# Charged AdS Black Holes in Presence of String Cloud and Cardy-Verlinde
Formula

Rishi Pokhrel E-Mail<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>Department of Physics, Sikkim Manipal Institute
of Technology, Sikkim Manipal University, Majitar, Rangpo, East Sikkim,
737136, India.  Tanay K. Dey E-mail<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>Department of Physics, Sikkim Manipal Institute of
Technology, Sikkim Manipal University, Majitar, Rangpo, East Sikkim, 737136,
India.

###### Abstract

In this study, charged AdS black holes with a string cloud are taken into
consideration. We identify three distinct black hole solutions within a
specific temperature range, which subsequently merge into a single black hole
beyond this range in the case of spherical space with low chemical potential
and string density. However, in the case of large chemical potential and
string density, only one black hole solution exists for all temperatures.
Notably, we observe that for flat and hyperbolic space, there is only one
black hole solution irrespective of the chemical potential and string density.
By subtracting the contribution of the extremal black hole from the on-shell
Euclidean action, the thermodynamical quantities associated with these black
holes are calculated. We also investigate the stability of these black holes
and find that medium-sized black hole is unstable. Finally, we establish the
Cardy-Verlinde formula for the corresponding dual boundary theory.

##  1 Introduction

The AdS/CFT correspondence [1, 2, 3] describes a duality between strongly
coupled $\mathcal{N}=4$, $SU(N)$ gauge theories on the boundary of Anti-de
Sitter spacetime and weakly coupled gravity theories in one higher dimension
in the bulk of AdS spacetime. This duality allows for the study of strongly
coupled gauge theories, which would otherwise be difficult to study. The
thermodynamics of the weakly coupled gravity theory can be mapped onto the
strongly coupled boundary $\mathcal{N}=4$, $SU(N)$ gauge theories. The
Hawking-Page phase transition in the weakly coupled gravity theory corresponds
to the confined and deconfined phase transition in the strongly coupled gauge
theory [4]. Furthermore, in [5] Verlinde showed that the entropy of the dual
gauge theory can be expressed as Cardy like formula for two dimensional CFT
[6]. Since then, this is commonly known as Cardy-Verlinde formula.
Subsequently, there has been several works reported in the literature to
establish a similar formula for the dual CFT theories for different types of
corresponding AdS black hole solutions [7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17].

In $\mathcal{N}=4$ $SU(N)$ gauge theory, quarks with fundamental
representation are not present. However, it is possible to incorporate these
quarks by introducing a large number of infinitely long strings in AdS
spacetime [18, 19, 20, 21, 22]. These strings extend from the boundary of AdS
spacetime and hang in the bulk space, representing the gluonic field. The
endpoints of these strings correspond to quarks and antiquarks for the
boundary gauge theory. Several studies have explored the coupling of the
string cloud with gravity theory [23, 24, 25, 26, 27, 28, 29]. In [30, 31,
32], charged AdS black hole solutions surrounded by quintessence in presence
of string cloud are considered in spherical spacetime.

This study examines charged AdS black hole solutions in the presence of a
string cloud for spherical, flat, and hyperbolic spaces in general dimensions.
It is observed that, regardless of the spacetime curvature, string density,
and chemical potential, an extremal black hole can exist at zero temperature.
The thermodynamical quantities related to the black hole solutions have been
computed from the Euclidean action by subtracting the contribution of the
extremal black hole since AdS space time is no longer a valid solution. The
analysis of the thermodynamic quantities demonstrates that the first law of
thermodynamics remains valid for these black hole solutions and provides the
number of possible black hole solutions and their nature for a given
temperature. At low temperatures and with small chemical potential, small
string density, and spherical curvature space, a small black hole exists.
Above a critical temperature, two more black holes nucleate, depending on
their size, which we categorize as small, medium, and large black holes.
Within a particular temperature range, these three black holes coexist. At the
upper limit of the temperature range, the medium and small black holes merge
and disappear, while only the large one remains above the temperature range.
The small and large black holes have positive specific heat, while the medium
black hole has negative specific heat. Consequently, the first two black holes
are thermodynamically stable, while the latter one is unstable. Additionally,
we calculate the free energy of these black holes and demonstrate that, at low
temperatures, the small black hole is thermodynamically favoured, while the
large black hole is preferred at high temperatures. At a critical temperature,
a phase transition similar to Hawking and Page between the small and large
black hole is possible, corresponding to the confined and deconfined phase
transition in the dual boundary gauge theory. For spherically curved space
with large chemical potential and string density, only one black hole exists,
which is also true for flat and hyperbolic space. Therefore, AdS space is not
a valid configuration in the system, indicating that there may not be any
confined state of quark-antiquark in the corresponding dual gauge theory. As a
preliminary step to investigate the gauge theory, we examine whether the
Cardy-Verlinde formula is valid in this system or not. Ultimately, we
establish the Cardy-Verlinde formula for the boundary gauge theory through two
distinct methods. The first approach derives the formula from the Casimir
energy of the boundary theory, while in the second approach the formula arises
naturally from the Landau function of the boundary gauge theory. It is
observed that the formula may not be valid for hyperbolic curvature space.

The work is organized as follows: in section (2) we first set up the action
with the addition of the string cloud into the system and then constructed the
Einstein equation of motion for the system. From the equation of motion, black
hole solutions are found. In order to study the thermodynamics of the black
holes, we compute the Euclidean action in section (3) and then calculate
entropy, energy, specific heat, free energy and Landau function. Subsequently,
we find the number of black hole solutions that can be possible in a
particular temperature and their stability in section (4) . In section (5),
the Cardy-Verlinde formula has been obtained for the dual boundary gauge
theory corresponding to the charged AdS black hole with string cloud. Finally,
in section (6), a summary and discussion have been presented.

##  2 Black Hole Solution

After the brief introduction in the previous section, this section discusses
the action, equation of motion, and black hole solutions for charged AdS black
holes in the presence of a string cloud. The string cloud represents the
introduction of string in the background of charged AdS black holes. These
strings are open and extend from the AdS space boundary to the black hole
horizon. According to the holographic principle, this system represents a
specific state in the gauge theory in the presence of quarks. Following [23,
24], we consider the action of charged AdS black holes in presence of string
cloud.

###  2.1 Action

To investigate the $(n+1)$-dimensional bulk theory resulting from the charged
AdS spacetime with a string cloud, we consider the following action

$I=-\frac{1}{16\pi
G_{n+1}}\int_{M}d^{n+1}x\sqrt{-g}[R-2\Lambda-F^{2}]+I_{SC}.$ (1)

Here $G_{n+1}$ is the Gravitational constant, $g$ is the determinant of the
spacetime metric tensor $g_{\mu\nu}$, $\Lambda$ represents the cosmological
constant term, $F^{2}=F_{\mu\nu}F^{\mu\nu}$ is the Maxwell charge term. We
have added the contribution of the string cloud as $I_{SC}$, where

$I_{SC}=-{\frac{1}{2}}\sum_{i}{\cal{T}}_{i}\int
d^{2}\xi{\sqrt{|h_{\alpha\beta}|}}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu}.$
(2)

Here ${\cal T}_{i}$ is the tension of the $i^{th}$ string and
$h^{\alpha\beta}$ is the world-sheet metric. With these setups we now move on
to set the equation of motion.

###  2.2 Equation of Motion

The equation of motion with respect to the spacetime metric $g_{\mu\nu}$ can
be written as,

$G_{\mu\nu}+\Lambda g_{\mu\nu}=T_{\mu\nu}^{EM}+8\pi G_{n+1}T_{\mu\nu}^{SC}.$
(3)

Here

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R.$ (4)

$T_{\mu\nu}^{EM}$ is the electromagnetic energy momentum tensor which is given
by

$T_{\mu\nu}^{EM}=2{F_{\mu}}^{\lambda}F_{\nu\lambda}-\frac{1}{2}g_{\mu\nu}F^{2}$
(5)

and $T_{\mu\nu}^{SC}$ is the energy momentum tensor corresponding to the
string cloud (SC) given by

$T^{(SC)\mu\nu}=-\sum_{i}{\cal{T}}_{i}\int
d^{2}\xi\frac{1}{\sqrt{|g_{\mu\nu}|}}\sqrt{|h_{\alpha\beta}|}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\delta_{i}^{n+1}(x-X).$
(6)

In equation (6), the delta function represents the source divergences due to
the presence of strings. Assuming the strings are uniformly distributed over
the $(n-1)$ spatial directions and write the density $a$ as,

$a(x)=T\sum_{i}\delta_{i}^{(n-1)}(x-X_{i}).$ (7)

###  2.3 Solution

The Maxwell equation (5) can be rewritten as,

$\nabla_{\alpha}F^{\alpha\beta}=0,$ (8)

with

$F_{rt}=\sqrt{\frac{(n-1)(n-2)}{2}}\frac{q}{r^{n-1}}.$ (9)

The parameter $q$ is related with the charge of the black hole,

$Q=\sqrt{2(n-1)(n-2)}\left(\frac{\omega_{n-1}}{8\pi G_{n+1}}\right)q.$ (10)

The chemical potential developed due to the presence of charge is

$\Phi=\sqrt{\frac{(n-1)}{2(n-2)}}\frac{q}{r_{+}^{n-2}}.$ (11)

With all these parameters, we now look for the solution of equation (3),
considering the following metric ansatz,

$ds^{2}=-V(r)dt^{2}+\frac{dr^{2}}{V(r)}+r^{2}d\Omega^{2}_{n-1}.$ (12)

Here $d\Omega_{n-1}^{2}$ denotes the metric on unit $(n-1)$ sphere. Under the
static gauge condition $X^{t}=\xi^{0},\quad X^{r}=\xi^{1},\quad
X^{\mu}=y^{\mu}\quad\rm{for}\quad\mu\neq t\quad\rm{and}\quad\mu\neq r$, the
non-vanishing components of the string cloud energy momentum tensor
$T^{(SC)\mu\nu}$ can be computed as,

$T^{(SC)tt}=-\frac{ag^{tt}}{r^{n-1}},\,\,\,T^{(SC)rr}=-\frac{ag^{rr}}{r^{n-1}},$
(13) ${T^{(SC)}}_{t}^{t}={T^{(SC)}}_{r}^{r}=-\frac{a}{r^{n-1}},\,\text{with
$a>0$}.$ (14)

Here, the density $a$ is assumed to be a constant. Following [23], we consider
$a$ as a positive constant to dominate the energy condition by the energy
momentum tensor.

The $V(r)$ of equation (12) can be computed and takes the form

$V(r)=k+\frac{r^{2}}{l^{2}}-\frac{m}{r^{n-2}}+\frac{q^{2}}{r^{2n-4}}-\frac{16\pi
G_{n+1}a}{(n-1)r^{n-3}}.$ (15)

Where $k=1,0,-1$ depends on whether the $(n-1)$ dimensional boundary is
spherical, flat or hyperbolic respectively. The $V(r)$ is written with the
parametrisation of cosmological constant $\Lambda=-\frac{n(n-1)}{2l^{2}}$,
where $l$ corresponds to the radius of the AdS spacetime, $m$ is the constant
of integration which represents the mass parameter of the black hole and is
related to the ADM mass $(M)$ of the black hole.

The metric tensor corresponding to the $V(r)$ of equation (15) has singularity
at $r=0$ for the generic value of $n\geq 3$. Irrespective of the nature of
spacetime curvature, from the Descartes’ sign rules, it is clear that $V(r)=0$
equation have either two or no real positive roots just like a charged black
hole. Above the extremal condition there are two black hole horizons– the
inner and the outer horizon. The singularity of the metric tensor at $r=0$ is
protected by these horizons. Below the extremal condition there is no horizon
and singularity is called naked singularity. At extremal condition, the two
horizons marge together and provide single horizon. The extremal condition
depends on the value of parameters like $m,q$ etc. as shown in the figure (1).

Figure 1:  $V(r)$ Vs $r$ plot, figure (a) for different values of $m$ and
figure (b) for different values of $q$. All the graphs are drawn for $k=1$.

The black hole mass parameter $m$ can be calculated from the equation
$V(r=r_{+})=0$ and it can be written as,

$m=\frac{(n-1)r_{+}^{2n-2}+(n-1)l^{2}kr_{+}^{2n-4}-16\pi
G_{n+1}al^{2}r_{+}^{n-1}+(n-1)l^{2}q^{2}}{(n-1)l^{2}r_{+}^{n-2}}.$ (16)

Here $r_{+}$ is the horizon radius. The integration constant $m$ is related
with the ADM mass $(M)$ of the black hole in the following way,

$M=\frac{(n-1)\omega_{n-1}}{16\pi G_{n+1}}m=\frac{(n-1)\omega_{n-1}}{16\pi
G_{n+1}}\left[\frac{r_{+}^{n}}{l^{2}}+kr_{+}^{n-2}+\frac{q^{2}}{r_{+}^{n-2}}-\frac{16\pi
G_{n+1}ar_{+}}{(n-1)}\right],$ (17)

where $\omega_{n-1}$ is the volume of the $(n-1)$ sphere.

##  3 Thermodynamical Quantities

It is well-established that black holes can be treated as thermodynamic
systems and therefore, we can calculate various thermodynamic quantities
associated with them, such as temperature, entropy, and free energy. To begin,
we will focus on calculating the temperature of the black hole.

###  3.1 Temperature

The temperature of the black hole is the inverse of the periodicity $\beta$ of
Euclidean time $(\tau\rightarrow it)$ of the black hole metric. The
temperature is given by,

$T=\frac{1}{4\pi}\left.\frac{dV(r)}{dr}\right|_{r=r_{+}}$ (18)

Using (15) it becomes,

$T=\frac{n(n-1)r_{+}^{2n+2}+kl^{2}(n-1)(n-2)r_{+}^{2n}-16\pi
G_{n+1}l^{2}ar_{+}^{n+3}-(n-1)(n-2)l^{2}q^{2}r_{+}^{4}}{4\pi
l^{2}(n-1)r_{+}^{2n+1}}.$ (19)

The Descartes’ sign rules indicate that there exists an extremal black hole
solution with a radius, denoted as $r_{\rm vac}$, even at zero temperature and
when the charge $q=0$, regardless the value of $k$. The extremal radius
$r_{vac}$ is the solution to the equation,

$(n-1)(n-2)kr_{\rm vac}^{n-2}-2\Lambda r_{\rm vac}^{n}-16\pi G_{n+1}ar_{\rm
vac}=0.$ (20)

When the string density disappears, $r_{\rm vac}$ becomes zero and the
background becomes the AdS spacetime. The mass parameter of the extremal black
hole is given by

$m_{0}=\left[\frac{r_{\rm vac}^{n}}{l^{2}}+kr_{\rm vac}^{n-2}-\frac{16\pi
G_{n+1}ar_{\rm vac}}{(n-1)}\right].$ (21)

In order to compute other thermodynamical quantities such as free energy,
entropy, etc. now we evaluate the on shell Euclidean action.

###  3.2 Action Calculation

In this section, we aim to calculate the on-shell Euclidean action for the
system described by equation (1). The on shell Euclidean action usually
diverges for infinite volume of the spacetime. The action must be finite to
get finite values of the thermodynamic quantities. Therefore, to have a finite
value of the on shell Euclidean action, we can either add a counter term [33]
with the action or subtract a contribution of the action of another spacetime
which has the same asymptotic geometry [3]. We opt for the latter method to
compute the on shell action.

To avoid the divergence of the Euclidean action due to the infinite volume of
spacetime, we set a limit on the radius of the spacetime up to a certain value
$r=r_{\rm max}$ and subtract the contribution of the action of the background.
Then we take the limit $r_{\rm max}\rightarrow\infty$.

Before going to compute the Euclidean action we take the trace of the equation
of motion (3) and rearrange the terms to match the structure of the action
given in equation (1) which will be useful later to calculate the action. The
resulting expression for the trace of the equation of motion is,

$R-\frac{(n+1)}{2}R+(n+1)\Lambda=2F^{\nu\lambda}F_{\nu\lambda}-\frac{(n+1)}{2}F^{2}+8\pi
G_{n+1}T^{SC}.$ (22)

After rearranging the terms, we get the above equation in the following form,

$R-2\Lambda-F^{2}=-\frac{2}{(n-3)}[R-4\Lambda+8\pi G_{n+1}T^{SC}].$ (23)

Now, we compute the on shell Euclidean action in the following way,

$I_{E}=\frac{1}{8\pi G_{n+1}(n-3)}\int d^{n+1}x\sqrt{-g}\left[R-4\Lambda+8\pi
G_{n+1}T^{SC}\right]+\int d\tau dra\omega_{n-1}.$ (24)

Here, the on shell string action is expressed as,

$I_{SC}=-\int d^{n+1}x\,a,$ (25)

since we have incorporated the delta function in the density function $a(x)$
in (7). The above equation (24) can be recasted as,

$I_{E}=\frac{\omega_{n-1}\beta}{8\pi G_{n+1}(n-3)}\int
drr^{n-1}\left[R-4\Lambda+8\pi G_{n+1}T^{SC}\right]+\omega_{n-1}\beta\int
dra.$ (26)

After replacing the value of $T^{SC}$ by using equation (13), we can write the
above equation as,

$I_{E}=\frac{\omega_{n-1}\beta}{8\pi G_{n+1}(n-3)}\int
dr\left[Rr^{n-1}-4\Lambda r^{n-1}+8\pi G_{n+1}(n-5)a\right].$ (27)

As mentioned in the beginning of this section, the value of the integral is
infinite for the infinite volume or infinite radius $r$ of the spacetime. In
order to avoid this infinity we take $r=r_{\rm max}$ and subtract the
contribution of the background called vacuum $(\rm vac)$ whose boundary is
similar to our spacetime boundary. Finally, we take $r_{\rm
max}\rightarrow\infty$. Since AdS spacetime is no longer a valid solution,
here we consider the extremal black hole as the background spacetime which is
defined by the extremal condition $T=q=0$. By doing this, we can write the
Euclidean action as,

$\displaystyle I_{E}=\frac{\omega_{n-1}}{8\pi G_{n+1}(n-3)}$
$\displaystyle\left(\beta\int_{r_{+}}^{r_{\rm max}}dr\left[Rr^{n-1}-4\Lambda
r^{n-1}+8\pi G_{n+1}(n-5)a\right]\right.$ $\displaystyle-\left.\beta_{\rm
vac}\int_{r_{\rm vac}}^{r_{\rm max}}dr\left[Rr^{n-1}-4\Lambda r^{n-1}+8\pi
G_{n+1}(n-5)a\right]\right).$ (28)

Here $\beta_{\rm vac}$ is the Euclidean time periodicity of the vacuum. In
order to have the same boundary between the black hole and vacuum spacetime,
the following constraint must be satisfied,

$\sqrt{V(r_{\rm max})}\beta=\sqrt{V_{\rm vac}(r_{\rm max})}\beta_{\rm vac}.$
(29)

Here $V(r_{\rm max})$ and $V_{\rm vac}(r_{\rm max})$ are the time component of
the spacetime metric at the boundary of the black hole and vacuum spacetime
respectively. Finally, the Euclidean action of equation (3.2) takes the form

$\displaystyle I_{E}=\frac{\beta\omega_{n-1}}{8\pi G_{n+1}(n-3)}$
$\displaystyle\left(\int_{r_{+}}^{r_{\rm max}}dr\left[Rr^{n-1}-4\Lambda
r^{n-1}+8\pi G_{n+1}(n-5)a\right]\right.$
$\displaystyle-\left.\sqrt{\frac{V(r_{\rm max})}{V_{\rm vac}(r_{\rm
max})}}\int_{r_{\rm vac}}^{r_{\rm max}}dr\left[Rr^{n-1}-4\Lambda r^{n-1}+8\pi
G_{n+1}(n-5)a\right]\right).$ (30)

We define the $V_{\rm vac}(r)$ from $V(r)$ by setting the extremal condition
$T=q=0$. We also consider the form of Ricci scalar as,

$R=-\frac{1}{r^{n-1}}(r^{n-1}V(r))^{\prime\prime}+\frac{(n-1)(n-2)k}{r^{2}}.$
(31)

After substituting the value of $R$ in the equation (3.2), the action can be
expressed as,

$\displaystyle I_{E}=\frac{\beta\omega_{n-1}}{8\pi
G_{n+1}(n-3)}\left(\int_{r_{+}}^{r_{\rm
max}}dr\left[-(r^{n-1}V(r))^{\prime\prime}+(n-1)(n-2)kr^{n-3}-4\Lambda
r^{n-1}+8\pi G_{n+1}(n-5)a\right]\right.$
$\displaystyle-\left(1-\frac{(m-m_{0})l^{2}}{2r_{\rm
max}^{n}}\right)\left.\int_{r_{\rm vac}}^{r_{\rm
max}}dr\left[-(r^{n-1}V(r))^{\prime\prime}+(n-1)(n-2)kr^{n-3}-4\Lambda
r^{n-1}+8\pi G_{n+1}(n-5)a\right]\right).$ (32)

After integration we get,

$\displaystyle I_{E}=\frac{\beta\omega_{n-1}}{8\pi
G_{n+1}(n-3)}\left(\frac{(n-1)}{2}(m-m_{0})+4\pi
Tr_{+}^{n-1}-(n-1)kr_{+}^{n-2}+\frac{4\Lambda}{n}r_{+}^{n}\right.$
$\displaystyle-\left.8\pi G_{n+1}(n-5)ar_{+}+(n-1)kr_{\rm
vac}^{n-2}-\frac{4\Lambda}{n}r_{\rm vac}^{n}+8\pi G_{n+1}(n-5)ar_{\rm
vac}\right).$ (33)

Using the extremal condition of equation (20), we can reduce the equation to,

$\displaystyle{}I_{E}$ $\displaystyle=\frac{\beta\omega_{n-1}}{16\pi
G_{n+1}(n-3)}\left[(n-1)(m-m_{0})+8\pi
Tr_{+}^{n-1}-2(n-1)kr_{+}^{n-2}+\frac{8\Lambda}{n}r_{+}^{n}\right.$
$\displaystyle-\left.16\pi G_{n+1}(n-5)ar_{+}+(n-1)(n-3)(n-4)kr_{\rm
vac}^{n-2}-\frac{2(n-1)(n-4)\Lambda}{n}r_{\rm vac}^{n}\right].$ (34)

Here note that for $n=4$, the last two terms of the action becomes zero and
for $n=3$, the action diverges. Therefore, the action is valid for $n\geq 4$
spatial dimensions though the analysis of equation (15) was demanding that to
exist a black hole $n$ should be $\geq 3$.

Now we are ready to compute the thermodynamical quantities like entropy,
specific heat, total internal energy and free energy etc.

###  3.3 Other Thermodynamical Quantities

In order to determine the state variables of the black hole spacetime we
consider potential $\Phi$ is fixed at the boundary. We also consider the
string density is fixed. Therefore, in the grand canonical ensemble, the free
energy is expressed as $F=E-TS-Q\Phi$. Following [34], we can calculate the
state variables of the system from the Euclidean action of equation (3.2) as;

  * •

Entropy:

$S=\beta\left.\left(\frac{\partial
I_{E}}{\partial\beta}\right)\right|_{\Phi}-I_{E}=\frac{r_{+}^{n-1}\omega_{n-1}}{4G_{n+1}}.$
(35)

  * •

Charge:

$Q=-\frac{1}{\beta}\left.\left(\frac{\partial
I_{E}}{\partial\Phi}\right)\right|_{\beta}=\sqrt{2(n-1)(n-2)}\left(\frac{\omega_{n-1}}{8\pi
G_{n+1}}\right)q.$ (36)

  * •

Energy:

$E=\left.\left(\frac{\partial
I_{E}}{\partial\beta}\right)\right|_{\Phi}-\frac{\Phi}{\beta}\left.\left(\frac{\partial
I_{E}}{\partial\Phi}\right)\right|_{\beta}=M+E_{0}.$ (37)

Where $E_{0}$ is the background energy contribution,

$E_{0}=\frac{(n-1)\omega_{n-1}r_{\rm vac}}{16\pi
G_{n+1}(n-3)l^{2}}\left[\\{n(n-5)+3\\}r_{\rm
vac}^{n-1}+kl^{2}\\{n(n-7)+11\\}r_{\rm vac}^{n-3}+\frac{16\pi
G_{n+1}al^{2}}{n-1}\right].$ (38)

For $r_{\rm vac}=0$ spacetime, $E_{0}$ becomes zero and for $n=4$ spatial
dimensions it turns out to be

$E_{0}=-\frac{3\,\omega_{3}}{16\pi G_{5}}\left[\frac{r_{\rm
vac}^{4}}{l^{2}}+kr_{\rm vac}^{2}-\frac{16\pi G_{5}ar_{\rm
vac}}{3}\right]=-\frac{3\,\omega_{3}}{16\pi G_{5}}m_{0}.$ (39)

For the latter purpose, we write $E_{0}$ in terms of only $r_{\rm vac}$ using
the extremal condition of equation (20) and it takes the form,

$E_{0}=\frac{(n-1)\omega_{n-1}r_{\rm vac}^{n-2}}{16\pi
G_{n+1}l^{2}}\left[(n-3)kl^{2}+(n-1)r_{\rm vac}^{2}\right].$ (40)

These state variables naturally satisfy the first law of thermodynamics
$dE=TdS+\Phi dQ$. Therefore one can compute the specific heat of the black
hole by using the relation,

$\displaystyle C$ $\displaystyle=T\left(\frac{\partial S}{\partial T}\right)$
$\displaystyle=\frac{(n-1)\omega_{n-1}r_{+}^{n-1}}{4G_{n+1}}\frac{[n(n-1)r_{+}^{n-1}+l^{2}[k(n-1)(n-2)-2(n-2)^{2}\Phi^{2}]r_{+}^{n-3}-16\pi
G_{n+1}l^{2}a]}{[n(n-1)r_{+}^{n-1}-l^{2}[k(n-1)(n-2)-2(n-2)^{2}\Phi^{2}]r_{+}^{n-3}+16\pi
G_{n+1}l^{2}(n-2)a]}.$ (41)

  * •

Free energy:

We compute the free energy of the black hole from the thermal partition
function Z of the black hole solution. The thermal partition function can be
calculated from the classical action of the Euclidean black hole metric
solution by using the saddle point approximation through the following
relation,

$Z=e^{-I_{E}}.$ (42)

Therefore, the free energy of the black hole can be written as,

$F=-k_{B}T\,{\rm ln}Z=k_{B}TI_{E},$ (43)

where $I_{E}$ is the Euclidean action of the black hole solution. If Boltzmann
constant $k_{B}$ is considered to be equal to one and $T=\frac{1}{\beta}$,
then equation (43) reduces as,

$F\beta=I_{E}.$ (44)

Therefore the free energy of the black hole takes the form

$\displaystyle F$ $\displaystyle=\frac{I_{E}}{\beta}=E-TS-\Phi Q$
$\displaystyle=\frac{\omega_{n-1}}{48\pi
G_{n+1}}\left(-\frac{3}{l^{2}}r_{+}^{n}+\frac{3[k(n-1)-2(n-2)\Phi^{2}]}{(n-1)}r_{+}^{n-2}-\frac{48\pi
G_{n+1}(n-2)a}{(n-1)}r_{+}\right)+E_{0}.$ (45)

  * •

Landau Function:

Now to calculate the Landau function, let us make the ansatz in terms of the
order parameter $r_{+}$, going from the highest to the lowest power of the
order parameter as given below,

$G(r_{+},T)=\frac{\omega_{n-1}}{48\pi
G_{n+1}}(b_{n}r_{+}^{n}-b_{n-1}Tr_{+}^{n-1}+b_{n-2}r_{+}^{n-2}-b_{n-3}r_{+}^{n-3}+\cdots+(-1)^{n-1}b_{1}r_{+})+b_{0}.$
(46)

Here $b_{n},b_{n-1},b_{n-2},b_{n-3},\cdots,b_{0}$ are constants and can be
fixed in the following way,

  1. 1.

By substituting the temperature expression of equation (19) in the Landau
function expression of equation (46), we should get the free energy expression
of equation (• ‣ 3.3). Therefore, the existing term in the Landau function can
be expressed as,

$\displaystyle G(r_{+},T)=\frac{\omega_{n-1}}{48\pi G_{n+1}}$
$\displaystyle\left[\left(b_{n}-\frac{nb_{n-1}}{4\pi
l^{2}}\right)r_{+}^{n}\right.$
$\displaystyle+\left.\left(\frac{(n-2)[k(n-1)-2(n-2)\Phi^{2}]b_{n-1}}{4\pi(n-1)}\right)r_{+}^{n-2}\right.$
$\displaystyle+\left.\left(\frac{4G_{n+1}b_{1}a}{(n-1)}+(-1)^{n-1}b_{1}\right)r_{+}\right]+b_{0}.$
(47)

  2. 2.

On the other hand, we take the derivative of the Landau function expression of
equation (46) with respect to horizon radius $r_{+}$ and equate it with zero.

$\frac{\partial G}{\partial r_{+}}=0.$ (48)

  3. 3.

The solution of this equation gives the expression of temperature in terms of
horizon radius $r_{+}$. Again the expected temperature expression can be
written as,

$T=\frac{nb_{n}}{(n-1)b_{n-1}}r_{+}+\frac{(n-2)b_{n-2}}{(n-1)b_{n-1}}\frac{1}{r_{+}}+\frac{(-1)^{n-1}b_{1}}{(n-1)b_{n-1}}\frac{1}{r_{+}^{n-2}}.$
(49)

  4. 4.

Now, comparing the temperature expression of equation (49) with the
temperature expression of equation (19) and Landau function expression of
equation (1) with the free energy expression of equation (19), we get the non
vanishing constants as,

$b_{1}=-\frac{48\pi G_{n+1}a}{(-1)^{n-1}},$ (50)
$b_{n-2}=3[k(n-1)-2(n-2)\Phi^{2}],$ (51) $b_{n-1}=12\pi,$ (52)
$b_{n}=\frac{3(n-1)}{l^{2}},$ (53) $b_{0}=E_{0}.$ (54)

Now using the values of non vanishing constant in (46) we get the Landau
function as,

$\displaystyle G(r_{+},T)=\frac{\omega_{n-1}}{48\pi
G_{n+1}}\left[\frac{3(n-1)}{l^{2}}r_{+}^{n}-12\pi
Tr_{+}^{n-1}+3\\{k(n-1)-2(n-2)\Phi^{2}\\}r_{+}^{n-2}\right.$
$\displaystyle-\Bigl{.}48\pi G_{n+1}ar_{+}\Bigr{]}+E_{0}.$ (55)

In the next section, we study the thermodynamic stability of the black hole
with the help of the above computed thermodynamical quantities.

##  4 Black Hole Phases

In this section, we study the different phases of black holes by analysing the
thermodynamical quantities derived in the previous section. First, we scale
the parameters as,

$8\pi
G_{5}=1,\,\bar{r}=\frac{r_{+}}{l},\,\bar{a}=\frac{a}{l^{2}},\,\bar{q}=\frac{q}{l^{2}}.$
(56)

With these scaled parameters and for $k=1$, in five dimensions the Hawking
temperature in equation (19) of the black hole can be expressed as,

$T=\frac{(3\bar{r}+6\bar{r}^{3}-4\bar{r}\Phi^{2}-\bar{a}l)}{6\pi
l\bar{r}^{2}}.$ (57)

Now we plot the temperature $T$ with respect to the black hole radius
$\bar{r}$ to get the information about the number of black hole solutions in a
particular temperature.

Figure 2: Both the graphs are Temperature ($T$) vs. scaled horizon radius
($\bar{r}$). In figure (a) $\bar{a}$ is constant but in figure (b) $\Phi$ is
constant.

Figure (2a) illustrates that if the potential is lower than a certain critical
value, there exists only one type of black hole solution upto a certain
temperature, referred to as the small black hole. This solution persists even
at zero temperature, known as the extremal black hole. At higher temperatures,
two additional black hole solutions nucleate, with one referred to as the
medium and the other as the large black hole, depending on their size. These
three types of black holes coexist within a specific temperature range until,
at a particular temperature, the small and medium black holes merge and
disappear, leaving only the large black hole solution at high temperatures. If
the potential exceeds the critical value, there is always one black hole
solution present for any temperature. Figure (2b) exhibits similar behaviour
but for different values of the scaled string cloud density $\bar{a}$.

In order to know the range of potential $\Phi$ and scaled string density
$\bar{a}$ for which three black hole solutions exist together, we compute the
discriminant of the cubic equation of the scaled horizon radius $\bar{r}$ from
equation (57) which turns out to be,

$\Delta=-972\bar{a}^{2}-24(3-4\Phi^{2})^{3}+648\bar{a}(3-4\Phi^{2})(T\pi)+36(3-4\Phi^{2})^{2}(T\pi)^{2}-864\bar{a}(T\pi)^{3}.$
(58)

When the discriminant is greater than zero, it indicates the existence of
three black hole solutions. Otherwise, there is only one black hole solution.
In figure (3), the parameter space for temperature, potential, and string
density is depicted, in which the discriminant becomes positive. The figure
highlights that for a specific value of $\bar{a}$ and $\Phi$, there are two
values of temperature for which the discriminant is positive. These
temperatures set the upper and lower boundaries of the temperature range where
three black hole solutions exist. If the potential $\Phi$ increases, the
critical value of $\bar{a}$ decreases and becomes zero when $\Phi$ equals the
critical value of $\frac{\sqrt{3}}{2}$ (for details refer to [34]). On the
other hand when $\Phi$ takes zero value then $\bar{a}$ reaches its maximum
value $\frac{1}{\sqrt{6}}$ (for details see [28, 24]).

Figure 3: Parameter space of $T$, $\Phi$ and $\bar{a}$ for positive
discriminant.

Conducting an analysis on the existence of various black hole solutions in a
specific parameter space, our next objective is to evaluate their stability.
To accomplish this, we focus on the three black hole solutions present in the
parameter space and investigate their specific heat, free energy, and Landau
function. We begin our analysis by calculating the specific heat using the
formula,

$C=\frac{6\pi\omega_{3}l^{3}\bar{r}^{3}\big{[}6\bar{r}^{3}+3\bar{r}(1-\frac{4}{3}\Phi^{2})-l\bar{a}\big{]}}{\big{[}6\bar{r}^{3}-3\bar{r}(1-\frac{4}{3}\Phi^{2})+2l\bar{a}\big{]}}.$
(59)

Figure (4a) illustrates the relationship between the temperature and the
number of black hole solutions. It shows that there are three black hole
solutions within a specific temperature range, whereas outside of this range,
only one black hole solution exists. In figure (4b), we study the specific
heat with respect to the temperature. We observe that the specific heat is
positive for small and large black holes, but negative for medium-sized black
holes. Consequently, we conclude that the medium-sized black hole is unstable,
whereas the small and large black holes are stable.

Figure 4: Figure $(a)$ and $(b)$ are drawn for the value of $\Phi$ and
$\bar{a}$ within their critical values. Figure $(a)$ is for number of solution
with respect to temperature $T$ and figure $(b)$ for the specific heat with
respect to temperature $T$.

Now we proceed to study the free energy with respect to the temperature and
radius of the black hole. The free energy can be calculated from the action in
equation (• ‣ 3.3) and with the shift of ground state energy by $E_{0}$, the
free energy takes the form as,

$F=\frac{\omega_{3}l^{2}\bar{r}}{2}\big{[}\bar{r}(1-\frac{4}{3}\Phi^{2})-\bar{r}^{3}-\frac{4}{3}l\bar{a}\big{]}.$
(60)

In figure (5a), we present a plot of the free energy against temperature for a
fixed value of scaled string cloud density $\bar{a}$ and potential $\Phi$. At
low temperatures, there is only one black hole solution, namely the small
black hole, and it has negative free energy, denoted by the red dashed line.
This indicates its stability at this low temperature. As the temperature
increases, two more black hole solutions appear, corresponding to the medium
and large black holes along with the small one. Initially, both the medium and
large black holes possess positive free energy, with the green dashed line
representing the large black hole and the blue dashed line representing the
medium black hole. As the temperature increases, the free energy of both the
black holes reduces, and eventually, at a critical temperature $T_{c}$, the
free energy of the large black hole becomes equivalent to the small black hole
and goes lower at higher temperatures. Furthermore, the free energy of the
medium black hole merges with the small black hole at another temperature,
which is higher than $T_{c}$. In figure (5b), we plot the free energy with
respect to the scaled black hole horizon for different $\bar{a}$ and $\Phi$.

Figure 5: Figure $(a)$ is for free energy with respect to temperature $T$ and
figure $(b)$ for the free energy with respect to scaled horizon radius
$\bar{r}$. Both the graphs drawn within the critical values of $\Phi$ and
$\bar{a}$.

Now we study the Landau function which can be expressed as,

$G=\frac{3\omega_{3}l^{2}}{2}\left[\bar{r}^{4}-\frac{4\pi
lT\bar{r}^{3}}{3}+\bar{r}^{2}\left(1-\frac{4\Phi^{2}}{3}\right)-\frac{2l\bar{a}\bar{r}}{3}\right].$
(61)

In figure (6), we plot the Landau function with respect to scaled horizon
$\bar{r}$ for different temperatures of the black hole. At low temperatures,
only one black hole solution corresponding to the small black hole exists,
while at higher temperatures, two more black holes nucleate, corresponding to
the medium and large black holes, respectively, with a higher value of the
Landau function. However, as the temperature continues to increase, the value
of the Landau function decreases, and finally, above the critical temperature
$T_{c}$, the Landau function of the large black hole becomes lower than that
of the small black hole. Therefore, at low temperatures, the small black hole
is stable, and at high temperatures, the large black hole becomes stable, with
a critical temperature $T_{c}$ marking the transition between the small and
large black holes. This is similar to the Hawking-Page like transition.

Figure 6: Landau function($G$) vs $\bar{r}$ plot for fixed $\bar{a}$, $\Phi$
and different values of $T$.

Now we consider the chemical potential $\Phi$ and scaled string density
$\bar{a}$ above their critical values so that only one black solution exist.
In figure (7a), we plot the specific heat with respect to temperature. It
shows that the specific heat is always positive. Therefore, the single black
hole is stable configuration against AdS spacetime. In figure (7b), free
energy is plotted with respect to temperature $T$, which shows that it is a
stable configuration. Therefore, we can demand that there is no valid AdS
configuration in the system.

Figure 7: Figure $(a)$ is drawn for the specific heat with respect to
temperature $T$ and figure $(b)$ is drawn for the free energy against
temperature. Both the graph are drawn above the critical values of $\Phi$ and
$\bar{a}$

Though we are not providing the data here for the flat and hyperbolic nature
of the curvature of spacetime, we can similarly conclude that there exists
only one stable black hole solution and further analysis is not necessary.

In the overall analysis, AdS space is no longer a valid state in this
configuration. Consequently, one can expect that in the corresponding dual
gauge theory there might not be any confined state of quark-antiquarks. One
should therefore study the dual theory side. As a preliminary step towards
studying the dual theory side, we establish the validity of the Cardy-Verlinde
formula in the next section.

##  5 Cardy-Verlinde Formula for Dual Gauge Theory

According to the AdS/CFT correspondence, the thermodynamical quantity of the
AdS bulk spacetime can be identified with the thermodynamical quantities of
boundary gauge theory at high temperature. Furthermore, the entropy of the
boundary gauge theory can be described by the Cardy-Verlinde formula [5, 7, 8,
9, 10, 11, 12, 13, 14, 14, 16, 17]. In this section, we are interested to see
whether the entropy of the corresponding boundary theory of our considered
spacetime satisfies the Cardy-Verlinde formula or not. In order to verify this
we write down the boundary metric from the bulk metric of equation (12). Here,
the boundary of the bulk geometry represents a hypersurface at a large $r$
distance. Therefore, the boundary metric can be written as,

$ds^{2}=\frac{r^{2}}{l^{2}}\big{[}-dt^{2}+l^{2}d\Omega_{n-1}^{2}\big{]}.$ (62)

By scaling the time parameter as $\tau=\frac{R}{l}t$ the boundary metric
recast as,

$ds^{2}=\frac{r^{2}}{R^{2}}\big{[}-d\tau^{2}+R^{2}d\Omega_{n-1}^{2}\big{]}.$
(63)

Which is conformally equivalent to $S^{n-1}\times\cal R$ with $S^{n-1}$ having
radius of $R$. Using the AdS/CFT correspondence the thermodynamical quantities
of the boundary theory can be computed. Certain quantities such as entropy,
charge, and string density remain unchanged when mapped from the bulk
spacetime to the boundary theory. However, other quantities such as energy,
temperature, pressure, chemical potential, Landau function, and the conjugate
of the string density ($\mathcal{A}^{a}$) can be modified in the mapping
process as,

$\tilde{E}=\tilde{M}+\tilde{E}_{0}=\frac{l}{R}E,\quad\tilde{T}=\frac{l}{R}T,\quad\tilde{P}=\frac{l}{R}P,\quad\tilde{\Phi}=\frac{l}{R}\Phi,\quad\tilde{G}=\frac{l}{R}G\,\,{\rm
and}\,\,{\cal\tilde{A}}^{a}=\frac{l}{R}{\cal A}^{a}.$ (64)

Here, the conjugate of string density takes the value ${\cal
A}^{a}=\frac{\partial E}{\partial a}=\frac{\partial M}{\partial
a}=-\omega_{n-1}r_{+}$ for $E_{0}$ independent of string density $a$ as
derived in the equation (40). Then the generalised first law of thermodynamics
of the boundary gauge theory can be expressed as,

$d\tilde{E}=d\tilde{M}=\tilde{T}dS-\tilde{P}dV+\tilde{\Phi}dQ+{\cal\tilde{A}}^{a}da.$
(65)

Here, we consider $r_{\rm vac}$ is fixed and $\tilde{P}$ is the pressure of
the boundary gauge theory and it takes the value

$\tilde{P}=-\left(\frac{\partial\tilde{E}}{\partial
V}\right)_{S,\,Q,\,a}=-\left(\frac{\partial\tilde{M}}{\partial
V}\right)_{S,\,Q,\,a}=\frac{\tilde{M}}{(n-1)\omega_{n-1}R^{n-1}}=\frac{\tilde{M}}{(n-1)V}.$
(66)

###  5.1 Cardy-Verlinde Formula from Casimir Energy

In [5], it has been discussed that the energy of the boundary gauge theory is
not an extensive quantity. It has two parts; extensive $(E_{E})$ and non-
extensive $(E_{C})$ which is called the Casimir energy. The total energy can
be expressed in terms of extensive and non-extensive energy as,

$\tilde{E}=E_{E}+\frac{1}{2}E_{C}.$ (67)

Following [5, 17] and with the help of equation (65) the Casimir energy can be
defined as,

$E_{C}=n\tilde{M}-(n-1)(\tilde{T}S+\tilde{\Phi}Q+\mathcal{\tilde{A}}^{a}a).$
(68)

Using the above values in equation (68), we get,

$E_{c}=\frac{l}{R}\frac{k(n-1)\omega_{n-1}}{8\pi G_{n+1}}r_{+}^{n-2}.$ (69)

Finally, using (67) and (69), we can deduce the entropy in the following form,

$S=\frac{4\pi
R}{(n-1)\sqrt{2}}\sqrt{\frac{E_{c}}{k}\left(\tilde{M}-\tilde{E}_{Q}-\tilde{E}_{a}-\frac{E_{c}}{2}\right)}.$
(70)

Here, $\tilde{E}_{Q}=\frac{\tilde{\Phi}}{2}Q$ is the energy of the
electromagnetic field outside the black hole horizon and
$\tilde{E}_{a}=\mathcal{\tilde{A}}^{a}a$ is the energy related with the string
hanging from the boundary of the AdS space to the bulk. This energy could be
related to the binding energy of the quark anti-quark pair represented by the
endpoints of the strings in the boundary CFT.

Though the Cardy-Verlinde formula takes the usual form for fixed $E_{0}$. But
for $k=-1$, the Casimir energy $E_{C}$ becomes negative, as observed in [35]
for the AdS spacetime with hyperbolic curvature. Therefore, the dual gauge
theory can not be unitary as discussed in [8, 36]. We may say that for fixed
$E_{0}$, the Cardy-Verlinde formula is only valid for charged AdS black hole
with string cloud for flat and spherical curvature space. If $E_{0}$ is not
constant then entropy can not be formulated as the Cardy-Verlinde formula.

Following [13], this formula can be rederived from the Landau function of the
boundary theory, which has been presented in the next section.

###  5.2 Cardy-Verlinde Formula from the Landau Function

Using the same scaling prescription of equation (64), we can also find the
Landau function for the boundary theory from the expression of bulk Landau
function and it takes the form as,

$\displaystyle\tilde{G}(r_{+},\tilde{T})=\frac{\omega_{n-1}}{48\pi
G_{n+1}R}\left[\frac{3(n-1)}{l}r_{+}^{n}-12\pi
R\tilde{T}r_{+}^{n-1}+3l[k(n-1)-2(n-2)\Phi^{2}]r_{+}^{n-2}\right.$
$\displaystyle-\Bigl{.}48\pi G_{n+1}lar_{+}\Bigr{]}+\frac{l}{R}E_{0}.$ (71)

The boundary Landau function (5.2) can be recasted as

$\tilde{G}(r_{+},\tilde{T})=\frac{1}{2}E_{c}\left[1-\frac{4\pi
R\tilde{T}}{kl(n-1)}r_{+}+\frac{r_{+}^{2}}{kl^{2}}-\frac{2(n-2)}{k(n-1)}\Phi^{2}-\frac{16\pi
G_{n+1}a}{k(n-1)r_{+}^{n-3}}\right]+\frac{l}{R}E_{0},$ (72)

where

$E_{c}=\frac{l}{R}\frac{k(n-1)\omega_{n-1}}{8\pi G_{n+1}}r_{+}^{n-2}.$ (73)

We also know that $\tilde{G}=\tilde{E}-\tilde{T}S-\tilde{\Phi}Q$, from which
we can say that

$S=\frac{2\pi R}{kl(n-1)}E_{c}r_{+}.$ (74)

Then we can also write,

$\tilde{M}-Q\tilde{\Phi}=\frac{1}{2}E_{c}\left[1+\frac{r_{+}^{2}}{kl^{2}}-\frac{2(n-2)}{k(n-1)}\Phi^{2}-\frac{16\pi
G_{n+1}a}{k(n-1)r_{+}^{n-3}}\right].$ (75)

Now using (74) and (75), $S$ can be written in terms of $\tilde{E}$ and
$E_{c}$ as,

$S=\frac{2\pi
R}{(n-1)\sqrt{k}}\left[E_{c}\left(2\tilde{M}-2Q\tilde{\Phi}+\frac{2(n-2)\Phi^{2}}{k(n-1)}E_{c}+\frac{16\pi
G_{n+1}a}{k(n-1)r_{+}^{n-3}}E_{c}-E_{c}\right)\right]^{\frac{1}{2}}.$ (76)

Simplifying the $2^{nd},\,3^{rd}$ and $4^{th}$ terms inside the parenthesis we
get,

$S=\frac{4\pi
R}{(n-1)\sqrt{2}}\sqrt{\frac{E_{c}}{k}\left(\tilde{M}-\tilde{E}_{Q}-\tilde{E}_{a}-\frac{E_{c}}{2}\right)},$
(77)

where,

$\displaystyle\tilde{E}_{Q}=\frac{Q\tilde{\Phi}}{2}$
$\displaystyle\tilde{E}_{a}=a\tilde{\mathcal{A}^{a}}$
$\displaystyle\tilde{\mathcal{A}^{a}}=\frac{l}{R}\mathcal{A}^{a}$
$\displaystyle\mathcal{A}^{a}=\frac{\partial M}{\partial
a}=-\omega_{n-1}r_{+}.$ (78)

The Cardy-Verlinde formula derived from the Landau function is similar to the
Cardy-Verlinde formula derived from Casimir energy in the last subsection.

Having the Cardy-Verlinde formula, one can study the phases of boundary
theory. We keep that study for our next report.

##  6 Summary and Discussion

In this work, we have studied the charged AdS black hole solutions in the
presence of strings cloud. Irrespective of the nature of curvature of
spacetime at zero temperature, there is one small black hole that exists.
Therefore AdS space is no longer a valid phase of the spacetime.
Thermodynamical quantities related to the black hole solutions are computed
from the Euclidean action of the system by subtracting the contribution of the
small black hole present at zero temperature. We confirmed that black hole
solutions maintained the first law of thermodynamics. For flat and hyperbolic
nature of the AdS space, there exist only one black hole solution for any
temperature, chemical potential and string density. For spherical curvature of
the AdS space, large chemical potential and large string density again one
black hole solution admits. However, for small enough chemical potential and
string density, three black hole solutions exist in a certain temperature
range. Beyond the temperature range, only one black hole solution exists.
Amongst the three, the medium sized black hole is unstable while the small and
large sized black hole are stable. Depending on the temperature one is
thermodynamically more favoured than the other. The Hawking-Page like phase
transition takes place between these two black holes at a critical
temperature. Since AdS space is no longer a valid phase in this system, one
can expect that in the corresponding dual gauge theory there might not be any
confined state of quarks and antiquarks. Which can be further explored. As a
first step, we map some of the bulk results into the boundary theory and
establish the validity of the Cardy-Verlinde formula.

##  7 Acknowledgement

We would like to thank S. Mukhopadhyay for the insightful discussions and S.
Mukherji for going through the manuscript and providing the valuable comments
on our work. RP thanks A. Lobo and B. Sharma for fruitful discussions.

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