Bhupendra Nath Tiwari

On the thermodynamic geometry of BTZ black holes

We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes. It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble. We further investigate the the contribution of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced. We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.

Thermodynamic geometry and extremal black holes in string theory

We study a generalisation of thermodynamic geometry to degenerate quantum ground states at zero temperatures exemplified by charged extremal black holes in type II string theories. Several examples of extremal charged black holes with non degenerate thermodynamic geometries and regular but vanishingly small state space scalar curvatures are established. These include black holes described by D1-D5-P and D2-D6-NS5-P brane systems and also two charged small black holes in Type II string theories. We also explore the modifications to the state space geometry and the scalar curvature due to the higher derivative contributions and string loop corrections as well as an exact entropy expression from quantum information theory. Our construction describes state space geometries arising out of a possible limiting thermodynamic characterisation of degenerate quantum ground states at zero temperatures.

On the Microscopic Perspective of Black Branes Thermodynamic Geometry

We study thermodynamic state-space geometry of the black holes in string theory and M-theory. For a large number of microstates, we analyze the intrinsic state-space geometry for (i) extremal and non-extremal black branes in string theory, (ii) multi-centered black brane configurations, (iv) small black holes with fractional branes, and (v) fuzzy rings in the setup of Mathur’s fuzzballs and subensemble theory. We extend our analysis for the black brane foams and bubbling black brane solutions in M-theory. We discuss the nature of state-space correlations of various black brane configurations, and show that the notion of state-space manifolds describes the associated coarse-grained interactions of the corresponding microscopic CFT data.

On the Thermodynamic Geometry of Hot QCD

We study the nature of the covariant thermodynamic geometry arising from the free energy of hot QCD. We systematically analyze the underlying equilibrium thermodynamic configurations of the free energy of two- and three-flavor hot QCD with or without the inclusion of thermal fluctuations in the neighborhood of the QCD transition temperature. We show that there exists a well-defined thermodynamic geometric notion for the QCD thermodynamics. The geometry thus obtained has no singularity as an intrinsic Riemannian manifold. We further show that there is a close connection of this geometric approach with the existing studies of correlations and quark number susceptibilities.

An exact fluctuating 1/2-BPS configuration

This work explores the role of thermodynamic fluctuations in the two parameter giant and superstar configurations characterized by an ensemble of arbitrary liquid droplets or irregular shaped fuzzballs. Our analysis illustrates that the chemical and state-space geometric descriptions exhibit an intriguing set of exact pair correction functions and the global correlation lengths. The first principle of statistical mechanics shows that the possible canonical fluctuations may precisely be ascertained without any approximation. Interestingly, our intrinsic geometric study exemplifies that there exist exact fluctuating 1/2-BPS statistical configurations which involve an ensemble of microstates describing the liquid droplets or fuzzballs. The Gaussian fluctuations over an equilibrium chemical and state-space configurations accomplish a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds for all physically admissible domains of black hole parameters. An explicit computation demonstrates that the underlying chemical correlations involve ordinary summations, whilst the state-space correlations may simply be depicted by standard polygamma functions. Our construction ascribes definite stability character to the canonical energy fluctuations and to the counting entropy associated with an arbitrary choice of excited boxes from an ensemble of ample boxes constituting a variety of Young tableaux.

State-space manifold and rotating black holes

We study a class of fluctuating higher dimensional black hole configurations obtained in string theory/M-theory compactifications. We explore the intrinsic Riemannian geometric nature of Gaussian fluctuations arising from the Hessian of the coarse graining entropy, defined over an ensemble of brane microstates. It has been shown that the state-space geometry spanned by the set of invariant parameters is non-degenerate, regular and has a negative scalar curvature for the rotating Myers-Perry black holes, Kaluza-Klein black holes, supersymmetric AdS5 black holes, D1-D5 configurations and the associated BMPV black holes. Interestingly, these solutions demonstrate that the principal components of the state-space metric tensor admit a positive definite form, while the off diagonal components do not. Furthermore, the ratio of diagonal components weakens relatively faster than the off diagonal components, and thus they swiftly come into an equilibrium statistical configuration. Novel aspects of the scaling property suggest that the brane-brane statistical pair correlation functions divulge an asymmetric nature, in comparison with the others. This approach indicates that all above configurations are effectively attractive and stable, on an arbitrary hyper-surface of the state-space manifolds. It is nevertheless noticed that there exists an intriguing relationship between non-ideal inter-brane statistical interactions and phase transitions. The ramifications thus described are consistent with the existing picture of the microscopic CFTs. We conclude with an extended discussion of the implications of this work for the physics of black holes in string theory.

State-space correlations and stabilities

The state-space pair correlation functions and the notion of stability of extremal and nonextremal black holes in string theory and M theory are considered from the viewpoint of thermodynamic Ruppeiner geometry. We have explicitly analyzed the state-space configurations for (i) the two- and three-charge extremal black holes, (ii) the four- and six-charge nonextremal black branes, which both arise from the string theory solutions. An extension is considered for the D{sub 6}-D{sub 4}-D{sub 2}-D{sub 0} multicentered black branes, fractional small black branes, and two-charge rotating fuzzy rings in the setup of Mathurs fuzzball configurations. The state-space pair correlations and the nature of stabilities have been investigated for three-charged bubbling black-brane foams, and thereby the M-theory solutions are brought into the present consideration. Novel aspects of the state-space interactions have been envisaged from the coarse graining counting entropy of underlying conformal field theory microstates.

THERMODYNAMIC GEOMETRIC STABILITY OF QUARKONIA STATES

We compute exact thermodynamic geometric properties of the non-Abelian quarkonium bound states from the consideration of one-loop strong coupling. From the general statistical principle, the intrinsic geometric nature of strongly coupled QCD is analyzed for the Colombic, rising and Regge rotating regimes. Without any approximation, we have obtained the nonlinear mass effect for the Bloch–Nordsieck rotating strongly coupled quarkonia. For a range of physical parameters, we show in each cases that there exists a well-defined, nondegenerate, curved, intrinsic Riemannian manifold. As the gluons become softer and softer, we find in the limit of the Bloch–Nordsieck resummation that the strong coupling obtained from the Sudakov form factor possesses exact local and global thermodynamic properties of the underlying mesons, kayons and Ds particles.

BLACK STRINGS, BLACK RINGS AND STATE-SPACE MANIFOLD

State-space geometry is considered, for diverse three and four parameter nonspherical horizon rotating black brane configurations, in string theory and M-theory. We have explicitly examined the case of unit Kaluza–Klein momentum D1D5P black strings, circular strings, small black rings and black supertubes. An investigation of the state-space pair correlation functions shows that there exist two classes of brane statistical configurations, viz. the first category divulges a degenerate intrinsic equilibrium basis, while the second yields a nondegenerate, curved, intrinsic Riemannian geometry. Specifically, the solutions with finitely many branes expose that the two charged rotating D1D5 black strings and three charged rotating small black rings consort real degenerate state-space manifolds. Interestingly, arbitrary valued M5-dipole charged rotating circular strings and Maldacena–Strominger–Witten black rings exhibit nondegenerate, positively curved, comprehensively regular state-space configurations. Furthermore, the state-space geometry of single bubbled rings admits a well-defined, positive definite, everywhere regular and curved intrinsic Riemannian manifold; except for the two finite values of conserved electric charge. We also discuss the implication and potential significance of this work for the physics of black holes in string theory.

A GEOMETRIC APPROACH TO CORRELATIONS AND QUARK NUMBER SUSCEPTIBILITIES

We study the thermodynamic geometry arising from the free energy for the two- and three-flavor finite temperature hot QCD near the critical temperature. We develop a geometric notion for QCD thermodynamics, relating it with the existing microscopic quantities, e.g. quark-number susceptibility, which appears naturally within an approximately self-consistent resummation of perturbative QCD. We further incorporate thermal fluctuations in the free energy, thus yielding the geometric properties of local and global chemical correlations. These investigations are perturbative in nature. Nevertheless, one could apply the same line of thought for the geometric realization of underlying quark susceptibilities, either in the fabric of lattice QCD or in that of nonperturbative QCD.