# Dual CFT on Nariai limit for Kerr-Sen-dS black holes

Muhammad Fitrah Alfian Rangga Sakti<EMAIL_ADDRESS>Piyabut Burikham
<EMAIL_ADDRESS>High Energy Physics Theory Group, Department of Physics,
Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

###### Abstract

In this work, we study the Kerr-Sen-de Sitter black hole (BH) in the Nariai
limit where the event and cosmological horizon coincide. We show that the
near-horizon Kerr-Sen-de Sitter black hole in Nariai limit is a fiber over
AdS2 with an appropriate coordinate transformation, instead of fiber over dS2.
Hence, we can compute the associated central charge and CFT temperature by
using the Kerr/CFT method. It is remarkably exhibited that through Cardy’s
growth of states, the Bekenstein-Hawking entropy on cosmological horizon is
reproduced. Moreover, we show that the radial equation of the quantum scalar
field in $J$\- and $Q$-pictures on this charged rotating background in Nariai
limit can be portrayed in quadratic Casimir operator form with $SL(2,R)\times
SL(2,R)$ isometry. We also compute the corresponding thermodynamic quantities
from CFT to find the absorption cross-section and real-time correlator in
$J$-picture. In $Q$-picture, we do not find a well-defined CFT description. We
then extend the study of quantum scalar field in Nariai limit for Kerr-Newman-
dS black hole solution and show that the hidden conformal symmetry on this
black hole’s background in in $J$\- and $Q$-pictures is well-defined.

††preprint: APS/123-QED

##  I Introduction

Establishing the holographic correspondence between de Sitter space (dS) and
quantum theory is a significant challenge in theoretical physics. The
holographic duality, most notably captured by the Anti-de Sitter/Conformal
Field Theory (AdS/CFT) correspondence, has provided profound insights into the
quantum theory of gravity. However, extending this framework to dS space has
proven to be more problematic. One of the main difficulties in the holographic
description of dS space arises from the absence of a global timelike Killing
vector and spatial infinity. In the AdS/CFT correspondence, the asymptotic
behavior of AdS space allows for a well-defined boundary at infinity, which
plays a crucial role in the holographic interpretation. This boundary serves
as a holographic screen, encoding the dynamics of the gravitational theory in
terms of a dual quantum field theory living on the boundary. In contrast, dS
space does not possess a similar asymptotic structure or boundary at infinity.
As a result, the standard techniques used in the AdS/CFT correspondence cannot
be straightforwardly applied to dS space. It is an intriguing and challenging
problem that requires a deeper understanding of the quantum nature of gravity,
as well as the dynamics of dS space itself.

Inspite of the challenges coming from the asymptotically de Sitter spacetime,
this solution may provide a correct interpretation to explain our physical
world in the context of cosmological model. The standard model of cosmology,
whose details are explained extensively e.g. in [1], is surprisingly
consistent with this de Sitter solution. The observed accelerated expansion of
our universe, supported by astronomical observations such as the cosmic
microwave background radiation and the distribution of galaxies, suggests that
our universe could be described by a dS spacetime [2, 3]. In particular, the
connection between the cosmological constant and the behavior of smaller
objects, such as black holes, is also of great interest. The presence of a
non-zero cosmological constant affects the configurations and properties of
black holes. In fact, the cosmological constant can influence the formation,
growth, and evaporation of black holes. Moreover, one can study the quantum
gravity from black hole’s properties through the gravitational waves
observations [4, 5, 6]. Hence, in the future, we expect that the gravitational
wave observations may give us more precise information about quantum structure
of black holes including when the cosmological constant is present.

Regarding the black hole solutions, one of the central challenges in quantum
gravity is to understand the microscopic origin of black hole entropy, which
is given by the famous Bekenstein-Hawking formula stating that the entropy is
one quarter of the black hole’s event horizon area. Remarkably, the AdS/CFT
correspondence provides a potential framework for addressing this problem. It
is assumed that the partition function of conformal field theory to be
identical to black hole’s partition function. This relation was firstly proven
by computing the entropy the extremal Kerr black hole [7]. That study is
inspired from the success of the investigation of the asymptotic symmetries of
Banados-Teitelboim-Zanelli (BTZ) black hole which contains two-dimensional
(2D) local conformal algebra. CFT techniques can then be used to calculate the
state degeneracy [8] leading to the agreement of CFT entropy with the
Bekenstein-Hawking entropy.

In the following, we consider a class of charged rotating black hole solution
in Einstein-Maxwell-Dilaton-Axion (EMDA) supergravity theory and Einstein-
Maxwell theory, namely dyonic Kerr-Sen-dS (KSdS) and dyonic Kerr-Newman-dS
(KNdS) black holes, respectively. Both black holes possess more than two
horizons because of the existence of the positive cosmological constant. The
largest horizon is the cosmological horizon for which the first thermodynamics
law on this horizon resembles the first thermodynamic law on black hole’s
horizon. In the cosmological horizon, one can also study the thermal spectrum
with non-zero entropy that is also proportional to the horizon area, likewise
in the event horizon. The existence of three horizons leads to different
limits. When the Cauchy and event horizons coincide, one can obtain an
extremal black hole. When the cosmological and event horizon coincide, the
resulting metric will be Nariai solution. Furthermore, when all three horizons
coincide, the ultracold solution will be produced.

The first holographic description of the cosmological entropy of the Nariai
solution obtained from Kerr-dS and KNdS solutions has been investigated in
Ref. [9]. The Nariai solution possesses $SL(2,R)\times U(1)$ isometry where
the spacetime metric is a fiber over dS2. We want to show that in Nariai
limit, we can write the solution with similar isometry, although with the
spacetime metric as a fiber over AdS2 by choosing different sign on the radial
near-horizon coordinate. This metric form on Nariai limit denotes the similar
form with near-horizon extremal metric [10, 11, 12, 13]. Furthermore, we want
to study the scalar wave equation on the geometry when we take the Nariai
limit. As has been studied for generic Kerr solution [14], we exhibit that in
Nariai limit, the scalar wave equation possesses $SL(2,R)\times SL(2,R)$
isometry. This isometry happens to appear in scalar wave equation of some
black hole solutions [15, 16, 17, 18] and also for higher-spin field [19, 20,
21]. Remarkably, one can also compute the absorption cross-section and real-
time correlator from 2D CFT of the solution on Nariai limit. We want to prove
that for both KNdS and KSdS black holes in Nariai limit, the conformal
symmetry can be shown on the spacetime metric directly and on the scalar wave
equation. Moreover, CFT dual will be constructed trough the matching between
the entropy, absorption cross-section, and the real-time correlator. We will
consider two different pictures, $J$\- and $Q$ pictures. $J$\- picture denotes
the CFT description when the scalar probe is neutral while $Q$-picture denotes
the CFT description when the scalar probe is electrically charged.

We organize the paper as follows. In Sec. II, we carry out the KSdS solution
and its thermodynamic quantities on the event and cosmological horizons. In
Sec. III, we show the ”fin” diagrams which portray the horizons and the
extremal limits. In the next section, we study the Nariai limit on KSdS
solution and find the entropy using the Kerr/CFT correspondece. In Sec. V, we
study the conformal symmetry on the scalar wave equation in $J$\- and
$Q$-pictures in the background of KSdS solution in Nariai limit. In Sec. VI,
we further extend the study of the scalar field on KNdS black hole in Nariai
limit. In Sec. VII, we summarize our findings for the whole paper.

##  II Dyonic Kerr-Sen-de Sitter solution and its thermodynamics

The dyonic KSdS black hole is the exact solution to EMDA supergravity theory
with positive cosmological constant. The line element of KSdS spacetime reads
as [46]

$ds^{2}=-\frac{\Delta}{\varrho^{2}}\hat{X}^{2}+\frac{\varrho^{2}}{\Delta}d\hat{r}^{2}+\frac{\varrho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\varrho^{2}}\hat{Y}^{2},\
$ (1)

where

$\hat{X}=d\hat{t}-a\sin^{2}\theta\frac{d\hat{\phi}}{\Xi},~{}\hat{Y}=ad\hat{t}-(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\frac{d\hat{\phi}}{\Xi},\
$ (2)
$\Delta=(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\left(1-\frac{\hat{r}^{2}-d^{2}-k^{2}}{l^{2}}\right)-2m\hat{r}+p^{2}+q^{2},\
\ $ (3)
$\Delta_{\theta}=1+\frac{a^{2}}{l^{2}}\cos^{2}\theta,~{}\Xi=1+\frac{a^{2}}{l^{2}},~{}\varrho^{2}=\hat{r}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta.\
$ (4)

Note that $m,a,d,k,p,q,l$ are the parameters of mass, spin, dilaton charge,
axion charge, magnetic charge, electric charge, and dS length, respectively.
The KSdS solution above is obtained from the gauged dyonic Kerr-Sen black hole
solution [46, 45] via analytic continuation [28]. There is an important
relation between charges of KSdS solution given by

$d=\frac{p^{2}-q^{2}}{2m},~{}~{}~{}~{}~{}k=\frac{pq}{m}.\ $ (5)

Therefore, when the magnetic charge vanishes, the axion charge also vanishes
while when $p=q$, the dilaton charge will vanish. One also can write

$d^{2}+k^{2}=\left(\frac{p^{2}+q^{2}}{2m}\right)^{2}.\ $ (6)

KSdS black hole possesses three positive horizons. The three positive horizons
are inner ($r_{-}$), outer/event ($r_{+}$) and cosmological ($r_{c}$) horizons
where $r_{c}\geq r_{+}\geq r_{-}\geq 0$. We can study the thermodynamics on
its horizons. Within this paper, we emphasize the study on thermodynamics on
the event horizon and cosmological horizon. On the event horizon, the KSdS
black hole satisfies the thermodynamic relation

$dM=T_{H}dS_{BH}+\Omega_{H}dJ+\Phi_{H}dQ+\Psi_{H}dP+Vd\mathcal{P},$ (7)

with the following quantities

$M=\frac{m}{\Xi},~{}~{}~{}J=\frac{ma}{\Xi},~{}~{}~{}Q=\frac{q}{\Xi},~{}~{}~{}P=\frac{p}{\Xi},\
$ (8)
$T_{H}=\frac{r_{+}(l^{2}-2r_{+}^{2}+2d^{2}+2k^{2}-a^{2})-ml^{2}}{2\pi(r_{+}^{2}-d^{2}-k^{2}+a^{2})l^{2}},$
(9)
$S_{BH}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2}),~{}~{}~{}\Omega_{H}=\frac{a\Xi}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
(10)
$\Phi_{H}=\frac{q(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},~{}\Psi_{H}=\frac{p(r_{+}+d-p^{2}/m)}{r_{+}^{2}-d^{2}-k^{2}+a^{2}},$
(11) $V=\frac{4}{3}r_{+}S_{BH},~{}~{}~{}\mathcal{P}=-\frac{3}{8\pi l^{2}}.$
(12)

where those are physical mass, angular momentum, electric charge, magnetic
charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity,
electric potential, magnetic potential, volume and pressure. We can also
consider the thermodynamic quantities on the cosmological horizon $r_{c}$.
Those thermodynamic quantities on $r_{c}$ are given as follows

$M_{c}=-\frac{m}{\Xi},~{}~{}~{}J_{c}=-\frac{ma}{\Xi},~{}~{}~{}Q_{c}=-\frac{q}{\Xi},~{}~{}~{}P_{c}=-\frac{p}{\Xi},\
$ (13)
$T_{c}=\frac{r_{c}(2r_{c}^{2}-2d^{2}-2k^{2}+a^{2}-l^{2})+ml^{2}}{2\pi(r_{c}^{2}-d^{2}-k^{2}+a^{2})l^{2}},$
(14)
$S_{c}=\frac{\pi}{\Xi}(r_{c}^{2}-d^{2}-k^{2}+a^{2}),~{}~{}~{}\Omega_{c}=\frac{a\Xi}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},$
(15)
$\Phi_{c}=\frac{q(r_{c}+d-p^{2}/m)}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},~{}\Psi_{c}=\frac{p(r_{c}+d-p^{2}/m)}{r_{c}^{2}-d^{2}-k^{2}+a^{2}},$
(16) $V_{c}=\frac{4}{3}r_{c}S_{c},~{}~{}~{}\mathcal{P}_{c}=\frac{3}{8\pi
l^{2}}.$ (17)

These thermodynamic quantities are obtained by considering the black hole’s
event horizon as the boundary [26]. This is in contrast with the previous
thermodynamic quantities on the black hole’s event horizon where the
cosmological horizon is considered as the boundary. For this black hole, we
consider the cosmological constant as dynamical variable as a consequence of
considering mass of the black hole as enthalpy of the spacetime [27, 28].

##  III Structure of Dyonic Kerr-Sen-de Sitter Spacetime

In order to gain insight into the structure of dyonic KSdS spacetime, we
explore the horizon solutions by plotting the “fin” diagram by slicing through
the parameter space with 4 planes; (a) $p=0$, (b) $q=0$, (c) $p=1$, (d) $q=1$,
for $a=0.5,l=10$ as shown in Fig. 1. The zeroes of metric function
$\Delta/\varrho^{2}$ give four roots for the horizons,
$\hat{r}=r_{i},~{}(i=1-4)$. We observe the continuity of the real part of
horizon ${\rm Re}[r_{i}]$ connecting solutions of merging horizons. In
addition, there is a ring singularity at $\varrho=0$ or
$\hat{r}=\sqrt{d^{2}+k^{2}}=\displaystyle{\frac{p^{2}+q^{2}}{2m}}\equiv
r_{s}$.

For zero-charge spinning KSdS spacetime at fixed $a,l$, there is an extremal
horizon where both inner (Cauchy) horizon $r_{-}$ and outer (black hole)
horizon $r_{+}$ emerge at the critical $m$, above which $r_{-}$ and $r_{+}$
move away from one another. Finally at certain mass, $r_{+}$ will merge with
cosmological horizon $r_{c}$ and the outer physical spacetime region
disappears leaving the naked singularity at $\hat{r}=0$.

When either electric ($q$) or magnetic ($p$) charge is turned on, the
spacetime structure becomes more complicated as depicted in Fig. 1. Region I
contains an outer horizon, naked singularity and cosmological horizon where
$r_{+}<r_{s}<r_{c}$. Region II, IV, and VII are spacetime with naked
singularity and cosmological horizon and no black holes. Region III and VI are
spacetime with black holes and cosmological horizon, there exists singulairty
hidden behind the Cauchy horizon. Region V intriguingly has black hole region
with inner and outer horizon behind the naked singularity and cosmological
horizon at the furthest.

The boundary lines between each Region where extremal horizons occur are
categorized as follows.

  1. (i)

I and II: $r_{-}=r_{+}<r_{s}$,

  2. (ii)

II and III+VI: $r_{-}=r_{+}>r_{s}$,

  3. (iii)

III+VI and IV: $r_{+}=r_{c}>r_{s}$, Nariai BH,

  4. (iv)

II and V: $r_{-}=r_{+}<r_{s}$,

  5. (v)

I and V: $r_{-}=0,r_{+}<r_{s}$,

  6. (vi)

IV and VII: $r_{1}=r_{4}\equiv r_{c}>r_{s}$,

  7. (vii)

II and IV, VII: $r_{1}=r_{2}\equiv r_{c}>r_{s}$.

The Nariai limit of all cases occurs in the high mass region where
$r_{+}=r_{c}$, the boundary line between region III+VI and IV. On the other
hand, the boundary line between region II and III+VI represents extremal
solutions where $r_{-}=r_{+}$. In comparison to the “shark fin” figure in Ref.
[29], the dependence of $d,k$ on $m$ and $a$ distort the fin shape in the
small mass and charge region in KSdS case. Interestingly, similar kinds of
“flag” diagram where extremal structures are explored for generalized
spherically symmetric metric are also presented in Ref. [30, 31].

|
---|---
(a) $p=0$ KSdS | (b) $q=0$ KSdS
|
(c) $p=1$ KSdS | (d) $q=1$ KSdS
Figure 1: Horizon structure of KSdS spacetime for $a=0.5,l=10$, color region
represents parameter space with the horizon $r_{i}\geq 0$ for $i=1-4$. Region
I consists of an outer horizon behind a naked singularity and cosmological
horizon. Region II, IV, and VII are spacetime with naked singularity and
cosmological horizon. Region III and VI are black hole spacetime with
singularity behind the Cauchy horizon, event horizon, and cosmological
horizon. Region V contains naked singularity and cosmological horizon, and
inner and outer horizon of black hole locating behind a naked singularity.

##  IV Charged Rotating Nariai/CFT Correspondence

The rotating Nariai solutions from the Kerr-dS and KNdS solutions haves been
obtained in Ref. [9]. In this section, we will exhibit that this rotating
solution in Nariai limit can also be expressed in the metric form with a fiber
over AdS2 as the common near-horizon metric form in the Kerr/CFT
correspondence [10, 11, 12, 13]. Another purpose of this section is to give a
more detailed derivation of the central charge, temperature and entropy for
solution in Nariai limit obtained from KSdS black hole. Furthermore, we will
also study the massless scalar wave equation on this rotating background in
Nariai limit which may possesses $SL(2,R)\times SL(2,R)$ isometry.

###  IV.1 Geometry of Nariai Limit

To find the rotating geometry in Nariai limit, we will assume two different
limits which are Nariai limit and near-horizon limit. The parameter that
parameterizes the Nariai limit is defined by

$\varepsilon=\frac{r_{c}-r_{+}}{\lambda r_{0}}.$ (18)

$r_{0}$ is a scaling constant that we define as
$r_{0}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}$ for KSdS solution. When
$\varepsilon\rightarrow 0$, we can obtain $r_{c}=r_{+}$. Furthermore, the
near-horizon coordinate transformations are given by [22]

$\hat{r}=r_{+}+\lambda
r_{0}r,~{}~{}\hat{t}=\frac{r_{0}}{\lambda}t,~{}~{}\hat{\phi}=\phi+\frac{\Omega_{c}r_{0}}{\lambda}t.\
$ (19)

The constant $\lambda$ is to parameterize the near-horizon limit.
$\lambda\rightarrow 0$ denotes the near-horizon limit. Using those coordinate
transformations, one can find the charged rotating solution in Nariai limit as
follows

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r(r-\epsilon)dt^{2}+\frac{dr^{2}}{r(r-\epsilon)}+\alpha(\theta)d\theta^{2}\right)$
(20) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $

where the metric functions are given by

$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (21)
$\varrho_{+}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta,~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.$
(22)

Note that we already approximate the function $\Delta$ as

$\Delta\simeq\upsilon(r-r_{c})(r-r_{+}),\ $ (23)

where $\upsilon=1-(6r_{+}^{2}-d^{2}-k^{2}+a^{2})/l^{2}$. The rotating solution
in Nariai limit (20) is a fiber over AdS2. This fact can obviously be seen
from the factor on time-like and radial coordinates of the metric in Nariai
limit. One can also write the spacetime metric (20) in Poincaré coordinates by
applying the following coordinate transformations [41]

$\displaystyle t$ $\displaystyle=$
$\displaystyle-\frac{2}{\varepsilon}\text{log}\frac{y}{\sqrt{\tau^{2}y^{2}-1}},$
(24) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\varepsilon}{2}(1+\tau y),$ (25) $\displaystyle\phi$
$\displaystyle=$ $\displaystyle\varphi+\frac{1}{2}\text{log}\frac{\tau
y+1}{\tau y-1}.\ $ (26)

From those transformations, one can find

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-y^{2}d\tau^{2}+\frac{dy^{2}}{y^{2}}+\alpha(\theta)d\theta^{2}\right)$
(27) $\displaystyle+\gamma(\theta)\left(d\varphi+eyd\tau\right)^{2},\ $

This metric form is obviously similar with the common near-horizon metric that
is used in the Kerr/CFT correspondence. This metric possesses $SL(2,R)\times
U(1)$ isometry generated by the following vector fields

$\zeta_{0}=\partial_{\varphi},\ $ (28)

which denote the rotational $U(1)$ isometry and

$X_{1}=\partial_{\tau},~{}~{}~{}X_{2}=\tau\partial_{\tau}-y\partial_{y},\ $
(29)
$X_{3}=\left(\frac{1}{2y^{2}}+\frac{\tau^{2}}{2}\right)\partial_{\tau}-\tau
y\partial_{y}-\frac{e}{y}\partial_{\varphi},$ (30)

denoting $SL(2,R)$ isometry.

As we have mentioned in the beginning of this section, in Ref. [9] they manage
to show another form of charged rotating Nariai geometry from dyonic KNdS
solution that is a fiber over dS2 in Eq. (B.2) in their appendix, which is
given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+\alpha(\theta)d\theta^{2}\right)$
(31) $\displaystyle+\gamma(\theta)\left(d\phi+krdt\right)^{2}.\ $

For the detail forms of $\Gamma(\theta),\alpha(\theta),\gamma(\theta),k$, one
can see in the reference that we have mentioned. This metric can be found by
using the near-horizon coordinate transformations (2.13) in Ref. [9] then
using the coordinate changes (2.18). In that paper, they use
$\hat{r}=r_{c}-\lambda r_{c}r$ in Eq. (2.13), instead of using positive new
radial coordinate, that will result in metric with a fiber over dS2. This is
different with our result which is a fiber over AdS2. Hence, the charged
rotating geometry in Nariai limit can be portrayed in both fibers over dS2 and
AdS2, depending on the sign of the near-horizon radial coordinate
transformation we choose. In the other words, we can also obtain the charged
rotating solution of KSdS black hole in Nariai limit with a fiber over dS2
when we use $\hat{r}=r_{c}-\lambda r_{c}r$.

###  IV.2 CFT Duals

For rotating Nariai geometry in Ref. [9], they basically apply the similar
Kerr/CFT method to study the relation between black hole’s thermodynamics
macroscopically and microscopically using 2D CFT. They propose the similar
fall-off conditions for the metric deviations although the spacetime structure
contains dS2 slice, unlike in the common dual CFT which contains AdS2 slice.

####  IV.2.1 Central charge

In the Kerr/CFT correspondence, the entropy is assumed to originate from
partition function of 2D CFT resulting in Cardy entropy formula. The main
ingredients for this formula are the central charges and temperatures. The
basic procedure to compute the central charges is to employ the approach of
Brown and Henneaux [32] where the asymptotic symmetry group (ASG) needs to
satisfy some certain boundary conditions. In this work, we consider the
similar boundary conditions for the metric deviations as given in Ref. [7],

$\displaystyle
h_{\mu\nu}\sim\left(\begin{array}[]{cccc}\mathcal{O}(r^{2})&\mathcal{O}\left(\frac{1}{r^{2}}\right)&\mathcal{O}\left(\frac{1}{r}\right)&\mathcal{O}(1)\\\
&\mathcal{O}\left(\frac{1}{r^{3}}\right)&\mathcal{O}\left(\frac{1}{r^{2}}\right)&\mathcal{O}\left(\frac{1}{r}\right)\\\
&&\mathcal{O}\left(\frac{1}{r}\right)&\mathcal{O}\left(\frac{1}{r}\right)\\\
&&&\mathcal{O}(1)\end{array}\right).$ (36)

in the basis $(t,r,\theta,\phi)$ for metric (20). These metric deviations are
subleading with respect to the background metric. These boundary conditions
are basically chosen to eliminate excitations above extremality. The
asymptotic symmetry of the general black hole family includes diffeomorphisms
$\xi$ that satisfy

$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}=\xi^{\sigma}(\partial_{\sigma}g_{\mu\nu})+g_{\mu\sigma}(\partial_{\nu}\xi^{\sigma})+g_{\sigma\nu}(\partial_{\mu}\xi^{\sigma}),\
$ (37)

where the metric deviation is denoted by $\delta_{\xi}g_{\mu\nu}=h_{\mu\nu}$.

The most general diffeomorphism symmetry that preserves such boundary
conditions (36) in the asymptotic infinity is generated by the following
Killing vector field

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\left[c_{t}+\mathcal{O}\left(r^{-3}\right)\right]\partial_{t}+\left[-r\epsilon^{\prime}(\phi)+\mathcal{O}(1)\right]\partial_{r}$
(38)
$\displaystyle+\mathcal{O}\left(r^{-1}\right)\partial_{\theta}+\left[\epsilon(\phi)+\mathcal{O}\left(r^{-2}\right)\right]\partial_{\phi},\
$

where $c_{t}$ is an arbitrary constant and the prime $(^{\prime})$ denotes the
derivative with respect to $\phi$. This ASG contains one copy of the conformal
group of the circle which is generated by

$\displaystyle\zeta_{\epsilon}=\epsilon(\phi)\partial_{\phi}-r\epsilon^{\prime}(\phi)\partial_{r},$
(39)

that will be the part of the near-horizon metric in Nariai limit. We know that
the azimuthal coordinate is periodic under the rotation $\phi\sim\phi+2\pi$.
Hence, we may define $\epsilon_{\hat{n}}=-e^{-i\hat{n}\phi}$ and
$\zeta_{\epsilon}=\zeta_{\epsilon}(\epsilon_{\hat{n}})$. By the Lie bracket,
the symmetry generator (39) satisfies the Witt algebra,

$\displaystyle
i[\zeta_{\hat{m}},\zeta_{\hat{n}}]_{LB}=({\hat{m}}-{\hat{n}})\zeta_{{\hat{m}}+{\hat{n}}}.\
$ (40)

${\hat{m}},{\hat{n}}$ are just integers. The zero mode is the azimuthal
translation or $U(1)$ isometry, $\zeta_{0}=-\partial_{\phi}$.

The associated conserved charge is [33]

$\displaystyle Q_{\xi}=\frac{1}{8\pi}\int_{\partial\Sigma}k^{g}_{\zeta}[h;g].$
(41)

This given integral is over the boundary of a spatial slice. The contribution
of the metric tensor on the central charge is given explicitly by

$\displaystyle k^{g}_{\zeta}[h;g]$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\epsilon_{\rho\sigma\mu\nu}\bigg{\\{}\zeta^{\nu}D^{\mu}h-\zeta^{\nu}D_{\lambda}h^{\mu\lambda}$
(42)
$\displaystyle+\frac{h}{2}D^{\nu}\zeta^{\mu}-h^{\nu\lambda}D_{\lambda}\zeta^{\mu}+\zeta_{\lambda}D^{\nu}h^{\mu\lambda}$
$\displaystyle\left.+\frac{h^{\lambda\nu}}{2}\left(D^{\mu}\zeta_{\lambda}+D_{\lambda}\zeta^{\mu}\right)\right\\}dx^{\rho}\wedge
dx^{\sigma}.\ $

We should note that the last two terms in Eq. (42) vanish for an exact Killing
vector and an exact symmetry, respectively. The charge $Q_{\zeta}$ generates
symmetry through the Dirac brackets. The ASG possesses algebra which is given
by the Dirac bracket algebra of the following charges [33]

$\displaystyle\\{Q_{\zeta},Q_{\bar{\zeta}}\\}_{DB}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int
k^{g}_{\zeta}\left[\mathcal{L}_{\bar{\zeta}}g;g\right]$ (43) $\displaystyle=$
$\displaystyle Q_{[\zeta,\bar{\zeta}]}+\frac{1}{8\pi}\int
k^{g}_{\zeta}\left[\mathcal{L}_{\bar{\zeta}}\bar{g};\bar{g}\right].\ \ $

By using (39) and upon quantization, we can transform the Dirac bracket
algebra into a commutation relation that allows us to interpret the classical
central charge as a quantum central charge of the dual CFT. For the
quantization, we replace the classical charges $Q_{\zeta}$ by their quantum
counterpart $L_{{\hat{n}}}$,

$\displaystyle Q_{\zeta}\equiv L_{{\hat{n}}}-x\delta_{{\hat{n}},0},$ (44)

so that we obtain the conserved charges algebra in quantum form, such that

$\displaystyle\left[L_{\hat{m}},L_{\hat{n}}\right]=({\hat{m}}-{\hat{n}})L_{{\hat{m}}+{\hat{n}}}+\frac{c_{L}}{12}\hat{m}({\hat{m}}^{2}-1)\delta_{{\hat{m}}+{\hat{n}},0}.$
(45)

$x$ is a free parameter to scale the central charge on the term $\sim\hat{m}$
(see Appendix B). The relation (40) and (44) have been used to find the
Virasoro algebra above. For the charged rotating solution in Nariai limit
obtained from KSdS solution, the corresponding central charge is then

$\displaystyle c_{L}$ $\displaystyle=$
$\displaystyle\frac{3e}{\Xi}\int^{\pi}_{0}d\theta\sqrt{\Gamma(\theta)\alpha(\theta)\gamma(\theta)}$
(46) $\displaystyle=$
$\displaystyle\frac{12ar_{+}}{1-\frac{6r_{+}^{2}-d^{2}-k^{2}+a^{2}}{l^{2}}}.\
$

The main difference between this central charge and the Nariai solution from
KNdS black hole [9] is the presence of dilaton and axion charges.

####  IV.2.2 Temperature

In the Nariai limit where $r_{c}\rightarrow r_{+}$, the variation of the
entropy can be expressed completely in terms of a variation of the angular
momentum, electric charge, magnetic charge, and the pressure. It takes the
following form

$\displaystyle
dS_{BH}=\frac{dJ}{T_{L}}+\frac{dQ}{T_{q}}+\frac{dP}{T_{p}}+\frac{d\mathcal{P}}{T_{\mathcal{P}}},\
$ (47)

where $T_{L},T_{q},T_{p},T_{\mathcal{P}}$ are the left-moving temperature,
conjugate temperature of electric charge, conjugate temperature of magnetic
charge, and conjugate temperature of pressure, respectively. Hence, we can
derive that

$\displaystyle T_{L}=-\frac{\partial T_{H}/\partial
r_{+}}{\partial\Omega_{H}/\partial
r_{+}}\bigg{|}_{r_{+}=r_{c}}\frac{\upsilon(r_{+}^{2}-d^{2}-k^{2}+a^{2})}{4\pi
ar_{+}\Xi}.\ $ (48)

Since we already take $r_{c}\rightarrow r_{+}$, so the right-moving
temperature is equal to zero which is proportional to the Hawking temperature
on the cosmological horizon. However, in the next section, we will see that
$T_{R}\sim\varepsilon$ generally which is fairly small.

####  IV.2.3 Cardy Entropy

We have calculated the central extension and the corresponding temperature
which are the main ingredients in Cardy entropy. So, the main upshot of this
section is to provide the derivation of Cardy entropy [7] for the charged
rotating solution in Nariai limit. Firstly, Cardy entropy formula is given as
follows

$S_{CFT}=\frac{\pi^{2}}{3}\left(c_{L}T_{L}+c_{R}T_{R}\right).$ (49)

The right-moving part is exactly zero, so only left-moving part contributes.
On the cosmological horizon, by using above formula, we find the following
entropy for charged rotating solution in Nariai limit

$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2})=S_{BH}=S_{c}.\ $ (50)

This result exhibits that the Kerr/CFT correspondence can be applied for
charged rotating solution in Nariai limit obtained from KSdS black hole which
precisely possesses $SL(2,R)\times U(1)$ isometry like the near-horizon
extremal black hole. The entropy on the event horizon is now similar with the
entropy on the cosmological horizon since we take $r_{c}\rightarrow r_{+}$.
So, for an observer in the visible region, the total entropy in Nariai limit
is [34]

$S_{tot}=S_{c}+S_{BH}=2S_{c}.\ $ (51)

Note that this is the total entropy for charged rotating solution in Nariai
limit with non-vanishing dyonic charge. So, when $p=0$, it is just the total
entropy for electrically charged one. Furthermore, when all electromagnetic
charges vanish, it will be the total entropy of rotating Nariai solution
obtained from Kerr-dS solution [9]. For the entropy on the event horizon for
extremal KSdS black hole, we calculate in Appendix A where we also prove the
result in Ref. [45] on KSdS solution that is mentioned therein. In case of
asymptotically AdS spacetime, it has been carried out in Ref. [45]. For
vanishing cosmological constant and $p$, this result recovers the result in
Ref. [47].

##  V Scattering of Scalar Field and Hidden Conformal Symmetry

In this section, we will further study the scattering of massless scalar field
on the background of charged rotating Nariai solution in $J$ (angular
momentum)- and $Q$ (electric charge)-pictures. We will also compute the
absorption cross-section as well as the real-time correlator on the
corresponding background. Yet, first we will show that there are hidden
conformal symmetries on this solution in Nariai limit. Lastly, we also
consider $Q$-picture where the scalar probe is assumed to be electrically
charged.

###  V.1 Scalar wave equation

To explore the hidden conformal symmetries in $J$-picture, we assume a
massless neutral scalar field in the background of charged rotating solution
in Nariai limit. The massless scalar wave equation for the scalar probe is
given by

$\nabla_{\alpha}\nabla^{\alpha}\Phi=0.$ (52)

We know that the charged rotating solution in Nariai limit is conserved under
time-like and azimuthal translations. Hence, we can separate the coordinates
in the scalar wave equation as

$\Phi(\hat{t},\hat{r},\theta,\hat{\phi})=\mathrm{e}^{-i\omega\hat{t}+in\hat{\phi}}R(\hat{r})S(\theta),$
(53)

where $\omega$ and $n$ are the asymptotic energy and angular momentum of the
scalar field. Plugging Eq. (53) into Eq. (52), leads to two differential
equations i.e., the angular $S(\theta)$ and radial $R(\hat{r})$ wave
functions,

$\displaystyle\left[\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta})-\frac{n^{2}\Xi^{2}}{\Delta_{\theta}\sin^{2}\theta}\right]S(\theta)$
$\displaystyle+\left[\frac{2an\omega\Xi-a^{2}\omega^{2}\sin^{2}\theta}{\Delta_{\theta}}\right]S(\theta)=-K_{l}S(\theta),\
$ (54)
$\bigg{[}\partial_{\hat{r}}(\Delta\partial_{\hat{r}})+\frac{\left[(\hat{r}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\Delta}-K_{l}\bigg{]}R(\hat{r})=0,\ $ (55)

where $K_{l}$ is the separation constant. Note that we still use the generic
KSdS metric (1). These equations are identical with the wave equations in the
Kerr-dS solutions. We do not consider the backreaction of the scalar field in
this case.

Firstly, we will study the radial equation (55) at the near region, which is
defined by $\omega\hat{r}\ll 1$. Furthermore, we assume that the frequency of
the scalar field to be very small $\omega M\ll 1$. Consequently, we also
impose $\omega a\ll 1,\,\omega d\ll 1,$ and $\omega k\ll 1$. On the other
hand, we also have $\omega q\ll 1,$ and $\omega p\ll 1$. Since we have
approximated $\Delta$ as (23), we may set up the radial equation in a suitable
form for exploring its hidden conformal symmetry. The radial equation (55)
reduces to

$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(56)

where

$A=\frac{\left[(r_{c}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (57)
$B=-\frac{\left[(r_{+}^{2}-d^{2}-k^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (58)

It is worth noting that the angular wave equation does not have $SL(2,R)\times
SL(2,R)$ isometry, yet $SU(2)\times SU(2)$ isometry [39]. Hence, we will not
consider to investigate further the angular part.

In order to reveal the hidden symmetries of Eq. (56), we need to perform the
following conformal coordinate transformations [14, 41]

$\displaystyle\omega^{c}=\sqrt{\frac{\hat{r}-r_{c}}{\hat{r}-r_{+}}}e^{2\pi
T_{R}\hat{\phi}+2n_{R}\hat{t}},$
$\displaystyle\omega^{+}=\sqrt{\frac{\hat{r}-r_{c}}{\hat{r}-r_{+}}}e^{2\pi
T_{L}\hat{\phi}+2n_{L}\hat{t}},$
$\displaystyle\hat{y}=\sqrt{\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}}e^{\pi(T_{L}+T_{R})\hat{\phi}+(n_{L}+n_{R})\hat{t}}.\
$ (59)

Then we may define three locally conformal operators in terms of the new
conformal coordinates $\omega^{c},\,\omega^{+}$ and $\hat{y}$ as

$\displaystyle H_{1}=i\partial_{c},$ $\displaystyle
H_{-1}=i\left(\omega^{c2}\partial_{c}+\omega^{c}\hat{y}\partial_{\hat{y}}-\hat{y}^{2}\partial_{+}\right),$
$\displaystyle
H_{0}=i\left(\omega^{c}\partial_{c}+\frac{1}{2}\hat{y}\partial_{\hat{y}}\right),$
(60)

as well as

$\displaystyle\bar{H}_{1}=i\partial_{+},$
$\displaystyle\bar{H}_{-1}=i\left(\omega^{+2}\partial_{+}+\omega^{+}\hat{y}\partial_{\hat{y}}-\hat{y}^{2}\partial_{c}\right),$
$\displaystyle\bar{H}_{0}=i\left(\omega^{+}\partial_{+}+\frac{1}{2}\hat{y}\partial_{\hat{y}}\right).$
(61)

Note that we have used
$\partial_{c}=\partial/\partial\omega^{c},\partial_{+}=\partial/\partial\omega^{+}$.
The set of operators (V.1) satisfies the $SL(2,R)$ Lie algebra

$\displaystyle\left[H_{0},H_{\pm 1}\right]=\mp iH_{\pm
1},~{}~{}~{}\left[H_{-1},H_{1}\right]=-2iH_{0},$ (62)

while a similar $SL(2,R)$ algebra exists for the set of operators (V.1). From
every set of operators, we can construct the quadratic Casimir operator as
given by

$\displaystyle\mathcal{H}^{2}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{H}}^{2}=-H_{0}^{2}+\frac{1}{2}(H_{1}H_{-1}+H_{-1}H_{1})$
(63) $\displaystyle=$
$\displaystyle\frac{1}{4}(\hat{y}^{2}\partial_{\hat{y}}^{2}-y\partial_{\hat{y}})+\hat{y}^{2}\partial_{c}\partial_{+}.$

We have found that the radial equation (V.1) could be re-written in terms of
the $SL(2,R)$ quadratic Casimir operator as
$\mathcal{H}^{2}R(r)=\bar{\mathcal{H}}^{2}R(r)=-CR(r)$ where, in $J$-picture,
we should identify the constants as

$n_{L}=-\frac{\upsilon}{2(r_{c}+r_{+})},~{}~{}~{}n_{R}=0,$ (64)
$T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}-2d^{2}-2k^{2}+2a^{2})}{4\pi
a(r_{c}+r_{+})\Xi},~{}~{}~{}T_{R}=\frac{\upsilon(r_{c}-r_{+})}{4\pi a\Xi}.$
(65)

$T_{R,L}$ are identified as the CFT temperatures that emerge as a result of
the spontaneously symmetry breaking of the partition function on
$SL(2,R)\times SL(2,R)$ theory to the partition function of $U(1)\times U(1)$
CFT. One can see that the periodic identification of the azimuthal coordinate
$\hat{\phi}\sim\hat{\phi}+2\pi$ causes the $SL(2,R)\times SL(2,R)$ symmetry to
spontaneously break down to $U(1)\times U(1)$ symmetry by temperatures
$T_{R},T_{L}$,

$\omega^{c}\sim e^{4\pi^{2}T_{R}\omega^{c}},~{}\omega^{+}\sim
e^{4\pi^{2}T_{L}\omega^{+}},~{}\hat{y}\sim e^{2\pi^{2}(T_{L}+T_{L})\hat{y}}.$
(66)

This identification is generated by the $SL(2,R)\times SL(2,R)$ group element,
$e^{-i4\pi^{2}T_{R}H_{0}-i4\pi^{2}T_{L}\bar{H}_{0}}$.

After finding the temperatures (65), we can also compute conjugate charges
$E_{L},E_{R}$ for this charged rotating solution. These conjugate charges can
be obtained from the entropy via [41]

$\delta S_{CFT}=\frac{\delta E_{L}}{T_{L}}+\frac{\delta E_{R}}{T_{R}}.$ (67)

In order to compute the conjugate charges, we consider the first law of
thermodynamics for the charged rotating solution in Nariai limit (7) with the
quantities (13)-(17). Since we consider the neutral scalar field, we have
$\delta Q=\delta P=\delta\mathcal{P}=0$. Now we can find the conjugate charges
via $\delta S_{BH}=\delta S_{CFT}$. Using the identification $\delta M$ as
$\omega$ and $\delta J$ as $n$ yields to the identification of $\delta
E_{L,R}$ as $\omega_{L,R}$. Hence, in $J$-picture, we obtain that

$\displaystyle\omega_{L}=\frac{r_{c}^{2}+r_{+}^{2}-2d^{2}-2k^{2}+2a^{2}}{2a\Xi}\omega,~{}\omega_{R}=\omega_{L}-n.\
$ (68)

So, we find the left and right frequencies of 2D CFT for the charged rotating
solution in Nariai limit for KSdS black hole in $J$-picture. For the study of
the scalar field in KSAdS black hole, one can see in Ref. [18].

###  V.2 Absorption cross-section in $J$-picture

In the previous section, we have derived one remarkable result of the dual
CFT, i.e. the entropy. Another realization of this duality is the equivalence
of the absorption cross-section of the scalar probe. In investigating the
absorption cross-section, we need to consider the asymptotic region. Since we
have approximated $\Delta$ in near-horizon region, this approximation in the
asymptotic region will break down, except we consider the Nariai limit. This
is similar in the near-extremal case of the scalar wave equation [15, 16, 20,
38, 42]. Beside the Nariai limit on (18), we consider the near-horizon
coordinate transformations (19). We also consider the scalar probe with
frequencies around the superradiant bound

$\omega=n\Omega_{H}+\hat{\omega}\frac{\lambda}{r_{0}},$ (69)

where $\Omega_{H}$ are given by (10). We can re-write the radial equation
(V.1) by

$\left(\partial_{r}\left[r\left(r-\varepsilon\right)\right]\partial_{r}+\frac{A_{s}}{r-\varepsilon}+\frac{B_{s}}{r}+C_{s}\right)R(r)=0,$
(70)

where

$\displaystyle
A_{s}=\frac{(\hat{\omega}+2nr_{+}\varepsilon\Omega_{H})^{2}}{\upsilon^{2}\varepsilon},~{}~{}~{}B_{s}=-\frac{\hat{\omega}^{2}}{\upsilon^{2}\varepsilon},~{}~{}~{}C_{s}=C_{s}(\hat{\omega}),\
$

and $C_{s}$ is the new separation constant that is dependent on the frequency
$\hat{\omega}$ that can be obtained by solving the angular wave equation. We
then apply the coordinate transformation $z=(r-\varepsilon)/r$. In this new
radial coordinate, the radial equation (70) becomes

$\left[z(1-z)\partial_{z}^{2}+(1-z)\partial_{z}+\frac{\hat{A_{s}}}{z}+\hat{B_{s}}+\frac{C_{s}}{1-z}\right]R(z)=0,$
(71)

where

$\displaystyle\hat{A_{s}}=\frac{(\hat{\omega}+2nr_{+}\varepsilon\Omega_{H})^{2}}{\upsilon^{2}\varepsilon^{2}},~{}~{}~{}\hat{B_{s}}=-\frac{\hat{\omega}^{2}}{\upsilon^{2}\varepsilon^{2}}.\
$

The ingoing solution to differential equation (71) is also given by
hypergeometric function

$R(z)=z^{-i\sqrt{\hat{A_{s}}}}(1-z)^{1+h}~{}_{2}F_{1}(a_{s},b_{s};c_{s};z),\ $
(72)

with the parameters $a_{s}=1+h-i(\sqrt{\hat{A}_{s}}+\sqrt{-\hat{B}_{s}})$,
$b_{s}=1+h-i(\sqrt{\hat{A}_{s}}-\sqrt{-\hat{B}_{S}})$, and
$c_{s}=1-2i\sqrt{\hat{A}_{s}}$. For this superradiant case, the relation
between $h$ and $C_{s}$ is given by

$h=\frac{1}{2}\left(-1+\sqrt{1-4C_{s}}\right).$ (73)

For the asymptotic region of the radial coordinate $\hat{r}$ (or equivalently
$r\gg\varepsilon$), where $z\sim 1$, the solution (72) reduces to

$\displaystyle R(y)\sim D_{0}r^{h}+D_{1}r^{-1-h},\ $ (74)

where

$D_{0}=\frac{\Gamma(c_{s})\Gamma(1+2h)}{\Gamma(a_{s})\Gamma(b_{s})},~{}~{}~{}D_{1}=\frac{\Gamma(c_{s})\Gamma(-1-2h)}{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{s})}.\
$ (75)

The conformal weight of the scalar field is equal to $h+1$.

For the coefficient (75), we find the absorption cross-section of the scalar
fields as

$P_{abs}\sim\left|D_{0}\right|^{-2}\sim\sinh\left({2\pi\hat{A}_{s}^{1/2}}\right){\left|{\Gamma\left(a_{s}\right)}{\Gamma\left(b_{s}\right)}\right|^{2}}.$
(76)

To be more supporting the correspondence between the charged rotating solution
in Nariai limit and 2D CFT, we show that the absorption cross-section for the
scalar fields (76) can be obtained from the absorption cross-section in a 2D
CFT [14]

$\displaystyle P_{abs}$ $\displaystyle\sim$
$\displaystyle{T_{L}}^{2h_{L}-1}{T_{R}}^{2h_{R}-1}\sinh\left({\frac{{\omega_{L}}}{{2{T_{L}}}}+\frac{{\omega_{R}}}{{2{T_{R}}}}}\right)$
(77) $\displaystyle\times$
$\displaystyle\left|{\Gamma\left({h_{L}+i\frac{{\omega_{L}}}{{2\pi{T_{L}}}}}\right)}{\Gamma\left({h_{R}+i\frac{{\omega_{R}}}{{2\pi{T_{R}}}}}\right)}\right|^{2}.\
$

The agreement between (76) and (77) can be shown when we choose proper left
and right frequencies $\omega_{L},\omega_{R}$. Since in the previous
subsection we have found the CFT temperatures and frequencies, we can directly
calculate the similar quantities for the frequency of the scalar field in the
near-superradiant bound by neglecting the second-order and higher corrections
from $\lambda$. Hence, in terms of Nariai limit parameter and in near-
superradiant bound, we obtain

$T_{L}=\frac{\upsilon}{4\pi\Omega_{H}r_{+}},~{}~{}~{}T_{R}=\frac{\upsilon\lambda
r_{0}}{4\pi a\Xi}\varepsilon.$ (78)

When we take the Nariai limit $\varepsilon\rightarrow 0$ the right-moving
temperature will vanish. Then we recover the result for $T_{R}$ and $T_{L}$ in
the previous section. The frequencies are now

$\omega_{L}=n,~{}~{}~{}\omega_{R}=\frac{r_{0}}{a\Xi}\left(\hat{\omega}+nr_{+}\varepsilon\Omega_{c}\right),\
$ (79)

The conformal weights in CFT are given by

$h_{L}=h_{R}=h+1.\ $ (80)

This gives rise to another nontrivial evidence to support the Kerr/CFT
correspondence for black holes in Nariai limit.

###  V.3 Real-time correlator in $J$-picture

Furthermore, one can also compute the real-time correlator. The asymptotic
behaviors of the scalar field with ingoing boundary condition on the
background of charged rotating solution in Nariai limit (74) indicate that two
coefficients possess different roles where $D_{1}$ indicates the response and
$D_{0}$ indicates the source. Hence, the two-point retarded correlator is
simply [43, 44]

$G_{R}\sim\frac{D_{1}}{D_{0}}=\frac{\Gamma(-1-2h)}{\Gamma(1+2h)}\frac{\Gamma(a_{s})\Gamma(b_{s})}{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{s})}.\
$ (81)

From Eq. (81), it is easy to check that

$G_{R}\sim\frac{\Gamma(h_{L}-i\frac{\omega_{L}}{2T_{L}})\Gamma(h_{R}-i\frac{\omega_{R}}{2T_{R}})}{\Gamma(1-h_{L}-i\frac{\omega_{L}}{2T_{L}})\Gamma(1-h_{R}-i\frac{\omega_{R}}{2T_{R}})}.$
(82)

Then by using the relation $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$, we can
write above two-point retarded correlator into

$\displaystyle G_{R}$ $\displaystyle\sim$ $\displaystyle\sin\left(\pi
h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(\pi
h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)$ (83)
$\displaystyle\bigg{|}\Gamma\left(h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\Gamma\left(h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)\bigg{|}^{2}.$

Since $h_{L},h_{R}$ are integers, so we have

$\displaystyle\sin\left(\pi
h_{L}+i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(\pi
h_{R}+i\frac{\omega_{R}}{2T_{R}}\right)$
$\displaystyle=(-1)^{h_{L}+h_{R}}\sin\left(i\frac{\omega_{L}}{2T_{L}}\right)\sin\left(i\frac{\omega_{R}}{2T_{R}}\right).\
$ (84)

In CFT, the Euclidean correlator is given by

$\displaystyle G_{E}(\omega_{EL},\omega_{ER})\sim
T_{L}^{2h_{L}-1}T_{R}^{2h_{R}-1}e^{i\omega_{EL}/2T_{L}}e^{i\omega_{ER}/2T_{R}}$
$\displaystyle\times\bigg{|}\Gamma\left(h_{L}+\frac{\omega_{EL}}{2T_{L}}\right)\Gamma\left(h_{R}+\frac{\omega_{EL}}{2T_{R}}\right)\bigg{|}^{2}.$
(85)

Where we can define the Euclidean frequencies $\omega_{EL}=i\omega_{L}$, and
$\omega_{ER}=i\omega_{R}$. It is important noting that $G_{E}$ corresponds to
the values of retarded correlator $G_{R}$. The retarded Green function $G_{R}$
is analytic on the upper half complex $\omega_{L,R}$-plane. The value of
$G_{R}$ along the positive imaginary $\omega_{L,R}$-axis gives the following
correlator

$G_{E}(\omega_{EL},\omega_{ER})=G_{R}(i\omega_{L},i\omega_{R}),~{}~{}~{}\omega_{EL,ER}>0.$
(86)

At finite temperature, $\omega_{EL,ER}$ should take discrete values of the
Matsubara frequencies, given by

$\omega_{EL}=2\pi m_{L}T_{L},~{}~{}~{}\omega_{ER}=2\pi m_{R}T_{R},$ (87)

where $m_{L}$ and $m_{R}$ are half integers for fermionic modes and integers
for bosonic modes. At these certain frequencies, the gravity computation for
correlator (83) matches precisely with CFT result trough Eq. (86) up to a
numerical normalization factor.

###  V.4 Hidden conformal symmetry in $Q$-Picture

In the previous subsections, we have shown the existence of hidden conformal
symmetry on the $J$-picture with the background charged rotating black hole in
Nariai limit. In this subsection, we will further study the scattering of
massless scalar field on the background of charged rotating black hole
solution in Nariai limit in $Q$-picture. On the other hand, we will consider
charged scalar field on the background solution. For some black hole
backgrounds, $Q$-picture has been explored well to exhibit the existence of
the conformal symmetries [15, 16, 37, 38, 35, 36]. Nonetheless, the
$Q$-picture is not well-defined for non-extremal Kerr-Sen black holes [35]. In
this section, we will investigate the $Q$-picture when the cosmological
constant exists. Since we will only consider $Q$-picture, for simplicity, we
may assume that $p=0$, so that $k=0$.

The hidden conformal symmetries can be explored by assuming a massless charged
scalar probe in the background of charged rotating black hole solution in
Nariai limit. It is given by

$(\nabla_{\alpha}-igA_{\alpha})(\nabla^{\alpha}-igA^{\alpha})\Phi=0,$ (88)

where $g$ is the electric charge of the scalar probe. In $Q$-picture, in
addition to the modes of asymptotic energy and angular momentum, we need to
add the charge $g$ as the eigenvalue of the operator $\partial_{\chi}$ that
denotes the additional internal direction $\chi$ to four dimensions. Actually,
$g$ has a natural geometrical interpretation as the radius of extra circle
when the the black hole solution is considered to be embedded into 5D. In
fact, this coordinate possesses similar $U(1)$ symmetry like the azimuthal
coordinate. However, so far, we do not find that 4D Kerr-Sen(-dS) black hole
can be uplifted to 5D solution. It is different with Kerr-Newman(-dS) black
hole that can be uplifted to 5D [22]. Here, we will just first assume the
fifth coordinate to reveal the conformal symmetries of the charged probe in
$Q$-picture. The ansatz in $Q$-picture is given by

$\Phi(\hat{t},\hat{r},\theta,\hat{\phi},\chi)=\mathrm{e}^{-i\omega\hat{t}+in\hat{\phi}+ig\chi}R(\hat{r})S(\theta).$
(89)

Beside the low-frequency assumption, we need to assume also that the probe’s
charge to be very small ($gq\ll 1$). By plugging Eq. (89) into Eq. (88) and
assuming small charge and low frequency, in the near-horizon region, we can
find the radial equation

$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(90)

where

$B=-\frac{\left[(r_{+}^{2}-d^{2}+a^{2})\omega-an\Xi-
gq(r_{+}+d)\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (91)
$A=\frac{\left[(r_{c}^{2}-d^{2}+a^{2})\omega-an\Xi-
gq(r_{c}+d)\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (92)

In $Q$-picture, we take $n=0$. As the previous computation for $J$-picture, we
can identify that

$n_{L}=-\frac{\upsilon(r_{c}+r_{+})}{4\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},\
$ (93)
$n_{R}=-\frac{\upsilon(r_{c}-r_{+})}{4\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},$
(94) $T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}-2d^{2}+2a^{2})}{4\pi
g\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]},\ $ (95)
$T_{R}=\frac{\upsilon(r^{2}_{c}-r^{2}_{+})}{4\pi
g\left[(r_{c}+d)(r_{+}+d)-a^{2}\right]}.$ (96)

The temperatures (96) are irregular. We cannot check directly that
$(r_{c}+d)(r_{+}+d)-a^{2}=0$ because we do not have the explicit analytical
form of $r_{c},r_{+}$. However, we can easily check that the form of CFT
temperatures is similar with the case when the cosmological constant vanishes.
So, when $1/l^{2}=0$, one can obtain

$(r_{+}+d)(r_{-}+d)-a^{2}=0,\ $ (97)

where

$r_{\pm}=M\pm\sqrt{M^{2}+d^{2}-a^{2}-q^{2}}.$ (98)

This is the reason that $Q$\- picture is not well-defined for Kerr-Sen black
hole. With the similar form of temperatures, we conclude also that $Q$-picture
for Kerr-Sen-dS black hole in Nariai limit is not well-defined.

##  VI Hidden Conformal Symmetry on Nariai limit for Kerr-Newman-de Sitter
solution

In the previous sections, we have carried out the calculation of the dual CFT
for charged rotating black hole in Nariai limit obtained from KSdS solution
and also the hidden conformal symmetry of the scalar probe in the black hole’s
background. In this section, we will study the scattering of scalar field on
Nariai limit for KNdS solution both in $J$\- and $Q$-pictures. Yet, we will
revisit the computation of the entropy from CFT for KNdS solution in Nariai
limit. The KNdS black hole solution is given by the following metric [22, 23]

$\displaystyle
ds^{2}=-\frac{\Delta}{\varrho^{2}}X^{2}+\frac{\varrho^{2}}{\Delta}d\hat{r}^{2}+\frac{\varrho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\varrho^{2}}Y^{2},\
$ (99)

where

$X=d\hat{t}-\frac{a\sin^{2}\theta}{\Xi}d\hat{\phi},~{}~{}~{}Y=ad\hat{t}-\frac{(\hat{r}^{2}+a^{2})}{\Xi}d\hat{\phi},\
$ (100)
$\Delta_{\theta}=1+\frac{a^{2}}{l^{2}}\cos^{2}\theta,~{}~{}\Xi=1+\frac{a^{2}}{l^{2}},~{}~{}~{}\varrho^{2}=\hat{r}^{2}+a^{2}\cos^{2}\theta,\\\
$ (101)
$\Delta=(\hat{r}^{2}+a^{2})\left(1-\frac{\hat{r}^{2}}{l^{2}}\right)-2m\hat{r}+p^{2}+q^{2}.\
$ (102)

The parameters $a$, $m$, $p$, $q$, and $l$ are spin, mass, magnetic charge,
electric charge, and de Sitter radius, respectively. The electromagnetic
potential and its dual are given by

$\displaystyle\textbf{A}=-\frac{q\hat{r}}{\varrho^{2}}X-\frac{p\cos\theta}{\varrho^{2}}Y,~{}~{}~{}\textbf{B}=-\frac{p\hat{r}}{\varrho^{2}}X+\frac{q\cos\theta}{\varrho^{2}}Y,$
(103)

At the event horizon, the thermodynamic quantities are given by [22, 23]

$M=\frac{m}{\Xi},~{}~{}~{}J=\frac{ma}{\Xi},~{}~{}~{}Q=\frac{q}{\Xi},~{}~{}~{}P=\frac{p}{\Xi},$
(104)
$T_{H}=\frac{r_{+}(l^{2}-2r_{+}^{2}-a^{2})-ml^{2}}{2\pi(r_{+}^{2}+a^{2})l^{2}},$
(105) $S_{BH}=\frac{\pi}{\Xi}(r_{+}^{2}+a^{2}),$ (106)
$\Omega_{H}=\frac{a\Xi}{r_{+}^{2}+a^{2}},~{}~{}\Phi_{H}=\frac{qr_{+}}{r_{+}^{2}+a^{2}},~{}~{}\Psi_{H}=\frac{pr_{+}}{r_{+}^{2}+a^{2}},$
(107) $V=\frac{4}{3}r_{+}S_{BH},~{}~{}~{}\mathcal{P}=-\frac{3}{8\pi l^{2}}.$
(108)

where those are physical mass, angular momentum, electric charge, magnetic
charge, Hawking temperature, Bekenstein-Hawking entropy, angular velocity,
electric potential, magnetic potential, thermodynamic volume and pressure.
These thermodynamic quantities also satisfy the relation (7). Furthermore, one
can find the similar relation for thermodynamics on the cosmological horizon
with the following quantities [24, 25]

$M_{c}=-\frac{m}{\Xi},~{}~{}~{}J_{c}=-\frac{ma}{\Xi},~{}~{}~{}Q_{c}=-\frac{q}{\Xi},~{}~{}~{}P_{c}=-\frac{p}{\Xi},$
(109)
$T_{c}=\frac{r_{c}(2r_{c}^{2}+a^{2}-l^{2})+ml^{2}}{2\pi(r_{c}^{2}+a^{2})l^{2}},$
(110) $S_{c}=\frac{\pi}{\Xi}(r_{c}^{2}+a^{2}),$ (111)
$\Omega_{c}=\frac{a\Xi}{r_{c}^{2}+a^{2}},~{}~{}\Phi_{c}=\frac{qr_{c}}{r_{c}^{2}+a^{2}},~{}~{}\Psi_{c}=\frac{pr_{c}}{r_{c}^{2}+a^{2}},$
(112) $V_{c}=\frac{4}{3}r_{c}S_{c},~{}~{}~{}\mathcal{P}_{c}=\frac{3}{8\pi
l^{2}}.$ (113)

These thermodynamic quantities are obtained also by considering the event
horizon of the black hole as the boundary. Likewise the KSdS solution, the
cosmological constant can be assumed as a dynamical quantity.

###  VI.1 Nariai limit on Kerr-Newman-de Sitter black hole revisited

On the Nariai limit, likewise the KSdS solution, we can find the geometry
which is fiber over AdS2. Using the similar transformations (18) and (19) on
the spacetime metric (1), we can obtain

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r(r-\epsilon)dt^{2}+\frac{dr^{2}}{r(r-\epsilon)}+\alpha(\theta)d\theta^{2}\right)$
(114) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $

where the metric functions are given by

$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (115)
$\varrho_{+}^{2}=r_{+}^{2}+a^{2}\cos^{2}\theta,~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.$
(116)

and now we have $r_{0}^{2}=r_{+}^{2}+a^{2}$ and
$\upsilon=1-(6r_{+}^{2}+a^{2})/l^{2}$. When the spin vanishes, the spacetime
becomes AdS${}_{2}\times S^{2}$.

Even we have different slice in the asymptotic region, by using similar
analysis of ASG, we can obtain the exact central charge as found in [9]. It is
precisely given by

$c_{L}=\frac{12ar_{+}}{1-\frac{6r_{+}^{2}+a^{2}}{l^{2}}}.\ $ (117)

The main different of this central charge with that of the solution in Nariai
limit obtained from KSdS solution is the presence of the dilaton and axion
charges. The temperature can be calculated directly as the previous result for
Nariai limit on KNdS solution. We can find that

$\displaystyle T_{L}=\frac{\upsilon(r_{+}^{2}+a^{2})}{4\pi
ar_{+}\Xi},~{}~{}T_{R}=0.\ $ (118)

We have obtained the corresponding central charge and the CFT temperature for
the solution in Nariai limit obtained from KNdS black hole. On the
cosmological horizon, by using Cardy formula, we find the following entropy
for the charged rotating black hole from KNdS black hole in Nariai limit,

$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}+a^{2})=S_{BH}=S_{c}.\ $ (119)

So, in this case, the total entropy will be $2S_{c}$.

###  VI.2 Hidden conformal symmetry in $J$-picture

For the charged rotating solution in Nariai limit obtained from KNdS black
hole, we also assume the massless neutral scalar probe. We also consider the
low-frequency limit for the scalar field to exhibit the conformal symmetry.
For the small frequency, we have $\omega M\ll 1$, $\omega a\ll 1,\,\omega q\ll
1,$ and $\omega p\ll 1$. The radial equation is then given by

$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(120)

where

$A=\frac{\left[(r_{c}^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (121)
$B=-\frac{\left[(r_{+}^{2}+a^{2})\omega-
an\Xi\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (122)

The investigation of hidden conformal symmetry for KNdS solution is also done
in Ref. [40], however, the author does not consider Nariai limit. The
conformal symmetry can be revealed using the conformal coordinates
transformations (59) which similarly result in $SL(2,R)\times SL(2,R)$
isometry for the set of operators (V.1) and (V.1). Using the similar
computation, we can calculate the temperatures. In this case, we obtain

$n_{L}=-\frac{\upsilon}{2(r_{c}+r_{+})},~{}~{}~{}n_{R}=0,$ (123)
$T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}+2a^{2})}{4\pi
a(r_{c}+r_{+})\Xi},~{}~{}~{}T_{R}=\frac{\upsilon(r_{c}-r_{+})}{4\pi a\Xi}.$
(124)

Note that those quantities are different to those of the Nariai solution
obtained from KSdS solution since there exist dilaton and axion charges. When
we take $\varepsilon\rightarrow 0$, the temperatures (124) reduce to (118).

After computing the CFT temperatures (124), we will compute the conjugate
charges $E_{L},E_{R}$ for this solution. These conjugate charges can be
obtained from the entropy via Eq. (67) by considering the first law of
thermodynamics for the charged rotating black hole solution (7) with the
quantities (109)-(113). For the neutral scalar field, we have $\delta Q=\delta
P=\delta\mathcal{P}=0$. Using again the identification $\delta M$ as $\omega$
and $\delta J$ as $n$ yields to the identification of $\delta E_{L,R}$ as
$\omega_{L,R}$. We obtain the following left and right frequencies,

$\omega_{L}=\frac{r_{c}^{2}+r_{+}^{2}+2a^{2}}{2a\Xi}\omega,~{}~{}~{}\omega_{R}=\omega_{L}-n.\
$ (125)

These are the left and right frequencies of 2D CFT for the charged rotating
solution for KNdS solution in Nariai limit.

Regarding the absorption cross-section and real-time correlator, we need to
employ the radial equation (120). Similarly with KNdS case, the approximation
of $\Delta$ in the asymptotic region will break down, except we consider the
Nariai limit which is identical to near-extremal limit. We also consider the
scalar probe with frequencies around the superradiant bound (69). With the
similar lengthy computation, we find the absorption cross-section of the
scalar fields as given in Eq. (76) which is precisely similar to (77) for the
charged rotating solution in Nariai limit obtained from KNdS solution. Once
again, the agreement between (76) and (77) for this solution can be shown when
we choose proper left and right frequencies $\omega_{L},\omega_{R}$ as given
in Eq. (125). In the superradiant bound, we also find the form of temperatures
(78) and the frequencies (79) with the conformal weights (80). Yet, we have to
note again that in the solution coming from Einstein-Maxwell theory, there is
no contribution from $d,k$ on $r_{+}$. We can also compute the real-time
correlator. This is given by Eq. (82). This further exhibits that the Kerr/CFT
correspondence for KNdS black hole in Nariai limit is valid likewise the
solution from KSdS black hole.

###  VI.3 Hidden conformal symmetry in $Q$-picture

In exploring the hidden conformal symmetries on the wave equation, again, we
assume a massless charged scalar probe in the background (99) in Nariai limit
as given by Eq. (88). By plugging Eq. (89) into Eq. (88) and assuming small
charge and low frequency, in the near-horizon region of (99) in Nariai limit,
we can find the radial equation

$\displaystyle\partial_{\hat{r}}\left[(\hat{r}-r_{c})(\hat{r}-r_{+})\partial_{\hat{r}}\right]R(\hat{r})+\frac{r_{c}-r_{+}}{\hat{r}-r_{c}}AR(\hat{r})=0,$
$\displaystyle+\left[\frac{r_{c}-r_{+}}{\hat{r}-r_{+}}B+C\right]R(\hat{r})=0,$
(126)

where

$B=-\frac{\left[(r_{+}^{2}+a^{2})\omega-an\Xi-
gqr_{+}\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},\ $ (127)
$A=\frac{\left[(r_{c}^{2}+a^{2})\omega-an\Xi-
gqr_{c}\right]^{2}}{\upsilon^{2}(r_{c}-r_{+})^{2}},~{}~{}~{}C=\frac{-K_{l}}{\upsilon}.\
$ (128)

For this charged rotating solution obtained from KNdS black hole in Nariai
limit, in $Q$-picture, we can identify that

$n_{L}=-\frac{\upsilon(r_{c}+r_{+})}{4\left(r_{c}r_{+}-a^{2}\right)},~{}n_{R}=-\frac{\upsilon(r_{c}-r_{+})}{4\left(r_{c}r_{+}-a^{2}\right)},$
(129) $T_{L}=\frac{\upsilon(r_{c}^{2}+r_{+}^{2}+2a^{2})}{4\pi
g\left(r_{c}r_{+}-a^{2}\right)},~{}T_{R}=\frac{\upsilon(r^{2}_{c}-r^{2}_{+})}{4\pi
g\left(r_{c}r_{+}-a^{2}\right)}.$ (130)

These CFT temperatures are regular, unlike to that of KSdS solution in Nariai
limit.

Since there exists electric charge of the scalar probe, in order to satisfy
the entropy relation (67), we need the conjugate of the electric charge,
namely the chemical potential ($\mu_{L,R}$), so that

$E_{L,R}=\hat{\omega}_{L,R}=\omega_{L,R}-\mu_{L,R}q_{L,R}.\ $ (131)

So, in $Q$-picture, the charged scalar probe on the charged rotating solution
obtained from KNdS black hole in Nariai limit is related to the following CFT
frequencies,

$\displaystyle\omega_{L}$ $\displaystyle=$
$\displaystyle\frac{(r_{c}+r_{+})(r_{c}^{2}+r_{+}^{2}+2a^{2})}{2g(r_{c}r_{+}-a^{2})}\omega,$
$\displaystyle\mu_{L}$ $\displaystyle=$
$\displaystyle\frac{r_{c}^{2}+r_{+}^{2}+2a^{2}}{2(r_{c}r_{+}-a^{2})},~{}~{}~{}q_{L}=q,$
$\displaystyle\omega_{R}$ $\displaystyle=$
$\displaystyle\omega_{L}-\frac{2a(r_{c}+r_{+})}{2g(r_{c}r_{+}-a^{2})}n,$
$\displaystyle\mu_{R}$ $\displaystyle=$
$\displaystyle\frac{(r_{c}+r_{+})^{2}}{2(r_{c}r_{+}-a^{2})},~{}~{}~{}q_{R}=q.\
$ (132)

One can apply the frequencies (131), temperatures (130), and conformal weights
(80) in order to compute the absorption cross-section and real-time
correlator. Note that to study the superradiant bound in $Q$-picture, we have
assume the frequency near the superradiant bound
$\omega=n\Omega_{H}+g\Phi_{H}+\hat{\omega}\frac{\lambda}{r_{0}}$.

##  VII Summary

In this work, we have shown the horizon solutions and extremal limits in Fig.
1. The Nariai limit could be achieved in some values of parameters. We then
have carried out the calculation of the entropy for KSdS solutions in Nariai
limit parameterized by a constant $\varepsilon$. When $\varepsilon\rightarrow
0$, the cosmological horizon and event horizon coincide. This solution in
Nariai limit can be written in the metric form with AdS2 structure. Hence, we
have shown that, in the Nariai limit, we could also portray the solution as a
fiber over AdS2, instead of a fiber over dS2 as exhibited in Ref. [9]. We have
computed the corresponding central charge and CFT temperature where the right-
moving temperature is proportional to $\varepsilon$ denoting that it will
vanish when $\varepsilon\rightarrow 0$. It is found that by employing Cardy
entropy formula, the Bekenstein-Hawking entropy on the cosmological constant
is reproduced. It denotes that the charged rotating solutions from KSdS black
hole in Nariai limit is holographically dual with 2D CFT.

To further support the CFT dual on KSdS black hole in Nariai limit, we have
investigated the neutral ($J$-picture) and charged ($Q$-picture) massless
scalar probes on that background. Similarly with generic rotating black hole,
we could exhibit the conformal symmetries on the radial wave equation. With
the appropriate locally conformal coordinate transformations, it has been
shown that the radial equation possesses $SL(2,R)\times SL(2,R)$ isometry.
This is similar with AdS3 space. The periodic identification of azimuthal
coordinate portrays the spontaneous symmetry breaking from $SL(2,R)\times
SL(2,R)$ to $U(1)\times U(1)$ by the left- and right-moving temperatures. In
$J$-picture, the temperatures produced on this symmetry breaking are precisely
similar with the temperatures that are obtained when the conformal symmetry
appears directly on the spacetime metric. Hence, if we employ the central
charges with the given temperatures, we can again reproduce the Bekenstein-
Hawking entropy on the cosmological horizon. Moreover, we also have computed
the absorption cross-section and the real-time correlator that correspond with
the KSdS solution in Nariai limit. However, in $Q$-picture, we coud not find a
well-defined CFT description because the temperatures are irregular.

We have also extended the calculation to the KNdS black hole in Nariai limit
both in $J$\- and $Q$-pictures. The CFT description in both pictures are well-
defined. The results from gravity calculation are also exactly in agreement
with the result from 2D CFT. So, this calculation is another proof that the
black hole solution in Nariai limit is holographically dual to 2D CFT.

##  Appendix A Entropy on the Event Horizon for Extremal KSdS solution

We will briefly derive the entropy of extremal KSdS solution in this section.
The entropy of the gauged dyonic Kerr-Sen black hole solution or KSAdS
solution has been calculated previously in Ref. [45]. For the asymptotically
de Sitter solution, we just need to follow the computation given in Ref. [45]
where therein, we have argued on the central charge for KSdS solution by
taking $l^{2}\rightarrow-l^{2}$. It has been noted in Ref. [45] that in the
extremal case of KSAdS solution, the mass can have more than two branches as
it should be similar to KSdS solution. Let start directly from the near-
horizon extremal form of KSdS solution,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Gamma(\theta)\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}+\alpha(\theta)d\theta^{2}\right)$
(133) $\displaystyle+\gamma(\theta)\left(d\phi+erdt\right)^{2},\ $

where the metric functions are given by

$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{\upsilon},~{}~{}~{}\alpha(\theta)=\frac{\upsilon}{\Delta_{\theta}},~{}~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\Delta_{\theta}\sin^{2}\theta}{\varrho_{+}^{2}\Xi^{2}},\
$ (134)
$\varrho_{+}^{2}=r_{+}^{2}-d^{2}-k^{2}+a^{2}\cos^{2}\theta,~{}~{}~{}e=\frac{2ar_{+}\Xi}{r_{0}^{2}\upsilon}.\
$ (135) $\upsilon=1-\frac{6r_{+}^{2}-2d^{2}-2k^{2}+a^{2}}{l^{2}}.$ (136)

In this case, we can precisely derive the corresponding central charge as

$\displaystyle c_{L}$ $\displaystyle=$
$\displaystyle\frac{3e}{\Xi}\int^{\pi}_{0}d\theta\sqrt{\Gamma(\theta)\alpha(\theta)\gamma(\theta)}$
(137) $\displaystyle=$
$\displaystyle\frac{12ar_{+}}{1-\frac{6r_{+}^{2}-d^{2}-k^{2}+a^{2}}{l^{2}}}.\
$

Note that although this central charge looks identical with that of in Nariai
limit, both central charges are differnt because we are considering different
limit on the horizons. When the magnetic charge vanishes, one can obtain the
central charge that corresponds to non-dyonic KSAdS solution. Furthermore,
when the cosmological constant vanishes, the central charge recovers the
central charge of Kerr-Sen black hole [47, 35]. For the CFT temperature, we
just need to follow Ref. [45]. For KSdS solution, we can find

$\displaystyle T_{L}=-\frac{\partial T_{H}/\partial
r_{+}}{\partial\Omega_{H}/\partial
r_{+}}\bigg{|}_{ex}=\frac{\upsilon(r_{+}^{2}-d^{2}-k^{2}+a^{2})}{4\pi
ar_{+}\Xi}.\ $ (138)

By implementing Cardy entropy formula and inserting (137) and (138) into it,
we successfully derive

$S_{CFT}=\frac{\pi}{\Xi}(r_{+}^{2}-d^{2}-k^{2}+a^{2}).\ $ (139)

This is precisely the Bekenstein-Hawking entropy of the KSdS black hole on the
event horizon. This result shows that we can change $l^{2}$ on those of KSAdS
solution into $-l^{2}$ by assuming analytical continuation in order to find
the result on those of KSdS solution. This is identical with the KNAdS(dS)
case [22].

##  Appendix B Virasoro algebra

The Dirac bracket of two classical conserved charges are given by

$\displaystyle\left\\{Q_{\zeta_{m}},Q_{\zeta_{n}}\right\\}=Q_{[\zeta_{m},\zeta_{n}]}+K[\zeta_{m},\zeta_{n}],$
(140)

where $K[\xi,\zeta]$ is the central term. Using the quantum version of charge
$Q_{\xi}$ (44), from Eq. (140) we can find

$\displaystyle[L_{m},L_{n}]$ $\displaystyle=$ $\displaystyle
i\\{Q_{\zeta_{m}},Q_{\zeta_{n}}\\}$ $\displaystyle=$ $\displaystyle
i\left(Q_{[\zeta_{m},\zeta_{n}]}+K[\zeta_{m},\zeta_{n}]\right)$
$\displaystyle=$
$\displaystyle(m-n)L_{m+n}-2mx\delta_{m+n}+iK[\zeta_{m},\zeta_{n}].\ $

In order to find the Virasoro algebra, we need to find the form of term
$K[\zeta_{m},\zeta_{n}]$. By comparing above equation with Virasoro algebra,

$\left[L_{m},L_{n}\right]=({m}-{n})L_{m+n}+\frac{c_{L}}{12}m(m^{2}-1)\delta_{m+n,0},$
(142)

we obtain

$K[\zeta_{m},\zeta_{n}]=-i\frac{c_{L}}{12}m\left(m^{2}-1+\frac{24x}{c_{L}}\right)\delta_{m+n,0}.$
(143)

So, the central charge is determined by the coefficient $m^{3}$ in term
$K[\zeta_{m},\zeta_{n}]$. The term linear in $m$ is not important because $x$
is a free parameter to scale the last term in the bracket of Eq. (143).

Acknowledgments

The authors thank the reviewer for the fruitful suggestions on this
manuscript. This work is supported by the Second Century Fund (C2F),
Chulalongkorn University, Thailand. P. B. is supported in part by National
Research Council of Thailand (NRCT) and Chulalongkorn University under Grant
N42A660500. This research has received funding support from the NSRF via the
Program Management Unit for Human Resources & Institutional Development,
Research and Innovation [grant number B39G660025].

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