By using (• ‣ 2.3), (A.1) and (A.1), we obtained:

$L_{9}^{r}=-\frac{0.0031({f_{r_{0}}}-1){g_{s}}N^{\frac{2{f_{r_{0}}}}{3}+\frac{4}{5}}}{({f_{r_{0}}}+1)\alpha_{\theta_{1}}^{3}}.$
(2.28)

Now, assuming:

${f_{r_{0}}}=1-\omega\alpha_{\theta_{1}}^{3},$ (2.29)

where

$\omega=\frac{4.6N^{-\frac{2{f_{r_{0}}}}{3}-\frac{4}{5}}}{{g_{s}}},$ (2.30)

hence we get a match with the phenomenological/experimental value $L_{9}^{\rm
exp}=6.9\times 10^{-3}$. Substituting (2.29) - (2.30) into (• ‣ 2.3), we
obtained:

$\left({\cal C}_{zz}^{(1)}-2{\cal C}_{\theta_{1}z}^{(1)}+2{\cal
C}_{\theta_{1}x}^{(1)}\right)\approx-\frac{987.4\lambda_{\epsilon}^{2}}{\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}}.$ (2.31)

Consistency of (2.29), (2.30) and (2.26), setting
$\sqrt[24]{\beta}\approx\beta^{0.04}$ to unity (because of the very small
exponent of $\beta$), requires:

$\displaystyle\alpha_{\theta_{1}}=\frac{11.33\sqrt[3]{{\cal C}^{\rm
IR}_{\phi_{0}}}N^{55/36}}{{g_{s}}\sqrt[12]{\lambda}_{\epsilon}\sqrt[3]{M}{N_{f}}^{2/3}\sqrt[3]{{\cal
C}^{\rm IR}_{\psi_{1}}}\log^{\frac{2}{3}}(N)}.$ (2.32)

  * •

Step 4: Matching $g_{\rm YM}^{2}(\Lambda_{\rm QCD}=0.4$GeV, $\Lambda=1.1$GeV,
$\mu=M_{\rho}$): Similarly, $g_{\rm YM}^{2}$ can be chosen to match the
experimental value $36$ (at $\Lambda_{\rm QCD}=0.3$GeV and the HLS-QCD
matching scale “$\Lambda^{\prime\prime}=1.1$GeV [97]) and renormalization
scale $\mu=M_{\rho}$ by imposing:

$\frac{12\delta
N}{\alpha_{\theta_{1}}^{2}\left(\frac{855.11\beta^{3/2}(\delta+1)\lambda_{\epsilon}^{3}{m_{0}}^{2}}{{f_{r_{0}}}^{2}\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}}+12N\right)}=36,$ (2.33)

which was affected by the following:

$\gamma=\frac{10^{6}\lambda_{\epsilon}^{3/2}{m_{0}}\beta^{3/4}\alpha_{\theta_{1}}\sqrt{8.4(\delta+1)}}{{f_{r_{0}}}{g_{s}}\left(\log
N\right)M\sqrt{N\left(3.27\times 10^{9}\delta-1.18\times
10^{11}\alpha_{\theta_{1}}^{2}\right)}}$ (2.34)

We obtained an upper bound on ${\cal C}_{\phi_{0}}^{\rm IR}$ from the
expression (2.34) by demanding that the argument of the square root in the
denominator of (2.34) be positive:

$\displaystyle{\cal C}_{\phi_{0}}^{\rm IR}<\frac{1.04\times
10^{-13}\sqrt[8]{\beta}({f_{r_{0}}}-1)^{2}{g_{s}}^{2}\sqrt[4]{\lambda_{\epsilon}}MN_{f}^{2}{\cal
C}_{\psi_{1}}^{\rm IR}\
N^{-3{f_{r_{0}}}-\frac{73}{20}}\log^{2}(N)}{{f_{r_{0}}}+1}.$ (2.35)

  * •

Step 5: Matching $L_{10}^{r}$: Now, using (A.1), (• ‣ 2.3), (A.1) and [98],
[99] (This also gives the UV-finite portion of the $rho-pi$ one-loop
correction through the use of a “$a(M_{\rho})$” factor with a value equal to
2), for $N_{f}=3$, we showed that:

$\displaystyle L_{10}^{r}=\frac{1}{4}\left(-\frac{1}{g_{\rm
YM}^{2}}+2z_{3}-2z_{2}+2z_{1}\right)+\frac{11N_{f}a(M_{\rho})}{96(4\pi)^{2}}$
$\displaystyle=\frac{0.47{g_{s}}N^{\frac{2{f_{r_{0}}}}{3}+1}}{\alpha_{\theta_{1}}}-\frac{8640.{g_{s}}N^{5/3}\alpha_{\theta_{1}}^{2}}{{g_{s}}N^{5/3}-1831.63\alpha_{\theta_{1}}^{3}}+4.4\times
10^{-3}.$ (2.36)

Since, $L_{10}^{r}=-5.5\times 10^{-3}$, the following is the value that we
were able to achieve for the $\theta_{1}$ delocalization parameter
$\alpha_{\theta_{1}}$:

$\displaystyle\alpha_{\theta_{1}}=-2.8\times
10^{-14}g_{s}N^{\frac{5}{3}}+\frac{\sqrt{4.2\times 10^{-6}-2.9\times
10^{-11}g_{s}^{2}N^{\frac{10}{3}}}}{2}.$ (2.37)

This indicates that $N<140$ for the case where $g_{s}=0.1$. For $g_{s}=0.1$,
this implies $N<140$. For the purpose of the numerical calculation, we set
$N=10^{2},g_{s}=0.1$, and $M=N_{f}=3$. We then obtained the following non-
linear relation between ${\cal C}_{\psi_{1}}^{\rm IR},\lambda_{\epsilon}$ and
${\cal C}_{\phi_{0}}^{\rm IR}$ (considered that they also satisfy (2.35):

$\frac{1.6\sqrt[3]{{\cal C}^{\rm
IR}_{\phi_{0}}}}{\lambda_{\epsilon}^{\frac{1}{12}}\sqrt[3]{{\cal C}^{\rm
IR}_{\psi_{1}}}}=10^{-7}.$ (2.38)

###  2.4 Summary

In this chapter, we computed the LECs of $SU(3)$ chiral perturbation theory
from the type IIA string dual inclusive of ${\cal O}(R^{4})$ corrections. We
constructed the $SU(3)$ Lagrangian and obtained the coupling constants from
the radial integrals quoted in appendix A. We already mention in the beginning
of this chapter that we have used the results of [2, 83] to do our
calculations in this chapter. My contribution has been presented in this
chapter.

On matching LECs computed holographically with their
phenomenological/experimental values, it turns out that a particular
combination of integration constants, $({\cal C}_{zz}^{\rm th}-{\cal
C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm th})$, is appearing in
all the LECs of $SU(3)$ chiral perturbation theory Lagrangian up to ${\cal
O}(p^{4})$. Here, ${\cal C}_{MN}^{\rm th}$ are integration constants appearing
in solutions to ${\cal O}(R^{4})$ corrections as worked out in [2] to the MQGP
metric for the thermal gravitational dual. This combination, $({\cal
C}_{zz}^{\rm th}-{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm
th})$, encodes information about the compact part of non-compact four cycle
around which flavor $D7$-branes are wrapping in type IIB setup. The key
results of this chapter are summarized below.

  1. 1.

To match our results of the one-loop renormalized $\chi$PT Lagrangian’s LECs
$L_{1,2,3,9,10}^{r},F_{\pi}^{2},g_{SU(3)}$ with the experimental values, we
had to fix the eight non-zero parameters listed below for the given values of
$N,M,N_{f}$ in the MQGP limit (1.39):

    1. (a)

a linear combination of integration constants ($({\cal C}_{zz}^{\rm th}-{\cal
C}_{\theta_{1}z}^{\rm th}+2{\cal C}_{\theta_{1}x}^{\rm th})$) appearing in the
solutions of the EOMs of the ${\cal O}(R^{4})$ ${\cal M}$-theory uplift’s
metric components $G^{\cal M}_{zz,\theta_{1}z,\theta_{1}x}$333It turns out
that there is a specific combination of only three costants of integration
that emerges while matching $\chi$PT LECs up to ${\cal O}(p^{4})$. Although in
theory there are additional constants of integration that exist in other
${\cal O}(R^{4})$ ${\cal M}$-theory metric components, it has been found that
there is only one such combination. Even though it is not clear the reason why
this particular combination, yet it is naturally obvious that it uses $G^{\cal
M}_{zz,\theta_{1}z,\theta_{1}x}$ as these simply belong to the $S^{3}$,
portion of the non-compact four-cycle wrapped around the flavor $D7$-branes in
the type IIB dual of [14]; these $D7$-flavor branes are (triple) T dualized to
the type IIA $D6$ flavor branes. In view of fact, this is a very non-trivial
signature of the four-cycle that is enveloped by the type IIB $D7$-branes. It
manifests itself as ${\cal O}(R^{4})$-corrections to the MQGP background,
which leads to an uplift of thermal QCD-like theories.;

    2. (b)

integral constants, ${\cal C}^{\rm IR}_{\psi_{1},\phi_{0}}$444It is possible
to self-consistently set the ${\cal C}^{\rm UV}_{\psi_{1},\phi_{0}}$ value to
zero occurring in the solutions of the EOMs in the IR of the $\rho$ meson and
the $\pi$ meson respectively.

    3. (c)

$\epsilon$ (or equivalently $\lambda_{\epsilon}$ since
$\epsilon=\lambda_{\epsilon}\frac{\sqrt{\beta}}{N}),\gamma$ appearing in
$a=\left(b+{\bf\gamma}{\cal O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{0}$;

    4. (d)

delocalization parameter $\alpha_{\theta_{1}}$ of $\theta_{1}$;

    5. (e)

tension of $D6$-brane or equivalently $\alpha^{\prime}$;

    6. (f)

$f_{r_{0}}$ as in $r_{0}=N^{-\frac{f_{r_{0}}}{3}}$

All the above points are summarised in the table 2.3.

S. No. | Quantities whose | Parameters of the holographic dual | Equation numbers
---|---|---|---
| experimental values | used for fitting |
| are fitted to |  |
1. | $L_{1,2,3}^{r}$ | Specific linear combination of |  (• ‣ 2.3)
|  | constants of integration appearing in |  [using (2.24)]
|  |  solutions to ${\cal O}(R^{4})$ corrections |
|  |  to ${\cal M}$-theory metric components |
|  |  $G^{\cal M}_{zz,\theta_{1}z,\theta_{1}x}$; $f_{r_{0}};\gamma;\lambda_{\epsilon}$; |
|  |  ${\cal O}(R^{4})-\frac{1}{N}$ connection: |  (2.24)
|  |  $a-r_{0}$ relation must have |
|  |  an $\epsilon\sim\frac{l_{p}^{3}}{N}$ contribution |
|  |  at ${\cal O}\left(\frac{g_{s}M^{2}}{N}\right)^{0}$ |
2. | $F_{\pi}^{2};\ L_{9}^{r}$ | $f_{r_{0}};\alpha_{\theta_{1}};{\cal C}^{\rm IR}_{\phi_{0}};{\cal C}^{\rm IR}_{\psi_{1}};\lambda_{\epsilon}$ |  (2.26), (2.28)-(2.32)[consistency
|  |  | check]
3. | $g_{SU(3)}$ |  $\gamma$; upper bound on ${\cal C}^{\rm IR}_{\phi_{0}}$ |  (2.34), (2.35)
4. | $L_{10}^{r}$ | $\alpha_{\theta_{1}};{\cal C}^{\rm IR}_{\phi_{0}};{\cal C}^{\rm IR}_{\psi_{1}}$ |  (2.37) - (2.38) [even though
|  |  |  specific values of $N,M,N_{f},g_{s}$
|  |  | chosen, but can find analog of
|  |  |  (2.38) $\forall N<140$ (2.37)
|  |  |  respecting (1.39)]
Table 2.3: An Overview of the agreement between the theoretical and
experimental Values of one-Loop Renormalized $\chi$PT LECs, $F_{\pi}^{2}$, and
$g_{SU(3)}^{2}$.

  2. 2.

In addition, the normalisation condition of the $\rho$-meson profile function
$\psi_{1}(Z)$ ($n=1$ mode) is used to determine the integration constants
${\cal C}_{\psi_{1}}^{\rm UV}$ appearing in the solution to $\psi_{1}(Z)$ EOM
in the UV, in terms of ${\cal C}^{\rm
IR}_{\psi_{1}},f_{r_{0}},\gamma,\epsilon,\lambda_{\epsilon}$ and $T_{D6}$ or
equivalently $\alpha^{\prime}$ (via “${\cal V}_{\Sigma_{2}}$”);

  3. 3.

Further, the normalization condition of the $\rho$-meson profile function
$\psi_{1}(Z)$ ($n=1$ mode) is used to determine the constant of integration
${\cal C}_{\psi_{1}}^{\rm UV}$ appearing in the solution to $\psi_{1}(Z)$ in
the UV, in terms of ${\cal C}^{\rm
IR}_{\psi_{1}},f_{r_{0}},\gamma,\epsilon,\lambda_{\epsilon}$ and $T_{D6}$ or
equivalently $\alpha^{\prime}$ (via “${\cal V}_{\Sigma_{2}}$”); it had been
shown that we were able to set ${\cal C}^{\rm UV}_{\psi_{1}}=0$ in a manner
that was self-consistent.

  4. 4.

If one were to take the expression for $T_{D6}/\alpha^{\prime}/{\cal
V}_{\Sigma_{2}}$ derived above from the normalisation condition of
$\psi_{1}(Z)$ and substitute it into the normalisation condition for the
profile function $\phi_{0}(Z)$, one would be able to self-consistently set
${\cal C}^{\rm UV}_{\phi_{0}}=0$.

##  Chapter 3 Deconfinement Phase Transition in Thermal QCD-Like Theories at
Intermediate Coupling in the Absence and Presence of Rotation

###  3.1 Introduction

We can compute the corrections to the infinite-’t Hooft-coupling limit as done
in [100] using gauge/gravity duality, but this time using an AdS background.
They used terms quartic in the Weyl tensor to include higher-derivative
corrections on the gravity side. From the gauge theory point of view, there is
a $SU(N)$ supersymmetric Yang-Mills plasma with ${\cal N}=4$ SUSY at an
intermediate ’t Hooft coupling. They provided an explanation for the transport
peak that appeared in the spectral function of the stress-energy tensor at
zero spatial momentum in the small frequency zone. This is a characteristic of
perturbative plasma generally.

Different bottom-up models built from gauge/gravity duality have been used to
study QCD at strong coupling, whereas perturbation theory has been used to
study it at weak coupling. The Sakai-Sugimoto model [90] is a well-known top-
down holographic type IIA dual, although one that only works in the infrared.
[14] and its ${\cal M}$-theory uplift [15] are the only UV-complete (type IIB)
top-down holographic dual of QCD-like theories at strong coupling that we are
aware of. Based on the hard-thermal-loop perturbation theory (HTLpt), authors
in [101] have investigated QCD thermodynamic functions at intermediate
coupling. Gauge/gravity duality can also be used to explore the intermediate
coupling regime of QCD. The authors obtained $O(l_{p}^{6})$ corrections to the
${\cal M}$-theory metric in the [1], as explained in chapter 1, in order to
proceed in this direction.

In general theories of gravity, Wald provided a formula to determine black
hole entropy [102]. He thought about the Lagrangian that results from the
diffeomorphism invariant classical theory of gravity in $n$ dimensions.
Noether charge is a $(n-2)$-form in these theories of gravity, and the
$(n-2)$-form Noether charge’s integral across its bifurcate killing horizon
will yield the black hole’s entropy. Therefore, for stationary black holes
with bifurcate killing horizons, entropy equals Noether charge. A formula to
calculate entropy for dynamical black holes was made in [103].

Every physical phenomenon has an energy/distance scale that defines it. RG
group flow is used to move from UV to IR. However, short-range physics begins
to interact with long-range physics in noncommutative field theories and
string theory, a process known as UV/IR mixing. UV divergences in real
$\phi^{4}$ theory defined on commutative space are changed into infrared poles
in the identical theory defined on noncommutative space as a result of UV/IR
mixing, for example. There are further examples that are nicely explained in
[104]. Theoretical models of gravity are nonlocal. As a result, UV/IR mixing
might be present in such theories. In the gauge/gravity duality, the energy
scale in the gauge theory side matches with the radial coordinate in
gravitational theory. In the context of the AdS/CFT correspondence, the
authors looked at the “I.R.-U.V.” connection and demonstrated how infrared
phenomena in bulk theory are turned into ultravoilet phenomena in boundary
theory [105]. In particular, the job of the ultraviolet regulator in the
${\cal N}=4$ super Yang Mills theory is played by the infrared regulator in
the bulk theory. It’s fascinating that in our work, matching at the
deconfinement temperature, ${\cal M}$-theory actions dual to the thermal and
black-hole backgrounds at the UV-cut-off, and obtaining a relationship between
the ${\cal O}(R^{4})$ metric corrections in the IR, a specific type of UV-IR
connection manifests itself.

Using a semiclassical analysis, we determined the deconfinement temperature of
QCD-like theory at the intermediate coupling using gauge/gravity duality, as
was first discussed in [4]. The confinement-deconfinement phase transition in
large $N_{c}$ gauge theories could also be discussed from the perspective of
entanglement entropy [106]. In this procedure, one must divide one of the
spatial coordinates into a segment of length ${\it l}$ and the other segment
in order to calculate the entanglement entropy among both of them. Ryu and
Takayanagi provided rules in [107] on how to calculate entanglement entropy
from AdS/CFT correspondence. There are two surfaces-connected and
disconnected-as explained in [106]. It has a critical value, which is
indicated by the symbol ${\it l_{crit}}$. It is the connected surface which
takes over entanglement entropy if we are below the critical point of ${\it
l}$, or ${\it l}<{\it l_{crit}}$, and it is the disconnected surface if we are
above the critical value of ${\it l}$, or ${\it l}>{\it l_{crit}}$. These two
regions correspond to the confining and deconfining phases of large $N_{c}$
gauge theories, respectively. Therefore, we can interpret this as a phase
transition for large $N_{c}$ gauge theories involving confinement-
deconfinement phases.

In noncentral relativistic heavy ion collisions (RHIC), a fluid known as
quark-gluon plasma (QGP) is created. According to experimental findings from
the RHIC collaboration, the angular momentum and angular velocity of this
rotating fluid are on the order of $10^{3}\hbar$ and $\omega\sim 6$ MeV,
respectively. In [108], the authors applied hydrodynamic simulations of heavy
ion collisions and found $\omega\sim 20-40$ MeV. Considering the results
mentioned above, a top-down analysis employing gauge-gravity duality of the
effect of rotation on the deconfinement temperature of thermal QCD-like
theories will be very interesting, with this motivation we obtained
deconfinement temperature of thermal QCD from a top-down approach in [5]. The
authors in [109] examined the consequences of rotating plasma on the
deconfinement temperature of QCD in Hard wall and Soft wall models, and they
discovered that deconfinement temperature of QCD decreases as angular velocity
of the plasma increases. Similar works using a bottom-up method are available
in [110, 111, 112].

###  3.2 Deconfinement Phase Transition in Thermal QCD-Like Theories in the
Absence of Rotation

In this section, we will discuss the deconfinement phase transition of thermal
QCD-like theories at intermediate coupling without including the effect of
rotation.

####  3.2.1 $T_{c}$ from Semi-classical Method

Here, we’ll detail how we implemented Witten’s proposal [4] to obtain the
deconfinement temperature thermal QCD by applying the gauge/gravity duality
and holographic renormalization of the eleven-dimensional supergravity action
via 3.2.1.1, 3.2.1.2, and 3.2.1.3.

The ${\cal M}$-theory dual for high temperatures, that is, temperatures that
are higher than the critical temperature $T_{c}$, will entail a black hole
with the metric presented by:

$\displaystyle\hskip-7.22743ptds_{11}^{2}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-g(r)dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\left(dx^{3}\right)^{2}\right)$
(3.1)
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{g(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where $A_{\rm IIA}^{F^{\rm IIB}_{i=1,3,5}}$ represent the type IIA RR 1-forms
that were derived using the triple T/SYZ-dual of the type IIB $F_{1,3,5}^{\rm
IIB}$ fluxes found in the type IIB holographic dual of [14], and the black
hole function $g(r)=1-\frac{r_{h}^{4}}{r^{4}}$111See appendix B.1 for the
explicit metric components.. For temperatures below the critical temperature,
denoted by the notation $T<T_{c}$, the metric is represented by a thermal
gravitational dual:

$\displaystyle\hskip-7.22743ptds_{11}^{2}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\tilde{g}(r)\left(dx^{3}\right)^{2}\right)$
(3.2)
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{\tilde{g}(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where $\tilde{g}(r)=1-\frac{r_{0}^{4}}{r^{4}}$ and $h(r,\theta_{1,2})$ is warp
factor in type IIB string theory [14, 15]. One notes that in (3.1),
$t\rightarrow x^{3},\ x^{3}\rightarrow t$, and that after performing a double
Wick rotation in the newly generated $x^{3},t$ coordinates, one finds (3.2).
In sum, this means: $-g_{tt}^{\rm BH}(r_{h}\rightarrow r_{0})=g_{x^{3}x^{3}}\
^{\rm Thermal}(r_{0}),$ $g_{x^{3}x^{3}}^{\rm BH}(r_{h}\rightarrow
r_{0})=-g_{tt}\ ^{\rm Themal}(r_{0})$ in the findings [113, 1]. For further
information on type IIA Euclidean/black $D4$-branes, see the work of
Kruczenski et al [29]. We assumed that world volume of the spatial part of the
solitonic $M3$ brane (equivalent to solitonic $M5$-brane wrapped around a
homologous sum of two-spheres [61]) is $\mathbb{R}^{2}(x^{1,2})\times
S^{1}(x^{3})$ where $S^{1}(x^{3})$ has the period $\frac{2\pi}{M_{\rm KK}}$
with $M_{\rm KK}=\frac{2r_{0}}{L^{2}}\left[1+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right]$ and $L=\left(4\pi
g_{s}N\right)^{\frac{1}{4}}$ because $r_{0}$ is very small IR cut-off for
thermal background. Hence, $\lim_{M_{\rm KK}\rightarrow
0}\mathbb{R}^{2}(x^{1,2})\times S^{1}(x^{3})=\mathbb{R}^{3}(x^{1,2,3})$
implies the four dimensional physics. We took $\tilde{g}(r)=1$ in (3.2) for
the computation of $T_{c}$.

If we assume that the periods of the thermal circles in the black hole and
thermal cal ${\cal M}$-theory backgrounds are $\beta_{\rm BH,Th}$, then at
$r={\cal R}_{\rm UV}$: $\beta_{\rm BH}\sqrt{G^{\rm BH}_{tt}}=\beta_{\rm
Th}\sqrt{G^{\rm Th}_{tt}}$, and at $T=T_{c}$ [4],

$\beta_{\rm BH}\not{\int}_{M_{11}}\left({\cal L}^{\rm BH}_{\rm EH}+{\cal
L}^{\rm BH}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal L}^{\rm BH}_{{\cal
O}(R^{4})}\right)=\beta_{\rm Th}\not{\int}_{M_{11}}\left({\cal L}^{\rm
Th}_{\rm EH}+{\cal L}^{\rm Th}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal
L}^{\rm Th}_{{\cal O}(R^{4})}\right),$ (3.3)

where $\not{\int}$ does not include the coordinate integral of $x^{0}$ which
implies:

$\displaystyle\left(\sqrt{1-\frac{r_{h}^{4}}{{\cal R}_{\rm
UV}^{4}}}\right)^{-1}\int_{M_{10}}\left({\cal L}^{\rm BH}_{\rm EH}+{\cal
L}^{\rm BH}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal L}^{\rm BH}_{{\cal
O}(R^{4})}\right)$ $\displaystyle=\int_{\tilde{M}_{10}}\left({\cal L}^{\rm
Th}_{\rm EH}+{\cal L}^{\rm Th}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal
L}^{\rm Th}_{{\cal O}(R^{4})}\right).$ (3.4)

Derivation of the On-Shell Action: The following is the structure of the
supergravity action in $\kappa_{11}^{2}=1$-units that corresponds to
$calM$-theory in the presence of ${\cal O}(R^{4})$ terms.

$\displaystyle S_{D=11}=\frac{1}{2}\Biggl{[}\int_{M_{11}}\sqrt{-G^{\cal
M}}\left(R-\frac{G_{4}^{2}}{2.4!}+\beta\left(J_{0}-\frac{E_{8}}{2}\right)\right)$
$\displaystyle+2\int_{\partial
M_{11}}\sqrt{-h}K-\frac{1}{6}\int_{M_{11}}C_{3}\wedge G_{4}\wedge
G_{4}+4\pi^{2}\int_{M_{11}}C_{3}\wedge X_{8}\Biggr{]},$ (3.5)

that results in an EOM:

$\displaystyle R_{MN}-\frac{G_{MN}}{2}R-\frac{1}{12}\left(G_{MPQR}G_{N}^{\ \
PQR}-\frac{G_{MN}}{8}G_{4}^{2}\right)$
$\displaystyle=-\beta\left[\frac{G_{MN}}{2}\left(J_{0}-\frac{E_{8}}{2}\right)+\frac{\delta}{\delta
G^{MN}}\left(J_{0}-\frac{E_{8}}{2}\right)\right].$ (3.6)

Using the trace of (3.2.1), we obtained:

$\displaystyle-\frac{9}{2}R+\frac{G_{4}^{2}}{32}=-\beta\left[\frac{11}{2}\left(J_{0}-\frac{E_{8}}{2}\right)+G^{MN}\frac{\delta}{\delta
G^{MN}}\left(J_{0}-\frac{E_{8}}{2}\right)\right]$
$\displaystyle\approx-\beta\left[\frac{11}{2}J_{0}+G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}\right],$ (3.7)

wherever there is a justification for the estimate in [1]. Let us write
$R=R^{(0)}+\beta R,K=K^{(0)}+\beta K,C_{3}=C_{3}^{(0)}+\beta C_{3}$, etc., in
[1], it is also demonstrated that one is capable of setting $C_{3}=0$ .
Therefore,

$\displaystyle G_{4}^{2}=144R^{(0)},$ $\displaystyle
J_{0}\approx\frac{9}{11}R^{(1)}-\frac{2}{11}G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}.$ (3.8)

Hence, on-shell action up to ${\cal O}(\beta)$ using $X_{8}(M_{11})=0$ [15] is
obtained as:

$S_{D=11}^{\rm on-shell}=-\frac{1}{2}\Biggl{[}-2S_{\rm EH}^{(0)}+2S_{\rm
GHY}^{(0)}+\beta\left(\frac{20}{11}S_{\rm
EH}-2\int_{M_{11}}\left(\sqrt{-G^{\cal M}}\right)^{(1)}R^{(0)}+2S_{\rm
GHY}-\frac{2}{11}\int_{M_{11}}\sqrt{-G^{\cal M}}G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}\right)\Biggr{]},$ (3.9)

where $\left(\sqrt{-G^{\cal M}}\right)^{(1)}=\frac{G_{\cal
M}^{(1)}}{2\sqrt{-G_{\cal M}^{(0)}}}$.

#####  3.2.1.1 Black Hole Background Uplift (Relevant to $T>T_{c}$) and
Holographic Renormalization of On-Shell $D=11$ Action

In this chapter, we worked near the Ouyang embedding implemented by the
following coordinate patches [66]:

$\theta_{1}=\frac{\alpha_{\theta_{1}}}{N^{\frac{1}{5}}},\
\theta_{2}=\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}.$ (3.10)

The following expression is what we computed the IR Einstein-Hilbert term:

$\displaystyle\left.\sqrt{-G^{\cal M}}R\right|_{\rm Ouyang}^{\rm IR}$
$\displaystyle\sim-\frac{b^{2}{g_{s}}^{3/4}{\log
N}^{3}M{N_{f}}^{3}{r_{h}}^{4}\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)(1-\beta({\cal C}_{zz}^{\rm bh}-2{\cal
C}_{\theta_{1}z}^{\rm bh}+3{\cal C}_{\theta_{1}x}^{\rm
bh}))}{{N^{\frac{5}{4}}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{3}(r-r_{h})\sin^{3}{\theta_{1}}\sin^{2}{\theta_{2}}}.$

Performing now the angular integration for the previously stated equation,

$\int\int
d\theta_{1}d\theta_{2}\frac{1}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}=\frac{1}{8}\cot({\theta_{2}})\left(\csc^{2}\left(\frac{{\theta_{1}}}{2}\right)-\sec^{2}\left(\frac{{\theta_{1}}}{2}\right)-4\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)+4\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)\right)$
(3.12)

which simplifies near (3.10) as given below:

$\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{d\theta_{1}d\theta_{2}}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}\sim\frac{N^{\frac{7}{10}}}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}+{\cal
O}\left(N^{\frac{3}{10}}\right).$ (3.13)

We derived the following expression for the action density (for Einstein-
Hilbert term) in the IR by utilizing equation (3.13) and carrying out radial
integration of (3.2.1.1):

$\displaystyle\frac{\left.\int_{r=r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-G^{\cal
M}}R\right|_{\rm Ouyang}}{{\cal V}_{4}}\sim$
$\displaystyle-\frac{b^{2}{g_{s}}^{3/4}{\log
N}^{3}M{N_{f}}^{3}{r_{h}}^{4}\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)(1-\beta({\cal C}_{zz}^{\rm bh}-2{\cal
C}_{\theta_{1}z}^{\rm bh}+3{\cal C}_{\theta_{1}x}^{\rm
bh}))}{{N}^{\frac{11}{20}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}},$

where ${\cal R}_{D5-\overline{D5}}^{\rm bh}\equiv\sqrt{3}a^{\rm bh}$, with the
resolution parameter, $a^{\rm bh}=\left(\frac{1}{\sqrt{3}}+\epsilon^{\rm
bh}+{\cal O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{h}$ of the blown up
$S^{2}$ [1]. In addition, there is one additional term that appears in the on-
shell action as stated in the equation (3.9), and the simplified version of
that term is as follows:

$\displaystyle\frac{\int_{r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\left(\sqrt{-G^{\cal
M}}\right)^{(1)}R^{(0)}}{{\cal V}_{4}}$ $\displaystyle\sim\beta{\cal
C}_{\theta_{1}x}^{\rm
bh}\frac{b^{2}g_{s}^{\frac{3}{4}}\log^{3}NMN_{f}^{3}r_{h}^{4}\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\log\left(1-\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (3.15)

The version of the Einstein-Hilbert term pertaining to the ultraviolet(UV)
region is:

$\displaystyle\hskip-43.36243pt\left.\sqrt{-G^{\cal M}}R\right|_{\rm
Ouyang}^{\rm UV}=-\frac{4\left(34{\log r}^{2}+14{\log r}-1\right){M_{\rm
UV}}r^{3}}{9\sqrt{3}\sqrt[4]{\pi}{g_{s}}^{9/4}N^{\frac{5}{4}}\sin^{3}{\theta_{1}}\sin^{2}{\theta_{2}}}$
$\displaystyle\times\frac{1}{\left(-{g_{s}}{N_{f}^{\rm UV}}(2{\log N}{\log
r}+{\log N}+3(1-6{\log r}){\log r})+2{g_{s}}(2{\log r}+1){N_{f}^{\rm
UV}}\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+8\pi{\log
r}\right)}$ $\displaystyle\hskip 36.135pt\approx\frac{4{M_{\rm
UV}}r^{3}\left(34\log^{2}(r)+14\log(r)-1\right)}{9\sqrt{3}\sqrt[4]{\pi}{g_{s}}^{13/4}{N_{f}^{\rm
UV}}N^{\frac{5}{4}}\sin^{3}{\theta_{1}}\sin^{2}{\theta_{2}}\log
N(2\log(r)+1)},$ (3.16)

where the correction terms, ${\cal O}\left(\frac{1}{\log N}\right)$, have been
removed to make the computation simpler. Since,
$\int\frac{r^{3}\left(34\log^{2}(r)+14\log(r)-1\right)}{2\log(r)+1}=\frac{1}{16}\left(\frac{4{Ei}(4\log(r)+2)}{e^{2}}+r^{4}(68\log(r)-23)\right)$
\+ constant. The following UV finite Einstein-Hilbert term was consequently
found:

$\displaystyle\left.\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\frac{\int_{{\cal R}_{D5-\overline{D5}}}^{{\cal R}_{\rm
UV}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-G^{\cal
M}}R}{{\cal V}_{4}}\right|_{\rm Ouyang}^{\rm UV-Finite}$
$\displaystyle\sim\frac{{M_{\rm UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{{g_{s}}^{13/4}{N}^{\frac{11}{20}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{N_{f}^{\rm
UV}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\log N}.$ (3.17)

The following is the part that make up the EH action’s contribution to UV
divergent:

$\displaystyle S_{\rm UV-div}^{\rm EH}\sim\frac{{M_{\rm UV}}{\cal R}_{\rm
UV}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{{g_{s}^{\rm UV}}^{13/4}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}{\log N}{N_{f}^{\rm
UV}}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (3.18)

Up to ${\cal O}(\beta)$, the UV-finite boundary Gibbons-Hawking-York
contribution is found to be:

$\displaystyle\hskip-32.52127pt\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\left.\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-h^{\cal
M}}K\right|_{\rm Ouyang}}{{\cal V}_{4}}\right|^{r={\cal R}_{\rm
UV}}\sim\frac{{M_{\rm UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}{{g_{s}}^{9/4}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}^{4}{N}^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$
(3.19)

The GHY term’s UV-divergent contribution is as follows:

$S_{\rm UV-div}^{\rm GHY}\sim\frac{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right){M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}}{\left(g_{s}^{\rm UV}\right)^{9/4}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}{N}^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$
(3.20)

We showed that the contribution from the higher derivative term is as follows,
up to ${\cal O}(\beta^{0})$:

$\displaystyle\hskip-28.90755pt\frac{\int_{r_{0}}^{{\cal
R}_{D5/\overline{D5}}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}}{{\cal
V}_{4}}$
$\displaystyle\hskip-28.90755pt\sim\frac{M{N_{f}}{r_{h}}\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{3}\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\left({\log
N}-9\left(\frac{{r_{h}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right)\left[\log N-3\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right]^{2}}{\epsilon^{5}{g_{s}}N^{39/20}\log^{2}(N)}$
$\displaystyle\hskip-28.90755pt\times\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}};$
(3.21)

utilizing:

$\displaystyle-\int dr\frac{r^{3}\log(r)}{(2\log(r)+1)^{7}}$
$\displaystyle=\frac{1}{360}\left(\frac{r^{4}\left(256\log^{5}(r)+704\log^{4}(r)+800\log^{3}(r)+488\log^{2}(r)+184\log(r)+19\right)}{(2\log(r)+1)^{6}}-\frac{16{Ei}(4\log(r)+2)}{e^{2}}\right);$
$\displaystyle-\int
dr\frac{r\log(r)\left(4\log^{2}(r)+15\log(r)+9\right)}{(2\log(r)+1)^{7}}$
$\displaystyle=\frac{2r^{2}\left(7504\log^{5}(r)+22512\log^{4}(r)+30016\log^{3}(r)+23384\log^{2}(r)+8799\log(r)+833\right)}{2880(2\log(r)+1)^{6}}-\frac{469{Ei}(2\log(r)+1)}{5760e};$
$\displaystyle-\int
dr\frac{\log(r)\left(45a^{4}\left(64\log^{3}(r)+208\log^{2}(r)+212\log(r)+57\right)-8{r_{h}}^{4}(2\log(r)+5)\right)}{r(2\log(r)+1)^{7}}$
$\displaystyle=\frac{15\left(3915a^{4}-4{r_{h}}^{4}\right)\log^{2}(r)+12\left(1620a^{4}-11{r_{h}}^{4}\right)\log(r)+21600a^{4}\log^{4}(r)+61200a^{4}\log^{3}(r)+1620a^{4}-11{r_{h}}^{4}}{30(2\log(r)+1)^{6}}.$

The higher derivative term that gives rise to the UV finite contribution was
found to be:

$\displaystyle\hskip-43.36243pt\left.\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}}{{\cal
V}_{4}}\right|^{\rm UV-finite}$
$\displaystyle\hskip-43.36243pt=\left.\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}}{{\cal
V}_{4}}\right|^{\rm UV-finite}$
$\displaystyle\hskip-43.36243pt\sim-\frac{{M_{\rm UV}}}{{g_{s}^{\rm
UV}}^{14/3}{\log N}^{11/3}N^{39/20}{N_{f}^{\rm
UV}}^{8/3}}\left(\frac{{r_{h}}^{2}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{2}}+1\right)$
$\displaystyle\times\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}.$
(3.23)

The following has been utilized. The most dominating term in $\sqrt{-G^{\cal
M}}G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}$ is
$-\frac{8{M_{\rm UV}}r^{3}\log(r)}{177147\pi{g_{s}^{\rm UV}}^{4}(2{\log
r}+1)^{7}N^{39/20}{N_{f}^{\rm
UV}}^{2}\log^{3}(N)}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$.
Since:
$\int
dr\frac{r^{3}\log(r)}{(2\log(r)+1)^{7}}=\frac{2{Ei}(4\log(r)+2)}{45e^{2}}-\frac{r^{4}\left(256\log^{5}(r)+704\log^{4}(r)+800\log^{3}(r)+488\log^{2}(r)+184\log(r)+19\right)}{360(2\log(r)+1)^{6}}$.
Therefore, it is important to highlight that the contribution of UV-finite to:
$\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm UV}^{4}}\right)\sqrt{-G^{\cal
M}}G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}$ is:
$-\frac{{M_{\rm UV}}}{{g_{s}^{\rm UV}}^{14/3}{\log
N}^{11/3}N^{39/20}{N_{f}^{\rm
UV}}^{8/3}}\frac{r_{h}^{4}}{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)}\left(\frac{{r_{h}}^{2}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}^{2}}+1\right)\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$,
which ends up being the case: ${\cal
O}\left(\frac{r_{h}^{4}}{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)}\right)$-suppressed in comparison to
$\left.\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\sqrt{-G^{\cal
M}}G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|^{\rm UV-finite}$.
Additionally, UV divergence contribution was found from higher derivative term
as:

$\displaystyle\hskip-21.68121pt\frac{\left.\int\sqrt{-G^{\cal
M}}G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm UV-div}}{{\cal
V}_{4}}$ (3.24)
$\displaystyle\hskip-21.68121pt\sim\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$
$\displaystyle\times\frac{{M_{\rm UV}}{{\cal R}_{\rm UV}}^{4}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{g_{s}^{\rm UV}}^{14/3}N^{39/20}{N_{f}^{\rm
UV}}^{8/3}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{4}\log^{\frac{11}{3}}(N)\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}.$

In addition, we observe that the surface counter term that neutralizes UV
divergence is emerging via higher derivative term is provided by the
expression that follows when r is held constant.

$\displaystyle\hskip-21.68121pt\beta\left.\frac{\left.\int\sqrt{-h}h^{mn}\frac{\delta
J_{0}}{\delta h^{mn}}\right|_{\rm UV-div}}{{\cal V}_{4}}\right|_{r=\rm
constant}\sim-\beta\frac{\left(\frac{1}{{g_{s}}^{\rm UV}}\right)^{2/3}{M_{\rm
UV}}{{\cal R}_{\rm UV}}^{4}}{{g_{s}^{\rm UV}}^{13/4}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{\log N}^{3}\left\\{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right\\}^{6}N^{11/5}{N_{f}^{\rm UV}}^{2}}$
$\displaystyle\hskip-21.68121pt\times\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}+{\cal
O}(\beta^{2}).$ (3.25)

Using:

$\displaystyle\hskip-21.68121pt\frac{1}{N^{\frac{7}{5}}}\int\int
d\theta_{1}d\theta_{2}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$
$\displaystyle\hskip-21.68121pt=\frac{1}{N^{\frac{7}{5}}}\Biggl{[}-2214\cot^{3}({\theta_{1}})\cot({\theta_{2}})+2214\cot({\theta_{1}})\left(2\csc^{2}({\theta_{1}})+1\right)\cot({\theta_{2}})+\frac{1}{32\sqrt{6}}\Biggl{\\{}20{\theta_{2}}\csc^{6}\left(\frac{{\theta_{1}}}{2}\right)+120{\theta_{2}}\csc^{4}\left(\frac{{\theta_{1}}}{2}\right)$
$\displaystyle\hskip-21.68121pt+600{\theta_{2}}\csc^{2}\left(\frac{{\theta_{1}}}{2}\right)+5{\theta_{2}}\sec^{6}\left(\frac{{\theta_{1}}}{2}\right)\Biggl{(}-150\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)+15\cos(3{\theta_{1}})\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)+150\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)$
$\displaystyle\hskip-21.68121pt+9\cos({\theta_{1}})\left(-25\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)+25\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)-8\right)+15\cos(2{\theta_{1}})\left(-6\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)+6\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)-1\right)$
$\displaystyle\hskip-21.68121pt-15\cos(3{\theta_{1}})\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)-61\Biggr{)}-1259712(\cos(2{\theta_{2}})-2)\cot({\theta_{2}})\csc^{2}({\theta_{2}})\left(\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)-\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)\right)\Biggr{\\}}\Biggr{]},$

We found that small values of $\theta_{1,2}$ produced the most significant
effect. Therefore,

$\displaystyle\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$
$\displaystyle\sim
N^{\frac{9}{10}}\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}\sim-\frac{\log
N}{\alpha_{\theta_{2}}^{3}}N^{\frac{9}{10}}.$ (3.27)

Because of this, the simpler version of the counter term is as follows:

$\displaystyle\hskip-21.68121pt\beta\left.\frac{\left.\int\sqrt{-h}h^{mn}\frac{\delta
J_{0}}{\delta h^{mn}}\right|_{\rm UV-div}}{{\cal V}_{4}}\right|_{r=\rm
constant}\sim\beta\frac{\left(\frac{1}{{g_{s}}^{\rm UV}}\right)^{2/3}{M_{\rm
UV}}{{\cal R}_{\rm UV}}^{4}\log N}{{g_{s}^{\rm UV}}^{13/4}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{\log N}^{3}\left\\{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right\\}^{6}N^{13/10}{N_{f}^{\rm
UV}}^{2}\alpha_{\theta_{2}}^{3}}.$ (3.28)

Further,

$\displaystyle\beta\left.\frac{\int_{r={\cal R}_{\rm UV}}\sqrt{-h}J_{0}}{{\cal
V}_{4}}\right|^{\rm UV}$ $\displaystyle\sim\frac{M_{\rm UV}}{g_{s}^{\rm
UV}\log^{\frac{8}{3}}N^{3}N_{f}^{\rm UV}\ {}^{\frac{8}{3}}}\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{d\theta_{1}d\theta_{2}}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}$
$\displaystyle\sim\frac{M_{\rm UV}}{g_{s}^{\rm
UV}\log^{\frac{8}{3}}N^{\frac{23}{10}}N_{f}^{\rm UV}\
{}^{\frac{8}{3}}}\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right).$ (3.29)

By matching the results of (3.24) with those of (3.2.1.1), one is able to
impose the following:

$\log^{3}N\left\\{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\right\\}^{5}{N_{f}^{\rm
UV}}^{2}\sim{N_{f}^{\rm UV}}^{8/3}\log^{\frac{11}{3}}(N),$ (3.30)

results in:

$N_{f}^{\rm UV}\sim\frac{\left(\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\right)^{\frac{15}{2}}}{\log N}.$
(3.31)

Holographic Renormalization When $T>T_{c}$: It has been found that the form of
the UV-divergent part of the on-shell action (3.9) for the black hole backdrop
is as follows:

$\displaystyle\hskip-25.29494ptS_{\rm D=11}^{\rm on-shellUV-
divergent}=-\frac{1}{2}\Biggl{[}\left(-2\kappa_{\rm EH}^{(0)}(g_{s}^{\rm
UV},N,M_{\rm UV},N_{f}^{\rm UV};\alpha_{\theta_{1,2}})+2\kappa^{(0)}_{\rm
GHY}(g_{s}^{\rm UV},N,M_{\rm UV},N_{f}^{\rm UV};\alpha_{\theta_{1,2}})\right)$
$\displaystyle\hskip-25.29494pt\times\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)+\beta\Biggl{\\{}-\frac{2}{11}\kappa^{(0)}_{\sqrt{-g}g^{MN}\frac{\delta
J_{0}}{\delta g^{MN}}}(g_{s}^{\rm UV},N,M_{\rm UV},N_{f}^{\rm
UV};\alpha_{\theta_{1,2}})\frac{\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}}{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}\Biggr{\\}}\Biggr{]}.$
(3.32)

As a result, the following are the necessary counter terms in order to cancel
out the UV-divergences:

$\displaystyle S_{\rm CT}^{\rm
BH}=-\frac{1}{2}\Biggl{[}-\frac{\kappa^{(0)}_{\rm EH+GHY}}{\kappa^{(0)}_{{\rm
EH}@{\cal R}_{\rm UV}}}\int_{r={\cal R}_{\rm UV}}\sqrt{-h}R$
$\displaystyle+\beta
N^{\frac{1}{4}}\frac{\left(\frac{2}{11}\kappa^{(0)}_{\sqrt{-g}g^{MN}\frac{\delta
J_{0}}{\delta g^{MN}}}(g_{s}^{\rm UV},M_{\rm UV},N_{f}^{\rm
UV};\alpha_{\theta_{1,2}})\right)\int_{r={\cal R}_{\rm
UV}}\sqrt{-h}g^{mn}\frac{\delta J_{0}}{\delta
g^{mn}}}{\kappa^{(0)}_{\sqrt{-h}g^{mn}\frac{\delta J_{0}}{\delta g^{mn}}@{\cal
R}_{\rm UV}}}\Biggr{]}_{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)=\left(N_{f}^{\rm UV}\log
N\right)^{\frac{2}{15}}},$

where the second term of (3.2.1.1) is worked out in (3.28), and

$\displaystyle\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-h}\right|_{r={\cal
R}_{\rm UV}}}{{\cal V}_{4}}\sim\frac{\left(\frac{1}{{g_{s}^{\rm
UV}}}\right)^{7/3}{M_{\rm UV}}{\cal R}_{\rm UV}^{4}\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{N^{\frac{3}{10}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}},$
$\displaystyle\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-h}R\right|_{r={\cal
R}_{\rm UV}}}{{\cal V}_{4}}\sim\frac{\left(\frac{1}{{g_{s}^{\rm
UV}}}\right)^{2/3}{M_{\rm UV}}{\cal R}_{\rm UV}^{4}\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}{{g_{s}^{\rm
UV}}^{3/2}N^{1/5}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (3.34)

#####  3.2.1.2 Thermal Background Uplift (Relevant to $T<T_{c}$) and
Holographic Renormalization of On-Shell $D=11$ Action

In the limit of large $N$, as well as in the infrared, the $f_{MN}$ EOMs
associated with the thermal background become algebraic. Writing below just
those components that receive a non-trivial ${\cal O}(\beta)$ contributions,
below is the ${\cal O}(\beta)$-corrected MQGP metric for the thermal
background in the IR in the $\psi=2n\pi,n=0,1,2$-coordinate patches.

$\displaystyle G^{\cal M}_{rr}=G^{\rm
MQGP}_{rr}\left(1-\frac{99\sqrt{\frac{3}{2}}\beta{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}N_{f}{r_{0}}\alpha_{\theta_{1}}^{6}f_{x^{10}x^{10}}({r_{0}})\log^{2}({r_{0}})}{2\pi^{3/2}\alpha_{\theta_{2}}^{5}}\right),$
$\displaystyle G^{\cal M}_{x\theta_{1}}=G^{\rm
MQGP}_{x\theta_{1}}\left(1-\beta f_{x^{10}x^{10}}(r_{0})\right),$
$\displaystyle G^{\cal M}_{y\theta_{1}}=G^{\rm
MQGP}_{y\theta_{1}}\left(1+\beta f_{\theta_{1}y}(r_{0})\right),$
$\displaystyle G^{\cal M}_{z\theta_{1}}=G^{\rm
MQGP}_{z\theta_{1}}\left(1+\frac{539\pi^{3}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})}{1728{g_{s}}^{3}M^{2}N_{f}^{2}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\cal M}_{y\theta_{2}}=G^{\rm
MQGP}_{y\theta_{2}}\left(1-2\beta f_{x^{10}x^{10}}(r_{0})\right),$
$\displaystyle G^{\cal M}_{yy}=G^{\rm MQGP}_{yy}\left(1-\frac{\pi^{3/2}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})\left(1617\sqrt{3}\pi^{3/2}{r_{0}}\alpha_{\theta_{1}}^{4}-32\sqrt{2}{g_{s}}^{3/2}MN_{f}\alpha_{\theta_{2}}\right)}{2592\sqrt{3}{g_{s}}^{3}M^{2}N_{f}^{2}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\cal M}_{yz}=G^{\rm MQGP}_{yz}\left(1+\frac{\pi^{3/2}\beta
N^{2/5}\alpha_{\theta_{2}}^{3}f_{x^{10}x^{10}}({r_{0}})}{81\sqrt{6}{g_{s}}^{3/2}MN_{f}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\cal M}_{zz}=G^{\rm MQGP}_{zz}\left(1+\frac{539\pi^{3}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})}{864{g_{s}}^{3}M^{2}N_{f}^{2}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\cal M}_{x^{10}x^{10}}=G^{\rm
MQGP}_{x^{10}x^{10}}\left(1+\beta f_{x^{10}x^{10}}(r_{0})\right).$ (3.35)

The Einstein-Hilbert term pertaining to the thermal background, which works in
the infrared close to (3.10), is:

$\displaystyle\left.\sqrt{-G^{\cal M}}R\right|_{\rm Ouyang}^{\rm
IR}\sim-\frac{{g_{s}}^{3/4}{\log N}^{2}\log\left(\frac{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)M{N_{f}}^{3}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}N^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}}\sum_{m=0}^{2}\kappa_{\rm
EH,\ IR}^{\beta^{0},m}\ {r_{0}}^{3-m}\left(r-r_{0}\right)^{m}$
$\displaystyle-\beta\frac{{\log
N}\left(243\sqrt{6}\sin^{3}{\theta_{1}}-8\sin^{2}{\theta_{2}}\right)f_{x^{10}x^{10}}({r_{0}})}{N^{\frac{1}{4}}\sin^{5}{\theta_{1}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{3}{g_{s}}^{9/4}\log\left(\frac{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)M({\log
N}{N_{f}})^{5/3}}\sum_{m=0}^{2}\kappa_{\rm EH,\ IR}^{\beta^{1},m}\
{r_{0}}^{3-m}\left(r-r_{0}\right)^{m},$

where ${\cal R}_{D5/\overline{D5}}^{\rm th}=\sqrt{3}a^{\rm th}$, the
resolution parameter $a^{\rm th}=\left(\frac{1}{\sqrt{3}}+\epsilon+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{0}$ [1] of the blown up $S^{2}$
associated with the thermal background, and we have calculated the numerical
prefactors $\kappa_{\rm EH,\ IR}^{\beta^{m},n}$, but we will not provide them
here because they do not shed any new light on the $\beta^{m}(r-r_{0})^{n}$
terms in $\sqrt{-G^{\cal M}}R$ that are constrained to the Ouyang embedding of
the flavor $D7$-branes in the type IIB dual and in the IR. From (3.2.1.2), we
evaluated $\frac{\left.\int_{r=r_{0}}^{{\cal
R}_{D5-\overline{D5}}}\sqrt{-G^{\cal M}}R\right|_{\rm Ouyang}}{{\cal V}_{4}}$.
At $O(\beta)$, the following terms also appeared in on-shell action:

$\displaystyle\frac{\int_{r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\left(\sqrt{-G^{\cal
M}}\right)^{(1)}R^{(0)}}{{\cal V}_{4}}$
$\displaystyle\sim\frac{\beta{g_{s}}^{3/4}{\log
N}^{2}M\left(\frac{1}{N}\right)^{11/20}{N_{f}}^{3}{r_{0}}{f_{x^{10}x^{10}}}({r_{0}})\left(1-\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (3.37)

In the ultraviolet(UV), the equivalent Einstein-Hilbert term is:

$\displaystyle\left.\sqrt{-G^{\cal M}}R\right|_{\rm Ouyang}^{\rm
UV}=\kappa_{\rm EH,\ UV}^{\beta^{0},0}\
\frac{\Biggl{[}11-8{\log\left(\frac{{r}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)}\Biggr{]}{M_{\rm UV}}{N_{f}^{\rm UV}}{r_{0}}^{4}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}{g_{s}}^{5/4}rN^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}}$
$\displaystyle+\kappa_{\rm EH,\ UV}^{\beta^{0},1}\
\frac{\Biggl{[}4{\log\left(\frac{{r}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)}-1\Biggr{]}{M_{\rm UV}}{N_{f}^{\rm UV}}r{r_{0}}^{2}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}{g_{s}}^{5/4}N^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}}+\kappa_{\rm
EH,\ UV}^{\beta^{0},2}\ \frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}r^{3}}{18\sqrt{3}\pi^{9/4}{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{3}{g_{s}}^{5/4}N^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}},$

implying:

$\displaystyle\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5-\overline{D5}}}^{{\cal R}_{\rm UV}}dr\sqrt{-G^{\cal M}}R\right|_{\rm
Ouyang}^{\rm UV-Finite}}{{\cal V}_{4}}$ $\displaystyle\sim-\kappa_{\rm EH,\
UV}^{\beta^{0},0}\
\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{{M_{\rm
UV}}{N_{f}^{\rm UV}}\left(-\frac{121{r_{0}}^{4}}{16{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}}-\frac{6{r_{0}}^{2}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{2}}+2\right)}{{g_{s}^{\rm
UV}}^{5/4}N^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}}.$ (3.39)

Hence, the UV finite Einstein-Hilbert term, after carrying out the radial and
angular integrals of the preceding equation is obtained as:

$\displaystyle\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5-\overline{D5}}}^{{\cal R}_{\rm UV}}dr\sqrt{-G^{\cal M}}R\right|_{\rm
Ouyang}^{\rm UV-Finite}}{{\cal V}_{4}}$ $\displaystyle\sim-\kappa_{\rm EH,\
UV}^{\beta^{0},0}\ \frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}\left(-\frac{121{r_{0}}^{4}}{16{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{4}}\frac{1}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}-\frac{6{r_{0}}^{2}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{2}}+2\right)}{{g_{s}^{\rm
UV}}^{5/4}N^{\frac{11}{20}}}\frac{1}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$
(3.40)

The Einstein-Hilbert divergent term for the thermal background in the UV
region is:

$\displaystyle\frac{S_{\rm EH-thermal}^{\rm UV-divergent}}{{\cal
V}_{4}}\sim\frac{\kappa_{\rm EH,\ UV}^{\beta^{0},2}}{4}\frac{{M_{\rm
UV}}{N_{f}^{\rm UV}}{\cal R}_{\rm UV}^{4}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{4}{g_{s}^{\rm
UV}}^{5/4}N^{\frac{11}{20}}}\frac{1}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$
(3.41)

For a thermal background up to ${\cal O}(\beta)$, the UV-finite portion of the
boundary Gibbons-Hawking-York term is calculated to be:

$\displaystyle\hskip-36.135pt\frac{\left.\int\sqrt{-h^{\cal M}}K\right|_{\rm
Ouyang}^{r={\cal R}_{\rm UV}\ {\rm UV-finite}}}{{\cal
V}_{4}}\sim\frac{{r_{0}}^{4}\left(\kappa_{\rm GHY}^{\beta^{0},{\rm UV-
finite}}\ \frac{{g_{s}}^{3}{M_{\rm UV}}^{2}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)}{N^{\frac{11}{20}}}\frac{1}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}\right)}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}{g_{s}}^{21/4}{M_{\rm UV}}}.$ (3.42)

The Gibbons-Hawking-York term in thermal background has an additional UV
divergence component, which is:

$\displaystyle\frac{S_{\rm GHY}^{\rm UV-divergent}}{{\cal V}_{4}}=-\kappa_{\rm
GHY}^{\beta^{0},\ {\rm UV-div}}\ \frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right){M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}^{4}{g_{s}^{\rm
UV}}^{9/4}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$
(3.43)

We also showed that the contribution from the higher derivative term in the
IR, up to ${\cal O}(\beta^{0})$, is:

$\displaystyle\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm
Ouyang}^{\rm
IR}\sim\frac{\left(19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}\right)}{\epsilon^{8}{g_{s}}^{9/4}{\log
N}^{2}N^{\frac{6}{5}}\sin^{4}\theta_{1}\sin^{2}\theta_{2}{N_{f}}^{2}\Biggl{[}{N_{f}}\left\\{{\log
N}-3\log\left(\frac{{r_{0}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\right\\}\Biggr{]}^{2/3}},$

implying:

$\displaystyle\frac{\int_{r_{0}}^{{\cal R}_{D5/\overline{D5}}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}}{{\cal
V}_{4}}\sim$
$\displaystyle\frac{M{N_{f}}{r_{0}}^{2}\left(1-\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)^{2}\log\left(\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\epsilon^{8}{g_{s}}{\log
N}^{2}N^{21/20}{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}$
$\displaystyle\times\kappa_{\footnotesize G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}}^{\beta^{0},\ \rm IR}\left\\{{\log N}-9\log\left(\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)\right\\}\Biggl{[}{\log
N}-3\log\left(\frac{{r_{0}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\Biggr{]}^{2}.$ (3.45)

In addition, the contribution arising from the higher derivative term in the
UV near (3.10) is:

$\displaystyle\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm
Ouyang}^{\rm UV}={g_{s}^{\rm UV}}^{1/4}{\log\tilde{r}}{M_{\rm
UV}}\frac{\left(19683\sin^{6}{\theta_{1}}+1107\sqrt{6}\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sin^{4}{\theta_{2}}\right)}{{g_{s}^{\rm
UV}}^{17/4}{\log N}^{3}(2{\log\tilde{r}}+1)^{7}N^{39/20}{N_{f}^{\rm
UV}}^{2}\tilde{r}\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$
$\displaystyle\times\Biggl{(}-\kappa_{\footnotesize G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}}^{\beta^{0},0}\
{a^{4}\left(64{\log\tilde{r}}^{3}+208{\log\tilde{r}}^{2}+212{\log\tilde{r}}+57\right)}$
$\displaystyle-\kappa_{\footnotesize G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}}^{\beta^{0},1}\
{8a^{2}\left(4{\log\tilde{r}}^{2}+15{\log\tilde{r}}+9\right)\tilde{r}}-\kappa_{\footnotesize
G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}}^{\beta^{0},2}\
{\tilde{r}^{3}}\Biggr{)},$ (3.46)

where $\tilde{r}\equiv\frac{{r}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}$,
Therefore, the UV-finite component of the aforementioned equation, after
angular and radial integration, found as:

$\displaystyle\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}^{\rm UV-
finite}}{{\cal V}_{4}}$ $\displaystyle\sim-\kappa_{\footnotesize
G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}}^{\beta^{0},\ \rm UV-
finite}\frac{{M_{\rm UV}}}{{g_{s}^{\rm UV}}^{4}{\log
N}^{3}N^{21/20}{N_{f}^{\rm UV}}^{2}}$
$\displaystyle\times\left(\frac{a^{2}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}+1\right)\left(\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}\right).$
(3.47)

The UV-differential portion of the equation (3.2.1.2) was further deduced as:

$\displaystyle\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\sqrt{-G^{\cal
M}}\left.G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}^{\rm UV-
divergent}}{{\cal V}_{4}}$ $\displaystyle=\kappa_{\footnotesize
G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}}^{\beta^{0},\ \rm UV-div}\
\frac{{\frac{1}{{g_{s}^{\rm UV}}}}{M_{\rm UV}}{{\cal R}_{\rm UV}}^{4}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}{g_{s}^{\rm UV}}^{9/2}N^{21/20}{N_{f}^{\rm
UV}}^{8/3}\log^{\frac{11}{3}}(N)\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)}$
$\displaystyle\times\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}.$
(3.48)

We found that:

$\displaystyle\left.\frac{\int\sqrt{-h}}{{\cal V}_{4}}\right|_{r={\cal R}_{\rm
UV}}=\kappa_{\sqrt{-h}}^{\beta^{0}}\ \frac{\left(\frac{1}{{g_{s}^{\rm
UV}}}\right)^{7/3}{M_{\rm UV}}{{\cal R}_{\rm UV}}^{4}\log\left(\frac{{\cal
R}_{\rm UV}}{{\cal R}_{D5/\overline{D5}}}\right)}{N{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}}$ $\displaystyle\hskip
86.72377pt\times\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{d\theta_{1}d\theta_{2}}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}$
$\displaystyle=\kappa_{\sqrt{-h}}^{\beta^{0}}\
\frac{\left(\frac{1}{{g_{s}^{\rm UV}}}\right)^{7/3}{M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}\log\left(\frac{{\cal R}_{\rm UV}}{{\cal
R}_{D5/\overline{D5}}}\right)}{N^{\frac{3}{10}}\alpha^{2}_{\theta_{1}}\alpha_{\theta_{2}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}}\equiv\kappa^{(0)}_{\sqrt{-h}@\partial
M_{11}}\ \left(\frac{{\cal R}_{\rm UV}}{{\cal R}_{D5/\overline{D5}}^{\rm
th}}\right)^{4}\log\left(\frac{{\cal R}_{\rm UV}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right),$
$\displaystyle\left.\frac{\int\sqrt{-h}R}{{\cal V}_{4}}\right|_{r={\cal
R}_{\rm UV}}^{\rm UV-div}\sim\frac{\left(\frac{1}{{g_{s}}}\right)^{2/3}{M_{\rm
UV}}{N_{f}^{\rm UV}}{{\cal R}_{\rm UV}}^{4}}{N^{\frac{11}{10}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}\sqrt{{g_{s}}^{\rm
UV}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}\equiv\kappa_{\sqrt{-h}R@\partial
M_{11}}^{\beta^{0}}\ {\cal R}_{\rm UV}^{4},$
$\displaystyle\left.\frac{\int\sqrt{-h}K}{{\cal V}_{4}}\right|_{r={\cal
R}_{\rm UV}}^{\rm UV-div}=-\kappa_{\rm GHY@\partial M_{11}}^{\beta^{0},\ {\rm
UV-div}}\ \frac{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right){M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}^{4}{g_{s}^{\rm
UV}}^{9/4}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}},$
$\displaystyle\beta\frac{\int\sqrt{-h}\left.G^{mn}\frac{\delta J_{0}}{\delta
G^{mn}}\right|^{\rm Ouyang}_{r={\cal R}_{\rm UV}}}{{\cal
V}_{4}}\sim{\footnotesize-\frac{{M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}\beta\left(-19683\sqrt{6}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{4}{g_{s}^{\rm UV}}^{47/12}{\log N}^{2}\left\\{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\right\\}^{6}N^{13/10}{N_{f}^{\rm
UV}}^{2}\alpha_{\theta_{2}}^{3}}}$
$\displaystyle\equiv\beta\kappa^{(0)}_{\sqrt{-h}G^{mn}@\partial
M_{11}\frac{\delta J_{0}}{\delta G^{mn}}}\frac{\left(\frac{{\cal R}_{\rm
UV}}{{\cal R}_{D5/\overline{D5}}}\right)^{4}}{\log^{6}\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}.$ (3.49)

Let us assume the following in relation to (3.2.1.2) and the fourth equation
in (3.2.1.2):

$\left[\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\right]^{6}N^{-\upsilon+\frac{4}{5}}=\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)N^{\frac{11}{20}},$ (3.50)

or

$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)=N^{\frac{4\upsilon-1}{20}}.$ (3.51)

Assuming additionally that $\upsilon=\frac{1}{4}+\epsilon$,

$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\sim 1+\frac{\epsilon}{5}\log N.$ (3.52)

Given (3.30), we can then say

$N_{f}^{\rm UV}=\frac{1}{\log N}+\epsilon_{1}:0<\epsilon_{1}\ll 1,$ (3.53)
$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\sim 1+\frac{2\epsilon_{1}}{15}\log N.$ (3.54)

Based on the values found in (3.52) and (3.54),
$\epsilon_{1}=\frac{3}{2}\epsilon$.
Holographic Renormalization When $T<T_{c}$: The UV-divergent contribution of
$\frac{1}{2}\Biggl{(}-\frac{7}{2}S_{\rm EH}^{(0)}+2S_{\rm GHY}^{(0)}\Biggr{)}$
was derived from the preceding discussion as follows:

$\displaystyle\tilde{\kappa}_{\rm UV-div}^{\beta^{0},1}\ \left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)^{4}+\tilde{\kappa}_{\rm UV-div}^{\beta^{0},2}\ \left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right).$ (3.55)

The following can be deduced from (3.2.1.2):

  * •

The counter term for $\tilde{\kappa}_{\rm UV-div}^{\beta^{0},1}\
\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)^{4}$ is $\int_{r={\cal R}_{\rm UV}}\sqrt{-h^{(0)}}R^{(0)}$;

  * •

Counter term to cancel the UV divergence, $\tilde{\kappa}_{\rm UV-
div}^{\beta^{0},2}\ \left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)$ is $\int_{r={\cal
R}_{\rm UV}}\sqrt{-h}$.

#####  3.2.1.3 $T_{c}$ Inclusive of ${\cal O}(R^{4})$ Corrections

Here, we’ll use the on-shell action comparison between black hole and thermal
backgrounds to determine the deconfinement temperature using information
gathered from 3.2.1.1 and 3.2.1.2. We will also describe a variant of the UV-
IR connection that emerges from the ${\cal O}(R^{4})$ terms.

We showed that leading order terms in $N,\log\left(\frac{{\cal R}_{\rm
UV}}{{\cal R}_{D5/\overline{D5}}}\right)$ and $\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}}$ corresponding to the black hole and thermal
backgrounds have the following forms:

$\displaystyle\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm UV}^{4}}\right)S_{D=11,\
{\rm on-shellUV-finite}}^{BH}\sim\frac{2\kappa_{\rm GHY}^{\rm bh}{M_{\rm
UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)}{{g_{s}}^{9/4}N^{11/20}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}+\Biggl{[}{2{\cal
C}_{\theta_{1}x}^{\rm bh}\kappa_{\left(\sqrt{-G^{\cal
M}}\right)^{(1)}R^{(0)}}^{\rm IR}}$ $\displaystyle+\frac{20\left(-{\cal
C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm bh}-3{\cal C}_{\theta_{1}x}^{\rm
bh}\right)\kappa_{\rm EH}^{\beta,\ \rm
IR}}{11}\Biggr{]}\frac{b^{2}{g_{s}}^{3/4}M{N_{f}}^{3}{r_{h}}^{4}\log^{3}(N)\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)}{N^{11/20}{{\cal
R}_{D5/\overline{D5}}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}\beta.$

Following is the on-shell action that corresponds to the thermal background
uplift:

$\displaystyle S_{D=11,\ {\rm on-shellUV-finite}}^{\rm
thermal}\sim\frac{2\kappa_{\rm GHY}^{{\rm th},\ \beta^{0}}{M_{\rm
UV}}{r_{0}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{g_{s}^{\rm UV}}^{9/4}N^{11/20}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}+\frac{2{g_{s}}^{3/4}\kappa_{\rm
EH,IR}^{{\rm th},\
\beta^{0}}M{N_{f}}^{3}{r_{0}}^{2}\log^{2}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{N^{11/20}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{2}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle-\frac{20\beta\kappa_{\rm EH,th}^{{\rm IR},\
\beta}{r_{0}}^{3}\left(2\alpha_{\theta_{2}}^{3}-729\sqrt{6}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}\right){f_{x^{10}x^{10}}}({r_{0}})}{11{g_{s}}^{9/4}MN^{7/20}{N_{f}}^{5/3}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}\alpha_{\theta_{1}}^{4}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}\beta.$ (3.57)

Now, the following equation must be solved for equality of ${\cal
O}(\beta^{0})$ terms in (3.2.1):

$\displaystyle\frac{2\kappa_{\rm GHY}^{\rm bh}{M_{\rm
UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{g_{s}}^{9/4}}=\frac{2\kappa_{\rm
GHY}^{{\rm th},\ \beta^{0}}{M_{\rm UV}}{r_{0}}^{4}\log\left(\frac{{{\cal
R}_{\rm UV}}}{{\cal R}_{D5/\overline{D5}}^{\rm th}}\right)}{{g_{s}^{\rm
UV}}^{9/4}{{\cal R}_{D5/\overline{D5}}^{\rm th}}^{4}}$
$\displaystyle+\frac{2{g_{s}}^{3/4}\kappa_{\rm EH,IR}^{{\rm th},\
\beta^{0}}M{N_{f}}^{3}{r_{0}}^{2}\log^{2}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}.$ (3.58)

Near
$(\theta_{1},\theta_{2})\sim\left(\frac{\alpha_{\theta_{1}}}{N^{\frac{1}{5}}},\frac{\alpha_{\theta_{1}}}{N^{\frac{3}{10}}}\right)$,
we found that $\frac{\kappa_{\rm GHY}^{{\rm th},\ \beta^{0}}}{\kappa_{\rm
EH,IR}^{{\rm th},\ \beta^{0}}}\sim 10^{5}$, therefore $\kappa_{\rm
EH,IR}^{{\rm th},\ \beta^{0}}$ term has been dropped. Hence,

$r_{h}=\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm th},\
\beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}.$ (3.59)

Since $\frac{r_{0}}{L^{2}}=\frac{m^{0^{++}}}{4}$ [114] and utilizing
$T_{c}=r_{h}/\pi L^{2}$ [66], the following equation describes the
deconfinement temperature:

$T_{c}=\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm th},\
\beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{m^{0^{++}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{4\pi{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}.$ (3.60)

The match at ${\cal O}(\beta)$ utilizing (3.2.1.3) and (3.2.1.3) produces:

$\displaystyle\Biggl{[}{2{\cal C}_{\theta_{1}x}^{\rm
bh}\kappa_{\left(\sqrt{-G^{\cal M}}\right)^{(1)}R^{(0)}}^{\rm
IR}}+\frac{20\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm
bh}-3{\cal C}_{\theta_{1}x}^{\rm bh}\right)\kappa_{\rm EH}^{\beta,\ \rm
IR}}{11}\Biggr{]}$
$\displaystyle\times\frac{b^{2}{g_{s}}^{3/4}M{N_{f}}^{3}{r_{h}}^{4}\log^{3}(N)\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)}{N^{11/20}{{\cal
R}_{D5/\overline{D5}}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle=\frac{20\kappa_{\rm EH,th}^{{\rm IR},\
\beta}{r_{0}}^{3}\left(2\alpha_{\theta_{2}}^{3}-729\sqrt{6}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}\right){f_{x^{10}x^{10}}}({r_{0}})}{11{g_{s}}^{9/4}MN^{7/20}{N_{f}}^{5/3}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}\alpha_{\theta_{1}}^{4}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)},$ (3.61)

this results in,

$\displaystyle\hskip-36.135ptf_{x^{10}x^{10}}({r_{0}})$
$\displaystyle\hskip-36.135pt\sim\left.-\frac{b^{2}{g_{s}}^{3}M^{2}{N_{f}}^{14/3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)^{4}\alpha_{\theta_{1}}^{2}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)\log\left(\frac{{r_{h}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\log\left(1-\frac{{r_{h}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{\beta{\kappa_{\rm EH}^{\beta,\ \rm
IR}}\sqrt[5]{N}\left(\frac{r_{0}}{{\cal R}_{D5/\overline{D5}}^{\rm
th}}\right)^{3}\alpha_{\theta_{2}}^{2}\left(729\sqrt{6}\alpha_{\theta_{1}}^{3}-2\alpha_{\theta_{2}}^{2}\right)}\right|_{(\ref{r_h-
r_0-relation})}$ $\displaystyle\hskip-36.135pt\times\left(11\pi^{17/4}{\cal
C}_{\theta_{1}x}^{\rm bh}{\kappa_{\left(\sqrt{-G^{\cal
M}}\right)^{(1)}R^{(0)}}^{\rm IR}}\log^{3}(N)-10{\kappa_{\rm EH}^{\beta,\ \rm
IR}}\log^{3}(N)\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm
bh}-3{\cal C}_{\theta_{1}x}^{\rm bh}\right)\right).$ (3.62)

In the context of this discussion, the term “UV-IR connection” refers to the
equations (3.2.1.3) and (3.59). Because this is providing a connection that
connects the integration constant appearing in the ${\cal O}(R^{4})$
corrections to the thermal background along the ${\cal M}$-theory circle and a
specific combination of integration constants appearing in the ${\cal
O}(R^{4})$ corrections to the black hole background. The combination figuring
in the black hole background is along the compact part $S^{3}$ of the non-
compact four cycle (locally) $\Sigma_{(4)}=\mathbb{R}_{+}\times S^{3}$
wrapping the flavor $D7$-branes of the type IIB dual [14], hence this has the
information of flavor branes in ${\cal M}$-theory uplift which has no branes
and this is known as “Flavor Memory” effect in this context.
Non-Renormalization of the Deconfinement Temperature ($T_{c}$) from
Semiclassical Method: We now turn to Green and Gutperle’s argument that a
$SL(2,\mathbb{Z})$ completion of an effective $R^{4}$ interaction results into
a fascinating non-renormalization theorem that prohibits perturbative
corrections to this term beyond one loop in the zero-instanton sector [73].
The term that occurs in (• ‣ 1.4.4) bilinear in the tensor $\hat{t}$ bears the
same structure just like terms that occur in the zero instanton sector via
both the one-loop four-graviton amplitude and a $(\alpha^{\prime})^{3}$ effect
at tree level [115]. Thus, in the Einstein frame, one gets the expression that
follows for the total effective $R^{4}$ action, which can be formally
represented by:

$S_{R^{4}}=(\alpha^{\prime})^{-1}\left[a\zeta(3)\tau_{2}^{3/2}+b\tau_{2}^{-1/2}+ce^{2\pi
i\tau}+\cdots\right]R^{4}\equiv(\alpha^{\prime})^{-1}f(\tau,\bar{\tau})R^{4},$
(3.63)

$a,b$ are understood to be numerical constants, $R^{4}$ represents the
contractions $\hat{t}\hat{t}R^{4}$ in (• ‣ 1.4.4), and $\cdots$ represents
perturbative and nonperturbative corrections to the coefficient of $R^{4}$
with $\tau=C_{\rm RR}+ie^{-\phi}$ (type IIB) and $\bar{\tau}$. The first term
of (3.63) reflects the $(\alpha^{\prime})^{3}$ tree-level contribution,
whereas the second term represents the one-loop compared to the first. The
entire equation for $S_{R^{4}}$, on the other hand, has to be invariant under
$SL(2,\mathbb{Z})$ transformations: $\tau\to(a\tau+b)(c\tau+d)^{-1}$
($a,b,c,d\in\mathbb{Z}:ad-bc=1$). It imposes severe constraints on its form.
Because the $R^{4}$ factor remains invariant, $f(\tau,\bar{\tau})$ in (3.63)
serves as a scalar under the $SL(2,\mathbb{Z})$ transformations, implying a
sum of all instantons and anti-instantons. The authors provided a simple
function which fulfills all of these conditions, such as [73]:

$f(\tau,\bar{\tau})=\sum_{(p,n)\neq(0,0)}{\tau_{2}^{3/2}\over|p+n\tau|^{3}},$
(3.64)

wherein the summation denotes the sum of all positive and negative $p,n$
values except $p=n=0$. Now:

$f=2\zeta(3)\tau_{2}^{3/2}+{\tau_{2}^{3/2}\over\Gamma(3/2)}\sum_{n\neq
0,p}\int_{0}^{\infty}dyy^{1/2}e^{-y|p+n\tau|^{2}}.$ (3.65)

The sum over $p$, utilizing the Poisson resummation formula
$\left(\sum_{n\rightarrow-\infty}^{\infty}f(n)=\sum_{m\rightarrow-\infty}^{\infty}{\rm
FT}[f](m);{\rm FT}[e^{-\pi
A(p+x)^{2}}]=\frac{e^{-\frac{M^{2}}{4A\pi}-iMx}}{2\pi A},M=2\pi m\right)$,
provides,

$\displaystyle f(\tau,\bar{\tau})=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over
3}\tau_{2}^{-1/2}+2\tau_{2}^{3/2}\sum_{m,n\neq
0}\int_{0}^{\infty}dy\exp\left(-{\pi^{2}m^{2}\over y}+2\pi
imn\tau_{1}-yn^{2}\tau_{2}^{2}\right),$
$\displaystyle=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over
3}\tau_{2}^{-1/2}+8\pi\tau_{2}^{1/2}\sum_{m\neq 0n\geq 1}\left|{m\over
n}\right|e^{2\pi imn\tau_{1}}K_{1}(2\pi|mn|\tau_{2})$
$\displaystyle=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over 3}\tau_{2}^{-1/2}$
$\displaystyle+4\pi^{3/2}\sum_{m,n\geq 1}\left({m\over
n^{3}}\right)^{1/2}(e^{2\pi imn\tau}+e^{-2\pi
imn\bar{\tau}})\left(1+\sum_{k=1}^{\infty}(4\pi
mn\tau_{2})^{-k}{\Gamma(k-1/2)\over\Gamma(-k-1/2)}\right).$ (3.66)

In (3.2.1.3), the asymptotic expansion for $K_{1}(z)$ for large $z$ is used
after carrying out a perturbative expansion in $\frac{1}{\tau_{2}}$ of the
non-perturbative instanton contribution of charge $mn$. The perturbative terms
in (3.2.1.3) end after the one-loop term, as previously proposed. The non-
perturbative terms are a sum of single multiply-charged instantons and anti-
instantons with action proportional to $|mn|$. The terms in parentheses in
(3.2.1.3) reflect a never-ending series of perturbative corrections near the
charge instantons $mn$. As a result, there aren’t any perturbative corrections
in the action up to ${\cal O}(R^{4})$, i.e., $S_{R^{4}}$, beyond one loop in
the zero-instanton sector. Hence, this suggests the “non-renormalization of
the deconfinement temperature ($T_{c}$)” beyond one-loop in the zero instanton
sector.

###  3.3 Wald Entropy, $T_{c}$ from Entanglement Entropy and M$\chi$PT
Compatibility

####  3.3.1 Wald Entropy

Lets now prove that the black hole entropy that was calculated using (3.2.1.3)
and Wald’s technique are consistent with one another. We takes away from the
(3.2.1.3) that the formula for calculating the BH entropy, ${\cal S}_{\rm
BH}$, up to ${\cal O}(\beta^{0})$, is as follows:

$\displaystyle{\cal S}_{\rm BH}=\beta_{\rm BH}\frac{\partial S^{E}_{\rm
BH}}{\partial\beta}-S^{E}_{\rm BH}\sim\frac{\sqrt{N}{M_{\rm
UV}}{r_{h}}^{3}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)}{{g_{s}}^{7/4}N^{11/20}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle\sim\frac{M_{\rm UV}\left(N_{f}^{\rm UV}\log
N\right)^{\frac{2}{15}}r_{h}^{3}}{N^{\frac{1}{20}}}.$ (3.67)

It’s interesting to note that the Wald entropy density that results from the
${\cal O}(\beta^{0})$ action may be expressed as:

$\displaystyle\frac{{\cal S}_{\rm BH}^{\rm Wald}}{{\cal
V}_{3}}=\oint_{M_{6}(\theta_{1,2},x,y,z,x^{10})}\frac{\partial R}{\partial
R_{mnpq}}\epsilon^{mn}\epsilon^{pq}\sqrt{g_{x^{1,2,3},\theta_{1,2},x,y,z,x^{10}}}d\theta_{1}d\theta_{2}dxdydzdx^{10}$
$\displaystyle\sim-\frac{\left(3b^{2}-1\right)\left(9b^{2}+1\right){g_{s}}^{5/4}MN^{-1/20}{N_{f}}{r_{h}}^{3}\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\left(\log
N-9\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)}{\left(6b^{2}+1\right)\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle\times\left[{N_{f}}\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)\right]^{2/3},$ (3.68)

where ${\cal V}_{3}$ represents $\mathbb{R}^{3}(x^{1,2,3}$ coordinates
volume). When replacing $b=\frac{1}{\sqrt{3}}+\epsilon,\epsilon\sim\left(|\log
r_{h}\right|)^{\frac{9}{2}}N^{-\frac{9}{10}-\alpha_{\epsilon}}$, (3.3.1)
implying:

$\displaystyle\frac{{g_{s}}^{5/4}M{N_{f}}{r_{h}}^{3}N^{-\frac{19}{20}-{\alpha_{\epsilon}}}\left(|\log
r_{h}\right|)^{\frac{11}{2}}\left(\log N-9\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle\times\left[{N_{f}}\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)\right]^{2/3}.$ (3.69)

Both approaches result in the same conclusion, which is that the black hole
entropy is proportional to $r_{h}^{3}$. The findings are a perfect match for
(for the sake of simplicity, we have disregarded in (3.70) numerical factors
and $M,N_{f}$ which, in the MQGP limit, are ${\cal O}(1)$):

$\displaystyle\hskip-21.68121pt\alpha_{\epsilon}=-\frac{9}{10}+\frac{\left|\log\left(M_{\rm
UV}\left(N_{f}^{\rm UV}\right)^{\frac{2}{15}}\right)\right|}{\log
N}+\frac{211\log\log N}{\log N}\sim\frac{\left|\log\left(M_{\rm
UV}\left(N_{f}^{\rm UV}\right)^{\frac{2}{15}}\right)\right|}{\log
N}-\frac{9}{10};$ (3.70)

$\alpha_{\epsilon}>0$ if $M_{\rm UV}\left(N_{f}^{\rm
UV}\right)^{\frac{2}{15}}<N^{-\frac{9}{10}}$. We demonstrated that in order to
compute the entropy of a black hole from ${\cal O}(R^{4})$ terms, one must
evaluate a total of four classes of terms independently while calculating the
$\frac{\partial J_{0}}{\partial R_{trtr}}$:

$\displaystyle\hskip-21.68121pt(i)\left(G^{rr}\right)^{2}\left(G^{tt}\right)^{2}\left(R_{PrtQ}+\frac{1}{2}R_{PQtr}\right)R_{t}^{\
\ RSP}R^{Q}_{\ \ RSr}$
$\displaystyle\hskip-21.68121pt\sim\left(G^{rr}\right)^{2}\left(G^{rr}\right)^{2}R_{trrt}\left(R_{t}^{\
\ z\theta_{1}t}R^{r}_{\ \ z\theta_{1}r}+R_{t}^{\ \ \theta_{1}zt}R^{r}_{\ \
\theta_{1}zr}\right)$ $\displaystyle\hskip-21.68121pt\sim-\frac{\left(\log
N-9\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{2}\sqrt[3]{{N_{f}}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)}}{{g_{s}}^{3/2}N^{3/2}{N_{f}}^{5}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{5}};$ $\displaystyle\hskip-21.68121pt(ii)R^{HrtK}R_{H}^{\
\ RSt}R^{r}_{\ \ RSK}+\frac{1}{2}R^{HKtr}R_{H}^{\ \ RSt}R^{Q}_{\ \ RSK}\sim
R^{trrt}\left(R_{t}^{\ \
z\theta_{1}t}R^{r}_{z\theta_{1}r}+R_{t}^{\theta_{1}zt}R^{r}_{\ \
\theta_{1}zr}\right)$ $\displaystyle\hskip-21.68121pt\sim-\frac{(\log
N-9\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right))^{2}}{{g_{s}}^{3/2}N^{3/2}{N_{f}}^{14/3}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{14/3}};$
$\displaystyle\hskip-21.68121pt(iii)\left(G^{rr}\right)^{2}G^{tt}\left(R_{PrtQ}+\frac{1}{2}R_{PQtr}\right)R_{t}^{\
\ RSP}R^{Q}_{\ \ RSr}\sim\frac{{g_{s}}^{9/2}M^{6}\log^{3}({r_{h}})(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right))^{3}}{{N_{f}}^{5/3}\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)^{14/3}};$
$\displaystyle\hskip-21.68121pt(iv)G^{tt}\left(R^{HMNr}R_{PMNt}+\frac{1}{2}R^{HrMN}R_{PtMN}\right)R_{H}^{\
\ rtP}\sim\frac{\sqrt{{g_{s}}}M^{2}\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{N_{f}}^{11/3}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{14/3}}$ $\displaystyle\hskip 252.94499pt\times\left(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right).$ (3.71)

As a result, we are able to notice that the contribution (iii) is the one that
predominates in the MQGP limit, and the black hole entropy from the higher
derivative terms is obtained as:

$\displaystyle\frac{{\cal S}_{E}^{{\cal O}(R^{4})}}{{\cal
V}_{3}}\sim\oint_{\theta_{1,2},x,y,z,x^{10})}\frac{\partial J_{0}}{\partial
R_{rtrt}}\sqrt{-G_{9}}d\theta_{1}d\theta_{2}dxdydzdx^{10}$
$\displaystyle\sim\frac{{g_{s}}^{23/4}M^{8}{N_{f}}^{4/3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)^{3}\log^{4}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\left(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{3}}{\sqrt[20]{N}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{8/3}}.$ (3.72)

One notes that (3.3.1) has the same dependency on $r_{h}^{3}$ and
$N^{-\frac{1}{20}}$, and semiclassical result is found at ${\cal O}(\beta)$.
This imposes the following restriction, which must be met in order for the
latter and the Wald entropy results to coincide exactly at ${\cal O}(\beta)$:

$\displaystyle\Biggl{[}{2{\cal C}_{\theta_{1}x}^{\rm
bh}\kappa_{\left(\sqrt{-G^{\cal M}}\right)^{(1)}R^{(0)}}^{\rm
IR}}+\frac{20\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm
bh}-3{\cal C}_{\theta_{1}x}^{\rm bh}\right)\kappa_{\rm EH}^{\beta,\ \rm
IR}}{11}\Biggr{]}\log^{3}(N)\left|\log\left(M_{\rm UV}\left(N_{f}^{\rm
UV}\right)^{\frac{2}{15}}\right)\right|$
$\displaystyle\sim\frac{\log^{3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\left(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{3}}{\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)^{8/3}}.$ (3.73)

As a result, we found that linear constraints must be placed on the constants
of integration that appear in the solutions to the EOMs of the metric
corrections at ${\cal O}(R^{4})$ in the directions that contain the “memory”
of the compact part of the non-compact four-cycle “wrapped” by the flavor
$D7$-branes in the type IIB dual [14] of the large-$N$ thermal QCD-like
theories.

####  3.3.2 Deconfinement from Entanglement Entropy

The confinement-deconfinement phase transition in thermal QCD has been
explored in this section from the perspective of entanglement entropy, which
relies on [106]. We have computed the entanglement entropy of regions that
have been defined as “connected” and “disconnected” in an appropriate manner.
We shall demonstrate that a phase transition from the confined phase to the
deconfined phase will take place at a critical value of the coordinate length
(l). The quantum entanglement entropy among the regions
$A\equiv\mathbb{R}^{d-1}\times l$ and
$B\equiv\mathbb{R}^{d-1}\times(\mathbb{R}-l)$, with $l$ representing a period
of length ${\it l}$, may be expressed as the following expression [107]:

$S_{A}=\frac{1}{4G_{N}^{(d+2)}}\int_{\gamma}d^{d}\sigma\sqrt{G_{ind}^{(d)}},$
(3.74)

where $G_{N}^{(d+2)}$ is the Newton constant in the dimension $(d+2)$, and
$G_{ind}^{(d)}$ is the value of the determinant corresponding to the induced
string frame metric on the co-dimension two minimal surface $\gamma$. The
equation (3.74) can be generalized to apply to non-conformal theories as shown
in the following [106]:

$S_{A}=\frac{1}{4G_{N}^{(d+2)}}\int_{\gamma}d^{d}\sigma
e^{-2\phi}\sqrt{G_{ind}^{(d)}},$ (3.75)

where $\phi$ denotes the profile of the dilaton. Due to the absence of a
dilaton in a ${\cal M}$-theory dual, the equation (3.75) will be considered to
be replaced with the following expression:

$S_{A}=\frac{1}{4G_{N}^{(11)}}\int_{\gamma}d^{9}\sigma\sqrt{G_{ind}^{(9)}}.$
(3.76)

Consider now the gravity dual’s metric in string frame, which may be stated
as:

$ds^{2}=\alpha(r)[\sigma(r)dr^{2}+dx_{\mu}dx^{\mu}]+g_{mn}dx^{m}dx^{n},$
(3.77)

where $x_{\mu}(\mu=0,1,2)$ denotes $(2+1)$ Minkowskian coordinates, $r$
represents the radial coordinate, and $x^{m}(m=3,5,6,7,8,9,10)$ is equivalent
to $x^{3}$ and has six angular coordinates
$(\theta_{1,2},\phi_{1,2},\psi,x^{10})$. Observing that
$g_{x^{3}x^{3}}(r=r_{0})=0$, $(x^{3},r)$ create a cigar-like shape. The
following expressions determine the volume of the seven-fold:

${\cal V}_{\rm int}=\int\prod_{m}dx^{m}\sqrt{g}.$ (3.78)

In this case, the QFT is defined on ${\rm I\\!R}^{2+1}$, which has a dimension
of $(2+1)$. Let us define two regions in two dimensions, labeled A and B, as
shown below:

$\displaystyle A={\rm I\\!R}\times l,$ $\displaystyle B={\rm I\\!R}\times({\rm
I\\!R}-l).$ (3.79)

Using the equation $(\ref{EE-def-M Theory})$, we will now determine the
entanglement entropy between $A$ and $B$ using the metric $(\ref{metric-Tc})$.
The following is an expression for the induced metric on $\gamma$ (note that
we replaced $x_{1}$ by $x$):

$ds^{2}|_{\gamma}=\alpha(r)\left[\left(\sigma(r)+\left(\frac{dx}{dr}\right)^{2}\right)dr^{2}+dx_{2}^{2}\right]+g_{mn}dx^{m}dx^{n}.$
(3.80)

Hence,

$\frac{S_{A}}{{\cal V}_{1}}=\frac{1}{4G_{N}^{(11)}}\int
dr\sqrt{H(r)}\sqrt{\sigma(r)+(\partial_{r}x(r))^{2}},$ (3.81)

where $H(r)$ is defined as:

$H(r)={\cal V}_{int}^{2}\alpha(r)^{2}.$ (3.82)

$H(r=r_{0})=0$ is the same as [106]. The equation of motion for $x(r)$ could
be derived from (3.81) as follows:

$\frac{dx}{dr}=\pm\frac{\sqrt{\sigma(r)}\sqrt{H(r_{*})}}{\sqrt{H(r)-H(r_{*})}},$
(3.83)

where $r_{*}$ is the radial distance when $dr/dx$ becomes zero. By integrating
the previous equation, we found:

$l(r_{*})=2\sqrt{H(r_{*})}\int_{r_{*}}^{\infty}dr\frac{\sqrt{\sigma(r)}}{\sqrt{H(r)-H(r_{*})}}.$
(3.84)

Entanglement entropy will be reduced to the following using equations (3.81)
and (3.83).

$\frac{S_{A}}{{\cal
V}_{1}}=\frac{1}{2G_{N}^{(11)}}\int_{r_{*}}^{r_{\infty}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r_{*})}}.$
(3.85)

The UV cut-off is denoted here by $r_{\infty}$. According to the explanation
in [106] that $r_{*}=r_{0}$, when $l>l_{max}$ the entanglement entropy of the
disconnected surface can be expressed as follows:

$\frac{S_{A}}{{\cal
V}_{1}}=\frac{1}{2G_{N}^{(11)}}\int_{r_{0}}^{r_{\infty}}dr\sqrt{\sigma(r)H(r)}.$
(3.86)

Connected and disconnected surfaces’ entanglement entropy differences have the
following expression for $l<l_{max}$:

$\frac{2G_{N}^{(11)}(S_{A}^{(conn)}-S_{A}^{(disconn)})}{{\cal
V}_{1}}=\int_{r_{*}}^{\infty}dr\left(\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r_{*})}}-\sqrt{\sigma(r)H(r)}\right)-\int_{r_{0}}^{r_{*}}dr\sqrt{\sigma(r)H(r)}.$
(3.87)

Let us discuss the implications of (3.87).

  * •

If ${\it l}$ is small then the entropy is lowest for a connected solution with
a big $r_{*}$.

  * •

There are two possible outcomes once ${\it l}$ begins to rise. First, until
${\it l}$ reaches the highest possible value, ${\it l}_{max}$, the connected
solution can keep being the lowest. Second, the disconnected solution may
become the dominant one above a certain value of ${\it l}$, denoted by ${\it
l}_{crit}$ and ${\it l}_{crit}<{\it l}_{max}$.

  * •

In the former, the phase transition will occur at ${\it l}={\it l}_{max}$,
while in the latter, it will occur at ${\it l}={\it l}_{crit}<{\it l}_{max}$.

Equations (3.2) and (3.77) allow us to deduce the following:

$\alpha(r)=\frac{e^{\frac{-2\phi^{IIA}}{3}}}{\sqrt{h(r,\theta_{1,2})}};\
\sigma(r)=\frac{e^{\frac{2\phi^{IIA}}{3}}}{\tilde{g(r)}},$ (3.88)

where $\phi^{IIA}$ represents the type IIA dilaton profile, which could be
deduced from the ${\cal M}$-theory metric as follows:

$G^{\cal M}_{x_{10}x_{10}}=e^{\frac{4\phi^{IIA}}{3}}.$ (3.89)

Based on the equation (3.78), the simpler version of the ${\cal V}_{int}$ for
the metric given in the equation (3.2) is as follows:

$\displaystyle{\cal V}_{\rm
int}\sim\frac{{g_{s}}^{3/4}MN^{9/20}\sqrt{1-\frac{{r_{0}}^{4}}{r^{4}}}\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{4/3}}{r\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}\left(\frac{1}{2}\beta({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th})+1\right)$ (3.90)
$\displaystyle\hskip-469.75499pt\times\Biggl{[}{g_{s}}\log
N{N_{f}}\left(3a^{2}-r^{2}\right)(2\log(r)+1)+\log(r)$
$\displaystyle\hskip-462.52756pt\times\left(4{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-24\pi
a^{2}+r^{2}(8\pi-3{g_{s}}{N_{f}})\right)$
$\displaystyle\hskip-462.52756pt+2{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+18{g_{s}}{N_{f}}\left(r^{2}-3a^{2}(6r+1)\right)\log^{2}(r)\Biggr{]},$

where $\Omega\equiv\left({g_{s}}(2\log
N+3){N_{f}}-4{g_{s}}{N_{f}}\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-8\pi\right)$.
$\alpha(r)$ and $\sigma(r)$ were derived from the equation (3.89) and
$h(r,\theta_{1,2})$ is provided in [14, 15]; a simpler version of them are
presented below:

$\displaystyle\alpha(r)=\frac{3^{2/3}\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}}{8\pi^{7/6}\sqrt{\frac{{g_{s}}N}{r^{4}}}}$
(3.91) $\displaystyle\hskip-252.94499pt-\frac{27\
3^{2/3}b^{10}\left(9b^{2}+1\right)^{4}\beta
M\left(\frac{1}{N}\right)^{3/4}r\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\left(6a^{2}+{r_{0}}^{2}\right)(r-2{r_{0}})}{16\pi^{13/6}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}\sqrt{N}{N_{f}}{r_{0}}^{4}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{0}}^{2}\right)\sqrt{\frac{{g_{s}}N}{r^{4}}}}$
$\displaystyle\hskip-252.94499pt\times\left(\log^{3}({r_{0}})\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}\right).$

and

$\displaystyle\hskip-57.81621pt\sigma(r)=\frac{4\left(\frac{\pi}{3}\right)^{2/3}\left(\frac{32\pi
a^{2}{g_{s}}M^{2}{N_{f}}({c_{1}}+{c_{2}}\log({r_{0}}))}{N\left(9a^{2}+r^{2}\right)\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right)}+1\right)}{\left(1-\frac{{r_{0}}^{4}}{r^{4}}\right)\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}}$
$\displaystyle\hskip-57.81621pt-\frac{18\sqrt[3]{\frac{3}{\pi}}b^{10}\left(9b^{2}+1\right)^{4}\beta
M\left(\frac{1}{N}\right)^{5/4}r^{5}\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)}{\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}{N_{f}}{r_{0}}^{4}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{0}}^{2}\right)\left({r_{0}}^{4}-r^{4}\right)}$
$\displaystyle\hskip-57.81621pt\times\frac{\left(6a^{2}+{r_{0}}^{2}\right)(r-2{r_{0}})\log^{3}({r_{0}})}{\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}},$
(3.92)

$c_{1,2}$ appeared from ${\cal O}\left(\frac{g_{s}M^{2}}{N}\right)$-correction
$\left(\frac{g_{s}M^{2}}{N}\right)(c_{1}+c_{2}\log r_{0})r_{0}$ from $a-r_{0}$
relationship (motivated by [60, 82]). We were able to get the following using
the equations (3.90) and (3.91):

$\displaystyle H(r)\equiv{\cal V}_{\rm int}^{2}\alpha(r)^{2}=h_{0}(r)+\beta
h_{1}(r),$ (3.93)

where:

$\displaystyle\hskip-50.58878pth_{0}(r)\sim\frac{\sqrt{\text{gs}}M^{2}r^{2}}{\sqrt[10]{N}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{4}}\left(1-\frac{{r_{0}}^{4}}{r^{4}}\right)\lambda_{1}^{2}\lambda_{2}^{4},$
(3.94)

and,

$\displaystyle\hskip-36.135pth_{1}(r)\sim\frac{{g_{s}}^{3/2}M^{2}N^{9/10}\left(1-\frac{{r_{0}}^{4}}{r^{4}}\right)}{r^{2}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{4}\left(9a^{2}+{r_{0}}^{2}\right)}\lambda_{1}^{2}\lambda_{2}^{8/3}\left(\frac{\lambda_{2}^{4/3}}{64\pi^{10/3}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}{g_{s}}\log
N^{4}N^{3/2}{N_{f}}{r_{0}}^{4}\alpha_{\theta_{2}}^{3}}\right)$
$\displaystyle\hskip-36.135pt\times\left(\frac{3\sqrt[3]{3}r^{4}({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th})}{64\pi^{7/3}{g_{s}}N}-81\sqrt[3]{3}b^{10}\left(9b^{2}+1\right)^{4}M\left(\frac{1}{N}\right)^{3/4}r^{5}\Sigma_{1}\left(6a^{2}+{r_{0}}^{2}\right)(r-2{r_{0}})\log^{3}({r_{0}})\right),$

wherein $\lambda_{1}$ and $\lambda_{2}$ have been defined as follows:

$\displaystyle\lambda_{1}=\Biggl{[}{g_{s}}\log
NN_{f}\left(3a^{2}-r^{2}\right)(2\log(r)+1)$
$\displaystyle+\log(r)\left(4{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-24\pi
a^{2}+r^{2}(8\pi-3{g_{s}}{N_{f}})\right)$
$\displaystyle+2{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+18{g_{s}}{N_{f}}\left(r^{2}-3a^{2}(6r+1)\right)\log^{2}(r)\Biggr{]},$
(3.96)
$\displaystyle\lambda_{2}=\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{8\pi}{{g_{s}}}-4{N_{f}}\log\left(\frac{\alpha_{\theta_{1}}\alpha_{\theta_{2}}}{4\sqrt{N}}\right)\right).$
(3.97)

In a similar manner, we were able to derive $h_{0}(r^{*})$ and $h_{1}(r^{*})$
by exchanging the value of $r$ with $r^{*}$. Hence,

$\displaystyle H(r^{*})=h_{0}(r^{*})+\beta h_{1}(r^{*}).$ (3.98)

After that, rewriting $\sigma(r)$ as follows:

$\displaystyle\sigma(r)=\sigma_{0}(r)+\beta\sigma_{1}(r).$ (3.99)

Utilizing equations (3.93), (3.98) and (3.99), the integrand that is seen in
(3.84) can also be expressed as:

$\displaystyle\sqrt{\frac{\sigma(r)}{H(r)-H(r^{*})}}=\sqrt{\frac{\sigma_{0}(r)}{h_{0}(r)-h_{0}(r^{*})}}+\beta\frac{\left(\sigma_{1}(r)(h_{0}(r)-h_{0}(r^{*}))-\sigma_{0}(r)(h_{1}(r)-h_{1}(r^{*}))\right)}{2\sqrt{\sigma_{0}(r)}(h_{0}(r)-h_{0}(r^{*}))^{\frac{3}{2}}};$

where,

$\displaystyle
h_{0}(r^{*})\sim\frac{M^{2}\left({r^{*}}^{4}-{r_{0}}^{4}\right)\left({g_{s}}{N_{f}}\log\left(9a^{2}{r^{*}}^{4}+{r^{*}}^{6}\right)-\Omega-16\pi\right){}^{4}}{{g_{s}}^{7/2}\sqrt[10]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{4}}$
(3.101)
$\displaystyle\hskip-325.215pt\times\Biggl{[}\log({r^{*}})\left(6a^{2}({g_{s}}\log
N{N_{f}}-4\pi)+4{g_{s}}{N_{f}}\left({r^{*}}^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+{r^{*}}^{2}(8\pi-{g_{s}}(2\log
N+3){N_{f}})\right)$
$\displaystyle\hskip-325.215pt+{g_{s}}{N_{f}}\left(3a^{2}-{r^{*}}^{2}\right)\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)+18{g_{s}}{N_{f}}\left({r^{*}}^{2}-3a^{2}(6{r^{*}}+1)\right)\log^{2}({r^{*}})\Biggr{]}.$

  * •

In $H(r^{*})$ if we consider

$\displaystyle\log r^{*}\to\frac{\sqrt{72{g_{s}}^{2}{N_{f}}^{2}\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)+\Omega{}^{2}}+\Omega}{36{g_{s}}{N_{f}}}$
(3.102) $\displaystyle\hskip-252.94499pt=\frac{{g_{s}}{N_{f}}\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)}{8\pi}+\frac{{g_{s}}^{2}{N_{f}}^{2}\left(-4\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+2\log
N+3\right)}{64\pi^{2}}$ $\displaystyle\hskip-252.94499pt\times\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)+\frac{{g_{s}}^{3}{N_{f}}^{3}\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)}{512\pi^{3}}$
$\displaystyle\hskip-252.94499pt\times\left(16\log^{2}\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+4\log
N^{2}-4(4\log
N-3)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-6\log
N+9\right)$ $\displaystyle\hskip-252.94499pt+O\left({N_{f}}^{4}\right),$

then,

$\displaystyle\hskip-14.45377pt\sqrt{h_{0}}(r^{*})=\frac{aM_{\rm
UV}{N_{f}^{\rm UV}}{r^{*}}\sqrt{\left|\log
N-2\log\left(\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+\log(16)\right|}}{24\sqrt{6}\pi^{9/4}{g_{s}}^{3/4}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}+O\left({N_{f}^{\rm
UV}}^{2}\right).$ (3.103)

The integral in the variable $l(r^{*})$ is now proportional to
$\frac{\sqrt[20]{N}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}{{g_{s}}^{5/4}M{N_{f}}^{10/3}r^{3}\log(r)(\log
N-3\log(r))^{7/3}|\log
N-9\log(r)|}\sim\frac{\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}\left(\frac{1}{\log(N)}\right)^{10/3}}{{g_{s}}^{5/4}M{N_{f}}^{10/3}r^{3}\log(r)}+O\left(\left(\frac{1}{\log
N}\right)^{13/3}\right)$. Since $\int\frac{dr}{r^{3}\log
r}={Ei}(-2\log(r))=\frac{-2\log^{2}(r)+\log(r)-1}{4r^{2}\log^{3}(r)}+O\left(\left(\frac{1}{r}\right)^{3}\right)\approx-\frac{1}{2r^{2}\log
r}$. Therefore, we obtained:

$\displaystyle\hskip-21.68121ptl(r^{*})\sim\frac{aM_{\rm UV}{N_{f}^{\rm
UV}}\left(\frac{1}{\log(N)}\right)^{10/3}\sqrt{-2\log\left(\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+\log(N)+\log(16)}}{{g_{s}}^{2}M{N_{f}}^{10/3}{r^{*}}\log({r^{*}})}\stackrel{{\scriptstyle
r^{*}\rightarrow\infty}}{{\longrightarrow}}0,$ (3.104)

similar to [106]. Since it was assumed that $r^{*}$ was in the UV, and we
found that $l(r^{*})$ is a function that monotonically decreases in the UV, we
found that $l(r^{*})$ gains a maximum value at $r^{*}={\cal
R}_{D5/\overline{D5}}^{\rm th}$. The solution of $l(r^{*})=\frac{C}{r^{*}\log
r^{*}}$ where
$C\sim\frac{a\left(\frac{1}{\log(N)}\right)^{10/3}\sqrt{-2\log\left(\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+{\log
N}+\log(16)}}{{g_{s}}^{2}{N_{f}^{\rm UV}}^{7/3}}$. implying

$r^{*}=\frac{C}{l(r^{*})W\left(\frac{C}{l(r^{*})}\right)}.$ (3.105)

Entanglement Entropy: The formula for the connected region entanglement
entropy that takes into account both tiny $l$ and large $r$ is as follows:

$\displaystyle\frac{S}{{\cal V}_{1}}=\left\\{\frac{\int_{{r^{*}}}^{{\cal
R}_{\rm
UV}}\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H({r^{*}})}}\,dr}{2{G_{N}}^{(11)}}\right\\}.$
(3.106)

For ${\cal M}$-theory background, we found the following contributions to the
entanglement entropies at ${\cal O}(\beta^{0})$ and ${\cal O}(\beta)$.

$\displaystyle\frac{S_{\rm connected}^{{\rm UV-finite},\ \beta^{0}}}{2{\cal
V}_{1}}\sim\frac{\log N^{4/3}M_{\rm UV}{N_{f}^{\rm
UV}}^{4/3}\alpha_{\theta_{2}}^{2}\left({r_{0}}^{4}{r^{*}}^{2}\log^{3}({r^{*}})+{r^{*}}^{6}\log^{2}({r^{*}})\right)}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\log({r^{*}})}$
$\displaystyle\approx\frac{\log N^{4/3}M_{\rm UV}{N_{f}^{\rm
UV}}^{4/3}{r^{*}}^{4}\alpha_{\theta_{2}}^{2}\log({r^{*}})}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{6}};$
$\displaystyle\frac{S_{\rm connected}^{{\rm UV-finite},\beta}}{2{\cal
V}_{1}}\sim\frac{\beta\sqrt[3]{\frac{1}{{g_{s}}}}M_{\rm UV}{r^{*}}^{4}({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th})\left(\frac{1}{\log(N)}\right)^{17/3}}{{g_{s}}^{7/4}\sqrt[20]{N}{N_{f}^{\rm
UV}}^{2/3}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}.$ (3.107)

Following is an expression that has been obtained for the disconnected region
entanglement entropy:

$\displaystyle\frac{S_{\rm disconnected}}{2{\cal
V}_{1}}=\left\\{\frac{\int_{{r_{0}}}^{{\cal R}_{\rm
UV}}\sqrt{\sigma(r)H(r)}\,dr}{2{G_{N}}{}^{(11)}}\right\\}$ (3.108)
$\displaystyle=\left\\{\frac{\left(\int_{{r_{0}}}^{{\cal
R}_{D5/\overline{D5}}}+\int_{D5/\overline{D5}}^{{\cal R}_{\rm
UV}}\right)\sqrt{\sigma(r)H(r)}\,dr}{2{G_{N}}{}^{11}}\right\\}$
$\displaystyle=\frac{1}{\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}g_{s}^{\frac{5}{4}}N_{f}^{\frac{8}{3}}\Biggl{[}\log^{2}{\cal
R}_{D5/\overline{D5}}{\cal R}_{D5/\overline{D5}}\left(-6a^{2}+{\cal
R}_{D5/\overline{D5}}^{2}\right)\left(\log N-3\log{\cal
R}_{D5/\overline{D5}}\right)^{\frac{5}{3}}$
$\displaystyle-\log^{2}r_{0}r_{0}^{2}\left(-6a^{2}+r_{0}^{2}\right)\left(\log
N-3\log r_{0}\right)^{\frac{5}{3}}\Biggr{]}\left(1+\frac{{\cal C}_{zz}^{\rm
th}-2{\cal C}_{\theta_{1}z}^{\rm th}}{2}\beta\right)+\frac{M_{\rm
UV}\sqrt[20]{\frac{1}{N}}{N_{f}^{\rm
UV}}^{4/3}}{\sqrt[12]{{g_{s}}}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}$
$\displaystyle\times\Biggl{[}{{\cal R}_{\rm UV}}^{4}\log({{\cal R}_{\rm
UV}})(\log N-3\log({{\cal R}_{\rm UV}}))^{4/3}$ $\displaystyle\hskip
21.68121pt-{\cal R}_{D5/\overline{D5}}^{4}\log({\cal
R}_{D5/\overline{D5}})(\log N-3\log({\cal
R}_{D5/\overline{D5}}))^{4/3}\Biggr{]}.$

Using $a=\left(\frac{1}{\sqrt{3}}+\epsilon\right)r_{0}$, focus in the range,
$r\in\left[r_{0},{\cal R}_{D5/\overline{D5}}\right]$, entanglement entropy is
obtained as:

$\displaystyle-\frac{\epsilon{g_{s}}^{5/4}M\sqrt[20]{\frac{1}{N}}{N_{f}}^{8/3}{r_{0}}^{4}(\beta({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th})+2)}{\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}.$ (3.109)

Due to the highly non-trivial cancellation of ${\cal O}(R^{4})$ corrections
${\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}=0$, see (• ‣ B.2), we
are able to deduce that there aren’t any ${\cal O}(R^{4})$-corrections to the
entanglement entropies of the connected and disconnected regions. It is
important to note that radial distances are always calculated in units of
${\cal R}_{D5/\overline{D5}}^{\rm th}$, i.e., $\log{\cal
R}_{D5/\overline{D5}}\equiv 0$, we obtained:

$\displaystyle\frac{S_{\rm disconnected}-S_{\rm UV}^{\rm disconnected}}{2{\cal
V}_{1}}\sim-\frac{{g_{s}}^{5/4}M\sqrt[20]{\frac{1}{N}}{N_{f}}^{8/3}r_{0}^{4}\log^{2}(r_{0})({\log
N}-3\log(r_{0}))^{5/3}}{64\sqrt[3]{2}3^{5/6}\pi^{41/12}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}};$
$\displaystyle\frac{S_{\rm connected}-S_{\rm UV}^{\rm connected}}{2{\cal
V}_{1}}\sim\frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}\alpha_{\theta_{2}}^{2}\log^{\frac{4}{3}}(N)\left(r_{0}^{4}{r^{*}}^{2}\log^{3}({r^{*}})-{r^{*}}^{6}\log^{2}({r^{*}})\right)}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\log({r^{*}})}$

$\displaystyle\frac{S_{\rm connected}-S_{\rm UV}^{\rm connected}}{2{\cal
V}_{1}}\approx-\frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}{r^{*}}^{4}\log^{\frac{4}{3}}(N)\log({r^{*}})}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}},$ (3.110)

wherein:

$\displaystyle S_{\rm UV}^{\rm disconnected}\sim\frac{M_{\rm UV}N_{f}^{\rm
UV}\ {}^{\frac{4}{3}}{\cal R}_{\rm UV}^{4}\left(\log N-3\log{\cal R}_{\rm
UV}\right)^{\frac{4}{3}}\log{\cal R}_{\rm
UV}}{N^{\frac{1}{20}}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{3}},$
$\displaystyle S_{\rm UV}^{\rm connected}\sim\frac{M_{\rm UV}N_{f}^{\rm UV}\
{}^{\frac{4}{3}}{\cal R}_{\rm UV}^{4}\left(\log
N\right)^{\frac{4}{3}}\log{\cal R}_{\rm
UV}\alpha_{\theta_{2}}^{2}}{N^{\frac{1}{20}}\alpha_{\theta_{1}}^{6}}.$ (3.111)

One should take attention that in the limit of large $N$, ${\cal R}_{\rm
UV}<\left(4\pi g_{s}N\right)^{\frac{1}{4}}$ and set
$\alpha_{\theta_{1}}^{3}=\alpha_{\theta_{2}}^{4}$, we found that $S_{\rm
UV}^{\rm disconnected}=S_{\rm UV}^{\rm connected}$. At $r^{*}=r_{\rm
criticial}$, $S_{\rm connected}=S_{\rm disconnected}$ and $r_{\rm critical}$
can be obtain from the equation given below:

$\displaystyle\left(r^{*}\right)^{4}\log r^{*}=\gamma,$ (3.112)

wherein
$\gamma\sim\frac{{g_{s}}^{4/3}M{N_{f}}^{8/3}\alpha_{\theta_{1}}^{3}\log^{2}(r_{0})(\log(N)-3\log(r_{0}))^{5/3}}{\pi^{4/3}{M_{\rm
UV}}{N_{f}^{\rm UV}}^{4/3}\alpha_{\theta_{2}}^{4}\log^{\frac{4}{3}}(N)}$.
Equation (3.112) has the solution,
${r^{*}}=e^{\frac{1}{4}W(4\gamma)}\approx\sqrt{2}\sqrt[4]{\gamma}$. Using
$M_{\rm UV},N_{f}\sim\frac{1}{\log N}$, $\log r_{0}=-\frac{f_{r_{0}}}{3}\log
N$, and $f_{r_{0}}\approx 1$ [2], we obtained:

$r_{\rm
critical}=\frac{\sqrt[3]{{g_{s}}}M{N_{f}}^{2/3}\alpha_{\theta_{1}}^{3/4}\log^{\frac{7}{6}}(N)}{\sqrt[4]{2}\sqrt[3]{\pi}\alpha_{\theta_{2}}}r_{0}.$
(3.113)

Choosing $M=N_{f}=3,g_{s}=0.1-1$; (3.113) implying $r_{\rm critical}\sim{\cal
O}(1)\frac{\alpha_{\theta_{1}}^{\frac{3}{4}}}{\alpha_{\theta_{2}}}$. By
considering the 4D-limit that is achieved through $M_{\rm KK}\rightarrow 0$,
or, analogously, $r_{0}\rightarrow 0$, one can therefore deduce the following
result from the (3.59):

$\displaystyle{\cal
O}(1)\frac{\alpha_{\theta_{1}}^{\frac{3}{4}}}{\alpha_{\theta_{2}}}\stackrel{{\scriptstyle\scriptsize
4D-{\rm limit}}}{{\longrightarrow}}\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm
th},\ \beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}},$ (3.114)

proportional to the deconfinement temperature. For the numerical purpose, we
choose $N=100,M_{\rm UV}=N_{f}^{\rm
UV}=0.01,M=N_{f}=3,\alpha_{\theta_{1,2}}:\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}=2$,
and $r_{0}=N^{-\frac{f_{r_{0}}}{3}},f_{r_{0}}\approx 1$ [2], and hence,
$l(r_{\rm critical})=3850$. For these values plot of entanglement entropy
versus $l(r^{*})$ is shown in Fig. 3.1.

Figure 3.1: Plot $S_{\rm connected}$ (blue) and $S_{\rm disconnected}$(orange)
versus $l(r^{*})$

In addition to this, let us consider the possibility that
$r^{*}\in[r_{0},{\cal R}_{D5/\overline{D5}}]$, i.e., IR-valued $r^{*}$, and
split the radial integral $\int_{r^{*}}^{{\cal R}_{\rm UV}}$ into
$\left(\int_{r^{*}}^{{\cal R}_{D5/\overline{D5}}}+\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}\right)$, we obtained:

$\displaystyle\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r^{*})}}$
$\displaystyle\sim 2\sqrt{{g_{s}}}M^{2}{N_{f}}^{13/3}\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}r^{6}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log^{2}(r)(\log
N-3\log(r))^{11/3}\,dr$ $\displaystyle=\int_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}\left(2\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{6}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log^{2}(r)+O\left(\left(\log
N\right)^{8/3}\right)\right)dr$
$\displaystyle=\Biggl{[}\frac{2}{343}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{7}\Biggl{(}7\log(r)\Biggl{(}7\log(r)\,_{2}F_{1}\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)$
$\displaystyle-2\,_{3}F_{2}\left(-\frac{7}{4},-\frac{7}{4},-\frac{1}{2};-\frac{3}{4},-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)\Biggr{)}+2\,_{4}F_{3}\left(-\frac{7}{4},-\frac{7}{4},-\frac{7}{4},-\frac{1}{2};-\frac{3}{4},-\frac{3}{4},-\frac{3}{4};\frac{r_{0}^{4}}{r^{4}}\right)\Biggr{)}\Biggr{]}_{r^{*}}^{{\cal
R}_{D5/\overline{D5}}}$ $\displaystyle=\frac{4}{343}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}-\frac{2r_{0}^{4}\left(\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}\right)}{27{{\cal
R}_{D5/\overline{D5}}}^{4}}+O\left(r_{0}^{5}\right)$
$\displaystyle-\frac{2}{7}\left(\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r^{*}\
{}^{7}\log^{2}(r^{*})\right)+\frac{1}{3}\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}r_{0}^{4}r^{*}\
{}^{3}\log^{2}(r^{*})+O\left(r_{0}^{5}\right),$

and,

$\displaystyle\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r^{*})}}$
$\displaystyle=\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}dr\frac{\sqrt[4]{\frac{1}{{g_{s}}}}\sqrt[6]{{g_{s}}}{M_{\rm
UV}}{N_{f}^{\rm
UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log(r)(\log
N-3\log(r))^{8/3}}{18\ 2^{2/3}3^{5/6}\pi^{25/12}\log
N^{4/3}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+O\left(\left({N_{f}^{\rm
UV}}\right)^{2}\right)$ $\displaystyle=\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}dr\frac{\log N^{4/3}{M_{\rm
UV}}{N_{f}^{\rm
UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\sqrt{1-\frac{r_{0}^{4}}{r^{4}}}\log(r)}{18\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+{\cal
O}\left(\sqrt[3]{\log N}\right)$ $\displaystyle=\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}dr\frac{\log N^{4/3}{M_{\rm
UV}}{N_{f}^{\rm UV}}^{4/3}r^{3}\alpha_{\theta_{2}}^{2}\log(r)}{18\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+O\left(\left(r_{0}^{4}\right)^{1}\right)$
$\displaystyle=\Biggl{[}\frac{\log N^{4/3}{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}r^{4}\alpha_{\theta_{2}}^{2}(4\log(r)-1)}{288\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}\Biggr{]}_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm UV}}$ $\displaystyle=\frac{\log
N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{{\cal
R}_{D5/\overline{D5}}}^{4}\alpha_{\theta_{2}}^{2}}{288\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}+\frac{\log
N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{\cal R}_{\rm
UV}^{4}\alpha_{\theta_{2}}^{2}\log({\cal R}_{\rm UV})}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}}.$
(3.116)

  * •

We found that in $S_{\rm connected}^{r^{*}\in[{\cal
R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]},S_{\rm
disconnected}^{r^{*}=r_{0}},S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}$, “UV-divergent” terms are same, represented by
$S_{\rm UV}\sim\frac{\log N^{4/3}{M_{\rm UV}}{N_{f}^{\rm UV}}^{4/3}{\cal
R}_{\rm UV}^{4}\log({\cal R}_{\rm UV})}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}}$.

  * •

Additionally, $S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}\sim\sqrt{{g_{s}}}\log
N^{11/3}M^{2}{N_{f}}^{13/3}$.

  * •

Hence, we obtained (in a vein of some of the examples in [106]):

$\displaystyle S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}>S_{\rm connected}^{r^{*}\in[{\cal
R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]}>S_{\rm
disconnected}^{r^{*}=r_{0}}-S_{\rm UV},\ l(r^{*})>l(r_{\rm critical}),$
$\displaystyle S_{\rm connected}^{r^{*}\in[r_{0},{\cal
R}_{D5/\overline{D5}}]}-S_{\rm UV}>S_{\rm connected}^{r^{*}\in[{\cal
R}_{D5/\overline{D5}},{\cal R}_{\rm UV}]}-S_{\rm UV}<S_{\rm
disconnected}^{r^{*}=r_{0}}-S_{\rm UV},\ l(r^{*})<l(r_{\rm critical}).$

####  3.3.3 ${\cal M}\chi$PT Compatibility

In chapter 2, we obtained one-loop renormalized LECs of the $SU(3)$ chiral
perturbation theory Lagrangian up to ${\cal O}(p^{4})$ from type IIA string
dual of large-$N$ thermal QCD like-theory, including the ${\cal O}(R^{4})$
corrections. Due to our chiral limit restriction, several terms in the $SU(3)$
chiral perturbation theory Lagrangian of Gasser and Leutwyler [87] were
missing. As a result, we were able to calculate $g_{YM}$,
$L_{1,2,3,9,10}^{r}$, and $F_{\pi}$. We compare our findings to the one-loop
renormalized LEC phenomenological values listed in [89]. In chapter 2, we
showed how to match the phenomenological values of the ${\cal O}(p^{4})$
$SU(3)$ $\chi$PT LECs ($g_{YM}$, $L_{1,9,10}^{r}$, and $F_{\pi}$) in five
stages, in addition to the order of the magnitude and signs of $L_{2,3}^{r}$.

We discovered that in order for $L_{9}^{r}$ to match the holographic
computation, a particular set of integration constants had to have a definite
sign [see equation (2.31)]. In appendix B, by taking the decompactification
limit (i.e. $M_{\rm KK}\rightarrow 0$ limit) of the spatial circle
$S^{1}(x^{3})$ appearing as part of the ${\cal M}$-theory metric utilized in
[2], i.e., in $S^{1}(x^{0})\times R^{2}\times
S^{1}\left(\frac{1}{M_{KK}}\right)$ to reconstruct a theory similar to 4D QCD
limit, we were able to derive values for these integration constants in an
explicit manner. In the matter of fact, we used those values to show that the
previously mentioned combination of integration constants, which appears in
all of the LECs of $SU(3)$ $\chi$PT Lagrangian up to ${\cal O}(p^{4})$,
actually could be rendered to have a negative sign as needed by comparing with
the phenomenological/experimental values of the one-loop renormalized LECs of
$SU(3)$ $\chi$PT Lagrangian as discussed previously. This can be seen from (•
‣ B.2), ${\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th}+2{\cal
C}_{\theta_{1}x}^{\rm th}=2{\cal C}_{\theta_{1}x}^{\rm
th}\sim\frac{\left(\frac{1}{N}\right)^{7/6}\Sigma_{1}}{\epsilon^{11}{g_{s}}^{9/4}\log
N^{4}{N_{f}}^{3}{r_{0}}^{5}\alpha_{\theta_{1}}^{7}\alpha_{\theta_{2}}^{6}}$
and this is negative when $\Sigma_{1}<0$.

###  3.4 Deconfinement Temperature of Rotating QGP at Intermediate Coupling
from ${\cal M}$-Theory

This part is based on [5]. In this paper, we have obtained the deconfinement
temperature of thermal QCD-like theories in the presence of rotation. We will
discuss the holographic construction of rotating QGP from a top-down approach
in 3.4.1. Using this holographic dual, we will discuss the calculation of
$T_{c}$ in the presence of rotation in 3.4.2 and UV-IR mixing, Flavor Memory
effect, and non-renormalization of $T_{c}$ in 3.4.3

####  3.4.1 Top-Down Holographic Dual of Rotating QGP

When $T>T_{c}$ on the gravity dual side and $T<T_{c}$ on the gauge theory
side, we could investigate the affect of rotation in thermal QCD-like theories
by introducing a rotating cylindrical black hole and thermal backgrounds on
the gravity dual side. In order to obtain the background of a rotating
cylinder black hole, we need to make $x^{3}$ periodic by replacing it with
$l\phi$, where $l$ represents the length of the cylinder and $0\leq\phi\leq
2\pi$. This will allow us to obtain the rotating cylinder black hole
background. After completing the following Lorentz transformation around the
cylinder of length $l$, one can obtain the holographic dual of rotating quark-
gluon plasma. This can be done by following [116, 117]:

$\displaystyle
t\rightarrow\frac{1}{\sqrt{1-l^{2}\omega^{2}}}\left(t+l^{2}\omega\phi\right);\
\phi\rightarrow\frac{1}{\sqrt{1-l^{2}\omega^{2}}}\left(\phi+\omega t\right),$
(3.117)

where $\omega$ represents the angular velocity of the rotating cylindrical
black hole in the gravity dual, which is connected to the rotation of quark-
gluon plasma by means of the gauge-gravity duality. Only when $\omega l<1$,
the Lorentz transformation, also known as the Lorentz-boost, considered to be
trustworthy. Following the application of the Lorentz transformation, which is
denoted by the equation (3.117), the equation (3.1) is rewritten as follows:

$\displaystyle\hskip-433.62ptds_{11}^{2}|_{BH}$ $\displaystyle=$
$\displaystyle e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\Biggl{(}-{\cal
Y}_{1}(r)dt^{2}+{\cal Y}_{2}(r)\left(d\phi+{\cal
Y}_{3}(r)dt\right)^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}\Biggr{)}$
(3.118)
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{g(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where ${\cal
Y}_{1}(r)=\frac{g(r)\left(1-l^{2}\omega^{2}\right)}{\left(1-g(r)l^{2}\omega^{2}\right)};\
{\cal
Y}_{2}(r)=\frac{l^{2}\left(1-g(r)l^{2}\omega^{2}\right)}{\left(1-l^{2}\omega^{2}\right)};\
{\cal
Y}_{3}(r)=\frac{\omega\left(1-g(r)\right)}{\left(1-g(r)l^{2}\omega^{2}\right)}$.
Note that we have the equation (3.118), we are able to determine the Hawking
temperature of the rotating cylindrical black hole by utilizing the formula
that appeared in [110]:

$\displaystyle\hskip-21.68121ptT_{H}(\gamma)=\Biggl{|}\frac{\kappa}{2\pi}\Biggr{|}=\Biggl{|}\frac{\lim_{r\rightarrow
r_{h}}-\frac{1}{2}\sqrt{\frac{G^{rr}}{-\hat{G_{tt}}}}\hat{G_{tt}},r}{2\pi}\Biggr{|}=\frac{r_{h}}{\sqrt{3}\pi^{3/2}\sqrt{N}\sqrt{g_{s}}}\left(\frac{1}{\gamma}+\frac{\beta}{2}\left(-{\cal
C}_{zz}^{\rm BH}+2{\cal C}_{\theta_{1}z}^{\rm BH}-3{\cal C}_{\theta_{1}x}^{\rm
BH}\right)\right),$ (3.119)

where $\hat{G_{tt}}=-{\cal Y}_{1}(r)$, $\hat{G_{tt}},r$ suggests that a
derivative of $\hat{G_{tt}}$ with respect to $r$ is being computed, and
Lorentz factor, $\gamma=\frac{1}{\sqrt{1-l^{2}\omega^{2}}}$. Considering that
$L^{4}=4\pi g_{s}N$, the Hawking temperature of the rotating cylindrical black
hole is calculated to be:

$\displaystyle T_{H}(\gamma)=\left(\frac{r_{h}}{\pi
L^{2}}\right)\left(\frac{1}{\gamma}+\frac{\beta}{2}\left(-{\cal C}_{zz}^{\rm
BH}+2{\cal C}_{\theta_{1}z}^{\rm BH}-3{\cal C}_{\theta_{1}x}^{\rm
BH}\right)\right)$
$\displaystyle=T_{H}(0)\left(\frac{1}{\gamma}+\frac{\beta}{2}\left(-{\cal
C}_{zz}^{\rm BH}+2{\cal C}_{\theta_{1}z}^{\rm BH}-3{\cal C}_{\theta_{1}x}^{\rm
BH}\right)\right),$ (3.120)

where $T_{H}(0)$ represents the Hawking temperature for a static black hole
that was obtained in [66], the ${\cal O}(\beta)$ term in equations (3.119) and
(3.4.1) is computed in small-$\omega$ limit with $\gamma=1$. The constants of
integration ${\cal C}_{zz/\theta_{1}z/\theta_{1}z}^{\rm BH}$ could be found in
the ${\cal O}(R^{4})$ correction to the black hole background metric [1]. As a
consequence of this, the Hawking temperature of the rotating cylindrical black
hole will be multiplied by the inverse of the Lorentz factor. Since ${\cal
C}_{zz}^{\rm BH}=2{\cal C}_{\theta_{1}z}^{\rm BH}$ and $|{\cal
C}_{\theta_{1}x}^{\rm BH}|\ll 1$ [3], the Hawking temperature does not get any
${\cal O}(R^{4})$ correction, which is the same thing as saying that the
Hawking temperature is unaffected by the presence of ${\cal O}(R^{4})$ terms,

$\displaystyle
T_{H}(\gamma)=\frac{T_{H}(0)}{\gamma}=T_{H}(0)\sqrt{1-l^{2}\omega^{2}}.$
(3.121)

In order to produce the rotating cylindrical thermal background, we substitute
$x^{3}$ by $l\phi$ and carry out the Lorentz transformation (3.117) of (3.2)
in the same manner as we did to the black hole background. As a result of
these steps, the metric of the rotating cylindrical thermal background
simplified as:

$\displaystyle\hskip-14.45377ptds_{11}^{2}|_{Thermal}$ $\displaystyle=$
$\displaystyle e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+l^{2}\left(d\phi\right)^{2}\right)$
(3.122)
$\displaystyle\hskip-36.135pt+\sqrt{h(r,\theta_{1,2})}\left(dr^{2}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2}.$

####  3.4.2 Deconfinement Temperature of Rotating QGP from ${\cal M}$-Theory
Dual

Using a semi-classical method [4], we will compute the deconfinement
temperature of a rotating QGP at intermediate coupling in this section similar
to 3.2.1. On-shell action, in this case, will be the same as (3.9), and hence
we computed the various terms appearing in (3.9) for the rotating cylindrical
black hole (3.118) and thermal (3.122) backgrounds, and integrated over the
angular coordinates similar to 3.2.1. In the upcoming pages, we will not
discuss every step, and we will write the final results because the procedure
is the same as 3.2.1. Only metric is different for the rotating cylindrical
black hole and thermal backgrounds compared to 3.2.1.

In a procedure analogous to that of 3.2.1, the UV-finite and holographic IR
regularized on-shell action density for the ${\cal M}$-theory rotating
cylindrical black hole background (3.118) was calculated as follows:

$\displaystyle\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\frac{S_{D=11,\ {\rm on-shell\ UV-finite}}^{\rm BH}}{{\cal
V}_{4}}=\lambda_{\rm EH,IR}^{{\rm BH}}\frac{\epsilon^{\rm
BH}\gamma^{8}\omega^{2}MN_{f}^{3}g_{s}^{3/2}r_{h}^{4}\log^{3}(N)\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}\right)\log\left(1-\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
BH}}^{4}N^{1/2}}$ $\displaystyle+\lambda_{\rm EH,UV}^{{\rm
BH}}\frac{\gamma^{8}lM^{\rm UV}r_{h}^{4}\log^{2}\left(\frac{{\cal R}_{\rm
UV}}{{\cal R}_{D5/\overline{D5}}^{\rm BH}}\right)}{N^{1/2}{{g_{s}^{\rm
UV}}^{3/2}}{{\cal R}_{D5/\overline{D5}}^{\rm BH}}^{4}}+\lambda_{\rm GHY}^{{\rm
BH}}\frac{lM^{\rm UV}r_{h}^{4}\log\left(\frac{{\cal R}_{\rm UV}}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}\right)}{N^{1/2}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}^{4}{{g_{s}^{\rm UV}}^{3/2}}},$ (3.123)

with numerical prefactors $\lambda_{\rm EH,IR}^{{\rm BH}},\ \lambda_{\rm
EH,UV}^{\rm BH}$ and $\lambda_{\rm GHY}^{{\rm BH}}$. ${\cal V}_{4}$ being the
coordinate volume of
$S^{1}(t)\times_{w}\mathbb{R}^{2}(x^{1,2})\times_{w}S^{1}(\phi)$ and ${\cal
R}_{D5-\overline{D5}}^{\rm{BH/th}}\equiv\sqrt{3}a^{\rm{BH/th}}$, where
$a^{\rm{BH/th}}$ represent the resolution parameter of the blown up $S^{2}$
and have the forms:
$a^{\rm{BH/th}}=\left(\frac{1}{\sqrt{3}}+\epsilon^{\rm{BH/th}}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{(h/0)}$ [1]. In addition, the UV-
finite and holographic IR regularized ${\cal O}(\beta^{0})$ contribution of
the on-shell action density for the ${\cal M}$-theory rotating cylindrical
thermal background (3.122) obtained as follows:

$\displaystyle\frac{S_{D=11,\ {\rm on-shell\ UV-finite}}^{\rm thermal}}{{\cal
V}_{4}}=\frac{\lambda_{\rm GHY}^{{\rm th}}{M_{\rm UV}}{\it
l}{r_{0}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{g_{s}^{\rm UV}}^{3/2}}N^{1/2}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}}+\frac{{g_{s}}^{3/2}\lambda_{\rm
EH,IR}^{{\rm th}}M{N_{f}}^{3}{\it
l}{r_{0}}^{2}\log^{2}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{N^{1/2}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{2}}$ $\displaystyle+\frac{\lambda_{\rm EH,\
UV}^{\rm th}{M_{\rm UV}}{N_{f}^{\rm UV}}{\it
l}\left(-\frac{121{r_{0}}^{4}}{16{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{4}}-\frac{6{r_{0}}^{2}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}+2\right)}{{g_{s}^{\rm UV}}^{1/2}N^{\frac{1}{2}}},$ (3.124)

wherein $r_{0}$ represents the Infrared cut-off of the theory when $T<T_{c}$
within QCD, and where $\lambda_{\rm GHY}^{{\rm th}},\ \lambda_{\rm
EH,IR}^{{\rm th}}$ and $\lambda_{\rm EH,\ UV}^{\rm th}$ are the numeric
factors. The UV-valued parameters are indicated in (3.4.2) and (3.4.2) by
parameters that include the superscript “${\rm UV}$” like $M^{\rm
UV},g_{s}^{\rm UV}$ etc. At the UV cut-off [3]:

$\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm UV}^{4}}\right)\frac{S_{D=11,\ {\rm
on-shell\ UV-finite}}^{\rm BH}}{{\cal V}_{4}}=\frac{S_{D=11,\ {\rm on-shell\
UV-finite}}^{\rm thermal}}{{\cal V}_{4}}.$ (3.125)

In (3.4.2), $\omega^{2}<1$, $\lambda_{\rm GHY}^{{\rm BH}}\sim{\cal
O}(10^{3})\lambda_{\rm EH,IR}^{{\rm BH}}$ and $\lambda_{\rm GHY}^{{\rm
BH}}\sim{\cal O}(10)\lambda_{\rm EH,UV}^{{\rm BH}}$, and in (3.4.2),
$\lambda_{\rm GHY}^{{\rm th}}\sim{\cal O}(10^{3})\lambda_{\rm EH,IR}^{{\rm
th}}$, $\lambda_{\rm GHY}^{{\rm th}}\sim{\cal O}(10^{2})\lambda_{\rm EH,\
UV}^{\rm th}$, Consequently, one has to solve the following equation:

$\displaystyle\lambda_{\rm GHY}^{{\rm BH}}\frac{lM^{\rm
UV}r_{h}^{4}\log\left(\frac{{\cal R}_{\rm UV}}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}\right)}{N^{1/2}{{\cal
R}_{D5/\overline{D5}}^{\rm BH}}^{4}{{g_{s}^{\rm
UV}}^{3/2}}}-\frac{\lambda_{\rm GHY}^{{\rm th}}{M_{\rm UV}}{\it
l}{r_{0}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{g_{s}^{\rm UV}}^{3/2}}N^{1/2}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{4}}=0.$ (3.126)

The following is a solution to the equation presented above:

$\displaystyle r_{h}=\frac{\sqrt[4]{\frac{\lambda_{\rm GHY}^{{\rm
th}}}{\lambda_{\rm GHY}^{{\rm BH}}}}r_{0}{{\cal R}_{D5/\overline{D5}}^{\rm
BH}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm BH}}}\right)}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}.$ (3.127)

With reference to the numerics: $\lambda_{\rm GHY}^{{\rm th}}=0.08$,
$\lambda_{\rm GHY}^{{\rm BH}}=0.23$; Since, ${{\cal R}_{D5/\overline{D5}}^{\rm
th/BH}}=\sqrt{3}\left(\frac{1}{\sqrt{3}}+\epsilon^{\rm
th/BH}\right)e^{-\kappa_{(h/0)}N^{-1/3}}$, where $\kappa_{(h/0)}\ll 1$ [82],
${\cal R}_{\rm UV}\approx L=(4\pi g_{s}N)^{1/4}$ [2] and assuming $g_{s}=0.1$,
$N=100$, $\kappa_{r_{h}}=0.01$, $\kappa_{r_{0}}=0.01$, $\epsilon^{\rm
BH}=0.36$ and $\epsilon^{\rm th}=0.1$, the equation (3.127) can be simplified
to:

$\displaystyle r_{h}\approx 1.16r_{0}.$ (3.128)

In the context of gauge-gravity duality, the Hawking Page phase transition,
which occurs at $T=T_{H}(\gamma)$ on the gravity side between a thermal and
black hole background is dual to the deconfinement phase transition, which
occurs at $T=T_{c}(\gamma)$ in gauge theories , i.e., $T_{H}(\gamma)\sim
T_{c}(\gamma)$ [4],

$\displaystyle\hskip-36.135ptT_{c}(\gamma)=\frac{1}{\gamma}\Biggl{(}\frac{r_{h}}{\pi
L^{2}}\Biggr{)}=\frac{1}{\gamma}\Biggl{(}\frac{1.16r_{0}}{\pi\sqrt{4\pi
g_{s}N}}\Biggr{)}=T_{c}(0)\sqrt{1-l^{2}\omega^{2}}.$ (3.129)

Taking $\frac{r_{0}}{\sqrt{4\pi g_{s}N}}=\frac{1700}{4}$MeV [71], we get
$T_{c}(0)\approx 157$ MeV, which is roughly the value found through
holographic research [118] and lattice calculations [119]. The ratio of the
deconfinement temperature of QCD in the presence and absence of rotation
versus the angular velocity of the rotating quark-gluon plasma for
$l=0.2,0.6,1\ GeV^{-1}$ is plotted in Fig. 3.2 which is justified by the
equation (3.129).

Figure 3.2: Plot of $\frac{T_{c}(\gamma)}{T_{c}(0)}$ versus $\omega$ for
$l=0.2,0.6,1$.

Comparison with Other Models: The authors found identical behavior from Nambu-
Jona-Lansinio model in [120] because of chiral condensate suppression, which
was investigated as well from lattice simulations in [121, 122]. Lattice
simulations with different kinds of boundary conditions (open, periodic, and
Dirichlet) have been carried out in rotating reference frames. In both
gluodynamics and the theory of dynamical fermions, the authors investigated
the impact of rotation on the critical temperature. The result found that the
decrease in deconfinement temperature of thermal QCD from a rise of rotation
of quark-gluon plasma was previously studied using gauge-gravity duality from
a bottom-up approach in [109, 110, 111, 112]. In contrast to our finding and
other holographic findings mentioned earlier, the critical temperature for
gludynamics exhibits the opposite behavior. However, the authors found that
the critical temperature for dynamical fermions decreases as rotation
increases. The hard-wall and soft-wall models were used for the bottom-up
holographic study in [109], and the Einstein-Maxwell-Dilaton system is
involved in the gravity dual of [110]. The authors of [111, 112] considered
the gravity dual of rotating QGP to be the Kerr-AdS black hole background in
five dimensions. The deconfinement temperature of rotating QGP from a top-down
model is not covered in any published article. Because top-down models
compactify the initial type IIB/IIA string theory or ${\cal M}$-theory on an
internal manifold to produce four-dimensional QCD at finite temperature, they
are more fundamental than bottom-up models. We successfully studied rotating
QGP from a top-down holographic dual for the first time and were able to
confirm the non-zero angular velocity of rotating QGP found in hydrodynamic
simulation [108] and STAR collaboration [123].

####  3.4.3 UV-IR Mixing, Flavor Memory Effect and Non-Renormalization of
$T_{c}$ in Rotating QGP

Focusing on the outcomes presented in the appendix B.4. At ${\cal O}(\beta)$,
the UV finite on-shell action densities for the rotating cylindrical black
hole and thermal backgrounds are as follows:

$\displaystyle\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\frac{S_{D=11,\ {\rm on-shell\ UV-finite}}^{\beta,\rm
BH}}{{\cal V}_{4}}=\Biggl{[}{-2{\cal
C}_{\theta_{1}x}\kappa_{\sqrt{G^{(1)}}R^{(0)}}^{\rm IR}}+\frac{20\left(-{\cal
C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm bh}-3{\cal C}_{\theta_{1}x}^{\rm
bh}\right)\kappa_{\rm EH}^{\beta,\ \rm IR}}{11}\Biggr{]}$
$\displaystyle\times\frac{b^{2}{g_{s}}^{3/2}M{N_{f}}^{3}{r_{h}}^{4}\log^{3}(N)\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)}{N^{1/2}{{\cal
R}_{D5/\overline{D5}}}^{4}}\beta,$ $\displaystyle\frac{S_{D=11,\ {\rm on-
shell\ UV-finite}}^{\beta,\rm thermal}}{{\cal V}_{4}}=-\frac{20\kappa_{\rm
EH,th}^{{\rm IR},\