Topological signatures in Kerr-Sen AdS black hole thermodynamics

Abstract

Black hole thermodynamics and topology have emerged as a strong foundation for a coordinate-independent understanding of phase transitions. Using both Duan's topological current theory and a novel complex residue method, we perform a topological study of the Kerr-Sen AdS black hole arising in heterotic string theory. In turn, we find the zero points corresponding to on-shell black hole states and calculate their winding numbers to find the global topological charge by building the generalized off-shell free energy and examining the corresponding vector field in a parametric space. Our analysis reveals that the Kerr-Sen AdS black hole exhibits three distinct thermodynamic phases -- small, intermediate, and large black hole branches -- characterized by critical points with winding numbers $+1$, $-1$, and $+1$ respectively, culminating in a total topological charge $W = +1$. Significantly, this topological number remains invariant under variations of the dilaton charge parameter, indicating that the dilaton field does not alter the fundamental topological class established for Kerr-AdS and RN-AdS black holes. However, the rotation parameter proves crucial in determining the phase structure and the emergence of multiple critical points. We systematically examine three limiting configurations: the full Kerr-Sen AdS spacetime, the GMGHS AdS limit ($a = 0$), and the asymptotically flat Kerr-Sen case ($Λ= 0$). In addition, we propose a novel approach that analytically continues the thermodynamic characterisation into the complex plane. The characterized complex function, derived from the off-shell Gibbs free energy, possesses isolated singular points whose residues directly encode the winding numbers. Our results indicate that topology offers deep insights into black hole phase transitions, with potential implications to holographic dualities.

Figures (13)

• Figure 1: Plot shows the unit vector field $n$ on the $r_h$--$\Theta$ plane for a four-dimensional Kerr-Sen AdS black hole with $\tau = 3.3$ and $a = 0.08$. The zero points (ZPs), marked with red dots, are located at $(r_h,\Theta)=(0.1012204,\pi/2)$ , $(0.34086,\pi/2)$, and $(0.86402,\pi/2)$ . The contours $C_1$, $C_2$, and $C_3$ surrounding these zero points carry winding numbers $w_1 = 1$, $w_2 = -1$, and $w_3 = 1$, respectively, whose sum defines the total topological charge $W = 1$.
• Figure 2: The behavior of the zero points of $\phi^{r_h}$ in the $r_h-\tau$ plane for the Kerr-Sen AdS black hole. The blue solid, black dashed, and red solid curves correspond to the large black hole (LBH), intermediate black hole (IBH), and small black hole (SBH) branches, respectively, for the Kerr--Sen AdS black hole with $a = 0.08$, $b = 0.01$, and $P = 0.1193$. Two critical temperatures, $\tau_{c1} = 2.4903$ and $\tau_{c2} = 3.60951$, characterize the phase transitions.
• Figure 3: The plot showing the behavior of $\Omega$ versus $\vartheta$ for the contours $C_1$ (Red solid curve), $C_2$ (Black dashed curve), and $C_3$ (Blue solid curve) for the Ker--Sen AdS black hole.
• Figure 4: The plot showing the unit vector field $n$ on the $r_h$-$\Theta$ plane for a four-dimensional Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) charged black holes in AdS spacetime with $\tau = 2$ when we switch off the rotation ($a = 0$). The zero points (ZPs), marked by red dots, are located at $(r_h,\Theta) = (0.16417,\pi/2)$ and $(1.903549,\pi/2)$. The orientations of the contours $C_1$ and $C_2$ are clockwise and anticlockwise, corresponding to the topological charges $Q_{c_1} = -1$ and $Q_{c_2} = +1$, respectively. The sum of the total topological charge $Q_\text{total} = 0$.
• Figure 5: The plot showing the behavior of the stable and unstable branches of the GMGHS AdS black hole phases in blue and red curves, respectively, for $a = 0,$$b = 0.01$, and $P = 0.1193$.