SCFT deformations via uplifted solitons

Dimitrios<EMAIL_ADDRESS>Ali
<EMAIL_ADDRESS>Carlos
<EMAIL_ADDRESS>and Peter<EMAIL_ADDRESS>

Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom

Abstract

A holographic method for implementing a particular supersymmetry-preserving
deformation to 4d SCFTs is presented. At the heart of the procedure is a
soliton solution of minimal $d=5$ gauged supergravity. Embedding this solution
into ten- and eleven-dimensional string theory backgrounds of the form
AdS${}_{5}\times M$, we systematically construct a range of new solutions.
Each holographically realizes a twisted compactification of the SCFT4 dual to
the original background. In the IR, the resulting SQFTs flow to gapped three-
dimensional systems. Using a variety of holographic observables, we give
evidence for this interpretation and for confinement in the deformed SQFTs.
Our method applies to any holographic solutions admitting a consistent
truncation to minimal $d=5$ gauged supergravity, and can likely be generalized
to solutions with other AdSd factors.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 The deformation
    3. 1.3 Summary of contents
  2. 2 Review of the supersymmetric AdS5 soliton
  3. 3 Deformed backgrounds in Type IIB
    1. 3.1 AdS${}_{5}\times S^{5}$ embedding
    2. 3.2 AdS${}_{5}\times Y^{p,q}$ embeddings
  4. 4 Deformed backgrounds in Type IIA
    1. 4.1 Embedding into IIA duals of $\mathcal{N}=1$ SCFTs
    2. 4.2 Application to reductions of 6d theories on $\Sigma_{2}$
      1. 4.2.1 AFPRT example
      2. 4.2.2 BPT example
  5. 5 Deformed backgrounds in 11d
    1. 5.1 Embedding via 5d Romans supergravity
    2. 5.2 LLM embedding
    3. 5.3 The electrostatic case
      1. 5.3.1 Rank function and Page charges
  6. 6 Observables
    1. 6.1 Wilson loop
    2. 6.2 ’t Hooft loop
    3. 6.3 Entanglement Entropy
    4. 6.4 Flow central charge
    5. 6.5 Holographic complexity
    6. 6.6 Gauge coupling
    7. 6.7 Global $U(1)$ symmetry breaking
  7. 7 Discussion
  8. A SUSY variations in 5d
  9. B $a$-anomaly match for reductions of 6d linear quivers
  10. C Gaiotto-Maldacena background Page charges

##  1 Introduction

In this rather long and dense paper we study the holographic description of UV
CFTs that are deformed by VEVs, flowing to confining and gapped systems. The
topic is motivated from both the holographic and field-theoretic viewpoints.
We start the introduction with our general motivation (in a historical
framework), describe the more particular deformation we aim to apply, and then
discuss the systems in which we study it, ending with a summary of the
contents of this work. The introduction is lengthy so that the reader
interested in just the basic idea can focus on section 1.2, while the level of
detail in the rest of the paper is intended for colleagues who may want to
work in this area. A shorter companion paper [1] presents the highlights.

###  1.1 Background and motivation

After the formulation of the AdS/CFT correspondence[2] and its initial
refinements [3, 4], extending the idea to Quantum Field Theories (QFTs) with
less symmetries than ${\cal N}=4$ Super Yang-Mills became a natural goal. The
extension to either gapped theories or conformal theories with less symmetries
allowed for the study of phenomena like confinement, symmetry breaking, and
the presence of condensates from the new holographic point of view.

Papers like [5] and [6] dealing with non-conformal field theories in different
dimensions initiated this line of research. These were followed by [7, 8, 9],
which studied QFTs with minimal or no-SUSY in four dimensions using their
geometric realisations. Soon after that, the works [10, 11, 12, 13, 14, 15,
16], produced a very clear geometric picture of non-perturbative aspects of a
two node quiver field theory with ${\cal N}=1$ SUSY (a quasi-marginal
deformation of the Klebanov-Witten ${\cal N}=1$ conformal field theory,
flowing to a confining IR ${\cal N}=1$ QFT). For a summary of these
developments, see [17].

A second line of work based on wrapped branes was pursued in [7, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29], extending the duality between gauge
fields and strings to non-conformal QFTs in diverse dimensions. See [30, 31,
32] for pedagogical reviews of this work, which gave a geometric
interpretation to various non-perturbative aspects of QFT. These two lines of
research were joined beautifully in the works [33, 34, 35, 36].

The addition of fields transforming in the fundamental representation of the
gauge group was discussed first in the quenched/probe approximation. See [37,
38, 39, 40, 41, 42, 43] for some representative papers and a review. This was
later improved by including the backreaction of these sources. The flavour
branes were ‘smeared’ so that one can work with BPS ODEs rather than BPS PDEs
[44, 45, 46, 47, 48, 49, 50, 51, 52].

However, some unwelcome features of these models are the following:

  * •

When the high-energy behaviour of the QFT is field theoretical and represented
by a deformed 4d UV-CFT, the IR part of the holographic dual background is
singular. This is the case e.g. for the models in [8, 9, 53, 54].

  * •

On the other hand, for the wrapped brane models [18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29] or for the two node quiver system described above [10, 11,
12, 13, 14, 15, 16], the high energy behaviour of the QFT does not strictly
reach a conformal point in the UV. This is reflected in the fact that the
holographic dual is not strictly asymptotically AdS, making the application of
holographic renormalisation technically challenging.

  * •

The addition of a large number of flavours, as explained above, is technically
cumbersome, requiring the smearing of branes. As a consequence, it is hard to
study CFTs with flavours in four or more dimensions.

Our motivation is to remedy some of these unwelcome features. To do this we
consider four-dimensional super-conformal field theories (SCFTs), whose dual
backgrounds have an AdS5 factor in the geometry. The SCFTs include a large
number of (localised) flavour branes. These SCFTs are deformed by VEVs and a
RG-flow is triggered, ending in a confining and gapped system. The holographic
description mirrors this, featuring an asymptotically AdS5 space that ends
smoothly at a fixed value of the radial coordinate. Non-perturbative aspects
of the RG-flow in the QFT can be consistently studied. To implement these
ideas, we consider a deformation on the holographic side, which we describe
below.

###  1.2 The deformation

The goal of this work is to establish a general holographic mechanism to
obtain confining, (2+1)-dimensional SQFTs from known (3+1)-dimensional SCFTs.
Our inspiration is [55] (see also [56, 57, 58, 59, 60, 61, 62] for related
work), in which a supersymmetry-preserving AdS5 soliton solution is
identified. From the perspective of AdS${}_{5}/$CFT4, the solution of [55]
realizes a compactification of $\mathcal{N}=4$ SYM, leading to an
$\mathcal{N}=2$ SYM theory (plus massive multiplets) in three dimensions.
Crucially, some supersymmetry remains unbroken, thanks to a mixing of the
R-symmetry of $\mathcal{N}=4$ SYM with the isometry for the compact circle. We
refer to this as a ‘twisted’ compactification throughout, although the compact
manifold is just a circle (and thus flat). The field theory story for
$\mathcal{N}=4$ SYM is discussed in detail in [63].

We have generalized this mechanism to a variety of SCFTs with holographic
duals of the form

$\text{AdS}_{5}\times M_{n},$ (1.1)

where $n=5$ for Type II and $n=6$ for 11d supergravity examples. In
particular, using known embeddings of $d=5$ minimal gauged supergravity
solutions into backgrounds the form (1.1), we have uplifted the soliton
solution of [55]. The result is new families of smooth string theoretic
backgrounds of the form

$\widehat{\text{AdS}_{5}}\times\widehat{M_{n}}$ (1.2)

where the hats denote a deformation which can be summarized as follows:

  * •

Compactify one of the AdS5 directions to a circle $S_{\phi}^{1}$ of size
$L_{\phi}/2\pi$.

  * •

Deform the AdS5 geometry by introducing a warping function $f(r)$ which
smoothly caps off the $S^{1}_{\phi}$ circle at a finite radius $r=r_{\star}$,
at which $f(r_{\star})=0$,

$\frac{r^{2}}{l^{2}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+d\phi^{2})+\frac{l^{2}}{r^{2}}dr^{2}\quad\rightarrow\quad\frac{r^{2}}{l^{2}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+f(r)d\phi^{2})+\frac{l^{2}}{r^{2}}~{}\frac{dr^{2}}{f(r)},$
(1.3)

with the solution parameters constrained by
$L_{\phi}=\frac{l^{2}}{r_{\star}^{2}}\frac{4\pi}{f^{\prime}(r_{\star})}$ to
avoid conical singularities.

  * •

Identify an appropriate $U(1)$ inside the isometry group of the internal
manifold $M_{n}$, and gauge it by

$\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{\star}^{2}}\right)d\phi\,,$
(1.4)

suitably modifying all the fluxes to ensure the full ten- or eleven-
dimensional supergravity equations of motion are satisfied.

Requiring $r_{\star}=(ql)^{1/3}$ ensures the preservation of four supercharges
for the $d=5$ gauged supergravity solution we are uplifting [55]. Thanks to
the embedding frameworks we are relying on [64, 65, 66], this ensures the
preservation of higher-dimensional supersymmetry. As a result, all of the
examples we will present preserve ten- or eleven-dimensional supersymmetry for
this choice of solution parameters.

Let us return to the field-theoretic interpretation of these new solutions.
The internal manifold $M_{n}$ of the undeformed backgrounds (1.1) encodes
various data of the dual SCFT4. In our examples, we will encounter not only
the case of $M_{5}=S^{5}$ and the $\mathcal{N}=4$ SYM dual, but also
$M_{5}=Y^{p,q}$ describing $\mathcal{N}=1$ toric quiver field theories,
internal manifolds describing $\mathcal{N}=2$ linear quivers, as well as some
for $\mathcal{N}=1$ non-Lagrangian SCFTs. When we deform each background in
the sense described above, we are implementing a particular twisted
compactification in the dual field theory.

When compactifying on $S_{\phi}^{1}$ one needs to specify boundary conditions
for the various fields in the SCFT. This generically breaks SUSY, as the
scalars and gauge fields are assigned periodic boundary conditions while the
fermions are anti-periodic. As alluded to earlier, SUSY can be preserved by
turning on a background gauge field $\mathcal{A}=\mathcal{A}_{\phi}\,d\phi$
which mixes a $U(1)$ inside the R-symmetry with the $U(1)_{\phi}$ isometry of
the compact circle.555While the background gauge field is constant in the
boundary field theory, it has a non-trivial holonomy, and thus cannot be
absorbed by a gauge transformation. The reason is that the background gauge
field modifies the covariant derivative. If its charge is tuned appropriately,
massless fermions can exist and enter into supermultiplets.

The introduction of the background gauge field and the scale $L_{\phi}$ into
the field theory break conformality while preserving some supersymmetry
(again, for an appropriate choice of parameters). At low energies, our
holographic construction of this twisted compactification takes the original
SCFT4 to a strongly coupled SQFT3, thanks to the closing of $S_{\phi}^{1}$ by
$f(r)$. Employing a variety of holographic observables, we find clear
indications of confinement in these (2+1)-dimensional IR theories.

###  1.3 Summary of contents

We will begin by reviewing $d=5$ minimal gauged supergravity and the soliton
solution of [55] in section 2. The remainder of the paper will concern uplifts
of this solution to ten- and eleven-dimensional supergravity and how
holography can tell us about the resulting QFT duals.

In section 3, we use the embedding prescriptions of [67, 64] to lift into
backgrounds in Type IIB. Specifically we review how to obtain the deformation
of AdS${}_{5}\times S^{5}$ first noted in [55], and introduce new deformed
AdS${}_{5}\times Y^{p,q}$ backgrounds. To uplift into massive Type IIA, in
section 4 we use the embedding scheme developed in [66]. This allows us to
write down our deformation for any AdS5 solution in massive IIA dual to an
$\mathcal{N}=1$ SCFT. We focus on two particular families, based on the work
of [68, 69, 70]. Section 5 utilizes the fact that minimal $d=5$ gauged
supergravity can be treated as a truncation of $d=5$ Romans’ $SU(2)\times
U(1)$ supergravity, allowing us to lift from there to eleven-dimensional
supergravity using [65]. Backgrounds based on the LLM family of solutions [71]
are presented. Restricting to the “electrostatic case” allows us to make
contact with the holographic solutions of Gaiotto-Maldacena [72].

Having presented a wide range of deformed backgrounds, in section 6 we study
the features of their dual QFTs. Wilson, ’t Hooft loop, and Entanglement
Entropy calculations reveal signatures of confinement in the IR three-
dimensional systems, as presented in sections 6.1, 6.2, and 6.3. Probing the
(deformed) AdS directions, these observables show universalities across all
the examples studied. In section 6.4, we present an observable capturing the
number of degrees of freedom along the full flow from UV SCFT4 to IR (S)QFT3,
known as the flow central charge. Holographic complexity is also studied.
Other probes engaging the internal manifold are presented to study
particularities of the SQFTs in section 6.6. The spontaneous symmetry breaking
of $U(1)_{\phi}$ is considered in 6.7, appearing as a massive vector mode in
the bulk.

We conclude with a discussion section and several appendices collecting more
technical details.

##  2 Review of the supersymmetric AdS5 soliton

In this section, we study the original 5d minimal gauged supergravity solution
of [55], which is used as a seed to obtain different backgrounds investigated
in this paper. The supersymmetry of the solution is discussed in appendix A

Consider the Einstein-Maxwell-AdS system in five dimensions, developed in
different works [73, 74, 75, 76, 77, 67]. The bosonic part of the action is

$S\left(g,A\right)=\frac{1}{16\pi G}\int
d^{5}x\sqrt{-g}\left(R+\frac{12}{l^{2}}-\frac{3}{4}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\right)+\frac{1}{16\pi
G}\int\sqrt{-g}\mathcal{F}\wedge\mathcal{F}\wedge\mathcal{A}\text{ },$ (2.1)

where $l$ is the AdS radius. The equations of motion are

$\displaystyle d\star\mathcal{F}+\mathcal{F}\wedge\mathcal{F}=0\,\,,$
$\displaystyle
R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R-\tfrac{3}{2}\big{[}\mathcal{F}_{\mu\rho}\,\mathcal{F}_{\nu}{}^{\rho}-\tfrac{1}{6}g_{\mu\nu}\mathcal{F}_{\rho\sigma}\mathcal{F}^{\rho\sigma}\big{]}-\frac{6}{l^{2}}\,g_{\mu\nu}=0\,.$
(2.2)

We restrict to solutions that satisfy $\mathcal{F}\wedge\mathcal{F}=0$, so the
Chern-Simons term will not play a role.

One can obtain the solution of [55] from a double wick rotation of an
electrically charged black hole with a flat boundary. The result reads

$\displaystyle ds_{5}^{2}=\frac{r^{2}}{l^{2}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2})$
$\displaystyle+\frac{l^{2}dr^{2}}{r^{2}f(r)}+\frac{r^{2}}{l^{2}}f(r)d\phi^{2},\qquad
f(r)=1-\frac{\mu l^{2}}{r^{4}}-\frac{q^{2}l^{2}}{r^{6}},$
$\displaystyle\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{\star}^{2}}\right)$
$\displaystyle d\phi,\qquad\mathcal{F}=d\mathcal{A}=-\frac{2q}{r^{3}}dr\wedge
d\phi,$ (2.3)

where $r_{\star}$ is the largest positive root of $f(r)$, $f(r_{\star})=0$.
One can write

$\mu=\frac{(r_{\star}^{6}-q^{2}l^{2})}{l^{2}r_{\star}^{2}}.$ (2.4)

Note that the $\phi$-coordinate parameterizes a compact circle which has a
finite size at $r\rightarrow\infty$, but shrinks at $r=r_{\star}$. In order to
have a smooth solution there, the periodicity of $\phi$ is fixed to

$L_{\phi}=\frac{4\pi l^{2}}{r_{\star}^{2}f^{\prime}(r_{\star})}.$ (2.5)

The magnetic flux through the $\phi$ direction at $r\rightarrow\infty$ is
$\Phi=-\oint\mathcal{A}=\frac{q}{r_{\star}^{2}}L_{\phi}$.

To understand the solution space, it is more natural to invert these relations
and write the bulk parameters $r_{\star}$, $q$, $\mu$ in terms of the
asymptotic boundary parameters $L_{\phi}$, $\Phi$. We have

$q=r_{\star}^{2}\frac{\Phi}{L_{\phi}},\qquad r_{\star}=\frac{\pi
l^{2}}{2L_{\phi}}\left(1\pm\sqrt{1-\frac{\Phi^{2}}{\Phi_{max}^{2}}}\right),$
(2.6)

where $\Phi_{max}=\frac{\pi l}{\sqrt{2}}$. The parameter $\mu$ can then be
written in terms of boundary data using (2.4). For $\Phi<\Phi_{max}$, there
are two branches of solutions. As $\Phi\rightarrow 0$, The $+$ branch of the
solution approaches the pure AdS soliton, and the $-$ branch approaches
Poincaré-AdS. We are interested in the solution of the $+$ branch. The two
branches coalesce at $\Phi=\Phi_{max}$.

As explained in appendix A, preserving supersymmetry requires $\mu=0$. This
occurs for $r_{\star}^{6}=q^{2}l^{2}$, that is, when

$r_{\star}^{2}=\frac{\Phi^{2}l^{2}}{L_{\phi}^{2}}\quad\Rightarrow\quad
2-3\frac{\Phi^{2}}{\Phi_{max}^{2}}\pm
2\sqrt{1-\frac{\Phi^{2}}{\Phi_{max}^{2}}}=0.$ (2.7)

On the $+$ branch one finds that the supersymmetric point corresponds to
$\Phi_{S}=\frac{2\pi}{3}l$. We will assume this condition is satisfied on any
occasion when a supersymmetric background is considered.

From the dual field theory perspective, it is convenient to introduce a new
parameter $Q$ such that the boundary gauge field is simply

$\mathcal{A}(r\to\infty)=Q\,d\phi$ (2.8)

at the supersymmetric point in the parameter space. This can be accomplished
by re-parameterizing as $q=-Q^{3}l^{2}$. The condition $\mu=0$ for the
solution to preserve supersymmetry is then $r_{\star}^{2}=(Ql)^{2}$. All
together, the supersymmetric soliton is given by

$\displaystyle ds_{5}^{2}=\frac{r^{2}}{l^{2}}$
$\displaystyle(-dt^{2}+dx_{1}^{2}+dx_{2}^{2})+\frac{l^{2}}{r^{2}}\,\frac{dr^{2}}{f(r)}+\frac{r^{2}}{l^{2}}f(r)d\phi^{2},\qquad
f(r)=1-\left(\frac{lQ}{r}\right)^{6},$ (2.9)
$\displaystyle\mathcal{A}=\left(Q-\frac{l^{2}Q^{3}}{r^{2}}\right)d\phi,\qquad\mathcal{F}=d\mathcal{A}=\frac{2l^{2}Q^{3}}{r^{3}}dr\wedge
d\phi.$

Note that the holonomy of the boundary gauge field is fixed as

$\Phi=\oint\mathcal{A}=\frac{2\pi}{3}l.$ (2.10)

##  3 Deformed backgrounds in Type IIB

As noted by Anabalon and Ross in [55], the supersymmetric AdS5 soliton
summarized in the previous section can be obtained from a dimensional
reduction of Type IIB supergravity. Here we review the reduction ansatz first
given in [55], in which the soliton is embedded into the background
AdS${}_{5}\times S^{5}$, before presenting a new infinite family of solutions
obtained from AdS${}_{5}\times Y^{p,q}$ backgrounds. As explained in section
1.2, the ten-dimensional solutions obtained in this fashion can be understood
as implementing a particular deformation in the corresponding SCFT duals.

###  3.1 AdS${}_{5}\times S^{5}$ embedding

The embedding of the solution (2) into the Type IIB background
AdS${}_{5}\times S^{5}$ yields the metric [67, 55]

$\displaystyle ds^{2}_{10}=ds^{2}_{5}$
$\displaystyle+l^{2}\biggl{\\{}d\theta^{2}+\sin^{2}\theta
d\varphi^{2}+\sin^{2}\theta\sin^{2}\varphi\left(d\varphi_{1}+\frac{\mathcal{A}}{l}\right)^{2}$
(3.1)
$\displaystyle+\sin^{2}\theta\cos^{2}\varphi\left(d\varphi_{2}+\frac{\mathcal{A}}{l}\right)^{2}+\cos^{2}\theta\left(d\varphi_{3}+\frac{\mathcal{A}}{l}\right)^{2}\biggr{\\}}.$

Note that the three $U(1)$ factors in the Cartan of the $S^{5}$ isometry group
are uniformly fibered with $\phi$-circle inside $ds_{5}^{2}$. In each example,
we will see a particular $U(1)$ inside the isometry group for the internal
manifold mixing with the $U(1)$ of the $S_{\phi}^{1}$. The smoothness and
supersymmetry-preserving features (for $\mu=0$) of the five-dimensional
solution are inherited by its embedding in ten dimensions. The range of the
various angular coordinates is

$0\leq\theta\leq\pi/2,\qquad 0\leq\varphi\leq\pi/2,\qquad 0\leq\varphi_{i}\leq
2\pi\quad i=1,2,3.$ (3.2)

It’s convenient to introduce,

$\displaystyle\mu_{1}=\sin\theta\sin\varphi,\quad\mu_{2}=\sin\theta\cos\varphi,\quad\mu_{3}=\cos\theta,$
(3.3)

satisfying $\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}=1$. We can then write the
five-form field strength solving the Type IIB supergravity equations of motion
for the metric (3.1) as

$\displaystyle F_{5}=G_{5}+\star G_{5},\qquad G_{5}=\frac{4}{l}\text{vol}_{5}$
$\displaystyle-l^{2}\sum_{i=1}^{3}\mu_{i}d\mu_{i}\wedge\left(d\varphi_{i}+\frac{\mathcal{A}}{l}\right)\wedge\star_{5}\mathcal{F}.$
(3.4)

Here and throughout this work, we use $\star_{5}$ to denote the Hodge star
with respect to the 5d metric in (2). Explicitly we have

$\text{vol}_{5}=\frac{r^{3}}{l^{3}}dt\wedge dx_{1}\wedge dx_{2}\wedge dr\wedge
d\phi,\qquad\star_{5}\mathcal{F}=-\frac{2q}{l^{3}}dt\wedge dx_{1}\wedge
dx_{2}.$ (3.5)

We can of course understand $F_{5}$ as sourced by a stack of D3-branes, whose
number is related to the quantization of flux by

$Q_{D3}=\frac{1}{(2\pi)^{4}}\int F_{5}=\frac{l^{4}}{4\pi},$ (3.6)

where we have suppressed factors of $g_{s}$ and
$\alpha^{\prime}=\ell_{s}^{2}$. The field theory dual of this construction is
a twisted compactification of $\mathcal{N}=4$ SYM, which still preserves four
supercharges when we set $\mu=0$ in (2).

###  3.2 AdS${}_{5}\times Y^{p,q}$ embeddings

Having reviewed the holographic realization of the deformation procedure for
the dual of $\mathcal{N}=4$ SYM, we now consider its application to duals of
SCFTs with less supersymmetry. In particular, we deform Type IIB backgrounds
with Sasaki-Einstein internal manifolds $Y^{p,q}$ [78],

$\displaystyle ds_{10}^{2}=ds_{\text{AdS}_{5}}^{2}+l^{2}ds_{Y}^{2},$
$\displaystyle\qquad ds_{Y}^{2}=ds_{B}^{2}+\frac{1}{9}(d\psi+A)^{2},$
$\displaystyle F_{5}=(1+\star)G_{5},$ $\displaystyle\qquad
G_{5}=\frac{4}{l}\text{vol}_{\text{AdS}_{5}}$ (3.7)

Here $B$ is a four-dimensional Kähler-Einstein base. Its Kähler form $J$ is
related to the one-form $A$ as $dA=6J$. This $S_{\psi}^{1}$ fibration over $B$
corresponds to the $\mathrm{U}(1)$ R-symmetry of the dual $\mathcal{N}=1$
superconformal field theories, which are the IR fixed points of quiver gauge
theories labelled by the integers $p$ and $q$. In each example we will provide
explicit expressions for $ds_{Y}^{2}$.

The deformation is accomplished by embedding the soliton solution (2) into
(3.2). The appropriate reduction ansatz of Type IIB to minimal 5d gauged
supergravity is given in [64], or in a more general form in [79], for example.
In this case it reads

$\displaystyle
ds_{10}^{2}=ds_{5}^{2}+l^{2}\left[ds_{B}^{2}+\frac{1}{9}(d\psi+A+\frac{3}{l}\mathcal{A})^{2}\right],$
$\displaystyle F_{5}=(1+\star)G_{5},\qquad
G_{5}=\frac{4}{l}\text{vol}_{5}-l^{2}\,J_{2}\wedge\star_{5}\mathcal{F},$ (3.8)

with $ds_{5}^{2}$, $\mathcal{A}$, and $\mathcal{F}$ given by (2). We provide
explicit examples for $T^{1,1}$ (equivalently $Y^{1,0}$) and generic $Y^{p,q}$
below.

####  $T^{1,1}$ example

For the case of $T^{1,1}$, the base manifold $B$ is $S^{2}\times S^{2}$, so
that

$ds^{2}_{Y}=\frac{1}{6}\sum_{i=1}^{2}\left(d\theta_{i}^{2}+\sin^{2}\theta_{i}d\phi_{i}^{2}\right)+\frac{1}{9}\left(d\psi+\sum_{i=1}^{2}\cos\theta_{i}d\phi_{i}\right)^{2}.$
(3.9)

The coordinates $(\theta_{i},\phi_{i})$ on the two-spheres have canonical
$\pi$ and $2\pi$ periodicities, respectively, while $\psi\in[0,4\pi]$. The
deformed ten-dimensional metric can then be expressed

$\displaystyle
ds^{2}_{10}=ds^{2}_{5}+l^{2}\left[\frac{1}{6}\sum_{i=1}^{2}\left(d\theta_{i}^{2}+\sin^{2}\theta_{i}d\phi_{i}^{2}\right)+\frac{1}{9}\left(d\psi+\sum_{i=1}^{2}\cos\theta_{i}d\phi_{i}+\frac{3}{l}\mathcal{A}\right)^{2}\right],$
(3.10)

again with $ds_{5}^{2}$ and $\mathcal{A}$ as in (2). The RR five-form field
strength can be computed from

$\displaystyle F_{5}=G_{5}+\star G_{5},\qquad G_{5}=\frac{4}{l}\text{vol}_{5}$
$\displaystyle+\frac{l^{2}}{6}(\sin\theta_{1}~{}d\theta_{1}\wedge
d\phi_{1}+\sin\theta_{2}~{}d\theta_{2}\wedge
d\phi_{2})\wedge\star_{5}\mathcal{F},$ (3.11)

where $\text{vol}_{5}$ and $\star_{5}\mathcal{F}$ are as in (3.5). The
quantized charge associated to $F_{5}$ is found by integrating over $T^{1,1}$,

$Q_{D3}=\frac{1}{(2\pi)^{4}}\int F_{5}=\frac{4l^{4}}{27\pi}.$ (3.12)

This can be interpreted as the number of D3-branes sitting at the tip of a
cone over $T^{1,1}$. The field theory we are deforming via this holographic
construction is the Klebanov-Witten CFT [10].

#### Generic $Y^{p,q}$ example

For the more general case in which $M_{5}$ is any $Y^{p,q}$ manifold, the
deformed metric reads

$\begin{split}ds_{10}^{2}=ds^{2}_{5}+&l^{2}\left[\frac{1-y}{6}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)+\frac{1}{w(y)v(y)}dy^{2}+\frac{w(y)v(y)}{36}\left(d\beta+\cos\theta
d\varphi\right)^{2}\right.\\\ &\left.+\frac{1}{9}\left(d\psi-\cos\theta
d\varphi+y\left(d\beta+\cos\theta
d\varphi\right)+\frac{3}{l}\mathcal{A}\right)^{2}\right],\\\ \end{split}$
(3.13)

where the functions $w(y)$, $v(y)$ are controlled by a single parameter $a$,

$w(y)=\frac{2(a-y^{2})}{1-y}\,\,,\,\,v(y)=\frac{a+2y^{3}-3y^{2}}{a-y^{2}}.$
(3.14)

The base space $B$ parameterized by coordinates

$0\leq\theta\leq\pi,\qquad 0\leq\varphi\leq 2\pi,\qquad y_{1}\leq y\leq
y_{2},\qquad 0\leq\beta\leq 2\pi,$ (3.15)

is ensured to be smooth and compact if one chooses

$0<a<1,$ (3.16)

and the endpoints $y_{1},~{}y_{2}$ are constrained appropriately— see e.g.
[78]. Note the integers $p,q$ are related to integrals of the Kähler form over
two-cycles in $B$. This two-form is given by

$J_{2}=\frac{1}{6}(1-y)\sin\theta~{}d\theta\wedge
d\varphi+\frac{1}{6}dy\wedge(d\beta+\cos\theta~{}d\varphi),$ (3.17)

and so the RR field strength $F_{5}$ can be expressed

$\displaystyle F_{5}=G_{5}+\star G_{5},\quad G_{5}=\frac{4}{l}\text{vol}_{5}$
$\displaystyle-\frac{l^{2}}{6}\left[(1-y)\sin\theta~{}d\theta\wedge
d\varphi+dy\wedge(d\beta+\cos\theta~{}d\varphi)\right]\wedge\star_{5}\mathcal{F}.$
(3.18)

The associated quantized charge is found to be

$Q_{D3}=\frac{1}{(2\pi)^{4}}\int
F_{5}=\frac{2l^{4}}{27\pi}(y_{1}-y_{2})(y_{2}+y_{1}-2).$ (3.19)

We have verified explicitly that the backgrounds presented above in
eqs.(3.13)-(3.18) solve the full Type IIB supergravity equations of motion.

Let us close this section with brief words about the ${\cal N}=1$ SCFTs
associated with any generic member of the $Y^{p,q}$ family. After the
construction of this family of Sasaki-Einstein manifolds [78], the dual SCFTs
were constructed in [80], see also [81] for a summary.

The quiver associated is constructed in terms of two basic ’unit-cells’
denoted by $\sigma,\tau$ that are put together like Legos. $Y^{p,q}$ has $p$
unit cells, of which $q$ are of the $\sigma$-type. There are bifundamental
fields $(U^{\alpha},V^{\alpha},Y,Z)$ with $\alpha=1,2$ is an $SU(2)$ index.
All gauge nodes are $SU(N)$. The superpotential is constructed by summing
cubic and quartic terms of the form

$W\sim
W_{3}+W_{4},~{}~{}~{}W_{3}\sim\epsilon_{\alpha\beta}(U_{L}^{\alpha}V^{\beta}Y+U_{R}^{\alpha}V^{\beta}Y),~{}~{}~{}W_{4}\sim\epsilon_{\alpha\beta}ZU_{R}^{\alpha}YU_{L}^{\beta}.$
(3.20)

The cubic terms are associated with $\sigma$-cells and the quartic ones with
$\tau$-cells. Imposing the vanishing of the NSVZ beta functions for the gauge
groups and the superpotentials (consequently the vanishing of R-symmetry
anomalies) leaves us with a two-dimensional space of anomalous dimensions.

there are some known deformations of the conformal quivers. For example, a
quasi-marginal deformation that triggers a cascade [81]. Another possibility
is to switch on a VEV represented by the parameter $q$, the R-symmetry
fibration and the replacement of AdS5 by $ds_{5}^{2}$ as indicated in (3.13).
Let us now discuss a different infinite family of backgrounds.

##  4 Deformed backgrounds in Type IIA

Let us begin this section by reviewing the general form of supersymmetric AdS5
backgrounds in massive Type IIA supergravity, dual to $\mathcal{N}=1$ SCFTs.
Given such a background, we can embed the five-dimensional supergravity
solution (2) using the framework of [66], which we will then summarize in a
slightly modified form. Finally, we will present examples of several deformed
backgrounds obtained in this way. The full massive IIA equations of motion and
Bianchi identities have been explicitly checked for each new background
discussed.

Supersymmetric AdS5 solutions in massive Type IIA supergravity were first
classified in [82]. Subsequent work by the authors of [70] reduced the number
of BPS equations controlling the classification, writing the metric in terms
of a pair of functions $D_{s},\,D_{u}$ as

$\displaystyle ds_{10}^{2}$
$\displaystyle=e^{2W}\left[ds_{\text{AdS}_{5}}^{2}+e^{2A}(dv_{1}^{2}+dv_{2}^{2})+\frac{1}{3}e^{-6\lambda}ds_{3}^{2}\right]$
(4.1) $\displaystyle ds_{3}^{2}$
$\displaystyle=-\frac{4}{\partial_{s}D_{s}}D\psi^{2}-\partial_{s}\widetilde{D}_{s}ds^{2}-2\partial_{u}D_{s}duds-\partial_{u}D_{u}du^{2},$

where $D_{s},\,D_{u}$ depend only on the coordinates $(v_{1},v_{2},s,u)$, and
we have defined

$\displaystyle\widetilde{D}_{s}=D_{s}-\frac{3}{2}\log s,\hskip
5.69054pt\text{and}\hskip
8.53581ptD\psi=d\psi-\frac{1}{2}\star_{2}d_{2}D_{s}.$ (4.2)

Note that the subscripts on the Hodge star $\star_{2}$ and total derivative
$d_{2}$ restrict these operators to the coordinates $(v_{1},v_{2})$. In the
examples we will discuss, this subspace describes a constant-curvature Riemann
surface. The angular coordinate $\psi$ parameterizes a U(1) isometry of the
spacetime, corresponding to an R-symmetry in the SCFT dual. We can express the
warp factors appearing in the metric in terms of $D_{s},\,D_{u}$ as

$\displaystyle e^{4W}=\frac{-\partial_{s}D_{s}}{3\text{det}(h)},\hskip
8.53581pte^{2A}=\frac{\text{det}(h)e^{D_{s}}}{24},\hskip
8.53581pte^{-6\lambda}=\frac{\text{det}(h)}{8s\,\text{det}(g)},$ (4.3)

where the determinants refer to the $(u,s)$ subspace,

$\displaystyle\text{det}(g)=\partial_{u}D_{u}\partial_{s}\widetilde{D}_{s}-(\partial_{u}D_{s})^{2},\hskip
8.53581pt\text{det}(h)=\partial_{u}D_{u}\partial_{s}D_{s}-(\partial_{u}D_{s})^{2}.$
(4.4)

The dilaton takes the form

$e^{2\Phi}=e^{6W}e^{-6\lambda},$ (4.5)

and the RR and NSNS field strengths are given in terms of $D_{s},\,D_{u}$ as

$\displaystyle
F_{0}=36\frac{\sqrt{2s}}{\partial_{2}D_{s}}\partial_{u}(\partial_{s}D_{u}-\partial_{u}D_{s}),$
(4.6) $\displaystyle F_{2}=\frac{1}{3}F_{0}\,\tilde{\xi}\wedge
D\psi-d\left[\star_{2}d_{2}D_{u}+2\frac{\partial_{u}D_{s}}{\partial_{s}D_{s}}D\psi\right]+\Delta_{2}D_{u}\,dv_{1}\wedge
dv_{2}$ $\displaystyle-\partial_{u}(se^{D_{s}}\text{det}(g))dv_{1}\wedge
dv_{2}+\star_{2}d_{2}(\partial_{u}D_{s}-\partial_{s}D_{u})\wedge ds,$
$\displaystyle F_{4}=\frac{1}{3}F_{2}\wedge\tilde{\xi}\wedge
D\psi-\frac{1}{36}d\left(\sqrt{2s}\,e^{D_{s}}\text{det}(h)dv_{1}\wedge
dv_{2}\wedge D\psi-2\sqrt{2s}\,d(\star_{2}d_{2}D_{s})\wedge D\psi\right)$
$\displaystyle+\frac{1}{36\sqrt{2s}}\frac{\partial_{u}D_{s}}{\partial_{s}D_{s}}\left[2\,du\wedge
d(\star_{2}d_{2}D_{s})+e^{D_{s}}\text{det}(h)du\wedge dv_{1}\wedge
dv_{2}+3\,d(\partial_{u}D_{s}e^{D_{s}})\wedge dv_{1}\wedge dv_{2}\right]\wedge
D\psi$ $\displaystyle H_{3}=\frac{1}{3}d\left[\tilde{\xi}\wedge
D\psi+\frac{1}{8}\frac{e^{D_{s}}}{\sqrt{2s}}\partial_{u}D_{s}\,dv_{1}\wedge
dv_{2}\right]+\frac{1}{36\sqrt{2s}}du\wedge
d\star_{2}d_{2}D_{s}-\frac{e^{D_{s}}}{12}\text{det}(g)\,\tilde{\xi}\wedge
dv_{1}\wedge dv_{2}.$

Borrowing from the notation in [66], we have written the field strengths in
terms of a one-form $\tilde{\xi}$,

$\displaystyle\tilde{\xi}=-\frac{1}{6\,\text{det}(g)\sqrt{2s}}\left(\frac{3}{2s}\partial_{u}D_{s}\,ds+\text{det}(h)\,du\right).$
(4.7)

The functions $D_{s}$, $D_{u}$ are constrained by the Bianchi identities for
the field strengths. The Bianchi identity for $F_{0}$ requires it to be piece-
wise constant, yielding a first differential equation for $D_{s}$, $D_{u}$.
The Bianchi identities for $H_{3}$ and $F_{2}$, respectively, then translate
to two additional equations,

$\displaystyle\Delta_{2}D_{s}$
$\displaystyle=\partial_{s}(s\text{det}(g)e^{D_{s}})+\frac{F_{0}}{24}\frac{\partial_{s}(e^{D_{s}})}{\sqrt{2s}},$
(4.8) $\displaystyle\Delta_{2}(\partial_{u}D_{u})$
$\displaystyle=\partial_{u}^{2}(s\text{det}(g)e^{D_{s}})+\frac{sF_{0}}{36}\frac{\partial_{s}(\text{det}(h)e^{D_{s}})}{\sqrt{2s}}.$
(4.9)

The Bianchi for $F_{4}$ is satisfied automatically.

###  4.1 Embedding into IIA duals of $\mathcal{N}=1$ SCFTs

As shown in [66], five-dimensional minimal gauged supergravity can be embedded
into the massive IIA system presented above by making the following
substitutions

$\displaystyle ds_{\text{AdS}_{5}}^{2}\,\rightarrow\,ds_{5}^{2},$ (4.10)
$\displaystyle
F_{4}\,\rightarrow\,F_{4}^{g}-\left(\star_{5}\mathcal{F}-\frac{1}{3}\mathcal{F}\wedge\mathcal{D}\psi\right)$
$\displaystyle\wedge\frac{ds}{\sqrt{2s}}+\frac{\sqrt{2s}}{3}\mathcal{F}\wedge\text{vol}_{\Sigma}$
$\displaystyle F_{2},\,H_{3}\,\rightarrow\,F_{2}^{g},\,H_{3}^{g},\qquad
F_{6},\,F_{8},$ $\displaystyle\,\rightarrow\,-\star F_{4}^{g},\,\star
F_{2}^{g},$

where $ds_{5}^{2}$ and $\mathcal{F}$ are the line element and two-form field
strength of a given solution to the five-dimensional action (2.1), and the
superscripts $g$ ($g$ for gauging) denote the substitutions

$D\psi\,\rightarrow\,\mathcal{D}\psi=D\psi-3\mathcal{A},$ (4.11)

taken inside of all potentials, before acting with total derivatives to
compute the corresponding field strengths. In other words, if in the initial
background one has

$H_{3}=dB_{2},\quad F_{2}=F_{0}B_{2}+dC_{1},\quad F_{4}=dC_{3}+B_{2}\wedge
F_{2}-\frac{1}{2}F_{0}\,B_{2}\wedge B_{2},$ (4.12)

then $H_{3}^{g},~{}F_{2}^{g},~{}F_{4}^{g}$ are computed here by first
implementing (4.11) on $B_{2},~{}C_{1},~{}C_{3}$ to obtain
$B_{2}^{g},~{}C_{1}^{g},~{}C_{3}^{g}$, and then constructing the gauged field
strengths from these new potentials. As an aside, note that we have chosen to
write (4.10) in the form $F_{4}\rightarrow F_{4}^{g}+(\cdots)$, at the cost of
introducing the term proportional to $\mathcal{F}\wedge\text{vol}_{\Sigma}$,
which is not present in [66, Eq. (3.17)].666In presenting the general system
in (4.1)-(4.6), we have also written $F_{4}$ differently from [66, Eqn.
(2.31)], to put it in a form where gauging inside all total derivatives
appearing reproduces $F_{4}^{g}$ obtained from gauged potentials.  The results
are the same. We are also using a different normalization of the five-
dimensional gauge vector $\mathcal{A}$, resulting in additional factors of 3
compared to [66].

The 5d solution we are interested in embedding is of course the $ds_{5}^{2}$,
$\mathcal{A}$, and $\mathcal{F}$ given in (2), as a mechanism to cap off the
geometry and deform the dual SCFT. It is worth emphasizing that the
calculations performed in [66] ensure that for any supersymmetric AdS5
solution in massive Type IIA, (4.1)-(4.8), one can make the substitutions
(4.10) to implement this deformation. For $\mu=0$ (in which case (2) can be
expressed as (2.9)) the resulting deformed SQFT dual will no longer be
conformal, but will preserve some supersymmetry, and become effectively three-
dimensional as it flows to the IR. We now consider several examples.

###  4.2 Application to reductions of 6d theories on $\Sigma_{2}$

For the solutions to the system (4.1)-(4.8) which we are interested in, the
coordinates $(\psi,s,u)$ describe a submanifold which is fibered over a
Riemann surface $\Sigma$ with coordinates $(v_{1},v_{2})$. This requires the
warp factor $e^{2A}$ to separate into a product of functions on these two
submanifolds, which we we can accomplish by adopting the ansatz

$D_{s}=F_{s}(s,u)+2A_{0}(v_{1},v_{2}),\qquad
D_{u}=F_{u}(s,u)+2\tilde{A}_{0}(v_{1},v_{2})$ (4.13)

The Bianchi identity (4.8) then implies

$\displaystyle\Delta_{2}A_{0}=-\kappa e^{2A_{0}},$ (4.14)

where $\Delta_{2}=\partial_{v_{1}}^{2}+\partial_{v_{2}}^{2}$, and $\kappa$ is
the curvature of the surface, which we parameterize as
$\kappa\in\\{-1,0,1\\}$. An appropriate solution for $A_{0}$ can be written

$\displaystyle e^{A_{0}}=\frac{2}{1+\kappa(v_{1}^{2}+v_{2}^{2})}.$ (4.15)

We will focus specifically on the hyperbolic case $\kappa=-1$. The function
$\tilde{A}_{0}(v_{1},v_{2})$ is actually unconstrained, since it does not
contribute to $s,u$ derivatives of $D_{u}$, and the instances in which $D_{u}$
is acted on only by $v_{1},v_{2}$ derivatives cancel against one another. The
local metric and volume form on the Riemann surface can then be written

$\displaystyle
ds_{\Sigma}^{2}=e^{2A_{0}}(dv_{1}^{2}+dv_{2}^{2}),\qquad\text{vol}_{\Sigma}=dA_{\Sigma}=e^{2A_{0}}\,dv_{1}\wedge
dv_{2},\qquad A_{\Sigma}\equiv\star_{2}d_{2}A_{0}.$ (4.16)

The one-form $A_{\Sigma}$ is precisely the connection appearing in $D\psi$,

$D\psi=d\psi-\frac{1}{2}\star_{2}d_{2}D_{s}=d\psi-A_{\Sigma}.$ (4.17)

Finally, note that for higher-genus examples, the volume $V_{\Sigma}$ of the
Riemann surface is given by

$V_{\Sigma}=\int\text{vol}_{\Sigma}=4\pi(g-1).$ (4.18)

The separable ansatz (4.13) encompasses many interesting solutions for
specific choice of $F_{s}$ and $F_{u}$, including not only those first
introduced in [82] and [70], but also the reductions to IIA of the GMSW
solution [83], the solution of BBBW [84], and that of Maldacena-Nuñez [85].

####  4.2.1 AFPRT example

First we consider deformations to an infinite family of backgrounds first
introduced by Apruzzi, Fazzi, Passias, Rota and Tomasiello in [82, 68]
(abbreviated AFPRT). These are dual to non-Lagrangian SCFTs, and can be
interpreted as twisted compactifications of a D6-D8-NS5 system on a negative
curvature Riemann surface $\Sigma$. The 6d ‘parent’ SCFTs should be UV
descriptions of linear quiver gauge theories. To obtain the AFPRT solutions
from the general AdS5 massive IIA system, in (4.13) take

$F_{s}=\log(2\alpha(z)^{2}\sin^{2}\theta),\qquad
F_{u}=\frac{\ddot{\alpha}(z)}{18^{2}\pi^{2}}\,\log\left(2\,\frac{1+\cos\theta}{1-\cos\theta}\right),$
(4.19)

and transform the coordinates $(s,u)$ as

$u=\alpha(z)\,\cos\theta,\qquad
s=\frac{1}{2}\left(\frac{\dot{\alpha}(z)}{18\pi}\right)^{2}.$ (4.20)

These solutions are controlled by a single function $\alpha(z)$, constrained
as

$\dddot{\alpha}(z)=-162\pi^{3}\,F_{0}.$ (4.21)

Recall $F_{0}$ is the Romans mass. We allow it to be piece-wise constant with
discontinuities at the positions of flavor D8 branes. The function $\alpha$ is
related to the rank function $\mathcal{R}$ in the associated 6d linear quiver
theory by

$\mathcal{R}(z)=-\frac{\ddot{\alpha}}{81\pi^{2}}.$ (4.22)

Note the coordinate $z$ can be thought of like a continuum version of the
direction along the quiver, in the sense developed in [86].

Following the embedding procedure to lift the five-dimensional solution (2)
into this background, we obtain the new deformed solution [1]

$\displaystyle\mathrm{d}s_{10}^{2}=$ $\displaystyle
18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}\left[\mathrm{d}s_{5}^{2}+\frac{1}{3}\mathrm{d}s_{\Sigma}^{2}-\frac{\ddot{\alpha}}{6\alpha}\mathrm{d}z^{2}-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathcal{D}\psi^{2}\right)\right],\qquad\mathrm{d}s_{\Sigma}^{2}=\frac{4(\mathrm{d}v_{1}^{2}+\mathrm{d}v_{2}^{2})}{(1-v_{1}^{2}-v_{2}^{2})^{2}},$
(4.23) $\displaystyle{\cal
D}\psi=\mathrm{d}\psi-3\mathcal{A}-A_{\Sigma},\qquad\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{*}^{2}}\right)\mathrm{d}\phi,\qquad
A_{\Sigma}=\frac{2(v_{1}~{}\mathrm{d}v_{2}-v_{2}~{}\mathrm{d}v_{1})}{1-v_{1}^{2}-v_{2}^{2}},$

The dilaton is given by

$e^{-4\Phi}=\frac{1}{2^{5}3^{17}\pi^{10}}\left(-\frac{\ddot{\alpha}}{\alpha}\right)^{3}\left(2\dot{\alpha}^{2}-3\alpha\ddot{\alpha}\right)^{2},$
(4.24)

and the fluxes are

$\displaystyle H_{3}=\mathrm{d}B_{2}^{g},$ $\displaystyle\qquad
F_{2}=F_{0}B_{2}^{g}+\mathrm{d}C_{1}^{g},$ (4.25) $\displaystyle
F_{4}=\left(\mathrm{d}C_{3}^{g}+B_{2}^{g}\wedge
F_{2}-\frac{1}{2}F_{0}\,B_{2}^{g}\wedge B_{2}^{g}\right)$
$\displaystyle-\frac{\ddot{\alpha}}{18\pi}\mathrm{d}z\wedge\left(\star_{5}\mathcal{F}-\frac{1}{3}\mathcal{F}\wedge\mathcal{D}\psi\right)-\frac{\dot{\alpha}}{54\pi}\mathcal{F}\wedge\text{vol}_{\Sigma},$
$\displaystyle B_{2}^{g}=\frac{1}{3}\tilde{\xi}\wedge\mathcal{D}\psi,\qquad
C_{1}^{g}=$
$\displaystyle\frac{\ddot{\alpha}}{162\pi^{2}}\cos\theta\,\mathcal{D}\psi,\qquad
C_{3}^{g}=\frac{\dot{\alpha}}{162\pi}\mathcal{D}\psi\wedge\text{vol}_{\Sigma}.$

We have expressed these in terms of the one-form

$\tilde{\xi}=3\pi\left(\cos\theta\mathrm{d}z-\frac{2\alpha\dot{\alpha}}{2\dot{\alpha}^{2}-3\alpha\ddot{\alpha}}\sin\theta\mathrm{d}\theta\right).$
(4.26)

The coordinate $z$ takes values in a finite interval, while the angles
$\theta$, $\psi$ range from $0$ to $\pi$ and $0$ to $2\pi$, respectively,
parameterizing a deformed two-sphere. Depending on the boundary conditions for
$\alpha$ on the $z$-interval, these solutions describe various brane systems,
including:

  * •

D6-branes at endpoints where $\alpha$ has a single zero while
$\ddot{\alpha}\neq 0$.

  * •

O6-D6 sources at endpoints where $\ddot{\alpha}$ has a single zero, with
$\alpha\neq 0$.

  * •

O8-D8 sources at single zeros of both $\ddot{\alpha}$ and
$2\dot{\alpha}^{2}-3\alpha\ddot{\alpha}$.

If both $\alpha$ and $\ddot{\alpha}$ have single zeros, the (deformed) $S^{2}$
shrinks smoothly and the metric is regular. For the moment, let us restrict
attention to setups with these regular boundary conditions. So

$\alpha=\ddot{\alpha}=0\quad\text{ at }\quad z=z_{i},z_{f},\qquad
z\in[z_{i},z_{f}].$ (4.27)

In this case, the quantized $H_{3}$ field strength evaluates to

$Q_{\text{NS5}}=\frac{1}{(2\pi)^{2}}\int
H_{3}=\left.\left(z-\frac{2\alpha\dot{\alpha}}{2\dot{\alpha}^{2}-3\alpha\ddot{\alpha}}\right)\right|_{z_{i}}^{z_{f}}=z_{f}-z_{i}$
(4.28)

Thus we restrict to $z$-intervals of integer-valued width. Moving to the Page
charge associated to $F_{2}$, there are two-cycles on both $\Sigma$ and the
$S^{2}$ defined by the coordinates $(\theta,\psi)$ to consider. Each are
defined at fixed values $z=z_{*}$, and at fixed points on the complementary
two-manifold. First, we compute

$Q_{\text{D6}}=\frac{1}{2\pi}\int_{\Sigma}(F_{2}-B_{2}F_{0})=-\frac{\ddot{\alpha}|_{z_{*}}}{162\pi^{2}}\cos\theta_{*}\,V_{\Sigma}.$
(4.29)

We also have

$Q_{\text{D6}}=\frac{1}{2\pi}\int_{S^{2}}(F_{2}-B_{2}F_{0})=-\frac{\ddot{\alpha}|_{z_{*}}}{81\pi^{2}}.$
(4.30)

The interpretation of this flux quanta in terms of the number of D6-branes in
the associated Hanany-Witten setup is somewhat subtle. Consider a linear
quiver consisting of some number of rank-$N_{k}$ gauge nodes, described by

$\mathcal{R}(z)=-\frac{\ddot{\alpha}}{81\pi^{2}}=N_{k}+(N_{k+1}-N_{k})(z-k)\quad\text{for}\quad
z\in[k,k+1],\quad k\in\mathbb{Z}.$ (4.31)

Equation (4.30) only appears to correctly count the number of D6-branes at the
integer interval boundaries, $z_{*}=k$. To exclude additional D6-charges
induced on D8 sources, we can implement the large gauge transformation

$B_{2}\rightarrow B_{2}-d[\pi(z-k)\cos\theta\mathcal{D}\psi],$ (4.32)

which, after using (4.21), has the effect of shifting (4.30) to

$Q_{\text{D6}}=-\frac{\ddot{\alpha}|_{z_{*}}}{81\pi^{2}}+\frac{\dddot{\alpha}|_{z_{*}}}{81\pi^{2}}(z_{*}-k).$
(4.33)

For the rank function in (4.31) this evaluates to $N_{k}$, for any $z_{*}$ the
$k^{\text{th}}$ interval. The Page charge associated to $F_{4}$ evaluates to
zero, signalling the absence of D4-branes. These remarks on the regularity and
Page charges apply both to the original AdS${}_{5}\times\Sigma$ backgrounds
and the deformed versions we have given in (4.23)-(4.25).

There are multiple sources of evidence that the undeformed backgrounds of this
class are dual to compactifications of six-dimensional $\mathcal{N}=(1,0)$
SCFTs— specifically UV-completions of linear quiver gauge theories.777This
applies to the case of regular boundary conditions for $\alpha$. These can,
however, be relaxed if the 6d quiver is suitably modified— see the next
example. First, the conjectured duality between corresponding AdS7 solutions
and the six-dimensional linear quivers has passed a variety of checks, see for
example [86], [87]. Second, very simple analytic maps between each
AdS${}_{5}\times\Sigma$ solution and a corresponding AdS7 solution exist [68],
as well as holographic solutions interpolating along such a flow [88]. Third,
direct comparison of observables between the AdS${}_{5}\times\Sigma$ solutions
and the reductions of the 6d linear quivers shows a match in the appropriate
limits (even though a quiver-like description of the 4d theories is not known)
[88].

As an example, in appendix B we show that the anomaly polynomial for the
reductions of 6d linear quiver theories reproduces the holographic central
charge result at leading order. Now, recall that the latter is essentially set
by the lower-dimensional Newton constant ($G_{5}$, here). This was hinted at
already in the 1980s— see [89]. If we simply had a direct product space one
would have $a$ proportional to $l^{3}/G_{5}$, with $1/G_{5}$ in turn
proportional to the volume of the internal manifold. For metrics of the form
(4.1), the warp factor $e^{W}$ instead acts as the effective AdS5 radius.
Following the reasoning of [86] and others, we can deal with the dependence on
internal coordinates by averaging over the space $M_{5}$. The resulting
formula for $a$ from gravity gives

$a=\frac{\pi}{8}\frac{1}{G_{10}}\int
e^{8W-2\Phi}\text{vol}_{M_{5}}=\frac{V_{\Sigma}}{(6\pi)^{5}}\int-\alpha\ddot{\alpha}dz,$
(4.34)

where the factors of $e^{\Phi}$ from the change between string and Einstein
frames. Note that one has $e^{8W}$ rather than $e^{3W}$ because we have
defined $\text{vol}_{M_{5}}$ as the volume form for $ds^{2}(M_{5})$ without
the overall $e^{2W}$ factor. For explicit choices of $\alpha(z)$ and the
associated quiver, we find that the reduction of the 6d anomaly polynomial
$I_{8}$ to the 4d polynomial $I_{6}$ gives the same answer for $a$ as (4.34)
(in the appropriate holographic, long-quiver limit).

This check may also tell us something about the 4d SCFTs we are deforming (or
at least what they are not). A heuristic 4d quiver proposal for these theories
was given in [88], but we find it fails to reproduce the leading-order
coefficients agreed on by both the anomaly polynomial and holography. It seems
likely to us that these 4d theories do not admit a quiver description, as
described in more detail in appendix B.

####  4.2.2 BPT example

The solutions found by Bah, Passias and Tomasiello in [70] (which we
abbreviate as BPT) can be obtained from (4.13) by choosing

$\displaystyle F_{s}=\log$
$\displaystyle\left[8F_{0}^{2}z^{3}(1-k^{3})p(z)\right],\qquad
F_{u}=\frac{F_{0}}{6}\,zk\log\left[\frac{2p(z)}{z^{3}(1-k^{3})}\right],$
(4.35) $\displaystyle p(z)=(z_{0}-z)[3\ell
z_{1}^{2}-\kappa(z^{2}+z_{0}z+z_{0}^{2})],$ (4.36)

where we have introduced new coordinates $(z,k)$ related to $(s,u)$ by

$u=F_{0}z^{3}(2-k^{3})\qquad s=\frac{1}{8}\,(F_{0}z^{2}k^{2})^{2}.$ (4.37)

Note $\ell=\pm 1$ and $z_{0},z_{1}$ are positive constants parameterizing this
family of backgrounds. The physics of the solutions is quite rich, as they
describe a system of D6-branes, D4-branes smeared in a continuous
distribution, and O8-planes with D8-branes on them. In the following, we will
restrict attention to the case

$\kappa=-1,\qquad\ell=+1$ (4.38)

for which the field-theoretic interpretation is best understood. 888Taking,
for example, $\ell=\kappa=1$, the scaling of the holographic $a$-anomaly
suggests instead a five-dimensional ‘parent’ theory rather than a six-
dimensional origin as seen here.

Applying the embedding procedure outlined earlier, we deform these BPT
backgrounds to obtain new solutions. The new metric is

$\displaystyle
ds_{10}^{2}=e^{2W}\left[ds_{5}^{2}-\frac{p^{\prime}(z)}{9z^{2}}ds_{\Sigma}^{2}+ds_{3}^{2}\right],\qquad\mathrm{d}s_{\Sigma}^{2}=\frac{4(\mathrm{d}v_{1}^{2}+\mathrm{d}v_{2}^{2})}{(1-v_{1}^{2}-v_{2}^{2})^{2}},$
(4.39) $\displaystyle
ds_{3}^{2}=-\frac{p^{\prime}(z)}{3zp(z)}\,dz^{2}-\frac{zp^{\prime}(z)}{3p(z)-zp^{\prime}(z)}\left[\frac{k\,dk^{2}}{1-k^{3}}+\frac{4}{3}\,\frac{p(z)(1-k^{3})}{3p(z)-zp^{\prime}(z)(1-k^{3})}\,\mathcal{D}\psi^{2}\right],$
(4.40) $\displaystyle{\cal
D}\psi=\mathrm{d}\psi-3\mathcal{A}-A_{\Sigma},\qquad\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{*}^{2}}\right)\mathrm{d}\phi,\qquad
A_{\Sigma}=\frac{2(v_{1}~{}\mathrm{d}v_{2}-v_{2}~{}\mathrm{d}v_{1})}{1-v_{1}^{2}-v_{2}^{2}},$
(4.41) $\displaystyle\qquad\qquad\qquad\qquad
e^{4W}=\frac{-z}{k\,p^{\prime}(z)}[3p(z)-zp^{\prime}(z)(1-k^{3})],$ (4.42)

where $ds_{5}^{2}$, $\mathcal{A}$ are as in (2), and the dilaton is given by

$\displaystyle
e^{-4\Phi}=F_{0}^{4}z^{3}k^{5}\,\frac{-p^{\prime}(3p-zp^{\prime})^{2}}{[3p-zp^{\prime}(1-k^{3})]^{3}}.$
(4.43)

The various mIIA fluxes are

$\displaystyle H_{3}=dB_{2}^{g},\qquad F_{2}=$ $\displaystyle
F_{0}B_{2}^{g}+dC_{1}^{g},$ (4.44) $\displaystyle
F_{4}=\left(dC_{3}^{g}+B_{2}^{g}\wedge F_{2}-\frac{1}{2}F_{0}\,B_{2}^{g}\wedge
B_{2}^{g}\right)$
$\displaystyle+\,F_{0}zkd(zk)\wedge\left(\star_{5}\mathcal{F}-\frac{1}{3}\mathcal{F}\wedge\mathcal{D}\psi\right)+\frac{F_{0}}{6}\,z^{2}k^{2}\,\mathcal{F}\wedge\text{vol}_{\Sigma},$
$\displaystyle B_{2}^{g}=\frac{1}{3}\tilde{\xi}\wedge\mathcal{D}\psi+$
$\displaystyle\frac{k}{9z}(3z^{2}\kappa-p^{\prime})\text{vol}_{\Sigma},$
$\displaystyle
C_{1}^{g}=F_{0}\frac{zk}{3}\frac{3p+(1-k^{3})zp^{\prime}}{3p-(1-k^{3})zp^{\prime}}\mathcal{D}\psi,\qquad
C_{3}^{g}$
$\displaystyle=\frac{F_{0}}{18}z^{2}k^{2}\left(\frac{4z^{-2}p^{\prime}p+9p+(1-k^{3})zp^{\prime}}{3p-(1-k^{3})zp^{\prime}}\right)\mathcal{D}\psi\wedge\text{vol}_{\Sigma},$

where the one-form $\tilde{\xi}$ in this case is

$\tilde{\xi}=-kdz-\frac{3p+zp^{\prime}}{3p-zp^{\prime}}\,zdk.$ (4.45)

Positivity of the metric can be ensured by restricting the range of the
coordinates as

$0\leq k\leq 1,\qquad 0\leq z\leq z_{0}.$ (4.46)

Note that for the choice of parameters (4.38), $z_{0}$ is the only real zero
of $p$, and $p^{\prime}=-3(z^{2}+z_{1}^{2})$ has no real zeros. In [70] a
careful study is made of the original backgrounds near the endpoints of the
$z$ and $k$ intervals. The system of branes identified can be summarized as
follows:

  * •

$2Q_{\text{D8}}$-many D8-branes inside an O8 plane positioned at $k=0$, with

$Q_{\text{D8}}=8-2\pi F_{0}$ (4.47)

  * •

$Q_{\text{D6}}$-many D6-branes, positioned at the point ($k=1$, $z=z_{0}$) and
extended on AdS5 and $\Sigma$. The integer $Q_{\text{D6}}$ is related to the
solution parameters by

$Q_{\text{D6}}=\frac{1}{2\pi}\int_{\mathcal{C}_{2}}(F_{2}-B_{2}F_{0})=\frac{2z_{0}}{3}F_{0},$
(4.48)

where $\mathcal{C}_{2}$ is the appropriate two-cycle, which is locally a
sphere near ($k=1$, $z=z_{0}$).

  * •

$Q_{\text{D4}}$-many D4-branes at $z=0$. They are extended on AdS5 and smeared
in a continuous distribution over the Riemann surface. In the dual field
theory, they play the role of simple punctures in the reduction on $\Sigma$.
In terms of the volume $V_{\Sigma}=4\pi(g-1)$ of the Riemann surface

$Q_{\text{D4}}=\frac{1}{(2\pi)^{2}}\int_{\mathcal{C}_{4}}(F_{4}-B_{2}\wedge
F_{2}+\frac{1}{2}F_{0}B_{2}\wedge
B_{2})=\frac{V_{\Sigma}z_{1}^{2}}{18\pi^{2}}F_{0},$ (4.49)

where $\mathcal{C}_{4}$ is a four-cycle at $z=0$ built from $\Sigma$ and a
local two-sphere.

Note that at the other boundaries, so $z=z_{0}$ away from $k=0$ or $1$, and
$k=1$ away from the $z=0$ or $z_{0}$, the $\psi$-circle shrinks smoothly. In
addition to the quantized charges given above there is the number
$Q_{\text{NS5}}$ of NS5-branes, whose near-horizon yields the AdS5 background.
This is found by integrating the NS-NS flux $H_{3}$ and is related to the
other flux quanta by $2\pi F_{0}Q_{\text{NS5}}=Q_{\text{D6}}$.

The regularity analysis and charge quantization is completely analogous for
the deformed backgrounds, (4.39)-(4.44).

Now, in the ‘punctureless’ case $Q_{\text{D4}}=0$ (obtained by setting the
parameter $z_{1}$ to zero) these BPT backgrounds are an instance of
(4.23)-(4.25). Renaming the coordinate $z$ appearing there to $\zeta$, the
function $\alpha$ required to get the BPT solution is

$\alpha=(3\pi)^{3}F_{0}(\zeta_{0}^{3}-\zeta^{3}),$ (4.50)

with constants identified as $z_{0}=3\pi\zeta_{0}$. To see this one should use
the change of coordinates

$\zeta=\frac{zk}{3\pi},\qquad\cos\theta=\frac{z_{0}^{3}-(2-k^{3})z^{3}}{z_{0}^{3}-z^{3}k^{3}}.$
(4.51)

Recall the six-dimensional origin of the AFPRT backgrounds. With $n$ non-zero,
the backgrounds of BPT describe an analogous reduction of a six-dimensional
$(1,0)$ theory on a Riemann surface, but with simple punctures. These are the
4d SCFTs which we are deforming via the solutions in (4.39)-(4.44). The parent
six-dimensional theories are E-string theories coupled to quiver gauge
theories, as described in [70]. The appearance of an additional, exceptional
flavour group in the 6d theories is related to the relaxation of regular
boundary conditions for $\alpha$, interpreted as O8-D8 sources at that
endpoint of the $z$-interval (at $\zeta=0$, in the example above).

##  5 Deformed backgrounds in 11d

In this section, we study the embedding of the solution in (2) into the family
of solutions introduced by Lin, Lunin and Maldacena (LLM) [71] and also those
of Gaiotto and Maldacena (GM) [72] in the electrostatic form. These embeddings
are accomplished using the fact that minimal, $D=5$ gauged supergravity is a
consistent truncation of Romans’ $D=5$ $SU(2)\times U(1)$ supergravity, which
can in turn be uplifted to $D=11$ supergravity using the results of [65].
Thus, we begin with a review of the bosonic content of the 5d Romans’
supergravity and the embedding of (2) into it. We will also describe the dual
field theory interpretation of these constructions in terms of quiver
theories.

###  5.1 Embedding via 5d Romans supergravity

The bosonic field content of Romans’ $D=5$ $SU(2)\times U(1)$ gauged
supergravity [90] is a scalar field $X$, a metric, $U(1)$ and $SU(2)$ gauge
fields $B$ and $A^{i}$, $i=1,2,3$, and a complex two-form $C$ that is charged
under the $U(1)$ gauge field. The corresponding field strengths for these
potentials are

$\displaystyle G$ $\displaystyle=$ $\displaystyle\differential B,$
$\displaystyle F^{i}$ $\displaystyle=$ $\displaystyle\differential
A^{i}-\tfrac{1}{\sqrt{2}}m\epsilon_{ijk}A^{j}\wedge A^{k},$ $\displaystyle F$
$\displaystyle=$ $\displaystyle\differential C+imB\wedge C,$ (5.1)

where $m$ is related to the gauge coupling of the non-Abelian field. The
equations of motion for the scalar and the gauge fields will read

$\displaystyle\differential(X^{-1}\,{\star\differential X})$ $\displaystyle=$
$\displaystyle\tfrac{1}{3}X^{4}\,{\star G}\wedge G-\tfrac{1}{6}X^{-2}\,({\star
F^{i}}\wedge F^{i}+{\star{C}}\wedge\bar{C})$
$\displaystyle-\tfrac{4}{3}m^{2}\,(X^{2}-X^{-1})\,{\star\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}},$ $\displaystyle\differential(X^{4}\,{\star
G})$ $\displaystyle=$ $\displaystyle-\tfrac{1}{2}F^{i}\wedge
F^{i}-\tfrac{1}{2}{\bar{C}}\wedge C,$ $\displaystyle D(X^{-2}\,{\star F^{i}})$
$\displaystyle=$ $\displaystyle-F^{i}\wedge G,$ $\displaystyle X^{2}\,{\star
F}$ $\displaystyle=$ $\displaystyle i\,m\,C\,,$ (5.2)

in which $D(X^{-2}\star F^{i})\equiv\differential(X^{-2}\star
F^{i})+{\sqrt{2}}m\epsilon_{ijk}A^{k}\wedge(X^{-2}\star F^{j})$, and
$\epsilon_{01234}=+1$ for the five-dimensional space. $\bar{C}$ is the complex
conjugate of $C$. The Einstein equation is

$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
3X^{-2}\,\partial_{\mu}X\,\partial_{\nu}X-\tfrac{4}{3}m^{2}\,(X^{2}+2X^{-1})\,g_{\mu\nu}$
(5.3)
$\displaystyle+\tfrac{1}{2}X^{4}\,(G_{\mu}{}^{\rho}G_{\nu\rho}-\tfrac{1}{6}g_{\mu\nu}\,G_{\rho\sigma}G^{\rho\sigma})+\tfrac{1}{2}X^{-2}\,(F^{i\
\rho}_{\mu}F^{i}_{\nu\rho}-\tfrac{1}{6}g_{\mu\nu}\,F^{i}_{\rho\sigma}F^{i\rho\sigma})$
$\displaystyle+\tfrac{1}{2}X^{-2}\,({C}_{(\mu}{}^{\rho}\,\bar{C}_{\nu)\rho}-\tfrac{1}{6}g_{\mu\nu}\,C_{\rho\sigma}\bar{C}^{\rho\sigma})\,.$

These equations of motion can be derived from the following 5d Lagrangian

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle R\,{\star\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}-3X^{-2}{\star\differential
X}\wedge\differential X-\tfrac{1}{2}X^{4}\,{\star G}\wedge
G-\tfrac{1}{2}X^{-2}\,({\star F^{i}}\wedge F^{i}+\star
C_{{\scriptscriptstyle(2)}}\wedge\bar{C})$ (5.4)
$\displaystyle-\frac{i}{2m}C\wedge\bar{F}-\tfrac{1}{2}F^{i}\wedge F^{i}\wedge
B+4m^{2}(X^{2}+2X^{-1})\,{\star\hbox to0.0pt{1\hss}\mkern 4.0mu{\rm l}}\,.$

Note that if we restrict the various fields as

$X=1,\qquad F^{1}=F^{2}=C=0,\qquad F^{3}={\sqrt{2}}G$ (5.5)

the equations of motion (5.1) truncate to the equations of motion for minimal
$D=5$ gauged supergravity, (2.2) with the U(1) gauge fields $\mathcal{A}$ and
$B$ identified. We are interested in the soliton solution in (2), which we
repeat here for convenience:

$\displaystyle ds^{2}=\frac{r^{2}}{l^{2}}$
$\displaystyle(-dt^{2}+dv_{1}^{2}+dv_{2}^{2})+\frac{l^{2}dr^{2}}{r^{2}f(r)}+\frac{r^{2}}{l^{2}}f(r)d\phi^{2},\qquad
f(r)=1-\frac{\mu l^{2}}{r^{4}}-\frac{q^{2}l^{2}}{r^{6}},$ $\displaystyle
B=\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{\star}^{2}}\right)d\phi,\qquad
G=\mathcal{F}=-\frac{2q}{r^{3}}dr\wedge d\phi.$ (5.6)

Now we present the uplift to the LLM setup.

###  5.2 LLM embedding

The geometry of the most general AdS5 solutions in $D=11$ supergravity dual to
4d $\mathcal{N}=2$ SCFTs was first presented in [71]. It was proven that these
supergravity solutions are determined by solutions to a continuous three-
dimensional Toda equation. In [91], the same conditions were found from a
different perspective.

Following [65], we consider the embedding of the solution of the previous
section (5.5, 5.6) into the LLM background in 11d. The result is

$\displaystyle\frac{ds_{11}^{2}}{\kappa^{2/3}}$
$\displaystyle=e^{2\lambda}\left[4ds_{5}^{2}+y^{2}e^{-6\lambda}\,D\mu_{i}\,D\mu^{i}+\frac{4}{1-y\partial_{y}D_{0}}D\chi^{2}-\frac{\partial_{y}D_{0}}{y}dy^{2}-\frac{\partial_{y}e^{D_{0}}}{y}(dv_{1}^{2}+dv_{2}^{2})\right]$
$\displaystyle
D\chi=d\chi+a_{1}+\mathcal{A},\qquad\mathcal{A}=q\left(\frac{1}{r^{2}}-\frac{1}{r_{\star}^{2}}\right)d\phi,\qquad
a_{1}=\frac{1}{2}\left(\partial_{v_{2}}D_{0}\,dv_{1}-\partial_{v_{1}}D_{0}\,dv_{2}\right).$
(5.7)

Where $\kappa$ is the Newton constant in 11d, $ds_{5}^{2}$ is the line element
in (5.6), the functions $\lambda(v_{2},v_{2},y)$ and $D_{0}(v_{1},v_{2},y)$
are related as

$e^{-6\lambda}=\frac{-\partial_{y}D_{0}}{y(1-y\partial_{y}D_{0})},$ (5.8)

and we have parameterized the deformed two-sphere by

$\displaystyle
D\mu_{1}=d\mu_{1}+\sqrt{2}\mu_{2}A^{(3)}=d\mu_{1}+2\mu_{2}\mathcal{A}$ (5.9)
$\displaystyle
D\mu_{2}=d\mu_{2}-\sqrt{2}\mu_{1}A^{(3)}=d\mu_{2}-2\mu_{1}\mathcal{A}$ (5.10)
$\displaystyle D\mu_{3}=d\mu_{3},$ (5.11)

with the $\mu_{i}$ are subject to the constraint
$\delta^{ij}\mu_{i}\mu_{j}=1$. For example, one can select

$\displaystyle\mu_{1}=\sin\theta\cos\varphi,\quad\mu_{2}=\sin\theta\sin\varphi,\quad\mu_{3}=\cos\theta.$
(5.12)

The four-form field strength is given by

$G_{4}=G_{4}^{g}+\beta_{2}\wedge\mathcal{F}+\beta_{1}\wedge\star_{5}\mathcal{F},$
(5.13)

where $G_{4}^{g}$ can be obtained from the four-form field strength of the LLM
background by taking $d\chi+a_{1}\rightarrow D\chi=d\chi+a_{1}+\mathcal{A}$
and $d\mu_{i}\rightarrow D\mu_{i}$. Namely

$\displaystyle G_{4}^{g}=2\kappa\,\text{vol
}\tilde{S}^{2}\wedge\left[D\chi\wedge
d(y^{3}e^{-6\lambda})+y(1-y^{2}e^{-6\lambda})da_{1}-\frac{1}{2}\partial_{y}e^{D_{0}}\,dv_{1}\wedge
dv_{2}\right]$ (5.14) $\displaystyle\text{ vol
}\tilde{S}^{2}=\frac{1}{2}\varepsilon^{ijk}\mu_{i}D\mu_{j}\wedge D\mu_{k},$
(5.15)

and we have defined

$\displaystyle\beta_{1}=4\kappa\,d(\mu_{3}y),$ (5.16)
$\displaystyle\beta_{2}=-4\kappa\left[\frac{y^{3}}{2}e^{-6\lambda}\text{vol
}\tilde{S}^{2}+\left(\mu_{3}dy+y(1-y^{2}e^{-6\lambda})d\mu_{3}\right)\wedge
D\chi+\frac{\mu_{3}}{2}\partial_{y}e^{D_{0}}\,dv_{1}\wedge dv_{2}\right].$
(5.17)

When our deformation is turned off ($\mu=q=\mathcal{A}=0$), the 4d $N=2$ SCFT
duals enjoy an $SU(2)\times U(1)$ R-symmetry, which is manifest as the
isometries of the internal metric. The isometries of the two-sphere
parameterised by the $\mu^{i}$ corresponds to the $SU(2)$ part of the
R-symmetry. The isometries $S^{1}_{\chi}$ corresponds to the $U(1)$
R-symmetry. The embedding we have presented twists a particular $U(1)$ inside
of this original $SU(2)\times U(1)$ symmetry with the $U(1)_{\phi}$ of the 5d
soliton solution. This is reflected in the coefficients by which the one-form
$\mathcal{A}$ contributes to $D\chi$ and the $D\mu_{i}$.

###  5.3 The electrostatic case

Let us move the new background in (5.2)-(5.16) to the electrostatic notation
of e.g. [72]. This entails a transformation from the variables
$[y,v_{1},v_{2},D_{0}(v_{1},v_{2},y)]$ to $[\sigma,\eta,V(\sigma,\eta)]$,
assuming an additional isometry in the $[v_{1},v_{2}]$ plane. In fact,
defining

$v_{1}=R\cos\beta,~{}~{}v_{2}=R\sin\beta,\qquad\tilde{\chi}=\chi+\beta,~{}~{}\tilde{\beta}=-\beta$
(5.18)

and imposing that $\beta$ is the isometry direction, we have the following
Backlünd transformation,

$R^{2}e^{D_{0}(R,y)}=\sigma^{2},~{}~{}y=\sigma\partial_{\sigma}V=\dot{V},~{}~{}\log
R=\partial_{\eta}V=V^{\prime}.$ (5.19)

Following the steps detailed in the appendix A of the paper [92] we find that
the eleven dimensional metric reads

$\displaystyle\frac{ds^{2}_{11}}{\kappa^{2/3}}=$ $\displaystyle
f_{1}\Big{[}4ds_{5}^{2}+f_{2}D\mu_{i}D\mu^{i}+f_{3}(d\tilde{\chi}+\mathcal{A})^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+f_{5}\left(d\tilde{\beta}+f_{6}d\tilde{\chi}+f_{6}\mathcal{A}\right)^{2}\Big{]},$
$\displaystyle
f_{1}=\bigg{(}\frac{\dot{V}\tilde{\Delta}}{2V^{\prime\prime}}\bigg{)}^{\frac{1}{3}},~{}~{}~{}~{}f_{2}=\frac{2V^{\prime\prime}\dot{V}}{\tilde{\Delta}},~{}~{}~{}~{}f_{3}=\frac{4\sigma^{2}}{\Lambda},~{}~{}~{}~{}f_{4}=\frac{2V^{\prime\prime}}{\dot{V}},$
$\displaystyle f_{5}=\frac{2\Lambda
V^{\prime\prime}}{\dot{V}\tilde{\Delta}},~{}~{}~{}f_{6}=\frac{2\dot{V}\dot{V}^{\prime}}{V^{\prime\prime}\Lambda},~{}~{}~{}f_{7}=-\frac{\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}},~{}~{}f_{8}=\frac{1}{2}\bigg{(}\frac{\dot{V}\dot{V}^{\prime}}{\tilde{\Delta}}-\eta\bigg{)},$
$\displaystyle\tilde{\Delta}=\Lambda(V^{\prime\prime})^{2}+(\dot{V}^{\prime})^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Lambda=\frac{2\dot{V}-\ddot{V}}{V^{\prime\prime}}.$
(5.20)

Defining $D\tilde{\chi}=d\tilde{\chi}+\mathcal{A}$ the four-form field
strength can be written

$\displaystyle G_{4}=$ $\displaystyle
4\kappa\,d\left[f_{7}D\tilde{\chi}+f_{8}d\tilde{\beta}\right]\wedge\text{vol}\tilde{S}^{2}+4\kappa\,d(\mu_{3}\dot{V})\wedge\star_{5}\mathcal{F}$
(5.21) $\displaystyle+8\kappa(f_{7}D\tilde{\chi}+f_{8}d\tilde{\beta})\wedge
d\mu_{3}\wedge\mathcal{F}-4\kappa\left[d(\mu_{3}\dot{V})\wedge
D\tilde{\chi}+d(\mu_{3}\eta)\wedge d\tilde{\beta}\right]\wedge\mathcal{F}.$

Then, reducing to Type IIA along the $\beta$-direction to preserve SUSY, as
explained in [92], we find the string frame background,

$\displaystyle ds^{2}$
$\displaystyle=f_{1}^{\frac{3}{2}}f_{5}^{\frac{1}{2}}\bigg{[}4ds^{2}_{5}+f_{2}D\mu_{i}D\mu^{i}+f_{4}(d\sigma^{2}+d\eta^{2})+f_{3}D\tilde{\chi}^{2}\bigg{]},$
(5.22) $\displaystyle e^{\frac{4}{3}\Phi}$
$\displaystyle=f_{1}f_{5},~{}~{}~{}~{}H_{3}=4\kappa~{}d\left[f_{8}\text{vol}\tilde{S}^{2}-\eta\mu_{3}\mathcal{F}\right],~{}~{}~{}~{}C_{1}=f_{6}D\tilde{\chi},~{}~{}~{}~{},$
$\displaystyle C_{3}$ $\displaystyle=4\kappa
f_{7}D\tilde{\chi}\wedge\text{vol}\tilde{S}^{2}+4\kappa~{}\mu_{3}\dot{V}\left(\star_{5}\mathcal{F}-D\tilde{\chi}\wedge\mathcal{F}\right).$

We have used the relation
$d[\text{vol}\tilde{S}^{2}]=-2d\mu_{3}\wedge\mathcal{F}$. Hence

$\displaystyle F_{4}$ $\displaystyle=dC_{3}-H_{3}\wedge
C_{1}=4\kappa~{}d\left[f_{7}D\tilde{\chi}\wedge\text{vol}\tilde{S}^{2}+\mu_{3}\dot{V}\left(\star_{5}\mathcal{F}-D\tilde{\chi}\wedge\mathcal{F}\right)\right]-H_{3}\wedge
C_{1}.$

In the $r\to\infty$ limit, the background asymptotes to the original GM
background. In this limit, the 5d subspace in 5.22 with metric $ds_{5}^{2}$
reduces to AdS5, and the fibered sphere $\tilde{S}^{2}$ can be written as a
round sphere $S^{2}$ by absorption of the constant factors in the fibered
metric. This procedure can be done locally in the $r\to\infty$ limit. Page
charges of the background solution can be calculated in this limit following
[72, 92], which we will review in the next section. These will be the only
relevant brane charges present in the background, as the deformation does not
introduce any new cycles carrying relevant fluxes.

####  5.3.1 Rank function and Page charges

Boundary conditions for the function $V(\sigma,\eta)$ encode the data of a
particular dual linear-quiver field theory, via a Rank function. The formalism
is described in [93] and [92], and we provide a very brief summary here. It is
in this logic that we have an infinite number of background solutions in this
family, each one of them associated with a distinct 4d linear quiver ${\cal
N}=2$ SCFT. In our configuration, these UV fixed points are deformed by
compactification on the $\phi$-circle into a confining QFT.

We are interested in solutions of the Laplace equation

$\frac{1}{\sigma}\partial_{\sigma}(\sigma\partial_{\sigma}V)+\partial_{\eta}^{2}V\equiv\ddot{V}+\sigma^{2}V^{\prime\prime}=0.$
(5.23)

Suitable boundary conditions must be imposed on $V(\sigma,\eta)$. We choose
the following conditions

$\dot{V}\Big{|}_{\eta=0,P}=0,~{}~{}~{}~{}\dot{V}|_{\sigma=0}=\mathcal{R}(\eta),~{}~{}~{}~{}V\Big{|}_{\sigma\to\infty}=0$
(5.24)

We can reproduce a linear quiver consisting of $P-1$ nodes associated to
$SU(N_{i})$ vector multiplets by choosing the rank function $\mathcal{R}$ [93,
92] as

$\mathcal{R}(\eta)=\begin{cases}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}N_{1}\eta&\eta\in[0,1]\\\
N_{i}+(N_{i+1}-N_{i})(\eta-i)&\eta\in[i,i+1]~{}~{}~{}i=1,\cdots,P-2\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}N_{P-1}(P-\eta)&\eta\in[P-1,P],\end{cases}$ (5.25)

so that the range of $\eta$ is between $0$ and the integer $P$. For this
choice of rank function, one has

$\displaystyle V(\sigma,\eta)$ $\displaystyle=-\sum_{n=1}^{\infty}{\cal
R}_{n}\sin\bigg(\frac{n\pi}{P}\eta\bigg{missing})K_{0}\bigg{(}\frac{n\pi}{P}\sigma\bigg{)},$
(5.26) $\displaystyle{\cal R}_{n}$
$\displaystyle=\frac{2}{P}\int_{0}^{P}\mathcal{R}(\eta)\sin\bigg(\frac{n\pi}{P}\eta\bigg{missing})d\eta=\frac{2P}{(n\pi)^{2}}\sum_{k=1}^{P}b_{k}\sin\left(\frac{n\pi
k}{P}\right),~{}~{}~{}b_{k}=2N_{k}-N_{k+1}-N_{k-1},$

with $K_{0}(\sigma)$ the modified Bessel function of the second kind and
$N_{i}=0$ for $i=0$ and $i=P$. Now we quote the Page charges associated to
different D-branes in the bulk geometry. The details of the calculation can be
found in Appendix C.

  * •

There are quantized charges of the NS5 branes in the background,

$Q_{\text{NS5}}=-\frac{1}{(2\pi)^{2}}\int_{S^{3}}H_{3}=P.$ (5.27)

with $S^{3}$ being parameterised by the $(\eta,\tilde{S}^{2})$ coordinates in
the metric of 5.22

  * •

There are D6 branes in each interval of $\eta=[i,i+1]$ with the corresponding
Page charges

$Q^{i}_{D6}=-\frac{1}{2\pi}\int_{\tilde{S}^{2}}F_{2}=2N_{i}-N_{i-1}-N_{i+1}.$
(5.28)

The integration contour in given in the Appendix C. These branes support the
flavour degrees of freedom in the dual quiver field theory.

  * •

We can also define a quantized Page charge for D4 branes

$Q^{i}_{D4}=-\frac{1}{(2\pi)^{3}}\int_{\text{S}^{2}\times\tilde{\text{S}}^{2}}\hat{F}_{4}=-\frac{1}{(2\pi)^{3}}\int_{\text{S}^{2}\times\tilde{\text{S}}^{2}}(F_{4}-B_{2}\wedge
F_{2})=N_{i}-N_{i-1},$ (5.29)

The choice of integration contour is discussed in Appendix C. This is the
total amount of charge of D4 branes, including the ‘true’ colour D4 present in
the background and the D4 charge induced on the D6 and NS branes. The ‘true’
D4 charge in the interval $[i,i+1]$, related to the colour degrees of freedom
in the QFT, is $N_{i}$. These are interpreted as colour branes and support the
vector multiplets in the dual QFT.

Let us provide a brief interpretation of the associated field theory. The
electrostatic backgrounds in (5.22) whose functions are written in terms of a
rank function, as in (5.26) admit a very simple field theory dual. In fact,
they can be thought as the field theory associated with the low-energy limit
of a Hanany-Witten set-up consisting of $P$ NS five branes, that bound $N_{k}$
D4 branes in the k-th interval. There are $F_{k}=(2N_{k}-N_{k+1}-N_{k-1})$ D6
branes in the same interval. The dual UV-SCFT is an ${\cal N}=2$ quiver with
$(P-1)$ gauge nodes of rank $N_{k}$ and $F_{k}$ flavours. These UV SCFTs are
deformed by the presence of a VEV (proportional to the parameter $q$). The
deformation introduces the fibration of a diagonal U(1) group inside the
R-symmetry with the compact $\phi$-direction. This is the interpretation of
the infinite family of backgrounds in (5.22).

The SCFTs associated with the more generic backgrounds in (5.2) is slightly
more elaborated, in the sense that the UV SCFT is non-Lagrangian. These QFTs
were uncovered by Gaiotto in [94]. The deformation on those SCFTs is again via
a VEV for an $U(1)$ R-symmetry current. This is holographically realised, as
above, by the fibration between a diagonal $U(1)$ group in the R-symmetry and
the compact $\phi$-direction. This triggers a flow ending in a confining and
gapped system.

Let us now study non-perturbative aspects of all of the RG-flows in QFTs
associated with the backgrounds in the previous sections.

##  6 Observables

In this section we will calculate a plethora of observables in the field
theories dual to the new families of supergravity backgrounds introduced in
sections 3 and 4. We will review how the calculations are realized
holographically and then present the results for each case. An important
feature highlighted here is the universality of many of the observables.
Although from the geometric point of view this is to be expected by
construction, we appreciate the fact that even though the dual $\mathrm{QFT}s$
are very different, they all bear certain common characteristics. A
significant example being the key signatures of confinement exhibited by all
of the QFTs obtained from our deformation procedure. This feature can be
attributed as a consequence of the Gauntlett and Varela conjecture [79] about
consistent Kaluza-Klein truncations of wrapped products of $\mathrm{AdS}$
spaces with Riemannian manifolds. We will comment further on this in the
discussion section.

###  6.1 Wilson loop

To compute the Wilson loop holographically, we embed a probe string with fixed
endpoints at $x_{1}=\pm L/2$ and $r=\infty$, which enters the bulk in a
$\mathrm{U}$-shaped fashion (see [95]) while keeping the rest of the
coordinates constant. Choosing to parameterise the worldsheet as $\tau=t$,
$x_{1}=\sigma$ while letting $r=r(x_{1})$, in each example the induced metric
on the string is:

$\mathrm{d}s^{2}_{\mathrm{ind}}=-\frac{r^{2}}{l^{2}}\mathrm{d}t^{2}+\left(\frac{r^{2}}{l^{2}}+\frac{l^{2}}{r^{2}f(r)}r^{\prime
2}\right)\mathrm{d}x_{1}^{2}.$ (6.1)

We emphasize that because the string embedding extends only in the
$\widehat{\mathrm{AdS}}_{5}$ subspace, common to all of the backgrounds, (6.1)
is going to be the same for every case covered in this paper, modulo an
overall constant factor depending on the internal coordinates. This is
responsible for the universal behaviour of the Wilson loop. The Nambu-Gotto
action for the probe string takes the form:

$S_{\mathrm{NG}}=\frac{\mathcal{T}}{2\pi\alpha^{\prime}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F^{2}(r)+G^{2}(r)r^{\prime
2}},$ (6.2)

where we introduce the functions:

$F(r)=\frac{r^{2}}{l^{2}}\,\,,\,\,G(r)=\frac{1}{\sqrt{f(r)}}.$ (6.3)

The fact that $F(r_{\star})=r_{\star}^{2}/l^{2}\neq 0$ reveals a nonzero
effective tension of the chromoelectric string and already hints at confining
behaviour [96, 97, 98, 99]. Following [96, 98] we can write an approximate
expression for the length of separation $\hat{L}$ of the quark anti-quark
pair, as a function of the turning point of the string (denoted by $r_{0}$):

$\hat{L}(r_{0})=\left.\frac{\pi
G(r)}{F^{\prime}(r)}\right|_{r_{0}}=\frac{l^{2}\pi}{2r\sqrt{f(r_{0})}}=\frac{\pi
l^{2}r_{0}^{2}}{2\sqrt{r_{0}^{6}-\mu l^{2}r_{0}^{2}-q^{2}l^{2}}}.$ (6.4)

Studying this expression we get further evidence for confinement: As the
string reaches the end of the spacetime, that is $r_{0}\to r_{\star}$,
$\hat{L}(r_{0})$ diverges, allowing for an infinite separation of the pair. We
can then calculate the effective potential, see [100], which controls the
equation of motion of the string999the equation of motion reads:
$\derivative{r}{x}=\pm V_{\mathrm{eff}}(r),$ where the $\pm$ sign refers to
the left and right branches of the string around $r_{0}$.:

$V_{\mathrm{eff}}(r)=\frac{F(r)}{F(r_{0})G(r)}\sqrt{F^{2}(r)-F^{2}(r_{0})}=\frac{\sqrt{r^{4}-r_{0}^{4}}}{r_{0}^{2}l^{2}r}\sqrt{r^{6}-l^{2}(\mu
r^{2}+q^{2})}.$ (6.5)

This diverges $\sim r^{4}$ as $r\to\infty$ and vanishes as $r\to r_{0}$,
satisfying the conditions for confinement [100]. Then one can write the
separation and the energy101010where we normalize the expression by
subtracting the contribution of an infinitely massive quark and antiquark. of
the pair as the following integrals:

$L_{QQ}(r_{0})=2\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{V_{\mathrm{eff}}(r)}=2l^{2}r_{0}^{2}\int_{r_{0}}^{\infty}\frac{\mathrm{d}rr}{\sqrt{\left(r^{4}-r_{0}^{4}\right)\left[r^{6}-l^{2}(q^{2}+r^{2}\mu)\right]}},$
(6.6)

$E_{QQ}(r_{0})=\frac{r_{0}^{2}}{l^{2}}L_{QQ}(r_{0})+2\int_{r_{0}}^{\infty}\mathrm{d}r\frac{\sqrt{r^{4}-r_{0}^{4}}}{r^{2}\sqrt{f(r)}}-2\int_{r_{\star}}^{\infty}\frac{\mathrm{d}r}{\sqrt{f(r)}},$
(6.7)

where in the case of $q=\mu=0$ we see that the length and energy reduce to the
standard $\mathrm{AdS}_{5}$ expressions:

$\begin{split}&L_{\mathrm{AdS}_{5}}(r_{0})=\frac{2l^{2}}{r_{0}}\int_{1}^{\infty}\frac{\mathrm{d}y}{y^{2}\sqrt{y^{4}-1}}=\frac{2l^{2}}{r_{0}}\frac{\sqrt{2}\pi^{3/2}}{\Gamma^{2}\left(\frac{1}{4}\right)},\\\
&E_{\mathrm{AdS}_{5}}(r_{0})=2r_{0}\left[\int_{1}^{\infty}\mathrm{d}y\left(\frac{y^{2}}{\sqrt{y^{4}-1}}-1\right)-1\right],\\\
&E_{\mathrm{AdS}_{5}}(L)=-\frac{8\pi^{2}\sqrt{2g_{\mathrm{YM}}^{2}N}}{\Gamma^{4}\left(\frac{1}{4}\right)}\frac{1}{L_{\mathrm{AdS}_{5}}}.\end{split}$
(6.8)

We make use of numerical methods in order to study (6.6) and (6.7) and
parametrically plot the energy as a function of the length in Figure 2.
Indeed, we see a linear behaviour for large separations that is characteristic
of confinement.

Figure 1: Plots of the length and energy of separation of the quark-anti-quark
system (6.6) as a function of the turning point of the string $r_{0}$, for the
deformed $\mathrm{AdS}\times\mathrm{S}^{5}$ case. The parameters are fixed to
$l=\mu=q=1$. We emphasize that the results depicted in these plots, as well as
Figure 2, are the same for all the families of solutions we presented.

One can address the question regarding the stability of the
$\mathrm{U}$-shaped embedding that gives (6.1). Although this question should
be addressed by studying the equations of motion of fluctuations around the
embedding, this being the most trusted way that can also detect instabilities
originating from internal directions of the background, see [101, 102, 103], a
usual and convenient way of extracting information about this is by
calculating the derivative of the length function with respect to the turning
point $r_{0}$ and looking at its sign [99, 104, 97, 98, 56]. Since here our
functions $F$ and $G$ do not depend on internal space coordinates, we can use
the approximate expression (6.4) to get:

$Z(r_{0}):=\frac{\mathrm{d}\hat{L}(r_{0})}{\mathrm{d}r_{0}}=-\frac{l^{2}\pi\left[2f(r_{0})+r_{0}f^{\prime}(r_{0})\right]}{4r^{2}f^{3/2}(r_{0})}=-\frac{l^{4}\pi}{4r_{0}^{3}f^{3/2}(r_{0})}\left(\frac{4q^{2}}{r_{0}^{5}}+\frac{2r_{0}}{l^{2}}+\frac{2\mu}{r_{0}^{3}}\right)<0,$
(6.9)

which is negative and therefore the embedding is deemed stable. The reasoning
behind this criterion is that there are two conditions that need to be
satisfied: That the derivative of $E(L)$ with respect to the length is
positive, producing an attractive force (this is satisfied, as it is
proportional to $F(r)>0$) and also that the force is a non increasing function
of the separation (concavity condition) [104]. The sign of
$E^{\prime\prime}(L)$ is governed by $L^{\prime}(r_{0})=Z(r_{0})$. Thus the
regions of $r_{0}$ where $Z(r_{0})>0$ are unphysical and destabilize the
embedding.

Figure 2: Parametric plot of the energy (6.7) with respect to the length of
the separation of the pair for the deformed $\mathrm{AdS}\times\mathrm{S}^{5}$
with $l=\mu=q=1$. This interpolates between a Coulomb-like behaviour dictated
by conformality and a linear behaviour for large values of $L$, signaling
confinement.

We make note that the universal character of the results presented here is
surprising from a purely QFT perspective. By following our deformation
procedure and tuning $\mu=0$, one obtains many families of supersymmetric QFTs
that are in general very different from each other (some of them not even
admitting Lagrangian descriptions). Yet we have shown they all have the same
confining behaviour. This showcases once more the power of holography, as from
the gravity perspective the common geometric origin of this behaviour is very
clear. One more thing to emphasize is that it would be interesting to probe
other directions in the Wilson loop embeddings, in particular those that span
$M_{5}$. In that we might extract new information about the QFTs and new
physical phenomena.

###  6.2 ’t Hooft loop

To calculate the ’t Hooft loop, that is the magnetic analogue of the Wilson
loop, we probe the background with either a $\mathrm{D}3$ or a $\mathrm{D}4$
brane, depending on whether the background is in Type $\mathrm{IIB}$ or
$\mathrm{IIA}$, that extends at least on a $\mathbb{R}^{1,2}[t,x,\phi]$
submanifold of the background with $r=r(x)$, where $x$ is a field theory
direction and $\phi$ the shrinking circle. Then, we calculate the Dirac-Born-
Infeld action for the $\mathrm{D}$-brane using the induced metric. After
integrating the compact coordinates, which give a different overall factor for
each background, we get a similar expression to the integral in (6.2) for the
effective $2\mathrm{d}$ string:

$S_{\mathrm{D}p}=T_{\mathrm{D}p}\int\mathrm{d}^{p+1}\sigma
e^{-\Phi}\sqrt{-\mathrm{det}(g_{\mathrm{ind}})}\propto\int_{-\frac{L}{2}}^{\frac{L}{2}}\mathrm{d}x\sqrt{F_{t}^{2}(r)+G_{t}^{2}(r)r^{\prime
2}},\quad p=3,4,$ (6.10)

but with different functions $F_{t}$ and $G_{t}$. By probing the circular
direction $\phi$ with the brane, we get a factor of the function $f(r)$ in the
determinant which leads to screening, expressed as the vanishing of the
function $F_{t}$ in the $\mathrm{IR}$:
$\mathrm{T}_{\mathrm{eff}}=F_{t}(r_{\star})=0$.

This fact is also true for the Entanglement Entropy studied in the next
section, which has a phase transition as well. This can be attributed as a
generic feature of cigar-like geometries [96].

Below we list the calculation of (6.10) carried out for each background,
emphasising the universal character given by the integral above.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{S}}^{\mathbf{5}}}$

We will study the dynamics of the ’t Hooft loop for the background (3.1) by
introducing a $\mathrm{D}3$ brane on the subspace spanned by the coordinates
$[t,x_{1},\phi,\varphi_{3}]$ keeping $\theta=0$ and $r=r(x_{1})$. The induced
metric on this subspace then reads:

$\mathrm{d}s^{2}_{\mathrm{ind}}=\frac{r^{2}}{l^{2}}\left[-\mathrm{d}t^{2}+\left(1+\frac{l^{4}}{r^{4}f(r)}r^{\prime
2}\right)\mathrm{d}x_{1}^{2}+f(r)\mathrm{d}\phi^{2}\right]+l^{2}\left(\mathrm{d}\varphi_{3}+\frac{\mathcal{A}}{l}\right)^{2}.$
(6.11)

The Dirac-Born-Infeld action for this metric and a trivial dilaton, reduces to
the following:111111here, $L_{\phi}$ and $L_{\varphi_{3}}$ are the periods of
each cycle.

$\displaystyle S_{\mathrm{D}3}$ $\displaystyle=\int\mathrm{d}^{4}\sigma
e^{-\Phi}\sqrt{-\mathrm{det}(g_{\mathrm{ind}})}=T_{\mathrm{D}3}L_{\phi}L_{\varphi_{3}}\mathcal{T}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{t}^{2}(r)+G_{t}^{2}(r)r^{\prime
2}}\,,$ (6.12)

where the functions $F_{t}$ and $G_{t}$ read:

$F_{t}(r)=\frac{r^{3}\sqrt{f(r)}}{l^{2}}\,\,,\,\,G_{t}(r)=r.$ (6.13)

As mentioned, $F_{t}$ vanishes as $r_{0}\to r_{\star}$, giving off a zero
effective tension. We then expect the system to screen. To prove this, we use
(6.13) to calculate the approximate length $\hat{L}_{\mathrm{MM}}$, effective
potential, length $L_{\mathrm{MM}}$ and energy $E_{\mathrm{MM}}$ functions of
$r_{0}$, in the same fashion as in the Wilson loop case. These are given
below:

$\displaystyle\hat{L}_{\mathrm{MM}}(r_{0})$ $\displaystyle=\frac{\pi
l^{2}r_{0}^{3}}{3r_{0}^{4}-l^{2}\mu}\sqrt{f(r_{0})},$ (6.14) $\displaystyle
V_{\mathrm{eff}}(r)$
$\displaystyle=\frac{r^{2}\sqrt{f(r)}}{l^{4}F_{t}(r_{0})}\sqrt{r^{6}-l^{2}(q^{2}+r^{2}\mu)-F_{t}^{2}(r_{0})l^{4}},$
(6.15) $\displaystyle L_{\mathrm{MM}}(r_{0})$
$\displaystyle=2l^{4}F_{t}(r_{0})\int_{r_{0}}^{\infty}\frac{\mathrm{d}r}{r^{2}\sqrt{f(r)\left[r^{6}-l^{2}(q^{2}+\mu
r^{2})-l^{4}F_{t}^{2}(r_{0})\right]}},$ (6.16) $\displaystyle
E_{\mathrm{MM}}(r_{0})$
$\displaystyle=\frac{\sqrt{f(r_{0})}r_{0}^{3}}{l^{2}}L_{\mathrm{MM}}(r_{0})+2\int_{r_{0}}^{\infty}\mathrm{d}r\frac{\sqrt{f(r)r^{6}-f(r_{0})r_{0}^{6}}}{r^{2}\sqrt{f(r)}}-2\int_{r_{\star}}^{\infty}\mathrm{d}rr.$
(6.17)

Figure 3: Plots of the length function, its approximate expression and the
energy as a function of the length, for the monopole-anti-monopole pair for
the ’t Hooft loop, with $l=\mu=q=1$ fixed. We see the double-valuedness of the
energy, which expresses screening as a phase transition.

Notice again the factor $\sqrt{f(r_{0})}$ multiplying the first term in (6.17)
which is responsible for the vanishing of the effective tension [99, 95]: The
monopole-antimonopole pair can be arbitrarily separated with no energy cost.
This can be also seen as a phase transition: The system favours the
disconnected embedding of two straight strings extending from $r=\infty$ to
$r=r_{*}$, over the $\mathrm{U}$-shaped one described in the Wilson loop
section. This embedding expresses two infinitely massive probe monopoles
moving freely.

To figure out the stability of this embedding, following the logic explained
around (6.14):

$\begin{split}Z(r_{0})=\derivative{\hat{L}_{\mathrm{MM}}}{r_{0}}&=-\frac{l^{2}\pi\left[12f^{2}(r_{0})-r_{0}^{2}f^{\prime
2}(r_{0})+2r_{0}f(r_{0})\left(5f^{\prime}(r_{0})+r_{0}f^{\prime\prime}(r_{0})\right)\right]}{r_{0}^{2}\sqrt{f(r_{0})}\left(6f(r_{0})+r_{0}f^{\prime}(r_{0})\right)^{2}}\\\
&=\frac{\pi
l^{2}\left[l^{4}\mu^{2}-3r_{0}^{8}+6l^{2}(2q^{2}r_{0}^{2}+r_{0}^{4}\mu)\right]}{(l^{2}r\mu-3r_{0}^{5})^{2}\sqrt{f(r_{0})}},\end{split}$
(6.18)

which changes sign around its single root. The root can be seen in Figure 3 as
the maximum of $L$: The left branch describes an unstable embedding
($Z(r_{0})>0$) while the right branch, which corresponds to the zero energy
configuration, is stable ($Z(r_{0})<0$). As was discussed in [101], the study
of small linear fluctuations agrees with these regions being unphysical, as
these are the regions of $r_{0}$ for which their spectrum has negative
eigenfrequencies.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{T}}^{\mathbf{1,1}}}$

We take a $\mathrm{D}3$ probe brane in (3.10) extended on the submanifold
spanned by $[t,x_{1},\phi,\psi]$ and keep $\theta_{1}=0=\theta_{2}$,
$\phi_{1}=\mathrm{const}$, $\phi_{2}=\mathrm{const}$, $r=r(x_{1})$. The
induced metric on the $\mathrm{D}3$ reads:

$\mathrm{d}s^{2}_{\mathrm{ind}}=-\frac{r^{2}}{l^{2}}\mathrm{d}t^{2}+\left[\frac{r^{2}}{l^{2}}+\frac{l^{2}r^{\prime
2}}{r^{2}f(r)}\right]\mathrm{d}x_{1}^{2}+\frac{r^{2}f(r)}{l^{2}}\mathrm{d}\phi^{2}+\frac{l^{2}}{9}\left(\mathrm{d}\psi+\frac{\mathcal{A}}{l}\right)^{2},$
(6.19)

and the action takes the form:

$S_{\mathrm{D}3}=\frac{T_{\mathrm{D}3}\mathcal{T}L_{\psi}L_{\phi}}{3}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{t}^{2}(r)+r^{\prime
2}G_{t}^{2}(r)},$ (6.20)

having exactly the same functions as in the
$\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ uplift (6.13), so the results for the
approximate length, length and energy functions will be the same as in
(6.14)-(6.17). We again have a vanishing $F_{t}(r_{\star})$ which signals
screening.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{Y}}^{\mathbf{p,q}}}$

We will use in (3.13) the same embedding for the $\mathrm{D}3$ brane as in the
$\mathrm{T}^{1,1}$ case that is extended in the coordinates
$[t,x_{1},\phi,\psi]$ while setting $r=r(x_{1})$, $\theta=0$ and
$\varphi,y,\beta$ constant. The induced metric is then:

$\mathrm{d}s^{2}_{\text{ind}}=\frac{r^{2}}{l^{2}}\left[-\mathrm{d}t^{2}+\left(1+\frac{l^{4}\,r^{\prime
2}}{r^{4}}\right)\mathrm{d}x_{1}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{l^{2}}{9}\left(\mathrm{d}\psi+\frac{A_{1}}{l}\right)^{2},$
(6.21)

which is the same as (6.19) and we therefore get the exact same expressions
for the ’t Hooft loop in the $\mathrm{Y}^{p,q}$ uplift.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{AFPRT}}}$

Now turning to the type $\mathrm{IIA}$ solutions, we consider a probe
$\mathrm{D}4$ brane in the background (4.23) extended in
$[t,x_{1},\phi,\theta,\psi]$ with $r=r(x_{1})$ and
$x_{2},z,v_{1},v_{2}=\mathrm{const}$. We can then write the induced metric:

$\begin{split}\mathrm{d}s^{2}_{\mathrm{ind}}&=18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}\left\\{\frac{r^{2}}{l^{2}}\left[-\mathrm{d}t^{2}+\left(1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right)\mathrm{d}x_{1}^{2}+f(r)\mathrm{d}\phi^{2}\right]-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta
D\psi^{2}\right)\right\\}.\end{split}$ (6.22)

In this case, the action takes the form:

$\begin{split}\mathrm{S}_{\mathrm{D}4}=\frac{4\pi}{3}\mathrm{T}_{\mathrm{D}4}\mathcal{T}L_{\phi}\sqrt{-\frac{\alpha^{3}(z_{0})\ddot{\alpha}(z_{0})}{6\dot{\alpha}^{2}(z_{0})-9\alpha(z_{0})\ddot{\alpha}(z_{0})}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{t}^{2}(r)+r^{\prime
2}G_{t}^{2}(r)},\end{split}$ (6.23)

where $z_{0}$ is a constant value and now $F_{t}$ and $G_{t}$ differ from
(6.13) by a factor of $l^{-1}$:

$F_{t}(r)=\frac{r^{3}\sqrt{f(r)}}{l^{3}}\,\,,\,\,G_{t}(r)=\frac{r}{l},$ (6.24)

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{GM}}}$

For the background (5.22), we will again consider a $\mathrm{D}4$ brane that
extends in the submanifold $[t,x_{1},\phi,\varphi,\chi]$ setting
$\theta=\frac{\pi}{2}$ and all the other coordinates constant, as well as
$r=r(x_{1})$. The induced metric on the $\mathrm{D}4$ is then121212where we
used that
$D\mu_{i}D\mu_{i}\Big{|}_{\theta=\pi/2}=\sin^{2}\varphi(\mathrm{d}\varphi-2\mathcal{A})^{2}+\cos^{2}\varphi(\mathrm{d}\varphi-2\mathcal{A})^{2}=(\mathrm{d}\varphi-2\mathcal{A})^{2}$
from (5.11) and (5.12).:

$\mathrm{d}s^{2}_{\mathrm{ind}}=f_{1}^{3/2}f_{5}^{1/2}\Bigg{\\{}4\frac{r^{2}}{l^{2}}\left[-\mathrm{d}t^{2}\left(1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right)\mathrm{d}x_{1}^{2}+f(r)\mathrm{d}\phi^{2}\right]+f_{2}(\mathrm{d}\varphi-2\mathcal{A})^{2}+f_{3}D\chi^{2}\Bigg{\\}}.$
(6.25)

From this, we get the following action, that is again indicating screening:

$\begin{split}S_{\mathrm{D}4}=32\mathrm{T}_{\mathrm{D}4}\mathcal{T}L_{\phi}\pi^{2}f_{1}(\sigma_{0},\eta_{0})^{3}\sqrt{f_{2}(\sigma_{0},\eta_{0})f_{3}(\sigma_{0},\eta_{0})f_{5}(\sigma_{0},\eta_{0})}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{t}^{2}(r)+G_{t}^{2}(r)r^{\prime
2}},\end{split}$ (6.26)

where $(\sigma_{0},\eta_{0})$ are constant values of the coordinates and
$F_{t}$ and $G_{t}$ are the same as (6.24).

###  6.3 Entanglement Entropy

For the calculation of the Entanglement Entropy, we will consider a
codimension 2 manifold $\Sigma_{8}$ connecting two entangled regions and take
one of them to be of width $L_{\mathrm{EE}}$ while the remaining space
comprises the other region [105, 96, 106]. Then we write down the action, that
is the square root of the determinant of the induced metric on $\Sigma_{8}$
multiplied by a power of the dilaton (in the cases where $\Phi\neq 0$). One
integrates out the 7 coordinates and ends up with an integral expression:

$S_{\mathrm{EE}}=\frac{1}{4G_{N}}\int_{\Sigma_{8}}\mathrm{d}^{8}\sigma\sqrt{e^{-4\Phi}\mathrm{det}\left(g_{\Sigma_{8}}\right)}\propto\int_{-\frac{L}{2}}^{\frac{L}{2}}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+G_{\mathrm{EE}}^{2}(r)r^{\prime
2}},$ (6.27)

where $G_{N}$ denotes Newton’s constant in $10$ dimensions.

We will now present the results for each background, calculating the action to
observe that $F_{\mathrm{EE}}$ and $G_{\mathrm{EE}}$ are the same functions as
in the ’t Hooft loop (6.24). This will entail a phase transition in the
Entanglement Entropy. This has been argued to be yet another signal for
confinement [106].

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{S}}^{\mathbf{5}}}$

To calculate the Entanglement Entropy for this background, we consider the 8
dimensional manifold
$\Sigma_{8}[x_{1},x_{2},\phi,\theta,\varphi,\varphi_{1},\varphi_{2},\varphi_{3}]$
and $r=r(x_{1})$. The induced metric is:

$\mathrm{d}s^{2}_{\mathrm{ind}}=\frac{r^{2}}{l^{2}}\left[\left(1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right)\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+l^{2}\mathrm{d}\tilde{\Omega}_{5},$
(6.28)

where $\mathrm{d}\tilde{\Omega}_{5}$ denotes the element on the fibered five-
sphere spanned by $[\varphi,\theta,\varphi_{1},\varphi_{2},\varphi_{3}]$.
After computing the determinant, the action (6.27) reads:

$\begin{split}S_{\mathrm{EE}}&=\mathrm{V}_{\tilde{\mathrm{S}}^{5}}\frac{L_{x_{2}}L_{\phi}l^{5}}{4G_{N}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}~{}\sqrt{\frac{r^{6}}{l^{6}}f(r)\left[1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right]}\\\
&=\frac{l^{5}L_{x_{2}}\pi^{3}L_{\phi}}{4G_{N}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}~{}\sqrt{\frac{r^{6}}{l^{6}}f(r)\left[1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right]}.\end{split}$ (6.29)

Here, $\mathrm{V}_{\tilde{\mathrm{S}}^{5}}$ denotes the volume of the five-
sphere mentioned above131313We make note that the volume of the internal
spaces is preserved by the fibration, in this case,
$\mathrm{V}_{\mathrm{S}^{5}}=\mathrm{V}_{\tilde{\mathrm{S}}^{5}}$. We also use
throughout this paper the convention that $\mathrm{vol}_{\mathcal{M}}$ denotes
the volume form of a manifold $\mathcal{M}$, while $\mathrm{V}_{\mathcal{M}}$
its integrated volume. and $L_{x_{2}}$ the extent of the variable $x_{2}$. We
see that one gets the same functions as in (6.24) (we chose to keep the extra
$l$ factor here by having $l^{5}$ in the coefficient), therefore the length
function and the behaviour will be the same. That is,
$\mathrm{T}_{\mathrm{eff}}=0$ in the $\mathrm{IR}$ and the system exhibits a
phase transition, which is a sign of confinement.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{T}}^{\mathbf{1,1}}}$

We define the codimension two submanifold spanned by all the compact and two
spatial directions of the five-dimensional solution and $\mathrm{T}^{1,1}$:
$\Sigma_{8}[x_{1},x_{2},\phi,\theta_{1},\theta_{2},\phi_{1},\phi_{2},\psi]$
with $r=r(x_{1})$. The metric induced on $\Sigma_{8}$ is:

$\begin{split}\mathrm{d}s^{2}_{\mathrm{ind}}&=\left[\frac{r^{2}}{l^{2}}+\frac{l^{2}r^{\prime
2}}{r^{2}f(r)}\right]\mathrm{d}x_{1}^{2}+\frac{r^{2}}{l^{2}}\mathrm{d}x_{2}^{2}+\frac{l^{2}f(r)}{r^{2}}\mathrm{d}\phi^{2}\\\
&+\frac{l^{2}}{6}\sum_{i=1}^{2}\left(\mathrm{d}\theta_{i}^{2}+\sin^{2}\theta_{i}\mathrm{d}\phi_{i}^{2}\right)+\frac{l^{2}}{9}\left(\mathrm{d}\psi+\sum_{i=1}^{2}\cos\theta_{i}\mathrm{d}\phi_{i}+\frac{\mathcal{A}}{l}\right)^{2}.\end{split}$
(6.30)

We now calculate the action, for which we find:

$S_{\mathrm{EE}}=\frac{l^{5}L_{\phi}L_{x_{2}}\mathrm{V}_{\mathrm{T^{1,1}}}}{4G_{N}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+G_{\mathrm{EE}}^{2}(r)r^{\prime
2}},$ (6.31)

where the volume is:

$\mathrm{V}_{\mathrm{T}^{1,1}}=\frac{1}{108}\int_{\mathrm{T}^{1,1}}\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}\mathrm{d}\phi_{1}\mathrm{d}\phi_{2}\mathrm{d}\psi\sin\theta_{1}\sin\theta_{2}=\frac{16\pi^{3}}{27},$
(6.32)

and once again we get the same functions as in (6.24). Using the same
arguments, we conclude that a phase transition takes place, signaling
confinement.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{Y}}^{\mathbf{p,q}}}$

In the same manner as previously, we define the eight dimensional manifold to
be $\Sigma_{8}[x_{1},x_{2},\phi,\theta,\varphi,y,\beta,\psi]$, $r=r(x_{1})$
and the induced metric reads:

$\begin{split}\mathrm{d}s^{2}_{\mathrm{ind}}&=\frac{r^{2}}{l^{2}}\left[\left(1+\frac{l^{4}\,r^{\prime
2}}{r^{4}}\right)\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{(1-y)l^{2}}{6}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right)\\\
&+\frac{l^{2}}{w(y)v(y)}\mathrm{d}y^{2}+\frac{w(y)v(y)l^{2}}{36}\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)^{2}\\\
&+\frac{l^{2}}{9}\left(\mathrm{d}\psi-\cos\theta\mathrm{d}\varphi+y\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)+\frac{A_{1}}{l}\right)^{2},\\\
\end{split}$ (6.33)

which yields the following action:

$\begin{split}S_{\mathrm{EE}}&=\frac{L_{x_{2}}L_{\phi}L_{\beta}L_{\psi}\pi
l^{5}}{108G_{N}}\int_{y_{1}}^{y_{2}}\mathrm{d}y(y-1)\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{\frac{r^{6}}{l^{6}}f(r)+\frac{r^{2}}{l^{2}}r^{\prime
2}}\\\
&=\frac{L_{x_{2}}L_{\phi}l^{5}}{4G_{N}}\mathrm{V}_{\mathrm{Y}^{p,q}}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+G_{\mathrm{EE}}^{2}(r)r^{\prime
2}},\end{split}$ (6.34)

which once again gives off (6.24). We can also substitute the formula of the
volume of the Sasaki-Einstein manifold for completeness, that is given in
terms of $p$ and $q$ [107]:

$\begin{split}\mathrm{V}_{\mathrm{Y}^{p,q}}&=\frac{1}{108}\int_{\mathrm{Y}^{p,q}}\mathrm{d}\theta\mathrm{d}\varphi\mathrm{d}y\mathrm{d}\beta\mathrm{d}\psi(y-1)\sin\theta=\frac{8\pi^{3}}{27}\int_{y_{1}}^{y_{2}}\mathrm{d}y(y-1)\\\
&=\frac{q^{2}\left(2p+\sqrt{4p^{2}-3q^{2}}\right)\pi^{3}}{3p^{2}\left(3q^{2}-2p^{2}+p\sqrt{4p^{2}-3q^{2}}\right)},\end{split}$
(6.35)

where the reader can check that this reproduces the volume of
$\mathrm{T}^{1,1}$ as $(p,q)\to(1,0)$. We can therefore write the general
expression:

$S_{\mathrm{EE}}=\frac{L_{x_{2}}L_{\phi}q^{2}\left(2p+\sqrt{4p^{2}-3q^{2}}\right)\pi^{3}l^{5}}{12G_{N}p^{2}\left(3q^{2}-2p^{2}+p\sqrt{4p^{2}-3q^{2}}\right)}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+G_{\mathrm{EE}}^{2}(r)r^{\prime
2}}.$ (6.36)

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{AFPRT}}}$

For this calculation, we choose the eight-manifold as
$\Sigma_{8}[x_{1},x_{2},\phi,\theta,\psi,v_{1},v_{2},z]$ with $r=r(x_{1})$.
The induced metric on $\Sigma_{8}$ is written as:

$\begin{split}\mathrm{d}s_{\mathrm{ind}}^{2}=18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}&\biggl{\\{}\frac{r^{2}}{l^{2}}\left[\left(1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right)\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{1}{3}\mathrm{d}s_{\Sigma}^{2}-\frac{\ddot{\alpha}}{6\alpha}\mathrm{d}z^{2}\\\
&-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathcal{D}\psi^{2}\right)\biggr{\\}},\end{split}$
(6.37)

and the action reads:

$S_{\mathrm{EE}}=\frac{L_{x_{2}}L_{\phi}}{486G_{N}}\mathrm{V}_{\Sigma}\int_{0}^{P}\mathrm{d}z\left(-\alpha\ddot{\alpha}\right)\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+r^{\prime
2}G_{\mathrm{EE}}^{2}(r)},$ (6.38)

where the volume of the Riemann surface is given by

$\mathrm{V}_{\Sigma}=4\int\frac{\mathrm{d}v_{1}\mathrm{d}v_{2}}{(v_{1}^{2}+v_{2}^{2}-1)^{2}}.$
(6.39)

Here $P$ denotes the number of nodes in the dual quiver description and the
functions $F_{\mathrm{EE}}$ and $G_{\mathrm{EE}}$ are again the same as in
(6.24). Note that the factor $\int\mathrm{d}z\alpha\ddot{\alpha}$ is seen also
in (4.34) and will make its appearance in the calculation of the flow central
charge and holographic complexity studied in the upcoming sections.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{GM}}}$

Here we take the 8-manifold to extend in
$\Sigma_{8}[x_{1},x_{2},\phi,\theta,\varphi,\sigma,\eta,\tilde{\chi}]$ and let
$r=r(x_{1})$ as usual. The metric on $\Sigma_{8}$ then reads:

$\begin{split}\mathrm{d}s^{2}_{\mathrm{ind}}=f_{1}^{3/2}f_{5}^{1/2}&\Bigg{\\{}4\frac{r^{2}}{l^{2}}\left[\left(1+\frac{l^{4}r^{\prime
2}}{r^{4}f(r)}\right)\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]\\\
&+f_{2}D\mu_{i}D\mu_{i}+f_{4}(\mathrm{d}\sigma^{2}+\mathrm{d}\eta^{2})+f_{3}D\tilde{\chi}^{2}\Bigg{\\}}.\end{split}$
(6.40)

Then, the action can be written using the definitions of the various $f_{i}$’s
from (5.20), in the following form:

$S_{\mathrm{EE}}=\frac{64\pi^{2}L_{\phi}L_{x_{2}}}{G_{N}}\int_{0}^{P}\mathrm{d}\eta\int_{0}^{\infty}\mathrm{d}\sigma\sigma\dot{V}V^{\prime\prime}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F_{\mathrm{EE}}^{2}(r)+G_{\mathrm{EE}}^{2}(r)r^{\prime
2}},$ (6.41)

where again, we end up with the same functions (6.24) in the $x_{1}$ integral.

Concluding this section, we emphasize that our results for the Entanglement
entropy are all of the form:

$S_{\mathrm{EE}}=\frac{\mathcal{N}_{\mathrm{i}}}{4}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{F^{2}_{\mathrm{EE}}(r)+G^{2}_{\mathrm{EE}}(r)r^{\prime
2}},$ (6.42)

where the functions $F_{\mathrm{EE}}$ and $G_{\mathrm{EE}}$ are found to be
the same as in the case of the ’t Hooft loop. This implies a phase transition
which following [106, 96] is indicative of confinement. We will see that the
constants $\mathcal{N}_{i}$, that depend on the details of each background,
are related to the counting of the degrees of freedom and complexity of the
system as analysed more closely in sections 6.4,6.5. These are given along
with formulas like (6.42) for the observables describing the aforementioned
properties in table 1 towards the end of subsection 6.5.

###  6.4 Flow central charge

In a generic $\mathrm{CFT}$ the central charge of the theory captures the
number of degrees of freedom and is equivalent to the free energy of the
system. The observable studied here called the holographic flow central
charge, is a monotonic function that extends this notion in holography for a
flow across dimensions. The quantity takes a constant value at the
$\mathrm{CFT}$ fixed points and describes the number of degrees of freedom as
the system flows between them [108, 109]. We make note that the dimension of
the QFT changes along the $\mathrm{RG}$ flow in the backgrounds studied here.
The flow central charge is capable of detecting if the theory is gapped as
well as the existence of any conformal fixed points.

At the $\mathrm{CFT}$ points, the central charge is proportional to the volume
of the internal space $M_{5}$. The flow central charge is defined following
this fact. We start by writing a background dual to a $(d+1)$-dimensional
$\mathrm{QFT}$ in the form [109]:

$\displaystyle\mathrm{d}s^{2}=-\alpha_{0}\mathrm{d}t^{2}+\alpha_{1}\mathrm{d}x_{1}^{2}+\alpha_{2}\mathrm{d}x_{2}^{2}+\dots+\alpha_{d}\mathrm{d}x_{d}^{2}+\left(\alpha_{1}\alpha_{2}\cdots\alpha_{d}\right)^{\frac{1}{d}}\beta(r)\mathrm{d}r^{2}+g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j}),$
$\displaystyle\Phi=\Phi(r,\theta^{i}),$ (6.43)

for some functions $\alpha_{i}(r),\beta(r)$. The last part expresses the
internal manifold. We then define the submanifold that includes the field
theory directions and the internal space:

$G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}=\alpha_{1}\mathrm{d}x_{1}^{2}+\alpha_{2}\mathrm{d}x_{2}^{2}+....+\alpha_{d}\mathrm{d}x_{d}^{2}+g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j}).$
(6.44)

For our purpose we have $d=3$. Then, the flow central charge can be calculated
by first working out the following quantity:

$H(r)=\left[\int\mathrm{d}^{5}\theta\sqrt{e^{-4\Phi}\mathrm{det}\left(G_{ij}\right)}\right]^{2}.$
(6.45)

Following [109, 88], we have that:

$c_{\mathrm{flow}}=\frac{d^{d}\beta_{0}^{d/2}H^{\frac{2d+1}{2}}}{G^{(10)}_{N}\left(H^{\prime}\right)^{d}}.$
(6.46)

We list the calculation of $c_{\mathrm{flow}}$ for all the different
backgrounds below, where once more we find the different dual $\mathrm{QFT}$s
to have a universal character for this observable as well.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{S}}^{\mathbf{5}}}$

We can rewrite the background in the form (6.43), where:

$\alpha_{0}=\frac{r^{2}}{l^{2}}=\alpha_{2}\,\,,\,\,\alpha_{1}=\frac{r^{2}}{l^{2}}f(r)\,\,,\,\,\beta_{0}=\frac{l^{4}}{r^{4}}f^{-4/3}(r),$
(6.47)

and the last part expresses the fibered five-sphere:

$\begin{split}g_{ij}\left(\mathrm{d}\theta^{i}-A^{(i)}\right)\left(\mathrm{d}\theta^{j}-A^{(j)}\right)=l^{2}\mathrm{d}\tilde{\Omega}^{2}_{5}&=l^{2}\left\\{\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}+\sin^{2}\theta\sin^{2}\varphi\left(\mathrm{d}\varphi_{1}+\frac{A_{1}}{l}\right)^{2}\right.\\\
+&\left.\sin^{2}\theta\cos^{2}\varphi\left(\mathrm{d}\varphi_{2}+\frac{A_{1}}{l}\right)^{2}+\cos^{2}\theta\left(\mathrm{d}\varphi_{3}+\frac{A_{1}}{l}\right)^{2}\right\\}.\end{split}$
(6.48)

We now define the subspace:

$\begin{split}G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}&=\alpha_{1}\mathrm{d}\phi^{2}+\alpha_{2}(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2})+g_{ij}(\mathrm{d}\theta^{i}-A^{(i)})(\mathrm{d}\theta^{j}-A^{(j)})\\\
&=\frac{r^{2}}{l^{2}}f(r)\mathrm{d}\phi^{2}+\frac{r^{2}}{l^{2}}\left(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}\right)+l^{2}\mathrm{d}\widehat{\Omega}_{5},\end{split}$
(6.49)

after taking the determinant and calculating (6.45) to be:

$H(r)=\left(\mathrm{V}_{\mathrm{S}^{5}}l^{2}r^{3}\sqrt{f(r)}\right)^{2}=l^{4}\pi^{6}r^{6}f(r),$
(6.50)

we find:

$c_{\mathrm{flow}}=\frac{l^{8}\mathrm{V}_{\tilde{\mathrm{S}}^{5}}}{8G_{N}}\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}=\frac{27l^{8}\pi^{3}r^{3}}{8G_{N}}\frac{\left(r^{6}-l^{2}r^{2}\mu-l^{2}q^{2}\right)^{3/2}}{\left(3r^{4}-l^{2}\mu\right)^{3}}.$
(6.51)

We immediately notice by power counting that this function takes a fixed value
at infinity:

$c_{\mathrm{UV}}=\lim_{r\to\infty}c(r)=\frac{l^{8}\pi^{3}}{8G_{N}},$ (6.52)

which is indicative of the conformal symmetry of the 4-dimensional QFT in the
$\mathrm{UV}$, and depends on the details of the internal space for each
background ($\tilde{\mathrm{S}}^{5}$, in the present example). We also notice
as the system flows towards the $\mathrm{IR}$, $c_{\mathrm{flow}}$ vanishes
due to the $f$ factor in the numerator. This expresses the absence of
dynamical degrees of freedom in the deep $\mathrm{IR}$, where the system is
gapped and governed by a $\mathrm{TQFT}$. These facts are true for all the
solutions studied here, all having a similar functional form as (6.51).

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{T}}^{\mathbf{1,1}}}$

Here, we have again

$\alpha_{0}=\frac{r^{2}}{l^{2}}=\alpha_{2}\,\,,\,\,\alpha_{1}=\frac{r^{2}}{l^{2}}f(r)\,\,,\,\,\beta_{0}=\frac{l^{4}}{r^{4}}f^{-4/3}(r),$
(6.53)

and the internal space is written as:

$g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j})=l^{2}\left\\{\frac{1}{6}\sum_{i=1}^{2}\left(\mathrm{d}\theta_{i}^{2}+\sin^{2}\theta_{i}\mathrm{d}\phi_{i}^{2}\right)+\frac{1}{9}\left(\mathrm{d}\psi+\sum_{i=1}^{2}\cos\theta_{i}\mathrm{d}\phi_{i}+\frac{3}{l}\mathcal{A}\right)^{2}\right\\}.$
(6.54)

Figure 4: Plot of the flow central charge normalized to its $\mathrm{UV}$
value for the deformed $\mathrm{AdS}_{5}\times\mathrm{T}^{1,1}$ (keeping
$l=\mu=q=1$): It interpolates between an IR gapped 3d system (expressing a
$\mathrm{TQFT}$ in the deep IR) and the 4d $\mathrm{SCFTs}$ in the UV, where
the value it takes depends on the details of the internal space of the theory.
This behaviour is similar for all the supergravity solutions in this work.

Then, we consider the following 8-dimensional subspace

$\begin{split}G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}&=\frac{r^{2}}{l^{2}}f(r)\mathrm{d}\phi^{2}+\frac{r^{2}}{l^{2}}\left(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}\right)+\frac{l^{2}}{6}\sum_{i=1}^{2}\left(\mathrm{d}\theta_{i}^{2}+\sin^{2}\theta_{i}\mathrm{d}\phi_{i}^{2}\right)\\\
&+\frac{l^{2}}{9}\left(\mathrm{d}\psi+\sum_{i=1}^{2}\cos\theta_{i}\mathrm{d}\phi_{i}\right)^{2},\end{split}$
(6.55)

and in a very similar calculation, we find $H$ to be:

$H(r)=\left(\mathrm{V}_{\mathrm{T}^{1,1}}l^{2}r^{3}\sqrt{f(r)}\right)^{2}=\mathrm{V}_{\mathrm{T}^{1,1}}^{2}l^{4}r^{6}f(r),$
(6.56)

This leads to the following flow central charge:

$c_{\mathrm{flow}}=\frac{l^{8}\mathrm{V}_{\mathrm{T}^{1,1}}}{8G_{N}}\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}=\frac{2l^{8}\pi^{3}r^{3}\left(r^{6}-l^{2}r^{2}\mu-l^{2}q^{2}\right)^{3/2}}{G_{N}(3r^{4}-l^{2}\mu)^{3}}.$
(6.57)

We emphasize that the form of this expression is nearly the same as in (6.51)
with the difference being the volume of the internal manifold. Here the
$\mathrm{UV}$ value reads:

$c_{\mathrm{UV}}=\frac{2l^{8}\pi^{3}}{27},$ (6.58)

and we present the plot of $c_{\mathrm{flow}}/c_{\mathrm{UV}}$ for this case
in Figure 4 where the previously described universal behaviour is apparent.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{Y}}^{\mathbf{p,q}}}$

For the $\mathrm{Y}^{p,q}$ calculation, the internal space is:

$\begin{split}g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j})=&l^{2}\left[\frac{1-y}{6}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right)+\frac{1}{w(y)v(y)}\mathrm{d}y^{2}+\frac{w(y)v(y)}{36}\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)^{2}\right.\\\
&\left.+\frac{1}{9}\left(\mathrm{d}\psi-\cos\theta\mathrm{d}\varphi+y\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)+\frac{3}{l}\mathcal{A}\right)^{2}\right],\\\
\end{split}$ (6.59)

and the functions $\alpha_{0},\alpha_{1},\alpha_{2},\beta$ are the same. Then,
by defining:

$\begin{split}G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}&=\frac{r^{2}}{l^{2}}f(r)\mathrm{d}\phi^{2}+\frac{r^{2}}{l^{2}}\left(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}\right)+l^{2}\left[\frac{1-y}{6}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right)+\frac{1}{w(y)v(y)}\mathrm{d}y^{2}\right.\\\
&\left.+\frac{w(y)v(y)}{36}\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)^{2}+\frac{1}{9}\left(\mathrm{d}\psi-\cos\theta\mathrm{d}\varphi+y\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)+\frac{3}{l}\mathcal{A}\right)^{2}\right],\end{split}$
(6.60)

we find

$H(r)=\left(\mathrm{V}_{\mathrm{Y}^{p,q}}l^{2}r^{3}\sqrt{f(r)}\right)^{2}=\mathrm{V}_{\mathrm{Y}^{p,q}}^{2}l^{4}r^{6}f(r),$
(6.61)

with the volume of $\mathrm{Y}^{p,q}$ given in (6.35). Then the expression for
the flow central charge is found to be the generalisation of (6.57) for a
generic $\mathrm{Y}^{p,q}$:

$\begin{split}c_{\mathrm{flow}}&=\frac{l^{8}\mathrm{V}_{\mathrm{Y}^{p,q}}}{8G_{N}}\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}=\frac{q^{2}\left(2p+\sqrt{4p^{2}-3q^{2}}\right)\pi^{3}}{3p^{2}\left(3q^{2}-2p^{2}+p\sqrt{4p^{2}-3q^{2}}\right)}\frac{l^{8}r^{3}\left(r^{6}-l^{2}r^{2}\mu-l^{2}q^{2}\right)^{3/2}}{8G_{N}(3r^{4}-l^{2}\mu)^{3}},\end{split}$
(6.62)

where we make note that this expression is also of the same form as (6.51) and
(6.57).

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{AFPRT}}}$

In this case we have the following functions:

$\alpha_{1}=\alpha_{2}=18\pi\sqrt{-\frac{\alpha}{\ddot{\alpha}}}\frac{r^{2}}{l^{2}}\,\,,\,\,\alpha_{3}=18\pi\sqrt{-\frac{\alpha}{\ddot{\alpha}}}\frac{r^{2}}{l^{2}}f(r)\,\,,\,\,\beta=\frac{l^{4}}{r^{4}f^{4/3}(r)},$
(6.63)

and the internal space metric is:

$g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j})=18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}\left[\frac{1}{3}\mathrm{d}s_{\Sigma}^{2}-\frac{\ddot{\alpha}}{6\alpha}\mathrm{d}z^{2}-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathcal{D}\psi^{2}\right)\right].$
(6.64)

We then take the subspace

$\begin{split}&G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}=\alpha_{1}\mathrm{d}\phi^{2}+\alpha_{2}(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2})+g_{ij}(\mathrm{d}\theta^{i}-A^{(i)})(\mathrm{d}\theta^{j}-A^{(j)})\\\
&=\alpha_{1}\mathrm{d}\phi^{2}+\alpha_{2}(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2})+18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}\left[\frac{1}{3}\mathrm{d}s_{\Sigma}^{2}-\frac{\ddot{\alpha}}{6\alpha}\mathrm{d}z^{2}-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathcal{D}\psi^{2}\right)\right],\end{split}$
(6.65)

and compute its determinant weighted by the dilaton to get:

$\begin{split}H(r)&=\left(\int_{\mathrm{S}^{2}}\mathrm{d}\theta\mathrm{d}\psi\sin\theta\int_{\Sigma}\mathrm{d}v_{1}\mathrm{d}v_{2}\int_{0}^{P}\mathrm{d}z\frac{2r^{3}\sqrt{f(r)}\sin\theta\alpha\ddot{\alpha}}{243l^{3}\pi(v_{1}^{2}+v_{2}^{2}-1)^{2}}\right)^{2}\\\
&=\left(\frac{2r^{3}\mathrm{V}_{\Sigma}\sqrt{f(r)}}{243l^{3}}\int_{0}^{P}\mathrm{d}z\alpha\ddot{\alpha}\right)^{2}.\end{split}$
(6.66)

This $H$ function leads to the flow central charge:

$c_{\mathrm{flow}}=\frac{l^{3}\mathrm{V}_{\Sigma}}{972G_{N}}\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}\int_{0}^{P}\mathrm{d}z\left(-\alpha\ddot{\alpha}\right).$
(6.67)

We notice the expression $\int\mathrm{d}z(-\alpha\ddot{\alpha})$ that was
foreshadowed in (4.34).

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{GM}}}$

In the Gaiotto-Maldacena case, (5.22) gives off:

$\alpha_{1}=\alpha_{2}=4(f_{1}^{3}f_{5})^{1/2}\frac{r^{2}}{l^{2}},\,\,\alpha_{3}=4(f_{1}^{3}f_{5})^{1/2}\frac{r^{2}}{l^{2}}f(r),\,\,\beta=\frac{l^{4}}{r^{4}f^{4/3}(r)},$
(6.68)

while the 5-dimensional internal metric has the form:

$g_{ij}(\mathrm{d}\theta^{i}-A^{i})(\mathrm{d}\theta^{j}-A^{j})=f_{1}^{\frac{3}{2}}f_{5}^{\frac{1}{2}}\left[f_{2}D\mu_{i}D\mu_{i}+f_{4}(\mathrm{d}\sigma^{2}+\mathrm{d}\eta^{2})+f_{3}D\tilde{\chi}^{2}\right].$
(6.69)

By defining

$\begin{split}G_{ij}\mathrm{d}\xi^{i}\mathrm{d}\xi^{j}=f_{1}^{3/2}f_{5}^{1/2}&\Bigg{\\{}4\frac{r^{2}}{l^{2}}\left[\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]\\\
&+f_{2}D\mu_{i}D\mu_{i}+f_{4}(\mathrm{d}\sigma^{2}+\mathrm{d}\eta^{2})+f_{3}(\mathrm{d}\tilde{\chi}+\mathcal{A})^{2}\Bigg{\\}},\end{split}$
(6.70)

and computing the $H$ function to be:

$\begin{split}H(r)&=\left(\int_{\mathrm{S}^{2}}\mathrm{d}\theta\mathrm{d}\phi\sin\theta\int\mathrm{d}\tilde{\chi}\int_{0}^{\infty}\mathrm{d}\sigma\int_{0}^{P}\mathrm{d}\eta\frac{8f_{1}^{9/2}f_{2}f_{4}r^{3}\sqrt{f_{3}f_{5}f(r)}}{l^{3}}\right)^{2}\\\
&=\left(\frac{256\pi^{2}r^{3}\sqrt{f(r)}}{l^{3}}\int_{0}^{\infty}\mathrm{d}\sigma\int_{0}^{P}\mathrm{d}\eta\sigma\dot{V}V^{\prime\prime}\right)^{2},\end{split}$
(6.71)

we find the central charge to be:

$c_{\mathrm{flow}}=\frac{32l^{3}\pi^{2}}{G_{N}}\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}\int_{0}^{\infty}\mathrm{d}\sigma\int_{0}^{P}\mathrm{d}\eta\sigma\dot{V}V^{\prime\prime}.$
(6.72)

As in the previous section, we will make note that all of the results for
$c_{\mathrm{flow}}$ are captured by the following formula using the factors
defined in the table 1:

$c_{\mathrm{flow}}=\frac{s_{i}l^{3}}{8}\left(\frac{\mathcal{N}_{i}}{L_{x_{2}}L_{\phi}}\right)\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3},$
(6.73)

where $s_{i}=1$ if $i\in\\{\mathrm{S}^{5},\mathrm{Y}^{p,q},\mathrm{AFPRT}\\}$
and $s_{i}=8$ for $i=\mathrm{GM}$.

###  6.5 Holographic complexity

Here we will calculate another quantity called holographic complexity, which
is very related to the holographic central charge. In general, given a quantum
circuit, computational complexity measures how many elementary gates one needs
in order to construct a specific state in the Hilbert space from a reference
one. Let us review the prescription to calculate this observable in
holography, given in [59], as for the type of backgrounds we work in here a
special prescription is needed. We will use the CV conjecture (a review of
different conjectures is given in [110]). The bottom line is that writing the
backgrounds in the form

$\mathrm{d}s_{10,\mathrm{st}}^{2}=-g_{tt}\mathrm{d}t^{2}+\mathrm{d}s_{9}^{2},$
(6.74)

we should compute, according to [59],

$\displaystyle\mathcal{C}_{V}=\frac{1}{G_{N}}\int\mathrm{d}^{9}x\sqrt{\frac{e^{-4\Phi}\det(g_{9})}{{\mathrm{A}}}},$
(6.75)

where $A$ denotes an overall conformal factor in the metric of each
background. Let us perform the explicit calculation for the three families of
solutions we have.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{S}}^{\mathbf{5}}}$

We start by writing a constant time slice of the 10-dimensional metric (3.1),
that is:

$\begin{split}\mathrm{d}s_{9}^{2}&=\frac{r^{2}}{l^{2}}\left[\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{l^{2}\mathrm{d}r^{2}}{r^{2}f(r)}+l^{2}\biggl{\\{}\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}+\sin^{2}\theta\sin^{2}\varphi\left(\mathrm{d}\varphi_{1}+\frac{\mathcal{A}}{l}\right)^{2}\\\
&+\sin^{2}\theta\cos^{2}\varphi\left(\mathrm{d}\varphi_{2}+\frac{\mathcal{A}}{l}\right)^{2}+\cos^{2}\theta\left(\mathrm{d}\varphi_{3}+\frac{\mathcal{A}}{l}\right)^{2}\biggr{\\}}.\end{split}$
(6.76)

while the conformal prefactor $\mathrm{A}$ and the dilaton factor are trivial:
$e^{-4\Phi}={\mathrm{A}}=1$. Then, the square root of the determinant yields:

$\sqrt{\mathrm{det}(g_{9})}=l^{3}r^{2}\cos\theta\sin^{3}\theta\sin\varphi$
(6.77)

$\mathcal{C}_{V}=\frac{l^{3}\pi^{3}L_{x_{1}}L_{x_{2}}L_{\phi}}{G_{N}}\int_{r_{\star}}^{\Lambda_{\mathrm{UV}}}\mathrm{d}rr^{2}=\frac{l^{3}\pi^{3}L_{x_{1}}L_{x_{2}}L_{\phi}}{3G_{N}}(\Lambda_{\mathrm{UV}}^{3}-r^{3}_{\star}).$
(6.78)

Compare this result with (6.29) and (6.51) to find the common features
mentioned previously.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{Y}}^{\mathbf{p,q}}}$

By looking at the metric (3.13) of the 10d background, we write:

$\begin{split}\mathrm{d}s_{9}^{2}&=\frac{r^{2}}{l^{2}}\left[\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{l^{2}\mathrm{d}r^{2}}{r^{2}f(r)}+l^{2}\left[\frac{1-y}{6}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right)+\frac{1}{w(y)v(y)}\mathrm{d}y^{2}\right.\\\
&\left.+\frac{w(y)v(y)}{36}\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)^{2}+\frac{1}{9}\left(\mathrm{d}\psi-\cos\theta\mathrm{d}\varphi+y\left(\mathrm{d}\beta+\cos\theta\mathrm{d}\varphi\right)+\frac{3}{l}\mathcal{A}\right)^{2}\right],\\\
\end{split}$ (6.79)

and again $A$ and the dilaton are trivial. We then have:

$\sqrt{\mathrm{det}(g_{9})}=\frac{l^{3}r^{2}(y-1)\sin\theta}{108},$ (6.80)

which gives off the following complexity:

$\mathcal{C}_{V}=\frac{L_{x_{1}}L_{x_{2}}L_{\phi}l^{3}}{G_{N}}\mathrm{V}_{\mathrm{Y}^{p,q}}\int_{r_{\star}}^{\Lambda_{\mathrm{UV}}}\mathrm{d}rr^{2}=\frac{L_{x_{1}}L_{x_{2}}L_{\phi}l^{3}}{3G_{N}}\mathrm{V}_{\mathrm{Y}^{p,q}}(\Lambda_{\mathrm{UV}}^{3}-r_{\star}^{3}),$
(6.81)

where we used a cutoff $\Lambda_{\mathrm{UV}}$ to regularize the expression.
This can also be compared with equations (6.36) and (6.62). The result for
$\mathrm{T}^{1,1}$ can be obtained by taking $(p,q)\to(1,0)$. The fact that
this is proportional to the $\mathrm{UV}$ cutoff reflects the infinite
dimensional space of states of the system and the difficulty of connecting
different states.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{AFPRT}}}$

In this case, comparing with (4.23) we have:

$\begin{split}\mathrm{d}s^{2}_{9}&=18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}}\left\\{\frac{r^{2}}{l^{2}}\left[\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{l^{2}\mathrm{d}r^{2}}{r^{2}f(r)}+\frac{1}{3}\mathrm{d}s^{2}_{\Sigma}-\frac{\ddot{\alpha}}{6\alpha}\mathrm{d}z^{2}-\frac{\alpha\ddot{\alpha}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathcal{D}\psi^{2}\right)\right\\},\\\
&\mathrm{A}=18\pi\sqrt{-\frac{\alpha}{6\ddot{\alpha}}},\,\,\,e^{-4\Phi}=\frac{1}{2^{5}3^{17}\pi^{10}}\left(-\frac{\ddot{\alpha}}{\alpha}\right)^{3}\left(2\dot{\alpha}^{2}-3\alpha\ddot{\alpha}\right)^{2}.\end{split}$
(6.82)

Given this, we find the holographic complexity to be:

$\begin{split}\mathcal{C}_{V}&=\frac{L_{x_{1}}L_{x_{2}}L_{\phi}\mathrm{V}_{\mathrm{S}^{2}}\mathrm{V}_{\Sigma}}{1458\pi
l^{2}G_{N}}\int_{0}^{P}\mathrm{d}z\left(-\alpha\ddot{\alpha}\right)\int_{r_{\star}}^{\Lambda_{\mathrm{UV}}}\mathrm{d}rr^{2}\\\
&=\frac{L_{x_{1}}L_{x_{2}}L_{\phi}\mathrm{V}_{\mathrm{S}^{2}}\mathrm{V}_{\Sigma}}{1458\pi
l^{2}G_{N}}\int_{0}^{P}\mathrm{d}z\left(-\alpha\ddot{\alpha}\right)\left(\Lambda_{\mathrm{UV}}^{3}-r_{\star}^{4}\right).\end{split}$
(6.83)

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{GM}}}$

For this solution, we have from (5.22):

$\begin{split}&\mathrm{d}s^{2}_{9}=(f_{1}^{3}f_{5})^{1/2}\left\\{\frac{4r^{2}}{l^{2}}\left[\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right]+\frac{4l^{2}\mathrm{d}r^{2}}{r^{2}f(r)}+f_{2}D\mu_{i}D\mu_{i}+f_{4}(\mathrm{d}\sigma^{2}+\mathrm{d}\eta^{2})+f_{3}D\tilde{\chi}^{2}\right\\},\\\
&\mathrm{A}=(f_{1}^{3}f_{5})^{1/2},\,\,\,e^{-4\Phi}=(f_{1}f_{5})^{-3},\end{split}$
(6.84)

and the complexity reads:

$\mathcal{C}_{V}=\frac{64L_{x_{1}}L_{x_{2}}L_{\phi}L_{\tilde{\chi}}\mathrm{V}_{\mathrm{S}^{2}}}{3l^{2}G_{N}}\left(\Lambda_{\mathrm{UV}}^{3}-r_{\star}^{3}\right)\int_{0}^{\infty}\mathrm{d}\sigma\int_{0}^{P}\mathrm{d}\eta\dot{V}V^{\prime\prime}\sigma,$
(6.85)

and can be compared with (6.41) and (6.72). Finally, we provide the following
formula that summarizes the results of this subsection:

$\mathcal{C}_{\mathrm{V}}=\frac{m_{i}}{3l^{2}}\left(L_{x_{1}}\mathcal{N}_{i}\right)\left(\Lambda^{3}-r_{\star}^{3}\right),$
(6.86)

where $m_{i}=1$ for $i\in\\{\mathrm{S}^{5},\mathrm{Y}^{p,q}\\}$, $m_{i}=1/2$
for $i=\mathrm{AFPRT}$ and $m_{i}=2$ for $i=\mathrm{GM}$ and $\mathcal{N}_{i}$
can be found in table 1.

Table 1: Summary of the results for the EE, flow central charge and complexity. |  $\displaystyle\rule{0.0pt}{21.52771pt}\widehat{\mathrm{AdS}}_{5}\times\widehat{\mathrm{S}}^{5}$ |  $\displaystyle\rule{-34.44434pt}{0.0pt}\widehat{\mathrm{AdS}}_{5}\times\widehat{\mathrm{Y}}^{p,q}$ |  $\displaystyle\rule{-150.69397pt}{0.0pt}\widehat{\mathrm{AdS}}_{5}\times\widehat{\mathrm{AFPRT}}$ |  $\displaystyle\rule{-64.58313pt}{0.0pt}\widehat{\mathrm{AdS}}_{5}\times\widehat{\mathrm{GM}}$
---|---|---|---|---
$\displaystyle\rule{6.0pt}{0.0pt}\mathcal{N}_{i}$ |  $\displaystyle\rule{0.0pt}{30.1388pt}\frac{l^{5}L_{x_{2}}L_{\phi}\mathrm{V}_{\mathrm{S}^{5}}}{G_{N}}$ |  $\displaystyle\rule{-34.44434pt}{0.0pt}\frac{l^{5}L_{x_{2}}L_{\phi}\mathrm{V}_{\mathrm{Y}^{p,q}}}{G_{N}}$ |  $\displaystyle\rule{-150.69397pt}{0.0pt}\frac{L_{x_{2}}L_{\phi}\mathrm{V}_{\mathrm{S}^{2}}\mathrm{V}_{\Sigma}}{486\pi G_{N}}\int_{0}^{P}\mathrm{d}z(-\alpha\ddot{\alpha})$ |  $\displaystyle\rule{-64.58313pt}{0.0pt}\frac{32L_{x_{2}}L_{\phi}L_{\tilde{\chi}}\mathrm{V}_{\mathrm{S}^{2}}}{G_{N}}\int_{0}^{P}\mathrm{d}\eta\int_{0}^{\infty}\mathrm{d}\sigma\sigma\dot{V}V^{\prime\prime}$
$\displaystyle\rule{6.0pt}{0.0pt}\mathrm{EE}$ |  |  $\displaystyle\rule{0.0pt}{34.44434pt}\frac{\mathcal{N}_{i}}{4}\int_{-L/2}^{L/2}\mathrm{d}x_{1}\sqrt{\frac{r^{6}}{l^{6}}\left[f(r)+\frac{l^{4}r^{\prime 2}}{r^{4}}\right]}$ |  |
$\displaystyle\rule{6.0pt}{0.0pt}c_{\mathrm{flow}}$ |  |  $\displaystyle\rule{0.0pt}{34.44434pt}\frac{s_{i}l^{3}}{8}\left(\frac{\mathcal{N}_{i}}{L_{x_{2}}L_{\phi}}\right)\left[\frac{\sqrt{f(r)}}{f(r)+\frac{r}{6}f^{\prime}(r)}\right]^{3}$ |  $\displaystyle\rule{43.05542pt}{0.0pt}\begin{aligned} s_{i}=\begin{cases}1,&i\in\\{\mathrm{S}^{5},\mathrm{Y}^{p,q},\mathrm{AFPRT}\\},\\\ 8,&i=\mathrm{GM}\end{cases}\end{aligned}$ |
$\displaystyle\rule{6.0pt}{0.0pt}\mathcal{C}_{V}$ |  |  $\displaystyle\rule{0.0pt}{34.44434pt}\frac{m_{i}}{3l^{2}}\left(L_{x_{1}}\mathcal{N}_{i}\right)\left(\Lambda_{\mathrm{UV}}^{3}-r_{\star}^{3}\right)$ |  $\displaystyle\rule{43.05542pt}{0.0pt}\begin{aligned} m_{i}=\begin{cases}1,&i\in\\{\mathrm{S}^{5},\mathrm{Y}^{p,q},\mathrm{AFPRT}\\},\\\ 2,&i=\mathrm{GM}\end{cases}\end{aligned}$ |

###  6.6 Gauge coupling

In this subsection, we attempt to define a gauge coupling and see how it flows
to the IR of the dual $\mathrm{QFT}$. All of our backgrounds correspond to a
dual $\mathrm{QFT}$ that at high energies extends in $[t,x_{1},x_{2},\phi]$,
being $\phi$ the periodic direction that we mix with the R-symmetry in a
twisting-like fashion. We will use the Dirac-Born-Infeld action for a probe
Dp-brane that extends along the field theory directions and contains a gauge
field on its worldvolume $\Sigma_{p+1}$. We will read the gauge coupling as
the coefficient of the gauge kinetic term $-\frac{1}{4}F_{\mu\nu}^{2}$, while
the $r$ dependence in our results will be interpreted as the running of the
coupling constant. We will briefly review the procedure which was nicely
summarized in [56], the main idea being that the general action for the Dp-
brane consists of two terms: The DBI141414we use the convention where there is
no $2\pi$ in front of $F$ and the Wess-Zumino term,

$S_{\mathrm{D}_{p}}=S_{\mathrm{D}_{p},\mathrm{DBI}}+S_{\mathrm{D}_{p},\mathrm{WZ}},$
(6.87)

which for a zero NS $B$ field read

$S_{\mathrm{D}_{p},\mathrm{DBI}}=T_{\mathrm{D}_{p}}\int_{\Sigma_{p+1}}\mathrm{d}^{p+1}\sigma\sqrt{-e^{-2\Phi}\mathrm{det}(h+F)},$
(6.88)
$S_{\mathrm{D}_{p},\mathrm{WZ}}=-T_{\mathrm{D}_{p}}\int_{\Sigma_{p+1}}\mathcal{C}\wedge
e^{-F},$ (6.89)

where $h$ is the induced metric on $\Sigma_{p+1}$,
$F=F_{\mu\nu}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}$ ($\mu,\nu=0,\dots,p+1)$
the field strength of the gauge field on $\Sigma_{p+1}$ and $\mathcal{C}$
denotes the RR polyform. Although for our purpose we will focus solely on the
$\mathrm{DBI}$ part of the action, we note that the $\mathrm{WZ}$ term
contains important information regarding the BPS status of the brane and the
$\theta$-term in the field theory. We will consider a small field strength
expansion which brings (6.88) to the form:

$S_{\mathrm{D}_{p},\mathrm{DBI}}=T_{\mathrm{D}_{p}}\int_{\Sigma_{p+1}}\mathrm{d}^{p+1}\sigma\sqrt{-e^{-2\Phi}\mathrm{det}(h)}\left(1+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathcal{O}(F^{4})\right),$
(6.90)

with the indices being raised with $h_{\mu\nu}$. For the cases studied in this
paper, we can without loss of generality turn on only the $t-x_{1}$ component
of $F_{\mu\nu}$ and define the part of the induced metric extending over the
field theory directions, which is of the form:

$h^{(3)}_{\mu\nu}\equiv h_{ij}=K(r,\theta^{a})\eta_{ij},$ (6.91)

where we introduced the flat indices $i,j\in\\{t,x_{1},x_{2}\\}$ and a
function $K$ of the holographic coordinate that for the type $\mathrm{IIA}$
solutions also depends on the internal space coordinates, denoted as
$\\{\theta^{a}\\}$. Then the quadratic in $F_{\mu\nu}$ term on $\Sigma_{p+1}$
gives off:

$F_{\mu\nu}F^{\mu\nu}\equiv
F_{ij}F^{ij}=h^{ik}h^{jl}F_{kl}F_{ij}=\frac{2}{K^{2}}F_{tx_{1}}F_{tx_{1}}$
(6.92)

By performing the $\phi$ integral and separating the remaining integral on the
worldvolume to the field theory part and the rest of the directions in (6.90),
we get the following Maxwell term included in the $\mathrm{DBI}$
action151515here $\Sigma_{\mathrm{int}}$ stands for the part of the
worldvolume consisting of internal coordinates, depending on the specific
probe.:

$\begin{split}S_{\mathrm{D}_{p},\mathrm{DBI}}&\supset
L_{\phi}\mathrm{T}_{\mathrm{D}_{p}}\int\mathrm{d}^{2+1}x\int_{\Sigma_{\mathrm{int}}}\mathrm{d}^{p-3}\sigma\sqrt{-e^{-2\Phi}\mathrm{det}(h)}\frac{1}{2}K^{-2}F_{tx_{1}}^{2}\\\
&=\frac{1}{4}\left[L_{\phi}\mathrm{T}_{\mathrm{D}_{p}}\int_{\Sigma_{\mathrm{int}}}\mathrm{d}^{p-3}\sigma\sqrt{-e^{-2\Phi}\mathrm{det}(h)}K^{-2}\right]2\int\mathrm{d}^{2+1}xF_{tx_{1}}^{2}.\end{split}$
(6.93)

From this term one can read off the object inside the square brackets tho be
the Yang-Mills gauge coupling of the 3-dimensional effective theory:

$\frac{1}{g_{\mathrm{YM}}^{2}}=L_{\phi}\mathrm{T}_{\mathrm{D}_{p}}\int_{\Sigma_{\mathrm{int}}}\mathrm{d}^{p-3}\sigma\sqrt{-e^{-2\Phi}\mathrm{det}(h)}K^{-2}(r,\theta^{a})$
(6.94)

Let us see the details of applying this formula in each of the backgrounds.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{S}}^{\mathbf{5}}},\mathrm{\mathbf{Y}^{p,q}}$

For the deformed uplifts of the five-sphere and Sasaki-Einstein manifolds, we
consider a $\mathrm{D}3$ probe brane extended along $[t,x_{1},x_{2},\phi]$ and
switch on a field strength in the $t-x_{1}$ directions as described above. The
induced metric on the brane for the $\mathrm{S}^{5}$ background can be read
from (3.1) to be161616by $\mathcal{A}_{\phi}$ we denote the component of the
one form $\mathcal{A}$.:

$\mathrm{d}s^{2}_{\mathrm{ind},\mathrm{D}3}=h_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\frac{r^{2}}{l^{2}}\left[-\mathrm{d}t+\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+(f(r)+\mathcal{A}_{\phi}^{2})\mathrm{d}\phi^{2}\right],$
(6.95)

where the quadratic term of the component of the gauge field originates from
the $\propto\mathrm{d}\phi^{2}$ part of the fibration. The 3d metric then
reads:

$h_{ij}=K(r)\eta_{ij},\,\,K(r)=\frac{r^{2}}{l^{2}},$ (6.96)

We see that in this case there are no remaining internal coordinates in the
worldvolume to integrate over and the dialton is trivial, therefore (6.94)
yields the following 3d coupling:

$\frac{1}{g_{\mathrm{YM}}^{2}}=L_{\phi}T_{\mathrm{D}_{3}}\sqrt{f(r)+\frac{l^{2}}{r^{2}}\mathcal{A}_{\phi}}.$
(6.97)

This expression reveals the strongly coupled nature governing the
$\mathrm{IR}$ dynamics of the $\mathrm{QFT}_{3}$ since for $r\to r_{\star}$
the coupling constant diverges. As for the $\mathrm{UV}$ limit $r\to\infty$ we
note that this takes the value of the classical gauge coupling of the circle
reduced $\mathrm{QFT}_{4}$. Finally, we comment that the case for the
$\mathrm{Y}^{p,q}$ uplift is extremely similar, and one can derive it by
setting $\mathcal{A}_{\phi}\to 3\mathcal{A}_{\phi}$ in (6.97).

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{AFPRT}}}$

Here we will use a $\mathrm{D}4$ brane to probe the
$[t,x_{1},x_{2},\phi,\psi]$ directions with
$\theta,r,v_{1},v_{2},z=\mathrm{const.}$ We switch on the same field strength
as before in the worldvolume ($F_{tx_{1}}\neq 0$) and the induced metric on
the $\mathrm{D4}$ is:

$\mathrm{d}s^{2}_{\mathrm{ind},\mathrm{D}4}=18\pi\sqrt{-\frac{\alpha}{\ddot{\alpha}}}\left\\{\frac{r^{2}}{l^{2}}\left(-\mathrm{d}t^{2}+\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2}+f(r)\mathrm{d}\phi^{2}\right)-\frac{\alpha\ddot{\alpha}\sin^{2}\theta_{0}}{6\dot{\alpha}^{2}-9\alpha\ddot{\alpha}}(\mathrm{d}\psi-3\mathcal{A})\right\\}.$
(6.98)

In this case the conformal factor of the 3d field theory metric reads:

$K(r)=18\pi\sqrt{-\frac{\alpha(z_{0})}{6\ddot{\alpha}(z_{0})}}\frac{r^{2}}{l^{2}}.$
(6.99)

Then, applying (6.94) we find the gauge coupling of the 3d field theory to be:

$\frac{1}{g^{2}_{\mathrm{YM}}}=-\frac{\sin\theta_{0}L_{\phi}L_{\psi}T_{\mathrm{D}4}\ddot{\alpha}(z_{0})}{162\pi^{2}}\sqrt{f(r)}.$
(6.100)

We notice that this also diverges in the $r\to r_{\star}$ limit and takes a
constant value in the $\mathrm{UV}$. This expression also depends on the node
of the quiver in the dual description, which amounts to focusing on one of the
colour groups.

##### Deformed
$\mathrm{\mathbf{AdS}_{\mathbf{5}}\mathbf{\times}\mathrm{\mathbf{GM}}}$