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

TL;DR

The paper addresses how ultraspinning Kerr-AdS black holes in arbitrary dimensions organize their thermodynamic phase structure. It applies a thermodynamic-topology framework, constructing the off-shell free energy $\mathcal{F}$ and a two-component field to locate zero points and define a conserved topological current, yielding a global invariant $W$ and local windings $w_i$. The main result is that only two topological structures appear: $W^{1+}$ for most configurations and a distinct subclass $\tilde{W}^{1+}$ for odd dimensions with maximal rotations; no new classes arise for $d\ge6$. This provides a dimension-independent, robust classification of ultraspinning Kerr-AdS black holes and clarifies how rotational saturation shapes thermodynamic topology, with potential extensions to charged or supergravity contexts.

Abstract

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.

Figures (2)

• Figure 1: Zero points of the vector $\phi^{r_h}$ on the $r_h$-$\tau$ plane for the four-dimensional ultraspinning Kerr-AdS black hole with $l = r_0$. The purple line denotes the thermodynamically stable branch ($w = 1$), corresponding to a topological number $W = 1$.
• Figure 2: Zero points of $\phi^{r_h}$ in the $\tau$-$r_h$ plane for the five-dimensional ultraspinning Kerr-AdS black hole with $l = r_0$: (a) doubly rotating case with $a_1/r_0 = 0.5$; (b) singly rotating case with $a_1 = 0$. The blue curve represents a thermodynamically unstable black hole branch with winding number $w = -1$, while the purple curve corresponds to a thermodynamically stable branch with winding number $w = +1$. The red dot denotes the annihilation point.