Thermodynamic topology of Kerr-Sen black holes via R´enyi statistics

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I. Introduction
The space-time of the Kerr-Sen black hole (BH) can be obtained from low energy limit of heterotic string theory.It exhibits different properties from those of general relativity, and its hidden conformal symmetries [1] and uniqueness [2] has been widely investigated.It is well known that Kerr-Sen black holes possess the rotation, charge and symmetry properties of asymptotically flat spacetime such as Kerr-Newman black holes.The physical properties of Kerr-sen solution exhibit certain similarities to those observed in the Einstein-Maxwell theory, yet it also demonstrates notable distinctions in various significant aspects.In addition to the U(1) vector field and metric tensor, this solution is characterized by two extra background fields: the 3-form H and the dilaton scalar field Φ.The recent studies demonstrate that the weak cosmic censorship exists in Kerr-Sen black holes, even when taking into account the second-order perturbation inequality [3]; whereas, a (near-)extremal Kerr-Sen BH can be compromised by disregarding the self-force and radiative effects [4].The Kerr-Newman spacetime and Kerr-Sen solution have been extensively compared in various aspects, including the shadows and evaporation of black holes [5][6][7], gravitational capture regions of photons [8], gyromagnetic ratios [9] and the superradiant instability of bound state with a rotating and electrically charged massive body [10,11].The dyonic Kerr-Sen black holes solution derived from the Einstein-Maxwell-Dilaton-Axion (EMDA) theory has been extensively investigated, including its AdS extension.This includes studies on the conformal invariance [12], thermodynamic properties [13,14], black holes Shadows [15] and Chaos [16].
On the other hand, base on the standard Gibbs-Boltzmann (GB) approach, a zero-charged BH in asymptotically flat space background in the phase with negative specific heat capacity, which means that it is unstable to reach thermal equilibrium [17].In addition, Bekenstein's pioneering work [18] and the four laws of BH mechanics have been shown BH entropy is proportional to its area rather than its volume, which indicates that BH entropy should be taken as a nonextensive quantity in the BH backgrounds.The standard GB statistics may not be sufficient to calculate thermodynamic properties for extreme cases, such as BH systems with strong gravitational characteristics.In other words, the GB entropy formula exhibits limitations in the case of nonextensive long-range interaction BH system.The BH entropy S BH should encode the BH information with non-extensive nature, making a non-Boltzmannian statistics crucial.Several approaches, including holography [19][20][21], string theory [22][23][24] and loop quantum gravity [25,26] have been proposed as potential explanations for the non-extensive nature of BH entropy.Currently, the functional form of the BH entropy can potentially be expressed with respect to Rényi statistics incorporating a non-extensive parameter, which reflects the non-extensivity exhibited by black holes.To address the stability issue, it is possible to achieve a stable BH by utilizing the Rényi entropy instead of the conventional GB entropy [27][28][29][30][31][32].In addition, A possible relation between nonextensive parameter λ and the cosmological constant Λ first proposed in [30], is strengthened in charged black holes [33,34], de Rham-Gabadadze-Tolley BH [35], and Bardeen BH [36], etc, further evidence is provided from the topological aspect [37,38].
Topology have emerged as an available mathematical tool for investigating physical systems, focusing on general characteristics of the system rather than specific structures.Defects play an important role in revealing certain properties of field configurations.Since Wei [39] identified BH solutions as thermodynamic defects through the establishment of topological numbers, the study of the thermodynamic topology of black holes has sparked research interest .This includes exploring the topological structures of black holes in rotating spacetime and their corresponding AdS extension [46][47][48][49][50], as well as examining complex BH context [57].From a topological perspective, the positive and negative topological charges, namely the positive and negative winding numbers, can serve as indicators of local stability and instability characteristics of black holes.Furthermore, by summing up these topological charges, we can categorize BH solutions into three distinct classes to reflect the overall properties of the system.The results show that different black holes or even same ones may be are in the same class under different ensemble and parameter background; alternatively, they may belong to different topological class.
In this paper, we employ the nonextensive Rényi statistics to study the thermodynamic topology of Kerr-Sen and dyonic Kerr-Sen black holes.We calculate the topological number under Boltzmannian statistics and non-Boltzmannian statistics, and we show that Rényi parameter has a significant effect on the topological number.The outline of our paper is as follows: in section II, we begin with a review of thermodynamic topological method in Boltzmann statistics and introduce the generalized free energy represented by Rényi entropy.Then we investigate the topological number of the Kerr-Sen BH in the absence of cosmological constant Λ by utilizing the Rényi statistics.In section III, we discuss the Kerr-Sen-AdS BH via conventional GB entropy.In section IV, the topological number of dyonic Kerr-Sen BH is calculated in the framework of Rényi statistics.In section V, we extend the dyonic Kerr-Sen solution in section IV to the case of AdS with the GB statistics.Finally, our remarks and conclusions are given in section VI.

II. Kerr-Sen BH via the Rényi statistics
Firstly, we investigate Rényi thermodynamic topology of Kerr-Sen BH.Written in terms of the standard Boyer-Lindquist coordinates (t, r, θ, ϕ), the Kerr-Sen metric can be expressed as [69][70][71] where in which b = Q 2 /2M is the twist parameter, when b = 0, the above solution reduces to the Kerr geometry.M is the BH mass, J = M a is the angular momentum, and Q is the U(1) charge.Then we list the thermodynamic quantities such as mass, Hawking temperature and entropy of this BH as follows In terms of GB statistics, the generalized off-shell Helmholtz free energy function can then be given by [39] This free energy appears its off-shell characteristics except at τ = 1/T BH .Based on the Rényi statistics, we can rewrite the generalized off-shell free energy as follows The Rényi entropy S R can be expressed as S R = 1 λ ln(1+λS BH ) [22].The acceptable range for the nonextensive parameter λ is −∞ < λ < 1 [72], otherwise the entropy function becomes ill-defined due to its convexity.In the current study on BH thermodynamics using Rényi statistics, the entropy S R is always well defined for the parameter λ within the range 0 < λ < 1, with λ exhibiting favorable thermodynamic properties in this interval, as demonstrated in [30].When the Rényi parameter λ → 0, the generalized off-shell free energy returns to the GB statistics case.We establish a vector ϕ as [39] where the parameter Θ is an auxiliary item for convenient and intuitive topological analysis, it's introduction is for the purpose of axial limit [73], and Θ obeys 0 ≤ Θ ≤ π.Notably, when Θ = 0, π, the component ϕ Θ is divergent and the direction of the vector points is outward.Utilizing Duan's theory of ϕ-mapping topological currents, we define a topological current as follows [74-76] where a, b = 1, 2 and . n is a unit vector defined by n = (n 1 , n 2 ), where The conservation of the topological current, i.e., ∂ µ J µ = 0, can be easily verified.Finally, the topological number in a parameter region is calculated as where w i is the winding number.In this section, we would like to calculate the topological number of the Kerr-Sen BH via Rényi statistics.According to Eqs. (3)(5) (7).We obtain the generalized free energy as we construct a vector field ϕ whose components read By considering the Eq. ( 12) to be zero, a curve on the r + − τ R plane can be obtained.In the case of a Kerr-Sen BH, one can reach the following: We plot the curve between r + and τ R with fixed a/r 0 = 0.1, b/r 0 = 0.2 and different values of λ in FIG. 1.Interestingly, annihilation/generation points occur at different values of λ.The critical nonextensive parameter λ c can be obtained with the condition By solving this equation, we obtain λ c = 0.57.We previously mentioned that when the parameter λ = 0, the generalized free energy returns to the GB statistical case, from FIG. 1(a), one generation point is located at τ c /r 0 = 6.6523.For τ > τ c , there are two BH branches, the solid red line is the stable BH branch, the dashed blue line is the unstable BH branch.In FIG.2(a), we plot the unit vector field n with τ /r 0 = 8, the zero points can be found at (r + /r 0 , Θ) = (0.15, π/2), and (r + /r 0 , Θ) = (0.40, π/2), respectively (in this paper, the zero points of all vector fields n are given from left to right).We have

III. Kerr-Sen-AdS BH via GB statistics
In Boyer-Lindquist coordinates, the Kerr-Sen-AdS BH solution [77] is expressed as where Through the horizon constraint can be derived by solving the equation ∆ r = 0. Then the mass and other thermodynamic variables can be expressed by the radius of the event horizon the generalized free energy in GB statistics takes the form where P = 3 8πl 2 is the thermodynamic pressure.According to the definition of the vector ϕ, its components are by solving ϕ r+ = 0 , we can obtain The graph of r + vs τ is plotted in FIG.3(a) for a/r 0 = 0.1, b/r 0 = 0.2 and P r 2 0 = 0.01 (below critical pressure).Interestingly, we find the curves of equation ( 24) look similar to the case of Kerr-Sen BH via Rényi statistics with λ < λ c .The generation and annihilation points are obtained at τ a /r 0 = 6.2078 and τ b /r 0 = 12.1509.FIG.3(b) represents the unit vector field n with τ /r 0 = 10.The zero points are at (r + /r 0 , Θ) = (0.13, π/2), (r + /r 0 , Θ) = (0.82, π/2) and (r + /r 0 , Θ) = (3.63,π/2), respectively.Thus the topological number of Kerr-Sen-AdS BH via GB statistics is 1, which is same as the Kerr-Sen BH from Rényi statistics.

IV. Dyonic Kerr-Sen BH via the Rényi statistics
The EMDA theory is considered to find dyonic Kerr-Sen BH, the corresponding BH spacetime metric can be written as follows: [14] where where parameters m, q, a, k, d, p represent mass, electric charge, spin, axion charge, dilaton charge, and magnetic (dyonic) charge of the BH, respectively.Here, d = p 2 −q 2 2m , and k = pq m , so when p = q, the dilaton charge d will vanish, and when p = 0, one have a vanishing axion charge k (k = 0), in this case the Kerr-Sen BH can be reduced.The Kerr BH solution can be obtained by considering p = q = 0.In the following calculation, we set p ̸ = q ̸ = 0.The expression for m can be derived by ∆ = 0.The other quantities are given by Next, we investigate the topological numbers of the dyonic Kerr-Sen BH via Rényi statistics.The definition of the generalized free energy is employed to obtain The following equation provides the zero points of the component ϕ r+ thus τ c /r 0 τ 1 /r 0 Similar to the analysis in section II.For λ = 0 (the resulting r + vs τ as show in FIG.4(a)), when we set a/r 0 = 0.1, d/r 0 = 0.1, p/r 0 = 0.2 and q/r 0 = 0.1, one have τ c /r 0 = 5.8606, when τ > τ c with τ /r 0 = 10, as show in FIG.5(a), the zero points of ϕ r+ are located at (r + /r 0 , Θ) = (0.31, π/2) and (r + /r 0 , Θ) = (0.85, π/2) and the corresponding topological number w 1 = 1, w 2 = −1, the total topological number of the dyonic Kerr-Sen BH via GB statistics is W = 0. From FIG. 4(b), when λ < λ c (λ c = 0.68 is calculated from equation ( 15)), one have τ a /r 0 = 5.7153 and τ b /r 0 = 11.3595.The unit vector field n at τ /r 0 = 9 is plotted in the FIG.5(b).The zero points from left to right are at (r + /r 0 , Θ) = (0.31, π/2), (r + /r 0 , Θ) = (0.92, π/2) and (r + /r 0 , Θ) = (3.67,π/2) and corresponding topological number w 1 = 1, w 2 = −1 and w 2 = 1, so W = 1.For λ > λ c , we can see from FIG. 4(c) and FIG.5(c), we have W = 1.Thus, the Kerr-Sen BH and dyonic Kerr-Sen BH are in the same topological class under both Rényi and GB statistics in this case.Now let's change at least one of the parameters, such as we set a/r 0 = 0.1, d/r 0 = 0.2, p/r 0 = 0.2 and q/r 0 = 0.1.When λ = 0, from FIG. 6(a) one find there is an unstable BH branch, FIG.7(a) shows the topological number W = −1.When we fix the parameter λ = 0.01, FIG.6(b) shows that there is an unstable BH branch (dashed blue line) and a stable BH branch (solid red line), they have topological numbers w 1 = −1 and w 1 = 1, respectively.A annihilation point is located at τ c /r 0 = 35.4486.The unit vector field n at τ /r 0 = 34 is plotted in the FIG.7(b).The zero points are (r + /r 0 , Θ) = (4.42,π/2) and (r + /r 0 , Θ) = (7.75,π/2).Thus, the topological number of the dyonic Kerr-Sen BH via Rényi statistics is W = 0.In such parameter space, the Kerr-Sen BH and dyonic Kerr-Sen BH belong to different topological class with both Rényi and GB statistics.V. Dyonic Kerr-Sen-AdS BH via GB statistics We extend the dyonic Kerr-Sen solution in previous section to the AdS case with GB statistics.The metric for the dyonic Kerr-Sen-AdS BH has the form [14] where The ADM mass M , and other thermodynamic quantities in AdS spacetimes are related as and we use the following generalized free energy Thus, the vector components can be written as where and so we have For a/r 0 = 0.1, d/r 0 = 0.1, p/r 0 = 0.2, q/r 0 = 0.1 and P r 2 0 = 0.01(P < P c ), from FIG. 8, we plot the r + vs τ graph and the unit vector field n.Similar to the Kerr-Sen-AdS BH via GB statistics, for τ a < τ < τ b , the stable branches of large and small black holes are represented by solid red and black lines, the zero points on these branches having a topological number of w = 1, the blue dotted curve represents an unstable intermediate BH branch, with the topological number of the zero point on that branch being w = −1.Thus, the topological number of dyonic Kerr-Sen-AdS BH is W = 1 + 1 − 1 = 1.The generation and annihilation points at τ a /r 0 = 5.6876 and τ b /r 0 = 12.6096, respectively.The zero points in FIG.8(b) from left to right are at (r + /r 0 , Θ) = (0.31, π/2), (r + /r 0 , Θ) = (1.04,π/2) and (r + /r 0 , Θ) = (4.11,π/2) with τ /r 0 = 10.Such a parameter space implies that the topological numbers of the dyonic Kerr-Sen-AdS BH and Kerr-Sen-AdS BH are the same.
Similarly, if we change the parameter space, i.e, a/r 0 = 0.1, d/r 0 = 0.2, p/r 0 = 0.2 and q/r 0 = 0.1 and P r 2 0 = 0.1.From FIG. 9 we find the curves of r + vs τ look similar to the case of dyonic Kerr-Sen BH via the Rényi statistics with λ = 0.01.We can obtain the topological number W for the case is equal to 0, which is different from the case of Kerr-Sen-AdS.

VI. CONCLUSIONS
In this paper, we investigated thermodynamic topology of Kerr-Sen BH and dyonic Kerr-Sen BH in the context of Rényi statistics approach.By representing the Hawking-Bekenstein entropy of BH horizons as a non-extensive Rényi entropy, one obtained generalized off-shell free energy on the Rényi entropy.The findings of our study as presented in this paper are now summarized in the TABLE I.In doing this study, we have taken into consideration the Kerr-Sen BH and dyonic Kerr-Sen BH within the framework of Rényi statistics, the corresponding topological number shows different value compared to the conventional GB statistics.It is worth noting that the non-extensive parameter λ, as an additional parameter, changes the topological number.Moreover, the topological classifications of Kerr-Sen and dyonic Kerr-Sen black holes remain consistent in both GB and Rényi statistics.Indeed, we have demonstrated that there are places in the parameter space where Kerr-Sen and dyonic Kerr-Sen black holes belong to the same (different) topology class in both GB and Rényi statistics.
Furthermore, we calculated the topological numbers of Kerr-Sen BH and dyonic Kerr-Sen BH in the presence of the cosmological constant Λ with GB statistics.The topological numbers of Kerr-Sen-AdS and dyonic Kerr-Sen-AdS black holes by using GB entropy were found to be consistent with these two types of black holes in the absence of cosmological constant with Rényi entropy processes.Such an analogy suggests a significant link between the nonextensive Rényi parameter λ and the cosmological constant Λ from topology perspective.One intriguing question is other black holes solutions and high-dimensional space-time also consider such an analysis in [38].There is no doubt that, the above mentioned equivalence require more scrutiny in other systems, possibly involving reentrant phase transitions and the triple point phenomenon.It may be worthwhile to explore the application of Rényi statistics in thermodynamic topology and investigate potential deviations from conventional GB statistics, which could provide insights into the nature of correlations within the system.Additionally, the topological analysis framework based on the nonextensive Rényi formalism is an essential approach for comprehending the nontrivial thermodynamical behavior from a thermodynamic topology perspective.We hope that issues beyond the present report will be addressed in future work.

FIG. 3 :
FIG. 3: Topological properties for Kerr-Sen-AdS BH via GB statistics.The left figure (a) reprensents the zero points of ϕ r+ in τ − r + plane.The right figure (b) reprensents the unit vector field n on the r + − Θ plane with τ /r 0 = 10.

FIG. 8 :
FIG. 8: Topological properties for dyonic Kerr-Sen-AdS BH via GB statistics.The left figure (a) reprensents of the zero points ϕ r+ in τ − r + plane.The right figure (b) reprensents the unit vector field n on the r + − Θ plane with τ /r 0 = 10.

FIG. 9 :
FIG. 9: Topological properties for dyonic Kerr-Sen-AdS BH via GB statistics.The left figure (a) reprensents the zero points of ϕ r+ in τ vs r + plane.The right figure (b) reprensents the unit vector field n on the r + − Θ plane with τ /r 0 = 2.

TABLE I :
In following table, we summarize the topological numbers associated with Kerr-Sen and dyonic Kerr-Sen black holes by using Rényi entropy processes, are well as the AdS cases of such two black holes via GB entropy, the annihilation point (AP) and generation point (GP) of black holes are also given.