#  ${\mathscr{M}}$cTEQ (${\mathscr{M}}$ chiral perturbation theory-compatible
deconfinement Temperature and Entanglement Entropy up to terms Quartic in
curvature) and FM (Flavor Memory)

Gopal Yadav111email-<EMAIL_ADDRESS>Vikas Yadav222email-
<EMAIL_ADDRESS>and Aalok Misra333email-<EMAIL_ADDRESS>
Department of Physics,
Indian Institute of Technology Roorkee, Roorkee 247667, India

###### Abstract

A (semiclassical) holographic computation of the deconfinement temperature at
intermediate coupling from (a top-down) ${\mathscr{M}}$-theory dual of thermal
QCD-like theories, has been missing in the literature. In the process of
filling this gap, we demonstrate a novel UV-IR connection, (conjecture and
provide evidence for) a non-renormalization beyond one loop of
$\boldsymbol{\mathscr{M}}-{\bf c}$hiral perturbation theory [1]-compatible
deconfinement Temperature, and show equivalence with an Entanglement (as well
as Wald) entropy [2] computation, up to terms Quartic in curvature ($R$). We
demonstrate a Flavor-Memory (FM) effect in the ${\mathscr{M}}$-theory uplifts
of the gravity duals, wherein the no-braner ${\mathscr{M}}$-theory uplift
retains the "memory" of the flavor $D7$-branes of the parent type IIB dual in
the sense that a specific combination of the aforementioned quartic
corrections to the metric components precisely along the compact part (given
by $S^{3}$ as an $S^{1}$-fibration over the vanishing two-cycle $S^{2}$) of
the non-compact four-cycle "wrapped" by the flavor $D7$-branes, is what
determines, e.g., the Einstein-Hilbert action at O$(R^{4})$. The
aforementioned linear combination of ${\cal O}(R^{4})$ corrections to the
${\mathscr{M}}$-theory uplift [3, 4] metric, upon matching the holographic
result from ${\mathscr{M}}\chi$PT [1] with the phenomenological value of the
coupling constant of one of the $SU(3)$ NLO $\chi$PT Lagrangian of [5], is
required to have a definite sign. Interestingly, in the decompactification (or
“$M_{\rm KK}\rightarrow 0$") limit of the spatial circle in [1] to recover a
QCD-like theory in four dimensions after integrating out the compact
directions, we not only derive this, but in fact obtain the values of the
relevant ${\cal O}(R^{4})$ metric corrections. Further, equivalence with Wald
entropy for the black hole in the high-temperature ${\mathscr{M}}$-theory dual
at ${\cal O}(R^{4})$ imposes a linear constraint on a similar linear
combination of the abovementioned metric corrections. Remarkably, when
evaluating the deconfinement temperature from an entanglement entropy
computation in the thermal gravity dual, due to a delicate cancellation
between the contributions arising from the metric corrections at ${\cal
O}(R^{4})$ in the ${\mathscr{M}}$ theory uplift along the $S^{1}$-fiber and an
$S^{2}$ (which too involves a similar $S^{1}$-fibration) resulting in a non-
zero contribution only along the vanishing $S^{2}$ surviving, one sees that
there are consequently no corrections to $T_{c}$ at quartic order in the
curvature supporting the conjecture made on the basis of a semiclassical
computation.

###### Contents

  1. 1 Introduction
  2. 2 Review of UV Complete Top-Down Type IIB String Dual of [6], Type IIA Mirror of [6] and its ${\mathscr{M}}$-Theory Uplift at Intermediate ’t Hooft Coupling
  3. 3 ${\mathscr{M}}$-Theory Duals at Low ($T<T_{c}$ on the gauge theory side) and High ($T>T_{c}$ on the gauge theory side) Temperatures
    1. 3.1 Black Hole Background Uplift (Relevant to $T>T_{c}$) and Holographic Renormalization of On-Shell $D=11$ Action
    2. 3.2 Thermal Background Uplift (Relevant to $T<T_{c}$) and Holographic Renormalization of On-Shell $D=11$ Action
  4. 4 $T_{c}$ Inclusive of ${\cal O}(R^{4})$ Corrections
  5. 5 Deconfinement from Entanglement Entropy
  6. 6 ${\mathscr{M}}\chi$PT Compatibility
  7. 7 Summary
  8. A ${\cal O}(R^{4})$ Corrections to the ${\mathscr{M}}$-theory metric of [3] in the MQGP limit near the $\psi=2n\pi,n=0,1,2$-branches
  9. B Thermal $f_{MN}$ EOMs, their Solutions in the IR, $4D$-Limit and ${\mathscr{M}}\chi$PT Compatibility

##  1 Introduction

AdS/CFT correspondence [7], in its original form, was a duality between
strongly coupled $\cal N$ = 4 $SU(N)$ supersymmetric Yang-Mills theory and
weakly coupled type IIB string theory on $AdS_{5}\times S^{5}$. All the
theories are not conformal therefore it is better to call this duality as
gauge/gravity duality to include non-conformal theories. Strong coupling
dynamics of non-abelian gauge theories at finite temperature have been studied
via gauge/gravity duality. In later years, this duality has been generalised
to many branches of Physics (e.g. Particle Physics, Condensed Matter Physics,
Cosmology etc.). We are interested in thermal QCD, which is a non-conformal
theory because its gauge coupling runs with energy. There are various
proposals to study QCD using gauge/gravity duality (e.g. AdS/QCD) but almost
all involve a conformal AdS background. There are two approaches to construct
holographic duals of thermal QCD-like theories - the bottom-up and the top-
down approach. In this paper we will work with the latter.

Gauge/gravity duality also allows us to compute the corrections to the
infinite-’t Hooft-coupling limit as done in [8], but again working with an AdS
background. These authors included higher-derivative terms on gravity side
using terms quartic in the Weyl tensor. On the gauge theory side one has $\cal
N$ = 4 $SU(N)$ supersymmetric Yang-Mills plasma at intermediate ’t Hooft
coupling. They explained the transport peak in the small frequency region of
the stress-energy tensor specatral function at zero spatial momentum. which is
a generic feature of perturbative plasma.

QCD at strong coupling has been studied by various bottom-up models
constructed from gauge/gravity duality(from both, bottom-up and top-down
approaches) and at weak coupling from perturbation theory. A popular top-down
holographic type IIA dual, though catering only to the IR, is the Sakai-
Sugimoto model [9]. The only UV-complete (type IIB) top-down holographic dual
of QCD-like theories at strong coupling that we are aware of is [6], and its
${\mathscr{M}}$-theory uplift [3]. Authors in [10] have studied QCD
thermodynamic functions at intermediate coupling based on hard-thermal-loop
perturbation theory (HTLpt). One can also study intermediate coupling regime
of QCD from gauge/gravity duality. In this direction, two of the authors (VY
and AM) worked out $O(l_{p}^{6})$ corrections to the ${\mathscr{M}}$-theory
metric in [11]. Starting with ${\mathscr{M}}$-theory dual of thermal QCD-like
theories in the ’MQGP’ limit as constructed in [3] and incorporating higher
derivative terms in eleven dimensional supergravity action which are quartic
in Riemann curvature tensor i.e. $\cal O$($R^{4}$), the corrections to the
supergravity background were then worked out in [11] and in fact, successfully
used in [1] in obtaining phenomenologically-compatible values of the NLO LECs
in $SU(3)\ \chi$PT Lagrangian of [5].

Wald proposed a method to calculate black hole entropy in general theories of
gravity [12]. He considered classical theory of gravity in $n$ dimensions
which arises from diffeomorphism invariant Lagrangian. In these theories of
gravity Noether charge is an $(n-2)$-form and black hole entropy will be given
by 2$\pi$ times integral over bifurcate killing horizon of the $(n-2)$-form
Noether charge. Therefore entropy is Noether Charge for stationary black holes
with bifurcate killing horizons. For dynamical black holes a proposal to
calculate entropy was given in [13].

Every physical phenomenon is characterised by their energy/distance scale. One
goes from UV to IR via RG group flow. But, e.g., in noncommutative field
theories and string theory, short distance physics becomes related to long
distance Physics, which is known as UV/IR mixing. The consequence of UV/IR
mixing [14] is, e.g., that UV divergences of real $\phi^{4}$ theory defined on
commutative space are transformed into infrared poles in the same theory
defined on noncommutative space. There are other more examples explained
nicely in [14].

Gravitational theories are nonlocal. Therefore UV/IR mixing could appear in
such theories. In gauge/gravity duality radial coordinate in gravitational
theory corresponds to energy scale in gauge theory side. In [15], authors have
studied "I.R.-U.V." connection in the context of AdS/CFT correspondance in
which they showed that infrared effects in bulk theory are transformed into
ultravoilet effects in the boundary theory. In particular, infrared regulator
in the bulk theory plays the role of ultraviolet regulator in $\cal N$ = 4
super Yang Mills theory. It is interesting that a certain form of UV-IR
connection manifests itself in our work when matching at the deconfinement
temperature, ${\mathscr{M}}$-theory actions dual to the thermal and black-hole
backgrounds at the UV-cut-off and obtain a relationship between the ${\cal
O}(R^{4})$ metric corrections in the IR

In this paper we have calculated deconfinement temperature of QCD-like theory
at intermediate coupling from gauge/gravity duality using a semiclassical
computation as first discussed in [16]. In our case gravity dual is
${\mathscr{M}}$-theory uplift of type IIB string dual [6] which includes
${\cal O}(R^{4})$ corrections [11] to the MQGP background [3]. One can also
discuss confinement deconfinement phase transition in large $N_{c}$ gauge
theories from entanglement entropy point of view based on [2]. In this process
one is required to calculate entanglement entropy between two regions by
dividing one of spatial coordinates into segment of length ${\it l}$ and its
complement. There is a prescription to calculate entanglement entropy from
AdS/CFT correspondence given by Ryu and Takayanagi in [17]. As discussed in
[2], there are two surfaces - connected and disconnected. There is a critical
value of ${\it l}$ which is denoted by ${\it l_{crit}}$. If one is below the
critical value of ${\it l}$ i.e. ${\it l}<{\it l_{crit}}$ then it is the
connected surface that dominates the entanglement entropy and if one is above
the critical value of ${\it l}$ i.e. ${\it l}>{\it l_{crit}}$ then it is the
disconnected surface that dominates entanglement entropy; ${\it l}<{\it
l_{crit}}$ corresponds to confining phase of large $N_{c}$ gauge theories
whereas ${\it l}>{\it l_{crit}}$ corresponds to deconfining phase of the same.
So we can interpret this as confinement deconfinement phase transition in
large $N_{c}$ gauge theories.

The rest of the paper is organized as follows. In Section 2, we review the
basic setup and results of [6, 3, 11] inclusive of a summary of the results of
[18, 19] on the conjectured $SL(2,\mathbb{Z})$ completion of the $D=11$
supergravity action by studying four-graviton scattering in a D-instanton
background and a non-renormalization beyond one loop at ${\cal O}(R^{4})$ in
the zero-instanton sector. Section 3, divided into two sub-sections ${\bf
3.1}$ and ${\bf 3.2}$, have to do with the holographic renormalization of the
on-shell action corresponding respectively to the gravity duals of the black-
hole and thermal backgrounds. Section 4 is on the computation of the
deconfinement temperature via a semiclassical computation, as well as on the
comparison of the black hole entropy computed with the Wald entropy up to
${\cal O}(R^{4})$. Section 5 is on the computation of the deconfinement
temperature from entanglement entropy between an interval along one of the
non-compact non-radial spatial coordinates and its complement, and comparison
with the semiclassical computation of the same in Section 4. Section 6 is on
the derivation of the sign and the individual values of the contributions that
figure in a linear combination of the same arising from the ${\cal O}(R^{4})$
corrections to the ${\mathscr{M}}$-theory uplift [3] of [6]. Sections 4 and 6
also include a discussion on the "Flavor Memory" effect. Section 7 has a
summary of the results obtained in the paper. There is a supplementary
appendix on the EOMs and solutions to the ${\cal O}(R^{4})$ metric corrections
to the MQGP background of [3, 4], as well as a discussion on taking the
decompactification-limit of a periodic spatial direction in the thermal
background ${\mathscr{M}}$-theory uplift used in [1] to obtain the
${\mathscr{M}}$-theory thermal background in this paper, and deriving the sign
of a linear combination of constants of integration in the solutions to the
EOMs of the ${\cal O}(R^{4}){\mathscr{M}}$-theory metric corrections as
required from matching of results of [1] with phenomenological values of the
LECs at NLO in the $SU(3)\ \chi$PT Lagrangian of [5].

##  2 Review of UV Complete Top-Down Type IIB String Dual of [6], Type IIA
Mirror of [6] and its ${\mathscr{M}}$-Theory Uplift at Intermediate ’t Hooft
Coupling

In this section we will start with a brief review type IIB string dual of
large $N$ QCD-like theory at finite temperature constructed by McGill group
[6]. Then we will briefly discuss type IIA SYZ mirror of [6] and its
${\mathscr{M}}$-theory uplift as constructed in [3]. We will also discuss the
origin of ${\cal O}(R^{4})$ terms in eleven dimensional supergravity action
and non-renormalization of type IIB action up to ${\cal O}(R^{4})$ beyond one
loop in the zero-instanton sector. In the last part of this section we will
explain that how the ${\mathscr{M}}$-theory metric of [3, 4] will be modified
by incorporating higher derivative terms in eleven dimensional supergravity
action.

To the best of our knowledge the only UV-complete type IIB string dual of
large $N$ thermal QCD-like theories from a top-down approach was given in [6]
in the infinite-’t Hooft-coupling limit. For strongly coupled systems like
QGP, the relavant coupling is finite/intermediate as explained in [20].
Therefore, it is required to construct string dual(s) of large $N$ thermal
QCD-like theories at finite/intermediate coupling. In this direction authors
in [3, 4] considered a particular limit which they called as ’MQGP limit’
which uses finite/intermdiate coupling. One can also study intermediate
coupling regime of gauge theory using gauge/gravity duality. To study thermal
QCD at intermediate coupling higher derivative corrections on the gravity
side, need to be incorporated - [8, 11].

Brane Setup used in [6]

  * •

The authors considered $N$ $D3$-branes placed at tip of the six-dimensional
conifold, $M\ D5$-branes wrapping the vanishing $S^{2}(\theta_{1},\phi_{1})$,
referred to as fractional $D3$-branes and $M\ \overline{D5}$-branes
distributed along the resolved $S^{2}_{a}(\theta_{2},\phi_{2})$ placed at
antipodal points relative to the $M$ $D5$-branes. Let average seperation
between $D5/\overline{D5}$-branes will be represented by ${\cal
R}_{D5/\overline{D5}}$. Then one can characterise the various regions using
radial coordinate on gravity side.

    1. 1.

$r_{0}<r<{\cal R}_{D5/\overline{D5}}$ : IR-UV interpolating region with
$r_{0}<r\ll{\cal R}_{D5/\overline{D5}}$ corresponding to the deep IR.

    2. 2.

$r>{\cal R}_{D5/\overline{D5}}$ : UV region.

  * •

To introduce quarks in the fundamental representation of the flavor group,
$N_{f}$ $D7$-branes were introduced via Ouyang embedding [21] in the resolved
conifold geometry, “smeared"/delocalized along the angular directions
$\theta_{1,2}$. The flavor $D7$-branes are embedded in the UV all the way into
the IR up to a certain minimum separation from the color $D3$-branes
determined by the modulus of the Ouyang embedding parameter $|\mu_{\rm
Ouyang}|$. To ensure UV conformality in the theory $N_{f}\
\overline{D7}$-branes were introduced in the UV and the UV-IR interpolaring
region but not the IR (to effect chiral symmetry breaking). The embedding
equation for the flavor $D7$-branes in the resolved conifold geometry is:

$\left(r^{6}+9a^{2}r^{4}\right)^{\frac{1}{4}}e^{\frac{i}{2}\left(\psi-\phi_{1}-\phi_{2}\right)}\sin\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\theta_{2}}{2}\right)=\mu_{\rm
Ouyang},$ (1)

effected by (32) for vanishingly small $|\mu_{\rm Ouyang}|$.

  * •

In the UV, the color gauge group is $SU(N+M)\times SU(N+M)$ and the flavor
gauge group is $SU(N_{f})\times SU(N_{f})$. When one goes RG flows from the UV
to the IR, two things happen,

    * –

The color gauge group $SU(N+M)\times SU(N+M)$ is partially Higgsed down to
$SU(N+M)\times SU(N)$ because in the IR $\overline{D5}$-branes are not present
due to which rank of one of the product gauge groups($SU(N+{\rm number\ of}\
D5-{\rm branes})\times SU(N+{\rm number\ of}\ \overline{D5}-{\rm branes})$)
decreases to ($SU(N+{\rm number\ of}\ D5-{\rm branes})\times SU(N)$.

    * –

Flavor gauge group $SU(N_{f})\times SU(N_{f})$ breaks to the diagonal subgroup
$SU(N_{f})$ because of absense of $\overline{D7}$-branes in the IR. This is
the analagoue of chiral symmetry breaking in this brane setup.

  * •

The pair of couplings corresponding to $SU(N+M)$ and $SU(N)$ gauge groups flow
in opposite directions in the IR.

$\displaystyle\hskip-21.68121pt4\pi^{2}\left(\frac{1}{g^{2}_{SU(N+M)}}+\frac{1}{g^{2}_{SU(N)}}\right)e^{\phi}\sim\pi;4\pi^{2}\left(\frac{1}{g^{2}_{SU(N+M)}}-\frac{1}{g^{2}_{SU(N)}}\right)e^{\phi}\sim\frac{1}{2\pi\alpha^{{}^{\prime}}}\int_{S^{2}}B_{2}$
(2)

From the above equation it is clear that $\int_{S^{2}}B_{2}$ is the reason for
introduction of non-conformality. If flux term on r.h.s. vanishes then
couplings for the both gauge groups will be same which is indication of a
conformal theory. This is the reason to include $M$ $\overline{D5}$-branes in
[6] to cancel the net $D5$-brane charge in the UV. Therefore we have a
holographic theory which, like QCD, is UV conformal.

  * •

Under the UV-to-IR RG flow, the higher rank gauge group($SU(N+M)$) flows
towards strong coupling and lower rank gauge group($SU(N)$) flows towards the
weak coupling. Since $SU(N+M)_{strong}\xrightarrow{\text{Sieberg-like
dual}}SU(N-(M-N_{f}))_{weak}$, after performing repeated Seiberg-like
dualities, in the IR the number of colors $N_{c}$ gets identified with $M$,
which in the ‘MQGP limit’ can be taken to be 3 (See [22]). ’MQGP limit’ have
been defined later in this paper.

  * •

Holographic gravity dual of the brane setup of [6]: If one is above the
deconfinement temperature i.e. $T>T_{c}$ then finite temperature on the
gauge/brane side corresponds to a black hole in the gravitational dual and if
one is below the deconfinement temperature i.e. $T<T_{c}$ then finite
temperature on the gauge/brane side corresponds to thermal background in the
gravitational dual [16]. Finite temperature on the brane/gauge side and finite
seperation between the $M\ D5$-branes and $M\ \overline{D5}$-branes (which is
denoted by ${\cal R}_{D5/\overline{D5}}$) requires to having a non-zero
resolution parameter of the conifold in gravitational dual side. Similarly, IR
confinement on the gauge theory side corresponds to deformation of the
conifold singular à la the Klebanov-Strassler model. Therefore holographic
dual of thermal QCD-like theories in this model [6] involves a resolved warped
deformed conifold; $D3$-branes and the $D5$-branes are replaced by fluxes in
the IR, and the back-reactions are included in the warp factor and fluxes.

  * •

Color-Flavor Enhancement of Length Scale in the IR:In the MQGP limit (10),
there is color-flavor enhancement of length scale as compared to a Planckian
length scale in Klebanov-Strassler(KS)-like model even for ${\cal O}(1)$ $M$,
in the IR . This is true when one includes terms higher order in $g_{s}N_{f}$
in the RR and NS-NS three-form fluxes and the NLO terms in $N$ in the metric.
which indicates suppression quantum corrections and validity of supergravity
calculations. For a detailed discussion of this issue, see [4, 22]. Now, the
effective number of color branes $N_{\rm eff}(r)$ is given by:

$\displaystyle N_{\rm eff}(r)=N\left[1+\frac{3g_{s}M_{\rm eff}^{2}}{2\pi
N}\left(\log r+\frac{3g_{s}N_{f}^{\rm eff}}{2\pi}\left(\log
r\right)^{2}\right)\right],$ $\displaystyle M_{\rm
eff}(r)=M+\frac{3g_{s}N_{f}M}{2\pi}\log r+\sum_{m\geq 1}\sum_{n\geq
1}N_{f}^{m}M^{n}f_{mn}(r),$ $\displaystyle N^{\rm
eff}_{f}(r)=N_{f}+\sum_{m\geq 1}\sum_{n\geq 0}N_{f}^{m}M^{n}g_{mn}(r).$ (3)

Type IIB axion is $C_{0}=N_{f}^{\rm
eff}\frac{\left(\psi-\phi_{1}-\phi_{2}\right)}{4\pi}$, $N_{\rm
eff}(r_{0}\in\rm IR)=0$ and writing the ten-dimensional warp factor
$h\sim\frac{L^{4}}{r^{4}}$, the length scale $L$ in the IR will be given by
the following equation:

$\displaystyle\hskip-65.04256ptL\sim\sqrt[4]{M}N_{f}^{\frac{3}{4}}\sqrt{\left(\sum_{m\geq
0}\sum_{n\geq 0}N_{f}^{m}M^{n}f_{mn}(r_{0})\right)}\left(\sum_{l\geq
0}\sum_{p\geq 0}N_{f}^{l}M^{p}g_{lp}(r_{0})\right)^{\frac{1}{4}}L_{\rm KS},$
(4)

$L_{KS}\sim\sqrt[4]{g_{s}M}\sqrt{\alpha^{\prime}}$. Equation (4) implies
enhancement of color-flavor length scale in the IR as compared to KS. Hence if
we consider number of colors $N_{c}^{\rm IR}=M=3$ and number of flavors
$N_{f}=2(u/d)+1(s)$, inclusion of $n,m>1$ terms in $M_{\rm eff}$ and
$N_{f}^{\rm eff}$ in (• ‣ 2) implies that $L\gg L_{\rm KS}(\sim L_{\rm
Planck})$ in the MQGP limit (10), which indicates validity of supergravity
calculations.

  * •

Obtaining ${\bf N_{c}=3}$: Now based on [22] we will briefly summarize how to
identify number of colors $N_{c}$ with $M$ which in the ‘MQGP limit’ (10) can
be tuned to equal 3. One can write $N_{c}$ as sum of the effective number
$N_{\rm eff}$ of $D3$-branes and the effective number $M_{\rm eff}$ of
$D5$-branes as:

$N_{C}=N_{\rm eff}(r)+M_{\rm eff}(r)$ (5)

where $N_{\rm eff}(r)$ is defined via the following relation,

$\tilde{F}_{5}\equiv dC_{4}+B_{2}\wedge F_{3}={\cal F}_{5}+*{\cal F}_{5}$ (6)

where, ${\cal F}_{5}\equiv N_{\rm eff}\times{\rm Vol}({\rm Base\ of\ Resolved\
Warped\ Deformed\ Conifold})$. Similarly, $M_{\rm eff}$ is defined via the
following relation,

$M_{\rm eff}=\int_{S^{3}}\tilde{F}_{3}$ (7)

In the above equation, $S^{3}$ being dual to $\
e_{\psi}\wedge\left(\sin\theta_{1}d\theta_{1}\wedge
d\phi_{1}-B_{1}\sin\theta_{2}\wedge d\phi_{2}\right)$, $B_{1}$ is an
‘asymmetry factor’ defined in [6]; $e_{\psi}\equiv d\psi+{\rm
cos}~{}\theta_{1}~{}d\phi_{1}+{\rm cos}~{}\theta_{2}~{}d\phi_{2}$) and [23]:
$\tilde{F}_{3}(\equiv F_{3}-\tau H_{3})\propto M(r)\equiv
M\frac{1}{1+e^{\alpha(r-{\cal R}_{D5/\overline{D5}})}},\alpha\gg 1.$

Let us denote UV and IR values of the resolution parameter "$a$" as
$[a_{UV},a_{IR}]$. In this notation,

$N_{\rm eff}\in[N,0]\hskip 14.45377ptand\hskip 14.45377ptM_{\rm eff}\in[0,M].$
(8)

From equations (5) and (8), it is clear that

$N_{c}\in[M,N].$ (9)

This implies that in the IR number of colors i.e. $N_{c}$ is $M$ which in the
MQGP limit we have taken to equal 3.

Therefore, we see that after application of repeated Seiberg-like duality $N\
D3$-branes are cascaded away in the IR and we have finite $M$ corresponding to
a strongly coupled IR-confining $SU(M)$ gauge theory; authors in [6]
considered finite temperature version this theory. The holographic dual of
large $N$ thermal QCD-like theories constructed in [6] exhibits UV
conformality (no Landau poles), IR confinement, quarks transform in the
fundamental representation of flavor and color groups, and is valid at all
temperatures.

  * •

Type IIA Strominger-Yau-Zaslow (SYZ) Mirror of [6], ${\mathscr{M}}$-Theory
Uplift and the MQGP Limit

Type IIA SYZ mirror of [6] and its ${\mathscr{M}}$-theory uplift using
Witten’s prescription at intermediate gauge coupling but large ’t Hooft
coupling, have been worked out in [3, 4]. The $O(R^{4})$ corrections to the
MQGP metric for the black hole background corresponding to intermediate ’t
Hooft coupling, were worked out by two of the authors (VY, AM) in [11], and
for the thermal background the $O(R^{4})$ corrections to the MQGP metric are
given in (B). Let us discuss how to obtain ${\mathscr{M}}$-theory uplift at
intermediate coupling.

    * –

To implement Strominger-Yau-Zaslow (SYZ) mirror symmetry one needs a
(delocalized) special Lagrangian (sLag) $T^{3}$ $-$ which could be identified
with the $T^{2}$-invariant sLag of [24] with a large base ${\cal
B}(r,\theta_{1},\theta_{2})$ (of a
$T^{3}(\phi_{1},\phi_{2},\psi)$-fibration444As there is no $\psi$-isometry in
the warped resolved deformed conifold, we in fact require a delocalized
$T^{3}(x,y,z)$, where $(x,y,z)$ are given by (78). over ${\cal
B}(r,\theta_{1},\theta_{2})$) [4, 25]. SYZ mirror symmetry is triple T duality
along three isometry directions (in our case T-duality along
$T^{3}(\phi_{1},\phi_{2},\psi)$). Let us see the effect of three T-dualities
along $(\phi_{1},\phi_{2},\psi)$ directions.

      * *

T-duality along $\psi$: First T-duality along the $\psi$ direction converts
$N\ D3$-branes into $N\ D4$-branes wrapping the $\psi$ circle, $M$ fractional
$D3(\overline{D3})$-branes into $M\ D4(\overline{D4})$-branes straddling a
pair of orthogonal $NS5$-branes and $N_{f}$ flavor $D7(\overline{D7})$-branes
into $N_{f}$ flavor $D6(\overline{D6})$-branes. Worldvolume coordinates for
$NS5$ branes are $NS5_{1}\ (x^{0,1,2,3},\theta_{1},\phi_{1})$ and $NS5_{2}\
(x^{0,1,2,3},\theta_{2},\phi_{2})$.

      * *

T-duality along $\phi_{1}$: Second T-duality along $\phi_{1}$ direction leaves
$NS5_{1}\ (x^{0,1,2,3},\theta_{1},\phi_{1})$ invariant but converts $NS5_{2}\
(x^{0,1,2,3},\theta_{2},\phi_{2})$ into a Taub-NUT
space$(r,\psi,\theta_{2},\phi_{1})$. Also $N\ D4$-branes, $M\
D4(\overline{D4})$-branes and $N_{f}$ flavor $D6(\overline{D6})$-branes will
be converted into $N\ D5$-branes, $M\ D5(\overline{D5})$-branes and $N_{f}$
flavor $D5(\overline{D5})$-branes.

      * *

T-duality along $\phi_{2}$: Third T-duality along $\phi_{2}$ direction leaves
$NS5_{2}\ (x^{0,1,2,3},\theta_{2},\phi_{2})$ invariant but converts $NS5_{1}\
(x^{0,1,2,3},\theta_{1},\phi_{1})$ into a Taub-NUT
space$(r,\psi,\theta_{1},\phi_{2})$. Also $N\ D5$-branes, $M\
D5(\overline{D5})$-branes and $N_{f}$ flavor $D5(\overline{D5})$-branes will
be converted into $N\ D6$-branes, $M\ D6(\overline{D6})$-branes and $N_{f}$
flavor $D6(\overline{D6})$-branes(which are "wrapping" a non-compact three-
cycle $\Sigma^{(3)}(r,\theta_{1},\phi_{2}$)).

Therefore we see that in SYZ type IIA mirror we have $N\ D6$-branes, $M\
D6(\overline{D6})$-branes and $N_{f}$ flavor $D6(\overline{D6})$-branes. Keep
in mind that in type IIA mirror ranges for the presence of anti-branes also
matter as in [6]. In actual calculation as done in [3], T-dualities are
performed along $T^{3}(x,y,z)$, where $(x,y,z)$ are toroidal analagoue of
$(\phi_{1},\phi_{2},\psi)$. For the convention we have just written
$(\phi_{1},\phi_{2},\psi)$. Upon uplifting the SYZ type IIA mirror as obtained
above to ${\mathscr{M}}$-theory using Witten’s prescription, $D6$-branes will
be converted into KK monopoles(variants of Taub-NUT spaces). Hence, all the
branes will be converted into geometry and fluxes. One can show that the
${\mathscr{M}}$-theory uplift involves a $G_{2}$-structure manifold [4, 11].

    * –

MQGP Limit
MQGP limit is defined as [3, 4]

$g_{s}\sim\frac{1}{{\cal O}(1)},M,N_{f}\equiv{\cal O}(1),\ g_{s}N_{f}<1,\ N\gg
1,\ \frac{g_{s}M^{2}}{N}\ll 1,$ (10)

wherein one considers intermediate/finite string coupling. This hence
necessitates addressing the MQGP limit from ${\mathscr{M}}$-theory.

  * •

In [11], two of the authors(VY and AM) worked out $O(l_{p}^{6})$ corrections
to the MQGP metric for blackhole background by incorporating $O(R^{4})$ terms
in eleven dimensional supergravity action. Since in gravitational dual side we
have higher derivative corrections which on gauge theory side corresponds to
intermediate gauge/’t Hooft coupling. Therefore using [11] we can explore the
intermediate coupling regime of thermal QCD-like theories. The
$SU(3)/G_{2}/SU(4)/Spin(7)$-structure torsion classes of the relevant six-,
seven- and eight-folds associated with the ${\mathscr{M}}$-theory uplift
worked out in [11].

There are two ways of understanding the origin of the ${\cal
O}(R^{4})$-corrections to the ${\cal N}=1,D=11$ supergravity action. One is in
the context of the effects of $D$-instantons in IIB supergravity/string theory
via the four-graviton scattering amplitude [18]. The other is $D=10$
supersymmetry [19]. Let us briefly discuss both.

  * •

Let us first look at interactions that are induced at leading order in a
$D$-instanton (closed-string states of type IIB superstring theory in which
the whole string is localized at a single point in superspace) background in
both, type IIB supergravity and the string descriptions, including a one-
instanton correction to the tree-level as well as one-loop $R^{4}$ terms [18]
\- both having the same tensorial structure. The bosonic zero modes are
parameterised by the coordinates corresponding to the position of the
$D$-instanton. The fermionic zero modes are generated by the broken
supersymmetries. The Grassmann parameters are fermionic supermoduli
corresponding to zero modes of the dilatino and must be integrated over
together with the bosonic zero modes. The simplest open-string world-sheet
that arises in a D-brane process is the disk diagram. An instanton carrying
some zero modes corresponds, at lowest order, to a disk world-sheet with open-
string states attached to the boundary. The one-instanton terms in the
supergravity effective action can be deduced by considering on-shell
amplitudes in the instanton background. The integration over the fermionic
moduli absorbs the independent fermionic zero modes. Consider now amplitudes
with four external gravitons. The leading term in supergravity is one in which
each graviton is associated with four fermionic zero modes. Integration over
the bosonic zero modes generates a nonlocal four-graviton interaction. In the
corresponding string calculation the world-sheet consists of four disconnected
disks to each of which is attached a single closed-string graviton vertex and
four fermionic open-string vertices. Writing the graviton polarization tensor
as $\zeta^{\mu_{r}\nu_{r}}=\zeta^{(\mu_{r}}\tilde{\zeta}^{\nu_{r})}$, and
evaluating the fermionic integral, it was shown in [18] that the four-graviton
scattering amplitude can be expressed in terms of:

$\displaystyle Ce^{2i\pi\tau_{0}}\int d^{10}ye^{i\sum_{r}k_{r}\cdot y}$
$\displaystyle\times\left(\hat{t}^{i_{1}j_{1}\cdots
i_{4}j_{4}}\hat{t}_{m_{1}n_{1}\cdots m_{4}n_{4}}-{1\over
4}\epsilon^{i_{1}j_{1}\cdots j_{4}j_{4}}\epsilon_{m_{1}n_{1}\cdots
m_{4}n_{4}}\right)R_{i_{1}j_{1}}^{m_{1}n_{1}}R_{i_{2}j_{2}}^{m_{2}n_{2}}R_{i_{3}j_{3}}^{m_{3}n_{3}}R_{i_{4}j_{4}}^{m_{4}n_{4}};$

$t_{8}$ symbol defined as:

$\displaystyle\hat{t}_{8}^{N_{1}\dots
N_{8}}=\frac{1}{16}\biggl{(}-2\left(G^{N_{1}N_{3}}G^{N_{2}N_{4}}G^{N_{5}N_{7}}G^{N_{6}N_{8}}+G^{N_{1}N_{5}}G^{N_{2}N_{6}}G^{N_{3}N_{7}}G^{N_{4}N_{8}}+G^{N_{1}N_{7}}G^{N_{2}N_{8}}G^{N_{3}N_{5}}G^{N_{4}N_{6}}\right)$
$\displaystyle+8\left(G^{N_{2}N_{3}}G^{N_{4}N_{5}}G^{N_{6}N_{7}}G^{N_{8}N_{1}}+G^{N_{2}N_{5}}G^{N_{6}N_{3}}G^{N_{4}N_{7}}G^{N_{8}N_{1}}+G^{N_{2}N_{5}}G^{N_{6}N_{7}}G^{N_{8}N_{3}}G^{N_{4}N_{1}}\right)$
$\displaystyle-(N_{1}\leftrightarrow N_{2})-(N_{3}\leftrightarrow
N_{4})-(N_{5}\leftrightarrow N_{6})-(N_{7}\leftrightarrow N_{8})\biggr{)}$
(12)

($N_{i}$ being valued in the ${\bf 8}_{v}$ of $SO(8)$ or covariantized to
$10D$ (or $11D$ to be used later in ${\mathscr{M}}$-theory)), from [26] the
light-cone 8D “zero mode" tensor $t_{8}$ is generalized to 10D:
$\hat{t}_{8}=t_{8}-1/4B\epsilon_{10}$ wherein assuming $B_{\rm
LCdirections}=1$, yields
$\hat{t}^{i_{1}j_{1}...i_{4}j_{4}}=t^{i_{1}j_{1}...i_{4}j_{4}}-\frac{1}{2}\epsilon^{i_{1}j_{1}...i_{4}j_{4}}$,
and the overall factor of $e^{2\pi i\tau_{0}}$, which is characteristic of the
stringy D-instanton, is evaluated at $\chi=\Re\tau_{0}=0$ in the stringy
calculation.

  * •

In [19], it is shown that the eleven-dimensional ${\cal O}(R^{4})$ corrections
have an independent motivation based on supersymmetry in ten dimensions. This
was shown to follow from its relation to the term $C^{(3)}\wedge X_{8}$ in the
${\mathscr{M}}$-theory effective action which is known to arise from a variety
of arguments, e.g. anomaly cancellation [27, 28]555We thank M.J.Duff for
bringing [27] to our attention.. The expression $X_{8}$ is the eight-form in
the curvatures that is inherited from the term in type IIA superstring theory
[29] which is given by

$-\int d^{10}xB\wedge X_{8}=-{1\over 2}\int
d^{10}x\sqrt{-g^{A(10)}}\epsilon_{10}BX_{8},$ (13)

where

$X_{8}={1\over 192}\left({\rm tr}\ R^{4}-{1\over 4}({\rm tr}\
R^{2})^{2}\right).$ (14)

There are two independent ten-dimensional $N=1$ super-invariants which contain
an odd-parity term ([26] and previous authors): $I_{3}=t_{8}{\rm tr}\
R^{4}-{1\over 4}\epsilon_{10}B{\rm tr}\ R^{4}$ and: $I_{4}=t_{8}({\rm tr}\
R^{2})^{2}-{1\over 4}\epsilon_{10}B({\rm tr}\ R^{2})^{2}.$ Using that
$t_{8}t_{8}R^{4}=24t_{8}{\rm tr}(R^{4})-6t_{8}({\rm tr}\ R^{2})^{2},$ it
follows that the particular linear combination,

$I_{3}-{1\over 4}I_{4}={1\over 24}t_{8}t_{8}R^{4}-48\epsilon_{10}B\ X_{8}$
(15)

contains both the ten-form $B\wedge X_{8}$ and $t_{8}t_{8}R^{4}$.

  * •

We will now summarize the argument of [18] that an $SL(2,\mathbb{Z})$
completion of the effective $R^{4}$ interactionleads to an interesting non-
renormalization theorem that forbids perturbative corrections to this term
beyond one loop in the zero-instanton sector. The term in (• ‣ 2) bilinear in
the tensor $\hat{t}$ has precisely the same form as terms that arise in the
zero instanton sector that come both from the one-loop four-graviton amplitude
and from an $(\alpha^{\prime})^{3}$ effect at tree level [30]. In the Einstein
frame one hence obtains the following expression for the complete effective
$R^{4}$ action in the Einstein’s frame, that can be expressed in the form,

$S_{R^{4}}=(\alpha^{\prime})^{-1}\left[a\zeta(3)\tau_{2}^{3/2}+b\tau_{2}^{-1/2}+ce^{2\pi
i\tau}+\cdots\right]R^{4}\equiv(\alpha^{\prime})^{-1}f(\tau,\bar{\tau})R^{4},$
(16)

where $a,b$ are known numerical constants, $R^{4}$ denotes the contractions
$\hat{t}\hat{t}R^{4}$ in (• ‣ 2) and $\cdots$ indicates possible perturbative
and nonperturbative corrections to the coefficient of $R^{4}$; the ‘constant’
$c$ can depend on $\tau=C_{\rm RR}+ie^{-\phi}$ (type IIB) and $\bar{\tau}$.
The first term in (16) represents the $(\alpha^{\prime})^{3}$ tree-level
contribution and the second the one-loop relative to the former.

However, the complete expression for $S_{R^{4}}$ must be invariant under
$SL(2,\mathbb{Z})$ transformations: $\tau\to(a\tau+b)(c\tau+d)^{-1}$
($a,b,c,d\in\mathbb{Z}:ad-bc=1$), which provides very strong constraints on
its structure. As the $R^{4}$ factor is already invariant,
$f(\tau,\bar{\tau})$ in (16) is a scalar under the $SL(2,\mathbb{Z})$
transformations, which hence implies a sum over all instantons and anti-
instantons.

There are some strong constraints which this term must satisfy and are spelt
out in [18], in which there is a simple function proposed by the authors that
satisfies all these criteria, namely,

$f(\tau,\bar{\tau})=\sum_{(p,n)\neq(0,0)}{\tau_{2}^{3/2}\over|p+n\tau|^{3}},$
(17)

where the sum indicates the sum is over all positive and negative values of
$p,n$ except $p=n=0$. Now:

$f=2\zeta(3)\tau_{2}^{3/2}+{\tau_{2}^{3/2}\over\Gamma(3/2)}\sum_{n\neq
0,p}\int_{0}^{\infty}dyy^{1/2}e^{-y|p+n\tau|^{2}}.$ (18)

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

$\displaystyle\hskip-36.135ptf(\tau,\bar{\tau})=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over
3}\tau_{2}^{-1/2}+2\tau_{2}^{3/2}\sum_{m,n\neq
0}\int_{0}^{\infty}dy\exp\left(-{\pi^{2}m^{2}\over y}+2\pi
imn\tau_{1}-yn^{2}\tau_{2}^{2}\right),$
$\displaystyle\hskip-36.135pt=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over
3}\tau_{2}^{-1/2}+8\pi\tau_{2}^{1/2}\sum_{m\neq 0n\geq 1}\left|{m\over
n}\right|e^{2\pi imn\tau_{1}}K_{1}(2\pi|mn|\tau_{2})$
$\displaystyle\hskip-36.135pt=2\zeta(3)\tau_{2}^{3/2}+{2\pi^{2}\over
3}\tau_{2}^{-1/2}$ $\displaystyle\hskip-36.135pt+4\pi^{3/2}\sum_{m,n\geq
1}\left({m\over n^{3}}\right)^{1/2}(e^{2\pi imn\tau}+e^{-2\pi
imn\bar{\tau}})\left(1+\sum_{k=1}^{\infty}(4\pi
mn\tau_{2})^{-k}{\Gamma(k-1/2)\over\Gamma(-k-1/2)}\right),$ (19)

where performing a perturbative expansion in $\frac{1}{\tau_{2}}$ of the non-
perturbative instanton contribution of charge $mn$, one uses the asymptotic
expansion for $K_{1}(z)$ for large $z$ in (• ‣ 2).

The perturbative terms in (• ‣ 2) terminate after the one-loop term as
suggested earlier. The non-perturbative terms have the form of a sum over
single multiply-charged instantons and anti-instantons with action
proportional to $|mn|$. Identifying $p$ with the discrete momentum (euclidean
energy) of a compactified $D$-particle of charge $n$ then the Poisson
resummation exchanges it with the winding number of the world-line and the
result is what is expected by T-duality from type IIA in nine dimensions. The
terms in parenthesis in (• ‣ 2) represent the infinite sequence of
perturbative corrections around the instantons of charge $mn$. It is hence
evident that in the zero-instanton sector, there are no perturbative
corrections figuring in the action up to ${\cal O}(R^{4})$, i.e., $S_{R^{4}}$,
beyond one loop. This will be important in demonstrating the "non-
renormalization" beyond one-loop of the deconfinement temperature $T_{c}$
later in the paper.

  * •

The ${\cal N}=1,D=11$ supergravity action inclusive of ${\cal O}(l_{p}^{6})$
terms (promoting the 10D $\hat{t}$ to 11D), is hence given by:

$\displaystyle\hskip-57.81621ptS=\frac{1}{2\kappa_{11}^{2}}\int_{M}\left[{\cal
R}*_{11}1-\frac{1}{2}G_{4}\wedge*_{11}G_{4}-\frac{1}{6}C\wedge G\wedge
G\right]+\frac{1}{\kappa_{11}^{2}}\int_{\partial M}d^{10}x\sqrt{h}K$
$\displaystyle\hskip-57.81621pt+\frac{1}{(2\pi)^{4}3^{2}2^{13}}\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)^{\frac{1}{3}}\int
d^{11}x\sqrt{-g}\left(J_{0}-\frac{1}{2}E_{8}\right)+\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)\int
C_{3}\wedge X_{8},$ (20)

where:

$\displaystyle\hskip-57.81621ptJ_{0}=3\cdot
2^{8}(R^{HMNK}R_{PMNQ}{R_{H}}^{RSP}{R^{Q}}_{RSK}+{1\over
2}R^{HKMN}R_{PQMN}{R_{H}}^{RSP}{R^{Q}}_{RSK})$
$\displaystyle\hskip-57.81621ptE_{8}={1\over 3!}\epsilon^{ABCM_{1}N_{1}\dots
M_{4}N_{4}}\epsilon_{ABCM_{1}^{\prime}N_{1}^{\prime}\dots
M_{4}^{\prime}N_{4}^{\prime}}{R^{M_{1}^{\prime}N_{1}^{\prime}}}_{M_{1}N_{1}}\dots
R^{M_{4}^{\prime}N_{4}^{\prime}}_{M_{4}N_{4}},$
$\displaystyle\hskip-57.81621pt\kappa_{11}^{2}=\frac{(2\pi)^{8}l_{p}^{9}}{2};$
(21)

$\kappa_{11}^{2}$ being related to the eleven-dimensional Newtonian coupling
constant, and $G=dC$ with $C$ being the ${\mathscr{M}}$-theory three-form
potential with the four-form $G$ being the associated four-form field
strength. The EOMS are:

$\displaystyle R_{MN}-\frac{1}{2}g_{MN}{\cal
R}-\frac{1}{12}\left(G_{MPQR}G_{N}^{\
PQR}-\frac{g_{MN}}{8}G_{PQRS}G^{PQRS}\right)$
$\displaystyle=-\beta\left[\frac{g_{MN}}{2}\left(J_{0}-\frac{1}{2}E_{8}\right)+\frac{\delta}{\delta
g^{MN}}\left(J_{0}-\frac{1}{2}E_{8}\right)\right],$ $\displaystyle
d*G=\frac{1}{2}G\wedge G+3^{2}2^{13}\left(2\pi\right)^{4}\beta X_{8},$

where [31]:

$\beta\equiv\frac{\left(2\pi^{2}\right)^{\frac{1}{3}}\left(\kappa_{11}^{2}\right)^{\frac{2}{3}}}{\left(2\pi\right)^{4}3^{2}2^{12}}\sim
l_{p}^{6},$ (23)

$R_{MNPQ},R_{MN},{\cal R}$ in (• ‣ 2)/(• ‣ 2) being respectively the elven-
dimensional Riemann curvature tensor, Ricci tensor and the Ricci scalar.

Now, one sees that if one makes an ansatz:

$\displaystyle\hskip-57.81621ptg_{MN}=g_{MN}^{(0)}+\beta g_{MN}^{(1)},$
$\displaystyle\hskip-57.81621ptC_{MNP}=C^{(0)}_{MNP}+\beta C_{MNP}^{(1)},$
(24)

then symbolically, one obtains:

$\displaystyle\beta\partial\left(\sqrt{-g}\partial
C^{(1)}\right)+\beta\partial\left[\left(\sqrt{-g}\right)^{(1)}\partial
C^{(0)}\right]+\beta\epsilon_{11}\partial C^{(0)}\partial C^{(1)}={\cal
O}(\beta^{2})\sim 0[{\rm up\ to}\ {\cal O}(\beta)].$

One can see that one can find a consistent set of solutions to (• ‣ 2) wherein
$C^{(1)}_{MNP}=0$ up to ${\cal O}(\beta)$. Assuming that one can do so,
henceforth we will define:

$\displaystyle\delta g_{MN}=\beta g^{(1)}_{MN}=G_{MN}^{\rm MQGP}f_{MN}(r),$
(26)

no summation implied.

  * •

Let us discuss how one can controls the higher derivative contributions to the
MQGP limit result of [3]. This was explained in [11] and we review the same
here. It is evident from (A) in the $\psi=2n\pi,n=0,1,2$-coordinate patch that
in the IR: $r=\chi r_{h},\chi\equiv{\cal O}(1)$, and up to ${\cal O}(\beta)$:

$f_{MN}\sim\beta\frac{\left(\log{\cal R}_{h}\right)^{m}}{{\cal
R}_{h}^{n}N^{\beta_{N}}},\ m\in\left\\{0,1,3\right\\},\
n\in\left\\{0,2,5,7\right\\},\ \beta_{N}>0,$ (27)

where ${\cal R}_{h}\equiv\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}}$. Now,
$|{\cal R}_{h}|\ll 1$. As estimated in [38], $|\log{\cal R}_{h}|\sim
N^{\frac{1}{3}}$, implying there is a competition between Planckian and
large-$N$ suppression and infra-red enhancement arising from $m,n\neq 0$ in
(27). Choosing a heirarchy: $\beta\sim
e^{-\gamma_{\beta}N^{\gamma_{N}}},\gamma_{\beta},\gamma_{N}>0:\gamma_{\beta}N^{\gamma_{N}}>7N^{\frac{1}{3}}+\left(\frac{m}{3}-\beta_{N}\right)\log
N$ (ensuring that the IR-enhancement does not overpower Planckian suppression
- we took the ${\cal O}(\beta)$ correction to $G^{\cal M}_{yz}$, which had the
largest IR enhancement, to set a lower bound on $\gamma_{\beta,N}$/Planckian
suppression). If $\gamma_{\beta}N^{\gamma_{N}}\sim 7N^{\frac{1}{3}}$, then one
will be required to go to a higher order in $\beta$. This hence answers the
question, when one can truncate at ${\cal O}(\beta)$.

##  3 ${\mathscr{M}}$-Theory Duals at Low ($T<T_{c}$ on the gauge theory side)
and High ($T>T_{c}$ on the gauge theory side) Temperatures

In this section we will explain how to calculate deconfinement temperature in
QCD-like theory via gauge/gravity duality and holographic renormalization of
the eleven dimensional supergravity action using a semiclassical computation
as advocated by Witten [16]. We have partitioned the calculation in two
subsections. In subsection 3.1, we will calculate various terms appearing in
on-shell supergravity action for blackhole background and discuss holographic
renormalization of the same. We do the same thing in subsection 3.2 for the
thermal background.

The ${\mathscr{M}}$-theory dual corresponding to high temperatures, i.e.,
$T>T_{c}$, will involve a black hole with the metric given by

$\displaystyle\hskip-7.22743ptds_{11}^{2}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-g(r)dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\left(dx^{3}\right)^{2}\right)$
(28)
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{g(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where $A_{\rm IIA}^{F^{\rm IIB}_{i=1,3,5}}$ are the type IIA RR 1-forms
obtained from the triple T/SYZ-dual of the type IIB $F_{1,3,5}^{\rm IIB}$
fluxes in the type IIB holographic dual of [6], and
$g(r)=1-\frac{r_{h}^{4}}{r^{4}}$. For low temperatures, i.e., $T<T_{c}$, is
given by the thermal gravitational dual:

$\displaystyle\hskip-7.22743ptds_{11}^{2}$ $\displaystyle=$ $\displaystyle
e^{-\frac{2\phi^{\rm
IIA}}{3}}\Biggl{[}\frac{1}{\sqrt{h(r,\theta_{1,2})}}\left(-dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\tilde{g}(r)\left(dx^{3}\right)^{2}\right)$
(29)
$\displaystyle\hskip-7.22743pt+\sqrt{h(r,\theta_{1,2})}\left(\frac{dr^{2}}{\tilde{g}(r)}+ds^{2}_{\rm
IIA}(r,\theta_{1,2},\phi_{1,2},\psi)\right)\Biggr{]}+e^{\frac{4\phi^{\rm
IIA}}{3}}\left(dx^{11}+A_{\rm IIA}^{F_{1}^{\rm IIB}+F_{3}^{\rm IIB}+F_{5}^{\rm
IIB}}\right)^{2},$

where $\tilde{g}(r)=1-\frac{r_{0}^{4}}{r^{4}}$. One notes that $t\rightarrow
x^{3},\ x^{3}\rightarrow t$ in (28) following by a Double Wick rotation in the
new $x^{3},t$ coordinates obtains (29); $h(r,\theta_{1,2})$ is the ten-
dimensional warp factor [6, 3]. This amounts to: $-g_{tt}^{\rm
BH}(r_{h}\rightarrow r_{0})=g_{x^{3}x^{3}}\ ^{\rm Thermal}(r_{0}),$
$g_{x^{3}x^{3}}^{\rm BH}(r_{h}\rightarrow r_{0})=-g_{tt}\ ^{\rm
Themal}(r_{0})$ in the results of [33, 11] (See [34] in the context of
Euclidean/black $D4$-branes in type IIA).

In (29), we will assume the spatial part of the solitonic $M3$ brane (which,
locally, could be interpreted as solitonic $M5$-brane wrapped around a
homologous sum of two-spheres [35]) world volume to be given by
$\mathbb{R}^{2}(x^{1,2})\times S^{1}(x^{3})$ where the period of
$S^{1}(x^{3})$ is given by a very large: $\frac{2\pi}{M_{\rm KK}}$, where the
very small $M_{\rm KK}=\frac{2r_{0}}{L^{2}}\left[1+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right]$, $r_{0}$ being the very small IR
cut-off in the thermal background (See also [36]) and $L=\left(4\pi
g_{s}N\right)^{\frac{1}{4}}$. So, $\lim_{M_{\rm KK}\rightarrow
0}\mathbb{R}^{2}(x^{1,2})\times S^{1}(x^{3})=\mathbb{R}^{3}(x^{1,2,3})$,
thereby recovering 4D Physics. The working metric used in this section and
section 4 for the thermal background corresponding to $T<T_{c}$ will involve
setting $\tilde{g}(r)$ to unity in (29).

Now, if $\beta_{\rm BH,Th}$ are respectively the periodicities of the thermal
circle in the black and thermal ${\mathscr{M}}$-theory backgrounds then at
$r={\cal R}_{\rm UV}$, $\beta_{\rm BH}\sqrt{G^{\rm BH}_{tt}}=\beta_{\rm
Th}\sqrt{G^{\rm Th}_{tt}}$. Now at $T=T_{c}$ [16],

$\beta_{\rm BH}\not{\int}_{M_{11}}\left({\cal L}^{\rm BH}_{\rm EH}+{\cal
L}^{\rm BH}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal L}^{\rm BH}_{{\cal
O}(R^{4})}\right)=\beta_{\rm Th}\not{\int}_{M_{11}}\left({\cal L}^{\rm
Th}_{\rm EH}+{\cal L}^{\rm Th}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal
L}^{\rm Th}_{{\cal O}(R^{4})}\right),$ (30)

where $\not{\int}$ excludes the coordinate integral with respect to $x^{0}$,
implying:

$\displaystyle\left(\sqrt{1-\frac{r_{h}^{4}}{{\cal R}_{\rm
UV}^{4}}}\right)^{-1}\int_{M_{10}}\left({\cal L}^{\rm BH}_{\rm EH}+{\cal
L}^{\rm BH}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal L}^{\rm BH}_{{\cal
O}(R^{4})}\right)$ $\displaystyle=\int_{\tilde{M}_{10}}\left({\cal L}^{\rm
Th}_{\rm EH}+{\cal L}^{\rm Th}_{\rm GHY}\delta(r-{\cal R}_{\rm UV})+{\cal
L}^{\rm Th}_{{\cal O}(R^{4})}\right).$ (31)

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

The ${\cal O}(\beta)$-corrected ${\mathscr{M}}$-theory metric of [3] in the
MQGP limit near the $\psi=2n\pi,n=0,1,2$-branches (whereat there is a
decoupling of $M_{5}(t,x^{1,2,3},r)$ and $M_{6}(\theta_{1,2},x,y,z,x^{11})$)
up to ${\cal O}((r-r_{h})^{2})$ [and up to ${\cal O}((r-r_{h})^{3})$ for some
of the off-diagonal components along the delocalized $T^{3}(x,y,z)$] - the
components which do not receive an ${\cal O}(\beta)$ corrections, are not
listed in (A) - was worked out in [11] and is given in (A).

Restricting to the Ouyang embedding of the flavor $D7$-branes in type IIB can,
for an extremely small (modulus of the) Ouyang embedding parameter, may be
effected by working near [4]:

$\theta_{1}=\frac{\alpha_{\theta_{1}}}{N^{\frac{1}{5}}},\
\theta_{2}=\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}.$ (32)

Einstein-Hilbert term in the IR will be given by the following expression:

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

Now performing angular integration of the above equation,

$\int\int
d\theta_{1}d\theta_{2}\frac{1}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}=\frac{1}{8}\cot({\theta_{2}})\left(\csc^{2}\left(\frac{{\theta_{1}}}{2}\right)-\sec^{2}\left(\frac{{\theta_{1}}}{2}\right)-4\log\left(\sin\left(\frac{{\theta_{1}}}{2}\right)\right)+4\log\left(\cos\left(\frac{{\theta_{1}}}{2}\right)\right)\right)$
(34)

implying:

$\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{d\theta_{1}d\theta_{2}}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}\sim\frac{N^{\frac{7}{10}}}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}+{\cal
O}\left(N^{\frac{3}{10}}\right),$ (35)

Using equation (35) and performing radial integration of equation (3.1), we
obtained the following expression for the action density (for Einstein-Hilbert
term) in the IR:

$\displaystyle\frac{\left.\int_{r=r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-G^{\mathscr{M}}}R\right|_{\rm
Ouyang}}{{\cal V}_{4}}\sim$ $\displaystyle-\frac{b^{2}{g_{s}}^{3/4}{\log
N}^{3}M{N_{f}}^{3}{r_{h}}^{4}\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)(1-\beta({\cal C}_{zz}^{\rm bh}-2{\cal
C}_{\theta_{1}z}^{\rm bh}+3{\cal C}_{\theta_{1}x}^{\rm
bh}))}{{N}^{\frac{11}{20}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}},$ (36)

where ${\cal R}_{D5-\overline{D5}}^{\rm bh}\equiv\sqrt{3}a^{\rm bh}$, $a^{\rm
bh}=\left(\frac{1}{\sqrt{3}}+\epsilon^{\rm bh}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{h}$ being the resolution
parameter of the blown up $S^{2}$ [11]. Further, there is one more term
appearing in on-shell action as given in equation (60), simplified form of
that term is:

$\displaystyle\frac{\int_{r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}R^{(0)}}{{\cal
V}_{4}}$ $\displaystyle\sim\beta{\cal C}_{\theta_{1}x}^{\rm
bh}\frac{b^{2}g_{s}^{\frac{3}{4}}\log^{3}NMN_{f}^{3}r_{h}^{4}\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\log\left(1-\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (37)

Similarly, Einstein-Hilbert term in the UV is:

$\displaystyle\hskip-43.36243pt\left.\sqrt{-G^{\mathscr{M}}}R\right|_{\rm
Ouyang}^{\rm UV}=-\frac{4\left(34{\log r}^{2}+14{\log r}-1\right){M_{\rm
UV}}r^{3}}{9\sqrt{3}\sqrt[4]{\pi}{g_{s}}^{9/4}N^{\frac{5}{4}}\sin^{3}{\theta_{1}}\sin^{2}{\theta_{2}}}$
$\displaystyle\times\frac{1}{\left(-{g_{s}}{N_{f}^{\rm UV}}(2{\log N}{\log
r}+{\log N}+3(1-6{\log r}){\log r})+2{g_{s}}(2{\log r}+1){N_{f}^{\rm
UV}}\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+8\pi{\log
r}\right)}$ $\displaystyle\hskip 36.135pt\approx\frac{4{M_{\rm
UV}}r^{3}\left(34\log^{2}(r)+14\log(r)-1\right)}{9\sqrt{3}\sqrt[4]{\pi}{g_{s}}^{13/4}{N_{f}^{\rm
UV}}N^{\frac{5}{4}}\sin^{3}{\theta_{1}}\sin^{2}{\theta_{2}}\log
N(2\log(r)+1)},$ (38)

where, for simplification of the computation, we have dropped the ${\cal
O}\left(\frac{1}{\log N}\right)$-correction terms. As
$\int\frac{r^{3}\left(34\log^{2}(r)+14\log(r)-1\right)}{2\log(r)+1}=\frac{1}{16}\left(\frac{4{Ei}(4\log(r)+2)}{e^{2}}+r^{4}(68\log(r)-23)\right)$
\+ constant.

Hence, one obtains:

$\displaystyle\left.\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\frac{\int_{{\cal R}_{D5-\overline{D5}}}^{{\cal R}_{\rm
UV}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-G^{\mathscr{M}}}R}{{\cal
V}_{4}}\right|_{\rm Ouyang}^{\rm UV-Finite}$ $\displaystyle\sim\frac{{M_{\rm
UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{{g_{s}}^{13/4}{N}^{\frac{11}{20}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{N_{f}^{\rm
UV}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\log N}.$ (39)

The UV-divergent contribution of the EH action is:

$\displaystyle S_{\rm UV-div}^{\rm EH}\sim\frac{{M_{\rm UV}}{\cal R}_{\rm
UV}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{{g_{s}^{\rm UV}}^{13/4}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}{\log N}{N_{f}^{\rm
UV}}N^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (40)

Now for the boundary metric:

$\displaystyle\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-h}\right|_{r={\cal
R}_{\rm UV}}\sim\frac{\left(\frac{1}{{g_{s}^{\rm UV}}}\right)^{7/3}{M_{\rm
UV}}{\cal R}_{\rm UV}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}{N^{\frac{3}{10}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}},$ (41)

and counter term to cancel the UV divergence appearing in Einstein-Hilbert
term is,

$\displaystyle\frac{\left.\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\sqrt{-h}R\right|_{r=r_{\Lambda}}}{{\cal
V}_{4}}\sim\frac{\left(\frac{1}{{g_{s}^{\rm UV}}}\right)^{2/3}{M_{\rm
UV}}{\cal R}_{\rm UV}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)}{{g_{s}^{\rm UV}}^{3/2}N^{1/5}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (42)

The UV-finite boundary Gibbons-Hawking-York contribution up to ${\cal
O}(\beta)$ turns to be:

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

The UV-divergent contribution to the GHY term is:

$S_{\rm UV-div}^{\rm GHY}\sim\frac{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right){M_{\rm UV}}{{\cal R}_{\rm
UV}}^{4}}{\left(g_{s}^{\rm UV}\right)^{9/4}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}^{4}{N}^{\frac{11}{20}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (44)

One can further show that up to ${\cal O}(\beta^{0})$ contribution from higher
derivative term will be:

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

using:

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

one can show that UV finite part coming from higher derivative term will be:

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

Use has been made of the following. The most dominant term in
$\sqrt{-G^{\mathscr{M}}}G^{MN}\frac{\delta J_{0}}{\delta G^{MN}}$ is
$-\frac{8{M_{\rm UV}}r^{3}\log(r)}{177147\pi{g_{s}^{\rm UV}}^{4}(2{\log
r}+1)^{7}N^{39/20}{N_{f}^{\rm
UV}}^{2}\log^{3}(N)}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$.
As:
$\int
dr\frac{r^{3}\log(r)}{(2\log(r)+1)^{7}}=\frac{2{Ei}(4\log(r)+2)}{45e^{2}}-\frac{r^{4}\left(256\log^{5}(r)+704\log^{4}(r)+800\log^{3}(r)+488\log^{2}(r)+184\log(r)+19\right)}{360(2\log(r)+1)^{6}}$,
one therefore notes that the UV-finite contribution to:
$\left(1+\frac{r_{h}^{4}}{2{\cal R}_{\rm
UV}^{4}}\right)\sqrt{-G^{\mathscr{M}}}G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}$ 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 turns out to be ${\cal O}\left(\frac{r_{h}^{4}}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)}\right)$-suppressed as compared to the UV-finite contribution
from $\left.\int_{{\cal R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}\sqrt{-G^{\mathscr{M}}}G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}\right|^{\rm UV-finite}$.
Also UV divergent contribution from higher derivative term will be:

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

We further note that at constant $r$ surface counter term to cancel UV
divergence appearing from higher derivative term will be given by the
following expression:

$\displaystyle\hskip-21.68121pt\beta\left.\frac{\left.\int\sqrt{-h}h^{mn}\frac{\delta
J_{0}}{\delta h^{mn}}\right|_{\rm UV-div}}{{\cal V}_{4}}\right|_{r=\rm
constant}\sim-\beta\frac{\left(\frac{1}{{g_{s}}^{\rm UV}}\right)^{2/3}{M_{\rm
UV}}{{\cal R}_{\rm UV}}^{4}}{{g_{s}^{\rm UV}}^{13/4}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{\log N}^{3}\left\\{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right\\}^{6}N^{11/5}{N_{f}^{\rm UV}}^{2}}$
$\displaystyle\hskip-21.68121pt\times\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}+{\cal
O}(\beta^{2}).$ (49)

From:

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

one notes that the dominant contribution arises from small values of
$\theta_{1,2}$, i.e., along Ouyang embedding for small Ouyang embedding
paramter(’s modulus). Hence,

$\displaystyle\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{19683\sqrt{6}\sin^{6}{\theta_{1}}+6642\sin^{2}{\theta_{2}}\sin^{3}{\theta_{1}}-40\sqrt{6}\sin^{4}{\theta_{2}}}{\sin^{7}{\theta_{1}}\sin^{4}{\theta_{2}}}$
$\displaystyle\sim
N^{\frac{9}{10}}\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}\sim-\frac{\log
N}{\alpha_{\theta_{2}}^{3}}N^{\frac{9}{10}}.$ (51)

Therefore simplified form of the counter term is:

$\displaystyle\hskip-21.68121pt\beta\left.\frac{\left.\int\sqrt{-h}h^{mn}\frac{\delta
J_{0}}{\delta h^{mn}}\right|_{\rm UV-div}}{{\cal V}_{4}}\right|_{r=\rm
constant}\sim\beta\frac{\left(\frac{1}{{g_{s}}^{\rm UV}}\right)^{2/3}{M_{\rm
UV}}{{\cal R}_{\rm UV}}^{4}\log N}{{g_{s}^{\rm UV}}^{13/4}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}^{4}{\log N}^{3}\left\\{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\right\\}^{6}N^{13/10}{N_{f}^{\rm
UV}}^{2}\alpha_{\theta_{2}}^{3}}.$ (52)

Also,

$\displaystyle\beta\left.\frac{\int_{r={\cal R}_{\rm UV}}\sqrt{-h}J_{0}}{{\cal
V}_{4}}\right|^{\rm UV}$ $\displaystyle\sim\frac{M_{\rm UV}}{g_{s}^{\rm
UV}\log^{\frac{8}{3}}N^{3}N_{f}^{\rm UV}\ {}^{\frac{8}{3}}}\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}}\right)\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\frac{d\theta_{1}d\theta_{2}}{\sin^{3}\theta_{1}\sin^{2}\theta_{2}}$
$\displaystyle\sim\frac{M_{\rm UV}}{g_{s}^{\rm
UV}\log^{\frac{8}{3}}N^{\frac{23}{10}}N_{f}^{\rm UV}\
{}^{\frac{8}{3}}}\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right).$ (53)

By comparing (3.1) with (3.1), one imposes:

$\log^{3}N\left\\{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\right\\}^{5}{N_{f}^{\rm
UV}}^{2}\sim{N_{f}^{\rm UV}}^{8/3}\log^{\frac{11}{3}}(N),$ (54)

implying:

$N_{f}^{\rm UV}\sim\frac{\left(\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)\right)^{\frac{15}{2}}}{\log N}.$ (55)

In $\kappa_{11}^{2}=1$-units, the $D=11$ supergravity action inclusive of
${\cal O}(R^{4})$ terms is given by:

$\displaystyle
S_{D=11}=\frac{1}{2}\Biggl{[}\int_{M_{11}}\sqrt{-G^{\mathscr{M}}}\left(R-\frac{G_{4}^{2}}{2.4!}+\beta\left(J_{0}-\frac{E_{8}}{2}\right)\right)$
$\displaystyle+2\int_{\partial
M_{11}}\sqrt{-h}K-\frac{1}{6}\int_{M_{11}}C_{3}\wedge G_{4}\wedge
G_{4}+4\pi^{2}\int_{M_{11}}C_{3}\wedge X_{8}\Biggr{]},$ (56)

which yields as an EOM:

$\displaystyle R_{MN}-\frac{G_{MN}}{2}R-\frac{1}{12}\left(G_{MPQR}G_{N}^{\ \
PQR}-\frac{G_{MN}}{8}G_{4}^{2}\right)$
$\displaystyle=-\beta\left[\frac{G_{MN}}{2}\left(J_{0}-\frac{E_{8}}{2}\right)+\frac{\delta}{\delta
G^{MN}}\left(J_{0}-\frac{E_{8}}{2}\right)\right].$ (57)

Taking the trace of (3.1) one obtains:

$\displaystyle-\frac{9}{2}R+\frac{G_{4}^{2}}{32}=-\beta\left[\frac{11}{2}\left(J_{0}-\frac{E_{8}}{2}\right)+G^{MN}\frac{\delta}{\delta
G^{MN}}\left(J_{0}-\frac{E_{8}}{2}\right)\right]$
$\displaystyle\approx-\beta\left[\frac{11}{2}J_{0}+G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}\right],$ (58)

where the approximation is justified in [11]. Writing $R=R^{(0)}+\beta
R,K=K^{(0)}+\beta K,C_{3}=C_{3}^{(0)}+\beta C_{3}$, etc., in [11] it is also
shown that one can self consistently set $C_{3}=0$. One thus notes:

$\displaystyle G_{4}^{2}=144R^{(0)},$ $\displaystyle
J_{0}\approx\frac{9}{11}R^{(1)}-\frac{2}{11}G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}.$ (59)

Therefore, the on-shell action up to ${\cal O}(\beta)$ using $X_{8}(M_{11})=0$
[3], becomes:

$S_{D=11}^{\rm on-shell}=-\frac{1}{2}\Biggl{[}-2S_{\rm EH}^{(0)}+2S_{\rm
GHY}^{(0)}+\beta\left(\frac{20}{11}S_{\rm
EH}-2\int_{M_{11}}\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}R^{(0)}+2S_{\rm
GHY}-\frac{2}{11}\int_{M_{11}}\sqrt{-G^{\mathscr{M}}}G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}\right)\Biggr{]},$ (60)

where
$\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}=\frac{G_{\mathscr{M}}^{(1)}}{2\sqrt{-G_{\mathscr{M}}^{(0)}}}$.
To see the holographic renormalization of (60), its UV-divergent portion is:

$\displaystyle S_{\rm D=11}^{\rm on-shellUV-
divergent}=-\frac{1}{2}\Biggl{[}\left(-2\kappa_{\rm EH}^{(0)}(g_{s}^{\rm
UV},N,M_{\rm UV},N_{f}^{\rm UV};\alpha_{\theta_{1,2}})+2\kappa^{(0)}_{\rm
GHY}(g_{s}^{\rm UV},N,M_{\rm UV},N_{f}^{\rm UV};\alpha_{\theta_{1,2}})\right)$
$\displaystyle\hskip-361.34999pt\times\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}}\right)+\beta\Biggl{\\{}-\frac{2}{11}\kappa^{(0)}_{\sqrt{-g}g^{MN}\frac{\delta
J_{0}}{\delta g^{MN}}}(g_{s}^{\rm UV},N,M_{\rm UV},N_{f}^{\rm
UV};\alpha_{\theta_{1,2}})\frac{\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)^{4}}{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}\Biggr{\\}}\Biggr{]}.$

Therefore, the counter terms required to cancel the UV-divergent terms are:

$\displaystyle\hskip-21.68121ptS_{\rm CT}^{\rm
BH}=-\frac{1}{2}\Biggl{[}-\frac{\kappa^{(0)}_{\rm EH+GHY}}{\kappa^{(0)}_{{\rm
EH}@{\cal R}_{\rm UV}}}\int_{r={\cal R}_{\rm UV}}\sqrt{-h}R$
$\displaystyle\hskip-21.68121pt+\beta
N^{\frac{1}{4}}\frac{\left(\frac{2}{11}\kappa^{(0)}_{\sqrt{-g}g^{MN}\frac{\delta
J_{0}}{\delta g^{MN}}}(g_{s}^{\rm UV},M_{\rm UV},N_{f}^{\rm
UV};\alpha_{\theta_{1,2}})\right)}{\kappa^{(0)}_{\sqrt{-h}g^{mn}\frac{\delta
J_{0}}{\delta g^{mn}}@{\cal R}_{\rm UV}}}\int_{r={\cal R}_{\rm
UV}}\sqrt{-h}g^{mn}\frac{\delta J_{0}}{\delta
g^{mn}}\Biggr{]}_{\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}}\right)=\left(N_{f}^{\rm UV}\log
N\right)^{\frac{2}{15}}}.$

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

In the large-$N$ limit and in the IR, the $f_{MN}$ EOMs for the thermal
background are algebraic. Writing below only those components which pick up a
non-trivial ${\cal O}(\beta)$ corrections, here is the ${\cal
O}(R^{4})$-corrected MQGP metric for thermal background in the IR in the
$\psi=2n\pi,n=0,1,2$-coordinate patches:

$\displaystyle G^{\mathscr{M}}_{rr}=G^{\rm
MQGP}_{rr}\left(1-\frac{99\sqrt{\frac{3}{2}}\beta{g_{s}}^{3/2}M\sqrt[5]{\frac{1}{N}}N_{f}{r_{0}}\alpha_{\theta_{1}}^{6}f_{x^{10}x^{10}}({r_{0}})\log^{2}({r_{0}})}{2\pi^{3/2}\alpha_{\theta_{2}}^{5}}\right),$
$\displaystyle G^{\mathscr{M}}_{x\theta_{1}}=G^{\rm
MQGP}_{x\theta_{1}}\left(1-\beta f_{x^{10}x^{10}}(r_{0})\right),$
$\displaystyle G^{\mathscr{M}}_{y\theta_{1}}=G^{\rm
MQGP}_{y\theta_{1}}\left(1+\beta f_{\theta_{1}y}(r_{0})\right),$
$\displaystyle G^{\mathscr{M}}_{z\theta_{1}}=G^{\rm
MQGP}_{z\theta_{1}}\left(1+\frac{539\pi^{3}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})}{1728{g_{s}}^{3}M^{2}N_{f}^{2}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\mathscr{M}}_{y\theta_{2}}=G^{\rm
MQGP}_{y\theta_{2}}\left(1-2\beta f_{x^{10}x^{10}}(r_{0})\right),$
$\displaystyle G^{\mathscr{M}}_{yy}=G^{\rm
MQGP}_{yy}\left(1-\frac{\pi^{3/2}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})\left(1617\sqrt{3}\pi^{3/2}{r_{0}}\alpha_{\theta_{1}}^{4}-32\sqrt{2}{g_{s}}^{3/2}MN_{f}\alpha_{\theta_{2}}\right)}{2592\sqrt{3}{g_{s}}^{3}M^{2}N_{f}^{2}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\mathscr{M}}_{yz}=G^{\rm
MQGP}_{yz}\left(1+\frac{\pi^{3/2}\beta
N^{2/5}\alpha_{\theta_{2}}^{3}f_{x^{10}x^{10}}({r_{0}})}{81\sqrt{6}{g_{s}}^{3/2}MN_{f}{r_{0}}\alpha_{\theta_{1}}^{4}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\mathscr{M}}_{zz}=G^{\rm
MQGP}_{zz}\left(1+\frac{539\pi^{3}\beta
N^{2/5}\alpha_{\theta_{2}}^{2}f_{x^{10}x^{10}}({r_{0}})}{864{g_{s}}^{3}M^{2}N_{f}^{2}\log^{2}({r_{0}})}\right),$
$\displaystyle G^{\mathscr{M}}_{x^{10}x^{10}}=G^{\rm
MQGP}_{x^{10}x^{10}}\left(1+\beta f_{x^{10}x^{10}}(r_{0})\right).$ (63)

Working in the IR and near (32), Einstein-Hilbert term for thermal background
is:

$\displaystyle\left.\sqrt{-G^{\mathscr{M}}}R\right|_{\rm Ouyang}^{\rm
IR}\sim-\frac{{g_{s}}^{3/4}{\log N}^{2}\log\left(\frac{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)M{N_{f}}^{3}}{{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}N^{\frac{5}{4}}\sin^{3}\theta_{1}\sin^{2}\theta_{2}}\sum_{m=0}^{2}\kappa_{\rm
EH,\ IR}^{\beta^{0},m}\ {r_{0}}^{3-m}\left(r-r_{0}\right)^{m}$
$\displaystyle-\beta\frac{{\log
N}\left(243\sqrt{6}\sin^{3}{\theta_{1}}-8\sin^{2}{\theta_{2}}\right)f_{x^{10}x^{10}}({r_{0}})}{N^{\frac{1}{4}}\sin^{5}{\theta_{1}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}^{3}{g_{s}}^{9/4}\log\left(\frac{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)M({\log
N}{N_{f}})^{5/3}}\sum_{m=0}^{2}\kappa_{\rm EH,\ IR}^{\beta^{1},m}\
{r_{0}}^{3-m}\left(r-r_{0}\right)^{m},$ (64)

where ${\cal R}_{D5/\overline{D5}}^{\rm th}=\sqrt{3}a^{\rm th}$, $a^{\rm
th}=\left(\frac{1}{\sqrt{3}}+\epsilon+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right)r_{0}$ [11] being the resolution
parameter for the blown up $S^{2}$ in the thermal background, and $\kappa_{\rm
EH,\ IR}^{\beta^{m},n}$ are the numerical prefactors - which we have worked
out but have not given here as they are not particularly illuminating in the
$\beta^{m}(r-r_{0})^{n}$ terms in $\sqrt{-G^{\mathscr{M}}}R$ restricted to the
Ouyang embedding of the flavor $D7$-branes in the type IIB dual and in the IR.
From (3.2), one can evaluate $\frac{\left.\int_{r=r_{0}}^{{\cal
R}_{D5-\overline{D5}}}\sqrt{-G^{\mathscr{M}}}R\right|_{\rm Ouyang}}{{\cal
V}_{4}}$ where ${\cal V}_{4}$ is the coordinate volume of
$S^{1}(x^{0})\times\mathbb{R}^{3}(x^{1,2,3})$.

Also additional term that appear in on-shell action (3.1) at $O(\beta)$ is:

$\displaystyle\frac{\int_{r_{h}}^{{\cal R}_{D5-\overline{D5}}^{\rm
bh}}\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}R^{(0)}}{{\cal
V}_{4}}$ $\displaystyle\sim\frac{\beta{g_{s}}^{3/4}{\log
N}^{2}M\left(\frac{1}{N}\right)^{11/20}{N_{f}}^{3}{r_{0}}{f_{x^{10}x^{10}}}({r_{0}})\left(1-\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}.$ (65)

Similarly Einstein-Hilbert term in the UV is:

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

implying:

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

Therefore, after performing radial as well as angular integral of the above
equation UV finite Einstein-Hilbert term is:

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

UV divergent Einstein-Hilbert term for thermal background is:

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

The UV-finite part of the boundary Gibbons-Hawking-York term for the thermal
background up to ${\cal O}(\beta)$ turns to be:

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

Further UV divergent part of the Gibbons-Hawking-York term for the thermal
background is:

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

One can further show that up to ${\cal O}(\beta^{0})$ contribution from higher
derivative term in the IR 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:

$\displaystyle\hskip-43.36243pt\frac{\int_{r_{0}}^{{\cal
R}_{D5/\overline{D5}}}\sqrt{-G^{\mathscr{M}}}\left.G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}\right|_{\rm Ouyang}}{{\cal V}_{4}}\sim$
$\displaystyle\hskip-43.36243pt\kappa_{\footnotesize G^{MN}\frac{\delta
J_{0}}{\delta G^{MN}}}^{\beta^{0},\ \rm IR}\
\frac{M{N_{f}}{r_{0}}^{2}\left(1-\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)^{2}\log\left(\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\epsilon^{8}{g_{s}}{\log
N}^{2}N^{21/20}{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}$
$\displaystyle\times\left\\{{\log N}-9\log\left(\frac{{r_{0}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)\right\\}\Biggl{[}{\log
N}-3\log\left(\frac{{r_{0}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\Biggr{]}^{2}.$

Further, near (32) contribution from higher derivative term in the UV is:

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

where $\tilde{r}\equiv\frac{{r}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}$,
thus after performing angular and radial integration of the above equation the
UV-finite part will be:

$\displaystyle\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}\sqrt{-G^{\mathscr{M}}}\left.G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}\right|_{\rm Ouyang}^{\rm UV-finite}}{{\cal V}_{4}}$
$\displaystyle\sim-\kappa_{\footnotesize G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}}^{\beta^{0},\ \rm UV-finite}\frac{{M_{\rm UV}}}{{g_{s}^{\rm
UV}}^{4}{\log N}^{3}N^{21/20}{N_{f}^{\rm UV}}^{2}}$
$\displaystyle\times\left(\frac{a^{2}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}+1\right)\left(\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}\right).$
(75)

Further, UV-divergent part from equation (3.2) will be:

$\displaystyle\frac{\int_{\theta_{1}={\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}^{\pi-{\alpha_{\theta_{1}}}{N^{-\frac{1}{5}}}}\int_{\theta_{2}={\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}^{\pi-{\alpha_{\theta_{2}}}{N^{-\frac{3}{10}}}}\int_{{\cal
R}_{D5/\overline{D5}}}^{{\cal R}_{\rm
UV}}\sqrt{-G^{\mathscr{M}}}\left.G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}\right|_{\rm Ouyang}^{\rm UV-divergent}}{{\cal V}_{4}}$
$\displaystyle=\kappa_{\footnotesize G^{MN}\frac{\delta J_{0}}{\delta
G^{MN}}}^{\beta^{0},\ \rm UV-div}\ \frac{{\frac{1}{{g_{s}^{\rm UV}}}}{M_{\rm
UV}}{{\cal R}_{\rm UV}}^{4}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{4}{g_{s}^{\rm UV}}^{9/2}N^{21/20}{N_{f}^{\rm
UV}}^{8/3}\log^{\frac{11}{3}}(N)\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}\right)}$
$\displaystyle\times\frac{\left(-6642{\alpha_{\theta_{1}}}^{3}{\alpha_{\theta_{2}}}^{2}+19683\sqrt{6}{\alpha_{\theta_{1}}}^{6}\log\left(\frac{{\alpha_{\theta_{1}}}}{2\sqrt[5]{N}}\right)-20\sqrt{6}{\alpha_{\theta_{2}}}^{4}\right)}{3{\alpha_{\theta_{1}}}^{6}{\alpha_{\theta_{2}}}^{3}}$
(76)

We note that:

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

One should note that in (3.1), (3.1), (3.1), (41), (42), (3.1), (3.1), (3.1),
(3.1), (3.1), (52), (3.1) (as well as (4) and (4)) for the supergravity dual
with a BH corresponding to $T>T_{c}$, and similarly (3.2), (3.2), (3.2),
(3.2), (3.2), (3.2), (3.2) and (3.2) for the supergravity dual with a thermal
background corresponding to $T<T_{c}$, given that the delocalized toroidal
coordinates $(x,y,z)$ are defined as [3]:

$x\sim h^{\frac{1}{4}}(r_{0},\theta_{10,20})r_{0}\sin\theta_{10}\phi_{1},\
y\sim h^{\frac{1}{4}}(r_{0},\theta_{10,20})r_{0}\sin\theta_{20}\phi_{2},\
z\sim h^{\frac{1}{4}}(r_{0},\theta_{10,20})r_{0}\psi,$ (78)

where the 10-D warp factor is given by:
$h(r,\theta_{1,2})=\frac{L^{4}}{r^{4}}\left[1+\frac{3g_{s}M_{\rm
eff}^{2}}{N}\log r\left\\{1+\frac{3g_{s}N_{f}^{\rm eff}}{2\pi}\left(\log
r+\frac{1}{2}\right)+\frac{g_{s}N_{f}^{\rm
eff}}{4\pi}\log\left(\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}\right)\right\\}\right]$,
upon promoting to global coordinates, and noting that all the aforementioned
equations already have an $N^{-\zeta},\zeta>0$ suppression implying that up to
${\cal O}\left(\frac{1}{N}\right)$, one can approximate
$h\approx\frac{L^{4}}{r^{4}}$, one sees that one will generate a
$\sin\theta_{1}\sin\theta_{2}$ contribution upon integration w.r.t. $x,y,z$.
As this is not accompanied with an $r$-dependence, it will only provide a
common multiplicative $N$-suppressed contribution to all the abovementioned
integrals. We have hence not considered the same.

Also, note there is a multiplicative factor of $\beta_{\rm
Th}\sim\frac{\sqrt{g_{s}N}}{r_{0}}$ in all terms in the action corresponding
to the thermal background, e.g., (3.2), (3.2), (3.2), (3.2), (3.2) and (3.2).

With reference to (3.2) and the fourth equation in (3.2), let us assume:

$\left[\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\right]^{6}N^{-\upsilon+\frac{4}{5}}=\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)N^{\frac{11}{20}},$ (79)

or

$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)=N^{\frac{4\upsilon-1}{20}}.$ (80)

Further assuming $\upsilon=\frac{1}{4}+\epsilon$,

$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\sim 1+\frac{\epsilon}{5}\log N.$ (81)

From (54), assuming,

$N_{f}^{\rm UV}=\frac{1}{\log N}+\epsilon_{1}:0<\epsilon_{1}\ll 1,$ (82)
$\log\left(\frac{{{\cal R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}}\right)\sim 1+\frac{2\epsilon_{1}}{15}\log N.$ (83)

From (81) and (83), $\epsilon_{1}=\frac{3}{2}\epsilon$.

One hence sees that the UV-divergent contribution of
$\frac{1}{2}\Biggl{(}-\frac{7}{2}S_{\rm EH}^{(0)}+2S_{\rm GHY}^{(0)}\Biggr{)}$
is given by:

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

From (3.2), one notices the following:

  * •

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

  * •

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

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

In this section using the results from 3.1, 3.2, we will calculate
deconfinement temperature by equating on-shell actions for blackhole and
thermal backgrounds. In the same process we will explain a version of UV-IR
connection which is arising from the ${\cal O}(R^{4})$ terms. We will also
calculate entropy of the black hole and discuss its consistentency with the
Wald entropy up to ${\cal O}(R^{4})$.

The ${\cal O}(\beta^{0,1})$ term in (60) $\left(1+\frac{r_{h}^{4}}{2{\cal
R}_{\rm UV}^{4}}\right)S_{D=11,\ {\rm on-shellUV-finite}}^{BH,\ \beta^{0,1}}$
and similarly $S_{D=11,\ {\rm on-shellUV-finite}}^{{\rm thermal},\
\beta^{0,1}}$ can now be easily computed.

One can show that the LO terms in $N,\log\left(\frac{{\cal R}_{\rm UV}}{{\cal
R}_{D5/\overline{D5}}}\right)$ and $\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}}$
of the aforementioned BH and thermal actions are given as under:

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

Writing ${\cal R}_{D5/\overline{D5}}^{\rm
bh}=\left(1+{\sqrt{3}}\epsilon\right)r_{h}$, and using (89),
$\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)=\log\left(\frac{\sqrt{3}\epsilon}{1+\epsilon}\right)\\\
\sim-\left|\log\left(M_{\rm UV}\left(N_{f}^{\rm
UV}\right)^{\frac{2}{15}}\right)\right|\sim-\log\log N$.

Let us show the consistency of the BH entropy calculated via (4) and the
Wald’s method. One notes from (4) that up to ${\cal O}(\beta^{0})$, the BH
entropy ${\cal S}_{\rm BH}$ is given by:

$\displaystyle{\cal S}_{\rm BH}=\beta_{\rm BH}\frac{\partial S^{E}_{\rm
BH}}{\partial\beta}-S^{E}_{\rm BH}\sim\frac{\sqrt{N}{M_{\rm
UV}}{r_{h}}^{3}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)}{{g_{s}}^{7/4}N^{11/20}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle\sim\frac{M_{\rm UV}\left(N_{f}^{\rm UV}\log
N\right)^{\frac{2}{15}}r_{h}^{3}}{N^{\frac{1}{20}}}.$ (86)

Note, there is a multiplicative factor of $\beta_{\rm
BH}\sim\frac{\sqrt{g_{s}N}}{r_{h}}$ in all terms in the action corrsponding to
the BH gravity dual, e.g., (3.1), (3.1), (3.1), (3.1), (3.1), (44), (3.1),
(3.1), (3.1), (52) and (3.1).

Interestingly, the Wald entropy density arising from the ${\cal O}(\beta^{0})$
action is given by:

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

where ${\cal V}_{3}$ is the $\mathbb{R}^{3}(x^{1,2,3}$ coordinate volume).
Upon substituting $b=\frac{1}{\sqrt{3}}+\epsilon,\epsilon\sim\left(|\log
r_{h}\right|)^{\frac{9}{2}}N^{-\frac{9}{10}-\alpha_{\epsilon}}$ yields:

$\displaystyle\frac{{g_{s}}^{5/4}M{N_{f}}{r_{h}}^{3}N^{-\frac{19}{20}-{\alpha_{\epsilon}}}\left(|\log
r_{h}\right|)^{\frac{11}{2}}\left(\log N-9\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)}{\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle\times\left[{N_{f}}\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)\right]^{2/3}.$ (88)

Both methods yield that the BH entropy goes like $r_{h}^{3}$. The results
match exactly for (for simplicity, we have disregarded in (89) numerical
factors and $M,N_{f}$ which in the MQGP limit are ${\cal O}(1)$):

$\displaystyle\alpha_{\epsilon}=-\frac{9}{10}+\frac{\left|\log\left(M_{\rm
UV}\left(N_{f}^{\rm UV}\right)^{\frac{2}{15}}\right)\right|}{\log
N}+\frac{211\log\log N}{\log N}\sim\frac{\left|\log\left(M_{\rm
UV}\left(N_{f}^{\rm UV}\right)^{\frac{2}{15}}\right)\right|}{\log
N}-\frac{9}{10};$ (89)

$\alpha_{\epsilon}>0$ if $M_{\rm UV}\left(N_{f}^{\rm
UV}\right)^{\frac{2}{15}}<N^{-\frac{9}{10}}$ .

At ${\cal O}(R^{4})$, one can show that to calculate one needs to evaluate the
following four class of terms while calculating $\frac{\partial
J_{0}}{\partial R_{trtr}}$:

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

One hence sees that it is the contribution (iii) which dominates in the MQGP
limit, and obtains:

$\displaystyle\frac{{\cal S}_{E}^{{\cal O}(R^{4})}}{{\cal
V}_{3}}\sim\oint_{\theta_{1,2},x,y,z,x^{10})}\frac{\partial J_{0}}{\partial
R_{rtrt}}\sqrt{-G_{9}}d\theta_{1}d\theta_{2}dxdydzdx^{10}$
$\displaystyle\sim\frac{{g_{s}}^{23/4}M^{8}{N_{f}}^{4/3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)^{3}\log^{4}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\left(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{3}}{\sqrt[20]{N}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\left(\log
N-3\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{8/3}}.$ (91)

One notes the identical $r_{h}^{3}$ and $N^{-\frac{1}{20}}$ dependence in (4)
and the corresponding semiclassical result at ${\cal O}(\beta)$. For the
latter and the Wald entropy result to match exactly at ${\cal O}(\beta)$
imposes the following constraint:

$\displaystyle\Biggl{[}{2{\cal C}_{\theta_{1}x}^{\rm
bh}\kappa_{\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}R^{(0)}}^{\rm
IR}}+\frac{20\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm
bh}-3{\cal C}_{\theta_{1}x}^{\rm bh}\right)\kappa_{\rm EH}^{\beta,\ \rm
IR}}{11}\Biggr{]}\log^{3}(N)\left|\log\left(M_{\rm UV}\left(N_{f}^{\rm
UV}\right)^{\frac{2}{15}}\right)\right|$
$\displaystyle\sim\frac{\log^{3}\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\left(\log
N-12\log\left(\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}^{\rm
bh}}\right)\right)^{3}}{\left(\log N-3\log\left(\frac{r_{h}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)\right)^{8/3}}.$ (92)

Hence, one sees that consistency with the Wald entropy computation up to
${\cal O}(R^{4})$, imposes a linear constraint on the constants of integration
appearing in the solutions to the EOMs of the metric corrections at ${\cal
O}(R^{4})$ along the directions that encode the "memory" of the compact part
of the non-compact four-cycle "wrapped" by the flavor $D7$-branes in the type
IIB dual [3] of the large-$N$ thermal QCD-like theories. This "Flavor
Memory"-effect is also displayed by the ${\mathscr{M}}$-theory uplift of the
corresponding thermal background, as seen in 3.2.

After that brief detour, returning back to the computation of $T_{c}$, the on-
shell action corresponding to the thermal background uplift is given below:

$\displaystyle S_{D=11,\ {\rm on-shellUV-finite}}^{\rm
thermal}\sim\frac{2\kappa_{\rm GHY}^{{\rm th},\ \beta^{0}}{M_{\rm
UV}}{r_{0}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{g_{s}^{\rm UV}}^{9/4}N^{11/20}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle+\frac{2{g_{s}}^{3/4}\kappa_{\rm EH,IR}^{{\rm th},\
\beta^{0}}M{N_{f}}^{3}{r_{0}}^{2}\log^{2}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{N^{11/20}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{2}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle-\frac{20\beta\kappa_{\rm EH,th}^{{\rm IR},\
\beta}{r_{0}}^{3}\left(2\alpha_{\theta_{2}}^{3}-729\sqrt{6}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}\right){f_{x^{10}x^{10}}}({r_{0}})}{11{g_{s}}^{9/4}MN^{7/20}{N_{f}}^{5/3}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}\alpha_{\theta_{1}}^{4}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}\beta.$ (93)

Now, equality at ${\cal O}(\beta^{0})$ in (3) requires the following equation
to be solved:

$\displaystyle\frac{2\kappa_{\rm GHY}^{\rm bh}{M_{\rm
UV}}{r_{h}}^{4}\log\left(\frac{{{\cal R}_{\rm UV}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\right)}{{g_{s}}^{9/4}}=\frac{2\kappa_{\rm
GHY}^{{\rm th},\ \beta^{0}}{M_{\rm UV}}{r_{0}}^{4}\log\left(\frac{{{\cal
R}_{\rm UV}}}{{\cal R}_{D5/\overline{D5}}^{\rm th}}\right)}{{g_{s}^{\rm
UV}}^{9/4}{{\cal R}_{D5/\overline{D5}}^{\rm th}}^{4}}$
$\displaystyle+\frac{2{g_{s}}^{3/4}\kappa_{\rm EH,IR}^{{\rm th},\
\beta^{0}}M{N_{f}}^{3}{r_{0}}^{2}\log^{2}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)}{{{\cal R}_{D5/\overline{D5}}^{\rm
th}}^{2}}.$ (94)

Now, one can show that near, e.g.,
$(\theta_{1},\theta_{2})\sim\left(\frac{\alpha_{\theta_{1}}}{N^{\frac{1}{5}}},\frac{\alpha_{\theta_{1}}}{N^{\frac{3}{10}}}\right)$,
$\frac{\kappa_{\rm GHY}^{{\rm th},\ \beta^{0}}}{\kappa_{\rm EH,IR}^{{\rm th},\
\beta^{0}}}\sim 10^{5}$, and hence, one can drop the $\kappa_{\rm EH,IR}^{{\rm
th},\ \beta^{0}}$ term. Therefore,

$r_{h}=\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm th},\
\beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{r_{0}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}.$ (95)

Identifying $\frac{r_{0}}{L^{2}}$ with $\frac{m^{0^{++}}}{4}$ [37] and using
$T_{c}=r_{h}/\pi L^{2}$ [4], deconfinement temperature will be given by the
following expression:

$T_{c}=\frac{\sqrt[4]{\frac{\kappa_{\rm GHY}^{{\rm th},\
\beta^{0}}}{\kappa_{\rm GHY}^{{\rm bh},\ \beta^{0}}}}{m^{0^{++}}}{{\cal
R}_{D5/\overline{D5}}^{\rm bh}}\sqrt[4]{\frac{\log\left(\frac{{{\cal R}_{\rm
UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm th}}}\right)}{\log\left(\frac{{{\cal
R}_{\rm UV}}}{{{\cal R}_{D5/\overline{D5}}^{\rm bh}}}\right)}}}{4\pi{{\cal
R}_{D5/\overline{D5}}^{\rm th}}}.$ (96)

Using (4) and (4), the matching at ${\cal O}(\beta)$ yields:

$\displaystyle\Biggl{[}{2{\cal C}_{\theta_{1}x}^{\rm
bh}\kappa_{\left(\sqrt{-G^{\mathscr{M}}}\right)^{(1)}R^{(0)}}^{\rm
IR}}+\frac{20\left(-{\cal C}_{zz}^{\rm bh}+2{\cal C}_{\theta_{1}z}^{\rm
bh}-3{\cal C}_{\theta_{1}x}^{\rm bh}\right)\kappa_{\rm EH}^{\beta,\ \rm
IR}}{11}\Biggr{]}$
$\displaystyle\times\frac{b^{2}{g_{s}}^{3/4}M{N_{f}}^{3}{r_{h}}^{4}\log^{3}(N)\log\left(\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)\log\left(1-\frac{{r_{h}}}{{{\cal
R}_{D5/\overline{D5}}}}\right)}{N^{11/20}{{\cal
R}_{D5/\overline{D5}}}^{4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}}$
$\displaystyle=\frac{20\kappa_{\rm EH,th}^{{\rm IR},\
\beta}{r_{0}}^{3}\left(2\alpha_{\theta_{2}}^{3}-729\sqrt{6}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}\right){f_{x^{10}x^{10}}}({r_{0}})}{11{g_{s}}^{9/4}MN^{7/20}{N_{f}}^{5/3}{{\cal
R}_{D5/\overline{D5}}^{\rm
th}}^{3}\alpha_{\theta_{1}}^{4}\log^{\frac{2}{3}}(N)\log\left(\frac{{r_{0}}}{{\cal
R}_{D5/\overline{D5}}^{\rm th}}\right)},$ (97)

which yields:

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

We thus see that by matching the thermal and black-hole ${\mathscr{M}}$-theory
background actions at the UV cut-off, respectively dual to large-$N$ QCD-like
theories at low and high temperatures, (4) gives a relationship between the
constants of integration in the IR in the respective metric perturbations as
${\cal O}(l_{p}^{6})$ corrections to the ${\mathscr{M}}$-theory uplift worked
out in [3] along respectively the ${\mathscr{M}}$-theory circle (for thermal
background dual to $T<T_{c}$) and the $S^{3}$-portion of the four cycle
(locally) $\Sigma_{(4)}=\mathbb{R}_{+}\times S^{3}$ "wrapped" by the flavor
$D7$-branes of the type IIB conifold geometry of [6] (for black hole
background dual to $T>T_{c}$)!! The equations (4) along with (96), are what we
refer to as UV-IR connection in this context. We also note that the
contributions arising from the ${\cal O}(R^{4})$ corrections to the BH
background metric retains the "Flavor Memory", referred to earlier, of the
compact three-cycle which is part of the non-compact four-cycle "wrapped" by
the parent type IIB flavor $D7$-branes.

##  5 Deconfinement from Entanglement Entropy

In this section we will discuss confinement-deconfinement phase transition in
QCD2+1-like theory from entanglement entropy point of view based on [2]. We
will calculate entanglement entropy for "connected" and "disconnected" regions
- suitably defined. We will show that at a critical value of ${\it l}$ of a
spatial interval which we are denoting here by ${\it l_{crit}}$, phase
transition will occur from confined phase to the deconfined phase. Contact
with results of the previous section is made by looking at the
$M_{KK}\rightarrow 0$\- or equivalently $r_{0}\rightarrow 0$-limit, i.e., the
4D-limit of results of this section.

Authors in [2] have investigated entanglement entropy between an interval and
its complement in the gravity dual of large $N_{c}$ gauge theories and they
found that there are two RT surfaces - disconnected and connected. Below a
critical value of ${\it l}$ connected surface dominates while above that
critical value of ${\it l}$ disconnected surface dominates. This is analogous
to finite temperature deconfinement transition in dual theories.

For $AdS_{d+2}/CFT_{d+1}$ correspondence, quantum entanglement entropy between
the regions $A\equiv\mathbb{R}^{d-1}\times l$ and
$B\equiv\mathbb{R}^{d-1}\times(\mathbb{R}-l)$, $l$ being an interval of length
$l$, is given by the following expression[17],

$S_{A}=\frac{1}{4G_{N}^{(d+2)}}\int_{\gamma}d^{d}\sigma\sqrt{G_{ind}^{(d)}}.$
(99)

where, $G_{N}^{(d+2)}$ is Newton constant in $(d+2)$ dimensions and
$G_{ind}^{(d)}$ is the determinant of the induced string frame metric on co-
dimension 2 minimal surface $\gamma$. Equation $(\ref{EE-def})$ can be
generalised to non-conformal theories as below [2]:

$S_{A}=\frac{1}{4G_{N}^{(d+2)}}\int_{\gamma}d^{d}\sigma
e^{-2\phi}\sqrt{G_{ind}^{(d)}},$ (100)

where $\phi$ is the dilaton profile. In an ${\mathscr{M}}$-theory dual since
there is no dilaton, therefore equation (100) will be assumed to be replaced
by the following expression,

$S_{A}=\frac{1}{4G_{N}^{(11)}}\int_{\gamma}d^{9}\sigma\sqrt{G_{ind}^{(9)}}.$
(101)

Now, consider gravity dual’s string frame metric (29), which can written as:

$ds^{2}=\alpha(r)[\sigma(r)dr^{2}+dx_{\mu}dx^{\mu}]+g_{mn}dx^{m}dx^{n},$ (102)

where $x_{\mu}(\mu=0,1,2)$ represents $(2+1)$ Minkowskian coordinates, $r$ is
the radial coordinate and $x^{m}(m=3,5,6,7,8,9,10)$ corresponds to $x^{3}$ and
six angular coordinates$(\theta_{1,2},\phi_{1,2},\psi,x^{10})$. Noting
$g_{x^{3}x^{3}}(r=r_{0})=0$, $(x^{3},r)$ form a cigar-like geometry. The
volume of the seven-fold is given by:

${\cal V}_{\rm int}=\int\prod_{m}dx^{m}\sqrt{g}.$ (103)

Here we have $(2+1)$ dimensional QFT defined on ${\rm I\\!R}^{2+1}$. Let us
define two regions $A$ and $B$ in two dimensions as below,

$\displaystyle A={\rm I\\!R}\times l,$ (104) $\displaystyle B={\rm
I\\!R}\times({\rm I\\!R}-l).$ (105)

Now, we are going to calculate enganglement entropy between $A$ and $B$ for
the metric $(\ref{metric})$ using equation $(\ref{EE-def-M Theory})$. Induced
metric on $\gamma$ is given below (here we have represented $x_{1}$ by $x$):.

$ds^{2}|_{\gamma}=\alpha(r)\left[\left(\sigma(r)+\left(\frac{dx}{dr}\right)^{2}\right)dr^{2}+dx_{2}^{2}\right]+g_{mn}dx^{m}dx^{n}.$
(106)

Therefore,

$\frac{S_{A}}{{\cal V}_{1}}=\frac{1}{4G_{N}^{(11)}}\int
dr\sqrt{H(r)}\sqrt{\sigma(r)+(\partial_{r}x(r))^{2}},$ (107)

where,

$H(r)={\cal V}_{int}^{2}\alpha(r)^{2}.$ (108)

As in [2], $H(r=r_{0})=0$. Equation of motion for $x(r)$ can be obtained from
equation $(\ref{EEC})$ as,

$\frac{dx}{dr}=\pm\frac{\sqrt{\sigma(r)}\sqrt{H(r_{*})}}{\sqrt{H(r)-H(r_{*})}},$
(109)

where, $r_{*}$ is the value of radial coordinate at which $dr/dx$ vanishes.
Integrating above equation we obtain,

$l(r_{*})=2\sqrt{H(r_{*})}\int_{r_{*}}^{\infty}dr\frac{\sqrt{\sigma(r)}}{\sqrt{H(r)-H(r_{*})}}.$
(110)

From Equations $(\ref{EEC})$ and $(\ref{dx/dr})$, entanglement entropy can be
simplified as,

$\frac{S_{A}}{{\cal
V}_{1}}=\frac{1}{2G_{N}^{(11)}}\int_{r_{*}}^{r_{\infty}}dr\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r_{*})}}.$
(111)

Here, $r_{\infty}$ is the UV cutoff.

For the disconnected surface we have $r_{*}=r_{0}$ as explained in [2], when
$l>l_{max}$ entanglement entropy for the disconnected surface will be given by
the following expression,

$\frac{S_{A}}{{\cal
V}_{1}}=\frac{1}{2G_{N}^{(11)}}\int_{r_{0}}^{r_{\infty}}dr\sqrt{\sigma(r)H(r)}.$
(112)

For $l<l_{max}$, difference of entanglement entropy for connected and
disconnected surface have the following form,

$\frac{2G_{N}^{(11)}(S_{A}^{(conn)}-S_{A}^{(disconn)})}{{\cal
V}_{1}}=\int_{r_{*}}^{\infty}dr\left(\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H(r_{*})}}-\sqrt{\sigma(r)H(r)}\right)-\int_{r_{0}}^{r_{*}}dr\sqrt{\sigma(r)H(r)}.$
(113)

  * •

Connected solution with large $r_{*}$ has the lower entropy when ${\it l}$ is
small.

  * •

When ${\it l}$ starts increasing, two things can happen. First, connected
solution can still remain the lower untill ${\it l}$ reaches it’s maximum
value i.e. ${\it l}_{max}$. Second, there could be a critical value of ${\it
l}$ which is denoted by ${\it l}_{crit}$ and ${\it l}_{crit}<{\it l}_{max}$
above which disconnected solution becomes dominant one.

  * •

There will be a phase transition at ${\it l}={\it l}_{max}$ in the first case
and at ${\it l}={\it l}_{crit}<{\it l}_{max}$ in the second case.

From equations (29) and (102) we have,

$\alpha(r)=\frac{e^{\frac{-2\phi^{IIA}}{3}}}{\sqrt{h(r,\theta_{1,2})}}$ (114)

and

$\sigma(r)=\frac{e^{\frac{2\phi^{IIA}}{3}}}{\tilde{g(r)}},$ (115)

where $\phi^{IIA}$ is type IIA dilaton profile, which can be read off from
${\mathscr{M}}$-theory metric component as,

$G^{\mathscr{M}}_{x_{10}x_{10}}=e^{\frac{4\phi^{IIA}}{3}}.$ (116)

From the equation (103), simplified form of the ${\cal V}_{int}$ for the
metric given in equation (29) is:

$\displaystyle{\cal V}_{\rm
int}\sim\frac{{g_{s}}^{3/4}MN^{9/20}\sqrt{1-\frac{{r_{0}}^{4}}{r^{4}}}\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{4/3}}{r\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}\left(\frac{1}{2}\beta({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm th})+1\right)$ (117)
$\displaystyle\hskip-469.75499pt\times\Biggl{[}{g_{s}}\log
N{N_{f}}\left(3a^{2}-r^{2}\right)(2\log(r)+1)+\log(r)$
$\displaystyle\hskip-462.52756pt\times\left(4{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-24\pi
a^{2}+r^{2}(8\pi-3{g_{s}}{N_{f}})\right)$
$\displaystyle\hskip-462.52756pt+2{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+18{g_{s}}{N_{f}}\left(r^{2}-3a^{2}(6r+1)\right)\log^{2}(r)\Biggr{]},$

(strictly speaking the periodicity of $x^{3}$, $\frac{2\pi}{M_{\rm KK}}$ where
$M_{\rm KK}=\frac{2r_{0}}{L^{2}}\left[1+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right]$, but this does not influence the
computation of $l_{\rm max}$ and $l_{\rm crit}$) 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 $\sigma(r)$ have been obtained from equation (116) and
$h(r,\theta_{1,2})$ given in [6, 3], simplified form of the same are given
below:

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

and

$\displaystyle\hskip-57.81621pt\sigma(r)=\frac{4\left(\frac{\pi}{3}\right)^{2/3}\left(\frac{32\pi
a^{2}{g_{s}}M^{2}{N_{f}}({c_{1}}+{c_{2}}\log({r_{0}}))}{N\left(9a^{2}+r^{2}\right)\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right)}+1\right)}{\left(1-\frac{{r_{0}}^{4}}{r^{4}}\right)\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}}$
$\displaystyle\hskip-57.81621pt-\frac{18\sqrt[3]{\frac{3}{\pi}}b^{10}\left(9b^{2}+1\right)^{4}\beta
M\left(\frac{1}{N}\right)^{5/4}r^{5}\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)}{\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}{N_{f}}{r_{0}}^{4}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{0}}^{2}\right)\left({r_{0}}^{4}-r^{4}\right)}$
$\displaystyle\hskip-57.81621pt\times\frac{\left(6a^{2}+{r_{0}}^{2}\right)(r-2{r_{0}})\log^{3}({r_{0}})}{\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{16\pi}{{g_{s}}}+\frac{\Omega}{g_{s}}\right){}^{2/3}},$
(119)

$c_{1,2}$ appearing as the ${\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)$-correction
$\left(\frac{g_{s}M^{2}}{N}\right)(c_{1}+c_{2}\log r_{0})r_{0}$ in the
$a-r_{0}$ relationship (inspired by [25, 38]). From equations (117) and (118)
we obtained,

$\displaystyle H(r)\equiv{\cal V}_{\rm int}^{2}\alpha(r)^{2}=h_{0}(r)+\beta
h_{1}(r),$ (120)

where,

$\displaystyle\hskip-50.58878pth_{0}(r)\sim\frac{\sqrt{\text{gs}}M^{2}r^{2}}{\sqrt[10]{N}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{4}}\left(1-\frac{{r_{0}}^{4}}{r^{4}}\right)\lambda_{1}^{2}\lambda_{2}^{4},$
(121)

and

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

where we have defined $\lambda_{1}$ and $\lambda_{2}$ as below:

$\displaystyle\lambda_{1}=\Biggl{[}{g_{s}}\log
NN_{f}\left(3a^{2}-r^{2}\right)(2\log(r)+1)$
$\displaystyle+\log(r)\left(4{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)-24\pi
a^{2}+r^{2}(8\pi-3{g_{s}}{N_{f}})\right)$
$\displaystyle+2{g_{s}}{N_{f}}\left(r^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+18{g_{s}}{N_{f}}\left(r^{2}-3a^{2}(6r+1)\right)\log^{2}(r)\Biggr{]},$
(123)
$\displaystyle\lambda_{2}=\left(-{N_{f}}\log\left(9a^{2}r^{4}+r^{6}\right)+\frac{8\pi}{{g_{s}}}-4{N_{f}}\log\left(\frac{\alpha_{\theta_{1}}\alpha_{\theta_{2}}}{4\sqrt{N}}\right)\right).$

Similarly, by replacing $r$ with $r^{*}$, we obtained $h_{0}(r^{*})$ and
$h_{1}(r^{*})$. Therefore,

$\displaystyle H(r^{*})=h_{0}(r^{*})+\beta h_{1}(r^{*}).$ (125)

Now rewritting $\sigma(r)$ as,

$\displaystyle\sigma(r)=\sigma_{0}(r)+\beta\sigma_{1}(r).$ (126)

From equations (120), (125) and (126), the integrand appearing in equation
(110) can be written 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,

$\displaystyle
h_{0}(r^{*})\sim\frac{M^{2}\left({r^{*}}^{4}-{r_{0}}^{4}\right)\left({g_{s}}{N_{f}}\log\left(9a^{2}{r^{*}}^{4}+{r^{*}}^{6}\right)-\Omega-16\pi\right){}^{4}}{{g_{s}}^{7/2}\sqrt[10]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{4}}$
(128)
$\displaystyle\hskip-325.215pt\times\Biggl{[}\log({r^{*}})\left(6a^{2}({g_{s}}\log
N{N_{f}}-4\pi)+4{g_{s}}{N_{f}}\left({r^{*}}^{2}-3a^{2}\right)\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+{r^{*}}^{2}(8\pi-{g_{s}}(2\log
N+3){N_{f}})\right)$
$\displaystyle\hskip-325.215pt+{g_{s}}{N_{f}}\left(3a^{2}-{r^{*}}^{2}\right)\left(\log
N-2\log\left(\frac{1}{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)\right)+18{g_{s}}{N_{f}}\left({r^{*}}^{2}-3a^{2}(6{r^{*}}+1)\right)\log^{2}({r^{*}})\Biggr{]}.$

  * •

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

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

then,

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

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

$\displaystyle\hskip-21.68121ptl(r^{*})\sim\frac{aM_{\rm UV}{N_{f}^{\rm
UV}}\left(\frac{1}{\log(N)}\right)^{10/3}\sqrt{-2\log\left(\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+\log(N)+\log(16)}}{{g_{s}}^{2}M{N_{f}}^{10/3}{r^{*}}\log({r^{*}})}\stackrel{{\scriptstyle
r^{*}\rightarrow\infty}}{{\longrightarrow}}0,$ (131)

as in [2]. As $r^{*}$ was assumed to be in the UV and as $l(r^{*})$ is a
monotonically dereasing function in the UV, one sees that $l(r^{*})$ attains a
maximum at $r^{*}={\cal R}_{D5/\overline{D5}}^{\rm th}$. One can invert (131)
by solving: $l(r^{*})=\frac{C}{r^{*}\log r^{*}}$ where
$C\sim\frac{a\left(\frac{1}{\log(N)}\right)^{10/3}\sqrt{-2\log\left(\alpha_{\theta_{1}}\alpha_{\theta_{2}}\right)+{\log
N}+\log(16)}}{{g_{s}}^{2}{N_{f}^{\rm UV}}^{7/3}}$. Hence,

$r^{*}=\frac{C}{l(r^{*})W\left(\frac{C}{l(r^{*})}\right)}.$ (132)

  * •

Even though we will not use this in the rest of the paper, but as an academic
curiosity, given that $r^{*}<{\cal R}_{\rm UV}<\left(4\pi
g_{s}N\right)^{\frac{1}{4}}$, if we do not require therefore
$l(r^{*}\rightarrow{\cal R}_{\rm UV})\rightarrow 0$, and expand first w.r.t.
$\log N>\log(r^{*})$ (continuing to assume: $r^{*}>\sqrt{3}a$), then :

$\displaystyle\sqrt{h_{0}}\sim\frac{M_{\rm UV}{N_{f}^{\rm
UV}}{r^{*}}^{3}\log({r^{*}})|2\log N-18\log({r^{*}})|({g_{s}}\log N{N_{f}^{\rm
UV}}-3{g_{s}}{N_{f}^{\rm
UV}}\log({r^{*}})+4\pi)^{2}}{{g_{s}}^{3/4}\sqrt[20]{N}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}.$

One can similarly show that
$l(r^{*})\sim{r^{*}}(\log(N)-9\log({r^{*}}))(\log(N)-3\log({r^{*}}))^{2}$ and
that $l_{\rm max}$ corresponds to:

$\displaystyle r^{*}_{l_{\rm
max}}=\exp\left(\frac{1}{18}\left(-\sqrt{4\log^{2}(N)-36\log(N)+729}+4\log(N)-27\right)\right)$
$\displaystyle=\exp\left(\frac{\log(N)}{9}-1-\frac{9}{\log(N)}-\frac{81}{2\log^{2}(N)}+O\left(\left(\frac{1}{\log(N)}\right)^{3}\right)\right)$
$\displaystyle\sim\sqrt[9]{N}e^{-\frac{81}{2\log^{2}(N)}-\frac{9}{\log(N)}-1}.$
(134)

As a numerical example, for $N=100$, the Fig. 1 has been obtained.

Figure 1: Plot of
${r^{*}}(\log(N)-9\log({r^{*}}))(\log(N)-3\log({r^{*}}))^{2}$-versus-$r^{*}$

Entropy

The connected region entropy catering to small $l$ and large $r$ is:

$\displaystyle\frac{S}{{\cal V}_{1}}=\left\\{\frac{\int_{{r^{*}}}^{{\cal
R}_{\rm
UV}}\frac{\sqrt{\sigma(r)}H(r)}{\sqrt{H(r)-H({r^{*}})}}\,dr}{2{G_{N}}^{(11)}}\right\\}$
(135)

$\displaystyle\frac{S_{\rm connected}^{{\rm UV-finite},\ \beta^{0}}}{2{\cal
V}_{1}}\sim\frac{\log N^{4/3}M_{\rm UV}{N_{f}^{\rm
UV}}^{4/3}\alpha_{\theta_{2}}^{2}\left({r_{0}}^{4}{r^{*}}^{2}\log^{3}({r^{*}})+{r^{*}}^{6}\log^{2}({r^{*}})\right)}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\log({r^{*}})}$
$\displaystyle\approx\frac{\log N^{4/3}M_{\rm UV}{N_{f}^{\rm
UV}}^{4/3}{r^{*}}^{4}\alpha_{\theta_{2}}^{2}\log({r^{*}})}{\sqrt[12]{{g_{s}}}\sqrt[20]{N}\alpha_{\theta_{1}}^{6}};$
$\displaystyle\frac{S_{\rm connected}^{{\rm UV-finite},\beta}}{2{\cal
V}_{1}}\sim\frac{\beta\sqrt[3]{\frac{1}{{g_{s}}}}M_{\rm UV}{r^{*}}^{4}({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th})\left(\frac{1}{\log(N)}\right)^{17/3}}{{g_{s}}^{7/4}\sqrt[20]{N}{N_{f}^{\rm
UV}}^{2/3}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}.$ (136)

From (• ‣ B), noting the extremely non-trivial cancelation of ${\cal
O}(R^{4})$ corrections ${\cal C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th}=0$, we note that there are no ${\cal O}\left({\cal
R}^{4}\right)$-corrections to the connected region (and similarly disconnected
region) entanglement entropy.

The disconnected region entropy will be given by the following expression:

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

Writing $a=\left(\frac{1}{\sqrt{3}}+\epsilon\right)r_{0}$, the contribution in
$r\in\left[r_{0},{\cal R}_{D5/\overline{D5}}\right]$ is given by:

$\displaystyle-\frac{\epsilon{g_{s}}^{5/4}M\sqrt[20]{\frac{1}{N}}{N_{f}}^{8/3}{r_{0}}^{4}(\beta({\cal
C}_{zz}^{\rm th}-2{\cal C}_{\theta_{1}z}^{\rm
th})+2)}{\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}}.$ (138)

Noting that all radial distances are measured in units of ${\cal
R}_{D5/\overline{D5}}^{\rm th}$, i.e., $\log{\cal R}_{D5/\overline{D5}}\equiv
0$, one obtains:

$\displaystyle\frac{S_{\rm disconnected}-S_{\rm UV}^{\rm disconnected}}{2{\cal
V}_{1}}\sim-\frac{{g_{s}}^{5/4}M\sqrt[20]{\frac{1}{N}}{N_{f}}^{8/3}r_{0}^{4}\log^{2}(r_{0})({\log
N}-3\log(r_{0}))^{5/3}}{64\sqrt[3]{2}3^{5/6}\pi^{41/12}\alpha_{\theta_{1}}^{3}\alpha_{\theta_{2}}^{2}};$
$\displaystyle\frac{S_{\rm connected}-S_{\rm UV}^{\rm connected}}{2{\cal
V}_{1}}\sim\frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}\alpha_{\theta_{2}}^{2}\log^{\frac{4}{3}}(N)\left(r_{0}^{4}{r^{*}}^{2}\log^{3}({r^{*}})-{r^{*}}^{6}\log^{2}({r^{*}})\right)}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}{r^{*}}^{2}\alpha_{\theta_{1}}^{6}\log({r^{*}})}$
$\displaystyle\approx-\frac{{M_{\rm UV}}{N_{f}^{\rm
UV}}^{4/3}{r^{*}}^{4}\log^{\frac{4}{3}}(N)\log({r^{*}})}{72\
2^{2/3}3^{5/6}\pi^{25/12}\sqrt[12]{{g_{s}}}\sqrt[20]{N}},$ (139)

where,

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