Dimensional structure of thermodynamic topology in ultraspinning Kerr-AdS black holes

In this paper, we apply the thermodynamic topology framework to ultraspinning Kerr-AdS black holes in arbitrary spacetime dimensions. By constructing the off-shell Helmholtz free energy and the associated vector field, black hole states are characterized as topological defects, and their phase structures are described through zero points, winding numbers, and asymptotic thermodynamic behavior. Analyses of the four- and five-dimensional cases highlight the differences between even- and odd-dimensional configurations, while representative higher-dimensional cases confirm that no additional topological classes or subclasses emerge. We find that only two thermodynamic topological structures appear: the standard class $W^{1+}$ for most configurations, and the distinct subclass $\tilde{W}^{1+}$ for odd-dimensional black holes with maximal rotations. These results establish a unified, dimension-independent classification scheme for ultraspinning Kerr-AdS black holes.

💡 Research Summary

The paper applies the recently developed thermodynamic topology framework to ultraspinning Kerr‑AdS black holes in arbitrary spacetime dimensions. Ultraspinning black holes are obtained by boosting one rotational angular velocity to the speed of light, yielding a non‑compact horizon with two punctures while preserving a finite horizon area. These objects violate the reverse isoperimetric inequality and are therefore termed “super‑entropic”. Because of their unusual thermodynamic behavior, a topological classification based on global invariants is both natural and powerful.

The authors begin by reviewing the thermodynamic topological method. The key ingredient is the off‑shell Helmholtz free energy (F = M - S \tau), where (M) is the mass, (S) the Bekenstein‑Hawking entropy, and (\tau) the inverse temperature of a cavity surrounding the black hole. By differentiating (F) with respect to the horizon radius (r_h) and an auxiliary angular variable (\Theta), a two‑component vector field (\phi = (\phi_{r_h}, \phi_{\Theta})) is constructed. The second component vanishes only at (\Theta = \pi/2); consequently, physical solutions are identified by the zeroes of (\phi_{r_h}), which directly enforce the on‑shell condition (\tau = 1/T). Using Duan’s (\phi)-mapping theory, a conserved topological current is defined, and each zero of (\phi) carries a winding number (w_i = \beta_i \eta_i) (Hopf index times Brouwer degree). A winding number (+1) signals a locally thermodynamically stable branch, while (-1) indicates instability. The total topological charge (W = \sum_i w_i) provides a global classification independent of the detailed metric.

The analysis proceeds with explicit calculations for four‑ and five‑dimensional ultraspinning Kerr‑AdS black holes, which serve as representative even‑ and odd‑dimensional cases. In (d=4) (even), the vector field has a single zero, its winding number is (+1), and the global charge is (W=1). The inverse temperature (\tau) diverges as the horizon radius approaches its minimal value (small black hole) and vanishes for large radii, indicating a stable small‑black‑hole branch at low temperature and a stable large‑black‑hole branch at high temperature. This behavior matches the previously identified topological class (W^{1+}).

In (d=5) (odd) two distinct rotation configurations are examined. With a single rotation parameter the situation mirrors the four‑dimensional case: one zero, (w=+1), (W=1), and the system belongs to (W^{1+}). When both independent rotation parameters are maximally turned on (the “maximal‑rotation” case), the vector field acquires three zeros. Their winding numbers follow the pattern (