Wilson loops for 5d and 3d conformal linear quivers

Ali Fatemiabhari<EMAIL_ADDRESS>and Carlos
<EMAIL_ADDRESS>

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

Abstract

Within the electrostatic formulation of holographic duals to (balanced)
conformal quivers in five and three dimensions, we study the expressions for
Wilson loops in antisymmetric representations. We derive general expressions
for various quantities participating in the formalism (VEV of Wilson loops,
representation, gauge-node) and apply these to examples, connecting some
results present in the bibliography. In the case of three dimensional quivers,
we present a relation between Wilson loops in an ’electric’ and in the
’magnetic/mirror’ descriptions. In a very detailed appendix, we relate the
electrostatic and holomorphic description of the holographic duals to these
SCFTS.

###### Contents

  1. 1 Introduction
  2. 2 Supergravity backgrounds
    1. 2.1 The Type IIB Backgrounds dual to 5d SCFTs
      1. 2.1.1 Resolution of the PDE and quantisation of charges
    2. 2.2 The Type IIB backgrounds dual to 3d SCFTs
      1. 2.2.1 Resolution of the PDE and quantisation of charges
    3. 2.3 Summary
  3. 3 Wilson loops in $d=5$ and $d=3$
    1. 3.1 Example 1
      1. 3.1.1 A numerical study
    2. 3.2 Example 2
  4. 4 Mirror symmetry and Wilson loops
  5. 5 Conclusions
  6. A Summary of the Matrix Model associated with the SCFTs
  7. B Map between holomorphic and electrostatic formalisms
    1. B.1 The DGKU solution dual to 5d SCFTs
      1. B.1.1 Matching the solutions
      2. B.1.2 Example: $T_{P}$ theory
      3. B.1.3 Wilson loops
    2. B.2 The DEGK background dual to 3d SCFTs
      1. B.2.1 Example: Generic balanced quivers
      2. B.2.2 Wilson loops
      3. B.2.3 An example of mapping S-dual backgrounds

##  1 Introduction

The Maldacena conjecture, or AdS/CFT [2] motivates the study of both gravity
and field theory topics. In particular, the study of supersymmetric and
conformal field theories in diverse dimensions. In relation to this, various
efforts have been dedicated to the classification of Type II or M-theory
backgrounds with AdSd+1 factors. These backgrounds are proposed as holographic
duals to (encoding semi-classically the highly quantum dynamics of) SCFTs in
$d$ dimensions with different amounts of SUSY. For the case in which the
solutions are half-maximally supersymmetric, important progress in classifying
string backgrounds and the mapping to families of quantum field theories has
been achieved.

A lot of work has been done along the lines described above. In this paper, we
focus our attention on the case of conformal and supersymmetric linear quiver
field theories in three and five dimensions preserving eight Poincare
supercharges. This is the framework in which this paper should be read.

In the case of three dimensional ${\cal N}=4$ SCFTs, the field theoretical
aspects of linear quivers presented in [3] were discussed holographically in
[4]-[10] among other works. The case of ${\cal N}=1$ five dimensional linear
quiver SCFTs (with eight Poincaré supercharges) was initially analysed
holographically in [11]. A non-exhaustive list of papers testing the
correspondence and analysing predictions derived for this case are [11]-[21].

In this work, we are mainly interested on Wilson loops. These gauge invariant
observables are of outmost importance and have been profusely studied in the
context of AdS/CFT. See [22]-[34], for a brief collection of papers on the
topic. In SUSY gauge theories, the Wilson loop is a particularly interesting
observable, as it can be computed exactly. Their relevance to AdS/CFT is the
addition they make to the already rich dictionary between gauge theory and
string theory. Studies in four dimensional ${\cal N}=4$ SYM have been done for
over twenty years. Less understood is the case of SUSY Wilson loops in three
or five dimensional SCFTs. This paper focuses on this particular problem.

In three dimensions, the Wilson loop in ${\cal N}=4$ supersymmetric field
theories is labelled by a representation $\mathbb{R}$ of a given gauge group,

$W_{\mathbb{R}}=\mbox{Tr}_{\mathbb{R}}{\cal
P}e^{i\oint\left(A_{\mu}\dot{x}^{\mu}+\sigma_{3}\sqrt{-\dot{x}^{2}}\right)d\tau}.$
(1.1)

Where $\sigma_{3}$ is one of the three scalars in the vector multiplet. The
original bosonic symmetries of the SCFT $SO(2,3)\times SU(2)_{L}\times
SU(2)_{R}$ is broken by the presence of the operator in eq.(1.1) into
$SU(1,2)\times U(1)\times SU(2)_{L}\times U(1)_{R}$. This is the algebra of
superconformal quantum mechanics. At low energies, when the three dimensional
QFT reaches a fixed point, the Wilson loop becomes a conformal line operator.
For further studies on three dimensional SCFTs and their Wilson loops see
[31]-[34].

The situation in five dimensional SCFTs is similar. The Wilson loop is given
by eq.(1.1), where $\sigma_{3}$ is in this case the adjoint scalar in the
vector multiplet. The Wilson loop in five dimensional SCFTs preserves the
$SU(2)_{R}$ of the theory and breaks $SO(2,5)\rightarrow SU(1,1)\times SO(4)$.
For further studies on Wilson loops in five dimensional SCFTs see [28],

In this paper we rely on the calculations with Wilson loops described in
[17],[27]. We use the electrostatic formalism described in [10], [21],
translating the results of [17], [27] into the the electrostatic formulation.
An advantage of the formalism presented here is that some other calculations
and the interpretation of the solutions may be easier to perform using our
electrostatic viewpoint. Also, the electrostatic formalism makes clear certain
analogies between systems in different dimensions. We discuss mirror symmetry
in the three dimensional case, proposing a relation between Wilson Loops in
both mirror descriptions.

The organisation of the material in this paper is the following: in Section 2
we summarise the electrostatic formalism to construct holographic duals to
balanced-quiver SCFTs in dimensions five and three, with emphasis on the
analogies between these two cases. The general characteristics described in
this section extend to SCFTs in 1,2,4 and 6 dimensions. In Section 3 we
summarise the result of the works [17], [27] in the electrostatic language,
relegating to an appendix the careful derivation of these results. We discuss
two examples in full detail, clarifying and connecting different results in
the bibliography. In Section 4 we discuss aspects of Mirror symmetry, as seen
by the electrostatic formalism. In particular, we derive an expression
relating the Wilson loops in a given representation in both electric and
magnetic description. In Section 5, we summarise and close this paper,
proposing some topics for further study.

In the appendixes, we briefly elaborate on the matrix model perspective of our
results and we describe precisely the translation between the ’holomorphic’
formalism of [4] and the electrostatic perspective pushed in this paper,
making clear the connection with S-duality.

##  2 Supergravity backgrounds

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

###  2.1 The Type IIB Backgrounds dual to 5d SCFTs

We present an infinite family of Type IIB backgrounds preserving eight
Poincaré supersymmetries with an AdS6 factor. The space also contains a two
sphere parameterised by coordinates $(\theta,\varphi)$. The isometries of this
manifold correspond with the $SO(2,5)\times SU(2)_{R}$ bosonic global symmetry
of the dual ${\cal N}=1$ five dimensional SCFTs.

The full configuration consists of a metric, dilaton, $B_{2}$-field in the NS
sector and $C_{2}$ and $C_{0}$ fields in the Ramond sector. The configuration
is written in terms of a potential function $V_{5}(\sigma,\eta)$ that solves a
linear partial differential equation written below. The type IIB background in
string frame is [21],

$\displaystyle
ds_{10,st}^{2}=f_{1}(\sigma,\eta)\Big{[}ds^{2}(\text{AdS}_{6})+f_{2}(\sigma,\eta)ds^{2}(S^{2})+f_{3}(\sigma,\eta)(d\sigma^{2}+d\eta^{2})\Big{]},\;\;e^{-2\Phi}=f_{6}(\sigma,\eta),$
$\displaystyle
B_{2}=f_{4}(\sigma,\eta)\text{Vol}(S^{2}),\;\;C_{2}=f_{5}(\sigma,\eta)\text{Vol}(S^{2}),\;\;\;C_{0}=f_{7}(\sigma,\eta),$
(2.1) $\displaystyle
f_{1}=\frac{3\pi}{2}\sqrt{\sigma^{2}+\frac{3\sigma\partial_{\sigma}V_{5}}{\partial^{2}_{\eta}V_{5}}},\;\;f_{2}=\frac{\partial_{\sigma}V_{5}\partial^{2}_{\eta}V_{5}}{3\Lambda},\;\;f_{3}=\frac{\partial^{2}_{\eta}V_{5}}{3\sigma\partial_{\sigma}V_{5}},\;\;\Lambda=\sigma(\partial_{\sigma}\partial_{\eta}V_{5})^{2}+(\partial_{\sigma}V_{5}-\sigma\partial^{2}_{\sigma}V_{5})\partial^{2}_{\eta}V_{5},$
$\displaystyle
f_{4}=\frac{\pi}{2}\left(\eta-\frac{(\sigma\partial_{\sigma}V_{5})(\partial_{\sigma}\partial_{\eta}V_{5})}{\Lambda}\right),\;\;\;\;f_{5}=\pi\left(V_{5}-\frac{\sigma\partial_{\sigma}V_{5}}{\Lambda}(\partial_{\eta}V_{5}(\partial_{\sigma}\partial_{\eta}V_{5})-3(\partial^{2}_{\eta}V_{5})(\partial_{\sigma}V_{5}))\right),$
$\displaystyle
f_{6}=12\frac{\sigma^{2}\partial_{\sigma}V_{5}\partial^{2}_{\eta}V_{5}}{(3\partial_{\sigma}V_{5}+\sigma\partial^{2}_{\eta}V_{5})^{2}}\Lambda,\;\;\;\;f_{7}=2\left(\partial_{\eta}V_{5}+\frac{(3\sigma\partial_{\sigma}V_{5})(\partial_{\sigma}\partial_{\eta}V_{5})}{3\partial_{\sigma}V_{5}+\sigma\partial^{2}_{\eta}V_{5}}\right).$

The function $V_{5}(\sigma,\eta)$ solves

$\partial_{\sigma}\left(\sigma^{2}\partial_{\sigma}V_{5}\right)+\sigma^{2}\partial^{2}_{\eta}V_{5}=0.$
(2.2)

The paper [21] proves that this infinite family of backgrounds is in exact
correspondence with the solutions discussed in [11]-[13].

Let us briefly summarise the study of [21] for the PDE, with boundary
conditions leading to a proper interpretation of the solutions, with quantised
Page charges and avoiding badly-singular behaviours.

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

We make the change
$V_{5}(\sigma,\eta)=\frac{\widehat{V}_{5}(\sigma,\eta)}{\sigma},$ which
implies that the PDE in (2.2) reads like a Laplace equation in flat space,

$\partial^{2}_{\sigma}\widehat{V}_{5}+\partial_{\eta}^{2}\widehat{V}_{5}=0.$
(2.3)

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

$\displaystyle\widehat{V}_{5}(\sigma\rightarrow\pm\infty,\eta)=0,\;\;\;\;\;\widehat{V}_{5}(\sigma,\eta=0)=\widehat{V}_{5}(\sigma,\eta=P)=0.$
$\displaystyle\lim_{\epsilon\rightarrow
0}\left(\partial_{\sigma}\widehat{V}_{5}(\sigma=+\epsilon,\eta)-\partial_{\sigma}\widehat{V}_{5}(\sigma=-\epsilon,\eta)\right)={\cal
R}(\eta).$ (2.4)

$\sigma$$\eta$$0$$P$$\mathcal{R}(\eta)$ Figure 1: Depiction of the
electrostatic problem for $\widehat{V}_{5}$. The two conducting planes at
$\eta=0,P$ have zero potential, while at $\sigma=0$ we have a charge
distribution equal to $\cal R(\eta)$.

These can be interpreted as the boundary conditions for the electrostatic
problem of two conducting planes (at zero electrostatic potential) as depicted
in Figure 1. The conducting planes extend over the $\sigma$-direction and are
placed at $\eta=0$ and $\eta=P$. We also have a charge density ${\cal
R}(\eta)$ at $\sigma=0$, extended along $0\leq\eta\leq P$, as indicated by the
difference of the normal components of the electric field. The function ${\cal
R}(\eta)$ can be taken to satisfy

${\cal R}(\eta=0)={\cal R}(\eta=P)=0.$ (2.5)

We refer to this in eq.(2.5) as a situation without offsets. Otherwise, if
${\cal R}(\eta)$ is non-zero at either $\eta=0$ or $\eta=P$ we refer to as a
situation with offsets.

The solution is found by separating variables; see [21] for the details. It is
convenient to Fourier expand the function ${\cal R}(\eta)$ as,

${\cal
R}(\eta)=\sum_{k=1}^{\infty}\mathcal{R}_{k}\sin\left(\frac{k\pi}{P}\eta\right),\;\;\;\;\mathcal{R}_{k}=\frac{2}{P}\int_{0}^{P}R(\eta)\sin\left(\frac{k\pi\eta}{P}\right)d\eta.$
(2.6)

Following [21], the solution reads,

$\widehat{V}_{5}(\sigma,\eta)=\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi}{P}\eta\right){e^{-\frac{k\pi}{P}|\sigma|}},\;\;\;\;a_{k}=\frac{P}{2\pi
k}\mathcal{R}_{k}.$ (2.7)

Notice that we can introduce a complex variable

$z=\sigma-i\eta,$

and write the potential $\widehat{V}_{5}=\sigma V_{5}$ as a harmonic function
for both $\sigma>0$ and $\sigma<0$

$\widehat{V}_{5}(\sigma,\eta)=\begin{cases}\sum_{k=1}^{\infty}\frac{a_{k}}{2i}\left(e^{-\frac{k\pi}{P}z}-e^{-\frac{k\pi}{{P}}\bar{z}}\right)&\sigma\geq
0,\\\\[5.69054pt]
\sum_{k=1}^{\infty}\frac{ia_{k}}{2}\left(e^{\frac{k\pi}{P}z}-e^{\frac{k\pi}{{P}}\bar{z}}\right)&\sigma<0.\end{cases}$
(2.8)

$\widehat{V}_{5}$ can therefore be expressed as the real part of a holomorphic
function, and regularity is broken at $\sigma=0$ due to the charge density in
the electrostatic problem. See [21] and Appendix B, for translation between
our formalism and the holomorphic one in [11, 12].

The reader can check that the potentials in eqs.(2.7)-(2.8) solve the
equations (2.2),(2.3) subject to the conditions in eq.(2.4).

Imposing the quantisation of the conserved Page charges in eq.(2.1), the
authors of [21] found that the function ${\cal R}(\eta)$ must be a convex
piecewise linear function.

${\cal R}(\eta)=\begin{cases}N_{0}+(N_{1}-N_{0})\eta&0\leq\eta\leq 1\\\
N_{l}+(N_{l+1}-N_{l})(\eta-l)&l\leq\eta\leq l+1,\;\;\;l:=1,....,P-2\\\
N_{P}+(N_{P-1}-N_{P})(P-\eta)&(P-1)\leq\eta\leq P.\end{cases}$ (2.9)

For $N_{0}=N_{P}=0$ this is a rank function without off-sets. Otherwise, it
has off-sets. In the case of no-offsets, the values of the quantised brane
charges in each interval $[k,k+1]$ and in the system as a whole have been
computed in [21],

$\displaystyle Q_{NS5,total}=P\,$ (2.10) $\displaystyle Q_{D7}[k,k+1]={\cal
R}^{\prime\prime}(k)=(2N_{k}-N_{k+1}-N_{k-1}),\;Q_{D7,total}=(N_{1}+N_{P-1})=\int_{0}^{P}{\cal
R}^{\prime\prime}(\eta)d\eta,$ $\displaystyle Q_{D5}[k,k+1]={\cal
R}(\eta)-{\cal
R}^{\prime}(\eta)(\eta-\Delta)=N_{k}\,,\;\;\;\;\;Q_{D5,total}=\int_{0}^{P}{\cal
R}\leavevmode\nobreak\ d\eta.$

For the generic rank function ${\cal R}(\eta)$ quoted in eq.(2.9), the
supergravity background is proposed to be dual to the strongly coupled, UV-
fixed point of the quiver in Figure 2 for which
$F_{i}=2N_{i}-N_{i+1}-N_{i-1}$. In other words, the quiver is balanced.

N1N2$\dots$NP-1NP-2F1F2$\dots$FP-1FP-2 Figure 2: Long quiver of length $P-1$
with gauge nodes $N_{i}$ and flavour nodes $F_{i}$. The quiver is balanced if
$F_{i}=2N_{i}-N_{i-1}-N_{i+1}$.

We now discuss the Type IIB backgrounds dual to three dimensional SCFTs
preserving eight SUSYs. The formalism is very much analogous to the five
dimensional one, hence we will be more sketchy. All the details can be found
in [10].

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

We are after solutions dual to 3d ${\cal N}=4$ super-conformal field theories.
Matching the global symmetries of the field theory implies that the background
must have isometries
$\text{SO}(2,3)\times\text{SU}(2)_{C}\times\text{SU}(2)_{H}$ and preserve
eight Poincaré supercharges. Our geometries must contain an AdS4 factor and a
couple of two spheres $S^{2}_{1}(\theta_{1},\varphi_{1})$ and
$S^{2}_{2}(\theta_{2},\varphi_{2})$. There are two extra directions labelled
by $(\sigma,\eta)$. The presence of
$\text{SO}(2,3)\times\text{SU}(2)_{C}\times\text{SU}(2)_{H}$ isometries allow
for warp factors that depend only on $(\sigma,\eta)$. The Ramond and Neveu-
Schwarz fields must also respect the above-mentioned isometries.

The preservation of eight Poincaré supersymmetries implies that the generic
type IIB background can be written in terms of a function
$V_{3}(\sigma,\eta)$. In string frame the solution reads [10],

$\displaystyle
ds_{10,st}^{2}=f_{1}(\sigma,\eta)\Big{[}ds^{2}(\text{AdS}_{4})+f_{2}(\sigma,\eta)ds^{2}(S^{2}_{1})+f_{3}(\sigma,\eta)ds^{2}(S^{2}_{2})+f_{4}(\sigma,\eta)(d\sigma^{2}+d\eta^{2})\Big{]},$
$\displaystyle
e^{-2\Phi}=f_{5}(\sigma,\eta),\;\;B_{2}=f_{6}(\sigma,\eta)\text{Vol}(S^{2}_{1}),\;\;C_{2}=f_{7}(\sigma,\eta)\text{Vol}(S^{2}_{2}),\;\;\;\tilde{C}_{4}=f_{8}(\sigma,\eta)\text{Vol(AdS}_{4}),$
$\displaystyle
f_{1}=\frac{\pi}{2}\sqrt{\frac{\sigma^{3}\partial^{2}_{\eta\sigma}V_{3}}{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}},\;\;f_{2}=-\frac{\partial_{\eta}V_{3}\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma\Lambda},\;\;f_{3}=\frac{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma\partial^{2}_{\eta\sigma}V_{3}},\;\;f_{4}=-\frac{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}{\sigma^{2}\partial_{\eta}V_{3}},$
$\displaystyle
f_{5}=-16\frac{\Lambda\partial_{\eta}V_{3}}{\partial^{2}_{\eta\sigma}V_{3}},\;\;\;\;f_{6}=\frac{\pi}{2}\left(\eta-\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\Lambda}\right),\;\;\;\;f_{7}=-2\pi\left(\partial_{\sigma}(\sigma
V_{3})-\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\partial^{2}_{\eta\sigma}V_{3}}\right),$
$\displaystyle
f_{8}=-\pi^{2}\sigma^{2}\left(3\partial_{\sigma}V_{3}+\frac{\sigma\partial_{\eta}V_{3}\partial_{\eta}^{2}V_{3}}{\partial_{\sigma}(\sigma\partial_{\eta}V_{3})}\right),\;\;\;\;\Lambda=\partial_{\eta}V_{3}\partial^{2}_{\eta\sigma}V_{3}+\sigma\left((\partial^{2}_{\eta\sigma}V_{3})^{2}+(\partial^{2}_{\eta}V_{3})^{2}\right).$
(2.11)

Where the fluxes are defined from the potentials as follows,

$F_{1}=0,\quad H_{3}=dB_{2}\quad F_{3}=dC_{2},\quad
F_{5}=d\tilde{C}_{4}+*d\tilde{C}_{4}.$ (2.12)

The configuration in eq.(2.11) is solution to the Type IIB equations of
motion, if the function $V(\sigma,\eta)$ satisfies,

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

As proven in [10] and in detail in Appendix B, this infinite family of
solutions is equivalent to the backgrounds described by D’Hoker, Estes and
Gutperle in [4].

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

Following [10], define
$V_{3}(\sigma,\eta)=\frac{\widehat{V}_{3}(\sigma,\eta)}{\sigma}$ and
$\widehat{V}_{3}(\sigma,\eta)=\partial_{\eta}\widehat{W}(\sigma,\eta)$.
Consider the coordinates to range in $0\leq\eta\leq P$, where $P$ is a real
number, and $-\infty<\sigma<\infty$. The differential equation (2.13) must be
supplemented by boundary and initial conditions. In terms of
$\widehat{W}(\sigma,\eta)$ the problem reads

$\displaystyle\partial^{2}_{\sigma}\widehat{W}(\sigma,\eta)+\partial^{2}_{\eta}\widehat{W}(\sigma,\eta)=0,\qquad\qquad\text{(almost
everywhere)}$ (2.14)
$\displaystyle\widehat{W}(\sigma,\eta=0)=0,\;\;\;\;\;\widehat{W}(\sigma,\eta=P)=0,$
$\displaystyle\partial_{\sigma}\widehat{W}(\sigma=0^{+},\eta)-\partial_{\sigma}\widehat{W}(\sigma=0^{-},\eta)=-{\cal
R}(\eta).$

In analogy with the five dimensional case discussed above, the function ${\cal
R}(\eta)$ is the input determined by the dual quiver field theory. Notice
that, since $\widehat{W}$ is a harmonic function, we have that also
$\widehat{V}$ is harmonic, which in turn implies (2.13).

Using a Fourier decomposition for the rank function ${\cal R}(\eta)$ as in the
five-dimensional case- see eq.(2.6), the solution to the problem in eq.(2.14)
is,

$\displaystyle\widehat{V}_{3}(\sigma,\eta)=\sum_{k=1}^{\infty}b_{k}\cos\left(\frac{k\pi\eta}{P}\right)e^{-\frac{k\pi|\sigma|}{P}},$
$\displaystyle\widehat{W}(\sigma,\eta)=\sum_{k=1}^{\infty}b_{k}\left(\frac{P}{k\pi}\right)\sin\left(\frac{k\pi\eta}{P}\right)e^{-\frac{k\pi|\sigma|}{P}}.$
(2.15) $\displaystyle b_{k}=\frac{1}{P}\int_{0}^{P}{\cal
R}(\eta)\sin\left(\frac{k\pi\eta}{P}\right)=\frac{\mathcal{R}_{k}}{2}.$

The study of the quantised charges for Neveu-Schwarz five branes, imposes that
the size of the interval $P$ is an integer–consistently with the boundary
conditions in eq.(2.14), exactly as it occurs in the five dimensional system.
Also, in analogy with the 5d-case, to have quantised numbers of D3 and D5
branes, the rank function must be a piece-wise linear and continuous function
of the exact same form as in the five dimensional case–see eq.(2.9).

In the case without offsets, $N_{0}=N_{P}=0$, the number of D3 (colour) branes
and D5 (flavour) branes in the interval $[k,k+1]$ and the total number of
branes are given in [10] ,

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

The rank function ${\cal R}(\eta)$ encodes the ‘kinematic data’ of the dual
conformal field theory. The presence of $P$ NS5 branes along the
$\eta$-direction suggests that we should place one NS5 at each integer value
of $\eta$. In between the $k^{th}$ and $(k+1)^{th}$ NS5-branes, we have
$N_{k}$ D3 branes and $F_{k}=2N_{k}-N_{k+1}-N_{k-1}$ D5 branes as indicated in
eq.(2.16).

The ${\cal N}=4$ quiver field theory, for the rank function without offsets is
the same as that in the five dimensional system, drawn below eq.(2.10). This
balanced QFT is proposed to reach a strongly coupled IR fixed point described
by the background in eq.(2.11).

###  2.3 Summary

Consider the balanced quiver field theory, preserving eight Poincare SUSYs
depicted in the Figure 2. In the case the field theory is five-dimensional, it
is conjectured to reach a strongly coupled fixed point at high energies
(compared with the scale defined by the gauge coupling). Conversely, if
defined in three dimensions the interacting fixed point will be at low
energies.

The quiver can be associated with a rank function. In this section, we
consider for generality the situation with offsets as indicated in eq.(2.9),

We calculate the Fourier coefficient of this rank function using eq.(2.6). We
find,

$\displaystyle\mathcal{R}_{k}=\frac{2}{k\pi}(N_{0}+(-1)^{k+1}N_{P})\left[1-\frac{P}{k\pi}\sin\left(\frac{k\pi}{P}\right)\right]+\frac{2P}{k^{2}\pi^{2}}\sum_{j=1}^{P-1}F_{j}\sin\left(\frac{k\pi
j}{P}\right).$ (2.17) $\displaystyle
F_{1}=2N_{1}-N_{2},\;\;\;F_{P-1}=2N_{P-1}-N_{P-2},\;\;\;\;F_{j}=2N_{j}-N_{j+1}-N_{j-1}.$

We use the expressions in eqs. (2.7) and (2.15) to calculate the potentials
$\widehat{V}_{5}(\sigma,\eta)$ and $\widehat{V}_{3}(\sigma,\eta)$ in terms of
which the supergravity backgrounds in eqs.(2.1),(2.11) are defined. Notice
that the input, namely the quiver field theory is the same in both cases
($d=5$ or $d=3$). The differences in the dynamics are encoded in the
coefficients $a_{k}$ and $b_{k}$, the potentials $\widehat{V}_{5}$ and
$\widehat{V}_{3}$ and the different expressions for the functions
$f_{i}(\sigma,\eta)$ in each of the backgrounds in eqs.(2.1), (2.11)
respectively.

It is convenient to define the complex variable

$\xi=\leavevmode\nobreak\
e^{-\frac{\pi}{P}\left[|\sigma|-i\eta\right]},\;\;\;-\frac{P}{\pi}\log|\xi|=|\sigma|,\;\;e^{-\frac{\pi}{P}\left[|\sigma|-i(\eta\pm
J)\right]}=\xi e^{\pm\frac{i\pi J}{P}}.$ (2.18)

In terms of this complex variable, we find for the five dimensional
$\widehat{V}_{5}(\sigma,\eta)$,

$\displaystyle\widehat{V}_{5}(\sigma,\eta)=\frac{N_{0}}{\pi^{2}}\Big{[}P\leavevmode\nobreak\
\text{Im}(\leavevmode\nobreak\ \text{Li}_{2}(\xi)\leavevmode\nobreak\
)+\frac{P^{2}}{2\pi}\text{Re}(\leavevmode\nobreak\ \text{Li}_{3}(\xi
e^{\frac{i\pi}{P}})-\text{Li}_{3}(\xi e^{-\frac{i\pi}{P}})\leavevmode\nobreak\
)\Big{]}$ $\displaystyle-\frac{N_{P}}{\pi^{2}}\Big{[}P\leavevmode\nobreak\
\text{Im}(\leavevmode\nobreak\ \text{Li}_{2}(-\xi)\leavevmode\nobreak\
)+\frac{P^{2}}{2\pi}\leavevmode\nobreak\ \text{Re}(\leavevmode\nobreak\
\text{Li}_{3}(-\xi e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi
e^{-\frac{i\pi}{P}})\leavevmode\nobreak\ )\Big{]}+$
$\displaystyle\frac{P^{2}}{2\pi^{3}}\sum_{J=1}^{P-1}F_{J}\text{Re}(\leavevmode\nobreak\
\text{Li}_{3}(\xi e^{-\frac{i\pi J}{P}})-\text{Li}_{3}(\xi e^{\frac{i\pi
J}{P}})\leavevmode\nobreak\ ).$ (2.19)

In the case $N_{0}=N_{P}=0$ (no offsets) this should be compared with the
particular expressions obtained in [21].

By comparing eqs.(2.7) and (2.15) we find that the three-dimensional quantity
$\widehat{W}(\sigma,\eta)$ is equal to the five dimensional potential
$\widehat{V}_{5}(\sigma,\eta)$ in eq.(2.19). The three dimensional potential
$\widehat{V}_{3}(\sigma,\eta)$ is,

$\displaystyle\widehat{V}_{3}(\sigma,\eta)=\frac{N_{0}}{4\pi^{2}}\Big{[}-2\pi\log[(1-\xi)(1-\bar{\xi})]-2P\leavevmode\nobreak\
\text{Im}(\leavevmode\nobreak\ \text{Li}_{2}(\xi
e^{\frac{i\pi}{P}})-\text{Li}_{2}(\xi e^{-\frac{i\pi}{P}})\leavevmode\nobreak\
)\Big{]}$
$\displaystyle-\frac{N_{P}}{4\pi^{2}}\Big{[}-2\pi\log[(1+\xi)(1+\bar{\xi})]-2P\leavevmode\nobreak\
\text{Im}(\leavevmode\nobreak\ \text{Li}_{2}(-\xi
e^{\frac{i\pi}{P}})-\text{Li}_{2}(-\xi
e^{-\frac{i\pi}{P}})\leavevmode\nobreak\ )\Big{]}$
$\displaystyle+\frac{P}{2\pi^{2}}\sum_{J=1}^{P-1}F_{J}\text{Im}(\leavevmode\nobreak\
\text{Li}_{2}(\xi e^{\frac{i\pi J}{P}})-\text{Li}_{2}(\xi e^{-\frac{i\pi
J}{P}})\leavevmode\nobreak\ ).$ (2.20)

The analogy observed in this section between the problems in five and three
dimensions is not a coincidence. An analysis for the free energy, reducing the
problem to matrix models, was performed by Uhlemann in the five dimensional
case [16] and by Coccia and Uhlemann in the three-dimensional one [8]. This
analysis also reveals the analogies between both problems. In Appendix A, we
briefly summarise these matrix models and link them to the electrostatic
problems we discussed in this section.
In the coming section, we briefly write the formulation of the Wilson Loops in
generic antisymmetric representations in the electrostatic language discussed
above.

##  3 Wilson loops in $d=5$ and $d=3$

In this section we re-state, in our electrostatic language, the result
obtained in [17], [27] for the Wilson loops in a given antisymmetric
representation. The precise mapping used in this translation is given in
Appendix B. After this, we discuss two examples in great detail. This makes
interesting connections between different results in the bibliography.

Consider first five dimensional linear quiver gauge theories flowing to a SCFT
in the UV. The field theory is realised in the low energy regime of stacks of
D5 branes (on which the gauge groups are realised) extending between NS five
branes with $(p,q)$ five branes. There are also stacks of D7 branes realising
the flavour symmetries of the quiver.

The Wilson loop is calculated using a D3 probe extending in the time direction
and a direction perpendicular to the stack of colour D5 branes. This D3 probe
preserves the $SU(2)$ R-symmetry of the SCFT. On this probe, charges of D1
brane and of fundamental string F1 are induced. The F1 extends between the D3
probe and $k$ of the $N_{l}$ D5 colour branes. The s-rule (Pauli principle)
indicates that either one or no F1 stretch between the probe D3 and a give D5.
The low energy description of such F1, as found in [28], is given by a one-
dimensional ${\cal N}=4$ conformal quantum mechanics in terms of a massive
Fermi multiplet. Integrating out these Fermi multiples as in [30] generates
the Wilson loop for the group $SU(N_{l})$ in the $k$-antisymmetric
representation. The charge of D1 brane induced on the D3 indicates the number
of NS-five branes the D3 probe has ’gone through’, hence is in correspondence
with the position of the $SU(N_{l}$)-colour group.

The holographic calculation of these Wilson loops in the $k$-antisymmetric
representation requires the calculation of the on-shell action of a D3 probe
that extends along AdS2 inside AdS6 in the background of eq.(2.1). The two
sphere $S^{2}(\theta,\varphi)$ representing the R-symmetry of the SCFT is also
wrapped, hence preserved. An $SO(4)$ isometry realised in the remaining
direction inside AdS6 is also preserved. There are fluxes switched on the
probe D3,

${\cal F}_{2}=f_{el}\text{vol}\text{AdS}_{2}+f_{mag}\text{vol}\text{S}^{2}.$
(3.1)

This flux induces the above mentioned charge of F1 and D1 on the D3 probe
world-volume. The D1 charge is in correspondence with the position of the
$l$-stack of D5 branes, hence the position in the $\eta$-coordinate is
$\eta_{*}=l$. The F1 charge is associated with the number $\mathbb{k}$,
labelling the representation. This can be thought of as the position
$\sigma_{*}$ in the $\sigma$-direction.

This probe was studied by Uhlemann in [17], finding the conditions to preserve
SUSY that are imposed on the charges in eq.(3.1). The on-shell action for the
probe D3 was found (after a Legendre transform).

The result for the VEV of Wilson loops for a given $\mathbb{k}$-antisymmetric
representation $\wedge$ in the electrostatic language is succinct,

$\displaystyle\ln\langle
W_{\wedge}\rangle=3\pi\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}\left(\frac{P}{k\pi}\right)\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+1).$
(3.2) $\displaystyle N_{\rm D1}=\eta^{*},\quad N_{\rm
F1}=\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}\operatorname{Sgn}(\sigma^{*}).$

The gauge node for which the Wilson loop is computed is labelled by the
position along the quiver $\eta^{*}=1,2,3,4....$. The antisymmetric
representation of the Wilson loop $\mathbb{k}$ coincides with the number of F1
strings. One should determine $|\sigma^{*}|$ by solving the equation for
$N_{F1}=\mathbb{k}$ for the particular value of $\eta^{*}$ given by the chosen
gauge node. These values used in eq.(3.2) give the VEV of the Wilson loop.

Let us now consider the same type of Wilson loop in the case of three
dimensional SCFTs. These 3d low energy SCFTs are realised on stacks of D3
branes that extend between NS five branes. There are also stacks of D5 branes,
representing the flavour groups. As found in [31], the Wilson loop in the
$\mathbb{k}$-antisymmetric representation is calculated by a probe D5’ brane,
extending along time, preserving one of the $SU(2)$ of the R-symmetry and also
preserving a $U(1)$ part of the R-symmetry. Hence, this probe does not extend
along the same directions of flavour D5 branes. The probe preserves
$SO(1,1)\times SU(2)_{R}\times U(1)_{L}$ inside the (bosonic part of) the
original symmetry group of the 3d SCFT $SO(2,3)\times SU(2)_{L}\times
SU(2)_{R}$. It should also preserve four SUSYs.

This probe D5’ is introduced in between the NS five branes that limit the
$l$-stack of colour D3 branes. Like in the five dimensional situation above
analysed, the parameter $\mathbb{k}$ labelling the antisymmetric
representation is realised by the charge of F1 induced on the D5’. A very
similar procedure—the integration of a one dimensional Fermi multiplet
describing the dynamics of these F1 strings leads to the insertion of a Wilson
operator in the $\mathbb{k}$-antisymmetric representation. See [30], [31].

In the holographic dual background of eq.(2.11) the VEV of the Wilson loop is
calculated by the on-shell action of a D5’ that wraps AdS2–to realise
$SO(1,1)$, the two sphere $S^{2}(\theta_{1},\varphi_{1})$–to realise the
$SU(2)_{R}$ and a circle inside $S^{2}(\theta_{2},\varphi_{2})$. This is
achieved by the D5’ extending along a direction parametrised by $y$ and
choosing $\eta(y)$, $\sigma(y)$, $\theta_{2}(y)$. Like in the five dimensional
case a world-volume flux on the D5’ is needed to preserve SUSY, see eq.(3.1).
Imposing SUSY preservation on this probe, Coccia and Uhlemann [27] found the
relation between the position of the D5’ probe in the $(\sigma,\eta)$ plane
and the induced charges of F1 and D3. From there the authors of [27]
calculated the Legendre transformed on-shell action for the D5’.

In our electrostatic language, we find calculating with the background in
eq.(2.11),

$\displaystyle\ln\langle W_{\wedge}\rangle=\pi\sum_{k=1}^{\infty}\frac{{\cal
R}_{k}}{2}\left(\frac{P}{k\pi}\right)\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+{1}),$
(3.3) $\displaystyle N_{D3}=\eta^{*},\quad
N_{\mathrm{F}1}=\sum_{k=1}^{\infty}\frac{{\cal
R}_{k}}{2}\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}\operatorname{Sgn}(\sigma^{*}).$

A similar explanation as in the five dimensional case applies here: the value
of $N_{D3}=\eta^{*}=1,2,3,4,....$ labels the gauge node along the quiver. The
number of fundamental strings, identified with the $\mathbb{k}$ labelling the
antisymmetric representation determine the value of $|\sigma^{*}|$. These two
values used in eq.(3.3) give the VEV of the Wilson loop. Below, we discuss
examples.

Notice that both expressions (3.2)-(3.3) are virtually identical. This is
confirmed by the matrix model treatment of these Wilson loops, that shows as
discussed in Appendix A, that from a field theory viewpoint both expression
differ only in proportionality factors. It would be interesting to learn about
sub-leading corrections to this result. Notice also that the result can be
written both in five and in three dimensions, using the fact that the three
dimensional quantity $\widehat{W}(\sigma,\eta)$ is identical to
$\widehat{V}_{5}(\sigma,\eta)$ in five dimensions,

$\displaystyle\mu^{-1}\ln\langle
W_{\wedge}\rangle=\widehat{V}_{5}(\sigma^{*},\eta^{*})+{|\sigma^{*}|}\operatorname{Sgn}(\sigma^{*})N_{F1}.$
(3.4) $\displaystyle\mu_{3d}=\pi,\;\;\;\;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mu_{5d}=3\pi.$

The expression for $\widehat{V}_{5}(\sigma^{*},\eta^{*})$ can be read from
eq.(2.19), whilst $N_{F1}\operatorname{Sgn}(\sigma_{*})$ can be computed using
eq.(2.17) to be,

$\displaystyle
N_{F1}\operatorname{Sgn}(\sigma^{*})=-\frac{P}{2\pi^{2}}\sum_{J=1}^{P-1}F_{J}\text{Re}\left[\text{Li}_{2}(\leavevmode\nobreak\
\xi_{*}e^{\frac{i\pi J}{P}}\leavevmode\nobreak\
)-\text{Li}_{2}(\leavevmode\nobreak\ \xi_{*}e^{-\frac{i\pi
J}{P}}\leavevmode\nobreak\ )\right]$ (3.5)
$\displaystyle+\frac{N_{0}}{2\pi}\Big{[}2\text{Im}[\leavevmode\nobreak\
\text{Li}_{1}(\xi_{*})\leavevmode\nobreak\
]+\frac{P}{\pi}\text{Re}[\leavevmode\nobreak\
\text{Li}_{2}(\leavevmode\nobreak\
\xi_{*}e^{\frac{i\pi}{P}}\leavevmode\nobreak\
)-\text{Li}_{2}(\leavevmode\nobreak\
\xi_{*}e^{-\frac{i\pi}{P}}\leavevmode\nobreak\ )\leavevmode\nobreak\ ]\Big{]}$
$\displaystyle-\frac{N_{P}}{2\pi}\Big{[}2\text{Im}[\leavevmode\nobreak\
\text{Li}_{1}(-\xi_{*})\leavevmode\nobreak\
]+\frac{P}{\pi}\text{Re}[\leavevmode\nobreak\
\text{Li}_{2}(-\xi_{*}e^{\frac{i\pi}{P}}\leavevmode\nobreak\
)-\text{Li}_{2}(-\xi_{*}e^{-\frac{i\pi}{P}}\leavevmode\nobreak\
)\leavevmode\nobreak\ ]\Big{]}$

We have defined $\xi_{*}=e^{-\frac{\pi}{P}[|\sigma^{*}|-i\eta^{*}]}$.

For the reader’s convenience we write explicitly eq.(3.4),

$\displaystyle\mu^{-1}\ln\langle W_{\wedge}\rangle=$ (3.6)
$\displaystyle\frac{P^{2}}{2\pi^{3}}\sum_{J=1}^{P-1}F_{j}\text{Re}\Big{[}\text{Li}_{3}(\xi_{*}e^{-\frac{i\pi
J}{P}})-\text{Li}_{3}(\xi_{*}e^{\frac{i\pi
J}{P}})+\log|\xi_{*}|\left(\text{Li}_{2}(\xi_{*}e^{\frac{i\pi
J}{P}})-\text{Li}_{2}(\xi_{*}e^{-\frac{i\pi J}{P}})\right)\Big{]}$
$\displaystyle+\frac{PN_{P}}{\pi^{2}}\Big{[}-\text{Im}[\leavevmode\nobreak\
\leavevmode\nobreak\
\text{Li}_{2}(-\xi_{*})-\log|\xi_{*}|\text{Li}_{1}(-\xi_{*})\leavevmode\nobreak\
\leavevmode\nobreak\ ]+\frac{P}{2\pi}\text{Re}\big{[}\leavevmode\nobreak\
\leavevmode\nobreak\
\text{Li}_{3}(-\xi_{*}e^{-\frac{i\pi}{P}})-\text{Li}_{3}(-\xi_{*}e^{\frac{i\pi}{P}})+$
$\displaystyle+\log|\xi_{*}|(\leavevmode\nobreak\ \leavevmode\nobreak\
\text{Li}_{2}(\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{2}(-\xi_{*}e^{-\frac{i\pi}{P}})\leavevmode\nobreak\
\leavevmode\nobreak\ )\leavevmode\nobreak\ \big{]}\Big{]}+$
$\displaystyle+\frac{PN_{0}}{\pi^{2}}\Big{[}\text{Im}[\leavevmode\nobreak\
\leavevmode\nobreak\
\text{Li}_{2}(\xi_{*})-\log|\xi_{*}|\text{Li}_{1}(\xi_{*})\leavevmode\nobreak\
\leavevmode\nobreak\ ]+\frac{P}{2\pi}\text{Re}\big{[}\leavevmode\nobreak\
\leavevmode\nobreak\
\text{Li}_{3}(\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{3}(\xi_{*}e^{-\frac{i\pi}{P}})+$
$\displaystyle-\log|\xi_{*}|(\leavevmode\nobreak\ \leavevmode\nobreak\
\text{Li}_{2}(\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{2}(\xi_{*}e^{-\frac{i\pi}{P}})\leavevmode\nobreak\
\leavevmode\nobreak\ )\leavevmode\nobreak\ \big{]}\Big{]}.$

The reader should compare this expression (in the case of no offsets
$N_{0}=N_{P}=0$), with the field theoretical expression (obtained with matrix
models calculation) in equation (4.53) of [27].
In the rest of this section we evaluate in two examples, the expressions for
the potentials $\widehat{V}_{5}=\widehat{W}$, $\widehat{V}_{3}$ and
$\ln\langle W_{\wedge}\rangle$ in eqs.(2.19),(2.20) and (3.6) respectively. We
focus on the cases of the $T_{M,N}$ and $+_{M,N}$ both in 3d and in 5d. Whilst
these are non-generic examples, they are very used in the existing
bibliography. We will work them out using rank functions with and without
offsets, finding relations between these cases that clarify previous results
in the bibliography.

###  3.1 Example 1

Let us consider a five dimensional gauge theory called $\tilde{T}_{N,P}$. The
gauge theory is described (in the IR) by the quiver

N2N3N$\dots$PN(P-1)N

In the case of this same quiver being considered in three dimensions, the
above quiver is the UV description of the QFT, flowing to a SCFT in the IR.
Let us study all the quantities discussed above.

First, we consider the case without offsets, the rank function associated with
this quiver is,

${\cal R}(\eta)=\begin{cases}N\eta&0\leq\eta\leq(P-1)\\\
N(P-1)(P-\eta)&(P-1)\leq\eta\leq P.\end{cases}$

If the quiver is five dimensional, the number of D7-branes can be read either
from $\mathcal{R}^{\prime\prime}=NP\delta(\eta-P+1)$, or from eq.(2.10) which
gives $Q_{D7}=PN$. The number of D5 branes at the positions
$\eta=1,2,3,4,\text{etc}$, is the value of $\mathcal{R}(\eta)$ at those
points. This coincides with the ranks of the first, second, third, fourth
node, etc. In total, we have $\int_{0}^{P}{\cal R}d\eta=\frac{NP(P-1)}{2}$ D5
branes. We also have a total of $P$ NS-five branes.

In the three dimensional case, we use eq.(2.16). The number of D5 branes
(flavours) is $NP$. The number of D3 branes in each interval coincides with
the values of the rank function and there are a total of $\frac{NP(P-1)}{2}$
D3 branes. The number of NS five branes is also $P$.

Given the rank function above, the coefficient $\mathcal{R}_{k}$ can be read
from eqs.(2.6),(2.17), with $N_{0}=N_{P}=0$ (no offsets) and
$F_{j}=NP\delta_{j,P-1}$. We find,

$\mathcal{R}_{k}=\frac{2NP^{2}}{k^{2}\pi^{2}}\sin\left(\frac{k\pi}{P}\right)(-1)^{k+1}.$
(3.7)

The five dimensional Fourier coefficient $a_{k}$ as defined in eq.(2.7) and
the Fourier coefficient of the three dimensional potential $b_{k}$ in
eq.(2.15) are,

$a_{k}=(-1)^{k+1}\frac{NP^{3}}{k^{3}\pi^{3}}\sin\left(\frac{k\pi}{P}\right),\;\;\;\;\;b_{k}=\frac{NP^{2}}{k^{2}\pi^{2}}\sin\left(\frac{k\pi}{P}\right)(-1)^{k+1}.$
(3.8)

Using eqs.(2.7) and (2.19) the potential $\widehat{V}_{5}(\sigma,\eta)$ is

$\widehat{V}_{5}(\sigma,\eta)=\frac{NP^{3}}{2\pi^{3}}\text{Re}\left(\text{Li}_{3}(-\xi
e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi e^{-\frac{i\pi}{P}})\right)\,.$ (3.9)

For the three dimensional case, we find
$\widehat{W}=\widehat{V}_{5}(\sigma,\eta)$ and using eqs.(2.15) and (2.20)

$\widehat{V}_{3}(\sigma,\eta)=\frac{NP^{2}}{2\pi^{2}}\text{Im}\left(\text{Li}_{2}(-\xi
e^{-\frac{i\pi}{P}})-\text{Li}_{2}(-\xi e^{\frac{i\pi}{P}})\right)\,.$ (3.10)

We now calculate the Wilson loop using eq.(3.4). The part corresponding to
$\widehat{V}_{5}(\sigma^{*},\eta^{*})$ is read from eq.(3.9). The part
corresponding to $N_{F1}$ is calculated explicitly from eqs.(3.2)-(3.3) or
read from eq.(3.5). We find,

$N_{F1}\operatorname{Sgn}(\sigma_{*})=\frac{NP^{2}}{\pi^{2}}\text{Re}\Big{[}\text{Li}_{2}\left(-\xi_{*}e^{\frac{i\pi}{P}}\leavevmode\nobreak\
)\right)-\text{Li}_{2}\left(-\xi_{*}e^{-\frac{i\pi}{P}}\leavevmode\nobreak\
)\right)\Big{]}.$ (3.11)

We write the full result for the Wilson loop, using eq.(3.6). In dimensions
three ($\mu=\pi$) and in five ($\mu=3\pi$),

$\mu^{-1}\ln\langle
W_{\wedge}\rangle=\frac{NP^{3}}{2\pi^{3}}\text{Re}\Big{[}\text{Li}_{3}(-\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi_{*}e^{\frac{-i\pi}{P}})-\log|\xi_{*}|\left(\leavevmode\nobreak\
\text{Li}_{2}(-\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{2}(-\xi_{*}ie^{\frac{-i\pi}{P}})\leavevmode\nobreak\
\right)\Big{]}.$ (3.12)

It is instructive to repeat the calculation for a rank function with offset.
In this case, we choose,

${\cal R}(\eta)=\begin{cases}N\eta&0\leq\eta\leq P.\end{cases}$

Implying $N_{0}=0,N_{P}=NP$. In this case the number of flavour branes will be
found from the derivative ${\cal R}^{\prime}(\eta)=NP\delta(\eta-P+1)$. In
other words $F_{j}=NP\delta_{j,P-1}$. The Fourier coefficient of the rank
function is computed using eq.(2.6) or equivalently, setting
$N_{0}=0,N_{P}=PN$ in eq.(2.17). The result is,

$\mathcal{R}_{k}=2\frac{NP}{k\pi}(-1)^{k+1},\quad
a_{k}=\frac{NP^{2}}{k^{2}\pi^{2}}(-1)^{k+1},\;\;\;\;\;b_{k}=\frac{NP}{k\pi}(-1)^{k+1}.$
(3.13)

The potential in five dimensions can be calculated using eq.(2.7) or read from
eq.(2.19). Using the variable $\xi$ defined in eq.(2.18), the result is,

$\widehat{V}_{5}(\sigma,\eta)=-\frac{P^{2}N}{\pi^{2}}\mathop{\rm
Im}\left[\operatorname{Li}_{2}(-{\xi})\right].$ (3.14)

For the three dimensional potential $\widehat{V}_{3}$ we calculate using
eq.(2.15) or read from eq.(2.20) the result,

$\widehat{V}_{3}(\sigma,\eta)=-\frac{NP}{2\pi}\log[(1+\xi)(1+\bar{\xi})].$
(3.15)

For the Wilson loop, both in 3d and in 5d, we find calculating from eqs.
(3.2)-(3.3), or using the generic result in eq.(3.6),

$\displaystyle\mu^{-1}\ln\langle W_{\wedge}\rangle=-\frac{NP^{2}}{\pi^{2}}$
$\displaystyle\\{\ln|\xi|[\operatorname{Arg}(1-\xi_{*})]+\mathop{\rm
Im}\left[\operatorname{Li}_{2}(-{\xi_{*}})\right]\\}.$ (3.16)

The goal of this carefully developed example is to show that if we perform the
limit of $P\rightarrow\infty$, keeping $\frac{|\sigma|}{P}$ and
$\frac{\eta}{P}$ fixed, the results without offsets —see
eqs.(3.9),(3.10),(3.12) at leading order, reproduce the results computed with
the rank function with offsets in eqs.(3.14),(3.15) and (3.16) respectively.

More precisely, eq.(3.9) would lead to eq.(3.14) in the the limit of
$P\rightarrow\infty$

$\frac{NP^{3}}{2\pi^{3}}\text{Re}\left(\text{Li}_{3}(-e^{-\frac{\pi}{P}(|\sigma|-i-i\eta)})-\text{Li}_{3}(-e^{-\frac{\pi}{P}(|\sigma|+i-i\eta)})\right)\longrightarrow-\frac{P^{2}N}{\pi^{2}}\mathop{\rm
Im}\left[\operatorname{Li}_{2}(-{\xi})\right],$ (3.17)

eq.(3.10) will give eq.(3.15)

$\frac{NP^{2}}{2\pi^{2}}\text{Im}\left(\text{Li}_{2}(-e^{-\frac{\pi}{P}(|\sigma|+i-i\eta)})-\text{Li}_{2}(-e^{-\frac{\pi}{P}(|\sigma|-i-i\eta)})\right)\longrightarrow-\frac{NP}{2\pi}\log[(1+\xi)(1+\bar{\xi})],$
(3.18)

and eq.(3.12) will result in eq.(3.16)

$\displaystyle\frac{NP^{3}}{2\pi^{3}}\text{Re}$
$\displaystyle\Big{[}\text{Li}_{3}(-\xi e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi
e^{\frac{-i\pi}{P}})-\log|\xi|\left(\leavevmode\nobreak\ \text{Li}_{2}(-\xi
e^{\frac{i\pi}{P}})-\text{Li}_{2}(-\xi
e^{\frac{-i\pi}{P}})\leavevmode\nobreak\ \right)\Big{]}\longrightarrow$
$\displaystyle-\frac{NP^{2}}{\pi^{2}}\Big{[}\ln|\xi|[\operatorname{Arg}(1-{\xi})]+\mathop{\rm
Im}\left[\operatorname{Li}_{2}(-{\xi})\right]\Big{]}.$ (3.19)

This observation is interesting as it explains differences in results for the
potentials $\widehat{V}_{5}$ found in [16] (that used rank functions with
offsets) compared with those in [21] (that used rank functions without
offsets). This also explains some differences for the Wilson loops VEV
computed in [17] (with offsets) compared with some of the field theoretical
results in [27] (without offsets). Interestingly, the free energy (or
holographic central charge) does not differ at leading order for rank
functions with or without offsets; the difference appears only at sub-leading
orders in the length of the quiver $P$.

####  3.1.1 A numerical study

For the benefit of the reader, we discuss an example of the calculation of
$\log<W_{\mathbb{k}}>$ in numerical detail. Let us consider the
$\tilde{T}_{N,P}$ example of this section (with no offsets) for the particular
values

$\displaystyle N=10,\;\;\;P=20,$ $\displaystyle\text{The quiver
is}\leavevmode\nobreak\ \leavevmode\nobreak\ SU(10)\times
SU(20)\times....\times SU(190)\leavevmode\nobreak\ \leavevmode\nobreak\
\text{with flavour group}\leavevmode\nobreak\ \leavevmode\nobreak\ SU(200).$

In the three dimensional case, the groups are unitary $U(N)$. We calculate the
Wilson loop for the $l$-gauge group (for $l=1,2,3,4,...,19$) in the
$\mathbb{k}$-antisymmetric representation for the values $\mathbb{k}=2,3,4,5$.
For this, we need to count $N_{F1}=2,3,4,5$ fundamental strings extending
between the probe and the stack of colour branes. The position of the probe in
the $\eta$-coordinate is $\eta^{*}=l$. The position in the $\sigma$-coordinate
is obtained by solving eq.(3.11) for $\sigma^{*}$. The numerical solution for
different values of $\eta^{*}=1,...19$ is plotted in Figure 3.

Figure 3: The values of $\sigma^{*}$ for Wilson loops in the representation
$\mathbb{k}=2,3,4,5.$.

Choosing a concrete $\eta^{*}$ we find the value of $\sigma^{*}$ making
$N_{F1}=2,3,4,5$ colour coded in the figure. The result for the VEV of the
Wilson loop is obtained by replacing these values $(\sigma^{*},\eta^{*})$ in
eq.(3.12)–remind that
$\xi_{*}=e^{-\frac{\pi}{P}\left(|\sigma^{*}|-i\eta^{*}\right)}$. The results
are shown in Figure 4.

Figure 4: The value of $\ln\langle W_{\wedge}\rangle$. The colour code is the
same as in Figure 3.

It is also instructive to plot $N_{F1}(\sigma^{*},\eta^{*})$ in terms of
$\eta^{*}$ for different values of $\sigma^{*}$. This is found in inset (a) of
Figure 5. In fact, for $\sigma^{*}=0$ we see the function $\frac{{\cal
R}(\eta^{*})}{2}$, while for $\sigma^{*}\rightarrow\infty$ we find a vanishing
value. This is in agreement with eqs.(3.2),(3.3). The inset (b) of the same
figure shows $N_{F1}(\sigma^{*},\eta^{*})$. Only the integer values of
$N_{F1}$ are acceptable (as this coincides with the representation) for
integer values of $\eta^{*}$ (indicating the gauge node).

Figure 5: The values of $\sigma^{*}$ for Wilson loops in the representation
$\mathbb{k}=2,3,4,5.$

Let us discuss a different example in a more succinct fashion.

###  3.2 Example 2

We consider a second example, known as the $+_{P,N}$ theory. We start, as
above with discussion without offsets. The rank function is,

${\cal R}(\eta)=\begin{cases}N\eta&0\leq\eta\leq 1\\\ N&1\leq\eta\leq(P-1)\\\
N(P-\eta)&(P-1)\leq\eta\leq P.\end{cases}$

In the five dimensional case, we have $N$ D7-branes localised at $\eta=1$ (the
first gauge group) and $N$ D7 branes at $\eta=P-1$ (the last gauge group).
This follows from
$\mathcal{R}^{\prime\prime}=N\delta(\eta-1)+N\delta(\eta-(P-1))$. There are a
total of $(P-1)N$ D5-branes, as calculated by $\int_{0}^{P}{\cal R}d\eta$. The
number comes from $N$ D5 branes for each integer value of $\eta$ between
$[1,P-1]$. In the three dimensional case, we have $N$ D5-branes localised at
$\eta=1$ and $N$ D5 branes at $\eta=P-1$, a total of $(P-1)N$ D3-branes and
$P$ NS five branes.

This is equivalent to a linear quiver field theory (in 5d it flows to a SCFT
in the UV, in 3d it flows to a SCFT in the IR),

NN$\dots$NNP-1

We calculate the Fourier coefficients,

$\displaystyle\mathcal{R}_{k}=2\frac{NP}{k^{2}\pi^{2}}\sin\left(\frac{k\pi}{P}\right)\left(1+(-1)^{k+1}\right),\;\;\;a_{k}=\frac{NP^{2}}{k^{3}\pi^{3}}\sin\left(\frac{k\pi}{P}\right)\left(1+(-1)^{k+1}\right),$
$\displaystyle
b_{k}=\frac{NP}{k^{2}\pi^{2}}\sin\left(\frac{k\pi}{P}\right)\left(1+(-1)^{k+1}\right).$
(3.20)

which leads to the potentials

$\displaystyle\widehat{V}_{5}(\sigma,\eta)=\frac{NP^{2}}{2\pi^{3}}\text{Re}\Big{[}\text{Li}_{3}(\xi
e^{-\frac{i\pi}{P}})-\text{Li}_{3}(\xi e^{\frac{i\pi}{P}})+\text{Li}_{3}(-\xi
e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi e^{-\frac{i\pi}{P}})\Big{]}\,,$ (3.21)
$\displaystyle\widehat{V}_{3}(\sigma,\eta)=\frac{NP}{2\pi^{2}}\text{Im}\Big{[}\text{Li}_{2}(\xi
e^{\frac{i\pi}{P}})-\text{Li}_{2}(\xi e^{-\frac{i\pi}{P}})+\text{Li}_{2}(-\xi
e^{-\frac{i\pi}{P}})-\text{Li}_{2}(-\xi e^{\frac{i\pi}{P}})\Big{]}\,.$

Finally, the result for the VEV of the Wilson loop is,

$\displaystyle\mu^{-1}\ln\langle
W_{\wedge}\rangle=\frac{NP^{2}}{2\pi^{3}}\text{Re}\Big{[}\text{Li}_{3}(\xi_{*}e^{-\frac{i\pi}{P}})-\text{Li}_{3}(\xi_{*}e^{\frac{i\pi}{P}})+\text{Li}_{3}(-\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{3}(-\xi_{*}e^{-\frac{i\pi}{P}})+$
$\displaystyle+\log|\xi|\left(\leavevmode\nobreak\
\text{Li}_{2}(\xi_{*}e^{\frac{i\pi}{P}})-\text{Li}_{2}(\xi_{*}e^{\frac{-i\pi}{P}})+\leavevmode\nobreak\
\text{Li}_{2}(-\xi_{*}e^{-\frac{i\pi}{P}})-\text{Li}_{2}(-\xi_{*}e^{-\frac{i\pi}{P}})\leavevmode\nobreak\
\right)\Big{]}.$ (3.22)

We now work out the same results for the situation with offsets. We have

${\cal R}(\eta)=\begin{cases}N&0\leq\eta\leq P.\end{cases}$

In this case we set $F_{j}=N\delta_{j,1}+N\delta_{j,P-1}$ and $N_{0}=N_{P}=N$.
The Fourier coefficients of the rank function and the potentials in 5d and 3d
are,

$\displaystyle\mathcal{R}_{k}=2\frac{N}{k\pi}\left(1+(-1)^{k+1}\right),\;\;a_{k}=\frac{PN}{k^{2}\pi^{2}}\left(1+(-1)^{k+1}\right),\;\;b_{k}=\frac{N}{k\pi}\left(1+(-1)^{k+1}\right).$
(3.23)

The potentials read

$\displaystyle\widehat{V}_{5}(\sigma,\eta)=\frac{PN}{\pi^{2}}\mathop{\rm
Im}\left[-\operatorname{Li}_{2}(-\xi)+\operatorname{Li}_{2}(\xi)\right],\;\;\;\;\widehat{V}_{3}(\sigma,\eta)=\frac{N}{2\pi}\log\left[\frac{(1+\xi)(1-\bar{\xi})}{(1-\xi)(1+\bar{\xi})}\right].$
(3.24)

For the Wilson loop, we find

$\displaystyle\mu^{-1}\ln\langle
W_{\wedge}\rangle=\frac{PN}{\pi^{2}}\\{\ln|\xi_{*}|[\operatorname{Arg}(1-\xi_{*})-\operatorname{Arg}(1+\bar{\xi}_{*})]+\mathop{\rm
Im}\left[\operatorname{Li}_{2}(\xi_{*})-\operatorname{Li}_{2}(-{\xi_{*}})\right]\\}.$
(3.25)

As pointed out in the first example, in the limit $P\rightarrow\infty$,
keeping the quotients $\frac{|\sigma|}{P}$ and $\frac{\eta}{P}$, we find that
the results without offsets in eqs.(3.21),(3.22) at leading order in $1/P$ are
approximated by the results with offsets in eqs.(3.24) and (3.25).

##  4 Mirror symmetry and Wilson loops

The material in this section applies primarily to the 3d SCFTs and their
string duals in eq.(2.11). The reader might want to extend these results also
to the five dimensional case.

It is known that three dimensional SUSY QFTs with eight supercharges enjoy a
symmetry called Mirror symmetry. The idea is that given two partitions $\rho$
and $\widehat{\rho}$ of the number $M$ the two theories
$T_{\rho}^{\widehat{\rho}}[SU(M)]$ and $T_{\widehat{\rho}}^{\rho}[SU(M)]$ are
conjectured to flow to the same IR SCFT. In the language of Hanany-Witten
setups, mirror symmetry is realized as an S-duality (we will discuss more
about this in Appendix B.2.3).

In contrast, in the electrostatic language described in Section 2.2, the
manifestation of mirror symmetry is in terms of a diffeomorphism; see the
paper [10] for explanations. In fact, as explained in [10] for balanced
quivers with one flavour node the mirror is also balanced and with one flavour
node, hence suitable to be described by the language in Section 2.2.

In what follows, we analyse the effect of a mirror symmetry transformation on
the VEV of a Wilson loop in the antisymmetric $\wedge$-representation We show
below that if the electric theory has $N_{f}^{el}$ flavours and the magnetic
theory has $N_{f}^{mag}$ flavours (in both cases with a single flavour node)
the calculation of the Wilson line in the same representation $\wedge$
satisfies,

$N_{f}^{el}\ln\langle W^{el}_{\wedge}\rangle=N_{f}^{mag}\ln\langle
W^{mag}_{\wedge}\rangle.$ (4.1)

To see this, let us consider a generic, balanced, one flavour node linear
quiver field theory. The quiver, rank function (with no offsets) and Fourier
coefficient of this electric theory are 111For the generic triangular rank
function the quotient $\frac{N}{(P-S)}$ is taken to be integer.,

$N$$2N$$\cdots$$SN$$\frac{SN(P-S-1)}{P-S}$$\frac{SN(P-S-2)}{P-S}$$\cdots$$\frac{SN}{P-S}$$\frac{NP}{P-S}$
${\cal R}_{e}(\eta)=\begin{cases}N\eta&0\leq\eta\leq S\\\
\frac{NS}{(P-S)}(P-\eta)&S\leq\eta\leq P,\end{cases}$

and the Fourier coefficient of the rank function,

$\mathcal{R}^{e}_{k}=\frac{2NP^{2}}{(P-S)\pi^{2}k^{2}}\sin\left(\frac{k\pi
S}{P}\right).$ (4.2)

Using eq.(3.3), the VEV of the Wilson line for this quiver is

$\ln\langle
W^{elec}_{\wedge}\rangle=\pi\sum_{k=1}^{\infty}\left(\frac{NP^{3}}{(P-S)\pi^{3}k^{3}}\right)\sin\left(\frac{k\pi
S}{P}\right)\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+1).$
(4.3)

Finally, the combination $N_{f}^{el}\ln\langle W^{el}_{\wedge}\rangle$ is

$N_{f}^{el}\ln\langle
W^{elec}_{\wedge}\rangle=\pi\sum_{k=1}^{\infty}\left(\frac{N^{2}P^{4}}{(P-S)^{2}\pi^{3}k^{3}}\right)\sin\left(\frac{k\pi
S}{P}\right)\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+1).$
(4.4)

Now, let us perform the same analysis for the mirror magnetic system.
Following the prescription in [10], the mirror system is described by a
magnetic quiver, rank function and Fourier coefficient,

$P-S$$2(P-S)$$\cdots$$S(N+1)-P$$SN$$S(N-1)$$\cdots$$S$$P$

$\mathcal{R}_{m}(\widehat{\eta})=\left\\{\begin{array}[]{cc}(P-S)\widehat{\eta}&\widehat{\eta}\in[0,\frac{SN}{P-S}]\\\
S\left(\frac{NP}{P-S}-\widehat{\eta}\right)&\widehat{\eta}\in[\frac{SN}{P-S},\frac{NP}{P-S}]\end{array}\right.$
(4.5)

$\mathcal{R}_{k}^{(m)}=\frac{2(P-S)}{NP}\int_{0}^{\frac{NP}{P-S}}\mathcal{R}^{m}(\widehat{\eta})\sin\left(\frac{k\pi(P-S)\widehat{\eta}}{NP}\right)d\widehat{\eta}=\frac{2NP^{2}}{(P-S)\pi^{2}k^{2}}\sin\left(\frac{k\pi
S}{P}\right).$ (4.6)

Notice that $\mathcal{R}_{k}^{(e)}=\mathcal{R}_{k}^{(m)}$. Also, note that the
range of the ’electric’ coordinate is $0\leq\eta\leq P$, whilst for the
’magnetic’ coordinate we have $0\leq\widehat{\eta}\leq\frac{NP}{P-S}$.
Finally, observe that in both mirror descriptions

$\displaystyle\text{electric description:}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
N_{NS5}=P,\;\;\;N_{D5}=\frac{NP}{(P-S)},$ $\displaystyle\text{magnetic
description:}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
N_{NS5}=\frac{NP}{(P-S)},\;\;\;\;N_{D5}=P.$

Following [10], we perform the identifications between electric variables
($\sigma,\eta$) and their magnetic counterparts
$(\widehat{\sigma},\widehat{\eta})$,

$\displaystyle\eta\leftrightarrow\frac{N_{NS5}}{N_{D5}}\widehat{\eta}=\frac{(P-S)}{N}\widehat{\eta},\leavevmode\nobreak\
\leavevmode\nobreak\
\sigma\leftrightarrow\frac{N_{NS5}}{N_{D5}}\widehat{\sigma}=\frac{(P-S)}{N}\widehat{\sigma}.$
(4.7)

The VEV of the magnetic Wilson loop is found by carefully using eq.(4.6) in
eq.(3.3),

$\ln\langle
W^{mag}_{\wedge}\rangle=\pi\sum_{k=1}^{\infty}\left(\frac{N^{2}P^{3}}{(P-S)^{2}\pi^{3}k^{3}}\right)\sin\left(\frac{k\pi
S}{P}\right)\sin\left(\frac{k\pi(P-S)}{PN}\widehat{\eta}^{*}\right){e^{-\frac{k\pi(P-S))}{PN}|\widehat{\sigma}^{*}|}}\left(\frac{k\pi(P-S)}{PN}|\widehat{\sigma}^{*}|+1\right).$
(4.8)

As above, the combination

$\displaystyle N_{f}^{mag}$ $\displaystyle\ln\langle W^{mag}_{\wedge}\rangle=$
$\displaystyle\pi\sum_{k=1}^{\infty}\left(\frac{N^{2}P^{4}}{(P-S)^{2}\pi^{3}k^{3}}\right)\sin\left(\frac{k\pi
S}{P}\right)\sin\left(\frac{k\pi(P-S)}{PN}\widehat{\eta}^{*}\right){e^{-\frac{k\pi(P-S))}{PN}|\widehat{\sigma}^{*}|}}\left(\frac{k\pi(P-S)}{PN}|\widehat{\sigma}^{*}|+1\right).$
(4.9)

For a given electric node (labelled by $\eta^{*}$) in a given antisymmetric
representation (labelled by $\wedge$), we find the Wilson loop in eq.(4.3). We
compare this with the magnetic Wilson loop in the same $\wedge$-representation
calculated for a different magnetic node labelled by $\widehat{\eta}^{*}$. If
these nodes satisfy

$\text{electric node}=\frac{P-S}{N}\times\text{magnetic
node}\longrightarrow\eta_{*}=\frac{P-S}{N}\widehat{\eta}_{*},$

then, using the identification in eq.(4.7), we find that eq.(4.4) equals
eq.(4.9), which is the relation in eq.(4.1).

Another way to arrive to eq.(4.1) is to start from eq.(3.6). Notice that for
the electric and magnetic quivers above, we have

$\displaystyle\xi_{*,el}=e^{-\frac{\pi}{P}\left(|\sigma^{*}_{el}|-i\eta^{*}_{el}\right)},\;\;\;F_{J}=\delta_{J,S}\frac{NP}{(P-S)},\;\;\;\;\;\;\;\text{quiver
length}=P.$ (4.10)
$\displaystyle\widehat{\xi}_{*,mag}=e^{-\frac{\pi(P-S)}{PN}\left(|\widehat{\sigma}^{*}_{mag}|-i\widehat{\eta}^{*}_{mag}\right)}=\xi_{*,el},\;\;\;F_{J}=\delta_{J,\frac{SN}{(P-S)}}P,\;\;\;\text{quiver
length}=\frac{NP}{(P-S)}.$

We have used the rescaling in eq.(4.7). Then, using eq.(3.6)–for the case of
no offsets, $N_{0}=N_{P}=0$, and multiplying the electric (magnetic) result by
the electric (magnetic) number flavours we get eq.(4.1).

Within the class of balanced linear quiver with one flavour node discussed in
this section, one special subclass is those quivers that are self-mirror. They
are characterised by the condition $N=(P-S)$. This implies that electric
result in eq.(4.3) and the magnetic result in eq.(4.8) are identical (same
node, same representation, same number of flavours) and eq.(4.1) is
automatically satisfied.

##  5 Conclusions

Let us start with a brief overview of the contents of this paper. In Section 2
we summarised the electrostatic description of an infinite family of Type IIB
backgrounds dual to SCFTs in five and three spacetime dimensions, preserving
eight Poincare supercharges. The electrostatic point of view is complementary
to the ’holomorphic’ one developed in [4],[11], that we review carefully in
Appendix B. The electrostatic description of Section 2 makes the connection
with quantum field theory (in particular, with the matrix-model/localisation)
more transparent. We discuss this briefly in Appendix A.

In Section 3 we write (using the formalism of Section 2) the result for the
VEV of Wilson loops for a given gauge group in a given antisymmetric
$\wedge$-representation. We make clear that this observable takes the same
expression in five and in three spacetime dimensions. Whilst this was already
clear from a purely field theoretical/matrix model perspective [27], from the
holographic point of view, this becomes more transparent, when written in the
electrostatic formalism. Two examples were worked out in full detail to
address a somewhat unclear situation in the existing bibliography. In fact, we
showed the relation between results when the rank function is taken with
(without) offsets is a limit procedure. This translates in field theory
language to the presence of offsets in the matrix model eigenvalue
distribution. We have resolved this potentially unclear issue in the
bibliography.

In Section 4, we have discussed the action of mirror symmetry on three
dimensional quiver field theories and how the holographic description of
balanced quivers with one flavour node realises this symmetry. This leads us
to propose a relationship between the Wilson loops; the one computed in a
given antisymmetric representation for a certain gauge node in the electric
description becomes equivalent (up to a precise multiplicative factor) with
the Wilson loop in the same representation for a different gauge node in the
magnetic description of the same system.

The very detailed Appendix B is of special note. There, we show many details
and derivations of the map between the holomorphic and the electrostatic
description of these systems, with worked out examples. This might prove
useful for colleagues working on these topics.

For the future, it would be interesting to extend the study of Wilson loops to
other systems in different dimensions, also admitting electrostatic
description. In fact, for SCFTs in dimension six–see [36], four–see [37],
two–see [38] and one–see [39] for a small sample of references, there is a
well developed electrostatic formalism. Also, the calculation of Wilson loops
in symmetric representations or products of various representations seems a
feasible problem to study. It should also be nice to further study the
relation between Wilson Loops in both mirror descriptions, expressed by
eq.(4.1).

We hope that the ’translation character’ of this work can show the analogies
between the problem in different dimensions. We anticipate that other
analogies, similar to those pointed out in this paper, will be encountered
when discussing different observables. We believe that the formalism of
Section 2, and references [36], [21], [37], [10], [38] and [39] is the best
suited to look for coincidences in observables for systems in diverse
dimensions. The analogies suggested by the holographic studies, in turn, may
find a field theoretical understanding of their own.

## Acknowledgments:

We would like to thank useful discussions with Andrea Legramandi and Christoph
Uhlemann. We are supported by STFC grant ST/T000813/1 and ST/V507143/1. The
authors have applied to a Creative Commons Attribution (CC BY) licence.

##  Appendix A Summary of the Matrix Model associated with the SCFTs

In this appendix, we discuss matrix model calculations of certain observables
in QFTs that flow to SCFTs. We will phrase quantities in the electrostatic
language using harmonic potentials. Five and three dimensional linear quiver
QFTs flowing to SCFTs are considered below. This appendix is a (very) brief
summary of certain aspects of the paper [40].

Various quantities in 5d SCFTs are studied using matrix model calculations in
[16, 17]. One can check that the free energy calculations of the theory in
terms of the function $\varrho(z,x)=N(z)\widehat{\rho}(z,x)$, satisfying the
saddlepoint equation

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

can be related to our language with the mapping (explained in [40]),

$N(z)\widehat{\rho}(z,x)=2P\partial^{2}_{\eta}\widehat{V}_{5}(\sigma,\eta)\quad
z=\eta/P\quad x=\sigma/(2P)\quad N(z)=\cal{R}(\eta).$ (A.2)

For 5d antisymmetric Wilson loops with association with gauge node at location
$z$ in the quiver in $\mathbb{k}$-antisymmetric rank, using these relations,
the matrix model calculation would read

$\displaystyle\ln\langle W_{\wedge}\rangle$ $\displaystyle=-6\pi
PN(z)\int_{b(0,z)}^{b(\mathbb{k},z)}b\widehat{\rho}(z,b)db\equiv-6\pi
P\int_{b(0,\eta)}^{b(\mathbb{k},\eta)}\sigma/(2P)2P\partial^{2}_{\eta}\widehat{V}(\sigma,\eta)d\sigma/2P$
(A.3)
$\displaystyle=-3\pi(\int_{b(0,\eta)}^{b(\mathbb{k},\eta)}-\partial_{\sigma}\widehat{V}(\sigma,\eta)d\sigma+\left.\sigma\partial_{\sigma}\widehat{V}(\sigma,\eta)\right|^{(\sigma*,\eta*)}_{\infty})=-3\pi(-\widehat{V}_{5}(\sigma,\eta)+\left.\sigma\partial_{\sigma}\widehat{V}_{5}(\sigma,\eta))\right|_{(\sigma*,\eta*)}.$
(A.4)

The Laplace equation and by part integration are used in the second line. The
above equation must be evaluated at the point
$\sigma^{*}=b(\mathbb{k},\eta^{*})$ which is determined by the integral
equation

$\mathbb{k}\equiv
N(z)y=N(z)\int_{b(z,y)}^{\infty}dx\widehat{\rho}(z,x)\equiv\int_{b(0,\eta)}^{b(\mathbb{k},\eta)}2P\partial^{2}_{\eta}\widehat{V}(\sigma,\eta)d\sigma/2P=\left.\partial_{\sigma}\widehat{V}_{5}(\sigma,\eta)\right|^{(\sigma*,\eta*)},$
(A.5)

Hence, by choosing the representation $\mathbb{k}$ and the gauge node,
$(\sigma*,\eta*)$ will be determined, and the Wilson loop can be calculated.

For the 3d field theory calculations [8, 27] with a similar map

$N(z)\widehat{\rho}(z,x)=2P\partial^{2}_{\eta}\widehat{V}(\sigma,\eta)\quad
z=\eta/P\quad x=\sigma/(2P)\quad N(z)=\cal{R}(\eta),$ (A.6)

we have

$\displaystyle\ln\langle W_{\wedge}\rangle$ $\displaystyle=2\pi
PN(z)\int_{b(0,z)}^{b(\mathbb{k},z)}b\widehat{\rho}(z,b)db\equiv 2\pi
P\int_{b(0,\eta)}^{b(\mathbb{k},\eta)}\sigma/(2P)2P\partial^{2}_{\eta}\widehat{W}(\sigma,\eta)d\sigma/2P$
(A.7)
$\displaystyle=\pi(-\widehat{W}(\sigma,\eta)+\left.\sigma\partial_{\sigma}\widehat{W}(\sigma,\eta))\right|^{(\sigma*,\eta*)},$
(A.8)

and a similar condition for evaluation point

$\mathbb{k}\equiv
N(z)y=N(z)\int_{b(z,y)}^{\infty}dx\widehat{\rho}(z,x)\equiv\int_{b(0,\eta)}^{b(\mathbb{k},\eta)}2P\partial^{2}_{\eta}\widehat{V}(\sigma,\eta)d\sigma/2P=\left.\partial_{\sigma}\widehat{W}(\sigma,\eta)\right|_{(\sigma*,\eta*)}.$
(A.9)

Again, by choosing the representation $\mathbb{k}$ and the gauge node,
$(\sigma*,\eta*)$ will be determined.

##  Appendix B Map between holomorphic and electrostatic formalisms

In this appendix, we will consider the details of the mapping between the
backgrounds of supergravity solutions in terms of holomorphic and
electrostatic real functions. The backgrounds dual to 5d and 3d SCFTs are
examined below.

###  B.1 The DGKU solution dual to 5d SCFTs

The mapping between the background eq.(2.1) to the DGKU solution [11] are
described in [21]. The DGKU solution parameterises the Riemann surface in the
internal space with a complex coordinate $w$, and it is entirely specified by
two holomorphic functions $\mathcal{A}_{\pm}(w)$. The metric in string frame

$ds^{2}_{10}=e^{\frac{\Phi}{2}}f_{1}(w,\bar{w})\left[ds^{2}(\text{AdS}_{6})+f_{2}(w,\bar{w})ds^{2}(S^{2})+f_{3}(w,\bar{w})dwd\bar{w}\right]$
(B.1)

is defined in terms of the following warping functions

$f_{1}=\frac{|\partial_{w}\mathcal{G}|\sqrt{1-R^{2}}}{\kappa\sqrt{R}},\quad
f_{2}=\frac{1}{9}\left(\frac{1-R}{1+R}\right)^{2}\,,\quad
f_{3}=\frac{4\kappa^{4}R}{|\partial_{w}\mathcal{G}|^{2}(1-R^{2})^{2}}\,$ (B.2)

where

$\begin{split}&\mathcal{G}=|\mathcal{A}_{+}|^{2}-|\mathcal{A}_{-}|^{2}+2\text{Re}\mathcal{B}\,,\qquad\kappa^{2}=-\partial_{w}\partial_{\bar{w}}\mathcal{G}=|\partial_{w}\mathcal{A}_{-}|^{2}-|\partial_{w}\mathcal{A}_{+}|^{2}\,,\\\
&\partial_{w}\mathcal{B}=\mathcal{A}_{+}\partial_{w}\mathcal{A}_{-}-\mathcal{A}_{-}\partial_{w}\mathcal{A}_{+}\,,\qquad\,R+R^{-1}=2+\frac{6\kappa^{2}\mathcal{G}}{|\partial_{w}\mathcal{G}|^{2}}\,,\end{split}$
(B.3)

while the fluxes are given by:

$\begin{split}\tau=C_{0}+ie^{-\Phi}=&-i\frac{\partial_{w}(\mathcal{A}_{+}+\mathcal{A}_{-})\partial_{\bar{w}}\mathcal{G}-R\partial_{\bar{w}}(\bar{\mathcal{A}}_{+}+\bar{\mathcal{A}}_{-})\partial_{w}\mathcal{G}}{\partial_{w}(\mathcal{A}_{+}-\mathcal{A}_{-})\partial_{\bar{w}}\mathcal{G}+R\partial_{\bar{w}}(\bar{\mathcal{A}}_{+}-\bar{\mathcal{A}}_{-})\partial_{w}\mathcal{G}}\,,\\\
B_{2}+iC_{2}=&\frac{2}{3}i\left(\left(\frac{1-R}{1+R}\right)^{2}\frac{\partial_{w}\mathcal{A}_{+}\partial_{\bar{w}}\mathcal{G}+\partial_{\bar{w}}\bar{\mathcal{A}}_{-}\partial_{w}\mathcal{G}}{3\kappa^{2}}-\bar{\mathcal{A}}_{-}-\mathcal{A}_{+}\right)\text{Vol}(S^{2})\,.\end{split}$
(B.4)

The AdS6 radius is set equal to one.

The general form of the functions ${\cal A}_{\pm}$ was derived in [35, 17],

$\displaystyle{\cal A}_{\pm}$ $\displaystyle={\cal
A}_{\pm}^{0}+\sum_{\ell=1}^{L}Z_{\pm}^{\ell}\ln(w-r_{\ell})\leavevmode\nobreak\
,$ $\displaystyle\overline{{\cal A}_{\pm}^{0}}$ $\displaystyle=-{\cal
A}_{\mp}^{0}\leavevmode\nobreak\ ,$ $\displaystyle\overline{Z_{\pm}^{\ell}}$
$\displaystyle=-Z_{\mp}^{\ell}\leavevmode\nobreak\ .$ (B.5)

The poles are on the real line at $r_{\ell}$, with residues $Z_{\pm}^{\ell}$.
These requirements of ${\cal G}$ to be single-valued and positive in the
interior of $\Sigma$, and vanishing on the boundary would lead to the
regularity conditions

$\displaystyle{\cal A}_{+}^{0}Z_{-}^{k}-{\cal
A}_{-}^{0}Z_{+}^{k}+\sum_{\stackrel{{\scriptstyle\ell=1}}{{\ell\neq
k}}}^{L}Z^{[\ell,k]}\ln|r_{\ell}-r_{k}|$ $\displaystyle=0\leavevmode\nobreak\
,$ $\displaystyle k$ $\displaystyle=1,\cdots,L\leavevmode\nobreak\ ,$ (B.6)

where $Z^{[\ell,k]}\equiv Z_{+}^{\ell}Z_{-}^{k}-Z_{+}^{k}Z_{-}^{\ell}$. The
residues $Z_{\pm}^{\ell}$ encodes the charges of the $(p,q)$ 5-brane emerging
at the pole $r_{\ell}$.

The entire solution is invariant under reparameterization of the complex
coordinate $w\rightarrow z(w)$. Then, we can use one of the holomorphic
functions (or a combination of them) as a definition of the complex
coordinate.

####  B.1.1 Matching the solutions

In this section we will show how to match eq.(2.1) with eq.(B.2)-eq.(B.4). By
equating the warping functions $f_{1},f_{2}$ we get the following conditions

$\mathcal{G}=\frac{9\pi^{2}}{4}\sigma^{2}\partial_{\sigma}V_{5}\,,\qquad\frac{\kappa^{2}}{|\partial_{w}\mathcal{G}|^{2}}=\frac{2}{9\pi^{2}\sigma^{2}}\frac{\partial_{\eta}^{2}V_{5}}{\Lambda-3\partial_{\eta}^{2}V_{5}\partial_{\sigma}V_{5}}\,.$
(B.7)

One needs to keep the metric factor in making a comparison for
$f_{3}ds^{2}(\mathbb{C})$. Using the definition of $R$ in eq.(B.3), we have

$\frac{2}{3}\frac{\kappa^{2}}{G}dwd\bar{w}=\frac{\partial_{\eta}^{2}V_{5}}{3\sigma\partial_{\sigma}V_{5}}(d\sigma^{2}+d\eta^{2}).$
(B.8)

Therefore, from equations $\eqref{eq:match1}$ we can write

$|\partial_{w}\mathcal{G}|^{2}dwd\bar{w}=\left(\left(\partial_{\eta}\mathcal{G}\right)^{2}+\left(\partial_{\sigma}\mathcal{G}\right)^{2}\right)(d\sigma^{2}+d\eta^{2})\,.$
(B.9)

By defining a complex variable $z=\sigma-i\eta$, this consistency relation is
automatically solved. Since the DGKU solution is defined up to a change of
complex variables, we identify $w=z$ from now on.

Considering the fluxes from eq.(2.1)

$B_{2}+iC_{2}=\frac{2}{3}i\left(\frac{6\pi}{4}V_{5}-i\frac{3\pi}{4}\eta-i\sigma\partial_{\sigma}V_{5}\frac{\left(-\frac{1}{3}-\frac{2}{3}i\partial_{\eta}V_{5}\right)\partial_{\sigma\eta}^{2}V_{5}+2i\partial_{\sigma}V_{5}\partial_{\eta}^{2}V_{5}}{\Lambda}\right)\text{Vol}(S^{2})\,$
(B.10)

and comparing it with eq.(B.4), the two expressions match if we set

$\mathcal{A}_{+}+\bar{\mathcal{A}}_{-}=i\frac{3\pi}{4}\eta-\frac{3\pi}{2}\partial_{\sigma}(\sigma
V_{5})\,.$ (B.11)

Where, $\sigma V_{5}=\widehat{V_{5}}$ is the harmonic function defined in
eq.(2.3) and, since it is also real, it defines just one holomorphic function
$\mathcal{V}(z)$

$\sigma V_{5}=\mathcal{V}(z)+\overline{\mathcal{V}(z)}.$ (B.12)

Using this condition and the fact that $\mathcal{A}_{+}$ is holomorphic while
$\bar{\mathcal{A}}_{-}$ is anti-holomorphic, we see that eq.(B.11) completely
defines $\mathcal{A}_{\pm}$ in terms of $\sigma V_{5}$ and the new coordinate
$z$

$\mathcal{A}_{\pm}=\mp\frac{3\pi z}{8}-\frac{3\pi}{2}\partial_{z}(\sigma
V_{5})\,.$ (B.13)

With these definitions, the reader can check that the axion-dilaton
expressions are identical.

In order to compare the two backgrounds, we had to impose $w=z$. As a
consequence, the two holomorphic functions $\mathcal{A}_{\pm}$ are defined
just in terms of one of the holomorphic function ($\mathcal{V}$) and the
coordinate $z$.

It should be noted that one can do the rescaling $V_{5}=\nu V_{5,\text{old}}$
and $(\sigma,\eta)=(\mu\sigma_{\text{old}},\mu\eta_{\text{old}})$ with
corresponding changes in the background to make quantised Page charges of the
branes integers. In [21], this rescaling is done to quantise charges properly.
In the case of holomorphic functions, this can also be done to get integer
charges which match with the ones obtained from the Rank function in the real
formalism. One can also perform the inverse of that rescaling again on our
potential and coordinates to match the results. From now on we do the change
$(\sigma,\eta)\rightarrow 2(\sigma,\eta)$ and $V_{5}\rightarrow V_{5}$ to
compare results.

####  B.1.2 Example: $T_{P}$ theory

In this section, we consider solutions with three poles, mapped in terms of
our potential $V_{5}$. We refer to [16] for the specific form of the solutions
we are considering.

The $T_{P}$ theory is given by the three-pole solution

$\mathcal{A}_{\pm}=\frac{3P}{4}(\pm\log(w-1)+(\mp 1-i)\log(w+1)+i\log(2w))\,,$
(B.14)

where the poles are at $w=1,0,-1$ and we set $\alpha^{\prime}=1$. The
coordinate $z=\sigma-i\eta$ is defined as

$z=-\frac{2}{3\pi}(\mathcal{A}_{+}-\mathcal{A}_{-})=\frac{N}{\pi}\log\left(\frac{1+w}{1-w}\right)\quad\Rightarrow\quad
w=\coth\left(\frac{\pi z}{2P}\right)\,.$ (B.15)

Notice that the imaginary axis for $w$ becomes the interval $\eta\in(0,N)$ at
$\sigma=0$, while the real axis, which is the space-time boundary, is mapped
as following

$\begin{split}w\in(-1,1)\quad&\Rightarrow\quad\eta=P\,,\,\sigma\in(-\infty,\infty)\,,\\\
w\in(-\infty,-1)\cup(1,\infty)\quad&\Rightarrow\quad\eta=0\,,\,\quad\sigma\in(-\infty,\infty)\,.\end{split}$

So the space-time boundary in the $w$ coordinate is consistently mapped in the
space-time boundary in $\sigma$ and $\eta$ coordinates.

The potential is defined by the following equation

$\partial_{z}(\sigma
V_{5})=-\frac{\mathcal{A}_{-}+\mathcal{A}_{+}}{3\pi}=-\frac{iP}{2\pi}\log\left(e^{-\frac{\pi
z}{P}}+1\right)$ (B.16)

which can be integrated leading to

$\displaystyle V_{5}=$
$\displaystyle\frac{iP^{2}}{2\pi^{2}\sigma}\left(\text{Li}_{2}\left(-e^{-\frac{\pi(|\sigma|+i\eta)}{P}}\right)-\text{Li}_{2}\left(-e^{\frac{-\pi(|\sigma|-i\eta)}{P}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{P^{2}}{\pi^{2}\sigma}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{2}}\sin\left(\frac{k\pi}{P}\eta\right)e^{-\frac{k\pi}{P}|\sigma|}\,;$
(B.17)

the integration constant is set to zero as required by the boundary
conditions. eq.(B.17) is exactly of the form eq.(2.7), and we can identify the
coefficient of the Fourier expansion:

$a_{k}=\frac{P^{2}}{\pi^{2}}\frac{(-1)^{k+1}}{k^{2}}\,.$ (B.18)

This result can be compared with eq.(3.13) for $N=1$, which exactly matches.

####  B.1.3 Wilson loops

Interestingly, the Wilson loop expectation value in $\mathbb{k}^{th}$ anti
symmetric representation is proportional to ${\cal G}$ [17] which in the real
formalism from eq.(B.7) is simply proportional to
$\sigma^{2}\partial_{\sigma}V_{5}$, evaluated at a point determined by gauge
node and anti symmetric representation chosen

$\ln\langle W_{\wedge}\rangle=-\frac{2}{3}T_{\rm D3}{\rm Vol}_{AdS_{2}}{\rm
Vol}_{S^{2}}{\cal
G}=3\pi\sigma^{2}\left.\partial_{\sigma}V_{5}\right|_{(\sigma*,\eta*)},$
(B.19)

with $T_{\rm D3}{\rm Vol}_{AdS_{2}}{\rm
Vol}_{S^{2}}=-4\pi/(2\pi\alpha^{\prime})^{2}$. So the Wilson loop is

$\ln\langle
W_{\wedge}\rangle=3\pi\left.(\sigma\partial_{\sigma}\widehat{V}-\widehat{V})\right|_{(\sigma*,\eta*)}=3\pi\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}(\frac{P}{k\pi})\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+1).$
(B.20)

The only calculation needed is the location on the $\Sigma$-plane, which this
function should be calculated. So

$N_{\rm F1}+iN_{\rm
D1}=\frac{2}{3\pi}\Big{[}\frac{i}{3}(\frac{9\pi}{2})\eta^{*}-6(\frac{\pi}{4})\left.\partial_{\sigma}(\widehat{V})\right|_{(\sigma*,\eta*)}\Big{]}$
(B.21)

$N_{\rm F1}$ and $N_{\rm D1}$ are related to the gauge node and anti symmetric
representation chosen. Hence,

$N_{\rm D1}=\eta^{*},\quad N_{\rm
F1}=\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}\operatorname{Sgn}(\sigma^{*}).$
(B.22)

###  B.2 The DEGK background dual to 3d SCFTs

The DEGK solutions [4] in the string frame is defined in terms of complex
functions in the variable $w$:

$\displaystyle
ds_{10,st}^{2}=f_{1}(w,\bar{w})\Big{[}ds^{2}(\text{AdS}_{4})+f_{2}(w,\bar{w})ds^{2}(S^{2}_{1})+f_{3}(w,\bar{w})ds^{2}(S^{2}_{2})+f_{4}(w,\bar{w})dwd\bar{w}\Big{]},$
$\displaystyle
e^{-2\Phi}=f_{5}(w,\bar{w}),\;\;\;\;B_{2}=f_{6}(w,\bar{w})\text{Vol}(S^{2}_{1}),\;\;\;\;C_{2}=f_{7}(w,\bar{w})\text{Vol}(S^{2}_{2}),$
(B.23)

where

$\displaystyle
f_{1}=2\sqrt{-\frac{N_{2}}{W}},\;\;\;\;f_{2}=-\frac{h_{1}^{2}W}{N_{1}},\;\;\;\;f_{3}=-\frac{h_{2}^{2}W}{N_{2}},\;\;\;\;f_{4}=-2\frac{W}{h_{1}h_{2}},\;\;\;\;f_{5}=\frac{N_{1}}{N_{2}}$
$\displaystyle
f_{6}=4\frac{h_{1}^{2}h_{2}\text{Im}(\partial_{w}h_{2}\partial_{\bar{w}}h_{1})}{N_{1}}+2h_{2}^{D},\;\;\;\;f_{7}=4\frac{h_{1}h_{2}^{2}\text{Im}(\partial_{w}h_{2}\partial_{\bar{w}}h_{1})}{N_{2}}-2h_{1}^{D}.$
(B.24)

The five-form field is given by

$F_{5}=\text{Vol(AdS}_{4})\wedge df_{8}+*(\text{Vol(AdS}_{4})\wedge df_{8}),$
(B.25)

and

$f_{8}=4\left(6\text{Re}(\mathcal{C})-3\mathcal{D}-2\frac{h_{1}h_{2}}{W}\text{Im}(\partial_{w}h_{1}\partial_{w}h_{2})\right)\,.$
(B.26)

All these functions could be defined in terms of two holomorphic functions
$\mathcal{A}_{1,2}(w)$, in particular $h_{1,2}$ and $h_{1,2}^{D}$ are the dual
real harmonic functions

$h_{1}=2\text{Im}(\mathcal{A}_{1})\,,\qquad
h_{1}^{D}=2\text{Re}(\mathcal{A}_{1})\,,\qquad
h_{2}=2\text{Re}(\mathcal{A}_{2})\,,\qquad
h_{2}^{D}=-2\text{Im}(\mathcal{A}_{2}).$ (B.27)

Also, we have the following definitions

$W=\partial_{w}\partial_{\bar{w}}(h_{1}h_{2})\,,\quad
N_{i}=2h_{1}h_{i}|\partial_{w}h_{i}|^{2}-h_{i}^{2}W\,,\quad\mathcal{D}=2\text{Re}(\mathcal{A}_{1}\bar{\mathcal{A}}_{2})\,,\quad\partial_{w}\mathcal{C}=\mathcal{A}_{1}\partial_{w}\mathcal{A}_{2}-\mathcal{A}_{2}\partial_{w}\mathcal{A}_{1}.$
(B.28)

A more detailed background solution description could be found in [27].

Solutions with different holographic interpretations can be constructed
depending on the choice of $h_{1/2}$ and $\Sigma$. We would be interested in
duals of $3\mathrm{\leavevmode\nobreak\ d}$ SCFTs. All solutions here describe
D3-branes suspended between, ending on, or intersecting combinations of D5 and
NS5 branes. For these solutions, the harmonic functions $h_{1},h_{2}$ on the
strip

$\Sigma=\left\\{w\in\mathbb{C}\mid
0\leq\operatorname{Im}(w)\leq\frac{\pi}{2}\right\\}$

would read

$\displaystyle h_{1}$
$\displaystyle=-\frac{\alpha^{\prime}}{4}\sum_{a=1}^{A}N_{\mathrm{D}5}^{(a)}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta_{a}}{2}\right)+\text{
c.c. }$ $\displaystyle h_{2}$
$\displaystyle=-\frac{\alpha^{\prime}}{4}\sum_{b=1}^{B}N_{\mathrm{NS}5}^{(b)}\ln\tanh\left(\frac{w-\delta_{b}}{2}\right)+\text{
c.c. }.$

These solutions describe $A$-groups of D5-branes with $N_{\text{D5 }}^{(a)}$
D5-branes in the $a^{\text{th }}$ group and $B$-groups of NS5-branes with
$N_{\text{NS5 }}^{(b)}$ NS5-branes in the $b^{\text{th }}$ group. D3-branes
are suspended between the 5 -branes for 3d SCFTs.

The background is invariant under conformal transformations $w\rightarrow
f(w)\equiv z$; specifically, one can choose one of the holomorphic functions
as a coordinate. The second holomorphic function can be defined in terms of an
auxiliary harmonic function $\widehat{V}_{3}(z,\bar{z})$ as follows

$\mathcal{A}_{1}=\pi\partial_{z}\widehat{V}_{3}\,,\qquad\mathcal{A}_{2}=\frac{\pi}{8}z\,.$
(B.29)

In order to match these backgrounds with those in eq.((2.11)), we can set

$z=\sigma-i\eta\,,\qquad\widehat{V}_{3}=\sigma V_{3}\,.$ (B.30)

With these identifications,

$\displaystyle h_{1}=\pi\sigma\partial_{\eta}V_{3}\,,\qquad
h_{1}^{D}=\pi\partial_{\sigma}(\sigma V_{3})\,,\qquad
h_{2}=\frac{\pi}{4}\sigma\,,\qquad h_{2}^{D}=\frac{\pi}{4}\eta\,,$ (B.31)
$\displaystyle
W=\frac{\pi^{2}}{8}\partial_{\sigma}(\sigma\partial_{\eta}V_{3})\,,\qquad
N_{1}=\frac{\pi^{4}}{8}\sigma^{3}\partial_{\eta}V_{3}\Lambda\,,\qquad
N_{2}=-\frac{\pi^{4}}{128}\sigma^{3}\partial^{2}_{\eta\sigma}V_{3}\,.$ (B.32)

These expressions will match eq.(B.24) with eq.(2.11). For the same reason as
the 5d case, we do the change $(\sigma,\eta)\rightarrow 2(\sigma,\eta)$ and
$V_{3}\rightarrow V_{3}$ to quantize the Page charges properly and match the
backgrounds.

####  B.2.1 Example: Generic balanced quivers

The backgrounds dual to generic balanced quivers are given by

$\displaystyle h_{1}$
$\displaystyle=-\frac{\alpha^{\prime}}{4}\sum_{a=1}^{A}N_{\mathrm{D}5}^{(a)}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta_{a}}{2}\right)+\text{
c.c. }$ (B.33) $\displaystyle h_{2}$
$\displaystyle=-\frac{\alpha^{\prime}}{4}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right)+\text{
c.c. }.$

The dual is a quiver with $N_{\mathrm{NS}5}-1$ nodes and $N_{\text{D5
}}^{(a)}$ flavors at gauge nodes $\mathrm{t}_{a}$ with

$\mathrm{t}_{a}=\frac{2}{\pi}N_{\text{NS5 }}\arctan e^{\delta_{a}}$

Since all nodes are balanced and $N_{0}=N_{L+1}=0$, the entire quiver can be
reconstructed from this information. These functions can be written in terms
of

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=-i\frac{\alpha^{\prime}}{4}\sum_{a=1}^{A}N_{\mathrm{D}5}^{(a)}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta_{a}}{2}\right)$
(B.34) $\displaystyle\mathcal{A}_{2}$
$\displaystyle=-\frac{\alpha^{\prime}}{4}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right).$

It must be noted that one can multiply the argument of logarithm in
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$functions with a phase $|c|=1$ (for
instance,
$\mathcal{A}_{2}\rightarrow-\frac{\alpha^{\prime}}{4}N_{\mathrm{D}5}\ln
c\tanh\left(-\frac{w}{2}\right)$ ), while $h_{1,2}$ functions remain the same.

If we take $z$ to be the new coordinate, our transformation will read (set
$\alpha^{\prime}=1$)

$z=-\frac{1}{\pi}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right).$ (B.35)

This change of coordinates could be done also in the following steps. First,
$w^{\prime}=e^{w}$ sends the strip to the upper right quadrant with NS5 on
reals and D5s on the imaginary axis. Second,
$\frac{w^{\prime}-1}{w^{\prime}+1}=-u\quad u^{\prime}=-u$ sending the upper
right quadrant to the upper half disk with NS5 on zero and D5s on the
circumference. Lastly, $z=-\frac{1}{\pi}N_{\mathrm{NS}5}\ln(u)$ which maps
upper half disk to an strip $0\leq\operatorname{Re}(z^{\prime})<\infty$ and
$0\leq\operatorname{Im}(z^{\prime})<-N_{NS5}$. The Neveu-Schwarz five barnes
are mapped to a vertical line at infinity, and D5s are on
$\operatorname{Re}(z^{\prime})=0$. By taking $z=\sigma-i\eta$ and the
condition $V(-\sigma,\eta)=-V(\sigma,\eta)$ the solution would be well defined
on $0\leq\eta\leq+N_{\mathrm{NS}5}$.

After the mapping, the holomorphic functions are

$\displaystyle\mathcal{A}_{1}(z)$
$\displaystyle=-\frac{i}{4}\sum_{a}N_{\mathrm{D}5}^{(a)}\left[\ln\left(1-e^{\frac{-\pi}{2\alpha^{\prime}N_{\mathrm{NS}5}}z^{\prime}}/\sigma_{a}\right)-\ln\left(1-\sigma_{a}e^{\frac{-\pi}{2\alpha^{\prime}N_{\mathrm{NS}5}}z^{\prime}}\right)\right]\equiv\pi\partial_{z}\widehat{V}_{3}$
(B.36) $\displaystyle\mathcal{A}_{2}(z)$ $\displaystyle=\frac{\pi}{4}z,$
(B.37)

with $\sigma_{a}=\frac{ie^{\delta_{a}}-1}{ie^{\delta_{a}}+1}=e^{\frac{-i\pi
t_{a}}{N_{\mathrm{NS}5}}}$. Then

$\int\mathcal{A}_{1}(z)\operatorname{d}z=\frac{i}{4}(\frac{-N_{\mathrm{NS}5}}{\pi})\sum_{a}N_{\mathrm{D}5}^{(a)}\left[\operatorname{Li_{2}}\left(\sigma_{a}e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}\right)-\operatorname{Li_{2}}\left(e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}/\sigma_{a}\right)\right]\equiv
f(z),$ (B.38)

hence $\widehat{V}_{3}=\frac{1}{\pi}f(z)+c.c.$. The integration constant is
chosen to make the $\widehat{V}_{3}$ function harmonic and the constants
$c_{a}=1/\sigma_{a}$ are chosen to keep the boundary conditions after mapping.
Then,

$\widehat{V}_{3}=(\frac{N_{\mathrm{NS}5}}{2\pi^{2}})\sum_{a}N_{\mathrm{D}5}^{(a)}\operatorname{Im}\left[\operatorname{Li_{2}}\left(\sigma_{a}e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}\right)-\operatorname{Li_{2}}\left(e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}/\sigma_{a}\right)\right],$
(B.39)

which matches with the last line of eq.(2.20) with identification
$N_{\mathrm{NS}5}\equiv P$, $N_{\mathrm{D}5}^{(a)}\equiv F_{J}$ and
$t_{a}\equiv J$.

####  B.2.2 Wilson loops

With $\widehat{V}_{3}^{D}$ dual to $\widehat{V}_{3}$ and
$\partial_{\eta}\widehat{W}=\widehat{V}_{3}$ the wilson loop computed in [27]
would read

$\displaystyle\ln\left\langle W_{\wedge}\right\rangle$
$\displaystyle=\frac{8}{\pi^{2}\alpha^{\prime 3}}\int d\xi
h_{1}h_{2}\left(\partial_{z}h_{2}\right)z^{\prime}=\pi\left[\int\sigma\partial_{\eta}\widehat{V}_{3}(\sigma,\eta)\mathrm{d}\sigma\right]_{(\infty,\eta*)}^{(\sigma*,\eta*)}$
(B.40)
$\displaystyle=\pi\left[\sigma\int\partial_{\eta}\widehat{V}_{3}(\sigma,\eta)\mathrm{d}\sigma-\int\int\partial_{\eta}\widehat{V}_{3}\mathrm{d}\sigma\mathrm{d}\sigma^{\prime}\right]_{(\infty,\eta*)}^{(\sigma*,\eta*)}=\pi\left[-\sigma\partial_{\sigma}\widehat{W}(\sigma,\eta)+\widehat{W}(\sigma,\eta)\right]_{(\infty,\eta*)}^{(\sigma*,\eta*)}.$

A by part integration is done in the second line and one can check that
$\partial_{\sigma}\widehat{W}(\sigma,\eta)=-\int\partial_{\eta}\widehat{V}_{3}(\sigma,\eta)\mathrm{d}\sigma$.
The limits can be computed easily from the rank and node of the Wilson loop in

$N_{\mathrm{F}1}=\frac{4}{\pi^{2}\alpha^{\prime
2}}\left[\operatorname{Im}\left(\mathcal{A}_{1}\mathcal{A}_{2}+\mathcal{C}\right)\right]_{(\infty,\eta*)}^{(\sigma*,\eta*)}=\left[\widehat{V}_{3}^{D}\right]_{(\infty,\eta*)}^{(\sigma*,\eta*)},\quad
N_{\mathrm{D}3}=\frac{4}{\pi\alpha^{\prime}}h_{2}^{D}\equiv\eta^{*}.$ (B.41)

$\alpha^{\prime}=1$ is chosen as above. It can be deduced from the second
equation that the integration limits are along constant $\eta$. Indeed, the
rank and node of the Wilson loop determine the $N_{\mathrm{F}1}$ and
$N_{\mathrm{D}3}$ on the probe D5 brane, which should be embedded in the
background geometry to calculate the expectation value. Hence, its trajectory
on the $\Sigma$ surface is calculated to satisfy the BPS conditions. ${z_{0}}$
and ${z_{1}}$ are endpoints of this trajectory, and integrals are along this
curve. More details can be found in the given references.

In terms of the Fourier expansion of potential fields with definitions

$\widehat{V}(\sigma,\eta)=\begin{cases}\sum_{k=1}^{\infty}\frac{a_{k}}{2}\left(e^{-\frac{k\pi}{P}z}+e^{-\frac{k\pi}{{P}}\bar{z}}\right)=\operatorname{Re}\sum_{k=1}^{\infty}a_{k}\left(e^{-\frac{k\pi}{P}z}\right)&\sigma\geq
0,\\\\[5.69054pt]
\sum_{k=1}^{\infty}\frac{a_{k}}{2}\left(e^{\frac{k\pi}{P}z}+e^{\frac{k\pi}{{P}}\bar{z}}\right)=\operatorname{Re}\sum_{k=1}^{\infty}a_{k}\left(e^{\frac{k\pi}{P}z}\right)&\sigma<0,\end{cases}$
(B.42)

and for $\widehat{V}^{D}$ we have
($\operatorname{Re}\mapsto\operatorname{Im}$)

$\widehat{V}^{D}(\sigma,\eta)=\begin{cases}\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi\eta}{P}\right)e^{-\frac{k\pi\sigma}{P}}&\sigma\geq
0,\\\\[5.69054pt]
-\sum_{k=1}^{\infty}a_{k}\sin\left(\frac{k\pi\eta}{P}\right)e^{\frac{k\pi\sigma}{P}}&\sigma<0.\end{cases}$
(B.43)

Hence the Wilson loop is

$\ln\langle
W_{\wedge}\rangle=\pi\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}(\frac{P}{k\pi})\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}(\frac{k\pi}{P}|\sigma^{*}|+1).$
(B.44)

The evaluation point is

$N_{D3}=\eta^{*},\quad
N_{\mathrm{F}1}=\widehat{V}^{D}=\sum_{k=1}^{\infty}\frac{\mathcal{R}_{k}}{2}\sin\left(\frac{k\pi}{P}\eta^{*}\right){e^{-\frac{k\pi}{P}|\sigma^{*}|}}\operatorname{Sgn}(\sigma^{*}).$
(B.45)

####  B.2.3 An example of mapping S-dual backgrounds

In this section, we consider mappings of the triangular quivers considered in
section 4. For the triangular quiver, one has

$\displaystyle h_{1}$
$\displaystyle=-\frac{i}{4}N_{\mathrm{D}5}\ln\left|\tanh\left(\frac{i\pi}{4}-\frac{w-\delta}{2}\right)\right|,$
$\displaystyle h_{2}$
$\displaystyle=-\frac{1}{4}N_{\mathrm{NS}5}\ln\left|\tanh\frac{w}{2}\right|,$
$\displaystyle\mathcal{A}_{1}$
$\displaystyle=-\frac{i}{4}N_{\mathrm{D}5}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta}{2}\right),$
$\displaystyle\mathcal{A}_{2}$
$\displaystyle=-\frac{1}{4}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right).$

Our mapping would be

$z=-\frac{1}{\pi}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right),$ (B.46)

giving

$\displaystyle\mathcal{A}_{1}(z^{\prime})$
$\displaystyle=-\frac{i\alpha^{\prime}}{4}\sum_{a}N_{\mathrm{D}5}^{(a)}\left[\ln\left(1-e^{\frac{-\pi}{2\alpha^{\prime}N_{\mathrm{NS}5}}z^{\prime}}/\tilde{\sigma}_{a}\right)-\ln\left(1-\tilde{\sigma}_{a}e^{\frac{-\pi}{2\alpha^{\prime}N_{\mathrm{NS}5}}z^{\prime}}\right)\right]\equiv\pi\partial_{z^{\prime}}\widehat{V},$
(B.47) $\displaystyle\mathcal{A}_{2}(z^{\prime})$
$\displaystyle=\frac{\pi}{8}z^{\prime},$ (B.48)

with $\sigma=\frac{ie^{\delta}-1}{ie^{\delta}+1}=e^{\frac{-i\pi
t}{N_{\mathrm{NS}5}}}$ and $t$ is the gauge node for which the flavour is
inserted. Here the constant $c=1/\tilde{\sigma}$ is chosen to keep the
boundary conditions after mapping. Then from eq.(B.39) one finds

$\widehat{V}^{elec}_{3}=(\frac{N_{\mathrm{NS}5}}{2\pi^{2}})N_{\mathrm{D}5}\operatorname{Im}\left[\operatorname{Li_{2}}\left(\sigma
e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}\right)-\operatorname{Li_{2}}\left(e^{\frac{-\pi}{N_{\mathrm{NS}5}}z}/\sigma\right)\right].$
(B.49)

The S-dual configuration can be obtained from exchanging $h_{1}$ and $h_{2}$

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=-\frac{i}{4}N_{\mathrm{NS}5}\ln\tanh\left(\frac{w}{2}\right)\,,$
$\displaystyle\mathcal{A}_{2}$
$\displaystyle=-\frac{1}{4}N_{\mathrm{D}5}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta}{2}\right).$

The desired transformation is

$z=-\frac{2\alpha^{\prime}}{\pi}N_{\mathrm{NS}5}\ln\tanh\left(\frac{i\pi}{4}-\frac{w-\delta}{2}\right).$
(B.50)

The transformation can be done in two steps

$\frac{z^{\prime}}{2}=\frac{i\pi}{4}-\frac{w-\delta}{2},\quad
z=-\frac{1}{\pi}N_{\mathrm{NS}5}\ln\tanh\left(\frac{z^{\prime}}{2}\right).$
(B.51)

After the first step, the holomorphic functions would be

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=-\frac{i}{4}N_{\mathrm{NS}5}\ln\tanh\left(\frac{i\pi}{4}-\frac{z^{\prime}-\delta}{2}\right),$
$\displaystyle\mathcal{A}_{2}$
$\displaystyle=-\frac{1}{4}N_{\mathrm{D}5}\ln\tanh\left(\frac{z^{\prime}}{2}\right),$

which is exactly the one before S-duality with
$N_{\mathrm{NS}5}\longleftrightarrow N_{\mathrm{D}5}$. The range of $\eta$
under final transformation would be $(0,N_{\mathrm{D}5})$ instead of
$(0,N_{\mathrm{NS}5})$ but the gauge node insertion relative to the range
would be the same as before S-duality,

$\widehat{V}^{mag}_{3}=(\frac{N_{\mathrm{D}5}}{2\pi^{2}})N_{\mathrm{NS}5}\operatorname{Im}\left[\operatorname{Li_{2}}\left(\sigma
e^{\frac{-\pi}{N_{\mathrm{D}5}}z}\right)-\operatorname{Li_{2}}\left(e^{\frac{-\pi}{N_{\mathrm{D}5}}z}/\sigma\right)\right].$
(B.52)

The relation between eqs.(B.49) and (B.52) can otherwise be obtained by
applying the generic expression for the potential $\widehat{V}_{3}$ in
eq.(2.20), in the case of no-offsets, and using eqs.(4.10). This makes the
point that the electrostatic version of mirror symmetry, encodes S-duality.

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