HU-EP-23/19

A holographic RG flow
from the squashed to the round $S^{7}$

Bastien Duboeuf ${}^{a}{\\!}$<EMAIL_ADDRESS>Michele Galli
${}^{b}{\\!}$<EMAIL_ADDRESS>Emanuel Malek ${}^{b}{\\!}$
<EMAIL_ADDRESS>Henning Samtleben ${}^{a,c}{\\!}$
<EMAIL_ADDRESS>

a ENSL, CNRS, Laboratoire de physique, F-69342 Lyon, France

b Institut für Physik, Humboldt-Universität zu Berlin,
IRIS Gebäude, Zum Großen Windkanal 2, 12489 Berlin, Germany

c Institut Universitaire de France (IUF)

Abstract

> We construct and analyze the domain wall solution in $D=11$ supergravity
> connecting the ${\cal N}=1$, AdS${}_{4}\times S_{\rm squashed}^{7}$ vacuum
> to the ${\cal N}=8$, AdS${}_{4}\times S_{\rm round}^{7}$ vacuum. This domain
> wall describes the holographic renormalization group flow from an ${\rm
> Sp}(2)\times{\rm Sp}(1)$ symmetric UV fixed point to the ${\rm SO}(8)$
> symmetric IR fixed point. It breaks all supersymmetries which are
> (partially) restored at its endpoints. We show how recent techniques from
> exceptional field theory allow us to compute the quadratic couplings of all
> Kaluza-Klein fluctuations around the domain wall background, encoding all
> two-point correlators along the holographic RG flow.

##  1 Introduction

The AdS/CFT correspondence is a powerful tool to analyse strongly-coupled
CFTs. In this paper, we will focus on holographic RG flows. On the gravity
side these correspond to domain-wall solutions with AdS vacua of string theory
at the endpoint. The existence of such a solution between two different AdS
vacua indicates that there is a renormalisation group flow between two
conformal fixed points in the UV and IR, dual to the AdS vacua, something
which may not even be directly visible in the QFT analysis. Moreover, the
gravitational action encapsulates the QFT correlators along the flow, giving
insight into strongly-coupled QFTs away from conformality, for example, into
confining gauge theories [1].

Despite the promising technology, in practice, the analysis of holographic RG
flows is limited to the use of lower-dimensional gauged supergravity models
due to technical difficulties in working with the 10-/11-dimensional
supergravity actions that capture the single-trace sector of the dual gauge
theories. In particular, domain wall solutions corresponding to holographic RG
flows are often constructed in such lower-dimensional gauged supergravities
[2, 3, 4], that can sometimes be uplifted via a consistent truncation to
10-/11-dimensional supergravity. However, in general, a consistent truncation
may not exist, and even when it does, it can obscure important properties of
the 10-/11-dimensional solutions. For example, the 10-/11-dimensional AdS
solutions may have more supersymmetries than the corresponding solutions
within the gauged supergravity.

Even more restrictively, the holographic computation of correlation functions
is limited to couplings between modes that are captured by the lower-
dimensional gauged supergravity, ignoring the higher Kaluza-Klein modes in the
supergravity. This is already true for 2-point functions. In fact, even the
masses of higher Kaluza-Klein modes around AdS vacua could until recently not
be studied, except for highly supersymmetric compactifications or those
corresponding to coset spaces, rendering the computation of 2-point functions
along an RG flow completely unfeasible.

The situation recently changed dramatically, thanks to the method of Kaluza-
Klein spectroscopy [5, 6] in Exceptional Field Theory (ExFT) [7, 8, 9, 10, 11,
12]. This allows the computation of the full Kaluza-Klein spectrum around
“generalised parallelisable” compactifications [13, 14]. These are
compactifications whose generalised tangent bundle in ExFT is trivial, i.e.
they admit a global frame for the generalised tangent bundle. Examples of such
compactifications include those that lead to consistent truncation of
maximally gauged supergravities, but also their deformations beyond maximal
gauged supergravity [15]. As a result, the Kaluza-Klein spectrum around a
variety of AdS vacua has been computed recently, including those with few or
no symmetries, or indeed supersymmetries. This has led to various insights,
such as infinitely many unprotected operators in strongly-coupled CFTs with
finite conformal dimensions [16], the compactness of conformal manifolds that
appear non-compact in gauged supergravity computations [17, 18, 19], as well
as instabilities triggered by higher Kaluza-Klein modes in non-supersymmetric
AdS vacua [20] that appear stable in gauged supergravity [21] and the proof of
perturbative stability of a number of AdS vacua in string theory [22, 18].

In this paper, we will extend these techniques, which have so far only dealt
with AdS vacua and their exactly marginal deformations, to compute the two-
point supergravity couplings along an RG flow triggered by a relevant
deformation. Crucially, this will be applicable to all Kaluza-Klein modes, not
just those present within a consistent truncation. Moreover, we will show how
this can be done for RG flows that do not even belong to a truncation to a
maximal gauged supergravity. Specifically, we focus on a holographic RG flow
from the ${\cal N}=1$ AdS${}_{4}\times S^{7}_{\rm squashed}$ vacuum [23] to
the ${\cal N}=8$ AdS${}_{4}\times S^{7}_{\rm round}$ vacuum of 11-dimensional
supergravity. The squashing of the $S^{7}$ preserves only
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$ isometries of the $\mathrm{SO}(8)$
isometries of the round $S^{7}$ and the corresponding 3-dimensional ${\cal
N}=1$ CFT was discussed in [24]. Interestingly, while the round $S^{7}$
compactification admits a consistent truncation to 4-dimensional ${\cal N}=8$
gauged supergravity [25], the ${\cal N}=1$ AdS${}_{4}\times S^{7}_{\rm
squashed}$ vacuum does not sit within this consistent truncation. Moreover,
the ${\cal N}=1$ Killing spinor does not belong to a subset of the ${\cal
N}=8$ ones, but instead corresponds to higher Kaluza-Klein modes on the round
$S^{7}$, in what is known as the space-invader scenario [26].

Since the round and squashed vacua are not both part of ${\cal N}=8$ gauged
supergravity, there has been some confusion in the literature about the
potential existence of an RG flow between them. A first attempt stems from
[27], where the relevant RG flow equations were written down, using an
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$-invariant consistent truncation Ansatz
[28]. However, [29] argued that these equations do not admit a first-order
formulation and, hence, that a flow is unlikely to exist. Later, consistent
truncations on the squashed $S^{7}$ were constructed more systematically in
[30], leading to a 4-dimensional ${\cal N}=4$ gauged supergravity. This can be
further truncated to a ${\cal N}=1$ subsector and even further to a 2-scalar
subsector, corresponding to that found in [28]. Crucially, none of these
truncations are subtruncations of the ${\cal N}=8$ gauged supergravity.
Moreover, within these truncations, while the squashed $S^{7}$ vacuum is
${\cal N}=1$, the round $S^{7}$ appears as an ${\cal N}=0$ vacuum. This
indicates that there is no supersymmetric
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$-invariant flow in 11 dimensions, which
can also be shown by an explicit computation.111We thank Nikolay Bobev for
private communications on this point.

This paper is laid out as follows. First, in section 2 we will review the
squashed $S^{7}$ in 11-dimensional supergravity, as well as its embedding into
the two-scalar truncation of[28]. We numerically solve the RG flow equations
and show the existence of a non-supersymmetric flow between the squashed and
round $S^{7}$ vacua, clarifying earlier claims in the literature. In section
3, we show how to embed the ${\cal N}=4$ [30], the ${\cal N}=1$ [30], and the
two-scalar truncations of [28] into ExFT. In particular, we will construct a
generalised parallelisation for these consistent truncations, which, however,
does not form an algebra under the generalised Lie derivative, reflecting the
fact that the consistent truncations do not lead to ${\cal N}=8$ gauged
supergravity. Nevertheless, the generalised parallelisation allows us to use
the Kaluza-Klein spectroscopy to compute the two-couplings of the higher
Kaluza-Klein modes along the RG flow from the squashed to round $S^{7}$, as we
demonstrate in section 4. Finally, we conclude with a discussion of our
results and open questions in section 5.

##  2 Squashed sphere, $S^{7}$ and domain wall in $D=11$ supergravity

In this section, we review the round and the squashed $S^{7}$ vacua of $D=11$
supergravity and construct the interpolating domain wall solution. Both
$S^{7}$ backgrounds are Freund-Rubin solutions of $D=11$ supergravity,
preserving ${\cal N}=8$ and ${\cal N}=1$ supersymmetry, respectively [23]. The
general ansatz for the $D=11$ metric with an internal space $S^{7}$ preserving
${\rm Sp}(2)\times{\rm Sp}(1)$ isometries can be put into the form [31]

$\begin{split}ds^{2}&=e^{-7u}\,ds^{2}_{(4)}+\frac{1}{4}\,e^{2u}\,\left(e^{3v}\,\Big{(}d\mu^{2}+\frac{1}{4}\,{\rm
sin}^{2}\mu\,\sum_{i}\omega_{i}^{2}\Big{)}+\frac{1}{4}\,e^{4v}\,\sum_{i}\left(\nu_{i}+{\rm
cos}\,\mu\,\omega_{i}\right)^{2}\right)\,,\end{split}$ (2.1)

with $S^{7}$ size parameter $u$ and squashing parameter $v$ that are taken as
scalar functions over the four-dimensional space-time. The one forms
$\omega_{i}=\sigma_{i}-\Sigma_{i}$, $\nu_{i}=\sigma_{i}+\Sigma_{i}$, satisfy

$d\sigma_{i}=-\frac{1}{2}\,\varepsilon_{ijk}\,\sigma_{j}\wedge\sigma_{k}\,,\quad
d\Sigma_{i}=-\frac{1}{2}\,\varepsilon_{ijk}\,\Sigma_{j}\wedge\Sigma_{k}\,.$
(2.2)

The 4-form flux for these solutions is of the form

$F_{\mu\nu\rho\sigma}=Q\,e^{-21u}\,\varepsilon_{\mu\nu\rho\sigma}\,,$ (2.3)

with conserved charge $Q$. The ansatz (2.1), (2.3) is, in fact, a consistent
truncation of $D=11$ supergravity as shown in [31]. More precisely, plugging
(2.1), (2.3), into the $D=11$ field equations leads to field equations that
are obtained from the $D=4$ Lagrangian

$\begin{split}|g|^{-1/2}\,{\cal
L}_{(4)}^{0}&=R_{(4)}-\frac{63}{2}\,\partial_{\mu}u\,\partial^{\mu}u-21\,\partial_{\mu}v\,\partial^{\mu}v-V_{\rm
pot}\,,\\\ V_{\rm
pot}&=-6\,e^{-9u+4v}-48\,e^{-9u-3v}+12\,e^{-9u-10v}+2\,Q^{2}\,e^{-21u}\,.\end{split}$
(2.4)

The parameter $Q$ may be absorbed into a shift of $u$ together with a
rescaling of the $D=4$ metric $g_{\mu\nu}$. In the following, we set $Q=3$ .

Extremisation of the potential $V_{\rm pot}$ from (2.4) yields two critical
points corresponding to ${\rm AdS}_{4}\times S^{7}$ solutions of $D=11$
supergravity, with the round and the squashed sphere located at

$\begin{split}\mbox{round }S^{7}&:\quad u=0\;,\;\;v=0\;,\quad\ell_{\rm
round}=\tfrac{1}{2}\,,\\\ \mbox{squashed }S^{7}&:\quad
u=u_{0}\equiv\tfrac{5}{42}{\rm ln}\,5-\tfrac{1}{6}{\rm
ln}\,3\;,\;\;v=v_{0}\equiv\tfrac{1}{7}{\rm ln}\,5\,,\quad\ell_{\rm
squashed}=\tfrac{5^{5/4}}{2\cdot 3^{7/4}}\,,\end{split}$ (2.5)

and the respective AdS lengths given by $\ell=\sqrt{-\frac{6}{V_{\rm pot}}}$.
With the standard ansatz for an interpolating domain wall solution

$u=u(r)\,,\;\;v=v(r)\,,\quad
ds_{(4)}^{2}=dr^{2}+e^{2A(r)}\,\eta_{ij}\,dx^{i}dx^{j}\,,\quad i,j=1,2,3\,,$
(2.6)

variation of (2.4) yields the equations

$\begin{split}u^{\prime\prime}+3\,A^{\prime}u^{\prime}&=-6\,e^{-21u}-\frac{12}{7}\,e^{-9u-10v}+\frac{48}{7}\,e^{-9u-3v}+\frac{6}{7}\,e^{-9u+4v}\,,\\\
v^{\prime\prime}+3\,A^{\prime}v^{\prime}&=-\frac{20}{7}\,e^{-9u-10v}+\frac{24}{7}\,e^{-9u-3v}-\frac{4}{7}\,e^{-9u+4v}\,,\\\
3\,(A^{\prime})^{2}-\frac{63}{4}\,(u^{\prime})^{2}-\frac{21}{2}\,(v^{\prime})^{2}&=-9\,e^{-21u}-6\,e^{-9u-10v}+24\,e^{-9u-3v}+3\,e^{-9u+4v}\,.\end{split}$
(2.7)

The existence of a domain wall solution to these equations, interpolating
between the two vacua (2.5), was discussed in [27], and later questioned in
[29]. Its holographic interpretation was further elaborated in [24]. The
domain wall represents a holographic renormalisation group flow from an ${\cal
N}=1$ superconformal UV fix point (dual to the squashed $S^{7}$) to an ${\cal
N}=8$, ${\rm SO}(8)$ symmetric IR fix point (dual to the round $S^{7}$).
Unlike most explicitly known domain wall solutions, there is no description of
the flow equations connecting (2.5) in terms of first order differential
equations and a superpotential. More precisely, [29] noted that the potential
(2.4) can be written in terms of a superpotential $W$ as

$\begin{split}V_{\rm
pot}&=\frac{16}{63}\,(\partial_{u}W)^{2}+\frac{8}{21}\,(\partial_{v}W)^{2}-12\,W^{2}\,,\\\
W&=-\frac{3}{\sqrt{8}}\,e^{-9u/2}\left(e^{2v}+2\,e^{-5v}-e^{-6u}\right)\,.\end{split}$
(2.8)

However only the squashed $S^{7}$ represents a critical point of this
superpotential. With hindsight, this is a manifestation of the fact that the
Lagrangian (2.4) lives within an ${\cal N}=1$ four-scalar truncation of $D=11$
supergravity [30], in which the round sphere appears as an ${\cal N}=0$
vacuum, since all the ${\cal N}=8$ massless gravitinos around this vacuum are
truncated out.222In the notation of [30], all this is on the $k<0$ branch.
Accordingly, the round $S^{7}$ does not correspond to a critical point of the
associated superpotential (2.8).

Figure 1: The domain wall (blue line) in the scalar potential of (2.4). The
green and the red dots represent the round and the squashed sphere (2.5),
respectively.

The interpolating domain wall solution can be found numerically, by solving
equations (2.7) and fine-tuning the initial conditions. To this end, it is
instructive to first study the general asymptotic behaviour of the scalar
fields. As usual, this is correlated with the conformal dimensions of the
associated dual operators, given by

$\begin{split}{\cal O}_{u}&:\quad\Delta_{\rm UV}=6=\Delta_{\rm IR}\,,\\\ {\cal
O}_{v}&:\quad\Delta_{\rm UV}=\tfrac{5}{3}\,,\;\;\Delta_{\rm
IR}=4\,,\end{split}$ (2.9)

for the scalar fields. Introducing the radial coordinate

$\rho=e^{-2r/{\ell_{\rm squashed}}}\;,$ (2.10)

and expanding the flow equations (2.7) near the UV boundary $\rho=0$, we find
the following asymptotic expansions of its general regular solution

$\begin{split}&u(\rho)=\tfrac{1}{6}\,{\rm
ln}\,\tfrac{5^{5/7}}{3}+\rho^{\frac{8}{6}}\,\Omega_{1,8}+\rho^{\frac{9}{6}}\,\Omega_{1,9}+\rho^{\frac{10}{6}}\,\Omega_{1,10}+\dots+\rho^{18/3}\,\underline{\Omega}_{1,18}+\dots\;,\\\
&v(\rho)=\tfrac{1}{7}\,{\rm
ln}\,5+\rho^{\frac{4}{6}}\,\underline{\Omega}_{2,4}+\rho^{\frac{5}{6}}\,\underline{\Omega}_{2,5}+\rho^{\frac{8}{6}}\,\Omega_{2,8}+\rho^{\frac{9}{6}}\,\Omega_{2,9}+\rho^{\frac{10}{6}}\,\Omega_{2,10}+\dots\;,\\\
&2A(\rho)=-\log\rho+A_{8}\,\rho^{\frac{8}{6}}+A_{9}\,\rho^{\frac{9}{6}}+A_{10}\,\rho^{\frac{10}{6}}+\dots\;.\end{split}$
(2.11)

Equations (2.7) fix all coefficients $\Omega_{i,j}$ in the above expressions,
except $\underline{\Omega}_{2,4}$, $\underline{\Omega}_{2,5}$, and
$\underline{\Omega}_{1,18}$ which source the others. Explicitly, we find that
the coefficients of the lowest powers are given by

$\begin{split}\Omega_{1,8}&=-\tfrac{6}{17}\,\underline{\Omega}_{2,4}^{2}\,,\quad\;\Omega_{1,9}=-\tfrac{20}{27}\,\underline{\Omega}_{2,4}\,\underline{\Omega}_{2,5}\,,\\\
\Omega_{2,8}&=-\tfrac{15}{4}\underline{\Omega}_{2,4}^{2}\,,\quad\;\,\Omega_{2,9}=-\tfrac{9}{2}\,\underline{\Omega}_{2,4}\,\underline{\Omega}_{2,5}\,,\\\
A_{8}&=-\tfrac{21}{4}\,\underline{\Omega}_{2,4}^{2}\,,\quad\;\;\;A_{9}=-\tfrac{280}{27}\,\underline{\Omega}_{2,4}\,\underline{\Omega}_{2,5}\,.\end{split}$
(2.12)

In (2.11), we have imposed regularity at $\rho=0$, which sets to zero a
potential $\rho^{-\frac{3}{2}}$ term in the $u$ expansion, allowed by
equations (2.7). The general solution of (2.7), regular at $\rho=0$, thus
carries three independent constants. It furthermore admits the scaling
symmetry $\rho\rightarrow\lambda\rho$, which can be used to set
$\underline{\Omega}_{2,4}=-1$ . For the interpolating solution, the remaining
two coefficients are then fixed by further demanding regularity at the other
end of the flow $\rho\rightarrow\infty$.

Figure 2: Numerical solutions for the two scalar fields, as well as the
derivative $A^{\prime}$. The horizontal green dashed line represent the
asymptotic values of the various fields. The UV boundary is located at
$r\rightarrow\infty$, the IR boundary is at $r\rightarrow-\infty$.

Imposing regularity at both ends of the flow, we have solved equations (2.7)
numerically, and plot the result in Figures 1, 2. As expected, the solution is
of kink type for the scalar fields $u$, $v$, as well as for the derivative
$\partial_{r}A$ . In particular, with $\underline{\Omega}_{2,4}=-1$, we find
for the coefficient $\underline{\Omega}_{2,5}$ the approximate numerical value

$\underline{\Omega}_{2,5}\approx-1.4\;.$ (2.13)

In Figure 3, we plot the asymptotics of the scalar fields, which confirms the
UV expansion (2.11), and in particular the fact that the leading coefficient
$\underline{\Omega}_{2,4}$ is non-vanishing. This is the expected power-law
behaviour

$v(\rho)-v_{0}\propto\rho^{\frac{3-\Delta}{2}}\,,$ (2.14)

consistent with the interpretation that the holographic dual of this domain
wall solution is an operator deformation (rather than a vev) of the UV CFT
[32, 3], by a relevant operator of conformal dimension $\Delta_{\rm
UV}=\frac{5}{3}$ .

Figure 3: Asymptotic behavior of the numerical solution. In the UV
($r\rightarrow\infty$), the green dashed line have respectively a slope of
$\frac{4}{3}$ for $u$ and $\frac{2}{3}$ for $v$. This matches the asymptotic
expansion (2.11). In the IR ($r\rightarrow-\infty$), the green dashed line
have respectively a slope of $1$ for $u$ and $\frac{1}{2}$ for $v$. This
matches the asymptotic expansion (2.15).

Similarly, one can work out the asymptotic behaviour in the IR. With the new
radial variable $\tilde{\rho}=e^{2r/{\ell_{\rm round}}}$, the expansion of a
general solution of (2.7), regular at the IR boundary $\tilde{\rho}=0$, is
given by

$\begin{split}&u(\tilde{\rho})={\omega}_{1,2}\,\tilde{\rho}+\underline{\omega}_{1,3}\,\tilde{\rho}^{3/2}+\psi_{1,3}\,\tilde{\rho}^{3/2}\,{\rm
log}\,\rho+\omega_{1,4}\,\tilde{\rho}^{2}+\dots\;,\\\
&v(\tilde{\rho})=\underline{\omega}_{2,1}\,\tilde{\rho}^{1/2}+\omega_{2,2}\,\tilde{\rho}+\omega_{2,3}\,\tilde{\rho}^{3/2}+\omega_{2,4}\,\tilde{\rho}^{2}+\psi_{2,4}\,\tilde{\rho}^{2}\,{\rm
log}\,\rho+\dots\;,\\\
&2A(\tilde{\rho})=\log\tilde{\rho}+\tilde{A}_{2}\,\tilde{\rho}+\tilde{A}_{3}\,\tilde{\rho}^{3/2}+\tilde{A}_{4}\,\tilde{\rho}^{2}+\dots\;,\end{split}$
(2.15)

in accordance with the IR conformal dimensions (2.9). Here, imposing
regularity at $\tilde{\rho}=0$ has set to zero two of the free coefficients of
the general solution of (2.7). All other coefficients in the expansion (2.15)
are then determined in terms of the remaining two free coefficients
$\underline{\omega}_{1,3}$, $\underline{\omega}_{2,1}$, e.g.

$\begin{split}\omega_{1,2}&=\tfrac{3}{2}\,\underline{\omega}_{2,1}^{2}\,,\quad\quad\,\psi_{1,3}=11\,\underline{\omega}_{2,1}^{3}\,,\quad\;\;\,\omega_{1,4}=-\tfrac{15525}{56}\,\underline{\omega}_{2,1}^{4}\,,\quad\\\
\omega_{2,2}&=-\tfrac{11}{2}\,\underline{\omega}_{2,1}^{2}\,,\quad\omega_{2,3}=\tfrac{5261}{168}\,\underline{\omega}_{2,1}^{3}\,,\quad\omega_{2,4}=-\tfrac{3}{2}\,\underline{\omega}_{2,1}\underline{\omega}_{1,3}-\tfrac{10583}{56}\,\underline{\omega}_{2,1}^{4}\,,\quad\psi_{2,4}=-\tfrac{33}{2}\,\underline{\omega}_{2,1}^{4}\,,\\\
\tilde{A}_{2}&=-\tfrac{21}{4}\,\underline{\omega}_{2,1}^{2}\,,\quad\,\;\tilde{A}_{3}=\tfrac{154}{3}\,\underline{\omega}_{2,1}^{3}\,,\;\;\,\,\quad\tilde{A}_{4}=-\tfrac{27081}{64}\,\underline{\omega}_{2,1}^{4}\,.\end{split}$
(2.16)

Again, the leading terms are confirmed by the plots of the numerical solution
in Figure 3. Regularity at the UV end of the flow finally fixes
$\underline{\omega}_{1,3}$ as a function of $\underline{\omega}_{2,1}$. From
the above numerical domain wall solution, we find the approximate value

$\frac{\underline{\omega}_{1,3}}{(\underline{\omega}_{2,1})^{3}}\leavevmode\nobreak\
\approx\leavevmode\nobreak\ -239\;,$ (2.17)

for the combination invariant under the scaling symmetry
$\tilde{\rho}\rightarrow\lambda\tilde{\rho}$ .

##  3 Generalised parallelisation of the domain wall in ExFT

In this section, we will identify the AdS${}_{4}\times S_{\rm squashed}^{7}$
background and the domain wall (2.1) within ExFT, i.e. within the duality-
covariant formulation of $D=11$ supergravity. This allows us to construct
consistent truncations around this vacuum as well as to compute the quadratic
couplings of Kaluza-Klein fluctuations around the domain wall background in
section 4.

The relevant ExFT is based on the exceptional group E7(7) and has been
constructed in [33] to which we refer for details. Its structures relevant to
our construction have been reviewed in [6]. From a most pragmatic point of
view, E7(7) ExFT can be viewed as a reformulation of $D=11$ supergravity (upon
splitting the coordinates into 11=4+7) in terms of variables that mimic the
field content of $D=4$ maximal supergravity. In particular, the internal
components of the $D=11$ metric, three-form, and six-form are assembled into
an ${\rm E}_{7(7)}$ group matrix, the generalized vielbein, defined as

$\begin{split}{\cal V}&\equiv{\rm
exp}\left[A_{klmnpq}\,t_{(+4)}^{klmnpq}\right]{\rm
exp}\left[A_{kmn}\,t_{(+2)}^{kmn}\right]\,V_{{\rm GL}(7)}\,,\end{split}$ (3.1)

i.e. as a coset representative of ${\rm E}_{7(7)}/{\rm SU}(8)$ in a particular
triangular gauge. Here, $V_{{\rm GL}(7)}\in{\rm GL}(7)\subset{\rm E}_{7(7)}$
is the internal block of the 11D vielbein (up to some power of its
determinant), while $A_{kmn}$ and $A_{klmnpq}$ denote the internal components
of the $D=11$ three-form and six-form, respectively, with $k,l,m=1,\ldots,7$.
The $t_{(+n)}$ refer to the E7(7) generators of positive grading in the
algebra decomposition

$\displaystyle\mathfrak{e}_{7(7)}$ $\displaystyle{\longrightarrow}$
$\displaystyle 7^{\prime}_{-4}\oplus 35_{-2}\oplus\mathfrak{gl}(7)_{0}\oplus
35^{\prime}_{+2}\oplus 7_{+4}\,.$ (3.2)

All generators in (3.1) are evaluated in the fundamental ${\bf 56}$
representation of ${\rm E}_{7(7)}$. For a Freund-Rubin (FR) background (i.e. a
solution with $A_{kmn}=0$), the parametrization (3.1) only involves generators
within the $\mathfrak{sl}(8)$ subalgebra of $\mathfrak{e}_{7(7)}$ and the
associated generalized vielbein can be represented as an $8\times 8$ matrix

${\cal V}_{\rm FR}\in{\rm SL}(8)\big{/}{\rm SO}(8)\,.$ (3.3)

For example, for the round $S^{7}$ with 7-form flux, the generalized vielbein
(3.1) takes the form

${\cal V}_{S^{7}}={\rm
exp}\left[-6\,t_{(+4)}^{lmnpqr}\,\zeta^{k}\mathring{\omega}_{klmnpqr}\right]\begin{pmatrix}\mathring{\omega}^{3/4}&0\\\
0&\mathring{\omega}^{-1/4}\,\mathring{e}_{m}{}^{i}\end{pmatrix}\in{\rm
SL}(8)\,,$ (3.4)

where $\mathring{e}_{m}{}^{i}$ is the $S^{7}$ vielbein,
$\mathring{\omega}_{klmnpqr}$ is the associated volume form,
$\mathring{\omega}\equiv{\rm det}\,\mathring{e}_{m}{}^{i}$, $\zeta^{k}$ is a
vector field with $\mathring{\nabla}_{k}\zeta^{k}=1$ and $t_{(+4)}$ is
evaluated in (3.4) in the $\mathbf{8}$ representation of $\mathrm{SL}(8)$.

###  3.1 Generalised Leibniz parallelisation and Kaluza-Klein fluctuations

The round $S^{7}$ with 7-form flux (3.4) has been extensively studied in the
ExFT framework due to its interesting properties. In particular, it is
generalised Leibniz parallelisable, i.e. the generalized vielbein (3.4) is
related by an ${\rm SO}(8)$ gauge transformation

${\cal V}_{S^{7}}=\mathring{U}\,S_{{\rm SO}(8)}\;,$ (3.5)

to a twist matrix $\mathring{U}$, such that the associated generalized frame
field ${\cal U}=\rho^{-1}\mathring{U}^{-1}$ (here,
$\rho=\mathring{\omega}^{-1/2}$) evaluated in the $\mathbf{56}$
representation, satisfies the condition

${\cal L}_{{\cal U}_{{{\underline{A}}}}}{\cal
U}_{{\underline{B}}}{}^{M}=X_{{\underline{AB}}}{}^{{\underline{C}}}\,{\cal
U}_{{\underline{C}}}{}^{M}\,,$ (3.6)

where $X_{{\underline{AB}}}{}^{{\underline{C}}}$ is the constant _intrinsic
torsion_. The generalized diffeomorphisms of the $\mathrm{E}_{7(7)}$ ExFT are
defined as [8, 11]

${\cal
L}_{\Lambda}V^{M}=\Lambda^{N}\partial_{N}V^{M}-12\,\partial_{K}\Lambda^{L}\,(t_{\alpha})^{K}{}_{L}(t^{\alpha})^{M}{}_{N}\,V^{N}+\tfrac{1}{2}V^{M}\partial_{N}\Lambda^{N}\,.$
(3.7)

Explicitly, the twist matrix $\mathring{U}$ for the round $S^{7}$ is given by
[13, 14]

$\mathring{U}_{\underline{m}}{}^{a}({\cal
Y})=\begin{pmatrix}\mathring{\omega}^{3/4}\left({\cal
Y}^{a}-6\,\zeta^{n}\partial_{n}{\cal Y}^{a}\right)\\\
\mathring{\omega}^{-1/4}\,\partial_{m}{\cal Y}^{a}\end{pmatrix}\in{\rm
SL}(8)\,,\qquad\underline{m}=\\{0,m\\}\,,\quad a=\\{1,\ldots,8\\}\,,$ (3.8)

in terms of the fundamental sphere harmonics ${\cal Y}^{a}{\cal Y}^{a}=1$.

As a consequence of (3.6), $D=11$ supergravity admits a maximally
supersymmetric consistent truncation around the round $S^{7}$ to $D=4$, ${\cal
N}=8$, ${\rm SO}(8)$ gauged supergravity [34, 35]. In particular, the
embedding of the 70 scalar fields of $D=4$, ${\cal N}=8$ supergravity into the
eleven-dimensional theory is compactly described in terms of (3.1) as

${\cal V}(x,y)=\mathring{U}(y)\,V(x)\,,$ (3.9)

with the twist matrix $\mathring{U}$ from (3.8), and the ${\rm E}_{7(7)}/{\rm
SU}(8)$ valued matrix $V(x)$ carrying the 70 scalar fields of the $D=4$
theory.

Moreover, the existence of a globally defined generalized frame allow us to
recover the full Kaluza-Klein spectrum around this background by virtue of the
universal formulae derived in [5, 6, 15]. Here, we just recall the result.
While (3.9) describes the embedding of the lowest Kaluza-Klein multiplet into
$D=11$, the embedding of the infinite Kaluza-Klein tower of scalar
fluctuations $j^{I,\Sigma}(x)$ is most compactly described as

${\cal V}(x,y)=\mathring{U}(y)\,\Big{(}\mathbb{I}+{\cal P}_{I}{\cal
J}^{I}(x,y)\Big{)}=\mathring{U}(y)\,\Big{(}\mathbb{I}+{\cal
P}_{I}\sum_{\Sigma}j^{I,\Sigma}(x)\,{\cal Y}_{\Sigma}(y)\Big{)}\;.$ (3.10)

Here, ${\cal Y}_{\Sigma}(y)$ label the scalar harmonics on $S^{7}$, and ${\cal
P}_{I}$, $I=1,\ldots,70$, are the 70 non-compact generators of ${\rm
E}_{7(7)}$. After subtraction of the non-physical Goldstone modes, the action
of the mass operator on the scalar fluctuations ${\cal J}^{I}(x,y)$ is given
by

$\begin{split}\Big{(}\mathbb{M}_{\rm
spin-0}\Big{)}{}^{I}{}_{J}&=\mathbb{M}_{(0)}{}^{I}{}_{J}+\left(\mathbb{N}^{I}{}_{J}{}^{{\underline{C}}}-\mathbb{N}_{J}{}^{I}{}^{{\underline{C}}}\right)\partial_{{\underline{C}}}+\partial_{{\underline{C}}}\mathbb{N}^{I}{}_{J}{}^{{\underline{C}}}+\delta^{I}_{J}\,\mathbb{M}_{\rm
spin-2}\,,\end{split}$ (3.11)

with

$\begin{split}\mathbb{M}_{(0)}{}^{I}{}_{J}&=X_{{\underline{A}}{\underline{E}}}{}^{{\underline{F}}}X_{{\underline{B}}{\underline{F}}}{}^{{\underline{E}}}\,({\cal
P}^{I}{\cal P}_{J})_{{\underline{A}}}{}^{{\underline{B}}}\\\
&\quad+\tfrac{1}{7}\left(X_{{\underline{A}}{\underline{E}}}{}^{{\underline{F}}}X_{{\underline{B}}{\underline{E}}}{}^{{\underline{F}}}+X_{{\underline{E}}{\underline{A}}}{}^{{\underline{F}}}X_{{\underline{E}}{\underline{B}}}{}^{{\underline{F}}}+X_{{\underline{E}}{\underline{F}}}{}^{{\underline{A}}}X_{{\underline{E}}{\underline{F}}}{}^{{\underline{B}}}\right)(\mathcal{P}^{I}{\cal
P}_{J})_{{\underline{A}}}{}^{{\underline{B}}}\\\
&\quad+\tfrac{2}{7}\left(X_{{\underline{A}}{\underline{C}}}{}^{{\underline{E}}}X_{{\underline{B}}{\underline{D}}}{}^{{\underline{E}}}-X_{{\underline{A}}{\underline{E}}}{}^{{\underline{C}}}X_{{\underline{B}}{\underline{E}}}{}^{{\underline{D}}}-X_{{\underline{E}}{\underline{A}}}{}^{{\underline{C}}}X_{{\underline{E}}{\underline{B}}}{}^{{\underline{D}}}\right)({\cal
P}^{I})_{{\underline{A}}}{}^{{\underline{B}}}\,({\cal
P}_{J})_{{\underline{C}}}{}^{{\underline{D}}}\\\
&\quad+\tfrac{1}{6}\,X_{{\underline{A}}}{}^{I}\,X_{{\underline{A}},J}\,,\\\\[4.30554pt]
\mathbb{N}^{I}{}_{J}{}^{{\underline{C}}}&=-2X_{{\underline{A}}}{}^{I}{\cal
P}_{J,{\underline{C}}}{}^{{\underline{A}}}-2X_{{\underline{A}}J}{\cal
P}^{I}{}_{{\underline{C}}}{}^{{\underline{A}}}-[{\cal P}^{I},\,{\cal
P}_{J}]_{{\underline{A}}}{}^{{\underline{B}}}\left(X_{{\underline{C}}{\underline{B}}}{}^{{\underline{A}}}+6\,X_{{\underline{A}}{\underline{B}}}{}^{{\underline{C}}}\right)\,,\\\\[4.30554pt]
\mathbb{M}_{\rm
spin-2}&=-\partial_{{\underline{A}}}\partial_{{\underline{A}}}\,.\end{split}$
(3.12)

Evaluating this formula with the intrinsic torsion
$X_{{\underline{A}}{\underline{B}}}{}^{{\underline{C}}}$ from (3.6) and an
explicit basis for ${\cal Y}_{\Sigma}(y)$ allows us to derive the Kaluza-Klein
spectrum for the full infinite tower of scalar fluctuations. Specifically, for
the round $S^{7}$ vacuum, this reproduces the result for the scalar Kaluza-
Klein towers [36, 37, 38]

$\sum_{\ell}\Big{(}[\ell+2,0,0,0]_{\frac{\ell}{2}+1}+[\ell-2,0,0,0]_{\frac{\ell}{2}+5}+[\ell-2,2,0,0]_{\frac{\ell}{2}+3}+[\ell,0,2,0]_{\frac{\ell}{2}+2}+[\ell-2,0,0,2]_{\frac{\ell}{2}+4}\Big{)}\,,$
(3.13)

in terms of SO(8) representations, labeled by their Dynkin weights, with
subscripts denoting conformal dimensions.

###  3.2 The squashed $S^{7}$ family in ExFT

The consistent truncation (3.9) to the lowest KK multiplet ($\ell=0$ in
(3.13)) contains the round $S^{7}$ vacuum (corresponding to $V(x)=\mathbb{I}$)
but no other solution of the family of squashed backgrounds (2.1). Rather,
these squashings correspond to excitations of scalars from higher KK levels.
Specifically, the squashing (2.1) preserves an ${\rm Sp}(2)\times{\rm Sp}(1)$
subgroup of isometries, which is embedded into the SO(8) isometry group of the
round $S^{7}$ such that the fundamental representations decompose as

$\begin{split}{\rm SO}(8)&\longrightarrow{\rm Sp}(2)\times{\rm Sp}(1)\,,\\\
[1,0,0,0]\rightarrow[1,0,1]\,,\quad[0,0,1,0]&\rightarrow[0,1,0]\oplus[0,0,2]\,,\quad[0,0,0,1]\rightarrow[1,0,1]\,.\end{split}$
(3.14)

A scan of the scalar spectrum (3.13) shows that the spectrum contains four
scalar fields that are singlet under ${\rm Sp}(2)\times{\rm Sp}(1)$, sitting
at KK levels 0, 2, and 4, respectively, descending from the SO(8)
representations

$\begin{split}\ell=0&:\;\;[0,0,2,0]_{2}\,,\\\
\ell=2&:\;\;[0,0,0,0]_{6}\oplus[0,2,0,0]_{4}\,,\\\
\ell=4&:\;\;[2,0,0,2]_{6}\,.\end{split}$ (3.15)

The squashed background (2.1) requires a non-vanishing contribution from the
$[0,2,0,0]$ at level $\ell=2$ [26] and, thus, lives beyond the consistent
truncation (3.9).

However, the embedding of the scalar KK fluctuations according to (3.10)
allows us to explicitly construct different alternative truncations to other
subsets of fields which in particular allow to embedd the solution (2.1). E.g.
the truncation of the full KK spectrum keeping all singlets under ${\rm
Sp}(2)\times{\rm Sp}(1)$ defines a consistent truncation by the standard
symmetry argument: by simple representation reasoning, singlet fields can
never define non-vanishing sources for non-singlet fields, it is thus
consistent to truncate out all the non-singlet fields. This truncation retains
one field from the KK tower of gravitino fields, and thus corresponds to a
$D=4$, ${\cal N}=1$ theory [30], further discussed in [39]. In the scalar
sector, the truncation to ${\rm Sp}(2)\times{\rm Sp}(1)$ singlets only keeps
the four scalar fields (3.15) and can be described in closed form by
integrating the corresponding fluctuations (3.10) to

${\cal V}(x,y)=\mathring{U}(y)\;{\rm exp}\Big{[}\sum_{\rm
singlets}\phi^{i}(x)\,s_{i}^{I,\Sigma}\,{\cal P}_{I}{\cal
Y}_{\Sigma}(y)\Big{]}\,.$ (3.16)

The index $i$ here labels the four ${\rm Sp}(2)\times{\rm Sp}(1)$ singlets
found in the tensor product of the 70 non-compact generators of E7(7) and the
scalar harmonics, thereby defining the constant tensor $s_{i}^{I,\Sigma}$. The
group theoretical structure underlying this truncation can be made more
transparent by representing the seven sphere as a coset space [40]

$S^{7}=\frac{{\rm G}}{{\rm H}}=\frac{{\rm Sp}(2)\times{\rm Sp}(1)_{0}}{{\rm
Sp}(1)_{\rm L}\times{\rm Sp}(1)_{\rm D}}\,,$ (3.17)

where the different ${\rm Sp}(1)$ factors are embedded as

${\rm Sp}(2)\supset{\rm Sp}(1)_{\rm L}\times{\rm Sp}(1)_{\rm R}\;,\qquad{\rm
Sp}(1)_{\rm D}=\big{(}{\rm Sp}(1)_{0}\times{\rm Sp}(1)_{\rm R}\big{)}_{\rm
diag}\,,$ (3.18)

and for clarity we have added a ‘0’ subscript to the ${\rm Sp}(1)$ subgroup of
(3.14). The seven sphere (3.17) can be represented by an $({\rm
Sp}(2)\times{\rm Sp}(1))$-valued coset representative $S(y)$, such that the
infinitesimal action of the ${\rm Sp}(2)\times{\rm Sp}(1)$ isometry group on
the coordinates is realized as

$\delta_{\Lambda}S(y)=\Lambda
S(y)-S(y)h_{\Lambda}(y)\;,\qquad\Lambda\in\mathfrak{sp}(2)\oplus\mathfrak{sp}(1)_{0}\;,\quad
h_{\Lambda}(y)\in\mathfrak{sp}(1)_{\rm L}\oplus\mathfrak{sp}(1)_{\rm D}\,.$
(3.19)

The consistent truncation (3.16) can then be given in more compact form as

${\cal V}(x,y)=\mathring{U}(y)\;S(y)\,W(x)\,S^{-1}(y)\;,$ (3.20)

with the $({\rm Sp}(2)\times{\rm Sp}(1))$ coset representative $S(y)$ and an
E7(7) matrix $W(x)$ defined to live in the commutant of the denominator group
${\rm H}={\rm Sp}(1)_{\rm L}\times{\rm Sp}(1)_{\rm D}$ in E7(7), i.e.

$W(x)\,h=h\,W(x)\quad\forall\,h\in\mathfrak{sp}(1)_{\rm
L}\oplus\mathfrak{sp}(1)_{\rm D}\;.$ (3.21)

Indeed, (3.19) and (3.21) imply that the factor $S(y)\,W(x)\,S^{-1}(y)$ in
(3.20) is invariant under the action of $({\rm Sp}(2)\times{\rm Sp}(1))$ up to
a compact gauge transformation acting from the right. The representation
(3.20) of the consistent truncation immediately reveals the geometry of its
scalar target space, given by the commutant of ${\rm H}={\rm Sp}(1)_{\rm
L}\times{\rm Sp}(1)_{\rm D}$ in E${}_{7(7)}/{\rm SU}(8)$ as

${\cal M}_{\rm scalar}=\frac{{\rm SL}(2)}{{\rm SO}(2)}\times\frac{{\rm
SL}(2)}{{\rm SO}(2)}\,,$ (3.22)

a Kähler manifold parametrized by the four scalar fields from (3.15).333 The
same reasoning shows that the analogous construction based on the coset
representation $S^{7}={{\rm Sp}(2)}\big{/}{{\rm Sp}(1)_{\rm L}}$ of the seven
sphere, yields an ${\cal N}=4$ consistent truncation to the ${{\rm Sp}(2)}$
singlets with scalar target space given by the commutant of the denominator
group ${\rm Sp}(1)_{\rm L}$ in ${\rm E}_{7(7)}/{\rm SU}(8)$:  ${\cal M}_{\rm
scalar}=\frac{{\rm SL}(2)}{{\rm SO}(2)}\times\frac{{\rm SO}(6,3)}{{\rm
SO}(6)\times{\rm SO}(3)}\,,$ (3.23) as explicitly found in [30].

The $D=4$ theory can be obtained by plugging the ansatz (3.20) into the ExFT
Lagrangian of [33]. For the bosonic sector, parametrizing the matrix $W$ by
four scalar fields $\\{u,v,c,\chi\\}$, this gives rise to the four-scalar
Lagrangian

$\begin{split}|g|^{-1/2}\,{\cal
L}_{(4)}&=R_{(4)}-\frac{63}{2}\,\partial_{\mu}u\,\partial^{\mu}u-21\,\partial_{\mu}v\,\partial^{\mu}v-3\,e^{-6u-2v}\partial_{\mu}c\,\partial^{\mu}c-\frac{1}{2}\,e^{-6u+12v}\,\partial_{\mu}\chi\,\partial^{\mu}\chi-
V_{\rm pot}\,,\\\\[4.30554pt] V_{\rm
pot}&=-6\,e^{-9u+4v}-48\,e^{-9u-3v}+12\,e^{-9u-10v}-72\,e^{-15u-12v}\,c^{2}\\\
&\quad-12\,e^{-15u+2v}\,(c+\chi)^{2}-18\,e^{-21u}\,\big{(}1+c^{2}+2\,c\,\chi\big{)}^{2}\,.\end{split}$
(3.24)

It is straightforward to verify that the intersection of the scalar target
space (3.22) with the ${\rm SL}(8)$ of (3.3) capturing the Freund-Rubin
solutions corresponds to a further consistent truncation to two scalar fields
which precisely reproduces the result (2.4) of [31]. The ${\cal N}=1$
consistent truncation to four scalars (3.24) has already been found in [30].
What is new in the present construction is its explicit embedding (3.20) via a
twist matrix in ExFT which allows us to apply the universal mass formulas such
as (3.11), (3.12) to any background within this truncation. With the frame
given by

${\cal V}=\mathring{U}\,S\,W\,S^{-1}\,,$ (3.25)

a general background will satisfy (3.6), but, in general, with $y$-dependent
intrinsic torsion $X_{{\underline{AB}}}{}^{{\underline{C}}}(y)$.444 Because of
the scalar fields in $W(x)$, the intrinsic torsion will also depend on $x$,
but this is standard already in gauged supergravity. In particular, it does
not interfere with the generalised (Leibniz) parallelisability.  Thus, it is
no longer generalised Leibniz parallelisable, but still generalised
parallelisable. As a consequence the mass formulae (3.11), (3.12) still apply.
This has been used in [15] in order to derive the full Kaluza-Klein spectrum
around the ${\cal N}=1$, AdS${}_{4}\times S_{\rm squashed}^{7}$ vacuum. In the
following, we will extend this analysis to also derive quadratic couplings
around the domain wall background (2.1), or any other background of the
consistent truncation.

##  4 Couplings around the domain wall background

As we reviewed in section 3.1, we can efficiently describe the linearised
fluctuations around the round $S^{7}$ using its generalised Leibniz
parallelisation in ExFT. Similarly, using the generalised parallelisation of
the family of squashed $S^{7}$’s of section 3.2, we obtain a simple expression
of the linearised fluctuations around the family of squashed $S^{7}$ that
describe the flow of section 2. While this was used in [15] to obtain the full
Kaluza-Klein spectrum of the squashed $S^{7}$, here we will further extend the
computation to obtain the quadratic couplings of the higher Kaluza-Klein modes
around the domain wall solution. These couplings encode all the information
about the holographic 2-point functions along the dual RG flow.

In computing the quadratic couplings, we have to choose a field basis for the
Kaluza-Klein fluctuations, or choice of “frame”. Two natural choices are to
multiply the generalised vielbein of the 2-scalar truncation, ${\cal V}(x,y)$
of (3.20), from the right or left, leading to different kinetic terms for the
scalars $u$, $v$ and the Kaluza-Klein fluctuations. The kinetic term of ExFT
is given by$D_{\mu}M^{MN}D^{\mu}M_{MN}$ with $M={\cal V}{\cal V}^{T}$, which
can be expressed in terms of the current $J_{\mu}({\cal
V})_{\underline{M}}{}^{{\underline{N}}}=({\cal V}^{-1}D_{\mu}{\cal
V})_{\underline{M}}{}^{\underline{N}}$ as

$D_{\mu}M^{MN}D^{\mu}M_{MN}=-2\left(\mathrm{Tr}\,J^{\mu}J_{\mu}^{T}+\mathrm{Tr}\,J^{\mu}J_{\mu}\right)\,.$
(4.1)

Let us denote the Kaluza-Klein fluctuations by the matrix $\delta=\exp[{\cal
P}_{I}\sum_{\Sigma}j^{I,\Sigma}(x)\,{\cal Y}_{\Sigma}(y)]$. Then, it is easy
to see that $J_{\mu}$ becomes

$\begin{split}J_{\mu}({\cal V}\delta)&=\delta^{-1}J_{\mu}({\cal
V})\delta+J_{\mu}(\delta)\,,\\\ J_{\mu}(\delta{\cal V})&={\cal
V}^{-1}J_{\mu}(\delta){\cal V}+J_{\mu}({\cal V})\,,\end{split}$ (4.2)

depending on whether we multiply the generalised ${\cal V}$ with the Kaluza-
Klein fluctuations from the left or the right. We clearly see that multiplying
${\cal V}$ from the left leaves the kinetic term for the background, i.e. the
kinetic term of the scalars $u$, $v$, invariant, but introduces $u$, $v$
factors into the kinetic term of the Kaluza-Klein fluctuations. On the other
hand, using ${\cal V}\,\delta$, we find that the Kaluza-Klein fluctuations
enter the kinetic terms of $u$, $v$ and hence introduce corrections to the
kinetic terms of (3.24). In order to simply retain the normalisation (3.24),
we will choose to define our Kaluza-Klein fluctuations as multiplying the
truncation ${\cal V}$ from the left. Note that this differs from the
conventions of the Kaluza-Klein spectrometry [5, 6, 15], used in (3.10). The
quadratic couplings around the domain wall solutions are then
straightforwardly obtained by evaluating the action of the mass operator
(3.11) in this basis.

Before presenting our results for the quadratic couplings, let us recall some
facts about the spectrum on the left-squashed $S^{7}$ from [41, 42, 43, 15,
44]. The Kaluza-Klein spectrum is organized into long ${\cal N}=1$
supermultiplets $L[J,\Delta]$ in various $\mathrm{Sp}(2)\times\mathrm{Sp}(1)$
representations. These multiplets are defined by the spacetime spin $J$ and
conformal dimension $\Delta$ of their superconformal primaries, given by the
universal formula [15]

$\Delta_{J,s}=1+\frac{5}{3}s+\frac{1}{3}\sqrt{(3J+2s^{2})^{2}+5\,{\cal
C}_{3}}\,.$ (4.3)

Here ${\cal C}_{3}$ is a combination of the $\mathrm{Sp}(2)$ and
$\mathrm{Sp}(1)$ quadratic Casimirs, which, for a generic representation with
Dynkin labels $[p,q,r]$, is given by

$\begin{split}{\cal C}_{3}&={\cal C}(p,q)+3{\cal C}(r)\\\
&=\frac{1}{2}(p^{2}+2q^{2}+4p+6q+2pq)+\frac{3}{4}r(r+2)\,.\end{split}$ (4.4)

The extra parameter $s$ in (4.3) is a half integer which, for most Kaluza-
Klein towers, surprisingly behaves like an $\mathrm{SL}(2)$ spin. This
motivates the following notational convention of [15]

$\begin{split}L[J]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![s]&=\bigoplus_{i=0}^{2s}L[J,\Delta_{J,i-s}]\,,\\\
L[J]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\!\\{s_{1},\cdots,s_{n}\\}&=\bigoplus_{i=1}^{n}L[J,\Delta_{J,s_{i}}]\,.\end{split}$
(4.5)

The Kaluza-Klein spectrum organises itself into towers of the form

$\begin{split}[k,q,k]_{k>1,q>1}\,&:\quad
L[\tfrac{3}{2}]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![0]\oplus
L[1]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}]\oplus
L[\tfrac{1}{2}]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\!\tfrac{1}{2}]\oplus
L[0]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\!1]\,,\\\
[k,q,k+2]_{k>0,q>1}\,\&\,[k+2,q,k]_{k>0,q>0}\,&:\quad
L[0]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}]\oplus
L[1]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}]\oplus
L[\tfrac{1}{2}]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\!\tfrac{1}{2}]\,,\\\
[k,q,k+4]_{q>1}\,\&\,[k+4,q,k]\,&:\quad
L[\tfrac{1}{2}]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![0]\oplus
L[0]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}]\,,\end{split}$
(4.6)

which degenerate for some small values of representations, such as

$[0,1,2]\,:\quad
L[1]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\![\tfrac{1}{2}]\oplus
L[\tfrac{1}{2}]\\!\mathbin{\raisebox{0.0pt}{$\scriptscriptstyle\otimes$}}\\!\\{-1,+1\\}\,.$
(4.7)

For more details and to see the remaining multiplets consult [15].

Since the consistent truncation (and thus domain wall) preserves
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$, different
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$ representations will not mix along the
flow. Therefore, we can restrict our attention to any fixed
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)$ representation to compute their quadratic
couplings. As an illustration of our method, we will evaluate the mass
operator (3.11) on the scalars fields in the $[0,0,0]$, the $[0,1,2]$ and
$[0,2,4]$ representations, which at the squashed $S^{7}$ sit in the long
multiplets (4.7) and (4.6) as follows

$\begin{split}L[\tfrac{1}{2},\Delta]:\lambda{\phantom{{}_{\mu}}}&\xrightarrow{\leavevmode\nobreak\
Q\leavevmode\nobreak\
}\,A_{\mu}\,\oplus\phi\phantom{{}_{\mu}}\xrightarrow{\leavevmode\nobreak\
Q\leavevmode\nobreak\ }\lambda\,,\\\
L[0,\Delta]:\phi{\phantom{{}_{\mu}}}&\xrightarrow{\leavevmode\nobreak\
Q\leavevmode\nobreak\
}\phantom{A_{\mu}}\;\;\lambda\phantom{+\,}\;\;\xrightarrow{\leavevmode\nobreak\
Q\leavevmode\nobreak\ }\phi\,.\end{split}$ (4.8)

###  4.1 $[0,0,0]$ sector

As a warm up, let us compute the quadratic couplings in the sector of scalar
fields singlet under $\mathrm{Sp}(2)\times\mathrm{Sp}(1)$, forming the bosonic
sector of the ${\cal N}=1$ truncation of [30, 39]. This sector carries four
scalar field fluctuations whose basis we label as $\varphi_{i}$. The
computation is straightforward: we first compute the intrinsic torsion from
(3.25), where $W$ is evaluated on the domain wall solution. Next we build the
associated mass operator (3.11) and evaluate its action on the scalar
fluctuations in this sector. The final result for the quadratic fluctuations
in this sector takes the form

$\displaystyle{\cal L}_{[0,0,0]}$
$\displaystyle=-\frac{1}{2}\,\left(\partial_{\mu}\varphi_{1}\partial^{\mu}\varphi_{1}+\partial_{\mu}\varphi_{2}\partial^{\mu}\varphi_{2}\right)-\frac{1}{2}\,e^{-6u-2v}\,\partial_{\mu}\varphi_{3}\partial^{\mu}\varphi_{3}-\frac{1}{2}\,e^{-6u+12v}\,\partial_{\mu}\varphi_{4}\partial^{\mu}\varphi_{4}$
$\displaystyle\quad-\left(54\,e^{-21u}+16\,e^{-9u-10v}-36\,e^{-9u-3v}-2\,e^{-9u+4v}\right)\varphi_{1}^{2}-12\,e^{-15u+2v}\,\varphi_{4}^{2}$
$\displaystyle\quad+2\sqrt{6}\left(9\,e^{-21u}-4\,e^{-9u-10v}-e^{-9u+4v}\right)\varphi_{1}\varphi_{2}-3\left(3\,e^{-21u}+2\,e^{-9u-10v}-e^{-9u+4v}\right)\varphi_{2}^{2}$
$\displaystyle\quad-2\left(3\,e^{-21u}+6\,e^{-15u-12v}+e^{-15u+2v}\right)\varphi_{3}^{2}-4\sqrt{6}\left(3\,e^{-21u}+e^{2v-15u}\right)\varphi_{3}\varphi_{4}\,,$
(4.9)

where $u$ and $v$ denote the domain wall solution constructed in section 2
above. One verifies that at the endpoints of the flow, the Lagrangian (4.1)
reproduces the masses of the scalar fields for the squashed and the round
$S^{7}$. In fact, since all four scalar singlets lie within the ${\cal N}=1$
truncation (3.22), we could have arrived at this result directly by
linearising the potential (3.24). It is a good consistency check of our
method, that the results indeed agree. In contrast, the scalar fluctuations at
higher KK levels do not lie within a consistent truncation and the respective
couplings can only be obtained by the mass operator (3.11), as we shall
discuss in the following. We finally note that in order to study the complete
set of fluctuation equations in this singlet sector, one also has to take into
account the fluctuations of the metric around the AdS4 background.

###  4.2 $[0,1,2]$ sector

Let us now extend the computation to scalar modes that do not lie within the
consistent truncation. As an example, we choose the sector in the $[0,1,2]$
representation of $\mathrm{Sp}(2)\times\mathrm{Sp}(1)$. This is the
representation of the vector fields that become massless on the round $S^{7}$.
As a consequence, the associated scalar Goldstone modes on the squashed sphere
become physical scalars on the round sphere. There are in total eight scalar
fluctuations $\phi_{i}$ transforming in the $[0,1,2]$ representation.
Repeating the above computation in this sector leads to the Lagrangian

$\displaystyle{\cal L}_{[0,1,2]}$
$\displaystyle=-\frac{1}{2}\,e^{-7v}\left(\partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1}+\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}\right)-\frac{1}{2}\,e^{-6u-9v}\,\partial_{\mu}\phi_{3}\partial^{\mu}\phi_{3}-\frac{1}{2}\,e^{-6u-2v}\,\partial_{\mu}\phi_{4}\partial^{\mu}\phi_{4}$
$\displaystyle\quad-\frac{1}{2}\,e^{-6u+5v}\left(\partial_{\mu}\phi_{5}\partial^{\mu}\phi_{5}+\partial_{\mu}\phi_{6}\partial^{\mu}\phi_{6}\right)-\frac{1}{2}\,e^{-12u-4v}\,\partial_{\mu}\phi_{7}\partial^{\mu}\phi_{7}-\frac{1}{2}\,e^{-12u+3v}\,\partial_{\mu}\phi_{8}\partial^{\mu}\phi_{8}$
$\displaystyle\quad-\left(10\,e^{-9u-10v}-2\,e^{-9u-3v}\right)\phi_{1}^{2}-4\,e^{-15u-5v}\,\phi_{345}^{2}-4\sqrt{2}\left(e^{-15u-5v}+3e^{-21u}\right)\phi_{6}\phi_{345}$
$\displaystyle\quad-\left(6\,e^{-15u-5v}+4\,e^{-15u+2v}+4\,e^{-15u+9v}+6\,e^{-21u}\right)\phi_{6}^{2}\,,$
(4.10)

with $\phi_{345}=\phi_{3}+\phi_{4}+\phi_{5}$. The result shows that the
potential only depends on three out of the eight $\phi_{i}$’s. Therefore, the
remaining five fluctuations are Goldstone modes along the flow. It is
interesting to compare this with the squashed sphere endpoint. From equations
(4.6) and (4.8) we see that at the squashed $S^{7}$ vacuum there are only two
physical scalars in the $[0,1,2]$ in the spectrum. Indeed, plugging in the
values of $u$ and $v$ for the squashed $S^{7}$ (2.5), we see that $\phi_{1}$
additionally drops out of the potential at the squashed $S^{7}$ endpoint, thus
reproducing the expected number of Goldstone scalars. Moreover, the potential
(4.2) reproduces the masses of the scalar fields at the squashed and round
$S^{7}$ points.

###  4.3 $[0,2,4]$ sector

As a last example, we study the six scalars $\psi_{i}$ in the $[0,2,4]$
representation. Their quadratic couplings are obtained by the analogous
computation and read

$\displaystyle{\cal L}_{[0,2,4]}$
$\displaystyle=-\frac{1}{2}\,e^{-7v}\,\partial_{\mu}\psi_{1}\partial^{\mu}\psi_{1}-\frac{1}{2}\,e^{-6u-2v}\left(\partial_{\mu}\psi_{3}\partial^{\mu}\psi_{3}+\partial_{\mu}\psi_{4}\partial^{\mu}\psi_{4}\right)-\frac{1}{2}\left(\partial_{\mu}\psi_{5}\partial^{\mu}\psi_{5}+\partial_{\mu}\psi_{6}\partial^{\mu}\psi_{6}\right)$
$\displaystyle\quad-\frac{1}{2}\,e^{-6u+5v}\,\,\partial_{\mu}\psi_{2}\partial^{\mu}\psi_{2}-6\,e^{-9u}\left(3\,e^{-10v}+e^{-3v}\right)\psi_{1}^{2}-8\,e^{-9u-10v}\left(\sqrt{5}\,\psi_{5}-\sqrt{3}\,\psi_{6}\right)\psi_{1}$
$\displaystyle\quad-4\,e^{-9u}\left(e^{-10v}+5\,e^{-3v}+e^{4v}\right)\psi_{5}^{2}-4\,e^{-9u}\left(e^{-10v}-e^{-3v}+3\,e^{4v}\right)\psi_{6}^{2}$
$\displaystyle\quad-\left(6\,e^{-15u-5v}+8\,e^{-15u+2v}+12\,e^{-21u}\right)\tilde{\psi}_{3}^{2}-\left(10\,e^{-15u-5v}+8\,e^{-15u+2v}-12\,e^{-21u}\right)\tilde{\psi}_{4}^{2}$
$\displaystyle\quad+4\sqrt{15}\,e^{-15u-5v}\,\tilde{\psi}_{3}\tilde{\psi}_{4}\,,$
(4.11)

where $\tilde{\psi}_{3}=\psi_{3}-\frac{1}{2}\sqrt{3}\,\psi_{2}$,
$\tilde{\psi}_{4}=\psi_{4}-\frac{1}{2}\sqrt{5}\,\psi_{2}$ . The potential only
depends on five out of the six scalars. Therefore, one scalar field will be a
Goldstone mode along the entire flow. Finally, as a consistency check, the
potential (4.3) again reproduces the masses at both of the endpoints.

###  4.4 Comments

We have shown how the mass operator (3.11) allows to determine the quadratic
couplings of scalar fluctuations around the domain wall background constructed
in section 2. We have restricted to spelling out three examples, but the
method of course extends to all higher KK levels. The resulting couplings
(4.1), (4.2), (4.3), carry the full information for the computation of the
holographic 2-point functions along the dual RG flow. The corresponding
computation requires a careful setup of the holographic renormalisation
procedure along the lines of [3, 45, 46], and will be interesting to take on
in the future.

##  5 Conclusions

In this paper, we revisited the problem of constructing a domain wall solution
interpolating between the ${\cal N}=1$ AdS${}_{4}\times S^{7}_{\rm squashed}$
and ${\cal N}=8$ AdS${}_{4}\times S^{7}_{\rm round}$ vacua of 11-dimensional
supergravity. There is no supersymmetric domain wall preserving the
$\mathrm{Sp}(2)\times\mathrm{Sp}(1)\subset\mathrm{SO}(8)$ isometry of the
squashed $S^{7}$, as can, for example, be seen by noting that the ${\cal N}=8$
gravitini of the round $S^{7}$ are not $\mathrm{Sp}(2)\times\mathrm{Sp}(1)$
singlets. Instead, we construct an explicit non-supersymmetric flow by using a
consistent truncation to 4-dimensional ${\cal N}=1$ supergravity [28, 30] and
solving the second-order flow equations numerically. Interestingly, within
this ${\cal N}=1$ truncation, the ${\cal N}=8$ round $S^{7}$ appears non-
supersymmetric, since its massless gravitini reside amongst the higher KK
modes.

Extending recent techniques from ExFT [5, 6, 15], we were able to compute the
quadratic couplings of KK fluctuations around the domain-wall solution (or any
other solution of the 2-dimensional truncation). This relied on the fact that
the family of squashed $S^{7}$ still admits a trivialisation of the
generalized tangent bundle [15], allowing us to construct a globally well-
defined generalised frame ${\cal V}(x,y)$. However, the intrinsic torsion
$X_{{\underline{M}}{\underline{N}}}{}^{\underline{P}}(y)$ of this generalised
frame is not constant, reflecting the fact that the squashed $S^{7}$ only
admits consistent truncations to ${\cal N}=4$ or ${\cal N}=1$ but not ${\cal
N}=8$. We presented the quadratic couplings of some low-lying KK scalars, but
the same method can be applied to any other KK tower. This encodes the
information needed to extract all the holographic two-point functions along
the flow.

Our work opens up several natural directions for future investigations. One
would be to understand the analytic structure underlying the quadratic
couplings of section 4, similar to the group-theoretic formula that appear in
the KK spectra of vacua with sufficient isometries, e.g. [6, 22]. For this it
would be useful to extend our computation of the quadratic couplings to other
KK towers. Another interesting question would be to push the calculations of
section 4 to cubic order, giving access to 3-point functions along the RG flow
on the field theory side. These results yield the relevant input for the
holographic renormalisation procedure [3, 45, 46], that would allow us to
precisely obtain the field theory correlators along the flow.

Finally, our results here, together with [15], provide the first extension of
the ExFT formalism of Kaluza-Klein spectroscopy [5, 6] beyond generalised
parallelisable spaces, i.e. vacua of maximally supersymmetric consistent
truncations. While the family of squashed $S^{7}$’s still admits a generalised
parallelisation, there is no longer an ${\cal N}=8$ consistent truncation that
the solutions belong to. Hence our method here provides a first window into
how to generalise the Kaluza-Klein spectroscopy to more general truncations
that break some supersymmetry.

All of these questions will allow us to address the ultimate question: how
does having a consistent truncation affect the structure of correlation
functions in the holographic dual? We leave these exciting research endeavours
for the future.

## Acknowledgements

We are grateful to Nikolay Bobev, Davide Cassani, Jerome Gauntlett, Joel
Karlsson, Gabriel Larios, Michela Petrini, and Dan Waldram for useful
discussions and correspondence. MG is grateful for the hospitality of ENS
Lyon. MG and EM are supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) via the Emmy Noether program “Exploring the
landscape of string theory flux vacua using exceptional field theory” (project
number 426510644).

## References

  * [1] L. Girardello, M. Petrini, M. Porrati, and A. Zaffaroni, The supergravity dual of ${N}=1$ super Yang-Mills theory, Nucl. Phys. B569 (2000) 451–469, [hep-th/9909047].
  * [2] D. Freedman, S. Gubser, K. Pilch, and N. Warner, Renormalization group flows from holography supersymmetry and a $c$-theorem, Adv. Theor. Math. Phys. 3 (1999) 363–417, [hep-th/9904017].
  * [3] M. Bianchi, D. Z. Freedman, and K. Skenderis, How to go with an RG flow, JHEP 0108 (2001) 041, [hep-th/0105276].
  * [4] M. Berg and H. Samtleben, An exact holographic RG flow between 2d conformal fixed points, JHEP 05 (2002) 006, [hep-th/0112154].
  * [5] E. Malek and H. Samtleben, Kaluza-Klein spectrometry for supergravity, Phys. Rev. Lett. 124 (2020) 101601, [arXiv:1911.12640].
  * [6] E. Malek and H. Samtleben, Kaluza-Klein spectrometry from exceptional field theory, Phys. Rev. D 102 (2020) 10, [arXiv:2009.03347].
  * [7] D. S. Berman and M. J. Perry, Generalized geometry and M theory, JHEP 1106 (2011) 074, [arXiv:1008.1763].
  * [8] A. Coimbra, C. Strickland-Constable, and D. Waldram, $E_{d(d)}\times\mathbb{R}^{+}$ generalised geometry, connections and M theory, JHEP 1402 (2014) 054, [arXiv:1112.3989].
  * [9] A. Coimbra, C. Strickland-Constable, and D. Waldram, Supergravity as generalised geometry II: $E_{d(d)}\times\mathbb{R}^{+}$ and M theory, JHEP 1403 (2014) 019, [arXiv:1212.1586].
  * [10] D. S. Berman, H. Godazgar, M. Godazgar, and M. J. Perry, The local symmetries of M-theory and their formulation in generalised geometry, JHEP 1201 (2012) 012, [arXiv:1110.3930].
  * [11] D. S. Berman, M. Cederwall, A. Kleinschmidt, and D. C. Thompson, The gauge structure of generalised diffeomorphisms, JHEP 1301 (2013) 064, [arXiv:1208.5884].
  * [12] O. Hohm and H. Samtleben, Exceptional form of ${D}=11$ supergravity, Phys. Rev. Lett. 111 (2013) 231601, [arXiv:1308.1673].
  * [13] K. Lee, C. Strickland-Constable, and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017), no. 10-11 1700048, [arXiv:1401.3360].
  * [14] O. Hohm and H. Samtleben, Consistent Kaluza-Klein truncations via exceptional field theory, JHEP 1501 (2015) 131, [arXiv:1410.8145].
  * [15] B. Duboeuf, E. Malek, and H. Samtleben, Kaluza-Klein spectrometry beyond consistent truncations: the squashed S7, JHEP 04 (2023) 062, [arXiv:2212.01135].
  * [16] N. Bobev, E. Malek, B. Robinson, H. Samtleben, and J. van Muiden, Kaluza-Klein spectroscopy for the Leigh-Strassler SCFT, JHEP 04 (2021) 208, [arXiv:2012.07089].
  * [17] A. Giambrone, E. Malek, H. Samtleben, and M. Trigiante, Global properties of the conformal manifold for S-fold backgrounds, JHEP 06 (2021) 111, [arXiv:2103.10797].
  * [18] A. Giambrone, A. Guarino, E. Malek, H. Samtleben, C. Sterckx, and M. Trigiante, Holographic evidence for nonsupersymmetric conformal manifolds, Phys. Rev. D 105 (2022), no. 6 066018, [arXiv:2112.11966].
  * [19] M. Cesàro, G. Larios, and O. Varela, $\mathcal{N}$ = 1 S-fold spectroscopy, JHEP 08 (2022) 242, [arXiv:2206.04064].
  * [20] E. Malek, H. Nicolai, and H. Samtleben, Tachyonic Kaluza-Klein modes and the AdS swampland conjecture, JHEP 08 (2020) 159, [arXiv:2005.07713].
  * [21] T. Fischbacher, K. Pilch, and N. P. Warner, New supersymmetric and stable, non-supersymmetric phases in supergravity and holographic field theory, arXiv:1010.4910.
  * [22] A. Guarino, E. Malek, and H. Samtleben, Stable non-supersymmetric Anti-de Sitter vacua of massive IIA supergravity, Phys. Rev. Lett. 126 (2021), no. 6 061601, [arXiv:2011.06600].
  * [23] M. A. Awada, M. J. Duff, and C. N. Pope, $N=8$ supergravity breaks down to $N=1$, Phys. Rev. Lett. 50 (1983) 294.
  * [24] H. Ooguri and C.-S. Park, Superconformal Chern-Simons theories and the squashed seven sphere, JHEP 11 (2008) 082, [arXiv:0808.0500].
  * [25] B. de Wit, H. Nicolai, and N. Warner, The embedding of gauged ${N}=8$ supergravity into $d=11$ supergravity, Nucl.Phys. B255 (1985) 29\.
  * [26] M. Duff, B. Nilsson, and C. Pope, Kaluza-Klein supergravity, Phys.Rept. 130 (1986) 1–142.
  * [27] C. Ahn and S.-J. Rey, Three-dimensional CFTs and RG flow from squashing M2-brane horizon, Nucl. Phys. B 565 (2000) 210–214, [hep-th/9908110].
  * [28] D. N. Page and C. N. Pope, Stability analysis of compactifications of $D=11$ supergravity with SU(3) $\times$ SU(2) $\times$ U(1) symmetry, Phys. Lett. B 145 (1984) 337–341.
  * [29] V. L. Campos, G. Ferretti, H. Larsson, D. Martelli, and B. E. W. Nilsson, A study of holographic renormalization group flows in ${D}=6$ and ${D}=3$, JHEP 06 (2000) 023, [hep-th/0003151].
  * [30] D. Cassani and P. Koerber, Tri-Sasakian consistent reduction, JHEP 01 (2012) 086, [arXiv:1110.5327].
  * [31] D. N. Page, Classical stability of round and squashed seven spheres in eleven-dimensional supergravity, Phys. Rev. D 28 (1983) 2976.
  * [32] I. R. Klebanov and E. Witten, AdS / CFT correspondence and symmetry breaking, Nucl.Phys. B556 (1999) 89–114, [hep-th/9905104].
  * [33] O. Hohm and H. Samtleben, Exceptional field theory II: E7(7), Phys. Rev. D89 (2014) 066017, [arXiv:1312.4542].
  * [34] B. de Wit and H. Nicolai, $N=8$ supergravity, Nucl. Phys. B208 (1982) 323.
  * [35] B. de Wit and H. Nicolai, The consistency of the $S^{7}$ truncation in $D=11$ supergravity, Nucl.Phys. B281 (1987) 211.
  * [36] E. Sezgin, The spectrum of the eleven-dimensional supergravity compactified on the round seven sphere, Phys. Lett. B 138 (1984) 57–62.
  * [37] B. Biran, A. Casher, F. Englert, M. Rooman, and P. Spindel, The fluctuating seven sphere in eleven-dimensional supergravity, Phys. Lett. 134B (1984) 179.
  * [38] A. Casher, F. Englert, H. Nicolai, and M. Rooman, The mass spectrum of supergravity on the round seven sphere, Nucl. Phys. B243 (1984) 173\.
  * [39] B. E. W. Nilsson and C. N. Pope, De-Higgsing in eleven-dimensional supergravity on the squashed $S^{7}$, arXiv:2302.03842.
  * [40] F. A. Bais, H. Nicolai, and P. van Nieuwenhuizen, Geometry of coset spaces and massless modes of the squashed seven sphere in supergravity, Nucl. Phys. B 228 (1983) 333–350.
  * [41] B. E. W. Nilsson, A. Padellaro, and C. N. Pope, The role of singletons in S7 compactifications, JHEP 07 (2019) 124, [arXiv:1811.06228].
  * [42] S. Ekhammar and B. E. W. Nilsson, On the squashed seven-sphere operator spectrum, JHEP 12 (2021) 057, [arXiv:2105.05229].
  * [43] J. Karlsson, Compactifications of string/M-theory and the swampland: A study of the AdS4 mass spectrum of eleven-dimensional supergravity on the squashed seven-sphere, arXiv:2110.09885.
  * [44] J. Karlsson and B. E. W. Nilsson, The complete Kaluza-Klein spectra of $\mathcal{N}=1$ and $\mathcal{N}=0$ M-theory on $AdS_{4}\times(\text{squashed }S^{7})$, arXiv:2305.00916.
  * [45] M. Bianchi, D. Z. Freedman, and K. Skenderis, Holographic renormalization, Nucl. Phys. B631 (2002) 159–194, [hep-th/0112119].
  * [46] I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075, [hep-th/0407071].


∎

11institutetext: Hamam Mokayed 22institutetext: 22email<EMAIL_ADDRESS>33institutetext: Palaiahnakote Shivakumara 44institutetext: 44email:
<EMAIL_ADDRESS>55institutetext: Lama Alkhaled 66institutetext: 66email:
<EMAIL_ADDRESS>77institutetext: Rajkumar Saini 88institutetext: 88email:
<EMAIL_ADDRESS>99institutetext: Muhammad Zeshan Afzal
1010institutetext: 1010email<EMAIL_ADDRESS>1111institutetext:
Yan Chai Hum 1212institutetext: 1212email<EMAIL_ADDRESS>1313institutetext: Marcus Liwicki 1414institutetext: 1414email:
<EMAIL_ADDRESS>
# Robust and Fast Vehicle Detection using Augmented Confidence Map

Hamam Mokayed  Palaiahnakote Shivakumara  Lama Alkhaled  Rajkumar Saini
Muhammad Zeshan Afzal  Yan Chai Hum  Marcus Liwicki

(Received: date / Accepted: date)

###### Abstract

Vehicle detection in real-time scenarios is challenging because of the time
constraints and the presence of multiple types of vehicles with different
speeds, shapes, structures, etc. This paper presents a new method relied on
generating confidence map-for robust and faster vehicle detection. To reduce
the adverse effect of different speeds, shapes, structures, and the presence
of several vehicles in a single image, we introduce the concept of
augmentation which highlights the region of interest containing the vehicles.
The augmented map is generated by exploring the combination of multiresolution
analysis and maximally stable extremal regions (MR-MSER). The output of MR-
MSER is supplied to fast CNN to generate a confidence map, which results in
candidate regions. Furthermore, unlike existing models that implement
complicated models for vehicle detection, we explore the combination of a
rough set and fuzzy-based models for robust vehicle detection. To show the
effectiveness of the proposed method, we conduct experiments on our dataset
captured by drones and on several vehicle detection benchmark datasets,
namely, KITTI and UA-DETRAC. The results on our dataset and the benchmark
datasets show that the proposed method outperforms the existing methods in
terms of time efficiency and achieves a good detection rate.

###### Keywords:

Vehicle detection Augmented confidence map rough set theory Fuzzy Logic
Maximally stable extremal regions

##  1 Introduction

With the increasing importance of vehicle-related industrial applications,
vehicle detection has been a very interesting research topic for the last two
decades Ref1 . The research started based on applying traditional nondeep
learning methods to accomplish the task. Because the implemented features of
these conventional algorithms are susceptible to image deformation, vehicle
occlusion, and illumination changes, these methods have lower accuracy and can
be deployed in limited scenarios Ref2 ; Ref4 ; Ref8 . With the revolution of
deep learning, many methods have been suggested to enhance vehicle detection
performance by proposing complex network architectures Ref5 ; Ref6 ; Ref7 .
The main drawback of these methods is the cost of the high computational power
machines and longer processing time requirements. Considerable research
related to object detection, in general, has been conducted, and many of these
generic classifiers cannot achieve competitive performance on vehicle
detection benchmarks because of the different problems related to vehicle
detection, such as large variations in light, dense occlusion, and size
differences. These factors motivated us to propose a new method that combines
a fast deep learning method for augmented feature maps with traditional
methods for vehicle detection. Fig. 1 illustrates the contribution of the
proposed model to detect multiple vehicles in drone images. The input drone
image containing multiple vehicles is shown in Fig. 1(a). The region of
interest is detected by the combination of multiresolution and maximally
stable extremal regions (MR-MSE) as shown in Fig. 1(b). The regions are
subjected to a CNN to generate an augmented confidence map (ACM), which
outputs saliency for the vehicle regions seen in Fig. 1(c). The combination of
rough set and fuzzy set is proposed to classify nonvehicle and vehicle blobs,
as shown in Fig. 1(d)-(e).

Figure 1: The proposed steps for vehicle detection in multivehicle environment

##  2 Related Work

Several methods have been developed in the past for addressing the challenges
of vehicle detection, especially for tiny vehicles and multiple vehicles in
images, based on a deep learning approach. Wang et al. Ref9  presented a
review of vehicle detection for intelligent vehicles. This paper discusses
various approaches for addressing different challenges of vehicle detection.
It is noted from the review that robust vehicle detection in crowded scenes
and when the image contains multiple vehicles are still considered open
challenges for vehicle detection. For example, He et al. Ref10  proposed a
method for vehicle detection with deep features and shape features. The
approach uses the combination of R-CNN and SVM for vehicle detection in
satellite images. Hu et al. Ref7  used a scale-intensive CNN for fast vehicle
detection. The technique targets small-scale vehicles and uses contextual
information for vehicle detection. Other researchers proposed to develop a
YOLO-based method for front-end vehicle detection Ref11 . The aim of the
method is to achieve time efficiency to meet the requirements of real-time
applications. Li et al. Ref13  explored the combination of several deep
learning models for detecting tiny vehicles in images. Since the target of the
method is to reduce the number of computations, the approach uses the YOLOv3
architecture for vehicle detection. Ojha et al.Ref14  proposed an instance-
based segmentation-based model for vehicle detection. To achieve this, the
method uses R-CNN transfer learning. Shan et al. Ref15  explored instance
segmentation for real-time vehicle detection. The method estimates the precise
loss for predicting the bounding box regression for vehicle detection. Shi et
al. Ref16  introduced orientation-aware vehicle detection based on an anchor-
free object detection approach for aerial images. The method splits transfer
learning into multitask learning for vehicle detection. Wang et al. Ref17
proposed a model for vehicle detection based on single- and two-stage deep
learning techniques. The method uses a soft weighted average method to achieve
the best results. Yao et al. Ref18  proposed YOLOv3 for traffic vehicle
detection. The approach uses a combination of clustering and deep learning to
overcome the challenges of traffic vehicle detection. It is observed from a
review of different vehicle detection methods that most of the methods focus
on high-resolution images and images captured in a head-on direction but not
drone images. Therefore, one can infer that the existing method may not be
effective and robust for drone images because of the adverse effects of height
distance variation, oblique angles, and weather conditions. Thus, this study
aims to develop a robust approach for accurately detecting vehicles in drone
images as well as normal images by exploring the combination of deep learning,
augmented confidence maps, rough sets and fuzzy logic. Hence, the
contributions are threefold: (i) generating augmented confidence maps with the
help of multiresolution and maximally stable extremal regions, (ii) exploring
the combination of rough set and fuzzy for vehicle blobs classification, and
(iii) the way the proposed work integrates the deep learning, augmentation,
rough set and fuzzy is new compared to the state-of-the-art methods.

##  3 Proposed Method

This research aims to propose a robust vehicle detection method by combining a
fast deep learning method with traditional steps. The work can be divided into
two main stages, namely, augmented confidence map (ACM) and a traditional
classifier based on ACM.

###  3.1 Augmented confidence map (ACM)

First, a robust algorithm, namely, maximally stable extremal regions (MSER),
has been proposed to provide a new set of image elements. These elements are
known as extremal regions, with two features extracted from the projective
transformation of image coordinates and the monotonic transformation of image
intensities Ref19 . Affine invariant feature descriptors are computed on a
grayscale image, and the robustness of MSER is different from multiple
measurement regions picked up from invariant constructions from extremal
regions Ref20 , but some of the regions show discriminative properties that
are significantly larger and possibly used to establish tentative
correspondences Ref19 . Since MSER can generate many blobs with different
sizes based on detection in an original image resolution, the different
resolutions that can yield from long distances or blurred (coarse image), the
loss of details in the image and different regions are connected to their
neighbor Ref20 . The MSER can work better in invariance to changing scale in
the scene image at different resolutions, which achieves stable vehicle
regions. The resolution of an image is obtained by a scale pyramid (without a
Gaussian filter) with one octave for a scale and a total of three scales from
the finest image (input image) to the coarser image (blurred image). This will
lead to generating multiresolution maximally stable extremal regions MR-MSER,
and this MR-MSER is applied to vehicle images with 640×480 resolution Ref19 .

Figure 2: The regions generated by applying MSER and MR-MSER over vehicle
images

An augmentation process is applied to the MR-MSER to enhance the detection
recall of the regions. The augmentation pipeline is adopted in each MR-MSER
region shown in Fig. 3. Each MR-MSER with a different size is in a rectangular
bounding box without moving its center. The bounding box is then squarified by
inflating 30% and 60% of its area in three different dimensions. The different
squared dimensions are resized to a 28×28 pixel image patch. Each image patch
applied online jitter is randomly rotated within $[-\pi/4,~{}\pi/4]$ for four
different times, which will be used in model training Ref21 .

Figure 3: The applied augmentation pipeline

The data generated by the augmented MR-MSER is used to generate the confidence
map. The data are extracted from 100 images randomly selected from the KITTI
dataset, UA-DETRAC dataset, and our collected dataset. The annotation of the
ground truth (foreground, which is the vehicle) is performed on 100 images.

Figure 4: The regions generated by applying augmented MR-MSER over vehicle
images

By using the augmented MR-MSER, the positive samples are obtained by the
annotated ground-truth vehicle and having an intersection-over-union IoU¿0.5,
no less than one of the ground-truth vehicle annotations. The negative samples
are equal to positive samples that are randomly selected with IoU=0 for every
ground-truth vehicle annotation. The proposed framework uses a slight variant
of LeNet for classifying image patches from augmented MR-MSER in two classes,
either foreground (vehicle) or background (nonvehicle). The input is a 28×28
pixel image. The framework used to generate the confidence map has a total of
three convolutional layers with 128,256,512 feature maps and two fully
connected layers that have 512 units. The kernel size and stride are fixed to
3 and 1, respectively, in each convolutional layer. Max pooling with a 2×2
pooling size is performed after each convolutional layer. The rectified linear
unit (ReLU) is performed as an activation function for all layers except the
prediction layer (last layer) using the sigmoid function. A dropout layer is
added with 0.5 rates before the prediction layer to increase the model
accuracy. The CNN model is trained with a training dataset that is extracted
from 100 images based on the MR-MSER that has the threshold on intersection-
over-union IoU such that the set for the foreground (vehicle) and the set for
the background (nonvehicle) is obtained with equal size. The training dataset
is split into a training set and validation set with an appropriate ratio, and
the model is trained in minibatches with the optimum batch size and a number
of epochs. The learning optimization in training the model uses the Adam
optimizer, which is straightforward in implementation, efficient in
computation with fewer memory requirements, invariant to diagonal rescaling of
the gradients, and well suited for larger data and/or parameters Ref22 . Since
the model is trained for binary classification, binary cross-entropy is the
loss function for the model. The confidence map is computed by using the
confidence values obtained from the stacking of each augmented MR-MSER
proposal. The confidence map described as higher intensity regions likely
denotes the interest of vehicle components. The threshold is used in the
confidence map to filter the region that likely does not belong to the
foreground regions. The remaining foreground regions are used for further
image denoising and segmentation.

Figure 5: Augmented confidence map

###  3.2 Traditional classifier based on ACM

The purpose of using the rough set approach is knowledge discovery and
approximation of sets using granular information. Given a confidence map, the
process for granularization splits the input image window into multiple
windows with a resolution of each subwindow (g = 4). The main purpose is to
classify pixel values into vehicle and nonvehicle approximations. Let a set of
objects be U. There is also an indiscernibility relation $R\subseteq~{}U*U$
that refers to the central concept of rough set theory. In the
indiscernibility relation, the values of the object are identical considering
a subset of the related attributes. In other words, it is an equivalence
relation where all identical values of the object are elementary. Hence, R can
also be considered an equivalence relation. Let X be a subset of U with two
possibilities, either is crisp which is explicit with respect to R if the
boundary region of X is empty, or is rough, which is inexplicit with respect
to R if the boundary region of X is nonempty using RST to characterize the set
U as possible for lower approximation and upper approximation, and boundary
region of set X.

$\underline{R}(X)=\cup_{x\in~{}U}R(X):R(X)\subseteq X$ (1)

R-upper approximation of X:

$\overline{R}(X)=\cup_{x\in~{}U}R(X):R(X)\cap X\neq\phi$ (2)

R-boundary region of X:

$RN_{R}(X)=\overline{R}(X)-\underline{R}(X)$ (3)

Rough entropy (RE) is introduced, which can avoid imprecision to find the
optimum threshold as precisely as possible. The rough entropy threshold (RET)
as the reference for the threshold in the binarization approach in the
grayscale image, which was obtained by using a sliding window with a
nonoverlapping granule window in m×n, is set as a 2×2 window size. RET can be
defined as Ref23

$RE_{T}=-\frac{e}{2}[R_{OT}\log(R_{OT})+R_{BT}\log(R_{BT})]$ (4)

where
$R_{OT}=1-\mid\overline{O}_{T}\mid/\mid\underline{O}_{T}\mid$ is the roughness
of the object
$R_{BT}=1-\mid\overline{B}_{T}\mid/\mid\underline{B}_{T}\mid$is the roughness
of the background
$\mid\overline{O}_{T}\mid$ and $\mid\underline{O}_{T}\mid$ are the cardinality
of the sets $\overline{O}_{T}$ and $\underline{O}_{T}$ for a given image
depending on the value T.
$\mid\overline{B}_{T}\mid$ and $\mid\underline{B}_{T}\mid$ are the cardinality
of the sets $\overline{B}_{T}$ and $\underline{B}_{T}$ for a given image
depending on the value T.

The principle of reducing the roughness of both the object and background and
maximizing $RE_{T}$ is computed for every T representing the object and
background regions, respectively (0,…,T) and (T+1,…,L-1). The optimum
threshold is selected for the maximum $RE_{T}$ to provide the object-
background segmentation given by the definition of $T^{*}$

$T^{*}=arg\>max_{T}RE_{T}$ (5)

Note that maximizing the rough entropy $RE_{T}$ to obtain the required
threshold basically implies minimizing both the object roughness and
background roughness such that this method is an object enhancement/extraction
method Ref23

Figure 6: Applying rough set theory to the ACM

Finally, the fuzzy classifier assigns a class label to an object based on the
object description. This can be known as predicting the class label where the
object description is in the form of a vector. This vector contains values of
the features deemed to be relevant information for the classification process.
The feature is the intensity distribution in the ACM.The classifier has
learned with a training dataset to predict class labels whether it is vehicle
or noise. A sample of the collected data used for training is shown in the
figure below.

Figure 7: Collected data used for the fuzzy logic training process

Fuzzy ruble-based classifiers have utilized soft labeling. A standard
assumption in pattern recognition is that the classes are mutually exclusive.
Soft labeling with the given dataset is split into 2 classes: vehicle set and
background set. Both the vehicle set and background set are computed with the
intensity distribution and soft labeling is performed for different sets. The
representation of the output image will be in bicolor, the “green” color is
labeled as a vehicle, and the “red” color is labeled as background, as shown
in Fig. 8.

Figure 8: Results of fuzzy logic over the nominated blobs

##  4 Experimental Results

To evaluate the proposed system, first, we used two standard datasets mainly
used for vehicle-related problems named KITTI and UA-DETRAC. In addition, a
self-collected dataset captured from drones is used for further evaluation.
The self-collected dataset consists of images collected at different times
(early morning, afternoon, and evening) to obtain vehicle images with multiple
densities from the open car parking area and highway. In addition, the dataset
includes images captured from varying height distances from 1-3, 3-5, and more
than 7 meters at different angles. The overall number of collected images is
1,000. A sample of our images is shown in the figure below.

Figure 9: Samples of our collected dataset for the vehicle detection problem

###  4.1 Ablation study

In this work, the combination of an ACM and a traditional classifier is
proposed to reach a promising level of accuracy and performance for
diversified situations. To show the effectiveness of the proposed steps in the
conducted work, the following experiments were implemented toward ablation
studies on our collected data and the other two benchmarked data. The results
reported in Table I show the importance of applying the augmentation process
on MR-MSER to enhance the detection performance of the regions. The three
experiments shown in Tables I, II, and III justify the need for ACM for better
generalization ability, flexibility, and robustness to different situations
compared to applying only MSER or MR-MSER at the ACM stage.

Table 1: EVALUATION OF VEHICLE DETECTION USING THE KITTI DATASET Measures | AP | Input image size
---|---|---
Method with MSER | 97.22 | 512*512
Method with MR-MSER | 84.45 | 512*512
ACM proposed work | 90.08 | 512*512
Figure 10: Results of applying MSER, MR-MSER, and augmented MR-MSER over KITTI vehicle images Table 2: EVALUATION OF VEHICLE DETECTION USING THE UA-DETRAC DATASET Measures | Easy(AP) | Med(AP) | Hard(AP)
---|---|---|---
Method with MSER | 68.48 | 50.22 | 31.87
Method with MR-MSER | 73.69 | 61.15 | 42.12
ACM proposed work | 89.22 | 77.43 | 60.22
Figure 11: Results of applying MSER, MR-MSER, and augmented MR-MSER over UA-DETRAC vehicle images Table 3: EVALUATION OF VEHICLE DETECTION USING OUR DATASET Measures | AP | Input image size
---|---|---
Method with MSER | 70.87 | 512*512
Method with MR-MSER | 81.65 | 512*512
ACM proposed work | 94.20 | 512*512
Figure 12: Results of applying MSER, MR-MSER, and augmented MR-MSER over our
dataset

###  4.2 Experiment on vehicle detection

To make a fair comparison between the proposed ACM method and existing
algorithms, average precision (AP0.7) is used for all the experiments in this
work [22]. The best match was calculated using ground truth and bounding boxes
with a threshold of 0.7 for the correct count (true positive). Otherwise, it
is considered a false negative. To evaluate the performance of the proposed
method over the KITTI dataset, the number of frames processed per second (FPS)
is measured. As the inference speed is related to the hardware equipment, the
inference test FPS in this paper is carried out on an NVIDIA RTX 2080Ti. Table
IV evaluates both the performance and the efficiency of ACM’s proposed method
over different methods.

Table 4: EVALUATION OF VEHICLE DETECTION USING THE KITTI DATASET Measures | AP0.7 | FPS | Image input size
---|---|---|---
MS-CNNRef24  | 87.42 | 8.13 | 1920*576
SINetRef28  | 89.82 | 23.98 | 1920*576
Faster R-CNN | 59.7 | - | -
YoloV3Ref25  | 79.49 | 43.57 | 512*512
YoloV4Ref26  | 90.5 | 52.14 | 512*512
ACM Proposed work | 90.08 | 60.42 | 512*512
Figure 13: Results of applying the proposed ACM to the KITTI sample image.

To evaluate the detection accuracy for different challenge scenarios, the UA-
DETRAC dataset is used with different levels of difficulty.

Table 5: EVALUATION OF VEHICLE DETECTION USING THE DETRAC DATASET Measures | Easy(AP) | Med(AP) | Hard(AP)
---|---|---|---
YoloV3Ref25  | 83.28 | 62.25 | 42,44
RFCNRef27  | 92.32 | 75.67 | 54.31
Yolov5sRef29  | 94.2 | - | -
Yolov5mRef29  | 95.6 | - | -
ACM Proposed work | 89.22 | 77.43 | 60.22
Figure 14: Results of applying the proposed ACM to the DETRAC sample image

We also implemented different models of YOLO on the collected dataset to
evaluate both the detection performance and the accuracy. The performance is
tested using the same hardware used for the KITTI dataset

Table 6: EVALUATION OF VEHICLE DETECTION USING OUR COLLECTED DATASET Measures | AP0.7 | FPS | Image input size
---|---|---|---
YoloV4Ref26  | 96.89 | 52.28 | 512*512
YoloV5SRef29  | 97.52 | 55.4 | 512*512
YoloV5MRef29  | 97.7 | 33.2 | 512*512
ACM Proposed work | 94.2 | 61.20 | 512*512

##  5 Conclusion

In this work, we proposed a new method for vehicle detection. We introduced
new concepts, namely, the augmented confidence map (ACM), which was fused with
traditional learning algorithms to achieve a faster classification method on
top of enhanced features generated by a simple CNN. The proposed system shows
the capability of detecting vehicles in different scenarios with faster
performance. Experimental results conducted on benchmark data called KITTI and
DETRAC and on our drone-captured datasets show that the proposed system
outperforms the existing methods in terms of timing and can achieve accurate
results near YOLOv4. Furthermore, the proposed system is consistent for
different datasets and situations.

## References

  * (1) K. Sakhare, T. Tewari and V. Vyas, “Review of vehicle detection systems in advanced driver assistant systems,” Arch. Comput. Methods Eng., vol. 27, no. 2, pp. 591-610, (2020).
  * (2) L. Tsai, J. Hsieh and K. Fan, “Vehicle detection using normalized color and edge map,” IEEE Trans. Image Process., vol. 16, no. 3, pp. 850-864 (2007).
  * (3) Xuezhi Wen, Ling Shao, Wei Fang, and Yu Xue. ”Efficient feature selection and classification for vehicle detection.” IEEE Transactions on Circuits and Systems for Video Technology, 25, no. 3, 508-517 (2015).
  * (4) Pedro F. Felzenszwalb, David A. McAllester, and Deva Ramanan. ”A discriminatively trained, multiscale, deformable part model.” In Computer Vision and Pattern Recognition (CVPR), vol. 2, no. 6, p. 7 (2008).
  * (5) A. Geiger, P. Lenz, R. Urtasun, “Are we ready for autonomous driving? The KITTI vision benchmark suite,” 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3354-3361 (2021).
  * (6) A. Howard, M. Zhu and B. Chen, “Mobilenets: Efficient convolutional neural networks for mobile vision applications,” arXiv preprint arXiv, 1704.04861 (2017).
  * (7) Hu, F., Venkatesh, G. M., O’Connor, N. E., Smeaton, A. F., and Little, S. Utilising Visual Attention Cues for Vehicle Detection and Tracking. In 2020 25th International Conference on Pattern Recognition (ICPR), pp. 5535-5542 (2021).
  * (8) Tian, K., Zhou, T., Xiang, S., and Pan, C. Forground-Guided Vehicle Perception Framework. In 2020 25th International Conference on Pattern Recognition (ICPR) pp. 8015-8020 (2021).
  * (9) Z. Wang, J. Zhan, C. Duan, X. Guan, P. Lu and K. Yang, “A review of vehicle detection techniques for intelligent vehicles”, IEEE Trans. NNLS (2022).
  * (10) K. He and L. Zhang, “Vehicle detecion in satellite images with deep neural network and vehicle shape features”, In Proc. IGARSS, pp 4212-4215 (2021).
  * (11) S. Du, P. Zhang, B. Zhang and H. Xu, ”Weak and Occluded Vehicle Detection in Complex Infrared Environment Based on Improved YOLOv4,” in IEEE Access, vol. 9, pp. 25671-25680 (2021).
  * (12) M. S. Junayed, B. Islam, A. Sadeghzadeh and T. Aydin, “Real-time YOLO-based heterogeneous front vehicle detection”, In Proc. INISTA (2021).
  * (13) L. Li and Y. Liang, “Deep learning target vehicle detection method based on YOLOv3-tiny”, In Proc. IMCEC, pp 1575-1579 (2021).
  * (14) A. Ojha, S. P. Sahu and D. K. Dewangan, “Vehicle detection through instance segmentation using Mask R-CNN for intelligent vehicle system”, In Proc. ICICCS, pp 954-959 (2021).
  * (15) Z. Shan and H. Tian, “A real-time vehicle detection algorithm based on instance segmentation”, In Proc. ICBAIE, pp 239-243 (2021).
  * (16) F. Shi, T. Zhang and T. Zhang, “Orientation-aware vehicle detection in aerial images via an achor-free object detection approach”, IEEE Trans. GRS, pp 5221-5233 (2021).
  * (17) H. Wang, Y. Yu, Y. Cai, X. Chen, L. Chen and Y. Li, “Soft-weighted-average ensemble vehicle detection method based on single-stage and two-stage deep leraning models”, IEEE Trans. IV, pp 100-109 (2021).
  * (18) X. Yao, Y. Zhang, Y. Yao, J. Tian, C. Yang, Z. Xu and Y. Guan, “ Traffic vehicle detection algorithm based on YOLOv3”, In Proc. ICITBS, pp 47-50 (2021).
  * (19) Matas, J., Chum, O., Urban, M. and Pajdla, T. Robust wide-baseline stereo from maximally stable extremal regions. Image and vision computing, 22(10), pp.761-767 (2004).
  * (20) Donoser, M. and Bischof, H, June. Efficient maximally stable extremal region (MSER) tracking. In 2006 IEEE computer society conference on computer vision and pattern recognition (CVPR’06) (Vol. 1, pp. 553-560) (2006).
  * (21) Zamberletti, A., Gallo, I. and Noce, L., 2015, November. Augmented text character proposals and convolutional neural networks for text spotting from scene images. In 2015 3rd IAPR Asian Conference on Pattern Recognition (ACPR), pp. 196-200 (2015).
  * (22) Kingma, D.P. and Ba, J. Adam: a method for stochastic Opoimization. (2017).
  * (23) Pal, S.K., Shankar, B.U. and Mitra, P. Granular computing, rough entropy and object extraction. Pattern recognition letters, 26(16), pp.2509-2517 (2005).
  * (24) Z. Cai, Q. Fan, R. S. Feris, and N. Vasconcelos, “A unified multi-scale deep convolutional neural network for fast object detection,” in Proceedings of the European conference on computer vision, pp. 354–370 (2016).
  * (25) R. Joseph and F. Ali, “YOLOv3: an incremental improvement,” https://arxiv.org/abs/1804.02767 (2018).
  * (26) A. Bochkovskiy, C.-Y. Wang, and H. Y. Mark Liao, “YOLOv4: optimal speed and accuracy of object detection,” https:// arxiv.org/abs/2004.10934 (2020).
  * (27) J. Dai , Y. Li , K. He , J. Sun , R-FCN: object detection via region-based fully con- volutional networks, in: Proceedings of the Advances in Neural Information Processing Systems, 2016, pp. 379–387 (2016).
  * (28) X. Hu, X. Xu, Y. Xiao, H. Chen, S. He, J. Qin and P. Heng, “SINet: A scale-insensitive convolutional neural network for fast vehicle detection”, IEEE Trans. TS, pp 1010-1019 (2019).
  * (29) Ultralytics: Yolov5. https://github.com/ultralytics/yolov5 (2021).