On Thermodynamics of AdS Black Holes in M-Theory

Motivated by a recent work on asymptotically AdS_4 black holes in M-theory, we investigate the thermodynamics and thermodynamical geometry of AdS black holes from M2 and M5-branes. Concretely, we consider AdS black holes in AdS_{p+2}\times S^{11-p-2}, where p=2,5 by interpreting the number of M2 (and M5-branes) as a thermodynamical variable. We study the corresponding phase transition to examine their stabilities by calculating and discussing various thermodynamical quantities including the chemical potential. Then, we compute the thermodynamical curvatures from the Quevedo metric for M2 and M5-branes geometries to reconsider the stability of such black objects. The Quevedo metric singularities recover similar stability results provided by the phase transition program.


Introduction
Recently, an increasing interest has been devoted to the study of the black hole physics in connection with many subjects including string theory and famous thermodynamical models. The interest has been explored to develop deeper relationships between the gravity theories and the thermodynamical physics using anti-De Sitter geometries. In this issue, laws of black holes can be identified with laws of thermodynamics [1,2,3,4,5]. More precisely, the phase transition and the critical phenomena for various AdS black holes have been extensively investigated using different approachs [6,7,8,9,10]. In this way, certain equations of state, describing rotating black holes, have been identified with some known thermodynamical ones. In particular, it has been remarked serious efforts discussing the behavior of the Gibbs free energy in the fixed charge ensemble. This program has led to a nice interplay between the behavior of the AdS black hole systems and the Van der Waals fluids [11,12,13,14,15,16,17,18]. In fact, it has been shown that P-V criticality, Gibbs free energy, first order phase transition and the behavior near the critical points can be associated with the liquid-gas systems.
More recently, a special focus has been put on the thermodynamics and thermodynamical geometry for a fivedimensional AdS black hole in type IIB superstring background known by AdS 5 × S 5 [19,20,21]. It is recalled that this geometry has been studied in many places in connection with AdS/CFT correspondence, providing a nice equivalence between gravitational theories in d-dimensional AdS geometries and conformal field theories (CFT) in a (d-1)-dimensional boundary of such AdS spaces [22,23,24,25]. In such black hole activities, the number of colors has been interpreted as a thermodynamical variable. In particular, the thermodynamic properties of black holes in AdS 5 × S 5 have been investigated by considering the cosmological constant in the bulk as the number of colors. In fact, many thermodynamical quantities have been computed to discuss the stability behaviors of such black holes.
Motivated by these activities and a recent work on asymptotically AdS 4 black holes in Mtheory [26,27,28,29], we investigate the thermodynamics and thermodynamical geometry of AdS black holes from the physics of M2 and M5-branes. Concretely, we study AdS black holes in AdS p+2 × S 11−p−2 , where p = 2, 5 by viewing the number of M2 and M5-branes as a thermodynamical variable. To discuss the stability of such solutions, we examine first the corresponding phase transition by computing the relevant quantities including the chemical potential. Then, we calculate the thermodynamical curvatures from the Quevedo metric for M2 and M5-brane geometries to reconsider the study of the stability.
The paper is organized as follows. We discuss thermodynamic properties and stability of the black holes in AdS p+2 × S 11−p−2 , where p = 2, 5 by viewing the number of M2 and M5-branes as a thermodynamical variable in section 2 and 3. Similar results which have been recovered using thermodynamical curvature calculations, associated with the Quevedo metric, are presented in section 4. The last section is devoted to conclusion.

Thermodynamics of black holes in AdS 4 × S 7 space
In this section, we investigate the phase transition of the AdS black holes in M-theory in the presence of solitonic objects. It is recalled that, at lower energy, M-theory describes an eleven dimensional supergravity. This theory, which was proposed by Witten, can produce some non perturbative limits of superstring models after its compactification on particular geometries [30].
It has been shown that the corresponding eleven supergravity involves a cubic R 4 oneloop UV divergence [31] which has been obtained using a specific cutoff motivated by string theory [32,33]. This calculation gives the following correction of the Einstein action with κ 11 is related to the Planck length by κ 2 11 = 2 4 π 5 ℓ 2 11 , ζ = 2π 2 3 . W can be given in terms of the Ricci tensor as follows W ∼ RRRR [34,35]. Roughly speaking, M-theory contains two fundamental objets called M2 and M5 branes coupled in eleven dimension to 3 and 6 forms respectively. The near horizon of such black objects is defined by the product of AdS spaces and spheres To start, let us consider the case of M2-brane. The corresponding geometry is AdS 4 × S 7 .
In such a geometric background, the line element of the black M2-brane metric is given by [36,34] where dΩ 2 7 is the metric of seven-dimensional sphere with unit radius. In this solution, the metric function reads as follows where L is the AdS radius and m is an integration constant. The cosmological constant is Λ = −6/L 2 . Form M-theory point of view, the eleven-dimensional spacetime eq.(3) can be interpreted as the near horizon geometry of N coincident configurations of M2-branes.
In this background, the AdS radius L is linked to the M2-brane number N via the relation [34,37] According to the proposition reported in [19,20,21], we consider the cosmological constant as the number of coincide M2-branes in the M theory background and its conjugate quantity as the associated chemical potential.
The event horizon r h of the corresponding black hole is determined by solving the equation f = 0. Exploring eq.(4), the mass of the black hole can be written as The Bekenstein-Hawking entropy formula of the black hole produces It is recalled that four-dimensional Newton gravitational constant is related to the elevendimensional one by For simplicity reason, we take in the rest of the paper G 11 = κ 11 = 1. In this way, the mass of the black hole can be expressed as a function of N and S Using the standard thermodynamic relation dM = TdS + µdN, the corresponding temperature takes the following form This quantity can be identified with the Hawking temperature of the black hole. Using eq.
(9)The chemical potential µ conjugate to the number of M2-branes is given by It defines the measure of the energy cost to the system when one increases the variable N. In . terms of these quantities, the Gibbs free energy reads as Having calculated the relevant thermodynamical quantities, we investigate the corresponding phase transition. To do so, we study the variation of the Hawking temperature as a function of the entropy. This variation is plotted in figure 1.
which corresponds to the minimal temperature T 4 min = . It is observed that for such a temperature no black hole can exist. Otherwise, two branches are shown up. Indeed, the first branch associated with small entropy S values is thermodynamically unstable.
However, the second phase corresponding to the large entropy S is considered as a thermodynamical stable one.
It is observed from Gibbs free energy, given in eq. (12), that the Hawking-Page phase transition occurs where the corresponding phase transition temperature is It is verified that this quantity is larger than the temperature T 4 min = T| S=S 4,1 . At the Hawking-Page transition, the associated entropy takes the following form In figure 2, we illustrate the Gibbs free energy as function of the Hawking temperature T for some fixed values of N. To study the phase transition, we vary the chemical potential in terms of the entropy. In figure 3, we plot such a variation for a fixed value of N.
For small values of S, the chemical potential is positive. However, it changes to be negative when S is large. Moreover, the chemical potential changes its sign at Figure 3: The chemical potential µ as function of the entropy for N = 3 It is easy to cheek the following constraint It turns out that the vanishing of the chemical potential appears in the unstable branch.
In figure 4, we plot the chemical potential as a function of temperature T 4 for a fixed N. To see the effect the number of the M2-branes, we discuss the behavior of the chemical potential µ in terms of such a variable. The calculation is illustrated in figure 5. It is observed that the chemical potential µ presents a maximum at It is noted that S 4,4 is also less than S 4,1 . It is remarked that this is quite different from the classical gas having a negative chemical potential. In the case where the chemical potential approaches to zero and becomes positive, quantum effects should be considered and be relevant in the discussion [21].
Having discussed the case of M2-branes, let us move a higher dimensional case provided by M-theory. It is shown that in eleven dimensions the dual magnetic of M2-branes are M5branes. In the following, we investigate the black holes in such magnetic brane backgrounds.

Thermodynamics of black holes in AdS 7 × S 4 space
In this section, we discuss the magnetic solution associated with the near geometry AdS 7 × S 4 . According to [36,34], the corresponding metric takes the following form where dΩ 2 4 is the metric of four-dimensional sphere with unit radius. As in the case of M2-branes, the metric function reads as In M-theory, the eleven-dimensional spacetime eq. (19) can be considered as the near horizon geometry of N coincident configurations of M5-branes. For this solution, the AdS radius L can be related to the number N via the relation [34,37] L 9 = N 3 κ 2 11 2 7 π 5 .
The mass of the black hole can be computed using eq. (20). The calculation gives the following expression It is found that the entropy is Similarly, the Gibbs free energy can be computed. It is given by  (27) As in the previous case, the stability discussion can be done by varying the two variables S and N. We first deal with the phases transition. Indeed, it can be studied in terms of monotony of the Hawking temperature in terms of the entropy. This variation is plotted in figure 6.
We can clearly see that the Hawking temperature is not a monotonic function. It involves  It follows from the Gibbs free energy, given in eq. (27), that the Hawking-Page phase transition occurs where the corresponding phase transition temperature This quantity is larger than the temperature T 7 min = T| S=S 7,1 . At the Hawking-Page transition, the corresponding entropy is given by The figure 7 illustrates the Gibbs free energy with respect to the Hawking Temperature T for some fixed values of N.
For a fixed N, it follows that the down branch Gibbs free energy changes its sign at the point S = S 7,2 , corresponding to the Hawking-Page transition point.
To study the phase transition, we vary the chemical potential in terms of the entropy. In As in the case of M2-branes, one has S 7,3 < S 7,2 < S 7,1 .
As we will see that the vanishing of the chemical potential appears in the unstable branch.
Similar behaviors appeared in the case of M2-branes. This implies that the vanishing of the chemical potential does not make any sense from the point of view of dual conformal field theory. This point deserves a deeper study. We hope to comeback to this point in future.
To see the effect of the temperature, figure 9 presents the chemical potential as a function of the temperature T 7 for fixed values of N. It is noted that one has similar behaviors appeared in the case of M2-branes. Figure 10 shows the chemical potential as a function of N in the case with a fixed entropy.
In the following section, we will study thermodynamical geometry of the M2 and M5branes black holes in the extended phase space to reconsider the study of the stability problem.

Geothermodynamics of AdS black holes in M-theory
In this section, we discuss the geothermodynamics AdS black holes in AdS p+2 × S 11−p−2 .
This study concerns singular limits of certain thermodynamical quantities including the heat capacity. This quantity is the relevant in the study of the stability of such black hole solutions.
To elaborate this discussion, the number of branes N should be fix to consider a canonical ensemble. For a fixed N, the heat capacity for M2 and M5-branes are given respectively by  To show the singularity of the corresponding heat capacity, the thermodynamical geometry of such black hole solutions should be discussed including the thermodynamical curvature. To compute such a quantity, one can use several metrics. However, we can explore the Quevedo metric which reads as [38,39,40,41] where M i j stands for ∂ 2 M/∂x i ∂x j , and x 1 = S, x 2 = N. The scalar curvature of this metric can be computed in a direct way. For M2 and M5-branes respectively, the calculation gives the following expressions and The quantities A, B and C are given by In figure 12, we plot the scalar curvature of the Quevedo metric as a function of the entropy for M2 and M5-branes. It follows from figure 12 that one has two divergent points located at S i,4 i∈{4,7} and S i,1 i∈{4,7} respectively. The first one coincides with the divergent point of C N . However, the second one is associated with the maximum of the chemical potential considered as a function of N.
It is worth noting that this result is in good agrement with the recent study reported in [43,44,42] saying that the divergences of scalar curvature for the Quevedo metric corresponds to the divergence or zero for the heat capacity. It has been suggested that these results can be explored to understand the link between the phase transition and the thermodynamical curvature.

Conclusion
In this paper, we have investigated the thermodynamics and thermodynamical geometry This work poses a question concerning a 9-dimensional AdS black holes associated with D7-branes on AdS 9 -space. In fact, it may be possible to consider a geometry of the form AdS 9 × S 1 × T 2 inspired by the recent work on black holes in F-theory [45]. This may support the results concerning the link between the phase transition and the thermodynamical curvature.