Thermodynamic topology of black Holes in f(R) gravity

In this work, we study the thermodynamic topology of a static, a charged static and a charged, rotating black hole in $f(R)$ gravity. For charged static black holes, we work in two different ensembles: fixed charge$(q)$ ensemble and fixed potential$(\phi)$ ensemble. For charged, rotating black hole, four different types of ensembles are considered: fixed $(q, J)$, fixed $(\phi, J)$, fixed $(q,\Omega)$ and fixed $(\phi,\Omega)$ ensemble, where $J$ and $\Omega$ denotes the angular momentum and the angular frequency respectively. Using the generalized off-shell free energy method, where the black holes are treated as topological defects in their thermodynamic spaces, we investigate the local and global topology of these black holes via the computation of winding numbers at these defects. For static black hole we work in three model. We find that the topological charge for a static black hole is always $-1$ regardless of the values of the thermodynamic parameters and the choice of $f(R)$ model. For a charged static black hole, in the fixed charge ensemble, the topological charge is found to be zero. Contrastingly, in the fixed $\phi$ ensemble, the topological charge is found to be $-1.$ For charged static black holes, in both the ensembles, the topological charge is observed to be independent of the thermodynamic parameters. For charged, rotating black hole, in fixed $(q, J)$ ensemble, the topological charge is found to be $1.$ In $(\phi, J)$ ensemble, we find the topological charge to be $1.$ In case of fixed $(q,\Omega)$ ensemble, the topological charge is $1$ or $0$ depending on the value of the scalar curvature($R$). In fixed $(\Omega,\phi)$ ensemble, the topological charge is $-1,0$ or $1$ depending on the values of $R,\Omega$ and $\phi.$

Black hole thermodynamics has advanced significantly over the last fifty years since its initiation in the 1970s [1][2][3][4][5][6][7][8][9][10][11][12][13], culminating in new frameworks of study such as extended black hole thermodynamics [14][15][16][17][18][19][20][21][22][23][24][25][26], restricted phase space thermodynamics and holographic thermodynamics [27][28][29][30][31][32][33][34][35][36][37].A relatively recent development in the context of understanding critical phenomena in black hole thermodynamics is the introduction of topology in black hole thermodynamics [38,39].In the approach shown in [39] and known as the off-shell free energy method, the topology of black hole thermodynamics can be studied by treating the black hole solutions as topological defects in their thermodynamic space.The local and global topology of the black hole are then analyzed by computing the winding numbers at these defects.Based on the total winding number or the topological charge, all black hole solutions are conjectured to be classified into three topological classes.Such studies of thermodynamic topology have been generalized to a variety of black hole solutions in different theories of gravity .
In the off-shell free energy method, one begins with an expression for the off-shell free energy of a black hole with arbitrary mass given by: Here, E and S are the energy and entropy of the black hole respectively.τ is the time scale parameter which can be considered as the reciprocal of the cavity's temperature that encloses the black hole : Here, T is the equilibrium temperature at the surface of the cavity and the time parameter τ is set freely to vary.Utilizing the generalized free energy, a vector field is constructed as follows [39] : Where θ = π/2 and τ = 1/T represent a zero point of the vector field ϕ.The topological property associated with the zero point of a field is its winding number or topological charge.The topological charge can be calculated by constructing a contour C around each zero point which is parametrized as: where ν ∈ (0, 2π) followed by calculating the deflection of vector field n along the contour C as: The unit vectors n 1 , n 2 are given by: Finally, the winding numbers w and topological charge W can be calculated as follows: In cases where the parameter region does not encompass any zero points, the overall topological number or charge is determined to be 0.This approach for computing the topological number or charge is referred to as Duan's ϕ mapping technique [84,85].
An alternative method used to calculate the winding number has been proposed in [74].In this approach, the winding number (w i ) for each solution can be calculated by using the residue theorem.First, a solution for τ is obtained for the following equation.

∂F ∂r +
= 0 the solution for τ , thus obtained, is a function of the horizon radius r + .This is followed by replacing r + with a complex variable z. and renaming the solution to equation 7 as G(z).Then a rational complex function R(z) is constructed as follows : In the final step, the residues at the poles of R(z) are computed to find the winding number at the defects.
The total winding number or the topological charge, W is given by W = i w i .
In this study, we investigate the thermodynamic topology of black holes within the framework of f (R) gravity.Applications of modified theories of gravity have recently garnered significant attention across several fields of theoretical physics.Among these alternatives to Einstein's gravity, f (R) theories which incorporate features of both cosmological and astrophysical significance, are particularly noteworthy [86][87][88][89][90].In f (R) gravity, the gravitational action is expressed as a general function of the scalar curvature R.Various facets associated with modified theories of gravity, including black hole solutions,cosmic inflation, cosmic acceleration, cosmic rays, dark matter, correction of solar system anomalies etc have been studied within the realm of f (R) gravity .
For our analysis, We have considered three black hole solutions in f (R) gravity.These are: static black holes, static charged black holes, and rotating charged black holes.For static black hole, we consider three different f (R) model.For the charged static black hole, we work in two ensembles.In one ensemble, the charge q is kept fixed, while in the other, its conjugate potential, ϕ is kept fixed.The rotating charged black hole is analyzed in four ensembles: fixed (q, J), fixed (ϕ, J), fixed (q, Ω), and fixed (ϕ, Ω).In this paper, we address the issues related to the dependence of thermodynamic topology on the f (R) model and the choice of ensemble.In particular, the choice of ensemble is often found to be a determining factor in the nature of thermodynamic properties and phase transitions of black holes [143][144][145][146][147][148].For all these black holes we study their thermodynamic topologies by computing the topological charge in different ensembles with different values of thermodynamic parameters.This paper is organized into the following sections: in Section II, we have studied the thermodynamic topology of static black holes in f (R) gravity where we have considered three f (R) models in the subsections Model I, Model II and Model III respectively.In Section III, subsection I, we have analyzed the thermodynamic topology of charged static black hole in fixed charge ensemble followed by extending the same in fixed ϕ ensemble in subsection II.In Section IV, we examine the rotating charged black hole in fixed (q, J)(Subsection I), fixed (ϕ, J)(Subsection II), fixed (q, Ω)(Subsection III), and fixed (ϕ, Ω)(subsection IV).Finally, the conclusions are presented in Section V.

II. STATIC BLACK HOLE IN f (R) GRAVITY
We consider a static black hole solution in f (R) gravity which appears as a solution to the following action [140] Varying the action with respect to the metric gives : for vacuum space where dR and □ = ∇ α ∇ α .The generic form of the metric for spherically symmetric space-time is : where M (r) = 1 N (r) .

MODEL I
The first model considered in this work is given by [105] : To find the static black hole solution using this model we will be using a technique showcased in [149,150].
Contracting the equation 11, we obtain : Differentiating the above equation we get the consistency relation as : Using equation 14, modified Einstein's field equation becomes : So, any solution of equation 16 and 11 must satisfy the relation 15.Equation 16 can be seen as a set of differential equations for F (r), M (r), and N (r) since the metric only depends on r.As only diagonal elements are non-zero for the metric, we get four equations.Considering we get two equation as follows : where Y = M (r)N (r) For the model we have considered here, we will have to take solution with a constant curvature.Hence taking F ′ = 0, F ′′ = 0, equation 17 and 18 takes the form : solving equation 19 we obtain The Schwarzschild-de Sitter-spacetime(SdS), which is the Schwarzschild solution in the presence of a cosmological constant has the form Here The relation between constant scalar curvature R and cosmological constant in this case is given by comparing equation 20 with equation 21, we get constant curvature R = R 0 can be obtain from equation 14 as : For the model we have considered in this section and N (r) of the solution 12 will take the form : where Solving the equation N (r + ) = 0 (r + is the event horizon radius), we calculate the mass as : The temperature is calculated as : and the entropy is given by : Using equation 24 and equation 25, the free energy F = M − S/τ is found to be: The components of the vector ϕ are found to be The unit vectors n 1 , n 2 are computed using the following prescription : The expression for τ corresponding to zero points is obtained by setting ϕ r = 0.
We plot τ vs r + plot for κ = 0.005 in figure .1a,where we observe one single black hole branch.In figure 1b, vector plot is shown for ϕ r and ϕ θ component taking τ = 100, where we observe the zero point of the vector field at r + = 4.9879.From figure.1c it is observed, that the winding number or the topological charge corresponding to r + = 4.9879 is found to be −1 which is represented by the black-colored solid line.Further analysis shows, that for all values of κ, the topological charge is always −1.

MODEL II
The second model we have considered is [104] : where Here, n = 1, 2, 3.... is a positive integer.R 0 is the current curvature (R 0 ∼ (10 −33 eV ) 2 ) and Λ i is the effective cosmological constant (Λ i ∼ 10 20∼38 ).The details about this model are available in [104].For simplicity, we take n = 1 and following the above-mentioned procedure, we find out the second order approximated metric solution of equation 12 : here, The mass is calculated as: The temperature is given by: And the entropy is calculated as : from equation 33 and 35, free energy is calculated as : The components of the vector ϕ are found to be The expression for τ corresponding to zero points is obtained by setting ϕ r = 0.
Next, we plot τ vs r + plot taking R 0 = (10 −33 ) 2 eV and Λ i = 10 26 in figure.2a,where we again observe one single black hole branch whose topological charge is −1.The Vector plot in Figure 2b and contourplot 4c shows the same.
In fact the topological charge is always −1 for all values of R 0 and Λ i .

MODEL III
The next model that will be used in this work is [150,151]: where R c is the integration constant and R 0 = 6α 2 /d 2 ;d and α are the free parameters of the action.Λ is the cosmological constant.
The metric solution of equation 12 is obtained as [150,151]: where β = α/d ≥ 0 is a real constant.From equation 41 ,mass M can be obtained by setting N (r = r + ) = 0,which gives: We have used Λ = 3 l 2 [152] in which l is the radius of curvature of the de sitter space.The entropy can be obtained by using the radius of the event horizon as: Using equation 42 and equation 43, the free energy F = M − S/τ ,for static black hole in this f (R) model is found to be: The components of the vector ϕ are found to be The expression for τ corresponding to zero points is obtained by setting ϕ r = 0.
It is to be noted that, not all values of r + and l are allowed for specific values of β.The allowed combination of values of r + and l 2 for β = 1 are shown in Figure 3

III. CHARGED STATIC BLACK HOLE IN f (R) GRAVITY
The second black hole solution in f (R) gravity that we have chosen to study is a charged static black hole solution originating from the following action [140] Varying the action with respect to the metric gives : dR and T µν is the stress-energy tensor of the electromagnetic field.
The trace of the above equation at R = R 0 results in : which eventually gives the constant curvature scalar as: Finally, the metric of the spherically symmetric spacetime is obtained as follows: Where, for the details about the metric see [141].
By putting G = 1 and in the above equation: where R 0 = 4Λ= 12 l 2 is the constant curvature

III.1. Fixed charge Ensemble
For charged static black hole, from the equation 50,mass M in canonical ensemble is obtained as [140] : In the context of the charged static black hole and rotating charged black hole, formulating the vector field component in terms of the horizon radius r + is difficult, particularly when working in different ensembles.Consequently, we have performed all calculations in terms of the entropy S. Using equation 51, free energy is computed as : The components of the vector field ϕ is obtained as : and The zero points of ϕ r is also obtained as: For all values of q and l, the static charged black hole has topological charge equal to 0. At q = 0.25 and l = 10, for example, we encounter two black hole branches with total topological charge equaling 1 − 1 = 0 as shown in figure.5.In figure.5a,τ vs S is plotted in the allowed range of S. For τ = 9, there are two zero points located at S = 0.38868 and S = 1.033022 as shown in figure 5b.From figure.5c, the winding numbers corresponding to S = 0.38868 and S = 1.033022 (represented by the black and the red colored solid line respectively) are found to be −1, and +1 respectively.The total topological charge of the black hole, therefore, is 1 − 1 = 0.If we set q = 0, then we encounter one single black hole branch with topological charge −1 as shown in figure.6.In Figure .6aτ vs S is plotted in the allowed range of S. For τ = 10, zero point is located at S = 1.91726 as shown in Figure 6b.From figure.6c, the winding number corresponding to S = 1.91726 (represented by the black-colored solid line) is found to be −1.So, when the charge q equals zero, the thermodynamic topology of static charged black holes becomes equivalent to that of static black holes as expected.

III.2. Fixed potential(ϕ) Ensemble
In fixed ϕ ensemble, we define a potential ϕ conjugate to q and keep it fixed.These two parameters are related by The mass in the fixed ϕ canonical ensemble is given by : Accordingly, the free energy is calculated using: or The components of the vector field are obtained as : and Finally, we obtain the expression for τ as : It is seen that for all the values of ϕ and l, static charged black hole in the fixed potential ensemble has a topological charge equal to −1.In Figure .7 τ vs S plot is shown where ϕ = 0.5 and l = 10.Here, we observe a single black hole branch.For τ = 6, the zero point is located at S = 0.39878..This is also confirmed from the vector plot of n in the S − θ plane as shown in Figure .7b.To find out the winding number/topological number associated with this zero point, we perform a contour integration around S = 0.39878.which is shown in Figure .7c.The topological charge, in this case, is equal to −1.We have explicitly verified that the topological charge of any zero point on the black hole branch remains the same and is equal to −1.Strikingly, in the fixed potential ensemble, the topological charge is different from that in the fixed charge ensemble.

IV. ROTATING CHARGED BLACK HOLE
In this section, we study the thermodynamic topology of a rotating charged black hole solution in f (R) gravity for four ensemble: fixed (q, J), fixed (ϕ, J), fixed (q, Ω) and fixed (ϕ, Ω) ensemble, where q, ϕ,J and Ω denotes the charge, potential, angular momentum and the angular frequency respectively.The black hole of our interest originates from the following action : where the first part represents the gravitational action and the second part represents the four dimensional Maxwell term.R is the scalar curvature and R + f (R) is a function of scalar curvature.The field equations in the metric formalism are [142] where, ∇ is the covariant derivative,R µν is the Ricci tensor, and T µν is the stress-energy tensor of the electromagnetic field given by: The trace of equation 63 gives the expression for R = R 0 as: The axisymmetric ansatz, utilizing Boyer-Lindquist type coordinates (t, r, θ, φ), derived from the Kerr-Newman-Ads black hole solution, as shown in [142] is: where in which R 0 = −4Λ, Q is the electric charge and a is the angular momentum per mass of the black hole.By setting dr = dt = 0 in equation 65, we can calculate the area of the two-dimensional horizon, which eventually gives the expression for area as [140]: where r + is the radius of the horizon.The expressions for total mass and the angular momentum are [142] : and From which the generalized Smarr formula of the rotating charged black hole is obtained as [142]: IV.1.Fixed (q, J) ensemble In the fixed (q, J) ensemble, we keep q and J as fixed parameters.The mass expression, given by equation 69, remains unchanged within this ensemble,which is : The off-shell free energy is calculated to be: The components of the vector field ϕ are obtained as : The expression for τ corresponds to the zero point of ϕ S can be obtained by setting ϕ S = 0.  8: τ vs S plots for a rotating charged black hole in the fixed (q, J) ensemble at q = 0.05, J = 1.5, l = 0.1.
where we have substituted R 0 = − 12 l 2 .We plot the entropy S against τ for a fixed length scale l = 0.1 while keeping J and q fixed at J = 1.5 and q = 0.05 which is shown in figure 8.Here we observed a single black hole branch.In this section, we have adopted another method to calculate the topological charge [74].The complex function R rc (z) defined in section I is given by : Considering the denominator of the equation as a polynomial function A(z) we can calculate the poles of R rc (z).
(73) To get the winding number, we put l = 0.1, J = 1.5 and q = 0.05 in equation 73.It is seen from figure 9, for τ = 0.008, the pole is at z = 86.108.The winding number can be calculated by finding the sign of the residue of the equation 72 around the pole z = 86.108.We find a positive valued residue in this case.Hence, the winding number or the total topological charge is 1.It is found that if the value of l is decreased below l = 0.1, no of branch and topological charge remains the same.
In figure.10,we have plotted S against τ when length scale is increased to l = 10 while keeping q = 0.05 and J = 1.5 fixed.Here we observe three black hole branches: a small, an intermediate, and a large black hole branch.We also According to the sign of the residue around these three poles, the winding numbers are found to be w 1 = +1, w 2 = −1 and w 3 = +1 respectively.Hence the topological charge is: Similarly, for a small black hole branch, taking τ = 33, the pole is at z = 17.9982 and according to the sign of residue around the pole, the winding number is found to be +1.We have checked that even if the value of l is increased further (i.e.beyond l = 10), the number of branches and topological charge remains the same.
We repeat the analysis for different values of J and q, keeping l constant.We observe no effect on the topological charge.The same is illustrated in figure 12.In figure 12a, figure 12b and figure 12c we show the effect of change in charge q when J and l are kept fixed at J = 1.5 and l = 10.In figure 12a the charge is changed to a significantly small value q = 0.0001.Here, we observe three black hole branches and the topological charge is found to be 1.In figure 12b where we set q = 1.3(figure 12b), again we find the number of black hole branches equal to three and topological charge to be one.When the value of the charge is changed to q = 2 in figure 12c, the number of branches becomes one but the topological charge remains one.For fixed values of l = 10 and q = 0.05 the effect of variation in J on FIG.
12: τ vs S plots for rotating charged black hole in fixed (q, J) ensemble when charge q is varied for a fixed length l = 10 while keeping J fixed at J = 1.5.Figure (a) shows τ vs S plot at q = 0.0001,J = 1.5, l = 10, figure (b) shows τ vs S plot at q = 1.3,J = 1.5, l = 10 and figure (c) shows the same at q = 2,J = 1.5, l = 10.W denotes total topological charge.
the thermodynamic topology is demonstrated in figure 13a and figure 13b.For J = 1, three black hole branches are observed with the sum of the corresponding winding number equal to 1. as shown in figure 13a.In figure 13b we set J = 7 and find a single black hole branch with topological charge equal to 1.We have explicitly verified that even for other values of J, the topological charge remains the same.
From our analysis, we conclude that the topological charge of the rotating charged black hole in fixed (q, J) ensemble is equal to +1 and is unaffected by the variation in the thermodynamic parameters l, q, J.

IV.2. Fixed (ϕ,J) ensemble
In fixed (ϕ, J) ensemble, the potential ϕ and angular momentum J are kept fixed.The potential ϕ is given by : Solving equation 74, we get an expression for q and find out the new mass(M ϕ ) in this ensemble as follows : The off-shell free energy is computed using: Following the same procedure as shown in the previous subsection, we calculate the expressions for ϕ S and τ .We plot τ vs S curve for two different values scalar curvature R as shown in figure 14.Here, J and ϕ are kept constant with J = 1.5 and ϕ = 0.05.In figure 14a, we set R = −0.1 and find three black hole phases phase with topological charge equaling one.In figure 14b for R = −4, we observed a single black hole branch with topological charge still equaling one.We have checked that for the other values of R, the topological charge remains equal to one.FIG.13: τ vs S plots for rotating charged black hole in fixed (q, J) ensemble when J is varied for a fixed length l = 10 while keeping the charge fixed at q = 0.05. Figure (a) shows τ vs S plot at q = 0.05,J = 1, l = 10 and figure (b) shows the same at q = 0.05,J = 7, l = 10.W denotes the corresponding topological charge.Next, we study the influence of changing J on the topological charge with R and ϕ kept fixed at R = −0.1 and ϕ = 0.05. in figure 15.In figure 15a we observe three black hole branches leading to a topological charge of 1.In figure 15b, we again encounter three branches and topological charge equal to 1 for J = 7.In figure 15c, with J = 10 we get one black hole branch with topological charges equaling 1.We have explicitly verified that for other values of J the topological charge remains the same.Finally, we analyzed the effect of ϕ on topological charge with fixed values of R and J., τ vs S plots for J = 1.5 and R = −0.1 are shown in figure 16a and figure 16b by setting ϕ = 0.005 and ϕ = 3 respectively.While in figure 16a we find a single black hole branch, in figure 16b) we see three branches.In both cases, the topological charge is found to be 1.The same is found to be true for other values of ϕ with J and R kept fixed.Therefore we infer that the topological charge of the rotating charged black hole under consideration in fixed (ϕ, J) ensemble is equal to one irrespective of the values of the thermodynamic parameters ϕ, J and R. IV.3.Fixed (Ω, q) ensemble Next, we work in the fixed (Ω, q) ensemble where the angular frequency Ω and charge q are kept fixed.We begin with solving the following equation for the angular momentum J From 76 we obtain the expressions for J as follows: The new mass (M Ω ) in this ensemble as : M ) Accordingly, the off-shell free energy is computed using: Repeating the procedure alluded to in the previous section, we obtained the expression for ϕ S and τ .First, we analyzed the dependence of topological charge on the scalar curvature R keeping Ω and q fixed.In figure 17a and 17b, τ vs S plots are shown with Ω = 0.1, q = 0.1 kept fixed and R = −0.01 and R = −4 respectively.In 17a two black hole branches and topological charge 0 are observed.In 17b, we have three black hole branches totaling a topological charge of 1.Therefore, depending on the value of the scalar curvature, the topological charge is found to be either 0 or 1.For other values of R, we arrived at the same topological charge(0 or 1) Now we want to understand the impact of change in charge q on the topological charge.In figure 18a and figure 18b, we set Ω = 0.1,R = −4 and q = 0.09 and 1 respectively.In figure 18a we find three black hole branches, in figure 18b we find a single black hole branch.The topological charge in both cases equals 1.
In figure 19a and 19b we repeat the same analysis with Ω = 0.1 R = −0.01 and q = 0.09 and q = 1 respectively.The number of branches and the topological charge in both cases are found to be identical(two black hole branches and topological charge 0).Hence it is found that the topological charge does not change with the charge q although the number of black hole branches may vary with a variation in q.FIG.19: τ vs S plots for rotating charged black hole in fixed (Ω, q) ensemble for different q value at R = −0.01 and Ω = 0.1.Figure (a) shows τ vs S plot at q = 0.09, figure (b) shows τ vs S plot at q = 1.The topological charge for all the cases is 1.W denotes the topological charge.
Finally, we examine the role of Ω in determining the topological charge.For that, we keep R fixed R = −0.01 and q = 0.1 in figures 20a and figure 20b.In Figure 20a and 20b we set Ω = 1 and Ω = 3 respectively.As seen from the figure the topological charge remains unaffected by the variation in Ω and is equal to 0. In conclusion our results indicate that the topological charge of the rotating charged black hole in fixed (Ω, q) is either 0 or 1 depending on the values of R. The other thermodynamic parameters Ω and q however, do not have any impact on the topological charge.The last ensemble in which we conduct our analysis is the fixed (Ω, ϕ) ensemble.In this ensemble Ω and ϕ are kept fixed.First, We substitute J from equation 77 in the expression for mass in equation 69 as follows: Now equation 80 becomes independent of variable J. From equation 80, we compute ϕ as: Accordingly, q is given by equation 81 as: The new expression for angular momentum(J) is given by: Finally, the modified mass in fixed (Ω,Φ) ensemble is written as : Using equation 83, F, ϕ S and τ are constructed following standard procedure.
We plot τ vs S curve for different values of R as shown in figure 21 keeping Ω = 0.1, ϕ = 0.1 constant.In figure 21a, with Ω0.1, ϕ = 0.1 and R = −0.01,one black hole branch with topological charge W = −1 is observed.With the same values of Ω and ϕ but for different values of R at R = −4, in figure 21b two black hole branches and topological charge 0 are found.Next in figures 22a and figure 22b, we fix ϕ = 0.1,R = −4 and vary Ω.In figure 22a setting Ω = 0.001 we get two black hole branches and topological charge W = 0.In figure 22b with Ω = 1 a single blackl hole branch with topological charge W = −1 is observed.
Finally, we probe the thermodynamic topology with reference to a variation in the potential ϕ in figure 23a and 23b we set Ω = 0.1, R = −4 and ϕ = 0.001 and ϕ = 2 respectively.While in the first case, we find two black hole branches and winding number W = 0, in the latter case a single black hole branch with topological charge 1 is found.
We continue our study at a different value of R equal to −0.01 at figure 24a and figure 24b.In figure 24a, Ω = 0.1 and ϕ = 0.001.In figure 24b, Ω is again fixed at 0.1 but ϕ is changed ϕ = 2 for ϕ = 0.001 a single black hole branch with topological charge W = −1 is seen.For ϕ = 2 we again encounter a single black hole branch, but this time with a topological charge of +1.
So to summarize, the topological charge for the rotating charged black hole in fixed (Ω, ϕ) ensemble is −1,0 or +1  depending on all the thermodynamic parameters R, Ω and ϕ.V. CONCLUSION In this study, we have analyzed the thermodynamic topology of a static black hole, a charged, static black hole, and a charged rotating black hole within the framework of f (R) gravity.We have considered two distinct ensembles for charged static black holes: the fixed charged(q) ensemble and the fixed potential(ϕ) ensemble.For charged, rotating black holes, we have explored four different ensembles: fixed (q, J), fixed (ϕ, J), fixed (q, Ω), and fixed (ϕ, Ω) ensembles.Considering these black holes as topological defects at thermodynamic spaces, We have computed the associated winding numbers or the topological charges to study the local and global topologies of these black holes.It has been observed that for the static black hole, the topological charge remains constant at −1, irrespective of the three models we have considered and the respective thermodynamic parameters of the model.
In the case of the static charged black hole in a fixed charge ensemble, the topological charge is computed to be zero and it does not change with variations in the charge q and the curvature radius l.However, in the fixed potential ensemble, for the charged static black hole, the topological charge is found to be −1.In this scenario as well, the topological charge remains unaffected by variations in the potential ϕ and the curvature radius l.
In the case of the rotating charged black hole, we have considered four ensembles and the results obtained in those ensembles can be summarized as follows : For fixed (q, J) ensemble, the topological charge is found to be 1 and it does not vary with charge q, angular momentum J and l.However, for different length scales, the number of branches in τ vs S plot change with variation of q, J and l.Although it does not result in a change of the topological charge.
In case of fixed (ϕ, J) ensemble, the topological charge is found to be 1 and it does not depend upon the values of potential ϕ, J and scalar curvature R. Again the number of branches in τ vs S plot varies with changes in ϕ, J and R values keeping the topological charge unchanged.
In the case of fixed (q, Ω) ensemble, the topological charge is 1 or 0 depending on the value of scalar curvature R.Although the number of branches of τ vs S plot changes with variation of q and Ω values for a fixed R, the topological charge remains unaltered.
Finally, in fixed (Ω, ϕ) ensemble, the topological charge is −1, 0 or 1 depending on the values of R, Ω and ϕ.The results for the charged, rotating black hole case are summarized in the following table.
Fixed (q, J) ensemble Therefore, we conclude that the thermodynamic topologies of the charged static black hole and charged rotating black hole are influenced by the choice of ensemble.In addition, the thermodynamic topology of the charged rotating black hole also depends on the thermodynamic parameters.

FIG. 1 :
FIG. 1: Plots for static black hole considering model I, where κ = 0.005.Figure (a) shows τ vs r + plot.Figure (b) is the plot of vector field n on a portion of r + − θ plane for τ = 100.The zero point is located at r + = 4.9879 and (c)shows the computation of the winding number for the contour around the zero point, r + = 4.9879.

FIG. 2 :
FIG. 2: Plots for static black hole considering model II, where R 0 = (10 −33 ) 2 eV and Λ i = 10 26 .Figure (a) shows τ vs r + plot.Figure (b) is the plot of vector field n on a portion of r + − θ plane for τ = 40.The zero point is located at r + = 3.18309 and (c) shows the computation of the winding number for the contour around the zero point, r + = 3.18309.

2 FIG. 3 :FIG. 4 :
FIG.3:The relation between l 2 and r for the positive temperature.Temperature is positive only on the shaded portions.We have taken β = 1.inFigure.4a,we observe one single black hole branch.The winding number is calculated by keeping β = 1, l 2 = 100 and τ = 20.For this combination of values, we observe the zero point of the vector field n at r + = 46.44048(Figure4b).From figure.4c it is observed, that the winding number or the topological charge corresponding to r + = 46.44048(represented by the black colored solid line) is found to be −1.Further analysis shows, that for all sets of values of β, l 2 , the topological charge remains constant, which is −1.

FIG. 5 :
FIG. 5: Plots for static charged black hole in f (R) gravity in fixed charge ensemble.Here q = 0.25 with l = 10. Figure (a) shows τ vs S plot. Figure (b) is the plot of vector field n on a portion of S − θ plane for τ = 40.The zero points are located at S = 0.38868 and S = 1.033022.In figure (c), the computation of the winding numbers for the contours around the zero points S = 0.38868 and S = 1.033022 are shown in black and red colored solid lines respectively.

FIG. 6 :
FIG.6: Plots for static charged black hole in f (R) gravity in fixed charge ensemble.Here, q = 0 with l = 10. Figure (a) shows τ vs S plot. Figure (b) is the plot of vector field n on a portion of S − θ plane for τ = 10.The zero point is located at S = 1.91726.In figure (c), the computation of the winding number for the contour around the zero point S = 2.5581 is shown in black-colored solid lines.

FIG. 7 :
FIG. 7: Plots for the static charged black hole in the fixed potential ensemble at ϕ = 0.5 with l = 10. Figure (a) shows τ vs S plot, figure (b) is the plot of vector field n on a portion of S − θ plane for τ = 6.The red arrows represent the vector field n.The zero point is located at S = 0.39878.In figure (c), computation of the contour around the zero point τ = 100 and S = 0.39878.is shown.

FIG. 10 :FIG. 11 :
FIG.10: Plots for zero points of ϕ r in the τ − S plane for rotating charged black hole with in fixed (q, J) ensemble at l = 10, J = 1.5 and q = 0.05.Two line are drawn at τ = 36.5182and τ = 33.8834which are corresponds to annihilation and generation point respectively.The solid red portion represents a large black hole branch, the blue dashed portion represents an intermediate black hole branch and the solid black portion represents a small black hole branch.

FIG. 20 :
FIG.20:τ vs S plots for rotating charged black hole in fixed (Ω, q) ensemble for different Ω value at R = −0.01 and q = 0.1.Figure (a) shows τ vs S plot at Ω = 1, figure (b) shows τ vs S plot at Ω = 3.The topological charge for all the cases is 0.