# QCD-Compatible Supermassive Inert Top-Down Holographic Mesinos at
Intermediate Coupling

Aalok Misra111email-<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India  Gopal
Yadav222email-<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand,
India  Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India

###### Abstract

A longstanding problem with the popular Sakai-Sugimoto holographic dual of
thermal QCD is that the “mesinos”, the (non-supersymmetric) fermionic partners
of the mesons, are nearly isospectral with mesons and have an unsuppressed
mesino-meson interaction, both being in contradiction with actual QCD. We
solve this problem in a UV complete (and different) type IIA string dual at
intermediate coupling of realistic thermal QCD, in which the mesinos are shown
to be much heavier than and non-interacting with mesons (the wave-
function/mass/interaction terms receiving no ${\cal M}$-theory ${\cal
O}(R^{4})$ corrections).

###### Contents

  1. 1 Introduction
  2. 2 Type IIA String Dual of Thermal QCD-Like Theories Inclusive of ${\cal O}(R^{4})$ Corrections
  3. 3 Supermassive Mesinos in Type IIA String Theory
  4. 4 Generation of $N$-hanced Mass Scale for $T<T_{c}$
  5. 5 Non-Interacting Mesinos
  6. 6 Top-Down $m_{\rm quark}\langle{\bar{q}}q\rangle$ Non-Renormalization up to ${\cal O}(R^{4})$
  7. 7 Universality in Particle Wave Functions in the IR
  8. 8 Summary
  9. A Finite Baryon Chemical Potential
  10. B EOM-Related for Massive Mesinos
  11. C $\tilde{z}=$Constant Embedding of Flavor $D6$-Branes Inclusive of ${\cal O}(\beta)$ Corrections
  12. D Constants appearing in the Solution to the Mesino Wave-Function for $T>T_{c}$
  13. E Summary of Applications of Top-Down Holographic QCD [1], [2]

##  1 Introduction

One can construct gauge theories from a stack of $D$-branes and various
configurations of the same. In this context in the spirit of (non-conformal
non-supersymmetric) gauge-gravity duality (inspired by [3]), mostly bosonic
fluctuations on the world volume of $D$-branes have been considered - as a
recent example, inclusive of higher derivative corrections to address the
finite-$N$/intermediate coupling regime, phenomenologically-compatible low
energy coupling constants up to NLO in the chiral expansion in $SU(3)$ chiral
perturbation theory (in the chiral limit) were obtained from the DBI action on
flavor $D6$-branes in [4]. Dirac-like action for the supersymmetric partners
of mesons, the mesinos, has been obtained from a top-down approach on
$Dp$-branes [5], see [6] for the bottom-up approach. However, using the same
for the Sakai-Sugimoto type IIA dual [7] of thermal QCD, it was shown that one
runs into a problem - the mesinos and mesons turn out to be approximately
isospectral and the their interaction is not large-$N$ suppressed [8] \- both
in contradiction with real QCD. This serves as the main motivation for this
paper - to see if this issue can be resolved in the type IIA mirror [1] at
intermediate coupling [2] of the non-supersymmetric UV-complete type IIB dual
[9] of thermal QCD-like theories. In short, we will show that the mesinos are
supermassive and do not interact with the vector/$\pi$ mesons which is why we
refer to them as W(eakly) I(nteracting) S(upermassive) P(articles), thereby
not being in conflict with realistic QCD.

The type IIA dual inclusive of the ${\cal O}(R^{4})$ corrections - to explore
the finite-$N$-limit/intermediate coupling regime of QCD - of the type IIB
dual [9] of thermal QCD-like theories, was worked out in [1] and [2]. In this
paper, we explicitly consider the mesino action on flavor $D6$-branes in the
aforementioned type IIA dual. We would also like to see the effect of higher
derivative terms on the fermions relevant to holographic thermal QCD in this
paper which was missing in [5] \- this leads us to a non-renormalization of
the mesino wave function and mass in the sense that both turn out to be
independent of the ${\cal O}(R^{4})$ terms up to
${\cal}(l_{p}^{6}/N^{\alpha}),\alpha\geq 1$ 333In [2], terms up to ${\cal
O}\left(\frac{\beta^{0}}{N}\right)$ and ${\cal
O}\left(\frac{\beta}{N^{\alpha}}\right),\ 0<\alpha<1,\ \beta\sim l_{p}^{6}$,
were considered., $l_{p}$ being the Planckian length.

The paper is organised as follows. In section 2, we discuss the type IIA
string dual construction of thermal QCD-like theories at intermediate
coupling. In section 3, we show that fermionic superpartner of mesons, i.e.,
mesionos are superheavy due to the generation of $N$-enhanced mass scale
discussed in section 4. Section 5 provides further evidence of superheavy
mesinos because of absence of mesino-mesino-meson interaction in type IIA
string dual. In section 6, we discuss the non-renormalization of the product
of quark mass and quark condensate up to ${\cal O}(R)^{4}$ . Section 7 has a
discussion of wave-function universality in the context of glueball, meson and
graviton wavefunctions. The summary of the paper has been provided in section
8.

There are five appendices. Appendix A contains the discussion of baryon
chemical potential. Appendix B consists of quantities appearing in the mesino
EOMs of section 3. In Appendix C, we compute the embedding of flavor
$D6$-branes in type IIA string theory inclusive of ${\cal O}(R^{4})$
corrections. We list the constants appearing in the wave function for the
black hole background in appendix D. Finally, we summarize the top-down
holographic QCD results obtained by our group in appendix E.

##  2 Type IIA String Dual of Thermal QCD-Like Theories Inclusive of ${\cal
O}(R^{4})$ Corrections

Thermal QCD-like theories refer to the equivalence class of theories that are
IR confining and UV conformal with the “quarks” transforming in the
fundamental representations of the symmetry group(color and flavor). The UV-
complete type IIB string dual of such large-$N$ thermal QCD-like theories was
constructed in [9]. The brane picture consists of $N$ space-time filling
$D3$-branes at the tip of a warped resolved conifold, $M$ space-time filling
$D5$ branes also at the tip of the aforementioned conifold wrapping the
vanishing squashed $S^{2}$ and at the North Pole of the resolved squashed
$S^{2}$ of radius $a$ (resolution parameter), and space-time filling
$\overline{D5}$-branes also at the tip of the conifold wrapping the
abovementioned vanishing squashed $S^{2}(\theta_{1},\phi_{1})$ and at the
South Pole of the resolved squashed $S^{2}(\theta_{2},\phi_{2})$. In addtion,
there are $N_{f}$ space-time filling flavor $D7$-branes wrapping the vanishing
sqaushed $S^{3}(\theta_{1},\phi_{1},\psi)$ as well as being at the North Pole
of the squashed resolved $S^{2}(\theta_{2},\phi_{2})$, dipping into the IR up
to $|\mu_{\rm Ouyang}|^{\frac{2}{3}},\ |\mu_{\rm Ouyang}|$ being the modulus
of the Ouyang embedding parameter in the Ouyang embedding of the flavor
$D7$-branes:
$\left(9a^{2}r^{4}+r^{6}\right)^{1/4}e^{\frac{i}{2}(\psi-\phi_{1}-\phi_{2})}\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}=\mu_{\rm
Ouyang}$. An equal number of $\overline{D7}$ wrapping the vanishing squashed
$S^{3}(\theta_{1},\phi_{1},\psi)$ and at the South Pole of the blown-up
squashed $S^{2}(\theta_{2},\phi_{2})$, are also present. Equal number of
$D5/D7$-branes and $\overline{D5}/\overline{D7}$-branes in the UV ensure UV
conformality. The presence of $N_{f}$ flavor $D7$ and $\overline{D7}$-branes
in the UV, implies a flavor gauge group $SU(N_{f})\times SU(N_{f})$ in the UV
which is broken to $SU(N_{f})$ due to absence of $\overline{D7}$-branes in the
IR 444On the gravity dual side we characterize UV($r>{\cal
R}_{D5/\overline{D5}}$) and IR ($r<{\cal R}_{D5/\overline{D5}}$) in term of
radial coordinate where ${\cal R}_{D5/\overline{D5}}$ is the boundary between
UV and IR, and separation between $D5$ and $\overline{D5}$-branes. (analog of
chiral symmetry breaking in this brane setup). IR confinement in the gravity
dual is effected by deforming the vanishing squashed $S^{3}$ in the conifold.
Since we are interested in finite temperature QCD, the same is effected via a
black hole ($T>T_{c}$) and thermal ($T<T_{c}$) backgrounds on the gravity dual
side. Due to finite temperature and finite separation of $D5$ and
$\overline{D5}$-branes on the brane side, the conifold further needs to also
possess an $S^{2}$-blow-up/resolution (with radius/resolution parameter $a$).
Further, the ten-dimensional warp factor and fluxes include the effect of
back-reaction. Therefore, we conclude that string dual of thermal QCD-like
theories in the large-$N$ limit, involves a warped resolved deformed conifold.
The additional advantage of the type IIB dual of [9] is that in the IR, at the
end of a Seiberg-like duality cascade, the number of colors $N_{c}$ gets
identified with $M$, which in the MQGP limit ($g_{s}\sim\frac{1}{{\cal
O}(1)},M,N_{f}\equiv{\cal O}(1),N\gg 1,\frac{g_{s}M^{2}}{N}\ll 1$) [1], can be
tuned to equal 3 [4].

Now to explore the intermediate coupling regime, the ${\cal O}(R^{4})$ terms
in eleven-dimensional supergravity action were considered in [2]. ${\cal
M}$-theory uplift was obtained in two steps: the type IIA Strominger-Yau-
Zaslow (SYZ)s mirror of type IIB setup was first obtained, and then the former
was uplifted to ${\cal M}$-theory. To obtain type IIA SYZ mirror of type IIB
setup, a triple T-duality was performed along a local special Lagrangian
(sLag) $T^{3}(x,y,z)$ where $(x,y,z)$ are the toroidal analogs of
$(\phi_{1},\phi_{2},\psi)$ \- which could be identified with the
$T^{2}$-invariant sLag of [10] \- in the large-complex structure limit
effected by making the base ${\cal B}(r,\theta_{1},\theta_{2})$ (of a
$T^{3}(\phi_{1},\phi_{2},\psi)$-fibration over ${\cal
B}(r,\theta_{1},\theta_{2})$) large [1], [11]. Hence, all the color and flavor
$D$-branes get T-dualized to color and flavor $D6$-branes. The ${\cal
M}$-theory uplift metric [1], [2] (finite-but-large-$N$/intermediate coupling)
of [9] (UV-complete type IIB holographic dual of large-$N$ thermal QCD-like
theories) is expressed in the following form:

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

where the type IIA RR 1-forms, $A_{\rm IIA}^{F^{\rm IIB}_{i=1,3,5}}$ are
obtained from type IIB $F_{1,3,5}^{\rm IIB}$ fluxes via the SYZ mirror of type
IIB string dual [9], $g(r)=1-\frac{r_{h}^{4}}{r^{4}}$, and $\phi^{\rm IIA}$ is
the type IIA dilaton profile. For low temperatures, i.e., $T<T_{c}$, the
thermal gravitational dual is given by:

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

where $\tilde{g}(r)=1-\frac{r_{0}^{4}}{r^{4}}$. One notes that $t\rightarrow
x^{3},\ x^{3}\rightarrow t$ in (1) followed by a Double Wick rotation in the
new $x^{3},t$ coordinates obtains (2); $h(r,\theta_{1,2})$ is the ten-
dimensional warp factor [9, 1]. This is equivalent 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 [12], [2] (See [13] in the context of
Euclidean/black $D4$-branes in type IIA). In (2), 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 $S^{2}_{\rm squashed}$
[14]) and their world volume given by $\mathbb{R}^{2}(x^{1,2})\times
S^{1}(x^{3})$ with the period of $S^{1}(x^{3})$ given by a very large:
$\frac{2\pi}{M_{\rm KK}}$, where the very small $M_{\rm KK}$ is given by
$\frac{2r_{0}}{L^{2}}\left[1+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)\right]$, $r_{0}$ being the very small IR
cut-off in the thermal background (see also [15]) 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 for the thermal background
corresponding to $T<T_{c}$ will involve setting $\tilde{g}(r)$ to unity in
(2).

Eleven dimensional supergravity action including ${\cal O}(R^{4})$ terms used
in [2] is:

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

where:

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

The equations of motion for metric and three form potential $C$ are:

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

where [16]:

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

$R_{MNPQ},R_{MN},{\cal R}$ in (2)/(2) are elven-dimensional Riemann curvature
tensor, Ricci tensor and the Ricci scalar. To solve (2), the following ansatz
was made:

$\displaystyle\hskip-57.81621ptg_{MN}=g_{MN}^{(0)}+\beta g_{MN}^{(1)},$
$\displaystyle\hskip-57.81621ptC_{MNP}=C^{(0)}_{MNP}+\beta C_{MNP}^{(1)}.$ (7)

EOM for $C_{MNP}$ symbolically can be written as:

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

It was shown in [2], that, $C^{(1)}_{MNP}=0$ up to ${\cal O}(\beta)$.
Therefore only the metric receives ${\cal O}(R^{4})$ corrections defined as:

$\displaystyle\delta g_{MN}=\beta g^{(1)}_{MN}=G_{MN}^{\rm MQGP}f_{MN}(r).$
(9)

In general, the ${\cal M}$ theory metric has the following form including
${\cal O}(R^{4})$ corrections:
$G_{MN}^{\cal M}=G_{MN}^{\rm MQGP}\left(1+\beta f_{MN}(r)\right).$ The EOMs
for $f_{MN}(r)$ were solved in [2]. The type IIA metric inclusive of ${\cal
O}(R^{4})$ corrections was obtained from the ${\cal M}$-theory metric by
descending back to type IIA string theory which has the following form:
$G_{mn}^{\rm IIA}=\sqrt{G_{x^{10}x^{10}}^{\cal M}}G_{mn}^{\rm
MQGP}\left(1+\frac{f_{x^{10}x^{10}(r)}}{2}+f_{mn}(r)\right).$

The type IIB dual of large-$N$ thermal QCD-like theories as constructed in [9]
and its type IIA mirror as constructed in [1], [11] were successfully used to
study a variety of issues in Condensed Matter Physics, lattice/PDG-compatible
particle phenomenology, doubly holographic extension and Page curves of
associated eternal black holes and $G$/(Almost)Contact(3)Metric structure
classification of underlying six-, seven- and eight-folds in differential
geometry (see E).

##  3 Supermassive Mesinos in Type IIA String Theory

The fermionic sector of the type IIA holographic dual of QCD as constructed in
[7] has the following problems. Not only are the mesinos approximately
isospetral with the mesons, the single-meson-mesino-mesino interaction terms
are not large-$N$ suppressed [8]. Evidently, this is in contradiction with
QCD/PDG as no mesino at the EW scale has thus far been observed. What we show
in this section is that Dirichlet/Neumann boundary condition at the the IR
cut-off (for the gravity dual corresponding to $T<T_{c}$) or the horizon
radius (for the gravity dual corresponding to $T>T_{c}$), is consistent with
having a supermassive mesino. Further, we show an $N$-enhancement of the
Kaluza-Klein mass scale via an $N$-enhancement of the resolution parameter for
the thermal background ($T<T_{c}$) hence providing the mechanism of generation
of the aforementioned supermassive mesino. Even though we have not been able
to provide in 3 an analog of the $N$-enhancement of the resolution parameter
(that was seen in the thermal background corresponding to $T<T_{c}$) for the
black hole background corresponding to $T>T_{c}$, the following should be
noted. In 3, what we were able to show for the gravity duals of both, the low
and high temperature QCD-like theories, is that Dirichlet/Neumann boundary
condition at the IR cut-off, horizon radius respectively in the gravity duals
for $T<T_{c},\ T>T_{c}$, do not fix the mesino mass. We can hence take the
same to be large and via the aforementioned $N$-enhancement of the resolution
parameter in the former, we had explicitly shown the mechanism of obtaining
supermassive mesinos in the thermal background. Given that we were able to
show the vanishing of meson-mesino-mesino interaction in 5, even if the
mesinos were of the EW scale, there still will be no contradiction with real
QCD.

The DBI action for the fermions on flavor D6-branes has the following
structute [5]:

$\displaystyle S_{D_{6}}^{f}=\frac{T_{D_{6}}}{2}\int d^{7}\xi e^{-\Phi^{\rm
IIA}}\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm IIA})}\
\overline{\Theta}\left(1-\Gamma_{D_{6}}\right)\left(\Gamma^{\alpha}D_{\alpha}-\Delta+L_{D_{6}}\right)\Theta,$
(10)

where $\Phi^{\rm IIA}$ is the type IIA dilaton. We can
define:${\mathcal{F}^{\rm
IIA}_{\alpha_{1}\alpha_{2}}}=i^{*}B_{\alpha_{1}\alpha_{2}}^{\rm
IIA}+F_{\alpha_{1}\alpha_{2}}^{\rm IIA}$ such that
$B_{\alpha_{1}\alpha_{2}}^{\rm IIA}$ and $F_{\alpha_{1}\alpha_{2}}^{\rm IIA}$
are NS-NS B field and gauge field restricted to the world volume of
$D6$-branes. Further, $\Gamma_{D_{6}}$ and $L_{D_{6}}$ appearing in (10) are
defined as 555Indices, $m,n,p$ correspond to type IIA bulk indices and
$\alpha_{i},\beta_{i},\gamma$ etc. correspond to indices on world-volume of
flavor $D6$-branes.:

$\displaystyle\Gamma_{D_{6}}=\sum_{q+r=3}\frac{(-)^{r+1}\left(\Gamma_{10}\right)^{r+1}\epsilon^{\alpha_{1}....\alpha_{2q}\beta_{1}.....\beta_{2r+1}}}{q!(2r+1)!2^{q}\sqrt{-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm IIA})}}{\mathcal{F}^{\rm
IIA}_{\alpha_{1}\alpha_{2}}}.....\mathcal{F}^{\rm
IIA}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}.....\beta_{2r+1}},$
$\displaystyle\Delta=\Delta^{(1)}+\Delta^{(2)},$ $\displaystyle
L_{D_{6}}=\sum_{q\geq
1,q+r=3}\frac{(-)^{r+1}\left(\Gamma_{10}\right)^{r+1}\epsilon^{\alpha_{1}....\alpha_{2q}\beta_{1}.....\beta_{2r+1}}}{q!(2r+1)!2^{q}\sqrt{-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}{\mathcal{F}_{\alpha_{1}\alpha_{2}}}.....\mathcal{F}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}.....\beta_{2r+1}}^{\
\ \ \ \ \ \ \ \ \ \gamma}D_{\gamma},$ (11)

where $D_{m}=D_{m}^{(0)}+W_{m}$, and

$\displaystyle
D_{m}^{(0)}=\nabla_{m}+\frac{1}{4.2!}H_{mnp}\Gamma^{np}\Gamma_{(10)},$
$\displaystyle W_{m}=-\frac{1}{8}e^{\Phi^{\rm
IIA}}\left(\frac{1}{2}F_{np}\Gamma^{np}\Gamma_{(10)}+\frac{1}{4!}F_{npqr}\Gamma^{npqr}\right)\Gamma_{m},$
$\displaystyle\Delta^{(1)}=\frac{1}{2}\left(\Gamma^{m}\partial_{m}\Phi^{\rm
IIA}+\frac{1}{2.3!}H_{mnp}\Gamma^{mnp}\Gamma_{(10)}\right),$
$\displaystyle\Delta^{(2)}=\frac{1}{8}e^{\Phi^{\rm
IIA}}\left(\frac{3}{2!}F_{mn}\Gamma^{mn}\Gamma_{(10)}-\frac{1}{4!}F_{mnpq}\Gamma^{mnpq}\right),$
(12)

where covariant derivative is defined as:
$\nabla_{m}=\partial_{m}+\frac{1}{4}\Omega_{m}^{\underline{np}}\Gamma_{\underline{np}}$.
$F_{mn}$ and $F_{mnpq}$ are field strength tensors corresponding to type IIA
$A_{n}$ and $A_{npq}$, and $H_{mnp}=\partial_{[m}B_{np]}$. For flavor
$D6$-branes in type IIA string theory

$\displaystyle\Gamma_{\rm
D6}=\frac{\epsilon^{\beta_{1}.....\beta_{7}}\Gamma_{\beta_{1}.....\beta_{7}}}{\sqrt{-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}-\frac{\Gamma_{(10)}\left(\epsilon^{\alpha_{1}\alpha_{2}\beta_{1}......\beta_{5}}{\cal
F}_{\alpha_{1}\alpha_{2}}^{\rm
IIA}\Gamma_{\beta_{1}......\beta_{5}}\right)}{5!\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm IIA})}}$
$\displaystyle+\frac{\epsilon^{\alpha_{1}...\alpha_{4}\beta_{1}\beta_{2}\beta_{3}}\mathcal{F}_{\alpha_{1}\alpha_{2}}^{\rm
IIA}\mathcal{F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}}{48\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm
IIA})}}-\frac{\Gamma_{(10)}\left(\epsilon^{\alpha_{1}....\alpha_{6}\beta_{1}}{\cal
F}_{\alpha_{1}\alpha_{2}}^{\rm IIA}{\cal F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}{\cal F}_{\alpha_{5}\alpha_{6}}^{\rm
IIA}\Gamma_{\beta_{1}}\right)}{48\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm IIA})}},$ $\displaystyle L_{\rm
D6}=-\frac{\Gamma_{(10)}\left(\epsilon^{\alpha_{1}\alpha_{2}\beta_{1}......\beta_{5}}{\cal
F}_{\alpha_{1}\alpha_{2}}^{\rm IIA}\Gamma_{\beta_{1}...\beta_{5}}^{\ \ \ \ \ \
\gamma}D_{\gamma}\right)}{240\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm
IIA})}}+\frac{\epsilon^{\alpha_{1}...\alpha_{4}\beta_{1}\beta_{2}\beta_{3}}\mathcal{F}_{\alpha_{1}\alpha_{2}}^{\rm
IIA}\mathcal{F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}^{\ \ \ \ \ \
\gamma}D_{\gamma}}{48\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}.$

The Dirac equation for the DBI action for the fermions on flavor $D6$-branes
appearing in type IIA string dual of thermal QCD-like theories turns out to
be:

$\displaystyle\hskip-14.45377pt\Biggl{[}{\cal
A}-\frac{\epsilon^{\alpha_{1}\alpha_{2}\beta_{1}.....\beta_{5}}\mathcal{F}^{\rm
IIA}_{\alpha_{1}\alpha_{2}}\Gamma_{\beta_{1}.....\beta_{5}}^{\ \ \ \ \ \ \ \
\gamma}D_{\gamma}\Gamma_{(10)}}{240\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm
IIA})}}-\frac{\epsilon^{\beta_{1}.....\beta_{7}}\Gamma_{\beta_{1}.....\beta_{7}}{\cal
A}}{7!\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{4\Gamma^{\beta_{1}.....\beta_{7}}\mathcal{F}^{\rm
IIA}_{\beta_{6}\beta_{7}}\Gamma_{\beta_{1}.....\beta_{5}}^{\ \ \ \ \ \ \ \
\gamma}D_{\gamma}\Gamma_{(10)}}{7!\left(-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm IIA})\right)}}$
$\displaystyle+\frac{\epsilon^{\alpha_{1}\alpha_{2}\beta_{1}.....\beta_{5}}\mathcal{F}^{\rm
IIA}_{\alpha_{1}\alpha_{2}}\Gamma_{\beta_{1}.....\beta_{5}}\Gamma_{(10)}{\cal
A}}{5!\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{7!\mathcal{F}^{\rm
IIA}_{\alpha_{1}\alpha_{2}}\mathcal{F}_{\rm
IIA}^{\alpha_{1}\alpha_{2}}\Gamma^{\beta_{1}.....\beta_{5}}\Gamma_{\beta_{1}.....\beta_{5}}^{\
\ \ \ \ \ \ \ \gamma}D_{\gamma}\Gamma_{(10)}}{5!240\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})\right)}}+\frac{\epsilon^{\alpha_{1}...\alpha_{4}\beta_{1}\beta_{2}\beta_{3}}\mathcal{F}_{\alpha_{1}\alpha_{2}}^{\rm
IIA}\mathcal{F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}{\cal A}}{48\sqrt{-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm IIA})}}$
$\displaystyle-{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{\Gamma_{(10)}{\cal
F}_{\rm IIA}^{2}{\cal F}_{\beta_{4}\beta_{5}}^{\rm
IIA}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}\Gamma^{\beta_{1}.....\beta_{5}\gamma}D_{\gamma}}{480\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})\right)}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{7!{\cal
F}_{\rm
IIA}^{4}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}\Gamma^{\beta_{1}\beta_{2}\beta_{3}\gamma}D_{\gamma}}{48\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})\right)}}-\frac{\Gamma_{(10)}\left(\epsilon^{\alpha_{1}....\alpha_{6}\beta_{1}}{\cal
F}_{\alpha_{1}\alpha_{2}}^{\rm IIA}{\cal F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}{\cal F}_{\alpha_{5}\alpha_{6}}^{\rm IIA}\Gamma_{\beta_{1}}\right){\cal
A}}{48\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm IIA})}}$
$\displaystyle\hskip-21.68121pt+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{\delta^{[\alpha_{3}}_{[\beta_{2}}\delta^{\alpha_{4}}_{\beta_{3}}\delta^{\alpha_{5}}_{\beta_{4}}\delta^{\alpha_{6}]}_{\beta_{5}]}{\cal
F}_{\rm IIA}^{2}{\cal F}_{\alpha_{3}\alpha_{4}}^{\rm IIA}{\cal
F}_{\alpha_{5}\alpha_{6}}^{\rm
IIA}\Gamma_{\beta_{1}}\Gamma^{\beta_{1}...\beta_{5}\gamma}D_{\gamma}}{5760\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})\right)}}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{{\cal
F}_{\rm IIA}^{4}{\cal F}_{\beta_{2}\beta_{3}}^{\rm
IIA}\Gamma_{\beta_{1}}\Gamma^{\beta_{1}\beta_{2}\beta_{3}\gamma}D_{\gamma}}{24\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})\right)}}+\frac{\epsilon^{\alpha_{1}...\alpha_{4}\beta_{1}\beta_{2}\beta_{3}}\mathcal{F}_{\alpha_{1}\alpha_{2}}^{\rm
IIA}\mathcal{F}_{\alpha_{3}\alpha_{4}}^{\rm
IIA}\Gamma_{\beta_{1}\beta_{2}\beta_{3}}^{\ \ \ \ \ \
\gamma}D_{\gamma}}{48\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm
IIA})}}$
$\displaystyle-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{\delta_{[\alpha_{1}}^{[\beta_{4}}\delta_{\alpha_{2}}^{\beta_{5}}\delta_{\alpha_{3}}^{\beta_{6}}\delta_{\alpha_{4}]}^{\beta_{7}]}{\cal
F}_{\rm IIA}^{2}{\cal F}_{\rm IIA}^{\alpha_{1}\alpha_{2}}{\cal F}_{\rm
IIA}^{\alpha_{3}\alpha_{4}}\Gamma^{\beta_{1}\beta_{2}\beta_{3}}\Gamma_{\beta_{1}...\beta_{7}}^{\
\ \ \ \ \ \ \gamma}D_{\gamma}}{48\left(-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}^{\rm
IIA})\right)}}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{{\cal
F}_{\rm IIA}^{2}{\cal F}_{\rm
IIA}^{\beta_{4}\beta_{5}}\Gamma_{\beta_{1}...\beta_{5}}\Gamma^{\beta_{1}\beta_{2}\beta_{3}\gamma}D_{\gamma}}{120\left(-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}^{\rm IIA})\right)}}\Biggr{]}\Theta=0,$ (14)

where

$\displaystyle{\cal
A}=\Gamma^{\alpha}D_{\alpha}-\frac{1}{2}\left(\Gamma^{m}\partial_{m}\Phi^{\rm
IIA}+\frac{1}{12}H_{mnp}\Gamma^{mnp}\Gamma_{(10)}\right)-\frac{1}{8}e^{\Phi^{\rm
IIA}}\left(\frac{3}{2}F_{mn}\Gamma^{mn}\Gamma_{(10)}-\frac{1}{4!}F_{mnpq}\Gamma^{mnpq}\right).$

In (3), $F_{mnpq}$ is the type IIA RR four-form field strength. This in our
computation is set to zero as one can show that the same can not be generated
by a triple T dual of the RR $F^{IIB}_{1,3,5}$ [1]. Type IIA NS-NS $B$ is
given by [17]:

$\displaystyle\hskip-28.90755ptB^{\rm
IIA}\left(\theta_{1}=\frac{\alpha_{\theta_{1}}}{N^{\frac{1}{5}}},\theta_{2}\sim\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}\right)=d\theta_{2}\wedge
d\tilde{x}\left(-\frac{2\sqrt[4]{\pi}\sqrt[4]{{g_{s}}}N^{3/4}\left(3\sqrt{6}\alpha_{\theta_{1}}^{3}-2\alpha_{\theta_{1}}^{2}\sqrt[5]{N}+2\alpha_{\theta_{2}}^{2}\right)}{27\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}}\right)$
$\displaystyle\hskip-28.90755pt+d\theta_{2}\wedge
d\tilde{y}\left(\frac{2\sqrt[4]{\pi}\sqrt[4]{{g_{s}}}N^{3/4}\left(3\sqrt{6}\alpha_{\theta_{1}}^{3}-2\alpha_{\theta_{1}}^{2}\sqrt[5]{N}+2\alpha_{\theta_{2}}^{2}\right)}{27\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}}\right)+d\theta_{2}\wedge
d\tilde{z}\left(-\frac{\sqrt[4]{\pi}\alpha_{\theta_{2}}\sqrt[4]{{g_{s}}}N^{3/20}\left(2\left(\sqrt[3]{3}-1\right)\alpha\sqrt[10]{N}+\sqrt[3]{3}\alpha_{\theta_{2}}\right)}{3^{5/6}\alpha\sqrt{\alpha_{\theta_{2}}^{2}}}\right).$

When we restrict to the world-volume of $D6$-branes then only non-trivial
component that survive will be $B^{\rm IIA}_{\theta_{2}\tilde{y}}$. Induced
metric on the world-volume of $D6$-branes can be obtained from target space
metric.

$\displaystyle ds_{\rm D6}^{2}=ds^{2}_{5}+g^{\rm
IIA}_{\theta_{2}\theta_{2}}d\theta_{2}^{2}+g^{\rm
IIA}_{\theta_{2}\tilde{y}}d\theta_{2}d\tilde{y}+g^{\rm
IIA}_{\tilde{y}\tilde{y}}d\tilde{y}^{2}.$ (17)

In general, type IIA metric is not diagonal in the basis $(x,y,z)$. Sine we
need the metric component along $\tilde{y}$-direction therefore we are writing
the metric in diagonal basis in subspace $(\tilde{x},\tilde{y},\tilde{z})$
[17].

$\displaystyle
ds^{2}=\frac{2d\tilde{x}^{2}\left(9\sqrt{2}\sqrt[6]{3}\alpha_{\theta_{1}}N^{4/5}-2\
3^{2/3}N\right)}{27\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{2}}+\frac{2{d\tilde{y}}^{2}\left(2\
3^{2/3}N-9\sqrt{2}\sqrt[6]{3}\alpha_{\theta_{1}}N^{4/5}\right)}{27\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{2}}$
$\displaystyle+\frac{2d\tilde{z}^{2}\left(3^{2/3}\alpha_{\theta_{1}}^{2}N^{3/5}+3^{2/3}\alpha_{\theta_{2}}^{2}N^{2/5}\right)}{27\alpha^{2}\alpha_{\theta_{2}}^{2}}.$
(18)

$ds^{2}_{5}$ in (17) is non-compact metric listed along $(t,x^{1,2,3},r)$
subspace, and from (3), $g^{\rm IIA}_{\theta_{2}\tilde{y}}=0$ and $g^{\rm
IIA}_{\tilde{y}\tilde{y}}=\frac{2\left(2\
3^{2/3}N-9\sqrt{2}\sqrt[6]{3}\alpha_{\theta_{1}}N^{4/5}\right)}{27\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{2}}$.

Consider the DBI action on the world-volume of flavor $D6$-branes
$S_{\rm DBI}^{\rm D6}=-T_{D6}N_{f}\int_{\Sigma_{(7)}}\sqrt{-{\rm
det}(i^{*}\left(g^{\rm IIA}+B^{\rm IIA}\right)+F^{\rm IIA})}$,
$i:\Sigma_{(7)}\cong S^{1}_{t}\times_{w}\mathbb{R}^{3}\times\mathbb{R}_{\geq
0}\times_{w}S^{2}_{\rm squashed}(a)\hookrightarrow M_{10}$ [the embedding of
the flavor $D6$-branes in the ten-dimensional background involving a warped
squashed resolved conifold] in the $\psi=2n\pi,n=0,1,2$-coordinate patches and
vanishingly small Ouyang embedding parameter in the parent type IIB dual.
Using the induced metric on the flavor $D6$-branes as given in (17), NS-NS
$B^{\rm IIA}$ as given in (3) and turning on a baryon chemical potential (by
looking at the DBI action in the UV and solving for $A_{t}(r)$ \- see (A))
corresponding to $U(1)$ sub-group of $U(N_{f})$ with the associated field
strength $F_{rt}=A_{t}^{\prime}(r)$, the background $A_{t}(r)$ can be obtained
(see appendix A). In the IR, ${\cal L}_{\rm DBI,\ on-shell}^{\rm D6}$, for
$N\sim 10^{2}$, can be shown to be infinitesimal.

The coefficient of the most dominant (quadratic) powers of $\frac{1}{{\cal
L}_{\rm DBI,\ on-shell}^{\rm D6}}$ in (3) marked in blue, is proportional to
$\Gamma^{\gamma}D_{\gamma}\Theta,\gamma\in\left.\left\\{t,x^{1,2,3},r,\theta_{2},\tilde{y}\right\\}\right|_{\left\\{\tilde{x}=0,\tilde{z}={\rm
constant}\right\\}}$ where $(\tilde{x},\tilde{y},\tilde{z})$ diagonalize
$T^{3}(x,y,z)$ of 2. One can further show:
$\left.E^{\gamma}_{\underline{a}}\Gamma^{\underline{a}}D_{\gamma}\Theta\right|_{N=100}\approx
0.$ The non-Kähler six-fold $M_{6}=S^{1}_{t}\times_{w}{\cal T}$ ($\times_{w}$
implying a warped product), $S^{1}_{t}$ being the thermal circle and ${\cal
T}$ \- deformed $T^{1,1}$ \- being the base of a warped non-Kähler squashed
resolved conifold, was shown to possess an $SU(3)$ structure in [2], with
another “transverse” $SU(3)$ structure induced from the (Almost) Contact
Metric Structure arising from the $G_{2}$ structure of warped product of the
${\cal M}$-theory circle and $M_{6}$. Further, the non-Kähler warped squashed
resolved conifold $\tilde{M}_{6}$ in the type IIA dual also possesses an
$SU(3)$ structure [11], [2]. Either way one is therefore guaranteed the
existence of a pair of globally defined spinors $\Theta_{1,0}$ and
$\Theta_{2,0}$ (either by looking at the $SU(3)$ and the “transverse” $SU(3)$
structures on $M_{6}/\tilde{M}_{6}$, or when considering the embedding of the
$D6$-brane world volume $\Sigma_{(7)}\cong
S^{1}_{t}\times_{w}(\mathbb{R}^{3}\times\mathbb{R}_{\geq
0})\times_{w}S^{2}_{\rm squashed}$ in $M_{10}$ considered either as
$(S^{1}_{t}\times_{w}\mathbb{R}^{3})\times_{w}\tilde{M}_{6}$ or
$\mathbb{R}^{3}\times_{w}(\mathbb{R}_{\geq 0}\times M_{6})$). Making an
ansatz:

$\displaystyle\Theta_{i}(x^{\mu},y^{m})=\Theta_{i}(t,x^{1},r,\theta_{2})=\sum_{n:-\infty}^{\infty}T_{n}(t)e^{-\sqrt{-1}px^{1}}R_{n,i}(r)\left(1+\beta
f_{i}(\theta_{2})\right)\Theta_{i,0},i=1,2,$ (19)

$\beta\sim l_{p}^{6}$ ($l_{p}$ being the Planckain length) and assuming
$T_{n}(t)=e^{i(2n+1)\pi Tt}$ (as one imposes anti-periodic boundary conditions
on the fermions along the thermal circle thereby breaking supersymmetry [18])
implying $\Theta(t+1/T,r)=-\Theta(t,r)$, (and after a double Wick rotation
along $t,x^{1}$, $p^{2}=-m_{\rm Mesino}^{2}$ with $m_{\rm Mesino}$ being the
non-supersymmetric mesino mass in the holographic dual of ${\rm QCD}_{\rm
Mesino}$) analogous to the relation between the killing spinors
$\epsilon_{1,2}$ for a supersymmetric $D6$-brane in flat space:
$\epsilon_{1}=\Gamma^{\underline{8}\underline{9}\
\underline{10}}\epsilon_{2}$, we will impose, by hand, and for our non-
supersymmetric model: $\Theta_{1,0}=\Gamma^{\underline{6}\underline{8}\
\underline{10}}\Theta_{2,0},$ in a curved space, where
$\Theta_{1,2}\equiv\frac{1}{2}\left({\bf 1}+/-\Gamma^{(10)}\right)\Theta$,
$\Gamma^{(10)}\equiv\prod_{\underline{A}=\underline{0}}^{\underline{9}}\Gamma^{\underline{A}}$;
$\underline{A}$, with $A=1,...,10$ denoting the ten-dimensional tangent space
indices.

The most dominant spin-connection terms in the IR is contained in
$E^{r}_{\underline{5}}\Gamma^{\underline{5}}D_{r}\Theta$, in particular
$\omega_{r}^{\underline{7}\ \underline{10}}/\omega_{r}^{\underline{8}\
\underline{10}}$ respectively for the thermal(“TH”), black-hole (“BH”)
backgrounds. Consequently, substituting (19) into $\Theta$’s EOM (details
given in this section and B), the same at ${\cal O}(\beta)$ is:

$\displaystyle\Biggl{[}i(2n+1)\pi
TR_{2,n}(r)f_{2}(\theta_{2})+\frac{E^{\theta_{2}}_{\underline{7}}}{E^{t}_{\underline{1}}}\Gamma^{\underline{1}\underline{7}}R_{2,n}(r)f_{2}\
^{\prime}(\theta_{2})+\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}\Gamma^{\underline{1}\underline{5}}R^{\prime}_{2,n}(r)f^{\rm
BH}_{2}(\theta_{2})-ip\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}\Gamma^{\underline{1}\underline{2}}R_{2,n}(r)f_{2}(\theta_{2})\Biggr{]}\Theta_{2,0}$
$\displaystyle-{\cal
J}(r)R_{1,n}(r)f_{1}(\theta_{2})\Gamma^{\underline{15678}}\Theta_{1,0}=0,$
(20)

with ${\cal J}\equiv\omega_{r}^{\underline{7}\
\underline{10}}\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}$
($E_{\underline{a}}^{M}$ being the frames:
$E_{\underline{a}}^{M}g_{MN}E^{N}_{\underline{b}}=\eta_{\underline{ab}}$) for
the TH background; for the BH background, $\Gamma^{\underline{15678}}$ in the
second line of (3) is to be replaced by $\Gamma^{\underline{156}}$ with ${\cal
J}\equiv\omega_{r}^{\underline{8}\
\underline{10}}\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}$. Note, we
have disregarded all ${\cal O}\left(\frac{\beta}{N^{\alpha}}\right),\
\alpha\geq 1$ terms (see footnote 1) and therefore there are no $\beta$
corrections in
$\frac{E^{\theta_{2}}_{\underline{7}}}{E^{t}_{\underline{1}}},\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}},\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}$.
One thus sees that the only consistent solution for $f_{i}(\theta_{2})$ is
$f_{i}(\theta_{2})=0$ for the TH/BH backgrounds.

Defining $u\equiv\sqrt{r-r_{0}}$, the EOM for $R_{n,2}(r)$ for the TH type IIA
background with $\Gamma^{\underline{15}}\Theta_{2,0}=\Theta_{2,0},\
\Gamma^{\underline{12}}\Theta_{2,0}=\Theta_{2,0},$ can be recast into a
Schrödinger-like equation (where, $a_{1},b_{1},{\cal A}_{\Theta_{2}},{\cal
B}_{\Theta_{2}},{\cal A}_{\Theta_{2}^{\prime}},{\cal B}_{\Theta_{2}^{\prime}}$
are defined in (• ‣ B)):

$\displaystyle\chi_{2,n}^{\prime\prime}(u)+V(u)\chi_{2,n}(u)=0,$ (21)

where,

$\displaystyle V(u)=-\frac{3}{4u^{2}}+\frac{{\cal
A}_{{\Theta_{2}}^{\prime}}}{a_{1}u}-\frac{{\cal
A}_{{\Theta_{2}}^{\prime}}}{a_{1}^{2}}+{\cal O}(u),$ (22)

and,

$R_{2,n}(u)=\sqrt{u}\left({a_{1}}+{b_{1}}u^{2}\right)^{-\frac{{\cal
B}_{{\Theta_{2}}^{\prime}}}{2{b_{1}}}}e^{-\frac{{\cal
A}_{{\Theta_{2}}^{\prime}}\tan^{-1}\left(\frac{\sqrt{{b_{1}}}u}{\sqrt{{a_{1}}}}\right)}{\sqrt{{a_{1}}}\sqrt{{b_{1}}}}}\chi_{2,n}(u).$
(23)

The solution of (21) is given by:

$\displaystyle\chi_{2,n}(u)=c_{1,n}M_{\frac{1}{2},1}\left(\frac{2{\cal
A}_{{\Theta_{2}}^{\prime}}u}{{a_{1}}}\right)+c_{2,n}W_{\frac{1}{2},1}\left(\frac{2{\cal
A}_{{\Theta_{2}}^{\prime}}u}{{a_{1}}}\right).$ (24)

One therefore obtains:

$\displaystyle\hskip-28.90755ptR_{2,n}(r\sim
r_{0})=\frac{c_{2,n}{a_{1}}^{\frac{1}{2}-\frac{{\cal
B}_{{\Theta_{2}}^{\prime}}}{2{b_{1}}}}}{\sqrt{2}\sqrt{{\cal
A}_{{\Theta_{2}}^{\prime}}}}-\frac{(r-r_{0}){a_{1}}^{-\frac{{\cal
B}_{{\Theta_{2}}^{\prime}}}{2{b_{1}}}-\frac{3}{2}}\left({a_{1}}{\cal
B}_{{\Theta_{2}}^{\prime}}c_{2,n}+{\cal
A}_{{\Theta_{2}}^{\prime}}^{2}(4c_{2,n}-8c_{1,n})\right)}{2\sqrt{2}\sqrt{{\cal
A}_{{\Theta_{2}}^{\prime}}}}+{\cal O}\left((r-r_{0})^{3/2}\right).$

From (• ‣ B), one sees the absence of ${\cal O}(R^{4})$ corrections in (3) .
One also sees from (3) that one can impose Dirichlet boundary condition at
$r=r_{0}$ (thereby setting $c_{2}=0$) for all and hence superheavy mesinos
($M_{\rm Mesino}$).

For the BH background assuming
$\Gamma^{\underline{15}}\Theta_{2,0}=\Theta_{2,0},\
\Gamma^{\underline{12}}\Theta_{2,0}=\Theta_{2,0},$ implying,
$\Gamma^{\underline{25}}\Theta_{2,0}=\Theta_{2,0},$ in the IR (i.e., near
$r=r_{h}$), redefining $u\equiv\sqrt{r-r_{h}}$, the solution of the EOM for
$R_{n,2}(r)$ is:

$\displaystyle
R_{2,n}(u)=u^{\Lambda}\left[c_{1}U\left(\mu_{1},\mu_{2},\mu_{3}u\right)+c_{2}L_{-\mu_{1}}^{\mu_{2}-1}\left(\mu_{3}u\right)\right],$
(26)

where $\Lambda,\mu_{1,2,3}$ are defined in (D), and $p=M_{\rm
Mesino}\frac{r_{h}}{\sqrt{g_{s}N}}$ 666 Glueball and meson masses at high
temperatures were obtained respectively in [20] and [17] in units of
$\frac{r_{h}}{\sqrt{g_{s}N}}$. is contained in the ${\cal
O}\left(\frac{\beta}{N}\right)$ term in $\mu_{3}$, which hence remains
undetermined as ${\cal O}\left(\frac{\beta}{N}\right)$ terms are dropped (see
footnote 1). One can show that $\lim_{u\rightarrow
0}u^{\Lambda}c_{1}U\left(\mu_{1},\mu_{2},\mu_{3}u\right)$ is singular. One
hence can not impose Dirichlet or Neumann boundary condition at $r=r_{h}$ if
$c_{2}=0$. Now,

$\displaystyle
L_{-\mu_{1}}^{\mu_{2}-1}(u)=\frac{\Gamma(\mu_{2}-\mu_{1})}{\Gamma(1-\mu_{1})\Gamma(\mu_{2})}-\frac{\Gamma(\mu_{2}-\mu_{1})}{\Gamma(-\mu_{1})\Gamma(\mu_{2}+1)}u+{\cal
O}(u^{2}),$ (27)

implying $\lim_{u\rightarrow
0}u^{\Lambda}c_{2}L_{-\mu_{1}}^{\mu_{2}-1}(\mu_{3}u)=0$, implying the
Dirichlet boundary condition is identically satisfied $\forall M_{\rm Mesino}$
including very large $M_{\rm Mesino}$. It is extremely non-trivial that the
$\mu_{i}$s receive no ${\cal O}(\beta)$ corrections up to ${\cal
O}\left(\frac{\beta}{N^{\alpha_{\mu_{i}}}}\right),\ \alpha_{\mu_{i}}\geq 1$ \-
see (D).

The absence of ${\cal O}(R^{4})$ corrections is essentially a reflection of
the fact that the $SL(2,Z)$ completion of the effective $R^{4}$ interaction
terms in type IIB supergravity, leads to an interesting non-renormalization
theorem that forbids perturbative corrections beyond one loop in the zero-
instanton sector [19].

What we now address in Sec. 4 is how an $N$-enhancement ($\equiv N$-hancement)
of the mass scale $M_{\rm KK}=\frac{r_{0}}{\sqrt{4\pi g_{s}N}}$ [4] is
obtained which therefore explains how one could obtain supermassive $M_{\rm
Mesino}$.

##  4 Generation of $N$-hanced Mass Scale for $T<T_{c}$

In this section, starting from the $D=11$ supergravity Einstein’s field
equations in the presence of four-form $G$ fluxes of ${\cal M}$-theory 777One
can show that “$E_{8}$”-dependent terms in the same are subdominant as
compared to the “$J_{0}$”-dependent terms [2]. \- the first in (2) (also given
in E) - we explicitly show the generation of an $N$-enhanced
($\equiv``N$-hanced”) mass scale thereby providing the mechanism of generation
of super-massive mesinos.

Replacing the resolution parameter “$a$” of the blown-up $S^{2}$ by $a(r)$,
substituting an ansatz: $a(r)=b+c^{\beta^{0}}(r-r_{0})+\beta c^{\beta}{\cal
A}(r)$ into ${\rm EOM}_{MN}$ in (2) ($b$ being a “bare” resolution parameter)
and estimating $r_{0}\sim e^{-\kappa_{r_{0}}N^{1/3}}$ [21], near the
$\psi=2n\pi,n=0,1,2$-coordinate patch, yields the following:

  1. 1.

$\displaystyle{\rm EOM}_{tt,x^{1}x^{1}/x^{2}x^{2}}:\
b\sim\kappa_{tt/x^{i}x^{i}/rr}N^{10/9}e^{-\kappa_{r_{0}}(N)\left(3+0.5\kappa_{r_{0}}(N)N^{1/3}\right)N^{1/3}}r_{0};$
$\displaystyle{\cal A}(r)\sim e^{\frac{c^{\beta}}{b}r}{\cal C}_{1},$ (28)

with $\kappa_{tt/x^{i}x^{i}/rr}\gg 1,\ \kappa_{r_{0}}(N=10^{2})=\frac{1}{{\cal
O}(1)}-{\cal O}(1)$, one obtains $b\gg r_{0}$ and in principle an
$r_{0}$-independent true bare resolution parameter proptional to $\beta$;
${\rm EOM}_{x^{3}x^{3}}$ near $r=r_{0}$ does not constrain $b$;

  2. 2.

$\displaystyle{\rm EOM}_{rr}:\
b\sim\tilde{\kappa}_{rr}N^{11/9}e^{\kappa_{r_{0}}N^{1/3}+1.25\sqrt{1.57+0.55\log
r_{0}-0.5\left(\log r_{0}\right)^{2}}}r_{0};$ (29)

and for an appropriate $\kappa_{r_{0}}\sim\frac{1}{{\cal O}(1)}:1.57+0.55\log
r_{0}-0.5\left(\log r_{0}\right)^{2}>0$, and $N\sim 10^{2}$, one regains the
result for $b$ as obtained in the first equation in (1).

  3. 3.

$\displaystyle\hskip-57.81621pt{\rm EOM}_{\theta_{1}\theta_{1}}:\ a(r)\ {\rm
determined\ by}:-$
$\displaystyle\hskip-57.81621pt\frac{1323\sqrt[5]{N}\alpha_{\theta_{1}}^{2}}{256\alpha_{\theta_{2}}^{2}}-\frac{729{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{6/5}{N_{f}}^{2}\left(2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}\right)a(r)^{3}\log^{3}({r_{0}})\left(2r{r_{0}}\log({r_{0}})a^{\prime}(r)+a(r)(r_{0}-r)\log({r_{0}})\right)}{16\pi^{3}r_{0}^{5}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{2}}=0,$

whose solution is given by:

$\displaystyle a(r)=\Biggl{(}\frac{864c_{1}{g_{s}}^{3}M^{2}{N_{f}}^{2}\Sigma
e^{\frac{2r}{r_{0}}}\log^{4}({r_{0}})-98\pi^{3}N^{7/5}r{r_{0}}^{5}\alpha_{\theta_{1}}^{4}-49\pi^{3}N^{7/5}{r_{0}}^{6}\alpha_{\theta_{1}}^{4}}{{g_{s}}^{3}M^{2}{N_{f}}^{2}r^{2}\Sigma\log^{4}({r_{0}})}\Biggr{)}^{1/4},$
$\displaystyle\sim
c_{1}\frac{e^{\frac{r}{2r_{0}}}}{\sqrt{r}}\sim\frac{c_{1}}{\sqrt{r_{0}}}\left[1+{\cal
O}\left(\frac{(r-r_{0})^{2}}{r_{0}^{2}}\right)\right],$ (31)

where
$\Sigma\equiv\left(2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}\right)$.
Recalling that $r_{0}\sim e^{-\kappa_{r_{0}}N^{1/3}}$, we reinterpret (3) as
$a(r\sim r_{0})\sim c_{1}e^{\frac{3\kappa_{r_{0}}N^{1/3}}{2}}r_{0}$, where for
compatibity with (1) and (29), one may choose an $N$-dependent $c_{1}\sim
N^{(10-11)/9}e^{-\gamma N^{1/3}}$ for an appropriate $\gamma$.

  4. 4.

${\rm EOM}_{\theta_{1}\theta_{2}}$

$\displaystyle\lambda_{3}a(r)^{3}\left(a(r)-a^{\prime}(r)\right)+\lambda_{1}a(r)^{4}+\frac{\lambda_{2}\left(36a(r)^{2}\log({r_{0}})+{r_{0}}\right)}{{r_{0}}^{2}-3a(r)^{2}}=0,$
(32)

where,

$\displaystyle\lambda_{1}\equiv-\frac{243{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{11/10}N_{f}^{2}\left(2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}\right)\log^{4}({r_{0}})}{8\pi^{3}{r_{0}}^{4}\alpha_{\theta_{1}}\alpha_{\theta_{2}}^{3}},$
$\displaystyle\lambda_{2}\equiv-\frac{1323N^{3/10}{r_{0}}\alpha_{\theta_{1}}^{3}}{256\alpha_{\theta_{2}}^{3}(\log
N-9\log({r_{0}}))},$
$\displaystyle\lambda_{3}\equiv-\frac{729{g_{s}}^{3}M^{2}\left(\frac{1}{N}\right)^{11/10}N_{f}^{2}\left(2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}\right)\log^{4}({r_{0}})}{8\pi^{3}{r_{0}}^{3}\alpha_{\theta_{1}}\alpha_{\theta_{2}}^{3}}.$
(33)

Defining,

$\displaystyle\Lambda\equiv\frac{2^{5/6}\sqrt{{g_{s}}}\sqrt[3]{M}\sqrt[3]{N_{f}}{r_{0}}^{2}\sqrt[6]{2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}}\log^{\frac{25}{6}}({r_{0}})}{9\sqrt[3]{7}\sqrt{3\pi}N^{7/30}\alpha_{\theta_{1}}^{2/3}},$
(34)

$a(r)$ is given by:

$\displaystyle\sqrt{{\Lambda}+\exp\left(\frac{2(r+\lambda_{3}c_{1})\left({\Lambda}(\lambda_{1}+\lambda_{3})\left(2{r_{0}}^{2}-9{\Lambda}\right)+36\lambda_{2}\log({r_{0}})\right)}{\lambda_{3}{\Lambda}\left({r_{0}}^{2}-3{\Lambda}\right)}\right)}$
$\displaystyle\sim\sqrt{\Lambda}\sim\frac{\sqrt[4]{{g_{s}}}\sqrt[6]{M}\sqrt[6]{N_{f}}\sqrt[12]{2187\alpha_{\theta_{1}}^{6}+270\sqrt{6}\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+50\alpha_{\theta_{2}}^{4}}\log^{\frac{25}{12}}({r_{0}})}{N^{7/60}\alpha_{\theta_{1}}^{1/3}}r_{0}.$
(35)

  5. 5.

${\rm EOM}_{\theta_{1}x}$:

$\displaystyle b\sim
N^{23/36}e^{\frac{1}{6}\kappa_{r_{0}}N^{1/3}\left(9\kappa_{r_{0}}N^{1/3}+\log
N\right)}r_{0}.$ (36)

  6. 6.

${\rm EOM}_{\theta_{1}y}$:

$\displaystyle b\sim Ne^{\frac{3}{2}\kappa_{r_{0}}^{2}N^{2/3}}r_{0}.$ (37)

  7. 7.

${\rm EOM}_{\theta_{2}x}$:

$\displaystyle
b\sim\kappa_{\theta_{2}y}N^{10/9}e^{-\kappa_{r_{0}}^{2}N^{2/3}+4\kappa_{r_{0}}N^{1/3}}r_{0},\
\kappa_{\theta_{2}x}\gg 1.$ (38)

  8. 8.

${\rm EOM}_{\theta_{2}y}$:

$\displaystyle b\sim
N^{10/9}e^{-3\kappa_{r_{0}}N^{1/3}+\kappa_{r_{0}}^{2}N^{2/3}}r_{0}$ (39)

  9. 9.

${\rm EOM}_{\theta_{2}z}$:

$\displaystyle b\sim
N^{10/9}e^{\kappa_{r_{0}}^{2}N^{2/3}-3\kappa_{r_{0}}N^{1/3}}r_{0}.$ (40)

  10. 10.

${\rm EOM}_{xz/yy/yz/zz}$:

$\displaystyle b\sim
N^{10/9}e^{\kappa_{r_{0}}^{2}N^{2/3}-6\kappa_{r_{0}}N^{1/3}}r_{0}.$ (41)

  11. 11.

${\rm EOM}_{x^{10}x^{10}}$:

$b\sim N^{10/9}r_{0}.$ (42)

We therefore see that the “bare resolution parameter” $b$ given by:

$\displaystyle b\sim N^{1+\frac{1}{{\cal O}(1)}}r_{0};\ \ \ a^{\beta}(r)={\cal
C}e^{\frac{c_{\rm linear}}{b}r},\ {\cal C}\equiv{\rm constant}.$ (43)

(One hence can not obtain an $r_{0}$-independent “$b$”.) The details are
given, for completeness, in section 4. One thus sees an $N$-hancement of the
effective KK mass scale $M_{KK}$ (from $M_{KK}$ to $M_{KK}^{\rm eff}\sim
N^{1+\frac{1}{{\cal O}(1)}}M_{KK}$) arising from the construction of SYZ type
IIA mirror of the non-Kähler type IIB dual [9] of thermal QCD-like theories,
as well as the generation of a one-parameter (${\cal C}$) family of
$r_{0}/M_{KK}$-independent bare resolution parameter at ${\cal O}(R^{4})$ in
the ${\cal M}$-theory uplift involving a $G_{2}$-structure wherein ${\cal C}$
can be made appropriately large. These are the pair of reasons for generating
super-massive mesinos in the fermionic sector in the string/${\cal M}$ theory
duals of thermal QCD at finite $N$ in [1], [2].

##  5 Non-Interacting Mesinos

Given that we have seen in 3 that supermassive mesinos, unlike [7] (see [8]),
are permissible in the type IIA holographic dual [1] at intermediate coupling
[2] of realistic thermal QCD-like theories, this already explains why mesinos
have thus far not been observed near the EW scale. In this section, we will
further show that mesino-mesino-single-($\rho/\pi$)meson interactions, unlike
[7] (see [8]), vanish identically in the aforementioned type IIA holographic
dual. Considering fluctuations of the vector mesons $A_{\mu,r}\rightarrow
A_{\mu,r}^{(0)}+\delta A_{\mu,r}$ with $A_{\mu=t}^{(0)}$ being the only non-
zero background value (see 3) which can be shown to be tunable so that
$|F_{rt}^{(0)}|\ll 1$, implying one need only consider terms linear in $F_{\rm
IIA}^{(0)}$ 888Use is made of
$i^{*}B_{\alpha_{1}\alpha_{2}}=\delta^{[\theta_{1}}_{\alpha_{1}}\delta^{\theta_{2}]}_{\alpha_{2}}B_{\theta_{1}\theta_{2}}$
and consequently, ${\cal F}_{rt}=F_{rt}$. which are contained (recalling from
section 3, $\sqrt{-{\rm det}(i^{*}g^{\rm IIA}+\mathcal{F}_{\rm IIA}^{(0)})}\ll
1,\ {\cal F}_{IIA}=i^{*}B_{IIA}+F$ in the large-$N$ limit) in :

$\displaystyle S_{D_{6}}^{f}=\frac{T_{D_{6}}}{2}\int d^{7}\xi e^{-\Phi^{\rm
IIA}}\
\overline{\Theta}\left({\frac{\Gamma^{\beta_{1}.....\beta_{7}}\mathcal{F}_{\rm
IIA}^{(0)}\ {}_{\beta_{6}\beta_{7}}\Gamma_{\beta_{1}.....\beta_{5}}^{\ \ \ \ \
\ \ \ \gamma}D_{\gamma}\Gamma_{(10)}}{\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}_{\rm IIA}^{(0)})}}}\right)\Theta.$ (44)

Considering fluctuations in the background gauge field in (44) and retaining
terms linear in the same yields:

$\displaystyle\delta S_{D_{6}}^{f}\sim
T_{D_{6}}\int_{\Sigma_{(7)}}d^{4}xdrd\theta_{2}d\tilde{y}e^{-\Phi^{\rm
IIA}}\overline{\Theta}\left(\frac{4\Gamma^{\beta_{1}.....\beta_{7}}\delta\mathcal{F}^{\rm
IIA}_{\beta_{6}\beta_{7}}\Gamma_{\beta_{1}.....\beta_{5}}^{\ \ \ \ \ \ \ \
\gamma}D_{\gamma}\Gamma_{(10)}}{\sqrt{-{\rm det}(i^{*}g^{\rm
IIA}+\mathcal{F}_{\rm IIA}^{(0)})}}\right)\Theta.$ (45)

The next step is to perform the KK expansion of $\delta\mathcal{F}^{\rm
IIA}_{\alpha\beta}$ and decompose spinors along $M_{4}$ and internal
directions, and by integrating over the $\theta_{2}$ and $\tilde{y}$ we will
get mesino-mesino-meson interaction action with couplings given in terms of
radial integrals of the radial profile functions of the mesino and mesons. The
usual KK expansion ansatz is: $\delta
A_{\mu}(x^{\mu},r)=\sum_{n=1}^{\infty}\rho_{\mu}^{(n)}(x)\psi_{n}(r)$ and
$\delta A_{r}(x^{\mu},r)=\sum_{n=0}^{\infty}\pi^{(n)}(x)\phi_{n}(r)$ implies
$\delta F_{\mu\nu}=\sum_{n=1}^{\infty}\tilde{F}_{\mu\nu}^{(n)}(x)\psi_{n}(r)$
and $\delta F_{\mu
r}=\sum_{n=0}^{\infty}\partial_{\mu}\pi^{(n)}(x^{\mu})\phi_{n}(r)-\sum_{n=1}^{\infty}\rho_{\mu}^{(n)}(x)\dot{\psi}_{n}(r)$
[4]. We will keep the $n=1$ term for the vector fluctuation and $n=0$ for the
$A_{r}(x^{\mu},r)$ and hence the degrees of freedom are $\rho$ vector meson
and $\pi$ meson. Using the KK decomposition of $\delta F_{\mu\nu}$ and $\delta
F_{\mu r}$, (45) simplifies as follows:

$\displaystyle\hskip-43.36243ptS_{D_{6}}^{int}\sim
T_{D_{6}}\int_{\Sigma_{(7)}}\Biggl{[}\frac{e^{-\Phi^{\rm IIA}}}{\sqrt{-{\rm
det}(i^{*}g^{\rm IIA}+\mathcal{F}_{\rm
IIA}^{(0)})}}\overline{\Theta}\Biggl{(}\Gamma^{\beta_{1}.....\beta_{5}\mu\nu}\delta{\tilde{F}}_{\mu\nu}\psi(r)\Gamma_{\beta_{1}.....\beta_{5}}^{\
\ \ \ \ \ \ \ \gamma}D_{\gamma}+\Gamma^{\beta_{1}.....\beta_{5}\mu
r}\left(\partial_{\mu}\pi(x^{\mu})\phi(r)-\rho_{\mu}(x^{\mu})\dot{\psi}(r)\right)\Gamma_{\beta_{1}.....\beta_{5}}^{\
\ \ \ \ \ \ \ \gamma}D_{\gamma}\Biggr{)}\Theta\Biggr{]}.$

Using the decomposition of the ten-dimensional gamma matrices [22]:

$\displaystyle\Gamma^{\underline{A}=\underline{t},\underline{x^{1,2,3}},\underline{r}}=\sigma_{y}\otimes{\bf
1}_{4}\otimes\gamma^{\underline{A}},\ \ \
\Gamma^{\underline{a}=5,...,9}=\sigma_{x}\otimes\gamma^{\underline{a}}\otimes{\bf
1}_{4},$ (47)

with

$\displaystyle\left\\{\gamma^{\underline{A}},\gamma^{\underline{B}}\right\\}=-2\eta^{\underline{AB}},$
$\displaystyle\left\\{\gamma^{\underline{a}},\gamma^{\underline{b}}\right\\}=-2\delta^{\underline{ab}}.$
(48)

The ten-dimensional chirality matrix is defined as:

$\displaystyle\Gamma^{(10)}=\sigma_{z}\otimes{\bf 1}_{4}\otimes{\bf 1}_{4}.$
(49)

The positive-chirality ten-dimensional $\Theta$ can hence be decomposed into

$\displaystyle\Theta=\uparrow\otimes\chi_{M_{5}(x^{0,1,2,3},r)}\otimes\psi_{\tilde{M}_{5}(\theta_{1,2},\phi_{1,2},\psi)},$
(50)

where $\psi_{\tilde{M}_{5}(\theta_{1,2},\phi_{1,2},\psi)}$ further splits into
$\psi_{\tilde{M}_{5}}=\psi_{S^{2}_{\rm squashed}}\otimes\psi_{S^{3}_{\rm
squashed}}$. Looking at the second fermionic bilinear in (5):

$\displaystyle{\bar{\Theta}}\Gamma^{\beta_{1}...\beta_{5}tr}\Gamma_{\beta_{1}...\beta_{5}}^{\
\ \ \ \
\gamma}D_{\gamma}\Theta(\beta_{i=1,...,5}=x^{0,1,2,3},\theta_{2},\tilde{y};\
\gamma=t,r)\sim{\bar{\Theta}}\Gamma^{t}D_{r}\Theta+{\bar{\Theta}}\Gamma^{r}D_{t}\Theta.$
(51)

Now, the non-vanishing
${\bar{\Theta}}\Gamma^{\underline{X}_{1}....\underline{X}_{p}}\Theta$
involving Majorana-Weyl spinor $\Theta$ requires $p=3,7$ [8]. One can further
show that the most dominant spin-connection component of the type
$\omega_{r}^{\underline{ab}}$ is $\omega_{r}^{\underline{7}\underline{9}}$ and
the only non-vanishing spin-connection component of the type
$\omega_{t}^{\underline{a}\underline{b}}$ is
$\omega_{t}^{\underline{x^{0}}\underline{r}}$. Therefore, using (47):

$\displaystyle{\bar{\Theta}}\Gamma^{\beta_{1}...\beta_{5}tr}\Gamma_{\beta_{1}...\beta_{5}}^{\
\ \ \ \
\gamma}D_{\gamma}\Theta\sim{\bar{\Theta}}\Gamma^{\underline{t}}\omega_{r}^{\underline{79}}\Gamma^{\underline{79}}\Theta\propto\langle\uparrow|\sigma_{y}|\uparrow\rangle=0.$
(52)

Also,

$\displaystyle\overline{\Theta}\Gamma^{\beta_{1}.....\beta_{5}\mu\nu}\delta{\tilde{F}}_{\mu\nu}\psi(r)\Gamma_{\beta_{1}.....\beta_{5}}^{\
\ \ \ \ \ \ \ \gamma}D_{\gamma}\Theta=0,$ (53)

as $\mu,\nu\in x^{1,2,3}$ and thus using (47):

$\displaystyle\hskip-21.68121pt\overline{\Theta}\Gamma^{kx^{0}r\theta_{2}\tilde{y}ij}\delta{\tilde{F}}_{ij}\psi(r)\Gamma_{kx^{0}r\theta_{2}\tilde{y}}^{\
\ \ \ \ \ \ \ i}D_{i}\Theta(i\neq j\neq
k=x^{1,2,3})\sim\delta{\tilde{F}}_{ij}\overline{\Theta}\Gamma^{ij}\Gamma^{l}\partial_{l}\Theta(l\neq
k)\ \propto\langle\uparrow|\sigma_{y}|\uparrow\rangle=0.$ (54)

Hence, no mesino-mesino-$\rho$/$\pi$-meson vertex is generated. Together with
what was argued earlier that one can have a super-massive mesino, this
suggests the “WISP”(Weakly Interating Supermassive Particle)y nature of the
non-supersymmetric mesino, and consequently resolves the tension between
actual QCD and top-down holographic QCD [8].

##  6 Top-Down $m_{\rm quark}\langle{\bar{q}}q\rangle$ Non-Renormalization up
to ${\cal O}(R^{4})$

The ${\cal O}(R^{4})$ corrections to the ${\cal M}$-theory dual’s metric are
vanishing small in the UV [23]. The EOM of the flavor $D6$-branes’ embedding,
$\tilde{z}=\tilde{z}(r)$ in the IR arising from the DBI action for the flavor
$D6$-branes with world volume
$\Sigma_{7}\left(S^{1}_{t}\times\mathbb{R}^{3}\times\mathbb{R}_{>0}\times
S^{2}_{\rm squashed}\right)$ embedded via $i:\Sigma_{7}\hookrightarrow
S^{1}_{t}\times\mathbb{R}^{3}\times_{w}M_{6}$ [$w\equiv$ warped product]
effected by $\tilde{z}=\tilde{z}(r)$ in a non-Kähler warped squashed resolved
conifold $M_{6}$ in the type IIA mirror of the UV-complete type IIB dual [9]
of thermal QCD-like theories, using the induced metric on flavor $D6$-branes
of (17), NS-NS $B^{\rm IIA}$ of (3), can be shown to yield:
$\tilde{z}=$constant, inclusive of ${\cal O}(\beta)$ corrections.

The DBI action in the UV is given by (disregarding overall $r$-independent
factors, and hence the $\sim$):

${\cal L}_{\rm DBI}^{\rm
D6}\sim\frac{r^{2}\sqrt{\frac{4\pi\sqrt{{g_{s}}}r^{2}\alpha_{\theta_{2}}^{3}\left(6a^{2}+r^{2}\right)}{9a^{2}+r^{2}}+3\sqrt{3\pi}N^{2/5}\left(r^{4}-{r_{h}}^{4}\right)\tilde{z}^{\prime}(r)^{2}}}{{g_{s}}^{3/4}},$
(55)

and consequently the $\tilde{z}(r)$ EOM: $\frac{\delta{\cal L}_{\rm DBI}^{\rm
D6}}{\delta\tilde{z}^{\prime}(r)}={\cal K}$ (constant) in the UV yields:
$\tilde{z}^{\prime}(r)=\frac{{\cal C}}{r^{5}}$, ${\cal C}$ being a constant
(subsuming $g_{s}$\- and $N$-dependent factors). One hence obtains:
$\tilde{z}(r)={\cal C}_{1}-\frac{\tilde{\cal
C}_{2}}{r^{4}}\stackrel{{\scriptstyle r\in{\rm UV}}}{{\longrightarrow}}{\cal
C}_{1}.$ As $\tilde{z}(r)$ is dimensionless, ${\cal C}_{1}$ will hence also be
so and ${\cal C}_{2}$ will have a mass dimension of four (in units of ${\cal
R}_{D5/\overline{D5}}=D5-\overline{D5}$-separation). By looking at
fluctuations: $\tilde{z}\rightarrow\tilde{z}+\delta\tilde{z}$ in the DBI
action (no mass term $\left(\delta\tilde{z}\right)^{2}$ is generated) one can
show that in the UV and in the $\psi=2n\pi,n=0,1,2$-coordinate patches and by
working near, e.g.,
$\left(\theta_{1},\theta_{2}\right)=\left(\frac{\alpha_{\theta_{1}}}{N^{1/5}},\frac{\alpha_{\theta_{2}}}{N^{3/10}}\right)$
[11], [21]:

$\delta\tilde{z}\stackrel{{\scriptstyle\rm
UV}}{{\longrightarrow}}c_{1}+\frac{{\cal C}_{2}}{r^{4}}+{\cal
O}\left(\left(\frac{{\cal C}_{2}}{r}\right)^{12}\right).$ (56)

Again, we see that the mass dimension of the coefficient ${\cal C}_{2}$ of
$\frac{1}{r^{4}}$ is four (and $c_{1}$ is dimensionless). Given that one
obtains an $AdS_{5}$ in the UV, the coefficent of $\frac{1}{r^{4}}$ for the
massless fluctuation $\delta\tilde{z}$ is identified wtih a chiral condensate
[24], we conjecture that ${\cal C}_{2}$ is the top-down holographic analog of
the mass-dimension-four $m_{q}\langle{\bar{q}}q\rangle$. As the ${\cal
O}(R^{4})$ corrections are vanishingly small in the UV [4], $c_{1}$ and ${\cal
C}_{2}$ receive no ${\cal O}(R^{4})$ corrections.

##  7 Universality in Particle Wave Functions in the IR

An intriguing universality in the wave functions of the following particle
spectroscopies is noticed.

Glueballs [20]:

  * •

$0^{-+}$ glueball: The EOM of the type IIA RR $A$ fluctuation (to which
tr$F\wedge\tilde{F}$ couples via the type IIA $D4$-brane with world volume
$\Sigma_{1,4}$) term WZ term $\int_{\Sigma_{1,4}}A\wedge{\rm
tr}F\wedge\tilde{F}$) $\partial_{\nu}\left(\sqrt{g^{\rm IIA}}g_{\rm
IIA}^{\mu\sigma}g_{\rm
IIA}^{\nu\rho}\left(\partial_{[\sigma}A_{\rho]}\right)\right)=0$ (where
$\mu,\nu,...=a(\equiv 0,1,2,3),r,\alpha(\equiv 5,...,9)$ and it was assumed
$A_{\mu}=\delta^{\theta_{2}}_{\mu}a_{\theta_{2}}(r)e^{ik\cdot x},k^{2}=-m^{2}$
as the fluctuation about the type IIA $A_{1}$ that was worked out in [1]).

  * •

$0^{--}$ glueball: The EOM for the fluctuation in the type IIB
$A_{MN}=B_{MN}+iC_{MN}$ (that figures in the Weiss-Zumino term
$A^{\mu\nu}d^{abc}{\rm Tr}\left(F_{\mu\rho}^{a}F^{b\ \rho}_{\lambda}F^{c\
\lambda}_{\nu}\right)$), $\delta A^{MN}=\delta^{M}_{2}\delta^{N}_{2}\delta
A_{23}$,
$\partial_{\mu}\left(\sqrt{-g}g^{22}g^{33}g^{\mu\nu}\partial\delta
A_{23}\right)=0$.

  * •

$1^{++}$ glueball: The EOM for the radial profile function of the vector-type
${\cal M}$-theory metric perturbation
$h_{ti}=h_{it}=g_{x^{1}x^{1}}G(r)e^{ikx^{1}},i=x^{2},x^{3}:R^{(1)}_{\mu\nu}\approx
0,\ R^{(1)}_{\mu\nu}$ denoting the first-order fluctuations in the Ricci
tensor as a consequence of linear metric perturbations.

Mesons [17]: Working with the redefined radial variable $Z:r=r_{h}e^{Z}$,
after integrating out the blown-up $S^{2}_{\rm squashed}$ in the DBI action of
the flavor type IIA $D6$-branes and KK reduction of the gauge field
$A_{\mu}(x^{\mu},Z)=\sum_{n=1}B_{\mu}^{(n)}(x^{\mu})\alpha^{\\{\mu\\}}_{n}(Z),\mu=t,x^{i=1,2,3}$,
the terms in the DBI action quadratic in the gauge field fluctuations are:
$\int d^{4}xdZ\left({\cal
V}_{2}(Z)F^{(n)}_{\mu\nu}F^{\mu\nu}_{(n)}\alpha_{m}^{\left\\{\mu\right\\}}(Z)\alpha_{n}^{\left\\{\mu\right\\}}(Z)+{\cal
V}_{1}(Z)B^{(m)}_{\mu}B^{(n)}_{\nu}\dot{\alpha}_{m}^{\left\\{\mu\right\\}}\dot{\alpha}_{n}^{\left\\{\mu\right\\}}\right)$.
The EOM for the radial profile $\alpha^{\\{i\\}}_{m}(Z)$ is :
$\frac{d}{dZ}\left({\cal V}_{1}(Z)\dot{\alpha}_{m}^{\\{i\\}}\right)+2{\cal
V}_{2}(Z){\cal M}_{(m)}^{2}\alpha_{m}^{\\{i\\}}=0$, where
${\cal V}_{1}(Z)=e^{-\Phi^{IIA}}\sqrt{h}g^{ZZ}\sqrt{{\rm
det}_{\theta_{2},\tilde{y}}\left(i^{*}(g+B)\right)}\sqrt{{\rm
det}_{\mathbb{R}^{1,3},Z}(i^{*}g)}$ and ${\cal
V}_{2}(Z)=e^{-\Phi^{IIA}}\frac{h}{2}\sqrt{{\rm
det}_{\theta_{2},\tilde{y}}\left(i^{*}(g+B)\right)}\sqrt{{\rm
det}_{\mathbb{R}^{1,3},|Z|}(i^{*}g)}$. The solution of $\alpha^{\\{i\\}}$ is
given in terms of the Tricomi Hypergeometric and associated Laguerre
functions.

Graviton [25]: In the context of obtaining the Page curve of an eternal black
hole from the ${\cal M}$-theory dual containing a black-hole in the ETW(End of
The World)-“brane” (a hypersurface
$AdS_{4}^{\infty}\times_{w}M_{6},\times_{w}$ implying a warped product, with
$G_{4}$ fluxes threading a homologous sum of four-cycles $S^{3}_{\rm
squashed}\times[0,1]$ and $S^{2}_{\rm squashed}\times S^{2}_{\rm squashed}$ in
$M_{6}=M_{5}(\theta_{1,2},\phi_{1,2},\psi)\times S^{1}(x^{10})\hookrightarrow
M^{SU(4)/Spin(7)}_{8}(t,r,\theta_{1,2},\phi_{1,2},\psi,x^{10})$, with a finite
“tension”) coupled to a non-conformal QCD bath in the doubly holographic
approach, the massless graviton wavefunction with the graviton localized on
the ETW-brane trapped in a “volcano”-like potential, is given in terms of the
Tricomi Hypergeometric and associated Laguerre functions.

Solutions to the EOMs for the aforementioned field fluctuations/radial profile
function are given in terms of the Tricomi Hypergeometric and associated
Laguerre functions. The reason is that the relevant near-$r_{h}$ EOMs for
$0^{-+},0^{--},1^{++}$-glueballs [20], and the radial profile function of the
graviton wave function [25] are all of the type:

$(r-r_{h})\xi^{\prime\prime}(r)+\left(b+c(r-r_{h})\right)\xi^{\prime}(r)+(f+(r-r_{h})G)\xi(r)=0,$
(57)

whose solution is given as:

$\displaystyle\hskip-28.90755pt\xi(r\sim
r_{h})=e^{-\frac{1}{2}r\left(\sqrt{c^{2}-4G}+c\right)}\Biggl{[}c_{1}U\left(\frac{b\left(c+\sqrt{c^{2}-4G}\right)-2f}{2\sqrt{c^{2}-4G}},b,\sqrt{c^{2}-4G}(r-r_{h})\right)+c_{2}L_{\frac{2f-b\left(c+\sqrt{c^{2}-4G}\right)}{2\sqrt{c^{2}-4G}}}^{b-1}\left(\sqrt{c^{2}-4G}(r-r_{h})\right)\Biggr{]}.$

In the context of the radial profile functions of vector mesons [17], and
mesinos at $T>T_{c}$ in equation (8), after appropriate coordinate
redefinitions, the near-horizon (IR) solutions are also given in terms of the
Tricomi Hypergeometric and Associate Laguerre functions.

##  8 Summary

The immensely popular holographic QCD dual of [7] suffered from the
longstanding problem that the Mesinos were nearly isospectral with the mesons,
with non-vanishing/un(large-$N$-)suppressed mesino-mesino-meson interaction
[8]\- both in direct conflict with real QCD. What we show is that using the
type IIA SYZ mirror of the UV-complete [9] (unlike [7] which caters only to
the IR) as constructed in [1] inclusive of ${\cal O}(R^{4})$ corrections
worked out in [2], not only is it possible to have super-massive mesinos that
do not interact with the mesons, the results obtained (mesino wave function,
mass, mesino-mesino-meson interaction) receive no ${\cal O}(R^{4})$
corrections up to ${\cal O}(\frac{l_{p}^{6}}{N^{\alpha}}),\alpha\geq 1$. The
“WISP”(Weakly Interacting Supermassive Particles)y mesinos and non-
renormalization of their wave functions and mass up to ${\cal O}(R^{4})$,
together, apart from solving a longstanding problem, also provide a major and
new insight into the fermionic sector of top-down holographic duals close to
real thermal QCD.

Further, the product of the quark mass and chiral condensate may be
conjectured to correspond to the coefficient of the leading non-constant term
in the flavor $D6$-branes’ embedding’s fluctuation, with the RG-invariance of
the former [26] corresponding to the non-renormalization up to ${\cal
O}(R^{4})$ of the latter - see section 6 for details. In the end, we would
also point out that there is a rather intriguing wave-function universality in
the form of appearance of (appropriate) Tricomi Hypergeometric and Associate
Laguerre functions in the glueball/meson/graviton (apart from mesinoic)
spectroscopies - see section 7.

## Acknowledgements

AM is partly supported by a Core Research Grant number SER-1829-PHY from the
Science and Engineering Research Board, Govt. of India. GY is supported by a
Senior Research Fellowship (SRF) from the Council of Scientific and Industrial
Research, Govt. of India. We thank Nick Evans for a useful clarification. GY
thanks the Infosys Foundation for the partial support at CMI.

##  Appendix A Finite Baryon Chemical Potential

We explicitly show the generation of a finite baryon chemical potential. From
equation (3) ($f(r)$ being valid $\forall r$), up to LO in N, $k^{\rm
UV}(r)=1-3\frac{a^{2}}{r^{2}},f(r)=\frac{2N^{2/5}r^{6}}{729\pi
g_{s}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{2}}$ and integrating
$\frac{\kappa\sqrt{k^{\rm UV}(r)}}{\sqrt{\kappa^{2}+f^{2}(r)}}$, one obtains:

$\displaystyle\hskip-36.135ptA_{t}(r\in{\rm
UV})\sim\frac{1}{\sqrt{i\left(r^{2}-3a^{2}\right)}}$
$\displaystyle\hskip-36.135pt\times\Biggl{\\{}(-1)^{2/3}\sqrt[4]{3}ar\sqrt{1-\frac{3a^{2}}{r^{2}}}\Biggl{(}F\left[\sin^{-1}\left(\frac{3^{3/4}\sqrt[3]{\kappa}\sqrt[6]{\frac{\pi}{2}}\sqrt{i\left(r^{2}-3a^{2}\right)}\sqrt[6]{g_{s}}\alpha_{\theta_{1}}^{2/3}\sqrt[3]{\alpha_{\theta_{2}}}}{{ar}\sqrt[15]{N}}\right)|\frac{1}{2}\left(1-i\sqrt{3}\right)\right]$
$\displaystyle\hskip-36.135pt-\Pi\left[\frac{\frac{3i2^{2/3}\kappa^{2/3}\sqrt[3]{\pi}\sqrt[3]{g_{s}}\alpha_{\theta_{1}}^{4/3}\alpha_{\theta_{2}}^{2/3}}{a^{2}N^{2/15}}-2(-1)^{5/6}}{2\sqrt{3}};\sin^{-1}\left(\frac{3^{3/4}\sqrt[3]{\kappa}\sqrt[6]{\frac{\pi}{2}}\sqrt{i\left(r^{2}-3a^{2}\right)}\sqrt[6]{g_{s}}\alpha_{\theta_{1}}^{2/3}\sqrt[3]{\alpha_{\theta_{2}}}}{{ar}\sqrt[15]{N}}\right)|\frac{1}{2}\left(1-i\sqrt{3}\right)\right]\Biggr{)}\Biggr{\\}}.$
$\displaystyle\hskip-36.135pt\sim\frac{3\kappa\sqrt{\pi}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\left(1-\frac{3a^{2}}{r^{2}}\right)^{3/2}\sqrt{g_{s}}}{\sqrt{2}a^{2}\sqrt[5]{N}},$
(A1)

$F(\phi|\mu)\equiv\int_{0}^{\phi}\frac{d\alpha}{\sqrt{1-m^{2}\sin^{2}\alpha}},$
being the incomplete elliptic integral of the first kind, and
$\Pi(\nu;\phi|\mu)\equiv\int_{0}^{\phi}\frac{d\alpha}{(1-\nu^{2}\sin^{2}\alpha)\sqrt{1-\mu^{2}\sin^{2}\alpha}}$
being the incomplete intregral of the fourth kind, generating a finite baryon
chemical potential:

$\mu=\frac{3\sqrt{\pi}\kappa\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}\sqrt{g_{s}}}{\sqrt{2}a^{2}\sqrt[5]{N}}.$
(A2)

##  Appendix B EOM-Related for Massive Mesinos

The EOM for the radial profile $R_{n,2}(r)$ of the Mesino $\Theta$, as defined
in equation (19), is given by:

$\displaystyle\frac{\Gamma^{\underline{1}\underline{5}}{E^{r}_{\underline{5}}}(r)R_{n,2}^{\prime\prime}(r)}{{E^{t}_{\underline{1}}}(r)}\Theta_{2,0}+R_{n,2}^{\prime}(r)\left(\Gamma^{\underline{1}\underline{5}}\left(\frac{{E^{r}_{\underline{5}}}(r)}{{E^{t}_{\underline{1}}}(r)}\right)^{\prime}-\frac{\Gamma^{\underline{1}\underline{2}}i{p}\
{E^{x^{1}}_{\underline{2}}}(r)}{{E^{t}_{\underline{1}}}(r)}-\frac{\Gamma^{\underline{1}\underline{5}}{\cal
J}^{\prime}(r)}{{\omega_{r}^{\underline{8}\ \underline{10}}}(r)}+2\pi
i(2n+1)T\right)\Theta_{2,0}$
$\displaystyle+R_{n,2}(r)\Biggl{(}-\frac{\pi^{2}\Gamma^{\underline{1}\underline{5}}(2n+1)^{2}T^{2}{E^{t}_{\underline{1}}}(r)}{{E^{r}_{\underline{5}}}(r)}+\frac{\pi\Gamma^{\underline{2}\underline{5}}(2n+1){p}\
T{E^{x^{1}}_{\underline{2}}}(r)}{{E^{r}_{\underline{5}}}(r)}+\frac{\Gamma^{\underline{1}\underline{2}}i{p}\
{E^{x^{1}}_{\underline{2}}}(r){\cal
J}^{\prime}(r)}{{E^{t}_{\underline{1}}}(r){\cal J}(r)}$
$\displaystyle+\frac{\Gamma^{\underline{1}\underline{5}}{p}\
^{2}{E^{x^{1}}_{\underline{2}}}(r)^{2}}{{E^{t}_{\underline{1}}}(r)^{2}}\omega_{r}^{\underline{a}\
\underline{10}}-\frac{\pi\Gamma^{\underline{2}\underline{5}}(2n+1)i{p}\
T{E^{x^{1}}_{\underline{2}}}(r)}{{E^{t}_{\underline{1}}}(r)}\frac{\omega_{r}^{\underline{a}\
\underline{10}}}{{\cal J}(r)}-\frac{\pi i(2n+1)T{\cal J}^{\prime}(r)}{{\cal
J}(r)}+{\cal J}(r){\omega_{r}^{\underline{a}\
\underline{10}}}(r)\Gamma^{\underline{1}\underline{5}}-i\pi\left(\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}\right)^{\prime}\Gamma^{\underline{1}\underline{2}}\Biggr{)}\Theta_{2,0}=0,$

with $a=7,8$ respectively for the TH, BH backgrounds with suitable
aforementioned definitions for ${\cal J}(r)$.

  * •

$T<T_{c}$: Writing $M_{\rm Mesino}=\tilde{M}_{\rm
Mesino}\frac{r_{0}}{\sqrt{g_{s}N}}$, the constants appearing in the
Schrödinger-like EOM (23)-(3), are given as under:

$\displaystyle\hskip-57.81621pta_{1}\equiv\frac{r_{0}^{2}}{\sqrt{3\pi
g_{s}N}};\ a_{2}\equiv\frac{23r_{0}}{12\sqrt{3\pi g_{s}N}};\
b_{1}\equiv\frac{23r_{0}}{12\sqrt{3\pi g_{s}N}};\
b_{6}\equiv-\frac{23\sqrt{\pi g_{s}N}}{4\sqrt{3}r_{0}^{3}},$
$\displaystyle\hskip-57.81621pt{\cal
A}_{\Theta_{2}^{\prime}}\equiv-\frac{1511.7\sqrt{{r_{0}}}\alpha_{\theta_{2}}}{{g_{s}}^{7/2}\kappa_{2}\log
NMN^{2/5}{N_{f}}^{2}\alpha_{\theta_{1}}^{2}},$
$\displaystyle\hskip-57.81621pt{\cal B}_{\Theta_{2}^{\prime}}\equiv
a_{2}+\frac{2^{3/2}\tilde{M}_{\rm
Mesino}\pi^{1/4}}{\left(g_{s}N\right)^{1/4}}+2i(2n+1)\pi T,$
$\displaystyle\hskip-57.81621pt{\cal
A}_{\Theta_{2}}\equiv\frac{-\frac{39.4\tilde{M}_{\rm
Mesino}}{\sqrt[4]{{g_{s}}}\sqrt[4]{N}}+T\left(n(-39.5{a_{6}^{\beta}}{r_{0}}T-(125i))-(62.5i)-39.5{a_{6}^{\beta}}n^{2}{r_{0}}T-9.9{a_{6}^{\beta}}{r_{0}}T\right)}{{r_{0}}},$
$\displaystyle\hskip-57.81621pt{\cal
B}_{\Theta_{2}}\equiv\frac{M\left(\lambda_{5}^{2}{g_{s}}^{15/4}\kappa_{2}^{2}\log
N^{2}\tilde{M}_{\rm
Mesino}(0.1n+0.1)N^{3/20}{N_{f}}^{2}{r_{0}}^{4}T\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{8}-1.1\lambda_{5}{g_{s}}^{3/2}\kappa_{2}\log
N\tilde{M}_{\rm
Mesino}^{2}{N_{f}}{r_{0}}^{2}\alpha_{\theta_{2}}\alpha_{\theta_{1}}^{4}+6.84N^{3/5}\right)}{\lambda_{5}^{2}\sqrt{{g_{s}}}\kappa_{2}\log
N{r_{0}}^{11/2}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{3}},$
$\displaystyle\hskip-57.81621pt{\cal
C}_{\Theta_{2}}\equiv\frac{\frac{786.1\tilde{M}_{\rm
Mesino}}{\sqrt[4]{{g_{s}}}\sqrt[4]{N}}+T\left(n\left(-39.5{b_{6}}{r_{0}}^{2}T+(0.\,+2485.6i)\right)+(1242.8i)-39.5{b_{6}}n^{2}{r_{0}}^{2}T-9.9{b_{6}}{r_{0}}^{2}T\right)}{{r_{0}}^{2}},$
(B2)

$\lambda_{5}$ being the parameter in terms of which the coframes of the
relevant non-Kähler six-folds were worked out in [2],
$g_{\theta_{2}\theta_{2}}^{\rm IIA}(r\sim r_{0})\sim\kappa_{2}\sqrt{g_{s}N}$
and,

$\displaystyle\hskip-57.81621pt{a_{6}^{\beta}}\equiv\frac{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}}{{r_{0}}^{2}}+\frac{\beta\sqrt{{g_{s}}}M\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{0}})}{4374\sqrt{\pi}\epsilon^{5}\log
N^{4}N^{3/4}{N_{f}}{r_{0}}^{4}\alpha_{\theta_{2}}^{3}}.$ (B3)

  * •

$T>T_{c}$: Based on

$\displaystyle\hskip-28.90755ptE^{r}_{\underline{5}}=\frac{\sqrt{\frac{9a^{2}+r^{2}}{6a^{2}+r^{2}}}\sqrt{r^{4}-{r_{h}}^{4}}\left(1-\frac{1}{2}\beta\left({\cal
C}_{zz}-2{\cal C}_{\theta_{1}z}+2{\cal
C}_{\theta_{1}x}\right)\right)}{\sqrt{2}\sqrt[4]{\pi}\sqrt{{g_{s}}}\sqrt{N}r},$
$\displaystyle\hskip-28.90755ptE^{t}_{\underline{1}}=\frac{\sqrt{2}\sqrt[4]{\pi}\sqrt[4]{{g_{s}}}\sqrt[4]{N}r}{\sqrt{r^{4}-{r_{h}}^{4}}}+\frac{27\left(9b^{2}+1\right)^{4}\beta
b^{10}\sqrt[4]{{g_{s}}}Mr^{2}\Sigma\left(6a^{2}+{r_{h}}^{2}\right)(r-2{r_{h}})\log^{3}({r_{h}})}{2\sqrt{2}\pi^{3/4}\left(3b^{2}-1\right)^{5}\left(6b^{2}+1\right)^{4}\log
N^{4}N{N_{f}}{r_{h}}^{4}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{h}}^{2}\right)\sqrt{r^{4}-{r_{h}}^{4}}},$
$\displaystyle\hskip-28.90755pt\omega_{r}^{\underline{8}\
\underline{10}}=-\frac{7N^{3/5}}{{\lambda_{5}}{g_{s}}^{3/2}\kappa_{2}\log
N{N_{f}}r^{2}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}\left(r^{2}-3.3a^{2}\right)}$
$\displaystyle\hskip-28.90755pt+\frac{\kappa_{\omega_{r}^{\underline{8}\
\underline{10}}}a^{8}\sqrt[4]{\beta}\sqrt{{\cal C}_{zz}}{\rm
const}{\lambda_{5}}{g_{s}}^{5/4}MN^{19/20}{N_{f}}\sqrt{\alpha_{\theta_{1}}}\sqrt{1-\frac{{r_{h}}^{4}}{r^{4}}}\log(r)}{r^{4}\alpha_{\theta_{2}}^{6}\left(r^{2}-3.a^{2}\right)^{2}\sqrt{\frac{6a^{2}+r^{2}}{9a^{2}+r^{2}}}},$
(B4)

with
$\Sigma\equiv-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}$,
and setting consistently the ${\cal O}(R^{4})$ corrections of ${\cal
M}$-theory’s three-form potential to zero requires: ${\cal C}_{zz}-2{\cal
C}_{\theta_{1}z}=0$ and $|{\cal C}_{\theta_{1}x}|\ll 1$ [2], we see that
$E^{r}_{\underline{5}}$ receives no ${\cal O}(\beta)$ corrections. Further,
the constants appearing in the EOM (13) for massive mesinos are therefore
given below:

$\displaystyle{\cal
C}_{\frac{E^{t}_{\underline{1}}}{E^{r}_{\underline{5}}}}^{M_{\rm
Mesino}}=\frac{\beta\sqrt{{g_{s}}}M\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{17496\sqrt{\pi}\epsilon^{5}\left(\log
N\right)^{4}N^{3/4}{N_{f}}{r_{h}}^{3}\alpha_{\theta_{2}}^{3}}+\frac{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}}{4{r_{h}}},$
$\displaystyle{\cal
C}_{\left(\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}\right)^{\prime}}^{M_{\rm
Mesino}}=\frac{2\beta
M\left(-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{6561\pi^{3/2}\epsilon^{5}\sqrt{{g_{s}}}\left(\log
N\right)^{4}N^{7/4}{N_{f}}{r_{h}}\alpha_{\theta_{2}}^{3}}+\frac{4{r_{h}}}{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}},$
$\displaystyle{\cal C}_{rt}^{M_{\rm
Mesino}}=-\frac{14}{3\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}},$ $\displaystyle{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}=\frac{4{r_{h}}}{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}}+\frac{4\beta
M\left(-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\sqrt{\frac{6a^{2}+{r_{h}}^{2}}{9a^{2}+{r_{h}}^{2}}}\log^{3}({r_{h}})}{6561\sqrt{3}\pi^{3/2}\epsilon^{5}\sqrt{{g_{s}}}\left(\log
N\right)^{4}N^{7/4}{N_{f}}{r_{h}}\alpha_{\theta_{2}}^{3}},$
$\displaystyle{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}=\frac{2\sqrt{2}\sqrt[4]{\pi}\sqrt[4]{{g_{s}}}\sqrt[4]{N}}{\sqrt{{r_{h}}^{3}}}-\frac{4\sqrt{\frac{2}{3}}\beta\sqrt[4]{{g_{s}}}M\left(-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\left(6a^{2}+{r_{h}}^{2}\right)\log^{3}({r_{h}})}{6561\pi^{3/4}\epsilon^{5}\left(\log
N\right)^{4}N{N_{f}}{r_{h}}^{7/2}\alpha_{\theta_{2}}^{3}\left(9a^{2}+{r_{h}}^{2}\right)},$
$\displaystyle{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{r}_{\underline{5}}}}^{M_{\rm
Mesino}}=\frac{\sqrt{\frac{3}{2}}\pi^{3/4}{g_{s}}^{3/4}N^{3/4}}{{r_{h}}^{3/2}}+\frac{\beta{g_{s}}^{3/4}M\sqrt{\frac{1}{N}}\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{2187\sqrt{2}\sqrt[4]{\pi}\epsilon^{5}\left(\log
N\right)^{4}{N_{f}}{r_{h}}^{7/2}\alpha_{\theta_{2}}^{3}},$ $\displaystyle
a_{2\ \beta}^{M_{\rm Mesino}}\equiv-\frac{21.9}{\sqrt{g_{s}N}}+{\cal
C}_{rt}^{M_{\rm Mesino}}$ $\displaystyle{\cal A}_{1}^{M_{\rm
Mesino}}=\pi^{2}\left(-(2n+1)^{2}\right)T^{2}\Biggl{(}\frac{\beta\sqrt{{g_{s}}}M\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{17496\sqrt{\pi}\epsilon^{5}\left(\log
N\right)^{4}N^{3/4}{N_{f}}{r_{h}}^{3}\alpha_{\theta_{2}}^{3}}$
$\displaystyle+\frac{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}}{4{r_{h}}}\Biggr{)}-(0.\,+3.3i)(2n+1)T,$
$\displaystyle{\cal A}_{2}^{M_{\rm Mesino}}=\frac{2\beta
M\left(-19683\sqrt{6}\alpha_{\theta_{1}}^{6}-6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{6561\pi^{3/2}\epsilon^{5}\sqrt{{g_{s}}}\left(\log
N\right)^{4}N^{7/4}{N_{f}}{r_{h}}\alpha_{\theta_{2}}^{3}}-\frac{1.3{r_{h}}}{\sqrt{{g_{s}}}\sqrt{N}}+\frac{4{r_{h}}}{\sqrt{3\pi}\sqrt{{g_{s}}}\sqrt{N}}+2i\pi(2n+1)T,$
$\displaystyle{\cal B}_{1}^{M_{\rm Mesino}}=1.1i{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}+(2n+1)T{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{r}_{\underline{5}}}}^{M_{\rm
Mesino}}.$ (B5)

##  Appendix C $\tilde{z}=$Constant Embedding of Flavor $D6$-Branes Inclusive
of ${\cal O}(\beta)$ Corrections

The EOM for the embedding of the flavor $D6$-branes in the warped squashed
resolved conifold
$\tilde{z}=\tilde{z}(r)=\tilde{z}_{(0)}+\beta\tilde{z}_{(1)}$, up to ${\cal
O}(\beta)$, is given by:

$\displaystyle\hskip-36.135pt\frac{N^{3/5}{N_{f}}r^{2}\left(r^{4}-{r_{h}}^{4}\right)(\log
N-3\log({r_{h}}))\left(\tilde{z}_{(0)}^{\prime}+\beta\tilde{z}_{(1)}^{\prime}\right)}{4\sqrt{6}\pi^{7/4}{g_{s}}^{3/4}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{5/2}\sqrt{\frac{4\sqrt{\pi}\sqrt{{g_{s}}}r^{2}\alpha_{\theta_{2}}^{3}\left(6a^{2}+r^{2}\right)}{9a^{2}+r^{2}}+3\sqrt{3}N^{2/5}\left(r^{4}-{r_{h}}^{4}\right)\left(\tilde{z}_{(0)}^{\prime}+\beta\tilde{z}_{(1)}^{\prime}\right)^{2}}}$
$\displaystyle\hskip-36.135pt-\frac{0.0005\beta
MN^{87/20}{r_{h}}^{5}\left(-492.1\alpha_{\theta_{1}}^{6}-67.8\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}+\alpha_{\theta_{2}}^{4}\right)(r-{r_{h}})^{2}\log^{3}({r_{h}})(\log
N-3\log({r_{h}}))\tilde{z}_{(0)}^{\prime}(r)}{\epsilon^{5}{g_{s}}^{5/2}\log
N^{4}\alpha_{\theta_{1}}^{6}\alpha_{\theta_{2}}^{6}\left(9a^{2}+{r_{h}}^{2}\right)}={K^{(0)}}+\beta{K^{(1)}}.$
(C1)

At ${\cal O}(\beta^{0})$, (C) yields:

$\displaystyle\tilde{z}_{(0)}^{\prime}=\pm\frac{8\sqrt{6}\pi^{2}{g_{s}}^{3/4}{K^{(0)}}\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{5/2}\sqrt{\frac{\sqrt{{g_{s}}}r^{2}\alpha_{\theta_{2}}^{3}\left(6a^{2}+r^{2}\right)}{9a^{2}+r^{2}}}}{\sqrt{N^{2/5}\left(r^{4}-{r_{h}}^{4}\right)\left(N^{4/5}{N_{f}}^{2}r^{4}\left(r^{4}-{r_{h}}^{4}\right)(\log
N-3\log({r_{h}}))^{2}-288\sqrt{3}\pi^{7/2}{g_{s}}^{3/2}{K^{(0)}}^{2}\alpha_{\theta_{1}}^{4}\alpha_{\theta_{2}}^{5}\right)}}.$
(C2)

From (C2), one obtains $\tilde{z}(r)\in\mathbb{R}$ if ${K^{(0)}}=0$
(irrespective of whether one performs first a large-$N$ followed by a
small-$r$ expansion or vice versa). In a similar manner, at ${\cal O}(\beta)$,

$\displaystyle\hskip-28.90755pt\tilde{z}_{(1)}=c_{1}$
$\displaystyle\hskip-28.90755pt-\frac{4\pi^{2}{g_{s}}^{5/4}{K^{(1)}}r\alpha_{\theta_{1}}^{2}\alpha_{\theta_{2}}^{11/2}\sqrt{6a^{2}+r^{2}}}{N^{3/5}{N_{f}}{r_{h}}^{4}\sqrt{9a^{2}+r^{2}}\sqrt{9a^{2}-{r_{h}}^{2}}\sqrt{9a^{2}+{r_{h}}^{2}}(\log
N-3\log({r_{h}}))\sqrt{\frac{\sqrt{{g_{s}}}r^{2}\alpha_{\theta_{2}}^{3}\left(6a^{2}+r^{2}\right)}{9a^{2}+r^{2}}}}$
$\displaystyle\hskip-28.90755pt\times\Biggl{(}\sqrt{6}\left(\sqrt{{r_{h}}^{2}-6a^{2}}\sqrt{9a^{2}+{r_{h}}^{2}}\tan^{-1}\left(\frac{\sqrt{6a^{2}+r^{2}}\sqrt{9a^{2}-{r_{h}}^{2}}}{\sqrt{9a^{2}+r^{2}}\sqrt{{r_{h}}^{2}-6a^{2}}}\right)+\sqrt{-6a^{2}-{r_{h}}^{2}}\sqrt{9a^{2}-{r_{h}}^{2}}\tan^{-1}\left(\frac{\sqrt{6a^{2}+r^{2}}\sqrt{9a^{2}+{r_{h}}^{2}}}{\sqrt{9a^{2}+r^{2}}\sqrt{-6a^{2}-{r_{h}}^{2}}}\right)\right)$
$\displaystyle\hskip-28.90755pt-4\sqrt{9a^{2}-{r_{h}}^{2}}\sqrt{9a^{2}+{r_{h}}^{2}}\tanh^{-1}\left(\frac{\sqrt{\frac{3}{2}}\sqrt{6a^{2}+r^{2}}}{\sqrt{9a^{2}+r^{2}}}\right)\Biggr{)}.$

Again, a finite real $\tilde{z}_{(1)}$ is obtained only for ${K^{(1)}}=0$.
This hence implies a constant $\tilde{z}$-embedding up to ${\cal O}(\beta)$.

##  Appendix D Constants appearing in the Solution to the Mesino Wave-Function
for $T>T_{c}$

The parameters $\mu_{1,2,3},\Lambda$ in equation (26), are defined as follows
(in the following, terms of ${\cal O}\left(\frac{\beta}{N^{\alpha}}\right),\
\alpha\geq 1$ have been dropped as the same were sub-dominant as compared to
the order considered in the [2]):

$\displaystyle\mu_{1}\equiv-\frac{{\cal A}_{2}^{M_{\rm Mesino}}}{{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
}+\frac{2i{{\cal B}_{1}^{M_{\rm Mesino}}}}{{{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}}}+\frac{\sqrt{{\cal A}_{2}^{M_{\rm Mesino}}\ {}^{2}-2{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
{\cal A}_{2}^{M_{\rm Mesino}}+{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
^{2}+4{{\cal A}_{1}^{M_{\rm Mesino}}}\ {{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
}}{{{\cal C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}}\ }+1$
$\displaystyle=2.5\sqrt{-(0.3\,+0.4i)n+(0.1\,-0.2i)-0.3n^{2}}+(0.9\,-0.9i)n-(0.7\,+0.4i)-\frac{0.6(n+0.5){r_{h}}}{\sqrt{{g_{s}}}\sqrt{N}}$
$\displaystyle+{\cal O}\left(\cancelto{0}{\frac{\beta}{N^{5/4}}}\right),$
$\displaystyle\mu_{2}\equiv\frac{2\sqrt{{\cal A}_{2}^{M_{\rm Mesino}}\
{}^{2}-2{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
{\cal A}_{2}^{M_{\rm Mesino}}+{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
^{2}+4{{\cal A}_{1}^{M_{\rm Mesino}}}\ {{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\
}}{{{\cal C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}}\ }+1$
$\displaystyle=1+4.9\sqrt{-(0.25\,+0.4i)n+(0.1\,-0.2i)-0.3n^{2}}+{\cal
O}\left(\cancelto{0}{\frac{\beta}{N^{5/4}}}\right),$
$\displaystyle\mu_{3}\equiv\frac{i\sqrt{6}\pi^{3/4}{g_{s}}^{3/4}N^{3/4}{p}u}{{r_{h}}^{5/2}}+\frac{2i{{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}}{p}}{{{\cal
C}_{\frac{E^{r}_{\underline{5}}}{E^{t}_{\underline{1}}}}^{M_{\rm Mesino}}}\ }$
$\displaystyle=\frac{i\sqrt{6}\pi^{3/4}{g_{s}}^{3/4}N^{3/4}{p}u}{{r_{h}}^{5/2}}+\frac{i\sqrt{2}\beta{g_{s}}^{3/4}M{p}u\left(19683\sqrt{6}\alpha_{\theta_{1}}^{6}+6642\alpha_{\theta_{2}}^{2}\alpha_{\theta_{1}}^{3}-40\sqrt{6}\alpha_{\theta_{2}}^{4}\right)\log^{3}({r_{h}})}{2187\sqrt[4]{\pi}\epsilon^{5}\log
N^{4}\sqrt{N}{N_{f}}{r_{h}}^{9/2}\alpha_{\theta_{2}}^{3}}$
$\displaystyle=\frac{i\sqrt{6}\pi^{3/4}{g_{s}}^{3/4}N^{3/4}{p}u}{{r_{h}}^{5/2}}+{\cal
O}\left(\cancelto{0}{\frac{\beta}{N}}\right);$
$\displaystyle\Lambda\equiv\mu_{1}-\frac{2i{{\cal B}_{1}^{M_{\rm
Mesino}}}}{{{\cal
C}_{\frac{E^{x^{1}}_{\underline{2}}}{E^{t}_{\underline{1}}}}^{M_{\rm
Mesino}}}}={2.5\sqrt{-(0.3\,+0.4i)n+(0.1\,-0.2i)-0.3n^{2}}+(-0.9i)n+(1\,-0.4i)+{\cal
O}\left(\cancelto{0}{\frac{\beta}{N^{5/4}}}\right)}.$

##  Appendix E Summary of Applications of Top-Down Holographic QCD [1], [2]

One of the authors (AM) is working on the top-down holographic QCD from the
past few years. The holographic dual of finite $N$ QCD was first constructed
in [1] and then ${\cal O}(R^{4})$ corrections to [1] was obtained in [2].
Following is the summary of results obtained in this direction.

  * •

Summary of Applications of [1]: In [14], transport coefficients such as shear
viscosity, diffusion constant, electrical conductivity, charge susceptibility,
etc., of black $M3$-banes (black $M5$-branes wrapping a homologous sum of two
cycles) in the MQGP limit were obtained, and it was found that the ratio of
shear viscosity-to-entropy density is $1/4\pi$. In [11], deconfinement
temperature and mass scale of the first generations quarks were obtained
without the inclusion of ${\cal O}(R^{4})$ corrections relevant to thermal
QCD. Further, thermodynamic stability and $G_{2}$ structure of [1] and
temperature dependence of electrical conductivity and charge susceptibility
were also discussed in [11], [27]. In this process, Einstein’s law was
verified by computing the ratio of electrical conductivity-to-charge
susceptibility. For the discussion on Wiedemann-Franz law by computing the
thermal and electrical conductivities up to LO in $N$ and NLO in $N$
correction to the aforementioned transport coefficients and speed of sound
from the gauge invariant metric perturbations, see [28]. The glueball and
meson spectra of finite $N$ QCD have been obtained in [20] and [17]
respectively, and the appearance of Hypergeometric function is discussed in
appendix I. Decay of glueballs into mesons ($\pi$ and $\rho$ mesons) has been
discussed in [12] and for the QCD trace anomaly from ${\cal M}$-theory
perspective, see [29].

  * •

Summary of Applications of [2]: The low energy coupling constants at the NLO
in chiral expansion of $SU(3)$ chiral perturbation theory (for simplicity in
the chiral limit) were obtained from the aforementioned type IIA dual, in [4]
where we observed a connection between higher derivative terms and large-$N$
expansion. In the process of computing the deconfinement temperature $(T_{c})$
in [23, 30], a novel “UV-IR” mixing, non-renormalization $T_{c}$ beyond one-
loop in the zero instanton sector and “Flavor Memory” effect were obtained.
Further, we constructed a doubly holographic setup with a non-conformal bath
in [25] to obtain the Page curve of the related eternal black hole from a top-
down approach. One of the interesting results that we obtained in [25] is the
Page curve of the relevant eternal black hole for massless gravity on the
Karch-Randall brane. Massless graviton was responsible for the exponential-
in-$N$ suppressed entanglement entropy coming from higher derivative terms in
eleven-dimensional supergravity action. This provided us the connection
between the mass of graviton and higher derivative terms. On the Math side
with the aim of classifying non-supersymmetric thermal geometries relevant to
realistic top-down holographic duals of thermal QCD-like theories,
mathematical structures on the underlying six-, seven- and eight-folds were
studied in [2] and [31].

## References

  * [1] M. Dhuria and A. Misra, Towards MQGP, JHEP 1311 (2013) 001 [arXiv:hep-th/1306.4339].
  * [2] A. Misra and V. Yadav, On ${\cal M}$-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to ${\cal O}(R^{4})$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries, to appear in Advances in Theoretical and Mathematical Physics (issue 26.10) [arXiv:2004.07259 [hep-th]].
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