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

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 new perspective of black hole vacuum fluctuations. For a given black hole entropy, we explicate the intrinsic state-space 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, \textit{viz.}, 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. Keywords: Intrinsic Geometry; String Theory; Physics of black holes; Classical black holes; Quantum aspects of black holes, evaporation, thermodynamics; Higher-dimensional black holes, black strings, and related objects; Statistical Fluctuation; Flow Instability. PACS: 02.40.Ky; 11.25.-w; 04.70.-s; 04.70.Bw; 04.70.Dy; 04.50.Gh; 5.40.-a; 47.29.Ky


Introduction
14. From the perspective of statistical configurations, we study stability properties of the charged and anticharged black holes in string theory. Specifically, we illustrate that the components of the vacuum fluctuations define a set of local pair correlations against the parameters, e.g., charges, anticharges, mass and angular momenta, if any.
15. Our consideration follows from the notion of the thermodynamic geometry, mainly introduced by Weinhold [1] and Ruppeiner [2,3]. Importantly, our framework provides a simple platform to geometrically understand the nature of local statistical pair correlations and underlying global statistical structures pertaining to the vacuum phase transitions.
16. In diverse contexts, this perspective offers an understanding of the phase structures of mixture of gases, black hole configurations [4,5], strong interactions, e.g., hot QCD [6], quarkonium configurations [7] and some other systems as well.
17. The main purpose of the present study is to determine the state-space properties of extremal and non-extremal black hole configurations in string theory, in general.
18. String theory, as the most promising framework to understand all possible fundamental interactions, celebrates the physics of black holes, both at the zero and non-zero temperature domains. Our consideration hereby plays a crucial role in understanding the possible phases and statistical stability of the string theory vacua.

Motivations from String Theory
19. N = 2 SUGRA is a low energy limit of the Type II string theory, admitting extremal black hole solutions with the zero Hawking temperature and a non-zero macroscopic entropy.
20. The entropy depends on a large number of scalar moduli arising from the compactification of the 10 dimension theory down to the 4 dimensional physical spacetime.
22. The macroscopic entropy exhibits a fixed point behavior under the radial flow of the scalar fields. The attractor mechanism, as introduced by Ferrara-Kallosh-Strominger, "N=2 Extremal Black Holes", Phys. Rev. D 52 (1995) 5412-5416, arXiv:hep-th/9508072, requires a validity from the microscopic/ statistical basis of the entropy.
23. In this work, we explore attractor fixed point structures in relation with the statistical properties and intrinsic state-space configurations.
24. We provide the statistical understanding to attractor mechanism, moduli space geometry and explain the vacuum fluctuations of black branes.

Definition of State-space Geometry
25. For any thermodynamic system, there exist equilibrium thermodynamic states given by the maxima of the entropy. These states may be represented by points on the state-space.
26. 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, [Ruppeiner, PRD 1978 ].
27. The invariant distance between two arbitrary equilibrium states is inversely proportional to the fluctuations connecting the two states. In particular, "less probable fluctuation" means "states are far apart".
28. For a given set of such states {X i }, the state-space metric tensor is defined by 29. A physical front of Eq. (1) can be given as follows: 30. Up to the second order, the Taylor expansion of the entropy S(X 1 , X 2 , . . . , X n ) gives where is called extended Ruppenier state-space metric tensor. As the limit, the relative coordinates ∆X i are defined as ∆X i := X i − X i 0 , for given {X i 0 } ∈ M n . 31. The probability distribution in the Gaussian approximation has the form: 32. With the normalization: ∫ ∏ i dX i P (X 1 , X 2 , . . . , X n ) = 1, we examine the nature of where g ij is defined as the inner product g( ∂ ∂X i , ∂ ∂X j ) on the tangent space T (M n )×T (M n ) with g(X) := g ij (7) as the determinant of the corresponding matrix [g ij ] n×n . For a given state-space M n , we shall think {dX i } n i=1 as the basis of the cotangent space T (M n ).
33. In subsequent analysis, we shall chose a neutral vacuum with X i 0 = 0. 34. A "good" question is to ask the following: Does Eqn. (4)  36. As the maxima of their macroscopic entropy S(q, p), the next step is to examine the statistical fluctuations about attractor fixed point configuration of the extremal black hole.
37. Later on, we shall analyze the state-space geometry of non-extremal counterparts. We have shown that the state-space correlations now modulate relatively swiftly to an equilibrium statistical basis than the corresponding extremal solutions.

Statistical Fluctuations
38. The Ruppenier metric on the state-space (M 2 , g) of two charge black hole is defined by 39. The Christoffel connections on the (M 2 , g) are defined by 40. The only non-zero component of the Riemann curvature tensor is where and 41. The scalar curvature and corresponding R ijkl of the two dimensional intrinsic state-space manifold (M 2 (R), g) is given by

Stability Conditions
42. For a given set of state-space variables {X 1 , X 2 , . . . , X n }, the local stability condition of the underlying statistical configuration demands 2, . . . , n} (14) 43. 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 comprehensibilities) whose positivity apprises that the black hole solution comply an underlying locally equilibrium statistical configuration.
44. The positivity of the principle components of the state-space metric tensor is not sufficient to insure the global stability of the chosen configuration, and thus one may only achieves a locally equilibrium statistical system.
45. Global stability condition constraint over 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, [Ruppeiner, RMP 1995 ].

Long Range Correlations
47. The thermodynamic scalar curvature of the state-space manifold is proportional to the correlation volume. Physically, the scalar curvature signifies an existence of interaction(s) in the underlying statistical system.
48. Ruppenier has in particular noticed for the black holes in general relativity that the scalar curvature where d is spatial dimension of the statistical system and the ξ fixes the physical scale, [Ruppeiner, RMP 1995 ].
49. The limit R(X) −→ ∞ indicates existence of certain critical points or phase transitions in the underlying statistical system. 50. "All the statistical degrees of freedom of a black hole live on the black hole event horizon" signifies that the scalar curvature indicates an average number of correlated Plank areas on the event horizon of the black hole, [Ruppeiner, PRD 1978 ]. 52. Ruppenier has further conjectured that (a) The zero state-space scalar curvature indicates certain bits of information on the event horizon, fluctuating independently of the each other.
(b) The diverging scalar curvature signals a phase transition, indicating an ensemble of highly correlated pixels of informations.

Extremal Black Holes
53. State-space of extremal (supersymmetric) black holes is a reduced phase-space comprising of the states respecting the extremality (BPS) condition.
54. The state-spaces of the extremal black holes possess an intrinsic geometric description.
55. Our intrinsic geometric analysis offers a possible zero temperature characterization of the limiting extremal black brane attractors.
56. From the perspective gauge/ gravity correspondence, we may think that the existence of state-space geometry could be relevant to the boundary gauge theories, namely, an ensemble of CFT states is parametrized by finitely many charges.

Non-extremal Black Holes
57. We shall analyze the state-space geometry of non-extremal black holes by an addition of the anti-brane charge(s) to the entropy of the corresponding extemal black holes.
58. To interrogate the stability of a chosen black hole system, we shall investigate the question that the underlying metric g ij (X i ) = −∂ i ∂ j S(X 1 , X 2 , . . . , X n ) should be a non-degenerate state-space manifold.
59. The exact dependence varies from case to case. In the next section, we shall proceed our analysis with an increasing number of the brane charges and antibrane charges.

Chemical Geometry
60. The thermodynamic configurations of non-extremal black holes in string theory with small statistical fluctuations in a "canonical" ensemble are stable if 61. The thermal fluctuations of non-extremal black holes, when considered in the canonical ensemble, give a closer approximation to the microcanonical entropy 62. In Eq. (18), the S 0 is the entropy in "canonical" ensemble and C is the specific heat of black hole statistical configuration.

Physics of Correlation
66. Geometrically, the positivity of the heat capacity C > 0 turns out to be the condition that g ij > 0. In many cases, this restriction on the parameters corresponds to the situation away from the extremality condition r + = r − . 67. Far from the extremality condition, even at the zero antibrane charge (or angular momentum), we find that there is a finite value of the state-space scalar curvature, unlike the non-rotating or only brane charged extremal configurations.
68. The Ruppenier geometry of the two charge extremal configurations turns out to be flat. So, the Einstein-Hilbert contributions lead to a non interacting statistical system. Some two derivative black hole configurations turn out to be ill-defined, as well.
69. The determinant(s) of the state-space tensor should be positive definite. If not, the configuration requires further higher order corrections ∈ {stringy, quantum}.
70. For non-extremal black branes, the global effects arise from the nature of the state-space scalar curvature R(S(X 1 , X 2 , . . . , X n )), and in fact, the statistical signature is kept intact under the limit of the extremality.
71. Given a non-extremal configuration, we find that R(S(X 1 , X 2 , . . . , X n ))| no antichagre = 0 gives statistical stability bound(s), and thus the state-space analysis offers sensible domain for the parameters of the black hole(s).

State-space Geometry of Extremal Black Holes
72. At the two derivative Einstein-Hilbert level, Ref. [Strominger and Vafa: arXiv:hepth/9601029v2 ] shows that the leading order entropy of the three charge D 1 -D 5 -P extremal black holes is 73. The components of state-space metric tensor are 74. For distinct i, j ∈ {1, 5} and p, list of relative correlation functions follow scalings 75. The local stabilities along the lines and on two dimensional surfaces of the state-space manifold are simply measured by 76. Local stability of the entire equilibrium phase-space configurations of the D 1 -D 5 -P extremal black holes are determined by the p 3 := g determinant of the state-space metric tensor 77. The universal nature of statistical interactions and the other properties concerning MSW rotating black branes are elucidated by the state-space scalar curvature 78. The constant entropy (or scalar curvature) curve defining state-space manifold is higher dimensional hyperbola where c takes respective value of (c S , c R ) = (S 0 /2π, 3/4πR 0 ).
79. Similar state-space results hold for the four charge tree level extremal black holes.
80. Let us examine the state-space configuration of the four charg-anticharge black holes. 82. For given brane charges and Kaluza-Klein momenta, the microscopic entropy and macroscopic entropy match with

Such a configuration is the non-extremal
83. State-space covariant metric tensor is defined as negative Hessian matrix of entropy with respect to number of D 1 , D 5 branes {n i | i = 1, 5} and clockwise-anticlockwise Kaluza-Klein momentum charges {n p , n p }.
84. The components of the metric tensor are 85. For distinct i, j ∈ {1, 5}, and k, l ∈ {p, p} describing four charge non-extremal D 1 -D 5 -P -P black holes, the statistical pair correlations consist of the following scaling relations 86. The list of other mix relative correlation functions concerning the non-extremal D 1 -D 5 -P -P black holes are 87. Local stability criteria on possible surfaces and hyper-surfaces of underlying state-space configuration are determined by the positivity of 88. Complete local stability of full non-extremal D 1 -D 5 black brane state-space configuration is acertained by positivity of the determinant of state-space metric tensor 89. Global state-space properties concerning four charge non-extremal D 1 -D 5 black holes are determined by the regularity of the state-space scalar curvature invariant where the function f (n p , n p ) of two momenta (n p , n p ) running in opposite directions of the KK circle S 1 has been defined as f (n p , n p ) := n 5/2 p + 10n 3/2 p n p + 5n 1/2 p n p 2 + 5n 2 p n p 1/2 + 10n p n p 3/2 + n p 90. Large charge non-extremal D 1 -D 5 black branes have non-vanishing scalar curvature function on the state-space manifold (M 4 , g), and thus imply an almost everywhere weakly interacting statistical basis.

The constant entropy curve is non-standard curve is
92. As in the case of two charge D 0 -D 4 extremal black holes and D 1 -D 5 -P extremal black holes, the constant c takes the same value of c := S 2 0 /4π 2 .
93. For given state-space scalar curvature k, the constant state-space curvature curves take the form of 94. Similar results hold for the six and eight charge-anticharge non-extremal black holes. are analyzed by considering the type IIA string theory compactified on the product 98. Entropy as a function the charge Γ corresponding to p0 D 6 branes on X, p D 4 branes on 100. The components of the covariant metric tensor are given by 101. Define a charge vector X a = (p0, p, q, q0) with a set of notations 1 102. The local stability condition of the underlying statistical configuration under the Gaussian fluctuations requires that all the principle components of the fluctuations should be positive definite, i.e. for given set of state-space variables Γ i := (p Λ i , q Λ,i ) one must demands that {g ii (Γ i ) > 0; ∀i = 1, 2}. The concerned state-space metric constraints are thus defined by where 103. For distinct i, j, k, l ∈ {1, 2, 3, 4}, the admissible statistical pair correlations are consisting of diverse scaling properties. The set of nontrivial relative correlations signifying possible scaling relations of state-space correlations are nicely depicted by 104. The local stability condition constraint the allowed domain of the parameters of black hole configurations, and requires positivity of the following simultaneous equations 105. The stability of full state-space configuration is determined by computing the determinant of the metric tensor 106. The determinant of the metric tensor takes positive definite value, and thus there exist positive definite volume form on the state-space manifold (M 4 , g) of concerned leading order multi-centered D 6 -D 4 -D 2 -D 0 black brane configurations.
107. Conclusive nature of the state-space interaction and the other global properties of the statistical configurations are analyzed by determining state-space scalar curvature invariant 108. For some given constant charge Γ 0 , both the constant entropy and constant scalar curvature curves are again defined as where the respective real constants c := (c S , c R ) are given by 110. The above state-space correlation functions reduce to 111. For all Λ, the concerned state-space metric constraints are 112. The relative correlations defined as c ijkl := g ij /g kl reduce to the following three set of constant values. 117. Possible stability of internal state-space configurations reduce to the positivity of 118. The scalar curvature remains non zero, positive and take the value of 119. Thus, the state-space correlation volume vary as an inverse function of the single center brane entropy.

Double Center
121. Subsequent notations of the relative state-space correlations are prescribed by defining c ijkl := g ij /g kl .
123. Following limiting values are achieved for the state-space pair correlation functions 124. The relative correlations of the state-space configuration concerning second center of the D 6 -D 4 -D 2 -D 0 system are similarly analyzed. 126. The entropies of both the two charge centers Γ 1 , Γ 2 match, and in particular we have

State-space Stability of Double Center
127. Apart from definite scaling in Λ, the above two center D 6 -D 4 -D 2 -D 0 configurations form two type of state-space pair correlation functions (1,3), (2,2), (2,4), (3,3), (3,4), (4,4)}} C 128. For both the Γ 1 and Γ 2 , the respective state-space metric constraints satisfy 129. Both the centers have the same principle minors 130. The general expression of determinant of the metric tensor implies well-defined state-space manifold (M 4 , g) 131. The state-space scalar curvature again remains non-zero, positive quantity and takes the same values for both of the two charge centers
134. We shall now consider the role of the statistical fluctuations, in the two parameter giant and superstar configurations, characterized by an ensemble of arbitrary liquid droplets or irregular shaped fuzzballs.
135. Covariant thermodynamic geometries are analyzed for the giant solutions in terms of the chemical configuration parameters and arbitrarily excited boxes of random Young tableaux.
136. Underlying moduli configurations appear horizonless and smooth, but one acquires an entropy associated with average horizon area of the black hole in the classical limit.
137. We shal work in the limit defined as (

Chemical Description
143. To analyze the chemical correlation of large number of excited free fermion states, we now consider Weinhold geometry.
144. Typical correlation of the statistical states are characterized by arbitrary Young diagrams.
145. The average canonical energy defined in terms of the effective canonical temperature T , and R-chemical potential λ is 146. To investigate the chemical fluctuations, we consider two neighboring statistical states characterized by (T, λ) and (T + δT, λ + δλ).
147. Chemical pair correlation functions are defined as 148. The components of Weinhold metric tensor find following series expansions 150. The stability of arbitrary chemical configurations is thence determined by the determinant of thermodynamic Weinhold metric tensor 151. Conclusive nature of the global chemical correlations are analyzed by scalar curvature invariant where the covariant Riemann tensor R T λT λ (T, λ) turns out to be 152. The Weinhold geometry allows dual entropy representation for the statistical correlations between the states characterizing arbitrary Young diagrams. The dual state-space geometry is defined by Legendre transform where the canonical temperature is defined by T = 1/β.

The Fluctuating Young Tableaux
153. Typical statistical fluctuations are divulged over an ensemble of states with large charge ∆ = J = N 2 in the limit N → ∞, → 0 such that N remains fixed.
155. An ensemble of degenerate microstates are described by Young tableaux characterizing arbitrary phase-space configurations having M 2 cells with at most n random excited cells.
156. The pictorial view of a typical Young diagram may be given as . . . 158. Therefore, the degeneracy in choosing random n maltese (excited boxes) out of the total 159. From the first principle of statistical mechanics, the canonical counting entropy is 160. The subsequent analysis do not exploit any approximation, such as Stirling's approximation or thermodynamic limit.
161. The present statistical fluctuations over the canonical ensemble offer exact expressions of the state-space pair correlations and global correlation length.
162. To demonstrate so, let n excited droplets are arbitrarily chosen among M 2 fundamental cells which form an ensemble of states.
163. Then, the state-space geometry describes correlations between two neighbouring statistical states (n, M ) and (n + δn, M + δM ) in random Young tableaux Y (N, N c ).
164. The statistical fluctuations (in the droplets or fuzzballs picture having a pair (n, M )) are defined via the state-space metric tensor 165. The components of covariant state-space metric tensor thus defined are where Ψ(n, x) is the n th polygamma function, defined as the n th derivative of the digamma function.
166. The digamma function Ψ(x) is defined as 167. State-space stability holds locally, if the metric satisfies 168. Modulus of the ratio of excited-excited and excited-unexcited statistical pair correlation functions determines selection parameter 169. For n > 1, state-space correlations involve ordinary rational number Ψ(n, x) = Ψ(n) + γ, where γ is the standard Euler's constant.
170. For small n, the Ψ(n) is computed as a sum of gamma, which is again a rational number.
171. To perform this computation for a larger value of n, we have used for given initial condition Ψ(0, x) = Ψ(x).
172. Stability of underlying statistical configurations is analyzed by computing the determinant of the state-space metric tensor 173. For a family of boxes and their excitations, this shows that there exists positive definite volume form on the (M 2 , g).

174
. Generic global properties of 1/2-BPS black holes state-space configurations are examined by the scalar curvature 176. Chemical configuration shows, non-trivially curved determinant and the scalar curvature, and surprisingly the results remain valid even for single component j = 1 configuration.
177. The Gaussian fluctuations over an equilibrium chemical and state-space configurations accomplish well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds for all physically admissible domains of parameters.

The Fuzzball Solutions
[With S. Bellucci: Phys. Rev. D (2010) where the electric-magnetic charges, (Q, P ) and angular momentum J form co-ordinate charts on the intrinsic state-space manifold (M 3 , g).
181. Explicitly, the components of the metric tensor are 182. ∀i = j ∈ {P, Q} and J, the relative pair correlation functions scale as 183. For all admissible parameters, the three parameter Fuzzball solutions stable if the following state-space minors are positive 184. For non-zero brane charges and angular momentum, the determinant of the metric tensor is non-zero 185. Thus, the Fuzzball black rings do not correspond to an intrinsic stable statistical basis, when all the configuration parameters fluctuate.
186. Important state-space global properties of the fuzzy black rings configurations are determined by the nature of state-space scalar curvature invariant 187. The state-space scalar curvature can be expressed as an inverse function of the entropy with a negative constant of proportionality, and thus Mathur's fuzzy ring is a regular and an attractive statistical configuration.
188. For all non-zero rotation, both the constant entropy and constant state-space scalar curvature curves are just some hyperbolic paraboloid on which the state-space geometry turns out to be well-defined, and in interacting statistical system.
189. In present case, the constants k := (k S , k R ) are respectively defined as 190. The vanishing angular momentum limit J → 0 makes an ill-defined state-space geometry, which in turn is the same case as that of the two charge small black holes. 193. For D 1 -D 5 -J solutions having total ring entropy S(n 1 , n 5 , J), if there are M number of subensembles with entropyS(n 1 , n 5 , J), then Mathur has shown that the entropy in each subensemble is given byS

Subensemble Theory
194. We have shown that the non-vanishing state-space scalar curvature indicates that the extremal D 1 D 5 J system corresponds to an interacting statistical basis.
197. Bubbling black brane solutions are considered as the black foams and axi-symmetric merger solutions [I. Bena,C. W. Wang, ].
198. State-space geometry of charge foamed black brane configurations in M -theory characterizes statistical correlations over an ensemble of equilibrium microstates.
199. The most general bubbling supergravity solutions possessing three brane charges corresponding to each GH center of the bubbled black brane foam configuration.
200. Considering all possible partitioning of the flux parameters {k 1 i , k 2 i , k 3 i }, the leading order topological entropy is given by 201. Characterized the coordinate chart of the state-space manifold in terms of the charges {Q i } of the equilibrium foam solution, we find that the components of covariant statespace metric tensor are 202. State-space metric constraints over the diagonal pair correlation functions are where 203. Precise scaling properties of possible ratios consisting of the components of metric tensor are visualized by considering C BB , as in the three charge toy model bubbling black branes.
204. To accomplish state-space stability, all the principle minors should be positive definite.
205. The local stability conditions on the one dimensional line, two dimensional surfaces and three dimensional hyper-surfaces of the state-space manifold are respectively measured by 206. The global stability on the full state-space configuration is achieved by demanding positivity of the determinant of the state-space metric tensor where the factor f 1 (Q 1 , Q 2 , Q 3 ) is defined by 207. Information about the global correlation volume of underlying statistical system is read-off form the intrinsic state-space scalar curvature where f 2 is defined by 216. In general, from the viewpoint of statistical correlations and stabilities, the black brane configurations in string theory are categorized as (a) The underlying sub-configurations turn out to be well-defined over possible domains, whenever there exist a respective set of positive (non-zero) state-space heat capacities.
(b) The underlying full configuration turns out to be everywhere well-defined, whenever there exist positive (non-zero) state-space principle minors.
(c) The underlying configuration corresponds to an interacting statistical system, whenever there exist a regular (non-zero) state-space scalar curvature.
217. The state-space manifold of extremal/ non-extremal and supersymmetric/ nonsupersymmetric string theory black holes may intrinsically be described by an embedding 218. 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 − .
219. For supersymmetric black holes, the restriction g →g| M =M 0 (P i ,Q i ) may be applied to the associated nonsupersymmetric black brane configuration.

Higher Derivative Corrections
In Ref. arXiv:0801.4087v2, we focus our attension on the followings: 220. Generalized Uncertainty Principle Corrections: Non-communative Geometry at Planck Length corrected state-space geometry of (a) Reissner-Nordström black holes and (b) magnetically charged black holes.
221. Stringy α Corrections: (i) We have examined the state-space geometry of Gauss-Bonnet corrected supersymetric dyonic extremal black holes in D = 4.
(ii) We have computed the correlation length of D = 4 non-supersymmetric extremal black holes at all orders in α .
(iii) We have explored the case of leading order α corrected state-space geomery for the non-extremal D 1 -D 5 and D 2 -D 6 -N S 5 black branes in D = 10.

Instanton Vacuum Moduli
In the intrinsic geometric study of instanton vacua with the consideration of M. Bianchi, F. Fucito, J. F. Morales, "D-brane Instantons on the T6/Z3 orientifold", JHEP 0707, 038, (2007), arxiv:hep-th/0704.0784, (i) we have introgated vacuun stability of stationary and non-stationary statistical configurations with and without the statistical fluctuations of the gauge and exotic instanton curves in the framework of Veneziano-Yankielowiz/ Affleck-Dine-Seiberg and nonperturbative instanton superpotentials.
(ii) We find that the Gaussian fluctuations over equilibrium (non)-stationary vacua yield a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds for statistically admissible domains of (a) one loop renormalized mass and vacuum expectation value of the chiral field for the stationary vacua and (b) the corresponding contributions of the instanton curves for the non-stationary vacua.
(iii) As a function of the vacuum expectation value of the chiral field, the global ensemble stability and phase transition criteria algebraically reduce to the invariance of the quadratic and quartic polynomials. 228. Physics at the Planck Scale: The thermodynamic state-space geometry may be explored with foam geometries, and empty space virtual black holes whose statistical correlations among the microstates would involve foam of two-spheres.

Geometric Perspective of Sen Entropy Function
229. The present exploration thus opens an avenue to give new insight into the promising vacuum structures of black brane space-time at very small scales. .