State-Space Geometry, Statistical Fluctuations, and Black Holes in String Theory

Abstract

We study the state-space geometry of various extremal and nonextremal black holes in string theory. From the notion of the intrinsic geometry, we offer a state-space perspective to the black hole vacuum fluctuations. For a given black hole entropy, we explicate the intrinsic geometric meaning of the statistical fluctuations, local and global stability conditions, and long range statistical correlations. We provide a set of physical motivations pertaining to the extremal and nonextremal black holes, namely, the meaning of the chemical geometry and physics of correlation. We illustrate the state-space configurations for general charge extremal black holes. In sequel, we extend our analysis for various possible charge and anticharge nonextremal black holes. From the perspective of statistical fluctuation theory, we offer general remarks, future directions, and open issues towards the intrinsic geometric understanding of the vacuum fluctuations and black holes in string theory.

1. Introduction

In this paper, we study statistical properties of the charged and anticharged black hole configurations in string theory. Specifically, we illustrate that the components of the vacuum fluctuations define a set of local pair correlations against the parameters, for example, charges, anticharges, mass, and angular momenta. Our consideration follows from the notion of the thermodynamic geometry, mainly introduced by Weinhold [1, 2] and Ruppeiner [3–9]. Importantly, this framework provides a simple platform to geometrically understand the statistical nature of local pair correlations and underlying structures pertaining to the vacuum phase transitions. In diverse contexts, the state-space geometric perspective offers an understanding of the phase structures of mixtures of gases, black hole configurations [10–26], generalized uncertainty principle [27], strong interactions, for example, hot QCD [28], quarkonium configurations [29], and some other systems, as well.

The main purpose of the present paper is to consider the state-space properties of various possible extremal and nonextremal black holes in string theory, in general. String theory [30], as the most promising framework to understand all possible fundamental interactions, celebrates the physics of black holes, in both the zero and the nonzero temperature domains. Our consideration hereby plays a crucial role in understanding the possible phases and stability of the string theory vacua. A further motivation follows from the consideration of the string theory black holes; namely, 𝒩=2 supergravity arises as a low energy limit of the Type II string theory solution, admitting extremal black holes with the zero Hawking temperature and a nonzero macroscopic attractor entropy.

A priori, the entropy depends on a large number of scalar moduli arising from the compactification of the 10-dimensional theory down to the 4-dimensional physical spacetime. This involves a 6-dimensional compactifying manifold. Interesting string theory compactifications involve 𝑇⁶, 𝐾₃×𝑇², and Calabi-Yau manifolds. The macroscopic entropy exhibits a fixed point behavior under the radial flow of the scalar fields. In such cases, the near horizon geometry of an extremal black hole turns out to be an 𝐴𝑑𝑆₂×𝑆² manifold which describes the Bertotti-Robinson vacuum associated with the black hole. The area of the black hole horizon is 𝐴 and thus the macroscopic entropy [31–42] is given as 𝑆_macro=𝜋|𝑍_∞|². This is known as the Ferrara-Kallosh-Strominger attractor mechanism, which, as the macroscopic consideration, requires a validity from the microscopic or statistical basis of the entropy. In this concern, there have been various investigations on the physics of black holes, for example, horizon properties [43, 44], counting of black hole microstates [45–47], spectrum of half-BPS states in 𝒩=4 supersymmetric string theory [48], and fractionation of branes [49]. From the perspective of the fluctuation theory, our analysis is intended to provide the nature of the statistical structures of the extremal and nonextremal black hole configurations. The attractor configurations exist for the extremal black holes, in general. However, the corresponding nonextremal configurations exist in the throat approximation. In this direction, it is worth mentioning that there exists an extension of Sen entropy function formalism for 𝐷₁𝐷₅ and 𝐷₂𝐷₆𝑁𝑆₅ nonextremal configurations [50–52]. In the throat approximation, these solutions, respectively, correspond to Schwarzschild black holes in 𝐴𝑑𝑆₃×𝑆³×𝑇⁴ and 𝐴𝑑𝑆₃×𝑆²×𝑆¹×𝑇⁴. In relation with the intrinsic state-space geometry, we will explore the statistical understanding of the attractor mechanism and the moduli space geometry and explain the vacuum fluctuations of the black brane configurations.

In this paper, we consider the state-space geometry of the spherical horizon topology black holes in four spacetime dimensions. These configurations carry a set of electric magnetic charges (𝑞_𝑖,𝑝_𝑖). Due to the consideration of Strominger and Vafa [53], these charges are associated with an ensemble of weakly interacting D-branes. Following [53–59], it turns out that the charges (𝑞_𝑖,𝑝_𝑖) are proportional to the number of electric and magnetic branes, which constitute the underlying ensemble of the chosen black hole. In the large charge limit, namely, when the number of such branes becomes large, we have treated the logarithm of the degeneracy of states of the statistical configuration as the Bekenstein-Hawking entropy of the associated string theory black holes. For the extremal black holes, the entropy is described in terms of the number of the constituent D-branes. For example, the two charge extremal configurations can be examined in terms of the winding modes and the momentum modes of an excited string carrying 𝑛₁ winding modes and 𝑛_𝑝 momentum modes. Correspondingly, the state-space geometry of the nonextremal black holes is described by adding energy to the extremal D-branes configurations. This renders as the contribution of the clockwise and anticlockwise momenta in the Kaluza-Klein scenarios and that of the antibrane charges in general to the black hole entropy.

From the perspective of black hole thermodynamics, we describe the structure of the state-space geometry of four-dimensional extremal and nonextremal black holes in a given duality frame. Thus, when we take arbitrary variations over the charges (𝑞_𝑖,𝑝_𝑖) on the electric and magnetic branes, the underlying statistical fluctuations are described by only the numbers of the constituent electric and magnetic branes. From the perspective of the intrinsic state-space geometry, if one pretends that the notion of statistical fluctuations applies to intermediate regimes of the moduli space, then the attractor horizon configurations require an embedding to the higher dimensional intrinsic Riemanian manifold. Physically, such a higher dimensional manifold can be viewed as a possible blow-up of the attractor fixed point phase-space to a nontrivial moduli space. From the perspective of thermodynamic Ruppenier geometry, we have offered future directions and open issues in the conclusion. We leave the explicit consideration of these matters open for further research.

In Section 2, we define the general notion of vacuum fluctuations. This offers the physical meaning of the state-space geometry. In Section 3, we provide a brief review of statistical fluctuations. In particular, for a given black hole entropy, we firstly explicate the statistical meaning of state-space surface and then offer the general meaning of the local and global stability conditions and long range statistical correlations. In Section 4, we provide a set of physical motivations pertaining to the extremal and nonextremal black holes, the meaning of Wienhold chemical geometry, and the physics of correlation. In Section 5, we consider state-space configurations pertaining to the extremal black holes and explicate our analysis for the two and three charge configurations. In Section 6, we extend the above analysis for the four, six, and eight charge-anticharge nonextremal black holes. Finally, Section 7 provides general remarks, conclusion and outlook, and future directions and open issues towards the application of string theory.

2. Definition of State-Space Geometry

Considering the fact that the black hole configurations in string theory introduce the notion of vacuum, it turns out for any thermodynamic system, that there exist equilibrium thermodynamic states given by the maxima of the entropy. These states may be represented by points on the state-space. Along with the laws of the equilibrium thermodynamics, the theory of fluctuations leads to the intrinsic Riemannian geometric structure on the space of equilibrium states [8, 9]. The invariant distance between two arbitrary equilibrium states is inversely proportional to the fluctuations connecting the two states. In particular, a less probable fluctuation means that the states are far apart. For a given set of states {𝑋_𝑖}, the state-space metric tensor is defined by 𝑔_𝑖𝑗(𝑋)=−𝜕_𝑖𝜕_𝑗𝑆𝑋₁,𝑋₂,…,𝑋_𝑛. (1) A physical motivation of (1) can be given as follows. Up to the second order approximation, the Taylor expansion of the entropy 𝑆(𝑋₁,𝑋₂,…,𝑋_𝑛) yields 𝑆−𝑆₀1=−2^𝑛_𝑖=1𝑔_𝑖𝑗Δ𝑋^𝑖Δ𝑋^𝑗, (2) where 𝑔_𝑖𝑗𝜕∶=−²𝑆𝑋₁,𝑋₂,…,𝑋_𝑛𝜕𝑋^𝑖𝜕𝑋^𝑗=𝑔_𝑗𝑖 (3) is called the state-space metric tensor. In the present investigation, we consider the state-space variables {𝑋₁,𝑋₂,…,𝑋_𝑛} as the parameters of the ensemble of the microstates of the underlying microscopic configuration (e.g., conformal field theory [60], black hole conformal field theory [61], and hidden conformal field theory [62, 63]), which defines the corresponding macroscopic thermodynamic configuration. Physically, the state-space geometry can be understood as the intrinsic Riemannian geometry involving the parameters of the underlying microscopic statistical theory. In practice, we will consider the variables {𝑋₁,𝑋₂,…,𝑋_𝑛} as the parameters, namely, charges, anticharges, and others if any, of the corresponding low energy limit of the string theory, for example, 𝒩=2 supergravity. In the limit, when all the variables, namely, {𝑋₁,𝑋₂,…,𝑋_𝑛}, are thermodynamic, the state-space metric tensor equation (1) reduces to the corresponding Ruppenier metric tensor. In the discrete limit, the relative coordinates Δ𝑋^𝑖 are defined as Δ𝑋^𝑖∶=𝑋^𝑖−𝑋^𝑖₀, for given {𝑋^𝑖₀}∈𝑀_𝑛. In the Gaussian approximation, the probability distribution has the following form: 𝑃𝑋₁,𝑋₂,…,𝑋_𝑛−1=𝐴exp2𝑔_𝑖𝑗Δ𝑋^𝑖Δ𝑋^𝑗. (4) With the normalization _𝑖𝑑𝑋_𝑖𝑃𝑋₁,𝑋₂,…,𝑋_𝑛=1, (5) we have the following probability distribution: 𝑃𝑋₁,𝑋₂,…,𝑋_𝑛=𝑔(𝑋)(2𝜋)^𝑛/2−1exp2𝑔_𝑖𝑗𝑑𝑋^𝑖⊗𝑑𝑋^𝑗, (6) where 𝑔_𝑖𝑗 now, in a strict mathematical sense, is properly defined as the inner product 𝑔(𝜕/𝜕𝑋^𝑖,𝜕/𝜕𝑋^𝑗) on the corresponding tangent space 𝑇(𝑀_𝑛)×𝑇(𝑀_𝑛). In this connotation, the determinant of the state-space metric tensor, 𝑔(𝑋)∶=𝑔_𝑖𝑗, (7) can be understood as the determinant of the corresponding matrix [𝑔_𝑖𝑗]_𝑛×𝑛. For a given state-space manifold (𝑀_𝑛,𝑔), we will think of {𝑑𝑋^𝑖} as the basis of the cotangent space 𝑇^⋆(𝑀_𝑛). In the subsequent analysis, by taking an account of the fact that the physical vacuum is neutral, we will choose 𝑋^𝑖₀=0.

3. Statistical Fluctuations

3.1. Black Hole Entropy

As a first exercise, we have illustrated thermodynamic state-space geometry for the two charge extremal black holes with electric charge 𝑞 and magnetic charge 𝑝. The next step has thence been to examine the thermodynamic geometry at an attractor fixed point(s) for the extremal black holes as the maxima of their macroscopic entropy 𝑆(𝑞,𝑝). Later on, the state-space geometry of nonextremal counterparts has as well been analyzed. In this investigation, we demonstrate that the state-space correlations of nonextremal black holes modulate relatively more swiftly to an equilibrium statistical basis than those of the corresponding extremal solutions.

3.2. State-Space Surface

The Ruppenier metric on the state-space (𝑀₂,𝑔) of two charge black holes is defined by 𝑔_𝑞𝑞𝜕=−²𝑆𝑞,𝑝𝜕𝑞²,𝑔_𝑞𝑝𝜕=−²𝑆𝑞,𝑝,𝑔𝜕𝑞𝜕𝑝_𝑝𝑝𝜕=−²𝑆𝑞,𝑝𝜕𝑝². (8) Subsequently, the components of the state-space metric tensor are associated with the respective statistical pair correlation functions. It is worth mentioning that the coordinates on the state-space manifold are the parameters of the microscopic boundary conformal field theory which is dual the black hole space-time solution. This is because the underlying state-space metric tensor comprises of the Gaussian fluctuations of the entropy which is the function of the number of the branes and antibranes. For the chosen black hole configuration, the local stability of the underlying statistical system requires both principle minors to be positive. In this setup, the diagonal components of the state-space metric tensor, namely, {𝑔_{𝑥𝑖𝑥𝑖}∣𝑥_𝑖=(𝑛,𝑚)}, signify the heat capacities of the system. This requires that the diagonal components of the state-space metric tensor 𝑔_{𝑥𝑖𝑥𝑖}>0,𝑖=𝑛,𝑚, (9) be positive definite. In this investigation, we discuss the significance of the above observation for the eight parameter nonextremal black brane configurations in string theory. From the notion of the relative scaling property, we will demonstrate the nature of the brane-brane pair correlations; namely, from the perspective of the intrinsic Riemannian geometry, the stability properties of the eight parameter black branes are examined from the positivity of the principle minors of the space-state metric tensor. For the Gaussian fluctuations of the two charge equilibrium statistical configurations, the existence of a positive definite volume form on the state-space manifold (𝑀₂(𝑅),𝑔) imposes such a global stability condition. In particular, the above configuration leads to a stable statistical basis if the determinant of the state-space metric tensor, 𝑔=𝑆_𝑛𝑛𝑆_𝑚𝑚−𝑆²_𝑛𝑚, (10) remains positive. Indeed, for the two charge black brane configurations, the geometric quantities corresponding to the underlying state-space manifold elucidate typical features of the Gaussian fluctuations about an ensemble of equilibrium brane microstates. In this case, we see that the Christoffel connections on the (𝑀₂,𝑔) are defined by Γ_𝑖𝑗𝑘=𝑔_{𝑖𝑗,𝑘}+𝑔_{𝑖𝑘,𝑗}−𝑔_{𝑗𝑘,𝑖}. (11) The only nonzero Riemann curvature tensor is 𝑅_{𝑞𝑝𝑞𝑝}=𝑁𝐷, (12) where 𝑁∶=𝑆_𝑝𝑝𝑆_𝑞𝑞𝑞𝑆_𝑞𝑝𝑝+𝑆_𝑞𝑝𝑆_𝑞𝑞𝑝𝑆_𝑞𝑝𝑝+𝑆_𝑞𝑞𝑆_𝑞𝑞𝑝𝑆_𝑝𝑝𝑝−𝑆_𝑞𝑝𝑆_𝑞𝑞𝑞𝑆_𝑝𝑝𝑝−𝑆_𝑞𝑞𝑆²_𝑞𝑝𝑝−𝑆_𝑝𝑝𝑆²_𝑞𝑞𝑝,𝐷∶=𝑆_𝑞𝑞𝑆_𝑝𝑝−𝑆²_𝑞𝑝². (13)(14) The scalar curvature and the corresponding 𝑅_{𝑖𝑗𝑘𝑙} of an arbitrary two-dimensional intrinsic state-space manifold (𝑀₂(𝑅),𝑔) may be given as 𝑅=2𝑞,𝑝𝑔𝑅_{𝑞𝑝𝑞𝑝}𝑞,𝑝. (15)

3.3. Stability Conditions

For a given set of variables {𝑋¹,𝑋²,…,𝑋^𝑛}, the local stability of the underlying state-space configuration demands the positivity of the heat capacities: 𝑔_𝑖𝑖𝑋^𝑖>0;∀𝑖=1,2,…,𝑛. (16) Physically, the principle components of the state-space metric tensor {𝑔_𝑖𝑖(𝑋^𝑖)∣𝑖=1,2,…,𝑛} signify a set of definite heat capacities (or the related compressibilities), whose positivity apprises that the black hole solution complies an underlying, locally in equilibrium, statistical configuration. Notice further that the positivity of principle components is not sufficient to insure the global stability of the chosen configuration and thus one may only achieve a locally stable equilibriumstatistical configuration. In fact, the global stability condition constraint over the allowed domain of the parameters of black hole configurations requires that all the principle components and all the principle minors of the metric tensor must be strictly positive definite [6]. The above stability conditions require that the following set of equations must be simultaneously satisfied: 𝑝₀𝑝∶=1,₁∶=𝑔₁₁𝑝>0,₂𝑔∶=¹¹𝑔₁₂𝑔₁₂𝑔₂₂𝑝>0,₃𝑔∶=¹¹𝑔₁₂𝑔₁₃𝑔₁₂𝑔₂₂𝑔₂₃𝑔₁₃𝑔₂₃𝑔₃₃⋮𝑝>0,_𝑛∶=𝑔>0. (17)

3.4. Long Range Correlations

The thermodynamic scalar curvature of the state-space manifold is proportional to the correlation volume [6]. Physically, the scalar curvature signifies the interaction(s) of the underlying statistical system. Ruppenier has in particular noticed for the black holes in general relativity that the scalar curvature 𝑅(𝑋)∼𝜉^𝑑, (18) where 𝑑 is the spatial dimension of the statistical system and the 𝜉 fixes the physical scale [6]. The limit 𝑅(𝑋)→∞ indicates the existence of certain critical points or phase transitions in the underlying statistical system. The fact that “all the statistical degrees of freedom of a black hole live on the black hole event horizon” signifies that the state-space scalar curvature, as the intrinsic geometric invariant, indicates an average number of correlated Plank areas on the event horizon of the black hole [8]. In this concern, [9] offers interesting physical properties of the thermodynamic scalar curvature and phase transitions in Kerr-Newman black holes. Ruppeiner [6] has further conjectured that the global correlations can be expressed by the following arguments: (a) the zero state-space scalar curvature indicates certain bits of information on the event horizon, fluctuating independently of each other; (b) the diverging scalar curvature signals a phase transition indicating highly correlated pixels of the information.

4. Some Physical Motivations

4.1. Extremal Black Holes

The state-space of the extremal black hole configuration is a reduced space comprising of the states which respect the extremality (BPS) condition. The state-spaces of the extremal black holes show an intrinsic geometric description. Our intrinsic geometric analysis offers a possible zero temperature characterization of the limiting extremal black brane attractors. From the gauge/gravity correspondence, the existence of state-space geometry could be relevant to the boundary gauge theories, which have finitely many countable sets of conformal field theory states.

4.2. Nonextremal Black Holes

We will analyze the state-space geometry of nonextremal black holes by the addition of antibrane charge(s) to the entropy of the corresponding extremal black holes. To interrogate the stability of a chosen black hole system, we will investigate the question that the underlying metric 𝑔_𝑖𝑗(𝑋_𝑖)=−𝜕_𝑖𝜕_𝑗𝑆(𝑋₁,𝑋₂,…,𝑋_𝑛) should provide a nondegenerate state-space manifold. The exact dependence varies case to case. In the next section, we will proceed in our analysis with an increasing number of the brane charges and antibrane charges.

4.3. Chemical Geometry

The thermodynamic configurations of nonextremal black holes in string theory with small statistical fluctuations in a “canonical” ensemble are stable if the following inequality holds: 𝜕_𝑖𝜕_𝑗𝑆𝑋₁,𝑋₂,…,𝑋_𝑛<0. (19) The thermal fluctuations of nonextremal black holes, when considered in the canonical ensemble, give a closer approximation to the microcanonical entropy: 𝑆=𝑆₀−12ln𝐶𝑇²+⋯. (20) In (20), the 𝑆₀ is the entropy in the “canonical” ensemble and 𝐶 is the specific heat of the black hole statistical configuration. At low temperature, the quantum effects dominate and the above expansion does not hold anymore. The stability condition of the canonical ensemble is just 𝐶>0. In other words, the Hessian function of the internal energy with respect to the chemical variables, namely, {𝑥₁,𝑥₂,…,𝑥_𝑛}, remains positive definite. Hence, the energy as the function of the {𝑥₁,𝑥₂,…,𝑥_𝑛} satisfies the following condition: 𝜕_𝑖𝜕_𝑗𝐸𝑥₁,𝑥₂,…,𝑥_𝑛>0. (21) The state-space coordinates {𝑋^𝑖} and intensive chemical variables {𝑥_𝑖} are conjugate to each other. In particular, the {𝑋^𝑖} are defined as the Legendre transform of {𝑥_𝑖}, and thus we have 𝑋^𝑖(𝑥)∶=𝜕𝑆𝜕𝑥_𝑖. (22)

4.4. Physics of Correlation

Geometrically, the positivity of the heat capacity 𝐶>0 turns out to be the positivity condition of 𝑔_𝑖𝑗>0, for a given 𝑖. In many cases, the state-space stability restriction on the parameters of the black hole corresponds to the situation away from the extremality condition; namely, 𝑟₊=𝑟₋. Far from the extremality condition, even at the zero antibrane charge or angular momentum, we find that there is a finite value of the thermodynamic scalar curvature, unlike the nonrotating or only brane-charged configurations. It turns out that the state-space geometry of the two charge extremal configurations is flat. Thus, the Einstein-Hilbert contributions lead to a noninteracting statistical system. At the tree level, some black hole configurations turn out to be ill-defined, as well. However, we anticipate that the corresponding state-space configuration would become well-defined when a sufficient number of higher derivative corrections [64–67] are taken into account with respect to the 𝛼^′-corrections and the string loop 𝑙_𝑠 corrections. For the BTZ black holes [13], we notice that the large entropy limit turns out to be the stability bound, beyond which the underlying quantum effects dominate.

For the black hole in string theory, the Ricci scalar of the state-space geometry is anticipated to be positive definite with finitely many higher order corrections. For nonextremal black brane configurations, which are far from the extremality condition, such effects have been seen from the nature of the state-space scalar curvature 𝑅(𝑆(𝑋₁,𝑋₂,…,𝑋_𝑛)). Indeed, [12, 14] indicate that the limiting state-space scalar curvature 𝑅(𝑆(𝑋₁,𝑋₂,…,𝑋_𝑛))|_noanticharge≠0 gives a set of stability bounds on the statistical parameters. Thus, our consideration yields a classification of the domain of the parameters and global correlation of a nonextremal black hole.

4.5. String Theory Perspective

In this subsection, we recall a brief notion of entropy of a general string theory black brane configuration from the viewpoint of the counting of the black hole microstates [53, 53–59, 68]. Given a string theory configuration, the choice of compactification [30] chosen is the factorization of the type ℳ_(3,1)×𝑀₆, where 𝑀₆ is a compact internal manifold. From the perspective of statistical ensemble theory, we will express the entropy of a nonextremal black hole as the function of the numbers of branes and antibranes. Namely, for the charged black holes, the electric and magnetic charges (𝑞_𝑖,𝑝_𝑖) form a coordinate chart on the state-space manifold. In this case, for a given ensemble of 𝐷-branes, the coordinate 𝑞_𝑖 is defined as the number of the electric branes and 𝑝_𝑖 as the number of the magnetic branes. Towards the end of this paper, we will offer further motivation for the consideration of the state-space geometry of large charged nonspherical horizon black holes in spacetime dimensions 𝐷≥5. In this concern, [68] plays a central role towards the formation of the lower dimensional black hole configuration. Namely, for the torus compatifications, the exotic branes play an important role concerning the physical properties of supertubes, the 𝐷₀-𝐹₁ system and associated counting of the black hole microstates.

In what follows, we consider the four-dimensional string theory black holes in a given duality basis of the charges (𝑞_𝑖,𝑝_𝑖). From the perspective of string theory, the exotic branes and nongeometric configurations offer interesting fronts for the black holes in three spacetime dimensions. In general, such configurations could carry a dipole or a higher pole charge, and they leave the four-dimension black hole configuration asymptotically flat. In fact, for the spacetime dimensions 𝐷≥4, [68] shows that a charge particle corresponds to an underlying gauge field, modulo 𝑈-duality transformations. From the perspective of nonextremal black holes, by taking appropriate boundary condition, namely, the unit asymptotic limit of the harmonic function which defines the spacetime metric, one can choose the spacetime regions such that the supertube effects arising from nonexotic branes can effectively be put off in an asymptotically flat space [68]. This allows one to compute the Arnowitt-Deser-Misner (ADM) mass of the asymptotic black hole. From the viewpoint of the statistical investigation, the dependence of the mass to the entropy of a nonextremal black hole comes from the contribution of the antibranes to the counting degeneracy of the states.

5. Extremal Black Holes in String Theory

5.1. Two Charge Configurations

The state-space geometry of the two charge extremal configurations is analyzed in terms of the winding modes and the momentum modes of an excited string carrying 𝑛₁ winding modes and 𝑛_𝑝 momentum modes. In the large charge limit, the microscopic entropy obtained by the degeneracy of the underlying conformal field theory states reduces to the following expression: 𝑆_micro=22𝑛₁𝑛_𝑝. (23) The microscopic counting can be accomplished by considering an ensemble of weakly interacting D-branes [54]. The counting entropy and the macroscopic attractor entropy of the two charge black holes in string theory which have a 𝑛₄ number of 𝐷₄ branes and a 𝑛₀ number of 𝐷₀ branes match and thus we have 𝑆_micro=2𝜋√𝑛₀𝑛₄=𝑆_macro. (24) In this case, the components of underlying state-space metric tensor are 𝑔_𝑛0𝑛0=𝜋2𝑛₀𝑛₄𝑛₀,𝑔_𝑛0𝑛4𝜋=−21√𝑛₀𝑛₄,𝑔_𝑛4𝑛4=𝜋2𝑛₄𝑛₀𝑛₄. (25) The diagonal pair correlation functions remain positive definite: 𝑔_{𝑛𝑖𝑛𝑖}>0∀𝑖∈{0,4}∣𝑛_𝑖>0,𝑔_𝑛4𝑛4>0,∀𝑛₀,𝑛₄. (26) For distinct 𝑖,𝑗∈{0,4}, the state-space pair correlation functions admit 𝑔_𝑖𝑖𝑔_𝑗𝑗=𝑛_𝑗𝑛_𝑖²,𝑔_𝑖𝑗𝑔_𝑖𝑖𝑛=−^𝑖𝑛_𝑗. (27) The global properties of fluctuating two charge 𝐷₀-𝐷₄ extremal configurations are determined by possible principle minors. The first minor constraint 𝑝₁>0 directly follows from the positivity of the first component of metric tensor: 𝑝₁=𝜋2𝑛₀𝑛₄𝑛₀. (28) The determinant of the metric tensor 𝑝₂∶=𝑔(𝑛₀,𝑛₄) vanishes identically for all allowed values of the parameters. Thus, the leading order large charge extremal black branes having (i) a 𝑛₀ number of 𝐷₀-branes and a 𝑛₄ number of 𝐷₄ or (ii) excited strings with a 𝑛₁ number of windings and a 𝑛_𝑝 number of momenta, where either set of charges forms local coordinates on the state-space manifold, find degenerate intrinsic state-space configurations. For a given configuration entropy 𝑆₀∶=2𝜋𝑐, the constant entropy curve can be depicted as the rectangular hyperbola 𝑛₀𝑛₄=𝑐². (29) The intrinsic state-space configuration depends on the attractor values of the scalar fields which arise from the chosen string compactification. Thus, the possible state-space Ruppenier geometry may become well-defined against further higher derivative 𝛼^′-corrections. In particular, the determinant of the state-space metric tensor may take positive/negative definite values over the domain of brane charges. We will illustrate this point in a bit more detail in the subsequent consideration with a higher number of charges and anticharges.

5.2. Three Charge Configurations

From the consideration of the two derivative Einstein-Hilbert action, [53] shows that the leading order entropy of the three charge 𝐷₁-𝐷₅-𝑃 extremal black holes is 𝑆_micro=2𝜋𝑛₁𝑛₅𝑛_𝑝=𝑆_macro. (30) The concerned components of state-space metric tensor are given in Appendix A. Hereby, it follows further that the local state-space metric constraints are satisfied as 𝑔_{𝑛𝑖𝑛𝑖}>0∀𝑖∈1,5,𝑝∣𝑛_𝑖>0. (31) For distinct 𝑖,𝑗∈{1,5} and 𝑝, the list of relative correlation functions is depicted in Appendix A. Further, we see that the local stabilities pertaining to the lines and two-dimensional surfaces of the state-space manifold are measured as 𝑝₁=𝜋2𝑛₁𝑛₅𝑛_𝑝𝑛₁,𝑝₂𝜋=−²4𝑛₁𝑛²₅𝑛_𝑝𝑛²_𝑝𝑛₁+𝑛³₅. (32) The stability of the entire equilibrium phase-space configurations of the 𝐷₁-𝐷₅-𝑃 extremal black holes is determined by the 𝑝₃∶=𝑔 determinant of the state-space metric tensor: 𝑔1=−2𝜋³𝑛₁𝑛₅𝑛_𝑝^−1/2. (33) The universal nature of statistical interactions and the other properties concerning Maldacena, Strominger, and Witten (MSW) rotating black branes [55] are elucidated by the state-space scalar curvature: 𝑅𝑛₁,𝑛₅,𝑛_𝑝=34𝜋𝑛₁𝑛₅𝑛_𝑝. (34) The constant entropy (or scalar curvature) curve defining the state-space manifold is the higher dimensional hyperbola: 𝑛₁𝑛₅𝑛_𝑝=𝑐², (35) where 𝑐 takes respective values of (𝑐_𝑆,𝑐_𝑅)=(𝑆₀/2𝜋,3/4𝜋𝑅₀). In [12, 14, 17, 18], we have shown that similar results hold for the state-space configuration of the four charge extremal black holes.

6. Nonextremal Black Holes in String Theory

6.1. Four Charge Configurations

The state-space configuration of the nonextremal 𝐷₁--𝐷₅ black holes is considered with nonzero momenta along the clockwise and anticlockwise directions of the Kaluza-Klein compactification circle 𝑆¹. Following [56], the microscopic entropy and the macroscopic entropy match for given total mass and brane charges: 𝑆_micro=2𝜋√𝑛₁𝑛₅𝑛_𝑝+𝑛_𝑝=𝑆_macro. (36) The state-space covariant metric tensor is defined as a negative Hessian matrix of the entropy with respect to the number of 𝐷₁, 𝐷₅ branes {𝑛_𝑖∣𝑖=1,5} and clockwise-anticlockwise Kaluza-Klein momentum charges {𝑛_𝑝,𝑛_𝑝}. Herewith, we find that the components of the metric tensor take elegant forms. The corresponding expressions are given in Appendix B. As in the case of the extremal configurations, the state-space metric satisfies the following constraints: 𝑔_{𝑛𝑖𝑛𝑖}>0,∀𝑖=1,5;𝑔_{𝑛𝑎𝑛𝑎}>0,∀𝑎=𝑝,𝑝. (37)

Furthermore, the scaling relations for distinct 𝑖,𝑗∈{1,5} and 𝑝, concerning the list of relative correlation functions, are offered in Appendix B. In this case, we find that the stability criteria of the possible surfaces and hypersurfaces of the underlying state-space configuration are determined by the positivity of the following principle minors: 𝑝₀=1,𝑝₁=𝜋2𝑛₅𝑛³₁𝑛_𝑝+𝑛_𝑝,𝑝₂=0,𝑝₃1=−2𝑛_𝑝𝜋³√𝑛₁𝑛₅𝑛_𝑝+𝑛_𝑝. (38) The complete local stability of the full nonextremal 𝐷₁--𝐷₅ black brane state-space configuration is ascertained by the positivity of the determinant of the metric tensor: 𝑔𝑛₁,𝑛₅,𝑛_𝑝,𝑛_𝑝1=−4𝜋⁴𝑛_𝑝𝑛_𝑝^3/2𝑛_𝑝+𝑛_𝑝². (39) The global state-space properties concerning the four charge nonextremal 𝐷₁--𝐷₅ black holes are determined by the regularity of the invariant scalar curvature: 𝑅𝑛₁,𝑛₅,𝑛_𝑝,𝑛_𝑝=94𝜋√𝑛₁𝑛₅𝑛_𝑝+𝑛_𝑝⁻⁶𝑓𝑛_𝑝,𝑛_𝑝, (40) where the function 𝑓(𝑛_𝑝,𝑛_𝑝) of two momenta (𝑛_𝑝,𝑛_𝑝) running in opposite directions of the Kaluza-Klein circle 𝑆¹ has been defined as 𝑓𝑛_𝑝,𝑛_𝑝∶=𝑛_𝑝^5/2+10𝑛_𝑝^3/2𝑛_𝑝+5𝑛_𝑝^1/2𝑛_𝑝²+5𝑛²_𝑝𝑛_𝑝^1/2+10𝑛_𝑝𝑛_𝑝^3/2+𝑛_𝑝^5/2. (41) By noticing the Pascal coefficient structure in (41), we see that the above function 𝑓(𝑛_𝑝,𝑛_𝑝) can be factorized as 𝑓𝑛_𝑝,𝑛_𝑝=𝑛_𝑝+𝑛_𝑝⁵. (42) Thus, (40) leads to the following state-space scalar curvature: 𝑅𝑛₁,𝑛₅,𝑛_𝑝,𝑛_𝑝=94𝜋√𝑛₁𝑛₅×1𝑛_𝑝+𝑛_𝑝. (43) In the large charge limit, the nonextremal 𝐷₁--𝐷₅ black branes have a nonvanishing small scalar curvature function on the state-space manifold (𝑀₄,𝑔). This implies an almost everywhere weakly interacting statistical basis. In this case, the constant entropy hypersurface is defined by the curve 𝑐²𝑛₁𝑛₅=𝑛_𝑝+𝑛_𝑝². (44) As in the case of two charge 𝐷₀-𝐷₄ extremal black holes and 𝐷₁-𝐷₅-𝑃 extremal black holes, the constant 𝑐 takes the same value of 𝑐∶=𝑆²₀/4𝜋². For a given state-space scalar curvature 𝑘, the constant state-space curvature curves take the following form: 𝑓𝑛_𝑝,𝑛_𝑝=𝑘√𝑛₁𝑛₅𝑛_𝑝+𝑛_𝑝⁶. (45)

6.2. Six Charge Configurations

We now extrapolate the state-space geometry of four charge nonextremal 𝐷₁--𝐷₅ solutions for nonlarge charges, where we are no longer close to an ensemble of supersymmetric states. In [57], the computation of the entropy of all such special extremal and near-extremal black hole configurations has been considered. The leading order entropy as a function of charges {𝑛_𝑖} and anticharges {𝑚_𝑖} is 𝑆𝑛₁,𝑚₁,𝑛₂,𝑚₂,𝑛₃,𝑚₃∶=2𝜋√𝑛₁+√𝑚₁√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃. (46) For given charges 𝑖,𝑗∈𝐴₁∶={𝑛₁,𝑚₁}; 𝑘,𝑙∈𝐴₂∶={𝑛₂,𝑚₂}; and 𝑚,𝑛∈𝐴₃∶={𝑛₃,𝑚₃}, the intrinsic state-space pair correlations are in precise accordance with the underlying macroscopic attractor configurations which are being disclosed in the special leading order limit of the nonextremal 𝐷₁--𝐷₅ solutions. The components of the covariant state-space metric tensor over generic nonlarge charge domains are not difficult to compute, and, indeed, we have offered their corresponding expressions in Appendix C.

For all finite (𝑛_𝑖,𝑚_𝑖),𝑖=1,2,3, the components involving brane-brane state-space correlations 𝑔_{𝑛𝑖𝑛𝑖} and antibrane-antibrane state-space correlations 𝑔_{𝑚𝑖𝑚𝑖} satisfy the following positivity conditions: 𝑔_{𝑛𝑖𝑛𝑖}>0,𝑔_{𝑚𝑖𝑚𝑖}>0. (47) The distinct {𝑛_𝑖,𝑚_𝑖∣𝑖∈{1,2,3}} describing six charge string theory black holes have three types of relative pair correlation functions. The corresponding expressions of the relative statistical correlation functions are given in Appendix C.

Notice hereby that the scaling relations remain similar to those obtained in the previous case, except that (i) the number of relative correlation functions has been increased, and (ii) the set of cross ratios, namely, {𝑔_𝑖𝑗/𝑔_𝑘𝑙,𝑔_𝑘𝑙/𝑔_𝑚𝑛,𝑔_𝑖𝑗/𝑔_𝑚𝑛} being zero in the previous case, becomes ill-defined for the six charge state-space configurations. Inspecting the specific pair of distinct charge sets 𝐴_𝑖 and 𝐴_𝑗, there are now 24 types of nontrivial relative correlation functions. The set of principle components denominator ratios computed from the above state-space metric tensor reduces to 𝑔_𝑖𝑗𝑔_𝑘𝑘=0,∀𝑖,𝑗,𝑘∈𝑛₁,𝑚₁,𝑛₂,𝑚₂,𝑛₃,𝑚₃. (48) For given 𝑖,𝑗∈𝐴₁∶={𝑛₁,𝑚₁}; 𝑘,𝑙∈𝐴₂∶={𝑛₂,𝑚₂}; 𝑚,𝑛∈𝐴₃∶={𝑛₃,𝑚₃}, and 𝑔_{𝑛𝑖𝑚𝑖}=0, there are the total 15 types of trivial relative correlation functions. There are five such trivial ratios in each family {𝐴_𝑖∣𝑖=1,2,3}. The local stability of the higher charged string theory nonextremal black holes is given by 𝑝₁=𝜋2𝑛₁^3/2√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃,𝑝₂=14𝜋²𝑛₁𝑚₁^3/2√𝑛₂+√𝑚₂²√𝑛₃+√𝑚₃²,𝑝₃=18𝜋³𝑛₁𝑚₁𝑛₂^3/2√𝑚₂√𝑛₃+√𝑚₃³×√𝑛₂+√𝑚₂√𝑛₁+√𝑚₁,𝑝₄=0. (49) The principle minor 𝑝₅ remains nonvanishing for all values of charges on the constituent brane and antibranes. In general, by an explicit calculation, we find that the hyper-surface minor 𝑝₅ takes the following nontrivial value: 𝑝₅1=−8𝜋⁵𝑛₁𝑚₁𝑛₂𝑚₂^3/2𝑛₃√𝑛₁+√𝑚₁³×√𝑛₂+√𝑚₂³√𝑛₃+√𝑚₃³. (50) Specifically, for an identical value of the brane and antibrane charges, the minor 𝑝₅ reduces to 𝑝₅(𝑘)𝜋=−64⁵𝑘^5/2. (51) The global stability on the full state-space configuration is carried forward by computing the determinant of the metric tensor: 𝑔1=−𝜋16𝑛₁𝑚₁𝑛₂𝑚⁶₂𝑛₃𝑚₃^3/2√𝑛₁+√𝑚₁⁴×√𝑛₂+√𝑚₂⁴√𝑛₃+√𝑚₃⁴. (52) The underlying state-space configuration remains nondegenerate for the domain of given nonzero brane antibrane charges, except for extreme values of the brane and antibrane charges {𝑛_𝑖,𝑚_𝑖}, when they belong to the set 𝐵∶=𝑛₁,𝑛₂,𝑛₃,𝑚₁,𝑚₂,𝑚₃∣𝑛_𝑖,𝑚_𝑖=(0,0),(∞,∞),some𝑖, (53) among the given brane-antibrane pairs {(𝑛₁,𝑚₁),(𝑛₂,𝑚₂),(𝑛₃,𝑚₃)}. The component 𝑅_{𝑛1𝑛2𝑚3𝑚4} diverges at the roots of the two variables polynomials defined as the functions of brane and antibrane charges: 𝑓₁𝑛₂,𝑚₂=𝑛⁴₂𝑚³₂+2𝑛₂𝑚₂^7/2+𝑛³₂𝑚⁴₂,𝑓₂𝑛₃,𝑚₃=𝑚₃^9/2𝑛⁴₃+𝑛⁴₃𝑚₃^9/2. (54) However, the component 𝑅_{𝑛3,𝑚3,𝑛3,𝑚3} with an equal number of brane and antibrane charges diverges at a root of a single higher degree polynomial: 𝑓𝑛₁,𝑚₁,𝑛₂,𝑚₂,𝑛₃,𝑚₃∶=𝑛⁴₂𝑚³₂𝑛₃^9/2𝑚⁴₃+𝑛⁴₂𝑚³₂𝑛⁴₃𝑚₃^9/2+2𝑛₂^7/2𝑚₂^7/2𝑛₃^9/2𝑚⁴₃+2𝑛₂^7/2𝑚₂^7/2𝑛⁴₃𝑚₃^9/2+𝑛³₂𝑚⁴₂𝑛₃^9/2𝑚⁴₃+𝑛³₂𝑚⁴₂𝑛⁴₃𝑚₃^9/2. (55) Herewith, from the perspective of state-space global invariants, we focus on the limiting nature of the underlying ensemble. Thus, we may choose the equal charge and anticharge limit by defining 𝑚_𝑖∶=𝑘 and 𝑛_𝑖∶=𝑘 for the calculation of the Ricci scalar. In this case, we find the following small negative curvature scalar: 𝑅(𝑘)=−15116𝜋𝑘^3/2. (56) Further, the physical meaning of taking an equal value of the charges and anticharges lies in the ensemble theory, namely, in the thermodynamic limit, all the statistical fluctuations of the charges and anticharges approach to a limiting Gaussian fluctuations. In this sense, we can take the average over the concerned individual Gaussian fluctuations. This shows that the limiting statistical ensemble of nonextremal nonlarge charge 𝐷₁--𝐷₅ solutions yields an attractive state-space configuration. Finally, such a limiting procedure is indeed defined by considering the standard deviations of the equal integer charges and anticharges, and thus our interest in calculating the limiting Ricci scalar in order to know the nature of the long range interactions underlying in the system.

For a given entropy 𝑆₀, the constant entropy hypersurface is again some nonstandard curve: √𝑛₁+√𝑚₁√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃=𝑐, (57) where the real constant 𝑐 takes the precise value of 𝑆₀/2𝜋.

6.3. Eight Charge Configurations

From the perspective of the higher charged and anticharged black hole configurations in string theory, let us systematically analyze the underlying statistical structures. In this case, the state-space configuration of the nonextremal black hole involves finitely many nontrivially circularly fibered Kaluza-Klein monopoles. In this process, we enlist the complete set of nontrivial relative state-space correlation functions of the eight charged anticharged configurations, with respect to the lower parameter configurations, as considered in [12, 14]. There have been calculations of the entropy of the extremal, near-extremal, and general nonextremal solutions in string theory; see, for instance, [58, 59]. Inductively, the most general charge anticharge nonextremal black hole has the following entropy: 𝑆𝑛₁,𝑚₁,𝑛₂,𝑚₂,𝑛₃,𝑚₃,𝑛₄,𝑚₄=2𝜋⁴_𝑖=1√𝑛_𝑖+√𝑚_𝑖. (58) For the distinct 𝑖,𝑗,𝑘∈{1,2,3,4}, we find that the components of the metric tensor are 𝑔_{𝑛𝑖𝑛𝑖}=𝜋2𝑛_𝑖^3/2_𝑗≠𝑖𝑛_𝑗+𝑚_𝑗,𝑔_{𝑛𝑖𝑛𝑗}𝜋=−2𝑛_𝑖𝑛_𝑗^{1/2𝑖≠𝑘≠𝑗}√𝑛_𝑘+√𝑚_𝑘,𝑔_{𝑛𝑖𝑚𝑖}𝑔=0,_{𝑛𝑖𝑚𝑗}𝜋=−2𝑛_𝑖𝑚_𝑗^{1/2𝑖≠𝑘≠𝑗}√𝑛_𝑘+√𝑚_𝑘,𝑔_{𝑚𝑖𝑚𝑖}=𝜋2𝑚_𝑖^3/2_𝑗≠𝑖𝑛_𝑗+𝑚_𝑗,𝑔_{𝑚𝑖𝑚𝑗}𝜋=−2𝑚_𝑖𝑚_𝑗^{1/2𝑖≠𝑘≠𝑗}√𝑛_𝑘+√𝑚_𝑘. (59) From the above depiction, it is evident that the principle components of the state-space metric tensor {𝑔_{𝑛𝑖𝑛𝑖},𝑔_{𝑚𝑖𝑚𝑖}∣𝑖=1,2,3,4} essentially signify a set of definite heat capacities (or the related compressibilities) whose positivity in turn apprises that the black brane solutions comply with an underlying equilibrium statistical configuration. For an arbitrary number of the branes {𝑛_𝑖} and antibranes {𝑚_𝑖}, we find that the associated state-space metric constraints as the diagonal pair correlation functions remain positive definite. In particular, ∀𝑖∈{1,2,3,4}; it is clear that we have the following positivity conditions: 𝑔_{𝑛𝑖𝑛𝑖}>0∣𝑛_𝑖,𝑚_𝑖>0,𝑔_{𝑚𝑖𝑚𝑖}>0∣𝑛_𝑖,𝑚_𝑖>0. (60) As observed in [12, 14], we find that the ratios of diagonal components vary inversely with a multiple of a well-defined factor in the underlying parameters, namely, the charges and anticharges, which changes under the Gaussian fluctuations, whereas the ratios involving off diagonal components in effect uniquely inversely vary in the parameters of the chosen set 𝐴_𝑖 of equilibrium black brane configurations. This suggests that the diagonal components weaken in a relatively controlled fashion into an equilibrium, in contrast with the off-diagonal components, which vary over the domain of associated parameters defining the 𝐷₁-𝐷₅-𝑃-𝐾𝐾 nonextremal nonlarge charge configurations. In short, we can easily substantiate, for the distinct 𝑥_𝑖∶=(𝑛_𝑖,𝑚_𝑖)∣𝑖∈{1,2,3,4} describing eight (anti)charge string theory black holes, that the relative pair correlation functions have distinct types of relative correlation functions. Apart from the zeros, infinities, and similar factorizations, we see that the nontrivial relative correlation functions satisfy the following scaling relations: 𝑔_{𝑥𝑖𝑥𝑖}𝑔_{𝑥𝑗𝑥𝑗}=𝑥_𝑗𝑥_𝑖^3/2𝑛_𝑗+𝑚_𝑗√𝑛_𝑖+√𝑚_𝑖,𝑔_{𝑥𝑖𝑥𝑗}𝑔_{𝑥𝑘𝑥𝑙}=𝑥_𝑖𝑥_𝑗𝑥_𝑘𝑥_𝑙^−1/2∏_{𝑖≠𝑝≠𝑗}𝑛_𝑝+𝑚_𝑝∏_{𝑘≠𝑞≠𝑙}𝑛_𝑞+𝑚_𝑞,𝑔_{𝑥𝑖𝑥𝑖}𝑔_{𝑥𝑖𝑥𝑘}=−𝑥_𝑘𝑥²_𝑖∏_𝑝≠𝑖𝑛_𝑝+𝑚_𝑝∏_{𝑖≠𝑞≠𝑘}𝑛_𝑞+𝑚_𝑞. (61) As noticed in [12, 14], it is not difficult to analyze the statistical stability properties of the eight charged anticharged nonextremal black holes; namely, we can compute the principle minors associated with the state-space metric tensor and thereby argue that all the principle minors must be positive definite, in order to have a globally stable configuration. In the present case, it turns out that the above black hole is stable only when some of the charges and/or anticharges are held fixed or take specific values such that 𝑝_𝑖>0 for all the dimensions of the state-space manifold. From the definition of the Hessian matrix of the associated entropy concerning the most general nonextremal nonlarge charged black holes, we observe that some of the principle minors 𝑝_𝑖 are indeed nonpositive. In fact, we discover a uniform local stability criteria on the three-dimensional hypersurfaces, two-dimensional surface, and the one-dimensional line of the underlying state-space manifold. In order to simplify the factors of the higher principle, we may hereby collect the powers of each factor (√𝑛_𝑖+√𝑚_𝑖) appearing in the expression of the entropy. With this notation, Appendix D provides the corresponding principle minors for the most general nonextremal nonlarge charged anticharged black hole in string theory involving finitely many nontrivially circularly fibered Kaluza-Klein monopoles.

Notice that the heat capacities, as the diagonal components 𝑔_𝑖𝑖, surface minor 𝑝₂, hypersurface minors 𝑝₃, 𝑝₅, 𝑝₆, and 𝑝₇, and the determinant of the state-space metric tensor, as the highest principle minor 𝑝₈ are examined as the functions of the number of branes 𝑛 and antibranes 𝑚. Thus, they describe the nature of the statistical fluctuations in the vacuum configuration. The corresponding scalar curvature is offered for an equal number of branes and antibranes (𝑛=𝑚), which describes the nature of the long range statistical fluctuations. As per the above evaluation, we have obtained the exact expressions for the components of the metric tensor, principle minors, determinant of the metric tensor, and the underlying scalar curvature of the fluctuating statistical configuration of the eight parameter black holes in string theory. Qualitatively, the local and the global correlation properties of the limiting vacuum configuration can be realized under the statistical fluctuations. The first seven principle minors describe the local stability properties, and the last minor describes the global ensemble stability.

The scalar curvature describes the corresponding phase space stability of the eight parameter black hole configuration. In general, there exists an akin higher degree polynomial equation on which the Ricci scalar curvature becomes null, and exactly on these points the state-space configuration of the underlying nonlarge charge nonextremal eight charge black hole system corresponds to a noninteracting statistical system. In this case, the corresponding state-space manifold (𝑀₈,𝑔) becomes free from the statistical interaction with a vanishing state-space scalar curvature. As in case of the six charge configuration, we find interestingly that there exists an attractive configuration for the equal number of branes 𝑛∶=𝑘 and antibranes 𝑚∶=𝑘. In the limit of a large 𝑘, the corresponding system possesses a small negative value of the state-space scalar curvature: 𝑅(𝑘)=−21132𝜋𝑘². (62) Interestingly, it turns out that the system becomes noninteracting in the limit of 𝑘→∞. For the case of the 𝑛=𝑘=𝑚, we observe that the corresponding principle minors reduce to the following constant values: 𝑝_𝑖⁸_𝑖=1=4𝜋,16𝜋²,32𝜋³,0,−2048𝜋⁵,−16384𝜋⁶,−163840𝜋⁷,−1048576𝜋⁸. (63) In this case, we find that the limiting underlying statistical system remains stable when at most three of the parameters, namely, {𝑛_𝑖=𝑘=𝑚_𝑖}, are allowed to fluctuate. Herewith, we find for the case of 𝑛∶=𝑘 and 𝑚∶=𝑘 that the state-space manifold of the eight parameter brane and antibrane configuration is free from critical phenomena, except for the roots of the determinant. Thus, the regular state-space scalar curvature is comprehensively universal for the nonlarge charge nonextremal black brane configurations in string theory. In fact, the above perception turns out to be justified from the typical state-space geometry, namely, the definition of the metric tensor as the negative Hessian matrix of the duality invariant expression of the black brane entropy. In this case, we may nevertheless easily observe, for a given entropy 𝑆₀, that the constant entropy hypersurface is given by the following curve: √𝑛₁+√𝑚₁√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃√𝑛₄+√𝑚₄=𝑐, (64) where 𝑐 is a real constant taking the precise value of 𝑆₀/2𝜋. Under the vacuum fluctuations, the present analysis indicates that the entropy of the eight parameter black brane solution defines a nondegenerate embedding in the viewpoints of intrinsic state-space geometry. The above state-space computations determine an intricate set of statistical properties, namely, pair correlation functions and correlation volume, which reveal the possible nature of the associated parameters prescribing an ensemble of microstates of the dual conformal field theory living on the boundary of the black brane solution. For any black brane configuration, the above computation hereby shows that we can exhibit the state-space geometric acquisitions with an appropriate comprehension of the required parameters, for example, the charges and anticharges {𝑛_𝑖,𝑚_𝑖}, which define the coordinate charts. From the consideration of the state-space geometry, we have analyzed state-space pair correlation functions and the notion of stability of the most general nonextremal black hole in string theory. From the perspective of the intrinsic Riemannian geometry, we find that the stability of these black branes has been divulged from the positivity of principle minors of the space-state metric tensor.

Herewith, we have explicitly extended the state-space analysis for the four charge and four anticharge nonextremal black branes in string theory. The present consideration of the eight parameter black brane configurations, where the underlying leading order statistical entropy is written as a function of the charges {𝑛_𝑖} and anticharges {𝑚_𝑖}, describes the stability properties under the Gaussian fluctuations. The present consideration includes all the special cases of the extremal and near-extremal configurations with a fewer number of charges and anticharges. In this case, we obtain the standard pattern of the underlying state-space geometry and constant entropy curve as that of the lower parameter nonextremal black holes. The local coordinate of the state-space manifold involves four charges and four anticharges of the underlying nonextremal black holes. In fact, the conclusion to be drawn remains the same, as the underlying state-space geometry remains well-defined as an intrinsic Riemannian manifold 𝑁∶=𝑀₈⧵𝐵, where 𝐵 is the set of roots of the determinant of the metric tensor. In particular, the state-space configuration of eight parameter black brane solutions remains nondegenerate for various domains of nonzero brane antibrane charges, except for the values, when the brane charges {𝑛_𝑖} and antibrane charges {𝑚_𝑖} belong to the set 𝐵∶=𝑛₁,𝑛₂,𝑛₃,𝑛₄,𝑚₁,𝑚₂,𝑚₃,𝑚₄∣𝑛_𝑖,𝑚_𝑖=(0,0),(∞,∞), (65) for a given brane-antibrane pair, 𝑖∈{1,2,3,4}. Our analysis indicates that the leading order statistical behavior of the black brane configurations in string theory remains intact under the inclusion of the Kaluza-Klein monopoles. In short, we have considered the eight charged anticharged string theory black brane configuration and analyzed the state-space pair correlation functions, relative scaling relations, stability conditions, and the corresponding global properties. Given a general nonextremal black brane configuration, we have exposed (i) for what conditions the considered black hole configuration is stable, (ii) how its state-space correlations scale in terms of the numbers of branes and antibranes.

7. Conclusion and Outlook

The Ruppenier geometry of two charge leading order extremal black holes remains flat or ill-defined. Thus, the statistical systems are, respectively, noninteracting or require higher derivative corrections. However, an addition of the third brane charge and other brane and antibrane charges indicates an interacting statistical system. The statistical fluctuations in the canonical ensemble lead to an interacting statistical system, as the scalar curvature of the state-space takes a nonzero value. We have explored the state-space geometric description of the charged extremal and associated charged anticharged nonextremal black holes in string theory.

Our analysis illustrates that the stability properties of the specific state-space hypersurface may exactly be exploited in general. The definite behavior of the state-space properties, as accounted in the specific cases, suggests that the underlying hypersurfaces of the state-space configuration include the intriguing mathematical feature. Namely, we find well-defined stability properties for the generic extremal and nonextremal black brane configurations, except for some specific values of the charges and anticharges. With and without the large charge limit, we have provided explicit forms of the higher principle minors of the state-space metric tensor for various charged, anticharged, extremal and nonextremal black holes in string theory. In this concern, the state-space configurations of the string theory black holes are generically well-defined and indicate an interacting statistical basis. Interestingly, we discover the state-space geometric nature of all possible general black brane configurations. From the very definition of the intrinsic metric tensor, the present analysis offers a definite stability character of string theory vacua.

Significantly, we notice that the related principle minors and the invariant state-space scalar curvature classify the underlying statistical fluctuations. The scalar curvature of a class of extremal black holes and the corresponding nonextremal black branes is everywhere regular with and without the stringy 𝛼^′-corrections. A nonzero value of the state-space scalar curvature indicates an interacting underlying statistical system. We find that the antibrane corrections modify the state-space curvature, but do not induce phase transitions. In the limit of an extremal black hole, we construct the intrinsic geometric realization of a possible thermodynamic description at the zero temperature.

Importantly, the notion of the state-space of the considered black hole follows from the corresponding Wald and Cardy entropies. The microscopic and macroscopic entropies match in the large charge limit. From the perspective of statistical fluctuations, we anticipate the intrinsic geometric realization of two point local correlation functions and the corresponding global correlation length of the underlying conformal field theory configurations. In relation to the gauge-gravity correspondence and extremal black holes, our analysis describes state-space geometric properties of the corresponding boundary gauge theory.

General Remarks. For distinct {𝑖,𝑗}, the state-space pair correlations of an extremal configurations scale as 𝑔_𝑖𝑖𝑔_𝑗𝑗=𝑋_𝑗𝑋_𝑖²,𝑔_𝑖𝑗𝑔_𝑖𝑖𝑋=−^𝑖𝑋_𝑗. (66) In general, the black brane configurations in string theory can be categorized as per their state-space invariants. The underlying subconfigurations turn out to be well-defined over possible domains, whenever there exists a respective set of nonzero state-space principle minors. The underlying full configuration turns out to be everywhere well-defined, whenever there exists a nonzero state-space determinant. The underlying configuration corresponds to an interacting statistical system, whenever there exists a nonzero state-space scalar curvature. The intrinsic state-space manifold of extremal/nonextremal and supersymmetric/nonsupersymmetric string theory black holes may intrinsically be described by an embedding: 𝑀_(𝑛),𝑔↪𝑀_(𝑛+1),𝑔. (67) The extremal state-space configuration may be examined as a restriction to the full counting entropy with an intrinsic state-space metric tensor 𝑔↦𝑔|_{𝑟+=𝑟−}. Furthermore, the state-space configurations of the supersymmetric black holes may be examined as the BPS restriction of the full space of the counting entropy with an understanding that the intrinsic state-space metric tensor is defined as 𝑔∶=𝑔|_𝑀=𝑀0. From the perspective of string theory, the restrictions 𝑟₊=𝑟₋ and 𝑀=𝑀₀(𝑃_𝑖,𝑄_𝑖) should be understood as the fact that it has been applied to an assigned entropy of the nonextremal/nonsupersymmetric (or nearly extremal/nearly supersymmetric) black brane configuration. This allows one to compute the fluctuations in ADM mass of the black hole. In the viewpoint of the present research on the state-space geometry, it is worth mentioning that the dependence of the mass to the entropy of a nonextremal black hole comes from the contribution of the antibranes, see, for instance, Section 4.5, and so we may examine the corresponding Weinhold chemical geometry, as mentioned in Section 4.3.

Future Directions and Open Issues. The state-space instabilities and their relation to the dual microscopic conformal field theories could open up a number of new realizations. The state-space perspective includes the following issues.(i)Multicenter Gibbons-Hawking solutions [69, 70] with generalized base space manifolds having a mixing of positive and negative residues, see [71, 72] for a perspective development of state-space geometry by invoking the role of foaming of black holes and plumbing the Abyss for the microstates counting of black rings.(ii)Dual conformal field theories and string duality symmetries, see [61] for a quantum mechanical perspective of superconformal black holes and [73, 74] for the origin of gravitational thermodynamics and the role of giant gravitons in conformal field theory.(iii)Stabilization against local and/or global perturbations, see [75–80] for black brane dynamics, stability, and critical phenomena. Thus, the consideration of state-space geometry is well-suited for examining the domain of instability. This includes Gregory-Laflamme (GL) modes, chemical potential fluctuations, electric-magnetic charges and dipole charges, rotational fluctuations, and the thermodynamic temperature fluctuations for the near-extremal and nonextremal black brane solutions. We leave this perspective of the state-space geometry open for a future research.

In general, various 𝐷 dimensional black brane configurations, see, for instance, [75–80] for black rings in 𝐷>5 spacetime dimensions with 𝑆¹×𝑆^𝐷−3 horizon topology, and the higher horizon topologies, for example, 𝑆¹×𝑆¹×𝑆², 𝑆³×𝑆³, and so forth offer a platform to extend the consideration of the state-space geometry.

On the other hand, the bubbling black brane solutions, namely, Lin, Lunin, and Maldacena (LLM) geometries [81], are interesting from the perspective of Mathur’s Fuzzball conjecture(s). From the perspective of the generalized hyper-Kähler manifolds, Mathur’s conjecture [82–85] reduces to classifying and counting asymptotically flat four-dimensional hyper Kähler manifolds [71] which have moduli regions of uniform signature (+,+,+,+) and (−,−,−,−).

Finally, the new physics at the length of the Planck scale anticipates an analysis of the state-space configurations. In particular, it materializes that the state-space geometry may be explored with the parameters of the foam geometries [71], and the corresponding empty space virtual black holes, see [81] for the notion of bubbling AdS space and 1/2 BPS geometries. In such cases, the local and global statistical correlations, among the parameters of the microstates of the black hole conformal field theory [60, 61], would involve the foams of two spheres. From the perspective of the string theory, the present exploration thus opens up an avenue for learning new insights into the promising structures of the black brane space-time configurations at very small scales.

Appendices

In these appendices, we provide explicit forms of the state-space correlation arising from the metric tensor of the charged (non)extremal (non)large black holes in string theory. In fact, our analysis illustrates that the stability properties of the specific state-space hypersurface may exactly be exploited in general. The definite behavior of state-space properties, as accounted in the concerned main sections, suggests that the various intriguing hypersurfaces of the state-space configuration include the nice feature that they do have definite stability properties, except for some specific values of the charges and anticharges.

As mentioned in the main sections, these configurations are generically well-defined and indicate an interacting statistical basis. Herewith, we discover that the state-space geometry of the general black brane configurations in string theory indicates the possible nature of the underlying statistical fluctuations. Significantly, we notice from the very definition of the intrinsic metric tensor that the related statistical pair correlation functions and relative statistical correlation functions take the following exact expressions.

A. Correlations for Three Charge Configurations

Following the notion of the fluctuations, we see from the Hessian of the entropy equation (30) that the components of state-space metric tensor are 𝑔_𝑛1𝑛1=𝜋2𝑛₁𝑛₅𝑛_𝑝𝑛₁,𝑔_𝑛1𝑛5𝜋=−2𝑛_𝑝𝑛₁𝑛₅,𝑔_{𝑛1𝑛𝑝}𝜋=−2𝑛₅𝑛₁𝑛_𝑝,𝑔_𝑛5𝑛5=𝜋2𝑛₅𝑛₁𝑛_𝑝𝑛₅,𝑔_{𝑛5𝑛𝑝}𝜋=−2𝑛₁𝑛₅𝑛_𝑝,𝑔_{𝑛𝑝𝑛𝑝}=𝜋2𝑛_𝑝𝑛₁𝑛₅𝑛_𝑝. (A.1)

For distinct 𝑖,𝑗∈{1,5} and 𝑝, the list of relative correlation functions follows the scaling relations: 𝑔_𝑖𝑖𝑔_𝑗𝑗=𝑛_𝑗𝑛_𝑖²,𝑔_𝑖𝑖𝑔_𝑝𝑝=𝑛_𝑝𝑛_𝑖²,𝑔_𝑖𝑖𝑔_𝑖𝑗𝑛=−^𝑗𝑛_𝑖,𝑔_𝑖𝑖𝑔_𝑖𝑝𝑛=−^𝑝𝑛_𝑖,𝑔_𝑖𝑝𝑔_𝑗𝑝=𝑛_𝑗𝑛_𝑖,𝑔_𝑖𝑖𝑔_𝑗𝑝𝑛=−^𝑗𝑛_𝑝𝑛²_𝑖,𝑔_𝑖𝑝𝑔_𝑝𝑝𝑛=−^𝑝𝑛_𝑖,𝑔_𝑖𝑗𝑔_𝑖𝑝=𝑛_𝑝𝑛_𝑗,𝑔_𝑖𝑗𝑔_𝑝𝑝𝑛=−^2𝑝𝑛_𝑖𝑛_𝑗. (A.2)

B. Correlations for Four Charge Configurations

For the given entropy as in (36), we find that the components of the metric tensor are 𝑔_𝑛1𝑛1=𝜋2𝑛₅𝑛³₁𝑛_𝑝+𝑛_𝑝,𝑔_𝑛1𝑛5𝜋=−2√𝑛₁𝑛₅𝑛_𝑝+𝑛_𝑝,𝑔_{𝑛1𝑛𝑝}𝜋=−2𝑛₅𝑛₁𝑛_𝑝,𝑔_{𝑛1𝑛𝑝}𝜋=−2𝑛₅𝑛₁𝑛_𝑝,𝑔_𝑛5𝑛5=𝜋2𝑛₁𝑛³₅𝑛_𝑝+𝑛_𝑝,𝑔_{𝑛5𝑛𝑝}𝜋=−2𝑛₁𝑛₅𝑛_𝑝,𝑔_{𝑛5𝑛𝑝}𝜋=−2𝑛₁𝑛₅𝑛_𝑝,𝑔_{𝑛𝑝𝑛𝑝}=𝜋2𝑛₁𝑛₅𝑛³_𝑝,𝑔_{𝑛𝑝𝑛𝑝}=0,𝑔_{𝑛𝑝𝑛𝑝}=𝜋2𝑛₁𝑛₅𝑛_𝑝³. (B.1)

For distinct 𝑖,𝑗∈{1,5}, and 𝑘,𝑙∈{𝑝,𝑝} describing four charge nonextremal 𝐷₁--𝐷₅-𝑃-𝑃 black holes, the statistical pair correlations consist of the following scaling relations: 𝑔_𝑖𝑖𝑔_𝑗𝑗=𝑛_𝑗𝑛_𝑖²,𝑔_𝑖𝑖𝑔_𝑘𝑘=𝑛_𝑘𝑛²_𝑖√𝑛_𝑘𝑛_𝑝+𝑛_𝑝,𝑔_𝑖𝑖𝑔_𝑖𝑗𝑛=−^𝑗𝑛_𝑖,𝑔_𝑖𝑖𝑔_𝑖𝑘√=−𝑛_𝑘𝑛_𝑖𝑛_𝑝+𝑛_𝑝,𝑔_𝑖𝑘𝑔_𝑗𝑘=𝑛_𝑗𝑛_𝑖,𝑔_𝑖𝑖𝑔_𝑗𝑘𝑛=−^𝑗𝑛²_𝑖√𝑛_𝑘𝑛_𝑝+𝑛_𝑝,𝑔_𝑖𝑘𝑔_𝑘𝑘𝑛=−^𝑘𝑛_𝑖,𝑔_𝑖𝑗𝑔_𝑖𝑘=√𝑛_𝑘𝑛_𝑗𝑛_𝑝+𝑛_𝑝,𝑔_𝑖𝑗𝑔_𝑘𝑘𝑛=−^𝑘𝑛_𝑖𝑛_𝑗√𝑛_𝑘𝑛_𝑝+𝑛_𝑝. (B.2) Notice that the list of other mixed relative correlation functions concerning the nonextremal 𝐷₁--𝐷₅-𝑃-𝑃 black holes read as 𝑔_𝑖𝑘𝑔_𝑖𝑙=𝑛_𝑙𝑛_𝑘,𝑔_𝑖𝑘𝑔_𝑗𝑙=𝑛_𝑗𝑛_𝑖𝑛_𝑙𝑛_𝑘,𝑔_𝑘𝑙𝑔_𝑖𝑗𝑔=0,_𝑘𝑙𝑔_𝑖𝑖𝑔=0,^𝑘𝑘𝑔_𝑙𝑙=𝑛_𝑙𝑛_𝑘^3/2,𝑔_𝑘𝑙𝑔_𝑘𝑘=0. (B.3)

C. Correlations for Six Charge Configurations

Over generic nonlarge charge domains, we find from the entropy equation (46) that the components of the covariant state-space metric tensor are given by the following expressions: 𝑔_𝑛1𝑛1=𝜋2𝑛₁^3/2√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃,𝑔_𝑛1𝑚1𝑔=0,_𝑛1𝑛2𝜋=−2√𝑛₁𝑛₂√𝑛₃+√𝑚₃,𝑔_𝑛1𝑚2𝜋=−2√𝑛₁𝑚₂√𝑛₃+√𝑚₃,𝑔_𝑛1𝑛3𝜋=−2√𝑛₁𝑛₃√𝑛₂+√𝑚₂,𝑔_𝑛1𝑚3𝜋=−2√𝑛₁𝑚₃√𝑛₂+√𝑚₂,𝑔_𝑚1𝑚1=𝜋2𝑚₁^3/2√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃,𝑔_𝑚1𝑛2𝜋=−2√𝑚₁𝑛₂√𝑛₃+√𝑚₃,𝑔_𝑚1𝑚2𝜋=−2√𝑚₁𝑚₂√𝑛₃+√𝑚₃,𝑔_𝑚1𝑛3𝜋=−2√𝑚₁𝑛₃√𝑛₂+√𝑚₂,𝑔_𝑚1𝑚3𝜋=−2√𝑚₁𝑚₃√𝑛₂+√𝑚₂,𝑔_𝑛2𝑛2=𝜋2𝑛₂^3/2√𝑛₁+√𝑚₁√𝑛₃+√𝑚₃,𝑔_𝑛2𝑚2=0,𝑔_𝑛2𝑛3𝜋=−2√𝑛₂𝑛₃√𝑛₁+√𝑚₁𝑔_𝑛2𝑚3𝜋=−2√𝑛₂𝑚₃√𝑛₁+√𝑚₁,𝑔_𝑚2𝑚2=𝜋2𝑚₂^3/2√𝑛₁+√𝑚₁√𝑛₃+√𝑚₃,𝑔_𝑚2𝑛3𝜋=−2√𝑚₂𝑛₃√𝑛₁+√𝑚₁,𝑔_𝑚2𝑚3𝜋=−2√𝑚₂𝑚₃√𝑛₁+√𝑚₁,𝑔_𝑛3𝑛3=𝜋2𝑛₃^3/2√𝑛₁+√𝑚₁√𝑛₂+√𝑚₂,𝑔_𝑛3𝑚3𝑔=0,_𝑚3𝑚3=𝜋2𝑚₃^3/2√𝑛₁+√𝑚₁√𝑛₂+√𝑚₂. (C.1)

In this case, from the definition of the relative statistical correlation functions, for 𝑖,𝑗∈{𝑛₁,𝑚₁}, and 𝑘,𝑙∈{𝑛₂,𝑚₂}, the relative correlation functions satisfy the following scaling relations: 𝑔_𝑖𝑖𝑔_𝑗𝑗=𝑗𝑖^3/2,𝑔_𝑖𝑖𝑔_𝑘𝑘=𝑘𝑖^3/2√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃,𝑔_𝑖𝑗𝑔_𝑖𝑖𝑔=0,_𝑖𝑖𝑔_𝑖𝑘√=−𝑘𝑖√𝑛₂+√𝑚₂,𝑔_𝑖𝑘𝑔_𝑗𝑘=𝑗𝑖,𝑔_𝑖𝑖𝑔_𝑗𝑘=−𝑗𝑘𝑖^3/2√𝑛₂+√𝑚₂,𝑔_𝑘𝑘𝑔_𝑖𝑘√=−𝑖𝑘√𝑛₂+√𝑚₂,𝑔_𝑖𝑗𝑔_𝑖𝑘𝑔=0,^𝑖𝑗𝑔_𝑘𝑘=0. (C.2) The other concerned relative correlation functions are 𝑔_𝑖𝑘𝑔_𝑖𝑙=𝑙𝑘,𝑔_𝑖𝑘𝑔_𝑗𝑙=𝑗𝑙,𝑔𝑖𝑘𝑔^𝑖𝑗_𝑘𝑙𝑔=n.d.,_𝑘𝑙𝑔_𝑖𝑖𝑔=0,^𝑘𝑘𝑔_𝑙𝑙=𝑙𝑘^3/2,𝑔_𝑘𝑙𝑔_𝑘𝑘=0. (C.3) For 𝑘,𝑙∈{𝑛₂,𝑚₂}, and 𝑚,𝑛∈{𝑛₃,𝑚₃}, we have 𝑔_𝑘𝑘𝑔_𝑚𝑚=𝑚𝑘^3/2√𝑛₃+√𝑚₃√𝑛₂+√𝑚₂,𝑔_𝑘𝑙𝑔_𝑘𝑘𝑔=0,^𝑘𝑘𝑔_𝑘𝑚√=−𝑚𝑘√𝑛₃+√𝑚₃,𝑔_𝑘𝑚𝑔_𝑙𝑚=𝑙𝑘,𝑔_𝑘𝑘𝑔_𝑙𝑚√=−𝑙𝑚𝑘^3/2√𝑛₃+√𝑚₃,𝑔_𝑚𝑚𝑔_𝑘𝑚√=−𝑘𝑚√𝑛₂+√𝑚₂,𝑔_𝑘𝑙𝑔_𝑘𝑚𝑔=0,^𝑘𝑙𝑔_𝑚𝑚=0. (C.4) The other concerned relative correlation functions are 𝑔_𝑘𝑚𝑔_𝑘𝑛=𝑛𝑚,𝑔_𝑘𝑚𝑔_𝑙𝑛=𝑙𝑛,𝑔𝑘𝑚𝑔^𝑘𝑙_𝑚𝑛𝑔=n.d.,_𝑚𝑛𝑔_𝑘𝑘𝑔=0,^𝑚𝑚𝑔_𝑛𝑛=𝑛𝑚^3/2,𝑔_𝑚𝑛𝑔_𝑚𝑚=0. (C.5) However, for 𝑖,𝑗∈{𝑛₁,𝑚₁}, and 𝑚,𝑛∈{𝑛₃,𝑚₃}, we have 𝑔_𝑖𝑖𝑔_𝑚𝑚=𝑚𝑖^3/2√𝑛₃+√𝑚₃√𝑛₁+√𝑚₁,𝑔_𝑖𝑗𝑔_𝑖𝑖𝑔=0,_𝑖𝑖𝑔_𝑖𝑚√=−𝑚𝑖√𝑛₃+√𝑚₃,𝑔_𝑖𝑚𝑔_𝑗𝑚=𝑗𝑖,𝑔_𝑖𝑖𝑔_𝑗𝑚=−𝑗𝑚𝑖^3/2√𝑛₃+√𝑚₃,𝑔_𝑚𝑚𝑔_𝑖𝑚√=−𝑖𝑚√𝑛₁+√𝑚₁,𝑔_𝑖𝑗𝑔_𝑖𝑚𝑔=0,^𝑖𝑗𝑔_𝑚𝑚𝑔=0,^𝑖𝑚𝑔_𝑖𝑛=𝑛𝑚,𝑔_𝑖𝑚𝑔_𝑗𝑛=𝑗𝑛,𝑔𝑖𝑚𝑔^𝑖𝑗_𝑚𝑛𝑔=n.d.,_𝑚𝑛𝑔_𝑖𝑖𝑔=0,^𝑚𝑛𝑔_𝑚𝑚=0. (C.6)

D. Principle Minors for Eight Charge Configurations

For the entropy equation (58) of the most general nonextremal nonlarge charged anticharged black hole in string involving finitely many nontrivially circularly fibered Kaluza-Klein monopoles, the principle minors take the following expressions: 𝑝₁=𝜋2𝑛₁^3/2√𝑛₂+√𝑚₂√𝑛₃+√𝑚₃√𝑛₄+√𝑚₄,𝑝₂=𝜋²4𝑛₁𝑚₁^3/2√𝑛₂+√𝑚₂²√𝑛₃+√𝑚₃²×√𝑛₄+√𝑚₄²,𝑝₃=𝜋³8𝑛₁𝑚₁𝑛₂^3/2√𝑛₃+√𝑚₃³√𝑛₄+√𝑚₄³×√𝑛₂+√𝑚₂√𝑚₂√𝑛₁+√𝑚₁,𝑝₄𝑝=0,₅𝜋=−⁵8𝑛₁𝑛₂𝑚₂𝑚₁^3/2𝑛₃√𝑛₂+√𝑚₂³×√𝑛₃+√𝑚₃³√𝑛₄+√𝑚₄⁵√𝑛₁+√𝑚₁³,𝑝₆𝜋=−⁶16𝑛₁𝑛₂𝑚₁𝑚₂𝑛₃𝑚₃^3/2√𝑛₂+√𝑚₂⁴×√𝑛₃+√𝑚₃⁴√𝑛₄+√𝑚₄⁶√𝑛₁+√𝑚₁⁴,𝑝₇𝜋=−⁷32𝑛₁𝑚₁𝑛₂𝑚₂𝑛₃𝑚₃𝑛₄^3/2√𝑛₂+√𝑚₂⁵×√𝑛₃+√𝑚₃⁵√𝑛₄+√𝑚₄⁵4√𝑛₄+√𝑚₄×√𝑛₁+√𝑚₁⁵,𝑝₈𝜋=−⁸16∏⁴_𝑖=1𝑛_𝑖𝑚_𝑖^3/2√𝑛₂+√𝑚₂⁶×√𝑛₃+√𝑚₃⁶√𝑛₄+√𝑚₄⁶√𝑛₁+√𝑚₁⁶. (D.1)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work has been supported in part by the European Research Council Grant no. 226455, “SUPERSYMMETRY, QUANTUM GRAVITY AND GAUGE FIELDS (SUPERFIELDS).” Bhupendra Nath Tiwari would like to thank Prof. V. Ravishankar for his support and encouragements towards the research in string theory. This work was conducted during the period Bhupendra Nath Tiwari served as a postdoctoral research fellow at the INFN-Laboratori Nazionali di Frascati, Roma, Italy.