# On Hawking/Unruh Process: Where does the Radiation Come from?

Tadas K. Nakamura

###### Abstract

The energy source of the radiation in Unruh/Hawking process is investigated
with emphasis on the particle number definition based on conservation laws. It
has been shown that the particle radiation is not the result of pair creation
by the gravitational force, but the result of difference in the conservation
laws to define the particle number. The origin of the radiated energy in the
distant future corresponds to the zero point oscillations with infinitely
large wave numbers. This result implies the need of reconsideration on the
scenario of black hole evaporation.

CFAAS, Fukui Prefectural University, Fukui 910-1195, Japan<EMAIL_ADDRESS>
##  1 Introduction

The theory of quantum particle radiation by gravitational force [1, 2] is
generally accepted as well established, however, several authors pointed out
essential problems that might brow up the whole story [3, 4, 5] (see [6] for a
review). What called trans-Plankian problem has been known from very early
years [7]. In the last two decades, this problem has been investigated
intensively [8, 9].

The derivation of Hawking/Unruh process is based on so called cis-Plankian
physics, i.e., the theories of gravitation and quantum field we know at the
present. It is believed these theories will break down beyond an extremely
small scale, the Plank scale presumably, and we do not know what happens
there. The unknown physics in that scale is called trans-Plankian physics. The
calculation of Hawking/Unruh effect inevitably requires the wave modes with
infinitesimally small wave length (see, e.g., [6]), therefore, we need to know
the trans-Plankian physics to understand the radiation mechanism; this is the
trans-Plankian problem. There has been attempts [9] to derive the radiation
within the cis-Plankian scale, however, they are based on _ad hoc_ assumptions
yet to be tested experimentally.

Helfer [4] pointed out another issue we have to consider before the trans-
Plankian problem. The radiation at later times must have plied up near the
horizon at the time of black hole formation, and its backreaction is far from
negligible to the black hole metric. This implies the theory of Hawking
radiation is intrinsically inconsistent even within the framework of cis-
Plankian physics, because such backreaction may completely destroy the black
hole formation.

This is a serious problem. Papers on the trans-Plankian problem so far assume
that the Hawking radiation is well predicted by the cis-Plankian physics, and
discuss what will happen if we have to consider trans-Plankian effects.
However, if cis-Plankian physics itself fails to derive the Hawking radiation,
then we do not have any reason to believe the existence of radiation.

The purpose of the present paper is to formulate the Helfer’s conclusion [4]
from a different point of view. We wish to show the theory of black hole
evaporation is inconsistent even if the cis-Plankian physics is valid up to
infinitesimally small scale. To this end, we need another unknown physics
within the cis-Plankian regime: the effect of the zero-point oscillation to
the gravity. We do not know the general theory for this, however, there can be
two possibilities for the Hawking radiation.

One is such that the radiation can carry away the black hole energy to cause
its evaporation. In this case, the backreaction of the quantum filed is so
large as to alter the black hole formation completely [4]. We will see in the
present paper there can be another possibility that the field has no
backreaction to the black hole geometry. In this case, however, there will be
no black hole evaporation at all even with the existence of Hawking radiation.
This may be plausible because there are considerable amount of observational
evidence to believe the existence of black holes.

We take an approach a little different from the conventional quantization with
creation/annihilation operators in the present paper; the mathematical
structure is equivalent, but its interpretation is not the same. The
quantization with creation/annihilation operators takes two different steps,
transition from the classic to quantum field and introduction of the particle
picture, namely, at the same time. The essential step in the canonical
quantization method is to replace a classical Poisson bracket with a quantum
commutation relation regarding the field as a collection of quantum operators.
If conserved quantities have discrete eigenvalues with equal intervals, then
we can construct the particle picture. It is well known the latter is not
always possible in a curved spacetime.

It also should be noted these two steps do not have to be done at the same
place even when we have the particle picture. The commutation relation must be
given on a Chaucy surface on which the Poisson bracket is defined. The
particle picture, in contrast, does not have to be on the same surface. It can
be on some other spacelike surface, which does not have to be a Chaucy surface
as long as there exist some conservation laws on it.

In the present paper, the essential quantization, i.e., definition of the
commutation relation, will be done on the surface of constant time in
Minkowski/Kruscal coordinates. Then the particle picture is introduced based
on that quantization, not only on the same surface but also on the surface of
constant time in Rindler/Schwarzschild coordinates.

The particle picture is based on conservation laws in general. What we
directly measure is not the particle number itself, but some conserved
quantity such as energy or electric charge. We imagine there are $n$
particles, each of which carries a certain amount of conserved quantity, if
the total of the quantity has discrete values proportional to an integer $n$.

This means the concept of “particle number” is defined by conserved
quantities. If all the conserved quantities share the same $n$, then we can
define one unique particle number, however, this is not the case. There can be
several different definitions of particle numbers because there can be several
different Killing vector fields that determine the conservation laws in a
relativistic spacetime.

Consequently one physical state can have different particle numbers, and this
is what is happening in the Unruh/Hawking process. Particles are not created
in the literal meaning of “creation”, which means the particle number
increases as time goes on. Rather, what takes place is just a difference of
particle numbers caused by the difference in their definitions. This is in
agreement with the result of Belinski [3] calculated from another viewpoint.

We will see in the present paper the radiation of particles comes from the
vacuum state, i.e., zero particle state, of another kind of particle number.
This is possible because a vacuum is not a completely empty space but has zero
point oscillations. The continuous particle radiation can take place because
the zero point oscillations exist up to infinitely large wave numbers, which
means infinite amount of energy source.

The present paper is organized in the following. In section 2 we first review
some basic concepts to clarify the procedure of quantization used in the later
sections. We examine in Section 3 the case of Unruh process in a flat
spacetime, since it has the two different types of conservation laws clearly
defined; we can understand the problem with this simple analogy. We apply the
results obtained in Section 3 to the case of Schwarzschild black holes in
Section 4, and a brief summary is given in Section 5.

##  2 Basics

###  2.1 Time and Energy

The concept of energy is often used in a sloppy way, which sometimes leads to
misconceptions. The integration of the energy-momentum tensor cannot be
carried out in the curved spacetime in general, however, there can be well
defined “energy” as a globally conserved quantity if there exists a Killing
vector field. If a Killing vector $\xi_{\nu}$ is timelike, then the
integration $\int_{\Sigma}\xi_{\nu}T^{\nu\mu}d\Sigma_{\mu}$ ($T^{\nu\mu}$:
energy-momentum tensor) over an appropriate spacelike surface $\Sigma$ is
conserved with respect to the time evolution in $\xi_{\nu}$. If there are
several different timelike Killing vector fields, there can be the same number
of corresponding energies; the energies defined by different Killing vector
fields are different physical entities.

Sometimes this difference in energies is not well understood and causes
confusion; one good example is an intuitive explanation of the Hawking
radiation found in popular science books. It goes like: (1) a vacuum is not an
empty space but filled with instantaneous pair production of virtual
particles; (2) a pair of virtual particles can exist within a short time
period of $\Delta t\sim\hbar/\Delta E$ because of the uncertain principle; (3)
when such virtual particles are created near the event horizon, one of the
pair may fall into the black hole across the horizon during the time interval
of $\Delta t$; (4) once a virtual particle crosses the horizon, its energy
becomes negative; (5) then the other particle of the pair can have positive
energy without violating the energy conservation law.

This explanation does not specify the Killing vector field with which the time
and energy are defined. If the Killing vector is something like the
Schwarzschild time, then a particle takes infinitely long time to reach the
horizon, and cannot cross the horizon during the period of $\Delta t$. If, on
the other hand, the Killing vector is such that a particle can cross the
horizon within a finite period, then the corresponding energy does not change
the sign on the other side of the horizon. The pair production near the
horizon is not likely to occur to cause the Hawking radiation in both cases.

###  2.2 Conservation Laws and Particle Numbers

Usually the quantization process to investigate Hawking/Unruh process is based
on the creation/annihilation operators defined by the negative/positive
frequency modes. In the present paper we take one step backwards and perform
the quantization by replacing the Poisson bracket with the commutation
relation. Hereafter, let us use the word “quantization” with the meaning of
the transition from the classical to quantum theory, and does not necessarily
mean the particle picture.

The particle picture is derived from conserved quantities after the
quantization. If the quantum observable of a conserved quantity has the
structure of a harmonic oscillator, its eigenvalues are proportional to
$n+\frac{1}{2}$ with $n=0,1,2,\cdots$. Usually the constant $\frac{1}{2}$ is
subtracted out by normal ordering, thus the quantity is proportional to $n$.
When there are several conserved quantities that share the same $n$ for the
same state, then we can interpret $n$ as the particle number.

Suppose we establish quantization somehow, and find an observable $\hat{a}$
(hat mark indicates a quantum operator) and its Hermite conjugate
$\hat{a}^{\dagger}$ have the following commutation relation

$[\hat{a},\hat{a}^{\dagger}]=\hat{a}\,\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}=1\,.$
(1)

We use the unit system with $G=\hbar=c=1$ throughout the present paper. The
general theory of quantum harmonic oscillators tells us (see., e.g., [10]) an
observable defined as

$\hat{A}=\frac{A_{0}}{2}\left(\hat{a}\,\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}\right)$
(2)

has the eigenstates $\left|n_{A}\right\rangle$ that satisfies

$\hat{A}\left|n_{A}\right\rangle=A_{0}\left(n+\frac{1}{2}\right)\left|n_{A}\right\rangle\;,(n=0,1,2\cdots,)\,.$
(3)

if $\hat{a}$ has the commutation relation of (1).

For the above argument the observable $\hat{A}$ does not have to be related to
the Hamiltonian explicitly (note: the Poisson bracket has something to do with
the Hamiltonian implicitly), and there can be several choices for such
observables. For example if we define a new observable $\hat{b}$ as

$\hat{b}=\alpha\,\hat{a}+\beta\,\hat{a}^{\dagger}$ (4)

with $\alpha^{2}-\beta^{2}=1$ then it also satisfies the commutation relation
like (1) and thus
$\hat{B}=\frac{1}{2}B_{0}(\hat{b}\,\hat{b}^{\dagger}+\hat{b}^{\dagger}\hat{b})$
has the eigenvalues $B_{0}(n+\frac{1}{2})$. It is easy to confirm its
eigenstates $\left|n_{B}\right\rangle$ are not the eigenstates of $\hat{A}$,
i.e.,
$\hat{A}\left|n_{B}\right\rangle\neq(n+\frac{1}{2})\left|n_{B}\right\rangle$,
and vice versa.

Both pairs $\hat{a}$, $\hat{a}^{\dagger}$ and $\hat{b}$, $\hat{b}^{\dagger}$
have the structure of annihilation/creation operators, however, it is not
enough for the particle picture. To construct the particle picture with
$\hat{a}$ and $\hat{a}^{\dagger}$, $\hat{A}$ must obey a conservation law in
time, i.e., $\partial\hat{A}/\partial t=0$ (here we employ the Heisenberg
picture) at least approximately. If $\hat{A}$ rapidly changes even without
interactions, so does $n$, and it is not appropriate to regard $n$ as a
particle number.

When the Hamiltonian does not depend on time explicitly, the condition of
$\partial\hat{A}/\partial t=0$ is equivalent to the following commutation
relation:

$[\hat{A},\hat{H}]=0\,.$ (5)

Obviously the Hamiltonian itself satisfies this condition, therefore, the
Hamiltonian is usually used to introduce the particle picture. Then the
operators $\hat{a}$ and $\hat{a}^{\dagger}$ become the amplitudes of wave
modes with positive and negative frequencies respectively, which are usually
used in the procedure of quantization as the annihilation and creation
operators. Therefore, the mathematical structure in the present paper is
equivalent to the one in the conventional method.

If there are other conserved observables with respect to the time $t$, then
they share the same set of eigenstates with the Hamiltonian $\hat{H}$ because
of the commutation relation (5). Therefore $n$ can be regarded as the particle
number without specifying the conserved quantities, as long as the
conservation laws are on the same time evolution of $t$. Especially, the
ground sate of the observables is uniquely determined, and we call it
“vacuum”. We can define the number operator as

$\hat{N}=\hat{a}^{\dagger}\hat{a},$ (6)

whose eigenvalue is the particle number $n$, and the vacuum means the
eigenstate with $n=0$.

However, in relativistic spacetimes there can be several different types of
time evolution with different sets of conservation laws because several
different Killing vector fields can exist; the Minkowski and Rindler times in
a flat spacetime are a good example.

If two different types of time evolution have their own conservation laws,
then the conserved quantities that belong to different time evolution may have
different sets of eigenstates. Consequently the ground state in one time
evolution is not the ground state in another, in other words, they have
different vacuum states. This is what causes Hawking/Unruh radiation as we
will see in the next section.

##  3 Unruh process

###  3.1 Minkowski Coordinates

Suppose the following real valued Klein-Goldon equation in a two dimensional
Minkowski spacetime where $t$ and $x$ are the time and space coordinates:

$\phi_{,tt}-\phi_{,xx}=0\,.$ (7)

We write $\partial\phi/\partial t=\phi_{,t}$ etc. in shorthand. We take the
Cauchy surface for the canonical dynamics as the one defined with
$t=\textnormal{constant}$, then the Hamiltonian may be written as

$H=\int_{-\infty}^{\infty}\frac{1}{2}\left[\phi_{,t}^{2}(t,x)+\phi_{,x}^{2}(t,x)\right]\,dx\,.$
(8)

We expand the field as

$\phi(x,t)=\int\left[a(k)\,u(k;x,t)+a^{*}(k)\,u^{*}(k;x,t)\right]dk\,,$ (9)

with mode functions

$u(k;x,t)=\frac{1}{\sqrt{4\pi\omega}}\,e^{-i\omega t+ikx}\,,$ (10)

where $\omega=|k|$ and the asterisk indicates complex conjugate. Precisely
speaking, $a_{k}$ diverges to infinity as we usually encounter in the Fourier
transform; we assume some appropriate prescription, such as the
distribution/hyperfunction formulation, has been applied to avoid this
difficulty in this paper.

The essential transition from the classical to the quantum field is done by
replacing $a(k)$ and $a^{*}(k)$ with the operators satisfying the commutation
relation of

$\left[\hat{a}(k),\hat{a}(k^{\prime})^{\dagger}\right]=\hat{a}(k)\,\hat{a}^{\dagger}(k^{\prime})-\hat{a}^{\dagger}(k)\,\hat{a}(k^{\prime})=\delta(k-k^{\prime})\,.$
(11)

The above quantization is based on the Cauchy surface of
$t=\textnormal{constant}$. It should be noted that the quantization in this
paper takes place only once at this point. Later we introduce the particle
picture on the surface of constant Rindler time, but it is expressed by a
linear superposition of $\hat{a}$ and $\hat{a}^{\dagger}$ and based on the
commutation relation defined here.

The Hamiltonian (8) can be expressed as a collection of quantum harmonic
oscillators:

$\hat{H}=\int\frac{\omega}{2}\left[\hat{a}(k)\,\hat{a}(k)^{\dagger}+\hat{a}^{\dagger}(k)\,\hat{a}(k)\right]dk\,.$
(12)

Therefore, we can define the particle number as explained in the previous
section, and ground sate of $\hat{H}$ is called “vacuum”.

Now that we have the quantized operators $\hat{a}(k)$ and
$\hat{a}^{\dagger}(k)$, we can calculate the field $\hat{\phi}(x,t)$ at any
point of the spacetime as a quantum observable. Any classical quantity defined
from the classical field $\phi$ can be quantized by replacing

$\phi\rightarrow\hat{\phi}=\int\left[\hat{a}(k)\,u(k;x,t)+\hat{a}^{\dagger}(k)\,u^{*}(k;x,t)\right]dk\,.$
(13)

###  3.2 Rindler Coordinates

The Hamiltonian $H$ in (8) is the energy with the conservation law based the
Killing vector field of Minkowski time $\partial_{t}$. We examine in the
following another conservation law resulting from another Killing vector field
$\kappa x\partial_{t}-\kappa t\partial_{x}$, where $\kappa$ is a real constant
that corresponds to the relativistic acceleration. The energy $M$ for this
Killing vector field is written in the classical field theory as

$M=\int_{\Sigma(\eta)}\frac{1}{\kappa(t^{2}-x^{2})}\left(t\phi_{x}^{2}+x\phi_{,t}^{2}\right)d\Sigma\,,$
(14)

where $\Sigma(\eta)$ is a surface specified by $t/x=\tanh(\kappa\eta)$ and
$x>0$. Clearly $M$ is not the same quantity as $H$, therefor, let us
distinguish $M$ and $H$ by calling them “Rindler energy” and “Minkowski
energy” respectively.

The density of the Rindler energy is conserved locally, and there is no
Rindler energy flow across the left and right Rindler wedges, thus we have
$\partial M/\partial\eta=0$; once we calculate $M$ on a surface $\Sigma(\eta)$
with a given $\eta$, then the result holds for all $\eta$. When we choose
$\eta=0$, we can express $M$ with the coefficients $a(k)$ of the Minkowski
modes. Since $M$ is quadratic in $\phi$ we can write

$M=\iint\left[A(k,k^{\prime})\,a(k)\,a(k^{\prime})+B(k,k^{\prime})\,a(k)\,a^{*}(k^{\prime})+C(k,k^{\prime})\,a^{*}(k)\,a^{*}(k^{\prime})\right]dk\,dk^{\prime}\,,$
(15)

with coefficients $A$, $B$, and $C$ that do not depend on $\eta$ or $t$.

The above quantity is quantized by replacing $a(k)\rightarrow\hat{a}(k)$ and
$a^{*}(k)\rightarrow\hat{a}^{\dagger}(k)$ as

$\hat{M}=\iint\left[A\,\hat{a}(k)\,\hat{a}(k^{\prime})+\frac{B}{2}\left(\hat{a}(k)\,\hat{a}^{\dagger}(k^{\prime})+\hat{a}^{\dagger}(k)\,\hat{a}(k^{\prime})\right)+C\,\hat{a}^{\dagger}(k)\,\hat{a}^{\dagger}(k^{\prime})\right]dk\,dk^{\prime}\,.$
(16)

As noted before, this quantization is based on the Cauchy surface of
$t=\textnormal{constant}$, not $\Sigma(\eta)$. Therefore, $\Sigma(\eta)$ does
not have to be a Cauchy surface.

What we wish to show in the following is that the particle numbers defined by
$\hat{M}$ and $\hat{H}$ are not the same. Before that, we have to show that
$\hat{M}$ surely can define the particle number. Suppose an operator $\hat{b}$
is defined as a linear superposition of $\hat{a}$ and $\hat{a}^{\dagger}$ as

$\hat{b}(p)=\int\left[\alpha(p,k)\,\hat{a}(k)+\beta(p,k)\,\hat{a}^{\dagger}(k)\right]dk\,.$
(17)

If $\hat{b}$ satisfies the commutation relation

$[\hat{b}(p),\hat{b}^{\dagger}(p)]=\delta(p-p^{\prime})\,,$ (18)

and $\hat{M}$ can be expressed with $\hat{b}$ as

$\hat{M}=\frac{1}{2}\int\sigma\left[\hat{b}(p)\,\hat{b}^{\dagger}(p)+\hat{b}^{\dagger}(p)\,\hat{b}(p)\right]dp\,,$
(19)

then we can define the particle number with $\hat{M}$ in the similar way as
done in the previous subsection with $\hat{H}$.

It is possible show the above two equations with direct calculation, however,
it is easier to use the following Rindler coordinates $(\eta,\rho)$ for the
right Rindler wedge i.e., region of $x>0,$ |x|>|t|:

$t=\rho\sinh(\kappa\eta),\;x=\rho\cosh(\kappa\eta)\,.$ (20)

Then the Rindler energy $\hat{M}$ may be written as

$\hat{M}=\int_{0}^{\infty}\frac{1}{\kappa\rho}\left(\hat{\phi}_{,\eta}^{2}+\kappa^{2}\rho^{2}\hat{\phi}_{,\rho}^{2}\right)d\rho\,.$
(21)

We introduce the eigenfunctions

$v(p;\eta,\rho)=\frac{1}{\sqrt{4\pi\sigma}}\,\exp(-i\sigma\eta+i\kappa^{-1}p\ln(\kappa\rho))\,,$
(22)

with $\sigma=|p|$. The wave function $\hat{\phi}$ can be expanded in the right
Rindler wedge as

$\hat{\phi}=\int\left[\hat{b}(p)\,v(p)+\hat{b}^{\dagger}(p)\,v^{*}(p)\right]dp\,,$
(23)

then $\hat{M}$ in (21) can be cast into (19). In this context $\alpha$ and
$\beta$ in (17) are equivalent to the Bogolubov coefficients that satisfy

$\int\left[\alpha(p_{1},k)\alpha^{*}(p_{2},k)-\beta(p_{1},k)\beta^{*}(p_{2},k)\right]dk=\delta(p_{1}-p_{2})\,,$
(24)

The above property combined with (11) yields the commutation relation of (18),
therefore we can define particle numbers with $\hat{M}$.

What we do next is to compare the eigenstates of $\hat{M}$ and $\hat{H}$. From
(16) it is clear that $\hat{M}$ and $\hat{H}$ do not share the same set of
eigenstates unless $A$ and $C$ vanishes. We can see from (17) and (15) that
$A$ and $C$ vanished when $\beta=0$, however, $\beta$ can be evaluated by a
straightforward integration (see, e.g., [11]), resulting

$\left|\beta(p,k)\right|^{2}=\frac{1}{2\pi\omega}\,\frac{1}{e^{2\pi\sigma/\kappa}-1}\neq
0\,.$ (25)

Therefore, the eigenstates of $\hat{M}$ are not the eigenstates of $\hat{H}$.
This means that the ground state of $\hat{H}$ is not the ground state of
$\hat{M}$, which can be stated in other words as “a vacuum defined by the
Minkowski energy is not a vacuum defined by the Rindler energy”. The expected
value of the particle number defined by Rindler energy (we call Rindler
particle number hereafter) in the Minkowski vacuum can be calculated using
(25), resulting the well known Plankian distribution [11].

What we have seen above is not surprising since different operators may have
different sets of eigenstates. However, it contradicts with the picture of the
pair production by the gravitational force often found in intuitive
explanations like the one in Section 2. The particle number defined by
Minkowski energy (Minkowski particle number hereafter) is zero for all $t$,
and the Rindler particle number has the fixed Plankian distribution for all
$\eta$. This is consistent with the time symmetry; the vacuum in the flat
spacetime must be symmetric in time, both in $t$ and $\eta$, but the particle
creation process is not symmetric.

###  3.3 Origin of the Radiation

The Rindler particle number in the Unruh process is calculated by the expected
value of the Rindler energy for the ground state of the Minkowski energy,
i.e., $\left\langle 0_{H}\right|\hat{M}\left|0_{H}\right\rangle$. If the
“vacuum” were a completely empty space, i.e.,
$\hat{a}\left|0_{H}\right\rangle=\hat{a}^{\dagger}\left|0_{H}\right\rangle=0$,
then $\left\langle 0_{H}\right|\hat{M}\left|0_{H}\right\rangle$ would vanish.
However, this is not true since the quantum ground state has zero point
oscillation, and hence $\left\langle
0_{H}\right|\hat{M}\left|0_{H}\right\rangle\neq 0$. This means the “particles”
found in the Unruh process comes from the zero point oscillation of the
Minkowski modes.

Then what we wish to know is the properties of zero point oscillations that
contribute to the continuous radiation of Rindler energy. In the present paper
we concentrate on right moving waves, i.e., $\omega k<0$ or $\sigma p<0$,
since their analog in the Schwarzschild spacetime play the key role in the
black hole evaporation. It should be noted, however, left moving waves are
also problematic and should be examined in the next step. We first examine the
properties of waves in the classical limit, and then apply the result to the
quantum vacua.

To begin with, we observe that the a eigenmode (22) has infinite Rindler
energy in a finite region of $0\leq\rho<\rho_{c}$ with arbitrary position
$\rho_{c}$ in the right Rindler wedge; this can be confirmed by the following
direct integration:

$\lim_{\varepsilon\rightarrow
0}\int_{\varepsilon}^{\rho_{c}}\frac{1}{\kappa\rho}\,\left(\hat{v}_{,\eta}^{2}+\kappa^{2}\rho^{2}\hat{v}_{,\rho}^{2}\right)\,d\rho\rightarrow\infty\,.$
(26)

Also it is easy to confirm there is a constant rightward outflux of the
Rindler energy at $\rho=\rho_{c}$ by direct calculation. This outflux comes
from the region of $0\leq\rho<\rho_{c}$, but the total Rindler energy can be
conserved because the amount of the Rindler energy in that region is
infinitely large. Belinski [3] considered this fact as physically unacceptable
and concluded the radiation results from $v(p)$ is just a mathematical
illusion.

The present paper takes a different interpretation. The infinite Rindler
energy can be physically real as long as we believe zero point oscillations
exist for any high frequency modes, because the collection of such
oscillations has infinitely large Rindler energy even in a finite volume.

To see this, we examine the behavior of the a wave packet in the following
form (“$c.c.$” means complex conjugate):

$\phi(\eta,\rho)=\exp\left(\frac{-(\rho-e^{\kappa\eta}\,\rho_{0})^{2}}{(e^{\kappa\eta}\,s_{0})^{2}}\right)\,\exp[-i\sigma\eta+i\kappa^{-1}p\ln(\kappa\rho)]+c.c.\,.$
(27)

This wave packet was initially localized around $\rho=\rho_{0}$ with width
$s_{0}$ at $\eta=0$, and propagates rightward. The width of the packet becomes
larger and the wave number becomes smaller as a result of wave propagation.

The wave packet can be expanded by the Minkowski modes $u(k;t,x)$ as

$\phi(t,x)=\int\left[(\phi,u(k))\,u(k;t,x)+(\phi,u^{*}(k))\,u^{*}(k;t,x)\right]\,dk$
(28)

with the Klein-Goldon inner products $(\phi_{1},\phi_{2})$, which can be
calculated at $t=\eta=0$ as

$(\phi_{1},\phi_{2})=\int\left[\phi_{1,t}\phi_{2}^{*}-\phi_{1}\phi_{2,t}^{*}\right]_{t=0}\,dx\,.$
(29)

When $s\ll\rho_{0}\ln(\kappa\rho_{0})$ then we can approximate

$\phi(\eta,\rho)\simeq\exp\left(\frac{-(\rho-\rho_{1})^{2}}{s_{1}^{2}}-ip\eta+\frac{ip}{\kappa}\left[\ln(\kappa\rho_{1})+\frac{1}{\rho_{1}}\,(\rho-\rho_{1})\right]\right)+c.c.\,,$
(30)

where $s_{1}=s_{0}\,e^{\eta}$ and $\rho_{1}=\rho_{0}\,e^{\eta}$ are the width
and center of the wave packet at a time $\eta$. Using the above approximation
we obtain

$\phi(t,x)=\frac{2}{s_{0}\sqrt{\pi}}\exp\left[i\kappa^{-1}p\ln(\kappa\rho_{0})\right]\int
e^{-ik\rho_{0}}\exp\left[-s_{0}^{2}(k-p/\kappa\rho_{0})^{2}\right]e^{-i\omega
t+ikx}dk+c.c.$ (31)

from (28) with (29). The above expression means that the wave packet comes
from the Minkowski modes with wave number around $k_{0}=p/\kappa\rho_{0}$ when
$s_{0}\gg\kappa\rho_{0}/p$.

Suppose we find a wave packet around $\rho_{1}$ at a given time
$\eta=\eta_{1}\,(>0)$ in the Rindler space then its position at $\eta=0$ was
$\rho_{0}=\rho_{1}\,e^{-\kappa\eta_{1}}$, therefore, the packet consists of
the Minkowski modes with $k\sim k_{0}=p\,e^{\kappa\eta_{1}}/\kappa\rho_{0}$.
When we regard the wave field at $\eta=\eta_{1}$ as a superposition of such
wave packets, we see that the waves in the region of $0<\rho<\rho_{1}$ at
$\eta=\eta_{1}$ consists of Minkowski modes with wave numbers larger than
$k_{0}=p\,e^{\kappa\eta_{1}}/\kappa\rho_{0}$.

Since $k_{0}\rightarrow\infty$ in the limit of $\eta_{1}\rightarrow\infty$, we
understand the Rindler energy radiation at the distant future in $\eta$ comes
from the Minkowski modes with infinitely large wave numbers. The Rindler
coordinates represent an observer with constant acceleration, and the relative
velocity of the accelerating observer to the rest frame becomes infinitely
large in the limit of $\eta\rightarrow\infty$. Waves with finite wave numbers
in this limit are infinitely red shifted, therefore its original wave number
must have been infinitely large.

Now let us apply the above observation to quantum vacua to see the origin of
the Rindler particles. Suppose the quantum state is Minkowski vacuum, i.e.,
the ground state of the Minkowski energy. Then the state has zero point
oscillation up to infinitely large wave numbers. Usually the energy of these
zero point oscillation is subtracted out by normal ordering, and we regard
there is no particle in the ground state. However, the ground state of the
Minkowski energy is not the ground state of the Rindler energy, which means
the existence of the Rindler particles. The continuous radiation of Rindler
particles is possible for any large $\eta$ because the zero point Minkowski
energy exist for modes with any large wave numbers. The radiated Rindler
energy must have been piled up near $\rho=0$ at the initial time of $\eta=0$,
since there is no particle creation as we have seen in the previous
subsection.

##  4 Hawking Radiation

Let us move on to the Hawking radiation from a Schwarzchild black hole in this
section. We introduce the Schwarzchild coordinates $(t,r,\theta,\varphi)$
whose metric is

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}\cos^{2}\theta
d\varphi\,.$ (32)

The Kruscal coordinates $(u,v,\theta,\varphi)$ are related to
$(t,r,\theta,\varphi)$ as

$\displaystyle u^{2}-v^{2}$ $\displaystyle=$ $\displaystyle
2M(2M-r)\exp\left(\frac{r}{2M}\right)\,,$ (33)
$\displaystyle\left|\frac{u-v}{u+v}\right|$ $\displaystyle=$
$\displaystyle\exp\left(\frac{t}{2M}\right)\,.$ (34)

The rest of the coordinates, $\theta$ and $\varphi$, are unchanged.

It is generally accepted that the quantum properties of vacuum near the
Schwarzchild event horizon is essentially the same as those in the Rindler
spacetime [2, 12], therefore, the results we obtained in the previous section
are basically valid by replacing Minkowski/Rindler coordinates with
Kruscal/Schwarzchild coordinates (note: $t$ in the Schwarzchild spacetime
corresponds to $\eta$, not $t$, in the flat spacetime). There are, however,
two fundamental differences. One is the definition of the energy in Kruscal
coordinates, and the other is the backreaction of the quantum fields to the
black hole metric. The latter causes the essential problem in the scenario of
the black hole evaporation.

The first difference is about the energy that corresponds to the Minkowski
energy. The Kruscal time $u$ is not a global Killing time, and thus there is
no global energy conservation law like for the Minkowski energy. However, $u$
can be approximately regarded as a Killing time near the horizons. As we have
seen in the previous section, the radiation in later time comes from the
infinitely high frequency modes infinitesimally near the horizon, therefore
the energy is conserved approximately for these waves.

This is in parallel to the approximation of geometrical optics used by Hawking
in his original paper [1]; geometrical optics assumes locally constant
frequency, which means locally constant energy. Hereafter we assume the energy
corresponds to the Kruscal time $u$ is approximately conserved, and treat it
in the same way as for the Minkowski energy in the previous subsection. We
call it Hartle-Hawking energy since its ground state is often called Hartle-
Hawking vacuum. The energy defined with the Schwarzchild time is called
Boulware energy hereafter for the same reason.

We have another problem in the definition of the Hartle-Hawking energy in a
Schwarzchild spacetime. Minkowski energy is defined as an integration over a
surface of $t=\textnormal{constant}$ in a flat spacetime. If we introduce a
similar definition for the Hartle-Hawking energy with Kruscal time
$u=\textnormal{constant}$, the energy would include the part of the white hole
in the extended Schwarzchild spacetime. This difficulty may be avoided by
analyzing the black hole formation process by a star collapse, or the
analytical continuation method proposed by Hartle and Hawking [13]. A detailed
analysis on this point will be given in a forthcoming paper of the author.

The second difference is far more serious; the energy of zero point
oscillations may change the metric. Usually the zero point energy is
subtracted out by normal ordering in the source term of the Einstein equation,
and the vacuum does not have effect on the metric. This means only the excited
state of the energy can cause gravitation. However, as we have seen in the
previous section, the ground state of Hartle-Hawking energy is not the ground
state of the Boulware energy, and vise versa.

In a flat spacetime we consider the ground state of the Minkowski energy is
the state of no gravitation, because the Minkowski coordinates are the
“natural” coordinate system. We have seen there is infinite accumulation of
the Rindler energy near $\rho=0$ for a Rindler mode $v(p;\eta,\rho)$. There
must be an infinitely strong source of gravitational force at $\rho=0$ if we
assume the ground state of the Rindler energy is the state of no gravitation,
since the vacuum state defined by the Minkowski energy is the excited state of
the Rindler energy. This is not plausible, and we can conclude the ground
state of the Minkowski energy has no effect on the metric.

In contrast, we do not know which coordinate system is “natural” to calculate
the energy (stress-energy tensor) for a curved spacetime in general (see,
e.g., [14]). The Schwarzchild coordinates are implicitly assumed to be
“natural” in the scenario of the black hole evaporation, in other words,
excited states of Boulware energy causes the gravity. The black hole
evaporation is believed to be the result of the Hawking radiation that carries
the energy away from the black hole, and the energy in this context is the
Boulware energy; this means the Boulware energy can have effects on the black
hole metric somehow.

If this is true, however, the Rindler energy radiated at later times must have
been exist just outside of the horizon from the beginning [3]. The energy does
not come from inside the black hole, but comes from the Minkowski modes with
extremely high wave numbers. This means the backreaction of the quantum field
is far from negligible to the black hole metric [4].

On the contrary, we can imagine the gravity is caused by the exited state of
Hartle-Hawking energy and its ground state has no effect on the metric. The
black hole can exist in this case, however, it cannot evaporate. There exists
a constant outflow of Boulware energy, but it is the ground state of the
Hartle-Hawking energy and does not have a backreaction on the metric.
Consequently the black hole metric is unchanged at all, just like the Unruh
process does not change the flat metric. There can be other possibilities for
the effect of the zero point energy to the metric, however, it is hard to
imagine there is an extremely convenient case which is favorable for the
scenario of black hole evaporation.

##  5 Summary

What we have seen in the present paper are:

  1. 1.

The ground state of Minkowski/Hartle-Hawking energy is not the ground states
of Rindler/Boulware energy, and this is what causes the Hawking/Unruh process.

  2. 2.

The quantum state is unchanged and particles pairs are not created in any
coordinate system; the number of Minkowski/Hartle-Hawking particles is zero
and number of Rindler/Boulware particles has time stationary Plankian
distribution all through the time, where “time” means the Rindler/Schwarzchild
time .

  3. 3.

The radiation of Rindler/Boulware energy in the distant future of
Rindler/Schwarzchild time comes from the zero point oscillation of
Minkowski/Hartle-Hawking energy with infinitely large wave frequencies.

  4. 4.

The effect of zero point energy to the metric is not known, however, we have
the following two possibilities for a Schwarzchild spacetime. The scenario of
black hole evaporation is inconsistent in both cases.

    1. (a)

If the Hawking radiation causes the black hole evaporation, it means the
excitation in the Boulware energy can cause the metric change. The Boulware
energy radiated later time was accumulated near the horizon at the initial
time, whose existence essentially alter the Schwarzchild metric from the
beginning.

    2. (b)

If, on the contrary, the Boulware energy of the Hawking radiation does not
affect the metric, then the Schwarzchild metric can exist as we expect, but
exists forever. There is no evaporation of the black hole.

We started the present study by assuming that the cis-Plankian physics is
valid for any small scale phenomena, and end up with the inconsistency of
black hole evaporation. This fact means we have no reason to believe the black
hole evaporation. A new theory of physics in trans-Plankian scale may save the
evaporation, but may not; we can imagine anything, but cannot believe. What we
can say for sure is that the physics we know at the present is not able to
predict the black hole evaporation, if the calculations in the present paper
are correct.

We see the scenario of black hole evaporation is inconsistent, however, we do
not know what is the consistent theory even within the cis-Plankian regime.
The problem deeply depends on the renormalization procedure in curved
spacetimes, to which we do not know the answer yet. It is often said Hawking
process can be a touchstone for the theory of quantum gravity. The author of
the present paper would like to say it also can be a touchstone for the
renormalization theory, or theory on what is avoided by renormalization at the
present, in curved spacetimes.

## References

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aainstitutetext: School of Physical Sciences, Indian Association for
Cultivation of Science,
Kolkata 700032, West Bengal, Indiabbinstitutetext: Department of Physics,
Indian Institute of Technology Roorkee,
Roorkee 247667, Uttarakhand, India

# Spin $2$ spectrum for marginal deformations of 4d $\mathcal{N}=2$ SCFTs

Sourav Roychowdhury b and Dibakar Roychowdhury<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute spin $2$ spectrum associated with massive graviton fluctuations in
$\gamma$-deformed Gaiotto-Maldacena background those are holographically dual
to marginal deformations of $\mathcal{N}=2$ SCFTs in four dimensions. Under
the special circumstances, we analytically estimate the spectra both for the
$\gamma$\- deformed Abelian T dual (ATD) as well as the non-Abelian T dual
(NATD) cases where we retain ourselves upto leading order in the deformation
parameter. Our analysis reveals a continuous spectra which is associated with
the breaking of the $U(1)$ isometry (along the directions of the internal
manifold) in the presence of the $\gamma$\- deformation. We also comment on
the effects of adding flavour branes into the picture and the nature of the
associated spin $2$ operators in the dual $\mathcal{N}=1$ SCFTs.

##  1 Introduction and summary

$\mathcal{N}=2$ SCFTs Gaiotto:2009we  those live in four dimensions and their
realization as dual supergravity solutions Gaiotto:2009gz -Petropoulos:2014rva
has witnessed as steady progress over the past one and half decade. In
particular, the realization of some of these Type IIA geometries Reid-
Edwards:2010vpm -Aharony:2012tz  (also known as the Gaiotto-Maldacena
background) as Abelian T dual (ATD) as well as non-Abelian T dual (NATD)
solutions of $AdS_{5}\times S^{5}$ has generated renewed interest in the
recent years Sfetsos:2010uq -Roychowdhury:2021eas .

In a typical Hanany-Witten set up Hanany:1996ie , the $\mathcal{N}=2$
dualities can be realised as an intersection of NS5-D4-D6 branes whose low
energy (IR) description is characterised by a quiver gauge theory containing
both color as well as flavour nodes. The color degrees of freedom are sourced
due to D4 branes while the flavour modes come through D6s.

In the IR, the quiver is represented by a (super)conformal fixed point which
preserves a global $SO(2,4)\times SU(2)\times U(1)$ symmetry. This is known as
the so called “holographic limit” where the dual supergravity description
makes sense.

In the present paper, however, we are interested in certain holographic
aspects associated to _marginal_ deformations of $\mathcal{N}=2$ SCFTs those
were introduced recently by authors in Nunez:2019gbg . The corresponding
quiver is conjectured to be describing a superconformal fixed point with
$\mathcal{N}=1$ supersymmetries. The dual supergravity ($\gamma$\- deformed)
background can be obtained following a sequence of TsT transformations
Lunin:2005jy -Gauntlett:2005jb  in the eleven dimensional description and
thereby following a Type IIA reduction.

The purpose of the present paper is to explore the properties of spin 2
operators associated to $\mathcal{N}=1$ superconformal quivers by probing the
structure of linearised gravitational perturbations in the dual Type IIA set
up. The spin 2 spectrum is what is identified through the massive graviton
equationPassias:2018swc -Speziali:2019uzn  in the dual supergravity
description.

The linearised perturbation equation is in principle difficult to solve
analytically, except for certain choices of the associated quantum numbers. In
fact, we are able to solve these equations for a particular choice of the
parameter $q(=1)$ where the graviton equation is analytically tractable and
the corresponding solutions turn out to be regular. This results in a non-
discrete mass spectrum for the graviton both in the deformed ATD as well as
NATD models. Here, $q$ is the quantum number associated with the residual
$U(1)$ isometry of the broken (as a results of the TsT transformation) $SU(2)$
of the parent Type IIA background.

Upon fixing one of the quantum numbers ($q=1$) we are finally able to solve
the massive graviton equation analytically considering the effects of the
$\gamma$\- deformation upto leading order in the perturbative expansion.
Contrary to the undeformed case Itsios:2019yzp , we notice that the effect of
the $\gamma$\- deformation is to change the spectrum of _zero_ (as well as non
zero) modes from discrete to continuous. These modes are associated to the
$\psi$\- direction of the internal manifold which turns out to be non-compact
and in particular unbounded in the presence of the $\gamma$-deformation.
However, for the ATD model, this turns out to be an isometry direction in the
absence of any deformation Itsios:2019yzp .

Therefore, to summarise, the non-discrete spectra is an artefact of the
breaking of the abelian isometry in the presence of the $\gamma$\-
deformation. One might expect a different scenario once the flavour D6 branes
are placed so that the $\psi$\- direction is bounded. However, as we outline
in the Appendix C, the corresponding massive graviton equations are quite
nontrivial to deal with and specially in the presence of the $\gamma$\-
deformation.

The rest of the paper is organised as follows. We briefly outline the
$\gamma$\- deformed set up in Section 2. In Section 3, we work out the
detailed structure of the linearised perturbation equations. In Section 4, we
compute the spin 2 spectrum taking specific examples of $\gamma$\- deformed
ATD as well as NATD models. Finally, we draw our conclusion in Section 5,
where we outline the possible structure of the spin 2 operator in the dual
field theory.

##  2 Deformed Gaiotto-Maldacena background in Type IIA

To begin with, we review in detail the marginal deformations of 4d
$\mathcal{N}=2$ SCFTs and their holographic duals in Type IIA. The marginal
deformation of these theories leads towards a new class of $\mathcal{N}=1$
SCFTs in 4d whose holographic dual (in Type IIA) has been obtained by authors
in Nunez:2019gbg . These class of geometries are termed as $\gamma$\- deformed
Gaiotto-Maldacena (GM) backgrounds those are constructed following an
algorithm developed in Lunin:2005jy -Gauntlett:2005jb .

The seed solution is considered to be an eleven dimensional background in M
theory. Typically, these class of geometries are obtained by applying an
$SL(3,R)$ transformation (along with a deformation parameter $\gamma$) in M
theory. Finally, the ten dimensional ($\gamma$\- deformed) Type IIA background
is constructed following a circle ($S^{1}$) reduction111See Nunez:2019gbg  for
details. Nunez:2019gbg

$\displaystyle d\hat{s}^{2}_{10}=e^{-\frac{\hat{\Phi}}{2}}f_{1}\Bigg{[}$
$\displaystyle
ds_{AdS_{5}}^{2}+\frac{f_{2}}{4f_{1}}\Big{(}d\sigma^{2}+d\eta^{2}\Big{)}+\frac{f_{3}}{4f_{1}}d\chi^{2}+\frac{f_{3}\sin^{2}\chi}{4f_{1}\big{(}1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi\big{)}}d\xi^{2}$
(2) $\displaystyle+\
\frac{f_{4}}{4f_{1}\big{(}1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi\big{)}}\Big{(}d\beta-\gamma
f_{5}\sin\chi d\chi\Big{)}^{2}\Bigg{]},$

where $\gamma$ is the deformation parameter such that the solution maps to the
standard $\mathcal{N}=2$ supersymmetric Gaiotto-Maldacena background Reid-
Edwards:2010vpm ; Aharony:2012tz ; Gaiotto:2009gz ; Lozano:2016kum ;
Nunez:2019gbg  in the limit of the vanishing deformation, $\gamma=0$.

The $AdS_{5}$ line element could be expressed as

$\displaystyle
ds_{AdS_{5}}^{2}=-\cosh^{2}rdt^{2}+dr^{2}+\sinh^{2}rd\Omega_{3}^{2},$ (3)

together with the metric functions $f_{i}(\sigma,\eta)$

$\displaystyle
f_{1}=\Bigg{(}\frac{2\dot{V}-\ddot{V}}{V^{\prime\prime}}\Bigg{)}^{\frac{1}{2}}~{};~{}f_{2}=f_{1}\frac{2V^{\prime\prime}}{\dot{V}}~{};~{}f_{3}=f_{1}\frac{2V^{\prime\prime}\dot{V}}{\Delta}~{};~{}f_{4}=f_{1}\frac{4V^{\prime\prime}}{2\dot{V}-\ddot{V}}\sigma^{2}$
(4) $\displaystyle
f_{5}=2\Big{(}\frac{\dot{V}\dot{V}^{\prime}}{\Delta}-\eta\Big{)}~{};~{}f_{6}=\frac{2\dot{V}\dot{V}^{\prime}}{2\dot{V}-\ddot{V}}~{};~{}f_{7}=-\frac{4\dot{V}^{2}V^{\prime\prime}}{\Delta}~{};~{}f_{8}=\Bigg{[}(2)^{12}\
\frac{4\big{(}2\dot{V}-\ddot{V}\big{)}^{3}}{V^{\prime\prime}\dot{V}^{2}\Delta^{2}}\Bigg{]}^{\frac{1}{2}}.$
(5)

The dot and the prime of the potential function $V(\sigma,\eta)$ are denoted
as

$\displaystyle\dot{V}=\sigma\partial_{\sigma}V\ ,\
V^{\prime}=\partial_{\eta}V\ ,\ \ddot{V}=\sigma\partial_{\sigma}\dot{V}\ ,\
V^{\prime\prime}=\partial_{\eta}^{2}V\ ,\
\dot{V}^{\prime}=\sigma\partial_{\sigma}V^{\prime}$ (6)
$\displaystyle\Delta=\big{(}2\dot{V}-\ddot{V}\big{)}V^{\prime\prime}+\big{(}\dot{V}^{\prime}\big{)}^{2}.$
(7)

The background NS plus RR fluxes together with the dilaton could be expressed
as Nunez:2019gbg

$\displaystyle\hat{B}_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\
\frac{1}{1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi}\bigg{(}f_{5}d\Omega_{2}(\chi,\xi)-\gamma
f_{3}f_{4}\sin^{2}\chi d\xi\wedge d\beta\bigg{)},$ (8)
$\displaystyle\hat{C}_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{16}\
\bigg{(}f_{6}d\beta+\gamma\big{(}f_{7}-f_{5}f_{6}\big{)}\sin\chi
d\chi\bigg{)},$ (9) $\displaystyle\hat{C}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\
\frac{1}{1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi}f_{7}d\beta\wedge
d\Omega_{2}(\chi,\xi),$ (10) $\displaystyle e^{2\hat{\Phi}}$ $\displaystyle=$
$\displaystyle\frac{f_{8}}{1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi}.$ (11)

##  3 Perturbations

Given the above set up (2)-(11), we are now in a position to examine the
spin-2 spectrum for the $\gamma$-deformed Gaiotto-Maldacena background those
correspond to marginal deformations of $\mathcal{N}=2$ SCFTs in four
dimensions.

Following the algorithm as in Itsios:2019yzp , we express the $\gamma$\-
deformed metric (2) (apart from the usual conformal factor) as sum of the AdS5
factor and a five-dimensional internal space $\mathcal{M}_{5}^{\gamma}$ in the
Einstein frame as

$\displaystyle ds^{2}_{10}=ds_{AdS_{5}}^{2}+ds_{\mathcal{M}_{5}^{\gamma}}^{2}\
.$ (12)

We set the notation below, where we use the indices $M$ to denote the
coordinates of the full ten-dimensional background. On the other hand, we use
the indices $x^{\mu}$ to label the AdS5 directions and $y^{m}$ for the
$\gamma$\- deformed five-dimensional internal space
$\mathcal{M}_{5}^{\gamma}$.

This results into a ten dimensional metric of the following form

$\displaystyle
ds^{2}_{10}=\tilde{g}_{MN}dx^{M}dx^{N}=\tilde{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+\tilde{g}_{mn}(y)dy^{m}dy^{n},$
(13)

where one can split the full ten dimensional metric as

$\displaystyle\tilde{g}_{MN}(x,y)=\begin{bmatrix}\tilde{g}_{\mu\nu}(x)&0\\\
0&\tilde{g}_{mn}(y)\\\ \end{bmatrix}.$ (14)

Next, we turn on the metric fluctuations along the AdS5

$\displaystyle\delta g_{\mu\nu}=e^{2A}h_{\mu\nu},$ (15)

while keeping the other background fluctuations to zero.

This allows us to express the metric schematically as

$\displaystyle
ds_{E}^{2}=e^{2A}\Bigg{[}\Big{(}\tilde{g}_{\mu\nu}(x)+h_{\mu\nu}\Big{)}dx^{\mu}dx^{\nu}+\tilde{g}_{mn}(y)dy^{m}dy^{n}\Bigg{]}.$
(16)

A straightforward comparison between equations (16) and (2) yields

$\displaystyle A=-\frac{\hat{\Phi}(\eta,\sigma,\chi)}{4}+\frac{1}{2}\ln
f_{1}(\eta,\sigma).$ (17)

Next, we decompose the metric perturbation

$\displaystyle h_{\mu\nu}(x,y)=\mathfrak{h}_{\mu\nu}^{[tt]}(x)F(y),$ (18)

along with the _transverse_ as well as the _traceless_ gauge
$\tilde{\nabla}^{\mu}\mathfrak{h}_{\mu\nu}^{[tt]}(x)=0=\tilde{g}^{\mu\nu}\mathfrak{h}_{\mu\nu}^{[tt]}(x)$.

Notice that, since the dilaton and the background fluxes do not change under
metric perturbations, therefore the corresponding equations in (120) are
trivially satisfied. On the other hand, the metric ($R_{MN}$) equation in
(120) could be rephrased as

$\displaystyle
R_{MN}-\frac{1}{2}\partial_{M}\Phi\partial_{N}\Phi-\frac{1}{2}\sum_{p=2}^{4}\gamma_{p}e^{\alpha_{p}\Phi}\Bigg{[}\Big{(}\mathcal{A}_{p}^{2}\Big{)}_{MN}-\beta_{p}g_{MN}\mathcal{A}_{p}^{2}\Bigg{]}=0,$
(19)

where we denote the remaining fields collectively along with other notations

$\displaystyle\mathcal{A}_{p}:=\\{F_{2},H_{3},F_{4}\\}\ ;\
\Big{(}\mathcal{A}_{p}^{2}\Big{)}_{MN}=\mathcal{A}_{MM_{1}..M_{p-1}}\mathcal{A}_{N}^{M_{1}..M_{p-1}}$
(20) $\displaystyle\alpha_{p}:=\Bigg{(}\frac{3}{2},-1,\frac{1}{2}\Bigg{)}\ ;\
\beta_{p}:=\Bigg{(}\frac{1}{16},\frac{1}{12},\frac{3}{32}\Bigg{)}\ ;\
\gamma_{p}:=\Bigg{(}1,\frac{1}{2},\frac{1}{6}\Bigg{)}.$ (21)

Next, we use (142) in order to simplify (19) while we simultaneously impose
the following conditions on the background fields:
$\bullet$ The only non zero part of $h_{MN}$ is $h_{\mu\nu}$ which means we
set $h_{\mu m}=h_{m\mu}=h_{mn}=0$.
$\bullet$ $h_{\mu\nu}$ is transverse and traceless.
$\bullet$ $A$ depends only on the internal coordinates $y^{m}$ namely,
$A=A(y^{m})$.
$\bullet$ Finally, the collective background fields
($\tilde{\mathcal{A}}_{p}$) also depend on the internal coordinates $y^{m}$.

Considering all the above points, one could finally express (19) as

$\displaystyle\Big{(}\tilde{g}^{\rho\sigma}\tilde{R}_{\sigma\mu}h_{\rho\nu}-\tilde{g}^{\kappa\sigma}\tilde{R}_{\nu\kappa\mu}^{\rho}h_{\sigma\rho}\Big{)}+\Big{(}\tilde{g}^{\rho\sigma}\tilde{R}_{\sigma\nu}h_{\rho\mu}-\tilde{g}^{\kappa\sigma}\tilde{R}_{\mu\kappa\nu}^{\rho}h_{\sigma\rho}\Big{)}-\tilde{\nabla}^{2}h_{\mu\nu}-8\tilde{\nabla}^{P}A\tilde{\nabla}_{P}h_{\mu\nu}$
(22)
$\displaystyle-2h_{\mu\nu}\tilde{\nabla}^{2}A-16h_{\mu\nu}\big{(}\tilde{\nabla}A\big{)}^{2}+h_{\mu\nu}\sum_{p=2}^{4}\beta_{p}\gamma_{p}e^{2(1-p)A+\alpha_{p}\Phi}\tilde{\mathcal{A}_{p}}^{2}=0.$
(23)

Next, considering $AdS_{5}$ to be of unit radius and using the fact

$\displaystyle\tilde{g}^{\rho\sigma}\tilde{R}_{\sigma\mu}h_{\rho\nu}-\tilde{g}^{\kappa\sigma}\tilde{R}_{\nu\kappa\mu}^{\rho}h_{\sigma\rho}$
$\displaystyle=$ $\displaystyle-4h_{\mu\nu}-h_{\mu\nu}=-5h_{\mu\nu}$ (24)
$\displaystyle\tilde{g}^{\rho\sigma}\tilde{R}_{\sigma\nu}h_{\rho\mu}-\tilde{g}^{\kappa\sigma}\tilde{R}_{\mu\kappa\nu}^{\rho}h_{\sigma\rho}$
$\displaystyle=$ $\displaystyle-4h_{\nu\mu}-h_{\nu\mu}=-5h_{\mu\nu},$ (25)

one could further simplify (22) as

$\displaystyle\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}h_{\mu\nu}+\tilde{\nabla}^{m}\tilde{\nabla}_{m}h_{\mu\nu}+8\tilde{\nabla}^{m}A\tilde{\nabla}_{m}h_{\mu\nu}+\mathcal{S}h_{\mu\nu}=0,$
(26)

where we denote the above function as

$\displaystyle\mathcal{S}=10+2\tilde{\nabla}^{2}A+16\big{(}\tilde{\nabla}A\big{)}^{2}-\sum_{p=2}^{4}\beta_{p}\gamma_{p}e^{2(1-p)A+\alpha_{p}\Phi}\tilde{\mathcal{A}_{p}}^{2}.$
(27)

The second and the third term of (26) could be further simplified as

$\displaystyle\tilde{\nabla}^{m}\tilde{\nabla}_{m}h_{\mu\nu}+8\tilde{\nabla}^{m}A\tilde{\nabla}_{m}h_{\mu\nu}=e^{-8A}\tilde{\nabla}^{m}\Big{[}e^{8A}\tilde{\nabla}_{m}h_{\mu\nu}\Big{]}\equiv\mathcal{L}^{(1)}\Big{[}h_{\mu\nu}\Big{]},$
(28)

which reveals (26) in its final form

$\displaystyle\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}h_{\mu\nu}+\mathcal{L}^{(1)}\Big{[}h_{\mu\nu}\Big{]}+\mathcal{S}h_{\mu\nu}=0.$
(29)

Here, the operator $\mathcal{L}^{(k)}$ is defined with respect to the internal
coordinates ($y^{m}$) as Passias:2018swc

$\displaystyle\mathcal{L}^{(k)}:=e^{-8kA}\tilde{\nabla}^{m}\Big{[}e^{8kA}\tilde{\nabla}_{m}\Big{]}.$
(30)

For a massive graviton of mass $M$ propagating in $AdS_{5}$, one satisfies the
Bachas Estes equation Itsios:2019yzp ; Passias:2018swc ; Bachas:2011xa ;
Passias:2016fkm ; Richard:2014qsa ; Gutperle:2018wuk ; Chen:2019ydk ;
Speziali:2019uzn  of the form

$\displaystyle\Bigg{[}\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}+\Big{(}2-M^{2}\Big{)}\Bigg{]}h_{\mu\nu}=0,$
(31)

which further simplifies (29) by replacing AdS5 derivatives
($\tilde{\nabla}^{\alpha}\tilde{\nabla}_{\alpha}$)

$\displaystyle\Bigg{[}\mathcal{L}^{(1)}+\mathcal{S}+\Big{(}M^{2}-2\Big{)}\Bigg{]}F(y)=0,$
(32)

which is entirely defined with respect to internal directions ($y^{m}$) where
we define

$\displaystyle\mathcal{L}^{(1)}F=\frac{1}{\sqrt{|\tilde{g}_{\mathcal{M}_{5}^{\gamma}}}|}\partial_{m}\Big{(}\sqrt{|\tilde{g}_{\mathcal{M}_{5}^{\gamma}}|}\
\tilde{g}^{mn}\partial_{n}F\Big{)}+8\tilde{g}^{mn}\partial_{m}A\partial_{n}F,$
(33)

where $\tilde{g}_{\mathcal{M}_{5}^{\gamma}}$ is the determinant of the
$\gamma$-deformed five dimensional metric associated with the internal space
(2)

$\displaystyle d\hat{s}^{2}_{{\mathcal{M}_{5}^{\gamma}}}$ $\displaystyle=$
$\displaystyle\frac{f_{2}}{4f_{1}}\Big{(}d\sigma^{2}+d\eta^{2}\Big{)}+\frac{f_{3}}{4f_{1}}d\chi^{2}+\frac{f_{3}\sin^{2}\chi}{4f_{1}\big{(}1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi\big{)}}d\xi^{2}$
(34) $\displaystyle+\
\frac{f_{4}}{4f_{1}\big{(}1+\gamma^{2}f_{3}f_{4}\sin^{2}\chi\big{)}}\Big{(}d\beta-\gamma
f_{5}\sin\chi d\chi\Big{)}^{2}.$

Using (34) together with (17) one can finally express (33) as

$\displaystyle\mathcal{L}^{(1)}F$ $\displaystyle=$
$\displaystyle\frac{2}{f_{2}}\Bigg{[}5\Big{(}\partial_{\eta}f_{1}\partial_{\eta}F+\partial_{\sigma}f_{1}\partial_{\sigma}F\Big{)}+\frac{1}{f_{3}f_{4}(\gamma^{2}f_{3}f_{4}+\csc^{2}\chi)}f_{1}\Big{(}2f_{2}\Big{(}\csc^{2}\chi
f_{3}\partial_{\beta}^{2}F$ (42)
$\displaystyle\gamma^{4}f_{3}^{2}f_{4}^{3}\partial_{\xi}^{2}F+f_{4}\Big{(}\csc^{2}\chi\Big{(}\Big{(}\cot\chi-2\partial_{\chi}\hat{\Phi}\Big{)}\partial_{\chi}F+\partial_{\chi}^{2}F+\csc^{2}\chi\partial_{\xi}^{2}F\Big{)}+$
$\displaystyle
2\gamma^{2}f_{3}^{2}\partial_{\beta}^{2}F+\gamma^{2}f_{5}^{2}\partial_{\beta}^{2}F+2\gamma\csc\chi
f_{5}\Big{(}\Big{(}\cot\chi-\partial_{\chi}\hat{\Phi}\Big{)}\partial_{\beta}F+\partial_{\chi}\partial_{\beta}F\Big{)}\Big{)}+$
$\displaystyle\gamma^{2}f_{3}f_{4}^{2}\Big{(}-\Big{(}2\partial_{\chi}\hat{\Phi}+\cot\chi\Big{)}\partial_{\chi}F+\partial_{\chi}^{2}F+2\csc^{2}\chi\partial_{\xi}^{2}F+\gamma^{2}\sin^{2}\chi
f_{3}^{2}\partial_{\beta}^{2}F+$ $\displaystyle\gamma^{2}\sin^{2}\chi
f_{5}^{2}\partial_{\beta}^{2}F+2\gamma\sin\chi
f_{5}\Big{(}\partial_{\chi}\partial_{\beta}F-\partial_{\chi}\hat{\Phi}\partial_{\beta}F\Big{)}\Big{)}\Big{)}+\csc^{2}\chi\Big{(}2f_{4}\Big{(}\partial_{\eta}f_{3}$
$\displaystyle\partial_{\eta}F+\partial_{\sigma}f_{3}\partial_{\sigma}F+f_{3}\Big{(}-2\partial_{\eta}\hat{\Phi}\partial_{\eta}F-2\partial_{\sigma}\hat{\Phi}\partial_{\sigma}F+\partial_{\eta}^{2}F+\partial_{\sigma}^{2}F\Big{)}\Big{)}$
$\displaystyle+f_{3}\Big{(}\partial_{\eta}f_{4}\partial_{\eta}F+\partial_{\sigma}f_{4}\partial_{\sigma}F\Big{)}\Big{)}-\gamma^{2}f_{3}^{2}f_{4}\Big{(}\partial_{\eta}f_{4}\partial_{\eta}F+\partial_{\sigma}f_{4}\partial_{\sigma}F+f_{4}\Big{(}4\partial_{\eta}\hat{\Phi}$
$\displaystyle\partial_{\eta}F-2\Big{(}-2\partial_{\sigma}\hat{\Phi}\partial_{\sigma}F+\partial_{\eta}^{2}F+\partial_{\sigma}^{2}F\Big{)}\Big{)}\Big{)}\Big{)}\Bigg{]}.$

It is quite trivial to notice that, in the limit $\gamma\rightarrow 0$,
together with proper redefinition of $f_{i}$’s, the expression (42) boils down
into the original Gaiotto-Maldacena background Itsios:2019yzp .

##  4 Spin $2$ spectrum

Below, we discuss in detail the spectrum associated with massive graviton
fluctuations taking two specific examples within the deformed Gaiotto-
Maldacena class of geometries. These are the $\gamma$\- deformed Abelian T
dual (ATD) and non-Abelian T dual (NATD) solutions those were obtained by
authors in Nunez:2019gbg . The addition of flavour branes complicates the
scenario and the corresponding equations of motion are hardly tractable
analytically. We outline these issues in the Appendix C.

###  4.1 $\gamma$\- deformed ATD

The potential function for the ATD case takes the form Lozano:2016kum

$\displaystyle
V_{\text{ATD}}(\sigma,\eta)=\ln\sigma-\frac{1}{2}\sigma^{2}+\eta^{2}\ .$ (43)

The associated functions $f_{i}(\sigma,\eta)$ in (4) turn out to be

$\displaystyle
f_{1}=1~{};~{}f_{2}=\frac{4}{1-\sigma^{2}}~{};~{}f_{3}=1-\sigma^{2}~{};~{}f_{4}=4\sigma^{2}$
(44) $\displaystyle
f_{5}=-2\eta~{};~{}f_{6}=0~{};~{}f_{7}=-2\big{(}1-\sigma^{2}\big{)}^{2}~{};~{}f_{8}=\frac{64}{1-\sigma^{2}}.$
(45)

Using (43), the expression in (42) takes the form

$\displaystyle\mathcal{L}^{(1)}F$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}\big{(}\sigma^{2}-1\big{)}}\Bigg{[}\Big{(}-4\gamma^{2}\sigma^{2}\Big{(}4\eta^{2}+\Big{(}\sigma^{2}-1\Big{)}^{2}\Big{)}\sin^{2}\chi+\sigma^{2}-1\Big{)}\partial_{\beta}^{2}F$
(49)
$\displaystyle\sigma\Big{(}4\sigma\Big{(}\Big{(}4\gamma^{2}\sigma^{2}\Big{(}\sigma^{2}-1\Big{)}-\csc^{2}\chi\Big{)}\partial_{\xi}^{2}F+4\gamma\eta\sin\chi\partial_{\chi}\partial_{\beta}F-\partial_{\chi}^{2}F$
$\displaystyle-\cot\chi\partial_{\chi}F\Big{)}-\sigma\Big{(}\sigma^{2}-1\Big{)}^{2}\partial_{\eta}^{2}F-\sigma\Big{(}\sigma^{2}-1\Big{)}^{2}\partial_{\sigma}^{2}F+\Big{(}-5\sigma^{4}+6\sigma^{2}-1\Big{)}$
$\displaystyle\partial_{\sigma}F\Big{)}+16\gamma\eta\sigma^{2}\cos\chi\partial_{\beta}F\Bigg{]}.$

Using the change of variables

$\displaystyle\eta=2\psi\ ;\ \sigma=\sin\alpha$ (50)

and expanding the L.H.S. of (32) upto $\mathcal{O}(\gamma)$ we find

$\displaystyle\partial_{\alpha}^{2}F+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F+\frac{1}{\sin^{2}\alpha}\partial_{\beta}^{2}F+\frac{\cos^{2}\alpha}{4}\partial_{\psi}^{2}F+\frac{4}{\cos^{2}\alpha}\nabla_{(2)}^{2}(\chi,\xi)F$
(51)
$\displaystyle-32\frac{\psi\gamma}{\cos^{2}\alpha}\sin\chi\Big{(}\partial_{\chi}\partial_{\beta}F+\cot\chi\partial_{\beta}F\Big{)}+M^{2}F=0,$
(52)

where
$\nabla_{(2)}^{2}(\chi,\xi)=\partial_{\chi}^{2}+\cot\chi\partial_{\chi}+\csc^{2}\chi\partial_{\xi}^{2}$,
is the Laplace operator corresponding to $S^{2}(\chi,\xi)$.

Next, we propose an ansatz of the form

$\displaystyle F=e^{im\beta}e^{iq\xi}\ \tilde{F}(\alpha,\chi,\psi),$ (53)

where $\beta$ and $\xi$ are the isometry directions of the internal manifold
and $\\{m,q\\}$ are respectively the associated quantum numbers.

This further simplifies (51) as

$\displaystyle\partial_{\alpha}^{2}\tilde{F}+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}\tilde{F}-\frac{1}{\sin^{2}\alpha}m^{2}\tilde{F}+\frac{\cos^{2}\alpha}{4}\partial_{\psi}^{2}\tilde{F}+\frac{4}{\cos^{2}\alpha}$
(54)
$\displaystyle\Big{(}\partial_{\chi}^{2}\tilde{F}+\cot\chi\partial_{\chi}\tilde{F}-q^{2}\csc^{2}\chi\tilde{F}\Big{)}$
$\displaystyle-32(im)\frac{\psi\gamma}{\cos^{2}\alpha}\sin\chi\Big{(}\partial_{\chi}\tilde{F}+\cot\chi\tilde{F}\Big{)}+M^{2}\tilde{F}=0.$
(55)

A consistent solution can be obtained by equating both the real and the
imaginary part to zero which are listed below

$\displaystyle\partial_{\alpha}^{2}\tilde{F}+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}\tilde{F}-\frac{1}{\sin^{2}\alpha}m^{2}\tilde{F}+\frac{\cos^{2}\alpha}{4}\partial_{\psi}^{2}\tilde{F}+\frac{4}{\cos^{2}\alpha}$
(56)
$\displaystyle\Big{(}\partial_{\chi}^{2}\tilde{F}+\cot\chi\partial_{\chi}\tilde{F}-q^{2}\csc^{2}\chi\tilde{F}\Big{)}+M^{2}\tilde{F}=0,$
(57)
$\displaystyle-32m\frac{\psi\gamma}{\cos^{2}\alpha}\sin\chi\Big{(}\partial_{\chi}\tilde{F}+\cot\chi\tilde{F}\Big{)}=0.$
(58)

Clearly, the effects of $\gamma$\- deformation enters through the solution
(60) of the imaginary part (58). In order to proceed further, we propose a
separation of variable of the form

$\displaystyle\tilde{F}(\alpha,\chi,\psi)=F_{\alpha}(\alpha)F_{\chi}(\chi)F_{\psi}(\psi),$
(59)

where the $\chi$\- equation reveals a simple solution of the form

$\displaystyle F_{\chi}(\chi)=\csc\chi.$ (60)

Using (60) and plugging (59) into (58) we finally obtain

$\displaystyle\partial_{\alpha}^{2}F_{\alpha}(\alpha)+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F_{\alpha}(\alpha)+\Bigg{[}M^{2}-\frac{m^{2}}{\sin^{2}\alpha}+\frac{\cos^{2}\alpha}{4}\frac{1}{F_{\psi}(\psi)}\partial_{\psi}^{2}F_{\psi}(\psi)$
(61)
$\displaystyle+\frac{4}{\cos^{2}\alpha}\Big{(}1-q^{2}\Big{)}\csc^{2}\chi\Bigg{]}F_{\alpha}(\alpha)=0,$
(62)

which turns out to be quite tricky to solve for generic values of $q$.
However, this problem simplifies a lot for a particular choice of the quantum
number which is the $q=1$ mode.

###  4.2 Special case: $q=1$

For $q=1$ mode, we have the following equation to solve

$\displaystyle\frac{1}{F_{\alpha}(\alpha)}\
\frac{1}{\cos^{2}\alpha}\Bigg{[}\partial_{\alpha}^{2}F_{\alpha}(\alpha)+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F_{\alpha}(\alpha)\Bigg{]}$
(63)
$\displaystyle+\frac{M^{2}}{\cos^{2}\alpha}-\frac{m^{2}}{\sin^{2}\alpha\cos^{2}\alpha}+\frac{1}{4F_{\psi}(\psi)}\partial_{\psi}^{2}F_{\psi}(\psi)=0,$
(64)

where the $\psi$\- equation has a solution of the form

$\displaystyle F_{\psi}(\psi)=A_{1}\sin(2n\psi)+A_{2}\cos(2n\psi).$ (65)

Notice that, unlike the previous example Itsios:2019yzp , the parameter $n$ is
_continuous_. This stems from the fact that, unlike the standard ATD case
Lozano:2016kum , the $\psi$ coordinate is no longer an isometry direction due
to the presence of the $\gamma$\- deformation.

Finally, by plugging (65) into (63) we find

$\displaystyle\partial_{\alpha}^{2}F_{\alpha}(\alpha)+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F_{\alpha}(\alpha)+\Bigg{[}M^{2}-\frac{m^{2}}{\sin^{2}\alpha}-n^{2}\cos^{2}\alpha\Bigg{]}F_{\alpha}(\alpha)=0.$
(66)

Next, we implement the change in variable $z=\sin^{2}\alpha$ which finally
yields

$\displaystyle
z\big{(}1-z\big{)}\partial_{z}^{2}F_{z}(z)+\big{(}1-3z\big{)}\partial_{z}F_{z}(z)+\Bigg{[}\frac{M^{2}}{4}-\frac{m^{2}}{4z}-\frac{n^{2}\big{(}1-z\big{)}}{4}\Bigg{]}F_{z}(z)=0.$
(67)

Notice that, the above equation clearly differs from Itsios:2019yzp  as we do
not have the contribution ($\sim\ell(\ell+1)$ ) due to the $S^{2}(\chi,\xi)$
of the internal manifold. This stems from the fact that the original $SU(2)$
symmetries of $S^{2}(\chi,\xi)$ are broken down to a single $U(1)$ isometry
(corresponding to the $\xi$ direction (see (34)) in the presence of the
$\gamma$\- deformation.

####  4.2.1 Zero modes: $n=0$

Setting $n=0$ for zero modes, one finds

$\displaystyle
z\big{(}1-z\big{)}\partial_{z}^{2}F_{z}(z)+\big{(}1-3z\big{)}\partial_{z}F_{z}(z)+\Bigg{[}\frac{M^{2}}{4}-\frac{m^{2}}{4z}\Bigg{]}F_{z}(z)=0.$
(68)

Let us now look for a particular solution of (68) in the small $z\sim 0$ limit
which is equivalent to the $\sigma\sim 0$ limit. A closer look reveals that
the corresponding solution can be expressed in terms of the Bessel functions
accompanied by Gamma functions of the form

$\displaystyle F_{z}(z)_{z\sim
0}=C_{1}\Gamma(1+m)J_{m}\big{(}M\sqrt{z}\big{)}+C_{2}\Gamma(1-m)J_{-m}\big{(}M\sqrt{z}\big{)}.$
(69)

Here, $m(=1,2,3,\cdots)$ takes positive integer values since it is quantum
number associated to $\beta$ symmetry direction. On the other hand, the Gamma
function of negative integers is ill-defined therefore we set the other
constant, $C_{2}=0$ in (69).

Combining all these facts together, the final solution takes the form

$\displaystyle F_{z}(z)|_{z\sim
0}=C_{1}\Gamma(1+m)J_{m}\big{(}M\sqrt{z}\big{)},$ (70)

where both $m>0$ and $M>0$.

The above analysis shows that $F_{z}(z)$ in (70) is indeed a smooth function
in the small $z$ limit without imposing any further constraints. Contrary to
the undeformed case Itsios:2019yzp , here we obtain a continuous spectrum for
the massive graviton. As mentioned previously, this is an artefact of the non-
compactness in the $\psi$-direction and the absence of the spherical symmetry
in the presence of the $\gamma$\- deformation222Similar observation was made
in the case of NATD of $AdS_{5}\times S^{5}$ where the $\psi$ direction
(originally identified as $\rho$-coordinate) was non-compact as well as non-
isometric Itsios:2019yzp ..

####  4.2.2 $n\neq 0$ case

The $n\neq 0$ case can be dealt through the WKB method of Minahan:1998tm
-Russo:1998by  where the graviton mass ($M$) is considered to be very large.

Introducing the following change in coordinates

$\displaystyle r=\frac{z}{1-z}~{};~{}r\in[0,+\infty)$ (71)

the graviton equation (67) could be rephrased as333Here we replace $F_{z}(z)$
by $\Psi(r)$ in the transformed coordinates.

$\displaystyle\partial_{r}\Bigg{[}p(r)\partial_{r}\Psi(r)\Bigg{]}+\Bigg{[}M^{2}w(r)+q(r)\Bigg{]}\Psi(r)=0,$
(72)

where we define the above functions as

$\displaystyle p(r)=\frac{r}{1+r}\ ;\ w(r)=\frac{1}{4\big{(}1+r\big{)}^{3}}\
;\
q(r)=-\Bigg{[}\frac{m^{2}}{4r\big{(}1+r\big{)}^{2}}+\frac{n^{2}}{4r\big{(}1+r\big{)}^{4}}\Bigg{]}.$
(73)

Expanding the above functions (73) near $r\sim 0$ one finds

$\displaystyle p(r)$ $\displaystyle\approx$ $\displaystyle
r+\mathcal{O}(r^{2}),$ (74) $\displaystyle w(r)$ $\displaystyle\approx$
$\displaystyle\frac{1}{4}+\mathcal{O}(r),$ (75) $\displaystyle q(r)$
$\displaystyle\approx$
$\displaystyle-\frac{m^{2}}{4r}+\frac{m^{2}}{2}-\frac{n^{2}}{4}-\frac{3m^{2}}{4}r+\mathcal{O}(r^{2}),$
(76)

which can be further compared to yield the associated exponents as

$\displaystyle p(r)$ $\displaystyle\approx$ $\displaystyle p_{1}r^{s_{1}}\ ;\
p_{1}=s_{1}=1,$ (77) $\displaystyle w(r)$ $\displaystyle\approx$
$\displaystyle w_{1}r^{s_{2}}\ ;\ w_{1}=\frac{1}{4}\ ,\ s_{2}=0,$ (78)
$\displaystyle q(r)$ $\displaystyle\approx$ $\displaystyle q_{1}r^{s_{3}}\ ;\
q_{1}=-\frac{m^{2}}{4}\ ,\ s_{3}=-1.$ (79)

On a similar note, an expansion near $r\rightarrow\infty$ reveals

$\displaystyle p(r)$ $\displaystyle\approx$ $\displaystyle
1+\mathcal{O}(r^{-1}),$ (80) $\displaystyle w(r)$ $\displaystyle\approx$
$\displaystyle\frac{1}{4r^{3}}+\mathcal{O}(r^{-4}),$ (81) $\displaystyle q(r)$
$\displaystyle\approx$
$\displaystyle-\frac{m^{2}}{4r^{3}}+\frac{m^{2}}{2r^{4}}-\frac{n^{2}}{4r^{4}}+\mathcal{O}(r^{-5}),$
(82)

which can be further compared to decode the associated exponents as

$\displaystyle p(r)$ $\displaystyle\approx$ $\displaystyle p_{2}r^{t_{1}}\ ;\
p_{2}=1\ ,\ t_{1}=0\ ,$ (83) $\displaystyle w(r)$ $\displaystyle\approx$
$\displaystyle w_{2}r^{t_{2}}\ ;\ w_{2}=\frac{1}{4}\ ,\ t_{2}=-3\ ,$ (84)
$\displaystyle q(r)$ $\displaystyle\approx$ $\displaystyle q_{2}r^{t_{3}}\ ;\
q_{2}=-\frac{m^{2}}{4}\ ,\ t_{3}=-3.$ (85)

Finally, we use the above information in order to express the graviton mass
Minahan:1998tm -Russo:1998by

$\displaystyle
M^{2}=\frac{\pi^{2}}{\zeta^{2}}\lambda\Bigg{[}\lambda-1+\frac{\alpha_{2}}{\alpha_{1}}+\frac{\beta_{2}}{\beta_{1}}\Bigg{]},$
(86)

where $\lambda(\geq 1)$ is a positive continuous parameter.

Here, we define the function $\zeta$ through the following integral

$\displaystyle\zeta=\int_{0}^{\infty}dr\sqrt{\frac{w}{p}}=\frac{1}{2}\int_{0}^{\infty}dr\frac{1}{\sqrt{r}(1+r)}=\frac{\pi}{2},$
(87)

together with the other parameters as

$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle s_{2}-s_{1}+2=1,$
(88) $\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\big{(}s_{1}-1\big{)}^{2}-4\frac{q_{1}}{p_{1}}}=m,$ (89)
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle t_{1}-t_{2}-2=1,$
(90) $\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle|t_{1}-1|=1.$
(91)

Plugging (88) into (86) we finally obtain

$\displaystyle M^{2}=4\lambda\Big{(}\lambda+m\Big{)}.$ (92)

Here, couple of points are to be noticed. First of all, the angular momentum
quantum number ($\sim\ell$) does not appear in (92) since the symmetries of
the two sphere $S^{2}(\chi,\xi)$ is broken (as a result of the $\gamma$\-
deformation) therefore it is not a conserved quantity anymore. Secondly, we
notice that the spectrum does not contain any information about the quantum
number $n$ as also shown previously by the authors in Itsios:2019yzp . This is
due to the fact that the $n^{2}$ term is always suppressed both in the $r\sim
0$ as well as $r\rightarrow\infty$ limits (see (79) and (82)) therefore does
not show up in the associated expressions for the exponents.

###  4.3 $\gamma$\- deformed NATD

Technically, the analysis for the $\gamma$\- deformed NATD is quite similar in
spirit to that with the $\gamma$\- deformed ATD example. The corresponding
potential function is given by Lozano:2016kum

$\displaystyle
V_{\text{NATD}}(\sigma,\eta)=\eta\Big{(}\ln\sigma-\frac{1}{2}\sigma^{2}\Big{)}+\frac{1}{3}\eta^{3},$
(93)

together with the associated metric functions $f_{i}(\sigma,\eta)$

$\displaystyle
f_{1}=1~{};~{}f_{2}=\frac{4}{1-\sigma^{2}}~{};~{}f_{3}=\frac{4\eta^{2}(1-\sigma^{2})}{4\eta^{2}+(1-\sigma^{2})^{2}}~{};~{}f_{4}=4\sigma^{2},$
(94) $\displaystyle
f_{5}=-\frac{8\eta^{3}}{4\eta^{2}+(1-\sigma^{2})^{2}}~{};~{}f_{6}=\big{(}1-\sigma^{2}\big{)}^{2}~{};~{}f_{7}=-\frac{8\eta^{3}(1-\sigma^{2})^{2}}{4\eta^{2}+(1-\sigma^{2})^{2}},$
(95) $\displaystyle
f_{8}=\frac{256}{(1-\sigma^{2})(4\eta^{2}+(1-\sigma^{2})^{2})}.$ (96)

Then corresponding expression in (42) takes the form

$\displaystyle\mathcal{L}^{(1)}F$ $\displaystyle=$
$\displaystyle-\frac{1}{\eta^{2}\sigma^{2}(\sigma^{2}-1)}\Bigg{[}\eta^{2}\Big{(}16\gamma^{2}\eta^{2}\sigma^{2}\sin^{2}\chi-\sigma^{2}+1\Big{)}\partial_{\beta}^{2}F-\sigma^{2}\csc^{2}\chi$
(102)
$\displaystyle\Big{(}16\gamma^{2}\eta^{2}\sigma^{2}\Big{(}\sigma^{2}-1\Big{)}\sin^{2}\chi-4\eta^{2}-\Big{(}\sigma^{2}-1\Big{)}^{2}\Big{)}\partial_{\xi}^{2}F-16\gamma\eta^{3}\sigma^{2}\sin\chi\partial_{\chi}\partial_{\beta}F$
$\displaystyle-16\gamma\eta^{3}\sigma^{2}\cos\chi\partial_{\beta}F+\eta^{2}\sigma^{6}\partial_{\eta}^{2}F+5\eta^{2}\sigma^{5}\partial_{\sigma}F-2\eta^{2}\sigma^{4}\partial_{\eta}^{2}F+6\eta^{2}\sigma^{3}$
$\displaystyle\partial_{\sigma}F+4\eta^{2}\sigma^{2}\partial_{\chi}^{2}F+\eta^{2}\sigma^{2}\partial_{\eta}^{2}F+\eta^{2}\Big{(}\sigma^{3}-\sigma\Big{)}^{2}\partial_{\sigma}^{2}F+\sigma^{2}\Big{(}4\eta^{2}+\sigma^{4}-2\sigma^{2}+1\Big{)}$
$\displaystyle\cot\chi\partial_{\chi}F+\eta^{2}\sigma\partial_{\sigma}F+\sigma^{6}\partial_{\chi}^{2}F+2\eta\sigma^{6}\partial_{\eta}F-2\sigma^{4}\partial_{\chi}^{2}F-4\eta\sigma^{4}\partial_{\eta}F$
$\displaystyle+\sigma^{2}\partial_{\chi}^{2}F+2\eta\sigma^{2}\partial_{\eta}F\Bigg{]}.$

Using the following change in variables

$\displaystyle\eta=2\psi\ ;\ \sigma=\sin\alpha,$ (103)

and retaining terms upto leading order in the deformation parameter one finds

$\displaystyle\partial_{\alpha}^{2}F+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F+\frac{1}{\sin^{2}\alpha}\partial_{\beta}^{2}F$
(104)
$\displaystyle+\frac{\cos^{2}\alpha}{4}\bigg{(}\partial_{\psi}^{2}F+\frac{2}{\psi}\partial_{\psi}F+\frac{1}{\psi^{2}}\nabla_{(2)}^{2}(\chi,\xi)F\bigg{)}+\frac{4}{\cos^{2}\alpha}\nabla_{(2)}^{2}(\chi,\xi)F$
(105)
$\displaystyle-32\frac{\psi\gamma}{\cos^{2}\alpha}\sin\chi\Big{(}\partial_{\chi}\partial_{\beta}F+\cot\chi\partial_{\beta}F\Big{)}+M^{2}F=0.$
(106)

Like before, we use an ansatz of the following form

$\displaystyle F=e^{im\beta}e^{iq\xi}\ \tilde{F}(\alpha,\chi,\psi),$ (107)

which splits the equation into the real and the imaginary components

$\displaystyle\partial_{\alpha}^{2}\tilde{F}+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}\tilde{F}-\frac{1}{\sin^{2}\alpha}m^{2}\tilde{F}+\frac{\cos^{2}\alpha}{4}$
(108)
$\displaystyle\Bigg{[}\partial_{\psi}^{2}\tilde{F}+\frac{2}{\psi}\partial_{\psi}\tilde{F}+\frac{1}{\psi^{2}}\Big{(}\partial_{\chi}^{2}\tilde{F}+\cot\chi\partial_{\chi}\tilde{F}-q^{2}\csc^{2}\chi\tilde{F}\Big{)}\Bigg{]}+$
(109)
$\displaystyle\frac{4}{\cos^{2}\alpha}\Big{(}\partial_{\chi}^{2}\tilde{F}+\cot\chi\partial_{\chi}\tilde{F}-q^{2}\csc^{2}\chi\tilde{F}\Big{)}$
$\displaystyle-32(im)\frac{\psi\gamma}{\cos^{2}\alpha}\sin\chi\Big{(}\partial_{\chi}\tilde{F}+\cot\chi\tilde{F}\Big{)}+M^{2}\tilde{F}=0.$
(110)

The next step would be to use the following ansatz

$\displaystyle\tilde{F}(\alpha,\chi,\psi)=F_{\alpha}(\alpha)F_{\chi}(\chi)F_{\psi}(\psi),$
(111)

and thereby solve the imaginary component namely the $\chi$-equation. We
substitute this solution into the real component to obtain

$\displaystyle\partial_{\alpha}^{2}F_{\alpha}(\alpha)+\Big{(}\cot\alpha-3\tan\alpha\Big{)}\partial_{\alpha}F_{\alpha}(\alpha)+\Bigg{[}M^{2}-\frac{m^{2}}{\sin^{2}\alpha}+\frac{\cos^{2}\alpha}{4}\frac{1}{F_{\psi}(\psi)}\bigg{(}\partial_{\psi}^{2}F_{\psi}(\psi)$
(112)
$\displaystyle+\frac{2}{\psi}\partial_{\psi}F_{\psi}(\psi)+\frac{1}{\psi^{2}}\big{(}1-q^{2}\big{)}\csc^{2}\chi\bigg{)}+\frac{4}{\cos^{2}\alpha}\big{(}1-q^{2}\big{)}\csc^{2}\chi\Bigg{]}F_{\alpha}(\alpha)=0,$
(113)

which like in the $\gamma$\- deformed ATD case, is quite nontrivial to solve
for generic values of the $q$ quantum number.

Like before, we simplify (112) by fixing $q=1$ which yields a solution for the
$\psi$\- equation

$\displaystyle F_{\psi}(\psi)=\frac{e^{-2n\psi}}{\psi}.$ (114)

Finally, using (114) and introducing the new coordinate $z=\sin^{2}\alpha$,
one arrives at an equation

$\displaystyle
z\big{(}1-z\big{)}\partial_{z}^{2}F_{z}(z)+\big{(}1-3z\big{)}\partial_{z}F_{z}(z)+\Bigg{[}\frac{M^{2}}{4}-\frac{m^{2}}{4z}+\frac{n^{2}\big{(}1-z\big{)}}{4}\Bigg{]}F_{z}(z)=0,$
(115)

which is almost identical to that of the $\gamma$\- deformed ATD (67) except
for the sign in front of the $n^{2}$ term. Notice that, the case with the zero
($n=0$) modes is exactly identical to what we have found for the deformed ATD
case. On the other hand, for non zero modes, one can show that the $n^{2}$
term is largely suppressed both in the $r\sim 0$ and $r\rightarrow\infty$
limits which reveals an identical spectra for the massive graviton as in the
ATD case.

##  5 Concluding remarks

Before we conclude, a couple of remarks are in order. First of all, it is
worthwhile to mention that the graviton spectrum does not receive any
correction due to $\mathcal{S}=2+\mathcal{O}(\gamma^{2})$ at leading order in
the deformation (see (32)). Therefore, at leading order, we have a nontrivial
cancellation $\mathcal{S}-2\sim 0$ like in the undeformed case Itsios:2019yzp
. This plays a crucial role while solving the graviton spectrum at leading
order in the $\gamma$\- deformation.

Finally, we would like comment on the dual spin 2 operator associated to the
$\mathcal{N}=1$ superconformal quiver. As mentioned previously, the $SU(2)$
charges of the original Gaiotto-Maldacena background are broken down to a
single $U(1)$ charge as result of the TsT transformation where the
corresponding charge is identified as the quantum number $q(=1)$ associated to
the $\xi$\- isometry direction of the internal space. The remaining $U(1)$
charge is identified as the quantum number $m$ corresponding to the $\beta$\-
isometry direction (see (53)).

Notice that, the spin 2 operator that we are going to propose is associated to
a dual superconformal quiver that is technically _unbounded_ (as a result of
marginal deformations) unless we add flavour degrees of freedom. The spin 2
operator has a continuous spectrum whose dimension could be typically
expressed in the form, $\Delta=4f(\lambda)+Q$, where $Q$ is the total $U(1)$
charge and $\lambda$ is some continuous parameter.

The dual spin 2 operator can be constructed taking scalars ($\phi^{k}$) from
tensor multiplets (those also connect different vector multiplets and thereby
carry color indices ($k$)) and the gauge fields ($A^{I}$) of the vector
multiplet those are sourced due to the color D4 branes. The resulting operator
could be schematically expressed as

$\displaystyle\mathcal{O}_{\mu\nu}\sim\text{Tr}(\phi^{k_{i}}A^{I_{i}})T_{\mu\nu}~{};~{}\sum_{i}k_{i}+\sum_{i}I_{i}=Q$
(116)

where the operator has a $U(1)$ R-charge of the form $(q,m)$.

It would be indeed an interesting project to repeat the above analysis in the
presence of flavours and construct the dual operator spectrum using a
holographic set up. However, as we outline in the Appendix C, the
corresponding massive graviton equations are quite involved and therefore
special numeric techniques must be adopted to solve the spectrum.

###### Acknowledgements.

Its a pleasure to thank G. Itsios for clarifying some issues. One of the
authors DR is indebted to the authorities of IIT Roorkee for their
unconditional support towards researches in basic sciences. DR also
acknowledges The Royal Society, UK for financial support.

##  Appendix A Type-IIA supergravity equations

In this Appendix, we briefly discuss the Type-IIA supergravity solutions of
Passias:2018swc . The field strengths of massive Type-IIA supergravity can be
expressed as

$\displaystyle H_{3}=dB_{2}\ ;\ F_{2}=dC_{1}+F_{0}B_{2}\ ;\
F_{4}=dC_{3}-H_{3}\wedge C_{1}+\frac{1}{2}F_{0}B_{2}\wedge B_{2},$ (117)

together with the Bianchi identities which take the following form

$\displaystyle dH_{3}=0\ ;\ dF_{2}=F_{0}H_{3}\ ;\ dF_{4}=H_{3}\wedge F_{2}.$
(118)

The corresponding equations of motion are given by

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
R_{MN}-\frac{1}{2}\partial_{M}\Phi\partial_{N}\Phi-\frac{1}{16}F_{0}^{2}e^{\frac{5\Phi}{2}}g_{MN}-\frac{1}{2}e^{\frac{3\Phi}{2}}\Bigg{(}F_{MP}F_{N}^{P}-\frac{1}{16}g_{MN}\big{(}F_{2}\big{)}^{2}\Bigg{)}$
(120)
$\displaystyle-\frac{1}{12}e^{\frac{\Phi}{2}}\Bigg{(}F_{MPQR}F_{N}^{PQR}-\frac{3}{32}g_{MN}\big{(}F_{4}\big{)}^{2}\Bigg{)}-\frac{1}{4}e^{-\Phi}\Bigg{(}H_{MPQ}H_{N}^{PQ}-\frac{1}{12}g_{MN}\big{(}H_{3}\big{)}^{2}\Bigg{)}\
,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{M}\nabla_{M}\Phi-\frac{5}{4}F_{0}^{2}e^{\frac{5\Phi}{2}}-\frac{3}{8}e^{\frac{3\Phi}{2}}\big{(}F_{2}\big{)}^{2}-\frac{1}{96}e^{\frac{\Phi}{2}}\big{(}F_{4}\big{)}^{2}+\frac{1}{12}e^{-\Phi}\big{(}H_{3}\big{)}^{2}\
,$ (121) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{M}\big{(}e^{-\Phi}H_{MNP})-F_{0}e^{\frac{3\Phi}{2}}F_{NP}-\frac{1}{2}e^{\frac{\Phi}{2}}F_{NPQR}F^{QR}+\frac{1}{2.4!4!}\epsilon_{M_{1}...M_{8}NP}F^{M_{1}...M_{4}}F^{M_{5}...M_{8}}\
,$ (122) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{M}\big{(}e^{\frac{3\Phi}{2}}F_{MN}\big{)}+\frac{1}{6}e^{\frac{\Phi}{2}}F_{PQRN}H^{PQR}\
,$ (123) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{M}\big{(}e^{\frac{\Phi}{2}}F_{MNPQ}\big{)}-\frac{1}{144}\epsilon_{M_{1}...M_{7}NPQ}F^{M_{1}...M_{4}}H^{M_{5}...M_{8}}\
,$ (124)

where $R_{MN}$ is the Ricci tensor and $\epsilon_{M_{1}...M_{10}}$ is totally
antisymmetric tensor.

Here, $\alpha_{p}$ is a $p$-form flux with
$\alpha_{p}^{2}=\alpha_{M_{1}...M_{p}}\alpha^{M_{1}...M_{p}}$. Taking the
trace of the first equation in (120) and by plugging it into the second
equation we have the simplified dilaton equation of the form

$\displaystyle\nabla^{M}\nabla_{M}\Phi-2R+g^{MN}\partial_{M}\Phi\partial_{N}\Phi+\frac{1}{6}e^{-\Phi}\big{(}H_{3}\big{)}^{2}=0,$
(125)

where $R$ is the Ricci scalar.

In addition to the bosonic sector, the massive Type-IIA supergravity also
contains the gravitino $\Psi_{M}$ as well as the dilatino $\Lambda$ sector
along with 32-component Majorana spinors. Their corresponding equations of
motion are expressed as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Gamma^{MNP}D_{N}\Psi_{P}-\frac{1}{4}d\Phi\ .\
\Gamma^{M}\Lambda+\frac{1}{4}F_{0}e^{\frac{5\Phi}{4}}\Gamma^{MN}\Psi_{N}+\frac{5}{16}F_{0}e^{\frac{5\Phi}{4}}\Gamma^{M}\Lambda$
(129) $\displaystyle-\frac{1}{8}e^{\frac{3\Phi}{4}}\Bigg{(}2\Gamma^{[M]}F_{2}\
.\ \Gamma^{[N]}\Gamma_{11}\Psi_{N}-\frac{3}{2}F_{2}\ .\
\Gamma^{M}\Gamma_{11}\Lambda\Bigg{)}$
$\displaystyle-\frac{1}{8}e^{-\frac{\Phi}{2}}\Bigg{(}2\Gamma^{[M]}H_{3}\ .\
\Gamma^{[N]}\Gamma_{11}\Psi_{N}-H_{3}\ .\
\Gamma^{M}\Gamma_{11}\Lambda\Bigg{)}$
$\displaystyle+\frac{1}{8}e^{\frac{\Phi}{4}}\Bigg{(}2\Gamma^{[M]}F_{4}\ .\
\Gamma^{[N]}\Psi_{N}+\frac{1}{2}F_{4}\ .\ \Gamma^{M}\Lambda\Bigg{)},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Gamma^{M}\nabla_{M}\Lambda-\frac{5}{16}e^{\frac{3\Phi}{4}}F_{2}\
.\ \Gamma_{11}\Lambda+\frac{3}{8}e^{\frac{3\Phi}{4}}\Gamma^{M}F_{2}\ .\
\Gamma_{11}\Psi_{M}$ (132)
$\displaystyle+\frac{1}{4}e^{-\frac{\Phi}{2}}\Gamma^{M}H_{3}\ .\
\Gamma_{11}\Psi_{M}+\frac{3}{16}e^{\frac{\Phi}{4}}F_{4}\ .\
\Lambda-\frac{1}{8}e^{\frac{\Phi}{4}}\Gamma^{M}F_{4}\ .\ \Psi_{M}$
$\displaystyle-\frac{1}{2}\Gamma^{M}d\Phi\ .\
\Psi_{M}-\frac{21}{16}F_{0}e^{\frac{5\Phi}{4}}\Lambda-\frac{5}{8}F_{0}e^{\frac{5\Phi}{4}}\Gamma^{M}\Psi_{M}.$

Here $\nabla_{M}$ is the spin-covariant derivative that acts on spinors and .
denotes Clifford product $\alpha_{p}\ .\
\Lambda=\frac{1}{p!}\alpha_{M_{1}..M_{P}}\Gamma^{M_{1}..M_{P}}\Lambda$. The
matrices $\Gamma_{M}$ generate the Clifford algebra $CL(1,9)$ with
$\\{\Gamma^{M},\Gamma^{N}\\}=2g^{MN}$. The chirality operator is defined as
$\Gamma_{11}=\Gamma_{0}...\Gamma_{9}$.

##  Appendix B Fluctuations of the Type IIA background

Below, we consider the fluctuations of the supergravity fields as

$\displaystyle g_{MN}$ $\displaystyle=$ $\displaystyle\bar{g}_{MN}+h_{MN},$
(133) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\bar{\Phi}+\varphi,$
(134) $\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle\bar{H}_{3}+\delta
H_{3},$ (135) $\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\bar{F}_{2}+\delta F_{2},$ (136) $\displaystyle F_{4}$
$\displaystyle=$ $\displaystyle\bar{F}_{4}+\delta F_{4},$ (137)

which yields the $R_{MN}$ equation at leading order as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{\nabla}^{B}\tilde{\nabla}_{M}h_{BN}+\frac{1}{2}\tilde{\nabla}^{B}\tilde{\nabla}_{N}h_{BM}-\frac{1}{2}\tilde{\nabla}^{2}h_{MN}-\frac{1}{2}\tilde{\nabla}_{N}\tilde{\nabla}_{M}\tilde{h}+4\tilde{\nabla}^{B}A\tilde{\nabla}_{M}h_{BN}$
(142)
$\displaystyle+4\tilde{\nabla}^{B}A\tilde{\nabla}_{N}h_{BM}-h_{MN}\tilde{\nabla}^{2}A-8h_{MN}\big{(}\tilde{\nabla}A\big{)}^{2}-4\tilde{\nabla}^{P}A\tilde{\nabla}_{P}h_{MN}-\frac{1}{2}\partial_{M}\varphi\partial_{N}\bar{\Phi}$
$\displaystyle-\frac{1}{2}\partial_{M}\bar{\Phi}\partial_{N}\varphi-\frac{1}{2}\sum_{p=2}^{4}\gamma_{p}e^{2(1-p)A+\alpha_{p}\bar{\Phi}}\Bigg{[}\big{(}\delta\mathcal{A}_{p}\big{)}_{M}\
.\
\big{(}\tilde{\mathcal{A}_{p}}\big{)}_{N}+\big{(}\tilde{\mathcal{A}_{p}}\big{)}_{M}\
.\ \big{(}\delta\mathcal{A}_{p}\big{)}_{N}$
$\displaystyle-\beta_{p}h_{MN}\tilde{\mathcal{A}_{p}}^{2}-\big{(}p-1\big{)}h_{PK}\tilde{\mathcal{A}}_{MA_{1}..A_{p-2}}^{P}\tilde{\mathcal{A}}_{N}^{A_{1}..A_{p-2}K}\Bigg{]}-\frac{\varphi}{2}\sum_{p=2}^{4}\alpha_{p}\gamma_{p}e^{2(1-p)A+\alpha_{p}\bar{\Phi}}\big{(}\tilde{\mathcal{A}_{p}}^{2}\big{)}_{MN}$
$\displaystyle+t\tilde{g}_{MN}.$

Here, we define the above function as

$\displaystyle t$ $\displaystyle=$
$\displaystyle\tilde{\nabla}^{B}h_{BP}\tilde{\nabla}^{P}A+h_{BP}\tilde{\nabla}^{B}\tilde{\nabla}^{P}A-\frac{1}{2}\tilde{\nabla}_{\Lambda}\tilde{h}\tilde{\nabla}^{\Lambda}A+8h_{PB}\tilde{\nabla}^{P}A\tilde{\nabla}^{B}A$
(145)
$\displaystyle+\frac{\varphi}{2}\sum_{p=2}^{4}\alpha_{p}\beta_{p}\gamma_{p}e^{2(1-p)A+\alpha_{p}\bar{\Phi}}\tilde{\mathcal{A}_{p}}^{2}+\frac{1}{2}\sum_{p=2}^{4}\beta_{p}\gamma_{p}e^{2(1-p)A+\alpha_{p}\bar{\Phi}}$
$\displaystyle\Bigg{[}2\big{(}\delta\mathcal{A}_{p}\big{)}\ .\
\big{(}\tilde{\mathcal{A}_{p}}\big{)}-ph_{PK}\tilde{\mathcal{A}}_{A_{1}..A_{p-1}}^{P}\tilde{\mathcal{A}}_{N}^{A_{1}..A_{p-1}K}\Bigg{]}\
,$

where $\tilde{h}=\tilde{g}^{MN}h_{MN}$ .

##  Appendix C Adding flavour branes

In this Appendix, we discuss effects of adding flavour D6 branes into the
above picture and the associated massive graviton equations. As we show below,
these equations are quite involved and almost impossible to solve
analytically. The purpose of the section is to outline the basic structure of
these equations which may be useful for future investigations.

We begin by considering the single kink profile whose corresponding potential
function is expressed as Roychowdhury:2021eas

$\displaystyle V(\sigma\sim 0,\eta)=\eta N_{6}\ln\sigma+\frac{\eta
N_{6}\sigma^{2}}{4}\Lambda_{k}(\eta,P)-\frac{\eta
N_{6}\sigma^{2}}{4}\frac{P+1}{P^{2}-\eta^{2}},$ (146)

where we define

$\displaystyle\Lambda_{k}(\eta,P)$ $\displaystyle=$
$\displaystyle\big{(}P+1\big{)}\sum_{m=1}^{k}\Bigg{[}\frac{1}{(2m+(2m-1)P)^{2}-\eta^{2}}-\frac{1}{(2m+(2m+1)P)^{2}-\eta^{2}}\Bigg{]}$
(148) $\displaystyle+\frac{P}{(2k+1)^{2}(1+P)^{2}-\eta^{2}}.$

The associated metric functions can be expressed as

$\displaystyle f_{1}(\sigma\sim 0,\eta)$ $\displaystyle\sim$
$\displaystyle\frac{4}{\sigma}\frac{1}{g(\eta)},$ (149) $\displaystyle
f_{2}(\sigma\sim 0,\eta)$ $\displaystyle\sim$ $\displaystyle\sigma g(\eta),$
(150) $\displaystyle f_{3}(\sigma\sim 0,\eta)$ $\displaystyle\sim$
$\displaystyle\eta^{2}\sigma g(\eta),$ (151) $\displaystyle f_{4}(\sigma\sim
0,\eta)$ $\displaystyle\sim$ $\displaystyle 0,$ (152)

along with the function

$\displaystyle
g(\eta)=\frac{\sqrt{2}}{\sqrt{\eta}(P^{2}-\eta^{2})^{\frac{3}{2}}}\Bigg{[}\big{(}P^{2}-n^{2}\big{)}^{3}\big{(}\eta\partial_{\eta}^{2}\Lambda+2\partial_{\eta}\Lambda\big{)}-2\eta\big{(}P+1\big{)}\big{(}\eta^{2}+3P^{2}\big{)}\Bigg{]}^{\frac{1}{2}}.$
(153)

Using the above information (149)-(153), it is straightforward to compute

$\displaystyle\mathcal{L}^{(1)}F$ $\displaystyle=$
$\displaystyle\frac{1}{\eta^{2}\sigma^{3}g(\eta)^{3}}16\Bigg{[}g(\eta)\Big{(}\sigma\Big{(}\eta\Big{(}-2\eta\partial_{\sigma}\Phi\partial_{\sigma}F-2\Big{(}\eta\partial_{\eta}\Phi-1\Big{)}\partial_{\eta}F+\eta\partial_{\sigma}^{2}F$
(156)
$\displaystyle+\eta\partial_{\eta}^{2}F\Big{)}+\partial_{\chi}F\Big{(}\cot\chi-2\partial_{\chi}\Phi\Big{)}+\csc^{2}\chi\Big{(}\partial_{\xi}^{2}F-2\partial_{\xi}\Phi\partial_{\xi}F\Big{)}+\partial_{\chi}^{2}F\Big{)}$
$\displaystyle-2\eta^{2}\partial_{\sigma}F\Big{)}-2\eta^{2}\sigma\partial_{\eta}g(\eta)\partial_{\eta}F\Bigg{]}.$

Next, we consider the Uluru profile, for which the potential function is
expressed as Roychowdhury:2021eas

$\displaystyle V(\sigma\sim 0,\eta)=-\eta N_{6}\ln\sigma+\frac{\eta
N_{6}\sigma^{2}}{4}\Lambda_{u}(\eta,K,P)+\frac{\eta
N_{6}\sigma^{2}}{4(P^{2}-\eta^{2})},$ (157)

where we define

$\displaystyle\Lambda_{u}(\eta,K,P)=\sum_{n=1}^{u}(-1)^{n+1}\Bigg{[}\frac{1}{(nK+(2n-1)P)^{2}-\eta^{2}}-\frac{1}{(nK+(2n+1)P)^{2}-\eta^{2}}\Bigg{]}.$
(158)

This results in a similar equation (156) together with a function

$\displaystyle
g(\eta)=\frac{\sqrt{2}}{\sqrt{\eta}(\eta^{2}-P^{2})^{\frac{3}{2}}}\Bigg{[}2\eta^{3}+\big{(}P^{2}-n^{2}\big{)}^{3}\big{(}\eta\partial_{\eta}^{2}\Lambda_{u}+2\partial_{\eta}\Lambda_{u}\big{)}+6\eta
P^{2}\Bigg{]}^{\frac{1}{2}}.$ (159)

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