Matrix Models and Holography: Mass Deformations of Long Quiver Theories in 5d
and 3d

Mohammad<EMAIL_ADDRESS>Andrea
<EMAIL_ADDRESS>Carlos
<EMAIL_ADDRESS>Leonardo<EMAIL_ADDRESS>and Lucas<EMAIL_ADDRESS>

†Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan
∗ Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
∘Pitaevskii BEC Center, CNR-INO and Physics Department, Universitá di Trento,
I-38123 Trento, Italy
‡ Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084,
China

Abstract

We enlarge the dictionary between matrix models for long linear quivers
preserving eight supercharges in $d=5$ and $d=3$ and type IIB supergravity
backgrounds with AdSd+1 factors. We introduce mass deformations of the field
theory that break the quiver into a collection of interacting linear quivers,
which are decoupled at the end of the RG flow. We find and solve a Laplace
problem in supergravity which realises these deformations holographically. The
free energy and expectation values of antisymmetric Wilson loops are
calculated on both sides of the proposed duality, finding agreement. With our
matching procedure, the free energy satisfies a strong version of the
F-theorem.

###### Contents

  1. 1 Introduction
    1. 1.1 General idea and organisation of this paper
  2. 2 Part I: Supergravity
    1. 2.1 Quivers, SCFTs, and rank functions
    2. 2.2 The Type IIB backgrounds dual to 5d SCFTs
      1. 2.2.1 Resolution of the PDE and quantisation of charges
    3. 2.3 The Type IIB backgrounds dual to 3d SCFTs
      1. 2.3.1 Resolution of the PDE and quantisation of charges
    4. 2.4 Holographic central charge
    5. 2.5 The case of two rank functions: two interacting linear quivers
    6. 2.6 Wilson loops in supergravity
    7. 2.7 On the reliability of the supergravity backgrounds
    8. 2.8 Matrix models for electrostatic problems
  3. 3 Part II: Matrix models
    1. 3.1 Coulomb branches and mass deformations
      1. 3.1.1 Mass parameter space
      2. 3.1.2 Coulomb branch geometry
    2. 3.2 Long quivers and their large $N$ limit
      1. 3.2.1 Notation
      2. 3.2.2 Mass deformations
      3. 3.2.3 Sphere partition functions
      4. 3.2.4 Setting up the large $N$ limit
      5. 3.2.5 Long quiver limit
      6. 3.2.6 Saddle point equation
      7. 3.2.7 Match with the supergravity solution
    3. 3.3 Free energies in massive deformations of 5d and 3d SCFTs
      1. 3.3.1 Free energies in 5d and 3d
      2. 3.3.2 Free energy in 5d
      3. 3.3.3 Free energy in 3d
      4. 3.3.4 Holographic match in 5d and 3d SCFTs
    4. 3.4 Two interacting linear quivers
      1. 3.4.1 Change of variables to probe the singularities
      2. 3.4.2 Change of variables, Coulomb branches and holography
      3. 3.4.3 Long quiver limit after change of variables
      4. 3.4.4 Free energy for two interacting linear quivers
      5. 3.4.5 Free energy for two interacting linear quivers in 5d
      6. 3.4.6 Free energy for two interacting linear quivers in 3d
    5. 3.5 Holographic match
      1. 3.5.1 Matching the effective free energy in 5d
      2. 3.5.2 Matching the effective free energy in 3d
      3. 3.5.3 Chern–Simons terms for background vector multiplets
    6. 3.6 On the F-theorem
    7. 3.7 Wilson loops in QFT
  4. 4 Conclusions
  5. A Potentials, harmonic and special functions
    1. A.1 Potentials and special functions in odd dimensions
  6. B Five-dimensional quivers in M-theory
  7. C Integrating out massive fields in long quivers
    1. C.1 Warm-up example: 3d SQCD
      1. C.1.1 Large mass: Coulomb branch considerations
      2. C.1.2 Large mass: free energy considerations
      3. C.1.3 Take-home lesson
      4. C.1.4 A remark on the F-theorem
    2. C.2 Massive fields in long quivers
      1. C.2.1 The contribution of massive fields in 5d
      2. C.2.2 The contribution of massive fields in 3d
  8. D Proof of the F-theorem
    1. D.1 Holographic F-theorem with two rank functions
    2. D.2 F-theorem in mass-deformed SCFTs

##  1 Introduction

The cross-fertilisation between AdS/CFT [1] and Matrix Models has a long
history. Matrix model techniques have been instrumental in checking
holographic results and suggesting ideas for string theory calculations. A
selection of early papers [2, 3, 4, 5, 6, 7, 8] shows various instances in
which exact results in field theory, calculated using matrix models, were
matched with calculations in supergravity. The field experienced a rapid
growth with the advent of supersymmetric localisation [9], with an impressive
amount of refined checks of dualities between AdS geometries and CFTs on a
sphere performed in three [10, 11, 12, 13, 14, 15, 16, 17], four [18, 19, 20,
21, 22, 23, 24, 25, 26] and five dimensions [27, 28, 29, 30, 31, 32, 33].

Let us focus on $d$-dimensional superconformal field theories (SCFTs)
preserving eight supercharges, whose dual supergravity solutions are
geometries containing AdSd+1 factors. The construction of the ten-dimensional
configurations, consisting of a metric, Ramond and Neveu–Schwarz background
fields, has been systematised for the case of eight preserved supercharges for
$d=1$ [34, 35, 36], $d=2$ [37, 38, 39, 40, 41, 42, 43], $d=3$ [44, 45, 46, 47,
48, 49, 50, 51, 52, 53], $d=4$ [54, 55, 56, 57, 58, 59, 60], $d=5$ [61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71] and $d=6$ [72, 73, 74, 75, 76, 77].

Based on localisation of the path integral [9], for every SCFT of the class
mentioned above which is connected to a gauge theory via Renormalization Group
(RG) flow, matrix models have been developed that accurately calculate various
observable quantities that preserve a fraction of the supersymmetry, most
notably the free energy and Wilson loop expectation values. These matrix
models encode exact information in principle, but their intricacies grow with
$d$ and with the complexity of the quiver. Whilst, in various cases, large
ranks or long quiver approximations are needed to solve the matrix model,
these usually coincide with the regime of validity of the dual supergravity
background.

Five-dimensional $\mathcal{N}=1$ SCFTs are inherently strongly coupled [78]
and do not admit a Lagrangian description. They therefore pose a challenge to
traditional approaches to calculating CFT observables. A fruitful strategy to
obtain information about their strongly coupled dynamics, is to deform the
theory away from the conformal point, where it may admit a quiver gauge theory
description, and compute their partition functions, possibly decorated with
Wilson loops.111Complementary approaches that address directly the conformal
point are geometric engineering of the SCFTs in M-theory [79, 80] or using
fivebrane webs in Type IIB string theory [81]. Three-dimensional
$\mathcal{N}=4$ SCFTs also enjoy many rich properties, chief amongst them the
infrared symmetry enhancement and mirror symmetry [82]. The method just
outlined applies to three-dimensional $\mathcal{N}=4$ theories as well. These,
though, have the advantage that the gauge kinetic term is $Q$-exact, thus the
$\mathbb{S}^{3}$ partition function can be evaluated directly in the SCFT.

In this paper, we focus on the matrix models and holographic backgrounds for
5d and 3d SCFTs. Remarkably, on both sides of the holographic duality, the
theory is characterized by a potential function which is the solution of a 2d
electrostatic problem. On the supergravity side, the two-dimensional space is
the unconstrained part of the internal space and the Laplace equation emerges
as a consequence of the BPS conditions. On the SCFT side, the electrostatic
problem arises as a saddle point equation of the matrix model. One of the two
dimensions has an immediate interpretation as the direction along the quiver.
The second direction is harder to interpret from the field theory perspective,
and it emerges at large $N$ from the spectrum of eigenvalues of the matrix
model, which effectively becomes a continuum.

With this dictionary, the supergravity solution is reliable when the dual SCFT
is given by a long quiver, whose solution was pioneered in [83] for 5d
$\mathcal{N}=1$ SCFTs and in [84, 85] for 3d $\mathcal{N}=4$ SCFTs. It has
been proven for all balanced linear quiver theories [70, 51] that the solution
to the aforementioned electrostatic problem, which is defined in terms of a
single charge distribution, gives rise to the supergravity background dual to
these long quivers.

In this paper, we aim to back up the expectation that more general
electrostatic problems provide holographic duals for a very large class of
theories. This approach is indeed amenable to extend the gauge/gravity
dictionary away from AdSd+1/CFTd, testing the correspondence at arbitrary
points of RG flows. In particular, our goal is to identify a holographic
manifestation of turning on real mass deformations.222In 5d, the only
supersymmetry-preserving relevant deformations are mass deformations,
including the deformations leading to gauge theory phases [86]. We will argue
that this operation corresponds, in the electrostatic setup, to introducing a
second charge density, separated in the 2d plane from the original charge
density by a finite distance proportional to the mass. Part of the results
were announced in [87].

###  1.1 General idea and organisation of this paper

In this work, we aim to provide another entry in the holographic dictionary
between supergravity and CFT, with the field theory side expressed via matrix
models. We study the situation in which a long balanced linear quiver is
deformed by one or more real mass parameters. Note that we do not simply give
mass to a single matter field, but suitably choose a configuration of masses
involving a large number of fields.

This setup also admits a description in terms of two or more interacting
linear quivers, with an effective interaction term controlled by the mass
parameter(s). These $d$-dimensional ($d=5$ or $d=3$ in this paper)
supersymmetric field theories describe the full RG flow from a single SCFT,
when the mass is switched off, to a collection of decoupled SCFTs when the
mass is very large compared to the inverse radius of the sphere
$\mathbb{S}^{d}$ on which the theory is placed. We attempted to capture this
information in Figure 1 and Figure 2 for five and three spacetime dimensions,
respectively.

The above mentioned conformal points are well captured by the new supergravity
solutions carefully derived and explained in this work. The matching of the
free energy, calculated for the field theory and in the holographic dual
background, gives credit to the proposal we put forward for the holographic
dual of the mass deformation.

$\bullet$SCFT$\bullet$SCFT${}_{1}\ \oplus$ SCFT2${\scriptstyle
m\to\infty}$$\bullet$Gauge theory$\bullet$Gauge${}_{1}\ \oplus$
Gauge2${\scriptstyle 1/g_{\text{\tiny YM}}^{2}}$${\scriptstyle
1/g_{\text{\tiny YM}}^{2}}$${\scriptstyle m}$UVIREnergy$d=5$ Figure 1:
Schematic representation of the conformal and gauge theories involved in our
construction, the corresponding energy scales, and RG flows among them in 5d.
$\bullet$Gauge theory$\bullet$Gauge${}_{1}\ \oplus$
Gauge2$\bullet$SCFT$\bullet$SCFT${}_{1}\ \oplus$ SCFT2${\scriptstyle
g_{\text{\tiny YM}}^{2}}$${\scriptstyle g_{\text{\tiny
YM}}^{2}}$${\scriptstyle m}$${\scriptstyle m\to\infty}$UVIREnergy$d=3$ Figure
2: Schematic representation of the conformal and gauge theories involved in
our construction, the corresponding energy scales, and RG flows among them in
3d.

The material is organised in two long sections, followed by conclusions and
appendices. A detailed account of the contents is given at the beginning of
each section.

Section 2, containing Part I of this work, describes the holographic side of
the problem. A convenient formalism based on a two dimensional Laplace
equation is reviewed and a new holographic situation is analysed. Calculations
of the holographic central charge (i.e., the quantity holographically dual to
the free energy) and expectation values of Wilson loops in arbitrary
antisymmetric representations in this new setup are presented. Finally, almost
as a curiosity, a simple matrix model is written that matches the holographic
results of some of the systems in Part I.

In Section 3, containing Part II of this work, we initiate the field theory
study of linear quivers deformed by a special choice of real mass. The
theories are equivalently presented as interacting quivers. The two
complementary viewpoints are explained in great detail using the matrix models
derived from localisation: as the one-matrix model of a single quiver deformed
by one (or more) real massive parameters, or as the multi-matrix model of two
(or more) linear quivers with an interaction term (see [88] for related
manipulations). The free energy is calculated and found to match with the
supergravity results of Part I, up to adding a local counterterm for
background fields in $d=5$. The matching gives support to our proposal for the
holographic duals of a mass deformation in a given quiver CFT. To add more
credit to this, the field theoretical calculation of Wilson loops in
antisymmetric representations is performed using the associated matrix model
and nicely reproduce the result of Part I. As a byproduct of our analysis, the
F-theorem is shown to hold in these types of RG flows.

Conclusions and prospects for future developments are collected in Section 4.
The technically intensive Appendices A and C-D complement the presentation of
Sections 2 and 3, respectively, while Appendix B establishes the dictionary
between our methods in $d=5$ and the M-theory engineering of the SCFTs.

We aim at a clear and pedagogical presentation, thus detail many steps of the
computations.

##  2 Part I: Supergravity

In this section, we discuss the supergravity solutions used in this paper. We
summarise the backgrounds preserving eight supercharges, i.e. ${\cal N}=1$
supersymmetry in five dimensions and ${\cal N}=4$ in three dimensions.
Supersymmetry is preserved subject to a linear PDE being satisfied. We solve
the PDE and briefly comment on the quantised charges and the associated dual
CFTs.

To make the section self-contained, we give a detailed account of its
contents. In Subsection 2.1 we present generalities about the problem under
study: the type of quivers (linear and balanced), the rank function formalism
and how it is used to calculate the relevant numbers of the quiver. In
Subsection 2.2, an infinite family of type IIB supergravity backgrounds dual
to 5d linear quivers with eight supercharges is presented. The problem of
finding these backgrounds boils down to the resolution of a two-dimensional
Laplace equation with suitable boundary conditions. The charges associated
with NS5 branes, D5 (colour) branes and D7 (flavour) branes are calculated.
Subsection 2.3 presents an analogous discussion for the case of 3d balanced
quivers with eight supercharges. The holographic problem is reduced to the
same Laplace equation as in the 5d case. The analogies between the
descriptions go along the rest of the paper. The two dimensions in which the
Laplace problem is set are denoted by $(\sigma,\eta)$. Interestingly, the same
Laplace problem arises in the matrix model treatment of Part II.

Subsection 2.4 discusses, for the backgrounds of the previous subsections, an
observable called the holographic central charge. This coincides with the free
energy of the dual CFTs. We present exact expressions for this observable, in
5d and 3d.

Subsection 2.5 poses a generalisation in the context of our Laplace-based
formalism. Namely, we consider the problem that consists of two (or more) rank
functions. We calculate the holographic central charge in this case, providing
exact expressions. This problem motivates some of the developments in Part II
of this work. Indeed, whilst there is a clean holographic interpretation of
the $\eta$-direction, the field theoretical significance of the
$\sigma$-direction is more elusive. This is one of the problems addressed in
the Part II of this work, building on the material in this section.

Subsection 2.6 studies the novel calculation of the vacuum expectation value
of Wilson loops in antisymmetric representations of a given gauge node, for
the system with two rank functions. Exact expressions are again given, and
will be recovered in Part II with a calculation in field theory. This gives
support to the field theoretical interpretation we present.

To close Part I of this work, Subsection 2.8 presents a very simple matrix
model matching the Laplace problem of Subsections 2.2-2.3 and their
holographic central charge found in Subsection 2.4. An extensive investigation
of the analogous matrix models for the new systems of Subsection 2.5 is left
for the future.

###  2.1 Quivers, SCFTs, and rank functions

Let us begin by presenting the problem from the perspective of Quantum Field
Theory (QFT). Consider field theories in five and three spacetime dimensions
preserving eight Poincaré supercharges. We restrict our attention to the class
of field theories whose dynamics will be encoded in framed linear quivers,
drawn in Figure 3.

$N_{1}$$N_{2}$$\dots$$N_{P-1}$$N_{P-2}$$F_{1}$$F_{2}$$\dots$$F_{P-1}$$F_{P-2}$
Figure 3: Linear quiver of length $P-1$ with gauge nodes $U(N_{j})$ in 3d or
$SU(N_{j})$ in 5d, and flavour nodes $F_{j}$. The quiver is balanced if
$F_{j}=2N_{j}-N_{j-1}-N_{j+1}$.

In five spacetime dimensions, there is by now a wealth of evidence that there
exist strongly coupled conformal fixed points which, when deformed by suitable
relevant operators, flow to these gauge theories. The conformal fixed point is
commonly referred to as UV SCFT, meaning that it sits at a higher energy
scale, compared with the scale set by $1/g^{2}_{\text{\tiny YM}}$. Different
quivers are encountered, in principle, along RG flows connected to different
UV CFTs, although a given CFT typically admits several gauge theory
deformations. A necessary condition for the five-dimensional gauge theories in
Figure 3 to admit a UV completion, is $F_{j}\leq 2N_{j}-N_{j-1}-N_{j+1}$ [79],
possibly augmented up to $F_{j}\leq 2N_{j}-N_{j-1}-N_{j+1}+4$ at certain gauge
nodes [89, 90].

Instead, if we work in three spacetime dimensions, the proposal is that each
member of the infinite family of field theories described by Figure 3 which is
ugly or good in the Gaiotto–Witten classification [91] flows at low energies
(compared with the scale set by $1/g^{2}_{\text{\tiny YM}}$) to a strongly
coupled conformal fixed point. Different quivers typically give rise to
distinct CFTs, although there exist dualities, most notably 3d mirror symmetry
[82], that relate distinct quivers flowing to the same CFT. In this work, we
aim to pursue a unified description of the linear quivers in five and three
dimensions. We will restrict the flavour ranks $F_{j}$ to

$F_{j}=2N_{j}-N_{j+1}-N_{j-1}$ (2.1)

from now on. In other words, the quiver is balanced [91].

In what follows, we present the holographic description of the infinite
families of conformal fixed points. We rely on previous work [70, 51], that we
summarise below. An important ingredient that we take from the field
theoretical description is the presence of a rank function. For the quiver in
Figure 3, this rank function is a convex, piecewise linear and continuous
function given by

${\cal R}(\eta)=\begin{cases}N_{1}\eta&0\leq\eta\leq 1\\\
N_{j}+(N_{j+1}-N_{j})(\eta-j)&j\leq\eta\leq j+1,\;\;\;j=1,....,P-2\\\
N_{P-1}(P-\eta)&(P-1)\leq\eta\leq P.\end{cases}$ (2.2)

The rank function evaluated at integer values of the coordinate $\eta$ gives
the ranks of each of the gauge groups,333To avoid verbosity, we slightly abuse
of nomenclature and refer to $N_{j}$ as the “rank” of both $U(N_{j})$ and
$SU(N_{j})$. that is, $\mathcal{R}(j)=N_{j}$. The second derivative of the
rank function,

$\displaystyle{\cal R}^{\prime\prime}(\eta)$
$\displaystyle=\sum_{j=1}^{P-1}F_{j}\delta(\eta-j),$ (2.3)

with $F_{j}$ as in (2.1), gives the ranks of the flavour groups that make the
quiver balanced. The convexity of the rank function guarantees that the ranks
of the flavour groups are non-negative. We emphasise that
$\mathcal{R}^{\prime\prime}(\eta)$ in (2.3) really means derivative from the
right, i.e.

$\mathcal{R}^{\prime\prime}(\eta)=\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon}\left[\partial_{\eta}\mathcal{R}|_{\eta-\frac{\varepsilon}{2}}-\partial_{\eta}\mathcal{R}|_{\eta+\frac{\varepsilon}{2}}\right].$

The ordinary derivative together with the balancing condition (2.1) would give
the opposite sign. This difference in the minus sign convention between
supergravity and QFT will resurface later on.

If the function ${\cal R}(\eta)$ is taken to satisfy ${\cal R}(\eta=0)={\cal
R}(\eta=P)=0$, we refer to this as a situation without offsets. Otherwise, if
${\cal R}(\eta)$ is non-zero at either $\eta=0$ or $\eta=P$ we refer to it as
a situation with offsets. We discuss the case with offsets in Subsection 2.4.

Below, we summarise the supergravity backgrounds in Type IIB string theory
which provide a holographic description of the SCFTs associated with these
quivers in five and three dimensions. This approach to the problem serves to
emphasise that the rank function encoding the kinematic data of the QFT (ranks
of the gauge and flavour groups), is common to five and three dimensions.
Next, we show that the rank function serves as initial condition to a boundary
value problem encoding the supergravity description.

###  2.2 The Type IIB backgrounds dual to 5d SCFTs

We present an infinite family of supergravity backgrounds in Type IIB string
theory preserving eight Poincaré supersymmetries with an AdS6 factor. The
space also contains a two-sphere parametrised by coordinates
$(\theta,\varphi)$. The isometries of this manifold are in correspondence with
the $SO(2,5)\times SU(2)_{R}$ bosonic subalgebra of the superconformal algebra
of the dual ${\cal N}=1$ five-dimensional SCFTs.

The full configuration consists of a metric, dilaton, $B_{2}$-field in the
Neveu–Schwarz sector and $C_{2}$ and $C_{0}$ fields in the Ramond sector. The
configuration is written in terms of a potential function $V_{5}(\sigma,\eta)$
that solves the linear partial differential equation

$\partial_{\sigma}\left(\sigma^{2}\partial_{\sigma}V_{5}\right)+\sigma^{2}\partial^{2}_{\eta}V_{5}=0.$
(2.4)

The type IIB background in string frame is [70],

$ds_{10,st}^{2}=f_{1}(\sigma,\eta)\Big{[}ds^{2}(\text{AdS}_{6})+f_{2}(\sigma,\eta)ds^{2}(\mathbb{S}^{2})+f_{3}(\sigma,\eta)(d\sigma^{2}+d\eta^{2})\Big{]},$
(2.5a) with fields $e^{-2\Phi}=f_{6}(\sigma,\eta),\qquad
B_{2}=f_{4}(\sigma,\eta)\text{Vol}(\mathbb{S}^{2}),\qquad
C_{2}=f_{5}(\sigma,\eta)\text{Vol}(\mathbb{S}^{2}),\qquad
C_{0}=f_{7}(\sigma,\eta),$ (2.5b) and warp factors $\displaystyle f_{1}$
$\displaystyle=\frac{3\pi}{2}\sqrt{\sigma^{2}+\frac{3\sigma\partial_{\sigma}V_{5}}{\partial^{2}_{\eta}V_{5}}},\qquad\qquad
f_{2}=\frac{\partial_{\sigma}V_{5}\partial^{2}_{\eta}V_{5}}{3\Lambda},\qquad\qquad
f_{3}=\frac{\partial^{2}_{\eta}V_{5}}{3\sigma\partial_{\sigma}V_{5}},$ (2.5c)
$\displaystyle f_{4}$
$\displaystyle=\frac{\pi}{2}\left(\eta-\frac{(\sigma\partial_{\sigma}V_{5})(\partial_{\sigma}\partial_{\eta}V_{5})}{\Lambda}\right),$
$\displaystyle f_{5}$
$\displaystyle=\frac{\pi}{2}\left(V_{5}-\frac{\sigma\partial_{\sigma}V_{5}}{\Lambda}(\partial_{\eta}V_{5}(\partial_{\sigma}\partial_{\eta}V_{5})-3(\partial^{2}_{\eta}V_{5})(\partial_{\sigma}V_{5}))\right),$
$\displaystyle f_{6}$
$\displaystyle=12\frac{\sigma^{2}\partial_{\sigma}V_{5}\partial^{2}_{\eta}V_{5}}{(3\partial_{\sigma}V_{5}+\sigma\partial^{2}_{\eta}V_{5})^{2}}\Lambda,\qquad\qquad
f_{7}=2\left(\partial_{\eta}V_{5}+\frac{(3\sigma\partial_{\sigma}V_{5})(\partial_{\sigma}\partial_{\eta}V_{5})}{3\partial_{\sigma}V_{5}+\sigma\partial^{2}_{\eta}V_{5}}\right),$
$\displaystyle\Lambda$
$\displaystyle=\sigma(\partial_{\sigma}\partial_{\eta}V_{5})^{2}+(\partial_{\sigma}V_{5}-\sigma\partial^{2}_{\sigma}V_{5})\partial^{2}_{\eta}V_{5}.$

The paper [70] proves that this infinite family of backgrounds is in exact
correspondence with the solutions discussed in [63, 64, 65, 66, 92, 93].

Let us briefly summarise the study of [70] for the PDE, with boundary
conditions leading to a proper interpretation of the solutions, with quantised
Page charges and avoiding badly-singular behaviours.

####  2.2.1 Resolution of the PDE and quantisation of charges

We make the change
$V_{5}(\sigma,\eta)=\frac{\hat{W}_{5}(\sigma,\eta)}{\sigma},$ which implies
that the PDE in (2.4) reads like a Laplace equation in flat space,

$\partial^{2}_{\sigma}\hat{W}_{5}+\partial_{\eta}^{2}\hat{W}_{5}=0.$ (2.6)

We choose the variable $\eta$ to be bounded in the interval $[0,P]$ and
$\sigma$ to range over the real axis $-\infty<\sigma<\infty$. We impose the
boundary conditions,

$\displaystyle\hat{W}_{5}(\sigma\to\pm\infty,\eta)=0,\;\;\;\;\;\hat{W}_{5}(\sigma,\eta=0)=\hat{W}_{5}(\sigma,\eta=P)=0,$
$\displaystyle\lim_{\epsilon\to
0}\left(\partial_{\sigma}\hat{W}_{5}(\sigma=+\epsilon,\eta)-\partial_{\sigma}\hat{W}_{5}(\sigma=-\epsilon,\eta)\right)=-{\cal
R}(\eta).$ (2.7)

$\sigma$$\eta$$0$$P$$\mathcal{R}(\eta)$ Figure 4: The electrostatic problem
for $\hat{W}_{5}$. The two conducting planes at $\eta=0,P$ have zero
potential, while at $\sigma=0$ we have a charge distribution equal to $\cal
R(\eta)$.

These can be interpreted as the boundary conditions for the electrostatic
problem of two conducting planes (at zero electrostatic potential) as depicted
in Figure 4. The conducting planes extend over the $\sigma$-direction and are
placed at $\eta=0$ and $\eta=P$. We also have a charge density ${\cal
R}(\eta)$ at $\sigma=0$, extended along $0\leq\eta\leq P$, as indicated by the
difference of the normal components of the electric field in (2.7).

The solution is found by separating variables, see [70] for the details. It is
convenient to Fourier expand the function ${\cal R}(\eta)$ as

${\cal
R}(\eta)=\sum_{k=1}^{\infty}R_{k}\sin\left(\frac{k\pi}{P}\eta\right),\;\;\;\;R_{k}=\frac{2}{P}\int_{0}^{P}{\cal
R}(\eta)\sin\left(\frac{k\pi\eta}{P}\right)\mathrm{d}\eta.$ (2.8)

Following [70], the solution reads,

$\hat{W}_{5}(\sigma,\eta)=\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi}{P}\eta\right){e^{-\frac{k\pi}{P}|\sigma|}},\qquad
a_{k}=\frac{P}{2\pi k}R_{k}.$ (2.9)

The potentials in (2.9) solve the equation (2.6) subject to the conditions
(2.7). Another way to encode the boundary condition at $\sigma=0$ is through a
source term that transforms the Laplace equation into a Poisson equation.
Indeed, the potential $\hat{W}_{5}(\sigma,\eta)$ in (2.9) satisfies

$\partial^{2}_{\eta}\hat{W}_{5}+\partial^{2}_{\sigma}\hat{W}_{5}=-{\cal
R}(\eta)\delta(\sigma)\,,\qquad\sigma\in\mathbb{R}\,,\quad\eta\in[0,P]\,.$
(2.10)

Imposing the quantisation of the conserved Page charges of the background
(2.5c), the authors of [70] found that the function ${\cal R}(\eta)$ must be a
convex piecewise linear function. If the rank function has no offsets
$\mathcal{R}(0)=\mathcal{R}(P)=0$, it takes the form (2.2). In this case, the
authors of [70] computed the values of the quantised brane charges in each
interval $\eta\in[j,j+1]$

$\displaystyle Q_{\text{\rm D7}}[j,j+1]$ $\displaystyle={\cal
R}^{\prime\prime}(j)=(2N_{j}-N_{j+1}-N_{j-1}),$ (2.11a) $\displaystyle
Q_{\text{\rm D5}}[j,j+1]$ $\displaystyle={\cal R}(\eta)-{\cal
R}^{\prime}(\eta)(\eta-j)=N_{j}$ (2.11b) (the second summand in (2.11b) can be
thought of as coming from a large gauge transformation of the $B_{2}$ field)
and in the whole system,444In this paper we define the ′-derivative
$\mathcal{R}^{\prime}$ as a derivative from the right — see below (2.3). This
flips the sign in front of the rank functions in the differential equation,
compared to [70].The integration domains in the Page charges are chosen
accordingly. $Q_{\text{\rm NS5}}^{\text{\rm total}}=P,\qquad Q_{\text{\rm
D7}}^{\text{\rm total}}=(N_{1}+N_{P-1})=\int_{0}^{P}{\cal
R}^{\prime\prime}(\eta)\mathrm{d}\eta,\qquad Q_{\text{\rm D5}}^{\text{\rm
total}}=\int_{0}^{P}{\cal R}(\eta)\mathrm{d}\eta.$ (2.11c)

For the generic rank function ${\cal R}(\eta)$ quoted in (2.2), the
supergravity background is proposed to be dual to the strongly coupled fixed
point in the UV of the quiver in Figure 3, with
$F_{i}=2N_{i}-N_{i+1}-N_{i-1}$. In other words, the quiver is balanced.

###  2.3 The Type IIB backgrounds dual to 3d SCFTs

We now discuss the Type IIB backgrounds dual to three dimensional SCFTs
preserving eight supercharges. The formalism is very much analogous to the
five dimensional one, hence we will be more sketchy. All the details can be
found in [51].

We are after solutions dual to 3d ${\cal N}=4$ superconformal field theories.
To match the $\mathcal{N}=4$ superconformal symmetry of the field theory, the
background must have isometries $SO(2,3)\times SU(2)_{C}\times SU(2)_{H}$ and
preserve eight Poincaré supercharges. Hence, our geometries must contain an
AdS4 factor and a pair of two-spheres
$\mathbb{S}^{2}_{1}(\theta_{1},\varphi_{1})$ and
$\mathbb{S}^{2}_{2}(\theta_{2},\varphi_{2})$. There are two extra directions
labelled by $(\sigma,\eta)$. The requirement of an isometry group that
contains $SO(2,3)\times SU(2)_{C}\times SU(2)_{H}$ allows for warp factors
that depend only on $(\sigma,\eta)$. The Ramond and Neveu–Schwarz fields must
also respect these isometries.

The preservation of eight Poincaré supersymmetries implies that the generic
type IIB background can be cast in terms of a function $V_{3}(\sigma,\eta)$.
In string frame the solution reads [51],

$ds_{10,st}^{2}=f_{1}(\sigma,\eta)\Big{[}ds^{2}(\text{AdS}_{4})+f_{2}(\sigma,\eta)ds^{2}(\mathbb{S}^{2}_{1})+f_{3}(\sigma,\eta)ds^{2}(\mathbb{S}^{2}_{2})+f_{4}(\sigma,\eta)(d\sigma^{2}+d\eta^{2})\Big{]},$
(2.12a) with $\displaystyle e^{-2\Phi}=f_{5}(\sigma,\eta),\qquad B_{2}$
$\displaystyle=f_{6}(\sigma,\eta)\text{Vol}(\mathbb{S}^{2}_{1}),$ (2.12b)
$\displaystyle
C_{2}=f_{7}(\sigma,\eta)\text{Vol}(\mathbb{S}^{2}_{2}),\qquad\tilde{C}_{4}$
$\displaystyle=f_{8}(\sigma,\eta)\text{Vol}(\text{AdS}_{4}),$ and
$\displaystyle f_{1}$
$\displaystyle=\frac{\pi}{2}\sqrt{\frac{\sigma^{3}\partial^{2}_{\eta\sigma}V_{3}}{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}},\qquad\qquad
f_{2}=-\frac{\partial_{\eta}V_{3}\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma\Lambda},\qquad\qquad
f_{3}=\frac{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma\partial^{2}_{\eta\sigma}V_{3}},$
(2.12c) $\displaystyle f_{4}$
$\displaystyle=-\frac{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma^{2}\partial_{\eta}V_{3}},\qquad\qquad
f_{5}=-16\frac{\Lambda\partial_{\eta}V_{3}}{\partial^{2}_{\eta\sigma}V_{3}},\qquad\qquad
f_{6}=\frac{\pi}{2}\left(\eta-\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\Lambda}\right),$
$\displaystyle f_{7}$ $\displaystyle=-2\pi\left(\partial_{\sigma}(\sigma
V_{3})-\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\partial^{2}_{\eta\sigma}V_{3}}\right),\qquad\qquad
f_{8}=-\pi^{2}\sigma^{2}\left(3\partial_{\sigma}V_{3}+\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}\right),$
$\displaystyle\Lambda$
$\displaystyle=\partial_{\eta}V_{3}\partial^{2}_{\eta\sigma}V_{3}+\sigma\left((\partial^{2}_{\eta\sigma}V_{3})^{2}+(\partial^{2}_{\eta}V_{3})^{2}\right).$

As usual, the fluxes are defined from the potentials as

$F_{1}=0,\quad H_{3}=\mathrm{d}B_{2}\quad F_{3}=\mathrm{d}C_{2},\quad
F_{5}=\mathrm{d}\tilde{C}_{4}+*\mathrm{d}\tilde{C}_{4}.$

The configuration in (2.12c) is solution to the Type IIB equations of motion,
if the function $V_{3}(\sigma,\eta)$ satisfies

$\partial_{\sigma}\left(\sigma^{2}\partial_{\sigma}V_{3}\right)+\sigma^{2}\partial^{2}_{\eta}V_{3}=0.$
(2.13)

As proven in [51], this infinite family of solutions is equivalent to the
backgrounds described by D’Hoker, Estes and Gutperle in [45].

####  2.3.1 Resolution of the PDE and quantisation of charges

Following [51], define
$V_{3}(\sigma,\eta)=\frac{\hat{V}_{3}(\sigma,\eta)}{\sigma}$ and
$\hat{V}_{3}(\sigma,\eta)=\partial_{\eta}\hat{W}_{3}(\sigma,\eta)$. Consider
the coordinates to range in $0\leq\eta\leq P$ and $-\infty<\sigma<\infty$. The
differential equation (2.13) must be supplemented by boundary and initial
conditions. In terms of $\hat{W}_{3}(\sigma,\eta)$ the problem reads

$\displaystyle\partial^{2}_{\sigma}\hat{W}_{3}(\sigma,\eta)+\partial^{2}_{\eta}\hat{W}_{3}(\sigma,\eta)=0,\qquad\qquad\text{(almost
everywhere)}$ (2.14)
$\displaystyle\hat{W}_{3}(\sigma\to\pm\infty,\eta)=0,\;\;\;\;\;\hat{W}_{3}(\sigma,\eta=0)=\hat{W}_{3}(\sigma,\eta=P)=0.$
$\displaystyle\lim_{\epsilon\to
0}\left(\partial_{\sigma}\hat{W}_{3}(\sigma=+\epsilon,\eta)-\partial_{\sigma}\hat{W}_{3}(\sigma=-\epsilon,\eta)\right)=-{\cal
R}(\eta).$

In the current conventions, we have ensured that $\hat{W}_{3}$ satisfies the
same electrostatic problem as $\hat{W}_{5}$ in the five dimensional case above
— see eqs. (2.6)-(2.2.1). The function ${\cal R}(\eta)$ is an input.

Using a Fourier decomposition for the rank function ${\cal R}(\eta)$ as in the
five-dimensional case, cf. (2.8), the solution to the problem in (2.14) is

$\hat{W}_{3}(\sigma,\eta)=\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi\eta}{P}\right)e^{-\frac{k\pi|\sigma|}{P}},\qquad\qquad
a_{k}=\frac{P}{2k\pi}R_{k}\,.$ (2.15)

The reader can check that

$\partial^{2}_{\eta}\hat{W}_{3}+\partial^{2}_{\sigma}\hat{W}_{3}=-{\cal
R}(\eta)\delta(\sigma).$

The study of the quantised charges for NS5 branes enforces the size of the
interval $P$ to be an integer, consistently with the boundary conditions in
(2.14), exactly as it occurs in the five dimensional system. Also in analogy
with the 5d case, to have quantised numbers of D3 and D5 branes, the rank
function must be a piecewise linear and continuous function of the exact same
form as in the five dimensional case — see (2.2).

Let us again consider the piecewise linear, convex and continuous function
(2.2) without offsets. For such a rank function, the number of D3 (colour)
branes and D5 (flavour) branes in the interval $[j,j+1]$ and the total number
of branes are given by

$\displaystyle Q_{\text{\rm D3}}[j,j+1]=N_{j},\;\;\;\;Q_{\text{\rm
D5}}[j,j+1]=2N_{j}-N_{j+1}-N_{j-1},$ (2.16) $\displaystyle Q_{\text{\rm
D3}}^{\text{\text{total}}}=\int_{0}^{P}{\cal
R}(\eta)d\eta,\;\;\;\;Q_{\text{\rm D5}}^{\text{\text{total}}}={\cal
R}^{\prime}(0)-{\cal R}^{\prime}(P),\;\;\;\;Q_{\text{\rm
NS5}}^{\text{\text{total}}}=P.$

As we emphasised, the supergravity backgrounds we have obtained,
holographically dual to five and three dimensional systems, have different
dynamics, but at their core are described by a function
$\hat{W}_{5}(\sigma,\eta)$ or $\hat{W}_{3}(\sigma,\eta)$ solving the same
Poisson equation. This translates into analogies between the quivers in
different odd dimensions.

We start by calculating the holographic central charge. To add to the already
existent bibliography, we study this observable in the case in which the rank
function has offsets.

###  2.4 Holographic central charge

In this subsection, we will compute the holographic central charges in the
AdS${}_{6}\times\mathbb{S}^{2}$ and
AdS${}_{4}\times\mathbb{S}^{2}\times\mathbb{S}^{2}$ cases. This can be done in
a slightly more general context in which we allow the rank function to have
offsets. That means we consider a rank function

$\mathcal{R}(\eta)=N_{j}+(N_{j+1}-N_{j})(\eta-j)\quad j\leq\eta\leq j+1,\quad
j=0,....,P-1\,,$ (2.17)

which satisfies $\mathcal{R}(j)=N_{j}$ for integers $j$, including $j=0,P$. In
the case $N_{0}=N_{P}=0$, the rank function (2.17) reduces to the one quoted
in eq. (2.2). This more general rank function was already considered in the 3d
case in [51]. We shall now present it in the 5d case. We calculate the Fourier
coefficient of the rank function (2.17):

$R_{k}=\frac{2}{k\pi}\left(N_{0}+(-1)^{k+1}N_{P}\right)+\frac{2P}{\pi^{2}k^{2}}\sum_{j=1}^{P-1}F_{j}\sin\left(\frac{k\pi
j}{P}\right)\,.$ (2.18)

They are found by first computing the integrals

$\displaystyle
R_{k}^{(\ell)}:=\frac{2}{P}\int_{\ell}^{\ell+1}\mathcal{R}(\eta)\sin\left(\frac{k\pi\eta}{P}\right)d\eta$
$\displaystyle=\frac{2}{\pi k}\left[N_{\ell}\cos\left(\frac{\pi
k\ell}{P}\right)-N_{\ell+1}\cos\left(\frac{\pi k(\ell+1)}{P}\right)\right]$
$\displaystyle+\frac{2P}{\pi^{2}k^{2}}(N_{\ell}-N_{\ell+1})\left[\sin\left(\frac{\pi
k\ell}{P}\right)-\sin\left(\frac{\pi k(\ell+1)}{P}\right)\right]\,,$

and then summing the results $R_{k}=\sum_{\ell=0}^{P-1}R_{k}^{(\ell)}$. The
terms with cosines vanish in the “bulk” but survive on the boundaries
$\ell=0,P-1$, whereas the terms with sines give the terms with flavour groups.

Notice that in the case $N_{0}=N_{P}=0$ in (2.18), the ranks for boundary
flavour groups were given by $F_{1}=2N_{1}-N_{2}$ and
$F_{P-1}=2N_{P-1}-N_{P-2}$. In (2.18), we have extended the definition
$F_{j}=2N_{j}-N_{j-1}-N_{j+1}$ to include $j=0,P$, which now incorporates
$N_{0}$ and $N_{P}$ into $F_{1}$ and $F_{P-1}$ respectively. Hence, $N_{0}$
and $N_{P}$ give additional flavour symmetry to make the quiver balanced. This
effect is clear from the Hanany–Witten brane configurations: stretching D$d$
colour branes to infinity on the left or on the right gives additional flavour
symmetry, exactly as if we let them end on D$(d+2)$ flavour branes. We
postpone a thorough analysis of the supergravity background needed to
accurately interpret the situation.

In the AdS6 geometry, the holographic central charge of the theory can be
computed as the volume of an internal manifold [70] which works out as

$\displaystyle c_{hol}$
$\displaystyle=\frac{2}{3\pi^{5}}\int_{-\infty}^{\infty}d\sigma\int_{0}^{P}d\eta\,\sigma^{3}(\partial_{\sigma}V_{5})(\partial_{\eta}^{2}V_{5})$
(2.19a)
$\displaystyle=\frac{2}{3\pi^{5}}\int_{-\infty}^{\infty}d\sigma\int_{0}^{P}d\eta\,\partial^{2}_{\eta}\hat{W}_{5}\left(\sigma\partial_{\sigma}\hat{W}_{5}-\hat{W}_{5}\right),$
(2.19b)

where $V_{5}=\frac{\hat{W}_{5}}{\sigma}$ and $\hat{W}_{5}$ is given by (2.9).
The integration over $\eta$ can be performed using the orthogonality of sine
functions which collapses the double summation into a single summation. After
the $\sigma$-integral is performed, one finds

$c_{hol}=\frac{1}{2\pi^{4}}\sum_{k=1}^{\infty}ka_{k}^{2}=\frac{P^{2}}{8\pi^{6}}\sum_{k=1}^{\infty}\frac{1}{k}R_{k}^{2}\,.$
(2.20)

Plugging in the Fourier coefficients (2.17) gives

$\displaystyle c_{hol}=$
$\displaystyle\frac{P^{2}}{4\pi^{8}}(2N_{0}^{2}+2N_{P}^{2}+3N_{0}N_{P})\zeta(3)$
(2.21)
$\displaystyle+\frac{P^{3}}{\pi^{9}}\sum_{l=1}^{P-1}F_{l}[N_{0}\text{Im}(\text{Li}_{4}(e^{\frac{il\pi}{P}}))+N_{P}\text{Im}(\text{Li}_{4}(e^{\frac{i(P-l)\pi}{P}}))]$
$\displaystyle-\frac{P^{4}}{4\pi^{10}}\sum_{l=1}^{P-1}\sum_{k=1}^{P-1}F_{l}F_{k}\mathop{\rm
Re}\left(\text{Li}_{5}\left(e^{i\frac{\pi(k+l)}{P}}\right)-\text{Li}_{5}\left(e^{i\frac{\pi(k-l)}{P}}\right)\right)\,.$

The first, second and third line of this expression stem from
$\frac{1}{k^{3}}$, $\frac{1}{k^{4}}$ and $\frac{1}{k^{5}}$ contributions
respectively. This result precisely matches the free energy of the 5d SCFT as
was obtained by using matrix model techniques [83, Eq.(3.17)].

In the AdS4 geometry dual to the three-dimensional case, a similar formula
holds [51]

$\displaystyle c_{hol}$
$\displaystyle=-\frac{1}{2}\int_{-\infty}^{\infty}d\sigma\int_{0}^{P}d\eta\,\sigma^{2}(\partial_{\eta}V_{3})\partial_{\sigma}(\sigma\partial_{\eta}V_{3})$
(2.22a)
$\displaystyle=-\frac{1}{2}\int_{-\infty}^{\infty}d\sigma\int_{0}^{P}d\eta\,\left(\sigma\partial^{2}_{\eta}\hat{W}_{3}\right)\left(\partial^{2}_{\eta}\partial_{\sigma}\hat{W}_{3}\right)\,.$
(2.22b)

In this case we use $V_{3}=\frac{\hat{V}_{3}}{\sigma}$ and
$\hat{V}_{3}=\partial_{\eta}\hat{W}_{3}$ before employing the solution (2.15).
We follow the same approach and use the orthogonality of sine functions to
arrive at

$c_{hol}=\frac{\pi}{32}\sum_{k=1}^{\infty}kR_{k}^{2}\,.$ (2.23)

To compute this series with offset we need to regularise the sums
$\sum_{k=1}^{\infty}\frac{1}{k}\rightarrow\gamma_{E}$ and
$\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\rightarrow-\frac{\log 2}{2}$. The
result is,

$\begin{split}c_{hol}=&\frac{1}{8\pi}(N_{0}^{2}+N_{P}^{2})\gamma_{E}+\frac{1}{8\pi}N_{0}N_{P}\log(2)\\\
&+\frac{P}{4\pi^{2}}\sum_{l=1}^{P-1}F_{l}[N_{0}\text{Im}(\text{Li}_{2}(e^{\frac{il\pi}{P}}))+N_{P}\text{Im}(\text{Li}_{2}(e^{\frac{i(P-l)\pi}{P}}))]\\\
&-\frac{P^{2}}{16\pi^{3}}\sum_{l=1}^{P-1}\sum_{k=1}^{P-1}F_{l}F_{k}\mathop{\rm
Re}\left(\text{Li}_{3}\left(e^{i\frac{\pi(k+l)}{P}}\right)-\text{Li}_{3}\left(e^{i\frac{\pi(k-l)}{P}}\right)\right)\,.\end{split}$
(2.24)

This formula is, broadly speaking, analogous to (2.21) with a polylogarithm of
2 degrees lower.

###  2.5 The case of two rank functions: two interacting linear quivers

In this section, we consider a natural extension of the Poisson problem
(2.10). We will study this problem in both backgrounds dual to the 5d and the
3d quivers. As we mentioned, the differential equations are identical, hence
we can study the solution simultaneously. We introduce a second rank function
$\mathcal{R}_{2}$ that places a new charge density at a line
$\sigma=\sigma_{0}$. The Poisson equation of such a problem is

$\partial_{\eta}^{2}\hat{W}(\sigma,\eta)+\partial_{\sigma}^{2}\hat{W}(\sigma,\eta)+\mathcal{R}_{2}(\eta)\delta(\sigma_{0}-\sigma)+\mathcal{R}_{1}(\eta)\delta(\sigma)\,=0,$
(2.25)

with boundary conditions

$\hat{W}(\sigma\rightarrow\pm\infty)=\hat{W}(\eta=0)=\hat{W}(\eta=P)=0\,.$
(2.26)

The electrostatic problem is depicted in Figure 5. The PDE is still linear,
which allows us a simple way to write a solution. We notice that if
$\hat{W}[\mathcal{R}_{1}](\sigma,\eta)$ is a solution to the electrostatic
problem with a source $\mathcal{R}_{1}$ at $\sigma=0$, this already solves
half of the source term in eq. (2.25). We need to add to this a solution which
takes care of the source $\mathcal{R}_{2}$ at $\sigma=\sigma_{0}$. This is
easily done by simply shifting a solution to the problem with a source
$\mathcal{R}_{2}$ at $\sigma=0$. The resulting solution is

$\hat{W}[\mathcal{R}_{1},\mathcal{R}_{2}](\sigma,\eta)=\sum_{k=1}^{\infty}\frac{P}{2k\pi}\sin\left(\frac{k\pi\eta}{P}\right)\left[R_{1,k}e^{-\frac{k\pi}{P}|\sigma|}+R_{2,k}e^{-\frac{k\pi}{P}|\sigma-\sigma_{0}|}\right]\,,$
(2.27)

with the Fourier coefficients of the two rank functions given by

$\mathcal{R}_{\alpha}(\eta)=\sum_{k=1}^{\infty}R_{\alpha,k}\sin\left(\frac{k\pi\eta}{P}\right),\qquad\qquad\alpha=1,2.$
(2.28)

$\sigma$$\eta$$\sigma_{0}$$0$$P$$\mathcal{R}_{1}(\eta)$$\mathcal{R}_{2}(\eta)$
Figure 5: The electrostatic problem for $\hat{W}$ in the two rank functions
setup. We have a charge distribution equal to $\mathcal{R}_{1}(\eta)$ at
$\sigma=0$ and a second charge distribution $\mathcal{R}_{2}(\eta)$ at
$\sigma=\sigma_{0}$.

In the AdS6 geometry, we compute the integral (2.19) with the potential in
(2.27) and substitute
$V_{5}(\sigma,\eta)=\frac{\hat{W}[\mathcal{R}_{1},\mathcal{R}_{2}](\sigma,\eta)}{\sigma}$.
We find

$\left.c_{hol}\right\rvert_{d=5}[\mathcal{R}_{1},\mathcal{R}_{2}]=\frac{P^{2}}{8\pi^{6}}\sum_{k=1}^{\infty}\frac{1}{k}\left[R_{1,k}^{2}+R_{2,k}^{2}+2R_{1,k}R_{2,k}e^{-\frac{k\pi\sigma_{0}}{P}}\left(1+\frac{k\pi\sigma_{0}}{P}\right)\right]\,.$
(2.29)

In the AdS4 case, the analogous integral to find the holographic central
charge is (2.22). Using the potential (2.27) and substituting
$V_{3}(\sigma,\eta)=\frac{\partial_{\eta}\hat{W}[\mathcal{R}_{1},\mathcal{R}_{2}](\sigma,\eta)}{\sigma}$,
we find

$\left.c_{hol}\right\rvert_{d=3}[\mathcal{R}_{1},\mathcal{R}_{2}]=\frac{\pi}{32}\sum_{k=1}^{\infty}k\left[R_{1,k}^{2}+R_{2,k}^{2}+2R_{1,k}R_{2,k}e^{-\frac{k\pi\sigma_{0}}{P}}\left(1+\frac{k\pi\sigma_{0}}{P}\right)\right]\,.$
(2.30)

We observe that, in both cases, the holographic central charge consists of
terms $R_{\alpha,k}^{2}$, which are familiar from the single rank function
setups. In addition to that, we have an “interaction” term that is controlled
by $R_{1,k}R_{2,k}$. Explicitly,

$c_{hol}[\mathcal{R}_{1},\mathcal{R}_{2}]-\left(c_{hol}[\mathcal{R}_{1}]+c_{hol}[\mathcal{R}_{2}]\right)\propto
2\sum_{k=1}^{\infty}k^{4-d}R_{1,k}R_{2,k}e^{-\frac{k\pi\sigma_{0}}{P}}\left(1+\frac{k\pi\sigma_{0}}{P}\right),$

where the symbol $\propto$ here means up to a $d$-dependent but otherwise
universal numerical coefficient.

It is interesting to study the limiting cases as we move the charge densities
very close to, or very far away from each other. The proper parameter for
these limiting procedures is $\frac{\sigma_{0}}{P}$. If we take
$\frac{\lvert\sigma_{0}\rvert}{P}\gg 1$, the exponential term suppresses the
interaction and we are left over with two copies of the holographic central
charges from the single rank function setup:

$c_{hol}[\mathcal{R}_{1},\mathcal{R}_{2}]=c_{hol}[\mathcal{R}_{1}]+c_{hol}[\mathcal{R}_{2}]\,,\qquad\frac{\sigma_{0}}{P}\rightarrow\infty\,.$
(2.31)

Instead, when the charge densities get very close together,
$0\leq\frac{\lvert\sigma_{0}\rvert}{P}\ll 1$, we have

$c_{hol}[\mathcal{R}_{1},\mathcal{R}_{2}]=c_{hol}[\mathcal{R}_{1}+\mathcal{R}_{2}]\,,\qquad\frac{\sigma_{0}}{P}\rightarrow
0\,.$ (2.32)

In the spirit of a holographic version of the F-theorem [94, 95], we argue in
Appendix D that both in $d=5$ and $d=3$, $c_{hol}$ satisfies

$\displaystyle\lim_{\sigma_{0}\to 0}c_{hol}>\lim_{\sigma_{0}\to\infty}c_{hol}$
$\displaystyle\partial_{\sigma_{0}}c_{hol}\leq 0\qquad\forall
0\leq\sigma_{0}<\infty.$

It is then tempting to think of this as a relevant deformation of a QFT with a
rank function $\mathcal{R}_{1}+\mathcal{R}_{2}$ that in a large
$\frac{\sigma_{0}}{P}$ limit factorises the theory into non-interacting
theories with rank functions $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$. In
Section 3 we motivate this claim further, in the context of matrix model
computations.

###  2.6 Wilson loops in supergravity

The holographic expectation value for a Wilson loop in the rank-$\ell$
antisymmetric representation $\wedge^{\ell}$ of $SU(N_{j})$ (respectively
$U(N_{j})$) in long quivers of the type depicted in Figure 3, were
investigated in [96, 97, 98]. In the usual Hanany–Witten brane setups
associated with the 5d (resp. 3d) quiver theories of our interest, insertion
of a Wilson loop in the $\wedge^{\ell}$ representation corresponds to a
configuration of $\ell$ fundamental strings stretched between a probe D3
(resp. D5′) brane and the stack of $N_{j}$ colour D5 (resp. D3) branes, in a
manner consistent with the s-rule. The Wilson loop vev is calculated by
evaluating the on-shell action of the probe D3 (resp. D5′) brane on which the
$\ell$ fundamental strings end.555Taking the probe brane limit has a nice
correspondence with the matrix model calculation, where the insertion of the
Wilson loop does not modify the saddle point equations.

The result can be written in terms of the potentials that appear in the
electrostatic problem

$\ln\left\langle\mathcal{W}_{\wedge^{\ell}}\right\rangle=(d-2)\left(\sigma\partial_{\sigma}\hat{W}(\sigma,\eta)-\hat{W}(\sigma,\eta)\right)_{(\sigma_{\ast},\eta_{\ast})}\;,$
(2.34)

where $\eta_{\ast}$ denotes the position in the interval $\eta\in[0,P]$ at
which the loop operator is inserted, and $\sigma_{\ast}$ fixes the
representation via the relation

$\ell=\left[\mathcal{R}(\eta)\partial_{\sigma}\hat{W}(\sigma,\eta)\right]_{(\sigma_{\ast},\eta_{\ast})}\;,$

which is the expression for the number of fundamental strings that end on the
probe mentioned above. In order to evaluate (2.34), we use (2.27) to obtain

$\sigma\partial_{\sigma}\hat{W}(\sigma,\eta)=-\frac{1}{2}\sum_{k=1}^{\infty}\sin\left(\frac{k\pi\eta}{P}\right)\left[|\sigma|R_{1,k}e^{-\frac{k\pi}{P}|\sigma|}+\sigma\text{sign}(\sigma-\sigma_{0})R_{2,k}e^{-\frac{k\pi}{P}|\sigma-\sigma_{0}|}\right]\;.$
(2.35)

Next we use (2.18), setting $N_{0}=N_{P}=0$, inside (2.35) and (2.27), and
substitute into (2.34) to obtain

$\displaystyle\begin{aligned}
\frac{\ln\left\langle\mathcal{W}_{\wedge^{\ell}}\right\rangle}{d-2}=\frac{P^{2}}{2\pi^{3}}\sum_{j=1}^{P-1}F_{1,j}\text{Re}&\left[\operatorname{Li}_{3}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}|+i(\eta_{*}+j)\right]}\right)-\operatorname{Li}_{3}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}|+i(\eta_{*}-j)\right]}\right)\right.\\\
&\left.+\frac{\pi}{P}|\sigma_{*}|\left(\operatorname{Li}_{2}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}|+i(\eta_{*}+j)\right]}\right)-\operatorname{Li}_{2}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}|+i(\eta_{*}-j)\right]}\right)\right)\right]\end{aligned}$
$\displaystyle\begin{aligned}
+\frac{P^{2}}{2\pi^{3}}\sum_{j=1}^{P-1}F_{2,j}\text{Re}&\left[\operatorname{Li}_{3}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}-\sigma_{0}|+i(\eta_{*}+j)\right]}\right)-\operatorname{Li}_{3}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}-\sigma_{0}|+i(\eta_{*}-j)\right]}\right)\right.\\\
&\left.+\frac{\pi}{P}\sigma_{*}\text{sign}(\sigma_{*}-\sigma_{0})\left(\operatorname{Li}_{2}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}-\sigma_{0}|+i(\eta_{*}+j)\right]}\right)-\operatorname{Li}_{2}\left(e^{-\frac{\pi}{P}\left[|\sigma_{*}-\sigma_{0}|+i(\eta_{*}-j)\right]}\right)\right)\right]\end{aligned}$
(2.36)

We note here that the Wilson loop expectation values are factorised; each of
the above two summands depends explicitly on one of the two rank functions in
our setup. This is unlike the expression for the holographic central charges
(2.29)-(2.30), which also include an interaction term between the Fourier
coefficients of the two rank functions. The reason is that, in the
electrostatic picture, the holographic central charge accounts for the
interaction between the two density of charges, hence the quadratic dependence
on $\mathcal{R}(\eta)$. On the contrary, a Wilson loop is a probe that
interacts with the two densities of charges individually, thus bearing a
linear dependence on $\mathcal{R}(\eta)$.

The cautious reader might question the representation carried by the Wilson
loops after the factorisation. It is slightly premature to answer this
question at this stage, seeing as we have not yet identified the field theory
interpretation of the double rank function setup. We will clarify this issue
in section 3.7, when we reproduce (2.36) from the gauge theory side.

###  2.7 On the reliability of the supergravity backgrounds

In the case of one rank function, the backgrounds in (2.5c) and (2.12c)
contain certain singularities. These singularities are localised around the
sources (flavour branes), becoming sharply localised in the limit of long
quivers with large ranks. In this case, the backgrounds are reliable in most
of the spacetime. Closer to the sources, the description should be
supplemented by the inclusion of the open string sector (realised on the
sources). In other words, the string-sigma model must include the presence of
boundaries.

The case of two (or more) rank functions separated by a distance $\sigma_{0}$
is qualitatively different. It can be noticed (see Figure 6 for an example in
AdS4) that the dilaton is singular in the region
$0<\lvert\sigma\rvert<\lvert\sigma_{0}\rvert$, indicating that the background
is not reliable there. This situation ameliorates, when either
$|\sigma_{0}|\to 0$ or $|\sigma_{0}|\to\infty$. The singularities are a
consequence of the function $\Lambda(\sigma,\eta)$ defined in equations (2.5c)
and (2.12c) vanishing. Hence, it is not only the dilaton that is problematic,
but the whole background.

This behaviour can be interpreted in the following way: spaces like the ones
studied here, containing an AdSd factor, are not reliable when the parameter
$\sigma_{0}$ is switched on. The case of very large $\frac{\sigma_{0}}{P}$ on
the other hand, improves the behaviour, keeping only the singularities
localised on the sources for $\frac{\sigma_{0}}{P}\to\infty$.

We speculate that this is the reflection that the isometries of AdSd are too
stringent for systems with finite $\frac{\sigma_{0}}{P}$. In other words, the
conformal symmetry of the dual field theory must be broken in this case. In
Section 3, we discuss the field theoretical interpretation of the deformation
by a finite $\sigma_{0}$, finding that an RG flow must take place, recovering
the conformal symmetry when $\frac{\sigma_{0}}{P}\to\infty$.

Figure 6: Plot of $e^{-2\Phi}$ at various values of $\sigma$ in the AdS4
background. Here we set $P=15$ and $\sigma_{0}=6$. The two quivers are given
by two triangular rank functions (see [51, Sec.4.1]), $\mathcal{R}_{1}$ has a
flavour group at $\eta=6$ while $\mathcal{R}_{2}$ at $\eta=9$. The presence of
flavour branes is signaled by one bump below ($\sigma=-2$) and one bump above
($\sigma=7$) the locations of the two rank functions in the $\sigma$-axis. In
the region between them, the dilaton is not well-defined since its exponential
can be negative.

Some natural questions arise: what is the value of calculations done with
backgrounds for which $\frac{\sigma_{0}}{P}$ is finite? Are the holographic
central charge or the Wilson loops vev calculations meaningful?

It has been observed that for holographic systems with singular behaviour,
certain observables become independent of the singularities. Indeed, the warp
factors and fields in the background conspire to produce a finite (and
physically meaningful) quantity. This is one of the hallmarks of the “good
singularities”, using the definitions developed in [99, 100]. This is the case
for the holographic central charge computed in Subsection 2.4 and the Wilson
loop vev of Subsection 2.5, whose results are reproduced by field theory
computations in Section 3. We emphasise that there should exist probes of the
backgrounds with two rank functions that will display the singular behaviour
in the case of finite $\frac{\sigma_{0}}{P}$.

Of course, it would be very interesting to find new backgrounds that
interpolate between the conformal field theories for very large and vanishing
values of $\frac{\sigma_{0}}{P}$. This is a difficult problem that we leave
for future investigation.

###  2.8 Matrix models for electrostatic problems

In this subsection, we set up a simple electrostatic problem whose large $P$
limit directly matches the holographic central charge of the geometry
containing the AdS6 factor. The derivation is based on constructing a unitary
matrix model for the electrostatic problem, as explained in [101, Sec.2.7],
without any knowledge of AdS/CFT. This model is distinct from, and enormously
simpler than, the matrix model obtained from localization on $\mathbb{S}^{d}$.

We begin with the simplest setting: consider the strip

$\left\\{(\eta,\sigma)\ :0\leq\eta\leq P,\
-\infty<\sigma<\infty\right\\}=[0,P]\times\operatorname{\mathbb{R}}$

and place $P-1$ unit charges on the interior points of the integral lattice
along the compact $\eta$-direction at $\sigma=0$. That is to say, the
$j^{\text{th}}$ particle is placed at position $(\eta_{j},0)$, with $\eta_{j}$
describing fluctuations centered at $\eta_{j}=j$, $\forall j=1,\dots,P-1$. The
electrostatic potential $W_{\text{el}}$ is subject to the boundary conditions

$\displaystyle W_{\text{el}}(\eta,\sigma\to\infty)$
$\displaystyle=0=W_{\text{el}}(\eta,\sigma\to-\infty),$ $\displaystyle
W_{\text{el}}(0,\sigma)$ $\displaystyle=0=W_{\text{el}}(P,\sigma).$ (2.37)

Using that the Coulomb potential is logarithmic in 1d, the Boltzmann factor
$e^{-\beta E}$ due to the interaction among the $P-1$ charges confined at
$\sigma=0$ is determined by

$E(\eta_{1},\dots,\eta_{P-1})=-\sum_{1\leq j\neq\ell\leq
P-1}\ln\left(\left\lvert
e^{i2\pi\eta_{j}/P}-e^{i2\pi\eta_{\ell}/P}\right\rvert\frac{P}{2\pi}\right).$
(2.38)

Furthermore, we introduce a background charge distribution, to compensate the
repulsion among the charges. Without this neutralising background charge
density, on an unbounded domain the particles would repel each other to
$\eta_{j}\to\pm\infty$. The presence of boundaries prevents this scenario, but
the equilibrium would simply be given by the charges being equi-spaced. To
allow more interesting configurations, we add the background charge. The
resulting potential energy of a charge at position $(\eta,0)$ is generically
written in Fourier expansions as

$V_{\text{el}}(z)=\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi\eta}{P}\right).$

The canonical partition function for this system is obtained as a mild
generalization of [101, Sec.2.7]. In units such that $\beta=2$, it reads

$\exp\left(\mathcal{F}^{\text{Dir}}_{\text{el}}\right)=\frac{1}{(P-1)!}\int_{[0,P]^{P-1}}\prod_{1\leq
j<\ell\leq
P-1}\left\lvert\left(e^{i2\pi\eta_{j}/P}-e^{i2\pi\eta_{\ell}/P}\right)\right\rvert^{2}\prod_{j=1}^{P-1}e^{-2V(z_{j})}\left(\frac{2\pi}{P}\right)\mathrm{d}\eta_{j}.$
(2.39)

We have expressed the partition function directly in terms of the free energy
$\mathcal{F}^{\text{Dir}}_{\text{el}}$, and dropped an $\eta$-independent
overall factor, that can be cancelled by a suitable choice of background
charge. The integral (2.39) is a unitary matrix model, equivalent to
integration over the unitary group with Haar measure,

$\exp\left(\mathcal{F}^{\text{Dir}}_{\text{el}}\right)=\int_{U(P-1)}\mathrm{d}U\leavevmode\nobreak\
\exp\left\\{\mathrm{Tr}V(U+U^{\dagger})\right\\}.$ (2.40)

The large $P$ limit of (2.40) is calculated by Szegő’s theorem [102]:

$\lim_{P\to\infty}\mathcal{F}^{\text{Dir}}_{\text{el}}=\sum_{k=1}^{\infty}ka_{k}^{2},$

which appears to be proportional to the holographic central charge (2.20).

The relation with $c_{hol}$ is only formal at this stage, because we have not
specified the Fourier coefficients $a_{k}$. First, let us notice that, in the
large $P$ limit, the $P-1$ discrete charges are replaced by an inhomogeneous
density of charge $\mathcal{R}_{\text{el}}(\eta)$, which grows linearly in
$P$. The electrostatic potential solves a continuous version of the previous
problem, in which the large number of discrete charges is replaced by the
continuous density. This leads to the Poisson equation

$\partial_{\eta}^{2}W_{\text{el}}+\partial_{\eta}^{2}W_{\text{el}}+\frac{2\pi}{P}\mathcal{R}_{\text{el}}(\eta)\delta(\sigma)=0,$

which is precisely (2.10), upon identification

$\mathcal{R}(\eta)=\frac{2\pi}{P}\mathcal{R}_{\text{el}}(\eta).$ (2.41)

Then, we determine the coefficients $a_{k}$ by requiring that, at equilibrium,
the electrostatic force on the position $(\eta,0)$ due to the density of
charges $\mathcal{R}_{\text{el}}$ is compensated by the interaction with the
background. The electrostatic potential on the position $\eta$ generated by
the distribution of charges at all positions $\tilde{\eta}\neq\eta$, in this
continuous version, is:

$\Phi(\eta)=\frac{1}{P}\int_{0}^{P}\mathrm{d}\tilde{\eta}\ln\sin\left(\frac{\pi}{P}\lvert\eta-\tilde{\eta}\rvert\right)\mathcal{R}_{\text{el}}(\tilde{\eta}).$

The corresponding Coulomb force is $\partial_{\eta}\Phi(\eta)$. Equating it
with the force $\partial_{\eta}V_{\text{el}}(\eta)$ results in:

$\frac{1}{P}\int_{0}^{P}\mathrm{d}\tilde{\eta}\cot\left(\frac{\pi}{P}\left(\eta-\tilde{\eta}\right)\right)\mathcal{R}_{\text{el}}(\tilde{\eta})=-2\sum_{k=1}^{\infty}ka_{k}\cos\left(\frac{k\pi\eta}{P}\right).$

Expanding the left-hand side of the equilibrium condition in Fourier modes,
with the identification (2.41), integrating over $\tilde{\eta}$ and imposing
the equality, we get

$ka_{k}=\frac{P}{2\pi}R_{k}.$

Thus, the coefficients $a_{k}$ we seek are exactly the ones defined in (2.9).
This completes the match with the single rank function supergravity setup,
concretely (2.10) and (2.20):

$\lim_{P\to\infty}\mathcal{F}^{\text{Dir}}_{\text{el}}=\sum_{k=1}^{\infty}ka_{k}^{2}=2\pi^{4}c_{hol}.$

This purely electrostatic matrix model predicts the correct holographic
central charge in the case of a single rank function. One may set up the
analogous problem in the two rank functions case. Two sets of charges are
placed along the $\eta$-direction at $\sigma=0$ and $\sigma=\sigma_{0}$. We
call their positions $\eta_{j},\hat{\eta}_{j}$, respectively. The Boltzmann
factor $e^{-\beta E}$ is given by

$E=E(\eta_{1},\dots,\eta_{P-1})+E(\hat{\eta}_{1},\dots,\hat{\eta}_{P-1})+E_{\text{int}}(\eta_{1},\dots,\eta_{P-1},\hat{\eta}_{1},\dots,\hat{\eta}_{P-1};\sigma_{0})$

where the first two pieces are as in (2.38) and [101, Eq.(2.72)]

$E_{\text{int}}=-\sum_{j=1}^{P-1}\sum_{\ell=1}^{P-1}\ln\left[\frac{P}{\pi}\left\lvert\sin\left(\frac{\pi}{P}(\eta_{j}-\hat{\eta}_{\ell}+i\sigma_{0})\right)\right\rvert\right].$

At $\sigma_{0}\to 0$ the two pairs of eigenvalues coalesce and we get a larger
matrix, containing the two sets of charges. At $\sigma_{0}\to\infty$, the
interaction term is exponentially suppressed and the matrix model breaks into
two copies of (2.39). The two limiting pictures agree with the supergravity
problem.

The purely electrostatic problem and the associated unitary matrix model allow
for several generalizations. An immediate one is to have distinct number of
charges at $\sigma=0$ and $\sigma=\sigma_{0}$. An exhaustive exploration of
the implications of these matrix models in holography is a mathematical
problem that we leave for future work.

##  3 Part II: Matrix models

In this section, we discuss the large $N$ limit of the free energy on the QFT
side. The field theories we are interested in are encoded in framed $A_{P-1}$
quivers, that is, linear quivers with $P-1$ gauge and flavour nodes, as in
Figure 3. The supergravity solution is reliable in the regime $P\gg 1$, that
we will consider below. Moreover, we focus on quivers that are balanced.

The long quiver limit of linear quivers was first solved by Uhlemann for 5d
SCFTs [83] and later extended to 3d in [84, 85]. Here we reformulate these
results in a unified notation and rederive them in a formalism consistent with
Section 2. More importantly, we extend the derivation away from the
superconformal fixed point, allowing mass deformations. Along the way, we
refine the notion of strong coupling limit for five-dimensional long quivers,
in a way that is invariant under fibre/base duality in M-theory.

On a technical level, the main results are the reformulation of [83], uniform
in $d$, and its extension along RG flows triggered by real masses. We compute
the free energies, identify the physical meaning of the results and discuss
related subtleties. The outcome is used to substantiate the AdS/CFT
correspondence and to identify the deformation holographically dual to the
problem studied in Subsection 2.5.

The outline of this section is as follows. Subsection 3.1 contains an
elementary presentation of the RG flows we consider, from the perspective of
the Coulomb branch of the QFT. In Subsection 3.2 we detail the general
procedure, following and extending [83, 84, 85]. The free energies in 5d and
3d are derived in Subsection 3.3 in this more general setting. The case of two
rank functions is discussed explicitly in Subsection 3.4, where we propose an
alternative derivation. In Subsection 3.5 we discuss the implications of the
RG flow on the QFT side, define an effective free energy and show the
agreement between the supergravity calculations and the matrix model. The
effective free energies satisfy the F-theorem, as detailed in Subsection 3.6.
Wilson loops in antisymmetric representations are studied in Subsection 3.7.

The main results of this section are: the deformed free energies of long
quivers in 5d and 3d (3.21) and (3.22), respectively; the explicit match with
the holographic central charges in (3.32) and (3.33); the match of the Wilson
loop vev in (3.41); the discussion on the effective free energy in Subsection
3.5.3; and finally the F-theorem in Subsection 3.6.

###  3.1 Coulomb branches and mass deformations

In this section, we study mass deformations of five- and three-dimensional
linear quivers by evaluating their sphere partition functions in the long
quiver and large $N$ limit. We begin with an overview of the mass deformations
and how they resolve the singularity of the Coulomb branch of the SCFT. For
definiteness, and only during the current subsection, we frame the discussion
in the 5d setting. It can be readily exported to the 3d $\mathcal{N}=4$ case,
because we only consider real masses.

The goal of the current subsection is to recast known facts in the framework
of the present work. This proves useful to shed light on the setup discussed
holographically in Section 2, but the experienced reader may safely skip this
subsection and refer to Figure 8 for a schematic summary.

####  3.1.1 Mass parameter space

The moduli space of massive deformations, called the parameter space of the
theory, is spanned by the vacuum expectation value (vev) of the real scalars
in the background vector multiplets for the global symmetry.

The fundamental hypermultiplets acquire a real mass by coupling them to a
background vector multiplet for the flavour symmetry. A non-vanishing vev for
the background vector multiplet thus activates an RG flow. The flows
holographically dual to the ones proposed in Subsection 2.5 are controlled by
a single mass parameter $m$: at every node $j=1,\dots,P-1$, the scalar in the
background vector multiplet for the flavour symmetry algebra
$\mathfrak{u}(F_{j})$ is given a vev

$\mathrm{diag}(\underbrace{0,\dots,0}_{F_{1,j}},\underbrace{m,\dots,m}_{F_{2,j}}).$

This is usually referred to as leaving $F_{1,j}$ of the hypermultiplets
massless, and giving a mass $m$ to the remaining $F_{2,j}$ hypermultiplets. As
we will show in Subsections 3.2-3.3, the argument extends to a larger number
of mass parameters without any conceptual difference.

In both 5d and 3d, the global symmetry of the SCFT contains the torus

$\mathbb{T}_{G}=\prod_{j=1}^{P-1}\left[U(1)_{f}^{F_{j}}\times
U(1)_{I}\right]/U(1)_{\mathrm{diag}}^{(5-d)/2},$

where the $U(1)_{f}$ factors come from the maximal torus of the flavour
symmetry, and $U(1)_{I}$ acting on the instantons in 5d, or on the monopoles
in 3d. There is always one such $U(1)_{I}$ per gauge node. Finally,
$U(1)_{\mathrm{diag}}$ acts diagonally on the flavour symmetry, and the
quotient by $U(1)_{\mathrm{diag}}^{(5-d)/2}$ conveniently encodes that the
center of the flavour symmetry is gauged in 3d, where the gauge groups are
unitary, but not in 5d.

The vevs of the scalars in the vector multiplets for the $U(1)_{f}$ are the
standard mass parameters, while the vevs of the scalar in the $U(1)_{I}$ are,
respectively, the inverse gauge couplings $1/g_{\text{\tiny YM},j}^{2}$ in 5d
and the FI parameters in 3d. For later reference, notice that, as opposed to
four-dimensional Yang–Mills theory, the gauge couplings of 5d $\mathcal{N}=1$
theories pertain to the mass parameter space, and shall be dealt with
accordingly in our analysis, as opposed to most of the existing large $N$
studies.

####  3.1.2 Coulomb branch geometry

Consider a five-dimensional linear quiver as in Figure 3, denoted by
$\mathcal{Q}$. The zero-mode of the real scalar $\vec{\sigma}$ in the Cartan
subalgebra of the gauge group parametrises the Coulomb branch of
$\mathcal{Q}$, CB$[\mathcal{Q}]$ for short. CB$[\mathcal{Q}]$ is fibered over
the mass parameter space of the theory [103]. At the origin of the parameter
space, CB$[\mathcal{Q}]$ has a conical singularity at its origin,
$\vec{\sigma}=0$. Moving on the parameter space, the singularity is fully
resolved at generic points, and partially resolved at positive-codimensional
loci in the parameter space.

Figure 7: Schematic view of the Coulomb branch fibered over the mass parameter
space. Moving along the RG flow triggered by the selected mass deformation
(orange) resolves the Coulomb branch singularity. It breaks in two disjoint
cones at the limiting point $m=\infty$.

As explained above, we are mainly interested in the scenario in which one
activates vevs for the background scalars controlled by a unique parameter
$m$. The hypermultiplets are grouped in two families, becoming massless at
separate points on CB$[\mathcal{Q}]$, whose distance increases along the RG
flow triggered by $m$. We are thus resolving the singularity of
CB$[\mathcal{Q}]$ in a minimal and controlled way, by moving along a one-real
dimensional locus on the parameter space. This idea is illustrated in Figure
7.

CB$[\mathcal{Q}]$ is expected to break in two smaller singularities at the end
of the RG flow, which we will identify with the Coulomb branches of two
quivers $\mathcal{Q}_{1},\mathcal{Q}_{2}$ with smaller gauge and flavour
ranks. Both CB$[\mathcal{Q}_{1}]$ and CB$[\mathcal{Q}_{2}]$ have a singularity
at their origin and another one at the point at infinity. The hypermultiplets
that would become massless at infinity are effectively decoupled from the
gauge theory.

Besides the geometry, another clear way to see the appearance of decoupled
modes is from the realization of the quiver theories using brane webs [81].
Our choice of mass deformation corresponds to splitting the color branes (D5
in 5d, D3 in 3d) into two separate stacks. The strings that connect branes
within the $\alpha^{\text{th}}$ stack give rise to the modes in
$\mathcal{Q}_{\alpha}$, $\alpha=1,2$, whilst the strings stretched between
branes belonging to different stacks give rise to heavy modes, that eventually
decouple.

Before moving on to the actual computation, we emphasise that the RG flow
activated by the mass deformation we introduce is independent of the RG flows
that deform the 5d SCFTs onto their gauge theory phases, as represented in
Figure 8. In the language of Subsection 3.1.1, the two deformations correspond
to moving along different directions in the mass parameter space, and can be
dealt with independently.

SCFT$[\mathcal{Q}]$SCFT$[\mathcal{Q}_{1}]\oplus$SCFT$[\mathcal{Q}_{2}]$$\oplus$
decoupled$m$Gauge theory $[\mathcal{Q}]$Gauge theory
$[\mathcal{Q}_{1}]$$\oplus$ Gauge theory $[\mathcal{Q}_{2}]$$\oplus$
decoupled$\left\\{1/g_{j}^{\text{\tiny
YM}}\right\\}$$\left\\{1/g_{1,j}^{\text{\tiny
YM}}\right\\}\cup\left\\{1/g_{2,j}^{\text{\tiny YM}}\right\\}$ Figure 8:
Illustration of the massive deformations and associated RG flows in our 5d
setup. The diagonal arrow labelled by $m$ is the flow that matches with
supergravity. Vertical arrows indicate RG flows from a SCFT to a gauge theory,
the relevant operator triggering it being the supersymmetric Yang–Mills
kinetic term.

###  3.2 Long quivers and their large $N$ limit

####  3.2.1 Notation

We denote the gauge ranks $N_{1},\dots,N_{P-1}$ and the flavour ranks
$F_{1},\dots,F_{P-1}$. We use a label $j=1,\dots,P-1$ running over the nodes
of the quiver, and labels $a,b=1,\dots,N_{j}$ running over the indices in the
Cartan of the gauge algebra $\mathfrak{u}(N_{j})$ at the node $j$.

Following [70, 51] and Section 2, we define a rank function
$\mathcal{R}(\eta)$, for a continuous index $0\leq\eta\leq P$, such that
$\mathcal{R}(\eta=j)=N_{j}$. Its Fourier expansion is

$\mathcal{R}(\eta)=\sum_{k=1}^{\infty}R_{k}\sin\left(\frac{k\pi\eta}{P}\right).$

In the following, we fix $N\in\mathbb{N}$. We will assume $N\gg 1$ later on.
It is convenient to introduce a continuous label $0\leq z\leq 1$,

$z=\frac{\eta}{P}$

and write

$\nu(z):=\frac{1}{N}\mathcal{R}(Pz),$

so that $N_{j}=N\nu(z)$. The Fourier expansion of the scaled rank function
$\nu(z)$ is

$\nu(z)=\sum_{k=1}^{\infty}\mathfrak{a}_{k}\sin\left(k\pi z\right),$

with coefficients $\mathfrak{a}_{k}$ related to $a_{k}$ defined in (2.15)
through $\mathfrak{a}_{k}=\frac{R_{k}}{N}=\frac{2\pi k}{NP}a_{k}$.

To set up the Veneziano limit, the flavour ranks must scale with $N$, so that
$F_{j}=N\zeta_{j}$, with $\zeta_{j}$ held fixed in the large $N$ limit. The
flavour ranks are collected into a flavour rank function $\zeta(z)$.

We work in units in which the radius of $\mathbb{S}^{d}$ is set to one. The
sphere free energy $\mathcal{F}$ is defined as [95]

$\mathcal{F}=(-1)^{\frac{d-1}{2}}\ln\mathcal{Z}_{\mathbb{S}^{d}}.$ (3.1)

####  3.2.2 Mass deformations

Throughout this section, the fundamental hypermultiplets are grouped into
$\mathscr{F}$ families of equal mass, such that $F_{\alpha,j}$ hypermultiplets
have the same mass $m_{\alpha}$, $\forall\alpha=1,\dots,\mathscr{F}$. This
corresponds to turning on the same mass for a fraction $\zeta_{\alpha}(z)$ of
$\zeta(z)$, at every $0\leq z\leq 1$. Clearly, for consistency we have

$\sum_{\alpha=1}^{\mathscr{F}}\zeta_{\alpha}(z)=\zeta(z).$

Let us stress that there are $\mathscr{F}$ different mass scales

$\left\\{m_{\alpha}\ :\ \alpha=1,\dots,\mathscr{F}\right\\}$

with a variable number of hypermultiplets with each mass at each node.
Hypermultiplets in the $\alpha^{\text{th}}$ family at every node have the same
mass $m_{\alpha}$. Previously, we demanded that $\zeta_{\alpha}(z)$ remains
finite in the large $N$ limit. In particular, $\mathscr{F}$ is held fixed. The
large $\mathscr{F}$ limit might be interesting for future investigation.

Recall that we are dealing with balanced quivers, and that the balancing
condition reads (see [83, 104] for the proof)666Recall the discussion below
(2.3) about sign conventions in the $\partial_{z}^{2}\nu$ versus
$\mathcal{R}^{\prime\prime}$ derivative.

$\displaystyle\zeta(z)$ $\displaystyle=-\frac{1}{P^{2}}\frac{\partial^{2}\
}{\partial z^{2}}\nu(z)$
$\displaystyle=\frac{\pi^{2}}{P^{2}}\sum_{k=1}^{\infty}\mathfrak{a}_{k}k^{2}\sin\left(k\pi
z\right).$

Because the function $\zeta(z)$ splits into the sum of $\zeta_{\alpha}(z)$, we
define the corresponding Fourier coefficients $\mathfrak{a}_{\alpha,k}$, such
that

$\zeta_{\alpha}(z)=\frac{\pi^{2}}{P^{2}}\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}k^{2}\sin\left(k\pi
z\right),\qquad\forall\alpha=1,\dots,\mathscr{F}.$ (3.2)

The corresponding “reduced” rank functions are

$\nu_{\alpha}(z)=\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}\sin\left(k\pi
z\right),\qquad\forall\alpha=1,\dots,\mathscr{F}.$ (3.3)

####  3.2.3 Sphere partition functions

To unify the 5d and 3d notation, we work on the round $d$-dimensional sphere
$\mathbb{S}^{d}$, $d\in\left\\{3,5\right\\}$. The $d$-sphere partition
function of a linear quiver with 8 supercharges is [105, 106, 107, 108, 109]

$\mathcal{Z}_{\mathbb{S}^{d}}=\int_{\mathbb{R}^{\lvert\vec{N}\rvert}}\mathrm{d}\vec{\sigma}\leavevmode\nobreak\
Z_{\text{cl}}(\vec{\sigma})Z_{\text{1-loop}}^{\text{vec}}(\vec{\sigma})Z_{\text{1-loop}}^{\text{hyp}}(\vec{\sigma},\vec{m})Z_{\text{non-
pert}}.$ (3.4)

Here $\vec{\sigma}$ is the zero-mode of the real scalar in the vector
multiplet, at the localization locus. We have used the shorthand notation
$\lvert\vec{N}\rvert:=\sum_{j=1}^{P}N_{j}$ and777For ease of notation, we
combine the Vandermonde factors coming from diagonalizing the scalar zero-mode
$\vec{\sigma}$ with the contribution of the vector multiplet.

$\mathrm{d}\vec{\sigma}:=\prod_{j=1}^{P-1}\prod_{a=1}^{N_{j}}\mathrm{d}\sigma_{a,j}\
\times\
\left[\prod_{j=1}^{P-1}\delta\left(\sum_{a=1}^{N_{j}}\sigma_{a,j}\right)\right]^{(d-3)/2}$

is the Lebesgue measure on the Cartan of the gauge algebra, isomorphic to
$\mathbb{R}^{\lvert\vec{N}\rvert}$ in 3d and to
$\mathbb{R}^{\lvert\vec{N}\rvert-P+1}$ in 5d. The term in square bracket
enforces that the gauge nodes in 5d are $SU(N_{j})$, while they are $U(N_{j})$
in 3d. The integrand of (3.4) comprises the following terms:

  * •

The classical contribution $Z_{\text{cl}}=e^{-S_{\text{CS}}-S_{\text{YM}}}$,
includes Chern–Simons and Yang–Mills terms, evaluated at the localization
locus. Hereafter we set the Chern–Simons couplings to zero, while

$S_{\text{YM}}=\begin{cases}0&d=3\\\
\sum_{j=1}^{P-1}\sum_{a=1}^{N_{j}}\frac{(2\pi)^{3}}{g_{\text{\tiny
YM},j}^{2}}\sigma_{a,j}^{2}&d=5.\end{cases}$

In flat space, the inverse 5d Yang–Mills coupling has the dimension of a mass
parameter. For later convenience, we define the corresponding mass parameter
at the node $j$ as

$m^{\text{\tiny YM}}_{j}=\frac{(2\pi)^{3}}{g_{\text{\tiny YM},j}^{2}}$

and collect all of them in the function $m^{\text{\tiny YM}}(z)=m^{\text{\tiny
YM}}_{zP}$

  * •

The one-loop contribution of the vector multiplet,
$Z_{\text{1-loop}}^{\text{vec}}$. It takes the form

$Z_{\text{1-loop}}^{\text{vec}}(\vec{\sigma})=\prod_{j=1}^{P-1}\prod_{1\leq
a\neq b\leq
N_{j}}\exp\left\\{-\mathsf{v}(\sigma_{a,j}-\sigma_{b,j})\right\\},$

where $\mathsf{v}$ is a real-valued function of a single variable

$\mathsf{v}(s)=\begin{cases}-\ln\left(2\sinh(\pi s)\right)\quad&d=3\\\
-\ln\left(2\sinh(\pi s)\right)-\frac{1}{2}f(is)&d=5.\end{cases}$

The function $f(is)$ in 5d is a real even function of $s$ [106, 107]:

$\displaystyle f(is)$
$\displaystyle\overbrace{=}^{\text{reg.}}\sum_{n=1}^{\infty}n^{2}\ln\left(1+\frac{s^{2}}{n^{2}}\right)$
$\displaystyle\Rightarrow\ f(is)$
$\displaystyle=\frac{\pi}{3}s^{3}-s^{2}\ln\left(1-e^{-2\pi
s}\right)+\frac{s}{\pi}\mathrm{Li}_{2}\left(e^{-2\pi
s}\right)+\frac{1}{2\pi^{2}}\mathrm{Li}_{3}\left(e^{-2\pi
s}\right)-\frac{\zeta(3)}{2\pi^{2}},$

the first line meaning that $f(is)$ is defined to be the $\zeta$-function
regularization of the divergent right-hand side, explicitly given in the
second line.

  * •

The one-loop contribution of the hypermultiplets. For the quivers we consider,
the hypermultiplets are of two types: fundamental and bifundamental. The
integrand factorises accordingly:

$Z_{\text{1-loop}}^{\text{hyp}}(\vec{\sigma},\vec{m})=Z^{\text{fund}}(\vec{\sigma},m)Z^{\text{bif}}(\vec{\sigma}).$

With the assumption above on the masses, we have

$\displaystyle Z^{\text{fund}}(\vec{\sigma},\vec{m})$
$\displaystyle=\prod_{j=1}^{P-1}\prod_{a=1}^{N_{j}}\prod_{\alpha=1}^{\mathscr{F}}\exp\left\\{-F_{\alpha,j}\mathsf{h}(\sigma_{a,j}-m_{\alpha})\right\\},$
$\displaystyle Z^{\text{bif}}(\vec{\sigma})$
$\displaystyle=\prod_{j=1}^{P-2}\prod_{a=1}^{N_{j}}\prod_{b=1}^{N_{j+1}}\exp\left\\{-\mathsf{h}(\sigma_{a,j}-\sigma_{b,j+1})\right\\},$

where $\mathsf{h}$ is a real-valued function of a single variable

$\mathsf{h}(s)=\begin{cases}\ln\left(2\cosh(\pi s)\right)\quad&d=3\\\
-\ln\left(2\cosh(\pi
s)\right)+\frac{1}{4}\left[f\left(\frac{1}{2}+is\right)+f\left(\frac{1}{2}-is\right)\right]&d=5.\end{cases}$

  * •

The non-perturbative contribution $Z_{\text{non-pert}}$ is identically 1 in 3d
[105], and accounts for codimension-4 field configurations in 5d [106, 107].
The mass of 5d instantons is $\propto 1/g^{2}_{\text{\tiny YM}}$, thus they
are non-perturbatively suppressed away from the superconformal point, but can
become massless in the conformal limit. As argued below, we will enforce a
procedure such that, schematically,

$Z_{\text{non-pert}}=1+e^{-P(\cdots)},$

where the dots are a strictly positive number as long as $g_{\text{\tiny
YM},j}^{2}>0$. In the following, we will work with finite gauge coupling in
5d, so that the SCFT flows to a gauge theory and the picture described so far
applies. Then, $Z_{\text{non-pert}}$ is safely neglected in the regime $P\gg
1$. The strong coupling limit is taken at the end of the computation.

Convergence of the perturbative $\mathbb{S}^{d}$ partition function, i.e.
setting $Z_{\text{non-pert}}\to 1$, imposes $F_{j}\geq 2N_{j}-N_{j-1}-N_{j+1}$
in $d=3$ and $F_{j}\leq 2N_{j}-N_{j-1}-N_{j+1}$ in $d=5$, making balanced
quivers a preferred choice.

Before taking the large $N$ limit, a remark is in order. Turning on massive
deformations of large $N$ field theories, there can be phase transitions
(typically of third order) when such mass parameters cross a given threshold.
It was proven in [104] that balanced linear quivers do not have such phase
transitions.

####  3.2.4 Setting up the large $N$ limit

The first step to take the large $N$ limit is to rewrite the partition
function in the form

$\mathcal{Z}_{\mathbb{S}^{d}}=\int_{\mathbb{R}^{\lvert\vec{N}\rvert}}\mathrm{d}\vec{\sigma}\leavevmode\nobreak\
e^{-S_{\text{eff}}(\vec{\sigma},\vec{m})}Z_{\text{non-pert}},$

where, with the above notation,

$\displaystyle S_{\text{eff}}$
$\displaystyle=\sum_{j=1}^{P-1}\sum_{a=1}^{N_{j}}\left[\delta_{d,5}m_{j}^{\text{\tiny
YM}}\sigma_{a,j}^{2}+\sum_{\alpha=1}^{\mathscr{F}}F_{\alpha,j}\mathsf{h}(\sigma_{a,j}-m_{\alpha})\right]$
(3.5) $\displaystyle+\sum_{j=1}^{P-1}\sum_{a=1}^{N_{j}}\left[\left(\sum_{b\neq
a}\mathsf{v}(\sigma_{a,j}-\sigma_{b,j})\right)+\left(\sum_{b=1}^{N_{j+1}}\mathsf{h}(\sigma_{a,j}-\sigma_{b,j+1})\right)\right]+\mathfrak{B}.$

The last term $\mathfrak{B}$ generically includes terms from the boundary of
the quiver. Here we are over-counting the contribution from a bifundamental
hypermultiplet at the last node, thus $\mathfrak{B}$ serves to correct this
extra counting. It follows from power counting, and is shown explicitly in
[83], that it is subleading in the large $P$ limit, thus we refrain from
writing down its explicit form.

It is customary to introduce the eigenvalue density at the $j^{\text{th}}$
node:

$\rho_{j}(\sigma)=\frac{1}{N}\sum_{a=1}^{N_{j}}\delta(\sigma_{a,j}-\sigma).$

Two caveats: first, notice that now $\sigma$ is a continuous real variable,
related to, but different from the $\lvert\vec{N}\rvert$-dimensional real
scalar $\vec{\sigma}$. Second, we are normalising all the densities $\rho_{j}$
with a unique (and so far, arbitrary) integer $N$. This implies

$\int_{\operatorname{\mathbb{R}}}\mathrm{d}\sigma\rho_{j}(\sigma)=\frac{N_{j}}{N}$

With the above definitions at hand, the sums over $j$ and $a$ can be replaced
by integrals:

$\sum_{a=1}^{N_{j}}f(\sigma_{a,j})\ \rightsquigarrow
N\int\mathrm{d}\sigma\rho_{j}(\sigma)f(\sigma)\;;\qquad\sum_{j=1}^{P-1}f_{j}\
\rightsquigarrow\ P\int_{0}^{1}\mathrm{d}zf(z)\;.$

for arbitrary expressions $f_{j}$. This allows to pack the eigenvalue
densities at each node into a single function of two variables

$\rho(z,\sigma):[0,1]\times\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}_{\geq
0},$

normalised as

$\int_{\operatorname{\mathbb{R}}}\mathrm{d}\sigma\rho(z,\sigma)=\nu(z),\qquad\forall
0\leq z\leq 1.$

Putting all together, (3.5) becomes

$\displaystyle S_{\text{eff}}$
$\displaystyle=PN^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}\sigma\rho(z,\sigma)\left[\delta_{d,5}\frac{m^{\text{\tiny
YM}}(z)}{N}\sigma^{2}+\sum_{\alpha=1}^{\mathscr{F}}\zeta_{\alpha}(z)\mathsf{h}(\sigma-
m_{\alpha})\right.$ (3.6a)
$\displaystyle+\int_{\phi\neq\sigma}\mathrm{d}\phi\rho(z,\phi)\mathsf{v}(\sigma-\phi)$
(3.6b) $\displaystyle+\left.\frac{1}{2}\int\mathrm{d}\phi\left(\rho(z+\delta
z,\phi)+\rho(z-\delta
z,\phi)\right)\mathsf{h}(\sigma-\phi)\right]+\mathfrak{B}.$ (3.6c)

In the last line, we have used the symmetry of the last term to slightly
rewrite it [83], with the understanding that, if $z=\frac{j}{P}$, then
$z\pm\delta z=\frac{j\pm 1}{P}$. The boundary piece $\mathfrak{B}$ now
contains half term from $j=P-1$ and half from $j=0$.

####  3.2.5 Long quiver limit

The long quiver limit, $P\gg 1$, was pioneered in [83]. Here we briefly sketch
their derivation, with a few variations to deal with mass deformations as well
as to treat $m^{\text{\tiny YM}}(z)$ in accordance with its geometric
engineering.

First, we add and subtract $\rho(z,\phi)\mathsf{h}(\sigma-\phi)$ in (3.6c):

$\displaystyle\frac{1}{2}\left(\rho(z+\delta z,\phi)+\rho(z-\delta
z,\phi)\right)\mathsf{h}(\sigma-\phi)$
$\displaystyle=\frac{1}{2}\left[\left(\rho(z+\delta
z,\phi)-\rho(z,\phi)\right)-\left(\rho(z,\phi)-\rho(z-\delta
z,\phi)\right)\right]\mathsf{h}(\sigma-\phi)+\rho(z,\phi)\mathsf{h}(\sigma-\phi).$

We combine the last summand with (3.6b) and define

$\mathsf{F}_{0}(s):=\left[\mathsf{v}(s)\cdot 1\\!\\!1_{s\neq
0}+\mathsf{h}(s)\right]/(d-2)^{2}$

where $1\\!\\!1_{s\neq 0}$ vanishes at $s=0$ and is 1 otherwise. Then, we
observe that, in the large $P$ limit [83]

$\left(\rho(z+\delta
z,\phi)-\rho(z,\phi)\right)-\left(\rho(z,\phi)-\rho(z-\delta
z,\phi)\right)\to\frac{1}{P^{2}}\frac{\partial\ }{\partial
z^{2}}\rho(z,\phi).$ (3.7)

Second, we assume that $\sigma$ grows with $P$. Since the mass parameters are
realised as background fields for the flavour symmetry, it is natural to put
them on an equal footing and assume the same scaling with $P$:

$\sigma=(d-2)P^{\chi}x,\quad m_{\alpha}=(d-2)P^{\chi}\mu_{\alpha},$

with $x,\mu$ independent of $P$. The exponent $\chi>0$ is determined
momentarily by self-consistency of the large $P$ limit. If we were to find
$\chi\leq 0$, we would have to reconsider our scaling ansatz.

The corresponding rescaled eigenvalue density is

$\varrho(z,x)\mathrm{d}x=\rho(z,\sigma)\mathrm{d}\sigma.$

Recall from Subsection 3.1.1 that, in 5d, the inverse gauge coupling has the
dimensions of a mass parameter. It corresponds to the vev of the background
scalar for the $U(1)_{I}$ global symmetry. Thus, we study the scaling

$m^{\text{\tiny YM}}(z)=P^{\chi}\varkappa(z),$

for some fixed function $\varkappa(z)$. Let us emphasise this point: we do not
consider a ’t Hooft scaling. Instead, we treat all the real scalars (dynamical
and background) equally. In 5d, this is the most natural standpoint, as
explained in Subsection 3.1.1. Moreover, on the Coulomb branch, the one-loop
corrected gauge coupling is shifted proportionally to $\lvert\sigma\rvert$,
which further advocates for scaling $m^{\text{\tiny YM}}$ precisely as
$\sigma$ does.

This choice of scaling, distinct from the ’t Hooft one, is also consistent
with the M-theory realization of the quivers. Indeed, vevs of the scalars,
regardless of dynamical or background, all are computed from the volumes of
certain two-cycles inside a Calabi–Yau threefold [79]. The only difference
between the inverse gauge couplings and the mass parameters, is that the two-
cycles giving rise to the former live in the base of a fibration, while the
two-cycles for the latter live in the fibre. With our prescription, as opposed
to the ’t Hooft limit considered in the existing literature, the volumes of
all the two-cycles scale equally.888LSa thanks Michele Del Zotto for
suggestions on this point. In particular, our procedure preserves the
fibre/base duality [110, 103, 111, 112], whenever the fibration mentioned in
the M-theory setup enjoys such duality.

In this approach, the Yang–Mills term is subleading in $N$ but dominant in
$P$. The two limits do not commute, and their order of limits is thus relevant
(as already pointed out in [104]).

  * $i$)

We first take the large $N$ limit at fixed gauge coupling. The Yang–Mills term
is subleading in the effective action. Non-perturbative contributions cannot
be neglected at this stage.

  * $ii$)

The large $P$ limit is taken afterwards. This suppresses the non-perturbative
contributions, and also treats all massive fields and parameters
democratically.

Introducing the functions [83, 84]

$\mathsf{f}_{0}(x)=\begin{cases}\frac{\pi}{4}\delta(x)&d=3\\\
-\frac{27\pi}{8}\lvert
x\rvert&d=5\end{cases}\qquad\qquad\mathsf{f}_{\mathrm{h}}(x)=\begin{cases}\pi\lvert
x\rvert&d=3\\\ -\frac{9\pi}{2}\lvert x\rvert^{3}&d=5\end{cases}$ (3.8)

the large argument behaviour of $\mathsf{F}_{0}$ and $\mathsf{h}$ is

$\mathsf{F}_{0}(P^{\chi}x)\approx
P^{\chi(d-4)}\mathsf{f}_{0}(x),\qquad\mathsf{h}(P^{\chi}x)\approx
P^{\chi(d-2)}\mathsf{f}_{\mathrm{h}}(x).$

These definitions, together with (3.7), allow us to rewrite the effective
action (3.6) as:

$\displaystyle
S_{\text{eff}}=P^{1+\chi(d-4)}N^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\varrho(z,x)$
$\displaystyle\left[P^{2\chi}\sum_{\alpha=1}^{\mathscr{F}}\zeta_{\alpha}(z)\mathsf{f}_{\text{h}}(x-\mu)\right.$
(3.9)
$\displaystyle+\left.\int\mathrm{d}y\left(\varrho(z,y)\mathsf{f}_{0}(x-y)+P^{2\chi-2}\frac{1}{2}\partial_{z}^{2}\varrho(z,y)\mathsf{f}_{\mathrm{h}}(x-y)\right)\right].$

Subleading contributions in $N$ and $P$ have been neglected. In the first
line, recall from (3.2) that $\zeta_{\alpha}(z)$ involves a factor $1/P^{2}$.
In order to reach an equilibrium configuration, at least two of the terms in
(3.9) must be of the same order in $P$ and compete. This requirement imposes

$2\chi-2=0\ \Longrightarrow\ \chi=1.$

Therefore $S_{\text{eff}}$, and hence $\mathcal{F}$, bear overall factors
$N^{2}P^{d-3}$.

####  3.2.6 Saddle point equation

At leading order in $N$ and $P$,

$\ln\mathcal{Z}_{\mathbb{S}^{d}}\approx\ln\int\mathrm{d}\sigma
e^{-S_{\text{eff}}}.$

From (3.9), the integrand is suppressed both in $N$ and $P$, meaning that the
leading order contribution comes from the saddle point configuration of
$S_{\text{eff}}$. We thus need to minimise it over the space of probability
densities $\varrho(z,x)$.

Taking the saddle point equation for (3.9) and acting onto it with
$\left(\frac{\partial\ }{\partial x}\right)^{d-2}$, we finally get

$\frac{1}{4}\partial_{x}^{2}\varrho(z,x)+\partial_{z}^{2}\varrho(z,x)+\sum_{\alpha=1}^{\mathscr{F}}P^{2}\zeta_{\alpha}(z)\delta(x-\mu_{\alpha})=0,$
(3.10)

both in 3d and 5d. The saddle point equation (3.10) is a Poisson equation, and
it is a modification of [83, Eq.(2.36)] by

$\zeta(z)\delta(x)\ \rightsquigarrow\
\sum_{\alpha=1}^{\mathscr{F}}\zeta_{\alpha}(z)\delta(x-\mu_{\alpha}),$

meaning that inserting mass deformations has a controlled effect on the long
quiver limit.

By linearity, (3.10) admits a solution

$\varrho(z,x)=\sum_{\alpha=1}^{\mathscr{F}}\varrho_{\alpha}(z,x),$ (3.11)

and, generalizing [70, 51], we get

$\varrho_{\alpha}(z,x)=\pi\sum_{k=1}^{\infty}k\mathfrak{a}_{\alpha,k}\sin(k\pi
z)e^{-2\pi k\lvert x-\mu_{\alpha}\rvert}.$ (3.12)

####  3.2.7 Match with the supergravity solution

We emphasise that the saddle point equation (3.10) agrees precisely with the
Poisson equation (2.10) derived in supergravity. To show this, we start with
the case of vanishing masses. We make the identifications

$z=\eta/P,\qquad x=\sigma/(2P)$ (3.13)

(here $\sigma$ is the holographic variable). It is convenient to switch to the
normalisation of [83], by introducing an eigenvalue density
$\hat{\varrho}(\eta,\sigma)=N\varrho(z,x)$ normalised to $N$. Analogously,
denote $F(\eta)=N\zeta(Pz)$ the flavour rank function without Veneziano
normalisation. Then, (3.10) becomes

$\partial_{\eta}^{2}\hat{\varrho}+\partial_{\sigma}^{2}\hat{\varrho}+2PF(\eta)\delta(\sigma)=0.$

Recall that $F(\eta)=\mathcal{R}^{\prime\prime}(\eta)$, in the conventions of
(2.3) for the ′-derivative, which yields opposite sign with respect to the
$\partial_{z}$-derivative. Dividing by $2P$ we get

$\partial_{\eta}^{2}\left(\frac{\hat{\varrho}}{2P}\right)+\partial_{\sigma}^{2}\left(\frac{\hat{\varrho}}{2P}\right)-\partial_{\eta}^{2}\mathcal{R}(\eta)\delta(\sigma)=0.$

Acting with the ′-derivative twice on the supergravity equation (2.10) and
identifying

$\hat{W}^{\prime\prime}=\frac{\hat{\varrho}}{2P},$ (3.14)

where the differentiation only involves the $\eta$-dependence, we find perfect
agreement with the saddle point equation derived in QFT.

The argument extends immediately away from the superconformal case. The
presence of masses simply splits the flavour and gauge rank functions, and we
identify the positions $\sigma_{\alpha}$ in supergravity with $2P\mu_{\alpha}$
in QFT. For instance, for $\mathscr{F}=2$, $\mu_{1}=0,\mu_{2}=\mu$, the
identifications above yield the supergravity equation (2.25) from the saddle
point equation (3.10).

It is instructive to rewrite the QFT effective action (3.9) in the
supergravity language, substituting (3.13)-(3.14). One finds:

$\displaystyle S_{\text{eff}}=P^{\chi(d-4)}N\int_{0}^{P}$
$\displaystyle\mathrm{d}\eta\int_{-\infty}^{+\infty}\mathrm{d}\sigma\partial_{\eta}^{2}\hat{W}(\sigma,\eta)\left\\{-P^{2\chi}\sum_{\alpha=1}^{\mathscr{F}}\partial_{\eta}^{2}\mathcal{R}(\eta)\frac{\mathsf{f}_{\text{h}}(\sigma-\sigma_{\alpha})}{(2P)^{\chi(d-2)}}\right.$
$\displaystyle+N\left.\int_{-\infty}^{+\infty}\mathrm{d}\tilde{\sigma}\left[\partial_{\eta}^{2}\hat{W}(\tilde{\sigma},\eta)\frac{\mathsf{f}_{0}(\sigma-\tilde{\sigma})}{(2P)^{\chi(d-4)}}+P^{2\chi}\frac{1}{2}\partial_{\eta}^{4}\hat{W}(\tilde{\sigma},\eta)\frac{\mathsf{f}_{\mathrm{h}}(x-y)}{(2P)^{\chi(d-2)}}\right]\right\\}.$

The powers of $P$ cancel out in this expression, regardless of the value of
$\chi$, while each power of $N$ accompanies a $\sigma$-integral. Simplifying
the expression, one arrives at

$\displaystyle S_{\text{eff}}=\frac{N}{2^{\chi(d-4)}}\int_{0}^{P}$
$\displaystyle\mathrm{d}\eta\int_{-\infty}^{+\infty}\mathrm{d}\sigma\partial_{\eta}^{2}\hat{W}(\sigma,\eta)\cdot\partial_{\eta}^{2}\left\\{-\sum_{\alpha=1}^{\mathscr{F}}\mathcal{R}(\eta)\cdot\frac{1}{2^{2\chi}}\mathsf{f}_{\text{h}}(\sigma-\sigma_{\alpha})\right.$
$\displaystyle+N\left.\int_{-\infty}^{+\infty}\mathrm{d}\tilde{\sigma}\left[\hat{W}(\tilde{\sigma},\eta)\mathsf{f}_{0}(\sigma-\tilde{\sigma})+\partial_{\eta}^{2}\hat{W}(\tilde{\sigma},\eta)\cdot\frac{1}{2^{2\chi+1}}\mathsf{f}_{\mathrm{h}}(x-y)\right]\right\\}.$

###  3.3 Free energies in massive deformations of 5d and 3d SCFTs

In this subsection, we compute the free energy for arbitrary $\mathscr{F}$ and
present the general solution in 5d and 3d. The SCFT result of [83, 85] is
recovered upon setting all masses to zero.

####  3.3.1 Free energies in 5d and 3d

The free energy $\mathcal{F}$ is defined in (3.1). It is computed at leading
order in $N,P$ plugging our solution (3.11)-(3.12) into $S_{\text{eff}}$ and
evaluating it on-shell.

The derivation is identical to [83, Sec.3] and [85, Sec.3], upon adjusting to
our conventions and replacing

$\displaystyle\varrho(z,x)\ $ $\displaystyle\rightsquigarrow\
\sum_{\alpha=1}^{\mathscr{F}}\varrho_{\alpha}(z,x)$
$\displaystyle\zeta(z)\mathsf{f}_{\mathrm{h}}(x)\ $
$\displaystyle\rightsquigarrow\
\sum_{\alpha=1}^{\mathscr{F}}\zeta_{\alpha}(z)\mathsf{f}_{\mathrm{h}}(x-\mu_{\alpha}).$

Using this and (3.9), we arrive at the result

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{(-1)^{\frac{d-3}{2}}}{2}N^{2}P^{d-3}\int_{0}^{1}\mathrm{d}z\int_{-\infty}^{+\infty}\mathrm{d}x\left(\sum_{\alpha=1}^{\mathscr{F}}\varrho_{\alpha}(z,x)\right)\left(P^{2}\sum_{\beta=1}^{\mathscr{F}}\zeta_{\beta}(z)\mathsf{f}_{\mathrm{h}}(x-\mu_{\beta})\right)$
$\displaystyle=\frac{(-1)^{\frac{d-3}{2}}}{2}N^{2}P^{d-3}\sum_{\alpha,\beta=1}^{\mathscr{F}}\int_{0}^{1}\mathrm{d}zP^{2}\zeta_{\beta}(z)\int_{-\infty}^{+\infty}\mathrm{d}x\varrho_{\alpha}(z,x)\mathsf{f}_{\mathrm{h}}(x-\mu_{\beta}).$
(3.15)

Importantly, by a straightforward generalization of the derivation in [83,
Sec.3] and denoting $S_{\text{fund}}$ the contribution of the fundamental
hypermultiplets to the action, we arrive at the identity

$S_{\text{eff}}\Big{|}_{\text{on
shell}}=\left.\frac{1}{2}S_{\text{fund}}\right\rvert_{\text{on shell}},$

which is in fact a general property of matrix models in the planar limit,
following from the equilibrium equation (see e.g. [113]).

Formula (3.15) yields the leading contribution to the free energy. One may
also use the eigenvalue densities (3.12) to evaluate the subleading
contributions, such as the boundary terms
$\left.\mathfrak{B}\right\rvert_{\text{on shell}}$.

Separating the cases $\alpha=\beta$ and $\alpha\neq\beta$, and shifting
variables, we rewrite (3.15) in the form

$\mathcal{F}=\sum_{\alpha=1}^{\mathscr{F}}\mathcal{F}_{\alpha}+\sum_{\alpha\neq\beta}\mathcal{F}^{\text{int}}_{\alpha,\beta}.$
(3.16)

Here $\mathcal{F}_{\alpha}$ is the free energy of a 5d or 3d balanced, linear
quiver $\mathcal{Q}_{\alpha}$ with reduced gauge rank function
$\nu_{\alpha}(z)$ and flavour rank function $\zeta_{\alpha}(z)$. From the
explicit form of $\varrho_{\alpha}$ in (3.12) and $\zeta_{\alpha}$ in (3.2),
each individual contribution reads

$\displaystyle\frac{(-1)^{\frac{d-3}{2}}}{N^{2}P^{d-3}}\mathcal{F}_{\alpha}$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\mathrm{d}z\left(\pi^{2}\sum_{\ell=1}^{\infty}\mathfrak{a}_{\alpha,\ell}\ell^{2}\sin\left(\ell\pi
z\right)\right)$ (3.17a) $\displaystyle\hskip
28.45274pt\int_{-\infty}^{+\infty}\mathrm{d}x\left(\pi\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}k\sin(k\pi
z)e^{-2\pi k\lvert
x-\mu_{\alpha}\rvert}\right)\mathsf{f}_{\mathrm{h}}(x-\mu_{\alpha})$
$\displaystyle=\frac{\pi^{3}}{2}\sum_{\ell,k=1}^{\infty}\mathfrak{a}_{\beta,\ell}\mathfrak{a}_{\alpha,k}k\ell^{2}\int_{0}^{1}\mathrm{d}z\sin\left(\ell\pi
z\right)\sin(k\pi z)\int_{-\infty}^{+\infty}\mathrm{d}xe^{-2\pi k\lvert
x\rvert}\mathsf{f}_{\text{h}}(x)$ (3.17b)
$\displaystyle=\frac{\pi^{3}}{4}\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}^{2}k^{3}\int_{-\infty}^{+\infty}\mathrm{d}xe^{-2\pi
k\lvert x\rvert}\mathsf{f}_{\text{h}}(x).$ (3.17c)

The terms in the second sum in (3.16) encode the pairwise coupling of the
quivers $\mathcal{Q}_{\alpha}$ and $\mathcal{Q}_{\beta}$. Its appearance
encapsulates the main technical novelty of the present analysis. Note that the
quivers $\mathcal{Q}_{\alpha}$ only interact pairwise, and the free energy
configuration may be drawn as a complete graph of $\mathscr{F}$ vertices, with
vertex set $\left\\{\mathcal{Q}_{\alpha}\right\\}$. For example:

$\mathcal{Q}_{1}$$\mathcal{Q}_{2}$$\mathcal{Q}_{3}$$\mathcal{Q}_{4}$$\mathcal{Q}_{5}$$\mathcal{Q}_{\alpha}$$\mathcal{F}_{\alpha}=\mathcal{F}^{\mathrm{int}}_{\alpha,\alpha}\
:\
$$\mathcal{Q}_{\alpha}$$\mathcal{Q}_{\beta}$$\mathcal{F}^{\mathrm{int}}_{\alpha,\beta}\
:\ $

From the Type IIB string theory viewpoint, the arrows in this picture encode
the free energy of strings stretched between stacks of D$d$ colour branes.
When the stacks are pairwise separated, the string excitations become massive.
An example is provided in Figure 9 in Appendix C.1.

To evaluate $\mathcal{F}^{\text{int}}_{\alpha,\beta}$, exploiting again (3.12)
and (3.2), we have

$\displaystyle\frac{(-1)^{\frac{d-3}{2}}}{N^{2}P^{d-3}}\mathcal{F}^{\text{int}}_{\alpha,\beta}$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\mathrm{d}z\left(\pi^{2}\sum_{\ell=1}^{\infty}\mathfrak{a}_{\beta,\ell}\ell^{2}\sin\left(\ell\pi
z\right)\right)$ (3.18a) $\displaystyle\hskip
28.45274pt\int_{-\infty}^{+\infty}\mathrm{d}x\left(\pi\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}k\sin(k\pi
z)e^{-2\pi k\lvert
x-\mu_{\alpha}\rvert}\right)\mathsf{f}_{\mathrm{h}}(x-\mu_{\beta})$
$\displaystyle=\frac{\pi^{3}}{2}\sum_{\ell,k=1}^{\infty}\mathfrak{a}_{\beta,\ell}\mathfrak{a}_{\alpha,k}k\ell^{2}\int_{0}^{1}\mathrm{d}z\sin\left(\ell\pi
z\right)\sin(k\pi z)\int_{-\infty}^{+\infty}\mathrm{d}xe^{-2\pi k\lvert
x\rvert}\mathsf{f}_{\text{h}}(x-\mu_{\beta\alpha})$ (3.18b)
$\displaystyle=\frac{\pi^{3}}{4}\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}\mathfrak{a}_{\beta,k}k^{3}\mathcal{I}_{n}(\mu_{\beta\alpha}).$
(3.18c)

In (3.18b) we have introduced the notation
$\mu_{\beta\alpha}:=\mu_{\beta}-\mu_{\alpha}$, and we have defined

$\mathcal{I}_{k}(\mu):=\int_{-\infty}^{+\infty}\mathrm{d}x\ e^{-2\pi k\lvert
x\rvert}\mathsf{f}_{\mathrm{h}}(x-\mu)$ (3.19)

in (3.18c), whose explicit form differs in 5d and 3d.

From (3.16)-(3.18c) we already learn an important lesson: because the
coefficients $\mathfrak{a}_{k}$ grow linearly in $P$, we predict the scaling:

$\mathcal{F}\propto(-1)^{\frac{d-1}{2}}N^{2}P^{d-1},$

which interpolates between and agrees with the scaling found in 5d [83] and 3d
[85]. This, however, will be the behaviour of a quiver with a _generic_
distribution of flavour ranks. Concrete examples, as for instance the $T_{N}$
theory, will show different scaling — see also Appendix B.

####  3.3.2 Free energy in 5d

To determine the free energy in 5d, it only remains to evaluate the integral
(3.19). Plugging $\mathsf{f}_{\mathrm{h}}(x)|_{d=5}$ from (3.8), we get

$\displaystyle\left.\mathcal{I}_{k}(\mu)\right\rvert_{d=5}$
$\displaystyle=-\frac{9\pi}{2}\int_{-\infty}^{+\infty}\mathrm{d}xe^{-2\pi
k\lvert x\rvert}\lvert x-\mu\rvert^{3}$
$\displaystyle=-\frac{27}{8\pi^{3}k^{4}}\left(e^{-2\pi k\lvert\mu\rvert}+2\pi
k\lvert\mu\rvert+\frac{4}{3}(\pi k\lvert\mu\rvert)^{3}\right).$ (3.20)

Inserting this expression in the general formula (3.15) yields the final
result

$\displaystyle\left.\mathcal{F}\right\rvert_{d=5}$
$\displaystyle=\sum_{\alpha=1}^{\mathscr{F}}\underbrace{\frac{27}{32}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{\alpha,k}^{2}}_{\left.\mathcal{F}_{\alpha}\right\rvert_{d=5}}$
(3.21)
$\displaystyle+2\sum_{1\leq\alpha<\beta\leq\mathscr{F}}\underbrace{\frac{27}{32}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{\alpha,k}\mathfrak{a}_{\beta,k}\left(e^{-2\pi
k\lvert\mu_{\beta\alpha}\rvert}+2\pi
k\lvert\mu_{\beta\alpha}\rvert+\frac{4}{3}(\pi
k\lvert\mu_{\beta\alpha}\rvert)^{3}\right)}_{\left.\mathcal{F}^{\text{int}}_{\alpha,\beta}\right\rvert_{d=5}}.$

####  3.3.3 Free energy in 3d

In 3d, the integral (3.19) is simpler. Recall from (3.8) that
$\mathsf{f}_{\mathrm{h}}(x)=\pi\lvert x\rvert$. Then

$\displaystyle\left.\mathcal{I}_{k}(\mu)\right\rvert_{d=3}$
$\displaystyle=\pi\int_{-\infty}^{+\infty}\mathrm{d}xe^{-2\pi k\lvert
x\rvert}\lvert x-\mu\rvert$ $\displaystyle=\frac{1}{2\pi k^{2}}\left(e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert\right).$

Plugging this expression in formula (3.15) we find the general result

$\displaystyle\left.\mathcal{F}\right\rvert_{d=3}$
$\displaystyle=\sum_{\alpha=1}^{\mathscr{F}}\underbrace{\frac{\pi^{2}}{8}N^{2}\sum_{k=1}^{\infty}k\mathfrak{a}_{\alpha,k}^{2}}_{\left.\mathcal{F}_{\alpha}\right\rvert_{d=3}}$
(3.22)
$\displaystyle+2\sum_{1\leq\alpha<\beta\leq\mathscr{F}}\underbrace{\frac{\pi^{2}}{8}N^{2}\sum_{k=1}^{\infty}k\mathfrak{a}_{\alpha,k}\mathfrak{a}_{\beta,k}\left(e^{-2\pi
k\lvert\mu_{\beta\alpha}\rvert}+2\pi
k\lvert\mu_{\beta\alpha}\rvert\right)}_{\left.\mathcal{F}^{\text{int}}_{\alpha,\beta}\right\rvert_{d=3}}.$

####  3.3.4 Holographic match in 5d and 3d SCFTs

Let us take the SCFT limit of our result, with $\mu_{\alpha}\to 0$, and show
the agreement with the supergravity result of Section 2.4. We do it first in
5d, by setting all masses to vanish in (3.21). The polynomial terms in
$\lvert\mu\rvert$ vanish, and the exponential dependence disappears.
Altogether, we have:

$\displaystyle\lim_{\vec{\mu}\to 0}\left.\mathcal{F}\right\rvert_{d=5}$
$\displaystyle=\frac{27}{32}N^{2}P^{2}\left[\sum_{\alpha=1}^{\mathscr{F}}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{\alpha,k}^{2}+\sum_{1\leq\alpha<\beta\leq\mathscr{F}}\sum_{k=1}^{\infty}\frac{2}{k}\mathfrak{a}_{\alpha,k}\mathfrak{a}_{\beta,k}\right]$
$\displaystyle=\frac{27}{32}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\left(\sum_{\alpha=1}^{\mathscr{F}}\mathfrak{a}_{\alpha,k}\right)^{2}$
$\displaystyle=\frac{27}{32}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{k}^{2}$

which is the free energy of the quiver with “reduced” rank function is
$\nu(z)$, whose Fourier coefficients are $\mathfrak{a}_{k}=\frac{1}{N}R_{k}$.
Comparing with (2.20), we obtain the relation:

$d=5\ :\ \qquad\qquad\mathcal{F}=\frac{27\pi^{6}}{4}c_{hol},$ (3.23)

We find agreement between the free energy and the holographic central charge,
up to an arbitrary numerical factor in the definition of $c_{hol}$. The
overall coefficient of $27$ is recurring in 5d, see for instance [114,
Eq.(5.11)], and appears to be due to the universal relation
$\mathcal{F}=\omega^{3}\mathcal{F}_{\text{univ.}}$. Here $\omega$ is related
to the equivariant parameters on a squashed $\mathbb{S}^{5}$, with $\omega=3$
on the round sphere.999$\mathcal{F}$ in (3.1) equals
$\mathcal{F}_{\vec{\omega}}$ of [83, Eq.(2.23)]. In [70], $c_{hol}$ was
compared with $\mathcal{F}_{\text{univ.}}=3^{-3}\mathcal{F}_{\vec{\omega}}$
read off from of [83]. Accounting for the factor $\omega^{3}=27$, our eq.
(3.23) agrees with [70, Eq.(3.9)].

In 3d, turning off all mass deformations in (3.22), the linear terms in
$\mathcal{F}^{\text{int}}_{\alpha,\beta}$ vanish, and the exponential terms go
to $1$. We get:

$\displaystyle\lim_{\vec{\mu}\to 0}\left.\mathcal{F}\right\rvert_{d=3}$
$\displaystyle=\frac{\pi^{2}}{8}N^{2}\left[\sum_{\alpha=1}^{\mathscr{F}}\sum_{k=1}^{\infty}k\mathfrak{a}_{\alpha,k}^{2}+\sum_{1\leq\alpha<\beta\leq\mathscr{F}}\sum_{k=1}^{\infty}2k\mathfrak{a}_{\alpha,k}\mathfrak{a}_{\beta,k}\right]$
$\displaystyle=\frac{\pi^{2}}{8}N^{2}\sum_{k=1}^{\infty}k\left(\sum_{\alpha=1}^{\mathscr{F}}\mathfrak{a}_{\alpha,k}\right)^{2}$
$\displaystyle=\frac{\pi^{2}}{8}N^{2}\sum_{k=1}^{\infty}k\mathfrak{a}_{k}^{2},$

thus recovering the reduced rank function $\nu(z)$, akin to the 5d case.
Comparing with (2.23), we find:

$d=3\ :\ \qquad\qquad\mathcal{F}=\frac{4}{\pi}c_{hol},$ (3.24)

so that the free energy matches the holographic central charge, up to an
overall normalisation.

###  3.4 Two interacting linear quivers

In this subsection we discuss the case $\mathscr{F}=2$ and relate it to the
holographic result of Subsection 2.5. Without loss of generality, we set
$m_{1}=0,m_{2}=m$.

The final result for the free energy can be read off from the general
procedure of Subsection 3.2. However, we will rederive it using a different
approach. We will reformulate the large $N$ limit in a way which probes the
two singularities of the Coulomb branch of $\mathcal{Q}$. The analysis of
Subsection 3.1, and especially the factorisation of the Coulomb branch into
two reduced branches plus decoupled massive zero-modes, will be manifest in
the ensuing procedure.

Despite detailing it only for the case of two ranks functions, the procedure
we show below extends to an arbitrary number $\mathscr{F}$ of mass scales and
rank functions.

####  3.4.1 Change of variables to probe the singularities

The data of the problem include the flavour rank functions

$\zeta_{\alpha}(z)=\frac{\pi^{2}}{P^{2}}\sum_{k=1}^{\infty}\mathfrak{a}_{\alpha,k}k^{2}\sin(k\pi
z),\qquad\alpha=1,2,$

from which one extracts the reduced gauge rank functions $\nu_{\alpha}(z)$, as
in (3.3). We then define the gauge ranks $N_{\alpha,j}=N\nu_{\alpha}(j/P)$,
$\alpha=1,2$. By construction, $N_{1,j}+N_{2,j}=N_{j}$. To probe the Coulomb
branch singularities at $0$ and $m$, we leave the variables

$\displaystyle\sigma_{a,j},\quad j=1,\dots,P-1,\ a=1,\dots,N_{1,j}$

untouched, and define the new variables

$\phi_{\dot{a},j}=\sigma_{\dot{a}+N_{1,j},j}-m,\quad j=1,\dots,P-1,\
\dot{a}=1,\dots,N_{2,j}.$

Undotted indices run over the gauge ranks $N_{1,j}$ while dotted indices run
over the gauge ranks $N_{2,j}$. With these new variables, the contributions to
the sphere partition function become:

  * •

from the vector multiplet,

$\displaystyle Z_{\text{1-loop}}^{\text{vec}}$
$\displaystyle=\prod_{j=1}^{P-1}\left[\prod_{1\leq a\neq b\leq
N_{1,j}}\exp\left\\{-\mathsf{v}(\sigma_{a,j}-\sigma_{b,j})\right\\}\right]\times\left[\prod_{1\leq\dot{a}\neq\dot{b}\leq
N_{2,j}}\exp\left\\{-\mathsf{v}(\phi_{\dot{a},j}-\phi_{\dot{b},j})\right\\}\right]$
$\displaystyle\times\left[\prod_{a=1}^{N_{j}}\prod_{\dot{b}=1}^{N_{j}}\exp\left\\{-\mathsf{v}(\sigma_{a,j}-\phi_{\dot{b},j}+m)\right\\}\right];$

  * •

from the fundamental hypermultiplets

$\displaystyle Z^{\text{fund}}$
$\displaystyle=\prod_{j=1}^{P-1}\left[\prod_{a=1}^{N_{1,j}}\exp\left\\{-F_{1,j}\mathsf{h}(\sigma_{a,j})\right\\}\right]\times\left[\prod_{\dot{a}=1}^{N_{2,j}}\exp\left\\{-F_{2,j}\mathsf{h}(\phi_{\dot{a},j})\right\\}\right]$
$\displaystyle\times\left[\prod_{a=1}^{N_{1,j}}\exp\left\\{-F_{2,j}\mathsf{h}(\sigma_{a,j}-m)\right\\}\right]\times\left[\prod_{\dot{a}=1}^{N_{2,j}}\exp\left\\{-F_{1,j}\mathsf{h}(\phi_{\dot{a},j}+m)\right\\}\right];$

and from the bifundamental hypermultiplets,

$\displaystyle Z^{\text{bif}}$
$\displaystyle=\prod_{j=1}^{P-2}\left[\prod_{a=1}^{N_{1,j}}\prod_{b=1}^{N_{1,j+1}}\exp\left\\{-\mathsf{h}(\sigma_{a,j}-\sigma_{b,j+1})\right\\}\right]\times\left[\prod_{\dot{a}=1}^{N_{2,j}}\prod_{\dot{b}=1}^{N_{2,j+1}}\exp\left\\{-\mathsf{h}(\phi_{\dot{a},j}-\phi_{\dot{b},j+1})\right\\}\right]$
$\displaystyle\times\left[\prod_{a=1}^{N_{1,j}}\prod_{\dot{b}=1}^{N_{2,j+1}}\exp\left\\{-\mathsf{h}(\sigma_{a,j}-\phi_{\dot{b},j+1}+m)\right\\}\right]\times\left[\prod_{\dot{a}=1}^{N_{2,j}}\prod_{b=1}^{N_{1,j+1}}\exp\left\\{-\mathsf{h}(\phi_{\dot{a},j}-\sigma_{b,j+1}-m)\right\\}\right].$

In the new variables, we readily find that the effective action splits
according to:

$S_{\text{eff}}=S_{\text{eff},1}+S_{\text{eff},2}+S_{\text{int}}(m),$

with $S_{\text{eff},\alpha}$ the effective action for the superconformal
quiver $\mathcal{Q}_{\alpha}$. Each such effective action does not depend on
the mass and is of the form analyzed in [83, 85]. The dependence on $m$ is
entirely captured by the interaction among the two quivers in
$S_{\text{int}}$.

The Hanany–Witten brane setup is easily read off from the sphere partition in
the adjusted variables. An example is drawn in Figure 9, in Appendix C.1.

####  3.4.2 Change of variables, Coulomb branches and holography

The variables $\vec{\sigma},\vec{\phi}$ are adjusted coordinates for the
Coulomb branches

  * $\vec{\sigma}:\ $

CB$[\mathcal{Q}_{1}]$ of the quiver $\mathcal{Q}_{1}$, with reduced gauge and
flavour rank functions $\nu_{1}$ and $\zeta_{1}$, and

  * $\vec{\phi}:\ $

CB$[\mathcal{Q}_{2}]$ of the quiver $\mathcal{Q}_{2}$, with reduced gauge and
flavour rank functions $\nu_{2}$ and $\zeta_{2}$.

In particular, the singularity expected at the origin of the Coulomb branch of
$\mathcal{Q}_{2}$ is indeed located at $\vec{\phi}=0$. See [88] for related
analysis.

This identification however is not exact at finite $m$. We have learnt from
Subsection 3.1.2 that there is a region, between the two conical singularities
of CB$[\mathcal{Q}_{1}]$ and CB$[\mathcal{Q}_{2}]$, in which a more
complicated geometry arises. The matrix model neatly sees this effect after
the change of variables. For $\sigma_{a,j}<0$ and $\phi_{\dot{a},j}>0$ there
are no massless modes and the approximate description as two disjoint Coulomb
branches is accurate. However, in the region $0<\sigma_{a,j}<m$ and
$-m<\phi_{\dot{a},j}<0$ there are additional massless states. These are read
off from the zeros of the argument in the functions $\mathsf{v},\mathsf{h}$.
In the limits $m\to 0$ and $m\to\infty$ we recover the conformal situation,
with massless states only at the origin.

This matrix model analysis is in perfect agreement with the supergravity
discussion of Subsection 2.7. It was noted there that the region where the
AdSd+1 solution is not reliable is precisely $0<\sigma<\sigma_{0}$.

####  3.4.3 Long quiver limit after change of variables

We now apply the large $N$ and large $P$ limits to the quiver, after the
change of variables. We define the eigenvalue densities

$\displaystyle\rho_{1,j}(\sigma)=\frac{1}{N}\sum_{a=1}^{N_{1,j}}\delta(\sigma-\sigma_{a,j})$
$\displaystyle\rho_{2,j}(\phi)=\frac{1}{N}\sum_{\dot{a}=1}^{N_{2,j}}\delta(\phi-\phi_{\dot{a},j}),$

corresponding to the two quivers $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$,
whose adjusted variables are $\sigma$ and $\phi$ respectively. From the
reasoning of Subsection 3.2, we introduce the scaled variables

$\sigma=(d-2)Px,\qquad\phi=(d-2)Py,\qquad m=(d-2)P\mu$

and the scaled eigenvalue densities $\varrho_{1}(z,x),\varrho_{2}(z,y)$.
Mimicking the long quiver limit of Subsection 3.2, now with two distinct
eigenvalue densities, the terms $S_{\text{eff},\alpha}$ are standard, while
for the interaction term we find

$S_{\text{int}}=S_{0}+S_{\text{der}}+S_{\text{fund},1}+S_{\text{fund},2},$
(3.25)

where

$\displaystyle S_{0}$
$\displaystyle=2P^{(d-3)}N^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\varrho_{1}(z,x)\int\mathrm{d}y\varrho_{2}(z,y)\mathsf{f}_{0}(x-y-\mu)$
$\displaystyle S_{\text{der}}$
$\displaystyle=\frac{1}{2}P^{(d-3)}N^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\int\mathrm{d}y\left[\varrho_{1}(z,x)\partial_{z}^{2}\varrho_{2}(z,y)+\varrho_{2}(z,y)\partial_{z}^{2}\varrho_{1}(z,x)\right]\mathsf{f}_{\mathrm{h}}(x-y-\mu)$
$\displaystyle S_{\text{fund},1}$
$\displaystyle=N^{2}P^{(d-1)}\int_{0}^{1}\mathrm{d}z\zeta_{2}(z)\int\mathrm{d}x\varrho_{1}(z,x)\mathsf{f}_{\mathrm{h}}(x-\mu)$
$\displaystyle S_{\text{fund},2}$
$\displaystyle=N^{2}P^{(d-1)}\int_{0}^{1}\mathrm{d}z\zeta_{1}(z)\int\mathrm{d}y\varrho_{2}(z,y)\mathsf{f}_{\mathrm{h}}(y+\mu).$

To solve the equilibrium problem, we must minimise the resulting effective
action, now with respect to the two sets of variables $\vec{\sigma}$ and
$\vec{\phi}$. We first minimise and then act on the two resulting equations
with $\frac{\partial^{d-2}\ }{\partial x^{d-2}}$ and $\frac{\partial^{d-2}\
}{\partial y^{d-2}}$, respectively. In this way, we arrive at the pair of
saddle point equations:

$\displaystyle\frac{1}{4}\partial_{x}^{2}\varrho_{1}(z,x)+\partial_{z}^{2}\varrho_{1}(z,x)+P^{2}\zeta_{1}(z)\delta(x)+\frac{1}{4}\partial_{x}^{2}\varrho_{2}(z,x-\mu)+\partial_{z}^{2}\varrho_{2}(z,x-\mu)+P^{2}\zeta_{2}(z)\delta(x-\mu)=0,$
(3.26a)
$\displaystyle\frac{1}{4}\partial_{y}^{2}\varrho_{1}(z,y)+\partial_{z}^{2}\varrho_{1}(z,y)+P^{2}\zeta_{2}(z)\delta(y)+\frac{1}{4}\partial_{x}^{2}\varrho_{1}(z,y+\mu)+\partial_{z}^{2}\varrho_{1}(z,y+\mu)+P^{2}\zeta_{1}(z)\delta(y+\mu)=0.$
(3.26b)

The pair of saddle point equations (3.26) has several expected properties.
First, (3.26a) and (3.26b) are related through exchanging the labels
$1\leftrightarrow 2$ and replacing $x=y-\mu$. Second, if we define the
eigenvalue density $\varrho(z,x)=\varrho_{1}(z,x)+\varrho_{2}(z,x-\mu)$,
(3.26a) becomes the saddle point equation (3.10) derived without change of
variables. Third, (3.26a) is the sum of a term which would yield the saddle
point equation for $\mathcal{Q}_{1}$ alone, plus an extra term coming from
$S_{\text{int}}$ and involving $\varrho_{2}(z,x-\mu)$, in which the
$\mu$-dependence is entirely contained (and likewise for (3.26b) and
$\mathcal{Q}_{2}$).

The solution to (3.26) is again given by (3.12).

####  3.4.4 Free energy for two interacting linear quivers

Equipped with the eigenvalue densities $\varrho_{1}(z,x),\varrho_{2}(z,y)$ we
can compute the free energy at leading order in $N$ and $P$ through

$\mathcal{F}=\left.(-1)^{\frac{d-3}{2}}\left(S_{\text{eff},1}+S_{\text{eff},2}+S_{\text{int}}\right)\right\rvert_{\text{on
shell}}.$

The terms $S_{\text{eff},\alpha}$ simply contribute a factor
$\pm\mathcal{F}_{\alpha}$, that is, the free energy for the linear quiver
$\mathcal{Q}_{\alpha}$. The novel contributions are:

$\displaystyle S_{\text{int}}\Big{|}_{\text{on shell}}$
$\displaystyle=\left(S_{0}+S_{\text{der}}+S_{\text{fund},1}+S_{\text{fund},2}\right)\Big{|}_{\text{on
shell}}$
$\displaystyle=:(-1)^{\frac{d-3}{2}}\left[\mathcal{F}_{0}+\mathcal{F}_{\text{der}}+\mathcal{F}_{\text{fund},1}+\mathcal{F}_{\text{fund},2}\right],$

with the first equality due to (3.25). The fundamental hypermultiplets
contribute:

$\displaystyle\frac{(-1)^{\frac{d-3}{2}}}{N^{2}P^{(d-3)}}\mathcal{F}_{\text{fund},1}$
$\displaystyle=\int_{0}^{1}\mathrm{d}zP^{2}\zeta_{2}(z)\mathrm{d}x\varrho_{1}(z,x)\mathsf{f}_{\mathrm{h}}(x-\mu)$
$\displaystyle=\int_{0}^{1}\mathrm{d}z\left(\pi^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{2,k}k^{2}\sin(k\pi
z)\right)\int\mathrm{d}x\left(\pi\sum_{\ell=1}^{\infty}\mathfrak{a}_{1,\ell}\ell\sin(\ell\pi
z)e^{-2\pi\ell\lvert x\rvert}\right)\mathsf{f}_{\mathrm{h}}(x-\mu)$
$\displaystyle=\frac{\pi^{3}}{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k^{3}\mathcal{I}_{k}(\mu),$

with the integral $\mathcal{I}_{k}(\mu)$ defined in (3.19). The contribution
from $\mathcal{F}_{\text{fund},2}$ is identical, due to the symmetries
$1\leftrightarrow 2$ and $\mu\leftrightarrow-\mu$ in the last line. Therefore

$(-1)^{\frac{d-3}{2}}\sum_{\alpha=1}^{2}\mathcal{F}_{\text{fund},\alpha}=N^{2}P^{(d-3)}\pi^{3}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k^{3}\mathcal{I}_{k}(\mu).$

The next contribution is

$\displaystyle\frac{(-1)^{\frac{d-3}{2}}}{N^{2}P^{(d-3)}}\mathcal{F}_{0}$
$\displaystyle=2\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\varrho_{1}(z,x)\int\mathrm{d}y\varrho_{2}(z,y)\mathsf{f}_{0}(x-y-\mu)$
$\displaystyle=2\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\left(\pi\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}k\sin(k\pi
z)e^{-2\pi k\lvert x\rvert}\right)$ $\displaystyle\hskip
42.67912pt\times\int\mathrm{d}y\left(\pi\sum_{\ell=1}^{\infty}\mathfrak{a}_{2,\ell}\ell\sin(\ell\pi
z)e^{-2\pi\ell\lvert y\rvert}\right)\mathsf{f}_{0}(x-y-\mu)$
$\displaystyle=\pi^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k^{2}\hat{\mathcal{I}}^{(0)}_{k}(\mu),$

where in the last line we have defined

$\hat{\mathcal{I}}^{(0)}_{k}(\mu):=\int_{-\infty}^{+\infty}\mathrm{d}x\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\
e^{-2\pi k\left(\lvert x\rvert+\lvert y\rvert\right)}\mathsf{f}_{0}(x-y-\mu).$
(3.27)

Likewise, for the last contribution to the free energy we get

$\displaystyle\frac{(-1)^{\frac{d-3}{2}}}{N^{2}P^{(d-3)}}\mathcal{F}_{\text{der}}$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\int\mathrm{d}y\left[\varrho_{1}(z,x)\partial_{z}^{2}\varrho_{2}(z,y)+\varrho_{2}(z,y)\partial_{z}^{2}\varrho_{1}(z,x)\right]\mathsf{f}_{\mathrm{h}}(x-y-\mu)$
$\displaystyle=\frac{\pi^{2}}{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\int\mathrm{d}y\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}k\sin(k\pi
z)e^{-2\pi k\lvert x\rvert}$ $\displaystyle\hskip
56.9055pt\times\sum_{\ell=1}^{\infty}\mathfrak{a}_{2,\ell}\ell\sin(\ell\pi
z)e^{-2\pi\ell\lvert
y\rvert}\left[-\pi^{2}\ell^{2}-\pi^{2}k^{2}\right]\mathsf{f}_{\mathrm{h}}(x-y-\mu)$
$\displaystyle=-\frac{\pi^{4}}{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k^{4}\hat{\mathcal{I}}^{(\text{der})}_{k}(\mu),$

with

$\hat{\mathcal{I}}^{(\text{der})}_{k}(\mu):=\int_{-\infty}^{+\infty}\mathrm{d}x\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\
e^{-2\pi k\left(\lvert x\rvert+\lvert
y\rvert\right)}\mathsf{f}_{\mathrm{h}}(x-y-\mu).$ (3.28)

The final answer for the free energy is obtained solving the integrals (3.19),
(3.27) and (3.28), whose explicit form depends on the dimension $d$, through
the functions (3.8).

####  3.4.5 Free energy for two interacting linear quivers in 5d

$\mathcal{I}_{k}(\mu)$ in 5d has already been computed in (3.20):

$\left.\mathcal{I}_{k}(\mu)\right\rvert_{d=5}=-\frac{27}{8\pi^{3}k^{4}}\left(e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert+\frac{1}{6}(2\pi
k\lvert\mu\rvert)^{3}\right).$

Using (3.8) at $d=5$, the integrals
$\hat{\mathcal{I}}^{(0)}_{k}(\mu),\hat{\mathcal{I}}^{(\text{der})}_{k}(\mu)$
are evaluated as:

$\displaystyle\hat{\mathcal{I}}^{(0)}_{k}(\mu)$
$\displaystyle=-\frac{27}{8}\pi\int_{-\infty}^{+\infty}\mathrm{d}x\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\
e^{-2\pi k(\lvert x\rvert+\lvert y\rvert)}\lvert x-y-\mu\rvert$
$\displaystyle=-\frac{27}{32\pi^{2}k^{3}}\left[3e^{-2\pi
k\lvert\mu\rvert}+(2\pi k\lvert\mu\rvert)\left(2+e^{-2\pi
k\lvert\mu\rvert}\right)\right]$

and

$\displaystyle\hat{\mathcal{I}}^{(\text{der})}_{k}(\mu)$
$\displaystyle=-\frac{9}{2}\pi\int_{-\infty}^{+\infty}\mathrm{d}x\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\
e^{-2\pi k(\lvert x\rvert+\lvert y\rvert)}\lvert x-y-\mu\rvert^{3}$
$\displaystyle=-\frac{27}{16\pi^{4}k^{5}}\left[5e^{-2\pi
k\lvert\mu\rvert}+(2\pi k\lvert\mu\rvert)\left(4+e^{-2\pi
k\lvert\mu\rvert}\right)+\frac{1}{3}\cdot(2\pi k\lvert\mu\rvert)^{3}\right].$

Plugging these expressions in the formula for $\mathcal{F}$, we observe from
direct computation that they satisfy

$\mathcal{F}_{\text{fund},1}+\mathcal{F}_{\text{fund},2}+\mathcal{F}_{0}+\mathcal{F}_{\text{der}}=-\frac{1}{2}\left[\mathcal{F}_{\text{fund},1}+\mathcal{F}_{\text{fund},2}\right].$

The latter is indeed a generic property, proved relying on the saddle point
condition, which was utilised in (3.15). We thus compute

$\displaystyle\left.\left[\mathcal{F}_{\text{fund},1}+\mathcal{F}_{\text{fund},2}\right]\right\rvert_{d=5}$
$\displaystyle=-\pi^{3}N^{2}P^{2}\left.\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k^{3}\mathcal{I}_{k}(\mu)\right\rvert_{d=5}$
$\displaystyle=\frac{27}{8}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}\left(e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert+\frac{4}{3}(\pi
k\lvert\mu\rvert)^{3}\right).$

The final expression for the free energy is

$\mathcal{F}=\mathcal{F}_{1}+\mathcal{F}_{2}+\frac{27}{16}N^{2}P^{2}\sum_{k=1}^{\infty}\frac{1}{k}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}\left(e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert+\frac{4}{3}(\pi
k\lvert\mu\rvert)^{3}\right),$ (3.29)

in agreement with (3.21).

####  3.4.6 Free energy for two interacting linear quivers in 3d

In 3d, we have

$\displaystyle\left.\mathcal{I}_{k}(\mu)\right\rvert_{d=3}=\frac{1}{2\pi
k^{2}}\left(e^{-2\pi k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert\right)$

and

$\displaystyle\left.\hat{\mathcal{I}}^{(0)}_{k}(\mu)\right\rvert_{d=3}$
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{d}x\leavevmode\nobreak\
e^{-2\pi k\lvert
x\rvert}\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\ e^{-2\pi
k\lvert y\rvert}\frac{\pi}{4}\delta(x-y-\mu)$
$\displaystyle=\frac{\pi}{4}\int_{-\infty}^{+\infty}\mathrm{d}x\leavevmode\nobreak\
e^{-2\pi k\left(\lvert x\rvert+\lvert x-\mu\rvert\right)}$
$\displaystyle=\frac{e^{-2\pi k\lvert\mu\rvert}}{8k}\left(1+2\pi
k\lvert\mu\rvert\right).$

The last integral we need is

$\displaystyle\left.\hat{\mathcal{I}}^{(\text{der})}_{k}(\mu)\right\rvert_{d=3}$
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{d}x\int_{-\infty}^{+\infty}\mathrm{d}y\leavevmode\nobreak\
e^{-2\pi k\left(\lvert x\rvert+\lvert y\rvert\right)}\pi\lvert x-y-\mu\rvert$
$\displaystyle=-\frac{2}{27}\left.\hat{\mathcal{I}}^{(0)}_{k}(\mu)\right\rvert_{d=5}$
$\displaystyle=\frac{1}{8\pi^{3}k^{3}}\left[3e^{-2\pi k\lvert\mu\rvert}+(2\pi
k\lvert\mu\rvert)\left(2+e^{-2\pi k\lvert\mu\rvert}\right)\right].$

Therefore, in 3d we have the contributions

$\displaystyle\sum_{\alpha=1}^{2}\mathcal{F}_{\text{fund},\alpha}$
$\displaystyle=\frac{1}{2}N^{2}\pi^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k\left(e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert\right)$ $\displaystyle\mathcal{F}_{0}$
$\displaystyle=\frac{1}{8}N^{2}\pi^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k\left(e^{-2\pi
k\lvert\mu\rvert}+e^{-2\pi k\lvert\mu\rvert}2\pi k\lvert\mu\rvert\right)$
$\displaystyle\mathcal{F}_{\text{der}}$
$\displaystyle=-\frac{1}{8}N^{2}\pi^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k\left[e^{-2\pi
k\lvert\mu\rvert}\left(3+2\pi k\lvert\mu\rvert\right)+2\left(2\pi
k\lvert\mu\rvert\right)\right].$

Summing all of them together we arrive at

$\mathcal{F}=\mathcal{F}_{1}+\mathcal{F}_{2}+\frac{\pi^{2}}{4}N^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}k\left[e^{-2\pi
k\lvert\mu\rvert}+2\pi k\lvert\mu\rvert\right].$ (3.30)

This expression agrees with the derivation without change of variables, eq.
(3.22).

###  3.5 Holographic match

Let us discuss the match of the free energy, computed in QFT, with the
holographic central charge computed in supergravity. Clearly, the free energy
(3.30) does not agree with the holographic central charge (2.30). But this was
to be expected, as we now explain.

The holographic flows we considered in Subsection 2.5 describe flows between
two SCFTs: a UV CFT at $\lvert\mu\rvert=0$ and an IR CFT at
$\lvert\mu\rvert=\infty$,101010For balanced quivers we find that the endpoints
$\mu=\pm\infty$ of the RG flow coincide. see Figure 8. Here, UV and IR refer
to the parameter $\mu$, not to a gauge coupling. More precisely, $\mu$ appears
in the coefficient of a relevant operator in the long quiver. By conformal
invariance, the radius $r$ of $\mathbb{S}^{d}$ does not enter the SCFT
computation, but enters in the combination $r\mu$ when $\mu\neq 0$,
effectively introducing a Wilsonian cutoff $\propto 1/r$. We have made the
customary redefinition $r\mu\to\mu$ in the matrix model, with $\mu$ being
dimensionless. Moving $\mu$ in the range $0\leq\mu\leq\infty$ describes the RG
flow between two SCFTs, with UV and IR referring to the Wilsonian approach
just mentioned.

The QFT picture, however, is slightly richer than the flow just sketched, and
this fact is captured by $\mathcal{F}$, as we now proceed to explain.

The free energy and holographic central charge agree at the UV fixed point
$\mu=0$, see Subsection 3.3.4. However, for $0<\lvert\mu\rvert<\infty$, the
free energy computes a deformation of the UV SCFT, which includes more fields
than the IR SCFT. This comes about because the breaking of gauge and flavour
groups

$N_{1}+N_{2}$$F_{1}$$F_{2}$$N_{1}$$F_{1}$$N_{2}$$F_{2}$

has a twofold effect:

  * $(i)$

to reduce the number of hypermultiplet modes at each node from
$(N_{1,j}+N_{2,j})(F_{1,j}+F_{2,j})$ to $N_{1,j}F_{1,j}+N_{2,j}F_{2,j}$;

  * $(ii)$

to lift the Higgsed W-bosons.

In addition, for a linear quiver,

  * $(iii)$

the Higgsing procedure also lifts the bifundamental hypermultiplets of
$U(N_{1,j})\times U(N_{2,j\pm 1})$.

The fields involved in these three effects are massive and decouple in the
$\lvert\mu\rvert\to\infty$ limit (that is, at the very end of the RG flow,
they become free fields at infinite distance from the interacting IR CFT).
Therefore, the correct statement is that, when the mass deformation of
interest is turned on, the UV SCFT flows to the pair of factorised SCFTs,
stacked with decoupled heavy fields. Schematically:

$\mathcal{Q}$UV:$\mathcal{Q}_{1}\sqcup\mathcal{Q}_{2}\sqcup\text{\small
decoupled}$.IR:${\scriptstyle\lvert\mu\rvert\to\infty}$

In conclusion, the holographic central charge is computed in supergravity and
only captures the features of the interacting SCFT at the end of the RG flow.
In contrast, the free energy is computed from a UV perspective. In order to
compare these two quantities, we must subtract the contributions elucidated in
the points $(i)$ to $(iii)$.

Let $\mathcal{F}_{\text{dec.}}$ denote the contributions of fields that
decouple at the end of the RG flow, properly defined in Appendix C. Enforcing
the subtraction, hence discarding $\mathcal{F}_{\text{dec.}}$, we define the
effective free energy

$\mathcal{F}_{\text{EFT}}=\mathcal{F}-\mathcal{F}_{\text{dec.}}.$ (3.31)

Intriguingly, we find that the effective free energy matches with the
holographic central charge, possibly up to a local counterterm involving only
the background vector multiplets. Furthermore, (3.31) interpolates precisely
between its UV value and the free energy of two decoupled quivers
$\mathcal{Q}_{1},\mathcal{Q}_{2}$. Namely:

$\displaystyle\lim_{\lvert\mu\rvert\to
0}\mathcal{F}_{\text{EFT}}=\mathcal{F}\left[\mathcal{Q}\right],\qquad\qquad\lim_{\lvert\mu\rvert\to\infty}\mathcal{F}_{\text{EFT}}=\mathcal{F}\left[\mathcal{Q}_{1}\right]+\mathcal{F}\left[\mathcal{Q}_{1}\right],$

which is the precise QFT analogue of the behaviour (2.31)-(2.32) of the
holographic central charge.

In summary, along the flow parametrised by $\mu$, we adopt an effective field
theory (EFT) approach, in which we separate the contributions of the fields
that would decouple in the IR limit $\mu\to\infty$. This EFT method, embodied
in the definition (3.31) and in the computations in Appendix C, leads to the
perfect holographic match all along the RG flow, up to a background
Chern–Simons counterterm in 5d. We elaborate on this last point in Subsection
3.5.3.

As a byproduct of our analysis, we show in Subsection 3.6 that the quantity
(3.31) satisfies a strong version of the F-theorem, both in 3d and 5d.

####  3.5.1 Matching the effective free energy in 5d

The computation of the contribution $\mathcal{F}_{\text{dec.}}$ from the
massive fields outlined in points $(i),(ii),(iii)$ is performed in Appendix C.

In 5d, the analysis of the regions of the 5d Coulomb branch onto which
fundamental hypermultiplets and W-bosons acquire a large mass and the
resolution of the ensuing integral leads to expression (C.6). The effective
free energy (3.31) is then readily evaluated by subtracting (C.6) from the 5d
$\mathcal{F}$ in eq. (3.29). The result is:

$\mathcal{F}_{\text{EFT}}\Big{|}_{d=5}=\mathcal{F}_{1}+\mathcal{F}_{2}+\frac{27}{16}N^{2}P^{2}\sum_{k=1}^{\infty}\mathfrak{a}_{1,k}\mathfrak{a}_{2,k}\frac{1}{k}e^{-2\pi
k\lvert\mu\rvert}\left(1+2\pi k\lvert\mu\rvert+\frac{4}{3}(\pi
k\lvert\mu\rvert)^{3}\right).$ (3.32)

This answer almost agrees with the holographic central charge (2.29), except
for the extra cubic dependence on $\mu$. This term is an effective
Chern–Simons coupling for the background vector multiplet, and can be removed
by adding a local counterterm only involving background fields. Similar
background Chern–Simons terms allow the match of the rank-$N$ $E_{n}$ theories
with its holographic dual [32]. A more thorough discussion is deferred to