From the perspective of black hole physics, the present research examines the thermodynamic geometries of extremal and non-extremal black hole configurations in string theory and M-theory with 2, 3, 4, 5 and 6 charges. We analyzed the structure of state-space geometry, such as regularity, existence of critical points, lines, surfaces, hypersurfaces and associated phase transitions for BPS black holes, rotating black holes, black strings, black rings, multi-centered black branes, brane fractionation, Mathur's fuzzballs, subensemble theory in string theory and bubbling black brane foam...

Given a Sen entropy function, the present research explores geometric and algebraic properties of a class of four and higher dimensional extremal black holes under arbitrary higher derivative stringy corrections. The notion of embedding theory offers generalized complex structures and mapping properties of associated differentiable manifolds. The convexity is realized in extended subfield of finitely many eigenvalues of the Hessian of Sen entropy function. The commutative algebra framework offers corresponding spectra and generalized spectra as Krull and convex hull of the given eigenvalues....

The present research offers thermodynamic geometric properties of black holes in string theory and M-theory. It systematically investigates the state-space and conformally related chemical geometries for extremal and non-extremal black holes, black strings, black rings and supertubes in four and higher spacetime dimensions. From the perspective of the intrinsic differential geometry, the questions of stability, regularity, existence of critical phenomena and phase transitions have been analyzed for charged anticharged black holes in a given basin of attractor. For the two parameter half BPS...

The present research explores the role of generalized uncertainty inequalities in the theory of quantum gravity. Motivated from the noncommutative nature of string theory, we show that there exists an ultraviolet/ infrared mixing dependent function. From the perspective of higher derivative stringy corrections, the uncertainty principle arises as the analyticity condition of a complex function. For a given ultraviolet cutoff, this observation non-trivially modifies the algebra of quantum observables. With the postulate that Planck length is the minimal length scale in nature, our analysis is...

The present research offers thesis of experiences, truth and false values, and thus the socio-scientific understanding of knowledge theories. From the philosophical journey of Heraclitus, Parmenidian, Plato, Aristotle, Descartes, Spinoza, Leibnitz, Hume and Kant, we construct the theory of goodwill, global wisdom and local experiences in the natural philosophy. Given a set of smooth experience functions, we have demonstrated that the Descarte's knowledge follows from the corresponding Hume's definition of the knowledge, as the late time effect. Following the notion that the reality consists...

From the perspective of moduli stabilization and physics of D-branes, we consider the role of the real intrinsic Riemannian geometry and describe the statistical nature of vacuum fluctuations. The issue of the wall (in)stabilities is analysed for the marginal and threshold like vacua for arbitrary component moduli configurations breaking down to finitely many U(1)'s. From the notion of the statistical fluctuation theory, we find for both the mariginal and threshold configurations that the Gaussian fluctuations over a given equilibrium vacuum yield a well-defined, non-degenerate, curved and...

From the perspective of black hole thermodynamics, this exposition provides the physical and mathematical account of thermodynamic geometries, higher derivative corrections, N= 1, 2, 4 configurations, Calabi-Yau compactification, string theory and M-theory. We illustrate the local and global statistical stabilities and phase transitions for an ensemble of Kerr black holes in general relativity, and thereby consider a class of macroscopic attractors, including (non)BPS black holes, (non)extremal black holes, and (non)supersymmetric black holes in supergravity theories. For a given black brane...

From the perspective of thermodynamic geometry, we study the hypersurface properties by considering the fluctuation theory and material formations for a class of finite parameter filters and arbitrary irregular shaped circuits. Given a constant mismatch factor, the Gaussian fluctuations over an equilibrium statistical basis accomplish a well-defined, non-degenerate, flat and regular intrinsic surface. For a variable mismatch factor ensemble, the long rang global correlation function is given by the ratio of two ordinary summations. Given covariant intrinsic description of a definite physical...

From the perspective of higher dimensional black holes in general relativity, we consider the role of the real Riemannian geometry and describe the (in)stabilities for statistical fluctuations of background black holes in arbitrary SU(N) gauge theories. For a given ensemble of non-abelian gauge theory vacuum extremal black holes carrying a nonzero pair of gauge charge and cosmological constant, we study the Gaussian fluctuations over an equilibrium ensemble and determine the criteria for the well-defined, non-degenerate, curved and regular state-space and chemical surfaces. From the notion of...