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.

• The paper introduces a topological analysis using Duan's φ-mapping and a complex residue approach to classify thermodynamic phase transitions.
• It identifies three critical black hole states with winding numbers, corresponding to stable small, intermediate, and unstable large branches.
• The study demonstrates that dilaton charge does not impact the topology, while rotation and negative cosmological constant crucially affect phase structure.

Topological Signatures and Phase Structure in Kerr-Sen AdS Black Hole Thermodynamics

Introduction

This work undertakes a rigorous analysis of the thermodynamic phase structure of Kerr-Sen anti-de Sitter (AdS) black holes, with an emphasis on characterizing their critical properties using topological methods. The Kerr-Sen AdS solution arises from the low-energy limit of heterotic string theory and generalizes the classical Kerr solution by incorporating both a dilaton field and a negative cosmological constant. The presence of charge, rotation, and scalar hair engenders a complex thermodynamic landscape, making the identification and classification of phase transitions particularly nontrivial. The authors utilize both Duan's $\phi$-mapping topological current framework and an analytically continued residue method to compute topological charges associated with thermodynamic critical points, providing a comprehensive perspective on universal properties and model-specific subtleties.

Thermodynamic Framework of Kerr-Sen AdS Black Holes

The Kerr-Sen AdS black hole embodies a paradigm wherein the mass $M$, angular momentum $J$, electric charge $Q$, and dilaton parameter $b$ are intertwined in the thermodynamic structure. The extension to AdS space introduces the pressure-volume conjugacy: $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$ The off-shell free energy

$\mathcal{F} = M - \frac{S}{\tau}$

facilitates the construction of a vector field in an auxiliary parameter space $(r_h, \Theta)$, where $r_h$ is the horizon radius and $\tau$ is an auxiliary inverse temperature. The zeros of this vector field coincide with physical (on-shell) black hole states.

The analysis specifically focuses on three parameter regimes:

• The generic Kerr-Sen AdS solution $M$0,
• The non-rotating GMGHS AdS limit $M$1,
• The asymptotically flat Kerr-Sen solution $M$2.

Thermodynamic quantities such as temperature, potential, and angular velocity are determined by standard methods, while the Smarr relation encodes the scaling behavior.

Topological Classification via $M$3-Mapping Theory

Duan's $M$4-mapping approach enables the identification of topological defects in the parametric $M$5 plane, with associated winding numbers computed from the vector field derived from $M$6.

(Figure 1)

Figure 1: The schematic of isolated singularities on the complex plane, where global and local contours enclose singular points $M$7; residues of these singularities determine the topological charges.

For the full Kerr-Sen AdS black hole, the analysis reveals three critical points: two with winding number $M$8 (corresponding to locally stable, positive heat capacity branches) and one with $M$9 (unstable, negative heat capacity branch). This yields a total topological charge $J$0, invariant under continuous deformation of the parameter space, including variations of the dilaton charge $J$1. The three critical points map onto small (SBH), intermediate (IBH), and large (LBH) black hole branches.

For the GMGHS AdS limit $J$2, the topological charges from two critical points $J$3 and $J$4 cancel, resulting in $J$5. Similarly, the asymptotically flat $J$6 case yields two critical points with charges $J$7 and $J$8, again giving $J$9.

The invariance of $Q$0 under changes in $Q$1, contrasted with its sensitivity to $Q$2 and $Q$3, indicates that the dilaton field does not affect the universality class of the black hole topology, while spin and AdS asymptotics do.

(Figure 2)

Figure 2: Plot of roots in the complex plane for the characteristic polynomial as parameters $Q$4, $Q$5, and $Q$6 are varied, revealing the structure of critical points and their merger or splitting.

Complex Residue Method and Analytic Continuation

Complementing the $Q$7-mapping analysis, the authors employ a complex analytic continuation, defining a function

$Q$8

where $Q$9 is the complexified horizon radius and $b$0 captures the thermodynamic equation. The topological defects are then mapped to singularities (poles) of $b$1 in the complex $b$2-plane. The residue at each singularity is interpreted as the winding number: $b$3 Summing these yields the total topological number, in agreement with the $b$4-mapping approach.

(Figure 3)

Figure 3: Evolution of the vector field and associated winding numbers on the $b$5 plane, demonstrating the topological nature and protection of zero points corresponding to distinct phases.

Through this analytic continuation, the residue method captures creation and annihilation of critical points, their coalescence at higher-order degeneracies, and provides a robust algebraic handle on phase transitions. Notably, the method demonstrates that at the critical temperature, a simple pole splits into two (corresponding to SBH and LBH branches) as parameters are tuned, matching the observed van der Waals-like behavior.

Theoretical and Practical Implications

The global topological number $b$6 serves as a coordinate-independent invariant that distinguishes universality classes of black hole phase structures. The observed invariance under changes in $b$7 underscores the insensitivity to certain matter sector deformations. In contrast, the critical dependence on the rotation parameter $b$8 manifests the essential role of angular momentum in enriching the phase structure (e.g., enabling the existence of IBH branches).

These results are consistent with previous topological thermodynamic studies of Reissner-Nordström and Kerr-AdS solutions, where $b$9 is also found for AdS settings supporting van der Waals analogies [Wei, Liu, Mann 2022 (Wei et al., 2022); Wu, Wu 2023 (Wu et al., 2023)].

The analytic continuation to the complex plane, with the corresponding residue theory, provides a powerful and mathematically rigorous alternative to classical vector field methods, extending the reach of topological classification to higher order and potentially nonlocal phenomena.

The framework holds significant implications for the gauge/gravity duality: the universality and invariance of $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$0 may have correspondents in dual quantum field theory phase structures. Furthermore, the coordinate-free nature of topological invariants supports their application to diverse gravitational backgrounds, including higher-curvature and non-Einsteinian theories.

Outlook and Future Developments

Potential research extensions include:

• Application of these topological tools to higher-dimensional black holes, or those involving additional fields and higher-derivative corrections, to probe the rigidity of universality classes.
• Exploration of the direct correspondence between topological invariants in black hole thermodynamics and order parameters in dual QFTs via AdS/CFT.
• Examination of topological changes under non-equilibrium processes such as quasinormal mode evolution, black hole mergers, or evaporation, utilizing the analytic machinery provided by the residue approach.
• Investigation into the manifestation of topological defects at the level of microstate counting and black hole entropy.

Conclusion

This work establishes that the thermodynamic phase structure of Kerr-Sen AdS black holes is governed by robust topological invariants, with the global charge $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$1 for the generic AdS case, and $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$2 in limiting regimes. The coordinate-independent nature of this classification, as well as the equivalence between $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$3-mapping and complex residue approaches, demonstrates the deep interplay between topology and black hole physics. The invariance under dilaton charge and critical dependence on spin and $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$4 delineate the landscape of possible phase structures and provide a template for future investigations into universality in gravitational thermodynamics.

(Figure 1)

Figure 1: Schematic diagram of contour integration in the complex plane, correlating thermodynamic defects with singularities of the analytically continued function $$P = -\frac{\Lambda}{8\pi}, \quad V = \left(\frac{\partial M}{\partial P}\right)_{S,J,Q}.$$5, whose residues encode the winding numbers and total topological charge.