}^{\prime})\,\rangle)}\
.$ (2.21)

Here we have split the contribution in the following:

$\displaystyle\langle\,\mathcal{O}^{4}\,\rangle_{c}$
$\displaystyle=2\,\text{Re}\,\langle\,\mathcal{O}_{1}(\vec{x}_{1}^{\prime})\,\mathcal{O}_{2}(\vec{x}_{2}^{\prime})\,\mathcal{O}_{3}(\vec{x}_{3}^{\prime})\,\mathcal{O}_{4}(\vec{x}_{4}^{\prime})\,\rangle_{c}\,,$
(2.22) $\displaystyle\langle\,\mathcal{O}^{4}\,\rangle_{d}$
$\displaystyle=\sum\int d^{d}\vec{y}_{1}d^{d}\vec{y}_{2}$ (2.23)
$\displaystyle\times\frac{2\text{Re}\,\langle\,\mathcal{O}_{1}(\vec{x}_{1}^{\prime})\,\mathcal{O}_{2}(\vec{x}_{2}^{\prime})\,J_{j_{1}\cdots
j_{s}}(\vec{y}_{1})\,\rangle\,\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\vec{y}_{12})\,2\text{Re}\,\langle\,J^{i_{1}\cdots
i_{s}}(\vec{y}_{2})\,\mathcal{O}_{3}(\vec{x}_{3}^{\prime})\,\mathcal{O}_{4}(\vec{x}_{4}^{\prime})\,\rangle}{2\text{Re}\,\langle\,J_{s}(\vec{y}_{1})\,J_{s}(\vec{y}_{2})\,\rangle}+\text{perm}\,.$

The term $\langle\,\mathcal{O}^{4}\,\rangle_{c}$, which is manifestly crossing
invariant without summing up different channels, corresponds to the
contributions exclusively from the four-point bulk contact diagrams. On the
other hand for $\langle\,\mathcal{O}^{4}\,\rangle_{d}$, we have to sum up all
intermediate exchange bulk fields for all channels. It should be noted that
only appropriate real combinations appear here as we take the product of the
wave functional $\Psi[\psi_{l}^{0}]$ and its complex conjugate
$\bar{\Psi}[\psi_{l}^{0}]$ when extracting bulk correlation functions.

Let us remark here on a subtlety to define the coefficient functions
$C_{m}(\\{\vec{x_{l}}\\})$ by an analytic continuation from correlation
functions of CFTd dual to gravity theory on AdSd+1. Here we assume that the
CFT has a large $N$ structure. We consider a four-point function of scalar
operators

$\displaystyle\mathcal{A}_{4}\equiv\left\langle\mathcal{O}_{1}^{\text{AdS}}(\vec{x}_{1})\mathcal{O}_{2}^{\text{AdS}}(\vec{x}_{2})\mathcal{O}_{3}^{\text{AdS}}(\vec{x}_{3})\mathcal{O}_{4}^{\text{AdS}}(\vec{x}_{4})\right\rangle\,.$
(2.24)

Its generalization to other correlation functions is straightforward. We
expand the four-point function by $1/N$ as

$\displaystyle\mathcal{A}_{4}=\mathcal{A}^{(0)}_{4}+\mathcal{A}^{(1)}_{4}+\cdots\,.$
(2.25)

The leading order contribution comes from the products of two-point functions

$\displaystyle\mathcal{A}^{(0)}_{4}$
$\displaystyle=\left\langle\mathcal{O}_{1}^{\text{AdS}}(\vec{x}_{1})\mathcal{O}_{2}^{\text{AdS}}(\vec{x}_{2})\right\rangle\left\langle\mathcal{O}_{3}^{\text{AdS}}(\vec{x}_{3})\mathcal{O}_{4}^{\text{AdS}}(\vec{x}_{4})\right\rangle+\text{perm}\,.$
(2.26)

This is a kind of definition of large $N$ structure. We would like to read off
coefficient functions $C_{4}(\\{\vec{x}_{l}\\})$ in (2.16) from the
computation of four-point function $\mathcal{A}_{4}$. However, the bulk
correlators are computed via (1.6), and the products of two-point functions
can already be obtained from two-point coefficients $C_{2}(\\{\vec{x}_{l}\\})$
in (2.16). Therefore, in order to obtain $C_{4}(\\{\vec{x}_{l}\\})$ from an
analytic continuation of $\mathcal{A}_{4}$, the leading order contribution
$\mathcal{A}_{N}^{(0)}$ should be extracted beforehand. The next leading order
can decomposed as

$\displaystyle\mathcal{A}^{(1)}_{4}=\mathcal{A}^{(1)}_{4,c}+\mathcal{A}^{(1)}_{4,d}\,.$
(2.27)

Here we define $\mathcal{A}^{(1)}_{4,d}$ by

$\displaystyle\mathcal{A}^{(1)}_{4,d}$ $\displaystyle=\sum\int
d^{d}\vec{y}_{1}d^{d}\vec{y}_{2}$ (2.28)
$\displaystyle\times\frac{\langle\,\mathcal{O}_{1}^{\text{AdS}}(\vec{x}_{1})\,\mathcal{O}_{2}^{\text{AdS}}(\vec{x}_{2})\,J^{\text{AdS}}_{j_{1}\cdots
j_{s}}(\vec{y}_{1})\,\rangle\,\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\vec{y}_{12})\langle\,J^{i_{1}\cdots
i_{s}}_{\text{AdS}}(\vec{y}_{2})\,\mathcal{O}_{3}^{\text{AdS}}(\vec{x}_{3})\,\mathcal{O}_{4}^{\text{AdS}}(\vec{x}_{4})\,\rangle}{\langle\,J_{s}^{\text{AdS}}(\vec{y}_{1})\,J_{s}^{\text{AdS}}(\vec{y}_{2})\,\rangle}+\text{perm.}\,,$

where the sum is over all intermediate operators. As expressed in (2.23),
$\langle\,\mathcal{O}^{4}\,\rangle_{d}$ can be computed only with two- and
three-point coefficient functions. Therefore, the four-point coefficient
functions $C_{4}(\\{\vec{x}_{l}\\})$ should be obtained by an analytic
continuation of the rest $\mathcal{A}_{4,c}^{(1)}$.

Figure 1: Bulk interpretation for the decompositions of four-point functions.

From the bulk viewpoints, the above procedure may be explained as follows. As
seen in fig. 1, the four-point function can be computed by the sum of Witten
diagrams. A type of diagrams consists of the product of two bulk propagators,
which correspond to the product of two-point functions. Another type consists
of the exchange diagrams of spin-$s$ fields, which roughly correspond to the
conformal partial waves of spin-$s$ current exchange. The rest is the four-
point contact diagram, which basically corresponds to the term we just
extracted. The precise relation can be read off from the decomposition of
conformal partial waves for bulk exchange diagrams as explained in appendix
B.1.

###  2.3 Relation to the in-in formulation

In the previous subsection, we explain our method to utilize the wave
functional of universe for evaluating correlation functions on bulk dS at late
time from dual CFT. Recently, there are developments on a different approach
to obtain correlation functions by making use of bulk Feynman diagram
computations [22, 23], see also [24]. In their analysis, they compute bulk
correlation functions in the in-in formulation, whose good review may be found
in [36]. They put an identical theory on AdS and dS and compare the
contributions coming from corresponding Feynman diagrams. In particular, they
concluded that only a phase factor should be associated with each contribution
along with the analytic continuation.

In this subsection, we compare the two formulations in order to clarify their
similarity and difference. In the in-in formulation, we compute correlation
function by

$\displaystyle\langle\Omega(t_{0})|\,\bar{T}\left(e^{i\int_{t_{0}}^{t_{\infty}}\mathrm{d}t^{\prime}H(t^{\prime})}\right)\psi_{1}^{0}(x_{1})\cdots\psi_{m}^{0}(x_{n})\,T\left(e^{-i\int_{t_{0}}^{t_{\infty}}\mathrm{d}t^{\prime}H(t^{\prime})}\right)\,|\Omega(t_{0})\rangle\,.$
(2.29)

Here $T$ and $\bar{T}$ denote time and anti-time orderings. Moreover,
correlators are among bulk fields at the late time $t=t_{\infty}$ and the
Bunch-Davies boundary condition is assigned at the early time $t=t_{0}$. The
vacuum corresponding to the boundary condition is denoted as $\Omega(t_{0})$.
Note that there are two types of interactions corresponding to time and anti-
time orderings. On the other hand, the formalism adopted in this paper utilize
the wave functional of universe as explained in subsection 2.2. The point here
is that we have to sum over the both contributions from the coefficient
functions of $\Psi[\psi_{l}^{0}]$ and $\bar{\Psi}[\psi_{l}^{0}]$.

We begin with three-point function at the tree level. In this case, the
correspondence between the two formalisms is rather clear. In the in-in
formulation, there are only two types of contact interactions, where one comes
in the anti-time ordered insertions and the other comes in the time ordered
insertions, see fig. 2.

Figure 2: Feynman diagrams for three-point correlators on dS at late time. The
$\pm$ indicates the (anti-)time ordering of inserted interaction terms in the
Hamiltonian.

As in [23], we put $+$ or $-$ in order to distinguish them. In the formulation
with wave functional, there are two types of contributions from $\Psi$ and
$\bar{\Psi}$ as mentioned above. In other words, a diagram in fig. 2
corresponds to $C_{3}(\\{\vec{x}\\})$, and the other corresponds to the
complex conjugate of $C_{3}(\\{\vec{x}\\})$. Precisely speaking, in order to
match with the scalar correlators on dS at late time, we have to multiply the
inverse of two-point functions, see subsection 2.2.

Interesting phenomena arise from four-point functions; here we only consider
four-point functions of external scalar operators and ignore the contributions
coming from the products of two-point functions. The sums of Feynman diagrams
in the first line of fig. 3

Figure 3: Feynman diagrams for four-point correlators on dS at late time.

come from different analytic continuations of bulk AdS computations, which
correspond to $\langle\mathcal{O}\mathcal{O}\mathcal{O}\mathcal{O}\rangle$ and
its complex conjugate, respectively. As discussed in subsection 2.2, the four-
point function of scalar operators can be obtained from the sum of contact and
exchange diagrams. The contact diagram with $\pm$ corresponds to
$C_{4}(\\{\vec{x}\\})$ or its complex conjugate. The remaining contributions
coming from the $\pm$ and $\pm$ exchange diagrams can be constructed from the
lower-point coefficient functions. The Feynman diagrams in the second line of
fig. 3 are made of bulk two-point functions between $\pm$ and $\mp$
interactions. We can interpret the various contributions when expanding (2.23)
as different combinations of analytic continuations from AdS space. The bulk
exchange diagrams correspond to conformal partial waves with exchange of
boundary spin-$s$ current. They correspond to the products of
$\langle\mathcal{O}\mathcal{O}J\rangle$ and its complex conjugate
$\overline{\langle\mathcal{O}\mathcal{O}J\rangle}$ divided properly by two-
point coefficient functions. We have thus listed the all Feynman diagrams
corresponding to the contributions after expanding the terms in (2.22) and
(2.23).

###  2.4 Higher order contributions in $G_{N}$

So far, we have restricted ourselves to the leading order contributions in the
Newton constant $G_{N}$ to bulk correlation functions. However, the
holographic formulation with the wave functional of universe can be applied to
higher order contributions as well. Note that, as in the case of four-point
functions, careful treatments have to be performed, which is explained in this
section with the example of one-loop corrections to two-point bulk correlator.
In subsection 6.1, we will evaluate it for an explicit example.

Let us compute two-point correlator of scalar field $\phi$ on dSd+1 at late
time by applying (1.6) with the wave functional of universe (2.16). At the
leading order in $G_{N}$, the two-point function is given by (2.17). There
could be several types of one-loop contributions to two-point correlator, but
here we only focus on one particular type. We define the following integral as

$\displaystyle\begin{aligned}
&\mathcal{F}_{2}(x_{1}^{\prime},x_{2}^{\prime})=\sum_{s=2}^{\infty}\int
d^{d}\vec{y}_{1}d^{d}\vec{y}_{2}d^{d}\vec{y}_{3}d^{d}\vec{y}_{4}\\\
&\quad\times\frac{2\,\text{Re}\,\langle\,\mathcal{O}(\vec{x}_{1}^{\prime})\,\mathcal{O}(\vec{y}_{1})\,J_{j_{1}\cdots
j_{s}}(\vec{y}_{2})\,\rangle\,\Pi^{j_{1}\cdots j_{s}}_{i_{1}\cdots
i_{s}}(\vec{y}_{23})\,2\,\text{Re}\,\langle\,J^{i_{1}\cdots
i_{s}}(\vec{y}_{3})\,\mathcal{O}_{3}(\vec{y}_{4})\,\mathcal{O}_{4}(\vec{x}_{2}^{\prime})\,\rangle}{(-2\,\text{Re}\,\langle\,J_{s}(\vec{y}_{2})\,J_{s}(\vec{y}_{3})\,\rangle)(-2\,\text{Re}\,\langle\,\mathcal{O}(\vec{y}_{1})\,\mathcal{O}(\vec{y}_{4})\,\rangle)}\,.\end{aligned}$
(2.30)

This integral naturally arises when considering so-called split representation
of one-loop correction to the bulk to bulk scalar propagator. The particular
type of contribution to the two-point function considered here can be
evaluated from the inversion formula:

$\displaystyle\int
d^{d}\vec{x}_{1}^{\prime}d^{d}\vec{x}_{2}^{\prime}\frac{\mathcal{F}_{2}(x_{1}^{\prime},x_{2}^{\prime})}{(-2\,\text{Re}\,\langle\,\mathcal{O}(\vec{x}_{1})\,\mathcal{O}(\vec{x}_{1}^{\prime})\,\rangle)(-2\,\text{Re}\,\langle\,\mathcal{O}(\vec{x}_{2})\,\mathcal{O}(\vec{x}_{2}^{\prime})\,\rangle)}\,.$
(2.31)

From (1.6) with (2.16), we can then read off other types of contributions.

The above expression implies another subtlety when $C_{2}(\\{\vec{x}_{l}\\})$
is obtained from an analytic continuation of AdS case. For CFTd dual to
AdSd+1, we may be able to obtain an exact expression of two-point function

$\displaystyle\langle\mathcal{O}^{\text{AdS}}(\vec{x}_{1})\mathcal{O}^{\text{AdS}}(\vec{x}_{2})\rangle$
(2.32)

in terms of $N$, which would be related to $1/N^{2}\propto G_{N}$ (or
$1/N\propto G_{N}$ in a higher-spin holography as our case, see (3.3)).
However, a part of contributions at the next leading order can be computed by

$\displaystyle\begin{aligned} &\sum_{s=2}^{\infty}\int
d^{d}\vec{y}_{1}d^{d}\vec{y}_{2}d^{d}\vec{y}_{3}d^{d}\vec{y}_{4}\\\
&\quad\times\frac{\langle\,\mathcal{O}^{\text{AdS}}(\vec{x}_{1}^{\prime})\,\mathcal{O}^{\text{AdS}}(\vec{y}_{1})\,J_{j_{1}\cdots
j_{s}}^{\text{AdS}}(\vec{y}_{2})\,\rangle\,\Pi^{j_{1}\cdots
j_{s}}_{i_{1}\cdots i_{s}}(\vec{y}_{23})\langle\,J^{i_{1}\cdots
i_{s}}_{\text{AdS}}(\vec{y}_{3})\,\mathcal{O}_{3}(\vec{y}_{4})\,\mathcal{O}_{4}(\vec{x}_{2}^{\prime})\,\rangle}{\langle\,J_{s}^{\text{AdS}}(\vec{y}_{2})\,J_{s}^{\text{AdS}}(\vec{y}_{3})\,\rangle\langle\,\mathcal{O}^{\text{AdS}}(\vec{y}_{1})\,\mathcal{O}^{\text{AdS}}(\vec{y}_{4})\,\rangle}\,,\end{aligned}$
(2.33)

which corresponds to evaluating the Witten diagram in fig. 4.

Figure 4: Split representation of a contribution to one-loop corrections in
two-point function.

If we define $C_{2}(\\{\vec{x}_{l}\\})$ from (2.32) with higher order
corrections in $1/N$, then the contribution in (2.33) would be double-counted
in (2.31) with (2.30). Thus we have to remove the contribution in (2.33) from
(2.32) to define $C_{2}(\\{\vec{x}_{l}\\})$ via an analytic continuation. We
can see that the situation is quite analogous to the case of four-point
correlator discussed above. In the same way, we can see that all higher order
contributions in $1/N$ to the two-point function (2.32) have to be removed in
order to avoid double-counting.

##  3 Higher-spin $\text{dS}_{3}/\text{CFT}_{2}$ correspondence

In the previous section, we explained the method to compute bulk correlation
functions at late-time from dual CFT using the formalism with the wave
functional of universe. In this section, we introduce a prescription to
construct a concrete example of dS3/CFT2 correspondence as an analytic
continuation of Gaberdiel-Gopakumar duality [19]. In the next subsection, we
review the known results on the higher-spin AdS3 holography. In subsection
3.2, we explain our prescription to perform the analytic continuation from the
AdS3 holography to dS3 holography. We also examine two- and three-point
correlators as simple examples. In the succeeding sections, we examine more
interesting examples including four-point correlators.

###  3.1 Review of Gaberdiel-Gopakumar duality

It was proposed in [19] that classical higher-spin gravity on AdS3 is dual to
the coset model (1.2) at the ’t Hooft limit (1.3). The higher-spin theory is
the one constructed by Prokushkin and Vasiliev in [20], which includes higher-
spin gauge fields with spin $s\,(=2,3,\ldots,\infty)$ and two massive complex
scalar fields. The gauge sector can be described by a Chern-Simons theory with
the action

$\displaystyle S=S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]\,,\quad
S_{\text{CS}}=\frac{k_{\text{CS}}}{4\pi}\int\text{tr}\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\right)\,.$ (3.1)

Here the gauge fields $A$ and $\bar{A}$ are holomorphic and anti-holomorphic
one-forms and they take values in higher-spin gauge algebra
$\mathfrak{hs}[\hat{\lambda}]$. The gauge algebra is characterized such as to
be truncated to $\mathfrak{sl}(N^{\prime})$ if we set $\hat{\lambda}=\pm
N^{\prime}$. We denote the generators by $V^{s}_{m}$. The index $s$ is related
to the spin of higher-spin gauge field as $s=2,3,\ldots$. The other index $m$
runs over $m=-s+1,-s+2,\ldots,s-1$. In particular, they satisfy

$\displaystyle[V^{2}_{m},V^{s}_{n}]=(m(s-1)-n)V^{s}_{m+n}\,,$ (3.2)

see, e.g., [37] for other commutation relations. The coupling is given by the
Chern-Simons level $k_{\text{CS}}$, and it is related to the gravity
parameters as

$\displaystyle k_{\text{CS}}=\frac{\ell_{\text{AdS}}}{4G_{N}}\,.$ (3.3)

The asymptotic symmetry near the AdS boundary was examined in [37, 38, 39,
40]. The symmetry algebra is identified with an infinite dimensional higher-
spin algebra denoted by $W_{\infty}[\hat{\lambda}]$ with the central charge

$\displaystyle c=6k_{\text{CS}}=\frac{3\ell_{\text{AdS}}}{2G_{N}}\,,$ (3.4)

which is the same as the famous Brown-Henneaux value [41]. The parameter
$\hat{\lambda}$ is the same as the one in (1.1), and the algebra can be
truncated to $W_{N^{\prime}}$-algebra with currents of spin
$s\,(=2,3,\ldots,N^{\prime})$ if we set $\hat{\lambda}=\pm N^{\prime}$. The
massive sector of the Prokushkin-Vasiliev theory consists of two scalar fields
$\phi_{\pm}$ with the same mass given in (1.1) as
$\ell^{2}_{\text{AdS}}\,m^{2}=-1+{\hat{\lambda}}^{2}$. Here we assign
$0<\hat{\lambda}<1$. The coupling to the gauge fields is described by the
equations of motion [20], see [42] for a good review.

The CFT dual to the higher-spin theory on AdS3 was proposed in [19] to be the
coset CFT (1.2), which describes a unitary minimal model with respect to
$W_{N}$-algebra. The central charge of the model is given by

$\displaystyle c=(N-1)\left(1-\frac{N(N+1)}{(N+k)(N+k+1)}\right)\,.$ (3.5)

The classical limit of higher-spin gravity corresponds to the ’t Hooft limit
of the CFT given by (1.3). The coset CFT (1.2) at the ’t Hooft limit is known
to possess the symmetry of $W_{\infty}[\lambda]$-algebra with the central
charge (3.5) [37, 43]. In such a limit the central charge becomes:

$\lim_{\rm{}^{\prime}tHooft}c\to N(1-\lambda^{2})+{\cal
O}(N^{0})\,,\quad\lambda=\frac{N}{N+k}\to\text{fixed}\,.$ (3.6)

The ’t Hooft parameter $\lambda$ defined above is identified with
$\hat{\lambda}$ appearing in the higher-spin theory, and in the following we
only use $\lambda$ instead of $\hat{\lambda}$. We will thus take (3.6) to be
the defining relation between the central charge $c$ and $N$ in the ’t Hooft
limit (1.3) and we will mostly be working in this limit throughout unless
otherwise stated. Furthermore using (3.4), we can directly relate the large
$c$/large $N$ expansion with the perturbative expansion in $G_{N}$ in AdS3.

The states of the coset CFT (1.2) are labeled by three Young diagrams
$(\Lambda_{+},\omega;\Lambda_{-})$. There are selection rules and field
identifications, and not all labels are allowed, see [44] for more details. In
this case, $\omega$ is uniquely fixed by the other two labels
$(\Lambda_{+};\Lambda_{-})$. The state labeled by $(\Lambda_{+};\Lambda_{-})$
has conformal weight

$\displaystyle
h_{(\Lambda_{+};\Lambda_{-})}=n+h^{(k)}_{\Lambda_{+}}+h^{(1)}_{\omega}-h^{(k+1)}_{\Lambda_{-}}\,,\quad
h_{\Lambda}^{(k)}=\frac{C_{2}(\Lambda)}{k+N}\,,\quad
C_{2}(\Lambda)=\frac{1}{2}(\Lambda,\Lambda+2\rho)\,.$ (3.7)

Here the positive integer $n$ represents how the denominator representation is
embedded into the numerator one. Moreover, $C_{2}(\Lambda)$ is the eigenvalue
of the quadratic Casimir operator and $\rho$ is the Weyl vector. The
fundamental states are given by those labeled by $(\square;0)$ and
$(0;\square)$ (and their conjugate representations) and other states may be
generated by fusing these fundamental states. We denote the corresponding
complex scalar operators by $\mathcal{O}_{\pm}(z,\bar{z})$, whose conformal
dimensions are

$\displaystyle\Delta_{\pm}^{(0)}=2h_{\pm}=1\pm\lambda$ (3.8)

at the ’t Hooft limit (1.3). They are dual to bulk scalar fields with standard
$(+)$ and alternative $(-)$ boundary conditions (see, e.g., [42]). The states
labeled by generic Young diagrams $(\Lambda_{+};\Lambda_{-})$ are interpreted
as bound states of these scalar fields. More precisely, to compare CFT results
with gravity computations later, let us introduce the expansion of the scaling
dimensions in terms of the inverse central charge $1/c$:

$\Delta_{\pm}=\sum_{r=0}^{\infty}\Delta_{\pm}^{(r)}c^{-r}\,,\quad
c\to\infty\,.$ (3.9)

As explained earlier, this is equivalent to an expansion in $G_{N}$ hence
(3.8) should strictly be taken as the leading classical value. We will apply
such an expansion when expanding the higher-point correlation functions in the
next section.

The coset CFT (1.2) has the symmetry of $W_{N}$-algebra as proven in [45] and
in the ’t Hooft limit, the algebra is generated by conserved currents
$J_{(s)}^{\text{AdS}}(z)$ (and anti-holomorphic currents
$\bar{J}_{(s)}^{\text{AdS}}(\bar{z})$) with $s=2,3,\ldots,\infty$. These
conserved currents are dual to higher-spin gauge fields $A$ (and $\bar{A}$) in
the bulk theory. The gauge configuration corresponding to the AdS background
is given by

$\displaystyle
A=e^{\rho}\,V_{1}^{2}\,dz+V^{2}_{0}\,d\rho\,,\qquad\bar{A}=e^{\rho}\,V_{-1}^{2}\,dz-V^{2}_{0}\,d\rho\,.$
(3.10)

Here $\rho$ relates to the radial coordinate $y$ in the Poincare metric (2.2)
via $y=e^{-\rho}$ and $(z,\bar{z})$ are the complex coordinates for the plane
parallel to the boundary. We consider a small fluctuation from the holomorphic
AdS background gauge field $A$, while keeping anti-holomorphic $\bar{A}$ fixed
as (see (3.20) of [42])

$\displaystyle\begin{aligned}
&A_{z}=e^{\rho}\,V_{1}^{2}+\frac{1}{cB^{(s)}}\,J_{(s)}^{\text{AdS}}(z)\,e^{-(s-1)\rho}\,V^{s}_{-(s-1)}\,,\\\
&A_{\bar{z}}=-\mu_{(s)}^{\text{AdS}}(z)\,e^{(s-1)\rho}\,V^{s}_{s-1}+\cdots\,.\end{aligned}$
(3.11)

Here we use the convention with

$\displaystyle\qquad
B^{(s)}=\frac{1}{2^{2s}\pi^{\frac{5}{2}}}\,\frac{\sin(\pi\lambda)}{\lambda(1-\lambda^{2})}\,\frac{\Gamma(s)\,\Gamma(s-\lambda)\,\Gamma(s+\lambda)}{\Gamma\left(s-\frac{1}{2}\right)}\,,$
(3.12)

where $\mu_{(s)}^{\rm AdS}(z)$ acts as the source term for $J_{(s)}^{\rm
AdS}(z)$. The generators of $W_{\infty}[\lambda]$ near the AdS boundary are
given by $J_{(s)}^{\text{AdS}}$, which can be identified with the generators
of symmetry algebra in the dual CFT.

Solving the bulk equations of motion, the authors in [42] obtained three-point
functions of scalar-scalar-higher spin current as

$\displaystyle\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,J_{(s)}^{\text{AdS}}(z_{3})\,\rangle=C_{\pm}^{(s)}\left(\frac{z_{12}}{z_{13}z_{23}}\right)^{s}\,\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,\rangle$
(3.13)

with

$\displaystyle
C_{\pm}^{(s)}=\frac{\eta^{s}_{\pm}}{2\pi}\,\frac{\Gamma(s)^{2}}{\Gamma(2s-1)}\frac{\Gamma(s\pm\lambda)}{\Gamma(1\pm\lambda)}\,.$
(3.14)

The phase factors can be chosen arbitrary but here we set $\eta^{s}_{+}=1$ and
$\eta^{s}_{-}=(-1)^{s}$ as in [46]. The three-point function with
$\bar{J}_{(s)}^{\text{AdS}}$ has an extra factor $(-1)^{s}$ along with $z$
replaced by $\bar{z}$. The higher-spin currents are normalized as

$\displaystyle\langle\,J_{(s)}^{\text{AdS}}(z)\,J_{(s)}^{\text{AdS}}(0)\,\rangle=\frac{cB^{(s)}}{z^{2s}}\,,\quad\langle\,\bar{J}_{(s)}^{\text{AdS}}(\bar{z})\,\bar{J}_{(s)}^{\text{AdS}}(0)\,\rangle=\frac{cB^{(s)}}{\bar{z}^{2s}}\,.$
(3.15)

They are evaluated by tree-level computations in the bulk theory, thus they
are valid at the leading order in $1/c$. It is also possible to confirm them
from the dual CFT viewpoints [42, 46], see also appendix A.2.

We should also mention about the two-point functions of scalar operators. The
coordinate dependence is fixed by conformal symmetry and the overall
normalization is just a convention. However, as in the case of two-point
functions of higher-spin currents (3.15), the canonical convention in the bulk
theory leads to

$\displaystyle\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,\rangle\sim\frac{c}{|z_{12}|^{4h_{\pm}}}\
,$ (3.16)

up to an multiplication of real number. We will use a different normalization
as defined below, but the original factor is quite important to perform an
analytic continuation from AdS3 to dS3, see [11] for a similar argument for
higher-spin dS4 holography.

###  3.2 Analytic continuation from AdS3 to dS3

We would like to introduce a dS3/CFT2 correspondence by taking a proper
analytic continuation of Gaberdiel-Gopakumar duality reviewed in the previous
subsection. For the bulk theory, we consider the Prokushkin-Vasiliev theory on
dS3, which is obtained by the map $\ell_{\text{AdS}}=-i\ell$ as in (2.8).
Because of this, we have to change the level of Chern-Simons theory as [47]

$\displaystyle
k_{\text{CS}}=\frac{\ell_{\text{AdS}}}{4G_{N}}=-i\frac{\ell}{4G_{N}}=-i\kappa$
(3.17)

with $\kappa\in\mathbb{R}$. The symmetry algebra at the future infinity of dS3
is $W_{\infty}[\lambda]$ with pure imaginary central charge as [4, 21]

$\displaystyle c=-i\frac{3\kappa}{2G_{N}}\equiv-ic^{(g)}\,.$ (3.18)

The bulk theory also includes two massive complex scalars with the mass

$\displaystyle\ell^{2}m^{2}=1-{\lambda}^{2}\,,$ (3.19)

which may be obtained from (1.1) by the map (2.8). We use $0<\lambda<1$ as
before, then we have positive $\ell^{2}m^{2}$. We assign the standard and
alternative boundary conditions near the future infinity, thus the conformal
dimensions of dual scalar operators are $\Delta^{(0)}_{\pm}=1\pm\lambda$ as
before.

We would like to claim that the dual CFT2 for dS3 is given by an analytic
continuation of the coset (1.2) as in [7]. The analytic continuation is
different from the one taken in [17, 18] and their differences will be
explained in section 5. As mentioned above in the ’t Hooft limit, the symmetry
organizing the CFT reduces to $W_{\infty}[\lambda]$ and has two parameters
$\lambda$ and $c$. The central charge is now taken to be purely imaginary
value $c=-ic^{(g)}$ with $c^{(g)}\in\mathbb{R}$. The original coset is labeled
by two parameters $N,k$, and these parameters can be written in terms of
$\lambda,c$ instead via (1.3) and (3.5). In practice, we compute physical
quantities by the coset CFT (or its dual description of Toda field theory as
derived in [48]), which may be expressed even with finite $N,k$. We then
rewrite them in terms of $\lambda,c$ via (1.3) and (3.5) and expand them in
terms of $1/c$.

####  3.2.1 Two-point correlators

We would like to begin by computing the two-point bulk correlators on dS3 at
late time. We will first consider the scalar fields and then move to the
symmetric tensor fields. For this, we use the formulas given in (2.17) and
(2.18). In the coordinate basis, taking the inverse of two-point function is
actually quite non-trivial. We first define the inverse function
$K_{h,\bar{h}}(x,y)$ which satisfies

$\displaystyle\int
d^{2}y\,K_{h,\bar{h}}(x,y)\,\frac{1}{(y-z)^{2h}(\bar{y}-\bar{z})^{2\bar{h}}}=\delta^{(2)}(x-z)\,.$
(3.20)

Explicitly we have,

$\displaystyle
K_{h,\bar{h}}(x,y)\equiv\frac{1}{\pi}k_{h,\bar{h}}\frac{1}{(x-y)^{2-2h}(\bar{x}-\bar{y})^{2-2h}}\,,\quad
k_{h,\bar{h}}=\frac{\Gamma(2-2\bar{h})}{\Gamma(2h-1)}=(-1)^{2(h-\bar{h})}\frac{\Gamma(2-2h)}{\Gamma(2\bar{h}-1)}\,,$
(3.21)

see, e.g., [49] for more extensive definitions. We have followed the notations
there. Notice that we can equivalently view $K_{h,\bar{h}}(x,y)$ as the two-
point correlation function of the so-called shadow operator, see (4.10) below.

Let us consider the two-point correlation functions of scalar fields at late
time in dS3. As discussed around (2.11), the scalar operators dual to the bulk
dS3 fields are related to those dual to the bulk AdS3 fields as

$\displaystyle\mathcal{O}_{\pm}=e^{i\frac{\pi}{2}(2h_{\pm}-2)}\mathcal{O}_{\pm}^{\text{AdS}}\,,\qquad\tilde{\mathcal{O}}_{\pm}=e^{i\frac{\pi}{2}(2h_{\pm}-2)}\bar{\mathcal{O}}_{\pm}^{\text{AdS}}\,,$
(3.22)

where we have set $\Delta_{\pm}^{(0)}=2h_{\pm}=1\pm\lambda$ and $d=2$. Note
that $\bar{\mathcal{O}}_{\pm}^{\text{AdS}}$ is complex conjugate of
$\mathcal{O}_{\pm}^{\text{AdS}}$, while $\tilde{\mathcal{O}}_{\pm}$ is not
complex conjugate of $\mathcal{O}_{\pm}$. The relations (3.22) and the
canonical normalization (3.16) imply that the two-point coefficient functions
are given by

$\displaystyle\begin{aligned}
\langle\,\mathcal{O}_{\pm}(z_{1})\,\tilde{\mathcal{O}}_{\pm}(z_{2})\,\rangle&=\left.e^{2i\pi
h_{\pm}}\,\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,\rangle\right|_{c\to-
ic^{(g)}}\\\ &=-\frac{ic^{(g)}}{c}\,e^{2i\pi
h_{\pm}}\,\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,\rangle\,.\end{aligned}$
(3.23)

In the following, we will set the overall normalization factors of the AdS/dS
two-point correlation functions to absorb $c/c^{(g)}$ respectively, such that
we are keeping only the overall phase after the analytic continuations (3.22):

$\displaystyle\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,\rangle=\frac{1}{|z_{12}|^{4h_{\pm}}}\,,\quad\langle\,\mathcal{O}_{\pm}(z_{1})\,\tilde{\mathcal{O}}_{\pm}(z_{2})\,\rangle=-ie^{2i\pi
h_{\pm}}\,\frac{1}{|z_{12}|^{4h_{\pm}}}\,.$ (3.24)

Now we can obtain the bulk two-point correlators at late time from the wave
functional formalism explicitly by applying the inversion formula (2.17). Here
the second expression in (3.24) plays the role of $C_{2}(\\{\vec{x}_{l}\\})$,
and we obtain

$\displaystyle\langle\,\phi_{\pm}(z_{1})\,\bar{\phi}_{\pm}(z_{2})\,\rangle=\frac{1}{2\sin(2\pi
h_{\pm})}K_{h_{\pm},h_{\pm}}(z_{1},z_{2})=a_{h_{\pm}}\frac{1}{\pi}\frac{\Gamma(2-2h_{\pm})}{\Gamma(2h_{\pm}-1)}\frac{1}{|z_{12}|^{4-4h_{\pm}}\,}\,.$
(3.25)

Here we have used $K_{h_{\pm},h_{\pm}}(z_{1},z_{2})$ defined in (3.21).
Compared with pure AdS3 case, we have extra phase factor

$\displaystyle a_{h_{\pm}}=\frac{1}{2\sin(2\pi h_{\pm})}\,.$ (3.26)

The factor of $2\sin(2\pi h_{\pm})$ in the denominator of (3.26) arises as we
need to include the contributions from $\Psi[\psi_{l}^{0}]$ and its complex
conjugate when extracting the bulk two-point correlators after the analytic
continuation. Note that the same factor $a_{h_{\pm}}=c_{2h_{\pm}}^{\text{dS-
AdS}}$ was inserted in order to relate bulk propagators on AdSd+1 and dSd+1 as
in (2.15) of [23]. In [23], the authors included the factor such that the both
two-point propagators have the correct normalization at small distance limit.
We also remark that the identical factor $c_{\Delta}^{\text{dS-AdS}}$ can be
used also for symmetric tensor fields with $\Delta$ and spin $s$.

We next move to the case of higher-spin gauge fields. The gauge configuration
corresponds to dS3 background is given by

$\displaystyle
A=i\eta^{-1}(V_{1}^{2}dz+iV_{0}^{2}d\eta)\,,\qquad\bar{A}=i\eta^{-1}(V_{-1}^{2}d\bar{z}-iV_{0}^{2}d\eta)\,,$
(3.27)

where we have set $y=e^{-\rho}$ and change as $y=-i\eta$ in (3.10). A small
fluctuation corresponding to the inclusion of holomorphic higher spin fields
can be expressed as

$\displaystyle\begin{aligned}
&A_{z}=i\eta^{-1}V_{1}^{2}+\left.\frac{i}{cB^{(s)}}\,J_{(s)}(z)(-\eta)^{s-1}V^{s}_{-(s-1)}\right|_{c\to-
ic^{(g)}}=i\eta^{-1}V_{1}^{2}-\frac{1}{c^{(g)}B^{(s)}}\,J_{(s)}(z)(-\eta)^{s-1}V^{s}_{-(s-1)}\,,\\\
&A_{\bar{z}}=-i\mu_{(s)}(z)\,(-\eta)^{-(s-1)}\,V^{s}_{s-1}+\cdots\,.\end{aligned}$
(3.28)

Assuming the identification of $A_{z}$ under the map from AdS3 to dS3, we have
the relation

$\displaystyle J_{(s)}(z)=e^{i\frac{\pi}{2}(s-2)}J_{(s)}^{\text{AdS}}(z)\,.$
(3.29)

The result is consistent with the one in (2.15) based on generic argument. In
a similar manner, we can obtain

$\displaystyle\tilde{J}_{(s)}(\bar{z})=e^{i\frac{\pi}{2}(s-2)}\bar{J}_{(s)}^{\text{AdS}}(\bar{z})\,.$
(3.30)

We again remark that while $\bar{J}_{(s)}^{\text{AdS}}(\bar{z})$ is complex
conjugate of $J^{\text{AdS}}_{(s)}(z)$ but $\tilde{J}_{(s)}(\bar{z})$ is not
complex conjugate of $J_{(s)}(z)$ due to the same phase assignments for the
both.

Using the relation (3.29), the two-point coefficient functions are written as

$\displaystyle\begin{aligned}
\langle\,J_{(s)}(z_{1})\,J_{(s)}(z_{2})\,\rangle&=\left.(-1)^{s}\,\langle\,J_{(s)}^{\text{AdS}}(z_{1})\,J_{(s)}^{\text{AdS}}(z_{2})\,\rangle\right|_{c\to-
ic^{(g)}}\\\
&=(-1)^{s+1}\frac{ic^{(g)}}{c}\,\,\langle\,J_{(s)}^{\text{AdS}}(z_{1})\,J_{(s)}^{\text{AdS}}(z_{2})\,\rangle\,.\end{aligned}$
(3.31)

We should add also the complex conjugate of the two-point function of
$\tilde{J}_{s}(\bar{z})$:

$\displaystyle\langle\,\tilde{J}_{(s)}(\bar{z}_{1})\,\tilde{J}_{(s)}(\bar{z}_{2})\,\rangle=(-1)^{s+1}\frac{ic^{(g)}}{c}\,\langle\,\bar{J}_{(s)}^{\text{AdS}}(\bar{z}_{1})\,\bar{J}_{(s)}^{\text{AdS}}(\bar{z}_{2})\,\rangle\,,$
(3.32)

where we have used (3.30). In contrast with the scalar two-point functions, we
do not change the normalization factors here to absorb $c/c^{(g)}$. As
suggested by the action of Chern-Simons theory, we need to multiply $(-1)$ to
convert left action to right action, see, e.g., [50], thus there is no overall
extra factor as $(-1)\cdot(-i)=i$. Applying the formula (2.18) and (3.21), the
bulk correlators of the higher-spin fields at late time are obtained as

$\displaystyle\begin{aligned}
\langle\mu_{(s)}(z_{1})\mu_{(s)}(z_{2})\rangle&=(-1)^{s+1}\frac{i}{2}\frac{1}{c^{(g)}B^{(s)}}K_{s,0}(z_{1},z_{2})\\\
&=a_{(s)}\frac{1}{c^{(g)}B^{(s)}\pi\Gamma(2s-1)}\frac{1}{(z_{12})^{2-2s}(\bar{z}_{12})^{2}}\,.\end{aligned}$
(3.33)

The factor

$\displaystyle a_{(s)}=(-1)^{s+1}\frac{i}{2}$ (3.34)

arises due to our prescription of analytic continuation as in the case of
scalar field. It does not match with the generic expression (3.26) since the
factor diverges at the massless limit with $\Delta=s+d-2=s\in\mathbb{N}$ for
$d=2$. This is however not a contradiction, since the massless limit is quite
delicate. For instance, let us consider a spin-one massive field on Lorentzian
dSd+1 spacetime. Its bulk two-point function may be found in (3.18)-(3.20) of
[51], but it diverges at the massless limit with $m\to 0$. In order to obtain
correct two-point function of massless spin-one gauge field, we may have to
work with the action of massless gauge field from the beginning as in section
4 of [51]. It is natural to expect that the factor (3.34) can be obtained by
working with massless higher-spin gauge fields from the beginning. It is an
important future problem to reproduce the factor (3.34) from the method of
[22, 23] for the case as well.

####  3.2.2 Three-point correlators

As in the AdS3 case, here we consider the basic three-point correlators in dS3
at the late time consisting of two scalar fields and one higher-spin gauge
field. Using the maps (3.22) and (3.29), the three-point coefficient functions
are related to (3.13) as:

$\displaystyle\begin{aligned}
\langle\,\mathcal{O}_{\pm}(z_{1})\,\tilde{\mathcal{O}}_{\pm}(z_{2})\,J_{(s)}(z_{3})\,\rangle&=-\left.e^{i\frac{\pi}{2}(4h_{\pm}+s)}\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(z_{2})\,J_{(s)}^{\text{AdS}}(z_{3})\,\rangle\right|_{c\to-
ic^{(g)}}\\\
&=i\,e^{i\frac{\pi}{2}(4h_{\pm}+s)}C_{\pm}^{(s)}\left(\frac{z_{12}}{z_{13}z_{23}}\right)^{s}\frac{1}{|z_{12}|^{4h_{\pm}}}\,.\end{aligned}$
(3.35)

In the same way, we find

$\displaystyle\begin{aligned}
\langle\,\mathcal{O}_{\pm}(z_{1})\,\tilde{\mathcal{O}}_{\pm}(z_{2})\,\tilde{J}_{(s)}(\bar{z}_{3})\,\rangle=i\,e^{i\frac{\pi}{2}(4h_{\pm}+s)}C_{\pm}^{(s)}\left(\frac{\bar{z}_{12}}{\bar{z}_{13}\bar{z}_{23}}\right)^{s}\frac{1}{|z_{12}|^{4h_{\pm}}}\end{aligned}$
(3.36)

from the maps (3.22) and (3.30). The sum of (3.35) and the complex conjugate
of (3.36) yields

$\displaystyle
2\sin\left((4h_{\pm}+s)\frac{\pi}{2}\right)C_{\pm}^{(s)}\left(\frac{z_{12}}{z_{13}z_{23}}\right)^{s}\frac{1}{|z_{12}|^{4h_{\pm}}}\,.$
(3.37)

Notice that the complex conjugation of (3.36) only depend on the holomorphic
coordinates $z_{ij}$ up to the scalar two-point function just like (3.35).

In order to obtain the corresponding dS bulk correlators from the coefficient
function $C_{3}(\\{\vec{x}_{l}\\})$, which is proportional to (3.35), we will
use the expression (2.20). For this, it is convenient to first consider the
inversion formula [49]

$\displaystyle\begin{aligned} \frac{k_{h,\bar{h}}}{\pi}\int
d^{2}y\frac{1}{(z-y)^{2-2h}(\bar{z}-\bar{y})^{2-2\bar{h}}}\mathcal{F}^{h,\bar{h}}_{12}(y,\bar{y})=\frac{\Gamma(1-h-h_{12})\Gamma(1-\bar{h}+\bar{h}_{12})}{\Gamma(\bar{h}+\bar{h}_{12})\Gamma(h-h_{12})}\mathcal{F}^{1-h,1-\bar{h}}_{12}(y,\bar{y}),\end{aligned}$
(3.38)

with

$\displaystyle\mathcal{F}^{h,\bar{h}}_{12}(z,\bar{z})=\frac{1}{z_{12}^{h_{1}+h_{2}-h}(z_{1}-z)^{h+h_{12}}(z_{2}-z)^{h-h_{12}}}\frac{1}{\bar{z}_{12}^{\bar{h}_{1}+\bar{h}_{2}-\bar{h}}(\bar{z}_{1}-\bar{z})^{\bar{h}+\bar{h}_{12}}(\bar{z}_{2}-\bar{z})^{\bar{h}-\bar{h}_{12}}}\,.$
(3.39)

Here we set $h_{12}=h_{1}-h_{2}$ and so on. Substituting kinematic dependence
in (3.37), we can now perform its inversion in turns. First we apply the two-
point inversion formula:

$\displaystyle\begin{aligned} &\int
d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}K_{h_{+},h_{+}}(z_{1},z_{1}^{\prime})K_{h_{+},h_{+}}(z_{2},z_{2}^{\prime})\left(\frac{z_{1^{\prime}2^{\prime}}}{z_{1^{\prime}3^{\prime}}z_{2^{\prime}3^{\prime}}}\right)^{s}\frac{1}{|z_{1^{\prime}2^{\prime}}|^{4h_{\pm}}}\\\
&\qquad\qquad=(-1)^{s+1}\frac{\Gamma(\mp\lambda)\Gamma(1\mp\lambda)}{\Gamma(1-s\pm\lambda)\Gamma(s\pm\lambda)}\left(\frac{z_{12}}{z_{13^{\prime}}z_{23^{\prime}}}\right)^{s}\frac{1}{|z_{12}|^{4h_{\mp}}}\,,\end{aligned}$
(3.40)

and next we evaluate

$\displaystyle\begin{aligned} \int
d^{2}z_{3}^{\prime}K_{s,0}(z_{3},z_{3}^{\prime})\left(\frac{z_{12}}{z_{13^{\prime}}z_{23^{\prime}}}\right)^{s}\frac{1}{|z_{12}|^{4h_{\mp}}}=\frac{(-1)^{s+1}}{\Gamma(s)^{2}}\left(\frac{z_{12}}{z_{13}z_{23}}\right)^{1-s}\left(\frac{\bar{z}_{12}}{\bar{z}_{13}\bar{z}_{23}}\right)\frac{1}{|z_{12}|^{4h_{\mp}}}\,.\end{aligned}$
(3.41)

Restoring the pre-factors in (3.37), as well as the normalization factors in
two-point functions (3.26) and (3.34), we finally obtain

$\displaystyle\begin{aligned}
&\langle\,\phi_{\pm}(z_{1})\,\bar{\phi}_{\pm}(z_{2})\,\mu_{(s)}(z_{3})\rangle\\\
&\quad=\lambda_{h_{\pm},h_{\pm},s}\frac{\Gamma(\mp\lambda)\Gamma(1\mp\lambda)}{\Gamma(s)^{2}\Gamma(1-s\pm\lambda)\Gamma(s\pm\lambda)}\frac{C_{\pm}^{(s)}}{c^{(g)}B^{(s)}}\left(\frac{z_{12}}{z_{13}z_{23}}\right)^{1-s}\left(\frac{\bar{z}_{12}}{\bar{z}_{13}\bar{z}_{23}}\right)\frac{1}{|z_{12}|^{4h_{\mp}}}\end{aligned}$
(3.42)

with

$\displaystyle\lambda_{h_{\pm},h_{\pm},s}=2a_{h_{\pm}}^{2}a_{(s)}\sin\left((4h_{\pm}+s)\frac{\pi}{2}\right)\,.$
(3.43)

Almost the same factor appears in (3.24) of [23] up to the subtlety associated
to the massless limit of higher-spin fields mentioned above, our result here
thus serves as a confirmation in an explicit model.

##  4 Four-point correlators dual to AdS3 and dS3

We next turn to computations of the bulk four-point correlation functions of
the following types:

$\displaystyle\langle\,\phi_{-}(\infty)\,\phi_{+}(1)\,\bar{\phi}_{+}(z)\,\bar{\phi}_{-}(0)\,\rangle\,,\quad\langle\,\phi_{\pm}(\infty)\,\bar{\phi}_{\pm}(1)\,\phi_{\pm}(z)\,\bar{\phi}_{\pm}(0)\,\rangle$
(4.1)

on dS3 at the late time. More precisely, we first consider the corresponding
four-point correlation functions computed exactly in the 2d coset CFTs dual to
higher-spin AdS3 gravity, and we then obtain their leading non-trivial
$1/c$-order contribution. In 2d CFTs, we normally consider the expansions of
correlation functions in terms of Virasoro conformal blocks, and in the limit
$c\to\infty$ while keeping fixed all the external scaling dimensions
(sometimes called as “LLLL” limit), it is well-known that these can be
replaced by global conformal blocks, see [52, 53] for recent progress. As
mentioned earlier, this also corresponds to the expansion in $G_{N}$ for the
bulk correlation functions, such that at the leading order we only need to
include tree-level interactions. We then perform the suitable analytic
continuations to obtain the corresponding results in dS3.

###  4.1 The case dual to AdS3

####  4.1.1 Four-point function $G_{-+}(z)$

Let us start with the first correlator in (4.1) which is of simpler structure.
This case was already analyzed in [7], but here we explain the analysis in
more details. We utilized the four-point function of operators dual to the
bulk scalars $\phi_{+}^{\text{AdS}}$ and $\phi_{-}^{\text{AdS}}$ on AdS3. They
are evaluated explicitly in [54] as

$\displaystyle\begin{aligned}
G_{-+}(z)&=\langle\,\mathcal{O}^{\text{AdS}}_{-}(\infty)\,\mathcal{O}^{\text{AdS}}_{+}(1)\,\bar{\mathcal{O}}^{\text{AdS}}_{+}(z)\,\bar{\mathcal{O}}^{\text{AdS}}_{-}(0)\,\rangle\\\
&=\frac{|z|^{\frac{2}{N}}}{|1-z|^{2\Delta_{+}}}\left|1+\frac{1-z}{Nz}\right|^{2}\,,\end{aligned}$
(4.2)

and the expression is exact for all finite $k,N$. Here we also list the exact
expressions for $\Delta_{\pm}$:

$\displaystyle\Delta_{+}=\frac{(N-1)(2N+1+k)}{N(N+k)}\,,\quad\Delta_{-}=\frac{(N-1)k}{N(N+k+1)}\,.$
(4.3)

Let us now consider the systematic $1/c$-expansion of $G_{-+}(z)$. As
explained above, the symmetry organizing the CFT is $W_{\infty}[\lambda]$,
which has two parameters $\lambda$ and $c$. As mentioned above, the coset
(1.2) has two parameters $N,k$, and we rewrite physical quantities instead in
terms $\lambda,c$. Explicitly we have

$\displaystyle
N=\frac{c}{1-\lambda^{2}}+\frac{1+\lambda+\lambda^{2}}{1+\lambda}+\mathcal{O}(c^{-1})\,.$
(4.4)

Therefore, taking $N$ large but keeping $\lambda=N/(N+k)$ finite and fixed in
the ’t Hooft limit, the $1/N$-expansion coincides with $1/c$-expansion at the
leading order, but there will be corrections at higher order expansions as we
will illustrate. The conformal dimensions $\Delta_{\pm}$ can be also expanded
in the form of (3.9) as:

$\displaystyle\Delta_{+}$
$\displaystyle=1+\lambda+\frac{\lambda^{2}-1}{c}+\mathcal{O}(c^{-2})=\Delta_{+}^{(0)}+\frac{\Delta^{(1)}_{+}}{c}+\mathcal{O}(c^{-2})\,,$
(4.5) $\displaystyle\Delta_{-}$
$\displaystyle=1-\lambda-\frac{(\lambda^{2}-1)^{2}}{c}+\mathcal{O}(c^{-2})=\Delta_{-}^{(0)}+\frac{\Delta^{(1)}_{-}}{c}+\mathcal{O}(c^{-2})\,.$
(4.6)

The leading $1/c$-expansion of $G_{-+}(z)$ (4.2) was obtained in [46] (see
also appendix A.2). In terms of global conformal block expansion, we have

$\displaystyle\begin{aligned}
G_{-+}(z)&=\frac{1}{|1-z|^{2\Delta_{+}^{(0)}+2\Delta_{+}^{(1)}/c}}\\\
&+\frac{1}{c}\frac{1}{|1-z|^{2\Delta_{+}^{(0)}}}\left[\sum_{s=2}^{\infty}(-1)^{s}\,\frac{C^{(s)}_{-}C^{(s)}_{+}}{B^{(s)}}\,(1-z)^{s}\,{}_{2}F_{1}(s,s;2s;1-z)+\text{c.c.}\right]+\mathcal{O}(c^{-2})\,.\end{aligned}$
(4.7)

Note that the hypergeometric function ${}_{2}F_{1}(s,s;2s,1-z)$ (and its
complex conjugate) corresponds to the global conformal block with intermediate
operator $J_{(s)}$ (and $\bar{J}_{(s)}$) which are conserved tensors
saturating the unitarity bound $\Delta=s$, as the result we have the
simplification or holomorphic factorization. This is also precisely what the
Virasoro blocks reduce to at this order in $1/c$-expansion.

As explained in subsection 2.2, we need to extract contributions which can be
obtained from lower-point functions. The first term in $\eqref{G-+exp}$
corresponds to the contribution from the products of disconnected two-point
functions,

$\displaystyle\langle\,\mathcal{O}_{-}^{\text{AdS}}(\infty)\bar{\mathcal{O}}_{-}^{\text{AdS}}(0)\,\rangle\langle\,\mathcal{O}_{+}^{\text{AdS}}(1)\bar{\mathcal{O}}_{+}^{\text{AdS}}(z)\,\rangle\,.$
(4.8)

More precisely at this order in $1/c$-expansion, the scaling dimension now
consists of the tree level $\Delta_{+}^{(0)}$ and the extra one-loop
correction $\Delta^{(1)}_{+}/c$. From the next line in (4.7), we can extract
the contribution of the form (2.28). For this, we evaluate

$\displaystyle\begin{aligned}
&\frac{1}{|1-z|^{2(1+\lambda)}}\hat{\mathcal{I}}_{s,0}(1-z,1-\bar{z})\\\
&\quad=\frac{k_{1-s,1}k_{s,0}}{\pi^{2}B^{(s)}}\int
d^{2}x\,d^{2}y\,\frac{\langle\,\mathcal{O}_{-}^{\text{AdS}}(\infty)\,\bar{\mathcal{O}}_{-}^{\text{AdS}}(0)\,J_{(s)}^{\text{AdS}}(x)\,\rangle\langle\,J_{(s)}^{\text{AdS}}(y)\,\mathcal{O}_{+}^{\text{AdS}}(1)\,\bar{\mathcal{O}}_{+}^{\text{AdS}}(z)\,\rangle}{{(x-y)^{2-2s}(\bar{x}-\bar{y})^{2}}}\,,\end{aligned}$
(4.9)

where we have used the fact that the bi-tensorial projector simplifies in two
dimensions. As in [55], it is convenient to define a shadow operator
$\tilde{\Phi}_{1-h,1-\bar{h}}$ for a primary operator $\Phi_{h,\bar{h}}$ by

$\displaystyle\tilde{\Phi}_{1-h,1-\bar{h}}(x)=\frac{k_{h,\bar{h}}}{\pi}\int
d^{2}y\,\frac{1}{(x-y)^{2-2h}(\bar{x}-\bar{y})^{2-2\bar{h}}}\,\Phi_{h,\bar{h}}(y)\,.$
(4.10)

The shadow operator $\tilde{\Phi}_{1-h,1-\bar{h}}$ satisfies the normalization
condition:

$\displaystyle\langle\,\tilde{\Phi}_{1-h,1-\bar{h}}(x)\,\Phi_{h,\bar{h}}(y)\,\rangle=\delta^{(2)}(x-y)$
(4.11)

for
$\langle\Phi_{h,\bar{h}}(z)\Phi_{h,\bar{h}}(0)\rangle=z^{-2h}\bar{z}^{-2\bar{h}}$
and $k_{h,\bar{h}}$ defined in (3.21). In terms of the shadow operator, the
integral (4.9) can be written as:

$\displaystyle\begin{aligned}
&\frac{1}{|1-z|^{2(1+\lambda)}}\hat{\mathcal{I}}_{s,0}(1-z,1-\bar{z})\\\
&\quad=\frac{\Gamma(2s)}{\pi B^{(s)}}\int
d^{2}x\,\langle\,\mathcal{O}_{-}^{\text{AdS}}(\infty)\,\bar{\mathcal{O}}_{-}^{\text{AdS}}(0)\,J_{(s)}^{\text{AdS}}(x)\,\rangle\,\,\langle\,\tilde{J}_{(s)}^{\text{AdS}}(x)\,\mathcal{O}_{+}^{\text{AdS}}(1)\,\bar{\mathcal{O}}_{+}^{\text{AdS}}(z)\,\rangle\,.\end{aligned}$
(4.12)

This is nothing but the definition of conformal partial wave up to an overall
factor given as:

$\displaystyle\hat{\mathcal{I}}_{s,0}(1-z,1-\bar{z})=(-1)^{s}\frac{C^{(s)}_{-}C^{(s)}_{+}}{B^{(s)}}\,\mathcal{I}_{s,0}(1-z,1-\bar{z})\,.$
(4.13)

The conformal partial wave $\mathcal{I}_{h,\bar{h}}(z,\bar{z})$ is introduced
in (A.26), and in the current case it is written explicitly as

$\displaystyle\begin{aligned}
\frac{\Gamma(s)^{2}}{\Gamma(2s)}\mathcal{I}_{s,0}(z,\bar{z})&=\frac{\Gamma(s)^{2}}{\Gamma(2s)}z^{s}{}_{2}F_{1}(s,s;2s;z)\\\
&+\frac{\Gamma(2s-1)}{\Gamma(s)^{2}}z^{1-s}{}_{2}F_{1}(1-s,1-s;2-2s;z)\bar{z}{}_{2}F_{1}(1,1;2;\bar{z})\,.\end{aligned}$
(4.14)

Replacing the conformal blocks by the conformal partial waves, we have

$\displaystyle G_{-+}(z)$
$\displaystyle=\frac{1}{|1-z|^{2\Delta_{+}^{(0)}+2\Delta_{+}^{(1)}/c}}$ (4.15)
$\displaystyle+\frac{1}{c}\frac{1}{|1-z|^{2{\Delta_{+}^{(0)}}}}\left[\sum_{s=2}^{\infty}(-1)^{s}\,\frac{C^{(s)}_{-}C^{(s)}_{+}}{B^{(s)}}\,\mathcal{I}_{s,0}(1-z,1-\bar{z})+(\lambda^{2}-1)\ln|z|^{2}\right]+\mathcal{O}(c^{-2})\,,$

see appendix A.3 for its derivation. The term proportional to $\ln|z|^{2}$ is
the contribution which cannot be obtained from lower-point functions, i.e.,
four-point contact diagrams.

####  4.1.2 Four-point functions $G_{\pm\pm}(z)$

We next move to the other correlators listed in (4.1). These have not been
dealt with in [7] since expressions become more complicated and contain two
possible exchange channels. However, we will see that similar analysis to the
previous case can be applied to these correlators as well. We use the four-
point functions of scalar operators dual to scalar fields
$\phi_{\pm}^{\text{AdS}}$ in Prokushkin-Vasiliev theory on AdS3. The
correlation functions are given by

$\displaystyle
G_{\pm\pm}(z)=\langle\,\mathcal{O}^{\text{AdS}}_{\pm}(\infty)\,\bar{\mathcal{O}}^{\text{AdS}}_{\pm}(1)\,\mathcal{O}^{\text{AdS}}_{\pm}(z)\,\bar{\mathcal{O}}^{\text{AdS}}_{\pm}(0)\,\rangle\,.$
(4.16)

They were also computed exactly in [54] with finite $N,k$ as

$\displaystyle\begin{aligned}
G_{++}(z)&=\frac{1}{|z(1-z)|^{2\Delta_{+}}}\left|(1-z)^{1+\lambda}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};-\lambda;z\right)\right|^{2}\\\
&+\frac{\mathcal{N}_{+}}{|z(1-z)|^{2\Delta_{+}}}\left|z^{1+\lambda}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;z\right)\right|^{2}\end{aligned}$
(4.17)

with the parameter:

$\displaystyle\mathcal{N}_{+}=-\frac{\Gamma\left(1+\lambda-\frac{\lambda}{N}\right)\,\Gamma(-\lambda)^{2}\,\Gamma\left(2+\lambda+\frac{\lambda}{N}\right)}{\Gamma\left(-1-\lambda-\frac{\lambda}{N}\right)\,\Gamma\left(\frac{\lambda}{N}-\lambda\right)\Gamma(2+\lambda)^{2}}\,,$
(4.18)

and

$\displaystyle\begin{aligned}
G_{--}(z)&=\frac{1}{|z(1-z)|^{2\Delta_{-}}}\left|(1-z)^{1-\lambda^{\prime}}\,{}_{2}F_{1}\left(1-\frac{\lambda^{\prime}}{N},\frac{\lambda^{\prime}}{N};\lambda^{\prime};z\right)\right|^{2}\\\
&+\frac{\mathcal{N}_{-}}{|z(1-z)|^{2\Delta_{-}}}\left|z^{1-\lambda^{\prime}}\,{}_{2}F_{1}\left(1-\frac{\lambda^{\prime}}{N},\frac{\lambda^{\prime}}{N};2-\lambda^{\prime};z\right)\right|^{2}\end{aligned}$
(4.19)

with the parameter:

$\displaystyle\mathcal{N}_{-}=-\frac{\Gamma\left(1-\lambda^{\prime}+\frac{\lambda^{\prime}}{N}\right)\,\Gamma(\lambda^{\prime})^{2}\,\Gamma\left(2-\lambda^{\prime}-\frac{\lambda^{\prime}}{N}\right)}{\Gamma\left(-1+\lambda^{\prime}+\frac{\lambda^{\prime}}{N}\right)\,\Gamma\left(-\frac{\lambda^{\prime}}{N}+\lambda^{\prime}\right)\,\Gamma(2-\lambda^{\prime})^{2}}\,.$
(4.20)

Here we have set

$\displaystyle\lambda^{\prime}=\frac{N}{k+N+1}\,.$

Note that it becomes identical to $\lambda$ in (1.3) at the leading ’t Hooft
limit. Even though it may not be obvious immediately, as shown in appendix
A.1, both $G_{\pm\pm}(z)$ are in fact invariant under the $s$-$t$ channel
crossing transformation $(z,\bar{z})\to(1-z,1-\bar{z})$. This also implies
that both correlators contain $s$\- and $t$-exchange channels to all orders of
$k$ and $N$.

In order to apply the method introduced in subsection 2.2, we expand the four-
point correlators in both of these channels. As in the previous case, we first
expand the four-point functions (4.16) in terms of the global conformal blocks
in both $s$\- and $t$-channels and here we do this as (see [46] and appendix
A.2) up to order $1/c$:

$\displaystyle\begin{aligned}
G_{\pm\pm}(z)&=\frac{1}{|z|^{2\Delta_{\pm}^{(0)}+2\Delta_{\pm}^{(1)}/c}}+\frac{1}{\left|1-z\right|^{{2\Delta_{\pm}^{(0)}+2\Delta_{\pm}^{(1)}/c}}}\\\
&+\frac{1}{c}\Biggl{[}\frac{1}{|z|^{2\Delta_{\pm}^{(0)}}}\sum_{s=2}^{\infty}\frac{(C^{(s)}_{\pm})^{2}}{B^{(s)}}\,z^{s}\,{}_{2}F_{1}(s,s;2s;z)\\\
&\quad\quad+\frac{1}{\left|1-z\right|^{2\Delta_{\pm}^{(0)}}}\sum_{s=2}^{\infty}\frac{(C^{(s)}_{\pm})^{2}}{B^{(s)}}\,(1-z)^{s}\,{}_{2}F_{1}(s,s;2s;1-z)+\text{c.c.}\Biggr{]}\\\
&+\frac{1}{c}\left[\frac{1}{|z|^{2\Delta_{\pm}^{(0)}}}g(\pm\lambda;\bar{z})+\text{c.c.}\right]+\mathcal{O}(c^{-2})\,,\end{aligned}$
(4.21)

with

$\displaystyle
g(\lambda;\bar{z})=-\frac{\epsilon}{1+\lambda}\left(\frac{\bar{z}}{1-\bar{z}}\right)^{1+\lambda}\,,\quad\epsilon=(1-\lambda^{2})\Gamma(2+\lambda)\,\Gamma(1-\lambda)\,.$
(4.22)

On the right hand side, the first and second terms are respectively the
products of disconnected two-point functions of scalar operators in $s$\- and
$t$-channels, while the third and fourth terms are respectively the exchange
of spin-$s$ tensor field in $s$\- and $t$-channels. The final line, when the
both terms are taken together, becomes crossing invariant and can be
attributed to the four point contact interactions. Notice that such a term
does not appear for $G_{-+}(z)$ correlation function. In order to precisely
extract the parts which cannot be obtained from lower-point coefficient
functions such as the contact interactions, we rewrite (4.21) in terms of
conformal partial waves instead. Here we introduce the corresponding conformal
partial wave:

$\displaystyle\begin{aligned}
&\frac{1}{|z|^{2(1\pm\lambda)}}\hat{\mathcal{I}}_{s,0}^{\pm}(z,\bar{z})\\\
&\quad=\frac{k_{1-s,1}k_{s,0}}{\pi^{2}B^{(s)}}\int
d^{2}x\,d^{2}y\,\frac{\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(\infty)\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(1)\,J_{(s)}^{\text{AdS}}(x)\,\rangle\,\langle\,J_{(s)}^{\text{AdS}}(y)\,\mathcal{O}_{\pm}^{\text{AdS}}(z)\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(0)\,\rangle}{(x-y)^{2-2s}(\bar{x}-\bar{y})^{2}}\\\
&\quad=\frac{\Gamma(2s)}{\pi B^{(s)}}\int
d^{2}x\,\langle\,\mathcal{O}_{\pm}^{\text{AdS}}(\infty)\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(1)\,J_{(s)}^{\text{AdS}}(x)\,\rangle\,\langle\,\tilde{J}_{(s)}^{\text{AdS}}(x)\,\mathcal{O}_{\pm}^{\text{AdS}}(z)\,\bar{\mathcal{O}}_{\pm}^{\text{AdS}}(0)\,\rangle\,,\end{aligned}$
(4.23)

where we have substituted the definition of shadow operator
$\tilde{J}_{(s)}^{\rm AdS}(x)$. The integral is proportional to the global
conformal partial wave:

$\displaystyle\hat{\mathcal{I}}_{s,0}^{\pm}(z,\bar{z})=\frac{(C^{(s)}_{\pm})^{2}}{B^{(s)}}\,\mathcal{I}_{s,0}(z,\bar{z})\,.$
(4.24)

Thus, we can rewrite (4.21) as

$\displaystyle\begin{aligned}
G_{\pm\pm}(z)&=\frac{1}{|z|^{2\Delta_{\pm}^{(0)}+2\Delta_{\pm}^{(1)}/c}}+\frac{1}{\left|1-z\right|^{2\Delta_{\pm}^{(0)}+2\Delta_{\pm}^{(1)}/c}}\\\
&+\frac{1}{c}\Biggl{[}\frac{1}{|z|^{2\Delta_{\pm}^{(0)}}}\left(\sum_{s=2}^{\infty}\frac{(C^{(s)}_{\pm})^{2}}{B^{(s)}}\mathcal{I}^{\pm}_{s,0}(z,\bar{z})+(\lambda^{2}-1)\ln(1-\bar{z})\right)\\\
&\quad\quad+\frac{1}{|1-z|^{2\Delta_{\pm}^{(0)}}}\left(\sum_{s=2}^{\infty}\frac{(C^{(s)}_{\pm})^{2}}{B^{(s)}}\mathcal{I}^{\pm}_{s,0}(1-z,1-\bar{z})+(\lambda^{2}-1)\ln\bar{z}\right)+\text{c.c.}\Biggr{]}\\\
&+\frac{1}{c}\left[\frac{1}{|z|^{2\Delta_{\pm}^{(0)}}}g(\pm\lambda;\bar{z})+\text{c.c.}\right]+\mathcal{O}(c^{-2})\,.\end{aligned}$
(4.25)

The contact interaction terms which cannot be obtained from lower-point
coefficient functions are thus summarized as

$\displaystyle\begin{aligned}
\mathcal{A}^{\pm}_{4,c}(z,\bar{z})&=\frac{1}{c}(\lambda^{2}-1)\left[\frac{1}{|z|^{2\Delta_{\pm}^{(0)}}}\ln|1-z|^{2}+\frac{1}{\left|1-z\right|^{2\Delta_{\pm}^{(0)}}}\ln|z|^{2}\right]\\\
&-\frac{1}{c}\frac{\epsilon}{1\pm\lambda}\left[\frac{1}{\left(z(1-\bar{z})\right)^{\Delta_{\pm}^{(0)}}}+\frac{1}{\left(\bar{z}(1-z)\right)^{\Delta_{\pm}^{(0)}}}\right]+\mathcal{O}(c^{-2})\end{aligned}$
(4.26)

as extracted from boundary CFT.

###  4.2 Analytic continuation from AdS3 to dS3

In the previous subsection, we obtain the terms which cannot be obtained from
lower-point coefficient functions, i.e., $n$-point AdS bulk contact diagrams.
In order to obtain dS four-point coefficient functions from these terms, we
have to perform an analytic continuation from AdS3 to dS3. We have already
defined how individual operators are mapped from AdS3 to dS3 as in (3.22) and
(3.29) in general. We have also observed that we have extra $-i$ factor if
correlation functions are proportional to $c$ due to the map which replaces
$c$ by $-ic^{(g)}$ with real $c^{(g)}$. Up to now, we have considered two-
point functions of scalar operators and conserved currents, and the three-
point functions of scalar-scalar-higher spin currents. We have observed that
all correlation functions are proportional to $c$, which induces $-i$ factor
by mapping from AdS3 to dS3.

Let us consider connected $n$-point functions of certain operators in the 2d
CFT dual to higher-spin AdS3 gravity in general. Applying an operator product
expansion (OPE), we can replace two neighboring local operators by another
one. The coefficients of OPE are thus given by three-point functions divided
by two-point function. Therefore, repeating OPEs, any connected $n$-point
function can be written as a product of $(n-2)$ three-point functions divided
by $(n-3)$ two-point functions, regardless of OPE channels. Thus, the
connected $n$-point function is always proportional to $c$. After mapping from
AdS3 to dS3, we have an extra phase factor $-i$ for all connected $n$-point
function. This is consistent with the fact that the coupling constant in the
bulk theory is organized by $k_{\text{CS}}$ or $-i\kappa$ after analytic
continuation.

####  4.2.1 Four-point correlator from $G_{-+}(z)$

We again begin with the first correlator in (4.1). According to the above
arguments, the connected part of four-point correlation functions are related
as

$\displaystyle\begin{aligned}
\langle\,\mathcal{O}_{-}(\infty)\,\mathcal{O}_{+}(1)\,\tilde{\mathcal{O}}_{+}(z)\,\tilde{\mathcal{O}}_{-}(0)\,\rangle&=-ie^{2\pi
i(h_{+}+h_{-})}\langle\,\mathcal{O}^{\text{AdS}}_{-}(\infty)\,\mathcal{O}^{\text{AdS}}_{+}(1)\,\bar{\mathcal{O}}^{\text{AdS}}_{+}(z)\,\bar{\mathcal{O}}^{\text{AdS}}_{-}(0)\,\rangle\\\
&=-i\langle\,\mathcal{O}^{\text{AdS}}_{-}(\infty)\,\mathcal{O}^{\text{AdS}}_{+}(1)\,\bar{\mathcal{O}}^{\text{AdS}}_{+}(z)\,\bar{\mathcal{O}}^{\text{AdS}}_{-}(0)\rangle\,.\end{aligned}$
(4.27)

We have used (3.22) and $h_{+}+h_{-}=1$ for $d=2$ to obtain the factor
$e^{2\pi i(h_{+}+h_{-})}=1$. Another factor $-i$ comes from the fact that the
connected four-point correlators are of order $\mathcal{O}(c)$ as explained
above. The term proportional to $\ln|z|^{2}$ in (4.15) is a real function.
Therefore, due to the $-i$ factor in (4.27), we conclude

$\displaystyle\langle\mathcal{O}^{4}\rangle_{c}=0$ (4.28)

in (2.22), which only contains real quantities by definition. Therefore, in
this case, there are no terms corresponding to the bulk contact diagram as
already mentioned in [7]. The remaining part
$\langle\mathcal{O}^{4}\rangle_{d}$ exclusively corresponds to the sum of bulk
exchange diagrams. Using the expression of conformal partial waves given in
(4.9), we can obtain the resultant phase factor after the map from AdS3 to
dS3. The result is

$\displaystyle\langle\mathcal{O}^{4}\rangle_{d}=\sum_{s=2}^{\infty}\frac{\lambda_{h_{+},h_{+},s}\lambda_{h_{-},h_{-},s}}{a_{h_{+}}^{2}a_{h_{-}}^{2}a_{(s)}c^{(g)}}\frac{1}{|1-z|^{2(1+\lambda)}}(\hat{\mathcal{I}}_{s,0}(1-z,1-\bar{z})+\hat{\mathcal{I}}_{0,s}(1-z,1-\bar{z}))\,,$
(4.29)

where $a_{h_{\pm}}$, $a_{(s)}$ and $\lambda_{h_{\pm},h_{\pm},s}$ are defined
in (3.26), (3.34) and (3.43) respectively. Applying (3.40) as in (2.21), we
find

$\displaystyle\begin{aligned}
&\langle\,\phi_{-}(\infty)\,\phi_{+}(1)\,\bar{\phi}_{+}(z)\,\bar{\phi}_{-}(0)\,\rangle_{c}\\\
&\quad=\sum_{s=2}^{\infty}\frac{\lambda_{h_{+},h_{+},s}\lambda_{h_{-},h_{-},s}}{a_{(s)}}\frac{\mathcal{B}_{-+}(s)}{|1-z|^{2(1-\lambda)}}(\hat{\mathcal{I}}_{s,0}(1-z,1-\bar{z})+\hat{\mathcal{I}}_{0,s}(1-z,1-\bar{z}))\end{aligned}$
(4.30)

with

$\displaystyle\mathcal{B}_{-+}(s)=\frac{\Gamma(-\lambda)\Gamma(1-\lambda)\Gamma(\lambda)\Gamma(1+\lambda)}{\Gamma(1-s+\lambda)\Gamma(s+\lambda)\Gamma(1-s-\lambda)\Gamma(s-\lambda)}\,.$
(4.31)

The result is consistent with the generic expression, say, in (4.39) of [23].
Here the overall factor $\mathcal{B}_{-+}(s)$ was inherited from the pre-
factor in three-point inversion formula (3.42).

####  4.2.2 Four-point correlator from $G_{\pm\pm}(z)$

We next consider the analytic continuation of the other more complicated
correlation functions in (4.1). Applying the similar procedures as above, the
relation among the connected part of four-point correlators is obtained as

$\displaystyle\langle\,\mathcal{O}_{\pm}(\infty)\,\tilde{\mathcal{O}}_{\pm}(1)\,\mathcal{O}_{\pm}(z)\,\tilde{\mathcal{O}}_{\pm}(0)\,\rangle=-ie^{4\,i\,\pi
h_{\pm}}\,\langle\,\mathcal{O}^{\text{AdS}}_{\pm}(\infty)\,\bar{\mathcal{O}}^{\text{AdS}}_{\pm}(1)\,\mathcal{O}^{\text{AdS}}_{\pm}(z)\,\bar{\mathcal{O}}^{\text{AdS}}_{\pm}(0)\,\rangle\,,$
(4.32)

which leads to the contributions of the four point contact diagram being:

$\displaystyle\langle\mathcal{O}^{4}\rangle_{c}=-2\sin(4\pi
h_{\pm})\mathcal{A}^{\pm}_{4,c}(z,\bar{z})\,,$ (4.33)

where $\mathcal{A}^{\pm}_{4,c}(z,\bar{z})$ is given in (4.26). The remaining
exchange part (2.23) can be obtained from the expression of conformal partial
waves in (4.23) as

$\displaystyle\begin{aligned}
\langle\mathcal{O}^{4}\rangle_{d}&=\Biggl{[}\frac{1}{|z|^{2(1\pm\lambda)}}\sum_{s=2}^{\infty}\frac{(\lambda_{h_{\pm},h_{\pm},s})^{2}}{a_{h_{\pm}}^{4}a_{(s)}c^{(g)}}(\hat{\mathcal{I}}_{s,0}^{\pm}(z,\bar{z})+\hat{\mathcal{I}}_{0,s}^{\pm}(z,\bar{z}))\\\
&+\frac{1}{|1-z|^{2(1\pm\lambda)}}\sum_{s=2}^{\infty}\frac{(\lambda_{h_{\pm},h_{\pm},s})^{2}}{a_{h_{\pm}}^{4}a_{(s)}c^{(g)}}(\hat{\mathcal{I}}_{s,0}^{\pm}(1-z,1-\bar{z})+\hat{\mathcal{I}}_{0,s}^{\pm}(1-z,1-\bar{z}))\Biggr{]}\,.\end{aligned}$
(4.34)

The bulk four-point correlators in late time dS3 can thus be divided into two
parts as

$\displaystyle\begin{aligned}
&\langle\,\phi_{\pm}(\infty)\,\bar{\phi}_{\pm}(1)\,\phi_{\pm}(z)\,\bar{\phi}_{\pm}(0)\,\rangle=\langle\phi^{4}\rangle_{c}+\langle\phi^{4}\rangle_{d}\,.\end{aligned}$
(4.35)

The first part can be rewritten in terms of the following inversion integral:

$\displaystyle\langle\phi^{4}\rangle_{c}=-2\sin(4\pi
h_{\pm})a_{h_{\pm}}^{4}\int\left[\prod_{i=1}^{4}d^{2}z_{i}^{\prime}K(z_{i},z_{i}^{\prime})\right]\frac{|z^{\prime}|^{2(1\pm\lambda)}}{|z_{1^{\prime}2^{\prime}}|^{2(1\pm\lambda)}|z_{3^{\prime}4^{\prime}}|^{2(1\pm\lambda)}}\mathcal{A}^{\pm}_{4,c}(z^{\prime})$
(4.36)

with cross ratio
$z^{\prime}=(z_{1^{\prime}2^{\prime}}z_{3^{\prime}4^{\prime}})/(z_{1^{\prime}3^{\prime}}z_{2^{\prime}4^{\prime}})$
and $(z_{1},z_{2},z_{2},z_{4})=(\infty,1,z,0)$. It is possible to perform the
integration over $z_{i}^{\prime}$, but we do not do so since it is not so
illuminating.

For the second part, the integration over $z_{i}^{\prime}$ in (2.21) can be
easily performed as

$\displaystyle\langle\phi^{4}\rangle_{d}$
$\displaystyle=\Biggl{[}\frac{1}{|z|^{2(1\mp\lambda)}}\sum_{s=2}^{\infty}\frac{(\lambda_{h_{\pm},h_{\pm},s})^{2}}{a_{(s)}c^{(g)}}\mathcal{B}_{\pm}(s)(\hat{\mathcal{I}}_{s,0}^{\pm}(z,\bar{z})+\hat{\mathcal{I}}_{0,s}^{\pm}(z,\bar{z}))$
$\displaystyle+\frac{1}{|1-z|^{2(1\mp\lambda)}}\sum_{s=2}^{\infty}\frac{(\lambda_{h_{\pm},h_{\pm},s})^{2}}{a_{(s)}c^{(g)}}\mathcal{B}_{\pm}(s)(\hat{\mathcal{I}}_{s,0}^{\pm}(1-z,1-\bar{z})+\hat{\mathcal{I}}_{0,s}^{\pm}(1-z,1-\bar{z}))\Biggr{]}$
(4.37)

with

$\displaystyle\mathcal{B}_{\pm}(s)=\left[\frac{\Gamma(\mp\lambda)\Gamma(1\mp\lambda)}{\Gamma(1-s\pm\lambda)\Gamma(s\pm\lambda)}\right]^{2}\,.$
(4.38)

This is again consistent with the result of [23]. Note that we have now the
explicit expression of the contribution from bulk four-point contact diagram
(4.36), which is not available from generic arguments previously.

Let us end this section about the four-point correlation functions up to the
first order in ${1}/{c}$\- or ${1}/{c^{(g)}}$-expansions by outlining the
explicit construction of their holographic dual configurations. In particular,
we focus on the terms, which are further expanded in terms of the conformal
partial waves for the exchange of holomorphic conserved tensor operators.
These terms correspond to the tree level bulk exchange diagrams. In [66],
inspired by the split representation of the AdS harmonic functions, so-called
geodesic three-point Witten diagram was introduced, whose interaction vertex
moves along the geodesic connecting two of the three boundary points. Such an
object is purely kinematical and differs from the usual three-point Witten
diagram by an overall factor which can be attributed to the strong coupling
OPE coefficient. We can thus use the result in appendix B.2, which applies the
bulk to boundary propagator in the holographic gauge to reproduce the three-
point correlation function of scalar-scalar-higher spin currents including the
desired OPE coefficient, see (B.28). In appendix B.3, the holomorphically
factorizable two-dimensional global conformal block was further expressed in
terms of an integral over two copies of three point geodesic Witten diagrams,
utilizing the holomorphic polarization vector, see (B.40). Combining these
results, along with the analytic continuation of the bulk to boundary
propagators in going from AdS to dS (c.f. [23]), we can directly reconstruct
the holographic configurations in both AdS and dS spacetimes for these
correlation functions.

##  5 Example of other late-time correlator

Up to now, we have only examined late-time correlation functions on the pure
dS3 spacetime at the leading order in the Newton constant $G_{N}$ or the
inverse of central charge $1/c$ by taking the ’t Hooft limit. Our explicit
formulation however is not restricted to these cases, and it can be applied to
more generic situations. In order to illustrate this, we examine two
interesting examples. In this section, we analyze one example that is a two-
point correlator of scalar field on dS3 conical geometry. In general, the
scalar two-point correlator on a geometry can be used to extract properties of
the geometry. In the next section, we consider another example that is a type
of one-loop correction to two-point correlator on pure dS3 spacetime. It is
quite important problem to deal with $1/c$-corrections systematically.

In order to compute such bulk correlators, it is convenient to utilize the
limit used in [17, 18]. So far, we have examined the ‘t Hooft limit where
$N,k\gg 1$ but keep $\lambda=N/(N+k)$ of order $\mathcal{O}(N^{0})$. Instead
of this, we consider dS3 holography at the regime where $k\sim-1-N$ in this
section.222The limit taken in [17, 18] is $k\to-N$, which is slightly
different from (5.5). If we write $k$ in terms of $c$, then there are two
branches and the difference appears due to the choice of branch, see, e.g.,
[56]. In AdS3 holography, a similar limit was considered in [25, 43, 56]. In
the case, the higher-spin gravity is described by
$\text{SL}(N)\times\text{SL}(N)$ Chern-Simons gauge theory. Due to the
truncation of spin, the gravity theory is more tractable than that with
infinite dimensional gauge algebra $\mathfrak{hs}[\lambda]$ appeared in the
case with ’t Hooft limit (1.3). Moreover, we can argue that some states of the
coset CFT (1.2) in the limit $k\to-1-N$ have conformal dimensions of order
$\mathcal{O}(c)$, which have been identified with conical defect geometry in
[25, 56] for AdS3 and in [17, 18] for dS3. In the next subsection, we review
the dS3/CFT2 correspondence with the critical limit of the coset (1.2)
introduced in [17, 18]. In subsection 5.2 we compute bulk two-point
correlators on dS3 conical defect geometry at late time.

###  5.1 Higher-spin dS3/CFT2 with truncated spin

We begin by reviewing the higher-spin AdS3 holography with the limit of [25,
43, 56] and its analytic continuation to the higher-spin dS3 holography as in
[17, 18]. We consider the coset model (1.2) with finite $N$, which possesses
$W_{N}$-algebra symmetry with spin-$s$ currents $(s=2,3,\ldots,N)$. The
central charge is bounded as $c<N-1$ for integer $k$, which can be seen from
(3.5). In order to be dual to classical gravity, the central charge should be
large. In [25, 43, 56], the large central charge is realized by performing an
analytic continuation of $k$ as

$\displaystyle k=-1-N+\frac{N(N^{2}-1)}{c}+\mathcal{O}(c^{-2})\,.$ (5.1)

Because of this analytic continuation, the unitarity of the theory is lost.

The conformal dimensions of operators $\mathcal{O}_{\pm}$ are given in (4.3).
In the limit of (5.1), they become

$\displaystyle\Delta_{+}=1-N-\frac{(-1+N^{2})^{2}}{c}+\mathcal{O}(c^{-2})\,,\quad\Delta_{-}=-\frac{c}{N^{2}}+\mathcal{O}(c^{0})\,.$
(5.2)

Note that the scaling dimensions are negative, which is a consequence of non-
unitarity. The operator $\mathcal{O}_{+}$ is dual to a perturbative matter,
while the other operator $\mathcal{O}_{-}$ is dual to a non-perturbative
gravity solution. More generically, the operators of the coset are labeled by
two Young diagrams $(\Lambda_{+};\Lambda_{-})$ as explained in subsection 3.1.
At the limit (5.1), the conformal dimension of operator labeled by
$(0;\Lambda_{-})$ become

$\displaystyle\Delta_{(0;\Lambda_{-})}=2h_{(0;\Lambda_{-})}=-\frac{2c\,C_{2}(\Lambda_{-})}{N(N^{2}-1)}+\mathcal{O}(c^{0})\,,$
(5.3)

where (3.7) has been used and $C_{2}(\Lambda_{-})$ defines the eigenvalue of
the quadratic Casimir operator.

For real $c$, the dual gravity is supposed to be
$\text{SL}(N)\times\text{SL}(N)$ Chern-Simons gauge theory described by the
action (3.1) with real $k_{\text{CS}}$. A complex matter with mass

$\displaystyle\ell^{2}_{\text{AdS}}m^{2}=N^{2}-1$ (5.4)

is coupled with the gauge fields. The coset operator labeled by
$(0;\Lambda_{-})$ is dual to conical defect geometry with the same label
$\Lambda_{-}$ [25, 56]. Generic coset operator labeled by
$(\Lambda_{+};\Lambda_{-})$ is then dual to a bound state of the conical
geometry labeled by $\Lambda_{-}$ and composites of matter fields
corresponding to $\Lambda_{+}$.

As in [17, 18], we perform further analytic continuation in order to map from
AdS3/CFT2 to dS3/CFT2. In the CFT side, we have proposed to take the limit
(5.1) but with $c=-ic^{(g)}$ $(c^{(g)}\in\mathbb{R})$. This is motivated by
the fact that higher-spin dS3 gravity can be obtained by taking
$k_{\text{CS}}=-i\kappa$ with $\kappa\in\mathbb{R}$ [47] and the asymptotic
symmetry near the boundary is given by $W_{N}$-algebra with central charge
$c=6k_{\text{CS}}=-i6\kappa$. In this way, we could propose a dS3/CFT2
correspondence. The gravity side is the $\text{SL}(N)\times\text{SL}(N)$
Chern-Simons gauge theory with level $-i\kappa$ $(\kappa\in\mathbb{R})$. dS3
is realized as a solution to the equations of motion, and the level is written
as $\kappa=\ell/4G_{N}$. The CFT side is given by the coset model (1.2) with
the limit

$\displaystyle k=-1-N+i\frac{N(N^{2}-1)}{c^{(g)}}+\mathcal{O}(c^{(g)-2})\,.$
(5.5)

As evidence, gravity partition functions were computed from the both sides and
the two computations were found to agree with each other [17, 18]. In these
papers, other supporting arguments were also provided.

###  5.2 Two-point correlator on dS3 conical defect

In this subsection, we compute bulk two-point function of scalar field

$\displaystyle\langle\phi_{+}(z_{1})\bar{\phi}_{+}(z_{2})\rangle_{\Lambda_{-}}$
(5.6)

at late time but on dS3 conical defect geometry instead from dual CFT (1.2).
The conical defects are obtained by the solutions of
$\text{SL}(N)\times\text{SL}(N)$ Chern-Simons gravity. The conical defects on
Euclidean AdS3 have been classified in [25], and those on Lorentzian dS3 could
be obtained by an analytic continuation.

We first consider dS3 solution corresponding to the principal embedding of
$\mathfrak{sl}(2)$ into $\mathfrak{sl}(N)$. Following the notation in [17,
18], the generators of $\mathfrak{su}(2)$ are expressed by

$\displaystyle
L_{3}=\sum_{i=1}^{N}\rho_{i}e_{i,i}\,,\quad\text{tr}\,L_{3}L_{3}=\frac{N(N^{2}-1)}{12}\,(\equiv\epsilon_{N})$
(5.7)

and $L_{1},L_{2}$ satisfying $[L_{a},L_{b}]=\frac{i}{2}\epsilon_{abc}L_{c}$.
Here we use $N\times N$ matrix
$(e_{i,j})_{k}^{~{}l}=\delta_{i,k}\delta_{j}^{~{}l}$. In this notation, the
level of Chern-Simons theory is related to Newton constant as
$\kappa=\ell/(8G_{N}\epsilon_{N})$. The metric can be read off from gauge
fields as

$\displaystyle
g_{\mu\nu}=-\frac{\ell^{2}}{4\epsilon_{N}}\text{tr}\,[(A_{\mu}-\bar{A}_{\mu})(A_{\nu}-\bar{A}_{\nu})]\,.$
(5.8)

Fixing a gauge, we can put the gauge fields into the form

$\displaystyle A=e^{-i\theta L_{0}}a(z)e^{i\theta
L_{0}}+iL_{0}d\theta\,,\quad\bar{A}=e^{i\theta
L_{0}}\bar{a}(\bar{z})e^{-i\theta L_{0}}-iL_{0}d\theta$ (5.9)

with $a=a_{+}(\mathrm{d}\phi+i\mathrm{d}\tau)$ and
$\bar{a}=\bar{a}_{-}(\mathrm{d}\phi-i\mathrm{d}\tau)$. With the gauge
configuration

$\displaystyle a_{+}=iL_{1}\,,\quad\bar{a}_{-}=iL_{1}\,,$ (5.10)

we find

$\displaystyle ds^{2}=\ell^{2}(d\theta^{2}-\cos^{2}\theta
d\tau^{2}+\sin^{2}\theta d\phi^{2})\,.$ (5.11)

This is a metric for a static patch of dS3. If we perform a Wick rotation as
$\tau\to-it_{\text{E}}$, then the metric is for three-sphere as utilized in
[17, 18].

We then consider more generic gauge fields but corresponding to geometry
without any rotation. Following [17, 18] (motivated by the AdS case in [25]),
we assign the condition of trivial holonomy around $\phi$-cycle around
$\theta=0$. We define the holonomy matrix by

$\displaystyle\text{Hol}_{\phi}(a_{+})=\mathcal{P}\exp\left(\int_{\theta=0}a_{+}\mathrm{d}\phi\right)\,,$
(5.12)

where $\mathcal{P}$ represents the path ordering. We would like to require the
trivial holonomy condition. The gauge group is now SL$(N,\mathbb{C})$, so the
center of the group is $Z_{N}$. Therefore, we require that

$\displaystyle\text{Hol}_{\phi}(a_{+})\propto e^{-\frac{2\pi
im}{N}}\mathbbm{1}\,,\quad m\in\mathbb{Z}_{N}\,.$ (5.13)

The situation is the same as in the case of Euclidean AdS3 analyzed in [25],
since the Lorentzian dS3 is related to the Euclidean AdS3 by a simple analytic
continuation $\ell^{\text{AdS}}=-i\ell$ (2.8) or $k_{\text{CS}}=-i\kappa$
(3.17). Note that the conical defects analyzed in [17, 18] are solutions in
$\text{SU}(2)\times\text{SU}(2)$ Chern-Simons theory, thus the trivial
holonomy condition is different from the current case. For the difference
between Lorentzian and Euclidean AdS3 cases, see [25].

Denoting the eigenvalues of $a_{+}$ as $in_{j}$, the triviality condition
becomes

$\displaystyle n_{j}=m_{j}-\frac{m}{N}\,,\quad m_{j}\in\mathbb{Z}\text{ for
all }j\,.$ (5.14)

Due to the diagonalizability, all $n_{j}$ should be distinct and we can set
$n_{1}>n_{2}>\cdots>n_{N}$. These numbers represent a Young diagram with
$r_{j}$ box in $j$-th row, and relation of parameters is

$\displaystyle n_{j}=r_{j}+\frac{N+1}{2}-\frac{B}{N}-j\,,\quad
B=\sum_{j}r_{j}\,.$ (5.15)

In this way, we have constructed conical defect geometry satisfying the
trivial holonomy condition as in the case of AdS analyzed in [25], and the
geometry is labeled by a Young diagram.

We would like to next evaluate bulk two-point correlator (5.6) on dS3 conical
geometry at late time. For this, we need the AdS counterpart of the
correlator. This is given by four-point function with insertions of
$\mathcal{O}_{+}^{\text{AdS}}$ and
$\mathcal{O}_{(0;\Lambda_{-})}^{\text{AdS}}$ and their conjugates, i.e.,

$\displaystyle
G(z,\bar{z})=\langle\mathcal{O}^{\text{AdS}}_{(0;\Lambda_{-})}(\infty)\bar{\mathcal{O}}^{\text{AdS}}_{+}(1)\mathcal{O}^{\text{AdS}}_{+}(z)\bar{\mathcal{O}}^{\text{AdS}}_{(0;\Lambda_{-})}(0)\rangle\,.$
(5.16)

In the limit (5.1), it becomes a heavy-heavy-light-light correlator.
Therefore, the correlator can be computed by the bulk scalar propagator on the
conical defect labeled by $\Lambda_{-}$ as [57]

$\displaystyle
G(z,\bar{z})=\left|(N-1)!z^{\frac{N-1}{2}}\sum_{j=1}^{N}\frac{z^{n_{j}}}{\prod_{l\neq
j}(n_{l}-n_{j})}\right|^{2}\,.$ (5.17)

We can obtain the above expression from dual CFT as well. Restoring the
coordinate dependence, we have:

$\displaystyle\begin{aligned}
G(z_{i},\bar{z}_{i})&=\langle\mathcal{O}^{\text{AdS}}_{(0;\Lambda_{-})}(z_{1})\bar{\mathcal{O}}^{\text{AdS}}_{+}(z_{2})\mathcal{O}^{\text{AdS}}_{+}(z_{3})\bar{\mathcal{O}}^{\text{AdS}}_{(0;\Lambda_{-})}(z_{4})\rangle\\\
&=\left|\frac{z}{z_{12}z_{34}}\right|^{\Delta_{-}+(1-N)}\left|\frac{z_{24}z_{13}}{z_{14}^{2}}\right|^{\Delta_{-}-(1-N)}G(z,\bar{z})\end{aligned}$
(5.18)

with

$\displaystyle
z=\frac{z_{12}z_{34}}{z_{13}z_{24}}\,,\quad\bar{z}=\frac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}\,.$
(5.19)

Now we would like to move to the case with dS3 conical defect. For this, we
take the limit (5.5). The gravity dual of $\mathcal{O}_{(0;\Lambda)}$ is given
by the dS3 conical defect. On the other hand, the gravity dual of
$\mathcal{O}_{+}$ is a bulk perturbative scalar $\phi_{+}$. Therefore, from
the four-point function (5.16), we can read off the information about the two-
point coefficient function on dS3 conical defect. Applying the inverse
functions (3.21), the correlator (5.6) can be expressed as

$\displaystyle\langle\phi_{+}(z_{2}^{\prime})\phi_{+}(z_{3}^{\prime})\rangle_{\Lambda_{-}}=\frac{a_{h_{+}}^{2}k_{h_{+},h_{+}}^{2}}{\pi^{2}}\int
d^{2}z_{2}d^{2}z_{3}\frac{1}{|z_{2^{\prime}2}|^{2(1+N)}|z_{3^{\prime}3}|^{2(1+N)}}G(z_{i},\bar{z}_{i})\,.$
(5.20)

We put the conical defect from $z_{4}\to 0$ to $z_{1}\to\infty$ and the scalar
propagator from $z_{2}^{\prime}\to z_{1}$ to $z_{3}^{\prime}\to z_{2}$. We may
observe divergence from $a_{h_{+}}^{2}$ at $2h_{+}=1-N$. However, the
conformal dimension $\Delta_{+}$ receives quantum corrections as mentioned
above, so we may slightly shift $N$ as $N+\epsilon$ with $\epsilon\to 0$ and
absorb $1/\epsilon$ by the normalization of scalar field. We are now examining
its bulk interpretation and would like to report on the result in near future.

##  6 Comments on the loops in dS spacetime

In this section, we present some partial new results on the loop computations
in the dS spacetime and higher order $1/c$-expansions of the dual CFT
correlation functions, which hopefully will become relevant in the subsequent
studies.

###  6.1 One-loop corrections to two-point correlator

In this subsection, we consider the two-point correlator of scalar field
$\phi_{+}$ on pure dS3 at late time,

$\displaystyle\begin{aligned}
\langle\phi_{+}(\vec{x})\bar{\phi}_{+}(0)\rangle&=\frac{\mathcal{C}_{0}+\delta\mathcal{C}}{|x^{2}|^{2-\Delta_{0}-\delta\Delta}}\\\
&=\frac{\mathcal{C}_{0}}{|x^{2}|^{2-\Delta_{0}}}+\delta\Delta\frac{\mathcal{C}_{0}}{|x^{2}|^{2-\Delta_{0}}}\ln|x^{2}|+\frac{\delta\mathcal{C}}{|x^{2}|^{2-\Delta_{0}}}+\mathcal{O}(c^{-2})\,,\end{aligned}$
(6.1)

where $\Delta_{0}=2h_{+}=1-N$ and $\delta\Delta,\delta\mathcal{C}$ is assumed
to be of order $\mathcal{O}(c^{-1})$. In the following, we only keep the term
proportional to $\delta\Delta$ and neglect the term proportional to
$\delta\mathcal{C}$. Typically, a divergence arises due to the infinite sum
over spin $s=2,3,\ldots,\infty$ for higher-spin gauge fields running over the
loop. In order to avoid the difficulty, we first work on the holography with
the limit (5.1), where there is an upper bound of spin as $s\leq N$ for tensor
fields. We then discuss the case with the limit (1.3).

For the current analysis, it is convenient to move to the momentum basis. We
introduce Fourier transform formula:

$\displaystyle\int\frac{d^{2}x}{|x^{2}|^{\Delta}}e^{ix\cdot
p}=\frac{\Gamma(1-\Delta)}{\Gamma(\Delta)}\left|\frac{4}{p^{2}}\right|^{1-\Delta}\,.$
(6.2)

In the momentum basis, the two-point function is thus written as

$\displaystyle\begin{aligned}
\langle\mathcal{O}^{\text{AdS}}_{+}(\vec{p})\bar{\mathcal{O}}^{\text{AdS}}_{+}(-\vec{p})\rangle^{\prime}&=\frac{\Gamma(1-\Delta)}{\Gamma(\Delta)}\left|\frac{4}{p^{2}}\right|^{1-\Delta}\\\
&=\frac{\Gamma(1-\Delta_{0})}{\Gamma(\Delta_{0})}\left|\frac{4}{p^{2}}\right|^{1-\Delta_{0}}+\delta\Delta\frac{\Gamma(1-\Delta_{0})}{\Gamma(\Delta_{0})}\left|\frac{4}{p^{2}}\right|^{1-\Delta_{0}}\ln|p^{2}|+\cdots\,.\end{aligned}$
(6.3)

Here $\langle*\rangle^{\prime}$ indicates that the divergent factor
$(2\pi)^{2}\delta^{(2)}(0)$ is removed. If we put the leading order value as
$\Delta_{0}=2h_{+}=1-N$, then there appears divergence from $\Gamma(1-N)$ in
the denominator for positive integer $N$. The divergence is harmless since it
could be removed by changing overall normalization of $\mathcal{O}_{+}$ (or
$\phi_{+}$) as discussed above. The two-point correlator of scalar field on
dS3 at late time is then obtained via the inversion formula as:333At the ’t
Hooft limit (1.3) with $h_{\pm}=(1\pm\lambda)/2$, we obtain
$\text{Re}\langle\mathcal{O}_{+}(\vec{p})\tilde{\mathcal{O}}_{+}(-\vec{p})\rangle^{\prime}=-\sin\pi(1\pm\lambda)\frac{\Gamma(\mp\lambda)}{\Gamma(1\pm\lambda)}|\frac{4}{p^{2}}|^{\mp\lambda}=-\frac{\pi}{\Gamma(1\pm\lambda)^{2}}|\frac{4}{p^{2}}|^{\mp\lambda}<0$.
This ensures the positivity of bulk two-point function and the convergence of
the Gaussian path-integral for (2.17), see footnote 1 as well.

$\displaystyle\begin{aligned}
\langle\phi_{+}(\vec{p})\bar{\phi}_{+}(-\vec{p})\rangle^{\prime}&=-\frac{1}{2\text{Re}\langle\mathcal{O}_{+}(\vec{p})\tilde{\mathcal{O}}_{+}(-\vec{p})\rangle^{\prime}}\\\
&=a_{h_{+}}\frac{\Gamma(1-N)}{\Gamma(N)}\left|\frac{p^{2}}{4}\right|^{N}+\delta\Delta\,a_{h_{+}}\frac{\Gamma(1-N)}{\Gamma(N)}\left|\frac{p^{2}}{4}\right|^{N}\ln|p^{2}|+\cdots\,.\end{aligned}$
(6.4)

We stress that the aim of this exploratory subsection is merely to show that
our method can be applied to one-loop corrections to two-point function and
not to provide a complete analysis of quantum corrections.

Here we only consider a single type of contribution as in fig. 4, where a
spin-$s$ field runs over the upper half of the loop and a scalar field runs
over the lower half of the loop. This thus implies that we only need to
consider scalar-scalar-spin vertices. More possible exchange loop diagrams
need to be included, and complete analysis is left as a future work. Applying
the split representation of bulk two-point functions as in (B.10) with (B.8)
together with the inversion formula for the two-point functions, a
$1/c$-contribution in (6.3) can be compute from

$\displaystyle\begin{aligned}
&\frac{k_{1-h_{+},1-h_{+}}k_{1-s,1}}{2\pi^{2}}\int
d^{2}z_{2}\,d^{2}z_{2}^{\prime}\,d^{2}z_{3}\,d^{2}z_{3}^{\prime}\,\langle\,\mathcal{O}_{+}^{\text{AdS}}(z_{1})\,\bar{\mathcal{O}}_{+}^{\text{AdS}}(z_{2})\,J_{(s)}^{\text{AdS}}(z_{3})\,\rangle\,\\\
&\quad\times
K_{h_{+},h_{+}}(z_{2},z_{2}^{\prime})\frac{K_{s,0}(z_{3},z_{3}^{\prime})}{cB^{(s)}}\,\langle\,J_{(s)}^{\text{AdS}}(z_{3}^{\prime})\,\mathcal{O}_{+}^{\text{AdS}}(z_{2}^{\prime})\,\bar{\mathcal{O}}_{+}^{\text{AdS}}(z_{4})\,\rangle\\\
&\quad=\frac{k_{1-h_{+},1-h_{+}}}{2\pi}\int
d^{2}z_{2}\,d^{2}z_{2}^{\prime}\,\frac{1}{|z_{12}|^{4h_{+}}|z_{2^{\prime}4}|^{4h_{+}}}\frac{(C_{+}^{(s)})^{2}}{cB^{(s)}}\mathcal{I}_{s,0}(z)K_{h_{+},h_{+}}(z_{2},z_{2}^{\prime})\,,\end{aligned}$
(6.5)

where the cross ratio is given by

$\displaystyle
z=\frac{z_{12}z_{2^{\prime}4}}{z_{12^{\prime}}z_{24}}\,,\quad\bar{z}=\frac{\bar{z}_{12}\bar{z}_{2^{\prime}4}}{\bar{z}_{12^{\prime}}\bar{z}_{24}}\,.$
(6.6)

We may express the result above through the following integral:

$\displaystyle\begin{aligned} D_{n,0}&\equiv\int
d^{2}z_{2}\,d^{2}z_{2}^{\prime}\,\frac{1}{|z_{12}|^{4h_{+}}|z_{2^{\prime}4}|^{4h_{+}}}z^{n}K_{h_{+},h_{+}}(z_{2},z_{2}^{\prime})\\\
&=\frac{k_{h_{+},h_{+}}}{\pi}\int
d^{2}z_{2}\,d^{2}z_{2}^{\prime}\,\frac{1}{(z_{42^{\prime}})^{2h_{+}-n}(z_{12^{\prime}})^{n}(z_{22^{\prime}})^{2-2h_{+}}}\frac{1}{(\bar{z}_{22^{\prime}})^{2-2h_{+}}(\bar{z}_{42^{\prime}})^{2h_{+}}}\\\
&\quad\times\frac{1}{(z_{12})^{2h_{+}-n}(z_{42})^{n}(\bar{z}_{12})^{2h_{+}}}\,.\end{aligned}$
(6.7)

The integration over $z_{2}^{\prime}$ can be performed by applying the
conformal integral formula [58]

$\displaystyle\frac{1}{\pi}\int
d^{2}z\prod_{i=1}^{3}\frac{1}{(z-z_{i})^{h_{i}}(\bar{z}-\bar{z}_{i})^{\bar{h}_{i}}}=K_{123}z_{12}^{h_{3}-1}z_{23}^{h_{1}-1}z_{31}^{h_{2}-1}\bar{z}_{12}^{\bar{h}_{3}-1}\bar{z}_{23}^{\bar{h}_{1}-1}\bar{z}_{31}^{\bar{h}_{2}-1}\,,$
(6.8)

where

$\displaystyle
K_{123}=\prod_{i=1}^{3}\frac{\Gamma(1-h_{i})}{\Gamma(\bar{h}_{i})}=\prod_{i=1}^{3}\frac{\Gamma(1-\bar{h}_{i})}{\Gamma(h_{i})}\,.$
(6.9)

Here we assumed

$\displaystyle\sum_{i=1}^{3}h_{i}=\sum_{i=1}^{3}\bar{h}_{i}=2\,,$ (6.10)

for the conformality condition to be satisfied. We find that:

$\displaystyle D_{n,0}$
$\displaystyle=\frac{\Gamma(1-2h_{+}+n)}{\Gamma(2h_{+})\Gamma(n)}\frac{1}{|z_{14}|^{4h_{+}-2}}\int
d^{2}z_{2}\,\frac{1}{|z_{1}-z_{2}|^{2}|z_{4}-z_{2}|^{2}}\,.$ (6.11)

Applying a formula (1.5) of [59] and neglecting the terms not including
$\log|z_{14}|^{2}$, the integration over $z_{2}$ leads to

$\displaystyle
D_{n,0}=2\pi\frac{\Gamma(1-2h_{+}+n)}{\Gamma(2h_{+})\Gamma(n)}\frac{1}{|z_{14}|^{4h_{+}}}\log|z_{14}|^{2}+\cdots\,.$
(6.12)

We thus have

$\displaystyle\frac{k_{1-h_{+},1-h_{+}}}{2\pi}\frac{(C_{+}^{(s)})^{2}}{cB^{(s)}}\frac{\Gamma(2s)}{\Gamma(s)^{2}}\sum_{n=0}^{\infty}\frac{\Gamma(s+n)^{2}}{\Gamma(2s+n)n!}D_{n+s,0}$
$\displaystyle\quad=\frac{N(1-N^{2})(2s-1)\Gamma(s-N)}{c\Gamma(1-N)\Gamma(s+N)}\frac{1}{|z_{14}|^{4h_{+}}}\log|z_{14}|^{2}\sum_{n=0}^{\infty}\frac{\Gamma(s+n)^{2}}{\Gamma(2s+n)n!}\frac{\Gamma(N+n+s)}{\Gamma(n+s)}+\cdots$
$\displaystyle\quad=\frac{(2s-1)(N^{2}-1)}{c}\frac{1}{|z_{14}|^{4h_{+}}}\log|z_{14}|^{2}+\cdots\,.$
(6.13)

Here we have used the identity:

$\displaystyle\sum_{n=0}^{\infty}\frac{\Gamma(s+n)^{2}}{\Gamma(2s+n)n!}\frac{\Gamma(N+n+s)}{\Gamma(n+s)}=\frac{\Gamma(-N)\Gamma(s+N)}{\Gamma(s-N)}\,.$
(6.14)

From the expression, we can read off the $1/c$-contribution from the loop
diagram in fig. 4 as

$\displaystyle\delta\Delta_{s}^{\text{AdS}}=-\frac{(2s-1)(N^{2}-1)}{c}+\mathcal{O}(c^{-2})\,.$
(6.15)

In the current case, there are conserved currents with spin $s=2,3,\ldots,N$.
This means that we need to take a sum over the spin as

$\displaystyle\delta\Delta^{\text{AdS}}=\sum_{s=2}^{N}\delta\Delta_{s}^{\text{AdS}}=-\frac{(N^{2}-1)^{2}}{c}+\mathcal{O}(c^{-2})\,.$
(6.16)

As one can read off from (5.2), this actually reproduces the anomalous
dimension at the next leading order in $1/c$. The result is consistent with
the analysis on loop-corrections from Wilson line networks as in discussion
section of [60]. However, as seen in fig. 12 of [59], there are many other
possible one-loop Witten diagrams contributing to the two-point function.
Currently, we do not have any good explanation why only the one-loop Witten
diagram of the form in fig. 4 can explain the anomalous dimension of scalar
operator at the order.

We would like to map the result from AdS3 to dS3. Denoting the one-loop
contribution to two-point coefficient function for scalar field by $\delta
C_{2}$, the sum of $\delta C_{2}$ and its complex conjugate becomes

$\displaystyle\delta C_{2}+(\delta
C_{2})^{*}=\sum_{s=2}^{\infty}\delta\Delta_{s}\frac{\lambda_{h_{+},h_{+},s}^{2}}{a_{h_{+}}a_{(s)}}\frac{1}{|x^{2}|^{1-N}}\log|x^{2}|+\cdots\,.$
(6.17)

Mapping to the momentum space, we can evaluate the bulk two-point function as
(6.4) and its one-loop correction is given by

$\displaystyle\frac{2(N^{2}-1)a_{h_{+}}^{3}}{c^{(g)}}\frac{\Gamma(1-N)}{\Gamma(N)}\sum_{s=2}^{N}(-1)^{s+1}(2s-1)\sin^{2}\left[(1-N+s/2)\pi\right]\left|\frac{p^{2}}{4}\right|^{N}\log|p^{2}|+\cdots\,.$
(6.18)

Thus the anomalous dimension represented by $\delta\Delta$ in (6.4) can be
read off as

$\displaystyle\delta\Delta=\begin{cases}\frac{1-N^{2}}{4c^{(g)}}\left(N(N-1)-(N^{2}+N-2)4a_{h_{+}}^{2}\cos^{2}\left[(1-N)\pi\right]\right)&\text{
for odd $N$}\,,\\\
\frac{1-N^{2}}{4c^{(g)}}\left(N(N+1)-(N^{2}-N-2)4a_{h_{+}}^{2}\cos^{2}\left[(1-N)\pi\right]\right)&\text{
for even $N$}\,.\end{cases}$ (6.19)

For integer $N$, $a_{h_{+}}^{2}$ diverges and a regularization has to be
taken. Again the investigation of its bulk interpretation in the dS3 spacetime
is postponed as a future work.

We would like to also comment on the case of $1/c$-expansion with the ’t Hooft
parameter $\lambda$ fixed as in (1.3). In this case, the anomalous dimension
is given by

$\displaystyle\delta\Delta_{+}=\frac{\lambda^{2}-1}{c}+\mathcal{O}(c^{-2})$
(6.20)

as in (4.5). In the ’t Hooft limit (1.3), the expression (6.15) becomes

$\displaystyle\delta\Delta_{s}^{\text{AdS}}=\frac{(2s-1)(1-\lambda^{2})}{c}+\mathcal{O}(c^{-2})\,.$
(6.21)

In (6.16), the sum over spin is finite as $s=2,3,\ldots,N$, while in the
current case the infinite sum over $s=2,3,\ldots$ looks divergent. Applying
the zeta functional regularization, we have

$\displaystyle\delta\Delta^{\text{AdS}}=\sum_{s=2}^{N}\delta\Delta_{s}^{\text{AdS}}=\frac{2(\lambda^{2}-1)}{3c}+\mathcal{O}(c^{-2})\,.$
(6.22)

The anomalous dimension is very close to (6.20) but there is an extra factor
$3/2$. Instead of this, we may first sum over $s=2,3,\ldots$. Then the
computation reduces to

$\displaystyle\begin{aligned}
&\frac{(1-\lambda^{2})\Gamma(1+\lambda)}{2c\pi\Gamma(-\lambda)}\sum_{n=2}^{\infty}\left[-\frac{1}{n}+\frac{\Gamma(1-\lambda)\Gamma(n)}{\Gamma(n-\lambda)}\right]D_{n,0}\\\
&\quad=\frac{(1-\lambda^{2})\Gamma(1+\lambda)}{c\Gamma(-\lambda)}\frac{1}{|z_{14}|^{4h_{+}}}\log|z_{14}|^{2}\sum_{n=2}^{\infty}\left[-\frac{1}{n}+\frac{\Gamma(1-\lambda)\Gamma(n)}{\Gamma(n-\lambda)}\right]\frac{\Gamma(-\lambda+n)}{\Gamma(1+\lambda)\Gamma(n)}+\cdots\,.\end{aligned}$
(6.23)

After the zeta functional regularization, we find

$\displaystyle\frac{(1-\lambda^{2})(2+\lambda)}{2c}\frac{1}{|z_{14}|^{4h_{+}}}\log|z_{14}|^{2}+\cdots\,.$
(6.24)

It again does not reproduce (6.20) but with an extra factor $-(2+\lambda)/2$.
Moreover, the sum over $s$ and the zeta functional regularization do not
commute with each other.

We may obtain the anomalous dimension for the ’t Hooft expansion in the AdS3
case by applying the triality relation [43]

$\displaystyle W_{\infty}[N]\simeq W_{\infty}\left[\frac{N}{N+k}\right]\simeq
W_{\infty}\left[-\frac{N}{N+k+1}\right]\,.$ (6.25)

As in (6.16), we have reproduced the conformal dimension with finite $N$ from
the gravity loop calculation,

$\displaystyle\Delta^{\text{AdS}}=1-N+\delta\Delta^{\text{AdS}}=1-N-\frac{(N^{2}-1)^{2}}{c}+\mathcal{O}(c^{-2})\,.$
(6.26)

If we replace $N$ by $N/(N+k)$, then (6.26) becomes

$\displaystyle\Delta^{\text{AdS}}=1-\lambda-\frac{(\lambda^{2}-1)^{2}}{c}+\mathcal{O}(c^{-2})$
(6.27)

in $1/c$-expansion with fixed $\lambda=N/(N+k)$. This reproduces $\Delta_{-}$
in (4.6). On the other hand, if we replace $N$ by $-N/(N+k+1)$, then (6.26)
becomes

$\displaystyle\Delta^{\text{AdS}}=1+\lambda+\frac{\lambda^{2}-1}{c}+\mathcal{O}(c^{-2})$
(6.28)

in $1/c$-expansion with fixed $\lambda=N/(N+k)$. This reproduces $\Delta_{+}$
in (4.5) (or (6.20)). In this way, we can reproduce the anomalous dimension
even for the ’t Hooft expansion. However, if we apply the triality relation
(6.25), then it becomes unclear how to map the AdS result to the dS one.

###  6.2 The expansion of four-point correlation functions at
$\mathcal{O}(c^{-2})$

In this subsection, we extend our analysis in section 4.1 to consider the
expansion of four-point CFT correlation functions at the order
$\mathcal{O}(c^{-2})$ in ’t Hooft limit. The purpose of including some partial
results here is to facilitate the comparisons with the various bulk loop
corrected four-point function in AdS3 hence dS3 in the subsequent work. Our
strategy here will be similar as previous sections; start with the
$1/N$-expansion and use the definition (3.5) to obtain the inverted relation:

$\frac{1}{N}=\frac{1}{c}(1-\lambda^{2})-\frac{1}{c^{2}}(1-\lambda^{2})(1-\lambda^{3})+O(c^{-3})$
(6.29)

and make the appropriate substitutions.

Starting with the relatively simple exact functional forms of $G_{-+}(z)$
given in (4.2), the direct expansion in $1/N$ yields:

$\displaystyle G_{-+}(z)$
$\displaystyle=\frac{1}{|1-z|^{2\Delta_{+}}}\Big{\\{}1+\frac{1}{N}\left(\frac{1-z}{z}+\log
z+{\rm c.c.}\right)$ (6.30)
$\displaystyle+\frac{1}{N^{2}}\left[\frac{1}{2}\left(\frac{1-z}{z}+\log z+{\rm
c.c.}\right)^{2}-\frac{1}{2}\left(\left(\frac{1-z}{z}\right)^{2}+{\rm
c.c.}\right)\right]+O(N^{-3})\Big{\\}}\,.$

We next apply the relation (6.29) to substitute $1/c$ for $1/N$, and expand
the correlation function into series of $1/c$ by the form:

$\displaystyle G_{-+}(z)$
$\displaystyle=\frac{1}{|1-z|^{2(\Delta_{+}^{(0)}+\Delta_{+}^{(1)}/c+\Delta_{+}^{(2)}/c^{2})}}Q_{-+}^{(0)}(1-z)+\frac{1}{c}\frac{1}{|1-z|^{2(\Delta_{+}^{(0)}+\Delta_{+}^{(1)}/c)}}Q_{-+}^{(1)}(1-z)$
(6.31)
$\displaystyle+\frac{1}{c^{2}}\frac{1}{|1-z|^{2\Delta_{+}^{(0)}}}Q_{-+}^{(2)}(1-z)+O(c^{-3})\,.$

As in (4.7), we single out the expansion of conformal dimension $\Delta_{+}$.
It corresponds to the loop corrections on the external legs of the diagrams,
and their dependence on the ’t Hooft coupling $\lambda$ at each loop order are
given by:

$\Delta_{+}^{(0)}=1+\lambda\,,\quad\Delta_{+}^{(1)}=\lambda^{2}-1\,,\quad\Delta_{+}^{(2)}=(1-\lambda^{2})(1-\lambda)\,,$
(6.32)

and for completeness we include the similar expansion for $\Delta_{-}$:

$\Delta_{-}^{(0)}=1-\lambda\,,\quad\Delta_{-}^{(1)}=-(1-\lambda^{2})^{2}\,,\quad\Delta_{-}^{(1)}=(1-\lambda^{2})^{2}(1+\lambda-2\lambda^{3})\,.$
(6.33)

Finally, in the expansion of $G_{-+}(z)$ (6.31) can be expressed by power
series around $z=1$ as

$\displaystyle Q_{-+}^{(0)}(1-z)$ $\displaystyle=1\,,$ $\displaystyle
Q_{-+}^{(1)}(1-z)$
$\displaystyle=(1-\lambda)^{2}\sum_{r=1}^{\infty}\frac{1}{r}(r-1)(1-z)^{r}+{\rm
c.c.}\,,$ (6.34) $\displaystyle Q_{-+}^{(2)}(1-z)$
$\displaystyle=\frac{1}{2}Q_{-+}^{(1)}(1-z)^{2}-(1-\lambda^{3})Q_{-+}^{(1)}(1-z)-\frac{(1-\lambda^{2})^{2}}{2}\left(\sum_{r=2}^{\infty}(r-1)(1-z)^{r}+\rm{c.c.}\right)\,.$

We can now comment on the identifications of some of the terms listed in
(6.31) with the possible Witten diagrams in AdS3 bulk, facilitating the
explicit quantitative comparisons in the subsequent work. The first term in
(6.31) corresponds to the contribution of a pair of boundary to boundary
propagators corrected up to two-loop order. As shown in section 4.1,
$Q_{-+}^{(1)}(1-z)$ can be further expanded into a summation of global
conformal blocks for the conserved tensor operators, and we can respectively
associate them with the four-point tree level exchange Witten diagrams in
AdS3. It is thus natural to expect that the second term in (6.31) comes from
replacing one of the external bulk to boundary propagators with the loop
corrected one in the corresponding tree level Witten diagrams respectively.
Finally the last term would be attributed to certain combination of the
exchange diagrams now with one-loop corrected three-point vertex, or with one-
loop corrected bulk to bulk exchange propagator. The appearance of
$Q_{-+}^{(1)}(1-z)^{2}$ can perhaps be explained by the fact that all of these
diagrams can be constructed by fusing a pair of tree level four-point exchange
diagrams using the split representation of the AdS Harmonic function. However
to make the identification precise at this level, it would be necessary to
combine the result from the higher order $1/c$ expansion of Virasoro conformal
block, see e.g. [53], as well as the actual AdS loop computations.

We then examine more complicated examples of $G_{\pm\pm}(z)$ in (4.17) with
(4.18) and (4.19) with (4.20). The main difference for expanding
$G_{\pm\pm}(z)$ to order $1/N^{2}$ is that they contain both $s$\- and
$t$-channels, i.e. the correlation function is invariant under the exchange
$z\leftrightarrow 1-z$ as shown in appendix A.1. This implies that we need to
expand around $z=0$ and $z=1$ separately. We expand $G_{++}(z)$ as a
demonstration of some key features. First before the expansion, we use the
identities of hypergeometric functions as in appendix A.2 to transform the
second term in (4.17) into:

$\displaystyle G_{++}(z)=$
$\displaystyle\frac{1}{|z(1-z)|^{2\Delta_{+}}}\big{|}(1-z)^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};-\lambda;z)\big{|}^{2}$
(6.35) $\displaystyle+{\cal
N}_{+}A^{2}\frac{1}{|z(1-z)|^{2\Delta_{+}}}\big{|}z^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N};-\frac{\lambda}{N};-\lambda;1-z)\big{|}^{2}$
$\displaystyle+{\cal
N}_{+}B^{2}\frac{1}{|z(1-z)|^{2\Delta_{+}}}\big{|}(1-z)^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;1-z)\big{|}^{2}$
$\displaystyle+{\cal
N}_{+}AB\frac{1}{|z(1-z)|^{2\Delta_{+}}}\big{[}z^{1+\lambda}(1-\bar{z})^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N};-\frac{\lambda}{N};-\lambda;1-z)$
$\displaystyle{}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;1-\bar{z})+\rm{c.c.}\big{]}$

with

$\displaystyle A$
$\displaystyle=\frac{\Gamma(2+\lambda)\Gamma(1+\lambda)}{\Gamma(1+\lambda-\frac{\lambda}{N})\Gamma(2+\lambda+\frac{\lambda}{N})}\,,$
(6.36) $\displaystyle B$
$\displaystyle=\frac{\Gamma(2+\lambda)\Gamma(-1-\lambda)}{\Gamma(1+\frac{\lambda}{N})\Gamma(-\frac{\lambda}{N})}=\frac{\sin(-\frac{\pi\lambda}{N})}{\sin\pi(-1-\lambda)}=\frac{-\sin(\frac{\pi\lambda}{N})}{\sin(\pi\lambda)}=\frac{-\pi\lambda}{\sin(\pi\lambda)}\frac{1}{N}+O(N^{-3})\,.$

Hence we can “symmetrize” the correlation function into manifestly $s-t$
crossing symmetric form suitable for the $1/c$-expansion in each channel:

$\displaystyle G_{++}(z)=$ $\displaystyle\frac{1}{2}[G_{++}(z)+G_{++}(1-z)]$
$\displaystyle=$
$\displaystyle\frac{1}{|z(1-z)|^{2\Delta_{+}}}\Big{\\{}(1-\frac{1}{2}B^{2})\big{|}(1-z)^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};-\lambda;z)\big{|}^{2}$
$\displaystyle+{\cal
N}_{+}\frac{B^{2}}{2}\big{|}(1-z)^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;1-z)\big{|}^{2}$
(6.37) $\displaystyle+\frac{1}{2}{\cal
N}_{+}AB\big{[}z^{1+\lambda}(1-\bar{z})^{1+\lambda}{}_{2}F_{1}(1+\frac{\lambda}{N};-\frac{\lambda}{N};-\lambda;z){}_{2}F_{1}(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;\bar{z})+{\rm
c.c.}\big{]}$ $\displaystyle+(z\leftrightarrow 1-z)\Big{\\}}\,.$

We can then expanding the above equation up to order $1/c^{2}$ and arrange it
into the following form:

$\displaystyle G_{++}(z)$
$\displaystyle=\left[\frac{1}{|z|^{2\Delta_{+}^{(0)}+2\Delta_{+}^{(1)}/c+2\Delta_{+}^{(2)}/c^{2}}}Q_{++}^{(0)}(z)+\frac{1}{c}\frac{1}{|z|^{2\Delta_{+}^{(0)}+2\Delta_{+}^{(1)}/c}}Q_{++}^{(1)}(z)+\frac{1}{c^{2}}\frac{1}{|z|^{2\Delta_{+}^{(0)}}}Q_{++}^{(2)}(1-z)\right]$
$\displaystyle\quad+\Big{[}\frac{1}{(\bar{z}(1-z))^{\Delta_{+}^{(0)}+\Delta_{+}^{(1)}/c+\Delta_{+}^{(2)}/c^{2}}}h^{(0)}_{++}(z)+\frac{1}{c}\frac{1}{(\bar{z}(1-z))^{\Delta_{+}^{(0)}+\Delta_{+}^{(1)}/c}}h^{(1)}_{++}(z)$
(6.38)
$\displaystyle\quad+\frac{1}{c^{2}}\frac{1}{(\bar{z}(1-z))^{\Delta_{+}^{(0)}}}h^{(2)}_{++}(z)+{\rm
c.c.}\Big{]}+(z\leftrightarrow 1-z)+O(c^{-3})\,.$

Again as in (6.31), we single out the loop corrections of conformal dimension,
such that the first term in the first line of (6.38) corresponds to the a pair
of disconnected loop corrected boundary to boundary propagators. Furthermore,
in this case, we separate terms which have distinct channel structure
$Q_{++}(z)$ with those related to the contact diagram $h_{++}(z)$. And the
final result can be found by

$\displaystyle Q^{(0)}_{++}(z)=1\,,\quad
Q^{(1)}_{++}(z)=(1-\lambda^{2})\left[\sum_{n=1}^{\infty}(\frac{\Gamma(1-\lambda)\Gamma(n)}{\Gamma(n-\lambda)}-\frac{1}{n})z^{n}+{\rm
c.c}\right]\,,$ $\displaystyle
Q^{(2)}_{++}(z)=\frac{Q^{(1)}_{++}(z)^{2}}{2}-(1-\lambda^{3})Q^{(1)}_{++}(z)+\frac{\pi^{2}\lambda^{2}(1-\lambda^{2})^{2}}{2\sin^{2}(\pi\lambda)}$
$\displaystyle\quad+\left\\{-\frac{(1-\lambda^{2})^{2}}{2}\left[\sum_{n=1}^{\infty}\frac{\Gamma(1-\lambda)\Gamma(n)}{\Gamma(n-\lambda)}z^{n}\right]^{2}+\lambda(1-\lambda^{2})^{2}\sum_{n=1}^{\infty}\frac{1}{n}\left[\frac{\Gamma(1-\lambda)\Gamma(n)}{\Gamma(n-\lambda)}-1\right]z^{n}+{\rm
c.c}\right\\}\,,$ $\displaystyle h^{(0)}_{++}(z)=0\,,\quad
h^{(1)}_{++}(z)=-\frac{\pi\lambda(1-\lambda^{2})}{2\sin(\pi\lambda)}\,,$
(6.39) $\displaystyle
h^{(2)}_{++}(z)=h^{(1)}_{++}(z)\big{[}(\lambda-1)(\lambda^{2}+1)+Q^{(1)}_{++}(z)+(1-\lambda^{2})\sum_{n=1}^{\infty}\frac{1}{n}[z^{n}-(1-z)^{n}]$
$\displaystyle\quad-(1-\lambda^{2})\sum_{n=1}^{\infty}[\frac{\lambda\Gamma(2+\lambda)\Gamma(n)}{\Gamma(2+\lambda+n)}+\frac{1}{n}]\bar{z}^{n}\big{]}\,.$

As noted in section 4.1, in contrast with the expansion for $G_{-+}(z)$, the
main additional feature here is the additional terms which can be attributed
to the manifestly crossing symmetric bulk four-point contact diagrams as
denoted by function $g(z)$ (4.22) earlier. We can thus readily identify
$h_{++}^{(1)}(z)$ as the four-point contact diagram with an external bulk to
boundary propagator replaced by loop corrected one, while $h_{++}^{(2)}(z)$
can be attributed to possible one loop correction to the four-point vertex or
so-called box diagram. It would be interesting and non-trivial to verify these
predictions from 2d CFT with explicit bulk loop computations in AdS3 and hence
dS3 spacetimes.

### Acknowledgements

We are grateful to Thomas Creutzig, Toshifumi Noumi, Kenta Suzuki, Tadashi
Takayanagi, Yusuke Taki, Seiji Terashima, and Takahiro Uetoko for useful
comments. We would like to particularly thank Tatsuma Nishioka for the
collaboration at the early stage. The work of H. Y. C. and S. C. are supported
in part by Ministry of Science and Technology (MOST) through the grant
110-2112-M-002-006-. This work of Y. H. was supported by JSPS Grant-in-Aid for
Scientific Research (B) No.19H01896, Grant-in-Aid for Scientific Research (A)
No. 21H04469, Grant-in-Aid for Transformative Research Areas (A) “Extreme
Universe” No. 21H05187.

##  Appendix A Partial wave (conformal block) expansions

In this appendix, we prove the crossing symmetric property of the
$G_{\pm\pm}(z)$ correlation functions and collect some detailed computations
associated with the expansions by conformal blocks or conformal partial waves.

###  A.1 Crossing symmetry under $s$-$t$ channel exchange

Starting with $G_{++}(z,\bar{z})$ given in (4.17), here we parameterize it in
terms of the following combinations for the convenience:

$G_{++}(z,\bar{z})=\frac{\left(|Q_{+}(z)|^{2}+{\cal
N}_{+}|P_{+}(z)|^{2}\right)}{|z(1-z)|^{2\Delta_{+}}}$ (A.1)

where we have defined the functions:

$Q_{+}(z)=(1-z)^{1+\lambda}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};-\lambda;z\right),~{}~{}P_{+}(z)=z^{1+\lambda}\,{}_{2}F_{1}\left(1+\frac{\lambda}{N},-\frac{\lambda}{N};2+\lambda;z\right),$
(A.2)

and the constant $\mathcal{N}_{+}$ is given in (4.18). Now under the $s$-$t$
channel crossing transformation $(z,\bar{z})\to(1-z,1-\bar{z})$, using the
hypergeometric function identities, we have:

$\displaystyle Q_{+}(1-z)$
$\displaystyle=+\frac{\sin\pi\frac{\lambda}{N}}{\sin\pi\lambda}Q(z)+{\cal
N}_{+}\frac{\Gamma(-\lambda)\Gamma(-(1+\lambda))}{\Gamma(-1-\lambda-\frac{\lambda}{N})\Gamma(-\lambda+\frac{\lambda}{N})}P(z)\,,$
(A.3) $\displaystyle P_{+}(1-z)$
$\displaystyle=-\frac{\sin\pi\frac{\lambda}{N}}{\sin\pi\lambda}P(z)+\frac{\Gamma(1+\lambda)\Gamma(2+\lambda)}{\Gamma(1+\lambda-\frac{\lambda}{N})\Gamma(2+\lambda+\frac{\lambda}{N})}Q(z)\,.$
(A.4)

Now if we evaluate:

$G_{++}(1-z,1-\bar{z})=\frac{\left(|Q_{+}(1-z)|^{2}+{\cal
N}_{+}|P_{+}(1-z)|^{2}\right)}{|z(1-z)|^{2\Delta_{+}}}$ (A.5)

and notice that the coefficients for $|Q_{+}(z)|^{2}$ and $|P_{+}(z)|^{2}$,
after repeatedly using the $\Gamma$-function identity
$\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin\pi x}$, are given by:

$\displaystyle|Q_{+}(z)|^{2}$ $\displaystyle:$
$\displaystyle\frac{\sin^{2}\pi\frac{\lambda}{N}}{\sin^{2}\pi\lambda}+\frac{\sin\pi(\lambda+\frac{\lambda}{N})\sin\pi(\lambda-\frac{\lambda}{N})}{\sin^{2}\pi\lambda}=1\,,$
(A.6) $\displaystyle|P_{+}(z)|^{2}$ $\displaystyle:$ $\displaystyle{\cal
N}_{+}\frac{\sin^{2}\pi\frac{\lambda}{N}}{\sin^{2}\pi\lambda}+{\cal
N}_{+}\frac{\sin\pi(\lambda+\frac{\lambda}{N})\sin\pi(\lambda-\frac{\lambda}{N})}{\sin^{2}\pi\lambda}={\cal
N}_{+}\,,$ (A.7)

while the coefficients for the crossed terms $Q(z)P(\bar{z})$ and
$Q(\bar{z})P(z)$ vanish identically. We thus established that:

$G_{++}(1-z,1-\bar{z})=G_{++}(z,\bar{z})$ (A.8)

to the all order of ${1}/{N}$ or $1/c$ expansions. We can similarly prove the
crossing symmetric property for $G_{--}(z,\bar{z})$.

###  A.2 Expansions of $G_{\pm\pm}(z)$

In this subsection, we derive (4.21) but focusing on the $G_{++}(z)$ case
(4.17). The first term has been analyzed in [46], but let us repeat the