# Linear Quivers at Large-$N$

Carlos<EMAIL_ADDRESS>Leonardo
<EMAIL_ADDRESS>Konstantin<EMAIL_ADDRESS>
( )

###### Abstract

Quiver theories constitute an important class of supersymmetric gauge theories
with well-defined holographic duals. Motivated by holographic duality, we use
localisation on $\mathbb{S}^{d}$ to study long linear quivers at large-$N$.
The large-$N$ solution shows a remarkable degree of universality across
dimensions, including $d=4$ where quivers are genuinely superconformal. In
that case we upgrade the solution of long quivers to quivers of any length.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results and organisation
  2. 2 Long quiver SCFTs in arbitrary $d$
    1. 2.1 Setup: Supersymmetric gauge theory in $d$ dimensions
    2. 2.2 Setup: Linear quivers in $d$ dimensions
    3. 2.3 Setup: Rank function
    4. 2.4 Long quiver limit
    5. 2.5 Long quiver solution in arbitrary $d$
    6. 2.6 Interpolating between $a$ and $F$
    7. 2.7 One-dimensional defects
    8. 2.8 Two-dimensional defects
    9. 2.9 Three-dimensional defects
    10. 2.10 Mirror-type identities
    11. 2.11 Mass deformation
  3. 3 Four-dimensional long quiver SCFTs
    1. 3.1 Anomaly coefficients from counting supermultiplets
    2. 3.2 Four-dimensional long quiver partition function
    3. 3.3 Recovering $a$ from the matrix model
    4. 3.4 Extremal correlators
    5. 3.5 Wilson loops
    6. 3.6 Solution at strong coupling
    7. 3.7 Free energy in four-dimensional long quivers
    8. 3.8 Four-dimensional linear quivers of arbitrary length
    9. 3.9 Towards comparison with supergravity: Simplified holography
  4. 4 Examples
    1. 4.1 Example 1
    2. 4.2 Example 2
  5. 5 Summary and outlook
  6. A Special functions
    1. A.1 Analytically continued one-loop determinants
    2. A.2 Lerch’s transcendent
  7. B Three perspectives on the saddle point equation

##  1 Introduction

Four-dimensional supersymmetric gauge theories with eight supercharges
($\mathcal{N}=2$) are an invaluable source of lessons for physicists and
mathematicians alike. The amount of supersymmetry allows to extract a wealth
of exact results, without forcing a rigid structure and thus including a wide
variety of phenomena. Renown examples of these powerful features are
Seiberg–Witten theory [1, 2] and the wall-crossing phenomenon for BPS states
[3, 4, 5]. We refer for instance to [6, 7, 8, 9] for reviews of the many
facets of four-dimensional $\mathcal{N}=2$ theories.

Embedding these quantum field theories in M-theory or Type II string theory
yields a top-down perspective on them [10]. If, on the one hand, the higher-
dimensional approach leads to the geometric engineering method [11, 12] to
explore strongly coupled dynamics in four dimensions, on the other hand it
entails the holographic principle at large-$N$ [13], which is especially well-
suited for explicit computations.

Building on [14, 15], a class of asymptotically-AdS5 supergravity solution was
constructed by Gaiotto and Maldacena [16], and further explored and
generalised in [17, 18, 19, 20, 21]. The holographic dual superconformal field
theories (SCFTs) are four-dimensional $\mathcal{N}=2$ quiver theories. As will
be manifest in our discussion, a necessary condition and a trademark for a
four-dimensional quiver to admit a weakly coupled supergravity dual is that
the number of gauge nodes of the quiver goes to infinity. The resulting SCFTs
are dubbed _long quiver theories_.

Supersymmetric localisation [22] guarantees that, placing the $\mathcal{N}=2$
theory on a four-sphere, the computation of several observables reduces to
evaluate ordinary, finite-dimensional integrals. The large-$N$ results thus
obtained are then instrumental to test the AdS/CFT correspondence, or to make
predictions on the holographic dual (see [23] for a review).

Long quiver theories are not exclusive to four dimensions: SCFTs with eight
supercharges that correspond to conformal fixed points of linear quiver gauge
theories exist in all possible dimensions ($d=1,\dots,6$). It is therefore
natural to pursue a field theoretical treatment that works uniformly in $d$
spacetime dimensions. This is one of the main achievements of this paper,
whose focus is on $d\geq 3$.

A formulation of AdSd+1/SCFTd in the Gaiotto–Maldacena electrostatic formalism
was provided in [24] for $d=3$ (see also [25] for related earlier work) and in
[26] for $d=5$ (see also [27, 28, 29, 30] for related work using different
methods). On the quantum field theory side, the large-$N$ limit of long linear
quivers in odd dimensions has been initiated by Uhlemann [31] and further
extensively analysed in the subsequent works [32, 33, 34, 35, 36]. The
intermediate case $d=4$ has not been addressed systematically so far, and we
amend the situation in this work.111Instances of four-dimensional
$\mathcal{N}=2$ conformal theories at large-$N$ have been considered in [37,
38, 39, 40, 41]. The $N_{f}=2N_{c}$ superconformal QCD, solved at large-$N$ in
[38], can be regarded as the simplest example of quiver CFT where the quiver
has but one node. We tackle long linear quivers comprehensively and extend the
existing bibliography in two major ways:

  * •

We solve the large-$N$ limit of arbitrary four-dimensional $\mathcal{N}=2$
linear quiver gauge theories;

  * •

We provide a uniform derivation in $d\geq 3$ of the large-$N$ and long quiver
limit.

In this way, we shed light on the mechanism governing the holographic
correspondence, its functioning across dimensions, and the relationship
between length of the quiver and supergravity dual.

###  1.1 Main results and organisation

This is a technically dense paper. Here we provide a roadmap of the main
results. The rest of this paper is organised in two main blocks:

  * •

Section 2 is devoted to long linear quivers in arbitrary dimension $d\geq 3$
(not necessarily integer). The main outcome of this first part is to show that
long quivers behave uniformly in $d\geq 3$.

    * $\triangleright$

After preliminaries to set up the analytic continuation in $d$ (Subsections
2.1-2.3), we consider the matrix models for the partition functions of linear
quiver SCFTs on the $d$-sphere.

    * $\triangleright$

The long quiver limit is solved for arbitrary $d$ in Subsections 2.4-2.5. We
find that the density of eigenvalues, which is a central tool to characterize
the matrix model, satisfies a Poisson equation (2.24) uniformly for $d\notin
2\operatorname{\mathbb{N}}$.

    * $\triangleright$

We then compute a quantity $\tilde{F}_{d}$ that interpolates between the
central charges ($d$ even) and free energies ($d$ odd) in Subsection 2.6. A
universal expression valid for real $d\geq 3$ is given in (2.31).

    * $\triangleright$

The interpolation across dimensions is extended to defect one-point functions
in the long quiver limit.

    * $\triangleright$

It is shown in Subsection 2.10 that a certain transformation of the gauge
ranks produces identities between the physical observables for the
corresponding long quiver gauge theories.

  * •

Section 3 is devoted to linear quivers in $d=4$. The main outcome of this
second part is to show that two different large-$N$ regimes exist. In the
infinite coupling regime, suitable for comparison with supergravity, we
recover the long quiver description.

    * $\triangleright$

$d=4$ is special in the sense that the theory is conformal at arbitrary gauge
coupling. We thus first solve the long quiver limit at arbitrary finite ’t
Hooft coupling (Subsections 3.2-3.3), finding that the procedure differs from
$d\neq 4$.

    * $\triangleright$

We compute the anomaly coefficient $a$ in $d=4$ obtained from this procedure
in Subsection 3.3, showing agreement both with the result from counting
supermultiplets as well as with the limit $d\to 4$ of the interpolating
quantity $\tilde{F}_{d}$ found previously in Subsection 2.6.

    * $\triangleright$

Then, in Subsection 3.6 we take the strong ’t Hooft coupling limit at
large-$N$. The ensuing procedure is distinct from the finite-coupling
solution, and it is much closer to the long quiver limit in $d\neq 4$. We show
that the $d=4$ eigenvalue density at infinite ’t Hooft coupling satisfies the
same Poisson equation found in Subsection 2.4.

    * $\triangleright$

In this strong coupling regime we are able to go beyond the computation of
$a$, and we obtain the free energy in Subsection 3.7. Despite the latter
quantity being scheme-dependent, we are able to identify and isolate a
universal dependence on the length of the quiver and on the gauge ranks.

    * $\triangleright$

In Subsection 3.9 we propose a comparison of the free energy with a
supergravity-inspired calculation, finding agreement. Instead of computing the
on-shell supergravity action, we take a shortcut in this subsection, which
makes the argument only heuristic.

    * $\triangleright$

Finally we provide a large-$N$ solution beyond the long quiver limit in $d=4$
in Subsection 3.8.

The results from both main parts are showcased in concrete examples in Section
4. We conclude in Section 5 with a summary of our main findings and an outlook
on avenues for future research. The main text is complemented with two
appendices.

##  2 Long quiver SCFTs in arbitrary $d$

In this section we study supersymmetric gauge theories in spacetime dimension
$d\geq 3$ and compute quantities that interpolate across dimensions.

Subsections 2.1-2.2 serve to set the stage and to introduce generalities on
balanced linear quivers, and in Subsection 2.3 we introduce the rank function,
which is a central quantity in the ensuing computations. The core of the
derivation is in Subsections 2.4 to 2.6, where the long quiver limit is
addressed in arbitrary $d\geq 3$. In the subsequent subsections, further
physical observables, such as defect one-point functions, are evaluated.

###  2.1 Setup: Supersymmetric gauge theory in $d$ dimensions

$d=$ | 3 | 4 | 5 | 6
---|---|---|---|---
$\mathcal{N}=$ | 4 | 2 | 1 | (1,0)
Table 1: Notation for theories with eight supercharges in diverse dimensions.
It is customary to indicate the amount of supersymmetry by the number
$\mathcal{N}$ of Killing spinors.

We consider a supersymmetric gauge theory in $d$ Euclidean dimensions with
eight supercharges. The notation for this amount of supersymmetry in diverse
dimensions is recalled in Table 1. The theory is coupled to the round
$d$-sphere $\operatorname{\mathbb{S}}^{d}$ preserving supersymmetry [42]. The
discussion that follows is physically meaningful in the range $3\leq d\leq 7$
but, as we will show, all the results for physical quantities can be continued
to $d\in\operatorname{\mathbb{R}}_{\geq 3}$.

The gauge group is assumed to be

$G=\mathrm{U}(N_{1})\times\mathrm{U}(N_{2})\times\cdots\times\mathrm{U}(N_{P-1})$
(2.1)

and the matter content consists in hypermultiplets in the (bi)fundamental
representation of the simple factors $\mathrm{U}(N_{j})$ of the gauge group.
The $\mathrm{U}(N_{j})$ gauge factors can be replaced with special unitary
gauge groups $\mathrm{SU}(N_{j})$,

$G=\mathrm{SU}(N_{1})\times\mathrm{SU}(N_{2})\times\cdots\times\mathrm{SU}(N_{P-1})$

without changing our results. We assume unitary groups for notational
simplicity throughout Section 2, and use special unitary groups later on in
Section 3, dedicated to $d=4$.

The gauge theory is conveniently encoded in a quiver. More precisely, the
gauge and matter content of a theory with eight supercharges is encoded in the
representation of a framed doubled quiver, in which the nodes indicate gauge
groups, the framing (drawn as square nodes) indicate the flavour groups, and
unoriented edges (which correspond to pairs of edges with opposite
orientation) indicate hypermultiplet in the bifundamental representation.

From the choice of gauge group (2.1) we can label the nodes of the quiver with
an index $j\in\left\\{1,\dots,P-1\right\\}$, and introduce the arrays

$\vec{N}=(N_{1},\dots,N_{P-1}),\qquad\vec{K}=(K_{1},\dots,K_{P-1}).$

We denote by $\mathsf{M}$ the Cartan matrix associated to the quiver, namely
twice the identity matrix minus the adjacency matrix.

######  Definition 2.1.

A quiver gauge theory with eight supercharges is said to be _balanced_ if

$\vec{K}=\mathsf{M}\vec{N}.$

We will refer to this condition as the _balancing condition_ , and interpret
it as a constraint on the flavour ranks for a given choice of gauge group
(2.1).

Balanced theories yield the unique choice of flavour ranks for which the gauge
theory is either conformal, or connected to a CFT via an RG flow controlled by
the $d$-dimensional gauge couplings $g_{d,j}$, in the whole window $3\leq
d\leq 6$. We now quickly recall how this comes about.222There exist many more
SCFTs which do not have a Lagrangian description, and in particular they are
not quiver gauge theories. In this work we only consider Lagrangian theories.

  * —

In $d=3$, a UV gauge theory flows at low energies to a strongly coupled fixed
point if it is ugly or good in the Gaiotto–Witten classification [43], meaning
$K_{j}\geq\sum_{k}\mathsf{M}_{jk}N_{k}-1$. The IR fixed point of an ugly
theory is the direct sum of the interacting SCFT of a good quiver and
decoupled free fields. Without loss of generality, we can thus consider good
theories only, for which balanced theories provide a lower bound.

  * —

In $d=4$, all the Lagrangian SCFTs must be balanced. Indeed, the beta
function, which is one-loop exact thanks to the $\mathcal{N}=2$ supersymmetry,
reads:

$\beta(g_{4,j})=\frac{g_{4,j}^{3}}{16\pi^{2}}\left(K_{j}-\sum_{k}\mathsf{M}_{jk}N_{k}\right)$
(2.2)

and vanishes as soon as the balancing condition is imposed.

  * —

In $d=5$, a UV complete gauge theory is obtained from a relevant deformation
of a strongly coupled SCFT. A sufficient condition for the gauge theory to sit
in the IR of one such SCFT is that $K_{j}\leq\sum_{k}\mathsf{M}_{jk}N_{k}$
[44]. This is equivalent to demand the convergence of the perturbative
partition function. Slightly less stringent conditions can be imposed in some
cases [45] (see [46] for lists of examples). However, due to the lack of
analytic control over instantons, we will have to work with the perturbative
partition function throughout, which forces the previous inequality.

  * —

In $d=6$, cancellation of gauge anomalies in any gauge theory obtained by a
relevant deformation of a SCFT requires the balancing condition.

Supersymmetric localisation reduces the partition function on
$\operatorname{\mathbb{S}}^{d}$ to a matrix model [22], schematically of the
form

$\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}=\int_{\mathfrak{t}}\mathrm{d}\vec{\phi}\leavevmode\nobreak\
\mathrm{e}^{-S_{\mathrm{cl}}(\vec{\phi})}\leavevmode\nobreak\
Z_{\mathrm{vec}}Z_{\mathrm{hyp}}Z_{\text{\rm non-pert}}.$ (2.3)

In this formula,

  * •

$\mathfrak{t}\cong\operatorname{\mathbb{R}}^{\mathrm{rk}(G)}$ denotes a Cartan
subalgebra of the Lie algebra of the gauge group $G$. The integration variable
$\vec{\phi}\in\mathfrak{t}$ is the zero-mode of a real scalar in the vector
multiplet, and $\mathrm{d}\vec{\phi}$ is the Lebesgue measure on
$\mathfrak{t}$.

  * •

$S_{\mathrm{cl}}$ is the classical action evaluated at the localisation locus.

  * •

$Z_{\mathrm{vec}}$ and $Z_{\mathrm{hyp}}$ are the one-loop determinants from
the vector and hypermultiplet.

  * •

In addition, there is a non-perturbative part $Z_{\text{\rm non-pert}}$ from
instantons. We set this last term to 1, because the instanton corrections are
non-perturbatively suppressed in the planar large-$N$ limit considered below.

We use the index $j=1,\dots,P-1$ to label the gauge node $\mathrm{U}(N_{j})$
and the index $a=1,\dots,N_{j}$ to label the basis of the Cartan of
$\mathfrak{u}(N_{j})$, so that the integration variables are

$\vec{\phi}=\left\\{\phi^{(j)}_{a}\right\\}^{j=1,\dots,P-1}_{a=1,\dots,N_{j}}.$

###  2.2 Setup: Linear quivers in $d$ dimensions

$N_{\scriptscriptstyle P-1}$$\cdots$$N_{\scriptscriptstyle
1}$$N_{\scriptscriptstyle 2}$$K_{\scriptscriptstyle
P-1}$$K_{\scriptscriptstyle 1}$$K_{\scriptscriptstyle 2}$ Figure 1: Linear
quiver of length $P-1$. Circular nodes indicate gauge groups, square nodes
indicate flavour symmetries.

From now on, we restrict our attention to _linear_ quivers. These correspond
to quiver gauge theories with gauge group (2.1), or its version with special
unitary instead of unitary groups, and hypermultiplets in the bifundamental
representation of $\mathrm{U}(N_{j})\times\mathrm{U}(N_{j+1})$, with
additional $K_{j}$ hypermultiplets in the fundamental representation of
$\mathrm{U}(N_{j})$. These data are encoded in a linear quiver of length
$P-1$, shown in Figure 1.

We moreover restrict our attention to _balanced_ quivers. Applying the
definition, a linear quiver is balanced if the numbers of flavours $K_{j}$
satisfy

$K_{j}=2N_{j}-N_{j+1}-N_{j-1}$ (2.4)

$\forall\leavevmode\nobreak\ j=1,\dots,P-1$.

We will consider Yang–Mills theories with eight supercharges and no
Chern–Simons term,333In $d=5$ a non-vanishing Chern–Simons term generically
requires the theory to be unbalanced. For unbalanced $d=5$ theories whose
Chern–Simons term is dynamically generated by integrating out hypermultiplets
from a balanced linear quiver, our procedure still applies with few
modifications. In $d=3$, only a suitably restricted class of Chern–Simons
terms preserves eight supercharges. Equalities of partition functions among a
subset of these Chern–Simons-matter theories and linear quivers have been
derived in [47, 48], thus our results will apply to that particular subset.
and study their sphere partition functions (2.3). The classical action
evaluated on the localisation locus reads

$S_{\mathrm{cl}}(\vec{\phi})=\frac{\pi^{(d-3)/2}}{\Gamma\left(\frac{d-3}{2}\right)}\sum_{j=1}^{P-1}\frac{8\pi^{2}}{g_{d,j}^{2}}\sum_{a=1}^{N_{j}}\left(\phi^{(j)}_{a}\right)^{2},$

where $g_{d,j}$ is the gauge coupling for the $j^{\text{th}}$ gauge node in
dimension $d$, and we understand $\frac{1}{g_{d<4}}=0$. This is because, while
the localised partition function is calculated in the gauge theory, it is a
protected quantity and the theory flows to strong coupling in the IR if $d<4$.
In integer dimension, $\frac{1}{g_{d<4}}=0$ is a consequence of the
$Q$-exactness of the Yang–Mills Lagrangian, and we extend this condition to
real $3\leq d<4$.

In view of the large-$N$ planar limit, we define the ’t Hooft couplings
$\left\\{t_{d,j}\right\\}$ in $d\geq 4$ according to

$t_{d,j}:=\frac{Ng_{d,j}^{2}}{16\pi^{2}},\qquad\forall\leavevmode\nobreak\
j=1,\dots,P-1.$

To lighten the formulae, we also adopt the shorthand notation

$u_{d}:=\frac{\pi^{(d-3)/2}}{\Gamma\left(\frac{d-3}{2}\right)}.$

Altogether we write

$\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}=\int_{\mathfrak{t}}\mathrm{d}\vec{\phi}\leavevmode\nobreak\
\exp\left\\{-S_{\mathrm{eff}}(\vec{\phi})\right\\}\leavevmode\nobreak\
Z_{\mathrm{non-pert}}$ (2.5)

where

$\displaystyle S_{\mathrm{eff}}(\vec{\phi})=\sum_{j=1}^{P-1}$
$\displaystyle\left\\{\frac{u_{d}}{2t_{j}}N\sum_{a=1}^{N_{j}}(\phi^{(j)}_{a})^{2}+\sum_{1\leq
a\neq b\leq
N_{j}}f_{\mathrm{v}}\left(\phi^{(j)}_{a}-\phi^{(j)}_{b}\right)\right.$
$\displaystyle\qquad\left.+K_{j}\sum_{a=1}^{N_{j}}f_{\mathrm{h}}\left(\phi^{(j)}_{a}\right)+(1-\delta_{j,P-1})\sum_{a=1}^{N_{j}}\sum_{b=1}^{N_{j+1}}f_{\mathrm{h}}\left(\phi^{(j)}_{a}-\phi^{(j+1)}_{b}\right)\right\\}.$

In this expression, $f_{\mathrm{h}}$ captures the one-loop contribution from
each hypermultiplet zero-mode and $f_{\mathrm{v}}$ includes the one-loop
contribution of the W-bosons and their fermionic superpartners, as well as a
Vandermonde term from the measure. These functions have been explicitly
derived by localisation for $d\in[3,7]\cap\operatorname{\mathbb{N}}$, and
expressions valid for $d\in\operatorname{\mathbb{R}}_{\geq 3}$ that
interpolates between the known functions on the integers were provided by
Minahan [49]. We collect them in Appendix A.1.

For later convenience, let us introduce the indicator function

$\theta_{\mathscr{S}}:=\begin{cases}1&\text{if $\mathscr{S}$ is true}\\\
0&\text{otherwise}\end{cases}$ (2.6)

for any condition $\mathscr{S}$. We define the function

$f_{0}(\phi):=f_{\mathrm{h}}(\phi)+f_{\mathrm{v}}(\phi)\theta_{\phi\neq 0}.$
(2.7)

It is also convenient to isolate the term proportional to $\delta_{j,P-1}$. It
is a contribution from the tail of the quiver, which avoids overcounting of
the bifundamental hypermultiplets:

$\mathfrak{B}:=-\sum_{j=1}^{P-1}\delta_{j,P-1}\sum_{a=1}^{N_{j}}\sum_{b=1}^{N_{j+1}}f_{\mathrm{h}}\left(\phi^{(j)}_{a}-\phi^{(j+1)}_{b}\right)=-\sum_{a=1}^{N_{P-1}}f_{\mathrm{h}}\left(\phi^{(P-1)}_{a}\right).$

###  2.3 Setup: Rank function

$\eta$$\eta$$\mathcal{R}(\eta)$$\mathcal{R}(\eta)$$\scriptstyle
N$$\scriptstyle NP$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
P$$\scriptscriptstyle P-1$$\scriptstyle 1$$\scriptstyle P$$\scriptscriptstyle
P-1$ Figure 2: Two sample plots of the rank function $\mathcal{R}(\eta)$.
Left: first example, also known as $+_{N,P}$ theory. Right: second example,
also known as $T_{P}$ theory if $N=1$.

We introduce a continuous variable $0\leq\eta\leq P$ and let
$\mathcal{R}(\eta)$ be the rank function.

######  Definition 2.2.

The rank function $\mathcal{R}(\eta)$ is the piece-wise linear function on
$[0,P]$ that satisfies

$\mathcal{R}(0)=0=\mathcal{R}(P),\qquad\mathcal{R}(j)=N_{j}.$

In terms of the rank function, the balancing condition (2.4) reads

$\partial_{\eta}^{2}\mathcal{R}(\eta)=-\sum_{j=1}^{P-1}K_{j}\delta\left(\eta-j\right).$
(2.8)

Two examples of rank function for balanced linear quivers are shown in Figure
2.

With the large-$N$ limit in mind, it is convenient to fix $N\in\mathbb{N}$ and
write $N_{j}=\nu_{j}N$ for some rational number $\nu_{j}$. We then introduce
the scaled variable

$z:=\frac{\eta}{P},\qquad 0\leq z\leq 1,$

and the scaled rank function

$\nu(z):=\frac{1}{N}\mathcal{R}(Pz).$ (2.9)

To work in the planar limit, we also introduce the Veneziano parameters

$\zeta_{j}:=\frac{K_{j}}{N},$

and gather them in the scaled flavour rank function

$\zeta(z)=\frac{1}{P}\sum_{j=1}^{P-1}\zeta_{j}\delta\left(z-\frac{j}{P}\right).$

Imposing the balancing condition (2.8), these redefinitions are subject to

$\partial_{z}^{2}\nu(z)=-P^{2}\zeta(z).$ (2.10)

The normalisation of $\nu(z)$ and $\zeta(z)$ is chosen so that all the
quantities that follow have a well-defined large-$N$ and large-$P$ limit. In
the same vein, we replace the ’t Hooft couplings $\left\\{t_{d,j}\right\\}$
with a continuous function $t_{d}(z)$ on $[0,1]$.

According to the definition, the Fourier expansion of a generic
$\mathcal{R}(\eta)$ is given by

$\mathcal{R}(\eta)=\sum_{k=1}^{\infty}R_{k}\sin\left(\frac{k\pi\eta}{P}\right).$
(2.11)

From here, it follows that

$\nu(z)=\frac{1}{N}\sum_{k=1}^{\infty}R_{k}\sin(k\pi z)$ (2.12)

and

$P^{2}\zeta(z)=\frac{\pi^{2}}{N}\sum_{k=1}^{\infty}k^{2}R_{k}\sin(k\pi z).$
(2.13)

We also define the Fourier expansion of the inverse gauge coupling:

$\frac{1}{g_{d}(z)}=\sum_{k=1}\left[\gamma_{k}\sin(k\pi
z)+\tilde{\gamma}_{k}\cos(k\pi z)\right].$ (2.14)

###  2.4 Long quiver limit

Our goal is to address the planar limit of (2.5), extending [31]. For the
linear quivers of Figure 1, we take a planar limit in which moreover
$P\to\infty$. It goes under the name of _long quiver_ limit.

######  Definition 2.3.

The long quiver limit of a linear quiver of $P-1$ nodes is the planar
large-$N$ and large-$P$ limit with ’t Hooft scaling of the gauge couplings. We
use the symbol $\simeq$ to indicate asymptotic equality in the long quiver
limit.

The goal of this subsection and the next one is to prove that the structure
observed in [26, 24, 36] for the long quiver limit in
$d\in\left\\{3,5\right\\}$ extends to arbitrary $d$.

######  Statement 2.4.

Let $d\in\operatorname{\mathbb{R}}_{\geq 3}\setminus
2\operatorname{\mathbb{Z}}$. The long quiver limit of a balanced linear quiver
is governed by a function

$\varrho\ :\ [0,1]\times\operatorname{\mathbb{R}}\ \to\
\operatorname{\mathbb{R}}_{\geq 0}$

that solves the Poisson equation

$\frac{1}{4}\partial_{x}^{2}\varrho(z,x)+\partial_{z}^{2}\varrho(z,x)-\partial_{z}^{2}\nu(z)\delta(x)=0,$

for $0<z<1$ and $x\in\operatorname{\mathbb{R}}$.

To prepare the stage, we introduce the eigenvalue densities

$\rho_{j}(\phi):=\frac{1}{N}\sum_{a=1}^{N_{j}}\delta\left(\phi-\phi^{(j)}_{a}\right),\qquad\phi\in\operatorname{\mathbb{R}}.$

Note that they carry an overall factor $1/N$, as opposed to the customary
$1/N_{j}$. Hence, they are not canonically normalised to 1. We can collect all
the eigenvalue densities for each gauge factor in the function of two
variables $\rho(z,\phi)$ with
$\rho\left(\frac{j}{P},\phi\right)=\rho_{j}(\phi)$. This new eigenvalue
density is normalised as

$\int_{\operatorname{\mathbb{R}}}\mathrm{d}\phi\rho(z,\phi)=\nu(z)\qquad\forall\leavevmode\nobreak\
0<z<1.$

In the latter expressions $\phi\in\operatorname{\mathbb{R}}$ is a one-
dimensional integration variable, related to but different from the
$\mathrm{rk}(G)$-dimensional vector $\vec{\phi}$.

With these definitions, the matrix model effective action in (2.5) is written
as a functional of $\rho$, in the form

$\displaystyle
S_{\mathrm{eff}}[\rho]=N^{2}P\int_{0}^{1}\mathrm{d}z\int\mathrm{d}\phi\rho(z,\phi)$
$\displaystyle\left\\{\frac{u_{d}}{2t_{d}(z)}\phi^{2}+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\rho(z,\sigma)f_{\mathrm{v}}(\phi-\sigma)\right.$
(2.15)
$\displaystyle\left.+\zeta(z)f_{\mathrm{h}}(\phi)+\int\mathrm{d}\sigma\rho(z+\delta
z,\sigma)f_{\mathrm{h}}(\phi-\sigma)\right\\}+\mathfrak{B}.$

Replacing each sum over $a=1,\dots,N_{j}$ with an integral over $\phi$ yields
an overall factor of $N$, and the factor of $P$ comes from replacing the sum
over $j=1,\dots,P-1$ with an integral over $z$. The symbol
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int$
stands for the principal value integral. Besides, we are using the notation
$\delta z=\frac{1}{P}$ in the contribution from bifundamental hypermultiplets.

We use the trivial identity $\delta_{j,P-1}=\delta_{j+1,P}$ and the
replacement $\delta_{j+1,P}\mapsto\frac{1}{P}\delta(z+\delta z)$ inside
$\mathfrak{B}$, to write

$(1-\delta_{j,P-1})\rho_{j+1}(\sigma)\ \mapsto\
\left(1-\frac{1}{P}\delta(z+\delta z)\right)\rho(z+\delta z,\sigma),$

suited for the planar limit. We then symmetrise this last term

$(1-\delta_{j,P-1})\rho(z+\delta z,\sigma)\ \mapsto\
\frac{1}{2}\left[(1-\delta_{j,P-1})\rho(z+\delta
z,\sigma)+(1-\delta_{j,1})\rho(z-\delta z,\sigma)\right].$

One may check directly the invariance of $S_{\mathrm{eff}}[\rho]$ under this
replacement, noting that $f_{\mathrm{h}}$ is an even function for every $d$. A
more straightforward way to see it, is that we have labelled the bifundamental
hypermultiplets with the $j$ of the gauge node at its left, leading to discard
the term at $j=P-1$. We could have simply labelled them with the $j$ from
right, which then would lead to not account for the term $j=1$. The more
symmetric option is to divide the contribution of each bifundamental by $2$
and label half from the left and half from the right, with the boundary term
written as

$\mathfrak{B}=-\frac{1}{2}\sum_{j=1}^{P-1}\left[\delta_{j,P-1}\sum_{a=1}^{N_{j}}\sum_{b=1}^{N_{j+1}}f_{\mathrm{h}}\left(\phi^{(j)}_{a}-\phi^{(j+1)}_{b}\right)+\delta_{j,1}\sum_{a=1}^{N_{j}}\sum_{b=1}^{N_{j-1}}f_{\mathrm{h}}\left(\phi^{(j)}_{a}-\phi^{(j-1)}_{b}\right)\right].$

Inserting the eigenvalue density and using $\rho(z,\sigma)=\delta(\sigma)$ if
$z\notin(0,1)$, this expression becomes

$\mathfrak{B}=-N\left[\lim_{z\to
0^{+}}\int\mathrm{d}\phi\rho(z,\phi)f_{\mathrm{h}}(\phi)+\lim_{z\to
1^{-}}\int\mathrm{d}\phi\rho(z,\phi)f_{\mathrm{h}}(\phi)\right].$

At leading order in the large-$P$ limit, we have [31]

$\frac{1}{2}\left[\rho(z+\delta z,\sigma)+\rho(z-\delta
z,\sigma)\right]-\rho(z,\phi)\ \to\
\frac{1}{2P^{2}}\partial_{z}^{2}\rho(z,\phi),$

and we arrive at

$\displaystyle S_{\mathrm{eff}}[\rho]\simeq N^{2}P\int_{0}^{1}\mathrm{d}z$
$\displaystyle\int\mathrm{d}\phi\rho(z,\phi)\left\\{\frac{u_{d}}{2t_{d}(z)}\phi^{2}+\zeta(z)f_{\mathrm{h}}(\phi)\right.$
(2.16)
$\displaystyle\left.+\int\mathrm{d}\sigma\left[\rho(z,\sigma)f_{0}(\phi-\sigma)+\frac{1}{2P^{2}}\partial^{2}_{z}\rho(z,\sigma)f_{\mathrm{h}}(\phi-\sigma)\right]\right\\}\
+\mathfrak{B},$

where $f_{0}$ was defined in (2.7), and $f_{0}=f_{\mathrm{v}}+f_{\mathrm{h}}$
except when the argument is zero. The piece $\mathfrak{B}$ from the boundaries
of the quiver is negligible compared to the $\mathcal{O}(N^{2}P)$ leading
contributions from the rest of nodes of the quiver. Stated more precisely, the
pieces proportional to $\delta_{j,P-1}$ and $\delta_{j,1}$ have measure 0 in
the long quiver limit, and therefore do not appear in the effective action as
long as we can neglect the more fine-grained discretised structure of the
$z$-interval.

The planar limit of the long linear quivers above has been studied in $d=5$
and $d=3$. Here, we leverage the knowledge of the analytic continuation in the
dimension $d$ of the main ingredients $f_{\mathrm{v}},f_{\mathrm{h}}$ [49].

Instead of writing the formulae of [49] for the one-loop determinants (cf.
Appendix A.1), we note that the functions $f_{\mathrm{h}}$ and $f_{0}$ behave
at large argument as

$f_{\mathrm{h}}(\phi)\simeq s_{d}\lvert\phi\rvert^{d-2},\qquad
f_{0}(\phi)\simeq\tilde{s}_{d}\partial_{\phi}^{2}f_{\mathrm{h}}(\phi),$ (2.17)

where the $d$-dependent coefficients are

$s_{d}=2\cos\left(\frac{\pi
d}{2}\right)\Gamma(2-d),\qquad\tilde{s}_{d}=\frac{1}{8}(d-2)^{2}.$

The coefficient $s_{d}$ diverges at even integer $d$, which signals that the
expression homogeneous in $d$ is not suitable there. Indeed, in $d=4$ the
naive $\phi^{2}$ scaling of $f_{\mathrm{h}}$ should be replaced by
$\phi^{2}\ln\lvert\phi\rvert$ [22], see for instance [39, Eq.(2.8)].

In the planar limit, we are led to study the saddle points of the effective
action. The saddle point equation is

$\int\mathrm{d}\sigma\leavevmode\nobreak\
\partial_{\phi}\left[f_{0}(\phi-\sigma)+\frac{1}{2P^{2}}f_{\mathrm{h}}(\phi-\sigma)\partial^{2}_{z}\leavevmode\nobreak\
\right]\rho(z,\sigma)=-\frac{u_{d}}{2t_{d}(z)}\phi-\frac{\zeta(z)}{2}\partial_{\phi}f_{\mathrm{h}}(\phi).$
(2.18)

The derivation of this equation is standard, thus we omit the details. See
Appendix B for additional comments on (2.18).

To proceed further, we make an ansatz about the long quiver behaviour of the
eigenvalues.

######  Ansatz.

In the long quiver limit, the leading contribution comes from the eigenvalues
that scale as

$\phi=(d-2)P^{\chi}x,\qquad\chi>0$ (2.19)

with $x$ fixed and $\mathcal{O}(1)$ in the limit. The overall factor $(d-2)$
is just for convenience, whereas the crucial aspect of the ansatz is $\chi>0$,
with strict inequality.

We utilise the ansatz (2.19), and also define a new density of eigenvalues
$\varrho(z,x)$ to make the substitution of measures

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

Inserting the ansatz (2.19) into the asymptotic expressions (2.17), we get
that the quantities in (2.18) scale as

$\displaystyle\partial_{\phi}f_{\mathrm{h}}(\phi-\sigma)$
$\displaystyle\approx s_{d}P^{\chi(d-3)}(d-2)^{d-3}\partial_{x}\lvert
x-y\rvert^{d-2}$ $\displaystyle\partial_{\phi}f_{0}(\phi-\sigma)$
$\displaystyle\approx\tilde{s}_{d}s_{d}P^{\chi(d-5)}(d-2)^{d-5}\partial_{x}^{3}\lvert
x-y\rvert^{d-2}=\frac{1}{8}s_{d}P^{\chi(d-5)}(d-2)^{d-3}\partial_{x}^{3}\lvert
x-y\rvert^{d-2}.$

For the ’t Hooft coupling, we impose the long quiver scaling

$t_{d}=P^{\tau}\tilde{t}_{d},\qquad\tau>0$

and get the contribution

$\frac{\phi}{2t_{d}(z)}=P^{\chi-\tau}(d-2)\frac{x}{2\tilde{t}_{d}(z)}.$

The powers $\chi,\tau$ will be fixed demanding self-consistency of the
equilibrium equation.

Integrating by parts twice the $f_{0}$ term in (2.18) after the substitution,
we arrive at

$\displaystyle\int\mathrm{d}y\leavevmode\nobreak$
$\displaystyle(d-2)^{d-3}s_{d}\partial_{x}\lvert
x-y\rvert^{d-2}\leavevmode\nobreak\
\left[P^{(d-3)\chi-2}\cdot\frac{1}{2}\partial_{z}^{2}\varrho(z,y)+P^{(d-5)\chi}\cdot\frac{1}{8}\partial_{y}^{2}\varrho(z,y)\right]$
(2.20)
$\displaystyle=-P^{\chi-\tau}(d-2)u_{d}\frac{x}{2\tilde{t}_{d}(z)}-s_{d}(d-2)^{d-3}P^{(d-3)\chi-2}\frac{[P^{2}\zeta(z)]}{2}\partial_{x}\lvert
x\rvert^{d-2}.$

We are using the fact that $P^{2}\zeta(z)$ is finite in the large-$P$ limit.
Note that we work in $d\geq 3$, thus we always have

$\partial_{x}\lvert x-y\rvert^{d-2}=(d-2)(x-y)^{d-3}\mathrm{sign}(x-y).$

We will loosely write the above as $\lvert x-y\rvert^{d-3}$, with the
understanding that in the limit $d\to 3^{+}$ we have the sign function.
Furthermore, working in $d\geq 3$, we can always divide both sides by
$\frac{1}{2}P^{\chi(d-3)}(d-2)^{d-2}$ and define

$w_{d}:=-(d-2)^{3-d}u_{d}=-(d-2)^{3-d}\frac{\pi^{(d-3)/2}}{\Gamma\left(\frac{d-3}{2}\right)}.$

With these elementary manipulations, the saddle point equation (2.18) is cast
in the form

$s_{d}\int\mathrm{d}y\lvert x-y\rvert^{d-3}\leavevmode\nobreak\
\left[\partial_{z}^{2}\varrho(z,y)+\frac{1}{4}P^{2-2\chi}\partial_{y}^{2}\varrho(z,y)\right]=P^{(4-d)\chi+2-\tau}w_{d}\frac{x}{\tilde{t}_{d}(z)}-s_{d}P^{2}\zeta(z)\lvert
x\rvert^{d-3}.$ (2.21)

We need to inspect this equation and determine $\chi,\tau$ such that it admits
a non-trivial solution. First, note that the first and last term are
$\mathcal{O}(1)$. Therefore, to cancel the powers of $P$, we demand

$2\chi-2=0,$

that is, $\chi=1$. It satisfies the self-consistency condition $\chi>0$. To
keep the ’t Hooft couplings as well, we impose $\tau=(4-d)\chi+2=6-d$,
consistent with $\tau>0$ in the region $d<6$. To sum up,

  * —

If $3\leq d<4$ we set $t_{d<4}(z)=\infty$;

  * —

If $4\leq d<6$ the long quiver admits a finite ’t Hooft coupling with scaling
$t_{4\leq d<6}(z)=P^{6-d}\tilde{t}_{d}(z)$;

  * —

If $d>6$ we set $t_{d>6}(z)=\infty$.

From now on, we set $\chi=1$ and $\tau=6-d$. We stress that, while
$t_{d<4}(z)=\infty$ is a natural consequence of the $Q$-exactness of the
Yang–Mills Lagrangian in lower dimensions, we have to impose
$t_{d>6}(z)=\infty$ for consistency with the procedure.

###  2.5 Long quiver solution in arbitrary $d$

The various terms in the saddle point equation (2.21) bear a coefficient
$s_{d}$, except for the Yang–Mills contribution, which has a coefficient
$w_{d}$. We thus need to study the zeros and poles of these coefficients as
functions of $d$ to understand the structure of the solution to (2.21).

The numerical coefficient $w_{d}$ of the gauge coupling is a smooth real
function on $d\geq 3$. It is negative on the whole $3<d<\infty$ and vanishes
at $d=3$ and $d\to\infty$; it has a unique minimum near $d\approx 4.2$ and
moreover $w_{4}=-\frac{1}{2}$.

Therefore, at $d=3$, we have $w_{3}=0$, the gauge coupling drops out, and all
the remaining terms in (2.21) have the same coeficient $s_{3}=\pi$, which can
be simplified. On the other hand, for every $3<d<\infty$, we can divide both
sides of (2.21) by $w_{d}$ and define $C_{d}:=\frac{s_{d}}{w_{d}}$,
explicitly444Recall that we have absorbed a coefficient $16\pi^{2}$ in the
definition of $t_{d}$. We might as well have defined $t_{d}=Ng_{d}$ and
eventually have reabsorbed the $16\pi^{2}$ in the definition of $w_{d}$ and
$C_{d}$.

$C_{d}=\left(\frac{d-2}{\sqrt{2\pi}}\right)^{d-3}\Gamma\left(\frac{d-3}{2}\right)\Gamma\left(2-d\right)\cos\left(\frac{\pi
d}{2}\right).$ (2.22)

The saddle point equation becomes

$C_{d}\int\mathrm{d}y\lvert x-y\rvert^{d-3}\leavevmode\nobreak\
\left[\partial_{z}^{2}\varrho(z,y)+\frac{1}{4}\partial_{y}^{2}\varrho(z,y)+P^{2}\zeta(z)\delta(y)\right]=\frac{x}{\tilde{t}_{d}(z)}.$
(2.23)

As a function of $d$, $C_{d}$ is singular at $d=d_{\star}$, for any
$d_{\star}\in 2\operatorname{\mathbb{N}}\cap\operatorname{\mathbb{R}}_{\geq
3}$. Therefore, we cannot analytically continue (2.23) across the even integer
values $d=d_{\star}$, which subdivide the $d$-axis in distinct regions.

######  Definition 2.5.

An admissible region is a connected component of
$\operatorname{\mathbb{R}}_{d\geq 3}\setminus 2\operatorname{\mathbb{N}}$. Two
distinct scenarios are identified:

  * i)

$d_{\star}<d<d_{\star}+2$, in which the long quiver limit of [31] can be
adapted.

  * ii)

$3\leq d<4$, where the adaptation of the long quiver limit [31] holds, with
moreover $\hat{t}_{3\leq d<4}=\infty$. Eq. (2.23) is understood with both
sides multiplied by $w_{d}$ if $d\to 3^{+}$.

Let us stress that localisation requires maximal supersymmetry in $d\geq 6$,
so it falls outside our framework. We can nevertheless study (2.23) for
arbitrary $d\geq 3$, although its field theoretical interpretation is unclear
for $d\geq 6$.

In the admissible regions, a solution to (2.23) is obtained by a direct
generalisation of [31, 36]. Let us define the operator $\mathsf{D}^{n}$ as
follows. Acting on a power $x^{m}$,

$\mathsf{D}^{n}x^{m}=\begin{cases}\frac{\Gamma(m+1)}{\Gamma(m-n+1)}x^{m-n}&n\leq
m\\\ 0&n>m\end{cases}$

where $n,m\in\operatorname{\mathbb{R}}_{\geq 0}$ are arbitrary non-negative
numbers. This operator is a generalisation of the ordinary derivative,
$\frac{\mathrm{d}\ }{\mathrm{d}x^{n}}x^{m}$ when
$n,m\in\operatorname{\mathbb{N}}$.

###### Digression.

One may give a more exhaustive mathematical definition of $\mathsf{D}^{n}$,
but we will not need it. Also, had we set $t_{d}(z)=\infty$ for all $0<z<1$
and all $d\geq 3$, we may use the textbook definition of fractional derivative
for $\mathsf{D}^{n}$, see for instance [50]. ∎

Now, we let $\mathsf{D}^{d-3}$ act on both sides of (2.23) and use
$\mathsf{D}^{d-3}x=0$ if $d>4$, while we set $\frac{1}{\tilde{t}_{d}}=0$ if
$d<4$, according to our previous assumption. We obtain:

$\frac{1}{4}\partial_{y}^{2}\varrho(z,y)+\partial_{z}^{2}\varrho(z,y)+P^{2}\zeta(z)\delta(y)=0.$
(2.24)

This is the Poisson equation already found in $d=3$ and $d=5$. We have proved
it for every admissible region (including admissible regions with $d>6$). We
use the balancing condition (2.10) and the Fourier expansion (2.13) of
$\zeta(z)$ in terms of the Fourier coefficients $\left\\{R_{k}\right\\}_{k\geq
1}$ of the rank function $\mathcal{R}(\eta)$. The solution $\varrho(x,z)$ to
(2.24) is

$\varrho(z,x)=\frac{\pi}{N}\sum_{k=1}^{\infty}kR_{k}\sin(\pi
kz)\mathrm{e}^{-2\pi k\lvert x\rvert}.$ (2.25)

In the rest of the current section, we will use this solution to compute
several physical observables at arbitrary $d$. The results will interpolate
and extend known expressions in the literature.

###  2.6 Interpolating between $a$ and $F$

We rewrite the effective action (2.16) after the substitution (2.19) and the
ensuing change of variables:

$\displaystyle S_{\mathrm{eff}}[\varrho]\simeq
N^{2}P^{d-3}(d-2)^{d-2}s_{d}\int_{0}^{1}\mathrm{d}z$
$\displaystyle\int\mathrm{d}x\varrho(z,x)\left\\{\frac{(d-2)}{2C_{d}\tilde{t}_{d}(z)}x^{2}+[P^{2}\zeta(z)]\lvert
x\rvert^{d-2}\right.$ $\displaystyle+$
$\displaystyle\int\mathrm{d}y\left[\varrho(z,y)\frac{\lvert
x-y\rvert^{d-4}}{8}+\frac{\lvert
x-y\rvert^{d-2}}{2}\partial^{2}_{z}\varrho(z,y)\right]$ $\displaystyle-$
$\displaystyle\left.\left[\delta(z)+\delta(z-1)\right]\frac{\lvert
x\rvert^{d-2}}{2NP}\right\\}.$

Here we have used (2.17), and recall that the numerical coefficient $C_{d}$
appearing in the first line was defined in (2.22) and satisfies
$\frac{(d-2)^{d-1}s_{d}}{C_{d}}=u_{d}$. Moreover, the last line proceeds from
the term $\mathfrak{B}$, which imposes $Pz\in\left\\{1,P-1\right\\}$ and is
manifestly suppressed compared to the rest. To evaluate $S_{\text{eff}}$ on-
shell in the planar limit we insert the solution (2.25) and discard all the
sub-leading contributions.

######  Statement 2.6.

Consider a balanced linear quiver. With the notation from Subsection 2.3, in
the long quiver limit it holds that

$\displaystyle\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}\simeq
P^{(d-3)}\sum_{k=1}^{\infty}\left[c_{\mathrm{univ.}}(d)\leavevmode\nobreak\
k^{4-d}R_{k}^{2}+c_{\mathrm{gauge}}(d)\leavevmode\nobreak\
k^{-2}R_{k}\gamma_{k}\right].$

with $d$-dependent coefficients

$\displaystyle c_{\mathrm{univ.}}(d)$
$\displaystyle=(d-2)^{d-2}\frac{\pi^{5-d}}{2^{d}\sin\left(\frac{\pi
d}{2}\right)},$ $\displaystyle c_{\mathrm{gauge}}(d)$
$\displaystyle=\begin{cases}-(d-2)^{2}\frac{\pi^{\frac{d-3}{2}}}{\Gamma\left(\frac{d-3}{2}\right)}&\
4\leq d<6\\\ 0&\ \text{otherwise}.\end{cases}$

###### Proof.

At leading order in the long quiver limit we have

$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}\simeq-\left.S_{\text{eff}}\right\rvert_{\text{on-
shell}},$

with the on-shell effective action evaluated using the solution (2.25). To
reduce clutter, let us introduce the shorthand notation

$\tilde{w}_{d}:=-(d-2)w_{d}=(d-2)^{4-d}u_{d}.$

A computation akin to [51, 36] (which is the adaptation to long quivers of the
standard calculation of the free energy in matrix models, reviewed for
instance in [52]) gives:

$\left.S_{\text{eff}}\right\rvert_{\text{on-
shell}}=\frac{1}{2}N^{2}P^{d-3}\cdot(d-2)^{d-2}\int_{0}^{1}\mathrm{d}z\int_{-\infty}^{+\infty}\mathrm{d}x\varrho(z,x)\left[P^{2}\zeta(z)s_{d}\lvert
x\rvert^{d-2}+\frac{\tilde{w}_{d}}{2\tilde{t}_{d}(z)}x^{2}\right].$

We now insert the Fourier expansions of $\varrho(z,x)$ from (2.25) and of
$P^{2}\zeta(z)$ and $\frac{1}{g_{d}(z)}$ from (2.13) and (2.14), respectively.
We get

$\displaystyle\left.S_{\text{eff}}\right\rvert_{\text{on-
shell}}=\frac{1}{2}P^{(d-3)}\cdot(d-2)^{d-2}\int_{0}^{1}\mathrm{d}z$
$\displaystyle\int_{-\infty}^{+\infty}\mathrm{d}x\sum_{k=1}^{\infty}\pi
kR_{k}\sin(\pi kz)\mathrm{e}^{-2\pi k\lvert x\rvert}$ (2.26)
$\displaystyle+\sum_{\ell=1}^{\infty}\sin(\pi\ell
z)\left[\ell^{2}\pi^{2}R_{\ell}\leavevmode\nobreak\ s_{d}\lvert x\rvert^{d-2}\
+\ 8\tilde{w}_{d}\pi^{2}\gamma_{\ell}\leavevmode\nobreak\ x^{2}\right]$

where the Fourier coefficients $\tilde{\gamma}_{\ell}$ of the cosine part of
the gauge coupling drop out due to the integration against sines. According to
the assumption we have made along the way, the part in (2.26) proportional to
the gauge coupling has to be multiplied by $\theta_{4\leq d<6}$, where the
characteristic function $\theta_{\mathscr{S}}$ introduced in (2.6) is $1$ if
the condition $\mathscr{S}$ is satisfied and vanishes otherwise. Besides, we
introduce the shorthand notation

$\hat{s}_{d}:=s_{d}\frac{\pi^{4-d}}{2^{d}}\Gamma(d-1)=-\frac{\pi^{5-d}}{2^{d}\sin\left(\frac{\pi
d}{2}\right)}.$

Equipped with these definitions, we solve the integrals in (2.26) and find

$\left.S_{\text{eff}}\right\rvert_{\text{on-
shell}}=P^{(d-3)}(d-2)^{(d-2)}\sum_{k=1}^{\infty}\left[k^{4-d}R_{k}^{2}\hat{s}_{d}+k^{-2}R_{k}\gamma_{k}\tilde{w}_{d}\theta_{4\leq
d<6}\right].$ (2.27)

∎

Let us briefly pause to remark on expression (2.27).

  * •

The appearance of the pre-factor $1/\sin\left(\frac{\pi d}{2}\right)$ in
$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}$ is consistent with the
expectations from [53], as we explain momentarily. This piece blows up at even
integer $d$, and is finite in the admissible regions.

  * •

The overall coefficient $(d-2)^{(d-2)}$ is unavoidable, and generalises the
number $27=(5-2)^{5-2}$ in the free energy at $d=5$, and the number $1$ in the
free energy at $d=3$.

  * •

If the gauge coupling is the same at all gauge nodes, $\gamma_{k}=0$ and the
dependence on the gauge coupling drops out.

  * •

Formula (2.27) is expressed in terms of Fourier coefficients of the rank
function $\mathcal{R}(\eta)$ and of the inverse gauge coupling. However, it
was derived in the planar long quiver limit. It should be understood as
factoring out $N^{2}$, with both $\frac{R_{k}}{N}$ and $\frac{\gamma_{k}}{N}$
having a well-defined large-$N$ limit, in which they are $\mathcal{O}(1)$.

Giombi and Klebanov proposed a function $\tilde{F}_{d}$ that interpolates
between the sphere free energy in odd dimensions and the Weyl anomaly
coefficient in even dimensions [53]. We introduce our notation and
normalisation and then define $\tilde{F}_{d}$.

######  Lemma 2.7.

Let $d\in\operatorname{\mathbb{N}}$. Consider a Euclidean SCFT on
$\operatorname{\mathbb{S}}^{d}$ of radius $r$ with cutoff $\Lambda$, and
denote $\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}$ its partition function.
It has the form [54]:

$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}=\begin{cases}c_{d-2}(\Lambda
r)^{d-2}+\cdots+c_{1}(\Lambda r)+(-1)^{(d-1)/2}F+\mathcal{O}((\Lambda
r)^{-1})&d\text{ odd}\\\ c_{d-2}(\Lambda r)^{d-2}+\cdots+c_{2}(\Lambda
r)^{2}-4a\ln(\Lambda r)-F+\mathcal{O}((\Lambda r)^{-1})&d\text{
even}\end{cases}$ (2.28)

where $c_{n}$ are numerical coefficients.

######  Definition 2.8.

Consider (2.28). The finite quantity $F$ is called _free energy_ — note the
coefficient $(-1)^{(d-1)/2}$ when $d$ is odd and $-1$ when $d$ is even.

The quantity $a$, that appears as $-\frac{1}{4}$ of the coefficient of
$\ln(\Lambda r)$ in $d\in 2\operatorname{\mathbb{N}}$, is called _Weyl
anomaly_ coefficient. Our definition of $a$ differs from [53] by the factor
$\frac{1}{4}$. In our conventions, a free scalar has $a=\frac{1}{360}$, whilst
in the conventions of [53] a free scalar has $a=\frac{1}{90}$.

######  Definition 2.9.

The interpolating free energy is [53]

$\tilde{F}_{d}:=\sin\left(\frac{\pi
d}{2}\right)\ln\mathcal{Z}_{\mathbb{S}^{d}}.$ (2.29)

When $d\in\operatorname{\mathbb{N}}$, it gives

$\tilde{F}_{d}=\begin{cases}F&d\text{ odd}\\\ (-1)^{d/2}2\pi a&d\text{
even}\end{cases}$ (2.30)

and, in particular, it regulates the poles of $c_{\mathrm{univ.}}(d)$ in even
dimensions.

We now state the main result of this section.

######  Statement 2.10.

For long balanced linear quivers, we find

$\displaystyle\tilde{F}_{d}$ $\displaystyle\simeq
P^{d-3}(d-2)^{(d-2)}\frac{\pi^{5-d}}{2^{d}}\sum_{k=1}^{\infty}k^{4-d}R_{k}^{2}$
(2.31) $\displaystyle-\theta_{4\leq
d<6}P^{d-3}(d-2)^{2}\pi^{(d-3)/2}\frac{\sin\left(\frac{\pi
d}{2}\right)}{\Gamma\left(\frac{d-3}{2}\right)}\sum_{k=1}^{\infty}\frac{1}{k^{2}}R_{k}\gamma_{k}.$

At integer $d$ the coefficients attain the values listed in Table 2.

Table 2: Coefficients in the first and second line of (2.31), respectively,
specialised at $d=3,4,5,6$.

Formula (2.31) correctly reproduces the known results in $d=3$ and $d=5$,
extending the $d=5$ result to finite gauge coupling along the way (the
previous results assumed $g_{5}(z)\to\infty$).

At $d=4$, (2.31) yields

$\lim_{d\to 4}\tilde{F}_{d}\simeq\frac{\pi}{4}P\sum_{k=1}^{\infty}R_{k}^{2}.$
(2.32)

Using (2.30), this expression matches the anomaly coefficient $a$ predicted
from the supergravity dual in [21].

One important corollary of our continuation in $d$ is to extract the value
$\tilde{F}_{d=6}$ for long linear quivers, which cannot be derived directly
from localisation. We find:

$\lim_{d\to
6^{-}}\tilde{F}_{d}\simeq\frac{4}{\pi}P^{3}\sum_{k=1}^{\infty}\frac{R_{k}^{2}}{k^{2}}.$

###### Digression.

We comment on the consistency of the derivation presented herein with three-
dimensional $\mathcal{N}=4$ mirror symmetry. Mirror symmetry in $d=3$ is the
statement that two different-looking gauge theories with eight supercharges
flow in the IR to the same SCFT [55]. The mirror of a linear balanced quiver
is not balanced in general. However, if a balanced linear quiver has
hypermultiplets in the fundamental representations charged under only one
gauge factor, say $\mathrm{U}(N_{J})$ for $J\in\left\\{1,\dots,P-1\right\\}$,
its mirror is linear, balanced and with fundamental hypermultiplets charged
under a single gauge node.

In practice, this leads to consider pairs of balanced linear quivers whose
rank functions $\mathcal{R}(\eta)$ and $\mathcal{R}^{\vee}(\eta^{\vee})$ are
triangular. If one quiver has length $P-1$, the gauge ranks $N_{j}$ are all
proportional to $N\in\operatorname{\mathbb{N}}$ and the hypermultiplets are
placed at the $J^{\text{th}}$ node, it mirror has length
$P^{\vee}-1=\frac{NP}{P-J}-1$, where $(P-J)$ always divides $NP$ as a
consequence of the balancing condition.

Quivers of this kind have been considered in [24, 56], where it was observed
that the Fourier coefficients $R_{k},R^{\vee}_{k}$ of the two rank functions
are equal, $R_{k}=R^{\vee}_{k}$. We see from (2.31) that the free energies of
the two quivers are equal exactly at $d=3$. However, the lengths $P,P^{\vee}$
of the two quivers differ, and hence their overall coefficients in (2.31)
mismatch if $d>3$. This underlines how mirror symmetry is characteristic of
$d=3$. ∎

###  2.7 One-dimensional defects

The next observable we compute in arbitrary $d$ is the expectation value of a
half-BPS Wilson loop in the rank-$\mathsf{k}$ antisymmetric representation
$\mathsf{A}_{\mathsf{k}}$ of the $j_{\ast}^{\text{th}}$ gauge group
$\mathrm{U}(N_{j_{\ast}})$. Correlation functions of such Wilson loops are
computed exactly inserting

$\frac{\mathsf{k}!(N_{j_{\ast}}-\mathsf{k})!}{N_{j_{\ast}}!}\mathrm{Tr}_{\mathsf{A}_{\mathsf{k}}}\mathrm{e}^{2\pi\phi^{(j)}}$

into the matrix model, where the binomial coefficient in front comes from
$1/\dim\mathsf{A}_{\mathsf{k}}$. We now study the expectation value of these
Wilson loop operators procedure closely follows and extends [57, 56].

######  Statement 2.11.

Let $\langle W_{\mathsf{A}_{\mathsf{k}}}\rangle_{d}$ denote the expectation
value of a Wilson loop in $d$ dimensions, with the dependence on $j_{\ast}$
implicit in the notation. Then, in the long quiver limit, the dependence on
$d$ appears only through a pre-factor. Namely,

$\ln\langle
W_{\mathsf{A}_{\mathsf{k}}}\rangle_{d}\simeq\frac{d-2}{3}\ln\langle
W_{\mathsf{A}_{\mathsf{k}}}\rangle_{d=5},$

with the right-hand side extensively analysed in [57].

###### Proof.

Let $z_{\ast}:=j_{\ast}/P$ and

$\kappa(z):=\frac{\mathsf{k}}{N_{j}},\qquad
0<\kappa(z)<1\quad\forall\leavevmode\nobreak\ 0<z<1,$

so that $\nu(z_{\ast})\kappa(z_{\ast})=\frac{\mathsf{k}}{N}$. We want to
isolate the $\mathsf{k}$ largest eigenvalues at the node $j_{\ast}$ [57].
First, let $0\leq\tilde{\kappa}\leq\kappa(z_{\ast})$ and define the number
$x_{\ast}\equiv x_{\ast}(z_{\ast},\tilde{\kappa})$ such that a fraction
$\tilde{\kappa}$ of eigenvalues is larger than $x_{\ast}$, and the remaining
fraction $1-\tilde{\kappa}$ of eigenvalues is smaller than $x_{\ast}$.
Explicitly, $x_{\ast}$ is defined via the relation

$\int_{x_{\ast}}^{\infty}\mathrm{d}x\leavevmode\nobreak\
\varrho(z_{\ast},x)=\nu(z_{\ast})\tilde{\kappa}.$ (2.33)

Note that $\tilde{\kappa}\to 1$ gives $x_{\ast}\to-\infty$ and
$\tilde{\kappa}\to 0$ gives $x_{\ast}\to+\infty$.

By a straightforward adaptation of [57] we find that the expectation value of
the Wilson loop is

$\ln\langle W_{\mathsf{A}_{\mathsf{k}}}(z_{\ast})\rangle\simeq
2\pi(d-2)PN\nu(z_{\ast})\int_{0}^{\kappa(z_{\ast})}x_{\ast}(z_{\ast},\tilde{\kappa})\mathrm{d}\tilde{\kappa}.$

The overall factor originates as follows: the coefficient $2\pi$ comes from
the definition of the Wilson loop; the factor $(d-2)P$ is due to the long
quiver scaling (2.19) to pass from the variable $\phi$ to the $\mathcal{O}(1)$
variable $x$; finally, passing from the sum over the $N_{j_{\ast}}$
eigenvalues to an integral produces the coefficient
$N_{j_{\ast}}=N\nu(z_{\ast})$, as appropriate in the planar limit.

Using the inverse function theorem we pass from an integral over
$\tilde{\kappa}$ to an integral over $x_{\ast}$:

$\displaystyle\ln\langle W_{\mathsf{A}_{\mathsf{k}}}(z_{\ast})\rangle$
$\displaystyle\simeq 2\pi(d-2)\nu(z_{\ast})\leavevmode\nobreak\
PN\int_{x_{\ast}(z_{\ast},0)}^{x_{\ast}(z_{\ast},\kappa(z_{\ast}))}\frac{\mathrm{d}\tilde{\kappa}}{\mathrm{d}x_{\ast}}\leavevmode\nobreak\
x_{\ast}\mathrm{d}x_{\ast}$ $\displaystyle=2\pi(d-2)\leavevmode\nobreak\
PN\int^{x_{\ast}(z_{\ast},0)}_{x_{\ast}(z_{\ast},\kappa(z_{\ast}))}\varrho(z_{\ast},x_{\ast})\leavevmode\nobreak\
x_{\ast}\mathrm{d}x_{\ast}$ (2.34)

where in the second line we have differentiated both sides of (2.33) with
respect to $x_{\ast}$ and utilised the ensuing expression for
$\frac{\mathrm{d}\tilde{\kappa}}{\mathrm{d}x_{\ast}}$ (a minus sign from the
derivative is used to invert the integration domain).

With the due identifications, (2.34) only differs from the $d=5$ computation
of [57, Eq.(2.18)] by the replacement of the coefficient
$3=(5-2)\mapsto(d-2)$. In other words, the expectation value of a half-BPS
Wilson loop in $d$ dimensions, in the allowed regions for $d$, always has the
same form except for the overall coefficient $(d-2)$. ∎

That the result must be uniform in $d$ could be predicted looking at the
functional dependence of the Wilson loop expectation value. It is linear in
$\varrho$ and does not involve any of the $d$-dependent functions
$f_{\mathrm{h}},f_{\mathrm{v}},f_{0}$. Therefore, the only $d$-dependence
should appear through overall scaling, entirely dictated by (2.19), whence the
$(d-2)$.

We can then go on and solve (2.34) uniformly in $d$.

  * —

Integrating the left-hand side of (2.33) explicitly, we fix
$x_{\ast}(z_{\ast},\kappa)$.

  * —

We then perform the integration in (2.34) with the thus-obtained boundary
values $x_{\ast}(z_{\ast},0)$ and $x_{\ast}(z_{\ast},\kappa(z_{\ast}))$.

This algorithm gives $\ln\langle W_{\mathsf{A}_{\mathsf{k}}}(z_{\ast})\rangle$
for every choice of $j_{\ast}$ and of representation $\mathsf{k}$, in
arbitrary $d$.

At this point, we continue the calculation following [56] and [36, Sec.3.7].
We plug the solution (2.25) for $\varrho(z,x_{\ast})$ into (2.33) and evaluate
the integral. The number $x_{\ast}=x_{\ast}(z_{\ast},\kappa)$ is fixed by
solving the equation

$\sum_{k=1}^{\infty}R_{k}\sin(\pi
kz_{\ast})\left[\theta(-x_{\ast})+\frac{1}{2}\mathrm{sign}(x_{\ast})\mathrm{e}^{-2\pi
k\lvert x_{\ast}\rvert}\right]=\mathsf{k}$ (2.35)

where $\theta(\cdot)$ is the Heaviside function and we have multiplied both
sides by $N$. We then integrate (2.34) explicitly.555The left-hand side of
(2.35) is continuous at $x_{\ast}=0$, with the square bracket evaluating to
$\frac{1}{2}$ from both sides $x_{\ast}\to 0^{-}$ and $x_{\ast}\to 0^{+}$.

######  Statement 2.12.

The defect free energy for a Wilson loop in the rank-$\mathsf{k}$
antisymmetric representation $\mathsf{A}_{\mathsf{k}}$ of
$\mathrm{U}(N_{j_{\ast}})$ supported on
$\operatorname{\mathbb{S}}^{1}\subset\operatorname{\mathbb{S}}^{d}$ is

$\ln\langle W_{\mathsf{A}_{\mathsf{k}}}(z_{\ast})\rangle\simeq
P\frac{(d-2)}{2}\sum_{k=1}^{\infty}k^{-1}R_{k}\sin(\pi
kz_{\ast})\mathrm{e}^{-2\pi k\lvert x_{\ast}\rvert}\left(2\pi k\lvert
x_{\ast}\rvert+1\right),$ (2.36)

where $x_{\ast}$ is fixed by (2.35).

We observe that (2.36) only depends on the rank $\mathsf{k}$ through the
right-hand side of (2.35). Moreover, using the representation
$\mathsf{k}=\frac{N_{j_{\ast}}}{2}$ in the right-hand side of (2.35) we get
$x_{\ast}=0$ and, plugging in (2.34), we find

$\displaystyle\ln\langle W_{\mathsf{A}_{N_{j_{\ast}}/2}}(z_{\ast})\rangle$
$\displaystyle\simeq 2\pi(d-2)\leavevmode\nobreak\
PN\int_{0}^{\infty}\varrho(z_{\ast},x)\leavevmode\nobreak\ x\mathrm{d}x$
(2.37)

###  2.8 Two-dimensional defects

Taking inspiration from the defects constructed and studied in [58], we
consider two-dimensional half-BPS defects consisting of a chiral and an anti-
chiral multiplets of equal R-charge $q$ localised on a
$\operatorname{\mathbb{S}}^{2}\subset\operatorname{\mathbb{S}}^{d}$. These
chiral multiplets interact with the $d$-dimensional long quiver through the
gauge coupling to the $j_{\ast}^{\text{th}}$ gauge group
$\mathrm{U}(N_{j_{\ast}})$.666Different two-dimensional holographic defects at
large-$N$ have been studied in [59].

The one-loop determinant for a chiral multiplet of R-charge $q$ in the
fundamental representation and coupled to a vector multiplet is [60, 61]

$\mathcal{Z}_{\text{\rm 2d
defect}}=\prod_{a=1}^{N_{j_{\ast}}}\frac{\Gamma\left(\frac{q}{2}-\mathtt{i}\phi_{a}^{(j_{\ast})}+\frac{b_{a}^{(j_{\ast})}}{2}\right)}{\Gamma\left(1-\frac{q}{2}+\mathtt{i}\phi_{a}^{(j_{\ast})}+\frac{b_{a}^{(j_{\ast})}}{2}\right)}$
(2.38)

where $\vec{b}=\left\\{b_{a}^{(j)}\right\\}$ is the flux of the gauge
connection through the $\operatorname{\mathbb{S}}^{2}$ on which the defect
lives. Using the property $H_{2}(\operatorname{\mathbb{S}}^{d})=\emptyset$ if
$d\in\operatorname{\mathbb{N}}_{\geq 3}$, we define the second homology of
$\operatorname{\mathbb{S}}^{d}$ to be trivial also for real $d\geq 3$. As a
consequence, there is no gauge flux, $b_{a}^{(j)}=0$ $\forall
a=1,\dots,N_{j}$, $j=1,\dots,P-1$. This would not be the case localising on
$d$-dimensional manifolds other than $\operatorname{\mathbb{S}}^{d}$.

We _assume_ that the defect contribution be accounted for by importing the
one-loop determinant (2.38) evaluated in the background of the $d$-dimensional
vector multiplet. This is a reasonable working assumption, analogous to and
compatible with all know defect localisation results in the literature, but we
do not have a derivation for it. The R-charge $q$ is taken with respect to the
defect R-symmetry.

######  Statement 2.13.

Let $F_{\text{\rm 2d defect},d}$ denote the two-dimensional defect free energy
in $d$ dimensions, with the dependence on $j_{\ast}$ implicit in the notation.
In the long quiver limit it is given by

$F_{\text{\rm 2d defect},d}\simeq 2(1-q)N_{j_{\ast}}\ln P$

###### Proof.

The computation is similar in spirit to [58]. The insertion of the 2d defect
does not modify the leading order contribution in the long quiver limit. Let
use define

$f_{\text{\rm 2d
defect}}(\phi_{a}):=-\ln\left[\frac{\Gamma\left(\frac{q}{2}-\mathtt{i}\phi_{a}\right)}{\Gamma\left(1-\frac{q}{2}+\mathtt{i}\phi_{a}\right)}\cdot\frac{\Gamma\left(\frac{q}{2}+\mathtt{i}\phi_{a}\right)}{\Gamma\left(1-\frac{q}{2}-\mathtt{i}\phi_{a}\right)}\right],$

which accounts for the contribution from both the chiral and the anti-chiral
multiplets. Using the asymptotic expression of the $\Gamma$ function

$\ln\Gamma(z)\simeq\left(z-\frac{1}{2}\right)\ln(z)-z,$

the behaviour of the defect contribution in the long quiver scaling is:

$f_{\text{\rm 2d defect}}((d-2)Px)\simeq
2(1-q)\left[\ln\left((d-2)P\right)+\ln\lvert x\rvert\right].$

We note that, had we kept the dependence on the fluxes $\vec{b}$ through
$\operatorname{\mathbb{S}}^{2}$, it would drop out at this stage, due to
simplifications at leading order in the planar limit. We can thus write

$\displaystyle F_{\text{\rm 2d defect},d}$
$\displaystyle\simeq-\left.\ln\mathcal{Z}_{\text{\rm 2d
defect}}\right\rvert_{\text{\rm on shell}}$
$\displaystyle=N\int\mathrm{d}x\varrho(z_{\ast},x)f_{\text{\rm 2d
defect}}((d-2)Px)$
$\displaystyle=2(1-q)N\left[\nu(z_{\ast})\ln\left((d-2)P\right)+\sum_{k=1}^{\infty}\frac{2\pi
k}{N}R_{k}\sin(\pi
kz_{\ast})\int_{-\infty}^{+\infty}\mathrm{d}x\leavevmode\nobreak\
\mathrm{e}^{-2\pi k\lvert x\rvert}\ln\lvert x\rvert\right]$
$\displaystyle=2(1-q)N_{j_{\ast}}\left[\ln\left((d-2)P\right)-\gamma_{\text{\tiny
EM}}\right]-2(1-q)\sum_{k=1}^{\infty}R_{k}\sin(\pi kz_{\ast})\ln(2\pi k),$

where $\gamma_{\text{\tiny EM}}$ is the Euler–Mascheroni constant. The last
term is ambiguous due to the $\ln P$ dependence of the leading term, and
likewise for the coefficients of $P$ inside the logarithm. Only the
coefficient of $\ln P$ is unambiguous and yields the leading behaviour. ∎

###  2.9 Three-dimensional defects

A class of three-dimensional defects in $d=5$ long quivers was studied in
[58]. The defects consist in a pair of 3d chiral and antichiral multiplets
localised on a
$\operatorname{\mathbb{S}}^{3}\subset\operatorname{\mathbb{S}}^{d}$ and
charged under the $j_{\ast}^{\text{th}}$ gauge group
$\mathrm{U}(N_{j_{\ast}})$. We now formally extend the computation of [58] by
promoting the $d=5$ gauge theory to arbitrary $d\geq 3$.

######  Statement 2.14.

Let $F_{\text{\rm 3d defect},d}$ denote the 3d defect free energy in $d$
dimensions, with the dependence on $j_{\ast}$ implicit in the notation. Then,
in the long quiver limit, the dependence on $d$ appears only through a pre-
factor. Namely,

$F_{\text{\rm 3d defect},d}\simeq\frac{d-2}{3}F_{\text{\rm 3d defect},d=5},$

with the right-hand side extensively analysed in [58].

###### Proof.

The sphere partition function in presence of the defect is modified by the
inclusion of the one-loop determinant of the chiral multiplets in the
background of the $d=5$ vector multiplet. This fact was derived from
localisation in [58] when $d=5$, and here we _assume_ that this form extends
to other values of $d$. We proceed directly to the evaluation of the defect
free energy

$F_{\text{\rm 3d
defect},d}=-\ln\frac{\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}^{\text{\rm 3d
defect}}}{\mathcal{Z}_{\operatorname{\mathbb{S}}^{d}}},$

where the argument of the logaritm is the ratio between the partition
functions with and without defects. This quantity isolates the contribution of
the 3d defect.

In the long quiver regime, we write $z_{\ast}=j_{\ast}/P$ and
$N_{j_{\ast}}=N\nu(z_{\ast})$. Let us also denote $q$ the defect R-charge of
the chiral and antichiral pair, which equals twice its scaling dimension. The
3d defect free energy is obtained as [58]

$F_{\text{\rm 3d defect},d}(z_{\ast})\simeq
2\pi\left(1-\frac{q}{2}\right)(d-2)PN\nu(z_{\ast})\int_{-\infty}^{\infty}\mathrm{d}x\leavevmode\nobreak\
\varrho(z_{\ast},x)\lvert x\rvert.$ (2.39)

∎

Plugging (2.25) into (2.39), we arrive at the following statement.

######  Statement 2.15.

The defect free energy for a chiral and an antichiral multiplets of R-charge
$q$ supported on
$\operatorname{\mathbb{S}}^{3}\subset\operatorname{\mathbb{S}}^{d}$ is

$F_{\text{\rm 3d
defect},d}(z_{\ast})\simeq\left(1-\frac{q}{2}\right)(d-2)P\nu(z_{\ast})\sum_{k=1}^{\infty}k^{-1}R_{k}\sin(\pi
kz_{\ast}).$

Comparing (2.39) with (2.37) we also observe the identity

$\frac{F_{\text{\rm 3d
defect},d}(z_{\ast})}{\nu(z_{\ast})}\simeq\left(2-q\right)\ln\langle
W_{\mathsf{A}_{N_{j_{\ast}}/2}}(z_{\ast})\rangle.$

For a fixed $j_{\ast}$ it is always possible to choose the normalisation
$N=N_{j_{\ast}}$, so that the left-hand side is divided by $\nu(z_{\ast})=1$.
The 1d and 3d defect free energies are then exactly equal at $q=1$,
corresponding to a conformal dimension $\frac{1}{2}$ for the defect chiral
multiplets.

###  2.10 Mirror-type identities

Recall the rank function $\mathcal{R}(\eta)$ with Fourier coefficients
$\left\\{R_{k}\right\\}_{k\geq 1}$, which takes the explicit form

$\mathcal{R}(\eta)=\begin{cases}N_{1}\eta&\quad 0\leq\eta<1\\\
N_{j}+(N_{j+1}-N_{j})(\eta-j)&\quad j\leq\eta<j+1\quad\forall j=1,\dots,P-2\\\
N_{P-1}(P-\eta)&\quad P-1\leq\eta\leq P.\end{cases}$

Let $K:=\sum_{j=1}^{P-1}K_{j}$ be the total number of fundamental flavours in
the quiver. Note that, using the balancing condition (2.4), this number is
easily obtained by

$\displaystyle K$
$\displaystyle=\int_{0}^{P}\mathrm{d}\eta\sum_{j=1}^{P-1}K_{j}\delta(\eta-j)$
$\displaystyle=-\int_{0}^{P}\mathrm{d}\eta\partial_{\eta}^{2}\mathcal{R}(\eta)=\partial_{\eta}\mathcal{R}(0)-\partial_{\eta}\mathcal{R}(P).$

Only throughout this subsection we impose the assumptions:

$(i)\ \frac{K}{P}\in\operatorname{\mathbb{N}}\qquad\text{ and }\qquad(ii)\
\frac{P}{K}N_{j}\in\operatorname{\mathbb{N}}\ \forall j=1,\dots,P-1.$ (2.40)

Let us now define the rank function $\mathcal{R}^{\vee}(\eta^{\vee})$ for a
quiver of length $P^{\vee}:=K$ as

$\mathcal{R}^{\vee}(\eta^{\vee})=\begin{cases}N_{1}^{\vee}\eta^{\vee}&\quad
0\leq\eta<\frac{K}{P}\\\
\frac{K}{P}N_{j}^{\vee}+(N_{j+1}^{\vee}-N_{j}^{\vee})(\eta^{\vee}-j)&\quad
j\frac{K}{P}\leq\eta^{\vee}<(j+1)\frac{K}{P}\quad\forall j=1,\dots,P-2\\\
N_{K-1}^{\vee}(K-\eta^{\vee})&\quad
K\left(1-\frac{1}{P}\right)\leq\eta^{\vee}\leq K\end{cases}$

where $N_{j}^{\vee}=N_{j}\frac{P}{K}$. We will denote the Fourier coefficients
of $\mathcal{R}^{\vee}(\eta^{\vee})$ by $\left\\{R_{k}^{\vee}\right\\}_{k\geq
1}$.

$\vee$$\vee$$\vee$$\vee$$\scriptscriptstyle 8$$\scriptscriptstyle
8$$\scriptscriptstyle 8$$\scriptscriptstyle 8$$\scriptscriptstyle
8$$\scriptscriptstyle 2$$\scriptscriptstyle 4$$\scriptscriptstyle
6$$\scriptscriptstyle 8$$\scriptscriptstyle
8$$\scriptstyle\cdots$$\scriptscriptstyle 8$$\scriptscriptstyle
2$$\scriptscriptstyle 4$$\scriptscriptstyle 6$$\scriptscriptstyle
2$$\scriptscriptstyle 2$$\scriptscriptstyle 6$$\scriptscriptstyle
4$$\scriptscriptstyle 2$$\scriptscriptstyle 8$$\scriptscriptstyle
1$$\scriptscriptstyle 2$$\scriptscriptstyle 3$$\scriptscriptstyle
4$$\scriptscriptstyle 5$$\scriptscriptstyle 6$$\scriptscriptstyle
3$$\scriptscriptstyle 4$$\scriptscriptstyle 4$$\scriptscriptstyle
8$$\scriptscriptstyle 4$$\scriptscriptstyle 8$$\scriptscriptstyle
2$$\scriptscriptstyle 4$$\scriptscriptstyle 6$$\scriptscriptstyle
8$$\scriptscriptstyle 6$$\scriptscriptstyle 4$$\scriptscriptstyle
2$$\scriptscriptstyle 4$$\scriptscriptstyle 6$$\scriptscriptstyle
12$$\scriptscriptstyle 12$$\scriptscriptstyle 12$$\scriptscriptstyle
6$$\scriptscriptstyle 6$$\scriptscriptstyle 6$$\scriptscriptstyle
3$$\scriptscriptstyle 6$$\scriptscriptstyle 9$$\scriptscriptstyle
12$$\scriptscriptstyle 12$$\scriptstyle\cdots$$\scriptscriptstyle
12$$\scriptscriptstyle 3$$\scriptscriptstyle 6$$\scriptscriptstyle
9$$\scriptscriptstyle 3$$\scriptscriptstyle 3$ Figure 3: Left: balanced linear
quivers that satisfy (2.40). Right: the corresponding quivers characterised by
$\mathcal{R}^{\vee}(\eta^{\vee})$. From top to bottom:
$(P,K)=(4,16),(4,8),(4,8),(6,12)$. In the right quiver of the first line the
dots omit 6 gauge nodes of rank 8; in the right quiver of the fourth line the
dots omit 2 gauge nodes of rank 12. In $d=3$ the second and third pairs of
quivers are mirror pairs.

The functions $\mathcal{R}(\eta)$ and $\mathcal{R}^{\vee}(\eta^{\vee})$ are in
general different, unless $K=P$, and provide different linear quivers.
Examples of quivers related by this transformation are given in Figure 3.
However, they have equal Fourier coefficients,

$\displaystyle R_{k}$
$\displaystyle=\frac{2}{P}\int_{0}^{P}\mathrm{d}\eta\mathcal{R}(\eta)\sin\left(\frac{\pi
k\eta}{P}\right)$
$\displaystyle=\frac{2}{K}\int_{0}^{K}\mathrm{d}\eta^{\vee}\mathcal{R}^{\vee}(\eta)\sin\left(\frac{\pi
k\eta^{\vee}}{P}\right)=R_{k}^{\vee},$

using the change of variables $\eta^{\vee}=\frac{K}{P}\eta$. From this simple
identity and the fact that the quantities we compute are expressed mainly in
terms of Fourier coefficients, we obtain the following mirror-type identities.

######  Statement 2.16.

Consider a balanced linear quiver that satisfies (2.40), and denote with a
symbol ∨ the quantities in the quiver constructed from
$\mathcal{R}^{\vee}(\eta^{\vee})$. It holds that

$\displaystyle\frac{\tilde{F}_{d}}{P^{d-3}}$
$\displaystyle=\frac{\tilde{F}^{\vee}}{K^{d-3}},$ (2.41)
$\displaystyle\frac{\ln\langle
W_{\mathsf{A}_{\mathsf{k}}}(z_{\ast})\rangle}{P}$
$\displaystyle=\frac{\ln\langle
W_{\mathsf{A}_{\mathsf{k}}}^{\vee}(z_{\ast})\rangle}{K},\qquad F_{\text{\rm 3d
defect},d}(z_{\ast})=F_{\text{\rm 3d defect},d}^{\vee}(z_{\ast}).$ (2.42)

At $d=3$ the relation (2.41) becomes an exact equality of free energies from
two distinct quiver gauge theories, first noted in [24]. When the initial
quiver has $K$ hypermultiplets at a single node, then the two rank functions
$\mathcal{R}(\eta)$ and $\mathcal{R}^{\vee}(\eta^{\vee})$ describe genuine
three-dimensional mirror theories and the equality (2.41) becomes a check of
mirror symmetry [24]. On the other hand, the identity (2.42) is not a
statement on mirror symmetry, since we are equating two Wilson loops, instead
of a Wilson and a vortex loop [62].

Let us now elaborate on aspects of (2.41)-(2.42). The equality of the free
energies divided by a power of the length of the quiver holds directly at the
infinite coupling point. Moreover, for every choice of gauge couplings
$g_{d,j}$ in the initial quiver, there always exists a suitable choice of
gauge couplings $g_{d,j}^{\vee}$ such that they have equal Fourier
coefficients, thus extending (2.41) to finite gauge couplings. We also stress
that in (2.42) the defects are located at the same relative position
$z_{\ast}$ along the quiver, which will correspond to the nodes
$j_{\ast}=\lfloor Pz_{\ast}\rfloor$ and $j_{\ast}^{\vee}=\lfloor
Kz_{\ast}\rfloor=\frac{K}{P}j_{\ast}$. The value of the parameter $x_{\ast}$
that appears in the evaluation of the Wilson loop expectation value is
uniquely fixed by $z_{\ast}$, $\left\\{R_{k}\right\\}$ and the representation
$\mathsf{k}$, thus it will be equal on the two sides of the map. Furthermore,
observe that the non-universal part of $F_{\text{3d defect},d}$ is contained
in the pre-factor $P\nu(z_{\ast})$. Using $P^{\vee}=K$ and
$N_{Kj_{\ast}/P}^{\vee}=\frac{P}{K}N_{j_{\ast}}$, this pre-factor is left
invariant under the transformation, namely
$P\nu(z_{\ast})=P^{\vee}\nu^{\vee}(z_{\ast})$.

###### Digression.

Partial Higgsing of a linear quiver preserves the balancing condition.
Therefore, one can in principle always start with pairs of quivers with
flavours at a single node, for which the graph of the rank function is
triangular, and perform subsequent partial Higgsing to reach other balanced
quivers, see Figure 4. It would be interesting to see if this observation can
be used to provide a physical derivation of (2.41). ∎

partial Higgsing$\scriptscriptstyle N$$\scriptscriptstyle
N$$\scriptscriptstyle N$$\scriptscriptstyle N$$\scriptscriptstyle
N$$\scriptscriptstyle N$$\scriptscriptstyle 2N$$\scriptscriptstyle
N$$\scriptscriptstyle 2N$ Figure 4: Two quivers that satisfy (2.40) are
related by partial Higgsing. Left: The graph of the rank function
$\mathcal{R}(\eta)$ is triangular, hence $\mathcal{R}^{\vee}(\eta^{\vee})$
describes the genuine mirror in $d=3$. Right: The rank functions
$\mathcal{R}(\eta)$ and $\mathcal{R}^{\vee}(\eta^{\vee})$ are not related by
duality.

###  2.11 Mass deformation

Next, we study the long quiver limit when the fundamental hypermultiplets are
given a real mass, closely following [36, Sec.3].

Let $\mathscr{F}\in\operatorname{\mathbb{N}}$ be an $\mathcal{O}(1)$ number,
in the sense that it does not scale with $N$ nor $P$ in the long quiver limit.
At every $j$, we split the number $K_{j}$ of fundamental hypermultiplets into
a collection

$\left\\{K_{j,\alpha}\right\\}_{\alpha=1,\dots,\mathscr{F}},\qquad\sum_{\alpha=1}^{\mathscr{F}}K_{j,\alpha}=K_{j}.$

The $\alpha^{\text{th}}$ collection at every $j$ is assigned a mass
$m_{\alpha}$, uniform along the quiver (that is, $m_{\alpha}$ does not depend
on the index $j=1,\dots,P$). The canonical way to introduce a real mass is to
couple the hypermultiplet to a background vector multiplet for the flavour
symmetry group. The real mass parameters $m_{\alpha}$ enter the matrix model
on the same footing as the dynamical scalar zero-mode $\vec{\phi}$, although
they are not integrated over. Therefore, by the long quiver scaling (2.19),
the masses scale as

$m_{\alpha}=(d-2)P\mu_{\alpha},$

where $\mu_{\alpha}$ are $\mathcal{O}(1)$ mass parameters. Below we employ the
notation $\mu_{\alpha\beta}:=\mu_{\alpha}-\mu_{\beta}$ and
$\vec{\mu}:=(\mu_{1},\dots,\mu_{\mathscr{F}})$.

To the $\alpha^{\text{th}}$ collection of hypermultiplets we associate a
Veneziano function $\zeta_{\alpha}(z)$ in the obvious way. Imposing that the
balancing condition (2.4) is satisfied by every collection of hypermultiplets,
the rank function splits into

$\mathcal{R}(\eta)\ \mapsto\
\sum_{\alpha=1}^{\mathscr{F}}\mathcal{R}_{\alpha}(\eta).$

We omit the details of the derivation, because they are identical to [36,
Sec.3]. The upshot is that the derivation in Subsections 2.4-2.5 goes through
with only mild modifications, and eventually the eigenvalue density takes the
form

$\varrho(z,x)=\sum_{\alpha=1}^{\mathscr{F}}\varrho_{\alpha}(z,x)\
,\qquad\varrho_{\alpha}(z,x)=\frac{\pi}{N}\sum_{k=1}^{\infty}kR_{\alpha,k}\sin(\pi
kz)\mathrm{e}^{-2\pi k\lvert x\rvert}.$

Evaluating the interpolating free energy with this solution we obtain

$\displaystyle\tilde{F}_{d}(\vec{\mu})\simeq-\frac{1}{2}N^{2}P^{(d-3)}\sin\left(\frac{\pi
d}{2}\right)(d-2)^{d-2}$
$\displaystyle\int_{0}^{1}\mathrm{d}z\int_{-\infty}^{+\infty}\mathrm{d}x\sum_{\alpha=1}^{\mathscr{F}}\varrho_{\alpha}(z,x)$
$\displaystyle\times\left[\sum_{\beta=1}^{\mathscr{F}}P^{2}\zeta_{\beta}(z)s_{d}\lvert
x-\mu_{\beta\alpha}\rvert^{d-2}+\frac{\lvert
w_{d}\rvert}{2\tilde{t}_{d}(z)}x^{2}\right].$

In the notation we are leaving implicit the fact that $\tilde{F}_{d}$ depends
not only on the values of the masses, but also on the way of assigning them to
the hypermultiplets, i.e. on the splitting
$\sum_{\alpha=1}^{\mathscr{F}}K_{j,\alpha}=K_{j}$. Using
$\sum_{\alpha=1}^{\mathscr{F}}R_{\alpha,k}=R_{k}$, the part of
$\tilde{F}_{d}(\vec{\mu})$ that depends on the gauge couplings is identical to
the second line of (2.31), thus it is not affected by the mass deformation.
Denoting

$\delta\tilde{F}_{d}(\vec{\mu}):=\tilde{F}_{d}(\vec{\mu})-\tilde{F}_{d}(0)$

the mass-dependent part of $\tilde{F}_{d}$, we get

$\displaystyle\delta\tilde{F}_{d}(\vec{\mu})$
$\displaystyle\simeq-P^{(d-3)}\sin\left(\frac{\pi
d}{2}\right)(d-2)^{(d-2)}\frac{\pi^{3}}{4}s_{d}$
$\displaystyle\times\sum_{\alpha,\beta=1}^{\mathscr{F}}\sum_{k=1}^{\infty}k^{3}R_{\alpha,k}R_{\beta,k}\int_{-\infty}^{+\infty}\mathrm{d}x\leavevmode\nobreak\
\mathrm{e}^{-2\pi k\lvert x\rvert}\left(\lvert
x-\mu_{\beta\alpha}\rvert^{d-2}-\lvert x\rvert^{d-2}\right).$

The integral can be performed explicitly for arbitrary $d$, but the resulting
expression is rather cumbersome:

$\displaystyle\delta\tilde{F}_{d}(\vec{\mu})$ $\displaystyle\simeq
P^{(d-3)}\frac{(d-2)^{(d-2)}\sin\left(\pi
d\right)\Gamma(2-d)}{4(d-1)}\sum_{\alpha,\beta=1}^{\mathscr{F}}\sum_{k=1}^{\infty}(\pi
k)^{4}R_{\alpha,k}R_{\beta,k}$
$\displaystyle\times\left[\frac{2\mu_{\beta\alpha}^{d}}{d}{}_{1}F_{2}\left(1;\frac{d+1}{2},\frac{d+2}{2};(\pi
k\mu_{\beta\alpha})^{2}\right)-\frac{\mu_{\beta\alpha}^{d-1}}{k\pi}{}_{1}F_{2}\left(1;\frac{d+1}{2},\frac{d}{2};(\pi
k\mu_{\beta\alpha})^{2}\right)+\cdots\right]$

with the ellipsis containing additional terms involving Meijer’s $G$-function
with argument $(\pi k\mu_{\beta\alpha})^{2}$. Evaluating
$\delta\tilde{F}_{d}(\vec{\mu})$ at integer $d$, we find

$\delta\tilde{F}_{d}(\vec{\mu})\simeq
P^{(d-3)}(d-2)^{(d-2)}\frac{\pi^{5-d}}{2^{d}}\sum_{\alpha,\beta=1}^{\mathscr{F}}\sum_{k=1}^{\infty}k^{4-d}R_{\alpha,k}R_{\beta,k}\leavevmode\nobreak\
\wp_{d}(2\pi k\mu_{\beta\alpha}),$

where $\wp_{d}(\cdot)$ is

$\displaystyle\wp_{d}(y)=\begin{cases}\mathrm{e}^{-y}+c_{1}y+c_{3}y^{3}+\cdots+c_{d-2}y^{d-2}&d\text{
odd }\\\ 1+c_{2}y^{2}+\cdots+c_{d-2}y^{d-2}&d\text{ even }.\end{cases}$

That is, we find a polynomial in $2\pi k\mu_{\alpha\beta}$, with the powers
having precisely the structure discussed in [54]. These polynomial pieces in
$\mu_{\alpha\beta}$ are scheme-dependent and can be cancelled by local
counterterms. Additionally, at odd $d$, we observe the exponential dependence
on the masses already found in [36].

##  3 Four-dimensional long quiver SCFTs

In this section we study four-dimensional $\mathcal{N}=2$ linear quivers shown
in Figure 5. The gauge group is a product of special unitary groups
$\mathrm{SU}(N_{j})$, $j=1,\dots,P-1$.

We assume that the gauge theory is balanced, which, in $d=4$, implies it is
conformal. One consequence of this fact is that there are two ways of taking
the limit:

  * •

the large-$P$ limit at fixed ’t Hooft coupling;

  * •

going to strong coupling first, and only then take the large-$P$ limit.

We will analyse both prescriptions and show that they explore different
regimes.

$\mathrm{SU}(\\!N_{\scriptscriptstyle
P-1}\\!)$$\cdots$$\mathrm{SU}(\\!N_{\scriptscriptstyle
1}\\!)$$\mathrm{SU}(\\!N_{\scriptscriptstyle
2}\\!)$$\mathrm{U}(\\!K_{\scriptscriptstyle
P-1}\\!)$$\mathrm{U}(\\!K_{\scriptscriptstyle
1}\\!)$$\mathrm{U}(\\!K_{\scriptscriptstyle 2}\\!)$ Figure 5: Linear quiver of
length $P-1$. Circular nodes indicate gauge groups, square nodes indicate
flavour symmetries. When the quiver is balanced, it describes a four-
dimensional $\mathcal{N}=2$ SCFT.

###  3.1 Anomaly coefficients from counting supermultiplets

Before delving into the study of the partition function of the gauge theory on
$\operatorname{\mathbb{S}}^{4}$, we derive the anomaly coefficients $a$ and
$c$ of the four-dimensional long quiver SCFTs from the relations [63]

$a=\frac{5n_{\rm v}+n_{\rm h}}{24},\qquad c=\frac{2n_{\rm v}+n_{\rm h}}{12}.$
(3.1)

In these expressions, $n_{\rm v}$ and $n_{\rm h}$ are, respectively, the
number of vector multiplet and hypermultiplet modes.

######  Statement 3.1.

In the long quiver limit, (3.1) yields

$a\simeq c\simeq\frac{P}{8}\sum_{k=1}^{\infty}R_{k}^{2}$ (3.2)

where we recall that $R_{k}$ are the Fourier coefficients of the rank
function.

###### Proof.

In the linear quivers we are considering, the number of vector multiplet and
hypermultiplet modes are given by

$n_{\rm v}=\sum_{j=1}^{P-1}\left(N_{j}^{2}-1\right),\qquad n_{\rm
h}=\sum_{j=1}^{P-2}N_{j}N_{j+1}+\sum_{j=1}^{P-1}N_{j}K_{j},$

where the first and second contributions in $n_{\rm h}$ come from
bifundamental and fundamental hypermultiplets. The contribution from the
bifundamental hypermultiplets is rewritten in a more symmetric form:

$\sum_{j=1}^{P-2}N_{j}N_{j+1}=\frac{1}{2}\sum_{j=1}^{P-1}N_{j}\left(N_{j+1}+N_{j-1}\right),\qquad
N_{0}:=0=:N_{P}.$

Imposing the balancing condition (2.4), we get

$n_{\rm v}=\sum_{j=1}^{P-1}\left(N_{j}^{2}-1\right),\qquad n_{\rm
h}=\sum_{j=1}^{P-1}N_{j}\left(2N_{j}-\frac{1}{2}N_{j-1}-\frac{1}{2}N_{j+1}\right).$

Computing the $a$ and $c$ anomaly coefficients with these values we obtain

$\displaystyle 48a$
$\displaystyle=\sum_{j=1}^{P-1}\left[12N_{j}^{2}-N_{j}\left(N_{j+1}-2N_{j}+N_{j-1}\right)\right]-10(P-1),$
$\displaystyle 48c$
$\displaystyle=\sum_{j=1}^{P-1}\left[12N_{j}^{2}-2N_{j}\left(N_{j+1}-2N_{j}+N_{j-1}\right)\right]-8(P-1).$

We now take the long quiver limit of these expressions, following the notation
of Section 2. We find

$\displaystyle 48a$
$\displaystyle=N^{2}P\int_{0}^{1}\mathrm{d}z\leavevmode\nobreak\
\left[12\nu(z)^{2}-\frac{1}{P^{2}}\nu(z)\partial_{z}^{2}\nu(z)\right],$
$\displaystyle 48c$
$\displaystyle=N^{2}P\int_{0}^{1}\mathrm{d}z\leavevmode\nobreak\
\left[12\nu(z)^{2}-\frac{2}{P^{2}}\nu(z)\partial_{z}^{2}\nu(z)\right].$

The scaled rank function $\nu(z)$ converges to a function of class
$C^{2}((0,1))$ in the limit $P\to\infty$, hence the second piece drops out. We
then expand $\nu(z)$ in Fourier series, recalling that
$N\nu(z)=\mathcal{R}(\eta=zP)$, and integrate over $z$. We thus derive the
identity

$a\simeq\frac{P}{8}\sum_{k=1}^{\infty}R_{k}^{2}\simeq c,$

for all four-dimensional linear quivers, at leading order in the planar and
long-quiver limit. ∎

The relation $a\simeq c$ is expected to hold in any SCFT with a supergravity
dual at large-$N$ [64], and we have checked it explicitly for linear quivers.
It matches the supergravity computation in [21, Eq.(2.41)]. Recalling that
$\tilde{F}_{d=4}=2\pi a$, (3.2) matches exactly with the derivation (2.32)
using the iterpolating function $\tilde{F}_{d}$ of Giombi–Klebanov [53].

We should stress that both large-$N$ and large-$P$ are needed to ensure $a=c$.
The simplest of all quivers, the $\mathcal{N}=2$ QCD, has $P=2$ and for it $a$
and $c$ differ by $\mathcal{O}(N^{2})$. It has been known for a while that the
string dual of $\mathcal{N}=2$ QCD must be a strongly coupled string theory,
even in the limit of infinite ’t Hooft coupling [65]. The central charge
computation above demonstrates, in a simple way, that only long quivers can
have a weakly coupled string dual well approximated by supergravity.

We can in fact compute the finite-$P$ corrections to $a-c$ while keeping the
large-$N$ limit, and get:

$a-c=\frac{\pi^{2}}{96P}\sum_{k=1}^{\infty}k^{2}R_{k}^{2}.$

This value should arise from stringy corrections in the holographic dual
background, and we have expressed it here using the rank function, which in
turn characterises the backgrounds of Gaiotto–Maldacena [16].

###  3.2 Four-dimensional long quiver partition function

The sphere partition function of the $d=4$, $\mathcal{N}=2$ linear quiver is
of the form introduced in Subsection 2.2. With the notation therein, we have
[22]

$\displaystyle f_{\mathrm{v}}(\phi)$ $\displaystyle=-\ln(\phi)-\ln H(\phi)$
$\displaystyle f_{\mathrm{h}}(\phi)$ $\displaystyle=\ln H(\phi).$

The function $H(\cdot)$ is expressed using Barnes’s $G$-function $G(\cdot)$
as777Our $H(\cdot)$ equals $H(\sqrt{-1}\cdot)$ of [22].

$H(\phi):=G(1+\mathtt{i}\phi)G(1-\mathtt{i}\phi),$

and it admits an infinite product representation [22, Eq.(4.49)] which, taking
the logarithm, reads

$\ln H(\phi)=(1+\gamma_{\text{\tiny
EM}})\phi^{2}+\sum_{n=1}^{\infty}\left[-\frac{\phi^{2}}{n}+n\ln\left(1+\frac{\phi^{2}}{n^{2}}\right)\right].$

Here $\gamma_{\text{\tiny EM}}$ is the Euler–Mascheroni constant.

After using the substitutions explained in full generality in Subsection 2.2,
the matrix model effective action becomes

$\displaystyle S_{\mathrm{eff}}\simeq
PN^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}\phi\rho(z,\phi)$
$\displaystyle\left\\{\frac{1}{2t_{4}(z)}\phi^{2}+\zeta(z)\ln H(\phi)\right.$
$\displaystyle-$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\rho(z,\sigma)\left[\ln\lvert
H(\phi-\sigma)\rvert+\ln\lvert\phi-\sigma\rvert\right]$ $\displaystyle+$
$\displaystyle\left.\int\mathrm{d}\sigma\left[\frac{1}{2P^{2}}\partial_{z}^{2}\rho(z,\sigma)+\rho(z,\sigma)\right]\ln\lvert
H(\phi-\sigma)\rvert\right\\}$

where
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int$
indicates the principal value integral. We have also used $u_{d=4}=1$. This
expression is the direct four-dimensional analogue of (2.15).

We combine part of the terms coming from vector multiplets and
hypermultiplets:

$\displaystyle-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\rho(z,\sigma)\ln\lvert
H(\phi-\sigma)\rvert+\int\mathrm{d}\sigma\rho(z,\sigma)\ln\lvert
H(\phi-\sigma)\rvert$
$\displaystyle=\int\mathrm{d}\sigma\rho(z,\sigma)\ln\lvert
H(\phi-\sigma)\rvert\delta(\sigma-\phi)$ $\displaystyle=\ln\lvert H(0)\rvert$
$\displaystyle=0,$

using that $H(0)=G(1)^{2}=1$. With this simplification we have

$\displaystyle S_{\mathrm{eff}}\simeq
PN^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}\phi\rho(z,\phi)$
$\displaystyle\left\\{\frac{1}{2t_{4}(z)}\phi^{2}+\zeta(z)\ln
H(\phi)-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\rho(z,\sigma)\ln\lvert\phi-\sigma\rvert\right.$
(3.3)
$\displaystyle\left.+\frac{1}{2P^{2}}\int\mathrm{d}\sigma\leavevmode\nobreak\
\ln\lvert H(\phi-\sigma)\rvert\leavevmode\nobreak\
\partial_{z}^{2}\rho(z,\sigma)\right\\}.$

Let us define the function

$\mathscr{K}(\phi):=-\partial_{\phi}\ln
H(\phi)=\phi\left[\psi(1+\mathtt{i}\phi)+\psi(1-\mathtt{i}\phi)-2\right],$

where $\psi(\cdot)$ is the digamma function.

The saddle point equation derived from the effective action (3.3) is

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\frac{\rho(z,\sigma)}{\phi-\sigma}+\frac{1}{2P^{2}}\int\mathrm{d}\sigma\mathscr{K}(\phi-\sigma)\partial_{z}^{2}\rho(z,\sigma)=\frac{\phi}{2t_{4}(z)}-\frac{1}{2P^{2}}[P^{2}\zeta(z)]\mathscr{K}(\phi).$
(3.4)

###### Digression.

Along the way, we observe that (3.4) is invariant under redefinitions

$\mathscr{K}(\phi)\ \mapsto\ \mathscr{K}(\phi)+q(z)\phi$

for every continuous function $q:[0,1]\to\operatorname{\mathbb{R}}$. Indeed,
gathering the two terms in (3.4) that bear the function $\mathscr{K}(\cdot)$
and performing the shift, we find

$\displaystyle\frac{1}{2P^{2}}\left[\int\mathrm{d}\sigma\mathscr{K}(\phi-\sigma)\partial_{z}^{2}\rho(z,\sigma)+[P^{2}\zeta(z)]\mathscr{K}(\phi)\right]$
$\displaystyle\hskip
71.13188pt+\frac{q(z)}{2P^{2}}\left[\phi\partial_{z}^{2}\int\mathrm{d}\sigma\rho(z,\sigma)-\partial_{z}^{2}\int\mathrm{d}\sigma\rho(z,\sigma)\sigma+[P^{2}\zeta(z)]\phi\right].$

We claim that the second square bracket vanishes. The matrix model we started
with is even in the eigenvalues, thus $\rho(z,\sigma)$ is even and has
vanishing first moment:

$\int\mathrm{d}\sigma\rho(z,\sigma)\sigma=0\qquad\forall\leavevmode\nobreak\
0<z<1.$

On the other hand, using the normalisation condition
$\int\mathrm{d}\sigma\rho(z,\sigma)=\nu(z)$, which stems from the definition
of $\rho$, we have that

$q(z)\phi\left[\partial_{z}^{2}\int\mathrm{d}\sigma\rho(z,\sigma)+P^{2}\zeta(z)\right]=q(z)\phi\left[\partial_{z}^{2}\nu(z)+P^{2}\zeta(z)\right]=0,$

which vanishes due to the balancing condition (2.10). ∎

###  3.3 Recovering $a$ from the matrix model

To solve (3.4) we need to understand the scaling of $\phi$ in the long quiver
limit. The behavior at large $P$ and strong coupling depends on the order of
limits. If $P$ is taken large first, the interaction between different nodes
cancels against the non-Abelian vector multiplet of each node and the solution
behaves as in the $\mathcal{N}=4$ super-Yang–Mills with the Wigner
distribution of eigenvalues, while taking the strong coupling limit first
results in the same solution as in $d<4$, even if the action has a ln
singularity as $d$ approaches $4$. Here we take $P\rightarrow\infty$ first and
return to the second case later, in Subsection 3.6.

######  Ansatz.

In $d=4$, $\phi$ remains $\mathcal{O}(1)$ as $P\to\infty$.

We will see that the computation of
$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}$ at finite coupling reproduces
the value of the Weyl anomaly coefficient. With this no-scaling ansatz, (3.4)
drastically simplifies. The terms that carry the function $\mathscr{K}(\cdot)$
drop out, as both of them are suppressed by $1/P^{2}$. The saddle point
equation reduces to

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\frac{\rho(z,\sigma)}{\phi-\sigma}=\frac{\phi}{2t_{4}(z)},$
(3.5)

The latter is precisely the saddle point equation of $\mathcal{N}=4$ super-
Yang–Mills theory on $\operatorname{\mathbb{S}}^{4}$, except for the
$z$-dependence. It appears in two ways:

  * •

The $z$-dependent normalisation

$\int_{-\infty}^{\infty}\mathrm{d}\phi\rho(z,\phi)=\nu(z).$

We denote $\rho(z,\phi)=\nu(z)\bar{\rho}(z,\phi)$, with $\bar{\rho}$
canonically normalised to 1.

  * •

The dependence of the ’t Hooft coupling $t_{4}(z)$ on the gauge node. We
introduce an effective ’t Hooft coupling

$\lambda(z):=\nu(z)t_{4}(z).$

With these redefinitions, (3.5) becomes

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\frac{\bar{\rho}(z,\sigma)}{\phi-\sigma}=\frac{\phi}{2\lambda(z)},$
(3.6)

which is now the canonically normalised saddle point equation for
$\mathcal{N}=4$ super-Yang–Mills at every $0<z<1$. The solution is well-known
to be the Wigner semicircle

$\bar{\rho}(z,\phi)=\frac{1}{2\pi\lambda(z)}\sqrt{4\lambda(z)-\phi^{2}},\qquad\qquad\phi\in\left[-2\sqrt{\lambda(z)},2\sqrt{\lambda(z)}\right],$

Equipped with the solution for the eigenvalue density we can compute the
leading order contribution to the partition function. We get

$\displaystyle\left.S_{\mathrm{eff}}\right\rvert_{\text{on-shell}}$
$\displaystyle=\frac{1}{2}N^{2}P\int_{0}^{1}\mathrm{d}z\int\mathrm{d}\phi\leavevmode\nobreak\
\rho(z,\phi)\frac{\phi^{2}}{2t_{d}(z)}$
$\displaystyle=\frac{1}{4}PN^{2}\int_{0}^{1}\mathrm{d}z\nu(z)^{2}$
$\displaystyle=\frac{P}{8}\sum_{k=1}^{\infty}R_{k}^{2}.$

We thus reproduce the evaluation of the Weyl anomaly coefficient already
obtained in (3.2),

$a\simeq\left.S_{\mathrm{eff}}\right\rvert_{\text{on-shell}}.$

Notice that there is no dependence on $t_{4}(z)$, which are marginal
parameters in the conformal gauge theory, despite $\rho(z,\phi)$ depends on
it. This result is consistent with the fact that $a$ is invariant under
marginal deformations.

###  3.4 Extremal correlators

Correlation functions of Coulomb branch chiral ring operators in any
$\mathcal{N}=2$ conformal gauge theory in four dimensions were considered in
[66]. There it was shown that the correlation functions of any number of
chiral primary and a single anti-chiral primary operators can be obtained from
a deformed matrix model for the 4d $\mathcal{N}=2$ theory on
$\operatorname{\mathbb{S}}^{4}$. Other recent studies of chiral operators
using the matrix model derived from localisation include [67, 68, 69, 70, 71,
72, 73, 74, 75, 76].

######  Definition 3.2.

Consider a balanced four-dimensional SCFT. Correlation functions of an
arbitrary number of chiral primary operators and one anti-chiral primary
operator are called _extremal correlators_.

The master formula of [66] states that the generating function of extremal
correlators is:

$\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\left[\left\\{\hat{\lambda}_{j,A}\right\\}\right]=\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\left\langle\exp\left(\sum_{j=1}^{P-1}\sum_{A=3}^{N_{j}}\pi^{A/2}\hat{\lambda}_{j,A}\sum_{a=1}^{N_{j}}\left(\phi_{a}^{(j)}\right)^{A}\right)\right\rangle_{\operatorname{\mathbb{S}}^{4}},$
(3.7)

where, on the right-hand side,
$\langle\cdot\rangle_{\operatorname{\mathbb{S}}^{4}}$ means the expectation
value taken in the matrix model derived from localisation on the sphere. The
couplings $\left\\{\hat{\lambda}_{j,A}=-2\Im(\tau_{j,A})\right\\}$ are
parameters which are assumed to live in a neighborhood of the origin, and
$\tau_{j,A}$ are the deformation parameters that accompany the deformation by
chiral operators in the Lagrangian. The correlation functions are taken by
differentiating (3.7) with respect to various $\tau,\bar{\tau}$ and eventually
setting $\hat{\lambda}_{j,A}=0$ for all $j,A$. The normalisation of the
couplings is chosen to match with [66, Sec.3.2].

We can evaluate (3.7) in the long quiver limit. To do so, we do not impose any
scaling on the additional coupling, so that they do not spoil the planar
limit, and evaluate the expectation value in the background of the long quiver
solution.

######  Statement 3.3.

Let

$f_{\text{\rm
extremal}}\left[\left\\{\hat{\lambda}_{A}(z)\right\\}\right]:=\lim\frac{1}{PN}\ln\frac{\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\left[\left\\{\hat{\lambda}_{j,A}\right\\}\right]}{\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}},$

where the limit operation on right-hand side means the long quiver limit. Then

$\frac{\delta f_{\text{\rm
extremal}}}{\delta\hat{\lambda}_{A}(z_{\ast})}\simeq\begin{cases}0&A\text{
odd}\\\
\pi^{(A-1)/2}2^{A}\frac{\Gamma\left(\frac{A+1}{2}\right)}{\Gamma\left(\frac{A}{2}+2\right)}t_{4}(z_{\ast})^{A/2}\nu(z_{\ast})^{1+A/2}&A\text{
even}.\end{cases}$ (3.8)

In particular, the generators associated to odd $A$ give a sub-leading
contribution to the extremal correlators.

###### Proof.

We use the fact that, at large-$N$, the generators of the chiral ring are not
subject to the trace relations. In the long quiver limit we write

$\displaystyle\sum_{j=1}^{P-1}\sum_{A=3}^{N_{j}}\pi^{A/2}\hat{\lambda}_{j,A}\sum_{a=1}^{N_{j}}\left(\phi_{a}^{(j)}\right)^{A}$
$\displaystyle\simeq PN\sum_{A\geq
3}\pi^{A/2}\int_{0}^{1}\mathrm{d}z\hat{\lambda}_{A}(z)\int\mathrm{d}\phi\rho(z,\phi)\phi^{A}$
$\displaystyle=PN\sum_{A\geq
3}\pi^{A/2}\int_{0}^{1}\mathrm{d}z\frac{\hat{\lambda}_{A}(z)\nu(z)}{2\pi\lambda(z)}\int_{-2\sqrt{\lambda(z)}}^{2\sqrt{\lambda(z)}}\mathrm{d}\phi\sqrt{4\lambda(z)-\phi^{2}}\phi^{A}$
$\displaystyle=PN\sum_{A\geq
3}\left(\frac{1+(-1)^{A}}{2}\right)\mathfrak{c}_{A}\int_{0}^{1}\mathrm{d}z\hat{\lambda}_{A}(z)t_{4}(z)^{A/2}\nu(z)^{1+A/2},$

where we recall that $\lambda(z)=\nu(z)t_{4}(z)$. The $A$-dependent
coefficient in the last line is

$\mathfrak{c}_{A}=\frac{\pi^{(A-1)/2}2^{A}\Gamma\left(\frac{A+1}{2}\right)}{\Gamma\left(\frac{A}{2}+2\right)}.$
(3.9)

Due to the parity of the eigenvalue density, the integral vanishes if $A\in
1+2\operatorname{\mathbb{Z}}$, while if $A\in 2\operatorname{\mathbb{Z}}$ the
coefficient is in $\pi^{A/2}\operatorname{\mathbb{N}}$, that is, it evaluates
to an integer times an integer power of $\pi$. The first few non-trivial
values of the coefficient $\mathfrak{c}_{A}$ are listed in Table 3.

$A$ | 4 | 6 | 8 | 10 | 12
---|---|---|---|---|---
$\mathfrak{c}_{A}$ | $2\pi^{2}$ | $5\pi^{3}$ | $14\pi^{4}$ | $42\pi^{5}$ | $132\pi^{6}$
Table 3: Explicit values of the numerical coefficient $\mathfrak{c}_{A}$ in
(3.9).

We then use the large-$N$ property of multi-trace correlators in matrix
models, which implies the standard relation

$\ln\left\langle\exp\left(\sum_{j=1}^{P-1}\sum_{A=3}^{N_{j}}\pi^{A/2}\hat{\lambda}_{j,A}\sum_{a=1}^{N_{j}}\left(\phi_{a}^{(j)}\right)^{A}\right)\right\rangle_{\operatorname{\mathbb{S}}^{4}}=\left.\left(\sum_{j=1}^{P-1}\sum_{A=3}^{N_{j}}\pi^{A/2}\hat{\lambda}_{j,A}\sum_{a=1}^{N_{j}}\left(\phi_{a}^{(j)}\right)^{A}\right)\right\rvert_{\text{saddle}}$

to evaluate

$f_{\text{\rm
extremal}}\left[\left\\{\hat{\lambda}_{A}(z)\right\\}\right]\simeq\sum_{\begin{subarray}{c}A\geq
3\\\ A\in
2\operatorname{\mathbb{N}}\end{subarray}}\frac{\pi^{(A-1)/2}2^{A}\Gamma\left(\frac{A+1}{2}\right)}{\Gamma\left(\frac{A}{2}+2\right)}\int_{0}^{1}\mathrm{d}z\hat{\lambda}_{A}(z)t_{4}(z)^{A/2}\nu(z)^{1+A/2}.$

The conclusion follows from functional differentiation of this expression. ∎

Construct the matrix of derivatives:

$\mathfrak{M}_{\vec{A},\vec{B}}:=\frac{1}{\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}}\frac{\partial\
}{\partial\tau_{j_{1},A_{1}}}\cdots\frac{\partial\
}{\partial\tau_{j_{m},A_{m}}}\frac{\partial\
}{\partial\bar{\tau}_{k_{1},B_{1}}}\cdots\frac{\partial\
}{\partial\bar{\tau}_{k_{m},B_{m}}}\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\left[\left\\{\hat{\lambda}_{j,A}\right\\}\right],$

where $\vec{A},\vec{B}$ are collections of pairs
$\vec{A}=\left\\{(j_{1},A_{1}),\dots,(j_{m},A_{m})\right\\},\vec{B}=\left\\{(k_{1},B_{1}),\dots,(k_{m},B_{m})\right\\}$
for some integer $m\in\operatorname{\mathbb{N}}$. We can introduce a
lexicographic order on the set of arrays $\vec{A}$, and use the map from this
ordered set to $\operatorname{\mathbb{N}}$ to treat each $\vec{A}$ as an
integer index. The extremal correlators are constructed algorithmically from
the Gram–Schmidt diagonalisation of $\mathfrak{M}$ [66].

Formula (3.8) allows to easily evaluate $\mathfrak{M}_{\vec{A},\vec{B}}$.
Using the leading order approximation

$\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\left[\left\\{\hat{\lambda}_{j,A}\right\\}\right]\simeq\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\mathrm{e}^{NPf_{\text{\rm
extremal}}\left[\left\\{\hat{\lambda}_{A}(z)\right\\}\right]},$

we obtain

$\mathfrak{M}_{\vec{A},\vec{B}}\simeq\prod_{s=1}^{m}\left[N_{j_{s}}\mathfrak{c}_{A_{s}}t_{j_{s}}^{A_{s}/2}\right]\times\left[N_{k_{s}}\mathfrak{c}_{B_{s}}t_{k_{s}}^{B_{s}/2}\right]$

if all $A_{s},B_{s}$ are even, and is sub-leading otherwise. Note that there
is a factor $\mathtt{i}^{m}$ in passing from $\frac{\partial\
}{\partial\tau_{j,A}}$ to $\frac{\partial\ }{\partial\hat{\lambda}_{j,A}}$,
and an analogous factor $(-\mathtt{i})^{m}$ from the derivatives
$\frac{\partial\ }{\partial\bar{\tau}_{j,A}}$, and the two cancel out.

We remark that the simplicity of this expression is a consequence of the
planar limit, and the lack of ’t Hooft scaling of the auxiliary couplings
$\left\\{\hat{\lambda}_{A}\right\\}_{A\geq 3}$. The extremal correlators in
the planar limit of $\mathcal{N}=4$ super-Yang–Mills with the additional ’t
Hooft scaling of $\left\\{\hat{\lambda}_{A}\right\\}_{A\geq 3}$ have been
studied in [67], and the results therein may be straightforwardly exported to
the present context.

###  3.5 Wilson loops

Consider a Wilson loop

$W_{\mathsf{R}}=\frac{1}{\dim\mathsf{R}}\mathrm{Tr}_{\mathsf{R}}\left(\mathrm{e}^{2\pi\vec{\phi}}\right)$

in the irreducible representation $\mathsf{R}$ of the gauge group. Fixing
$j_{\ast}\in\left\\{1,\dots,P-1\right\\}$, $\mathsf{R}$ is the tensor products
of an arbitrary $\mathrm{SU}(N_{j_{\ast}})$ representation and the trivial
representation of $\mathrm{SU}(N_{j})$ for $j \neq j_{\ast}$. We will focus on
fundamental $\Box$, rank-$\mathsf{k}$ antisymmetric $\mathsf{A}_{\mathsf{k}}$
and rank-$\mathsf{k}$ symmetric $\mathsf{S}_{\mathsf{k}}$ representations of
$\mathrm{SU}(N_{j_{\ast}})$. We denote $z_{\ast}:=\frac{j_{\ast}}{P}$.

The expectation value $\langle W_{\mathsf{R}}\rangle$ in the planar limit is
obtained as an immediate corollary of the previous result. Indeed, we have
found that, at every $z$, the normalised eigenvalue density
$\bar{\rho}(z,\phi)$ equals the one of $\mathcal{N}=4$ super-Yang–Mills with
’t Hooft coupling $\lambda(z)$. Therefore, in the planar limit, the result for
$\langle W_{\mathsf{R}}\rangle$ is exported from $\mathcal{N}=4$ super-
Yang–Mills.

For the fundamental representation we have the classical result

$\displaystyle\langle W_{\Box_{j_{\ast}}}\rangle$
$\displaystyle\simeq\int\mathrm{d}\phi\bar{\rho}(z_{\ast},\phi)\leavevmode\nobreak\
\mathrm{e}^{2\pi\phi}$
$\displaystyle=\frac{1}{2\pi\sqrt{\lambda(z_{\ast})}}\,I_{1}\left(4\pi\sqrt{\lambda(z_{\ast})}\right)=\sum_{n=0}^{\infty}\frac{\left(4\pi^{2}\lambda(z_{\ast})\right)^{n}}{(n+1)!n!},$

where in the second line $I_{1}(\cdot)$ is the modified Bessel function. This
expression only differs from $\mathcal{N}=4$ super-Yang–Mills in the
dependence $\lambda(z_{\ast})=t_{4}(z_{\ast})\nu(z_{\ast})$, which varies as
we vary the gauge node under which the line operator is charged.

The computation for the $\mathsf{A}_{\mathsf{k}}$ and
$\mathsf{S}_{\mathsf{k}}$ Wilson loops also follows from $\mathcal{N}=4$, and
can be exported step by step from [77].

###  3.6 Solution at strong coupling

We wish to go beyond the computation of the coefficient $a$ and extract more
interesting behaviour from the sphere free energy. To do so, we consider the
strong ’t Hooft coupling limit $t_{4}(z)\to\infty$. This is the regime in
which we expect a more precise match with supergravity, beyond the purely
kinematical anomaly coefficient $a$.

In the strong coupling limit, (3.5) trivialises. This indicates that the
ansatz that $\phi$ does not scale with $P$ misses all the information at
strong ’t Hooft coupling. We therefore abandon the no-scaling ansatz, and plug
the long quiver scaling ansatz (2.19) into (3.4).

The saddle point equation at strong coupling reads

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\frac{\rho(z,\sigma)}{\phi-\sigma}+\frac{1}{2P^{2}}\int\mathrm{d}\sigma\mathscr{K}(\phi-\sigma)\partial_{z}^{2}\rho(z,\sigma)+\frac{1}{2P^{2}}[P^{2}\zeta(z)]\mathscr{K}(\phi)=0.$
(3.10)

For large argument,

$\mathscr{K}(\phi)\approx 2\phi\ln\lvert\phi\rvert-2\phi+\frac{1}{6\phi},$

and therefore, replacing $\phi$ with $2P^{\chi}x$ as prescribed by (2.19), we
have

$\frac{1}{2P^{2}}\mathscr{K}(\phi)\approx 2\chi
P^{\chi-2}\ln(P)x+2P^{\chi-2}\left(x\ln\lvert
2x\rvert-x\right)+\frac{1}{24x}P^{-\chi-2}.$ (3.11)

With the substitutions (2.19)-(3.11), eq. (3.10) becomes

$\displaystyle 0=2\chi P^{\chi-2}\ln(P)$
$\displaystyle\left[\int\mathrm{d}y\partial_{z}^{2}\varrho(z,y)(x-y)+[P^{2}\zeta(z)]x\right]$
$\displaystyle+2P^{\chi-2}$
$\displaystyle\left[\int\mathrm{d}y\partial_{z}^{2}\varrho(z,y)(x-y)\left(\ln\lvert
2x-2y\rvert-1\right)+[P^{2}\zeta(z)]x\left(\ln\lvert 2x\rvert-1\right)\right]$
$\displaystyle+\frac{1}{2}P^{-\chi}$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y\frac{\varrho(z,y)}{x-y}+\mathcal{O}(P^{-\chi-2}).$

The first line vanishes as a consequence of the balancing condition (2.10).
The shifts by $\ln(2)-1$ in the second line cancel for the same reason. To
have competition between the terms in the second and third line, so to reach a
non-trivial saddle point configuration, the matrix model requires
$\chi-2=-\chi$. We thus recover in $d=4$ the value $\chi=1$, that we have
found in Section 2 $\forall\leavevmode\nobreak\
d\in\operatorname{\mathbb{R}}_{\geq 3}\setminus 2\operatorname{\mathbb{Z}}$.

We arrive at

$\frac{1}{4}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y\frac{\varrho(z,y)}{x-y}+\int\mathrm{d}y(x-y)\ln\lvert
x-y\rvert\partial_{z}^{2}\varrho(z,y)+[P^{2}\zeta(z)]x\ln\lvert x\rvert=0.$

We now use the trick of writing

$\frac{1}{x-y}=\partial_{y}^{2}\left((x-y)\ln\lvert x-y\rvert\right)$

and integrate twice by parts,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y\frac{\varrho(z,y)}{x-y}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y(x-y)\ln\lvert
x-y\rvert\partial_{y}^{2}\varrho(z,y).$

Besides, we use the trivial equalities

$\displaystyle x\ln\lvert x\rvert$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y(x-y)\ln\lvert
x-y\rvert\delta(y),$ $\displaystyle\int\mathrm{d}y(x-y)\ln\lvert x-y\rvert
f(y)$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y(x-y)\ln\lvert
x-y\rvert f(y),$

with the second identity valid for every smooth function
$f:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$. The saddle point
equation is thus recast in the form

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y(x-y)\ln\lvert
x-y\rvert\leavevmode\nobreak\
\left[\frac{1}{4}\partial_{y}^{2}\varrho(z,y)+\partial_{z}^{2}\varrho(z,y)+[P^{2}\zeta(z)]\delta(y)\right]=0,$
(3.12)

that must hold $\forall\leavevmode\nobreak\ x\in\operatorname{\mathbb{R}}$ and
$0<z<1$. Therefore, the solution to (3.12) satisfies the Poisson equation

$\frac{1}{4}\partial_{x}^{2}\varrho(z,x)+\partial_{z}^{2}\varrho(z,x)+P^{2}\zeta(z)\delta(x)=0.$

This exact same equation was already found in (2.24) for $d\neq 4$, and we
have derived it directly in $d=4$ at strong coupling.

###### Digression.

The effect for which, at large-$N$, the eigenvalues scale differently at
finite versus infinite ’t Hooft coupling already appears in $\mathcal{N}=4$
super-Yang–Mills.

The partition function of $\mathcal{N}=4$ super-Yang–Mills on
$\operatorname{\mathbb{S}}^{4}$ is the Gaussian Unitary Ensemble. In the
planar limit with ’t Hooft coupling $\lambda=Ng_{4}$, one obtain the saddle
point equation (3.6), solved by the Wigner semicircle. The finite part of the
partition function is

$F_{\mathcal{N}=4}=N^{2}\ln\frac{1}{\lambda}.$ (3.13)

However, one may also take the limit with fixed gauge coupling $g_{4}$. This
corresponds to $\lambda\to\infty$ linearly with $N\to\infty$. The only way for
the action to admit an equilibrium configuration is if the eigenvalues grow as
$\phi=\sqrt{N}x$ with $x$ being $\mathcal{O}(1)$. One then proceeds to the
computation with fixed $g_{4}$ and find the same answer as directly replacing
$\lambda=Ng_{4}$ in (3.13). ∎

###  3.7 Free energy in four-dimensional long quivers

######  Statement 3.4.

In the long quiver limit and at infinite ’t Hooft coupling,

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\simeq\frac{1}{2}P\sum_{k=1}^{\infty}R_{k}^{2}\,\ln\frac{\pi
k\mathrm{e}^{\gamma_{\text{\tiny EM}}}}{P}+\mathcal{O}(\ln P).$ (3.14)

###### Proof.

We compute

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\simeq\left.S_{\mathrm{eff}}\right\rvert_{\text{on-
shell}}.$

The on-shell effective action is given by

$\displaystyle\left.S_{\mathrm{eff}}\right\rvert_{\text{on-
shell}}=\frac{1}{2}PN^{2}\int_{0}^{1}\mathrm{d}z\int\mathrm{d}x\varrho(z,x)$
$\displaystyle\left[\zeta(z)\ln
H(2Px)-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y\varrho(z,y)\ln\lvert
2P(x-y)\rvert\right.$
$\displaystyle\left.-\frac{1}{8P^{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}y\varrho(z,y)\partial_{y}^{2}\left(\ln\lvert
H(2P(x-y))\rvert\right)\right],$

where we have used the fact that $\varrho(z,y)$ solves the saddle point
equation to manipulate the expression. At large values of the argument,

$\ln H(\phi)\simeq-\frac{\phi^{2}}{2}\left(\ln\phi^{2}-3\right),$

and $\partial^{2}_{\phi}\ln H(\phi)\simeq-\ln\phi^{2}$. We thus find
cancellations between the double integrals in $S_{\mathrm{eff}}$. The
remaining single integral can be computed with the help of the Fourier series
(2.13) and (2.25):

$\left.S_{\mathrm{eff}}\right\rvert_{\text{on-
shell}}=-\frac{\pi^{3}P}{2}\sum_{k=1}^{\infty}k^{3}R_{k}^{2}\int_{\operatorname{\mathbb{R}}}\mathrm{d}x\,x^{2}\left[\ln\left(2Px\right)^{2}-3\right]\mathrm{e}^{-2\pi
k|x|}.$

An integration over $x$ gives (3.14).

Before concluding, let us stress that, using $a\propto
P\sum_{k=1}^{\infty}R_{k}^{2}$, the $\mathcal{O}(P)$ constant in
$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}$ is scheme-dependent. Recall
from (2.28) that the partition function on $\operatorname{\mathbb{S}}^{4}$
contains a UV divergence controlled by the central charge:
$\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=-4a\ln(\Lambda r)+{\rm
finite}$, for $r$ the radius of $\operatorname{\mathbb{S}}^{4}$ and $\Lambda$
the UV cutoff. Changing renormalisation scheme shifts the free energy by
$\,{\rm const}\,P\sum_{k=1}^{\infty}R_{k}^{2}$, effectively multiplying the
the argument of the logarithm in (3.14) by a constant. The constant we
computed ($\pi\mathrm{e}^{\gamma_{\text{\tiny EM}}}$) corresponds to a
particular scheme used in [22] and in practice has no particular meaning. On
the other hand, the dependence on $P$ and on the mode number $k$, does not
change with the scheme and can be faithfully compared to supergravity, which
we will do later on in Subsection 3.9.

∎

###  3.8 Four-dimensional linear quivers of arbitrary length

It is possible to solve the planar limit of any linear $d=4$ $\mathcal{N}=2$
quiver for arbitrary $P$. We omit the details of the derivation of the saddle
point equation, since they are conceptually analogous to what we have done so
far, and only highlight the differences.

  * •

The discrete index $j$ labelling the nodes of the quiver is not replaced by a
parameter $z=j/P$.

  * •

The gauge ranks $N_{j}$ scaled into rational numbers
$\left\\{\nu_{j}\right\\}_{j},$ $\nu_{j}=N_{j}/N$ are not collected into a
function $\nu(z)$.

  * •

Likewise, the Veneziano parameters $\left\\{\zeta_{j}\right\\}$ are not
collected into a function $\zeta(z)$. In the planar limit, $\zeta_{j}$ are
$\mathcal{O}(1)$.

  * •

The eigenvalue densities are normalised as

$\int\mathrm{d}\phi\rho_{j}(\phi)=\nu_{j},\qquad\forall j=1,\dots,P-1.$

  * •

Related to the above, being $P$ finite, the contribution from the
bifundamental hypermultiplets is a difference

$\frac{1}{2}\left[\rho_{j+1}-2\rho_{j}+\rho_{j-1}\right],$

and does not produce a differential $\partial_{z}^{2}$.

  * •

To encode the above fact, we define the $(P-1)\times(P-1)$ matrix

$\mathsf{M}:=\left(\begin{matrix}2&-1&&&&&\\\ -1&2&-1&&&&\\\ &-1&2&-1&&&&\\\
&&&&\ddots&&&\\\ &&&&-1&2&-1\\\ &&&&&-1&2\\\ \end{matrix}\right)$

That is, $\mathsf{M}$ is the Cartan matrix of the $A_{P-1}$ Lie algebra, which
equals twice the unit matrix minus the adjacency matrix of the linear quiver.

  * •

The balancing condition is written in matrix form as
$K_{j}=\mathsf{M}_{ji}N_{i}$, or more compactly

$\vec{K}=\mathsf{M}\vec{N}.$

  * •

As a notational reminder, the indices $i,j\in\left\\{1,\dots,P-1\right\\}$
will indicate the nodes of the quiver. Moreover, sum over repeated indices
will be understood to lighten the formulae. The imaginary unit is denoted
$\mathtt{i}=\sqrt{-1}$ to avoid confusion with the index $i$.

To diagonalise the Cartan matrix we introduce its eigenvectors
$\vec{v}^{(p)}=(v_{1}^{(p)},\cdots,v_{P-1}^{(p)})$, $p=1,\dots,P-1$. To be
more explicit, the components are

$v_{j}^{(p)}=\sin\frac{\pi pj}{P},\qquad j=1,\dots,P-1,$ (3.15)

and, after elementary trigonometric manipulations, the corresponding
eigenvalues are

$\mathsf{M}_{ji}v_{i}^{(p)}=\left(2\sin\frac{\pi p}{2P}\right)^{2}v_{j}^{(p)}$

or, in matrix notation,

$\mathsf{M}\vec{v}^{(p)}=\left(2\sin\frac{\pi
p}{2P}\right)^{2}\vec{v}^{(p)},\qquad\forall p=1,\dots,P-1.$

These vectors form a basis that satisfies orthogonality and completeness
conditions

$\sum_{j=1}^{P-1}v_{j}^{(p)}v_{j}^{(p^{\prime})}=\frac{P}{2}\,\delta^{pp^{\prime}},\qquad\sum_{p=1}^{P-1}v_{j}^{(p)}v_{i}^{(p)}=\frac{P}{2}\,\delta_{ji}.$
(3.16)

We can express the vector of gauge ranks $\vec{N}$ in this basis:

$\vec{N}=\sum_{p=1}^{P-1}\tilde{N}_{p}\vec{v}^{(p)},$

which in components reads

$N_{j}=\sum_{p=1}^{P-1}\tilde{N}_{p}\sin\frac{\pi pj}{P}\,.$

That is to say, $\vec{N}$ has components $N_{i}$ in the canonical basis
$\left\\{\vec{e}^{(i)}\right\\}_{i=1,\dots,P-1}$ with
$e^{(i)}_{j}=\delta_{ji}$, and it has components $\tilde{N}_{p}$ in the
eigenbasis of $\mathsf{M}$ $\left\\{\vec{v}^{(p)}\right\\}_{p=1,\dots,P-1}$.
The balancing condition (2.4) results in

$K_{j}=\sum_{p=1}^{P-1}\tilde{N}_{p}\left(2\sin\frac{\pi
p}{2P}\right)^{2}\sin\frac{\pi pj}{P}.$ (3.17)

######  Statement 3.5.

Consider a balanced linear quiver in $d=4$, and take the planar limit with
arbitrary $P\in\operatorname{\mathbb{N}}_{\geq 2}$. At infinite ’t Hooft
coupling, the densities of eigenvalues are

$\rho_{j}(\phi)=\frac{1}{\sinh\pi\phi}\sum_{p=1}^{P-1}\frac{\sinh\left(\frac{\pi\phi(P-p)}{P}\right)\sin\left(\frac{\pi
pj}{P}\right)}{\sin\left(\frac{\pi p}{P}\right)}\leavevmode\nobreak\
\frac{1}{P}\sum_{\ell=1}^{P-1}\sin\left(\frac{\pi
p\ell}{P}\right)\zeta_{\ell},$ (3.18)

$\forall\leavevmode\nobreak\ j=1,\dots,P-1$.

A corollary and cross-check of this statement is that for $P=2$, namely
$\mathrm{SU}(N)$ gauge theory and $2N$ fundamental hypermultiplets, this
expression reproduces the strong coupling solution of conformal
$\mathcal{N}=2$ QCD [38, Eq.(4.4)]:

$\rho_{{\rm SQCD}}(\phi)=\frac{1}{2\cosh\frac{\pi\phi}{2}}.$

###### Proof.

The finite-$P$ version of the saddle point equation (3.4) reads

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\mathrm{d}\sigma\frac{\rho_{j}(\sigma)}{\phi-\sigma}-\frac{1}{2}\mathsf{M}_{ji}\int\mathrm{d}\sigma\mathscr{K}(\phi-\sigma)\rho_{i}(\sigma)+\frac{\zeta_{j}}{2}\mathscr{K}(\phi)=0,$
(3.19)

where again we restrict ourselves to the infinite ’t Hooft coupling limit.

We take the Fourier transform of (3.19) and denote $\tilde{\rho}_{j}(\omega)$
the Fourier transform of $\rho_{j}(\phi)$. We then simplify the expression
using that $(2+\gamma_{\text{\tiny EM}})(\mathsf{M}_{ji}\nu_{i}-\zeta_{j})=0$
by virtue of the balancing condition. After these manipulations we obtain

$\int\mathrm{d}\omega\mathrm{e}^{-\mathtt{i}\omega(\phi-\sigma)}\mathrm{sign}(\omega)\left[\tilde{\rho}_{j}(\omega)+\frac{1}{4}\mathsf{M}_{ji}\tilde{\rho}_{i}(\omega)\frac{1}{\left(\sinh\frac{\omega}{2}\right)^{2}}-\frac{\zeta_{j}}{4}\frac{1}{\left(\sinh\frac{\omega}{2}\right)^{2}}\right]=0,$

which is valid if and only if

$\left[\left(2\sinh\frac{\omega}{2}\right)^{2}\delta_{ji}+\mathsf{M}_{ji}\right]\tilde{\rho}_{i}(\omega)=\zeta_{j},$
(3.20)

$\forall\leavevmode\nobreak\ j=1,\dots,P-1$, where now
$P\in\operatorname{\mathbb{N}}_{\geq 2}$ is arbitrary and finite.

We now expand the vector of densities
$(\tilde{\rho}_{1}(\omega),\dots,\tilde{\rho}_{P-1}(\omega))$ in the basis
(3.15),

$\tilde{\rho}_{j}(\omega)=\sum_{p=1}^{P-1}\tilde{r}_{p}(\omega)v_{j}^{(p)}=\sum_{p=1}^{P-1}\tilde{r}_{p}(\omega)\sin\frac{\pi
pj}{P}.$

Plugging this formula in (3.20), we obtain:

$\tilde{r}_{p}(\omega)=\frac{1}{2P}\,\,\frac{1}{\left(\sinh\frac{\omega}{2}\right)^{2}+\left(\sin\frac{\pi
p}{2P}\right)^{2}}\,\sum_{\ell=1}^{P-1}\zeta_{\ell}\sin\frac{\pi p\ell}{P}\,.$

Therefore, the Fourier transform of the eigenvalue densities at large-$N$ but
finite length of the quiver is

$\tilde{\rho}_{j}(\omega)=\sum_{p=1}^{P-1}\left(\frac{1}{2P}\sum_{\ell=1}^{P-1}\frac{\zeta_{\ell}\sin\frac{\pi
p\ell}{P}}{\left(\sinh\frac{\omega}{2}\right)^{2}+\left(\sin\frac{\pi
p}{2P}\right)^{2}}\right)\sin\frac{\pi pj}{P}\,.$

After the inverse Fourier transform, we find the eigenvalue densities (3.18).
Using (3.17) we can also write the solution as

$\tilde{\rho}_{j}(\omega)=\sum_{p=1}^{P-1}\frac{\left(\sin\frac{\pi
k}{2P}\right)^{2}}{\left(\sinh\frac{\omega}{2}\right)^{2}+\left(\sin\frac{\pi
p}{2P}\right)^{2}}\frac{\tilde{N}_{p}}{N}\sin\frac{\pi pj}{P},$ (3.21)

with $\left\\{\frac{\tilde{N}_{p}}{N}\right\\}_{p=1,\dots,P-1}$ being just a
rotation of $\left\\{\nu_{j}\right\\}_{j=1,\dots,P-1}$. ∎

Equipped with the strong coupling solution (3.18), we can now evaluate the
free energy.

######  Statement 3.6.

Consider a balanced linear quiver in $d=4$, and take the planar limit with
arbitrary $P\in\operatorname{\mathbb{N}}_{\geq 2}$. At infinite ’t Hooft
coupling we have:

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\frac{P}{2}\sum_{p=1}^{P-1}\tilde{N}_{p}^{2}\left[2\tan\frac{\pi
p}{2P}\,\sum_{a=1}^{2P-1}\sin\frac{\pi
pa}{P}\,\ln\Gamma\left(\frac{a}{2P}\right)-\ln(2P)\right].$ (3.22)

###### Proof.

We compute

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\simeq\left.S_{\mathrm{eff}}\right\rvert_{\text{on-
shell}},$

which in the strict infinite coupling limit becomes (sum over repeated indices
understood)

$\displaystyle-\frac{1}{N^{2}}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=-\frac{1}{2}$
$\displaystyle\int\mathrm{d}\phi\int\mathrm{d}\sigma\,\rho_{j}(\phi)\rho_{i}(\sigma)\left[\mathsf{M}_{ji}\ln
H(\phi-\sigma)+\delta_{ji}\ln(\phi-\sigma)^{2}\right]$ $\displaystyle+$
$\displaystyle\int\mathrm{d}\phi\rho_{j}(\phi)\zeta_{j}\ln H(\phi).$

Mimicking our derivation of the eigenvalue density, we take the Fourier
transform of the integrals:

$\displaystyle-\frac{1}{N^{2}}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}$
$\displaystyle=\int\frac{\mathrm{d}\omega}{|\omega|\left(2\sinh\frac{\omega}{2}\right)^{2}}\left\\{\frac{1}{2}\,\tilde{\rho}_{j}(\omega)\tilde{\rho}_{i}(\omega)\left[\left(2\sinh\frac{\omega}{2}\right)^{2}\delta_{ji}+\mathsf{M}_{ji}\right]-\zeta_{j}\tilde{\rho}_{j}(\omega)\right\\}$
$\displaystyle=-\int\frac{\mathrm{d}\omega}{2|\omega|\left(2\sinh\frac{\omega}{2}\right)^{2}}\zeta_{j}\tilde{\rho}_{j}(\omega),$
(3.23)

where in the last step we have used the saddle point equations in the form
(3.20). This expression is ill-defined because of a bad divergence at
$\omega=0$. The divergence arises because $\ln H(\phi)$ and $\ln\phi^{2}$ are
growing too fast and their Fourier images, strictly speaking, do not exist.
Substractions are needed to define them properly.

More accurate definition of the Fourier transform involves
regularisation:888We mention in passing that regularisation was not necessary
in the saddle point equations. The latter contain once-differentiated kernels,
that brings in an extra $\omega$ in the Fourier space softening the
$1/\omega^{3}$ behavior of the integrand. The remaining $1/\omega^{2}$
divergence is cancelled by the balancing condition. Because of these
cancellations we could use the Fourier transforms without subtractions in
deriving the saddle point eigenvalue densities.

$\displaystyle\ln\phi^{2}$ $\displaystyle=\lim_{\alpha\to
0}\left(-\int\mathrm{d}\omega\,|\omega|^{\alpha-1}\mathrm{e}^{-\mathtt{i}\omega\phi}+\frac{2}{\alpha}-2\gamma_{\text{\tiny
EM}}\right)$ $\displaystyle\ln H(\phi)$
$\displaystyle=-\int\frac{\mathrm{d}\omega}{|\omega|}\frac{\mathrm{e}^{-\mathtt{i}\omega\phi}-1+\frac{\omega^{2}\phi^{2}}{2}}{\left(2\sinh\frac{\omega}{2}\right)^{2}}\,.$

Using these formulae we get a regularised version of (3.23):

$\displaystyle-\frac{1}{N^{2}}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\lim_{\alpha\to
0}$
$\displaystyle\left\\{-\int\frac{\mathrm{d}\omega}{2|\omega|^{1-\alpha}\left(2\sinh\frac{\omega}{2}\right)^{2}}\left[\zeta_{j}\left(\tilde{\rho}_{j}(\omega)-\nu_{j}+\frac{\omega^{2}}{2}\,\left\langle\phi^{2}\right\rangle_{j}\right)\right.\right.$
$\displaystyle\
\left.\left.-\frac{\omega^{2}}{2}\left(2\mathsf{M}_{ji}\nu_{i}-\zeta_{j}\right)\left\langle\phi^{2}\right\rangle_{j}\right]+\left(\gamma_{\text{\tiny
EM}}-\frac{1}{\alpha}\right)\nu_{j}\nu_{j}\right\\},$

with the obvious notation for the second moment of the density:

$\left\langle\phi^{2}\right\rangle_{j}:=\int\mathrm{d}\phi\,\rho_{j}(\phi)\phi^{2}=-\tilde{\rho}^{\prime\prime}_{j}(0).$

The latter can be actually found from the saddle point equation (3.20). When
expanded to the second order in $\omega$, it gives:

$\mathsf{M}_{ji}\left\langle\phi^{2}\right\rangle_{i}=2\nu_{j}.$

This relation and the balancing condition simplify the second line:

$\displaystyle-\frac{1}{N^{2}}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}$
$\displaystyle=-\int\frac{\mathrm{d}\omega}{2|\omega|\left(2\sinh\frac{\omega}{2}\right)^{2}}\zeta_{j}\left(\tilde{\rho}_{j}(\omega)-\tilde{\rho}_{j}(0)-\frac{\omega^{2}}{2}\,\tilde{\rho}^{\prime\prime}_{j}(0)\right)$
$\displaystyle+\lim_{\alpha\to
0}\left(\int\frac{\mathrm{d}\omega|\omega|^{1+\alpha}}{2\left(2\sinh\frac{\omega}{2}\right)^{2}}+\gamma_{\text{\tiny
EM}}-\frac{1}{\alpha}\right)\nu_{j}\nu_{j}.$

This expression has a finite $\alpha\rightarrow 0$ limit:

$-\frac{1}{N^{2}}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=-\int\frac{\mathrm{d}\omega}{2|\omega|\left(2\sinh\frac{\omega}{2}\right)^{2}}\zeta_{j}\left(\tilde{\rho}_{j}(\omega)-\tilde{\rho}_{j}(0)-\frac{\omega^{2}}{2}\,\tilde{\rho}^{\prime\prime}_{j}(0)\right)+(1+\gamma_{\text{\tiny
EM}})\nu_{j}\nu_{j}.$ (3.24)

This is a well-defined, properly regularised version of (3.23) with the
integral in the first term manifestly convergent.

It remains to substitute the solution for the density (3.21). That gives rise
to an integral of the form

$\mathcal{I}(m):=\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega\left(\sinh\frac{\omega}{2}\right)^{2}}\,\left[\frac{1}{\left(\sinh\frac{\omega}{2}\right)^{2}+m^{2}}-\frac{1}{m^{2}}+\frac{\omega^{2}}{4m^{4}}\right].$

This integral can be calculated in closed form. After lengthy but
straightforward manipulations we find:

$\mathcal{I}\left(\sin\frac{\pi p}{2P}\right)=\frac{1+\gamma_{\text{\tiny
EM}}+\ln(2P)}{\left(\sin\frac{\pi
p}{2P}\right)^{4}}-\frac{4}{\left(\sin\frac{\pi
p}{2P}\right)^{2}}\sum_{a=1}^{2P-1}\frac{\sin\frac{\pi pa}{P}}{\sin\frac{\pi
p}{P}}\,\ln\Gamma\left(\frac{a}{2P}\right).$ (3.25)

Substitution of (3.21) into (3.24) and the use of the basis (3.15) result in

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\frac{P}{2}\sum_{p=1}^{P-1}\tilde{N}_{p}^{2}\left[1+\gamma_{\text{\tiny
EM}}-\left(\sin\frac{\pi p}{2P}\right)^{4}\mathcal{I}\left(\sin\frac{\pi
p}{2P}\right)\right],$

which finally gives (3.22) upon using the explicit expression (3.25) for
$\mathcal{I}$. ∎

The thermodynamic limit of the free energy (3.14) can be recovered from the
exact expression (3.22) by taking the $P\rightarrow\infty$ limit. Indeed,

$\displaystyle 2\tilde{N}_{p}^{2}\tan\frac{\pi
p}{2P}\sum_{a=1}^{2P-1}\sin\frac{\pi
pa}{P}\,\ln\Gamma\left(\frac{a}{2P}\right)$ $\displaystyle\simeq 2\pi
p\tilde{N}^{2}_{p}\int_{0}^{1}d\alpha\,\sin(2\pi p\alpha)\ln\Gamma(\alpha)$
$\displaystyle=\tilde{N}^{2}_{p}\ln\left(2\pi p\mathrm{e}^{\gamma_{\text{\tiny
EM}}}\right),$

and the exact free energy reduces to our previous result. For the first
approximation we have used that the rank function converges to a smooth
function in the limit, thus either $p\ll P$ and $2\sin\frac{\pi
p}{2P}\simeq\frac{\pi p}{P}$, or otherwise $\tilde{N}_{p}$ is vanishingly
small for large $p$.

So far we have expressed the free energy in terms of $\tilde{N}_{p}$, but it
can be also written in the ‘coordinate’ representation. A suggetive form valid
in the long quiver limit is

$-\frac{2}{P}\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}\stackrel{{\scriptstyle
P\rightarrow\infty}}{{\simeq}}\left\langle\vec{N}\right|\gamma_{\text{\tiny
EM}}+\ln\mathsf{M}\left|\vec{N}\right\rangle.$

Indeed, at large $P$,

$\displaystyle\left\langle\vec{N}\right|\frac{1}{2}\ln\mathsf{M}\left|\vec{N}\right\rangle$
$\displaystyle=\frac{1}{2}\sum_{p=1}^{P-1}\tilde{N}_{p}^{2}\ln\left(2\sin\frac{\pi
p}{2P}\right)^{2}$ $\displaystyle\stackrel{{\scriptstyle
P\rightarrow\infty}}{{\simeq}}\frac{1}{2}\sum_{p=1}^{\infty}\tilde{N}_{p}^{2}\ln\frac{\pi
p}{P},$

where we have used that the eigenvalues of the matrix $\mathsf{M}$ are
$\left(2\sin\frac{\pi p}{2P}\right)^{2}$ and $\tilde{N}_{p}$ are the
components of the vector $\vec{N}$ in the basis where $\mathsf{M}$ is
diagonal. To pass to the second line, we have used once again that only values
of $p\ll P$ contribute at leading order in the limit. Therefore, using
$\left\\{\tilde{N}_{p}\right\\}_{1\leq p\leq P-1}\stackrel{{\scriptstyle
P\rightarrow\infty}}{{\simeq}}\left\\{R_{k}\right\\}_{k\geq 1}$ in the long
quiver limit, the free energy (3.14) coincides with the matrix element above.
In the same approximation,

$\mathsf{M}\stackrel{{\scriptstyle
P\rightarrow\infty}}{{\simeq}}\frac{\Delta}{P^{2}}\,,$ (3.26)

where $\Delta$ is the Laplace operator
$\Delta=-\frac{\mathrm{d}^{2}}{\mathrm{d}z^{2}}$ on the interval $[0,1]$ with
Dirichlet boundary conditions. The free energy can thus be expressed through
the Green’s function of the operator $\ln\Delta$, properly defined.

It is possible to compute the exact Green’s function without taking the long
quiver limit. By examining (3.22) we arrive at the following result:

######  Statement 3.7.

For every $\ell\in\operatorname{\mathbb{Z}}$ let

$\Xi_{\ell}:=\sum_{r=-\infty}^{+\infty}\left(N_{\ell+2Pr}-N_{-\ell-2Pr}\right),$

with the understanding that $N_{j}:=0$ if
$j\notin\left\\{1,\dots,P-1\right\\}$. The sum is formally infinite, but in
practice only one term contributes for any $\ell$. In other words,
$\Xi_{\ell}$ is a continuation of $N_{\ell}$ outside of its natural domain of
defintion $1\leq\ell\leq P-1$ that satisfies

  * (i)

$\Xi_{j}=N_{j}$ and $\Xi_{-j}=-N_{j}$ for $j=1,\dots,P-1$; and

  * (ii)

$\Xi_{j+2P}=\Xi_{j}$.

The free energy can then be written as

$\displaystyle-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\sum_{j=1}^{P-1}$
$\displaystyle\left[K_{j}\sum_{a=1}^{2P-1}\ln\Gamma\left(\frac{a}{2P}\right)\sum_{s=1}^{a}\Xi_{j+2s-a-1}-\ln(2P)N_{j}^{2}\right].$
(3.27)

###### Proof.

Using completeness relations (3.16) for the basis (3.15) we can re-write
(3.22) as

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\frac{2}{P}\,K_{j}N_{\ell}\sum_{p=1}^{P-1}\sin\frac{\pi
pj}{P}\,\,\sin\frac{\pi p\ell}{P}\,\sum_{a=1}^{2P-1}\frac{\sin\frac{\pi
pa}{P}}{\sin\frac{\pi
p}{P}}\,\ln\Gamma\left(\frac{a}{2P}\right)-N_{j}N_{j}\ln(2P),$ (3.28)

with the summation over $i$ and $j$ implied. Now,

$\frac{\sin\frac{\pi pa}{P}}{\sin\frac{\pi
p}{P}}=\sum_{s=1}^{a}\cos\frac{(2s-a-1)\pi p}{P}\,,$

and

$\displaystyle\sum_{p=1}^{P-1}\sin\frac{\pi pj}{P}\sin\frac{\pi
p\ell}{P}\cos\frac{\pi pm}{P}$
$\displaystyle=-\frac{1}{8}\sum_{p=1}^{P-1}\sum_{\epsilon_{1},\epsilon_{2},\epsilon_{3}\in\left\\{\pm
1\right\\}}\epsilon_{1}\epsilon_{2}\mathrm{e}^{\frac{\mathtt{i}\pi
p}{P}(\epsilon_{1}j+\epsilon_{2}\ell+\epsilon_{3}m)}$
$\displaystyle=-\frac{P}{4}\sum_{\epsilon_{1},\epsilon_{3}\in\left\\{\pm
1\right\\}}\epsilon_{1}\delta_{\epsilon_{1}j+\ell+\epsilon_{3}m,0}$
$\displaystyle=\frac{P}{4}\left(\delta_{j-\ell,m}+\delta_{-j+\ell,m}-\delta_{j+\ell,m}-\delta_{-j-\ell,m}\right).$

with all Kronecker deltas understood $\mod 2P$. To get the second line we have
rearranged the terms in the sum and note a symmetry $p\mapsto-p$, then added
the $p=0$ term for free since the summand vanishes, and finally observed that
the non-trivial phases cancel out in the sum. Applying these formulae to
(3.28) gives (3.27). ∎

The free energy is a quadratic form in $K_{j}$ and $N_{j}$. When it is written
as (3.27), however, a given monimial $K_{j}N_{i}$ appears more than once in
the sum. One may want to collect all the terms with the same $K_{j}N_{i}$.
This leads to the following result.

######  Statement 3.8.

Define

$\mathbf{I}_{\ell}:=\ln\left(\prod_{m=0}^{\left\lfloor
P-\frac{\ell}{2}-1\right\rfloor}\Gamma\left(\frac{2m+1+\ell}{2P}\right)\prod_{m=0}^{\left\lfloor\frac{\ell}{2}\right\rfloor-1}\Gamma\left(\frac{2m+2P+1-\ell}{2P}\right)\right)$

and

$\mathsf{G}_{jk}:=\mathbf{I}_{|j-k|}-\mathbf{I}_{j+k}.$

Then

$-\ln\mathcal{Z}_{\operatorname{\mathbb{S}}^{4}}=\vec{K}^{\top}\mathsf{G}\vec{N}-\vec{N}^{\top}\vec{N}\ln(2P).$

###  3.9 Towards comparison with supergravity: Simplified holography

The supergravity dual of a 4d quiver CFT is the Gaiotto–Maldacena background
[16], with the supergravity approximation valid at infinite ’t Hooft coupling
and in the long quiver limit.999See however [18] for comments on finite gauge
couplings. The sphere free energy is holographic dual to the on-shell
Einstein–Hilbert action, evaluated on the classical solution and properly
regularised. The regularisation unavoidably introduces scheme dependence, and
both in CFT and supergravity the free energy is defined up to a finite
counterterm proportional to the central charge:

$S_{\text{\rm counterterm}}=\text{\rm const}\cdot a.$

The supergravity action and the free energy of the matrix model thus agree
possibly up to a counterterm of this form. Since $a$ is a fairly simple
function of the parameters, this ambiguity is easy to factor out and
faithfully compare localisation to supergravity.

We now quickly review the generalities of the supergravity solution, and then