HyperKhaler Metrics Building and Integrable Models

Methods developed for the analysis of integrable systems are used to study the problem of hyperK\"ahler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation $\beta\partial^{++2}\omega -\xi^{++2}\exp 2\beta\omega =0$ and its Toda like generalizations are integrable. Explicit solutions together with the conserved currents generating the symmetry responsible of the integrability of these equations are given. Other features are also discussedComment: Latex file, 12 page

On Thermodynamics of AdS Black Holes in M-Theory

Motivated by a recent work on asymptotically AdS_4 black holes in M-theory, we investigate the thermodynamics and thermodynamical geometry of AdS black holes from M2 and M5-branes. Concretely, we consider AdS black holes in AdS_{p+2}\times S^{11-p-2}, where p=2,5 by interpreting the number of M2 (and M5-branes) as a thermodynamical variable. We study the corresponding phase transition to examine their stabilities by calculating and discussing various thermodynamical quantities including the chemical potential. Then, we compute the thermodynamical curvatures from the Quevedo metric for M2 and M5-branes geometries to reconsider the stability of such black objects. The Quevedo metric singularities recover similar stability results provided by the phase transition program.Comment: 16 pages, 12 figures. Late

Maxwell's equal-area law for Gauss-Bonnet Anti-de Sitter black holes

Interpreting the cosmological constant \Lambda as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we study the Maxwell's equal area law of higher dimensional Gauss-Bonnet-AdS black holes in extended phase space. These black hole solutions critically behave like Van der Waals systems. It has been realized that below the critical temperature T_c the stable equilibrium is violated. We show through calculations that the critical behaviors for the uncharged black holes only appear in d=5. For the charged case, we analyse solutions in d = 5 and d = 6 separately and find that, up to some constrains, the critical behaviors only appear in the spherical topology. Using the Maxwell's construction, we also find the isobar line for which the liquid-gas-like phases coexist.Comment: 18 pages, 4 figures, latex. References added and typos corrected. Minor changes. Final version accepted in European Physical Journal

Ehrenfest Scheme of Higher Dimensional Topological AdS Black Holes in The Third Order Lovelock-Born-Infeld Gravity

Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the investigation of P-V critical behaviors of (1+n)-dimensional topological AdS black holes in Lovelock-Born-Infeld gravity. In particular, we give an explicit expression of the universal number \chi=\frac{P_c v_c}{T_c} in terms of the space dimension $n$. Then, we examine the phase transitions at the critical points of such topological black holes for 6 \leq n \leq 11 as required by the physical condition of the thermodynamical quantities. More precisely, the Ehrenfest equations have been checked revealing that the black hole system undergoes a second phase transition at the critical points.Comment: 18 pages, latex with 10 figures, titled modified, section added, typos corrected, accepted for publication in International Journal of Geometric Methods in Modern Physics (2015