In this paper, we extend our previous work [D. Wu, Phys. Rev. D 107, 024024 (2023)] to the more general cases with a negative cosmological constant, and investigate the topological numbers for the singly rotating Kerr-AdS black holes in all dimensions and the four-dimensional Kerr-Newman-AdS black hole as well as the three-dimensional Ba\~nados-Teitelboim-Zanelli black hole. We find that the topological numbers of black holes are remarkably influenced by the cosmological constant. In addition, we also demonstrate that the dimension of spacetimes has an important effect on the topological number for rotating anti--de Sitter (AdS) black holes. Furthermore, it is interesting to observe that the difference between the topological number of the AdS black hole and that of its corresponding asymptotically flat black hole is always unity. This new observation leads us to conjure that it might be valid also for other black holes. Of course, this novel conjecture needs to be further verified by examining the topological numbers of many other black holes and their AdS counterparts in the future work. 

http://link.aps.org/pdf/10.1103/PhysRevD.107.084002

Topological classes of thermodynamics of rotating AdS black holes

pdf/consistent-thermodynamics-and-topological-classes-for-the-bk2hpstp.pdf

Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes

In the recent proposal [Phys. Rev. Lett. 129, 191101 (2022)], the black holes were viewed as topological thermodynamic defects by using the generalized off-shell free energy. In this paper, we follow such proposal to study the local and global topological natures of the Gauss-Bonnet black holes in AdS space. The local topological natures are reflected by the winding numbers, where the positive and negative winding numbers correspond to the stable and unstable black hole branches. The global topological natures are reflected by the topological numbers, which are defined as the sum of the winding numbers for all black hole branches and can be used to classify the black holes into different classes. When the charge is present, we find that the topological number is independent on the values of the parameters, and the charged Gauss-Bonnet AdS black holes can be divided into the same class of the RNAdS black holes with the same topological number 1. However, when the charge is absent, we find that the topological number has certain dimensional dependence. This is different from the previous studies, where the topological number is found to be a universal number independent of the black hole parameters. Furthermore, the asymptotic behaviors of curve {\tau}(r_h) in small and large radii limit can be a simple criterion to distinguish the different topological number. We find a new asymptotic behavior as {\tau}(r_h \to 0) = 0 and {\tau}(r_h \to \infty) = 0 in the black hole system, which shows topological equivalency with the asymptotic behaviors {\tau}(r_h \to 0)=\infty and {\tau}(r_h \to \infty)=\infty. We also give an intuitional proof of why there are only three topological classes in the black hole system under the condition (\partial_{r_h} S)P > 0. 

Topological natures of the Gauss-Bonnet black hole in AdS space

We study the topology of the thermodynamic space of four dimensional dyonic anti--de Sitter (AdS) black hole in three different ensembles: canonical, mixed, and grand canonical ensemble. While canonical ensemble refers to the ensemble with fixed electric and magnetic charges, mixed ensemble is an ensemble where we fix magnetic charge and electric potential. In the grand canonical ensemble, potentials corresponding to both electric and magnetic charges are kept fixed. In each of these ensembles, we first compute the topological charges associated with critical points. We find that while in both canonical and mixed ensembles, there exists one conventional critical point with topological charge $\ensuremath{-}1$, in the grand canonical ensemble, we find no critical point. Then, we consider the dyonic AdS black hole as topological defects in thermodynamic space and study its local and global topology by computing the winding numbers at the defects. We observe that while the topologies of the black hole in canonical and mixed ensembles are identical with total topological charge equaling 1, in the grand canonical ensemble, depending on the values of potentials, the total topological charge is either equal to 0 or 1. In canonical and mixed ensembles, either one generation and one annihilation points or no generation/annihilation points are found. In the grand canonical ensemble, depending on the values of potentials, we find either one generation point or no generation/annihilation point. Thus, we infer that the topological class of 4D dyonic AdS black hole is ensemble dependent. 

pdf/thermodynamic-topology-of-4d-dyonic-ads-black-holes-in-hbxsyzofps.pdf

Thermodynamic topology of 4D dyonic AdS black holes in different ensembles

Setting the cosmological constant to be dynamical, we study the bulk and boundary thermodynamics of charged Anti-de Sitter black holes. We develop mass/energy formulas in terms of thermodynamic state functions for the extended thermodynamics, mixed thermodynamics, and boundary conformal field theory thermodynamics. We employ the residue method to study the topological properties of the phase transitions. Our analysis reveals that the bulk and boundary thermodynamics are topologically equivalent for both criticalities and first-order phase transitions in the canonical ensembles, as well as for the Hawking-Page(-like) phase transitions in the grand canonical ensembles. Additionally, those three kinds of phase transitions are shown to be distinguished by their unique topological charges. Our results exemplify the gravity-gauge duality in terms of topology. 

/pdf/bulk-boundary-thermodynamic-equivalence-a-topology-viewpoint-cs9peumt.pdf

Bulk-boundary thermodynamic equivalence: a topology viewpoint

In this work, we develop a universal picture from topology in the thermodynamic parameters space to describe first order phase transitions of black holes. By employing an off-shell internal energy, we find two types of topological defects. The first is normal, describing black holes which move with a nonzero velocity and acceleration, in a central force field. The second type of defects are exotic: they are static in the space, not describing black holes but encoding information about first order transitions. For each defect, we assign a winding number and an inertial mass. By studying neutral and charged black holes in asymptotically anti--de Sitter space, we show that first order transitions can be viewed as a once or twice interchange of winding numbers between black holes and the exotic defects, through wired action at a distance. This corresponds to the usual notion: a smaller black hole grows into a larger black hole or vice versa. However, our topological analysis illustrates that the transition can also be locally interpreted as virtual collisions between black holes and the exotic defects. In this interpretation, a smaller black hole first grows into a new exotic defect whereas an original exotic defect grows into a larger black hole. All the defects simply change their positions and momentums, rather than interchanging the winding numbers. The critical point of the transition can be extracted when all the defects meet in the parameters space. Certain quantities, such as the Jacobians and the velocities of normal defects show universal behaviors near the critical point. 

/pdf/topological-interpretation-for-phase-transitions-of-black-gcl498w3.pdf

Topological interpretation for phase transitions of black holes

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Topological interpretation for phase transitions of black holes