Confinement in $(1+1)$ dimensions: a holographic perspective from I-branes

Carlos<EMAIL_ADDRESS>Marcelo<EMAIL_ADDRESS>and Ricardo<EMAIL_ADDRESS>
aDepartment of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
bDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción,
Chile

Abstract

In this paper we holographically study the strongly coupled dynamics of the
field theory on I-branes (D5 branes intersecting on a line). In this regime,
the field theory becomes $(2+1)$ dimensional with 16 supercharges. The
holographic dual background has an IR singularity. We solve this singularity
by compactifiying the theory on a circle, preserving 4 supercharges. We study
various aspects of the QFT: confinement, symmetry breaking, Entanglement
Entropy, etc. We also discuss a black hole solution and make some comments on
the string $\sigma$-model on our backgrounds.

###### Contents

  1. 1 Introduction
  2. 2 The supergravity backgrounds
    1. 2.1 Background I
    2. 2.2 Background II
    3. 2.3 Conserved charges
    4. 2.4 Bonus: a black hole solution
  3. 3 A proposal for the dual field theory and its observables
    1. 3.1 The holographic dual to the Background I
    2. 3.2 The field theory dual to the Background II.
    3. 3.3 Gauge coupling
    4. 3.4 Theta angle
    5. 3.5 $U(1)_{A,B}$ symmetry breaking pattern
    6. 3.6 Holographic central charge
  4. 4 Maldacena-Wilson, ’t Hooft loops and Entanglement Entropy
    1. 4.1 General comments on Maldacena-Wilson loops and similar probes
    2. 4.2 Maldacena-Wilson loops in our background
    3. 4.3 ’t Hooft loops
    4. 4.4 Entanglement Entropy
  5. 5 Sigma Model
  6. 6 Conclusions and future research
  7. A Details of the Supergravity Backgrounds
    1. A.1 Type IIB Supergravity
    2. A.2 Checking SUSY for the Fibered Background
  8. B How are the backgrounds obtained
    1. B.1 4D $\mathcal{N}=4\,SU(2)\times SU(2)$ Gauged Supergravity
    2. B.2 Lift to 10D Supergravity
  9. C On the Unfibered Geometry
  10. D Charges of the black hole background
  11. E R-Symmetry Breaking
    1. E.1 $U(1)$ R-Symmetry of $\psi_{A}$ and $\psi_{B}$
    2. E.2 $U(1)$ R-Symmetry of $\varphi$
  12. F Maldacena-Wilson, ’t Hooft loops and EE. Detailed calculations
    1. F.1 Wilson loop
    2. F.2 t’ Hooft loop
    3. F.3 Entanglement entropy

##  1 Introduction

Shortly after the conjectured duality between super conformal field theories
and string/M theory on spaces with an AdS factor [2] -[4], these ideas were
extended to non-conformal situations [5]-[12].

In this work, we holographically study one particular two dimensional field
theory that at weak coupling is defined on the intersection of two stacks of
D5 branes (these are called I-branes). Dynamical features of these theories
imply that, as the coupling is increased the field theory turns
$(2+1)$-dimensional and doubles the amount of SUSY preserved. This dynamics is
very well explained in [13].

The string background, holographic dual to this strongly coupled QFT is well
understood at large values of the radial coordinate, where it can be written
as an intersection of NS five branes. Good control over the string
$\sigma$-model in such background has been developed [13], [14]. Nevertheless,
this background is singular in the IR (at small values of a suitable radial
coordinate). In this paper we propose a ’completion’ of this dual background,
making it trustable at low energies. The new solution is very explicit and
simple. It preserves four supercharges and can be thought of as the dual to a
$(2+1)$-dimensional field theory that is compactified to $(1+1)$-dimensions.
The QFT is a two nodes quiver with Chern-Simons terms, connected by
bifundamental matter.

In the bulk of this paper, we study holographically various aspects of the
strongly coupled dynamics of this QFT. We define a suitable gauge coupling,
that suggest a low energy confining behaviour. Theta angles and the breaking
of $U(1)$-R symmetries are discussed, together with an estimate of a density
of degrees of freedom as a function of the energy. Maldacena-Wilson loops are
calculated (again indicating confinement). Also, ’t Hooft loops and
Entanglement Entropy on a strip are calculated, discussing how the non-local
UV dynamics of the system impacts on these observables.

We also briefly touch upon two aspects that will be further developed in
future publications: we present a black hole solution, obtained as analytic
continuation of our new background and calculate some characteristic
quantities. Also, we shortly discuss some aspects of the string $\sigma$-model
on our backgrounds.

The paper is organised as follows. In Section 2 we present the supergravity
backgrounds studied in the rest of the paper. In Section 3 we propose the QFT
dual to these backgrounds, with various characteristic observables calculated.
This proposal is sharpened in Section 4, where the IR confining behaviour is
determined and the influence of the high energy LST dynamics on observables
like ’t Hooft loops and Entanglement Entropy is discussed. Some aspects of the
string $\sigma$-model are discussed in Section 5. The conclusions and future
lines of research suggested by this paper are written in Section 6. Various
appendices are included, these discuss in great detail the very many
interesting technical aspects needed in the main body of this work.

##  2 The supergravity backgrounds

In this section we write the supergravity backgrounds studied. The first
background already appears in the bibliography [15], the second background is
new. We refer the reader to Appendices A and B for detailed derivations. The
associated charges are studied. A black hole solution is obtained as a bonus,
by performing analytic continuations. Some characteristic observables of the
black hole are calculated.

###  2.1 Background I

To describe the backgrounds we use the coordinates
$\left(t,x,\varphi,r,{\theta_{A}},{\phi_{A}},{\psi_{A}},{\theta_{B}},{\phi_{B}},{\psi_{B}}\right)$.
We set $\alpha^{\prime}=g_{s}=1$ and define two sets of left-invariant forms
of $SU(2)$,

$\displaystyle\hat{\omega}_{1}=\cos{\psi_{A}}d{\theta_{A}}+\sin{\psi_{A}}\sin\theta_{A}d{\phi_{A}},\;\;\;\;\;\tilde{\omega}_{1}=\cos{\psi_{B}}d{\theta_{B}}+\sin{\psi_{B}}\sin\theta_{B}d{\phi_{B}},$
$\displaystyle\hat{\omega}_{2}=-\sin{\psi_{A}}d{\theta_{A}}+\cos{\psi_{A}}\sin\theta_{A}d{\phi_{A}},\;\;\tilde{\omega}_{2}=-\sin{\psi_{B}}d{\theta_{B}}+\cos{\psi_{B}}\sin\theta_{B}d{\phi_{B}},$
$\displaystyle\hat{\omega}_{3}=d{\psi_{A}}+\cos{\theta_{A}}d{\phi_{A}},\;\;\;\;\;\;\;\;~{}~{}~{}~{}~{}~{}~{}~{}~{}\;\;\;\;\;\tilde{\omega}_{3}=d{\psi_{B}}+\cos{\theta_{B}}d{\phi}_{B}.$

In terms of these we present the first background. The string frame metric,
the Ramond three form $F_{3}$, the potential $C_{2}$ and the dilaton $\Phi$
read,

$\displaystyle ds^{2}_{st}$ $\displaystyle=$ $\displaystyle
r\left\\{-dt^{2}+dx^{2}+\frac{(e_{A}^{2}+e_{B}^{2})d{\varphi}^{2}}{2}+\frac{8~{}dr^{2}}{r^{2}(e_{A}^{2}+e_{B}^{2})}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\hat{\omega}_{3}^{2}\right]+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2}\right]\right\\}\
,$ $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
dC_{2}=-\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\hat{\omega}_{3}-\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\tilde{\omega}_{3}\
,$ (2.1) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{e_{A}^{2}}\psi_{A}\sin{\theta_{A}}d{\theta_{A}}\wedge
d{\phi_{A}}-\frac{2}{e_{B}^{2}}\psi_{B}\sin{\theta_{B}}d{\theta_{B}}\wedge
d{\phi_{B}}.$ $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\log r\ .$

Here $\left(e_{A},e_{B}\right)$ are parameters which are fixed when imposing
charge quantisation. The coordinate $\varphi$ is non-compact and could be
rescaled to absorb the prefactor $\frac{(e_{A}^{2}+e_{B}^{2})}{2}$.

There are two three-spheres labelled by $\hat{S}^{3}$ and $\tilde{S}^{3}$
respectively parametrised by the Euler angles
$[{\theta_{A}},{\phi_{A}},{\psi_{A}}]$ and
$[{\theta_{B}},{\phi_{B}},{\psi_{B}}]$. The range for these angles are
$\theta_{A,B}\in[0,\pi[$ $\psi_{A,B}\in\left[0,4\pi\right]$, and
$\phi_{A,B}\in\left[0,2\pi\right]$.

The Ricci scalar for the metric in eq.(2.1) is

$R=-\frac{3\left(e_{A}^{2}+e_{B}^{2}\right)}{2r},$ (2.2)

indicating a singularity at $r=0$. This is the singular behaviour found close
to a stack of D5 branes–see equation (3.38) in the paper [16]. In fact, the
background can be understood as the backreaction of two stacks of D5 branes
that intersect along the non-compact coordinates $(t,x)$ and extend
respectively over $(y_{1},y_{2},y_{3},y_{4})$ and $(w_{1},w_{2},w_{3},w_{4})$,
which can be written as radial coordinates and three spheres. After
backreaction (at strong coupling), the two stacks share the directions
$[t,x,\varphi]$ wrap the spheres $\hat{S}^{3}[\theta_{A},\phi_{A},\psi_{A}]$
and $\tilde{S}^{3}[\theta_{B},\phi_{B},\psi_{B}]$, as we find in the
background (2.1). See details in Appendix C.

The singular behaviour at $r\sim 0$ indicates the need of a description in
terms of other variables. On the other hand for large values of the radial
coordinate $r$, the growth of the dilaton and the string coupling $(g_{s}\sim
e^{\Phi})$ requires an S-duality and the description of the system is in terms
of an intersection of NS five branes. The system is then dual to two Little
String Theories (LST) that intersect along $(t,x,\varphi)$ each one wrapping
the spheres $\hat{S}^{3}$ and $\tilde{S}^{3}$. We further elaborate on this
background in Section 5 and Appendix C.

We are interested in resolving the singular behaviour at $r=0$ of the
background in eq.(2.1), making the solution in terms of D5 branes trustable in
the ”IR-regime”. The goal is to write a trustable dual description for a
strongly coupled QFT (that is UV-completed by a LST). We are also interested
in preserving some amount of SUSY for stability purposes.

###  2.2 Background II

We write below a solution to the equations of motion of Type IIB supergravity
that resolves the singular behaviour by compactifying the coordinate
$\varphi$, with a precise period. A fibration between the spheres and the
coordinate $\varphi$ is also needed. This solution reads,

$\displaystyle ds^{2}_{st}$ $\displaystyle=$ $\displaystyle
r\left\\{-dt^{2}+dx^{2}+f_{s}\left(r\right)d\varphi^{2}+\frac{4}{r^{2}f_{s}\left(r\right)}dr^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)d\varphi\right)^{2}\right]\right.$
(2.3)
$\displaystyle\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}-e_{B}Q_{B}\zeta\left(r\right)d\varphi\right)^{2}\right]\right\\}\
,$ $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
dC_{2}=2\zeta^{\prime}(r)dr\wedge
d\varphi\wedge\left(\frac{Q_{A}}{e_{A}}\hat{\omega}_{3}+\frac{Q_{B}}{e_{B}}\tilde{\omega}_{3}\right)+\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\left(e_{A}Q_{A}\zeta(r)d\varphi-\hat{\omega}_{3}\right)$
$\displaystyle+\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\left(e_{B}Q_{B}\zeta(r)d\varphi-\tilde{\omega}_{3}\right)\
,$ $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle{\psi_{A}}\left(\frac{2Q_{A}}{e_{A}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{A}^{2}}\sin{\theta_{A}}d{\theta_{A}}\wedge
d{\phi_{A}}\right)+\frac{2}{e_{A}}\cos{\theta_{A}}Q_{A}\zeta\left(r\right)d\varphi\wedge
d{\phi_{A}}$
$\displaystyle+{\psi_{B}}\left(\frac{2Q_{B}}{e_{B}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{B}^{2}}\sin{\theta_{B}}d{\theta_{B}}\wedge
d{\phi_{B}}\right)+\frac{2}{e_{B}}\cos{\theta_{B}}Q_{B}\zeta\left(r\right)d\varphi\wedge
d{\phi_{B}}\ .$ $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\log r\ .$

Here $\left(e_{A},Q_{A},e_{B},Q_{B}\right)$ are parameters. The functions
$f_{s}(r),\zeta(r)$ are given by

$\displaystyle f_{s}(r)$ $\displaystyle=$
$\displaystyle\frac{e_{A}^{2}+e_{B}^{2}}{2}-\frac{m}{r^{2}}-\frac{2\left(Q_{A}^{2}+Q_{B}^{2}\right)}{r^{4}}\equiv\frac{e_{A}^{2}+e_{B}^{2}}{2r^{4}}(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\,,$
(2.4) $\displaystyle\zeta\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}-\frac{1}{r_{+}^{2}}\
,\;\;\;r_{\pm}^{2}=\frac{m\pm\sqrt{m^{2}+4(Q_{A}^{2}+Q_{B}^{2})(e_{A}^{2}+e_{B}^{2})}}{e_{A}^{2}+e_{B}^{2}}.$
(2.5)

If the parameter $m=0$ and $e_{A}Q_{B}=\pm e_{B}Q_{A}$, the background
preserves four supercharges. For the SUSY study and the details of the
construction of the background in eqs.(2.3)-(2.5), see Appendices A and B.

Note that the circle parametrised by the angle $\varphi$ shrinks smoothly at
$r=r_{+}$ if we choose its periodicity to be

$\displaystyle\varphi\sim\varphi+L_{\varphi},~{}~{}L_{\varphi}\\!=\\!\frac{8\pi}{r_{+}f^{\prime}_{s}(r_{+})}\\!=\\!\frac{4\pi}{e_{A}^{2}+e_{B}^{2}}\left(1+\frac{m}{\sqrt{m^{2}+4(e_{A}^{2}+e_{B}^{2})(Q_{A}^{2}+Q_{B}^{2})}}\right).$
(2.6)

In the BPS limit, the Ricci scalar associated with the geometry in eq.(2.3) is

$R=-\frac{(e_{A}^{2}+e_{B}^{2})}{2e_{A}^{2}r^{5}}\left(4Q_{A}^{2}+3e_{A}^{2}r^{4}\right).$
(2.7)

That is bounded for all the range of the radial coordinate $[r_{+},\infty)$.

Notice that in the case $Q_{A}=Q_{B}=m=0$ (conversely, in the strict limit
$r\to\infty$), the background in eq.(2.3) becomes that in eq.(2.1). In some of
the observables we discuss below, we can perform a regularisation that takes
away the effects on the observable that come from the background (2.1) from
the same observable computed in the background (2.3). Also, in what follows,
for any object, $\xi_{A,B}$, (like a D-brane) extended along the spheres we
use the notation $\xi_{A}\equiv\hat{\xi}$ and $\xi_{B}\equiv\tilde{\xi}$.

###  2.3 Conserved charges

To calculate the D5 brane charges associated with the Ramond Field $F_{3}$ in
eq.(2.3), we define the three-cycles,

$\mathcal{M}_{A}=(\psi_{A},\theta_{A},\phi_{A}),~{}~{}~{}~{}\mathcal{M}_{B}=(\psi_{B},\theta_{B},\phi_{B}).$
(2.8)

Let us call the A-stack of branes to be the one extended along the coordinates
$[t,x,\varphi,\theta_{A},\phi_{A},\psi_{A}]$ and analogously for the B-stack.
To calculate the number of branes in the A-stack, we need to integrate $F_{3}$
over the three cycle $\mathcal{M}_{B}$–as this is orthogonal to the A-brane
stack. Analogously, the number of branes in the B-stack will be obtained by
integrating $F_{3}$ over $\mathcal{M}_{A}$.

Setting $\alpha^{\prime}=g_{s}=1$, the quantisation condition for Dp-branes is

$\displaystyle{(2\pi)^{7-p}g_{s}\alpha^{\prime\frac{7-p}{2}}}N_{Dp}=\int_{\Sigma_{8-p}}F_{8-p},~{}~{}\text{leads
to}~{}~{}N^{i}_{D5}=\frac{1}{(2\pi)^{2}}\int_{\mathcal{M}_{i}}F_{3}.$

After choosing a convenient orientation for the three-cycles (equivalently,
changing the sign of $F_{3}$) we find for the D5 charges,

$N_{A}=\frac{8}{e^{2}_{B}},~{}~{}~{}~{}N_{B}=\frac{8}{e^{2}_{A}}.$ (2.9)

This implies a quantisation condition for the parameters $(e_{A},e_{B})$. We
could have chosen different three cycles, leading to the same conditions. The
result is the same for the either of the backgrounds in eqs.(2.1), (2.3).

###  2.4 Bonus: a black hole solution

Let us consider our new background in eq.(2.3)111The material in this section
arose in discussion with Juan Maldacena, whom we gratefully acknowledge.. We
perform a double Wick rotation

$\displaystyle\varphi\rightarrow it\ ,\qquad Q_{A,B}\rightarrow-iQ_{A,B}\
,~{}~{}t\rightarrow iy,$ (2.10)

we find the black hole configuration, which we present in Einstein frame

$\displaystyle ds_{E}^{2}$ $\displaystyle=$
$\displaystyle\sqrt{r}\left\\{dy^{2}+dx^{2}-f_{bh}\left(r\right)dt^{2}+\frac{4}{r^{2}f_{bh}\left(r\right)}dr^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta\left(r\right)dt\right)^{2}\right]\right.$
$\displaystyle+\left.\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}-e_{B}Q_{B}\zeta\left(r\right)dt\right)^{2}\right]\right\\}\
,$ $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
dC_{2}=2\zeta^{\prime}\left(r\right)dr\wedge
dt\wedge\left(\frac{Q_{A}}{e_{A}}\hat{\omega}_{3}+\frac{Q_{B}}{e_{B}}\tilde{\omega}_{3}\right)+\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\left(e_{A}Q_{A}\zeta\left(r\right)dt-\hat{\omega}_{3}\right)$
$\displaystyle+\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\left(e_{B}Q_{B}\zeta\left(r\right)dt-\tilde{\omega}_{3}\right)\
,$ $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\log\left(r\right)\ .$
(2.11)

where

$\displaystyle f_{bh}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{e_{A}^{2}+e_{B}^{2}}{2}-\frac{m}{r^{2}}+\frac{2\left(Q_{A}^{2}+Q_{B}^{2}\right)}{r^{4}}\equiv\frac{e_{A}^{2}+e_{B}^{2}}{2r^{4}}\left(r^{2}-r_{+}^{2}\right)\left(r^{2}-r_{-}^{2}\right),$
$\displaystyle\zeta\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}-\frac{1}{r_{+}^{2}}\ ,\quad
r_{\pm}^{2}=\frac{m\pm\sqrt{m^{2}-4\left(Q_{A}^{2}+Q_{B}^{2}\right)\left(e_{A}^{2}+e_{B}^{2}\right)}}{e_{A}^{2}+e_{B}^{2}}\
.$ (2.12)

This configuration $F_{3}$ has both ”electric” and magnetic parts.
Nevertheless, the integrating of the magnetic part of $F_{7}=\star F_{3}$ does
not lead to charge of D1 brane since the 7-cycle in which one integrates is
not closed, thus is not possible to apply Gauss’s law.

In general $f_{bh}\left(r\right)$ has two real roots $r_{\pm}$. The extremal
black hole is obtained when $r_{+}=r_{-}$ that is,

$\displaystyle
m^{2}=4\left(Q_{A}^{2}+Q_{B}^{2}\right)\left(e_{A}^{2}+e_{B}^{2}\right),\,~{}~{}r_{+}^{2}=r_{-}^{2}=2\sqrt{\frac{Q_{A}^{2}+Q_{B}^{2}}{e_{A}^{2}+e_{B}^{2}}},$
$\displaystyle
f_{bh}\left(r\right)=\frac{e_{A}^{2}+e_{B}^{2}}{2r^{4}}\left(r^{2}-r_{+}^{2}\right)^{2}=\frac{e_{A}^{2}+e_{B}^{2}}{2r^{4}}\left(r^{2}-2\sqrt{\frac{\left(Q_{A}^{2}+Q_{B}^{2}\right)}{\left(e_{A}^{2}+e_{B}^{2}\right)}}\right)^{2}\
,$ (2.13)

The preservation of SUSY imposes extremality and $e_{A}Q_{A}\pm e_{B}Q_{B}=0\
.$ The BPS background with $Q_{B}=\frac{e_{A}}{e_{B}}Q_{A}$ reads,

$\displaystyle
ds_{E}^{2}=\sqrt{r}\left\\{dy^{2}+dx^{2}+\frac{4}{r^{2}\frac{e_{A}^{2}+e_{B}^{2}}{2}r_{+}^{4}\zeta\left(r\right)^{2}}dr^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\hat{\omega}_{3}^{2}-2\hat{\omega}_{3}e_{A}Q_{A}\zeta
dt\right]\right.$
$\displaystyle\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2}-2\tilde{\omega}_{3}e_{A}Q_{A}\zeta
dt\right]\right\\}\ .$ $\displaystyle
F_{3}=2d\left[\zeta\left(r\right)dt\wedge\left(\frac{Q_{A}}{e_{A}}\hat{\omega}_{3}+\frac{Q_{B}}{e_{B}}\tilde{\omega}_{3}\right)\right]-\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\hat{\omega}_{3}-\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\tilde{\omega}_{3},$
$\displaystyle\Phi=\log(r).$ (2.14)

Note that $g_{tt}=0$, thus the vector $\partial_{t}$ is null.

In what follows we consider the non-BPS black hole background (2.11). This
configuration is rotating along the directions $\partial_{\psi_{A}}$ and
$\partial_{\psi_{B}}$. Since the fibrations do not decay at infinity, the
coordinate system used in (2.11) correspond to a rotating frame, i.e. an
observer at infinity is rotating with the system. We can move to a non-
rotating frame via large gauge transformation which cancels the constant term
of $\zeta(r)$ in (2.12). Also for simplicity in the computation of the charges
we shift the Dilaton by a constant and the $F_{3}$ by a factor

$\zeta(r)=\frac{1}{r^{2}},\qquad\Phi\to\Phi-2\log\left(\frac{e_{A}^{2}+e_{B}^{2}}{2}\right),\qquad
F_{3}\to\frac{e_{A}^{2}+e_{B}^{2}}{2}F_{3}.$ (2.15)

The geometry is asymptotically locally flat and has the same causal structure
as the Reissner-Nordström spacetime. Nevertheless, the spacetime is
asymptotically conformal to $\mathbb{R}^{1,3}\times S^{3}\times S^{3}$.

We compute the conserved charges associated to the spacetime by using the
Noether-Wald method [17]. The expression for the charges are derived in detail
in Appendix D. The energy (mass), angular momentum associated to
$\partial_{\psi_{A}}$ and $\partial_{\psi_{B}}$, the temperature and the
entropy for this configuration are given by 222The asymptotic form of the
metric leaves an ambiguity in the normalisation of the time-like killing
vector at infinity that appears in the computation of the energy and the
temperature. Therefore, the temperature in ten dimensions is the same as in
four dimensions up to a numerical factor.

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{2m}{e_{A}^{3}e_{B}^{3}r_{+}^{2}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,$ (2.16) $\displaystyle J_{A}$ $\displaystyle=$
$\displaystyle\frac{8Q_{A}}{e_{A}^{4}e_{B}^{3}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,\quad
J_{B}=\frac{8Q_{B}}{e_{B}^{4}e_{A}^{3}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,$ (2.17) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{e_{A}^{2}+e_{B}^{2}}{16\pi}-\frac{4\left(Q_{A}^{2}+Q_{B}^{2}\right)}{16\pi
r_{+}^{4}}\ ,\quad
S=\frac{2r_{+}^{2}}{e_{A}^{3}e_{B}^{3}G_{10}}\left(16\pi\right)^{2}L_{x}L_{y}\,\,\
.$ (2.18)

According to our normalisation $\kappa^{2}=8\pi G_{10}$ where $G_{10}$ is the
Newton constant in ten dimensions. These quantities satisfy the first law of
thermodynamics as expected

$dE=TdS+\Omega_{A}dJ_{A}+\Omega_{B}dJ_{B}\ ,$ (2.19)

where the angular velocities are

$\Omega_{A}=\frac{e_{A}Q_{A}}{r_{+}^{2}}\,\,,\qquad\Omega_{B}=\frac{e_{B}Q_{B}}{r_{+}^{2}}\,\,.$
(2.20)

This background can be understood as the lift to Type IIB of the four
dimensional planar black hole with electric charges, found in [18]. The
presence of the electric charges in four dimensions corresponds to rotations
of the branes in ten dimensions.

We leave this black hole background here, as it is not the focus of the rest
of this work. We move into the study of the dual Field Theories to the
backgrounds I and II in eqs.(2.1) and (2.3)

##  3 A proposal for the dual field theory and its observables

Here we present a proposal for the field theory dual to our new background in
eq.(2.3). It is convenient to first discuss the field theory dual to the D5-D5
intersection and the background in eq.(2.1).

###  3.1 The holographic dual to the Background I

We start discussing the field theory on I-branes. The result in eq.(2.9)
indicates the presence of two stacks of D5 branes, with $N_{A}$ and $N_{B}$
being the number of branes on each stack. When taken at weak coupling these
stacks intersect over two dimensions. In [19] it was shown that when two
stacks of branes intersect along $(4k+2)$, being the transverse dimensions a
multiple of four (in our case $k=0$, D5 stacks intersect in two dimensions and
have eight transverse directions) the massless spectrum contains chiral
fermions, arising from the open strings connecting the branes. These fermions
give rise to gauge (and gravitational) anomalies on the intersection. The
anomalies are cancelled by anomaly inflow from the ’bulk of the brane’. This
implies that the D-branes world-volume action must contain a Chern-Simons
term.

When studied at weak coupling the D5-D5 system preserve chiral supercharges.
We have two gauge groups $SU(N_{A})\times SU(N_{B})$ with chiral fermions
transforming in the $({\bf N_{A},\bar{N}_{B}})$ representation, the system has
$SO(1,1)$ Poincare symmetry. The anomaly is cured by inflow from the bulk of
the D5 branes. In other words, the dynamics of the intersection is not
decoupled from the brane dynamics. The system preserves eight SUSYs [19]. The
weakly coupled field theory is summarised by the quiver in Figure 1.

$N_{A}$$N_{B}$ Figure 1: Dual Theory in ($1+1$) dimensions with chiral
fermions running in the links. This encodes the field theory at weak coupling.
The inflow from the bulk of the branes is understood.

As the couplings grow large, the above description breaks down and is replaced
by a description in terms of the background in eq.(2.1). This is carefully
described in [13]. In the strong coupling regime, the system preserves
$SO(2,1)$ with $SO(4)\times SO(4)$ R-symmetry and sixteen SUSYs. The three
dimensional field theory has gauge groups with Chern Simons terms
$SU(N_{A})_{N_{B}}\times SU(N_{B})_{N_{A}}$ [13], [20]. At strong coupling,
these stacks intersect in the coordinates $(t,x,\varphi)$. One of the stacks
extends along $(\theta_{A},\phi_{A},\psi_{A})$ whilst the other does it over
$(\theta_{B},\phi_{B},\psi_{B})$. The papers [13], [20], argue that the field
theory is gapped.

Notice that the background in eq. (2.1) is not trustable for all the range of
the radial coordinate. In fact, for large values of $r$, the dilaton becomes
large the type IIB system is better described by performing an S-duality and
describing the dynamics in terms of the Little String Theory on the two stacks
of NS five branes. That is, the field theory above described has a non-field
theoretical UV completion.

On the other hand, for $r\to 0$ the Ricci scalar in eq.(2.2) diverges and the
background is not trustable. We then need to replace the description by the
one given by the configuration in eq.(2.3).

###  3.2 The field theory dual to the Background II.

Here, we analyse the background in eq.(2.3). The supergravity solution is
smooth, hence the dual QFT is strongly coupled.

In fact, the Ricci scalar in eq.(2.7) is finite for $r>r_{+}$ and the string
coupling (proportional to $e^{\Phi}$) is bounded below some value $r<r_{*}$.
The value of $r_{*}$ is determined by observing that the dilaton in eq.(2.3)
can be changed by $\Phi=\Phi_{0}+\log r$ at the cost of rescaling the Ramond
form $F_{3}\to e^{-\Phi_{0}}F_{3}$. These scalings make the string coupling
$g_{s}=e^{\Phi}=re^{\Phi_{0}}$. The value $r_{*}\sim e^{-\Phi_{0}}$ (for which
$g_{s}\sim 1$), can be made arbitrarily large by suitably choosing the
parameter $\Phi_{0}$. Notice that a chosen large and negative $\Phi_{0}$,
makes the charges of the D5 branes larger.

We then conclude that the background is trustable in a large region of the
radial coordinate $[r_{+},r_{*})$. Consequently, in a large regime of energies
the dual field theory is strongly coupled. At very high energies, when the
string coupling becomes large we should S-dualise arriving to a configuration
of intersecting and wrapped NS-five branes. The high energy behaviour of the
field theory is UV-completed in terms of a Little String Theory.

We now discuss the flowing to lower energies. In this case, the coordinate
$\varphi$ is compactified, shrinks to zero size and one ends with a QFT with
less SUSY, smaller R-symmetry and effectively in ($1+1$) dimensions. Whilst
for large values of the radial coordinate, the backgrounds I and II coincide
at leading order, a relevant operator is deforming the field theory dual to
background II. This deformation is associated with the subleading terms,
proportional to the parameters $Q_{A},Q_{B}$. As we lower the energy (still at
strong coupling) a Kaluza-Klein spectrum of massive modes arises due to the
compactification of the branes on $\varphi$.

At energies around the scale set by $r_{+}$, the QFT dual to the background in
eq.(2.3) should be a $(1+1)$ dimensional QFT. This should be the reduction of
the Yang-Mills-Chern-Simons $SU(N_{A})_{N_{B}}\times SU(N_{B})_{N_{A}}$ to
($1+1$), preserving four supercharges. The QFT is expected to be gapped,
confine and break part of the R-symmetry.

In what follows, we start the study of this interesting field theory. We do so
by calculating observables of the two-dimensional QFT using probes of the
background in eqs. (2.3)-(2.5). These probes inform us about gauge couplings,
theta-angles, symmetry breaking, confinement, etc. We present a quantity that
indicates the number of degrees of freedom (density of states in terms of the
energy).

###  3.3 Gauge coupling

To study the background using D-branes probes, it is first useful to set our
conventions for the Dirac-Born-Infeld-Wess-Zumino (DBIWZ) action, describing
the dynamics of branes in our background.

Consider a ten dimensional manifold $M_{10}$ equipped with a metric tensor
$G_{\mu\nu}$, Neveu-Schwarz two form $B_{\mu\nu}$, dilaton $\Phi$ and Ramond
potentials encoded in the poly-form ${\cal C}$. In this space, there is an
embedded manifold $\Sigma$ of dimension $(p+1)$ with $(9-p)$ space-like normal
vectors. This embedded manifold hosts a Dp-brane. We denote the coordinates on
the Dp-brane as $X^{M}$ with $M=0,1,\dots,p$. and the induced metric $g_{MN}$.
The action of a single Dp-brane is the Dirac-Born-Infeld-Wess-Zumino action
given by

$\displaystyle S_{Dp,DBI}\left[g_{MN},{\cal F}_{MN}\right]=T_{p}\int
d^{p+1}xe^{-\Phi}\sqrt{-\det\left(g_{MN}+{\cal F}_{MN}\right)}\ .$ (3.1)
$\displaystyle S_{Dp,WZ}\left[{\cal C},{\cal
F}_{MN}\right]=-T_{p}\int_{\Sigma}{\cal C}\wedge e^{-{\cal F}_{MN}}$ (3.2)

Here ${\cal F}_{MN}=B_{MN}+2\pi\alpha^{\prime}F_{MN}$, where $B_{MN}$ is the
pull-back of the background Neveu-Schwarz two-form on $\Sigma$, and $F_{MN}$
is an Abelian gauge field strength defined on the brane. The tension of the Dp
brane is $T_{p}=\frac{1}{(2\pi)^{7-p}}$, in our chosen units. For the case of
our backgrounds, we have $B_{MN}=0$ and the poly-form ${\cal C}=C_{2}$ given
in eq.(2.3), and/or its electric dual $C_{6}$, given in Appendix B.

With our choice of units ($g_{s}=\alpha^{\prime}=1$) we can perform a small
field $F_{MN}$ expansion of the action in eq.(3.1), equivalent to a
small-$\alpha^{\prime}$ expansion. We obtain an effective action for the Dp-
brane,

$S_{Dp,DBI}=T_{p}\int d^{p+1}xe^{-\Phi}\sqrt{-\det
g_{MN}}\left[1-\frac{1}{4}\left(2\pi\right)^{2}F^{BC}F_{BC}+\mathcal{O}\left(F^{3}\right)\right]\
$ (3.3)

where $F^{BC}=g^{BM}g^{CN}F_{MN}$. The Wess-Zumino part of the action in
eq.(3.2) contains a finite number of terms. Specialising for our backgrounds
with $B_{MN}=0$ we have for any Dp brane probe,

$\displaystyle
S_{Dp,WZ}\\!=\\!\\!-\\!T_{p}\\!\\!\int_{\Sigma_{p+1}}\\!\\!C_{p+1}\\!-\\!2\pi
C_{p-1}\\!\wedge\\!F_{2}+\\!\frac{(2\pi)^{2}}{2}\\!C_{p-3}\\!\wedge\\!F_{2}^{2}-\\!\frac{(2\pi)^{3}}{6}\\!C_{p-5}\wedge\\!F_{2}^{3}+\\!\frac{(2\pi)^{4}}{24}\\!C_{p-7}\wedge\\!F_{2}^{4}.$
(3.4)

In what follows we study the backgrounds in eqs. (2.1), (2.3)-(2.5) with
various probe branes in Type IIB.

The first probe is a D5 brane that extends on the directions
$[t,x,\varphi,\theta_{A},\phi_{A},\psi_{A}]$. This is like a probe that
extends where the A-stack originally was. We will switch on an electric field
on its worldvolume.

Calculation in the Background I

Let us start by performing the probe calculation in the background of
eq.(2.1). The dual QFT is $(2+1)$-dimensional, as the coordinate $\varphi$ is
not compact. By expanding the Born-Infeld action, we find a Maxwell term, with
coupling (the details of this calculation are spelled out below)

$\frac{1}{g_{YM,A}^{2}}=\frac{16\pi^{4}}{e_{A}^{3}}T_{5}\sqrt{e_{A}^{2}+2e_{B}^{2}}.$
(3.5)

Similarly, the Wess-Zumino term gives,

$S_{D5,WZ}=-2\pi^{2}T_{5}\int C_{2}\wedge F_{2}\wedge
F_{2}=-\int_{S^{3}_{A}}F_{3}\int_{t,x,\varphi}A_{1}\wedge
F_{2}=-N_{B}\int_{t,x,\varphi}A_{1}\wedge F_{2}.$ (3.6)

We have performed an integration by parts, used the quantisation condition in
eq.(2.9) and set $T_{5}=\frac{1}{(2\pi)^{2}}$ (in our units). There is a
similarly symmetric calculation for a D5 probe along
$[t,x,\varphi,\theta_{B},\phi_{B},\psi_{B}]$. In agreement with the field
theory picture discussed above, we find two gauge groups with Yang-Mills
Chern-Simons dynamics, $SU(N_{A})_{N_{B}}\times SU(N_{B})_{N_{A}}$, with fixed
gauge couplings. This is exactly in agreement with the field theory
expectations [13], that we summarised in the previous section.

Calculation in the Background II

Let us now study the case for which the $(2+1)$ QFT has been compactified
along the $\varphi$-direction and we are dealing with a $(1+1)$ dimensional
QFT. We work with the background in eqs.(2.3)-(2.5), and follow the
calculation above, by first writing the induced metric on the probe D5 brane,

$ds_{\hat{D}_{5}}^{2}=r\left\\{-dt^{2}+dx^{2}+\left[f_{s}\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}\right]d\varphi^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)d\varphi\right)^{2}\right]\right\\}\
.$ (3.7)

From here we calculate

$e^{-\Phi}\sqrt{-\det
g_{MN}^{\left(\hat{D}_{5}\right)}}=r^{2}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}\sin{\theta_{A}}\sqrt{f\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}}.$

The effective action for the brane in eq. (3.3) reads,

$\displaystyle S_{\hat{D}_{5},BI}$ $\displaystyle=$ $\displaystyle-
T_{D_{5}}\int dtdxd\varphi
d{\theta_{A}}d{\phi_{A}}d{\psi_{A}}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}r^{2}\sin{\theta_{A}}\sqrt{f\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}}\left(1-\frac{(2\pi)^{2}}{4}F^{BC}F_{BC}\right)\
,$ (3.8) $\displaystyle=$ $\displaystyle
T_{D_{5}}L_{\varphi}\left(4\pi\right)^{2}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}r^{2}\sqrt{f\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}}\int
dtdx\left(1-\frac{(2\pi)^{2}}{4}F^{BC}F_{BC}\right)\ .$

We turn on $F_{tx}$ so that

$F^{BC}F_{BC}=2F_{tx}F_{tx}g^{tt}g^{xx}=2\frac{1}{r^{2}}F_{tx}F_{tx}\eta^{tt}\eta^{xx}=\frac{1}{r^{2}}F_{\mu\nu}^{2}\
.$ (3.9)

From here we identify the Yang-Mills coupling for this probe D5,

$\frac{1}{g_{YM,A}^{2}}=\left(2\pi\right)^{4}T_{\hat{D}_{5}}L_{\varphi}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}\sqrt{f\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}}.$
(3.10)

If the D5 brane probes the SUSY preserving background, we impose
$Q_{B}=\frac{e_{B}}{e_{A}}Q_{A}$ and $m=0$ on the parameters appearing in
eqs.(2.5)-(2.6). This implies $r_{\pm}^{2}=\pm\frac{2Q_{A}}{e_{A}}$ and
$L_{\varphi}=4\pi/(e_{A}^{2}+e_{B}^{2})$. Together with eq.(2.4) and the
quantisation condition (2.9), the gauge coupling reduces to333For large values
of the radial coordinate $r$ and decompactifying $\varphi$, this result
reduces to that in eq.(3.5).

$\frac{1}{\hat{g}_{YM,A}^{2}}=\frac{16\pi^{4}}{e_{A}^{3}}T_{\hat{D}_{5}}L_{\varphi}\sqrt{\left(1-\frac{2Q_{A}}{e_{A}r^{2}}\right)\left(\frac{2e_{A}Q_{A}}{r^{2}}+e_{A}^{2}+2e_{B}^{2}\right)}\
,$ (3.11)

with limiting values

$\frac{1}{\hat{g}_{YM,A}^{2}}=\left\\{\begin{array}[]{cc}\frac{16\pi^{4}}{e_{A}^{3}}T_{\hat{D}_{5}}L_{\varphi}\sqrt{e_{A}^{2}+2e_{B}^{2}}&,\quad
r\rightarrow\infty\\\ 0&,\quad r\rightarrow r_{+}\end{array}\right.\ .$ (3.12)

In other words, the gauge coupling grows very large at low energies and
asymptotes to a constant value for high energies. As discussed, at very high
energies the field theory is best described in terms of a Little String Theory
(LST). This calculation above refers to the gauge coupling $g_{YM,A}$. Other
interactions in the QFT may become large at high energies, in such a way that
the field theory is strongly coupled in the UV. This is in agreement with the
background in eq.(2.3) being weakly curved for all values of the radial
coordinate $[r_{+},\infty)$.

Had we studied a D5 probe extended along
$[t,x,\varphi,\theta_{B},\phi_{B},\psi_{B}]$, with an electric field $F_{tx}$
switched on the brane, the result would be,

$\frac{1}{g_{YM,B}^{2}}=\left(2\pi\right)^{4}T_{D_{5}}L_{\varphi}\left(\frac{2}{e_{B}^{2}}\right)^{3/2}\sqrt{f\left(r\right)+2Q_{A}^{2}\zeta\left(r\right)^{2}}.$
(3.13)

###  3.4 Theta angle

From the viewpoint of the $(2+1)$ dimensional QFT, represented by Background I
in eq.(2.1), we can consider dimensionally reducing the Chern-Simons term
obtained in eq.(3.6). We then obtain a theta-term proportional to
$N_{B}\oint_{\varphi}A_{\varphi}$ for the QFT on the A-stack.

For the $(1+1)$ viewpoint, additional probes calculate the $\Theta$-angle of
each gauge group. Let us use a D3 probe, extended along
$[t,x,\theta_{A},\phi_{A}]$ with an electric field $F_{tx}$ switched on. We
study the Wess-Zumino term following eq.(3.4) and using the two-form potential
pulled-back on this D3 probe

$C_{2}|_{D_{3}}=-\frac{2}{e_{A}^{2}}{\psi_{A}}\sin{\theta_{A}}d{\theta_{A}}\wedge
d{\phi_{A}}\ .$ (3.14)

Replacing this $C_{2}$ in eq.(3.4) and using that $F_{2}=F_{tx}dt\wedge dx$ we
find the Wess-Zumino term for this probe is (note that $C_{4}=0$ in the
background),

$S_{WZ,D3}=-T_{{3}}2\pi\int C_{2}\wedge
F_{2}=\frac{T_{{3}}16\pi^{2}}{e_{A}^{2}}{\psi_{A}}\int F_{tx}dt\wedge dx\ \ .$
(3.15)

The $\Theta$-angle associated with the gauge group should be identified
according to ,

$\displaystyle S_{WZ,D3}=\frac{\Theta_{A}}{4\pi^{2}}\int
dtdxF_{tx},~{}~{}\longrightarrow~{}~{}\Theta_{A}=\frac{T_{D_{3}}64\pi^{4}}{e_{A}^{2}}{\psi_{A}}=\frac{\psi_{A}N_{B}}{2}\
.$ (3.16)

We have used $16\pi^{4}T_{D3}=1$ (in our units) and the quantisation condition
in eq.(2.9). Notice that the periodic identification
$\Theta_{A}\sim\Theta_{A}+2k\pi$ implies that the angle $\psi_{A}$ gets
quantised to the values

$\Delta\psi_{A}=\frac{4k\pi}{N_{B}},~{}~{}\text{with
}~{}~{}k=0,1,2....,N_{B}-1.$ (3.17)

For $k=N_{B}$ we have $\Delta\psi_{A}=4\pi$, covering the full circle.

Had we considered the D3 probe extended along $[t,x,\theta_{B},\phi_{B}]$ with
all other coordinates fixed, we would have found (the calculation is
identical), $\Theta_{B}=\frac{\psi_{B}N_{A}}{2}.$

Another way of understanding the R-symmetry breaking would be to consider
euclidean D1 branes wrapping $[\theta_{A},\phi_{A}]$. The Wess-Zumino action
contributes to the partition function via

$Z_{D1}\sim e^{\frac{i}{2\pi}\int C_{2}}.$ (3.18)

This contribution should not depend on a (large) gauge transformation
parameter $\epsilon_{A}$, that appears as we change
$\psi_{A}\to\psi_{A}+\epsilon_{A}$. We enforce

$\left(\frac{i}{2\pi}\right)\left(\frac{-2\epsilon_{A}}{e_{A}^{2}}\right)\int\sin\theta_{A}d\theta_{A}d\phi_{A}=2ik\pi,~{}~{}\text{leads
to}~{}~{}\epsilon_{A}=\frac{4k\pi}{N_{B}}.$ (3.19)

These results indicate that the background’s continuous isometries
transforming $\psi_{A,B}\to\psi_{A,B}+\epsilon_{A,B}$, with ($\epsilon_{A,B}$
being constants), are actually broken. In fact, the allowed changes are
$\Delta\psi_{A,B}=\frac{4k\pi}{N_{B,A}}$, with $k=0,1,2....,N_{B,A}-1$, which
should be interpreted as the breaking of the two field theory global
symmetries $U(1)_{A,B}$ into discrete subgroups.

The argument used to derive eqs.(3.17)-(3.19) is not airtight. It uses a two
manifold that has a boundary and at the same time a gauge choice is made for
the potential $C_{2}$ in eq.(3.14). It would be better to have an argument for
R-symmetry breaking that is explicitly gauge invariant. In the coming section
we present a different holographic perspective on the $U(1)_{A,B}$ breaking
with this property.

###  3.5 $U(1)_{A,B}$ symmetry breaking pattern

Some supersymmetric field theories exhibit a classical $U(1)$ R-symmetry that
is quantum mechanically broken to a discrete subgroup. The symmetry breaking
can be understood diagrammatically or in terms of instantons. The supergravity
dual to the given field theory should encode this, but the mechanism should
not involve instantons (as that are very suppressed in supergravity). The fact
that the Ramond potentials are not gauge invariant under the $U(1)$ R-symmetry
is key. The breaking of the global R-symmetry in the field theory manifest as
spontaneous breaking in supergravity. The vector field in the bulk, dual of
the R-symmetry current acquires a mass. We find this below for our background
of eq.(2.3). The argument that follows is gauge invariant at all steps. All
along this section, we set $m=0$ and focus only in the BPS case.

The $U(1)$ symmetry of the metric is perturbed, the Lagrangian for this
fluctuation is described by the usual $F_{\mu\nu}^{2}$-term. For the
perturbation to be consistent, the Ramond fields must be also perturbed, this
contributes to the mass term for the fluctuation. The massive gauge field is
understood as symmetry breaking in supergravity.

In the previous section, we hinted at a breaking of the isometries represented
by the Killing vectors $\partial_{\psi_{A,B}}$. To better understand the
breaking of the field theory global $U(1)$ symmetries associated with the
translations in $\psi_{A,B}$ we proceed as explained in [21], [22]. We give
full details in Appendix E. In the holographic background, we gauge the
isometry by replacing, both in the metric and in the Ramond potential of
eq.(2.3)

$d\psi_{A,B}\rightarrow
d\psi_{A,B}+A_{A,B},\quad\psi_{A,B}\rightarrow\psi_{A,B}+\epsilon_{A,B}.$
(3.20)

Such that a change $\psi_{j}\rightarrow\psi_{j}+\epsilon_{j}$ is compensated
by $A_{j}\rightarrow A_{j}+d\epsilon_{j}$. We then study the Lagrangian for
the gauge fields $A_{A,B}$, by replacing these changes in the string frame
Lagrangian. The metric perturbation changes the Ricci scalar to

$e^{-2\Phi}R\rightarrow
e^{-2\Phi}\left(R-\frac{1}{4}\frac{2r}{e^{2}_{A}}F^{2}_{A}-\frac{1}{4}\frac{2r}{e^{2}_{B}}F^{2}_{B}\right).$
(3.21)

The kinetic term for $F_{3}$ changes as,

$\displaystyle\frac{1}{12}F^{\mu\nu\lambda}F_{\mu\nu\lambda}\rightarrow$
$\displaystyle\frac{1}{12}F^{\mu\nu\lambda}F_{\mu\nu\lambda}+\frac{1}{2}\frac{4Q^{2}_{A}+e^{2}_{A}r^{4}}{e^{2}_{A}r^{6}}\left(A_{A}-d\epsilon_{A}\right)^{2}+\frac{1}{2}\frac{4Q^{2}_{B}+e^{2}_{B}r^{4}}{e^{2}_{B}r^{6}}\left(A_{B}-d\epsilon_{B}\right)^{2}$
(3.22)
$\displaystyle+\frac{4Q_{A}Q_{B}}{e_{A}e_{B}r^{6}}\left(A_{A}-d\epsilon_{A}\right)\cdot\left(A_{B}-d\epsilon_{B}\right).$

These perturbations around an isometry direction are usually known to be
consistent. It is interesting to note that the kinetic term for the gauge
field $A_{(A,B)}$ comes only from the Ricci scalar. The information of
$\epsilon_{A,B}$ on the other hand, comes only from $F^{2}_{3}$. Defining

$W_{(A,B)}=A_{A,B}-d\epsilon_{A,B},$ (3.23)

and imposing the BPS condition $e_{A}Q_{B}=e_{B}Q_{A}$, the Lagrangian for the
perturbation reads,

$\mathcal{L}=-\frac{1}{4}\frac{2}{re^{2}_{A}}F^{2}_{A}-\frac{1}{4}\frac{2}{re^{2}_{B}}F^{2}_{B}+\frac{1}{2r^{2}}(W^{(A)}_{\mu}W^{(A)\mu}+W^{(B)}_{\mu}W^{(B)\mu})+\frac{1}{2}\frac{4Q^{2}_{A}}{e^{2}_{A}r^{6}}(W^{(A)}_{\mu}+W^{(B)}_{\mu})^{2}.$
(3.24)

We interpret this result as follows. There are two $U(1)$ global symmetries in
the QFT, holographically they are represented by the invariance of the metric
under changes in $\psi_{A}$ and $\psi_{B}$. These global symmetries are broken
to discrete groups $\mathbb{Z}_{N_{B}}$ and $\mathbb{Z}_{N_{A}}$, as indicated
by eq.(3.17). This breaking is an effect of the lack of invariance of the
gauge potential $C_{2}$. The breaking of the global symmetries is addressed in
this section without appeal to the Ramond potentials, by observing that
gauging the metric isometries $\partial_{\psi_{A,B}}$ leads to a breaking of
the gauge symmetry, by a mass term. These mass terms are dependent on the
radial coordinate.

Contrary to what happens for the duals to ${\cal N}=1$ SYM, the metric in
eq.(2.3) does not break these discrete isometries to $\mathbb{Z}_{2}$. In
other words, there is not a radial-regime in the metric that explicitly breaks
the isometry $\partial_{\psi_{A,B}}$. We interpret this result as the
(presumably anomalous) breaking of the two $U(1)_{A}\times
U(1)_{B}\rightarrow\mathbb{Z}_{N_{B}}\times\mathbb{Z}_{N_{A}}$ in the QFT not
being followed by a further spontaneous breaking. One might argue that VEVs
are not allowed in a two dimensional QFT, hence no further breaking can take
place by the formation of a condensate. This argument is not completely
rigorous, as our QFT is two dimensional in the far IR, but get UV completed
around the confining scale to a higher dimensional QFT.

We study now a different observable that gives an approximate idea of the
number of degrees of freedom as a function of the energy (a density of
states).

###  3.6 Holographic central charge

Consider a generic holographic background dual to a QFT in $(d+1)$ spacetime
dimensions, with metric and dilaton given by

$ds^{2}=a(r,y^{i})\left[dx^{2}_{1,d}+b(r)dr^{2}\right]+g_{ij}(r,y^{i})dy^{i}dy^{j},\quad\Phi(r,y^{i}).$
(3.25)

Following [23], we define quantities $V_{int},{H}$ according to

$V_{int}=\int
dy^{i}\sqrt{e^{-4\Phi}a(r,y^{i})^{d}\det[g_{ij}]},\quad{H}=V_{int}^{2}.$

From these we define the holographic central charge (or free energy),

$c_{hol}=d^{d}\frac{b(r)^{\frac{d}{2}}H^{\frac{2d+1}{2}}}{G^{(10)}_{N}(H^{\prime})^{d}},$
(3.26)

where $G^{(10)}_{N}=8\pi^{6}$ is (in our conventions), the ten-dimensional
Newton constant.

The holographic central charge of eq.(3.26) makes perfect sense for
backgrounds with an AdS-factor. In those cases the quantity in eq.(3.26) is a
number depending on the parameters of the background and it was successfully
matched with the free energy of the CFT. In contrast for our case, without an
AdS-factor we use eq.(3.26) to give an indication of the number of degrees of
freedom of the QFT.

Let us first compute the quantities in eqs.(3.25)-(3.26) for the solution in
eq.(2.1), dual to a $(2+1)$ dimensional QFT. We find,

$\displaystyle
d=2,~{}~{}a(r,y^{i})=r,~{}~{}b(r)=\frac{8}{(e_{A}^{2}+e_{B}^{2})r^{2}},~{}~{}V_{int}={\cal
N}r^{3},~{}~{}{\cal N}=\frac{8(4\pi)^{4}}{e_{A}^{3}e_{B}^{3}},$ $\displaystyle
H={\cal N}^{2}r^{6},~{}~{}~{}~{}c_{hol}=\frac{8{\cal
N}}{9G_{N}(e_{A}^{2}+e_{B}^{2})}r^{3}.$ (3.27)

This quantity diverges at large energies, hinting at a UV completion in terms
of a system in higher dimensions (a LST). Also, it vanishes for $r=0$,
indicating a gapped system. Notice nevertheless that the calculation should
not be trusted close to $r=0$, as the background (2.1) is singular there.

Let us now calculate for the background in eq.(2.3), as a dual to a
($1+1$)-dim QFT. We find

$\displaystyle d=1,~{}~{}a(r,y^{i})=r,~{}~{}~{}b(r)=\frac{4}{r^{2}f_{s}(r)},$
$\displaystyle V_{int}={\cal N}r^{2}\sqrt{f_{s}(r)},\;\;\;\;\hat{H}={\cal
N}^{2}r^{4}f_{s}(r),~{}~{}~{}{\cal
N}=(4\pi)^{4}\frac{8}{e_{A}^{3}e_{B}^{3}}L_{\varphi},$ $\displaystyle
c_{hol}=\frac{{\cal
N}}{2G_{N}}\frac{f_{s}(r)r^{2}}{\left(f_{s}(r)+\frac{r}{4}f^{\prime}_{s}(r)\right)}.$
(3.28)

At high energies the number of degrees of freedom grows unbounded (as
$r^{2}$), signalling the UV completion in terms of a decompatified QFT in
higher dimensions. At very low energies the number of degrees of freedom
vanish, as $f_{s}(r_{+})=0$. In this case, the calculation is trustable, hence
the gapped character of the system is clear.

A related calculation can be done that encodes the fact that we can think our
field theory as a three dimensional QFT with anisotropies (or a QFT with a
flow across dimensions). We follow the treatment described in section 8.2 of
the paper [24], see also [25]. Using the notation of [24] we have,

$\displaystyle
d=2,~{}\alpha_{0}=\alpha_{1}=r,~{}~{}\alpha_{2}=rf_{s}(r),~{}~{}\beta=\frac{4}{r^{2}f_{s}^{3/2}},~{}~{}H={\cal
N}^{2}r^{4}f_{s}(r),~{}{\cal
N}=\left(\frac{2}{e_{A}e_{B}}\right)^{3}(4\pi)^{4},$ $\displaystyle
c_{flow}=\frac{{\cal
N}}{G_{N}}\frac{r^{2}f_{s}}{(f_{s}(r)+\frac{r}{4}f_{s}(r)^{\prime})^{2}}.$
(3.29)

The result in eq.(3.29) has a similar interpretation. At low energies we have
no degrees of freedom, at high energies an unbounded growth in the degrees of
freedom indicates the UV completion. Note that the growth at high energies for
the flow-central-charge–eq.(3.29), is slower than it is for the case in which
we consider the QFT to be two dimensional, see eq. (3.28). This same feature
occurs when considering flows between conformal points in different
dimensions.

These results suggest that our QFT generates a mass gap at low energies. The
behaviour of the gauge couplings at low energies, see eq.(3.12) also suggest
that the QFT is confining. To ascertain the confining behaviour we calculate
Maldacena-Wilson and ’t Hooft loops, that provide order parameters for
confinement. We also investigate the Entanglement Entropy (EE), which gives
information about the interplay between a confining IR and the non-local UV
dynamics of the QFT.

##  4 Maldacena-Wilson, ’t Hooft loops and Entanglement Entropy

In this section we calculate different observables to learn more about the
proposed field theory. We start calculating the Maldacena-Wilson loops [26],
in order to test the above proposal that at low energies the QFT presents a
mass gap and confines. We start with a summary of the formalism to compute
Maldacena-Wilson loops. This same formalism is then adapted for the study of
’t Hooft loops and Entanglement Entropy.

###  4.1 General comments on Maldacena-Wilson loops and similar probes

We start summarising general results pertaining holographic Wilson loops. We
follow the treatment of [27], [28]. This generic treatment is also useful for
the study of other probes that reduce to an ’effective string’ in the
background. Hence it will apply to ’t Hooft loops, Entanglement Entropy, as we
discuss below.

Consider a generic holographic background of the form

$ds^{2}=-g_{tt}dt^{2}+g_{xx}d\vec{x}^{2}+g_{rr}dr^{2}+g_{ij}d\theta^{i}d\theta^{j}\
.$ (4.1)

We assume that $g_{tt}$, $g_{xx}$, $g_{rr}$ depend only on the radial
coordinate $r$. As usual, we propose a string embedding (parametrised in terms
of $(\tau,\sigma)$ the worldsheet coordinates), which leads to a Nambu-Goto
action for the F1-string of the form,

$\displaystyle t=\tau,~{}~{}~{}x=x(\sigma),~{}~{}~{}r=r(\sigma).$
$\displaystyle S_{NG}=T_{F1}\int d\tau
d\sigma\sqrt{g_{tt}(r)g_{xx}(r)x^{\prime 2}+g_{tt}(r)g_{rr}(r)r^{\prime 2}}.$
(4.2)

From this action, the equations for the string moving in the generic
background reduce to (see [28] for a detailed derivation)

$\frac{dr}{d\sigma}=\pm\frac{dx}{d\sigma}V_{eff}\left(r\right)\,.$ (4.3)

We defined the effective potential

$V_{eff}\left(r\right)=\frac{F\left(r\right)}{CG\left(r\right)}\sqrt{F^{2}\left(r\right)-C^{2}}\
,~{}~{}F^{2}\left(r\right)=g_{tt}g_{xx},~{}~{}G^{2}\left(r\right)=g_{tt}g_{rr}.$
(4.4)

The constant $C=\frac{F^{2}x^{\prime}}{\sqrt{F^{2}x^{\prime 2}+G^{2}r^{\prime
2}}}$ is obtained from one of the equations of motion. In the simpler case in
which we take $x(\sigma)=\sigma$ we find eq.(4.3) from the conserved
Hamiltonian. In that case $C=F(r_{0})$, being $r_{0}$ the turning point of the
string satisfying $r^{\prime}(\sigma)=0$ (these are called U-shaped
embeddings). We set $C=F(r_{0})$ in what follows. We enumerate below a set of
properties of the U-shaped embeddings.

  1. 1.

This formalism applies to an open string whose end points are at $r\to\infty$,
where we add a D-brane. Dirichlet boundary conditions for the string at
$r\rightarrow\infty$ require that $V_{eff}|_{r\rightarrow\infty}\sim\infty$.

  2. 2.

We compute the separation between the two ends of the string on the D-brane,
which can be thought as the separation between a quark-antiquark pair. The
energy of the pair of quarks calculated from the Nambu-Goto action needs
regularisation, implemented by subtracting the mass of two non-dynamical
strings extended along the whole range of the radial coordinate
$[r_{+},\infty)$.

The separation and energy are given as functions of $r_{0}$ (the distance from
the origin of the radial coordinate, $r_{+}$, to the position of the turning
point of the string). The expressions for these quantities are,

$\displaystyle L_{QQ}\left(r_{0}\right)$
$\displaystyle=2\int_{r_{0}}^{+\infty}\frac{dz}{V_{eff}(z)}\,,$ (4.5)
$\displaystyle E_{QQ}\left(r_{0}\right)$
$\displaystyle=F\left(r_{0}\right)L_{QQ}\left(r_{0}\right)+2\int_{r_{0}}^{+\infty}dz\frac{G\left(z\right)}{F\left(z\right)}\sqrt{F\left(z\right)^{2}-F\left(r_{0}\right)^{2}}-2\int_{r_{+}}^{+\infty}dz\
G\left(z\right)\,.$ (4.6)

  3. 3.

To obtain a finite contribution coming from the upper limit of the $Q\bar{Q}$
pair separation in (4.5), a further restriction on the behaviour of the
effective potential at infinity is needed. See [28] for a derivation,

$V_{eff}|_{r\rightarrow+\infty}\sim r^{\beta}\,,\text{\quad with }\beta>1\,.$
(4.7)

  4. 4.

When expanded close to the end of the space, which is at $r=r_{+}$ in the case
of the background in eq.(2.3), we find $V_{eff}\sim(r-r_{+})^{\gamma}$. If
$1\leq\gamma$, the separation between the pair becomes infinite–see [28].
Otherwise (if $\gamma<1$) we have screening behaviour.

  5. 5.

There is an analytic relation between $E_{QQ}$ and $L_{QQ}$ which is

$\frac{dE_{QQ}}{dr_{0}}=F(r_{0})\frac{dL_{QQ}}{dr_{0}}~{}~{}\longrightarrow~{}~{}\frac{dE_{QQ}}{dL_{QQ}}=F(r_{0}).$
(4.8)

We have inverted the relation (4.5) as $r_{0}=r_{0}(L_{QQ})$.

  6. 6.

For generic backgrounds, the evaluation of the integral in eq.(4.5) need not
be simple nor have an expression in terms of elementary functions.
Nevertheless, the quantity

$\hat{L}_{QQ}(r_{0})=\pi\frac{G}{F^{\prime}}\Big{|}_{r_{0}},$ (4.9)

provides a reasonable approximation to eq.(4.5). We check this approximate
expression for different observables studied below.

  7. 7.

Following [29] we define

$Z(r_{0})=\frac{d}{dr_{0}}\hat{L}_{QQ}(r_{0})=\pi\frac{d}{dr_{0}}\left(\frac{G(r_{0})}{F^{\prime}(r_{0})}\right).$
(4.10)

The stability of the U-shaped string embedding in eq.(4.2) is guaranteed if
$Z(r_{0})<0$ [29]. This is valid for any observable that can be reduced to an
effective string action of the form (4.2)444 It was shown in [29],[30] that
embeddings for which $Z(r_{0})\geq 0$ do not satisfy two physically well
motivated criteria [31]: The force between the quark and the antiquark is
always attractive and positive $\frac{dE_{QQ}}{dL_{QQ}}>0$. It is also a non-
increasing function of the separation $\frac{d^{2}E_{QQ}}{dL_{QQ}^{2}}\leq 0$.
The proposal in [29] is that the two criteria above are equivalent to the
stability of the U-shaped embedding or conversely $Z(r_{0})<0$.

  8. 8.

In calculations like those in eqs.(4.5)-(4.6) we can introduce a cutoff
$r_{UV}$ to regulate divergences coming from the upper limit in the integrals.
We define a quantity analog to (4.5),

$L_{QQ}(r_{0},r_{UV})=2\int_{r_{0}}^{r_{UV}}\frac{dz}{V_{eff}(z)}.$ (4.11)

Following [29] we calculate

$L_{a}=\lim_{r_{0}\to\infty}\lim_{r_{UV}\to\infty}L_{QQ}(r_{0},r_{UV})~{}~{}\text{and}~{}~{}L_{b}=\lim_{r_{UV}\to\infty}\lim_{r_{0}\to
r_{UV}}L_{QQ}(r_{0},r_{UV}).$ (4.12)

Whilst $L_{b}=0$ by definition, it is sometimes the case that $L_{a}$ is
nonzero. If in this case the U-shaped configuration is unstable
($Z(r_{0})>0$), there exist ’short configurations’ (that appear very close to
the cutoff). These short configurations are energetically favoured and induce
a phase transition in the observable calculated [29], [32].

The existence of these short configurations that appear when introducing the
UV-cutoff is not an artefact of the cutoff. They indicate that the string
embedding we proposed in eq.(4.2) is not capturing the dynamically favoured
configuration. In the study of the ’t Hooft loop and the Entanglement Entropy
we encounter these short configurations, that cure the problem of the
instability of the embedding and introduce a phase transition in the
observable. We take the instability of the configuration (cured by the
introduction of a cutoff) as an indication that the dynamics of the LST is
driving the observable.
Let us now apply this general treatment to our background in eq.(2.3).

###  4.2 Maldacena-Wilson loops in our background

Let us apply the expressions in eqs.(4.1)-(4.9) to our background in eq.(2.3).
The relevant functions are

$F(r)=r,~{}~{}~{}~{}G(r)=\frac{2}{\sqrt{f_{s}(r)}},~{}~{}~{}~{}~{}V_{eff}(r)=\sqrt{\frac{e_{A}^{2}+e_{B}^{2}}{8}}\left(\frac{1}{r_{0}~{}r}\right)\sqrt{(r^{2}-r_{0}^{2})(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}.$
(4.13)

The condition in eq.(4.7) to have Dirichlet boundary conditions for the string
at $r\to\infty$, is satisfied. When expanded close to $r_{0}=r_{+}$,
$V_{eff}\sim(r-r_{+})$, indicating that $L_{QQ}$ diverges when $r_{0}$
approaches the end of the space $r_{+}$. This points to confining behaviour.

Another quick way of determining a confining behaviour is to check the
function $F(r_{+})$, that intuitively represents the tension of a QCD-string
at low energies in the QFT (close to the end of the space of the geometry). In
this case we find $F(r_{+})=r_{+}$, pointing to confining behaviour. In fact,
a finite QCD-string tension leads to an energy growing with the separation of
the quark pair. Note also that the approximate formula for the separation of
the quark pair eq.(4.9), gives for the functions in eq.(4.13)

$\hat{L}_{QQ}(r_{0})=\frac{2\pi}{\sqrt{f_{s}(r_{0})}}.$ (4.14)

For $r_{0}\to\infty$ this gives
$\hat{L}_{QQ}=\pi\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}$, which refers to a
characteristic length of the UV completion (the Little String Theory scale).
On the other hand, for $r_{0}\sim r_{+}$, the approximate length diverges and
$\frac{1}{\sqrt{r_{0}-r_{+}}}$. This indicates that the quark-anti-quark pair
can be infinitely separated. According to eq.(4.6), this gives an energy that
scales linearly with the separation, another signal of a confining behaviour.

The quantity $Z(r_{0})$ in eq.(4.10) reads,

$Z(r_{0})=\frac{2\sqrt{2}\pi
r_{0}}{\sqrt{e_{A}^{2}+e_{B}^{2}}}\frac{\left(-r_{0}^{2}(r_{+}^{2}+r_{-}^{2})+2r_{+}^{2}r_{-}^{2}\right)}{\left[(r_{0}^{2}-r_{+}^{2})(r_{0}^{2}-r_{-}^{2})\right]^{3/2}}.$
(4.15)

By inspection, one find that $Z(r_{0})<0$ in all the range. Hence the U-shaped
embeddings are stable and no phase transition is expected for the Maldacena-
Wilson loops.

The rigorous way of determining the low energy behaviour of the QFT (either
confining or screening) is to analyse the expressions for the distance and the
energy of the quarks pair. The distance between the quark-antiquark pair and
its energy are written from eqs.(4.5)-(4.6)

$\displaystyle
L_{QQ}\left(r_{0}\right)=\sqrt{\frac{32}{e_{A}^{2}+e_{B}^{2}}}r_{0}\int_{r_{0}}^{\infty}z\sqrt{\frac{1}{\left(z^{2}-r_{+}^{2}\right)\left(z^{2}-r_{-}^{2}\right)\left(z^{2}-r_{0}^{2}\right)}}dz\
,$ (4.16) $\displaystyle
E_{QQ}\left(r_{0}\right)=r_{0}L_{QQ}\left(r_{0}\right)+\sqrt{\frac{32}{e_{A}^{2}+e_{B}^{2}}}\Big{[}\int_{r_{0}}^{\infty}dz\sqrt{\frac{z^{2}\left(z^{2}-r_{0}^{2}\right)}{\left(z^{2}-r_{+}^{2}\right)\left(z^{2}-r_{-}^{2}\right)}}-$
$\displaystyle\int_{r_{+}}^{\infty}dz\frac{z^{2}}{\sqrt{\left(z^{2}-r_{+}^{2}\right)\left(z^{2}-r_{-}^{2}\right)}}\Big{]}\
.$ (4.17)

The integrals can be performed analytically and expressed in terms of in terms
of Elliptic integrals of the first kind. It is interesting to study this in
the BPS limit, when $m=0$ and $r_{+}^{2}=-r_{-}^{2}$. See Appendix F for a
detailed study. Let us quote the explicit results for the separation and
energy of the quark pair in eqs.(4.16)-(4.17). Defining the elliptic
integrals,

$\displaystyle{\bf
K}(x)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-x\sin^{2}\theta}},\;\;\;\;{\bf
E}(x)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-x\sin^{2}\theta}d\theta,$ (4.18)

we can write the explicit expressions for the separation and energy of the
quark-antiquark pair, which read

$L_{QQ}\left(r_{0}\right)=2\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\frac{r_{0}/r_{+}}{\sqrt{r_{0}^{2}/r_{+}^{2}-1}}\mathbf{K}\left(\frac{1-r_{-}^{2}/r_{+}^{2}}{1-r_{0}^{2}/r_{+}^{2}}\right)\
,$ (4.19) $\displaystyle E_{QQ}\left(r_{0}\right)$ $\displaystyle=$
$\displaystyle
2r_{+}\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\left[\frac{r_{0}^{2}/r_{+}^{2}}{\sqrt{r_{0}^{2}/r_{+}^{2}-1}}\mathbf{K}\left(\frac{1-r_{-}^{2}/r_{+}^{2}}{1-r_{0}^{2}/r_{+}^{2}}\right)\right.$
(4.20)
$\displaystyle\qquad\qquad\left.-\mathbf{E}\left(\frac{1-r_{-}^{2}/r_{+}^{2}}{1-r_{0}^{2}/r_{+}^{2}}\right)\sqrt{r_{0}^{2}/r_{+}^{2}-1}+C\left(r_{-}/r_{+}\right)\right],$

where

$\displaystyle
C\left(r_{-}/r_{+}\right)=\mathbf{E}\left(r_{-}^{2}/r_{+}^{2}\right)+\lambda_{-}\mathbf{K}\left(1-r_{-}^{2}/r_{+}^{2}\right)+i\lambda_{-}\mathbf{K}\left(r_{-}^{2}/r_{+}^{2}\right)-\left(1-r_{-}^{2}/r_{+}^{2}\right)\mathbf{K}\left(r_{-}^{2}/r_{+}^{2}\right)\
.$ $\displaystyle\lambda_{-}^{2}=-\frac{r_{-}^{2}}{r_{+}^{2}}.$

See Appendix F for a careful derivation of these expressions. We plot these
results in Figure 2. The various panels of Figure 2 show some conventional and
other less conventional behaviours. First, note that the expression for
$\hat{L}_{QQ}$ in eq.(4.14) very well approximates the exact expression in
eq.(4.19). As is usual, the concavity of the curve $E_{QQ}(L_{QQ})$ is
’downwards’, indicating that the Nambu-Goto string configuration of eq.(4.13)
is stable, as confirmed by the $Z(r_{0})$ in eq.(4.15). Note also that for
large separations between the quark pair $L_{QQ}$, the energy grows linearly
(signalling confinement). What is less conventional is that there is a minimal
separation, given by the Little String Theory scale. This indicates that the
far UV of the QFT dynamics is not field theoretical (but has the dynamics of
the LST). The plot of the strings profiles confirms this. Indeed, strings that
barely explore the bulk (with $\frac{r_{0}}{r_{+}}$ large) show a minimal
fixed separation between the quark pair. On the other hand, the strings that
explore deeper into the bulk display a bigger quark separation and carry
higher energy.

Let us now focus on a second interesting observable, the ’t Hooft loop. In the
next section we propose a string-like object (for the gauge theory observer)
with magnetic charge. This characteristically is represented by a Dp brane
that wraps a $(p-1)$ cycle in the internal space. Once the Born-Infeld action
for this Dp brane is written and integrals over the internal space are
performed, we arrive at an action for the ’effective string’. This action is
studied with the same formalism as that used for Wilson loops, described in
Section 4.1.

Figure 2: Upper left: Plot comparing the exact expression for the quark
separation $L_{QQ}$ in eq.(4.19) with the approximate one $\hat{L}_{QQ}$ in
eq.(4.14). The minimal separation as $r_{0}$ grows large hints at a LST
behaviour. Upper right: Plot of $E_{QQ}(r_{0})$ with $r_{0}$ in units of
$r_{+}$. The plot is made in the BPS limit with $m=0$. Bottom left: Parametric
plot of $E_{QQ}(L_{QQ})$ in the BPS bound $m=0\,,\,r_{+}=e_{B}=e_{A}=1$.
Bottom right: the profiles of different strings as they explore the bulk. The
longer the separation $L_{QQ}$, the more the string approaches
$\frac{r_{0}}{r_{+}}\sim 1$. This is usual of the backgrounds dual to a
confining QFT behaviour.

###  4.3 ’t Hooft loops

The ’t Hooft loop can be calculated by proposing an object with magnetic
charge that effectively appears like a string for the (1+1) dimensional QFT.
The ends of this magnetic string appear as a pair of monopoles of oposite
charge. To study this object, we adapt the formulas summarised in
eqs.(4.1)-(4.10) for the effective magnetic string.

We propose to calculate the ’t Hooft loop by studying the effective magnetic
string obtained when extending a D5 brane along the directions
$[t,x,\varphi,\theta_{A},\phi_{A},\psi_{A}]$, with $r(x)$. There is an analog
magnetic string for the second gauge group, for the D5 in the configuration
$[t,x,\varphi,\theta_{B},\phi_{B},\psi_{B}]$, with $r(x)$. We do not discuss
the latter object, as its result is analog to the one obtained below. Note
that this object becomes string-like when we consider the size of the
$S^{1}_{\varphi}$ to be small enough.

The induced metric for the above D5 is,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
r\left\\{-dt^{2}+\left(1+\frac{4}{r^{2}f_{s}\left(r\right)}r^{\prime
2}\right)dx^{2}+\left(f_{s}\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}\right)d\varphi^{2}\right.$
(4.21)
$\displaystyle\left.+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)d\varphi\right)^{2}\right]\right\\}\
.$

Then we calculate,

$e^{-\Phi}\sqrt{-\det
g_{MN}}=\left(\frac{2}{e_{A}^{2}}\right)^{3/2}r^{2}\sin{\theta_{A}}\sqrt{\left(1+\frac{4r^{\prime
2}}{r^{2}f\left(r\right)}\right)\left(f\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}\right)}\
.$ (4.22)

The action for this D5 is,

$S_{\hat{D}_{5}}\left[r\right]=T_{{D}_{5}}L_{\varphi}\left(4\pi\right)^{2}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}\int
dtdx\sqrt{\left(f_{s}\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}\right)r^{4}+4r^{2}\left(1+\frac{2Q_{B}^{2}\zeta\left(r\right)^{2}}{f_{s}\left(r\right)}\right)r^{\prime
2}}\ .$

Comparing with (4.2)-(4.4) we identify

$\displaystyle{F}^{2}=\left(f_{s}(r)+2Q_{B}^{2}\zeta(r)^{2}\right)r^{4}\,\
,~{}~{}~{}{G}^{2}=\frac{4}{r^{2}f_{s}\left(r\right)}{F}^{2}\
,~{}~{}T_{eff}=T_{{D}_{5}}L_{\varphi}L_{t}\left(4\pi\right)^{2}\left(\frac{2}{e_{A}^{2}}\right)^{3/2}\
.$ (4.23)

Where $T_{eff}$ is the effective tension of the magnetic string. Following
eq.(4.4), the effective potential

$V_{eff}\left(r\right)=\frac{r^{3}}{2C}\sqrt{f_{s}\left(r\right)\left(f_{s}\left(r\right)+2Q_{B}^{2}\zeta\left(r\right)^{2}-\frac{C^{2}}{r^{4}}\right)}.$
(4.24)

The constant
$C=F(r_{0})=\sqrt{\left(f_{s}(r_{0})+2Q_{B}^{2}\zeta^{2}(r_{0})\right)}r_{0}^{2}$.
In the asymptotic region the effective potential in eq.(4.24) scales as
$V_{eff}\left(r\rightarrow\infty\right)\sim r^{3}$, satisfying the condition
in eq.(4.7).

The intuitive criteria discussed in the previous section applied for this case
indicate a screening behaviour. In fact $F(r_{+})=0$ and
$V_{eff}\sim(r-r_{+})^{1/2}$, signalling a vanishing tension of the magnetic
QCD-string in the IR, and a finite maximal separation, after which the
monopoles are screened (more about this is discussed below).

We perform a similar analysis using the approximate expression for the
separation between the monopole-anti-monopole pair $\hat{L}_{MM}$. Replacing
in eq.(4.9) the functions in eq.(4.23), we find an involved expression that
asymptotes as,

$\hat{L}_{MM}(r_{0}\to\infty)\sim\pi\sqrt{\frac{2}{e_{A}^{2}+e_{B}^{2}}},\;\;\;\;\;\hat{L}_{MM}(r_{0}\to
r_{+})\sim
8\pi\sqrt{\frac{r_{+}(r_{0}-r_{+})}{(e_{A}^{2}+e_{B}^{2})(r_{+}^{2}-r_{-}^{2})}}.$
(4.25)

These asymptotic behaviours indicate that at high energies in the field
theory, the pair is separated by a maximum distance characteristic of the UV
completion (Little String Theory). In this sense, the magnetic string behaves
oppositely to the electric one used to compute the Wilson loop (which shows a
minimum separation). On the other hand, at low energies the separation
decreases to zero (again, oppositely to the electric string case). We
calculate $Z(r_{0})$ in eq.(4.10) for this configuration and find that is
positive in all the range. This indicates the instability of the U-shaped
embeddings. A more general embedding that the one proposed here should drive
the dynamics. Instead of finding this more complicated embedding, below we
introduce a UV-cutoff. New short configurations appear close to a cutoff, that
dominate the dynamics and produce a phase transition (to a deconfining
behaviour).

More formally, we write expressions for the separation between the monopole-
anti-monopole pair as those in eqs.(4.5)-(4.6). A careful analysis of these
integrals is performed in Appendix F. Let us quote the exact expression for
the separation for the monopole-anti-monopole pair. We leave the study of the
expression for the energy between the pair of monopoles for Appendix F. In
terms of the elliptic integral of the first kind

${\bf
F}\left(y|x\right)=\int_{0}^{y}\frac{d\theta}{\sqrt{1-x\sin^{2}\theta}},$
(4.26)

and the definition for $\mathbf{K}(x)$ in eq.(4.18), working in the BPS limit
and defining $\eta=e_{A}/e_{B}$, we find

$\displaystyle L_{MM}^{BPS}\left(r_{0}\right)$ $\displaystyle=$
$\displaystyle\frac{2}{e_{B}}\sqrt{\frac{2\left(r_{0}^{2}/r_{+}^{2}-1\right)\left(\left(\eta^{2}+2\right)r_{0}^{2}/r_{+}^{2}+\eta^{2}\right)}{\left(\eta^{2}+1\right)\left(r_{0}^{2}/r_{+}^{2}\left(\eta^{2}+2\right)-1\right)}}$
(4.27)
$\displaystyle\times\left[\mathbf{F}\left(\left.\arcsin\sqrt{\frac{2\left(-1+\left(2+\eta^{2}\right)r_{0}^{2}/r_{+}^{2}\right)}{\left(2r_{0}^{2}/r_{+}^{2}+\eta^{2}\left(1+r_{0}^{2}/r_{+}^{2}\right)\right)}}\right|\frac{\left(1+r_{0}^{2}/r_{+}^{2}\right)\left(2r_{0}^{2}/r_{+}^{2}+\eta^{2}\left(1+r_{0}^{2}/r_{+}^{2}\right)\right)}{-4+4\left(2+\eta^{2}\right)r_{0}^{2}/r_{+}^{2}}\right)\right.$
$\displaystyle\qquad\left.+i\mathbf{K}\left(1-\frac{\left(1+r_{0}^{2}/r_{+}^{2}\right)\left(2r_{0}^{2}/r_{+}^{2}+\eta^{2}\left(1+r_{0}^{2}/r_{+}^{2}\right)\right)}{-4+4\left(2+\eta^{2}\right)r_{0}^{2}/r_{+}^{2}}\right)\right]\
.$

In Figure 3 we compare this exact expression with the approximating function
whose asymptotics we write in eq.(4.25). As stated above, opposite to the
Wilson loop case, we encounter a maximal separation for the monopole pair.
This maximal separation is associated with the Little String Theory scale. We
also plot the string profiles as they enter the bulk. We observe a different
behaviour to that found in the case of the Wilson loop–compare with the lower
right panel of Figure 2.

For magnetic strings that barely explore the bulk, the separation between the
monopole pair is large (equal to the LST scale). As we decrease the separation
between the monopole pair, the magnetic string dives into $r_{0}\to r_{+}$.
These unconventional behaviours, together with the instability of the string
embedding–note that $E_{MM}(L_{MM})$ has upwards concavity (and $Z(r_{0})$ is
positive), indicate the presence of a second ’disconnected’ configuration for
which the pair of monopoles separate without energy expense, which is
indicative of screening. The transition to a disconnected configuration is
dynamically favoured.

To avoid the ’t Hooft loop to be driven by the UV (LST) dynamics and to
realise explicitly these short configurations, we introduce a hard cutoff in
the radial direction and recalculate things. Doing so, the behaviour changes
qualitatively.

In fact, after introducing the UV cut-off, the separation between the pair of
monopoles does not show a maximum value, instead we find a ’double valued’
behaviour as displayed in the left panel of Figure 4. This leads to a phase
transition in the curve $E_{MM}(L_{MM})$. Note that this curve now has the
correct ’downwards’ concavity, indicating that the configuration is stable.
This phase transition is the physical manifestation of the magnetic string
suddenly changing into two disconnected magnetic strings that move without
energy expense. This is deconfinement for the pair monopole anti-monopole. We
are finding confinement for the quark-anti-quark pair and screening for a pair
of monopoles. These behaviours are consistent with the (electric) confining
behaviour of the dual QFT.

The introduction of the UV cutoff might seem unsatisfactory. Here we use it as
a device to show that the correct five brane embedding must be more elaborated
than the one we proposed above. It is also used to avoid the LST overtaking
the dynamics. The effect of the cutoff is clear considering the integral
needed to calculate $L(r_{0})$. This integral vanishes for $r_{0}\to r_{MAX}$.
This produces a double-valued $L(r_{0})$ and a consequent phase transition.

For the Entanglement Entropy a very similar behaviour occurs. We study this
next.

Figure 3: Left: Plot comparing the exact expression for the quark separation
$L_{MM}$ in eq.(4.27) with the approximate one $\hat{L}_{MM}$ in eq.(4.25),
both in the BPS limit. Right: the profiles of different strings as they enter
the bulk. Strings with small separation of the monopole pair penetrate deeper
into the bulk. There is a maximum separation for the pair of monopoles,
associated with the Little String Theory scale.

Figure 4: Left: after introducing a cutoff at $r_{MAX}$, the figure shows the
double-valued character of $L_{MM}$. Removing the cutoff recovers the LST
behaviour (on the left panel of Figure 3). Right: with the UV cutoff at
$r_{MAX}=20$, we see that $E_{MM}$ as a function of $L_{MM}$ has the ’upwards’
concavity (indicating stability). The presence of the phase transition to a
disconnected configuration is observed.

###  4.4 Entanglement Entropy

The Entanglement Entropy (EE) between two regions for field theories with a
string dual can be calculated as shown in [33]. The method is to find a
minimal area eight-surface ($\Sigma_{8}$, a codimension-two surface to which
we refer below as RT surface) such that the boundary of the surface coincides
with the two entangled regions. We focus on the case in which one of the
regions is a strip of size $L_{EE}$ and the other region is the complement.
The EE between these regions is given in [33], [34], minimising the quantity

$S_{EE}=\frac{1}{4G_{N}}\int_{\Sigma_{8}}d^{8}\sigma\sqrt{e^{-4\Phi}\det[g_{\Sigma_{8}}]}.$
(4.28)

There are various eight-surfaces that minimise $S_{EE}$ in eq.(4.28). Due to
this, in some cases there is a phase transition between different extremal
surfaces. It was suggested in [34] that a criterium for confinement is the
presence of a phase transition in the EE. This proposal was critically
analysed in [32]. It was found that for the case of field theories that
confine, but have a non-local high energy behaviour, the phase transition in
the EE is absent. The point is subtle, as introducing a UV cutoff or UV-
completing the QFT to avoid the non-locality, recovers the phase transition.

It is in this way that the EE can serve as an order parameter for confinement,
but also as a tool to diagnose non-locality in the UV-behaviour of the QFT
(when used together with a confining Wilson loop). In [32] it was found that
introducing a UV-cutoff implies the existence of new configurations realising
the phase transition (and resolving a stability issue with the original eight-
surface). Below, we perform an analysis of these features in our background of
eq.(2.3).

We follow the approach of [33], in particular the treatment for non-AdS
backgrounds developed in [34], [32]. We calculate the Entanglement Entropy on
a strip by computing the area of an eight-surface
$[x,\varphi,\theta_{A},\phi_{A},\psi_{A},\theta_{B},\phi_{B},\psi_{B}]$ with
$r=r(x)$ in the background of eq.(2.3). The induced metric on the RT eight-
surface, its determinant and the Entanglement Entropy are,

$\displaystyle ds^{2}_{st}$ $\displaystyle=$ $\displaystyle
r\left\\{dx^{2}(1+\frac{4r^{\prime
2}}{r^{2}f_{s}(r)})+f_{s}\left(r\right)d\varphi^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)d\varphi\right)^{2}\right]\right.$
(4.29)
$\displaystyle\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}^{2}-e_{B}Q_{B}\zeta\left(r\right)d\varphi\right)^{2}\right]\right\\}\
,$ $\displaystyle\sqrt{e^{-4\Phi}\det[g_{8}]}$ $\displaystyle=$
$\displaystyle\left(\frac{8}{e_{A}^{3}e_{B}^{3}}\right)\sqrt{r^{4}f_{s}(r)+4r^{2}r^{\prime
2}}\sin\theta_{A}\sin\theta_{B}.$ $\displaystyle S_{EE}$ $\displaystyle=$
$\displaystyle\frac{1}{4G_{N}}\int
d^{8}x\sqrt{e^{-4\Phi}\det[g_{8}]}=\left(\frac{2(4\pi)^{4}L_{\varphi}}{e_{A}^{3}e_{B}^{3}G_{N}}\right)\int_{-L/2}^{L/2}dx\sqrt{r^{4}f_{s}(r)+4r^{2}r^{\prime
2}}.$ (4.30)

From eqs.(4.2)-(4.4), this implies

$F(r)=r^{2}\sqrt{f_{s}(r)},~{}~{}~{}~{}G(r)=2r.$ (4.31)

To minimise the $S_{EE}$ above, we follow the usual conserved Hamiltonian
treatment. The Entanglement Entropy needs to be regularised by the area of two
eight-surfaces that hang straight from infinity. Then, computing the regulated
area for a surface that turns around at $r_{0}$, we find for the length of the
interval and the Entanglement Entropy,

$\displaystyle
L=4r_{0}^{2}\sqrt{f_{s}(r_{0})}\int_{r_{0}}^{\infty}\frac{dr}{\sqrt{r^{2}f_{s}(r)\left(r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})\right)}},$
(4.32) $\displaystyle S_{EE}=\frac{{\cal
N}}{G_{N}}\left[\int_{r_{0}}^{\infty}\sqrt{\frac{r^{6}f_{s}(r)}{r^{4}f_{s}(r)-r_{0}^{4}f_{s}(r_{0})}}dr-\int_{r_{+}}^{\infty}rdr\right].$
(4.33)

As in eq.(4.9), we can write a simple expression that approximates $L_{EE}$ in
eq.(4.32)—see [32],

$\hat{L}_{EE}=\frac{\pi
G(r_{0})}{F^{\prime}(r_{0})}=2\pi\frac{H(r)\sqrt{\beta(r)}}{H^{\prime}(r)}\Big{|}_{r_{0}},~{}~{}\text{with}~{}~{}H(r)={\cal
N}^{2}r^{4}f_{s}(r),~{}~{}\beta(r)=\frac{4}{r^{2}f_{s}(r)}.$ (4.34)

Using eq.(4.34) we find

$\hat{L}_{EE}=\left(\pi\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\right)\frac{\sqrt{(r_{0}^{2}-r_{+}^{2})(r_{0}^{2}-r_{-}^{2})}}{\left(2r_{0}^{2}-r_{+}^{2}-r_{-}^{2}\right)}.$
(4.35)

This function is monotonous, going from a vanishing value at $r_{0}=r_{+}$ to
a constant value at $r_{0}\to\infty$. This behaviour prevents the possibility
of phase transitions, which require that for a given $L_{EE}$ there are two
possible values of $r_{0}$. In fact, the conditions for the presence of a
phase transition (see section 2.4 of the work [32]) are not satisfied, in
particular equations (2.26)-(2.29) of [32] imply $j=2$ preventing a phase
transition. The absence of a phase transition in a confining model was
interpreted in [32] as an effect of the non-locality of the completion of the
QFT, in this case, by a LST.

Since $Z(r_{0})$ defined in eq.(4.10) gives

$Z(r_{0})=\frac{\sqrt{8}\pi(r_{+}^{2}-r_{-}^{2})^{2}}{r_{0}f_{s}(r_{0})(r_{+}^{2}+r_{-}^{2}-2r_{0}^{2})^{2}}>0,$
(4.36)

the proposed embedding is unstable. Upon the introduction of a cutoff, new
surfaces appear as found in [32], [35]. These cure the instability problem of
the embedding and give place to the phase transition, in agreement with
confinement.

The treatment in the papers [32], [35] applies to our background, even when
the IR dynamics is different, the UV dynamics is similarly driven by a Little
String Theory.

Before discussing the presence (or absence) of phase transitions we write the
analytic expressions for the values of the separation between the two
entangled regions $L_{EE}(r_{0})$ and the Entanglement Entropy
$S_{EE}(r_{0})$. These expressions are explicitly derived in Appendix F. Using
the definitions in eqs.(4.18),(4.26) and recalling that
$\lambda_{-}^{2}=-\frac{r_{-}^{2}}{r_{+}^{2}}$, we find

$\displaystyle L_{EE}\left(r_{0}\right)$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{8}{e_{A}^{2}+e_{B}^{2}}}\sqrt{\frac{r_{0}^{2}/r_{+}^{2}-1}{r_{0}^{2}/r_{+}^{2}+\lambda_{-}^{2}}}\left[i\mathbf{K}\left(\frac{\left(r_{0}^{2}/r_{+}^{2}-1\right)^{2}}{\left(r_{0}^{2}/r_{+}^{2}+\lambda_{-}^{2}\right)^{2}}\right)\right.$
(4.37)
$\displaystyle\left.+\mathbf{F}\left(\left.\arcsin\sqrt{\frac{r_{0}^{2}/r_{+}^{2}-r_{-}^{2}/r_{+}^{2}}{1-r_{-}^{2}/r_{+}^{2}}}\right|\frac{\left(1-r_{-}^{2}/r_{+}^{2}\right)\left(-r_{-}^{2}/r_{+}^{2}+2r_{0}^{2}/r_{+}^{2}-1\right)}{\left(r_{0}^{2}/r_{+}^{2}-r_{-}^{2}/r_{+}^{2}\right)^{2}}\right)\right]\
,$

and

$S_{EE}^{BPS}\left(r_{0}\right)=\frac{\mathcal{N}}{G_{N}}r_{+}^{2}\left[\frac{r_{+}^{2}}{2r_{0}^{2}}\left(-r_{0}^{4}/r_{+}^{4}\mathbf{E}\left(r_{+}^{4}/r_{0}^{4}\right)-\mathbf{K}\left(r_{+}^{4}/r_{0}^{4}\right)+r_{0}^{4}/r_{+}^{4}\mathbf{K}\left(r_{+}^{4}/r_{0}^{4}\right)\right)+\frac{1}{2}\right]\
.$ (4.38)

To analyse these expressions, it is useful to show some plots. First, we check
that the approximate $\hat{L}_{EE}(r_{0})$ in eq.(4.35) approximates well the
analytic expression in eq.(4.37), see the left panel of Figure 5.

Figure 5: Left: Plot comparing the exact expression for the separation of the
entangled regions $L_{EE}$ in eq.(4.37) with the approximate one
$\hat{L}_{EE}$ in eq.(4.35). Right: The plot of $S_{EE}(r_{0})$.

We also plot $S_{EE}(r_{0})$, see the right panel of Figure 5. The plot of
$S_{EE}$ in terms of $L_{EE}$ in the left panel of Figure 6, shows an upwards
concavity indicating that the configuration is unstable. This follows the
prediction of [32] that indicates that new configurations should appear as we
introduce a UV-cutoff in the geometry. The profiles of the effective strings
shown in the right panel of Figure 6, display a behaviour similar to the one
we encountered in the study of ’t Hooft loops (and opposite to that of the
Wilson loop), again suggesting the need for a phase transition.

In analogy with the case of the ’t Hooft loop, if we introduce a UV-cutoff,
the separation between the two entangled regions becomes multiple-valued, as
shown in Figure 7 (right panel). This is at the root of the phase transition.
The plot of $S_{EE}(L_{EE})$ shows the correct concavity and the presence of a
transition to the disconnected configuration is clearly displayed. See Figure
7

Figure 6: Left: The plot of $S_{EE}(L_{EE})$ showing the ’downwards’ concavity
(instability). Right: The profile of the effective strings used in computing
the EE. Note the analogous behaviour with the ’t Hooft loops strings.

Figure 7: Left: Plot of $L_{EE}$ in eq.(4.37) imposing the presence of a UV-
cutoff. Right: The plot of $S_{EE}(L_{EE})$. Notice that the concavity has
changed, indicating the stability of the configuration. The phase transition
becomes apparent thanks to the cutoff, avoiding the non-local behaviour

Following the findings of [32], we state that if a field theoretical UV
completion to our system (that is completed by a LST) were found, the phase
transitions for the ’t Hooft loops and the EE would become apparent. In this
sense the UV-cutoff captures the correct dynamics.

##  5 Sigma Model

In this section we review some results [13], [14], concerning the string
$\sigma$-model on the background in eq.(2.1)555We gratefully acknowledge
conversations with Lewis Cole and Gastón Giribet on the topics discussed
here.. To properly study the string action we move to the S-dual frame, and
work in terms of NS5 branes. After suitable coordinate changes detailed in
Appendix B, the background reads,

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+d\varphi^{2}+d\rho^{2}+N_{B}ds^{2}(S^{3}_{A})+N_{A}ds^{2}(S^{3}_{B}),$
(5.1) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\,\rho.$

For a careful derivation we refer the reader to Appendices B and C. The
background is a product space of the form

$\mathbb{R}^{2,1}\times\mathbb{R}_{\rho}\times S^{3}_{N_{A}}\times
S^{3}_{N_{B}},$ (5.2)

where $\mathbb{R}_{\rho}$ denotes the direction (with the linear dilaton), and
the subscript on the $S^{3}$ denotes the square of their radius. The metric on
the spheres together with the $H_{3}$ flux on each of them allows us to write
the $\sigma$-model on them as a WZW model on $SU(2)$. This is due to the fact
the the $S^{3}$ is a group manifold. Naively, the contribution of the $\rho$
coordinate to the string action is (for clarity we reinstate the
$\alpha^{\prime}$-factor)

$S_{\rho}=\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{-h}\left(h^{ab}\partial_{a}\rho\partial_{b}\rho-\alpha^{\prime}\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}R^{(2)}\rho\right),$
(5.3)

where $R^{(2)}$ is the world-sheet Ricci scalar. However when
$\rho\rightarrow-\infty$ the theory becomes non-perturbative, since $g_{s}\sim
e^{-\rho}$. In order to avoid the strong coupling region, it is necessary to
add the tachyon operator $e^{2b\rho}$ to the action, so that

$S_{\rho}=\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{-h}\left(h^{ab}\partial_{a}\rho\partial_{b}\rho-\alpha^{\prime}QR^{(2)}\rho+\Lambda
e^{2b\rho}\right).$ (5.4)

Here, $b$ is related to the background charge
$Q=\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}$ as

$Q=b+\frac{1}{b},$

such that it does not have a strong coupling region. Thus the contribution of
the $\rho$ direction to the $\sigma$-model corresponds to a Liouville field.
The complete $\sigma$-model on this geometry is

$U(1)^{3}\times\text{Liouville}\times SU(2)_{N_{A}}\text{WZW}\times
SU(2)_{N_{B}}\text{WZW}$ (5.5)

where the subscripts denote the WZW level. One can check that this is indeed a
good $\sigma$-model by computing the central charge. Here we use

$c(U(1))=1,\quad c(\text{Liouville})=1+6Q^{2},\quad
c(SU(2)_{k})=\frac{3(k-2)}{k}.$ (5.6)

In total we have

$c_{\text{total}}=3+1+6\left(\frac{1}{N_{A}}+\frac{1}{N_{B}}\right)+\frac{3(N_{A}-2)}{N_{A}}+\frac{3(N_{B}-2)}{N_{B}}=10.$
(5.7)

To this we should add the ten free fermions, that contribute to the central
charge $c_{ferm}=5$. The central charge of the SUSY system $c_{SUSY}=15$, is
then cancelled by the $b-c$ and $\beta-\gamma$ ghosts.

We now study the interesting case $Q_{A}=Q_{B}=0$, but with $m>0$. The
configuration of interest is obtained by S-dualising eq.(2.3),

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+f_{s}(r)d\varphi^{2}+\frac{4dr^{2}}{r^{2}f_{s}(r)}+N_{B}\,ds^{2}(S^{3}_{A})+N_{A}\,ds^{2}(S^{3}_{B}),$
(5.8) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$ $\displaystyle=-\log(r).$

Recall that here $\varphi\sim\varphi+\pi/Q^{2}$, see eq.(2.6), with
$Q^{2}=\frac{1}{N_{A}}+\frac{1}{N_{B}}$, as above. Before proceeding to the
$\sigma$-model analysis, it is convenient to perform some changes of
variables. First, we want to rewrite the background (5.8) in such a way that
it reduces to (5.1) when $m=0$. For this use

$r=e^{Q\rho},\quad\tilde{m}=\frac{m}{4Q^{2}},$ (5.9)

which leads to

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+4Q^{2}\left(1-\tilde{m}e^{-2Q\rho}\right)d\varphi^{2}+\frac{d\rho^{2}}{1-\tilde{m}e^{-2Q\rho}}+N_{B}\,ds^{2}(S^{3}_{A})+N_{A}\,ds^{2}(S^{3}_{B}),$
(5.10) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$ $\displaystyle=-Q\rho.$

We change coordinates as,

$\tanh^{2}(\lambda)=1-\tilde{m}e^{-2Q\rho},\quad\varphi=\frac{\phi}{2Q^{2}},$
(5.11)

which puts the geometry in the usual cigar form (note that $\phi$ has period
$2\pi$)

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+\frac{1}{Q^{2}}\left(\tanh^{2}(\lambda)d\phi^{2}+d\lambda^{2}\right)+N_{B}\,ds^{2}(S^{3}_{A})+N_{A}\,ds^{2}(S^{3}_{B}),$
(5.12) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$
$\displaystyle=-\log\left(\cosh(\lambda)\right)-\frac{1}{2}\log(\tilde{m}).$

As explained in [36], for more details see Section 2 of [14], the above
backgrounds leads to an exact $\sigma$-model

$U(1)^{2}\times\frac{SL(2,\mathbb{R})_{k}}{U(1)}\times
SU(2)_{N_{B}}~{}\text{WZW}\times SU(2)_{N_{A}}~{}\text{WZW},$ (5.13)

where $k^{-1}=Q^{2}=\frac{1}{N_{A}}+\frac{1}{N_{B}}$. To check that this is
also a good string $\sigma$-model we use

$c\left(SL(2,\mathbb{R})_{k}\right)=\frac{3(k+2)}{k},\quad
c\left(G/H\right)=c(G)-c(H),$ (5.14)

so that

$c\left(\frac{SL(2,\mathbb{R})_{k}}{U(1)}\right)=2+\frac{6}{k}=2+\frac{6}{N_{A}}+\frac{6}{N_{B}},$
(5.15)

from where is easy to see that $c_{\text{total}}=10$. We leave for future
research the study of the $\sigma$-model in the background with
$m,Q_{A},Q_{B}$ arbitrary.

##  6 Conclusions and future research

The I-brane QFT, defined as the $(1+1)$ field theory on the intersection of
two stacks of D5 branes, was studied in [13]. The field theory has the
remarkable behaviour that as the coupling is increased, the system gains one
more dimension and enhances its SUSY (with a peculiar SUSY algebra in $(2+1)$
dimensions [20]). The background dual to this strongly coupled QFT was written
in [13], see our eq.(2.1). This presents a singular behaviour for large values
of the radial coordinate, where the dilaton diverges and string coupling
effects cannot be neglected. This is solved by performing an S-duality and
working with the NS branes system. The background of eq.(2.1) is also singular
for small values of the radial coordinate, $r\to 0$, as indicated by eq.(2.2).
This ill-defined IR behaviour is amended by our background in eq.(2.3). Our
simple and explicit solution describes the holographic dual to a $(2+1)$ QFT
that gets compactified to $(1+1)$ dimensions, preserving four supercharges and
ending the flow with a confining and gapped behaviour.

We holographically studied different aspects of this peculiar QFT. Maldacena-
Wilson loops, ’t Hooft loops, Entanglement entropy were discussed in dedicated
sections, with emphasis on the effects of the UV-completion in terms of LST.
R-symmetry and its breaking, a suitably defined gauge coupling and a quantity
measuring the number of degrees of freedom as a function of the radial
coordinate (the energy) are presented and discussed. By a double Wick
rotation, a black hole solution is found. Also, some of the NS string
$\sigma$-model aspects are briefly mentioned.

It would be interesting to dedicate future efforts to

  * •

The careful study of the black hole solution in Section 2.4. In particular if
its entropy can be computed in terms of a $(2+1)$ field theory compactified on
a torus.

  * •

The study of the string $\sigma$-model for the full solution in eq.(2.3), in
the NS5 branes frame.

  * •

To achieve a cleaner understanding of the R-symmetry breaking in terms of
anomalies in two dimensional QFT. To relate this to the Chern Simons
coefficients discussed in eq.(3.6). Note that while we find an anomalous
breaking $U(1)_{A}\times
U(1)_{B}\rightarrow\mathbb{Z}_{N_{B}}\times\mathbb{Z}_{N_{A}}$, we do not find
a further spontaneous breaking to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, as it
normally occurs in holographic models to four dimensional ${\cal N}=1$
dynamics. Understanding the reason of the absence of the spontaneous breaking
would be of interest.

  * •

Geometrically, it would be interesting to generalise the metric and fluxes in
eq.(2.3) adding warp factors in front of the $\omega_{i}$’s and more general
fibrations. Finding a more general classification is of interest. It may be
possible to relate this to the material in [20].

  * •

A fair amount of papers have been written studying the background in eq.(2.1).
See for example [37]-[38]. It would be interesting to understand the effects
of the resolution provided in eq.(2.3) on some of these observables.

We hope to report on some of these problems in the near future.

## Acknowledgements:

The contents and presentation of this work much benefitted from extensive
discussion with various colleagues. We are very happy to thank: Andres
Anabalon, Adi Armoni, Fabrizio Canfora, Lewis Cole, Gaston Giribet, Nicolas
Grandi, Nabil Iqbal, Prem Kumar, Juan Maldacena, Anibal Neira, Leo Pando
Zayas, Julio Oliva, Niels Obers, Dibakar Roychowdhury, Kostas Skenderis,
Christoph Uhlemann who shared their knowledge with us. We are supported by
STFC grant ST/T000813/1. The work of M.O. is partially funded by Beca ANID de
Doctorado 21222264. The work of R.S. is supported by STFC grant ST/W507878/1.
The authors have applied to a Creative Commons Attribution (CC BY) licence.

##  Appendix A Details of the Supergravity Backgrounds

In this appendix we set some of the conventions used in this paper and study
the SUSY preserved by the background in eq.(2.3).

###  A.1 Type IIB Supergravity

We start this appendix by explicitly writing the Type IIB Supergravity action
and its SUSY variations. The field content of Type IIB is split into two
sector. In the NS-NS sector we have: the metric $g_{\mu\nu}$, 2-form potential
$B_{2}$ with field strength $H_{3}$, and the Dilaton $\Phi$. In the R-R sector
we have a set of Abelian p-form gauge fields: $C_{0}$, $C_{2}$, $C_{4}$. By
defining

$F_{1}=dC_{0},\quad F_{3}=dC_{2}-C_{0}\wedge H_{3},\quad
F_{5}=dC_{4}-C_{2}\wedge H_{3}.$ (A.1)

The bosonic part of the Type IIB action in String frame is

$\displaystyle S_{\text{IIB}}$ $\displaystyle=\frac{1}{2\kappa^{2}}\int
d^{10}x\sqrt{-g}\left[e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}|H_{3}|^{2}\right)-\frac{1}{2}|F_{1}|^{2}-\frac{1}{2}|F_{3}|^{2}-\frac{1}{4}|F_{5}|^{2}\right]$
(A.2) $\displaystyle\phantom{=}-\frac{1}{4\kappa^{2}}\int\,C_{4}\wedge
H_{3}\wedge F_{3}$

where $|F_{p}|^{2}=F_{\mu_{1}...\mu_{p}}F^{\mu_{1}...\mu_{p}}/p!$ and
analogously for $H_{3}$. On the solutions of this theory we need to impose
666To be precise, there is no covariant action for the effective theory of the
Type IIB Superstring, but the presented here is close enough. The issue is
that is not possible to implement the self-duality condition for $F_{5}$ at
the level of the action. self-duality of the $F_{5}=\star F_{5}$. The
equations of motion of this theory are

$\displaystyle\nabla^{2}\Phi-\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{1}{4}R-\frac{1}{8}|H_{3}|^{2}=0,$
(A.3) $\displaystyle d\left(e^{-2\Phi}\star H_{3}\right)=-F_{5}\wedge
F_{3}-F_{1}\wedge F_{7},$ $\displaystyle dF_{5}-H_{3}\wedge F_{3}=0,$
$\displaystyle dF_{7}-H_{3}\wedge F_{5}=0,$ $\displaystyle dF_{9}-H_{3}\wedge
F_{7}=0,$ $\displaystyle
R_{\mu\nu}+2\partial_{\mu}\partial_{\nu}\Phi-\frac{1}{2}|H_{3}|^{2}_{\,\mu\nu}-\frac{e^{2\Phi}}{2}\left(\frac{1}{2}|F_{1}|^{2}_{\,\mu\nu}+\frac{1}{2}|F_{3}|^{2}_{\,\mu\nu}+\frac{1}{2}|F_{5}|^{2}_{\,\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(|F_{1}|^{2}+|F_{3}|^{2}\right)\right)=0$

where
$|F_{p}|^{2}_{\mu\nu}=F_{\mu\nu_{1}...\nu_{p-1}}F_{\nu}^{\phantom{\mu}\nu_{1}...\nu_{p-1}}/(p-1)!$,
similarly for $H_{3}$, and

$F_{7}=-\star F_{3},\quad F_{9}=\star F_{1}.$ (A.4)

The equations of motion are complemented by the Bianchi Identities

$dF_{1}=0,\quad dF_{3}-H_{3}\wedge F_{1}=0.$ (A.5)

Due to the self-duality of $F_{5}$, its equation of motion and its the Bianchi
identity are the same.

Solutions of the purely bosonic part of Type IIB has all the fermionic
partners, the dilatino $\lambda$ and gravitino $\Psi_{\mu}$, set to zero. If
we are interested in finding SUSY solutions, we need to be consistent with the
fact that we turn off the fermions by asking for the SUSY variations of these
fields to vanish. In string frame, the SUSY variations of the fermionic fields
are [39],

$\displaystyle\delta\lambda=\frac{1}{2}\left(\Gamma^{\mu}\partial_{\mu}\Phi+\frac{1}{2\cdot
3!}H_{\mu\nu\lambda}\Gamma^{\mu\nu\lambda}\sigma^{3}-e^{\Phi}\left(F_{\mu}\Gamma^{\mu}(i\sigma_{2})+\frac{1}{2\cdot
3!}F_{\mu\nu\lambda}\Gamma^{\mu\nu\lambda}\sigma^{1}\right)\right)\epsilon,$
(A.6) $\displaystyle\begin{aligned}
\delta\Psi_{\mu}&=\partial_{\mu}\epsilon+\frac{1}{4}\omega^{\phantom{\mu}ab}_{\mu}\Gamma_{ab}\epsilon+\frac{1}{4\cdot
2!}H_{\mu\nu\lambda}\Gamma^{\nu\lambda}\sigma^{3}\epsilon\\\
&\phantom{=}+\frac{e^{\Phi}}{8}\left(F_{\nu}\Gamma^{\nu}(i\sigma_{2})+\frac{1}{3!}F_{\nu\lambda\rho}\Gamma^{\nu\lambda\rho}\sigma^{1}+\frac{1}{2\cdot
5!}F_{\nu\lambda\rho\sigma\tau}\Gamma^{\nu\lambda\rho\sigma\tau}(i\sigma_{2})\right)\Gamma_{\mu}\epsilon.\end{aligned}$
(A.7)

Here $\omega^{\phantom{\mu}ab}_{\mu}$ is the spin connection of the 10D
background, where the $a,b$ indexes are flat space ones, and $\sigma^{1}$,
$\sigma^{2}$ and $\sigma^{3}$ are Pauli matrices. Also

$\Gamma^{\mu_{1}...\mu_{p}}=\Gamma^{[\mu_{1}}...\Gamma^{\mu_{p}]}$ (A.8)

Here $\epsilon$ is a 64 component spinor,

$\epsilon=\begin{pmatrix}\epsilon_{1}\\\ \epsilon_{2}\end{pmatrix}$ (A.9)

where both 32-component parts are left-handed.

###  A.2 Checking SUSY for the Fibered Background

Here we aim to compute how many supercharges are preserved by the the
backgrounds presented in this paper. For the un-fibered background in eq.(2.1)
we refer the reader to [13], [20], where it is shown that this solution
preserves 16 Supercharges in an interesting way: the anti-commutator of two
supercharges includes the $R$-Symmetry generators. Now we present the analysis
for the fibered background in eq.(2.3). We perform all the analysis in the
S-dual system, in terms of NS5 branes, where we only have $H_{3}$ flux.

First, note that the dilatino variation is a matrix equation of the form
$M\epsilon=0$. In order to have non-trivial solutions to this equation, we
require $M$ to be non-invertible, for which we need to impose
$\text{det}(M)=0$. It is also possible to obtain a matrix equation from the
gravitino variation. Noting that we can write the gravitino variation as a
covariant derivative, for which we define the connection

$W_{\mu}=\frac{1}{4}\omega^{\phantom{\mu}ab}_{\mu}\Gamma_{ab}+\frac{1}{4\cdot
2!}H_{\mu\nu\lambda}\Gamma^{\nu\lambda}\sigma^{3}+\frac{e^{\Phi}}{8}\left(F_{\mu}\Gamma^{\mu}(i\sigma_{2})+\frac{1}{3!}F_{\mu\nu\lambda}\Gamma^{\mu\nu\lambda}\sigma^{1}+\frac{1}{2\cdot
5!}F_{\mu\nu\lambda\rho\sigma}\Gamma^{\mu\nu\lambda\rho\sigma}(i\sigma_{2})\right)\Gamma_{\mu},$
(A.10)

then we can write the gravitino variation as

$\delta\psi_{\mu}dx^{\mu}=\left(\partial_{\mu}\epsilon+W_{\mu}\epsilon\right)dx^{\mu}\equiv\mathcal{D}\epsilon.$
(A.11)

We can get rid of the partial derivative of the spinor by acting with
$\mathcal{D}$ a second time

$\mathcal{D}\wedge\mathcal{D}\epsilon=\left(dW+W\wedge
W\right)\epsilon=\frac{1}{2}\Theta_{\mu\nu}dx^{\mu}\wedge dx^{\nu}\epsilon.$
(A.12)

Each of the components of $\Theta_{\mu\nu}$ defines a matrix equation, giving
a total of 45 independent equations. We need to make sure that
$\text{det}(\Theta_{\mu\nu})=0$ for each of the components. The equations

$M\epsilon=0,\quad\Theta_{\mu\nu}\epsilon=0,$ (A.13)

constrain the number of independent components of the spinor. After this
procedure we use the gravitino variation to solve the dependence of the spinor
on the spacetime coordinates.

Specialising to our background, the determinant of the Dilatino variation for
the background in eq.(2.3) reads

$\text{det}(M)\sim\left(4(e_{B}Q_{A}-e_{A}Q_{B})^{2}+m^{2}\right)^{8}\left(4(e_{B}Q_{A}+e_{A}Q_{B})^{2}+m^{2}\right)^{8}.$
(A.14)

In order to have non-trivial solutions we need to impose the following BPS
conditions on the parameters of the background

$e_{A}Q_{B}=\pm e_{B}Q_{A},\quad m=0.$ (A.15)

With this conditions it is possible to check that
$\text{det}(\Theta_{\mu\nu})=0$ is also satisfied. Solving these matrix
equations shows that the spinor has 8 independent components. Then, solving
for the gravitino variation shows that these components are not independents,
and in fact, the total number of independent components its reduced to 4. The
solution for the spinor is

$\epsilon_{1}=\vec{0}$ (A.16)

and

$\epsilon_{2}=\begin{pmatrix}\frac{c_{1}e^{-\frac{1}{4}i\text{$\varphi$}\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}+2Q_{A}}}{r}\\\
0\\\ 0\\\
\frac{c_{1}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}+2Q_{A}}}{r}\\\
0\\\
\frac{ic_{2}e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}-2Q_{A}}}{r}\\\
\frac{ic_{2}e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}-2Q_{A}}}{r}\\\
0\\\ 0\\\
\frac{c_{1}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\left(e_{A}^{2}+e_{B}^{2}\right)\left(e_{A}r^{2}-2Q_{A}\right)}}{r(e_{A}-ie_{B})}\\\
\frac{c_{1}(e_{A}+ie_{B})e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\left(2Q_{A}-e_{A}r^{2}\right)}{r\sqrt{\left(e_{A}^{2}+e_{B}^{2}\right)\left(e_{A}r^{2}-2Q_{A}\right)}}\\\
0\\\
\frac{ic_{2}(e_{A}+ie_{B})e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\frac{e_{A}r^{2}+2Q_{A}}{e_{A}^{2}+e_{B}^{2}}}}{r}\\\
0\\\ 0\\\
\frac{c_{2}(e_{B}-ie_{A})e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\frac{e_{A}r^{2}+2Q_{A}}{e_{A}^{2}+e_{B}^{2}}}}{r}\\\
0\\\
\frac{c_{3}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}+2Q_{A}}}{r}\\\
-\frac{c_{3}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}+2Q_{A}}}{r}\\\
0\\\
\frac{ic_{4}e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}-2Q_{A}}}{r}\\\
0\\\ 0\\\
-\frac{ic_{4}e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{e_{A}r^{2}-2Q_{A}}}{r}\\\
-\frac{c_{3}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\left(e_{A}^{2}+e_{B}^{2}\right)\left(e_{A}r^{2}-2Q_{A}\right)}}{r(e_{A}+ie_{B})}\\\
0\\\ 0\\\
-\frac{c_{3}e^{-\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\left(e_{A}^{2}+e_{B}^{2}\right)\left(e_{A}r^{2}-2Q_{A}\right)}}{r(e_{A}+ie_{B})}\\\
0\\\
-\frac{ic_{4}(e_{A}-ie_{B})e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\frac{e_{A}r^{2}+2Q_{A}}{e_{A}^{2}+e_{B}^{2}}}}{r}\\\
-\frac{ic_{4}(e_{A}-ie_{B})e^{\frac{1}{4}i\varphi\left(e_{A}^{2}+e_{B}^{2}\right)}\sqrt{\frac{e_{A}r^{2}+2Q_{A}}{e_{A}^{2}+e_{B}^{2}}}}{r}\\\
0\end{pmatrix}$ (A.17)

We have found a spinor with four arbitrary constants
$(c_{1},c_{2},c_{3},c_{4})$. Being the spinor complex, we count four preserved
supercharges.

##  Appendix B How are the backgrounds obtained

In this appendix we describe the procedure followed to obtain the backgrounds
in eqs.(2.1) and (2.3). These solutions are originally obtained in gauged
supergravity, together with a lift procedure. Below, we review these steps.

###  B.1 4D $\mathcal{N}=4\,SU(2)\times SU(2)$ Gauged Supergravity

The action of the bosonic part of the Freedman-Schwarz (FS) gauged
Supergravity is

$\displaystyle S_{\text{FS}}$ $\displaystyle=\int
d^{4}x\sqrt{-g_{(4)}}\left(-\frac{R^{(4)}}{4}+\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}e^{4\phi}(\partial{\textbf{a}})^{2}-V(\phi)\right.$
(B.1)
$\displaystyle\phantom{=}\left.-\frac{e^{-2\phi}}{4}\text{Tr}\left(F_{(A)mn}F_{(A)}^{mn}+F_{(B)mn}F_{(B)}^{mn}\right)-\frac{\textbf{a}}{2}\,\text{Tr}\left(\tilde{F}_{(A)mn}F_{(A)}^{mn}+\tilde{F}_{(B)mn}F_{(B)}^{mn}\right)\right).$

Here $g_{(4)}$ and $R^{(4)}$ are the determinant of the 4D metric and the 4D
Ricci scalar, $\phi$ is the 4D dilaton, a is a pseudo-scalar called axion and
$F_{(A)mn}$ and $F_{(B)mn}$ are the field strengths of two $SU(2)$ gauge
fields $A_{m}$ and $B_{m}$,

$\displaystyle F^{i}_{(A)mn}$
$\displaystyle=\partial_{m}A^{i}_{n}-\partial_{n}A^{i}_{m}+e_{A}\epsilon_{ijk}A^{j}_{m}A^{k}_{n}$
(B.2) $\displaystyle F^{i}_{(B)mn}$
$\displaystyle=\partial_{m}B^{i}_{n}-\partial_{n}B^{i}_{m}+e_{B}\epsilon_{ijk}B^{j}_{m}B^{k}_{n}$
(B.3)

where $e_{A}$ and $e_{B}$ are the gauge couplings of $A_{m}$ and $B_{m}$
respectively. Here the index $i=1,2,3$ transform in the adjoint of each of the
copies of $SU(2)$. Also, the duals of the field strengths are

$\tilde{F}_{(A)mn}=\frac{1}{2\sqrt{-g}}\epsilon_{mn\lambda\rho}F_{(A)mn},\quad\tilde{F}_{(B)mn}=\frac{1}{2\sqrt{-g}}\epsilon_{mn\lambda\rho}F_{(B)mn},$
(B.4)

####  B.1.1 A BPS Solution

This theory admits a series of BPS solution preserving some amount of
supersymmetry. Our main focus is the 1/4 BPS soliton presented in [40]. As we
will review, this solution is particularly interesting because it manages to
resolve a singularity by introducing a thermal cycle that preserves some SUSY
(in the usual case, the non-extremal factor completely breaks SUSY). The field
configuration of the Soliton is777In [40] the solution was presented in the
mostly-minus signature. We write the solution in the mostly-plus one.

$\displaystyle ds^{2}_{4D}$ $\displaystyle=-\rho
dt^{2}+\frac{d\rho^{2}}{g(\rho)}+g(\rho)d\varphi^{2}+\rho dx^{2},$ (B.5)
$\displaystyle\phi(\rho)$ $\displaystyle=-\frac{1}{2}\log(\rho),$ (B.6)
$\displaystyle A^{1}$ $\displaystyle=0,\quad A^{2}=0,\quad
A^{3}=Q_{A}\zeta(\rho)d\varphi,$ (B.7) $\displaystyle B^{1}$
$\displaystyle=0,\quad B^{2}=0,\quad B^{3}=Q_{B}\zeta(\rho)d\varphi,$ (B.8)

where

$g(\rho)=\frac{e^{2}_{A}+e^{2}_{B}}{2}\rho-m-2\frac{Q^{2}_{A}+Q^{2}_{B}}{\rho}$
(B.9)

which has a zeros

$\rho_{\pm}=\frac{m\pm\sqrt{4(e^{2}_{A}+e^{2}_{B})(Q^{2}_{A}+Q^{2}_{B})+m^{2}}}{(e^{2}_{A}+e^{2}_{B})}.$
(B.10)

Also

$\zeta(\rho)=\frac{1}{\rho}-\frac{1}{\rho_{+}},$ (B.11)

where the last term ensures that both of the gauge fields vanish at
$\rho=\rho_{+}$.

In order for the cycle $\varphi$ to close smoothly at $\rho=\rho_{+}$, the
period of $\varphi$ needs to be

$\beta_{\varphi}=\frac{4\pi}{g^{\prime}(\rho_{+})}=\frac{8\pi\rho^{2}_{+}}{(e^{2}_{A}+e^{2}_{B})\rho^{2}_{+}+4(Q^{2}_{A}+Q^{2}_{B})}.$
(B.12)

This solution was shown to preserve 4 supercharges [40], when the parameters
$(e_{A},e_{B},Q_{A},Q_{B},m)$ satisfy

$e_{A}Q_{B}=\pm e_{B}Q_{A},\quad m=0.$ (B.13)

In what follows it is convenient to perform the change of coordinates
$\rho=r^{2}$. The background configuration now reads

$\displaystyle ds^{2}_{4D}$
$\displaystyle=-r^{2}dt^{2}+\frac{4dr^{2}}{f_{s}(r)}+r^{2}f_{s}(r)d\varphi^{2}+r^{2}dx^{2},$
(B.14) $\displaystyle\phi(r)$ $\displaystyle=-\log(r),$ (B.15) $\displaystyle
A^{1}$ $\displaystyle=0,\quad A^{2}=0,\quad A^{3}=Q_{A}\zeta(r)d\varphi,$
(B.16) $\displaystyle B^{1}$ $\displaystyle=0\quad B^{2}=0,\quad
B^{3}=Q_{B}\zeta(r)d\varphi,$ (B.17)

with

$f_{s}(r)=\frac{e^{2}_{A}+e^{2}_{B}}{2}-\frac{m}{r^{2}}-2\frac{Q^{2}_{A}+Q^{2}_{B}}{r^{4}},\quad\zeta(r)=\frac{1}{r^{2}}-\frac{1}{r^{2}_{+}}.$
(B.18)

###  B.2 Lift to 10D Supergravity

It was shown in [41] that the FS Supergravity has a Kaluza-Klein
interpretation as a compactification of $\mathcal{N}=1$ Supergravity in 10D on
the group manifold $S^{3}\times S^{3}$. The action of the 10D theory is

$S_{10D}=-\frac{1}{4}\int
d^{10}x\sqrt{-g}\left(R-2\partial_{\mu}\tilde{\Phi}\partial^{\mu}\tilde{\Phi}-\frac{e^{2\tilde{\Phi}}}{3}\tilde{H}_{\mu\nu\lambda}\tilde{H}^{\mu\nu\lambda}\right),$
(B.19)

where $R$ is the 10D Ricci scalar, $\tilde{\Phi}$ the 10D Dilaton and
$\tilde{H}_{\mu\nu\lambda}$ is a 3-form field strength
$\tilde{H}_{3}=d\tilde{B}_{2}$. We split the indexes as

$x^{\mu}=\left\\{x^{m}=t,r,\varphi,x\,;\,z_{A}^{i}=\psi_{A},\theta_{A},\phi_{A}\,;\,z_{B}^{i}=\psi_{B},\theta_{B},\phi_{B}\right\\}\
.$ (B.20)

The of the lift to 10D is

$ds^{2}_{10D}=e^{3\phi/2}ds^{2}_{4D}+2e^{-\phi/2}\left(\Theta_{A}^{i}\Theta_{A}^{i}+\Theta_{B}^{i}\Theta_{B}^{i}\right)$
(B.21)

where the 1-forms

$\displaystyle\Theta^{i}_{A}$
$\displaystyle=A^{i}+\frac{1}{e_{A}}\omega_{A}^{i},$ (B.22)
$\displaystyle\Theta^{i}_{B}$
$\displaystyle=B^{i}+\frac{1}{e_{B}}\omega_{B}^{i},$ (B.23)

with $\omega_{A}^{i}$ and $\omega_{B}^{i}$ are the Maurer-Cartan forms of the
two different $SU(2)$, is so that $\Theta^{i}_{A}$ and $\Theta^{i}_{B}$
realise a fibration of the two 3-spheres.

The 10D Dilaton $\tilde{\Phi}$ is written in terms of the four dimensional one
$\phi$

$\tilde{\Phi}=-\frac{\phi}{2},$ (B.24)

while the 3-form field strength is written in term of the non-Abelian gauge
fields and the $SU(2)$ Maurer-Cartan forms as

$\tilde{H}_{3}=-\sum^{3}_{i=1}F_{A}^{i}\wedge\Theta_{A}^{i}-\sum^{3}_{i=1}F_{B}^{i}\wedge\Theta_{B}^{i}+e_{A}\Theta_{A}^{1}\wedge\Theta_{A}^{2}\wedge\Theta_{A}^{3}+e_{B}\Theta_{B}^{1}\wedge\Theta_{B}^{2}\wedge\Theta_{B}^{3}.$
(B.25)

####  B.2.1 A Note on Conventions

We are interested in lifting the theory to Type II Supergravity, when the
field content is purely of the NS-NS sector. The action in eq.(B.19) can be
mapped to Type II, after the field redefinitions

$\tilde{\Phi}\rightarrow\Phi=-2\tilde{\Phi},\quad\tilde{H}_{3}\rightarrow
H_{3}=2\tilde{H}_{3}.$ (B.26)

The action corresponds to the Type II in Einstein frame

$S_{\text{Type II, E}}=-\frac{1}{4}\int
d^{10}x\sqrt{-g}\left(R-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{e^{-\Phi}}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right).$
(B.27)

We move to String frame by
$g^{(S)}_{\mu\nu}=e^{\frac{1}{2}\Phi}g^{(E)}_{\mu\nu}$, then the action reads

$S_{\text{Type II, S}}=-\frac{1}{4}\int
d^{10}x\sqrt{-g}e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2\cdot
3!}H_{\mu\nu\rho}H^{{\mu\nu\rho}}\right).$ (B.28)

In this frame, the lift of the 4D FS Supergravity reads

$\displaystyle ds^{2}_{st}$
$\displaystyle=e^{2\phi}ds^{2}_{4D}+2\left(\Theta_{A}^{i}\Theta_{A}^{i}+\Theta_{B}^{i}\Theta_{B}^{i}\right),$
(B.29) $\displaystyle H_{3}$
$\displaystyle=2\left(-\sum^{3}_{i=1}F_{A}^{i}\wedge\Theta_{A}^{i}-\sum^{3}_{i=1}F_{B}^{i}\wedge\Theta_{B}^{i}+e_{A}\Theta_{A}^{1}\wedge\Theta_{A}^{2}\wedge\Theta_{A}^{3}+e_{B}\Theta_{B}^{1}\wedge\Theta_{B}^{2}\wedge\Theta_{B}^{3}.\right),$
(B.30) $\displaystyle\Phi$ $\displaystyle=\phi(x^{m}).$ (B.31)

It is convenient to write the lift in the S-dual frame, where instead of
$H_{3}$ flux, we have a $F_{3}$ flux, the Dilaton is $\Phi^{\prime}=-\Phi$ and
the metric now is $g_{\mu\nu}^{\prime}=e^{-\Phi}g_{\mu\nu}$, explicitly this
is

$\displaystyle ds^{2}_{st}$
$\displaystyle=e^{\phi}ds^{2}_{4D}+2e^{-\phi}\left(\Theta_{A}^{i}\Theta_{A}^{i}+\Theta_{B}^{i}\Theta_{B}^{i}\right),$
(B.32) $\displaystyle F_{3}$
$\displaystyle=2\left(-\sum^{3}_{i=1}F_{A}^{i}\wedge\Theta_{A}^{i}-\sum^{3}_{i=1}F_{B}^{i}\wedge\Theta_{B}^{i}+e_{A}\Theta_{A}^{1}\wedge\Theta_{A}^{2}\wedge\Theta_{A}^{3}+e_{B}\Theta_{B}^{1}\wedge\Theta_{B}^{2}\wedge\Theta_{B}^{3}.\right),$
(B.33) $\displaystyle\Phi$ $\displaystyle=-\phi(x^{m}).$ (B.34)

####  B.2.2 Lift of the BPS Solution

Following the explicit construction of the lift, we now read the lift of the
4D solution that preserves 4 Supercharges. In the S-dual frame of eqs. (B.32),
we have

$\displaystyle ds^{2}_{st}$ $\displaystyle=$ $\displaystyle
r\left\\{-dt^{2}+dx^{2}+f_{s}\left(r\right)d\varphi^{2}+\frac{4}{r^{2}f_{s}\left(r\right)}dr^{2}+\frac{2}{e_{A}^{2}}\left[\hat{\omega}_{1}^{2}+\hat{\omega}_{2}^{2}+\left(\hat{\omega}_{3}-e_{A}Q_{A}\zeta(r)d\varphi\right)^{2}\right]\right.$
(B.35)
$\displaystyle\left.+\frac{2}{e_{B}^{2}}\left[\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\left(\tilde{\omega}_{3}-e_{B}Q_{B}\zeta\left(r\right)d\varphi\right)^{2}\right]\right\\}\
,$ $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
dC_{2}=2\zeta^{\prime}(r)dr\wedge
d\varphi\wedge\left(\frac{Q_{A}}{e_{A}}\hat{\omega}_{3}+\frac{Q_{B}}{e_{B}}\tilde{\omega}_{3}\right)+\frac{2}{e_{A}^{2}}\hat{\omega}_{1}\wedge\hat{\omega}_{2}\wedge\left(e_{A}Q_{A}\zeta(r)d\varphi-\hat{\omega}_{3}\right)$
$\displaystyle+\frac{2}{e_{B}^{2}}\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\left(e_{B}Q_{B}\zeta(r)d\varphi-\tilde{\omega}_{3}\right)\
,$ $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle{\psi_{A}}\left(\frac{2Q_{A}}{e_{A}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{A}^{2}}\sin{\theta_{A}}d{\theta_{A}}\wedge
d{\phi_{A}}\right)+\frac{2}{e_{A}}\cos{\theta_{A}}Q_{A}\zeta\left(r\right)d\varphi\wedge
d{\phi_{A}}$
$\displaystyle+{\psi_{B}}\left(\frac{2Q_{B}}{e_{B}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{B}^{2}}\sin{\theta_{B}}d{\theta_{B}}\wedge
d{\phi_{B}}\right)+\frac{2}{e_{B}}\cos{\theta_{B}}Q_{B}\zeta\left(r\right)d\varphi\wedge
d{\phi_{B}}\ .$ $\displaystyle C_{6}$ $\displaystyle=$
$\displaystyle-\frac{2e_{A}r^{2}}{e^{3}_{B}}dt\wedge dx\wedge
d\varphi\wedge\text{Vol}(S^{3}_{B})+\frac{2e_{B}r^{2}}{e^{3}_{A}}dt\wedge
dx\wedge d\varphi\wedge\text{Vol}(S^{3}_{A}),$
$\displaystyle-\frac{8Q_{B}}{e^{3}_{A}e^{2}_{B}}\cos(\theta_{B})dt\wedge
dx\wedge\text{Vol}(S^{3}_{A})\wedge
d\phi_{B}+\frac{8Q_{A}}{e^{3}_{B}e^{2}_{A}}\cos(\theta_{A})dt\wedge
dx\wedge\text{Vol}(S^{3}_{B})\wedge d\phi_{A},$ $\displaystyle\Phi$
$\displaystyle=$ $\displaystyle\log r\ .$

This is the background in eq.(2.3). In the case for which $Q_{A}=Q_{B}=m=0$,
the background fields read (note that we S-dualise moving to the NS5 brane
frame),

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+\frac{e^{2}_{A}+e^{2}_{B}}{2}d\varphi^{2}+\frac{8}{e^{2}_{A}+e^{2}_{B}}\frac{dr^{2}}{r^{2}}+\frac{8}{e^{2}_{A}}ds^{2}(S^{3}_{A})+\frac{8}{e^{2}_{B}}ds^{2}(S^{3}_{B}),$
(B.36) $\displaystyle H_{3}$
$\displaystyle=-\frac{8}{e^{2}_{A}}\text{Vol}(S^{3}_{A})-\frac{8}{e^{2}_{B}}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$ $\displaystyle=-\log(r).$

This is the background in eq.(2.1). We can rescale $\varphi$ to absorb the
prefactor. Also, it is convenient to set as in eq.(2.9)

$N_{A}=\frac{8}{e^{2}_{B}},\quad N_{B}=\frac{8}{e^{2}_{A}},$ (B.37)

and perform the change of coordinates

$r=e^{\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\,\rho},$ (B.38)

after changing $H_{3}\rightarrow-H_{3}$, the background reads

$\displaystyle ds^{2}_{10D}$
$\displaystyle=-dt^{2}+dx^{2}+d\varphi^{2}+d\rho^{2}+N_{B}ds^{2}(S^{3}_{A})+N_{A}ds^{2}(S^{3}_{B}),$
(B.39) $\displaystyle H_{3}$
$\displaystyle=N_{A}\text{Vol}{S^{3}_{A}}+N_{B}\text{Vol}{S^{3}_{B}},$
$\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\,\rho.$

This is the background written in Section 5 to study the string $\sigma$-model
on this field configuration.

##  Appendix C On the Unfibered Geometry

Here, we review a different derivation of the background (5.1). The S-dual of
this background (2.1) was first introduced in [13], [15], here we review the
derivation of the pure NS-NS frame for simplicity. Let us consider two stacks
of $NS5$-branes, the first extended in $(t,x,y_{1},y_{2},y_{3},y_{4})$ and
while the second one spans $(t,x,w_{1},w_{2},w_{3},w_{4})$. These stacks
intersect in the $(t,x)$ directions, thus in the weak coupling regime, the
effective theory on the intersection is 1+1 dimensional and preserves 8
Supercharges.

We now move to the strong coupling regime. For this, we write the space
$\mathbb{R}^{4}_{y}=(y_{1},y_{2},y_{3},y_{4})$ in spherical coordinates
$(r_{A},S^{3}_{A})$, and similarly for $\mathbb{R}^{4}_{w}$ we use
$(r_{B},S^{3}_{B})$. In terms of the harmonic functions

$H_{A}(r_{A})=1+\frac{N_{B}}{r^{2}_{A}},\quad
H_{B}(r_{B})=1+\frac{N_{A}}{r^{2}_{B}},$ (C.1)

the backreacted fields are given by

$\displaystyle ds^{2}_{st}$
$\displaystyle=dx^{2}_{1,1}+H_{A}(r_{A})\left(dr^{2}_{A}+r^{2}_{A}ds^{2}(S^{3}_{A})\right)+H_{B}(r_{B})\left(dr^{2}_{B}+r^{2}_{B}ds^{2}(S^{3}_{B})\right),$
(C.2) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$ $\displaystyle=\frac{1}{2}\log(H_{A}(r_{A})H_{B}(r_{B})).$

By taking the near-horizon geometry we are led to

$\displaystyle ds^{2}_{st}$
$\displaystyle=dx^{2}_{1,1}+N_{B}\frac{dr^{2}_{A}}{r^{2}_{A}}+N_{A}\frac{dr^{2}_{B}}{r^{2}_{B}}+N_{B}ds^{2}(S^{3}_{A})+N_{A}ds^{2}(S^{3}_{B}),$
(C.3) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$
$\displaystyle=-\log(r_{A})-\log(r_{B})+\frac{1}{2}\log(N_{A}N_{B}).$

Under the change of coordinates $r_{A}=e^{\rho_{A}/\sqrt{N_{B}}}$ and
$r_{B}=e^{\rho_{B}/\sqrt{N_{A}}}$ we obtain

$\displaystyle ds^{2}_{st}$
$\displaystyle=dx^{2}_{1,1}+d\rho^{2}_{A}+d\rho^{2}_{B}+N_{B}ds^{2}(S^{3}_{A})+N_{A}ds^{2}(S^{3}_{B}),$
(C.4) $\displaystyle H_{3}$
$\displaystyle=N_{B}\text{Vol}(S^{3}_{A})+N_{A}\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1}{N_{B}}}\rho_{A}-\sqrt{\frac{1}{N_{A}}}\rho_{B}+\frac{1}{2}\log(N_{A}N_{B}).$

Finally, taking the linear combination

$\displaystyle\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\rho_{A}$
$\displaystyle=\sqrt{\frac{1}{N_{B}}}\rho-\sqrt{\frac{1}{N_{A}}}\varphi,$
(C.5) $\displaystyle\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\rho_{B}$
$\displaystyle=\sqrt{\frac{1}{N_{A}}}\rho+\sqrt{\frac{1}{N_{B}}}\varphi,$

we reach the background fields in (5.1)

$\displaystyle ds^{2}_{st}$
$\displaystyle=-dt^{2}+dx^{2}+d\varphi^{2}+d\rho^{2}+N_{B}\,ds^{2}(S^{3}_{A})+N_{A}\,ds^{2}(S^{3}_{B}),$
(C.6) $\displaystyle H_{3}$
$\displaystyle=N_{B}\,\text{Vol}(S^{3}_{A})+N_{A}\,\text{Vol}(S^{3}_{B}),$
$\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1}{N_{A}}+\frac{1}{N_{B}}}\,\rho,$

where we cancelled the constant term of the Dilaton by a suitable shift of its
zero mode. Note that the Dilaton in (C.4) has functional dependence in two of
the coordinates of the background, while the one in (C.6) only depends only on
one coordinate. This allows us to interpret the extra flat direction of (C.6)
as being part of the Field Theory ones. In this way, we see that in the strong
coupling regime, the theory on the intersection acquires an extra dimension,
becoming (2+1) dimensional. There is also a SUSY enhancement from 8 to 16
supercharges. We refer the reader to [13] to cover this matter.

##  Appendix D Charges of the black hole background

We compute the charges of the configuration considering the Noether-Wald
method [17]. The bulk action principle in string frame of IIB in the metric-
dilaton-$F_{3}$ sector is

$S_{\text{IIB,bulk}}=\frac{1}{2\kappa^{2}}\int
d^{10}x\sqrt{-g}\left(R-\frac{1}{2}\left(\partial\Phi\right)^{2}-\frac{1}{12}e^{\Phi}F_{\mu\nu\rho}F^{\mu\nu\rho}\right)=\int
d^{10}x\sqrt{-g}\mathcal{L}\ ,$ (D.1)

A general variation of the action gives

$\delta S_{\text{IIB,bulk}}=\int d^{10}x\sqrt{-g}\left[\delta
g^{\mu\nu}\mathcal{E}_{\mu\nu}^{\left(g\right)}+\delta\Phi\mathcal{E}^{\left(\Phi\right)}+\delta
C_{\nu\rho}\mathcal{E}_{\left(F_{3}\right)}^{\nu\rho}+\nabla_{\mu}\Theta^{\mu}\left(\mathbf{f},\delta\mathbf{f}\right)\right]\
.$ (D.2)

where $\mathbf{f}$ denotes the fields collectively and

$\displaystyle\Theta^{\mu}\left(\mathbf{f},\delta\mathbf{f}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\left(g^{\delta\eta}\delta\Gamma_{\
\eta\delta}^{\mu}-g^{\delta\mu}\delta\Gamma_{\
\lambda\delta}^{\lambda}-\delta\Phi\partial^{\mu}\Phi-e^{\Phi}\delta
C_{\nu\rho}F^{\mu\nu\rho}\right)\ ,$ (D.3)
$\displaystyle\mathcal{E}_{\mu\nu}^{\left(g\right)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\left[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right.$
$\displaystyle-\frac{1}{2}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\partial_{\rho}\Phi\partial^{\rho}\Phi\right)$
$\displaystyle\left.-\frac{1}{2}e^{\Phi}\left(\frac{1}{2}F_{\mu\delta\rho}F_{\nu}^{\
\delta\rho}-\frac{1}{12}g_{\mu\nu}F_{\delta\rho\sigma}F^{\delta\rho\sigma}\right)\right]\
,$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\left(G_{\mu\nu}-\frac{1}{2}T_{\mu\nu}^{\left(\Phi\right)}-\frac{1}{2}T_{\mu\nu}^{\left(F_{3}\right)}\right)\
,$ $\displaystyle\mathcal{E}^{\left(\Phi\right)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\left(\nabla^{\rho}\nabla_{\rho}\Phi-\frac{1}{12}e^{\Phi}F_{\mu\nu\rho}F^{\mu\nu\rho}\right)\
,$ (D.5) $\displaystyle\mathcal{E}_{\left(F_{3}\right)}^{\nu\rho}$
$\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\nabla_{\mu}\left(e^{\Phi}F^{\mu\nu\rho}\right)\
.$ (D.6)

The Noether current is defined by

$J^{\mu}=\Theta^{\mu}\left(\mathbf{f},\mathcal{L}_{\xi}\mathbf{f}\right)-\xi^{\mu}\mathcal{L}\mathbb{\
},$ (D.7)

where $\mathcal{L}$ is the Lagrangian scalar under diffeomorphisms in (D.1)
and $\mathcal{L}_{\xi}$ is the Lie derivative along the vector $\xi$. The
Noether current is conserved on-shell, thus it can be written locally as
$J^{\mu}=\nabla_{\nu}q^{\mu\nu}$. The Noether current (D.7) for our system
gives

$J^{\mu}=-\frac{1}{\kappa^{2}}\nabla_{\nu}\left(\nabla^{[\mu}\xi^{\nu]}+\frac{1}{2}e^{\Phi}2C_{\lambda\rho}\xi^{\lambda}F^{\mu\nu\rho}\right)+2\xi^{\lambda}\mathcal{E}_{\left(g\right)\lambda}^{\mu}-2\xi^{\lambda}C_{\lambda\rho}\mathcal{E}_{\left(F_{3}\right)}^{\mu\rho}\
,$ (D.8)

on-shell it defines the Noether pre-potential:

$q^{\mu\nu}\left(\xi\right)=-\frac{1}{\kappa^{2}}\left(\nabla^{[\mu}\xi^{\nu]}+\frac{1}{2}e^{\Phi}2C_{\lambda\rho}\xi^{\lambda}F^{\mu\nu\rho}\right)\
.$ (D.9)

The Hodge dual of the Noether pre-potential gives the 8-form

$\boldsymbol{Q}\left[\xi\right]=\frac{1}{2}\frac{1}{8!}\sqrt{-g}\epsilon_{\mu\nu\rho_{1}\dots\rho_{8}}q^{\mu\nu}dx^{\rho_{1}}\wedge\dots\wedge
dx^{\rho_{8}}\ ,$ (D.10)

that in differential forms is

$\boldsymbol{Q}\left[\xi\right]=-\frac{1}{\kappa^{2}}\left(\star
d\xi+e^{\Phi}\xi\lnot C_{2}\wedge\star F_{3}\right)\ .$ (D.11)

$\lnot$ stands for the contraction operator.

The boundary term that allow us to have a well posed action principle and
finite mass is

$S_{\text{full}}=S_{\text{IIB,bulk}}+\int_{\partial
M}d^{9}x\sqrt{-h}\frac{1}{\kappa^{2}}\left(\mathcal{K}-e^{-\frac{1}{4}\Phi}\right)\
,$ (D.12)

where the first term in the integral is the Gibbons-Hawking-York term and the
last term corresponds to a counter term, that depends only on intrinsic
quantities, that allow us to renormalize the mass term. The extrinsic
curvature is defined in terms of the normal unit outwards vector $n^{\mu}$ to
the boundary of the spacetime by

$\mathcal{K}_{\mu\nu}=h_{\ \mu}^{\rho}h_{\
\nu}^{\sigma}\nabla_{\rho}n_{\sigma}\ ,$ (D.13)

and the induced metric is $h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$ for our case.

Following [17] the energy, angular momentum and entropy are defined by

$\displaystyle\mathcal{E}\left[\boldsymbol{t}\right]$ $\displaystyle=$
$\displaystyle\int_{\infty}\left(\boldsymbol{Q}\left[\boldsymbol{t}\right]-\xi\lnot\boldsymbol{B}\right)\
,$ (D.14) $\displaystyle\mathcal{J}\left[\boldsymbol{\psi}\right]$
$\displaystyle=$
$\displaystyle-\int_{\infty}\boldsymbol{Q}\left[\boldsymbol{\psi}\right]\ ,$
(D.15) $\displaystyle S\left[\boldsymbol{\xi}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{\mathcal{H}}\boldsymbol{Q}\left[\boldsymbol{\xi}\right]\
.$ (D.16)

The boundary terms are in the 9-form

$\boldsymbol{B}=-\frac{1}{\kappa^{2}}\left(\mathcal{K}-e^{-\frac{1}{4}\Phi}\right)\star
n\ .$ (D.17)

$\boldsymbol{t}$ is the time-like killing vector at infinity properly
normalized, $\boldsymbol{\psi}$ is the rotation generator and
$\boldsymbol{\xi}$ is the horizon generator

$\boldsymbol{\xi}=\boldsymbol{t}+\Omega\boldsymbol{\psi\ .}$ (D.18)

$\boldsymbol{\xi}$ is null at the Horizon which defines the angular velocity
$\Omega$ and satisfies the geodesic equation at the horizon

$\xi^{\mu}\nabla_{\mu}\xi^{\nu}=\kappa_{s}\xi^{\nu}\ ,$ (D.19)

defining the surface gravity $\kappa_{s}$ that is related to the temperature
as $T=\frac{\kappa_{s}}{2\pi}$. In this case we are in general relativity,
therefore the entropy give one-quarter of the horizon area.

Let us consider the black hole configuration in Einstein frame (2.11), with

$\zeta(r)=\frac{1}{r^{2}},\qquad\Phi\to\Phi-2\log\left(\frac{e_{A}^{2}+e_{B}^{2}}{2}\right),\qquad
F_{3}\to\frac{e_{A}^{2}+e_{B}^{2}}{2}F_{3}\,\,.$ (D.20)

The in-going Eddington-Finkelstein coordinates are well-defined at the
horizon, which are defined by

$\displaystyle dt$ $\displaystyle=$ $\displaystyle
dv-\frac{2dr}{rf_{bh}\left(r\right)}\ ,$ (D.21) $\displaystyle d\psi_{A}$
$\displaystyle=$ $\displaystyle
d\psi_{A}^{\prime}-\frac{2e_{A}Q_{A}\zeta\left(r\right)}{rf_{bh}\left(r\right)}dr\
,$ (D.22) $\displaystyle d\psi_{B}$ $\displaystyle=$ $\displaystyle
d\psi_{B}^{\prime}-\frac{2e_{B}Q_{B}\zeta\left(r\right)}{rf_{bh}\left(r\right)}dr\
.$ (D.23)

Then, the metric becomes

$\displaystyle ds_{E}^{2}$ $\displaystyle=$
$\displaystyle\sqrt{r}\left\\{dy^{2}+dx^{2}-f_{bh}\left(r\right)dv^{2}+\frac{4}{r}drdv\right.$
$\displaystyle+\frac{2}{e_{A}^{2}}\left[d\theta_{A}^{2}+\sin^{2}\theta_{A}d\phi_{A}^{2}+\left(d\psi_{A}^{\prime}+\cos\theta_{A}d\phi_{A}-e_{A}Q_{A}\zeta\left(r\right)dv\right)^{2}\right]$
$\displaystyle\left.+\frac{2}{e_{B}^{2}}\left[d\theta_{B}^{2}+\sin^{2}\theta_{B}d\phi_{B}^{2}+\left(d\psi_{B}^{\prime}+\cos\theta_{B}d\phi_{B}-e_{B}Q_{B}\zeta\left(r\right)dv\right)^{2}\right]\right\\}\
,$

In these coordinates we consider the vector

$\boldsymbol{\xi}=\boldsymbol{t}+\Omega_{A}\boldsymbol{\psi}_{A}+\Omega_{B}\boldsymbol{\psi}_{B}$
(D.25)

where

$\boldsymbol{t}=\frac{1}{2}\frac{\partial}{\partial v}\
,\qquad\boldsymbol{\psi}_{A}=\frac{\partial}{\partial\psi_{A}}\
,\qquad\boldsymbol{\psi}_{B}=\frac{\partial}{\partial\psi_{B}}\ .$ (D.26)

The vector $\boldsymbol{\xi}$ is null at the horizon located at $r_{+}$ when

$\Omega_{A}=\frac{e_{A}Q_{A}}{r_{+}^{2}}\ ,\qquad
Q_{B}=\frac{e_{B}Q_{B}}{r_{+}^{2}}\ .$ (D.27)

Due to the fact that the spacetime that we are considering is not
asymptotically Minkowski times $S^{3}\times S^{3}$, instead is conformal to
Minkowski times $S^{3}\times S^{3}$, it is not clear how we should normalize
the vector $\boldsymbol{t}$ time-like at infinity. This ambiguity propagates
to the energy and the temperature. Therefore we expect to obtain the
temperature in 4D up to a factor. The energy (D.14), angular momentum (D.15),
temperature defined through (D.19) and the entropy give

$\displaystyle E$ $\displaystyle=$
$\displaystyle\mathcal{E}[\boldsymbol{t}]=\frac{2m}{e_{A}^{3}e_{B}^{3}r_{+}^{2}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,$ (D.28) $\displaystyle J_{A}$ $\displaystyle=$
$\displaystyle\mathcal{J}[\boldsymbol{\psi_{A}}]=\frac{8Q_{A}}{e_{A}^{4}e_{B}^{3}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,$ (D.29) $\displaystyle J_{B}$ $\displaystyle=$
$\displaystyle\mathcal{J}[\boldsymbol{\psi_{B}}]=\frac{8Q_{B}}{e_{B}^{4}e_{A}^{3}\kappa^{2}}\left(16\pi^{2}\right)^{2}L_{x}L_{y}\
,$ (D.30) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{e_{A}^{2}+e_{B}^{2}}{16\pi}-\frac{4\left(Q_{A}^{2}+Q_{B}^{2}\right)}{16\pi
r_{+}^{4}}\ ,\quad
S=\frac{2r_{+}^{2}}{e_{A}^{3}e_{B}^{3}G_{10}}\left(16\pi\right)^{2}L_{x}L_{y}\
.$ (D.31)

They satisfy the first law of thermodynamics

$dE=TdS+\Omega_{A}dJ_{A}+\Omega_{B}dJ_{B}\ .$ (D.32)

##  Appendix E R-Symmetry Breaking

In this appendix we give a detailed derivation of the symmetry breaking
pattern of three $U(1)$ directions present in our background. In order to do
this, we gauge this symmetries by introducing a gauge field $A$ and a scalar
$\epsilon$. The presence of a mass term in the effective action of the gauge
field, i.e. an explicit symmetry breaking of the gauge symmetry, signals the
breaking of this $U(1)$ symmetry on the dual field theory. In the QFT, the
breaking of the global symmetry can be either spontaneous or anomalous.

###  E.1 $U(1)$ R-Symmetry of $\psi_{A}$ and $\psi_{B}$

Let us start by recalling the R-R $C_{2}$ potential

$\displaystyle
C_{2}={\psi_{A}}\left(\frac{2Q_{A}}{e_{A}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{A}^{2}}\sin{\theta_{A}}d{\theta_{A}}\wedge
d{\phi_{A}}\right)+\frac{2}{e_{A}}\cos{\theta_{A}}Q_{A}\zeta\left(r\right)d\varphi\wedge
d{\phi_{A}}$
$\displaystyle+{\psi_{B}}\left(\frac{2Q_{B}}{e_{B}}\zeta^{\prime}\left(r\right)dr\wedge
d\varphi-\frac{2}{e_{B}^{2}}\sin{\theta_{B}}d{\theta_{B}}\wedge
d{\phi_{B}}\right)+\frac{2}{e_{B}}\cos{\theta_{B}}Q_{B}\zeta\left(r\right)d\varphi\wedge
d{\phi_{B}}\ .$ (E.1)

Since this potential is not invariant under
$\psi_{A,B}\rightarrow\psi_{A,B}+4\pi$, we expect this symmetry to be broken
in the dual field theory. We gauge these isometries by doing the following
replacements in the R-R potential and the metric

$d\psi_{A,B}\rightarrow
d\psi_{A,B}+A_{A,B},\quad\psi_{A,B}\rightarrow\psi_{A,B}+\epsilon_{A,B}.$
(E.2)

Where $A_{A,B}$ is a $U(1)$ gauge field and $\epsilon_{A,B}$ is a scalar
charged under the gauged $U(1)$ symmetry, which makes the combination
$D_{A,B}\epsilon=\partial_{A,B}\epsilon_{A,B}-A_{A,B}$ is gauge invariant
(here $\partial_{A,B}=\partial/\partial\psi_{A,B}$). These fields only depend
on the coordinates of the field theory directions, i.e. $(t,x)$. After these
replacements the metric and the R-R 3-form read

$\displaystyle\begin{aligned}
ds^{2}=&ds^{2}_{(0)}+\frac{4r}{e^{2}_{A}}\left(\hat{\omega}_{3}-e_{A}Q_{A}\xi(r)d\varphi\right)A_{A\,\mu}dx^{\mu}+\frac{2r}{e^{2}_{A}}A_{A\,\mu}A_{A\,\nu}dx^{\mu}dx^{\nu}\\\
&+\frac{4r}{e^{2}_{B}}\left(\tilde{\omega}_{3}-e_{B}Q_{B}\xi(r)d\varphi\right)A_{B\,\mu}dx^{\mu}+\frac{2r}{e^{2}_{B}}A_{B\,\mu}A_{B\,\nu}dx^{\mu}dx^{\nu}\end{aligned}$
(E.3) $\displaystyle\begin{aligned}
F_{3}=&F^{(0)}_{3}-2d\epsilon_{A}\wedge\left(-\frac{Q_{A}}{e_{A}}\xi^{\prime}(r)dr\wedge
d\varphi+\frac{1}{e^{2}_{A}}\text{Vol}(S^{2}_{A})\right)\\\
&-2d\epsilon_{B}\wedge\left(-\frac{Q_{B}}{e_{B}}\xi^{\prime}(r)dr\wedge
d\varphi+\frac{1}{e^{2}_{B}}\text{Vol}(S^{2}_{B})\right)\end{aligned}$ (E.4)

where $ds^{2}_{(0)}$ and $F^{(0)}_{3}$ denotes the metric and the 3-form of
the configuration before gauging the $U(1)$ symmetries and
$\text{Vol}(S^{2}_{A,B})=\sin(\theta_{A,B})d\theta_{A,B}\wedge d\phi_{A,B}$.
Now we want to obtain an effective lagrangian for $A_{A,B}$ and
$\epsilon_{A,B}$. In order to do this, we consider how the Ricci scalar and
the kinetic term of $F_{3}$ change under the gauging of the symmetry.
Explicitly, the Ricci scalar transforms as

$R=R^{(0)}-\frac{1}{4}\frac{2r}{e^{2}_{A}}F^{2}_{A}-\frac{1}{4}\frac{2r}{e^{2}_{B}}F^{2}_{B},$
(E.5)

where $F^{2}_{A,B}=F_{A,B\,\mu\nu}F^{\mu\nu}_{A,B}$ and $F_{A,B\,\mu\nu}$ is
the field strengh of $A_{A,B\,\mu}$, while the kinetic term of the R-R
potential reads

$\displaystyle\frac{1}{12}F_{\mu\nu\lambda}F^{\mu\nu\lambda}=$
$\displaystyle\frac{1}{12}F^{(0)}_{\mu\nu\lambda}F_{(0)}^{\mu\nu\lambda}+\frac{1}{2r^{2}}\left(\frac{Q^{2}_{A}r^{2}}{e^{2}_{A}}\xi^{\prime}(r)^{2}+1\right)\left(A_{A}-d\epsilon_{A}\right)^{2}$
(E.6)