# Stability of Wilson loops and other observables in various type IIB
Backgrounds

Dimitrios Chatzis<EMAIL_ADDRESS>
###### Abstract

This work involves the stability study of various observables (Wilson loop, ’t
Hooft loop and Entanglement Entropy) under linear fluctuations of the
coordinates for certain ten-dimensional solutions of type $\mathrm{IIB}$
Supergravity that have appeared in the literature recently. These backgrounds
are defined using intersecting and wrapped $\mathrm{D}5$ branes, their
holographically dual field theories are speculated to confine and they have a
non-local $\mathrm{UV}$ completion that is governed by Little String Theory.
The present study confirms previous claims on the stability of the solutions
made using the concavity condition for the energy of the probe strings, by
studying the eigenvalue problem for each case and also providing a numerical
analysis.

###### Contents

  1. 1 Introduction
  2. 2 The setup
  3. 3 Type $\mathrm{IIB}$ Supergravity solutions and their dual QFTs
    1. 3.1 Background I
    2. 3.2 Background II
    3. 3.3 Background III
    4. 3.4 Background IV
  4. 4 Observables of interest
    1. 4.1 Wilson loops
    2. 4.2 ’t Hooft loops
    3. 4.3 Entanglement Entropy
  5. 5 Perturbative study
    1. 5.1 Perturbations and their equations of motion
    2. 5.2 Boundary conditions
  6. 6 Stability analysis for the geometries of interest
    1. 6.1 Backgrounds dual to (1+1)-dimensional QFTs
      1. 6.1.1 Wilson Loop
      2. 6.1.2 ’t Hooft loop
      3. 6.1.3 Entanglement Entropy
    2. 6.2 Background dual to (4+1)-dimensional QFTs
      1. 6.2.1 Wilson loop
      2. 6.2.2 ’t Hooft loop
      3. 6.2.3 Entanglement Entropy
    3. 6.3 Background dual to (2+1)-dimensional QFTs
      1. 6.3.1 Wilson loop
      2. 6.3.2 ’t Hooft loop
  7. 7 Conclusions
  8. A Zero modes of longitudinal perturbations $\Leftrightarrow L^{\prime}(r_{0})=0$
  9. B Schrödinger description of the problem

##  1 Introduction

Since the early days of the gauge/gravity correspondance [1, 2] and its
generalisation to non-conformal settings with no $\mathrm{AdS}$ factor [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13], there have been many new insights into strong
coupling phenomena of Quantum Field Theories. Various works have been done in
this direction in the past, see [14, 15, 16, 17, 18, 19, 20, 21] and many
more, in order to study non-perturbative effects of some phenomenological
resemblance to the theory of strong interactions. In particular, there has
been a standard way of computing the Wilson loop, represented as the boundary
of the worldsheet of a string exploring the holographic dimension [22],
through which one can examine the confinement of the dual theory using the
usual arguments for the potential energy [23].

It was realised that in some cases the classical probe string exploring the
bulk, whose endpoints at radial infinity represent the static quark-anti-quark
pair, could admit more than one embedding for a given set of boundary
conditions. This allows for a phase transitions to occur between different
classical configurations of the system, as the energy of the pair is a
multivalued function of the length of separation. While there has been a
criterion based on [24, 15] for the stability of a given embedding that agrees
with past examples, namely $\mathrm{d}^{2}E(L)/\mathrm{d}L^{2}\leq 0$ where
$E(L)$ is the energy of the pair as a function of their separaration, the most
trusted way of determining stability is through the study of small
fluctuations around a classical solution.

In the current work, we will first present the four Type $\mathrm{IIB}$
Supergravity backgrounds of interest appearing in [25, 26], and the behaviour
of their dual field theories in Sections 2 and 3. These are ten-dimensional
gravity backgrounds describing different setups of $\mathrm{D}5$ branes with
corresponding dual theories of various dimensions, that have a non field-
theoretic UV completion. Section 4 will concern the observables which we are
going to study, as we will also be interested in ’t Hooft loops and
Entanglement Entropy, and review how they are calculated using holography. In
Section 5 we will analyse the dynamics of small linear fluctuations and their
eigenvalue problems, using techniques that were developed in [27, 28, 29, 30].
The goal of this study is to investigate if their spectrum includes negative
eigenvalues for some range of values for the tip of the string, thus producing
instabilities. Finally, the stability analysis will be carried out for
backgrounds $\mathrm{I}-\mathrm{IV}$ in Section 6: The equations of motion for
the perturbations are studied in the form of a Schrödinger equation and their
potentials are calculated in each case. We also include a numerical
calculation of the eigenvalues when necessary, in order to further support our
claims.

##  2 The setup

Consider the general non-diagonal metric of the form:

$\mathrm{d}\mathrm{s}^{2}=G_{tt}\mathrm{d}t^{2}+G_{x_{i}x_{j}}\mathrm{d}x^{i}\mathrm{d}x^{j}+G_{rr}\mathrm{d}r^{2}+G_{x_{i}a}\mathrm{d}x^{i}\mathrm{d}\theta^{a}+G_{ab}\mathrm{d}\theta^{a}\mathrm{d}\theta^{b},$
(1)

where $\\{x^{i}\\}$ are some spatial coordinates, $r$ is the holographic
radial coordinate and $\\{\theta^{a}\\}$ are compact coordinates of the
internal part of the space. All the metric components carry $r$-dependence and
we let the components $G_{x_{1}a}$ and $G_{ab}$ to depend on one of the
internal coordinates $\theta$. The backgrounds that will be considered in this
paper are part of ten-dimensional solutions of Type $\mathrm{II}$B
supergravity. They will have spatial coordinates $x_{1},x_{2},\dots$ and a
compact part consisting of spheres. More specifically, we will consider
background $\mathrm{I}$ [31, 32, 33], background $\mathrm{II}$ introduced in
[25], as well as backgrounds $\mathrm{III}$ and $\mathrm{IV}$ introduced in
[26]. The first two solutions share a $\mathbb{R}^{1,1}$ Minkowski subspace
and their geometry is of the form:

$\mathrm{I},\mathrm{II}:\,\,\mathbb{R}^{1,1}\times\mathbb{R}_{r}\times\mathrm{S}^{3}\times\mathrm{S}^{3}\times\mathcal{M}_{\varphi},$
(2)

where $\mathcal{M}_{\varphi}=\mathbb{R}_{\varphi}$ and $\varphi\in\mathbb{R}$
for background $\mathrm{I}$, while
$\mathcal{M}_{\varphi}=\mathrm{S}^{1}_{\varphi}$ for background $\mathrm{II}$,
in which case $\varphi$ is taken to be a compact cigar like direction that
shrinks smoothly to a point $r_{+}$ where the space terminates. The other two
backgrounds have a single three-sphere factor and the remaining 3 coordinates
are either spanning real lines or a two sphere:

$\begin{split}\mathrm{III}:&\,\,\mathbb{R}^{1,4}\times\mathrm{S}^{1}_{\varphi}\times\mathbb{R}_{r}\times\mathrm{S}^{3},\\\
\mathrm{IV}:&\,\,\mathbb{R}^{1,2}\times\mathrm{S}^{1}_{\mu}\times\mathbb{R}_{r}\times\mathrm{S}^{3}\times\mathrm{S}^{2},\end{split}$
(3)

where again, $\varphi$ and $\mu$ express shrinking cycles. We will also define
the following functions of the metric components for later convenience:

$f_{x_{1}}(r)=-G_{tt}G_{x_{1}x_{1}},\,\,f_{ij}(r)=-G_{tt}G_{ij},$ (4)
$g(r)=-G_{tt}G_{rr},$ (5) $h(r)=G_{rr}G_{x_{1}x_{1}},$ (6)
$f_{ab}(r,\theta)=-G_{tt}G_{ab},\,\,f_{ia}(r,\theta)=-G_{tt}G_{x_{i}a},$ (7)

Lastly, the backgrounds share another common feature, that is they all have
logarithmic dilatons and thus the string coupling constants grow linearly for
large values of the radial coordinate111 The coupling constants of the dual
Quantum Field Theories were calculated in [25, 26]..

##  3 Type $\mathrm{IIB}$ Supergravity solutions and their dual QFTs

###  3.1 Background I

We consider the following solution of Type $\mathrm{IIB}$ supergravity,
introduced in [33, 31] describing the intersection of two stacks of
$\mathrm{D}5$ branes along $\mathbb{R}^{1,1}[t,x]$, while each stack extends
further on a four-dimensional space $\mathcal{M}_{4}\cong\mathbb{R}^{4}$.
Considering the backreaction of the stacks at strong coupling forces the
branes to also share the non-compact $\varphi$ direction, where each stack
wraps around a three-sphere
$\hat{\mathrm{S}}^{3}[\theta_{A},\phi_{A},\psi_{A}]$,
$\tilde{\mathrm{S}}^{3}[\theta_{B},\phi_{B},\psi_{B}]$. After setting
$\alpha^{\prime}=1=g_{s}$ the metric is expressed in the string frame as:

$\begin{split}\mathrm{d}\mathrm{s}^{2}_{(\mathrm{I})}=r&\left\\{-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\frac{(e_{A}^{2}+e_{B}^{2})}{2}\mathrm{d}\varphi^{2}+\frac{8}{r^{2}\left(e_{A}^{2}+e_{B}^{2}\right)}\mathrm{d}r^{2}\right.\\\
&\left.+\frac{2}{e_{A}^{2}}\left(\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\hat{\omega}_{3}^{2}\right)+\frac{2}{e_{B}^{2}}\left(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2}\right)\right\\},\end{split}$
(8)

where $e_{A}$ and $e_{B}$ are fixed parameters and by $\hat{\omega}_{i}$ and
$\tilde{\omega}_{i}$ we note the two sets of Maurer-Cartan $\mathrm{SU}(2)$
one-forms for the three-spheres $\mathrm{\hat{S}}^{3},\mathrm{\tilde{S}}^{3}$
parametrized by $(\theta_{A},\phi_{A},\psi_{A})$ and
$(\theta_{B},\phi_{B},\psi_{B})$
respectively222$\theta_{A,B}\in[0,\pi),\psi_{A,B}\in[0,4\pi],\phi_{A,B}\in[0,2\pi]$:

$\begin{matrix}\begin{split}&\hat{\omega}_{1}=\cos\psi_{A}\mathrm{d}\theta_{A}+\sin\psi_{A}\sin\theta_{A}\mathrm{d}\phi_{A},\\\
&\hat{\omega}_{2}=-\sin\psi_{A}\mathrm{d}\theta_{A}+\cos\psi_{A}\sin\theta_{A}\mathrm{d}\phi_{A},\\\
&\hat{\omega}_{3}=\mathrm{d}\psi_{A}+\cos\theta_{A}\mathrm{d}\phi_{A},\end{split}&&\begin{split}&\tilde{\omega}_{1}=\cos\psi_{B}\mathrm{d}\theta_{B}+\sin\psi_{B}\sin\theta_{B}\mathrm{d}\phi_{B},\\\
&\tilde{\omega}_{2}=-\sin\psi_{B}\mathrm{d}\theta_{B}+\cos\psi_{B}\sin\theta_{B}\mathrm{d}\phi_{B},\\\
&\tilde{\omega}_{3}=\mathrm{d}\psi_{B}+\cos\theta_{B}\mathrm{d}\phi_{B},\end{split}\end{matrix}$
(9)

For completeness, we also present the Ramond-Ramond three-form, its two-form
potential and the dilaton:

$F_{3}=\mathrm{d}C_{2}=-\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\hat{\omega}_{3}-\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\tilde{\omega}_{3},$
(10)

$C_{2}=-\frac{2}{e_{A}^{2}}\psi_{A}\sin\theta_{A}\mathrm{d}\theta_{A}\wedge\mathrm{d}\phi_{A}-\frac{2}{e_{B}^{2}}\psi_{B}\sin\theta_{B}\mathrm{d}\theta_{B}\wedge\mathrm{d}\phi_{B},$
(11)

$\Phi=\log r\,\,,\,\,g_{s}\sim r.$ (12)

The number of branes of each stack is given by the flux of $F_{3}$ over the
sphere the other stack is extending, which gives:

$N_{A}=\frac{8}{e_{B}^{2}}\,\,,\,\,N_{B}=\frac{8}{e_{A}^{2}},$ (13)

and thus the parameters $e_{A},e_{B}$ are quantized.

Low energy behaviour: As $r\to 0$ the background is singular (the Ricci scalar
diverges) and the description using (8)-(12) is not trustable. In the IR, the
dual field theory is described by a $(1+1)$-dimensional two node quiver gauge
theory with gauge group
$\mathrm{SU}(\mathrm{N}_{\mathrm{A}})\times\mathrm{SU}(\mathrm{N}_{\mathrm{B}})$,
which also contains massless chiral fermions in the bifundamental333these
belong to the spectrum as one can consider open strings with their endpoints
constrained on the branes. $(\textbf{N}_{A},\overline{\textbf{N}}_{B})$. We
make note that the theory on the intersection is not independent of the
worldvolume theory of the D-branes, as there is an inflow from the Chern-
Simons terms from the bulk in order to cancel the anomalies on the
intersection.

High energy behaviour: In the strong coupling regime of the theory, that is
large values of the holographic coordinate $r$, the $\varphi$ coordinate is
included to the effective theory on the intersection ($[t,x,\varphi]$) making
it $(2+1)$-dimensional, while the amount of supersymmetry preserved also
increases, see [25] for more details. For even higher values of the dilaton,
which diverges in the far UV, in order to have a trustable description of the
system one needs to use $\mathrm{S}$-duality, after which the system is
described by $\mathrm{NS}5$ branes and two Little String Theories intersecting
along the same coordinates as the branes do.

We therefore conclude that the field theory dual to the singular background
(8)-(12), while initially describing an effective $(1+1)$-dimensional theory
on the intersection, as one approaches the strong coupling regime an extra
dimension emerges and the theory becomes $(2+1)$-dimensional. While initially
the solution preserves eight supercharges, the addition of a spatial dimension
enhances the Poincaré group from $\mathrm{SO}(1,1)$ to $\mathrm{SO}(2,1)$ and
the number of preserved supercharges grows to sixteen. Then for very large
coupling the theory is UV completed by a non field theoretic description of
Little String Theories.

###  3.2 Background II

We now turn to the second background of interest, introduced in [25], which is
to be thought of as a smooth version of background I, resolving the
singularity at $r=0$ by compactifying $\varphi$ as a cigar-like coordinate on
a smoothly shrinking circle and introducing a fibration of this circle and the
three-spheres:

$\begin{split}\mathrm{d}\mathrm{s}^{2}_{(\mathrm{II})}=r&\left\\{-\mathrm{d}t^{2}+\mathrm{d}x^{2}+f_{s}(r)\mathrm{d}\varphi^{2}+\frac{4}{r^{2}f_{s}(r)}\mathrm{d}r^{2}\right.\\\
&+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\\\
&\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}-e_{B}Q_{B}\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\right\\},\end{split}$
(14)

where the functions appearing in the metric are:

$\begin{split}&f_{s}(r):=\frac{e_{A}^{2}+e_{B}^{2}}{2}-\frac{m}{r^{2}}-\frac{2(Q_{A}^{2}+Q_{B}^{2})}{r^{4}}\equiv\frac{e_{A}^{2}+e_{B}^{2}}{2r^{4}}\left(r^{2}-r_{+}^{2}\right)\left(r^{2}-r_{-}^{2}\right),\\\
&\zeta(r):=\frac{1}{r^{2}}-\frac{1}{r_{+}^{2}},\\\
&r_{\pm}^{2}:=\frac{m\pm\sqrt{m^{2}+4\left(Q_{A}^{2}+Q_{B}^{2}\right)\left(e_{A}^{2}+e_{B}^{2}\right)}}{e_{A}^{2}+e_{B}^{2}}.\end{split}$
(15)

This background preseves four supercharges if the parameters satisfy $m=0$ and
$e_{A}Q_{B}=\pm e_{B}Q_{A}$, while for $m=0=Q_{A}=Q_{B}$, background I is
recovered. The rest of the fields read:

$\begin{split}F_{3}=\mathrm{d}C_{2}=&2\zeta^{\prime}(r)\mathrm{d}r\wedge\mathrm{d}\varphi\wedge\frac{Q_{A}}{e_{A}}\hat{\omega}_{3}+\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\left(e_{A}Q_{A}\zeta(r)\mathrm{d}\varphi-\hat{\omega}_{3}\right)\\\
&+(\hat{\omega}_{i}\leftrightarrow\tilde{\omega}_{i},A\leftrightarrow
B)\end{split},$ (16)

$\begin{split}C_{2}=&\psi_{A}\left(\frac{2Q_{A}}{e_{A}}\zeta^{\prime}(r)\mathrm{d}r\wedge\mathrm{d}\varphi-\frac{2}{e_{A}^{2}}\sin\theta_{A}\mathrm{d}\theta_{A}\wedge\mathrm{d}\phi_{A}\right)+\frac{2}{e_{a}}\cos\theta_{A}Q_{A}\zeta(r)\mathrm{d}\varphi\wedge\mathrm{d}\phi_{A}\\\
&+(\hat{\omega}_{i}\leftrightarrow\tilde{\omega}_{i},A\leftrightarrow
B)\end{split},$ (17)

$\Phi=\log r.$ (18)

In order for $\mathrm{S}^{1}[\varphi]$ to shrink smoothly at the point
$r=r_{+}$, the period of $\varphi$ is fixed to be:

$L_{\varphi}=\frac{8\pi}{r_{+}f_{s}^{\prime}(r_{+})}.$ (19)

Low energy behaviour: The choice of periodicity $L_{\varphi}$ forces the space
to be smooth and free from conical singularities, as the Ricci scalar does not
diverge in the range $[r_{+},\infty)$ and the dilaton is bounded from below.
The role of $\varphi$ is now different, as it is compact, and as $r\to r_{+}$
(and $f_{s}(r)\approx 0$) the effective description is again a
$(1+1)$-dimensional
$\mathrm{SU}(\mathrm{N}_{\mathrm{A}})_{\mathrm{N}_{\mathrm{B}}}\times\mathrm{SU}(\mathrm{N}_{\mathrm{B}})_{\mathrm{N}_{\mathrm{A}}}$
gauge theory, that is dependent on the D-brane bulk theory as in the case of
background $\mathrm{I}$. The compactification of $\varphi$ also results in the
appearance of a massless Kaluza-Klein spectrum on the branes.

High energy behaviour: At higher values of the coupling, $\varphi$
decompactifies, thus the effective theory is $(2+1)$-dimensional, while for
even higher energies the dual theory is again completed by the non local
description of Little String Theories.

As was the case with the first background, the increase in spatial dimensions
of the effective theory on the intersection for higher energies also increases
the amount of supersymmetry preserved: While for lower energies this system
preserves four supercharges, the addition of $\varphi$ increases this number
to eight. The reader can review [25] for a detailed discussion on
supersymmetry preservation of backgrounds $\mathrm{I}$ and $\mathrm{II}$.

###  3.3 Background III

We will now consider the following background that was recently introduced in
[26] describing a single stack of $N$ $\mathrm{D}5$ branes wrapped around a
circle $\mathrm{S}^{1}[\varphi]$, which is again shrinking smoothly to zero at
the point $r=r_{+}$ as in (14). This geometry also includes a fibration of
$\mathrm{S}^{1}[\varphi]$ and $\mathrm{S}^{3}[\theta,\phi,\psi]$, but now we
have only one three-sphere and 4 spatial directions444where
$\mathrm{d}x^{2}_{1,\mathrm{n}}$ is an abbreviation for the Minkowski metric
$-\mathrm{d}t^{2}+\mathrm{d}x_{1}^{2}+\dots+\mathrm{d}x_{\mathrm{n}}^{2}$:

$\mathrm{d}\mathrm{s}^{2}_{(\mathrm{III})}=r\left\\{\mathrm{d}x^{2}_{1,4}+f_{s}(r)\mathrm{d}\varphi^{2}+\frac{N\mathrm{d}r^{2}}{r^{2}f_{s}(r)}+\frac{N}{4}\left[\omega_{1}^{2}+\omega_{2}^{2}+\left(\omega_{3}-\sqrt{\frac{8}{N}}Q\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\right\\},$

(20)

where $\omega_{i}$ are again the Maurer-Cartan one-forms of $\mathrm{SU}(2)$.
Here, we define a rescaled version of $f_{s}$ from background II, such that
$f_{s}(r)\to 1$ as $r\to\infty$:

$f_{s}(r)=1-\frac{m}{r^{2}}-\frac{2Q^{2}}{r^{4}}=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{4}},$
(21)

$2r_{\pm}^{2}=m\pm\sqrt{m^{2}+8Q^{2}},$ (22)

and $\zeta(r)$ is as before. The period of $\varphi$ is now:

$L_{\varphi}=\frac{2\pi\sqrt{N}r_{+}^{2}}{(r_{+}^{2}-r_{-}^{2})}=\sqrt{N}\pi\left(1+\frac{m}{\sqrt{m^{2}+8Q^{2}}}\right).$
(23)

Low energy behaviour: The space is smooth in the range $[r_{+},\infty)$ and
the dual theory to this background is $(4+1)$-dimensional for small values of
the holographic coordinate near $r_{+}$.

High energy behaviour: The behaviour is similar to background II, as the
coordinate $\varphi$ is non compact ($f_{s}(r)\approx 1$) in the UV and the
dual Quantum Field Theory is $(5+1)$-dimensional while for even higher
energies the UV completion is achieved by moving to the NS5 brane setup, just
as is the case with the previous geometries.

We note that the dual field theory for this gravity background preserves eight
supercharges when the parameter $m=0$. The supersymmetry preservation
mechanism is again interesting, as it uses the technique developed in [34] and
can be read in [26] where a thorough analysis is given.

###  3.4 Background IV

The last background we work with in this paper, also introduced in [26],
describes a stack of $\mathrm{D}5$ branes wrapping a two-sphere in a non
supersymmetry-preserving manner. The bosonic fields are555the solution also
admits a six-form potential $\mathrm{C}_{6}$ which we will not write here
explicitly (see [26]).:

$\mathrm{d}\mathrm{s}^{2}_{(\mathrm{IV})}=r\left\\{\mathrm{d}x^{2}_{1,2}+\left(1-\frac{m}{r^{2}}\right)\mathrm{d}\mu^{2}+\frac{2\mathrm{d}r^{2}}{r^{2}\left(1-\frac{m}{r^{2}}\right)}+\mathrm{d}\vartheta^{2}+\sin^{2}\vartheta\mathrm{d}\varphi^{2}+\sum_{i=1}^{3}\left(\Theta^{i}\right)^{2}\right\\},$
(24)

$F_{3}=2N\mathrm{Vol}(\mathrm{S}^{3})+\frac{N}{4}\mathrm{d}\left[\omega_{1}\wedge
A^{(1)}+\omega_{2}\wedge A^{(2)}+\omega_{3}\wedge A^{(3)}\right],$ (25)

$\Phi=\log\left(\frac{4r}{N}\right),$ (26)

where the coordinates $x_{2},x_{3}$ are non compact, $\mu$ is expressing a
shrinking $S^{1}[\mu]$ whose period is $L_{\mu}=\sqrt{8}\pi$, $\vartheta$ and
$\varphi$ parametrize a two-sphere $S^{2}[\vartheta,\varphi]$ and there is
also a three-sphere $S^{3}[\theta,\phi,\psi]$ that is expressed using the one-
forms $\Theta^{i}$. The definition of $\Theta^{i}$ is done by introducing a
gauge field $A$ as666This satisfies:
$\left(A^{1}\right)^{2}+\left(A^{2}\right)^{2}+\left(A^{3}\right)^{2}=\mathrm{d}\mathrm{s}^{2}\left(\mathrm{S}^{2}[\vartheta,\varphi]\right)=\mathrm{d}\vartheta^{2}+\sin^{2}\vartheta\mathrm{d}\varphi^{2}$
:

$\begin{split}&A^{1}=\sin\vartheta\cos\vartheta\cos\varphi\mathrm{d}\varphi,\\\
&A^{2}=-\cos\varphi\mathrm{d}\vartheta+\cos\vartheta\sin\vartheta\sin\varphi\mathrm{d}\varphi\,\,,\,\,A^{3}=-\sin^{2}\vartheta\mathrm{d}\varphi.\end{split}$
(27)

Then the $\Theta$ one-forms are defined as:

$\Theta^{1}=\omega_{1}-A^{1}\,\,,\,\,\Theta^{2}=\omega_{2}-A^{2}\,\,,\,\,\Theta^{3}=\omega_{3}-A^{3},$
(28)

where $\omega_{i}$ are the usual Maurer-Cartan forms of $\mathrm{SU}(2)$
defined in (9).

Low energy behaviour: The space of this background ends at $r=\sqrt{m}$ in the
far $\mathrm{IR}$, where $\mathrm{S}^{1}[\mu]$ shrinks. The theory living on
the $\mathrm{D}5$ branes extended in $[x_{1},x_{2},\mu,\vartheta,\varphi]$ is
$(5+1)$-dimensional, but due to the compactification on
$\mathrm{S}^{2}[\vartheta,\varphi]$ and the shrinking cycle, the dual theory
is $(2+1)$-dimensional in low energies and for777The case $m=0$ is not
trustable in this description, as both the Ricci scalar and dilaton diverge
$m>0$. As was noted in [26], the two-sphere
$\mathrm{S}^{2}[\vartheta,\varphi]$ is not shrinking, therefore this theory is
really six-dimensional.

High energy behaviour: In large values of the radial coordinate, this
background has the same behaviour as the previous one, with the difference
being that three directions are compactified on a circle and a two-sphere.
This theory is UV completed by Little String Theory as well.

##  4 Observables of interest

We will probe various objects to get the Wilson loop, ’t Hooft loop and also
the Entanglement Entropy for each case, and study the stability of the
embeddings under small coordinate perturbations, in order to deduce if the
observables have been calculated with the correct/stable way in [25, 26].

###  4.1 Wilson loops

For the classical embedding of the string, we choose to fixing its end points
at $x_{1}=\pm L/2$ and $r=\infty$ (interpreted as the probe quark and anti-
quark) and also consider static solutions. Thus we parametrize the embedding
as $\left(t(\tau),x_{1}(\sigma),r(\sigma)\right)$ and keep the rest of the
coordinates constant. Then the induced metric, after using reparametrization
invariance to let $t=\tau$ (static gauge), is written:

$\mathrm{d}\mathrm{s}^{2}_{\mathrm{ind}}=G_{tt}\mathrm{d}\tau^{2}+\left(G_{rr}r^{\prime
2}+G_{x_{1}x_{1}}x_{1}^{\prime 2}\right)\mathrm{d}\sigma^{2},$ (29)

where the prime denotes the derivative with respect to $\sigma$. Then, the
Nambu-Gotto action reads888where by $\mathcal{T}$ we denote the temporal
extent of the loop.:

$S=-\frac{\mathcal{T}}{2\pi}\int\mathrm{d}\sigma\sqrt{f_{x_{1}}(r)x_{1}^{\prime
2}+g(r)r^{\prime
2}}=-\frac{\mathcal{T}}{2\pi}\int\sqrt{f_{x_{1}}(r)\mathrm{d}x_{1}^{2}+g(r)\mathrm{d}r^{2}}.$
(30)

The quantity under the square root defines an effective two-dimensional space
on which geodesics correspond to Wilson loop configurations. The Euler-
Lagrange equation for $x_{1}$ is:

$\frac{f_{x_{1}}(r)x_{1}^{\prime}}{\sqrt{f_{x_{1}}(r)x_{1}^{\prime
2}+g(r)r^{\prime 2}}}=\mathrm{const.}\equiv\sqrt{f_{x_{1}}(r_{0})}$ (31)

$\Rightarrow
x_{1}^{\prime}(\sigma)=\pm\sqrt{\frac{g(r)f_{x_{1}}(r_{0})}{f_{x_{1}}(r)\left[f_{x_{1}}(r)-f_{x_{1}}(r_{0})\right]}}r^{\prime}(\sigma),$
(32)

and using reparametrization invariance we can choose $\sigma=r$, so the static
solution reads:

$x_{1\mathrm{cl}}^{\prime}(r)=\pm\frac{\sqrt{f_{x_{1}}(r_{0})F(r)}}{f_{x_{1}}(r)}.$
(33)

where $F$ is defined to be999It is apparent that the expression under the
square root that appears in the Lagrangian is exactly $F$. Using (33):
$f_{x_{1}}(r)x_{1\mathrm{cl}}^{\prime
2}+g(r)=\frac{f_{x_{1}}(r_{0})g(r)}{f_{x_{1}}(r)-f_{x_{1}}(r_{0})}+g(r)=\frac{f_{x_{1}}(r)g(r)}{f_{x_{1}}(r)-f_{x_{1}}(r_{0})}=F(r)$
:

$F(r):=\frac{f_{x_{1}}(r)g(r)}{f_{x_{1}}(r)-f_{x_{1}}(r_{0})}$ (34)

and $r_{0}$ is the turning point of the string101010In our parametrization,
this satisfies $x_{1\mathrm{cl}}^{\prime}(r_{0})=\infty$ where the two
branches of the solution (33) join. Then by integrating (33), one gets the
separation $L$ of the quark anti-quark pair:

$L(r_{0})=2\sqrt{f_{x_{1}}(r_{0})}\int_{r_{0}}^{\infty}\mathrm{d}r\frac{\sqrt{F(r)}}{f_{x_{1}}(r)}.$
(35)

Also, plugging the classical solution into (30), we can read off the
expression for the potential energy of the system (using that $S\sim T\cdot
E$), which we need to regularize by subtracting the infinite contribution due
to the masses of strings extending from $\pm L/2$ at $r=\infty$ down to the
point $r^{\star}$ where the space terminates:

$E(r_{0})=\frac{1}{\pi}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{F(r)}-\frac{1}{\pi}\int_{r^{\star}}^{\infty}\mathrm{d}r\sqrt{g(r)}$
(36)

We need to make note of the two conditions the energy as a function of the
length must satisfy, that is, to produce an attractive force:

$\frac{\mathrm{d}E(L)}{\mathrm{d}L}=\frac{\mathrm{d}E(L)}{\mathrm{d}L}=\frac{\mathrm{d}E(r_{0})}{\mathrm{d}r_{0}}\frac{1}{L^{\prime}(r_{0})}=\frac{\sqrt{f_{x_{1}}(r_{0})}}{2\pi}>0,$
(37)

which holds for every Supergravity background we are interested in
($f^{\prime}_{x_{1}}(r_{0})>0$), and also the concavity condition [24]:

$\frac{\mathrm{d}^{2}E(L)}{\mathrm{d}L^{2}}=\frac{1}{L^{\prime}(r_{0})}\frac{\mathrm{d}}{\mathrm{d}r_{0}}\left(\frac{\sqrt{f_{x1}(r_{0})}}{2\pi}\right)=\frac{f^{\prime}_{x1}(r_{0})}{4\pi\sqrt{f_{x}(r_{0})}L^{\prime}(r_{0})}\leq
0,$ (38)

namely, to be a non increasing function of the quark-anti-quark distance. We
also know $\sqrt{f^{\prime}_{x1}(r_{0})}>0$ to be true for all backgrounds,
therefore this condition is violated in the (potential) regions in which
$L^{\prime}(r_{0})>0$. These are unphysical regions in which one might find
the configuration of the string to be unstable under perturbations.

Finally, we can rewrite the energy to find an expression which also contains
the length:

$\begin{split}E(r_{0})=\frac{1}{\pi}&\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{\frac{g(r)}{f_{x_{1}}(r)}}\left(\frac{f_{x_{1}}(r_{0})}{\sqrt{f_{x_{1}}(r)-f_{x_{1}}(r_{0})}}+\sqrt{f_{x_{1}}(r)-f_{x_{1}}(r_{0})}\right)\\\
&-\frac{1}{\pi}\int_{r^{\star}}^{\infty}\mathrm{d}r\sqrt{g(r)}\\\ \end{split}$
(39)

$\Rightarrow
E(r_{0})=\frac{\sqrt{f_{x_{1}}(r_{0})}}{2\pi}L(r_{0})+\frac{1}{\pi}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{\frac{g(r)(f_{x_{1}}(r)-f_{x_{1}}(r_{0}))}{f_{x_{1}}(r)}}-\frac{1}{\pi}\int_{r^{\star}}^{\infty}\mathrm{d}r\sqrt{g(r)}$
(40)

In the large $L$ limit when the turning point of the U-shaped string explores
the bulk deeply, reaching the end of the space $r_{0}\to r^{\star}$, the
difference of the two integrals in (40) approaches zero and the potential is
proportional to the length. For backgrounds which $f_{x_{1}}(r^{\star})=0$, we
can already see from (31) that the differential equation for the string
embedding changes its form and admits constant solutions, corresponding to the
disconnected configuration of two straight strings extending from $r=\infty$
to the end of the space whose endpoints are moving in $x_{1}$ with zero
energy. We remind the reader that the function appearing in (40) that defines
the slope of the energy with respect to the length is interpreted as the
effective tension of the dual chromoelectric string [15]:

$T_{\mathrm{eff}}\equiv\frac{\mathrm{d}E(L)}{\mathrm{d}L}=\sqrt{f_{x_{1}}(r^{\star})},$
(41)

which gives a rather physical argument about confinement and screening: If
$f_{x_{1}}(r^{\star})\neq 0$ there exist a non-zero tension and the system
exibits confinement, while for a zero tension system, the quark and anti-quark
can move away from each other at no energy cost, that is, the potential is
screened. We also comment that we expect all of the Wilson loop configurations
in this paper to be stable, since the concavity condition seems to be
satisfied in all cases and moreover, the length as a function of $r$ is
monotonically decreasing and thus there is no possibility of $E(L)$ to be
multivalued [15]. That being said, the study of small fluctuations around a
certain classical configuration is the most trusted method to reach
conclusions about stability.

###  4.2 ’t Hooft loops

In order to study the ’t Hooft loop, that is the magnetic analogue of the
Wilson loop, we probe a $\mathrm{D}5$ brane extending at least in the
$\mathbb{R}^{1,1}[t,x_{1}]$ Minkowski subspace in order to obtain the
effective string, while the remaining four directions can extend into compact
or non compact coordinates depending on the background. After the induced
metric on the $\mathrm{D}5$ brane is inserted in the Dirac-Born-Infeld action,
and after computing the integral on the internal space, one can reduce the
expression to a similar integral as in the Wilson loop case:

$S_{\hat{\mathrm{D}}5}\left[x(r)\right]=\int\mathrm{d}^{6}\sigma
e^{-\Phi}\sqrt{-\mathrm{det}(g_{\mathrm{ind}})}=\mathrm{A}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f^{\mathrm{t}}_{x_{1}}(r)x^{\prime
2}(r)+g^{\mathrm{t}}(r)},$ (42)

where the constant $\mathrm{A}$ is dependent on the details of the background.
Moreover, the functions $f^{\mathrm{t}}_{x_{1}}(r)$ and $g^{\mathrm{t}}(r)$
are read directly through the above expression for each case and are no longer
defined using the metric components. Nevertheless,
$T_{\mathrm{eff}}=\sqrt{f^{\mathrm{t}}_{x_{1}}(r^{\star})}$ is still
interpreted as the effective tension of the chromomagnetic string. In the
examples of backgrounds I-IV the ’t Hooft loop configurations are expected to
be unstable as $L^{\prime}(r_{0})>0$ and $T_{\mathrm{eff}}=0$, signifying that
the monopole-anti-monopole pair screens: It is preferable for the system to
transition to the disconnected configuration of the two strings extending in
the entirety of the bulk. This is a typical behaviour of the magnetic analogue
of the Wilson line for confining systems.

###  4.3 Entanglement Entropy

For the definition of Entanglement Entropy we consider two entangled regions
with a minimal codimension 2 manifold connecting them. We take one of the
regions to be the strip of width $L_{\mathrm{EE}}$ and the remaining space to
be the other one and define the action for the Entanglement Entropy:

$S_{\mathrm{EE}}=\frac{1}{4G_{N}}\int_{\Sigma_{8}}\mathrm{d}^{8}\sigma\sqrt{e^{-4\Phi}\mathrm{det}(g_{\Sigma_{8}})}$
(43)

As with the Wilson and ’t Hooft loops, one can have a phase transition between
different embeddings (8-manifolds in this case). Moreover, there is evidence
to support that a phase transition in Entanglement Entropy signals confinement
of the dual theory and therefore this observable can serve as an order
parameter for confinement [35, 36, 37, 38]. As was pointed out in [37, 38,
25], there is a subtlety for theories that admit non-local $\mathrm{UV}$
completions: In such cases one must introduce a $\mathrm{UV}$ cutoff in order
to make the phase transition apparent. The 8-manifolds that we are going to
embed are extending either in four non compact spatial directions and four
compact internal coordinates, or two spatial and six compact ones (in the case
of background II where there are two three-spheres). After inducing the metric
on the 8-manifold in the aforementioned gauge, the action (43) can also be
brought, after computing the integrals, to the usual integral form:

$S_{\mathrm{EE}}=\mathrm{B}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f^{\mathrm{EE}}_{x_{1}}(r)x^{\prime
2}(r)+g^{\mathrm{EE}}(r)},$ (44)

again as in the ’t Hooft loop case, the functions $f^{\mathrm{EE}}_{x_{1}}$
and $g^{\mathrm{EE}}$ are to be read for the expression of each background
separately and the constant $\mathrm{B}$ in front of the integral depends on
the background.

##  5 Perturbative study

In this section we are going to present the derivation of the equations of
motion that control the dynamics of small linear perturbations of the
spacetime coordinates. In particular, we will derive them in the form of a
Sturm-Liouville problem and study the eigenvalues for each case, having the
turning point $r_{0}$ as a parameter.

###  5.1 Perturbations and their equations of motion

The perturbative study will be done in the spirit of [27, 28, 29, 30] by
considering small fluctuations in the coordinates of (1) as:

$\left.\Big{(}t,x_{1},x_{i},r,\theta_{a}\right.\Big{)}=\left.\Big{(}t,x_{1\mathrm{cl}}(r)+\delta
x_{1}(r,t),\delta
x_{i}(r,t),r,\theta_{a0}+\delta\theta_{a}(r,t)\right.\Big{)}$ (45)

This leads to an expansion of the action with respect to the perturbations,
where the $\mathcal{O}(\delta x^{\mu 0})$ term is just (30):

$\mathcal{O}(\delta x^{\mu
1}):\,\,S_{1}=-\frac{1}{2\pi}\int\mathrm{d}t\mathrm{d}r\frac{f_{x_{1}}x_{1}^{\prime}}{F^{1/2}}\delta
x_{1}^{\prime},$ (46)

$\begin{split}\mathcal{O}(\delta x^{\mu
2}):S_{2}=-\frac{1}{2\pi}\int\mathrm{d}t\mathrm{d}r&\left\\{\frac{f_{x_{1}}g}{2F^{3/2}}\delta
x_{1}^{\prime
2}-\frac{h}{2F^{1/2}}\delta\dot{x}_{1}^{2}+\frac{f_{ij}}{2F^{1/2}}\delta
x_{i}^{\prime}\delta x_{j}^{\prime}\right.\\\
&-\frac{f_{ij}hF^{1/2}}{2f_{x_{1}}g}\delta\dot{x}_{i}\delta\dot{x}_{j}+\frac{f_{ab}}{2F^{1/2}}\delta\theta_{a}^{\prime}\theta_{b}^{\prime}\\\
&\left.-\frac{f_{ab}hF^{1/2}}{2f_{x_{1}}g}\delta\dot{\theta}_{a}\delta\dot{\theta}_{b}-\frac{f_{ia}hF^{1/2}}{gf_{x_{1}}}\delta\dot{x}_{i}\delta\dot{\theta}_{a}\right\\}.\end{split}$
(47)

The first order contribution, using (33), is a total derivative and thus can
be omitted111111
$S_{1}=-\frac{1}{2\pi}\int\mathrm{d}t\mathrm{d}r\frac{f_{x_{1}}(r)x_{1\mathrm{cl}}^{\prime}(r)}{F^{1/2}(r)}\delta
x_{1}^{\prime}(r,t)=\mp\frac{\sqrt{f_{x_{1}}(r_{0})}}{2\pi}\int\mathrm{d}t\mathrm{d}r\frac{\mathrm{d}}{\mathrm{d}r}\delta
x_{1}(r,t).$ One can expect this to be true in general, since we consider
first order corrections of the action around a classical solution of the
equation of motion.. The Lagrangian at second order, when imposing harmonic
time dependence on the perturbations: $\delta x^{\mu}(r,t)=e^{-i\omega
t}\delta x^{\mu}(r)$, is of the form:

$\mathcal{L}^{(2)}=\frac{1}{2}\Omega_{ij}(r)\delta\Phi^{\prime
i}\delta\Phi^{\prime
j}-\frac{\omega^{2}}{2}M_{ij}(r)\delta\Phi^{i}\delta\Phi^{j},$ (48)

where $\delta\Phi^{i}=(\delta x_{1},\delta
x_{i},\delta\theta_{a})^{\intercal}$. The equations of motion for the system
are:

$\frac{\mathrm{d}}{\mathrm{d}r}\left(\Omega_{ij}\delta\Phi^{\prime
j}\right)=\omega^{2}M_{ij}\delta\Phi^{j},$ (49)

where the matrices $\Omega$ and $M$ have in general non zero non-diagonal
elements, that is the perturbations couple to each other. This fact together
with the $r$-dependence of the matrix $\Omega$ makes the system difficult to
diagonalize. One can usually consider the simpler problem of zero mode
solutions ($\omega^{2}=0$) where we can write:

$\Omega_{ij}(r)\delta\Phi^{(0)\prime
j}=c_{i}\Rightarrow\delta\Phi^{(0)i}=c_{j}\int_{r_{0}}^{r}\mathrm{d}\xi\Omega^{-1}_{ij}(\xi)+\tilde{c}_{i},$
(50)

for some constants $c_{i},\tilde{c}_{i}$. The existence of zero modes
signifies the boundary between solutions with $\omega^{2}(r_{0})>0$
(stability) and $\omega^{2}(r_{0})<0$ (instability) [27], the goal being to
find if there exists a range of $r_{0}$ values for which the later is true.

The fact that the zero mode solutions can be written in the integral form (50)
would not be possible in the case where the metric components $G_{tt}$,
$G_{x_{1}x_{1}}$ also had explicit dependence on a certain non-cyclic
coordinate, eg an angular coordinate as happens in other backgrounds121212see
[27] for interesting examples where the orientation in the internal space
produces an angular instability.. This happens due to their expansions
producing terms $\propto\delta\Phi$ that do not originate from time
derivatives, and thus have no factor of $\omega^{2}$ in the equations of
motion and would survive in the zero mode problem. This type of angular
perturbations as well as the transverse ones, are coupled among themselves due
to the quadratic in $\mathrm{d}x_{i}$ and $\mathrm{d}\theta_{a}$ terms. In the
case of the metric introduced in (1) representing all four gravity
backgrounds, the nonzero $G_{x_{i}a}$ forces $\delta x_{i}$ and
$\delta\theta_{a}$ to couple, while in the case where $G_{tx_{i}}\neq 0$ and
also in the presence of a nonzero $B$ field, the longitudinal, transverse and
angular perturbations would all couple to each other, see [28]. The equations
of motion for our cases of interest can be written in the following form from
(47):

$\left\\{\frac{\mathrm{d}}{\mathrm{d}r}\left[\begin{pmatrix}\frac{f_{x_{1}}g}{F^{3/2}}&&0&&0\\\
0&&\frac{f_{ij}}{\sqrt{F}}&&0\\\
0&&0&&\frac{f_{ab}}{\sqrt{F}}\end{pmatrix}\frac{\mathrm{d}}{\mathrm{d}r}\right]+\omega^{2}\begin{pmatrix}\frac{h}{\sqrt{F}}&&0&&0\\\
0&&\frac{f_{ij}h\sqrt{F}}{f_{x_{1}}g}&&\frac{f_{ib}h\sqrt{F}}{f_{x_{1}}g}\\\
0&&\frac{f_{ij}h\sqrt{F}}{f_{x_{1}}g}&&\frac{f_{ab}h\sqrt{F}}{f_{x_{1}}g}\end{pmatrix}\right\\}\begin{pmatrix}\delta
x_{1}\\\ \delta x_{j}\\\ \delta\theta_{b}\end{pmatrix}=0,$ (51)

where $r_{0}\leq r<\infty$ and $r_{0}\geq r^{\star}$. We can see that there
are no real $"$angular$"$ perturbations, as the equations for
$\delta\theta_{a}$ are of the same form as the $\delta x_{j}$ ($j\neq 1$).

It is in general more illuminating to use a change of variables to map (51) to
a Schrödinger equation, as it captures the information about the spectrum of
the eigenvalues best. This formulation of the problem is reviewed in Appendix
B, however for convenience, we quote the main equations here. Considering the
Sturm-Liouville problem for the longitudinal fluctuations $\delta x_{1}$,
although the same technique holds for any decoupled fluctuation, that is read
from (51) as:

$\begin{split}-\frac{\mathrm{d}}{\mathrm{d}r}&\left[P(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W(r;r_{0})\delta x_{1}(r)\,\,,\,\,r^{\star}\leq r_{0}\leq
r<\infty,\\\
&P(r;r_{0}):=\frac{f_{x_{1}}(r)g(r)}{F^{3/2}(r;r_{0})}\,\,,\,\,W(r;r_{0}):=\frac{h(r)}{\sqrt{F(r;r_{0})}},\end{split}$
(52)

we can get a Schrödinger equation for $\Psi(y):=(PW)^{1/4}(r;r_{0})\delta
x_{1}(r)$ that has the usual form131313where it is understood that
$y=y(r;r_{0})$, as defined in the Appendix.

$\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+V(y,r_{0})\right]\Psi(y)=\omega^{2}\Psi(y)\,\,,\,\,0\leq
y\leq y_{0}.$ (53)

The potential, which we are going to study for each case, is given by the
expression:

$V(r,r_{0})=\frac{P^{1/4}}{W^{3/4}}\frac{\mathrm{d}}{\mathrm{d}r}\left[\left(\frac{P}{W}\right)^{1/2}\frac{\mathrm{d}}{\mathrm{d}r}\left(PW\right)^{1/4}\right].$
(54)

Two final important remarks on the types of fluctuations, that hold for any
type of metric, are the following: The transverse ones do not admit zero
modes141414Proof of this can be found in section 3.3.1 of [27]. and the
longitudinal zero modes are in one-to-one correspondance with the critical
points of the length function (see Appendix A for a proof of this). Since in
our cases the fluctuations of $x_{j}$ and $\theta_{a}$ are all treated as
transverse, do not coupled to the longitudinal ones and also their lowest
eigenvalue is found to be positive, they will not be able to produce
instabilities.

###  5.2 Boundary conditions

We will now state the boundary conditions needed in order to formulate the
Sturm-Liouville problem. As was described in previous works [27, 28, 39, 30],
one needs to consider two boundary conditions: One at the endpoints of the
string and one at the tip of the $\mathrm{U}$-shaped embedding at $r=r_{0}$.
In order for the endpoints to stay fixed and also to obtain normalizable
solutions, we impose the vanishing of the perturbations at $r=\infty$:

$\left.\delta x^{\mu}(r)\right|_{r=\infty}=0.$ (55)

The second boundary condition is more subtle and forces the two branches of
(33) to glue together at the turning point, which in the $r$-gauge is a
singular point of the differential equation (51). The equations of motion near
$r=r_{0}$ take the form:

$\frac{\mathrm{d}}{\mathrm{d}r}\left[(r-r_{0})^{3/2}\frac{\mathrm{d}\delta
x_{1}(r)}{\mathrm{d}r}\right]+\frac{\omega^{2}h(r_{0})}{f^{\prime}_{x_{1}}(r_{0})}\sqrt{r-r_{0}}\delta
x_{1}(r)=0,$ (56)

$\frac{\mathrm{d}}{\mathrm{d}r}\left[\sqrt{r-r_{0}}\frac{\mathrm{d}\delta\theta_{a}(r)}{\mathrm{d}r}\right]+\frac{\omega^{2}h(r_{0})}{f^{\prime}_{x_{1}}(r_{0})}\frac{\delta
x_{i}(r)}{\sqrt{r-r_{0}}}=0,$ (57)

$\frac{\mathrm{d}}{\mathrm{d}r}\left[\sqrt{r-r_{0}}\frac{\mathrm{d}\delta
x_{i}(r)}{\mathrm{d}r}\right]+\frac{\omega^{2}h(r_{0})}{f^{\prime}_{x_{1}}(r_{0})}\frac{\delta\theta_{a}(r)}{\sqrt{r-r_{0}}}=0,$
(58)

where again, we emphasize that for the metric (1) the angular perturbations
obey the same equations of motion as the transverse ones and therefore will be
treated as such, that is, they do not have any zero modes as well. The
solutions of the above have the following form near $r_{0}$, for some
constants $\alpha_{0},\alpha_{1},\beta_{0},\beta_{1},\gamma_{0},\gamma_{1}$:

$\delta x_{1}(r)\approx\alpha_{0}+\frac{\alpha_{1}}{\sqrt{r-r_{0}}},$ (59)

$\delta x_{m}(r)\approx\beta_{0}+\beta_{1}\sqrt{r-r_{0}},$ (60)

$\delta\theta(r)\approx\gamma_{0}+\gamma_{1}\sqrt{r-r_{0}}.$ (61)

One can see that $\delta x_{1}(t,r_{0})$ is not physical at $r=r_{0}$ in this
gauge, which is further supported by the fact that this fluctuation is
oriented parallel to the worldvolume of the string, while $\delta r(t,x)$ is
transverse to it and thus finite at $r_{0}$. The way one overcomes the
singularity of the perturbation at this point and the way to do it is to
define the variable:

$u:=r+\Delta(t,r)\,\,,\,\,\Delta(t,r)=\frac{\delta
x_{1}(r,t)}{x^{\prime}_{\mathrm{cl}(r)}}.$ (62)

Then, one can see that the classical solution remains unperturbed:

$\begin{split}x_{1}(r)&=x_{1\mathrm{cl}}(r)+\delta
x_{1}(r,t)=x_{1\mathrm{cl}}(u-\Delta(r,t))+\delta x_{1}(r)\\\ &\approx
x_{1\mathrm{cl}}(u)-x^{\prime}_{1\mathrm{cl}}(u)\Delta(r,t)+\delta
x_{1}(r,t)\\\
&=x_{1\mathrm{cl}}(u)-x^{\prime}_{1\mathrm{cl}}(r+\Delta(r,t))\Delta(r,t)+\delta
x_{1}(r,t)\\\ &\approx
x_{1\mathrm{cl}}(u)-x^{\prime}_{1\mathrm{cl}}(r)\frac{\delta
x_{1}(r,t)}{x^{\prime}_{1\mathrm{cl}}(r)}+\delta
x_{1}(r,t)=x_{1\mathrm{cl}}(u).\end{split}$ (63)

We can also write,

$r=u-\Delta(r,t)=u-\Delta(u-\Delta(r,t),t)\approx u-\Delta(u,t)=u-\frac{\delta
x_{1}(u,t)}{x^{\prime}_{1\mathrm{cl}}(u)},$ (64)

to see that the physical $\delta r(u,t)$ perturbation at $u\to r_{0}$ is
finite151515the first approximate equation gives us the interpretation of
$\Delta(u,t)=\delta r(u,t)$.:

$r\approx r_{0}-\frac{\delta
x_{1}(u,t)}{x^{\prime}_{1\mathrm{cl}}(u)}=r_{0}-C\left(\alpha_{0}+\frac{\alpha_{1}}{\sqrt{u-r_{0}}}\right)\sqrt{u-r_{0}}$
(65)

$\Rightarrow\delta r\approx-C\left(C_{0}\sqrt{u-r_{0}}+C_{1}\right),$ (66)

where $C$ is a possible constant in front of (59). Then, we can write the
boundary condition

$\left.\frac{\mathrm{d}\delta r(r,t)}{\mathrm{d}x_{1}}\right|_{r\to r_{0}}=0,$
(67)

which leads to:

$\left(\frac{\delta x^{\prime}_{1}}{x^{\prime}_{1\mathrm{cl}}}-\frac{\delta
x_{1}}{x^{\prime
2}_{1\mathrm{cl}}}x^{\prime\prime}_{1\mathrm{cl}}\right)\frac{1}{x^{\prime}_{1\mathrm{cl}}}=0$
(68)

$\Rightarrow\delta x_{1}(r)+2\left(r-r_{0}\right)\frac{\mathrm{d}\delta
x_{1}(r)}{\mathrm{d}r}=0\,\,\,\,,\mathrm{as}\,\,r\to r_{0}.$ (69)

##  6 Stability analysis for the geometries of interest

Having presented the study of small fluctuations of the background, we are
ready to apply this to the solutions presented in Section 5 and examine the
stability of the embeddings of the observables displayed in Section 4. We will
calculate the Schrödinger potential in each case and also provide numerical
results to back up the arguments when necessary.

###  6.1 Backgrounds dual to (1+1)-dimensional QFTs

We will categorize the different backgrounds with respect to the-
dimensionality of their dual theories, in order to stress the fact that even
though we are dealing with theories that do not even have the same number of
dimensions, as far as confinement is concerned they have similar behaviours.
In the present subsection we deal with the first two backgrounds, dual to two-
dimensional theories.

####  6.1.1 Wilson Loop

The functions of the metric components for the first background read:

$\begin{split}&f^{(\mathrm{I})}_{x}(r)=-G_{tt}G_{xx}=r^{2}\,\,,\,\,f_{\varphi}^{(\mathrm{I})}(r)=-G_{tt}G_{\varphi\varphi}=\frac{r^{2}\left(e_{A}^{2}+e_{B}^{2}\right)}{2}\\\
&g^{(\mathrm{I})}(r)=-G_{tt}G_{rr}=\frac{8}{e_{A}^{2}+e_{B}^{2}}\,\,,\,\,h^{(\mathrm{I})}(r)=G_{rr}G_{xx}=g^{(\mathrm{I})}(r),\\\
\end{split}$ (70)

and the classical solution can be computed form (33):

$x_{\mathrm{cl}}^{(\mathrm{II})\prime
2}(r)=\frac{8r_{0}^{2}}{r^{2}\left(r^{2}-r_{0}^{2}\right)\left(e_{A}^{2}+e_{B}^{2}\right)}.$
(71)

From this, the length of separation is found to be:

$L^{(\mathrm{I})}(r_{0})=2r_{0}\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r\sqrt{r^{2}-r_{0}^{2}}}=\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\pi\equiv
L^{(\mathrm{I})}_{\mathrm{LST}},$ (72)

where we emphasize the fact that it is constant and independent of $r_{0}$,
which is a particularly interesting result, as even the concavity condition
cannot help as an indication for stability
($L^{(\mathrm{I})\prime}(r_{0})=0$). This is interpreted as the characteristic
length of the Little String Theory that UV completes the dual Quantum Field
theory and suggests there exist only one configuration for the system. The
energy is from (36):

$\begin{split}E^{(\mathrm{I})}(r_{0})=\frac{1}{\pi}\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr^{2}}{\sqrt{r^{2}-r_{0}^{2}}}-\frac{1}{\pi}\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\int_{0}^{\infty}\mathrm{d}r.\end{split}$
(73)

We now consider perturbing the $x_{1}$ coordinate of the embedding as:
$x_{1}=x_{\mathrm{cl}}(r)+\delta x_{1}(t,r)$, since every other perturbation
is irrelevant to the stability. The Sturm-Liouville equation for the
longitudinal perturbation $\delta x_{1}$ reads:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[\frac{(r^{2}-r_{0}^{2})^{3/2}}{r}\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}\left(\frac{8}{e_{A}^{2}+e_{B}^{2}}\right)^{2}\sqrt{1-\frac{r_{0}^{2}}{r^{2}}}\delta
x_{1}(r),$ (74)

where we note that this is independent of $r_{0}$, as can be seen by changing
variables to the dimensionless $\rho:=r/r_{0}$ as:

$-\frac{\mathrm{d}}{\mathrm{d}\rho}\left[\frac{(\rho^{2}-1)^{3/2}}{\rho}\frac{\mathrm{d}}{\mathrm{d}\rho}\right]\delta
x_{1}(\rho)=\omega^{2}\left(\frac{8}{e_{A}^{2}+e_{B}^{2}}\right)^{2}\sqrt{1-\frac{1}{\rho^{2}}}\delta
x_{1}(\rho).$ (75)

The variable to map (74) to a Schrödinger equation can be read from (B2) and
in this case is:

$\begin{split}y(r;r_{0})&=\frac{8}{e_{A}^{2}+e_{B}^{2}}\int_{r_{0}}^{r}\mathrm{d}\xi\sqrt{\xi}\left(1-\frac{r_{0}^{2}}{\xi^{2}}\right)^{1/4}\left(\xi^{2}-r_{0}^{2}\right)^{-3/4}\\\
&=\frac{8}{e_{A}^{2}+e_{B}^{2}}\mathrm{arcoth}\left(\frac{r}{\sqrt{r^{2}-r_{0}^{2}}}\right)=\frac{8}{e_{A}^{2}+e_{B}^{2}}\mathrm{arcoth}\left(\frac{\rho}{\sqrt{\rho^{2}-1}}\right),\end{split}$
(76)

which defines the problem on the non-compact interval $[0,\infty)$ for
$r\in[r_{0},\infty)$ and is independent of $r_{0}$. The wavefunction of the
Schrödinger problem is related to $\delta x_{1}$ from (B3) as:

$\Psi(y)=\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\sqrt{\frac{\rho^{2}-1}{\rho}}\sqrt{r_{0}}\delta
x_{1}(\rho),$ (77)

which vanishes for $r\to r_{0}$ ($\rho\to 1$) due to the prefactor, and for
$r\to\infty$ satisfies $\Psi(y)\sim\sqrt{r}\delta x_{1}(r)$. The potential
(54) for this problem as a function of $r$ can be computed from (75) and the
result is:

$V^{(\mathrm{I})}(\rho)=\left(\frac{e_{A}^{2}+e_{B}^{2}}{8}\right)^{2}\frac{\rho^{2}-3}{4\rho^{2}},$
(78)

which asymptotes to
$(e_{A}^{2}+e_{B}^{2})^{2}/256=\pi^{2}/(4L^{(\mathrm{I})}_{\mathrm{LST}})$ for
large values of $\rho$ and is independent of $r_{0}$ as well. We can also look
at the parametric graph of $V^{(\mathrm{I})}(y)$, using $\rho=\rho(y)$ from
(76). Both graphs of the potential as a function of the radial and the
Schrödinger variable are presented in Figure 1.

(a)

(b)

Figure 1: Schrödinger potential of $\delta x_{1}$ for background $\mathrm{I}$
as a function of $\rho$ and $y(r)$ ($e_{A}=e_{B}=1$). We see that the
potentials start from a negative value and after $\rho\approx 1.73$ become
positive, being independent of $r_{0}$.

The fact that (72) was found to be a constant characteristic length,
independent of the point on the tip of the string entering the bulk, is
reflected in the equations of motion and the potentials of the longitudinal
perturbations being also $r_{0}$-independent. Since there does not seem to be
another configuration the system can change into, we expect no instabilities
to exist for this background. In order to check the analysis, we also perform
a numerical study of the eigenvalues for (74) using the shooting method: We
solve the equation separately in the intervals
$[r_{\mathrm{IR}},r_{\mathrm{M}}]$ and $[r_{\mathrm{M}},r_{\mathrm{UV}}]$
where we took $r_{\mathrm{IR}}$ to be very close to $r_{0}$ and the middle
point $r_{\mathrm{M}}$ to also be near $r_{\mathrm{UV}}$. The boundary
conditions imposed were that the solution vanishes at $r=r_{\mathrm{UV}}$ and
that (69) is satisfied at $r_{\mathrm{IR}}$, while the two solutions are then
$"$matched$"$ with respect to the eigenvalue, in order to get the full
solution on $[r_{\mathrm{IR}},r_{\mathrm{UV}}]$. The results in Figure 2 show
the lowest eigenvalue $\omega^{2}=\omega^{2}(r_{0})$, the first one of which
is positive, therefore no instability occurs.

Figure 2: Plot of the lowest eigenvalues of the Schrödinger problem for
background I as a function of $r_{0}$, using the shooting method. We can see
the independence of $r_{0}$ as the eigenvalue is constant, and also positive.

Moving on to background II, the functions of the metric components are:

$\begin{split}&f^{(\mathrm{II})}_{x}(r)=r^{2},\\\
&f_{\varphi}^{(\mathrm{II})}(r)=r^{2}f_{s}(r)+2\left(Q_{A}^{2}+Q_{B}^{2}\right)r^{2}\zeta^{2}(r),\\\
&g^{(\mathrm{II})}(r)=\frac{4}{f_{s}(r)}\,\,,\,\,h^{(\mathrm{II})}(r)=g^{(\mathrm{II})}(r),\\\
&F^{(\mathrm{II})}(r)=\frac{4r^{2}}{f_{s}(r)\left(r^{2}-r_{0}^{2}\right)}.\end{split}$
(79)

Having again the same parametrization for the embedding, the classical
solution reads:

$x^{(\mathrm{II})\prime}_{\mathrm{cl}}(r)=\pm\frac{2r_{0}}{r}\sqrt{\frac{1}{f_{s}(r)(r^{2}-r_{0}^{2})}}=\pm\frac{2\sqrt{2}rr_{0}}{\sqrt{(e_{A}^{2}+e_{B}^{2})(r^{2}-r_{0}^{2})(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}},$
(80)

and the length of the separation and energy are:

$\begin{split}L^{(\mathrm{II})}(r_{0})&=4r_{0}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r\sqrt{f_{s}(r)(r^{2}-r_{0}^{2})}}=\frac{4\sqrt{2}r_{0}}{\sqrt{e_{A}^{2}+e_{B}^{2}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr}{\sqrt{(r^{2}-r_{0}^{2})(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}\\\
&=\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\frac{2}{\sqrt{1-\left(\frac{r_{-}}{r_{0}}\right)^{2}}}\textbf{K}\left(\frac{r_{+}^{2}-r_{-}^{2}}{r_{0}^{2}-r_{-}^{2}}\right)\end{split}$
(81)

$E^{(\mathrm{II})}(r_{0})=\frac{2}{\pi}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr}{\sqrt{f_{s}(r)(r^{2}-r_{0}^{2})}}-\frac{2}{\pi}\int_{r_{+}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f_{s}(r)}},$
(82)

where $\textbf{K}(x)$ is the complete elliptic integral of the first
kind161616This is defined using the incomplete elliptic integral of the first
kind as:
$\textbf{K}(x):=\textbf{F}\left(\left.\frac{\pi}{2}\right|x\right)\,\,,\,\,\textbf{F}(\phi|x)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-x^{2}\sin^{2}\theta}}.$
. Notice how the insertion of the function $f_{s}(r)$ in the square root of
the denominator of (81) forces the length to be non constant and have $r_{0}$
dependence. Also, for $r_{0}$ close to $r_{+}$ the length diverges as
$(r-r_{+})^{-1}$ which means that string configurations reaching the end of
the geometry give off infinite separation between the quark and anti-quark.
This is yet another signal for confinement. As for the perturbations along the
longitudinal direction, they obey the following equations of motion:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[\frac{(r^{2}-r_{0}^{2})^{3/2}\sqrt{f_{s}(r)}}{2r}\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x^{(\mathrm{II})}(r)=\omega^{2}2\frac{\sqrt{r^{2}-r_{0}^{2}}}{r\sqrt{f_{s}(r)}}\delta
x^{(\mathrm{II})}(r).$ (83)

Once again, we formulate the Schrödinger problem for $\delta
x_{1}^{(\mathrm{II})}$, for which the Schrödinger potential in terms of $r$
reads:

$\begin{split}V^{(\mathrm{II})}(r;r_{0})&=\frac{(r^{2}-3r_{0}^{2})f_{s}(r)+r(r^{2}+r_{0}^{2})f_{s}^{\prime}(r)}{16r^{2}}\\\
&=\frac{e_{A}^{2}+e_{B}^{2}}{32r^{6}}\left\\{r^{6}-7r_{0}^{2}r_{-}^{2}r_{+}^{2}+r^{4}(r_{+}^{2}+r_{-}^{2}-3r_{0}^{2})+r^{2}\left[5r_{0}^{2}(r_{+}^{2}+r_{-}^{2})-3r_{+}^{2}r_{-}^{2}\right]\right\\},\end{split}$
(84)

and its asymptotic values are:

$\lim_{r\to\infty}V^{(\mathrm{II})}(r;r_{0})=\frac{e_{A}^{2}+e_{B}^{2}}{32},$
(85)

$\begin{split}\lim_{r_{0}\to r_{+}}\lim_{r\to
r_{0}}V^{(\mathrm{II})}(r;r_{0})=\frac{e_{A}^{2}+e_{B}^{2}}{32r_{+}^{6}}&\Big{[}r_{+}^{6}-7r_{+}^{4}r_{-}^{2}+r_{+}^{4}\left(r_{-}^{2}-2r_{+}^{2}\right)\\\
&+r_{+}^{2}\left(5r_{+}^{2}(r_{+}^{2}+r_{-}^{2})-3r_{+}^{2}r_{-}^{2}\right)\Big{]}\end{split}$
(86)

where we make note that $V(r_{0};r_{0})$ approaches the negative value
$-(e_{A}^{2}+e_{B}^{2})/16$ as we increase $r_{0}$ and the behaviour of the
potential is controlled by the parameters $e_{A},e_{B},r_{-}$ and $r_{+}$. One
can also notice that in the case where $r_{+}^{2}=r_{-}^{2}$, for which four
supercharges are preserved, $V(r_{0};r_{0})$ becomes zero171717Interestingly,
the choice $r_{+}^{2}=-r_{-}^{2}$ gives off
$V(r_{0};r_{0})=(e_{A}^{2}+e_{B}^{2})/4$..

(a)

(b)

Figure 3: Schrödinger potential for the perturbations $\delta x$ in background
II, for two values of $r_{0}$ ($e_{A}=e_{B}=1,r_{-}=0.2$) as $r_{+}$ also
varies.

While the potential starts off with negative values for small $r$, which grow
as $r_{0}$ gets larger, the numerical analysis shows that there are no
negative eigenvalues.

Figure 4: Lowest eigenvalues as a function of $r_{0}$ for the potential (84)
of background II (with $e_{A}=1=e_{B}$, $r_{-}=0.3,r_{+}=0.8$).

We can also plot $L^{\prime}(r_{0})$ (Figure 5) from (81), in order to see
that it is manifestly negative and the concavity condition is satisfied.

Figure 5: Plot of the derivative of the length function with respect to the
turning point $r_{0}$, for various values of $r_{+}$
($e_{A}=e_{B}=1,r_{-}=0.2$).

There is also an approximate expression for the length of the separation,
introduced in [25] which has the following form:

$\hat{L}(r_{0})=\pi\left.\sqrt{g(r)}\left[\frac{\mathrm{d}}{\mathrm{d}r}\left(\sqrt{f_{x}(r)}\right)\right]^{-1}\right|_{r\to
r_{0}}.$ (87)

For background ($\mathrm{II}$) we have:

$\hat{L}^{(\mathrm{II})}(r_{0})=\frac{2\pi}{\sqrt{f_{s}(r_{0})}}=\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\frac{\pi
r_{0}^{2}}{\sqrt{(r_{0}^{2}-r_{+}^{2})(r_{0}^{2}-r_{-}^{2})}},$ (88)

where we immediately see that for $r_{0}\to\infty$ we get the expression (72)
from background $\mathrm{I}$, indicating that the dual field theory of this
background is also UV-completed by Little String Theory with this
characteristic length, but due to the shrinking effect of the $\varphi$
circle, this theory confines. We also comment that this behaviour of the
length asymptoting to a constant nonzero value is similar to the Maldacena-
Nu$\tilde{\mathrm{n}}$ez [9] solution, which also is UV completed by the
theory dual to $\mathrm{NS}5$ branes. This phenomenon acts as an indicator for
non-local completions in general: A field theoretic UV completion would not be
able to prevent the quark-anti-quark distance from becoming arbitrarily small
as there is no characteristic scale, contrary to a theory of strings.

(a)

(b)

Figure 6: Approximate form of the length for background II ($e_{A}=e_{B}=1$)
for various values of $r_{+}$ and $r_{-}$.

Finally, we note that for the case where $m=0$ and $e_{A}Q_{B}=\pm
e_{B}Q_{A}$, that is when the background preserves four supercharges, the
potentials look very similar to Figure 3 and the numerical method also found
the eigenvalues to be positive.

####  6.1.2 ’t Hooft loop

We will now study the ’t Hooft loop in background $\mathrm{II}$ by embedding a
$\mathrm{D}5$ brane along181818as before, we are using the $r$-gauge: $x=x(r)$
and $r=\sigma$. $[t,r,\varphi,\theta_{A},\phi_{A},\psi_{A}]$ (we take
$\hat{\mathrm{D}}5$ to wrap $\hat{\mathrm{S}}^{3}$ although the same is true
for $\tilde{\mathrm{S}}^{3}$). Effectively, as the $\varphi$ cycle vanishes
smoothly the object becomes $(1+1)$-dimensional, that is a string. The induced
metric for the ’t Hooft loop is:

$\begin{split}\mathrm{d}\mathrm{s}^{2}_{\mathrm{ind}}=r&\left\\{-\mathrm{d}t^{2}+\left[x^{\prime
2}(r)+\frac{4}{r^{2}f_{s}(r)}\right]\mathrm{d}r^{2}\right.\\\
&+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\\\
&+\left[f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right]\mathrm{d}\varphi^{2}\bigg{\\}},\end{split}$
(89)

and the Born-Dirac-Infeld action can be written in the form:

$S_{\hat{\mathrm{D}}5}\left[x(r)\right]=\int\mathrm{d}^{6}\sigma
e^{-\Phi}\sqrt{-\mathrm{det}(g_{\mathrm{ind}})}$ (90)
$\begin{split}&=\int_{\hat{\mathrm{S}}^{3}}\mathrm{d}\Omega_{3}\int\mathrm{d}t\mathrm{d}r\mathrm{d}\varphi
r^{2}\sin\theta_{A}\sqrt{\left(x^{\prime
2}(r)+\frac{4}{r^{2}f_{s}(r)}\right)\left(f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right)}\\\
&=\left(\frac{8}{e_{A}^{2}}\right)^{3/2}L_{\varphi}T(4\pi)^{2}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f_{x}(r)x^{\prime
2}(r)+g(r)},\end{split}$ (91)

where191919$T=\int\mathrm{d}t$. in this case, the functions read:

$\begin{split}&f^{(\mathrm{II})}_{t}(r)=r^{4}\left[f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right]\\\
&g^{(\mathrm{II})}_{t}(r)=4r^{2}\left[1+\frac{2Q_{B}^{2}\zeta^{2}(r)}{f_{s}(r)}\right]\end{split}$
(92)

and we also take:

$h^{(\mathrm{II})}_{t}(r)=g^{(\mathrm{II})}_{t}(r)=4r^{2}\left[1+\frac{2Q_{B}^{2}\zeta^{2}(r)}{f_{s}(r)}\right]$
(93)

The classical solution then is

$\begin{split}x^{\prime}_{\mathrm{cl}}(r)&=\pm\frac{2r_{0}^{2}}{r}\sqrt{\frac{f_{s}(r_{0})+2Q_{B}^{2}\zeta^{2}(r_{0})}{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})+2Q_{B}^{2}\left(r^{4}\zeta^{2}(r)-r_{0}^{4}\zeta^{2}(r_{0})\right)\right]}}\\\
\end{split}$ (94)

where $r_{0}$ is the turning point and $F$ is defined as in (34). The length
of the separation of the monopole-anti-monopole pair and energy in this case,
are:

$\begin{split}L^{(\mathrm{II})}_{\mathrm{MM}}(r_{0})=&4r_{0}^{2}\sqrt{f_{s}(r_{0})+2Q_{B}^{2}\zeta^{2}(r_{0})}\times\\\
&\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r\sqrt{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})+2Q_{B}^{2}\left(r^{4}\zeta^{2}(r)-r_{0}^{4}\zeta^{2}(r_{0})\right)\right]}},\end{split}$
(95)

$\begin{split}E^{(\mathrm{II})}_{\mathrm{MM}}(r_{0})&=\frac{2}{\pi}\int_{r_{0}}^{\infty}\mathrm{d}r\frac{r^{3}\left[f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right]}{\sqrt{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})+2Q_{B}^{2}\left(r^{4}\zeta^{2}(r)-r_{0}^{4}\zeta^{2}(r_{0})\right)\right]}}\\\
&-\frac{2}{\pi}\int_{r_{+}}^{\infty}\mathrm{d}rr\sqrt{1+\frac{2Q_{B}^{2}\zeta^{2}(r)}{f_{s}(r)}}\end{split}$
(96)

One can then check that $L^{\prime}(r_{0})>0$ for every $r_{0}$, thus the
embedding is deemed unstable. We can then formulate the Sturm-Liouville
problem for the longitudinal perturbations of the magnetic string:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P^{(\mathrm{II})}_{\mathrm{MM}}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W^{(\mathrm{II})}_{\mathrm{MM}}(r;r_{0})\delta x_{1}(r),$
(97)

where:

$\begin{split}&P^{(\mathrm{II})}_{\mathrm{MM}}(r;r_{0})=\frac{\sqrt{f_{s}(r)}\left[r^{4}(f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r))-r_{0}^{4}(f_{s}(r_{0})+2Q_{B}^{2}\zeta^{2}(r_{0}))\right]^{3/2}}{2r^{3}\left[f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right]}\\\
&W^{(\mathrm{II})}_{\mathrm{MM}}(r;r_{0})=\frac{2\sqrt{r^{4}(f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r))-r_{0}^{4}(f_{s}(r_{0})+2Q_{B}^{2}\zeta^{2}(r_{0}))}}{\sqrt{f_{s}(r)}r^{3}\left[f_{s}(r)+2Q_{B}^{2}\zeta^{2}(r)\right]}.\end{split}$
(98)

Using (54) we can calculate (omitting the analytic expression of $V(r)$ since
it is too complicated to write here explicitly) and plot the Schrödinger
potential of the magnetic string for various values of $r_{0}$ and $r_{+}$:

(a)

(b)

Figure 7: Schrödinger potential for the longitudinal perturbations of the ’t
Hooft loop, with various values of $r_{0}$ and $r_{+}$ (setting
$r_{-}=0.2,r_{+}=0.5,Q_{B}=1=Q_{A}$).

The potential asymptotes to $(e_{A}^{2}+e_{B}^{2})/8$ for large values of the
radial coordinate and as $r_{0}$ approaches the end of the space at $r_{+}$,
it diverges to negative infinity for $r\to r_{0}$ for all values of the
parameters. The later should indicate that there exist modes with negative
eigenvalues, which means that the system is unstable under longitudinal
fluctuations.

Studying the approximate length (87) is also interesting, as the expression we
get:

$\begin{split}\hat{L}^{(\mathrm{II})}_{\mathrm{MM}}(r_{0})=&2\sqrt{2}\pi\frac{r_{+}^{2}\sqrt{(r_{0}^{2}-r_{+}^{2})\left[(e_{A}^{2}+e_{B}^{2})(r_{0}^{2}-r_{-}^{2})r_{+}^{4}+4Q_{B}^{2}(r_{0}^{2}-r_{-}^{2})\right]}}{8Q_{B}^{2}(r_{0}^{2}-r_{+}^{2})-(e_{A}^{2}+e_{B}^{2})r_{+}^{4}(r_{+}^{2}+r_{-}^{2}-2r_{0}^{2})}\\\
&\times\sqrt{1+\frac{4Q_{B}^{2}(r_{0}^{2}-r_{-}^{2})}{(e_{A}^{2}+e_{B}^{2})(r_{0}^{2}-r_{-}^{2})r_{+}^{4}}},\end{split}$
(99)

satisfies:

$\lim_{r_{0}\to\infty}\hat{L}^{(\mathrm{II})}_{\mathrm{MM}}(r_{0})=\pi\sqrt{\frac{2}{e_{A}^{2}+e_{B}^{2}}},$
(100)

$\hat{L}^{(\mathrm{II})}_{\mathrm{MM}}(r_{0})\sim
8\pi\sqrt{\frac{r_{+}(r_{0}-r_{+})}{(e_{A}^{2}+e_{B}^{2})(r_{+}^{2}-r_{-}^{2})}}\to
0\,\,\,,\,\,\text{for}\,\,r_{0}\to r_{+}.$ (101)

We see that for the ’t Hooft loop we get a maximum separation in the UV (since
in this case the length which is monotonous increases from zero) again
indicated by Little String Theory, that is half of the the minimum length we
found for the Wilson loop. Also, as the string approaches the end of the space
$r_{+}$, delving deeper into the bulk towards the far IR, the length goes to
zero. This together with the fact that the effective tension for the dual
string satisfies
$T_{\mathrm{eff}}(r_{+})=r_{+}^{2}\sqrt{f_{s}(r_{+})+2Q_{B}^{2}\zeta^{2}(r_{+})}=0$
indicate screening of the monopole-anti-monopole pair: The system favors the
disconnected configuration of the two magnetic strings extending infinitely
into the bulk, whose endpoints at $r=\infty$ can move away from each other
with no energy cost and therefore if one consideres the $U$ shaped embedding
for the magnetic string, small fluctuations will destabilize the system.

(a)

(b)

Figure 8: Plot of the approximate length function for various $r_{+}$, whose
first derivative is positive. It is apparent that the concavity condition does
not hold, as the derivative is negative and then becomes zero.

To support the above argumets, we solve the equation numerically in order to
get the $r_{0}$ dependence of the lowest eigenvalues using the shooting
method, which indeed shows there are negative eigenvalues for small $r_{0}$:

(a)

(b)

Figure 9: Lowest eigenvalues for the ’t Hooft loop as a function of $r_{0}$
for two values of $r_{+}$ (keeping $r_{-}=0.2$ and
$e_{A}=e_{B}=1=Q_{A}=Q_{B}=1$).

This behaviour is typical for backgrounds dual to confining theories, that is
the Wilson loop is stable and confines while the ’t Hooft loop is unstable and
screens [40]. This is in fact also the case for the Maldacena-
Nu$\tilde{\mathrm{n}}$ez and Klebanov-Strassler solutions, which were studied
using the same methods in [30].

####  6.1.3 Entanglement Entropy

We now move to studying the perturbative stability of the Entangelment
Entropy, for which we will choose the 8-manifold on which the metric (14) is
induced to be spanned by the coordinates
$[r,\varphi,\theta_{A},\phi_{A},\psi_{A},\theta_{B},\phi_{B},\psi_{B}]$:

$\begin{split}\mathrm{d}\mathrm{s}^{2}_{\Sigma_{8}}&=r\left\\{\left[x^{\prime
2}(r)+\frac{4}{r^{2}f_{s}(r)}\right]\mathrm{d}r^{2}+f_{s}(r)\mathrm{d}\varphi^{2}\right.\\\
&+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\\\
&\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}-e_{B}Q_{B}\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\right\\},\end{split}$
(102)

and the square root of the determinant of the induced metric is:

$\sqrt{e^{-4\Phi}\mathrm{det}(g_{\Sigma_{8}})}=\frac{8}{e_{A}^{3}e_{B}^{3}}\sqrt{r^{4}f_{s}(r)x^{\prime
2}(r)+4r^{2}}\sin\theta_{A}\sin\theta_{B}.$ (103)

Therefore the action is again reduced the the following integral:

$S^{(\mathrm{II})}_{\mathrm{EE}}=\frac{2(4\pi)^{2}L_{\varphi}}{e_{A}^{3}e_{B}^{3}G_{N}}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f^{(\mathrm{II})}_{\mathrm{EE}}(r)x^{(\mathrm{II})\prime
2}(r)+g^{(\mathrm{II})}_{\mathrm{EE}}(r)},$ (104)

where now,

$\begin{split}f^{(\mathrm{II})}_{\mathrm{EE}}(r)=r^{4}f_{s}(r)\\\
g^{(\mathrm{II})}_{\mathrm{EE}}(r)=4r^{2}=h^{(\mathrm{II})}_{\mathrm{EE}}(r).\end{split}$
(105)

The classical solution and the length read:

$x_{\mathrm{cl}}^{(\mathrm{II})\prime}(r)=\pm
2\frac{r_{0}^{2}}{r}\sqrt{\frac{f_{s}(r_{0})}{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right]}},$
(106)

$L^{(\mathrm{II})}_{\mathrm{EE}}(r_{0})=4r_{0}^{2}\sqrt{f_{s}(r_{0})}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r\sqrt{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right]}},$
(107)

where the derivative of the later is found to be manifestly positive, as in
the ’t Hooft loop, which is an indication of instability. The Sturm-Liouville
equation for the longitudinal perturbation of this problem is then:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P^{(\mathrm{II})}_{\mathrm{EE}}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W^{(\mathrm{II})}_{\mathrm{EE}}(r;r_{0})\delta x_{1}(r),$
(108)

where:

$\begin{split}&P^{(\mathrm{II})}_{\mathrm{EE}}(r;r_{0})=\frac{\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right]^{3/2}}{2r^{3}\sqrt{f_{s}(r)}}\\\
&W^{(\mathrm{II})}_{\mathrm{EE}}(r;r_{0})=\frac{2\sqrt{r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})}}{r\sqrt{f_{s}(r)}}.\end{split}$
(109)

Again, we can get the Schrödinger potential from (54) and plot it for various
$r_{0}$ to find that it too expresses instability:

Figure 10: Schrödinger potential for longitudinal perturbations of the
Entanglement Entropy for various $r_{0}$ (fixing
$r_{-}=0.2,r_{+}=0.5,e_{A}=1=e_{B}$).

The approximate length reads:

$\hat{L}^{(\mathrm{II})}_{\mathrm{EE}}(r_{0})=\pi\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\frac{\sqrt{(r_{0}^{2}-r_{-}^{2})(r_{0}^{2}-r_{+}^{2})}}{2r_{0}^{2}-r_{-}^{2}-r_{+}^{2}}.$
(110)

This approaches the same maximum length with the ’t Hooft loop as
$r_{0}\to\infty$ , that is half of (72) and $\hat{L}_{\mathrm{EE}}\to 0$ as
$r_{0}\to r_{+}$. We thus have a similar behaviour as the monopole-anti-
monopole scenario, namely the system screens and the preferable configuration
is the disconnected one. We note that for the ’t Hooft loop and Entanglement
Entropy, due to the fact that the functions expressing the shrinking
$\mathrm{S}^{1}[\varphi]$ and the fibration with
$\mathrm{S}^{3}[\theta,\phi,\psi]$ both vanish at $r=r_{+}$:
$f_{s}(r_{+})=0=\zeta(r_{+})$, the effective tension of the dual magnetic flux
tube vanishes as $r_{0}\to r_{+}$, which agrees with our results.

###  6.2 Background dual to (4+1)-dimensional QFTs

We will now study the stability of our observables in background
$\mathrm{III}$, which is dual to five-dimensional theory at low energies.

####  6.2.1 Wilson loop

The functions of the metric read:

$\begin{split}f^{(\mathrm{III})}_{x_{1}}(r)=r^{2},\,\,g^{(\mathrm{III})}(r)=\frac{N}{f_{s}(r)}=h^{(\mathrm{III})}(r),\\\
f^{(\mathrm{III})}_{\varphi}(r)=r^{2}\left[f_{s}(r)+2Q^{2}\zeta^{2}(r)\right],\end{split}$
(111)

and the classical solution is:

$x^{(\mathrm{III})\prime}_{1\mathrm{cl}}(r)=\pm\frac{r_{0}}{r}\sqrt{\frac{N}{f_{s}(r)(r^{2}-r_{0}^{2})}}=\pm
rr_{0}\sqrt{\frac{N}{(r^{2}-r_{0}^{2})(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}},$
(112)

or, after integration:

$\begin{split}x^{(\mathrm{III})}_{1\mathrm{cl}}(r)=\mp&\frac{Nr_{0}}{(r_{0}^{2}-r_{-}^{2})(r_{0}^{2}-r_{+}^{2})(r_{-}^{2}-r_{+}^{2})}\left[r_{0}^{3}(r_{-}^{2}-r_{+}^{2})\operatorname{arctanh}\left(\frac{r}{r_{0}}\right)+\right.\\\
&\left.r_{-}^{4}(r_{+}^{2}-r_{0}^{2})\operatorname{arctanh}\left(\frac{r}{r_{-}}\right)+r_{+}^{3}(r_{0}^{2}-r_{-}^{2})\operatorname{arctanh}\left(\frac{r}{r_{+}}\right)\right],\end{split}$
(113)

which gives off the expression for the length:

$L^{(\mathrm{III})}(r_{0})=2\sqrt{N}r_{0}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr}{\sqrt{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})(r^{2}-r_{0}^{2})}}.$
(114)

The approximate expression (87) for this case is:

$\hat{L}^{(\mathrm{III})}(r_{0})=\frac{\sqrt{N}\pi}{\sqrt{f_{s}(r_{0})}}=\frac{\pi\sqrt{N}r_{0}^{2}}{\sqrt{(r_{0}^{2}-r_{+}^{2})(r_{0}^{2}-r_{-}^{2})}},$
(115)

which, up to a constant, is the same as the case of background $\mathrm{II}$,
signifying that the Wilson loop in this background has the same behaviour: We
get a configuration with minimum separation dictated by Little String Theory
for $r_{0}\to\infty$ and a divergence for $r_{0}\to r_{+}$, that is stable
($L^{\prime}(r_{0})<0$). The equation of motion for the longitudinal
fluctuations in this case is:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P^{(\mathrm{III})}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W^{(\mathrm{III})}(r;r_{0})\delta x_{1}(r),$ (116)

where:

$\begin{split}&P^{(\mathrm{III})}(r;r_{0})=\frac{(r^{2}-r_{0}^{2})^{3/2}\sqrt{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}}{r^{3}\sqrt{N}},\\\
&W^{(\mathrm{III})}(r;r_{0})=\frac{\sqrt{N}r\sqrt{r^{2}-r_{0}^{2}}}{\sqrt{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}},\end{split}$
(117)

and the potential reads:

$V^{(\mathrm{III})}(r;r_{0})=\frac{r^{6}-7r_{0}^{2}r_{+}^{2}r_{-}^{2}+r^{4}\left(r_{+}^{2}+r_{-}^{2}-3r_{0}^{2}\right)+r^{2}\left[5r_{0}^{2}\left(r_{+}^{2}+r_{-}^{2}\right)-3r_{+}^{2}r_{-}^{2}\right]}{4Nr^{6}}.$
(118)

It is clear that for large radial coordinates this asymptotes to the value
$\frac{1}{4N}$ which gets smaller as the number of branes grows, while for $r$
close to $r_{0}$ as $r_{0}$ approaches $r_{+}$, we get:

$\lim_{r_{0}\to
r_{+}}V^{(\mathrm{III})}(r_{0};r_{0})=\frac{r_{+}^{6}+r_{+}^{4}(r_{-}^{2}-2r_{+}^{2})+r_{+}^{2}\left[5r_{+}^{2}(r_{+}^{2}+r_{-}^{2})-3r_{+}^{2}r_{-}^{2}\right]-7r_{+}^{4}r_{-}^{2}}{4Nr_{+}^{6}}$
(119)

(a)

(b)

Figure 11: Schrödinger potential for longitudinal perturbations in background
$\mathrm{III}$, for $r_{0}=1.2$ and $r_{0}=1.8$ as $r_{+}$ varies
($r_{-}=0.2$,$N=1$). Figure 12: Lowest eigenvalues as a function of $r_{0}$
for the Wilson loop of background $\mathrm{III}$ ($r_{-}=0.2$, $r_{+}=0.2$,
$N=1$).

Even though the potential explores a small negative region for larger values
of $r_{0}$ (Figure 11) it is evident from the numerical results in Figure 12
that no negative eigenvalue modes exist.

####  6.2.2 ’t Hooft loop

For the ’t Hooft loop, we use the same procedure as in Section 6.1.2 and we
find:

$S_{\hat{D}5}=L_{\varphi}(4\pi)^{2}T\left(\frac{N}{4}\right)^{3/2}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f_{x}(r)x^{\prime
2}(r)+g(r)},$ (120)

and the functions are:

$\begin{split}&f^{(\mathrm{III})}_{t}(r)=r^{4}f_{s}(r),\,\,g^{(\mathrm{III})}_{t}(r)=r^{2}N=h^{(\mathrm{III})}_{t}(r),\\\
&F^{(\mathrm{III})}_{t}(r)=\frac{r^{6}f_{s}(r)N}{r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})}.\end{split}$
(121)

The length of the monopole-anti-monopole separation and its approximate
expression are then found to be:

$L^{(\mathrm{III})}_{\mathrm{MM}}(r_{0})=2r_{0}^{2}\sqrt{Nf_{s}(r_{0})}\int_{r_{0}}^{\infty}\mathrm{d}r\frac{1}{r\sqrt{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right]}},$
(122)

$\hat{L}^{(\mathrm{III})}_{\mathrm{MM}}(r_{0})=\pi\sqrt{N}\frac{\sqrt{(r_{0}^{2}-r_{+}^{2})(r_{0}^{2}-r_{-}^{2})}}{2r_{0}^{2}-r_{+}^{2}-r_{-}^{2}}.$
(123)

From this we see that again, as in background II, we get a maximal length as
$r_{0}\to\infty$ and a vanishing one in the IR as $r_{0}\to r_{+}$. Indeed, we
find the plot in Figure 13 to be of the same form as the one in background II,
that is, there exist negative eigenvalues $\omega^{2}$ and the configuration
is unstable: It is preferable for the system to move into the disconnected
solution of the two $\mathrm{D}5$ branes which they can move away from each
other freely, therefore we expect a screening scenario.

The perturbations obey:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P_{\mathrm{MM}}^{(\mathrm{III})}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W_{\mathrm{MM}}^{(\mathrm{III})}(r;r_{0})\delta x_{1}(r),$
(124)

whith:

$P_{\mathrm{MM}}^{(\mathrm{III})}(r;r_{0})=\frac{\left[(r^{2}-r_{0}^{2})(r^{2}+r_{0}^{2}-r_{-}^{2}-r_{+}^{2})\right]^{3/2}}{r\sqrt{N}\sqrt{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}},$
(125)

$W_{\mathrm{MM}}^{(\mathrm{III})}(r;r_{0})=\frac{r\sqrt{N}\sqrt{(r^{2}-r_{0}^{2})(r^{2}+r_{0}^{2}-r_{-}^{2}-r_{+}^{2})}}{\sqrt{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}.$
(126)

(a)

(b)

Figure 13: Schrödinger potential for longitudinal perturbations of the ’t
Hooft loop in background $\mathrm{III}$ as $r_{0}$ and $r_{+}$ vary, for fixed
values of parameters ($N=1$ and $r_{-}=0.2$).

The potentials depicted in Figure 13 and the numerical solution in Figure 14
confirm the claims about screening, as the system is unstable under small
perturbations and transitions to the disconnected configuration.

Figure 14: Plot of the lowest eigenvalues $\omega^{2}(r_{0})$ of the
Schrödinger equation for background III ($r_{-}=0.2$, $r_{+}=0.8$ and $N=1$).

####  6.2.3 Entanglement Entropy

The induced metric on the 8-manifold $\Sigma_{8}$ spanned by
$\left[x_{1},x_{2},x_{3},x_{4},\varphi,\theta,\phi,\psi\right]$ after letting
$x_{1}=x_{1}(r)$ is:

$\begin{split}\mathrm{d}s^{2}_{\Sigma_{8}}&=r\left\\{\mathrm{d}x_{2}^{2}+\mathrm{d}x_{3}^{2}+\mathrm{d}x_{4}^{2}+\left(\frac{N}{r^{2}f_{s}(r)}+x_{1}^{\prime
2}\right)\mathrm{d}r^{2}\right.\\\
&\left.+\frac{N}{4}\left[\omega_{1}^{2}+\omega_{2}^{2}+\left(\omega_{3}-\sqrt{\frac{8}{N}}Q\zeta(r)\mathrm{d}\varphi\right)^{2}\right]\right\\}.\end{split}$
(127)

The action is then found to be:

$S_{\mathrm{EE}}^{(\mathrm{III})}=\frac{(4\pi)^{2}N^{3/2}}{8G_{N}}L_{\varphi}L_{x_{2}}L_{x_{3}}L_{x_{4}}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f_{\mathrm{EE}}^{(\mathrm{III})}(r)x^{\prime
2}(r)+g_{\mathrm{EE}}^{(\mathrm{III})}(r)},$ (128)

where

$f_{\mathrm{EE}}^{(\mathrm{III})}=r^{4}f_{s}(r),\,\,g_{\mathrm{EE}}^{(\mathrm{III})}=r^{2}N,\,\,F_{\mathrm{EE}}^{(\mathrm{III})}=\frac{r^{6}Nf_{s}(r)}{r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})}$
(129)

The length in this case is given by the following integral, which we notice is
the exact same expression with the ’t Hooft loop (122):

$L_{\mathrm{EE}}^{(\mathrm{III})}(r_{0})=2r_{0}^{2}\sqrt{Nf_{s}(r_{0})}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r\sqrt{f_{s}(r)\left[r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right]}},$
(130)

so the same screening behaviour will take place, as the equations of motion
(and thus the potential) of the fluctuations are also the same as in the ’t
Hooft loop case (124).

###  6.3 Background dual to (2+1)-dimensional QFTs

Lastly, we will study background $\mathrm{IV}$ which produces at low energies
a $(2+1)$-dimensional effective theory.

####  6.3.1 Wilson loop

The functions from the metric of this geometry are:

$\begin{split}&f^{(\mathrm{IV})}_{x_{1}}(r)=r^{2}\,\,,\,\,g^{(\mathrm{IV})}(r)=\frac{2r^{2}}{r^{2}-m}=h^{(\mathrm{IV})}(r)\\\
&f_{\varphi}^{(\mathrm{IV})}(r)=-r^{2}\sin^{2}\vartheta\,\,,\,\,f_{\vartheta}^{(\mathrm{IV})}(r)=-r^{2}\,\,,\,\,f^{(\mathrm{IV})}_{\mu}(r)=(r^{2}-m^{2}),\end{split}$
(131)

and the classical solution:

$x^{(\mathrm{IV})\prime}_{1\mathrm{cl}}(r)=\pm
r_{0}\sqrt{\frac{2}{(r^{2}-r_{0}^{2})(r^{2}-m)}}.$ (132)

The length of the separation and energy of the quark-anti-quark pair are then
given by the integrals:

$\begin{split}L^{(\mathrm{IV})}(r_{0})&=2\sqrt{2}r_{0}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{(r^{2}-r_{0}^{2})(r^{2}-m)}}\\\
&=2\sqrt{2}\textrm{{F}}\left(\arcsin\left(\frac{r_{0}}{\sqrt{m}}\right)\Big{|}\frac{m}{r_{0}^{2}}\right)+2i\sqrt{2}\mathrm{\textbf{K}}\left(1-\frac{m}{r_{0}^{2}}\right),\end{split}$
(133)

$E^{(\mathrm{IV})}(r_{0})=\frac{\sqrt{2}}{\pi}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr^{2}}{\sqrt{(r^{2}-r_{0}^{2})(r^{2}-m)}}-\frac{\sqrt{2}}{\pi}\int_{m}^{\infty}\frac{\mathrm{d}rr}{\sqrt{r^{2}-m}},$
(134)

and the perturbations obey the following equation of motion:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P^{(\mathrm{IV})}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W^{(\mathrm{IV})}(r;r_{0})\delta x_{1}(r),$ (135)

where:

$\begin{split}&P^{(\mathrm{IV})}(r;r_{0})=\frac{\sqrt{r^{2}-m}(r^{2}-r_{0}^{2})^{3/2}}{\sqrt{2}r^{2}},\\\
&W^{(\mathrm{IV})}(r;r_{0})=\sqrt{\frac{2(r^{2}-r_{0}^{2})}{r^{2}-m}}.\end{split}$
(136)

The Schrödinger potential for the above reads:

$V^{(\mathrm{IV})}(r;r_{0})=\frac{r^{4}+(m-3r_{0}^{2})r^{2}+5mr_{0}^{2}}{8r^{4}},$
(137)

which has the limiting behaviour:

$\begin{split}&\lim_{r\to\infty}V^{(\mathrm{IV})}(r;r_{0})=\frac{1}{8},\\\
&\lim_{r_{0}\to\sqrt{m}}V^{(\mathrm{IV})}(r_{0};r_{0})=\frac{1}{2}.\end{split}$
(138)

The plot of the potential can be seen in Figure 15 and the numerical solution
for the lowest eigenvalues in 16, where we deduce that the embedding is stable
under longitudinal fluctuations.

(a)

(b)

Figure 15: Schrödinger potential for longitudinal perturbations of the Wilson
loop in background $\mathrm{IV}$ as $r_{0}$ and $m$ vary. Figure 16: Plot of
the lowest eigenvalues of the Schrödinger problem for background IV as a
function of $r_{0}$ for $m=0.2$.

####  6.3.2 ’t Hooft loop

We now move to the ’t Hooft loop for this background, for which we embed a
probe $\mathrm{D}5$ brane in the submanifold spanned by
$[t,r,\mu,\theta,\phi,\psi]$, yielding the following effective action after
performing the integrals:

$S^{(\mathrm{IV})}_{\mathrm{D}5}=\mathrm{T}L_{\mu}16\pi^{2}\int_{r_{0}}^{\infty}\mathrm{d}r\sqrt{f_{x_{1}}(r)x^{\prime
2}+g(r)},$ (139)

where $\mathrm{T}$ is the temporal length of the loop and the functions
$f_{x_{1}}$ and $g$ read in this case:

$\begin{split}&f^{(\mathrm{IV})}_{t}(r)=r^{2}(r^{2}-m),\\\
&g^{(\mathrm{IV})}_{t}(r)=2r^{2},\end{split}$ (140)

where it is evident that the monopole-anti-monopole string tension vanishes at
the end of the space: $f^{(\mathrm{IV})}_{t}(\sqrt{m})=0$ and therefore we
expect this embedding to be unstable, much like the other cases covered. The
length and its approximate expression for this system are:

$L^{(\mathrm{IV})}_{\mathrm{MM}}(r_{0})=\sqrt{2}r_{0}\sqrt{r_{0}^{2}-m}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{(r^{2}-m)(r^{2}-r_{0}^{2})(r^{2}+r_{0}^{2}-m)}},$
(141)

$\hat{L}^{(\mathrm{IV})}_{\mathrm{MM}}(r_{0})=\frac{\sqrt{2}\pi
r\sqrt{r_{0}^{2}-m}}{2r_{0}^{2}-m},$ (142)

where we can easily read that the last expression approaches zero as
$r_{0}\to\sqrt{m}$. The fluctuations obey:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P_{\mathrm{MM}}^{(\mathrm{IV})}(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}\right]\delta
x_{1}(r)=\omega^{2}W_{\mathrm{MM}}^{(\mathrm{IV})}(r;r_{0})\delta x_{1}(r),$
(143)

where:

$\begin{split}&P_{\mathrm{MM}}^{(\mathrm{IV})}(r;r_{0})=\frac{\left[(r^{2}-r_{0}^{2})\left(r^{2}+r_{0}^{2}-m\right)\right]^{3/2}}{r^{2}\sqrt{2(r^{2}-m)}},\\\
&W_{\mathrm{MM}}^{(\mathrm{IV})}(r;r_{0})=\frac{\sqrt{2(r^{2}-r_{0}^{2})(r^{2}+r_{0}^{2}-m)}}{\sqrt{r^{2}-m}}.\end{split}$
(144)

(a)

(b)

Figure 17: Schrödinger potential for longitudinal perturbations of the ’t
Hooft loop in background $\mathrm{IV}$ as $r_{0}$ and $m$ vary. Figure 18:
Plot of the lowest eigenvalues of the ’t Hooft loop for background IV as a
function of $r_{0}$ ($m=1$).

The potentials, which asymptote to $1/2$ for large $r$ values and have the
same form as in the case of background $\mathrm{III}$, are ploted in Figure
17. We also performed the numerical analysis producing similar results shown
in Figure 18, therefore proving that (143) admits negative eigenvalues and the
system screens.

##  7 Conclusions

Summarizing the contents presented in this paper:

In Sections 2,3 we considered four ten-dimensional Type $\mathrm{IIB}$
Supergravity solutions that have been studied in [25, 26], which describe
stacks $\mathrm{D}5$ branes that either intersect or wrap a compact manifold.
These backgrounds have very interesting holographically dual theories that are
confining and exhibit the following behaviour: At low energies there is some
effective Quantum Field Theory that in all cases except background
$\mathrm{IV}$ preserves some amount of supersymmetry. As the energy becomes
larger, the dimension of the dual theory is increased by one, supersymmetry
(where preserved) is enhanced and the system is eventually described, after
performing an S-duality, by the holographic dual of an $\mathrm{NS}5$ brane
setup that is Little String Theory.

Section 4 defines the Wilson loop, ’t Hooft loop and Entanglement Entropy, as
well as how they are calculated using holography in this setting. These are
the observables whose embeddings are tested for instabilities.

In Section 5 we studied in detail the problem of small fluctuations around a
classical solution, based on [27, 28, 30]. Although the problem is described
in general by a coupled Sturm-Liouville system of the longitudinal, transverse
and angular perturbations, for the cases under consideration the only relevant
perturbation is the longitudinal one which is in fact decoupled. We reviewed
how it is more convenient to map the Sturm-Liouville equation into a
Schödinger one and study the spectrum of the eigenvalues using the later. The
existence of a negative eigenvalue mode implies an instability of the
embedding under the corresponding perturbation.

We finally apply the stability analysis to the four Supergravity backgrounds
in Section 6, studying the differential equations and ploting the Schrödinger
potentials for each case, in order to determine if there are any negative
eigenvalues. This is done both by examining the plots and the length function,
as well as using numerical methods where the eigenvalues are ploted with
respect to the tip of the string.

The results found in the present work agree with the claims in [25, 26] where
the stability was speculated using the concavity condition. In particular, the
Wilson loop U-shaped embeddings were stable in all backgrounds while the ’t
Hooft loops and Entanglement entropy (where calculated) were found to be
unstable under longitudinal fluctuations. Furthermore, our findings reflected
the fact that the dual field theories are not UV completed by a field-
theoretical system and also validated the general behaviour that when the
Wilson loops confine the ’t Hooft loops exhibit screening [40], which is
evident as small fluctuations are found to destabilize the embedding into the
disconnected one of a monopole and an anti-monopole moving freely.

There has been a new family of asymptotically AdS Type $\mathrm{IIB}$
backgrounds recently proposed in [41], that is achieved by uplifting a six-
dimensional Romans $F_{4}$ gauged Supergravity [42]. These ten-dimensional
backgrounds model a five-dimensional family of superconformal field theories,
that is expressed using balanced linear quivers. It was noted that even though
one can compute the Wilson loop for a gauge node with no flavour group
attached, much in the same way as we have described here, there are still
massless flavour quarks at some node in the quiver that need to be taken into
account. The presence of dynamical flavour quarks is crucial in the study of
confinement, as these quarks can break the flux tube and cause screening. This
is not captured by the usual embedding in the $(x_{1}(\sigma),r(\sigma))$
plane, but one needs to probe the string in the $\eta=\eta(\sigma)$ direction
that labels the gauge nodes of the quiver as well. It will be interesting in
the future to apply the stability analysis for embeddings in this setting, and
in particular investigate the role of fluctuations around the $\eta$
direction.

## Acknowledgements

The author would like to thank Carlos Nu$\tilde{\mathrm{n}}$ez for his
guidance, support and continuous conversations on topics relevant but not
limited to holography and string theory. Conversations with Ricardo Stuardo
and Ali Fatemiabhari were also vital for understanding much of the background
needed for this work, while a special gratitude is expressed to A.F. for the
help he provided with the shooting method code. This work is supported by STFC
DTP grant No. ST/Y509644-1. For the purpose of open access, the author has
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

##  Appendix A Zero modes of longitudinal perturbations $\Leftrightarrow
L^{\prime}(r_{0})=0$

Up to a constant, the solution for the zero modes of the equation of motion
for the longitudinal fluctuation, namely202020We make use of the
abbreviations: $g(r)\equiv
g,f_{x_{1}}(r)\equiv,f_{x_{1}},f_{x_{1}}(r_{0})\equiv f_{x_{1}0}$.,

$\begin{split}&\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{f_{x_{1}}g}{F^{3/2}}\delta
x_{1}^{\prime(0)}(r)\right)=0\Rightarrow\delta
x^{(0)}_{1}(r)=\int_{r}^{\infty}\mathrm{d}\xi\frac{F^{3/2}(\xi)}{f_{x_{1}}(\xi)g(\xi)}\\\
&\Rightarrow\delta
x^{(0)}_{1}(r)=\int_{r}^{\infty}\mathrm{d}\xi\frac{\sqrt{f_{x_{1}}g}}{\left(f_{x_{1}}-f_{x_{1}0}\right)^{3/2}},\end{split}$
(A1)

where we understand everything inside the integral as a function of $\xi$. We
can now use the following identity:

$(f_{x_{1}}-f_{x_{1}0})^{-3/2}=-\frac{2}{f^{\prime}_{x_{1}}}\frac{\partial}{\partial\xi}\left(f_{x_{1}}-f_{x_{1}0}\right)^{-1/2}$
(A2)

and perform integration by parts to get another expression for $\delta
x^{(0)}$:

$\begin{split}&\delta
x^{(0)}(r)=-\int_{r}^{\infty}\mathrm{d}\xi\sqrt{f_{x_{1}}g}\frac{2}{f^{\prime}_{x_{1}}}\frac{\partial}{\partial\xi}\left(\frac{1}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\right)\\\
&=-\left.\frac{2}{f^{\prime}_{x_{1}}}\sqrt{\frac{f_{x_{1}}g}{f_{x_{1}}-f_{x_{1}0}}}\right|^{\infty}_{r}+2\int_{r}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\\\
&=\lim_{\xi\to
r}\frac{2}{f^{\prime}_{x_{1}}}\sqrt{\frac{f_{x_{1}}g}{f_{x_{1}}-f_{x_{1}0}}}+2\int_{r}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\\\
&=2\sqrt{\frac{f_{x_{1}0}g(r_{0})}{f^{\prime
3}_{x_{1}0}}}(r-r_{0})^{-1/2}+2\int_{r}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right),\end{split}$
(A3)

where in the third line we used that the first term vanishes at $r\to\infty$.
In order for the zero mode to be a solution of the problem, we need to impose
the boundary condition (69) for $r\to r_{0}$, namely:

$\delta x_{1}^{{0}}(r)+2(r-r_{0})\delta x^{\prime(0)}_{1}(r)=0,$ (A4)

$\begin{split}\Rightarrow&2\sqrt{\frac{f_{x_{1}0}g(r_{0})}{f^{\prime
3}_{x_{1}0}}}(r-r_{0})^{-1/2}+2\int_{r_{0}}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\\\
&+2(r-r_{0})\left[-\sqrt{\frac{f_{x_{1}0}g(r_{0})}{f^{\prime
3}_{x_{1}0}}}(r-r_{0})^{-3/2}+\frac{2}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\right]=0,\end{split}$
(A5)

$\Rightarrow\frac{4(r-r_{0})}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)+2\int_{r_{0}}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)=0.$
(A6)

As $r\to r_{0}$, the first term goes like $\sim\sqrt{r-r_{0}}$ so it vanishes.
This leaves us with the following condition in order for a longitudinal zero
mode to exist:

$\int_{r_{0}}^{\infty}\frac{\mathrm{d}\xi}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial\xi}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)=0.$
(A7)

Let us now calculate the derivative of the length from (35):

$L^{\prime}(r_{0})=2\int_{r_{0}}^{\infty}\mathrm{d}r\left[\frac{f^{\prime}_{x_{1}0}}{2\sqrt{f_{x_{1}0}}}\frac{\sqrt{F}}{f_{x_{1}}}+\frac{\sqrt{f_{x_{1}0}}}{2f_{x_{1}}\sqrt{F}}\frac{\partial
F}{\partial
r_{0}}\right]+2\left.\frac{\sqrt{f_{x_{1}0}F}}{f_{x_{1}}}\right|^{\infty}_{r_{0}}$
(A8)

$=\frac{f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\int_{r_{0}}^{\infty}\mathrm{d}r\frac{\sqrt{f_{x_{1}}g}}{\left(f_{x_{1}}-f_{x_{1}0}\right)^{3/2}}-2\lim_{r\to
r_{0}}\frac{\sqrt{f_{x_{1}0}F}}{f_{x_{1}}},$ (A9)

again, using (A2),

$\begin{split}=&-\left.\frac{2f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\frac{1}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\right|^{\infty}_{r_{0}}+\frac{2f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\\\ &-2\lim_{r\to
r_{0}}\frac{\sqrt{f_{x_{1}0}F}}{f_{x_{1}}}\frac{f_{x_{1}0}}{f_{x_{1}}},\end{split}$
(A10)

$\begin{split}=&2\lim_{r\to
r_{0}}\frac{\sqrt{F}}{\sqrt{f_{x_{1}0}}}\frac{f^{\prime}_{x_{1}0}}{f^{\prime}_{x_{1}}}+\frac{2f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)\\\ &-2\lim_{r\to
r_{0}}\frac{\sqrt{f_{x_{1}0}F}}{f_{x_{1}}}\frac{f_{x_{1}0}}{f_{x_{1}}},\end{split}$
(A11)

where we multiplied and devided by the last factor on the third term, which is
1 inside the limit. We can then join the two limits which cancel each other,
to get:

$=2\lim_{r\to
r_{0}}\frac{\sqrt{F}}{\sqrt{f_{x_{1}0}}}\left(\frac{f^{\prime}_{x_{1}0}}{f^{\prime}_{x_{1}}}-\frac{f_{x_{1}}}{f_{x_{1}0}}\right)+\frac{2f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right)$ (A12)

$\Rightarrow
L^{\prime}(r_{0})=\frac{2f^{\prime}_{x_{1}0}}{\sqrt{f_{x_{1}0}}}\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f_{x_{1}}-f_{x_{1}0}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{f_{x_{1}}g}}{f^{\prime}_{x_{1}}}\right).$ (A13)

Thus, the condition (A7) can be written as:

$\frac{\sqrt{f_{x_{1}}}}{f^{\prime}_{x_{1}0}}L^{\prime}(r_{0})=0,$ (A14)

from which we are lead to the conclusion that the zero modes of the
longitudinal perturbations are in 1-1 correspondance with the critical points
of the length. This informs us that for our examples in this paper, where the
length functions are all monotonous, there are no zero eigenvalue modes. One
more fact to note is that due to (37), namely,

$E^{\prime}(r_{0})=\frac{\sqrt{f_{x0}}}{2\pi}L^{\prime}(r_{0})$ (A15)

we see that the critical points of the length and the energy coincide.

##  Appendix B Schrödinger description of the problem

Here we present the change of variables that maps the Sturm-Liouville problem
for the fluctuations to a Schrödinger equation in some more detail [27, 30].
The general form of the Sturm-Liouville equation satisfied by any decoupled
fluctuation $\delta\Phi$ is:

$-\frac{\mathrm{d}}{\mathrm{d}r}\left[P(r;r_{0})\frac{\mathrm{d}}{\mathrm{d}r}+Q(r;r_{0})\right]\delta\Phi(r)=\omega^{2}W(r;r_{0})\delta\Phi(r)\,\,,r^{\star}\leq
r_{0}\leq r<\infty,$ (B1)

where $Q\neq 0$ expresses the restoring force term originating from possible
dependence of the metric components on $\Phi$, if $\Phi$ is an angular
coordinate (which in the cases we are going to study is zero). The only
$"$angular$"$ dependence of the metric components is seen in backgrounds
$\mathrm{II}$ and $\mathrm{III}$ but it does not produce such a restoring
force term in the equations of motion. Using the following coordinate
transformation, this Sturm-Liouville problem can be translated to a
Schrödinger equation:

$y(r;r_{0}):=\int_{r_{0}}^{r}\mathrm{d}\xi\sqrt{\frac{W(\xi,r_{0})}{P(\xi,r_{0})}}=\int_{r_{0}}^{r}\mathrm{d}\xi\sqrt{\frac{h(\xi)}{f_{x_{1}}(\xi)-f_{x_{1}}(r_{0})}},$
(B2)

Then the Schrödinger wavefunction satisfying the equation is:

$\Psi(y):=\left[P(r;r_{0})W(r;r_{0})\right]^{1/4}\delta\Phi(r),$ (B3)

and (52) is turned into:

$\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+V(y,r_{0})\right]\Psi(y)=\omega^{2}\Psi(y)\,\,,0\leq
y\leq y_{0},$ (B4)

where $y_{0}$ is defined as:

$y_{0}(r_{0})=\int_{r_{0}}^{\infty}\mathrm{d}\xi\sqrt{\frac{h(\xi)}{f_{x_{1}}(\xi)-f_{x_{1}}(r_{0})}}.$
(B5)

If the space has an asymptotically AdS factor $y_{0}$ is finite in both
integration limits, but in general it can be either finite or infinite
depending on the details of the geometry. The interval $[r_{0},\infty)$ is
mapped into $[0,y_{0}]$ under (B2) and the boundary conditions for (53) are:

$\begin{split}&\Psi(y_{0})=0\,\,,\,\,\text{from the fixed points of the string
at}\,r=\infty\\\
&\frac{\mathrm{d}\Psi}{\mathrm{d}y}\Big{|}_{y_{0}}=0\,\,,\,\,\text{from the
condition at the tip}\end{split}$ (B6)

The Schrödinger potential in (B4) is then found to be:

$V(y;r_{0})=-\frac{Q}{W}+\left(PW\right)^{-1/4}\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}\left(PW\right)^{1/4},$
(B7)

or, expressed in terms of $r$:

$V(r,r_{0})=-\frac{Q}{W}+\frac{P^{1/4}}{W^{3/4}}\frac{\mathrm{d}}{\mathrm{d}r}\left[\left(\frac{P}{W}\right)^{1/2}\frac{\mathrm{d}}{\mathrm{d}r}\left(PW\right)^{1/4}\right].$
(B8)

Although it is not often that one can invert $y=y(r)$ in order to get (B7),
the expression in terms of $r$ which follows from the equations of motion can
provide enough evidence for the study of the spectrum. We also note that for
all Wilson loops of backgrounds $(\mathrm{I}-\mathrm{IV})$ we have the
asymptotic behaviour: As $r\to\infty$, $\Psi(y)\sim\sqrt{r}\delta\Phi(r)$,
while for the ’t Hooft loops and Entanglement Entropy: $\Psi(y)\sim
r\delta\Phi(r)$.

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