Quantum corrections to a finite temperature BIon

TL;DR

This paper investigates quantum fluctuations in a finite temperature BIon, revealing logarithmic entropy corrections and their impact on thermodynamic stability and critical points using Euclidean quantum gravity.

Contribution

It introduces a detailed analysis of quantum corrections to a finite temperature BIon, including entropy, energy, and stability modifications.

Findings

Quantum fluctuations produce logarithmic entropy corrections.

Quantum corrections affect internal energy and specific heat.

Stability and critical points are influenced by quantum effects.

Abstract

In this paper, we will analyze a finite temperature BIon, which is a finite temperature brane-anti-brane wormhole configuration. We will analyze the quantum fluctuations to this BIon solution using the Euclidean quantum gravity. It will be observed that these quantum fluctuations produce logarithmic corrections to the entropy of this finite temperature BIon solution. These corrections to the entropy also correct the internal energy and the specific heat for this finite temperature BIon. We will also analyze the critical points for this finite temperature BIonic system, and analyze the effects of quantum corrections on the stability of this system.

Equations

$$Z=\int [\mathcal{D}]e^{-{\mathcal{I}}_{\text{E}}}={\int }_0^{\infty }\rho (E)e^{-\beta E}\text{d}E,$$

$$\rho (E)=\frac{1}{2\pi i}{\int }_{a-i\infty }^{a+i\infty }{e}^{S(\beta )}\text{d}\beta .$$

$$S(\beta )=S_0+\frac{1}{2}(\beta -{\beta }_0{)}^2{\left[\frac{{\partial }^{2}S(\beta )}{\partial {\beta }^2}\right]}_{\beta ={\beta }_0}+\cdots .$$

$$\rho (E)=\frac{e^{S_0}}{\sqrt{2\pi }}{\left\{{\left[\frac{{\partial }^{2}S(\beta )}{\partial {\beta }^2}\right]}_{\beta ={\beta }_0}\right\}}^{-\frac{1}{2}},$$

$$\mathcal{S}=\ln \rho (E)=S_0-\frac{1}{2}\ln \left\{{\left[\frac{{\partial }^{2}S(\beta )}{\partial {\beta }^2}\right]}_{\beta ={\beta }_0}\right\}.$$

$$\mathcal{S}=S_0-\frac{\gamma }{2}\ln (S_0{T}^2)\mathcal{Y}\sim S_0-\frac{\gamma }{2}\ln (S_0{T}^2)-\frac{\gamma }{2}\mathcal{Y},$$

$$M$$

$$S_0$$

$$\mathcal{F}$$

$${\mu }_{D3}$$

$${\mu }_{F1}$$

$$U_0$$

$$C_0$$

$$S$$

$${\cosh }^2\alpha =\frac{3}{2}\frac{\cos \frac{\delta }{3}+\sqrt{3}\cos \frac{\delta }{3}}{\cos \delta },$$

$$U$$

$$-\frac{\gamma T}{2}\ln \left[\frac{4T_{D3}^2}{\pi T^3}{\int }_{{\sigma }_0}^{\infty }d\sigma \frac{4{\sigma }^{2}F(\sigma )}{\sqrt{F^2(\sigma )-F^2({\sigma }_0)}{\cosh }^4\alpha }\right].$$

$$C=T\frac{d}{dT}\left(S-\frac{\gamma }{2}\ln [S{T}^2]\right),$$

Full text

1 Introduction

In string theory, it is possible to analyze certain physical objects in a region of spacetime in terms of very different objects. Thus, it is possible to analyze a system of many coincident strings in terms of D-brane geometry, and this is done in the BIon solution [1, 2]. So, this BIon solution can describe an F-string coming out of the D3-brane or a D3-brane parallel to an anti-D3-brane, such that they are connected by a wormhole with F-string charge. This configuration is called as the brane-antibrane-wormhole configuration. It is also possible to use such a solution to analyze D-branes probing a thermal background [3, 4]. This can be done using the blackfold approach [5, 6, 7, 8]. In this method, a large number of coincident D-branes form a brane probe. Furthermore, as this probe is in thermal equilibrium with the background, this method has been used to heat up a BIon. This was done by putting it in a hot background. It is also possible to analyze the thermodynamics of this finite temperature BIon solution [3, 4]. In this paper, we will analyze the effects of thermal fluctuations on the thermodynamics of this system.

The entropy-area law of black holes thermodynamics [9, 10], is expected to get modified near Planck scale due to quantum fluctuations [11]. These quantum fluctuations in the geometry of any black object are expected to produce thermal fluctuations in its associated thermodynamics. It is interesting to note that the thermal fluctuations produce a logarithmic correction term to the thermodynamics of black objects [12, 13, 14, 15]. The consequences of such logarithmic correction have been studied for a charged AdS black hole [18], charged hairy black hole [19], a black saturn [20], a Hayward black hole [21] a small singly spinning Kerr-AdS black hole [22], and a dyonic charged AdS black hole [23]. In non-perturbative quantum general relativity, the density of microstates were associated with the conformal blocks has been used to obtain logarithmic corrections to the entropy [24]. It has also been demonstrated that the Cardy formula can produce logarithmic correction terms for all black objects whose microscopic degrees of freedom are characterized by a conformal field theory [25]. The logarithmic correction has also been studied from the black hole in the presence of matter fields [26] and dilatonic black holes [27]. Leading order quantum corrections to the semi-classical black hole entropy have been obtained [28], and applied to Gödel black hole [29, 30]. The logarithmic corrections were also used to study different aspects of regular black holes satisfying the weak energy condition [31], three-dimensional black holes with soft hairy boundary conditions [32], and certain aspects of Kerr/CFT correspondence [33]. The logarithmic corrected entropy also corrects some hydrodynamical quantities, and so the the field theory dual to such corrected solutions has also been studied [38, 34, 35, 37, 36].

The logarithmic corrections to the entropy of various black holes have also been obtained using the Euclidean Quantum Gravity [39, 40, 41]. In this approach, Euclidean Quantum Gravity [42] is used to obtain the partition function for the black hole, which is then used to obtain the logarithmic corrections to the thermodynamics of that black hole. As the logarithmic corrections occur almost universally in the thermodynamics of black objects, in this paper we will analyze the consequences of such corrections for a thermal BIon. We compute the quantum correction to the black hole entropy, internal energy, specific heat using the Euclidean Quantum Gravity [42]. We find that the logarithmic correction affects the critical points, and the corrections significantly change the stability of this system.

2 Euclidean Quantum Gravity

Now, we start with the Euclidean Quantum Gravity that is obtained by performing a Wick rotation on the temporal coordinates in the path integral. Thus, we obtain gravitational partition function in Euclidean Quantum Gravity [42],

[TABLE]

where $\mathcal{I}_{\text{E}}$ is the Euclidean action for the BIon solution [3, 4], and $\beta\propto 1/T$. The density of states $\rho(E)$ is easily obtained from (2.1) by performing an inverse Laplace transform, so that one obtains

[TABLE]

Here, $S(\beta)$ is the entropy and its exact form is given in terms of the partition function and the total energy as $S(\beta)=\beta E+\ln Z$, where $S_{0}=S(\beta_{0})$. The integral (2.2) can be evaluated using the method of steepest decent, around the saddle point $\beta_{0}$, so that $[\partial S(\beta)/\partial\beta]_{\beta=\beta_{0}}$ vanishes, and the equilibrium relation $E=-[\partial\ln Z(\beta)/\partial\beta]_{\beta=\beta_{0}}$ is satisfied. Therefore, the equilibrium temperature is given by $T_{0}=1/\beta_{0}$, and we can expand the entropy $S(\beta)$ around the equilibrium point $\beta_{0}$ as follows

[TABLE]

Here, the first term $S_{0}=S(\beta_{0})$ denotes the entropy at the equilibrium, the second term represents the first order correction over it. If we restrict ourselves to this first order and replace (2.3) into (2.2), we obtain

[TABLE]

for $[\partial^{2}S(\beta)/\partial\beta^{2}]_{\beta=\beta_{0}}>0$, where we choose $a=\beta_{0}$ and $\beta-\beta_{0}=ix$ with $x$ being a real variable. Thus, the expression of the microcanonical entropy $\mathcal{S}$ turns out to be [12, 13]

[TABLE]

Note that the entropy $S(\beta)$ given in (2.3) is different from $\mathcal{S}$ as given by (2.5), the former $S(\beta)$ being the entropy at any temperature, whereas the latter one, $\mathcal{S}$ is the corrected microcanonical entropy at equilibrium, which is computed by incorporating small fluctuations around thermal equilibrium. However, the result obtained in (2.5) is completely model independent and, it can be applied to any canonical thermodynamical system including a BIon solution. Thus, the first order correction is solely governed by the term $\ln[\partial^{2}S(\beta)/\partial\beta^{2}]_{\beta=\beta_{0}}$. This can be simplified to a generic form of the entropy correction given by $\ln(CT^{2})$ [12, 13].

Now for a system with equilibrium temperature $\beta_{0}$, and the equilibrium entropy $S_{0}$, the fluctuation around this equilibrium entropy, $\ln[\partial^{2}S(\beta)/\partial\beta^{2}]_{\beta=\beta_{0}}$ do not depend on $\beta_{0}$ or $S_{0}$. However, the exact form of these fluctuation can be obtained by assuming that this system is dual to a conformal field theory, and using the modular invariance of the conformal field theory [13, 14]. This is done by assuming that $S(\beta)=a\beta^{m}+b\beta^{-n},$ with $m,n,a/b>0$, and observing that at equilibrium this function has an extremum, with $\beta_{0}=(nb/am)^{1/m+n}$. Using this observation, it can be demonstrated that this term $[\partial^{2}S(\beta)/\partial\beta^{2}]_{\beta=\beta_{0}}$ (which represents first order correction around the equilibrium) has to be proportional to $\ln[S_{0}\beta_{0}^{2}]$ [12, 13, 14, 15]. As the Hawking temperature $T$ for the black object is obtained at the equilibrium, so we identified the equilibrium temperature $\beta_{0}$ with $T$. Thus, the thermal fluctuations around the equilibrium entropy can expressed in terms of the equilibrium entropy $S_{0}$ and equilibrium temperature $T$. It may be noted that such corrections to the entropy of AdS black holes have been obtained from the entropy of the boundary theory using AdS/CFT correspondence [43, 44, 45, 46], and it has been observed that the corrections to entropy can be expressed as a logarithmic function of the original equilibrium temperature $T$.

As this correction term is proportional to the logarithmic function of the area, it is of the universal form of the leading order corrections to the entropy of the black object [52, 53, 54, 55, 56, 57]. It may be noted that even thought the form of these corrections is universal, the exact value of the constant of proportionality to these corrections is model dependent [52, 53, 54, 55, 56, 57]. Hence, as the constant of proportionality is model dependent, we will use a free parameter $\gamma=[0,1]$ [47, 48, 49, 50, 51]. Now it is obvious that, when we neglect these thermal fluctuations $\gamma\to 0$, and we obtain the original equilibrium results. Furthermore, for $\gamma\neq 0$, these corrections are proportional to the logarithmic function of the area, and hence are the leading order corrections to the entropy [52, 53, 54, 55, 56, 57]. Furthermore, as these are the leading order corrections to the equilibrium entropy, so they are expected to be less the original equilibrium entropy of the black object (for the perturbative expansion to be valid) [47, 48, 49, 50, 51]. It may be noted that when the black object is large, we can neglect thermal fluctuations, and at that stage, the original equilibrium entropy can describe the system. Thus, $S_{0}$ is a good approximation to entropy when $S_{0}>>ln(S_{0}T^{2})$. However, as the black object reduces in size due to Hawking radiation, we need to consider the corrections to entropy due to thermal fluctuations. Thus, we need to consider $\gamma\ln(S_{0}T^{2})$, when $S_{0}>\ln(S_{0}T^{2})$. These corrections are still smaller than the original equilibrium entropy, but they are large enough to change the behavior of the system, and so they cannot be neglected. At first we can use the leading order corrections, and then we have to use higher order corrections. These corrections can be obtained by considering higher order perturbative corrections around the equilibrium [58, 59]. However, as the black hole approaches Planck scale and $S_{0}\sim\ln(S_{0}T^{2})$, the perturbative expansion around equilibrium breakdown, and this expansion cannot be used to obtain the corrections to the entropy. This corresponds to the breaking of spacetime manifold by quantum fluctuations, which occur near Planck scale [60, 61]. So, this perturbative approximation cannot be used for analyzing Planck scale black objects.

Now, as this expansion for leading order corrections, holds for any black object whose degrees of freedom can be analyzed using a CFT [13, 14, 15], and it has been argued that degrees of freedom of a BIon can also be analyzed using a CFT [16, 17], we can use these corrections for analyzing a BIon. So, we propose that the quantum correction to the entropy of a BIon can be expressed as

[TABLE]

where $S_{0}$ is the original entropy of the BIon solution [3, 4], and $\mathcal{Y}$ in general, is a function of other quantities (such as the properties of branes and string charges). Thus, a full analysis of this system should incorporate such a quantities, but as a toy model, we will only analyze the corrections produced by $\gamma\ln{(S_{0}T^{2})}/2$ on the thermodynamics of such a system, and neglect the effect of $\mathcal{Y}$. This can possible be justified by fixing certain quantities in the system, and analyzing it as a toy model. So, here the last term of (2.6), along with the higher order corrections to entropy [62] are neglected, and the leading order corrections to the entropy from thermal fluctuations are considered.

It may be noted that such logarithmic corrections terms are universal, and occur in almost all approaches to quantum gravity. However, the coefficient of such logarithmic correction term is model dependent [52, 53, 54, 55, 56, 57]. As the corrected expression used in this paper involves a free parameter $\gamma=[0,1]$ [47, 48, 49, 50, 51], it will hold even using different approaches. As any other approaches to this problem can only change the value of this coefficient $\gamma$, which is not fixed in this paper. Thus, the validity of the (2.6) can be argued on general grounds, and the main aim of the paper is to analyze the effects of such a logarithmic corrections on the thermodynamics of a BIon solution.

So, to obtain quantum corrections to the entropy of a BIon solution, we need to use the original entropy $S_{0}$ and original equilibrium temperature $T$ of the BIon solution [3, 4]. Now a BIonic system is a flat spacetime configuration of a D-brane parallel to an anti-D-brane, connected by a wormhole, which has a F-string charge. Geometrically, it is composed of $\mathcal{N}$ coincident D-branes which are infinitely extended, and has $\mathcal{K}$ units of F-string charge, ending in a throat, with minimal radius $\sigma_{0}$ (at temperature $T$). To construct a wormhole solution from this, all we have to do, is to attach a mirror solution at the end of the throat.

It is well known that the blackfold action can be used to describe the D-brane for probing the zero temperature background. However, it was shown that one can also use DBI action for probing the thermal backgrounds [4], where it is ensured that the brane is not affected by the thermal background, but the degrees of freedom living on the brane are ’warmed up’ due to the temperature of thermal background. Thus, the thermal background acts as a heat bath to the D-brane probe, and due to this, the probe stays in thermal equilibrium with the thermal background, which is a ten dimensional hot flat space. This is constructed in the blackfold approach. Thus the thermal generalization of BIon solution has been carried out, and the thermodynamic quantities for this configuration are given by [3, 4]

[TABLE]

where $M$ is the total mass, $S_{0}$ is the entropy and $\mathcal{F}$ is free energy of the BIon. Here, $T_{D3}$ is the D3-brane tension, $z$ is a transverse coordinate to the branes and $F(\sigma)=\sigma^{2}(4\cosh^{2}\alpha-3)/\cosh^{4}\alpha$, with $\sigma$ being the world volume coordinate and $\sigma_{0}$ being the minimal (sphere) radius of the throat or wormhole. Here $\alpha$ is a function of the temperature. The chemical potentials for the D3-brane and F-string are as follows

[TABLE]

It should be noted that these relations satisfy the first law of thermodynamics $dM=TdS_{0}+\mu_{D3}d\mathcal{N}+\mu_{F1}d\mathcal{K}$ as well as the Smarr relation, $4(M-\mu_{D3}\mathcal{N}-\mu_{F1}\mathcal{K})-5TS_{0}=0$. One can also calculate internal energy and the specific heat of the BIon solution as

[TABLE]

which indicates that the system has a negative specific heat.

3 Corrected Thermodynamics for the BIon

Let us now look for the thermal corrections to the above equations by considering logarithmic correction to the entropy $S$ given by the equation (2.6). The entropy (2.8) of $\mathcal{N}$ coincident D-branes, with a throat solution gets corrected as

[TABLE]

In order to analyze the expression for the corrected entropy, we can assume $\cosh^{2}\alpha(\sigma_{0})=\frac{3}{4}$, which means that $F(\sigma_{0})=0$ produces a relatively easy solution. It is indeed a possible solution of the equation of motion at $\sigma_{0}$ [3]. However, we would like to work on the regime where the branch is connected to the extremal BIon [3]. In this formulation one obtains

[TABLE]

where $\cos\delta={\bar{T}}^{4}\sqrt{1+\mathcal{K}^{2}/\sigma^{4}}$, with ${\bar{T}}^{4}=9\pi^{2}T^{4}\mathcal{N}/(4\sqrt{3}T_{D3})$. We further assume an infinitesimal $\delta$ (corresponding to $\sigma^{2}>\mathcal{K}$ at low temperature $\bar{T}\approx 1$), so that $\cos\delta\approx 1$ and $\sin\delta\approx\delta\approx{\bar{T}}^{4}[1+1/(2\sigma^{4})]$. Here we have considered $\mathcal{K}=1$, as it can always be absorbed in $\sigma$ by means of a re-scaling. Now the remaining parameter is $T_{D3}$, which we can set to one, $T_{D3}=1$. So, we can plot the corrected entropy (3.1) by varying the $\bar{T}$, $\sigma_{0}$ and $\gamma$ as depicted in Figs. 1, 2 and 3. In Fig. 1, we draw the corrected entropy in terms of $\bar{T}$ for different values of $\sigma_{0}$. We should note that there is a minimal radius $\sigma_{min}={\bar{T}}^{2}(1-{\bar{T}}^{8})^{-\frac{1}{4}}$ corresponding to each plots, with $\sigma_{0}\geq\sigma_{min}$. In the panel (a) of Fig. 1, we have considered $\sigma_{0}=0.2$, such that ${\bar{T}}\geq 0.45$ and, in such a situation, we can see that the corrected entropy is larger than the uncorrected one. This means that the logarithmic corrections have increased the value of the entropy. Similar thing happens in the panel (b), where $\sigma_{0}=1$, thus, ${\bar{T}}\geq 0.9$. However, the system behaves differently, when we consider $\sigma_{0}=2$ and ${\bar{T}}\geq 0.99$, as shown in Fig. 1(c). Here we see that corrected entropy is smaller than the uncorrected entropy. Therefore, the behavior of the entropy with the logarithmic correction depends on both the parameters, namely, the temperature and $\sigma_{0}$.

In Fig. 2, we plot the corrected entropy as a function of $\sigma_{0}$ for ${\bar{T}}\geq 0.5$, i.e. $\sigma_{min}=0.25$. In this case, we see that there exists a critical $\sigma_{c}$ for each of the plots, and when the value of $\sigma_{0}$ is less than the value of $\sigma_{c}$, the corrected entropy is larger than the uncorrected one. Whereas, when $\sigma_{0}>\sigma_{c}$, the corrected entropy is less. However, the critical points depend on the temperature, for instance, in the case when ${\bar{T}}=0.5$, i.e. in Fig. 2(a), we notice that $\sigma_{c}\approx 0.7$, while in Fig. 2(b), i.e. for ${\bar{T}}\approx 1$, $\sigma_{c}\approx 1.6$. We should note that the region compatible with our assumption is $\sigma_{0}>1$. Fig. 3 demonstrates the behavior of the entropy with respect to $\gamma$, which is the coefficient that determines the amount of corrections on the system. By choosing ${\bar{T}}\geq 1$, we can see that the entropy is a decreasing function of $\gamma$ for $\sigma_{0}=2$, while it is an increasing function of $\gamma$ for $\sigma_{0}=1$. It means that for the small throat (smaller than $\sigma_{c}$), the thermal fluctuation increase the entropy, which may yield more stability to the system, with maximum value of the entropy. On the other hand, a bigger throat may make the system unstable.

The logarithmic correction also modifies the internal energy and the specific heat. Let us analyze the effects for the modification of the internal energy first, which we compute as follows

[TABLE]

We can perform a graphical analysis similar to the entropy to give similar results. However, we focus only on the behavior of the internal energy with the variation of the parameter $\gamma$. Fig. 4 shows that, although the variation of the internal energy is smaller with the logarithmic corrections, however, the slope of the increasing or decreasing functions depends on the value of the temperature and the radius of the throat, as expected. For larger radius, the entropy is decreased due to the thermal fluctuations, while for the smaller radius the entropy is increased. Let us now see the effects on the specific heat. The exact expression of the corrected specific heat is given by

[TABLE]

which has been analyzed in Fig. 5. Here, we show the effect of the logarithmic correction on the specific heat inside the allowed region $1\leq\sigma_{0}$ for Fig. 5(a), and for the whole range in Fig. 5(b) in order to explore the general approximate behavior. In the case of $\gamma=0$, we find that the specific heat is entirely negative, however, in the presence of the thermal fluctuations there are some regions where it is positive. This means that in the presence of the logarithmic correction, there is a special radius $\sigma_{s}$ for which the specific heat is negative, $\sigma_{0}>\sigma_{s}$, while it is positive for $\sigma_{0}<\sigma_{s}$. Here again the value of the $\sigma_{s}$ depends on the temperature, for instance, in Fig. 5(a) it is obvious that when we choose ${\bar{T}}=0.9$, we obtain $\sigma_{s}\approx 1.03$. On the other hand, in Fig. 5(b), when we increase $\sigma_{0}$, the difference between the corrected and uncorrected case slowly vanishes. It indicates that the thermal fluctuation becomes relevant for the smaller radius. Also, we can see from the Fig. 5(b) an asymptotic behavior which may be interpreted as a first order phase transition as found in [3]. Here we show that it occurs due to the thermal fluctuations. We can confirm this by analyzing the free energy ($\mathcal{F}$) and find that $\frac{\partial\mathcal{F}}{\partial{\bar{T}}}=0$ at the phase transition point. It is indicated that we have a first order phase transition in this system.

Also, we demonstrate the variation of the specific heat with the parameter $\gamma$ in Fig. 6. It is obvious that the effect of the logarithmic correction is to increase the specific heat and, for $\gamma>0.65$ (approximately) the specific heat is completely positive, while for $\gamma=0$ it is completely negative. Finally in the Fig. 7, we can analyze the variation of the specific heat with ${\bar{T}}$ to plot its asymptotic behavior. At this asymptotic point, the first derivative of the free energy with respect to the temperature is zero, and so it is a first order phase transition.

4 Conclusion

Quantum fluctuations are important when dealing with objects of very small length scales. They can be neglected when the object is large compared to the Planck scale, but for small objects, the quantum fluctuation become important. Now at Planck scale the background spacetime breaks down, and it is difficult to analyze this system. However, there is a stage before such a total breakdown, when the quantum fluctuations can be not be neglected, but can be analyzed as perturbations around a fixed spacetime. This corresponds to analyzing thermal fluctuations around equilibrium for a black hole. We analyze the quantum corrections for the BIonic systems using such fluctuations around equilibrium. We explicitly include the correction terms produced by such thermal fluctuations. We also analyze the behavior of system at the critical points, after including these correction terms. Moreover, we demonstrate that these correction terms are important by analyzing the stability conditions for this system. In fact, we can demonstrate numerically that these quantum fluctuation affects the critical points, thus affecting the stability of the system. The stability increases under certain conditions. For instance, when the throat is smaller, the inclusion of fluctuations increases the stability. Our analysis explicitly shows how the quantum fluctuation terms become important, with the decrease of the radius. In Fig. 5(b), we observe that increasing the radius $\sigma_{0}$, the correction term slowly vanishes, and the corrected result become identical with the uncorrected one. Apart from the stability analysis, we have computed the corrections to the internal energy and specific heat due to the quantum fluctuation. We confirmed that the first order phase transition occurs in this system, by analyzing the free energy. We have also analyzed the change in the behavior of the corrected system with the temperature.

It may be noted that here we have only analyzed the system numerically. However, to demonstrate that the system actually have phase transition, it is important to analyze it analytically. It would thus be important to analyze this system analytically. However, as this system is very complicated, it might be interesting to study a simpler model of black branes analytically. Such an analysis can give a better understating of such a system. It is possible to construct a BIon solution in M-theory using a system of M2-branes and M5-branes [63]. It would be interesting to analyze the system at a finite temperature. Then the thermodynamics of this system can be studied. It would be possible to study the quantum fluctuations to the geometry of a BIon in M-theory, and these could produce thermal fluctuations in the thermodynamics of this system. It would be interesting to analyze the critical points for such a system, and study the effects of these fluctuations on the stability of this system. It may also be noted that the thermodynamics of the AdS black hole have been studied in M-theory [64, 65]. It is possible to analyze the quantum corrections to these black holes, and this can also be done in Euclidean Quantum Gravity. It would also be interesting to generalize the work of this paper to such AdS black holes in M-theory.

Acknowledgments: S.D. acknowledges the support of research grant (DST/INSPIRE/04 /2016/001391) from the Department of Science and Technology, Govt. of India. S.C. is funded by Conicyt grant 21181211. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.