# Topology of restricted phase space thermodynamics in Kerr-Sen-Ads black
holes

Bidyut Hazarika1<EMAIL_ADDRESS>Prabwal Phukon1,2
<EMAIL_ADDRESS>$1.$Department of Physics, Dibrugarh University,
Dibrugarh, Assam,786004.
$2.$Theoretical Physics Division, Centre for Atmospheric Studies, Dibrugarh
University, Dibrugarh, Assam,786004.

## Abstract

In this study, we investigate the thermodynamic topology of the Kerr-Sen-Ads
black hole in restricted phase space. In the restricted phase space, a new
parameter, central charge $C$, and its conjugate parameter $\mu$ are
introduced, omitting the well-known $PdV$ term in the first law of black hole
thermodynamics. We study the local and global topology of the black hole by
considering the black hole solution as topological defects in the free energy
landscape. We compute the winding number and the total topological number at
the thermodynamic defects. For our analysis, we have considered five ensembles
of Kerr-Sen-Ads black holes in restricted phase space: fixed $(Q,J,C)$, fixed
$(\phi,J,C)$, fixed $(Q,\Omega,C)$, fixed $(Q,J,\mu)$, and fixed
$(\phi,\Omega,C)$, where $Q$ is the electric charge, $J$ is the angular
momentum, $C$ is the central charge, $\phi$ is the electric potential
conjugate to charge, $\Omega$ is the angular frequency conjugate to $J$, and
finally, $\mu$ is the chemical potential. In the fixed $(Q,J,C)$, fixed
$(\phi,J,C)$, and fixed $(Q,J,\mu)$ ensembles, we find a topological charge of
$+1$. In the fixed $(Q,\Omega,C)$ and fixed $(\phi,\Omega,C)$ ensembles,
depending on the values of the thermodynamic parameters, we find topological
charges of $-1$, $0$, and $+1$. Interestingly, in ensembles where we find the
topological charge to be $0$, we observe both Hawking-Page and Davies type
phase transitions. We show that both types of these phase transitions can be
studied using a common vector field, and the topological charges associated
with Davies type and Hawking-Page phase transitions are $-1$ and $+1$,
respectively.

##  I Introduction:

The study of black hole thermodynamics has made substantial strides since its
inception in the 1970s [5, 6, 7, 8]. Further studies have lead a multitude of
intriguing findings which contribute to a better understanding of this area[9,
10, 11, 12, 13, 14, 15, 16, 17]. One captivating aspect of black hole
thermodynamics is the study of black holes phase transitions. Black holes
exhibit various forms of these transitions, including the Davies-type phase
transition [18], the Hawking–Page phase transition [19], extremal phase
transitions [20, 21, 22, 23, 24, 25, 26, 27, 28], van der Waals-type phase
transitions [29, 30, 31, 32, 33, 34, 35, 36] etc. The study of black hole
phase transition behavior has been given a new dimension through the
introduction of extended phase-space thermodynamics [37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49]. This framework introduces a novel pair of
variables (P, V) into the domain of thermodynamic parameters, where pressure
can be expressed in terms of the negative cosmological constant as :

$P=\frac{-\Lambda}{8\pi G}$

Subsequently, applying AdS/CFT correspondence [50], Visser introduced a new
formalism [51] (which is an extension of extended thermodynamic phase space),
by adding CFT central charge $C$ and its conjugate parameter $\mu$ to the
first law of black hole thermodynamics. Moreover in Visser’s formalism the
pair ($P,V$), is transformed into a new pair $(\mathcal{P},\mathcal{V})$. Here
the CFT volume $\mathcal{V}~{}L^{d-2}$ and the corresponding pressure term
$\mathcal{P}$ is related through the CFT equation of state
$E=(d-2)\mathcal{P}\mathcal{V}$ where $L$ is the Ads radius. $d$ is the
spacetime dimension in the bulk. So, for a charged rotating black hole the
first law of black hole thermodynmics takes the form :

$dE=TdS-\mathcal{P}d\mathcal{V}+\tilde{\phi}\tilde{Q}+\Omega J+\mu C$

where, $\tilde{Q}$ and $\tilde{\phi}$ are the rescaled electric charge and its
conjugate electric potential respectively.

A restricted version of Visser’s formalism known as the restricted phase space
thermodynamics (RPST) for Ads black holes was proposed [52]. These formalism
has been applied to a number of black holes [53, 54, 55, 56, 57, 58, 60, 61,
59, 62, 63]. Studying black hole thermodynamics in RPST formalism has the
potential to provide better understanding of AdS/CFT correspondence. Apart
from extended phase space and RPST formlism, holographic thermodynamics [64,
65, 66, 51], geometric interpretation of phase transitions[67, 68, 69, 70, 71,
72, 73, 74, 75, 76] are a few of the techniques used to study critical
phenomenon in black hole thermodynamics.

A novel avenue in the context of understanding critical phenomena in black
hole thermodynamics is the incorporation of topology in black hole
thermodynamics[79, 80]. In one of these approaches, to investigate the local
and global thermodynamic topology, black hole solutions are considered as
topological defects within their thermodynamic phase space[80]. At these
defects, a fundamental quantity known as the topological charge can be
assigned. The topological charge signifies the distinct topological class
present fo a particular black hole solution.The study of thermodynamic
topology has been extended to a range of black hole solutions found within
various gravitational theories [81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,
92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 125,
126, 123, 124, 127, 128, 129, 130]. The following paragraph provides a brief
overview of the off-shell method and outlines the procedure for computing the
topological charge.

The expression for the off-shell free energy of a black hole with arbitrary
mass is given by:

$\displaystyle\mathcal{F}=E-\frac{S}{\tau}$ (1)

Here, $E$ and $S$ are the energy and entropy of the black hole respectively.
$\tau$ is the time scale parameter which equals to :

$\tau=\frac{1}{T}$ (2)

Here, $T$ is the equilibrium temperature at the surface of the cavity that
encloses the black hole. The time parameter $\tau$ is set freely to vary
instead of the mass. Utilizing the generalized free energy, a vector field is
constructed as follows :

$\displaystyle\phi=\left(\phi^{S},\phi^{\Theta}\right)=\left(\frac{\partial\mathcal{F}}{\partial
S},-\cot\Theta~{}\csc\Theta\right)$ (3)

Where $\theta=\pi/2$ and $\tau=1/T$ represent a zero point of the vector field
$\phi$. In the $\theta$ component, a particular combination of trigonometric
functions is chosen because at $\theta=\frac{\pi}{2}$, the vector field
diverges, allowing one to easily identify the zero point of the $S$ component
by simply examining the vectors along the $\theta=\frac{\pi}{2}$ line. While
constructing the vector field the Duan’s $\phi$ mapping current theory has
been taken into consideration [77, 78]. Using Duan’s theory, the topological
current can be written as

$j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b}\hskip
22.76228pt\mu,\nu,\rho=0,1,2$ (4)

Here $\partial_{\nu}=\frac{\partial}{\partial x^{\nu}}$. It is important to
note that the topological current is conserved or

$\partial_{\mu}j^{\mu}=0$ (5)

The normalized vector is calculated as :

$n^{a}=\frac{\phi^{a}}{||\phi||};\quad
a=1,2\quad\text{with}\quad\phi^{1}=\phi^{S},\quad\phi^{2}=\phi^{\theta}.$ (6)

The two conditions that has to be stisfied by $n^{a}$ are :

$n^{a}n^{a}=1\quad\text{and}\quad n^{a}\partial_{\nu}n^{a}=0.$ (7)

To calculate the topological charge we use the residue method [115]. This
method involves irst determining the winding number $(w_{i})$ for each pole
using the residue theorem and by adding the winding numbers for each pole, the
topological charge can be computed. Initially, we find a solution for $\tau$
by solving the equation:

$\frac{\partial\mathcal{F}}{\partial r_{+}}=0$ (8)

Our aim is to create a two-dimensional space where we can introduce a real
contour. This contour will encircle the thermodynamic imperfections described
by equation 8. Thus the resulting solution, which depends on the horizon
radius $r_{+}$, is then expressed as a function of a complex variable $z$ and
denoted as $\mathcal{G}(z)$. Subsequently, a rational complex function
$\mathcal{R}(z)$ is constructed as follows:

$\mathcal{R}(z)=\frac{1}{\tau-\mathcal{G}(z)}$ (9)

The complex function $R(z)$ is analytic except for a limited number of
specific points $z_{1},z_{2},z_{3}.....z_{m}$. To enclosed these points, a
real closed contour $C$ is introduced where the complex function is analytic
within and on the contour. This is followed by employing the residue theorem
to compute the integral of function $R(z)$ over the path $C_{i}$ surrounding
$z_{i}$ or

$\oint\frac{R(z)}{2\pi i}dz=\sum_{k=1}^{m}Res[R(z_{k})]$ (10)

Finally, local topology can be studied by computing the winding number at
thermodynamic defects. We do that by just taking normalized residues at the
poles of $\mathcal{R}(z)$

$\displaystyle
w_{i}=\frac{Res\mathcal{R}(z_{i})}{|Res\mathcal{R}(z_{i})|}=Sign[Res\mathcal{R}(z_{i})]$
(11)

Again, the Cauchy-Goursat theorem says that the integral around contour $C$,
winding around the singular points, is equal to the integrals of $R(z)$ along
$C_{i}$ which $z_{i}$ is enclosed. Applying this concept we can either
consider a bigger contour which encompasses all thermodynamic defects or we
can simply add the individual winding numbers $w_{i}$, computed around small
individual contours. Hence the total winding number or topological charge,
$W$, is then computed as

$W=\sum_{i}w_{i}$

In this study, we investigate the thermodynamic topology of Kerr-Sen-Ads black
hole in restricted phase space. We consider the black hole solution as
topological defects in the thermodynamic phase space. To study the local and
the global topology of the black hole, we calculate the winding number and the
total topological charge at the thermodynamic defects. In our analysis, we
have considered five ensemble of Kerr-Sen-Ads black hole in restricted phase
space. These are: fixed $(Q,J,C)$, fixed $(\phi,J,C)$, fixed $(Q,\Omega,C)$,
fixed $(Q,J,\mu)$ and fixed $(\phi,\Omega,C)$ where $Q$ is the electric
charge, $J$ is the angular momentum, $C$ is the central charge, $\phi$ is the
electric potential conjuagate to charge, $\Omega$ is the angular frequency
conjugate to $J$ and finally $\mu$ is the chemical potential. The selection of
an ensemble frequently emerges as a decisive element influencing the
thermodynamic characteristics and phase transitions exhibited by black
holes[131, 132, 133, 134, 135, 136]. For all these ensembles we study their
thermodynamic topologies by computing the topological charge.

This paper is organized into the following sections: in Section II, we briefly
review the Kerr-Sen-Ads black hole in restricted phase space. We analyse the
thermodynamic topology of fixed ($Q,J,C$) ensemble in subsection II.I., fixed
($\phi,J,C$) in subsection II.II., fixed ($Q,\Omega,C$) in subsection II.III.
, fixed ($Q,J,\mu$) in subsection II.IV. and finally in subsection II.V, we
analyze the same for fixed $(\phi,\Omega,C)$ ensemble. The conclusions are
presented in Section III.

##  II Kerr-Sen Ads black hole in restricted phase space

The Kerr-Sen Ads black hole is the charged rotating black hole solution to the
low-energy limit of heterotic string theory discovered by Sen [137].The Kerr-
Sen black holes in four dimensions have been examined from various
perspectives[59, 138, 139, 140, 141, 142, 143, 144, 145, 146]. The
thermodynamics of phase space in the extended phase space was studied in [139]
and the thermodynamic of Kerr-Sen Ads black hole was studied in [59]. In this
section a brief review of the Kerr-Sen-Ads black hole in restricted phase
space is provided[59].
The metric for Kerr-Sen-Ads black hole in Boyer-Lindquist coordinate is given
by :

$ds^{2}=-\frac{\Delta_{r}}{\rho^{2}}\left[dt-\frac{asin^{2}\theta
d\varphi}{\Xi}\right]^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}sin^{2}\theta}{\rho^{2}}\left[adt-\frac{r^{2}+a^{2}}{\Xi}d\varphi\right]^{2}$
(12)

where

$\Delta_{r}=(r^{2}+2br+a^{2})\left(1+\frac{r^{2}+2br}{l^{2}}\right)-2Gmr$
$\Xi=1-\frac{a^{2}}{l^{2}}\hskip 14.22636pt,\hskip
14.22636pt\rho^{2}=r^{2}+2br+a^{2}cos^{2}\theta,$
$\Delta_{\theta}=1-\frac{a^{2}}{l^{2}}cos^{2}\theta$

here $b$ is the diatonic charge which can be express in terms of black hole
mass $m$ and electric charge $q$ as $b=q^{2}/2m.$ The expressions for ADM mass
$M$ and the angular momentum $J$ and charge $Q$ are :

$M=\frac{m}{\Xi^{2}}\hskip 14.22636pt,\hskip
14.22636ptJ=\frac{am}{\Xi^{2}}\hskip 14.22636pt\text{and}\hskip
14.22636ptQ=\frac{q}{\Xi}$ (13)

using the condition $\Delta_{r}(r=r_{+})=0$ we can write $M$ and $J$ as

$M=\frac{1}{2\Xi^{2}Gr_{+}}(r^{2}+2br+a^{2})\left(1+\frac{r^{2}+2br}{l^{2}}\right)$
(14)

and

$J=\frac{a}{2\Xi^{2}Gr_{+}}(r^{2}+2br+a^{2})\left(1+\frac{r^{2}+2br}{l^{2}}\right)$
(15)

where $r_{+}$ is the event horizon radius and $l$ is the Ads radius. To
introduce RPST formalism two new quantities are defined which is the central
charge $C$ and its conjugate parameter $\mu$[51, 52, 58].

$C=\frac{l^{2}}{G},\hskip 28.45274pt\mu=\frac{M-TS+\Omega
J+\hat{\phi}\hat{Q}}{C}$ (16)

Here $\hat{Q}$ and $\hat{\phi}$ is the rescaled charge and electric potential
expressed as:

$\hat{Q}=\frac{Ql}{\sqrt{G}},\hskip
28.45274pt\hat{\phi}=\frac{\phi\sqrt{G}}{l}$ (17)

Using these defined quantities we write the two most important equations in
RPST formalism. First one is the modified first law of black hole
thermodynamic which is incorporated with a new term $\mu dC$ and made free
from the usual $\mathcal{P}d\mathcal{V}$ term :

$dM=TdS+\Omega dJ+\phi dQ+\mu dC$ (18)

and followed by the Euler-like relation :

$M=TS+\Omega J+\hat{\phi}\hat{Q}+\mu C$ (19)

Now, rewritting $a,G$ as follows:

$a=\frac{J}{M}\hskip 28.45274ptG=\frac{l^{2}}{C}$ (20)

we can expressed the horizon radius $r_{+}$ as [59]:

$r_{+}=\frac{1}{2M}\left(\sqrt{\frac{-4J^{2}(\pi C+S)+\pi
CQ^{4}+4l^{2}M^{2}S}{\pi C}}-Q^{2}\right)$ (21)

Putting equation.20 and equation.21 into equation.13, the ADM mass is obtained
as [59] :

$M=\frac{\sqrt{\pi C+S}\sqrt{\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)+2\pi^{2}Q^{2}S+S^{3}}}{2\pi^{3/2}l\sqrt{CS}}$
(22)

rest of the thermodynamic quantities can be computed using the first law of
thermodynmics. For details of the RPST formlism of Kerr-Sen-Ads black hole see
[59]. Now we study the thermodynamic topology in five different ensemble in
the following subsections.

###  II.1 Fixed (Q,J,C) Ensemble

For Kerr sen Ads black hole,mass $M$ in this ensemble is :

$M=\frac{\sqrt{\pi C+S}\sqrt{\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)+2\pi^{2}Q^{2}S+S^{3}}}{2\pi^{3/2}l\sqrt{CS}}$
(23)

Using equation 23,free energy is computed as :

$\mathcal{F}=M-S/\tau=-\frac{2\pi^{3/2}lS\sqrt{CS}-\tau\sqrt{\pi
C+S}\sqrt{4\pi^{3}CJ^{2}+\pi
CS^{2}+2\pi^{2}Q^{2}S+S^{3}}}{2\pi^{3/2}l\tau\sqrt{CS}}$ (24)

The components of the vector field $\phi$ is obtained as :

$\phi^{r}=-\frac{4\pi^{4}C^{3}J^{2}\tau-4\pi
C^{2}S^{3}\tau-\pi^{2}C^{3}S^{2}\tau+4\pi^{3/2}l\sqrt{\pi
C+S}(CS)^{3/2}\sqrt{4\pi^{3}CJ^{2}+\pi
CS^{2}+2\pi^{2}Q^{2}S+S^{3}}-2\pi^{2}CQ^{2}S^{2}\tau-3CS^{4}\tau}{4\pi^{3/2}l\tau(CS)^{3/2}\sqrt{\pi
C+S}\sqrt{4\pi^{3}CJ^{2}+\pi CS^{2}+2\pi^{2}Q^{2}S+S^{3}}}$ (25)

and

$\phi^{\Theta}=-\cot\Theta~{}\csc\Theta$ (26)

The zero points of $\phi^{r}$ is also obtained as:

$\tau=\frac{4\pi^{3/2}l(CS)^{3/2}\sqrt{\pi C+S}\sqrt{4\pi^{3}CJ^{2}+\pi
CS^{2}+2\pi^{2}Q^{2}S+S^{3}}}{-4\pi^{4}C^{3}J^{2}+4\pi
C^{2}S^{3}+\pi^{2}C^{3}S^{2}+2\pi^{2}CQ^{2}S^{2}+3CS^{4}}$ (27)

The entropy is plotted against $\tau$ at $C=0.2,Q=1.5,J=0.02$ and $l=1$ in
Figure.1. Here we observe three black hole branches : a small, an
intermediate, and a large black hole branch. We also see a generation point at
$\tau=1.84076,S=2.02107$ and an annihilation point at $\tau=1.27059,S=0.07658$
which are shown as blue dot and black dot respectively.

Figure 1: $\tau$ vs $S$ plot for Kerr Sen Ads black hole at
$C=0.2,Q=1.5,J=0.02$ and $l=1$. The $LBH$ and $SBH$ regions represent the
large and small black hole branches respectively.

We now calculate the topological charge of the black hole using the residue
method. The complex function $\mathcal{R}(z)$ mentioned in section I is given
by :

$\mathcal{R}(z)=-\frac{-4\pi^{4}C^{2}J^{2}+\pi^{2}C^{2}z^{2}+4\pi
Cz^{3}+2\pi^{2}Q^{2}z^{2}+3z^{4}}{4\pi^{4}C^{2}J^{2}\tau-\pi^{2}C^{2}\tau
z^{2}+4\pi^{3/2}lz\sqrt{Cz}\sqrt{\pi C+z}\sqrt{4\pi^{3}CJ^{2}+\pi
Cz^{2}+2\pi^{2}Q^{2}z+z^{3}}-4\pi C\tau z^{3}-2\pi^{2}Q^{2}\tau z^{2}-3\tau
z^{4}}$ (28)

The denominator of the complex function is considered as a polynomial function
$\mathcal{A}(z)$

$\mathcal{A}(z)=4\pi^{4}C^{2}J^{2}\tau-\pi^{2}C^{2}\tau
z^{2}+4\pi^{3/2}lz\sqrt{Cz}\sqrt{\pi C+z}\sqrt{4\pi^{3}CJ^{2}+\pi
Cz^{2}+2\pi^{2}Q^{2}z+z^{3}}-4\pi C\tau z^{3}-2\pi^{2}Q^{2}\tau z^{2}-3\tau
z^{4}$ (29)

To calculate the topological charge, we put $C=0.2,Q=1.5,J=0.02$ and $l=1$ in
equation 29.

Figure 2: Plot for polynomial function $\mathcal{A}(z)$ for $\tau=2.2,2.4$ and
$2.6$

Figure 2 represents the plot for $\mathcal{A}(z)$. This demonstrates that
there is one pole for $\tau<1.27059$, three poles for $1.27059<\tau<1.84075$,
and one pole for $\tau>1.84075$. Using $\tau=1.2$ for the large black hole
branch, the pole is located at $z=8.16619$ as suggested by the blue solid line
in figure 2. The winding number is $w=+1$ around the pole $z=8.16619$. Using
$\tau=1.6$, the poles for the intermediate black hole branch are located at
$z_{1}=0.0235863,z_{2}=0.676384$, and $z_{3}=4.39265$, which is represented by
the solid green line. The winding numbers around these three poles are
$w_{1}=+1,w_{2}=-1$, and $w_{3}=+1$, respectively, based on the sign of the
residue around them. Thus,

$W=w_{1}+w_{2}+w_{3}=1-1+1=1$

is the topological charge. Similarly, as represented by the solid red line,
for a small black hole branch, taking $\tau=2$, the pole is located at
$z=0.0186124$, and the winding number is determined to be $+1$ based on the
sign of residue surrounding the pole.
We now analyze the effect of thermodynamic parameter on the topological
charge. We start with analysing the effect of Ads radius $l$ keeping
$C=0.2,Q=1.5,J=0.02$ constant. We observe no effect of change in $l$ on the
topological charge. The same is illustrated in figure 3. Here we have
considered two values of $l$ keeping $C=0.2,Q=1.5,J=0.02$ constant. In figure
13(a), $l=0.1$ and in figure 13(b), $l=100$. For both cases, we find the
number of black hole branches equal to three and topological charge to be one.

(a) W=1

(b) W=1

Figure 3:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(q,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for two
different $l$ values while keeping $C=0.2,Q=1.5,J=0.02$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $l=0.1$ and in figure
$\left(b\right)$ shows $\tau$ vs $S$ plot at $l=100$.$W$ denotes topological
charge.

Our above analysis is followed by studying the dependence of topological
charge on the central charge $C$ keeping $l=1,Q=1.5,J=0.02$ fixed. It is seen
that topological charge does not depends on variation in central charge $C$.
Although for a specific range of $C$ the three phases of the black hole merges
to a single branch.However it does not alter the topological number. In figure
4(a) three black hole branches and topological charge $1$ is observed. In
4(b), we have only a single black hole branches with a topological charge of
$1.$ In figure 4(a) the number of branch is again three and topological charge
is $1.$

(a) W=1

(b) W=1

(c) W=1

Figure 4:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(Q,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for
three different $C$ values while keeping $l=1,Q=1.5,J=0.02$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $C=0.5$,figure $\left(b\right)$
shows $\tau$ vs $S$ plot at $C=1$ and in figure $\left(c\right)$ shows $\tau$
vs $S$ plot at $C=10$.$W$ denotes topological charge.

Next, we want to vary $q$ while holding $l=1,C=1,J=0.02$ constant in order to
see how the change in charge affects the topological charge. In Figures 5(a),
5(b), and 5(b), respectively, we have set $Q=0.1$, $Q=1.5$, and $Q=6$. Three
black hole branches are found in figure 5(a), one black hole branch is found
in picture 5(b), and three black hole branches are found once again in figure
5(c). In each of the three scenarios, the topological charge is $1$.
Therefore, it is noted that charge likewise has no effect on the topological
number.

(a) W=1

(b) W=1

(c) W=1

Figure 5:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(q,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for
three different $Q$ values while keeping $l=1,C=1,J=0.02$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $Q=0.1$,figure $\left(b\right)$
shows $\tau$ vs $S$ plot at $Q=1.5$ and in figure $\left(c\right)$ shows
$\tau$ vs $S$ plot at $Q=6$.$W$ denotes topological charge.

Lastly, we perform the same study in figures 6(a), 6(b), and 6(c) to examine
how $J$ affects topological charge while maintaining constant $l,Q,J$ at
$l=1,Q=1.5,C=0.2$. J is equal to 0.02 in figure 6(a), 0.1 in figure 6(b), and
1 in figure 6(c).Topological charge in each of the three scenarios is $1$.
Thus, it is concluded that while the number of black hole branches may change
in response to changes in angular momentum $J$, the topological charge remains
constant at $W=1$.

(a) W=1

(b) W=1

(c) W=1

Figure 6:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(q,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for
three different $J$ values while keeping $l=1,C=0.2,Q=1.5$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $J=0.02$,figure $\left(b\right)$
shows $\tau$ vs $S$ plot at $J=0.1$ and in figure $\left(c\right)$ shows
$\tau$ vs $S$ plot at $J=1$.$W$ denotes topological charge.

In conclusion, our findings show that the Kerr-Sen-Ads black hole in a fixed
$(Q,J,C)$ ensemble, the topological charge is $1$. The topological charge is
independent of the thermodynamic parameters $Q,J,C$.

###  II.2 Fixed ($\phi$,J,C) Ensemble

The potential $\phi$, angular momentum $J$ and central charge $C$ is kept
fixed in fixed ($\phi$,J,C) ensemble. The potential $\phi$ is given by :

$\phi=\frac{\partial M}{\partial q}=\frac{\sqrt{\pi}QS\sqrt{\pi
C+S}}{l\sqrt{CS}\sqrt{\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)+2\pi^{2}Q^{2}S+S^{3}}}$ (30)

To get an expression for $q$ we solve equation 30 and get the expression as :

$Q=\frac{\sqrt{\pi}\sqrt{C}l\phi\sqrt{4\pi^{3}CJ^{2}+\pi
CS^{2}+S^{3}}}{\sqrt{S}\sqrt{-2\pi^{2}Cl^{2}\phi^{2}+\pi C+S}}$ (31)

Now the new mass($M_{\phi}$) in this ensemble is calculated using equation 31
as follows :

$M_{\phi}=M-q\phi\\\ =\frac{\sqrt{\pi
C+S}\sqrt{\frac{2\pi^{3}Cl^{2}\phi^{2}\left(4\pi^{3}CJ^{2}+\pi
CS^{2}+S^{3}\right)}{-2\pi^{2}Cl^{2}\phi^{2}+\pi C+S}+\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)+S^{3}}}{2\pi^{3/2}l\sqrt{CS}}-\frac{\sqrt{\pi}\sqrt{C}l\phi^{2}\sqrt{4\pi^{3}CJ^{2}+\pi
CS^{2}+S^{3}}}{\sqrt{S}\sqrt{-2\pi^{2}Cl^{2}\phi^{2}+\pi C+S}}$ (32)

The off-shell free energy is computed using:

$\mathcal{F}=M_{\phi}-S/\tau$

We compute the expressions for $\phi^{S}$ and $\tau$ using the same method as
in the preceding part.
In Figure.7, the entropy $S$ vs $\tau$ is plotted at $C=1,\phi=1.5,J=0.02$,
and $l=0.1$. Here also three black hole branches are visible. We see two
points that are : a generation point at $\tau=0.35285,S=0.9876$ and an
annihilation point at $\tau=0.342307,S=0.4037$, shown as blue and black dots,
respectively

Figure 7: $\tau$ vs $S$ plot for Kerr Sen Ads black hole at
$C=1,\phi=1.5,J=0.02$ and $l=0.1$. The $LBH$ and $SBH$ regions represent the
large and small black hole branches respectively.

To analyze the effect of thermodynamic parameter on the topological charge we
vary one particular parameter keeping other parameter fixed. For this ensemble
we observe no effect of change in any thermodynamic parameter on the
topological charge. The same is illustrated in figure 8,9,10 and 11.
In figure 8, central charge $C$ is varied keeping $l=0.1,\phi=1.5,J=0.02$
fixed. For a specific range of $C$ the three phases of the black hole is
visible and it merges to a single branch for other values, through out the
process the topological charge remains constant at $1.$ For $C=10$, in figure
8(a) three black hole branches and topological charge $1$ is observed. In
4(b), we have only a single black hole branches with a topological charge of
$1$ when $C=0.1$.

(a) W=1

(b) W=1

Figure 8:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(\phi,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for
two different $C$ values while keeping $l=0.1,\phi=1.5,J=0.02$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $C=10$,figure $\left(b\right)$
shows $\tau$ vs $S$ plot at $C=0.1$. $W$ denotes topological charge.

In figure 9, effect of variation of $\phi$ topological charge is shown while
keeping $l=0.1,C=1,J=0.02$ constant. In Figures 9(a), 9(b) and 9(c)
respectively, we have set $\phi=0.5$, $\phi=1$, and $\phi=5$. Three black hole
branches are found in figure 9(b) on the other hand only one black hole branch
is found in figure 9(a) and figure 9(c). In each of the three scenarios, the
topological charge is $1$. Therefore, it is noted that potential $\phi$ has no
effect on the topological number.

(a) W=1

(b) W=1

(c) W=1

Figure 9:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted phase
space in fixed $(\phi,J,C)$ ensemble. Here $\tau$ is plotted against $S$ for
three different $\phi$ values while keeping $l=0.1,C=1,J=0.02$ fixed. Figure
$\left(a\right)$ shows $\tau$ vs $S$ plot at $\phi=0.5$,figure
$\left(b\right)$ shows $\tau$ vs $S$ plot at $\phi=1$ and in figure
$\left(c\right)$ shows $\tau$ vs $S$ plot at $\phi=5$.$W$ denotes topological
charge.

Lastly, in figures 10(a), 10(b), and 10(c) $\tau$ vs $S$ is plotted to check
how $J$ affects topological charge while keeping constant $l,\phi,C$ at
$l=1,\phi=1.5,C=1$. J is equal to 0.02 in figure 10(a), 0.1 in figure 10(b),
and 1 in figure 10(c).In all three cases, topological charge is $1$.
Consequently, it is inferred that the topological charge at $W=1$ remains
constant, however the number of black hole branches may vary in response to
variations in angular momentum $J$.

(a) W=1

(b) W=1

(c) W=1

Figure 10:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(\phi,J,C)$ ensemble. Here $\tau$ is plotted against $S$
for three different $J$ values while keeping $l=1,C=0.2,\phi=1.5$ fixed.
Figure $\left(a\right)$ shows $\tau$ vs $S$ plot at $J=0.02$,figure
$\left(b\right)$ shows $\tau$ vs $S$ plot at $J=0.1$ and in figure
$\left(c\right)$ shows $\tau$ vs $S$ plot at $J=1$.$W$ denotes topological
charge.

In conclusion, our findings show that the Kerr-Sen-Ads black hole in a fixed
$(Q,J,C)$ ensemble, the topological charge is $1$. The topological charge is
independent of the thermodynamic parameters $Q,J,C$.

###  II.3 Fixed (Q,$\Omega$,C) Ensemble

In this ensemble, the angular frequency $\Omega$, charge $Q$ and central
charge $C$ is kept fixed. The angular frequency $\Omega$ is given by :

$\Omega=\frac{2\pi^{3/2}CJ\sqrt{\pi C+S}}{l\sqrt{CS}\sqrt{\pi
C\left(4\pi^{2}J^{2}+S^{2}\right)+2\pi^{2}Q^{2}S+S^{3}}}$ (33)

By solving equation 33, we get the expression for $J$ as :

$J=\frac{lS\Omega\sqrt{\pi
CS+2\pi^{2}Q^{2}+S^{2}}}{\pi^{3/2}\sqrt{C}\sqrt{4\pi C-4l^{2}S\Omega^{2}+4S}}$
(34)

The newly mass($M_{\Omega}$) in this ensemble :

$M_{\Omega}=M-J\Omega\\\ =-\frac{l^{2}S\Omega^{2}\sqrt{CS}\sqrt{\pi
CS+2\pi^{2}Q^{2}+S^{2}}-\sqrt{C}\sqrt{\pi C+S}\sqrt{\pi
C-l^{2}S\Omega^{2}+S}\sqrt{\frac{S(\pi C+S)\left(\pi
CS+2\pi^{2}Q^{2}+S^{2}\right)}{\pi
C-l^{2}S\Omega^{2}+S}}}{2\pi^{3/2}\sqrt{C}l\sqrt{CS}\sqrt{\pi
C-l^{2}S\Omega^{2}+S}}$ (35)

The off-shell free energy is computed using:

$\mathcal{F}=M_{\Omega}-S/\tau$

We compute the expressions for $\phi^{S}$ and $\tau$ using the same method as
in the preceding part.
In Figure.11, the entropy $S$ vs $\tau$ is plotted at
$C=10,Q=1.5,\Omega=0.02$, and $l=0.1$. We observe two black hole branches with
a generation point at $(\tau,S)$ = $(0.37635,8.2427)$ represented by the black
dot. We observe a large black hole branch shown as red solid line and a small
black hole branch shown as black solid line. The topological number of the
smaller black hole branch is $-1$ and that of the large black hole branch is
$+1$. The total topological number is found to be $-1+1=0.$

Figure 11: $\tau$ vs $S$ plot for Kerr Sen Ads black hole at
$C=1,\phi=1.5,J=0.02$ and $l=0.1$. The red and the black colored solid
represent the large and small black hole branches respectively.

Taking a cue from figure.11, study of Hawking-Page transition and Davies type
of phase transition also can be studied by constructing a noval vector field
given by [123]

$\phi=\left(\frac{1}{S}\frac{\partial F^{2}}{\partial S},-cot\theta
csc\theta\right)$ (36)

Here $F$ is the on-shell free energy with the conditon $\tau=T=\frac{\partial
M}{\partial S}$ From the vectorplot shown in figure.12

(a)

(b)

Figure 12: A portion of the vector field $\phi$ on $S-\theta$ plane. The black
dot of figure $(a)$ represents Hawking page phase transition point and the
blue dot in figure $(b)$ represents Davies point.

By solving $F=0$ and $C_{\Omega}=T\frac{\partial S}{\partial T}=0$, the
Hawking Page phase transition point is found to be $S=33.3908$ and the Davies
point if found to be $S=8.2427$ which exactly matches with the critical points
of the vector field in equation 45. The black dot ($S=8.2427$) in figure 12(a)
is a Hawking page transition point and blue dot ($S=33.3908$) is a Davies
point.
The topological charge can be calculated by constructing a contour $C$ around
each zero point($C_{0}$ and $C_{1}$ as shown in figure 12) which is
parametrized as:

$\begin{cases}S=acos\nu+r_{0}\\\ \theta=bsin\nu+\frac{\pi}{2}\\\ \end{cases}$
(37)

where $\nu\in(0,2\pi)$. To calculate the deflection $\Omega$ of the vector
field $n$ along the contour $C_{1}$ and $C_{2}$, we choose $a=b=0.2$ and
$r_{0}=\text{the zero point itself enclosed by the contours}.$ We use the
formula [77, 78]:

$\Omega(\nu)=\int_{0}^{\nu}\epsilon_{12}n^{1}\partial_{\nu}n^{2}d\nu$ (38)

where the unit vectors $\left(n^{1},n^{2}\right)$ are given by:

$n^{1}=\frac{\phi^{S}}{\sqrt{(\phi^{S})^{2}+(\phi^{\Theta})^{2}}}\hskip
14.22636pt\text{and}\hskip
14.22636ptn^{2}=\frac{\phi^{\theta}}{\sqrt{(\phi^{S})^{2}+(\phi^{\Theta})^{2}}}$

Finally, the winding numbers $w$ and topological charge $W$ can be calculated
as follows:

$\begin{cases}w=\frac{1}{2\pi}\Omega(2\pi)\\\ W=\sum_{i}w_{i}\\\ \end{cases}$
(39)

In cases where the parameter region does not encompass any zero points, the
topological charge equals to $0$. Following the above procedure, we plot
deflection $\Omega$ vs $\theta$ plot in figure.13. From the figure, the
topological charge of the Hawking page phase transition(blue colored solid
line) is $+1$ and that of Davis point(black colored solid line) is found to be
$-1.$

(a)

(b)

Figure 13: Kerr Sen AdS black hole in fixed ($\phi$,J,C) ensemble : Figure (a)
represents topological number calculation for Davies point around the $C_{0}$
contour in Figure.2 and Figure (b) represents the topological number
calculation for Hawking-Page phase transition point around the $C_{1}$ contour

To analyze the effect of thermodynamic parameter on the topological charge we
vary one particular parameter keeping other parameter fixed. In figure 14, the
value of central charge $C$ is varied keeping $l=0.1,Q=1.5,\Omega=0.02$ fixed.
For a different value of $C$ the topological charge is found to be $0$ or $1.$
For $C=0.01$, in figure 14(a) there are two black hole branches and
topological charge is $0$. In 14(b), we have only a single black hole branch
with a topological charge of $1$ when $C=1$. In 14(c) again we observe two
black hole branch with a topological charge of $0$ when $C=15$.

(a) W=0

(b) W=1

(c) W=0

Figure 14:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(Q,\Omega,C)$ ensemble. Here $\tau$ is plotted against
$S$ for two different $C$ values while keeping $l=0.1,Q=1.5,\Omega=0.02$
fixed. Figure $\left(a\right)$ shows $\tau$ vs $S$ plot at $C=0.01$, figure
$\left(b\right)$ shows $\tau$ vs $S$ plot at $C=0.1$, figure $\left(c\right)$
shows the same at $C=15$ $W$ denotes topological charge.

In figure 15, effect of variation of $Q$ on topological charge is shown while
keeping $l=0.1,C=1,J=0.02$ constant. In Figures 15(a), 15(b) and 15(c)
respectively, we have set $Q=0.05$, $Q=0.5$, and $Q=3.5$. For $Q=0.05$ and
$Q=3.5$, two black hole branches are found in figure 15(a) and figure.15(c)
with topological charge $0$. On the other hand only one black hole branch is
found with topological charge $1$ in figure 15(b) when $Q=0.5$ is kept fixed.

(a) W=0

(b) W=1

(c) W=0

Figure 15:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(\phi,J,C)$ ensemble. Here $\tau$ is plotted against $S$
for three different $\phi$ values while keeping $l=0.1,C=1,J=0.02$ fixed.
Figure $\left(a\right)$ shows $\tau$ vs $S$ plot at $\phi=0.5$,figure
$\left(b\right)$ shows $\tau$ vs $S$ plot at $\phi=1$ and in figure
$\left(c\right)$ shows $\tau$ vs $S$ plot at $\phi=5$.$W$ denotes topological
charge.

Lastly, in figures 16(a), 16(b), and 16(c) $\tau$ vs $S$ is plotted to check
how $\Omega$ affects topological charge while keeping constant $l,Q,C$ at
$l=1,Q=0.05,C=1$. $\Omega$ is equal to 0.01 in figure 16(a), 5 in figure
16(b), and 12 in figure 16(c). For $\Omega=5$ and $\Omega=0.05$, the
topological charge is $0$ and for $\Omega=12$, the topological charge is found
to be $-1$.

(a) W=0

(b) W=0

(c) W=-1

Figure 16:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(Q,\Omega,C)$ ensemble. Here $\tau$ is plotted against
$S$ for three different $\Omega$ values while keeping $l=1,C=0.2,Q=0.05$
fixed. Figure $\left(a\right)$ shows $\tau$ vs $S$ plot at
$\Omega=0.05$,figure $\left(b\right)$ shows $\tau$ vs $S$ plot at $\Omega=5$
and in figure $\left(c\right)$ shows $\tau$ vs $S$ plot at $\Omega=12$.$W$
denotes topological charge.

In conclusion, our findings show that the Kerr-Sen-Ads black hole in a fixed
$(Q,\Omega,C)$ ensemble, the topological charge is $-1,0$ or $1$ depending
upon the values of the thermodynamic parameters $Q,\Omega,C$.

###  II.4 Fixed (Q,J,$\mu$) Ensemble

In this ensemble, the chemical potential $\mu$, charge $Q$ and angular
momentum $J$ is kept fixed. The chemical potential $\mu$ is given by :

$\mu=\frac{S\left(\pi^{2}C^{2}\left(4\pi^{2}J^{2}+S^{2}\right)-S^{2}\left(2\pi^{2}Q^{2}+S^{2}\right)\right)}{4\pi^{3/2}l(CS)^{3/2}\sqrt{\pi
C+S}\sqrt{\pi C\left(4\pi^{2}J^{2}+S^{2}\right)+2\pi^{2}Q^{2}S+S^{3}}}$ (40)

By solving equation 40, we get the expression for $C$ and substituting $C$ in
the following expression, newly mass($M_{\mu}$) in this ensemble as:

$M_{\mu}=M-C\mu$ (41)

The off-shell free energy is computed using:

$\mathcal{F}=M_{\mu}-S/\tau$

We compute the expressions for $\phi^{S}$ and $\tau$ using the same method as
in the preceding part.
It is seen that, for this ensemble the topological charge is always $1$
irrespective of the values of the thermodynamic parametrs $\mu,Q$ and $J.$
Moreover, only one branch is observed in the entropy $S$ vs $\tau$ plot for
all allowed values of the thermodynamic parameters. For example, in figure.17,
the entropy $S$ vs $\tau$ is plotted for $\mu=1,Q=1.5,J=1.2$, and $l=1$. We
observe a single black hole branch with topological charge equals to $1$.

Figure 17:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(Q,J,\mu)$ ensemble. Here $\tau$ is plotted against $S$
while keeping $l=1,\mu=1,Q=1.5$ and $J=1.2$ fixed. The topological charge is
found to be $1$.

###  II.5 Fixed ($\phi,\Omega$,C) ensemble

The next ensemble we consider is the fixed $(\Omega,\phi,C)$ ensemble. In this
ensemble angular frequency $\Omega$ and electric potential $\phi$ are kept
fixed. First, We substitute J from equation 33 in the expression for mass in
equation 23 as follows:

$M_{J}=\frac{\sqrt{S(\pi C+S)^{2}\left(\pi
CS+2\pi^{2}Q^{2}+S^{2}\right)}}{2\pi^{3/2}l\sqrt{CS\left(\pi
C-l^{2}S\Omega^{2}+S\right)}}$ (42)

Now equation 42 becomes independent of variable $J$. Next we find out $\phi$
fromequation 42 as follows:

$\phi=\frac{\partial M_{J}}{\partial Q}=\frac{\sqrt{\pi}Q\sqrt{S(\pi
C+S)^{2}\left(\pi CS+2\pi^{2}Q^{2}+S^{2}\right)}}{l\sqrt{CS\left(\pi
CS+2\pi^{2}Q^{2}+S^{2}\right)\left(\pi C-l^{2}S\Omega^{2}+S\right)}}$ (43)

Accordingly, $Q$ is given by equation 43 as:

$q=\frac{\sqrt{C}l\sqrt{S}\phi\sqrt{\pi C+S}\sqrt{\pi
C-l^{2}S\Omega^{2}+S}}{\sqrt{\pi}\sqrt{-\pi^{2}C^{2}\left(2l^{2}\phi^{2}-1\right)+2\pi
CS\left(l^{4}\Omega^{2}\phi^{2}-l^{2}\phi^{2}+1\right)+S^{2}}}$

Finally, we substitute $Q$ in the following expression for the modified mass
in fixed ($\Omega$,$\Phi$) ensemble, which is written as :

$\tilde{M}=M-q\phi-J\Omega$ (44)

Now $\tilde{M}$ is a function of $\phi,\Omega,C$ and $l.$ Using equation 44,
$\mathcal{F}$, $\phi^{S}$ and $\tau$ are constructed following standard
procedure as shown in the previous sections.
We plot $\tau$ vs $S$ curve for different values of $C$ as shown in figure 18
keeping $\Omega=1.2,\phi=3.5,l=0.1$ constant. In figure 18(a) and figure
18(b), with $C=0.05$ and $C=0.5$ respectively, one black hole branch with
topological charge $W=1$ is observed. At $C=10$, in figure 18(c) two black
hole branches and topological charge $0$ are found.

(a) W=1

(b) W=1

(c) W=0

Figure 18:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(\phi,\Omega,C)$ ensemble. Here $\tau$ is plotted
against $S$ for three different $C$ values while keeping
$l=0.1,\phi=3.5,\Omega=1.2$ fixed. Figure $\left(a\right)$ shows $\tau$ vs $S$
plot at $C=0.05$,figure $\left(b\right)$ shows $\tau$ vs $S$ plot at $C=0.5$
and in figure $\left(c\right)$ shows the same at $C=10$.$W$ denotes
topological charge.

Next we investigate the topological charge with reference to a variation in
the potential $\phi$ while keeping $C=10,\Omega=12,l=0.1$ fixed. In figure
19(a) and 19(b) we set $\phi=1$ and $\phi=10$ respectively where we find one
black hole branch and topological charge $W=-1$. In the latter case, where
$\phi=15$, we find two black hole branches with topological charge $0.$

(a) W=-1

(b) W=-1

(c) W=0

Figure 19:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(\phi,\Omega,C)$ ensemble. Here $\tau$ is plotted
against $S$ for three different $\phi$ values while keeping
$l=0.1,C=10,\Omega=12$ fixed. Figure $\left(a\right)$ shows $\tau$ vs $S$ plot
at $\phi=1$,figure $\left(b\right)$ shows $\tau$ vs $S$ plot at $\phi=10$ and
in figure $\left(c\right)$ shows the same at $\phi=15$.$W$ denotes topological
charge.

Finally,in figures 20, we fix $\phi=0.01$,$C=10,l=0.1$ and vary $\Omega.$ In
figure 20(a) and figure 20(b), setting $\Omega=5$ and $\Omega=10$
respectively, we get two black hole branches and topological charge $W=0.$ In
figure 20(c), with $\Omega=15$ a single blackl hole branch with topological
charge $W=-1$ is observed.

(a) W=0

(b) W=0

(c) W=-1

Figure 20:  $\tau$ vs $S$ plots for Kerr-Sen-Ads black hole in restricted
phase space in fixed $(\phi,\Omega,C)$ ensemble. Here $\tau$ is plotted
against $S$ for three different $\Omega$ values while keeping
$l=0.1,C=1,\phi=0.01$ fixed. Figure $\left(a\right)$ shows $\tau$ vs $S$ plot
at $\Omega=5$,figure $\left(b\right)$ shows $\tau$ vs $S$ plot at $\Omega=10$
and in figure $\left(c\right)$ shows the same at $\Omega=15$.$W$ denotes
topological charge.

So to summarize, the topological charge in fixed $(\phi,\Omega,C)$ ensemble is
$-1$,$0$ or $+1$ depending on all the thermodynamic parameters $C,\Omega$ and
$\phi.$

Like fixed ($Q,\Omega,C$) ensemble, here also we find a class of black hole,
where topological charge is found to be zero. Here also both Hawking-Page
transition and Davies type of phase transition are observed. For example,
taking $C=10,\phi=0.01,\Omega=5$ and $l=0.1$ we observe two black hole branch
as shown in figure.21

Figure 21: $\tau$ vs $S$ plot for Kerr Sen Ads black hole at
$C=10,\phi=0.01,\Omega=5$ and $l=0.1$. The red and the black colored solid
represent the large and small black hole branches respectively.

Using the following equation we construct the vector field

$\phi=\left(\frac{1}{S}\frac{\partial F^{2}}{\partial S},-cot\theta
csc\theta\right)$ (45)

The zero poins of the vector field is found to be $(12.0763,\frac{\pi}{2})$
and $(36.276,\frac{\pi}{2})$. The vectorplot is shown in figure.22

(a)

(b)

Figure 22: A portion of the vector field $\phi$ on $S-\theta$ plane. The black
dot of figure $(a)$ represents Hawking page phase transition point and the
blue dot in figure $(b)$ represents Davies point.

The black dot ($S=12.0763$) in figure 22(a) is a Davies transition point and
blue dot ($S=36.276$) is a Hawking-Page phase transition point. We have
confirmed that, these zero points exactly matches with the critical points of
the Hawking-page and Davies type phase transition. The topological charge is
calculated by constructing a contour $C$ around each zero point($C_{0}$ and
$C_{1}$ as shown in figure 22). As expected, the topological charge of the
Hawking page phase transition is $+1$ and that of Davis point is found to be
$-1$ as shown in figure.23

(a)

(b)

Figure 23: Kerr Sen AdS black hole in fixed ($\phi$,$\Omega$,C) ensemble :
Figure (a) represents topological number calculation for Davies point around
the $C_{0}$ contour in Figure.2 and Figure (b) represents the topological
number calculation for Hawking-Page phase transition point around the $C_{1}$
contour

##  III Conclusion

In this study, thermodynamic topology of the Kerr-Sen-Ads black hole in
restricted phase space is analyzed. For our analysis, we have considered five
ensembles of Kerr-Sen-Ads black holes in restricted phase space: fixed
$(Q,J,C)$, fixed $(\phi,J,C)$, fixed $(Q,\Omega,C)$, fixed $(Q,J,\mu)$, and
fixed $(\phi,\Omega,C)$. We studied the local and global topology in each
ensemble by computing the topological number. In the fixed $(Q,J,C)$, fixed
$(\phi,J,C)$, and fixed $(Q,J,\mu)$ ensembles, we found a topological charge
of $+1$. In the fixed $(Q,\Omega,C)$ and fixed $(\phi,\Omega,C)$ ensembles, we
found topological charges of $-1$, $0$, and $+1$. The results are summarized
below.

| Ensembles | Topological Charge | Annihilation Point | Generation Point
---|---|---|---|---
| Fixed ($Q,J,C$) | 1 | 1 or 0 | 1 or 0
| Fixed ($\phi,J,C$) | 1 | 1 or 0 | 1 or 0
| Fixed ($Q,\Omega,C$) | -1, 0 or 1 | 0 | 1 or 0
| Fixed ($Q,J,\mu$) | 1 | 0 | 0
| Fixed ($\phi,\Omega,C$) | -1, 0 or 1 | 0 | 1 or 0

Notably, within ensembles where the topological charge is determined to be
$0$, occurrence of both Hawking-Page and Davies-type phase transitions is
observed. We have shown that these distinct types of phase transitions can be
comprehensively investigated employing a common vector field. Furthermore,
both types of phase transitions can be distinguished by the associated
topological charges. The topological charge linked with Davies-type and
Hawking-Page phase transitions are found to be $-1$ and $+1$, respectively.

It is important to mention that three more possible ensembles exist in this
system: fixed ($\phi,J,\mu$), fixed ($Q,\Omega,\mu$), and fixed
($\phi,\Omega,\mu$). Mathematically, we found it very challenging to conduct
our analysis in these ensembles. We will endeavor to address this issue in our
future work. It will be interesting to study the thermodynamic topology of
various black hole systems in more challenging ensembles, which exist in other
phase space formalisms or different modified theories of gravity.

##  IV Acknowledgments

BH would like to thank DST-INSPIRE, Ministry of Science and Technology
fellowship program, Govt. of India for awarding the DST/INSPIRE
Fellowship[IF220255] for financial support.

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