aainstitutetext: INFN, Sezione di Trieste, SISSA, via Bonomea 265, 34136
Trieste, Italybbinstitutetext: Dipartimento di Fisica, Università di Milano-
Bicocca, Piazza della Scienza 3, I-20126 Milano, Italyccinstitutetext: INFN,
Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano,
Italyddinstitutetext: Mathematical Institute, University of Oxford, Andrew-
Wiles Building, Woodstock Road, Oxford, OX2 6GG, United Kingdom

# Deconfinements, Kutasov–Schwimmer dualities and $D_{p}[SU(N)]$ theories

Sergio Benvenuti b,c Riccardo Comi b,c Sara Pasquetti d Matteo Sacchi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Kutasov–Schwimmer (KS) dualities involve a rank-$2$ field with a polynomial
superpotential. We derive KS-like dualities via deconfinement, that is
assuming only Seiberg-like dualities, which instead just involve fundamental
matter. Our derivation is split into two main steps. The first step is the
construction of two families of linear quivers with $p\\!-\\!1$ nodes that
confine into a rank-$2$ chiral field with degree-$(p\\!+\\!1)$ superpotential.
Such chiral field is an $U(N)$ adjoint in 3d and an $USp(2N)$ antisymmetric in
4d. In the second step we use these linear quivers to derive, via
deconfinement, in a relatively straightforward fashion, two classes of KS-like
dualities: the Kim–Park duality for $U(N)$ with adjoint in 3d and the
Intriligator duality for $USp(2N)$ with antisymmetric in 4d. We also discuss
the close relation of our 3d family of confining unitary quivers to the 4d
$\mathcal{N}\\!=\\!2$ $D_{p}[SU(N)]$ SCFTs by circle compactification and
various deformations.

##  1 Introduction and summary

Infrared (IR) dualities are an interesting phenomenon that can characterize
the low energy dynamics of quantum field theories (QFT). They involve a pair
of different microscopic QFTs that describe the same low energy physics.
Several instances of IR dualities are known, with the most important example
being the Seiberg duality Seiberg:1994rs  for 4d ${\cal N}=1$ supersymmetric
quantum chromodynamics (SQCD) with $SU(N)$ gauge group and fundamental matter.
Soon after its discovery, Kutasov and Schwimmer (KS) 1Kutasov_1995 ;
Kutasov_1995  proposed a duality for $SU(N)$ with $F$ fundamental matter
fields and an adjoint $\Phi$, with polynomial superpotential ${\cal
W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. The dual theory theory has
$SU(pF-N)$ gauge group, $F$ flavors and one adjoint $\phi$ with the same
${\cal W}=\mathop{\text{Tr}}\nolimits(\phi^{p+1})$ superpotential, plus $p$
sets of singlets flipping the mesons dressed with the adjoint $\phi$ . The KS
duality can be understood as a generalization of the Seiberg duality to which
it reduces for $p=1$, since the superpotential becomes a mass term for the
adjoint field which is then integrated out in the IR.

There exist other variants of this duality, to which we will sometime refer as
_KS-like dualities_. In Intriligator:1995ff  Intriligator proposed a version
of the duality for symplectic gauge group (IKS). In this case we have a
$USp(2N)$ theory with $2F$ fundamental chirals and one antisymmetric $\Phi$
with $\mathcal{W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$
superpotential.111Throughout this paper, traces over $USp(2N)$ groups are
taken with the insertion of the $2N\times 2N$ totally antisymmetric tensor
$J_{N}=i\sigma_{2}\otimes\mathbb{I}_{N}$. Moreover, there exist also three-
dimensional $\mathcal{N}=2$ analogues of the KS duality: the Kim–Park (KP)
duality for a $U(N)$ theory with $F$ fundamental flavors and an adjoint chiral
Kim_2013 , and its $SU(N)$ variant by Park–Park Park_2013 .

These KS-like dualities and their tests are subtle in many ways. The most
straightforward test is anomaly matching. The mapping of chiral gauge
invariant operators can also be worked out, however it is more intricate than
in the ordinary Seiberg duality because of the truncation in the chiral ring
of dressed operators due to the polynomial superpotential. The most powerful
test of supersymmetric dualities is usually the matching of supersymmetric
partition functions such as the supersymmetric index Kinney:2005ej ;
Romelsberger:2005eg ; Dolan:2008qi ; Bhattacharya:2008zy ;
Bhattacharya:2008bja ; Kim:2009wb ; Imamura:2011su ; Kapustin:2011jm ;
Dimofte:2011py , but this presents various difficulties for the KS-like
dualities. Contrary to other dualities no index identities are known in the
math literature and perturbative tests are hard to push to high order since
the dual gauge group grows quickly (however see Dolan:2008qi  for the matching
of the index in the KS duality in the Venziano limit).

During the last few years, various works managed to prove many dualities
involving gauge theories with rank-$2$ matter fields assuming only the
validity of _Seiberg-like dualities_ , that is dualities for simple classical
gauge groups and only fundamental matter. This technique goes under the name
of _deconfinement_ and it schematically works as follows. We trade the rank-2
field for an additional gauge node using some known s-confining duality in the
sense of Csaki:1996zb  for which the dual theory is just a Wess–Zumino (WZ)
model of chiral fields with no gauge group, such as the Seiberg duality for
$F=N+1$. This gives an auxiliary quiver gauge theory that can be further
dualized using Seiberg-like dualities until we reach the desired dual frame.

The idea of the deconfinement first appeared in Berkooz:1995km ;
Pouliot:1995me ; Luty:1996cg , but it has been recently revamped in many ways
and in various dimensions (see e.g. Pasquetti:2019uop ; Sacchi:2020pet ;
Benvenuti:2020gvy ; Etxebarria:2021lmq ; Benvenuti:2021nwt ; Bajeot:2022kwt ;
Bottini:2022vpy ; Bajeot:2022lah ; Bajeot:2022wmu ; Bajeot:2023gyl ;
Amariti:2023wts ; Amariti:2024sde ; Amariti:2024gco ). Examples of successes
of the deconfinement program are the derivation Bajeot:2022kwt  of most of the
s-confining dualities with classical gauge group of Csaki:1996zb , and the
derivation of a self-duality for $USp(2N)$ with one antisymmetric and $8$
fundamental chirals Benvenuti:2020gvy ; Bajeot:2022lah .

The deconfinement technique fits into the larger program of trying to derive
all IR dualities from a small set of fundamental Seiberg-like dualities. Along
this line of research, a related approach is the _local dualization algorithm_
, which allows us to prove mirror dualities with eight Hwang:2021ulb ;
Comi:2022aqo ; Giacomelli:2023zkk ; Giacomelli:2024laq  and four supercharges
Benvenuti:2023qtv , assuming only Seiberg-like dualities that are either the
Intriligator–Pouliot (IP) duality Intriligator:1995ne  in 4d or the Aharony
duality Aharony:1997gp  in 3d.

In this paper we tackle the problem of proving KS-like dualities using the
deconfinement. In particular, we focus on the 3d KP duality for $U(N)$ gauge
group with adjoint matter and the 4d IKS duality for $USp(2N)$ gauge group
with antisymmetric matter, and show how it is possible to derive them assuming
only Seiberg-like dualities. These are again the IP duality in 4d and the
Aharony duality in 3d. In fact in three dimensions we will also make use of
dualities with monopole superpotentials Benini:2017dud , which however can
still be obtained as a deformation of the Aharony duality. Let us point out
that the derivation of the 3d KP duality provides also strong evidence for the
validity of the 4d KS $SU(N)$ duality. Indeed, as shown in Amariti:2014iza ;
Nii_2015 , the 3d KP duality follows from the 4d $SU(N)$ KS duality via
dimensional reduction plus deformations.222The 3d KP duality can also be
obtained by a different dimensional reduction from the 4d IKS duality along
the lines of Benini:2017dud ; Amariti:2018wht ; Pasquetti:2019hxf ;
Bajeot:2022lah ; Bottini:2021vms ; Comi:2022aqo ; Bajeot:2023gyl . From this
perspective, our 4d derivation already contains the 3d one since we can take
such dimensional reduction limit at any of the steps. However, for clarity of
exposition we will discuss the 3d and the 4d cases separately.

It is important to emphasize again that in the mathematical literature there
is no proof of the equality of supersymmetric partition functions and indices
associated to KS dual theories. Since the integral identities corresponding to
Seiberg-like dualities have instead been proven 2003math……9252R ; Fokko ;
Spiridonov:2009za ; Spiridonov:2011hf , our arguments amount to a proof of the
integral identities corresponding to KS-like dualities.

Our derivation is somewhat of a tour-de-force, the strategy involves two main
steps. The first one consists of constructing families of linear quivers with
$p-1$ nodes, which are infrared dual to a chiral superfield $\Phi$ in a
rank-$2$ representation of the flavor symmetry with polynomial superpotential
${\cal W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. We build two such classes
of theories: the 3d ${\cal N}=2$ $\mathcal{C}_{p}[SU(N)]$ family deconfines an
$SU(N)$ adjoint, while the 4d ${\cal N}=1$ $\mathsf{C}_{p}[USp(2N)]$ family
deconfines a $USp(2N)$ antisymmetric. As an example, the
$\mathcal{C}_{p}[SU(pr+l)]$ ($0<l<p$) theories are given by the following
quiver:

where $\mathcal{W}_{\text{cub}}$ consists of the cubic couplings between all
the adjoint fields and the adjacent bifundamentals, while the non-perturbative
monopole terms break supersymmetry to $\mathcal{N}=2$ and are responsible for
the confinement of the theory.

The proof that these linear quivers are IR dual to the WZ model with a single
tensor field and degree-$(p+1)$ superpotential is highly non-trivial and works
by induction. Each inductive step involves taking the mirror quiver, and using
sequentially the appropriate Seiberg-like dualities.

The second step is relatively straightforward and consists in using our new
confining dualities to deconfine the rank-$2$ field in the KS-like dualities.
We start from the electric side of the KS-like duality and deconfine the
rank-$2$ field into the $p-1$ nodes quiver, obtaining a linear quiver of
lenght $p$. Now we apply iteratively the Seiberg-like duality from one side to
the other of the quiver. This process involves $p$ dualizations, hence it
produces precisely $p$ sets of flipping singlets. Moreover, the final quiver
is precisely the magnetic gauge group predicted by the KS-like duality
attached to the $p-1$ nodes quiver that deconfines the rank-$2$ field $\phi$
in the appropriate representaiton and with ${\cal
W}=\mathop{\text{Tr}}\nolimits(\phi^{p+1})$ interaction. Hence, we can
reconfine the whole quiver into the magnetic theory. In this way, we manage to
derive the 3d ${\cal N}=2$ KP duality for $U(N)$ with adjoint and $F$
fundamental flavors and the 4d ${\cal N}=1$ IKS duality for $USp(2N)$ with
antisymmetric and $2F$ fundamentals.

One important aspect of our story is the relation between the 3d ${\cal N}=2$
$\mathcal{C}_{p}[SU(N)]$ family and the 4d ${\cal N}=2$ $D_{p}[SU(N)]$
superconformal field theories (SCFTs), which are non-Lagrangian theories that
can be realized either via geometric engineering or in class $\mathcal{S}$
Cecotti:2012jx ; Cecotti:2013lda ; Wang:2015mra . Recently Maruyoshi:2023mnv ,
building on some earlier results Xie:2021omd ; Kang:2023dsa ; Bajeot:2023gyl ,
it was argued that a $D_{p}[SU(N)]$ theory with $\gcd(p,N)=1$ deformed by a
Coulomb Branch (CB) operator of dimension $p/(p+1)$ flows in the IR to an
$SU(N)$ adjoint chiral with
$\mathcal{W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$ superpotential. This
result is closely related to our new confining dualities, in particular our 3d
$\mathcal{N}=2$ ${\cal C}_{p}[SU(N)]$ quiver theories are very similar to the
3d $\mathcal{N}=4$ quivers describing the dimensional reduction of
$D_{p}[SU(N)]$, as already pointed out in Bajeot:2023gyl  for $p=2$ and
$N=2r+1$. Indeed, despite being non-Lagrangian in 4d, the circle
compactification of $D_{p}[SU(N)]$ admits a Lagrangian description in terms of
a linear quiver Closset:2020afy ; Giacomelli:2020ryy . When $\gcd(p,N)=1$ this
has exactly the same matter content and perturbative superpotential, i.e.
excluding the monopole terms, of our $\mathcal{C}_{p}[SU(N)]$ confining
theory. Interestingly for other values of $p$ and $N$ our 3d
$\mathcal{C}_{p}[SU(N)]$ confining theories are still very closely related (by
simple deformations) to the circle reduction of $D_{p}[SU(N)]$. Specifically,
compactifying on $S^{1}$ the non-Lagrangian 4d ${\cal N}=2$ $D_{p}[SU(N)]$
theory for $\gcd(p,N)\neq 1$ and $N\neq pr$, one still gets a linear 3d ${\cal
N}=4$ quiver of length $p-1$, but with mixed unitary and special unitary gauge
groups. One then has to perform a combination of a deformation breaking the
special unitary groups to unitary, dualizations and deformations by monopole
superpotential to land on $\mathcal{C}_{p}[SU(N)]$.

The rest of the paper is organized as follows. In Section 2 we introduce the
3d $\mathcal{N}=2$ $\mathcal{C}_{p}[SU(N)]$ theory, discuss various of its
properties and show that it confines to an $SU(N)$ adjoint with ${\cal
W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. In Section 3 we use this new
confining duality to provide a derivation of the 3d KP duality via
deconfinement and sequential Aharony dualizations. Section 4 deals with the
relation between our 3d $\mathcal{N}=2$ $\mathcal{C}_{p}[SU(N)]$ theories and
the 4d ${\cal N}=2$ $D_{p}[SU(N)]$ SCFTs. In Section 5 we discuss the 4d
uplift of the $\mathcal{C}_{p}[SU(N)]$ theory, which we call
$\mathsf{C}_{p}[USp(2N)]$, and its confining duality into a $USp(2N)$
antisymmetric with ${\cal W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. In
Section 6 we use this to provide a derivation of the 4d IKS duality via
deconfinement and sequential IP dualizations. We conclude in Section 7 with
some comments on various open questions and possible future directions. The
main text is supplemented by Appendix A, in which we review some important
facts about the Aharony duality and the variants with monopole
superpotentials.

##### Note added:

While this manuscript was being finalized we learned about the work of Hwang
which has a substantial overlap with our manuscript. We thank the authors of
Hwang  for sharing their draft with us and for coordinating the submission.

##### Notation for flippers.

Flipping fields $f_{O}$ are gauge singlet chiral fields that enter the
superpotential only once, linearly and multiplying some operator $O$. In other
words the superpotential term is $\mathcal{W}_{\text{flip}}=f_{O}\,O$. As a
consequence, the F-term equation of $f_{O}$ set to zero $O$ in the chiral
ring. We denote such singlet fields simply as $F[O]$. Moreover, we write the
superpotential term as ${\cal W}_{\text{flip}}=\text{Flip}[O]$.

##  2 The 3d ${\cal N}=2$ $\mathcal{C}_{p}[SU(N)]$ family

In this section we introduce a class of 3d $\mathcal{N}=2$ theories named
$\mathcal{C}_{p}[SU(N)]$, where $p$ and $N$ are integers such that $N\geq
p+1$. As we shall see, these theories are constructed by turning on an
appropriate monopole superpotential in certain 3d $\mathcal{N}=4$ theories
which explicitly breaks supersymmetry down to $\mathcal{N}=2$. We will show
that the $\mathcal{C}_{p}[SU(N)]$ theories confine to a Wess–Zumino (WZ) model
of a chiral $\Phi$ in the adjoint representation of a $SU(N)$ flavor symmetry
with ${\cal W}\sim\Phi^{p+1}$. Along the way, we also construct the mirror
description of the $\mathcal{C}_{p}[SU(N)]$ theories.

###  2.1 The $\mathcal{C}_{p}[SU(N)]$ family

The $\mathcal{C}_{p}[SU(N)]$ theories are 3d ${\cal N}=2$ theories whose
global symmetry,on top of the $U(1)_{R}$ R-symmetry, is333In the following we
will only focus on the $SU(N)$ Lie algebra of the non-abelian flavor symmetry
and ignore the global structure. We will however pay attention to the finite
$\mathbb{Z}_{p+1}$ symmetry.

$PSU(N)\times\mathbb{Z}_{p+1}\,.$ (1)

They are defined as a quiver theory with $p-1$ unitary gauge nodes of the
form444In the drawing each circle node is a gauge symmetry, the square node is
the flavor symmetry, the double lines between nodes represent pairs of
bifundamental chirals in conjugate representations, and arcs are adjoint
chirals.

(2)

The ${\cal C}_{p}[SU(N)]$ theory is also depicted in short as given on the
right. This notation is motivated by the fact that, as we shall see, the basic
operator charged under the $SU(N)$ symmetry transforms in the adjoint
representation and more generally the only representations that appear in the
spectrum of gauge invariant operators are uncharged under the $\mathbb{Z}_{N}$
center, so that the faithful symmetry is actually $SU(N)/\mathbb{Z}_{N}$.

It is convenient to define

$N=pr+l\,,\qquad l=0,\cdots,p-1\,,$ (3)

so that the ranks of all the gauge nodes can be obtained via the formula

$\displaystyle n_{j}=jr+[l-p+j]^{+}\,,\qquad j=1,\cdots,p-1\,,$ (4)

where $[k]^{+}=k$ if $k>0$ and zero otherwise. From (4), it is possible to
compute that all the gauge nodes are balanced, i.e. the number of flavors is
twice their ranks, except for the $j=p-l$ node which is overbalanced with
excess number $1$, i.e. the number of flavors is twice the gauge rank plus
one.

The superpotential contains two types of terms. The first one ${\cal
W}_{\text{cub}}$ consists of the standard cubic couplings between all the
adjoint fields and the adjacent bifundamentals that characterizes
$\mathcal{N}=4$ theories when presented in terms of $\mathcal{N}=2$ fields.
The second one ${\cal W}_{\text{monopoles}}$, which we shall specify
momentarily, involves monopole operators and breaks supersymmetry to
$\mathcal{N}=2$. In particular, it only preserves one combination of the two
abelian factors in the Cartan subgroup of the $\mathcal{N}=4$ $SU(2)_{H}\times
SU(2)_{C}$ R-symmetry, which is identified with the $\mathcal{N}=2$
R-symmetry. For any generic $l$, the R-charges of the chiral fields under such
symmetry are

$\displaystyle R[\text{bifundamental}]=\frac{1}{p+1}\,,$ $\displaystyle
R[\text{adjoint}]=2-\frac{2}{p+1}\,.$ (5)

The remaining combination is instead broken to a finite subgroup. As we will
see more in details later, this is a $\mathbb{Z}_{p+1}$ symmetry under which
the charges of the chirals are555Despite the fact that the bifundamental
chirals have fractional charge, as we shall see below the minimal charge
appearing in the spectrum of gauge invariant operators is the unit one in this
parametrization.

$\displaystyle q[\text{bifundamental}]=\frac{1}{2}\text{ mod }(p+1)\,,$
$\displaystyle q[\text{adjoint}]=-1\text{ mod }(p+1)=p\text{ mod }(p+1)\,.$
(6)

This finite symmetry is also a residue of the $p-1$ topological symmetries, so
that the charge of the monopoles under it also has a bare contribution on top
of the 1-loop one due to the fermions in the chirals Borokhov:2002ib ;
Borokhov:2002cg ; Gaiotto:2008ak ; Benna:2009xd ; Bashkirov:2010kz .
Specifically, the remnant of the topological symmetry is the one that mixes
with coefficient $+1$ for the balanced nodes and with coefficient
$\frac{3}{2}$ for the unbalanced $(p-l)$-th node in the $l\neq 0$ case. In
this way, a monopole with magnetic fluxes $m^{(j)}_{a_{j}}$ under the $j$-th
node, with $j=1,\cdots,p-1$ and $a_{j}=1,\cdots,n_{j}$, has bare charge (in
this convention $n_{p}=0$ when $l=0$)

$\displaystyle
q_{0}\left[m^{(j)}_{a_{j}}\right]=\sum_{\begin{subarray}{c}{j=1}\\\ j\neq
p-l\end{subarray}}^{p-1}\sum_{a_{j}=1}^{n_{j}}m^{(j)}_{a_{j}}+\frac{3}{2}\sum_{a_{p-l}=1}^{n_{p-l}}m^{(p-l)}_{a_{p-l}}\,,$
(7)

to which one should add the contribution of the fermions in the chirals with
charges (2.1) with the usual formula.

Let us now discuss in more details the two possible cases $l\neq 0$ and $l=0$.

#####  $l\neq 0$.

In this case ${\cal W}_{\text{monopoles}}$ consists of a linear monopole
superpotential, involving the monopoles $\mathfrak{M}_{i}^{+}$
($i=1,\cdots,p-1$) with magnetic fluxes $+1$ for each gauge node separately
and a monopole $\mathfrak{M}^{-\ldots-}$ with magnetic flux $-1$ for all gauge
nodes simultaneously. The theory can be depicted as

(8)

This theory starts with a balanced tail of $p-l-1$ nodes, then we have an
overbalanced node and then a sequence of $l-1$ balanced nodes.

#####  $l=0$.

In this case ${\cal W}_{\text{monopoles}}$ is similar to that of the $l\neq 0$
case in (8), with the difference being that the $\mathfrak{M}^{-\ldots-}$
monopole is now dressed with an adjoint chiral,666This can be taken to be any
of the adjoint chirals, since monopoles with the same level of dressing are
identified due to quantum relations. denoted by $\mathfrak{M}_{A}^{-\ldots-}$,
and there is also a superpotential flipping the $\mathfrak{M}^{-\ldots-}$
monopole. The theory can be depicted as

(9)

Notice that now all the nodes are balanced.

##### Effects of the monopole superpotential ($l\neq 0$).

If we turn off all the monopole superpotentials in ${\cal C}_{p}[SU(N)]$, then
this coincides with the 3d ${\cal N}=4$ $T^{\sigma}_{\rho}[SU(N)]$ theory
Gaiotto:2008ak  with

$\sigma=[1^{N}]\,,\qquad\rho=[(r+1)^{l},r^{p-l}]\,,$ (10)

in particular for $l=0$ we just have $\rho=[r^{p}]$. The theory is

(11)

where $Y_{i}-Y_{i+1}$ is the FI parameter for the topological symmetry of the
$i$-th node, $W_{1}\ldots W_{N}$ the mass parameters for the flavor symmetry
and we pick the following convention for the R-charges and charges under the
$U(1)_{A}$ axial symmetry (that is the commutant of the $\mathcal{N}=2$
R-symmetry inside the $\mathcal{N}=4$ R-symmetry):

$\displaystyle R[\text{hypers}]=\frac{1}{2}\quad,\quad A[\text{hypers}]=1\,,$
$\displaystyle R[\text{adjoint}]=1\quad,\quad A[\text{adjoint}]=-2\ \,.$ (12)

From the partition $\rho$ we see an $S[U(l)\times U(p-l)]$ symmetry acting on
the Coulomb branch operators, which is enhanced from the topological
symmetries of the two tails of balanced nodes. In this theory the monopoles
are charged under the various abelian symmetries, as it can be computed using
the usual monopole formula. For convenience, we encode these charges as the
coeffients in the linear combination of the parameters in the $S^{3}$
partition function for the corresponding symmetries. Focusing on the monopoles
appearing in the superpotential for $l\neq 0$, we have

$\displaystyle\mathfrak{M}_{i}^{+}:\quad
2\left(i\frac{Q}{4}-m_{A}\right)+Y_{i}-Y_{i+1}\,,\qquad
i=1,\cdots,p-1\,,\,\,i\neq p-l\,,$ $\displaystyle\mathfrak{M}_{p-l}^{+}:\quad
3\left(i\frac{Q}{4}-m_{A}\right)+Y_{p-l}-Y_{p-l+1}\,,$
$\displaystyle\mathfrak{M}^{-\ldots-}:\quad
3\left(i\frac{Q}{4}-m_{A}\right)+Y_{p}-Y_{1}\,,$ (13)

where $m_{A}$ is the real mass parameter for $U(1)_{A}$ and $i\tfrac{Q}{2}$ is
related to the background field for the ${\cal N}=2$ R-symmetry, meaning that
the R-charge can be read as the coefficient of $i\tfrac{Q}{2}$. Turning on
${\cal W}_{\text{monopoles}}$ requires all of these monopoles to have charge
two, which corresponds to setting the combinations of parameters in (14) equal
to $iQ$. From the positive monopoles we see that all the $Y_{i}$ parameters
are then specialized in terms of the axial mass $m_{A}$ as

$\displaystyle Y_{i}-Y_{i+1}=i\frac{Q}{2}+2m_{A}\,,\qquad
i=1,\cdots,p-1\,,\,\,i\neq p-l\,,$ $\displaystyle
Y_{p-l}-Y_{p-l+1}=i\frac{Q}{4}+3m_{A}\,.$ (14)

Furthermore, from the negative monopole we then get the further constraint

$\displaystyle\ 2(p+1)m_{A}=i\frac{Q}{2}(1-p)\,,$ (15)

which consistently yields the R-charge assignment in (5). From this
perspective one can also show that there is a finite $\mathbb{Z}_{p+1}$
subgroup of the axial and the topological symmetries that is preserved. We
will come back to this point around equation (41). The $l=0$ case is
analogous.

##### Background BF couplings.

If we treat the flavor symmetry as $U(N)$ rather than $SU(N)$, as we will do
in the later section where such symmetry will be gauged, then we should also
include in the definition of the theories a BF coupling between the R-symmetry
and the $U(1)$ part of $U(N)$. This can be encoded in its contribution to the
$S^{3}$ partition function as follows:

$\displaystyle\begin{cases}\mathrm{e}^{2\pi
i\frac{iQ}{2}\frac{1-l}{p+1}\sum_{i=1}^{pr+l}W_{i}}&l\neq 0\\\
\mathrm{e}^{2\pi
i\frac{iQ}{2}\frac{1-p}{p+1}\sum_{i=1}^{pr}W_{i}}&l=0\end{cases}$ (16)

Notice that these only appear in the combination $\sum_{i=1}^{N}W_{i}$
corresponding to the $U(1)$ part of $U(N)$, so the BF couplings trivialize if
we treat the symmetry as $SU(N)$ setting $\sum_{i=1}^{N}W_{i}=0$.

##### Spectrum of gauge invariant operators.

Below we discuss some interesting gauge invariant operators of the ${\cal
C}_{p}[SU(N)]$ theory which we will later map to the singlets in the confined
dual frame (23). First of all we have the operator $\mu$ in the adjoint of
$SU(N)$. From the quiver theory in (2) this operator is constructed as

$\mu=Q\tilde{Q}\,,$ (17)

where $Q$ and $\tilde{Q}$ are the last bifundamental chirals, charged under
$SU(N)$. Notice that this coincides with the moment map of the flavor $SU(N)$
in the $\mathcal{N}=4$ theory obtained by turning on only ${\cal
W}_{\text{cub}}$ and not ${\cal W}_{\text{monopoles}}$, and by some abuse of
notation we will sometimes refer to it as “moment map” even in the
$\mathcal{N}=2$ ${\cal C}_{p}[SU(N)]$ theory. This operator has R-charge

$\displaystyle R[\mu]=\frac{2}{p+1}$ (18)

and it has charge under the finite $\mathbb{Z}_{p+1}$ symmetry

$\displaystyle q[\mu]=1\text{ mod }(p+1)\,.$ (19)

In addition to this operator we also have a set of gauge invariant chiral
operators that are singlets under the $SU(N)$ flavor symmetry. In (2) we can
construct monopoles that have $-1$ magnetic flux under a string of $j$
consequent balanced gauge nodes, which we denote by
$\mathfrak{M}^{\ldots-^{j}\ldots}$. We recall that for $l\neq 0$ the theory
has two sequences of balanced nodes of length $p-l-1$ and $l-1$ respectively,
as explained below (8). For $l=0$ all the nodes are balanced, however the
monopole with magnetic flux $-1$ under all the gauge nodes is flipped, so that
in total we have $p-2$ such monopoles. The R-charge of a monopole charged
under a string of $j$ balanced gauge nodes is777This can be easily calculated
starting from the ${\cal N}=4$ theory with the same strategy used around
(2.1). A monopole with $-1$ magnetic flux under a string of consequent
balanced nodes starting from the node $k$ and extending to the node $k+j$ has
the following charges:  $\mathfrak{M}^{\ldots-^{j}\ldots}:\quad
2\left(i\frac{Q}{4}-m_{A}\right)-Y_{k}+Y_{k+j}=\left(2-\frac{2(j+1)}{p+1}\right)i\frac{Q}{2}\,,$
when using the constraints in eqs. (14) and (15). Notice that the result does
not depend on the choice of $k$ but only on the length $j$.

$\displaystyle
R\left[\mathfrak{M}^{\ldots-^{j}\ldots}\right]=2-\frac{2(j+1)}{p+1}\,.$ (20)

Instead, the charge under the $\mathbb{Z}_{p+1}$ symmetry is

$\displaystyle q\left[\mathfrak{M}^{\ldots-^{j}\ldots}\right]=-(j+1)\text{ mod
}(p+1)=p-j\text{ mod }(p+1)\,.$ (21)

Indeed, in the ${\cal C}_{p}[SU(N)]$ theory there are many such monopoles. We
claim that monopoles with the same length constructed from the same balanced
tail are degenerate. This leaves us with two sets

$\displaystyle l\neq 0\quad:\quad$
$\displaystyle\begin{cases}&\mathfrak{M}_{L}^{\ldots-^{j}\ldots}\qquad
j=1,\ldots,p-l-1\qquad\text{from the left tail}\\\
&\mathfrak{M}_{R}^{\ldots-^{j}\ldots}\qquad j=1,\ldots,l-1\qquad\text{from the
right tail}\end{cases}\,,$ $\displaystyle l=0\quad:\quad$
$\displaystyle\mathfrak{M}^{\ldots-^{j}\ldots}\qquad
j=1,\ldots,p-2\qquad\text{from the only tail}\,.$ (22)

Finally, for $l=0$ we also have the flipper $F[\mathfrak{M}^{-\ldots-}]$,
whose R-charge is $\tfrac{2p}{p+1}$ and whose $\mathbb{Z}_{p+1}$ charge is
$p$.

The operators that we have just discussed are the most elementary ones,
however the theory also possesses composite operators, some of which in
particular parametrize a conformal manifold. Indeed the operator obtained as
the product of $\mathfrak{M}_{L/R}^{\ldots-^{j}\ldots}$ and
$\mathfrak{M}_{L/R}^{\ldots-^{p-j-1}\ldots}$ has R-charge equal to 2 and is
not charged under any global symmetry for every $j$, therefore it represents
an exactly marginal deformation. One may also consider other exactly marginal
deformations given by the product of more than two
$\mathfrak{M}_{L/R}^{\ldots-^{j}\ldots}$ operators, such that the R-charge of
the result is 2.

####  2.1.1 Statement of the confining duality

We claim that these theories confine into an adjoint $SU(N)$ chiral with
${\cal W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$ plus extra singlets

(23)

where in the figure we are depicting the ${\cal C}_{p}[SU(N)]$ theory in short
as the label ${\cal C}_{p}$ embedded in an arc to recall that its spectrum
contains only one operator charged under the $SU(N)$ global symmetry, which is
in the adjoint representation.

On the r.h.s. of (23) we have flipping fields for traces of $\Phi$ from power
2 up to $k(l)$, which is given by (recall $N=pr+l$)

$\displaystyle k(l)=\text{min}(l,p-l)\,.$ (24)

Notice that the above formula enjoys the property $k(l)=k(p-l)$. This will
turn out to be important in the next section when we will use these theories
to provide a derivation of the Kim–Park duality, which maps the gauge rank
$N=pr+l$ to $N^{\prime}=pr^{\prime}+(p-l)$.

In the duality (23), the superpotential ${\cal W}_{\text{multi-monopole}}$ on
the l.h.s. contains all the possible exactly marginal deformations given by
the product of many $\mathfrak{M}_{L/R}^{\ldots-^{j}\ldots}$ operators, as
described in the end of the previous subsection. Correspondingly on the r.h.s.
we are turning on all the exactly marginal deformations given as multi trace
operators, encoded in ${\cal W}_{\text{multi-trace}}$. The reason why the
duality holds at such point of the conformal manifold is motivated by the
$S^{3}$ partition function perspective. The derivation of the duality that we
will discuss in field theory in the next section can indeed be rigorously
applied at the level of the partition function. However, the downside of this
approach is that the partition function is not sensitive to the precise
structure of the superpotential at each step of the derivation, but only to
the charges of the fields induced by it. In particular, in the partition
function computation one finds contributions of gauge singlet fields with
charges that are compatible with exactly marginal massive deformations. In
order to find the duality (23) one has to simplify the contribution of such
fields, which should be interpreted as being on a point of the conformal
manifold where the mass deformations are activated and the singlets are
integrated out. Additionally, some of the steps of our derivation will involve
confining gauge nodes and in this process all possible superpotential terms
compatible with the charge assignments can be generated. A similar situation
has been discussed in Amariti:2021lhk ; Amariti:2022dyi .

For these reasons, we only claim that the duality (23) holds on a generic
point of the conformal manifold. See Section 2.2.3 for more comments on this
point in the case $N=p+1$.

The superpotential $\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$ in the WZ theory
uniquely determines the R-charge of $\Phi$ to be

$\displaystyle R[\Phi]=\frac{2}{p+1}$ (25)

and it also preserves a finite $\mathbb{Z}_{p+1}$ symmetry such that

$\displaystyle q[\Phi]=1\text{ mod }(p+1)\,.$ (26)

Such charge assignment is compatible with the following operator map across
the duality. The operator $\mu$ in the adjoint of $SU(N)$ on the electric side
gets mapped to the adjoint chiral in the dual WZ model

$\displaystyle\mu\quad$ $\displaystyle\longleftrightarrow\quad\Phi\,.$ (27)

We can also map the two set of monopoles
$\mathfrak{M}_{L/R}^{\ldots-^{j}\ldots}$ of the ${\cal C}_{p}[SU(N)]$ theory
to flavor singlets in the WZ model. Suppose that
$\mathfrak{M}_{L}^{\ldots-^{j}\ldots}$ is the biggest of the two sets, that is
$p-l\geq l$. Then the map is

$\left\\{\begin{tabular}[]{c}$\mathfrak{M}_{L}^{\ldots,-^{j}\ldots}$\\\
$\mathfrak{M}_{R}^{\ldots,-^{j}\ldots}$\end{tabular}\right\\}\qquad\longleftrightarrow\qquad\left\\{\begin{tabular}[]{c}$\mathop{\text{Tr}}\nolimits(\Phi^{p-j})$\\\
${\cal F}[\mathop{\text{Tr}}\nolimits(\Phi^{j+1})]$\end{tabular}\right\\}$
(28)

If $p-l<l$ instead the map is

$\left\\{\begin{tabular}[]{c}$\mathfrak{M}_{L}^{\ldots,-^{j}\ldots}$\\\
$\mathfrak{M}_{R}^{\ldots,-^{j}\ldots}$\end{tabular}\right\\}\qquad\longleftrightarrow\qquad\left\\{\begin{tabular}[]{c}${\cal
F}[\mathop{\text{Tr}}\nolimits(\Phi^{j+1})]$\\\
$\mathop{\text{Tr}}\nolimits(\Phi^{p-j})$\end{tabular}\right\\}$ (29)

The operator map also tells that some of the monopoles are actually composite
because they are mapped to traces of powers of $\Phi$, so there are quantum
relations in the electric theory that identify such monopoles with traces of
powers of $\mu$. For $l=0$ the electric theory also has the flipper
$F[\mathfrak{M}^{-\ldots-}]$, however looking at its charges we claim that
this is also involved in a relation with
$\mathop{\text{Tr}}\nolimits(\mu^{p})$ so that the independent combination is
mapped to $\mathop{\text{Tr}}\nolimits(\Phi^{p})$.

####  2.1.2 The mirror theories $\check{{\cal C}}_{p}[SU(N)]$

To prove the confinement we need the mirror dual theories $\check{{\cal
C}}_{p}[SU(pr+l)]$ which are given for $l\neq 0$ and $l=0$ by

(30) (31)

where we are indicating with ${\cal W}_{\text{cub}}$ all the cubic
superpotential terms coupling the bifundamentals with the adjacent adjoint
chirals. In the second theory, notice that we can equivalently use the
bifundamental on the right of the flavor to construct the
$sb\tilde{b}\tilde{s}$ operator due to F-term relations.

Both for $l\neq 0$ and $l=0$, the R-charges of the bifundamentals and adjoint
chirals are fixed by the superpotential to be

$\displaystyle R[\text{bifundamental}]=1-\frac{1}{p+1}\,,$ $\displaystyle
R[\text{adjoint}]=\frac{2}{p+1}\,.$ (32)

The superpotential also constraints the R-charge of the flavors to be

$\displaystyle l\neq
0\quad:\quad\begin{cases}R[p,\tilde{p}]=1-\frac{l}{p+1}\\\
R[q,\tilde{q}]=1-\frac{p-l}{p+1}\end{cases}\,,$ $\displaystyle l=0\quad:\quad
R[s,\tilde{s}]=1-\frac{p}{p+1}\,.$ (33)

For any generic $l$, the map between chiral ring generators relates the $\mu$
operator of the ${\cal C}_{p}[SU(N)]$ theory with a matrix where on the
diagonal there are the traces of the adjoint fields, while the other entries
are filled by extended mopoles (as is customary for mirror dualities). In the
mirror theory we can also construct dressed mesonic operators. These are
different depending on whether $l\neq 0$ or $l=0$888For $l\neq 0$ the
operators $qA^{l-1}\tilde{q}$ and $pA^{p-l-1}\tilde{p}$ are related to
$b\tilde{b}$ due to the F-terms of the adjoint chiral fields. Therefore there
is only one independent operator that we map to the trace of any adjoint
chiral in the ${\cal C}_{p}[SU(N)]$ theory, that are all identified in the
chiral ring. Similarly, for $l=0$ the operator $sA^{p-1}\tilde{s}$ is set to
zero in the chiral ring due to F-term relations.

$\displaystyle l\neq
0\quad:\quad\begin{cases}R[pA_{L}^{j}\tilde{p}]=2-2\frac{l-j}{p+1}\quad\text{for}\quad
j=0,\ldots,l-2\\\
R[qA_{R}^{j}\tilde{q}]=2-2\frac{p-l-j}{p+1}\quad\text{for}\quad
j=0,\ldots,p-l-2\end{cases}\,,$ $\displaystyle l=0\quad:\quad
R[sA^{j}\tilde{s}]=2\frac{j+1}{p+1}\quad\text{for}\quad j=1,\ldots,p-2\,.$
(34)

These operators are mapped to the set of monopoles in ${\cal C}^{p}[SU(N)]$ as

$\displaystyle pA_{L}^{j}\tilde{p}\quad$
$\displaystyle\leftrightarrow\quad\mathfrak{M}_{R}^{\ldots-^{l-1-j}\ldots}\qquad
j=0,\ldots,l-2\,\,,$ $\displaystyle qA_{R}^{j}\tilde{q}\quad$
$\displaystyle\leftrightarrow\quad\mathfrak{M}_{L}^{\ldots-^{p-l-1-j}\ldots}\qquad
j=0,\ldots,p-l-2\,.$ (35)

For $l=0$ the map is given by

$\displaystyle sA^{j}\tilde{s}\quad$
$\displaystyle\leftrightarrow\quad\mathfrak{M}^{\ldots-^{p-1-j}\ldots}\qquad
j=0,\ldots,p-2\,.$ (36)

If we consider the ${\cal C}_{p}[U(N)]$ theories, that are defined as the
${\cal C}_{p}[SU(N)]$ theory with the extra BF coupling defined in (16), the
mirror $\check{{\cal C}}_{p}[U(N)]$ coincides exactly with $\check{{\cal
C}}_{p}[SU(N)]$, without extra BF couplings. We shall show this momentarily.

#### Deriving the mirror theories

The $\check{{\cal C}}_{[}SU(N)]$ mirror theories are derived as a deformation
of some known 3d $\mathcal{N}=4$ mirror pair. Indeed, as we have already
mentioned, if turn off all the monopole superpotentials in ${\cal
C}_{p}[SU(N)]$, then this coincides with some $T^{\sigma}_{\rho}[SU(N)]$
theory. We can then proceed by taking the mirrors of these $\mathcal{N}=4$
theories (which are simply obtained by swapping the partitions
$\sigma\leftrightarrow\rho$) and studying the effect of the monopole
superpotential across the duality.

#####  $l\neq 0$.

In this case the procedure to obtain the mirror theory is schematically
depicted below.

(37)

We start from the theory $T^{\rho}_{\sigma}[SU(N)]$ theory with
$\sigma=[1^{N}]$ and $\rho=[(r+1)^{l},r^{p-l}]$ in the first line of the
figure above. In the figure we have also specified the parameterization of the
global symmetries of the theory. The FI of the $j$-th gauge node is
$W_{j+1}-W_{j}$ while the labels beside the flavor symmetries are the
corresponding mass parameters. We pick the following assignation of charges:

$\displaystyle R[\text{hypers}]=\frac{1}{2}\quad,\quad
A[\text{hypers}]=-1\quad,\quad\eta[\text{hypers}]=-\frac{1}{2}\,,$
$\displaystyle R[\text{adjoint}]=1\quad,\quad
A[\text{adjoint}]=2\quad,\quad\eta[\text{adjoint}]=1\,.$ (38)

We have also introduced a redundant symmetry $U(1)_{\eta}$ which can be
reabsorbed by $U(1)_{A}$ transformations. This symmetry will be useful to
observe the discrete $\mathbb{Z}_{p+1}$ symmetry in the resulting theory after
the $\mathcal{N}=2$ deformation. Notice that the charge assignment is such
that the moment map on the Coulomb branch has charge $+1$ under $U(1)_{\eta}$.

In the mirror we have two sets of respectively $l$ and $p-l$ flavors rotated
by a flavor $S[U(l)\times U(p-l)]$ symmetry. We now want to turn on a
superpotential deformation which is the mirror of the monopole superpotential
in (8). It is convenient to first study the effect of turning on the monopoles
with magnetic fluxes $+1$ under all the balanced nodes, that is all nodes
except the $(p-l)$-th one. In the mirror, this corresponds to the next-to-
maximal nilpotent mass for both the mesons in the adjoint of $U(l)$ and
$U(p-l)$, with the effect of breaking them down to two $U(1)$ symmetries. The
result of the deformation is depicted in the second line of (37).

The breaking of symmetries due to this deformation is encoded in the following
specialization of the parameters:

$\displaystyle
Y_{j}=\frac{p-l+1-2j}{2}\left(i\frac{Q}{2}+2m_{A}+\eta\right)+V_{1}\qquad\text{for}\quad
j=1,\ldots,p-l\,,$ $\displaystyle
Y_{p-l+j}=\frac{l+1-2j}{2}\left(i\frac{Q}{2}+2m_{A}+\eta\right)+V_{2}\qquad\text{for}\quad
j=1,\ldots,l\,,$ (39)

where we recall that the set of $Y_{j}$ for $j=1,\ldots,p-l$ are the real mass
parameters for the $U(l)$ symmetry and the $Y_{j}$ for $j=p-l+1,\ldots,p$ are
those of the $U(p-l)$ symmetry. Moreover, $\eta$ is the real mass parameters
for the $U(1)_{\eta}$ symmetry, where notice that this can be completely
reabsorbed by a shift of $m_{A}$, the parameter for the axial $U(1)_{A}$
symmetry. Finally, $V_{1,2}$ are the parameters for the two $U(1)$’s left
after the nilpotent mass, which breaks $U(l)\to U(1)_{V_{2}}$ and $U(p-l)\to
U(1)_{V_{1}}$. However, we can always reabsorb an overall flavor $U(1)$
symmetry by a compensating gauge transformation, which in this case allows us
to get rid of the diagonal combination $V_{1}+V_{2}$, so that only the off-
diagonal $V_{1}-V_{2}$ is physical.

In the theory in the second line of (37) all the bifundamentals and adjoints
have the same R-charges as stated in (2.1), while the two flavors surviving
the nilpotent mass deformation have

$\displaystyle R[p,\tilde{p}]=1-\frac{l}{2}\quad,\quad
A[p,\tilde{p}]=-l\quad,\quad\eta[p,\tilde{p}]=-\frac{l}{2}\,,$ $\displaystyle
R[q,\tilde{q}]=1-\frac{p-l}{2}\quad,\quad
A[q,\tilde{q}]=-(p-l)\quad,\quad\eta[q,\tilde{q}]=-\frac{p-l}{2}\,.$ (40)

Turning on the monopole with magnetic flux $+1$ for the $(p-l)$-th node and
the one with magnetic flux $-1$ under all the gauge node simultaneously in the
electric theory corresponds to adding two triangle superpotentials in the
mirror, which break the aforementioned $U(1)_{V_{1}-V_{2}}$ and one
combination of the $\mathcal{N}=2$ R-symmetry and of the $U(1)_{A}$ axial
symmetry. We can compute how these two symmetries are broken starting from the
charges given in (2.1) and (2.1). The breaking condition written in term of
$S^{3}$ partition function variables is999Recall that the variable $\eta$ is
redundant, as in can be reabsorbed into $m_{A}$. However, we do not do this to
explicitly isolate the residual $\mathbb{Z}_{p+1}$ symmetry.

$\displaystyle V_{1}=V_{2}=0\,,$
$\displaystyle(p+1)(2m_{A}+\eta)=i\frac{Q}{2}(1-p)\,.$ (41)

From the above condition we see that $m_{A}$ is fixed as

$\displaystyle m_{A}=i\frac{Q}{2}\frac{1-p}{2(p+1)}\,,$ (42)

with $\eta$ a $\mathbb{Z}_{p+1}$ variable satisfying

$\displaystyle(p+1)\eta=0\,.$ (43)

We thus end up with the $\mathcal{N}=2$ theory depicted in the last line of
(37), with enhanced $SU(N)$ topological symmetry and a $\mathbb{Z}_{p+1}$
finite symmetry, that is the mirror $\check{{\cal C}}_{p}[SU(N)]$ theory.

#####  $l=0$.

In this case the procedure to obtain the mirror theory is schematically
depicted below.

(44)

Here we start from the $T^{\rho}_{\sigma}[SU(N)]$ theory with $\sigma=[1^{N}]$
and $\rho=[r^{p}]$ depicted in the first line of (44). From the partition
$\rho$ we see an $SU(p)$ symmetry which in the electric theory acts on the
Coulomb branch operators, which is enhanced from the topological symmetries of
the nodes which are all balanced. In the mirror we therefore have $p$ flavors
rotated by a flavor $SU(p)$ symmetry. Turning on the positive monopoles of all
the nodes in the electric theory maps to a next-to-maximal nilpotent mass for
the meson in the adjoint of such $SU(p)$, therefore breaking it completely.
The resulting theory is the one in the second line of (44), where the naive
$U(1)$ flavor symmetry can be as usual reabsorbed with a gauge transformation.
Now the dressed monopole with magnetic flux $-1$ under all nodes maps to the
quartic coupling between the flavor and the bifundamental $\delta{\cal
W}=q\tilde{q}b\tilde{b}$, which breaks a combination of the $\mathcal{N}=2$
R-symmetry and of the axial symmetry. Finally, the superpotential
$\text{Flip}[\mathfrak{M}^{-\ldots-}]$ is mapped to $\text{Flip}[q\tilde{q}]$
in the mirror theory. We thus reach the last line of (44).

As a final comment, one can observe that the actual global symmetry of the
resulting theory has a $\mathbb{Z}_{p+1}$ factor whose presence can be
observed using the same strategy implemented in the previous paragraph for the
$l\neq 0$ case.

##### Comments on the BF couplings.

We now comment on the origin of the BF couplings (16) in the
$\mathcal{C}_{p}[U(N)]$ theories. In particular, we show that such a choice of
BF couplings implies that there are none in their mirror duals $\check{{\cal
C}}_{p}[U(N)]$. As we will see in the next section, this is the only step in
the derivation of the confining duality (23) of $\mathcal{C}_{p}[U(N)]$ in
which the BF couplings can change, so that there will be none on the WZ side
as well.

Let us start from the case $l\neq 0$. We follow the steps of (37) and trace
how the BF couplings change after these steps. We start from the ${\cal N}=4$
mirror pair between the theory in figure (11) and the top quiver in figure
(37). More precisely for this mirror duality to hold, one should include the
following BF couplings on the two sides:

$\displaystyle\text{electric:}\qquad\mathrm{e}^{2\pi
iY_{p}\sum_{j=1}^{pr+l}W_{j}}\,,$
$\displaystyle\text{magnetic:}\qquad\mathrm{e}^{2\pi
i\left((\sum_{a=1}^{l}Y_{p-l+a})(\sum_{j=1}^{(p-1)r+l-1}W_{j})+(\sum_{a=1}^{p-l}Y_{a})(\sum_{j=1}^{(p-1)r+l}W_{j})\right)}\,.$
(45)

These can be derived using the mirror dualization algorithm Hwang:2021ulb ;
Comi:2022aqo . The general rule is that in the electric theory the flavor mass
parameters couple to the FI’s for the nodes on their right, while in the
mirror we have the opposite. Applying the specializations in (2.1) and (42),
the BF couplings become

$\displaystyle\text{electric:}\qquad\mathrm{e}^{2\pi
i\frac{iQ}{2}\frac{1-l}{p+1}\sum_{j=1}^{pr+l}W_{j}}\,,$
$\displaystyle\text{magnetic:}\qquad\mathrm{e}^{0}\,.$ (46)

For the $l=0$ case the ${\cal N}=4$ mirror pair involves the BF couplings

$\displaystyle\text{electric}\qquad\mathrm{e}^{2\pi
iY_{p}\sum_{j=1}^{pr}W_{j}}\,,$
$\displaystyle\text{magnetic}\qquad\mathrm{e}^{2\pi
i\left((\sum_{a=1}^{l}Y_{a})(\sum_{j=1}^{(p-1)r}W_{j})\right)}\,.$ (47)

Implementing the specializations we obtain

$\displaystyle\text{electric}\qquad\mathrm{e}^{2\pi
i\frac{iQ}{2}\frac{1-p}{p+1}\sum_{j=1}^{pr}W_{j}}\,,$
$\displaystyle\text{magnetic}\qquad\mathrm{e}^{0}\,.$ (48)

From this analysis we thus see that including the BF couplings (16) in the
definition of the $\mathcal{C}_{p}[U(N)]$ theories results in having no BF
coupling in their mirrors $\check{\mathcal{C}}_{p}[U(N)]$.

###  2.2 Derivation of the 3d confining duality for ${\cal C}_{p}[SU(N)]$

In this section we provide a derivation of the confining duality (23) for the
${\cal C}_{p}[SU(N)]$ theory.

####  2.2.1 Inductive derivation

The derivation is inductive and is based on the following two preliminary
results, which we will in turn derive in the following sections.

##### Inductive step.

If $N=pr+l>p+1$, the ${\cal C}_{p}[SU(N)]$ theory enjoys a duality in terms of
a model composed of a ${\cal C}_{p}[SU(N-1)]$ theory plus singlets

(49)

where the $SU(N)$ moment map and the monopole operators on the r.h.s. are
primed in order to distinguish them from those on the l.h.s. On the r.h.s. of
(49) the superpotential contains all the possible couplings between $a$, $b$,
$\tilde{b}$, $\mu^{\prime}$ of degree $p+1$, such that $b$ and $\tilde{b}$
appear with the same powers. Moreover, we have a flipping term for the
${\mathfrak{M}^{\prime}}_{L}^{\ldots-^{p-l}\ldots}$ operator. Notice that for
$l=0,1$ such operator does not exists in the ${\cal C}_{p}[SU(N-1)]$
theory,101010More precisely, for $l=0$ it does not exist since the length of
the quiver is only $p-1$, while for $l=1$ it exists but it is not part of the
chiral ring since it is already turned on in the superpotential. therefore we
do not have such flipping term and thus there is no extra singlet on the
r.h.s. of (49). We also point out that the action of the duality on the gauge
ranks $N=pr+l$ and $N^{\prime}=N-1=pr^{\prime}+l^{\prime}$ implies that for
$l\neq 0$ we have $r^{\prime}=r$ and $l^{\prime}=l-1$, while for $l=0$ we have
$r^{\prime}=r-1$ and $l^{\prime}=p-1$.

For $l\neq 0$ the map of the operators in the duality (49) is

$\displaystyle\mu$
$\displaystyle\quad\longleftrightarrow\quad\left(\begin{array}[]{c|c}\mu^{\prime}&b\\\
\hline\cr\tilde{b}&a\end{array}\right)\,,$ (52)
$\displaystyle\mathfrak{M}_{L}^{\ldots-^{j}\ldots}$
$\displaystyle\quad\longleftrightarrow\quad{\mathfrak{M}^{\prime}}_{L}^{\ldots-^{j}\ldots}\,,$
$\displaystyle\mathfrak{M}_{R}^{\ldots-^{j}\ldots}$
$\displaystyle\quad\longleftrightarrow\quad\\{{\mathfrak{M}^{\prime}}_{R}^{\ldots-^{j}\ldots},{\cal
F}[{\mathfrak{M}^{\prime}}_{L}^{\ldots-^{p-l}\ldots}]\\}\,.$ (53)

Let us comment on the last two lines of the map. The set of
$\mathfrak{M}_{L}^{\ldots-^{j}\ldots}$ monopoles in the second line always
contains one less element with respect to the dual
${\mathfrak{M}^{\prime}}_{L}^{\ldots-^{j}\ldots}$ set, however the extra
operator ${\mathfrak{M}^{\prime}}_{L}^{\ldots-^{p-l}\ldots}$ is flipped and so
$j=1,\ldots,p-l-1$. Conversely the set $\mathfrak{M}_{R}^{\ldots-^{j}\ldots}$
in the third line is always bigger by one unit than the dual set
${\mathfrak{M}^{\prime}}_{R}^{\ldots-^{j}\ldots}$, which is then completed
with the flipper the flipper ${\cal
F}[{\mathfrak{M}^{\prime}}_{L}^{\ldots-^{j}\ldots}]$ and so $j=1,\ldots,l-1$
where $j=l-1$ corresponds to the flipper. Notice in particular that this third
line is absent for $l=1$. Moreover, we stress that some of the monopole
operators involved in this map are composite, as we have already commented
below (29).

For $l=0$ the situation is simpler, since on the electric side we have a
single set of monopole operators $\mathfrak{M}^{\ldots-^{j}\ldots}$. The
operator map is then simply

$\displaystyle\mu$
$\displaystyle\quad\longleftrightarrow\quad\left(\begin{array}[]{c|c}\mu^{\prime}&b\\\
\hline\cr\tilde{b}&a\end{array}\right)\,,$ (56)
$\displaystyle\mathfrak{M}^{\ldots-^{j}\ldots}$
$\displaystyle\quad\longleftrightarrow\quad{\mathfrak{M}^{\prime}}_{R}^{\ldots-^{j}\ldots}\,,$
(57)

where $j=1,\ldots,p-2$ since on the l.h.s. the monopole
$\mathfrak{M}^{\ldots-^{p-1}\ldots}=\mathfrak{M}^{-\ldots-}$ is flipped and,
as we have already commented below (29), its flipper is not independent from
$\mathop{\text{Tr}}\nolimits(\mu^{p})$ and so it does not need to be mapped.
Notice also that on the r.h.s. the set $\mathfrak{M}_{L}^{\ldots-^{j}\ldots}$
is empty and therefore does not need to be mapped.

##### Inductive basis.

The basis of the inductive derivation consists in the confining duality (23)
for the ${\cal C}_{p}[SU(p+1)]$ theory, that is $N=p+1$ or equivalently $r=1$,
$l=1$

(58)

We will provide a derivation of the two dualities (49) and (58) in Sections
2.2.2 and 2.2.3, respectively, by iteration of the Aharony duality only, while
now we will use them to derive inductively the general confining duality (23).

##### Inductive derivation.

Let us start from a ${\cal C}_{p}[SU(N)]$ theory with $N=pr+l>p+1$. Applying
(49) we reach ${\cal C}_{p}[SU(N-1)]$ coupled to $SU(N-1)$ fundamentals and
flavor singlets. Applying (49) again to ${\cal C}_{p}[SU(N-1)]$ and
decomposing the other fields appropriately lowers the rank of the flavor
symmetry by another unit and produces fields which can be recombined with
those produced at the previous step into $SU(N-2)\times SU(2)$ bifundamentals
and an $SU(2)$ adjoint

(59)

From now on, to avoid cluttering, we do not give the exact superpotential at
each step. One can reconstruct it from that given in (49), recalling that all
the fields, including the adjoint operator of the ${\cal C}_{p}$ theories,
have R-charge $\tfrac{2}{p+1}$ by writing the most general superpotential
involving all the operators with R-charge 2 that are singlets under the
$SU(N)$ global symmetry. After $k$ iterations of (49) we reach

(60)

For $k=N-p-1$ we reach ${\cal C}_{p}[SU(p+1)]$ coupled to an $SU(p+1)\times
SU(N-p-1)$ bifundamental and an $SU(N-p-1)$ adjoint. The confined frame is
then obtained using the inductive basis (58): we are left with $SU(p+1)\times
SU(p(r-1)+l-1)$ bifundamentals $b,\tilde{b}$ and two adjoints $a,a^{\prime}$
which recombine in the $SU(N)$ adjoint $\Phi$

(61)

All the superpotential terms of degree $p+1$ reconstruct the superpotential
$\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$ plus all exactly marginal multi-
trace terms. The adjoint $\Phi$ in the confined frame is a $N\times N$ matrix
and it can be mapped across each step of the inductive derivation by
collecting together the adjoint operator $\mu$ of the ${\cal C}_{p}[SU(N-k)]$
theory, the $SU(N-k)\times U(k)$ bifundamentals $b,\tilde{b}$ and the $U(k)$
adjoint $a$ as:

$\displaystyle\Phi_{N\times
N}=\left(\begin{array}[]{c|c}\mu_{(N-k)\times(N-k)}&b_{k\times(N-k)}\\\
\hline\cr\tilde{b}_{(N-k)\times k}&a_{k\times k}\end{array}\right)\,.$ (64)

##### Comments on the flipping fields.

The derivation we have just presented was schematic and missed some important
details, in particular the flipping fields
$F[\mathop{\text{Tr}}\nolimits(\Phi^{j})]$ that are produced at each step.
These can be efficiently tracked (including their charges under the global
symmetries) by performing the derivation at the level of the $S^{3}$ partition
function, but here we will only make some comments on their pattern.

Each inductive step produces, in addition to the bifundamentals $b,\tilde{b}$
and the adjoint $a$ fields discussed above, an extra singlet
$F[{\mathfrak{M}^{\prime}}_{L}^{\ldots-^{p-l}\ldots}]$ that has R-charge
$2-\frac{2l}{p+1}$ for $l\neq 0,1$. After a generic number of iterations some
of the singlets have R-charges that sum up to 2. Hence, there are quadratic
deformations that are exactly marginal, so if we are on a generic point of the
conformal manifold these will be activated with the effect of making the
singlets massive. We claim that starting from $N=pr+l$ and reaching $N=p+1$
(the base of the induction) we produce exactly the flipping terms in the
superpotential as in (23).

More precisely, we can see the following general pattern. The singlet produced
for a given $l>1$ becomes massive together with the singlet produced for
$l^{\prime}=p+1-l$ which has R-charge $2-\frac{2(p+1-l)}{p+1}=\frac{2l}{p+1}$.
We can then see that if we do $p$ iterations we produce the set of singlets

$\displaystyle\\{2-\frac{2j}{p+1}\,|\,j=2,\ldots,p-1\\}=\\{2-\frac{4}{p+1},2-\frac{6}{p+1},\ldots,\frac{6}{p+1},\frac{4}{p+1}\\}\,.$
(65)

The first singlet of the set acquires a mass together with the last one, same
for the second singlet and the second-to-last, and so on. Therefore we
conclude that $p$ iterations do not produce any singlet, after taking into
account mass relations. Suppose now that we are trying to prove the
confinement for $N=pr+l$. To reach the base of the induction ${\cal
C}_{p}[SU(p+1)]$ we first perform $p(r-1)$ iterations to reach ${\cal
C}_{p}[SU(p+l)]$ which do not produce any singlet as explained above, and then
perform $l-1$ iterations to get to ${\cal C}_{p}[SU(p+1)]$. The produced
singlets will be of the form

$\displaystyle\\{2-\frac{2j}{p+1}\,|\,j=2,\ldots,l\\}\,.$ (66)

If $l<\lfloor p/2\rfloor$ one can notice that there are no massive singlets in
the set. However if $l\geq\lfloor p/2\rfloor$, then the $j=\lfloor
p/2\rfloor+k$ singlet will become massive together with $j^{\prime}=\lfloor
p/2\rfloor-k$. Therefore for generic $l$ we are left with the following set of
singlets:

$\displaystyle\\{2-\frac{2j}{p+1}\,|\,j=2,\ldots,\,\text{min}(l,p-l)\,\\}\,,$
(67)

which are exactly those presented in (23), both in number and in their charges
which are compatible with them flipping the $j$-th power of the adjoint chiral
in the confined frame.

####  2.2.2 Derivation of the inductive step

To prove the inductive step (49) we start from the mirror of the ${\cal
C}^{p}[SU(N)]$ theory, as depicted in (30). The leftmost node is $U(1)$ whose
adjoint chiral is simply a singlet, hence we can dualize it with the Aharony
duality. The main effects of the dualization are listed below, for more
details see Appendix A.1.

  * •

The dual gauge rank is $F-N_{c}$, where $F$ is the total number of flavors
seen by the dualized node whose rank is $N_{c}$. Notice that when we dualize a
balanced node $F=2N_{c}$ the rank does not change.

  * •

$F^{2}$ singlets are produced, which flip the adjoint mesonic operator
constructed from the flavors and maps to the meson $Q\tilde{Q}$ of the theory
before the dualization. If the $SU(F)$111111Notice that actually in Aharony
duality the global symmetry of the $U(N)$ SQCD with ${\cal W}=0$ is
$SU(F)^{2}$ rotating independently the fundamentals and anti-fundamentals (see
Appendix A.1). However, in all the calculations that we will perform in the
following sections the $SU(F)^{2}$ symmetries are always broken down to the
vector-like diagonal $SU(F)$ subgroup due to gaugings or superpotentials.
symmetry is broken, for example down to $SU(F_{1})\times SU(F_{2})$ with
$F_{1}+F_{2}=F$, one has to decompose the adjoint operator according to the
relevant branching rule, for example the $SU(F)$ adjoint decomposes in
$SU(F_{1})$ and $SU(F_{2})$ adjoints and two bifundamentals of
$SU(F_{1})\times SU(F_{2})$.

  * •

If the flavors are involved in a cubic superpotential, such as the
$\mathcal{N}=4$ superpotential $\mathcal{W}_{\text{cub}}=\Phi Q\tilde{Q}$,
after the dualization we get a quadratic superpotential for the singlet in the
adjoint of $SU(F)$ and the third field $\Phi$ involved in the interaction.
These fields are then massive and integrated out.

  * •

Two extra singlets that do not transform under the non-abelian global symmetry
are produced, these are flippers for the two monopoles of the dualized node.

After dualizing the first $U(1)$ node, this stays the same as it has two
flavors. However, from the second and third bullet point we observe that the
adjoint chiral of the $U(2)$ gauge node that is next to the one we dualized
gets flipped. Therefore we can now apply the Aharony duality again to this
second node and keep iterating this procedure. At each dualization, the
adjoint chiral of the following node is removed, while the one at the previous
node is restored. After we have dualized all the nodes in the balanced tail,
starting from $U(1)$ up to $U((p-1)r+l-2)$, we reach the following theory:

(68)

where the last dualized node is highlighted in orange. As usual, ${\cal
W}_{\text{cub}}$ encodes the cubic couplings between adjoint chirals and
bifundamentals on its sides. In addition we have ${\cal W}_{\text{quart}}$
which contains quartic superpotentials between the bifundamentals on the sides
of a gauge node without an adjoint chiral. These superpotential terms are
generated by Aharony duality when the adjoint chiral is removed. The mapping
of the monopoles in the quiver after each dualization can be worked out using
the results of Benvenuti:2020wpc  (see Appendix A.2 for a review). In
particular $\mathcal{W}_{\text{monopole-flip}}$ contains the contribution of
all the flippers of the monopoles generated by the iterations of the Aharony
duality.

We now dualize the following node with rank $(p-1)r+l-1$. This is not balanced
and its rank changes as

$\displaystyle[(p-1)r+l-2+(p-1)r]-[(p-1)r+l-1]=(p-1)r-1\,.$ (69)

The result of this dualization is

(70)

The flippers produced by the Aharony duality have the effect, as usual, of
flipping the adjoint of the following node and restoring the adjoint on the
previous one. In addition the $q,\tilde{q}$ fields in (68) are flipped and the
$p,\tilde{p}$ fields in (70) are generated.

We then dualize the following node with rank $(p-1)r$. The rank of this node
changes as

$\displaystyle[(p-1)r-1+(p-1)(r-1)]-[(p-1)r]=(p-1)(r-1)\,,$ (71)

meaning that its rank decreased by $p-1$ units. We keep iterating the Aharony
duality towards the right, each time decreasing the rank of the dualized node
by $p-1$ units. Therefore when we perform the Aharony duality on the last
$U(p-1)$ gauge node it confines. We thus reach the final frame

(72)

All in all the effect of iterating the Aharony duality is to shorten the
quiver by one node, resulting in the mirror $\check{{\cal C}}_{p}[SU(N-1)]$
theory with the addition of an extra singlet $F[q\tilde{q}]$ which corresponds
to the flipper $F[\mathfrak{M}^{\ldots-^{p-l}\ldots}]$ on the r.h.s. of figure
(49) obtained by mirroring back to ${\cal C}_{p}[SU(N-1)]$, as can be read
from the operator map in (2.1.2). In addition all the monopole flippers
generated by the iterations of the Aharony duality reconstruct the
$SU(N-1)\times U(1)$ bifundamentals $b,\tilde{b}$. Finally, the adjoint of the
first dualized node, which is a gauge singlet, is the $a$ field. Notice that
when the last node is confined, all the possible superpotential couplings will
be generated in $\mathcal{W}_{\text{inductive}}$, reconstucting the
superpotential in (49).

For $l=0,1$ there are subtleties related to the fact that for $l=0$ the
$\check{{\cal C}}^{p}[SU(N)]$ theory is different. However, a slight
modification of the same strategy we just presented provides the expected
result also for these cases.

####  2.2.3 Derivation of the inductive basis

We are left to prove the base of the induction, which is the case $r=l=1$ of
(23). The sketch of the derivation is given below in (73) and it involves the
one- and two-monopole Aharony dualities introduced in Benini:2017dud  (see
also Appendix A.1).

(73)

We start from the ${\cal C}_{p}[SU(p+1)]$ theory, whose quiver description is
depicted in the first line of (73). Focusing on the first $p-2$ nodes we see
that each of them has the fundamental positive monopole linearly turned on.
Starting from the leftmost $U(1)$ node we can use iteratively the one-monopole
duality in the confining case relating $U(N)$ with $N+1$ flavors and
$\mathcal{W}=\mathfrak{M}^{+}$ to a WZ model with $(N+1)^{2}$ singlets
$M^{i}{}_{j}$ and a singlet $\gamma$ with $\mathcal{W}=\gamma\det M$, to
sequentially confine the first $p-2$ nodes. This is because, similarly to the
previous derivation of the inductive step, the $U(1)$ adjoint is a singlet and
we can indeed use the one-monople duality which confines it in a WZ removing
the adjoint of the next $U(2)$ node, so that the duality can be iterated.

The sequential confinement of the first $p-2$ nodes, due to the positive
monopoles at each node, is explained in great details in Section $3.1$ of
Aprile:2018oau  (see also Hwang:2020wpd ) and yields a $U(p-1)$ theory with
$p+1$ flavors and
$\mathcal{W}=\sum_{j=1}^{p-1}\gamma_{j}\mathop{\text{Tr}}\nolimits((q\tilde{q})^{j})+\mathfrak{M}^{+}+\mathfrak{M}^{-}$
. The flippers of the powers of the meson are produced by the $p-2$
applications of the one-monopole duality and their effect is to break the non-
abelian flavor symmetry from $SU(N+1)\times SU(N+1)$ to $SU(N+1)$, while the
monopole superpotential is the residual of the $\mathcal{C}_{p}[SU(p+1)]$
theory original superpotential and breaks the abelian axial and topological
symmetries. Notice in particular that as explained in Benvenuti:2020wpc  (see
Appendix A.2) the monopole with negative charge under all nodes is shortened
to $\mathfrak{M}^{-}$.

The last step is performed by applying the confining two-monople duality
relating $U(N)$ with $N+2$ flavors and
$\mathcal{W}=\mathfrak{M}^{+}+\mathfrak{M}^{-}$ to a WZ model with $(N+2)^{2}$
singlets $M^{i}{}_{j}$ and $\mathcal{W}=\det M$. Hence, we obtain that the
$\mathcal{C}_{p}[SU(p+1)]$ theory confines to an $SU(p+1)$ adjoint with ${\cal
W}=\det\Phi+\sum_{j=1}^{p-1}\gamma_{j}\mathop{\text{Tr}}\nolimits(\Phi^{j})$.

One can rewrite the superpotential in terms of the traces of $\Phi$ and use
the F-term relations to simplify the superpotential expression. Indeed, in
(58) we are considering the ${\cal C}_{p}[SU(p+1)]$ theory at a generic point
of the conformal manifold, where ${\cal W}_{\text{multi-monopole}}$ is turned
on. Correspondingly, on the r.h.s. we turn on all the exactly marginal
deformations. These terms include the product of two flipping fields
$F[\mathop{\text{Tr}}\nolimits(\Phi^{j})]F[\mathop{\text{Tr}}\nolimits(\Phi^{p-1-j})]$.
Using the F-term equations all the flipping fields can be integrated out and
in turn we obtain the ${\cal W}_{\text{multi-trace}}$ superpotential, which
contains all the possible exactly marginal deformations obtained from the
product of two or more traces of the adjoint $\Phi$. Notice that still the
flipping field $F[\mathop{\text{Tr}}\nolimits(\Phi)]$ is present, so that
$\Phi$ is traceless even after solving the F-term equations. With all these
considerations taken into account we obtain the duality in (58) for $r=l=1$,
which is the inductive basis (58).

##  3 Derivation of the 3d ${\cal N}=2$ $U(N)$ Kim–Park duality via
deconfinement

In this section we use the confining duality of the 3d ${\cal N}=2$ theories
$\mathcal{C}_{p}[SU(N)]$ to provide a derivation the Kim–Park (KP) duality
Kim_2013  for $U(N)$ gauge group. Since this was derived using only the
Aharony duality and a mirror duality which can also be obtained from the
Aharony duality via the mirror dualization algorithm Hwang:2021ulb ;
Comi:2022aqo , it means that our derivation of the KP duality assumes only the
validity of the Aharony duality for $U(N)$ gauge group.

The KP duality relates 3d ${\cal N}=2$ unitary gauge theories with adjoint and
fundamental matter as follows:121212The theory has a continuous non-abelian
$SU(F)\times SU(F)$ flavor symmetry where each factor acts on $Q$ and
$\tilde{Q}$ independently. In order to simplify the notation in the quivers of
this section we identify these two $SU(F)$ into a single one and represent it
with a square box. However our discussion equally applies considering the full
$SU(F)\times SU(F)$ symmetry.

(74)

Notice that on the r.h.s. the flippers $F[q\phi^{j}\tilde{q}]$ are not singlet
under the non-abelian global symmetries and, indeed, they flip all the
components of the $q\phi^{j}\tilde{q}$ matrix. These singelts are also
included in the picture as arcs. In the case $p=1$, the adjoint chirals are
massive on both sides and we recover the Aharony duality. The mapping of the
chiral ring generators reads

$\left\\{\begin{tabular}[]{c}$Q\Phi^{j}\tilde{Q}$\\\
$\mathfrak{M}^{\pm}_{\Phi^{j}}$\\\
$\mathop{\text{Tr}}\nolimits(\Phi^{j})$\end{tabular}\right\\}\qquad\longleftrightarrow\qquad\left\\{\begin{tabular}[]{c}$F[q\phi^{p-j}\tilde{q}]$\\\
$F[\mathfrak{M}^{\mp}_{\phi^{p-j}}]$\\\
$\mathop{\text{Tr}}\nolimits(\phi^{j})$\end{tabular}\right\\}\qquad\begin{tabular}[]{c}$j=0,1,\ldots,p-1$
\,,\\\ $j=0,1,\ldots,p-1$ \,,\\\ $j=2,3,\ldots,p-1$ \,.\end{tabular}$ (75)

where $\mathfrak{M}^{\pm}_{\Phi^{j}}$ denotes the monopole with magnetic flux
$\pm 1$ dressed with the $j$-th power of the adjoint field of the residual
$U(1)\times U(N-1)$ gauge group in the monopole backgroud, see e.g.
Cremonesi:2013lqa .

Given our construction of the ${\cal C}_{p}[SU(N)]$ theories, the proof of the
duality (74) assuming Aharony duality is rather straightforward. We first
deconfine the adjoint of the electric theory into a ${\cal C}_{p}[SU(N)]$,
then we sequentially dualize all the $p$ gauge nodes, this builds a quiver
which is a $U(pF-N)$ node attached to a ${\cal C}_{p}[SU(pF-N)]$, so we can
reconfine the ${\cal C}_{p}[SU(pF-N)]$ theory into an adjoint, obtaining the
magnetic side of the KP duality. We are going to discuss derivation in the
simpler case of $p=2$, $l=1$ first and then in full generality.

###  3.1 A simple example: $p=2$, $l=1$

Let us first discuss the simplest case, namely cubic superpotential and odd
rank of the gauge groups. Here only the S-confining duality for $U(r)$ with
$2r+1$ flavors, found in Bajeot:2023gyl , is needed. The 3d KP duality in this
case reads

(76)

Starting from the electric theory, we can use the confining duality (23) to
replace the adjoint $\Phi$ plus the $\mathop{\text{Tr}}\nolimits(\Phi^{3})$
interaction with a ${\cal C}_{2}[SU(2r+1)]$, which in turn is given by a
$U(r)$ theory with adjoint and $2r+1$ fundamentals. So the electric theory in
(76) is dual to the following quiver131313Notice that the $p=2$ case has no
conformal manifold. Therefore the deconfinement of the adjoint through the
${\cal C}_{2}[SU(N)]$ theory does not involve subtleties about exactly
marginal deformations or the presence of flipping fields. This will not be the
case for $p>2$, as we will discuss in the next subsection.

(77)

Now we dualize the right node, killing the $U(r)$ adjoint and producing the
first set of flippers

(78)

Notice that the monopole terms in the superpotential of (77)
$\mathfrak{M}^{+0}+\mathfrak{M}^{-0}$ _extended_ to
$\mathfrak{M}^{++}+\mathfrak{M}^{--}$, following the rules of
Benvenuti:2020wpc  reviewed in Appendix A.2.

Dualizing the left node in (78) produces the second set of flippers and
_shortens_ the two monopoles in the superpotential

(79)

We can at this point recognize the quiver of the theory ${\cal
C}_{2}[SU(2F-2r-1)]$ attached to a $U(2F-2r-1)$ node. Replacing the ${\cal
C}_{2}[SU(2F-2r-1)]$ theory with an adjoint $\phi$ with
$\mathop{\text{Tr}}\nolimits(\phi^{3})$ interaction by means of the confining
duality (23), we arrive at

(80)

which is the magnetic side of the duality (76).

In the last step, from (79) to (80), we claim that the monopoles
$\mathfrak{M}^{\pm 0}$ and $\mathfrak{M}^{\pm\pm}$ map to $\mathfrak{M}^{\pm}$
and $\mathfrak{M}^{\pm}_{\phi}$ respectively, leading to the superpotential in
(80). We have checked this claim by carefully keeping track of all the
specialized FI parameters and BF couplings discussed in Section 2.

###  3.2 General derivation

In this subsection we go through the steps involved in the proof of (74) for
generic $p$ and $N$.

Let us start from the electric theory, the $U(N)$ theory with one adjoint
$\Phi$, $F$ flavors $Q$, $\tilde{Q}$ and ${\cal
W}=\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. The first step is to use (23) to
deconfine the adjoint into a ${\cal C}_{p}[SU(N)]$ theory with the last flavor
node gauged as a $U(N)$ with $F$ flavors

(81)

In order to deconfine the adjoint chiral $\Phi$ with a ${\cal C}_{p}[SU(N)]$
theory we have added all the exactly marginal deformations given by multi
trace operators on the l.h.s. and by multi monopole operators on the r.h.s.,
as discussed below the confining duality in (23). In the following steps we
will not write the ${\cal W}_{\text{multi-monopole}}$ terms anymore to avoid
cluttering. In the last step, when we will reconfine the ${\cal C}_{p}[SU(N)]$
theory we will reintroduce them. We also added the correct flipping terms
required by the duality (23), recalling that given $N=pr+l$ we have

$\displaystyle k(l)=\min(l,p-l)\,.$ (82)

We then use the definition of the ${\cal C}_{p}[SU(N=pr+l)]$ theory, so that
the r.h.s. of (76) becomes141414Here we discuss the case $l\neq 0$. The case
$l=0$ is similar.

(83)

We recall that $\mathfrak{M}_{i}^{+}=\mathfrak{M}^{0^{i-1}+0^{p-i}}$ is the
monopole with charge $+1$ under the topological symmetry of the $i$-th node,
counting the nodes from the left.

Starting from the rightmost gauge node, which does not have the adjoint and
does not have the monopole superpotential, we dualize sequentially the quiver
using the Aharony duality. After the first dualization we get

(84)

where ${\cal W}_{\text{triangle}}$ encodes the cubic couplings associated to
the triangle. Also, the term ${\cal W}_{\text{quart}}$ encodes the quartic
superpotential between bifundamentals on the sides of the gauge nodes without
an adjoint chiral. The dualization extended the rightmost
$\mathfrak{M}^{+}_{i}$ monopole and the $\mathfrak{M}^{-\ldots-0}$ monopole,
according to the rules of Benvenuti:2020wpc  (see Appendix A.2), and produced
$F^{2}+2$ flippers.

The adjoint for the next to last node became massive, so now we can use the
Aharony duality on this node. Defining $s=F-r-1$ we get

(85)

We keep dualizing sequentially $l$ nodes in the quiver, from right to left.
This builds a tail on the right with nodes $U(js)$, $j=l,\ldots,2,1$. At each
step the flavor node moves left, and we produce a new set of $F^{2}+2$
flippers. Let us also comment about the monopole superpotential. Each
$\mathfrak{M}^{+}_{i}$ term in the superpotential will be extended at one step
and shortened at the next step. The $\mathfrak{M}^{-\ldots-}$ stays the same
until the last step, when it is shortened to $\mathfrak{M}^{0-\ldots-}$. The
flipped monopoles are extended at each step after the step where they have
been created. After $l$ iterations we reach the following quiver:

(86)

Now we have to dualize the node which was overbalanced in the original
definition of the ${\cal C}^{p}[SU(N)]$ theory. After $p$ more iterations of
the Aharony duality, we reach the following quiver:

(87)

The leftmost node has rank $ps+p-l=pF-pr-l=pF-N$. Moreover, we can recognize
the quiver and superpotential of ${\cal C}_{p}[SU(pF-N)]$ with its global
symmetry gauged as $U(pF-N)$ with $F$ flavors, plus $p(F^{2}+2)$ flippers.
When can then reconfine the ${\cal C}_{p}[SU(pF-N)]$ to an $SU(pF-N)$ adjoint
using (23)

(88)

In the last step we have also reintroduced explicitly the exactly marginal
deformations that were present in the starting point (81).

Some comments regarding the result are in order. First of all let us remark
that KP duality relates a $U(N)$ gauge group with a $U(pF-N)$. If we take
$N=pr+l$ then $pF-N=pr^{\prime}+l^{\prime}$ with $r^{\prime}=F-r-1$ and
$l^{\prime}=p-l$. The formula (82), which determines the set of flippers, is
invariant under $l\to p-l$, therefore the flippers that we have on the l.h.s.
in the first step (81) are the same as those on the r.h.s. of the last step
(88). Moreover, one can check that the list of all the possible exactly
marginal deformations generated by multi-monopole operators is the same for a
${\cal C}_{p}[SU(N=pr+l)]$ and ${\cal
C}_{p}[SU(N^{\prime}=pr^{\prime}+l^{\prime})]$ if $l^{\prime}=p-l$. We
therefore claim that starting from the ${\cal C}_{p}[SU(N)]$ theory on the
r.h.s. of (81) at the most generic point of the conformal manifold, after
performing the steps described above, we get to the ${\cal C}_{p}[SU(pF-N)]$
theory on the l.h.s. of (88) which, again, has all the exactly marginal
deformations turned on.

To conclude, let us stress one more time that our derivation of the KP duality
assumes that we are at a generic point of the conformal manifold. It would be
interesting to understand if our technique could be used to probe how the
conformal manifolds of the two dual theories are related. However, we leave
this problem to future investigation.

##  4 Relation to the 4d $\mathcal{N}=2$ $D_{p}[SU(N)]$ SCFTs

In this section we discuss the relation between the family of 3d
$\mathcal{N}=2$ $\mathcal{C}_{p}[SU(N)]$ theories and the dimensional
reduction of a class of 4d $\mathcal{N}=2$ SCFTs, called $D_{p}[SU(N)]$
theories Cecotti:2012jx ; Cecotti:2013lda ; Wang:2015mra . In particular, we
will discuss how the confining duality (23) enjoyed by the
$\mathcal{C}^{p}[SU(N)]$ theories that we presented in Section 2 can be
understood as the three-dimensional version of the result of Maruyoshi:2023mnv
, that a 4d $\mathcal{N}=1$ chiral in the adjoint $\Phi$ of an $SU(N)$
symmetry with superpotential
$\mathcal{W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$ can be deconfined
into a $D_{p}[SU(N)]$ theory with a suitable CB deformation when
$\gcd(p,N)=1$.

###  4.1 Review of $D_{p}[SU(N)]$ theories and their dimensional reduction

In this section we review some properties of the $D_{p}[SU(N)]$ theories and
of the Lagrangians for their circle reduction to 3d. The $D_{p}[SU(N)]$
theories are 4d $\mathcal{N}=2$ SCFTs that admit various different
realizations Cecotti:2012jx ; Cecotti:2013lda ; Wang:2015mra . They can for
example be obtained via geometric engineering of Type IIB string theory on
certain Calabi–Yau 3-fold isolated hypersurface singularities, or by
compactifying the 6d $(2,0)$ $A_{N}$-type SCFT on a sphere with one irregular
and one regular puncture, along the lines of the class $\mathcal{S}$
construction Gaiotto:2009we . Each realization has its utility for
understanding various properties of these theories.

All of these theories possess at least an $SU(N)$ flavor symmetry, which in
the construction via the 6d to 4d compactification is manifest in the regular
puncture. However, generically there can also be additional abelian flavor
symmetries, as well as sporadic symmetry enhancements. In general, the flavor
symmetry is at least Giacomelli:2017ckh

$\displaystyle G_{F}=SU(N)\times U(1)^{m-1}\,,\qquad m=\gcd(p,N)\,.$ (89)

The flavor central charge of the $SU(N)$ factor, which is related to the
anomaly with the $U(1)_{r}$ factor of the 4d $\mathcal{N}=2$ R-symmetry, is
given by Xie:2016evu

$\displaystyle k_{SU(N)}=-2\,\mathrm{Tr}\,r\,SU(N)^{2}=2N-\frac{2N}{p}\,.$
(90)

One can also consider the SCFTs obtained via a HB RG flow triggered by a
nilpotent vev, labelled by a partition $[Y]$ of $N$, for the $SU(N)$ moment
map operator $\mu$. This can be understood as partially closing the regular
puncture in the class $\mathcal{S}$ construction. Following the notation of
Beem:2023ofp , we denote these theories by $D_{p}(SU(N),[Y])$ and simplify
this notation to $D_{p}(SU(N),[Y=1^{N}])=D_{p}[SU(N)]$ when the puncture is
maximal.

The CB spectrum of the theory is given by all the operators $u_{i,j}$ with
$0\leq i\leq p$ and $0\leq j\leq N-2$ whose scaling dimensions

$\displaystyle\Delta(u_{i,j})=N-j-\frac{N}{p}i$ (91)

are strictly greater than two. When $m=\gcd(p,N)=1$ this is simplified to
Maruyoshi:2023mnv

$\displaystyle\Delta(u_{i})=\frac{p+1+i}{p}\,,\qquad 0\leq i\leq p-2\,.$ (92)

In particular, the operator $u_{\text{min}}$ with the minimal scaling
dimension has

$\displaystyle\Delta(u_{\text{min}})=\frac{p+1}{p}\,.$ (93)

Another important piece of data that characterize these theories are the
conformal central charges. The $c$ central charge is given by Cecotti:2013lda
; Giacomelli:2017ckh

$\displaystyle
c_{\mathcal{N}=2}=\frac{1}{2}\mathrm{Tr}\,r\,I_{3}^{2}-\frac{1}{24}\mathop{\text{Tr}}\nolimits\,r^{3}=\frac{p-1}{12}(N^{2}-1)\,.$
(94)

where $I_{3}$ is the Cartan generator of the $SU(2)_{R}$ factor of the 4d
$\mathcal{N}=2$ R-symmetry. Together with the knowledge of the CB spectrum,
this allows us to determine also the $a$ central charge via the
Shapere–Tachikawa relation Shapere:2008zf

$\displaystyle
2c_{\mathcal{N}=2}-a_{\mathcal{N}=2}=\frac{1}{4}\sum_{i=1}^{r}(2\Delta_{i}-1)\,,$
(95)

where $r$ denotes the rank of the theory, i.e. the dimension of the CB, and
$\Delta_{i}$ is the scaling dimension of the $i$-th CB operator. For
$m=\gcd(p,N)=1$ one finds more explicitly

$\displaystyle
a_{\mathcal{N}=2}=\frac{1}{2}\mathrm{Tr}\,r\,I_{3}^{2}-\frac{1}{48}\mathop{\text{Tr}}\nolimits\,r^{3}=\frac{(4p-1)(p-1)}{48p}(N^{2}-1)\,.$
(96)

Finally, let us consider the dimensional reduction of the $D_{p}[SU(N)]$
theory on a circle, which yields a 3d $\mathcal{N}=4$ SCFT. Even though the 4d
theory is generically non-Lagrangian, it turns out that the 3d SCFT obtained
via the $S^{1}$ compactification admits a Lagrangian description in terms of a
linear quiver gauge theory. In particular, in Closset:2020afy ;
Giacomelli:2020ryy  it has been shown that when $m=\gcd(p,N)=1$ and focusing
on the case $p\leq N$ that is relevant to us, such quiver corresponds with the
one in (23) for the $\mathcal{C}_{p}[SU(N)]$ theory, but with the
superpotential given only by the cubic interaction $\mathcal{W}_{\text{cub}}$
without the monopole superpotential, so to have $\mathcal{N}=4$ supersymmetry.
As we have already pointed out previously, these coincide with the
$T^{\sigma}_{\rho}[SU(N)]$ theories Gaiotto:2008ak  for

$\displaystyle\sigma=[1^{N}]\,,\qquad\rho=[(r+1)^{l},r^{p-l}]\,,$ (97)

where we recall the notation $N=pr+l$ with $0\leq l\leq p-1$. The quiver for
the theory $D_{p}(SU(N),[Y])$ with a partially closed regular puncture, again
for $m=\gcd(p,N)=1$ and $p<N$, is simply obtained by taking $\sigma=[Y]$ in
the definition of $T^{\sigma}_{\rho}[SU(N)]$.

When $m=\gcd(p,N)\neq 1$ the situation is instead more complicated for the
following reason. In such case, the $D_{p}[SU(N)]$ theory generically has
(except for some special cases) $m-1$ exactly marginal deformations, and one
can find a point of the conformal manifold where the theory can be described
as a partially weakly coupled gauge theory, consisting of a linear sequence of
conformal gaugings via special unitary groups of theories of the same family
$D_{p_{i}}(SU(N_{i}),[Y_{i}])$ with $m_{i}=\gcd(p_{i},N_{i})=1$ and thus no
exactly marginal deformations. This is captured by the following expression
(which actually holds for any $p$ and $N$) Closset:2020afy ;
Giacomelli:2020ryy ; Beem:2023ofp :

$\displaystyle D_{p}[SU(N)]=D_{(m-1)q}(SU(N-n))\longleftarrow
SU(N-n)\longrightarrow D_{q}(SU(qN-n),[(q-1)^{N},1^{N-n}])\,,$ (98)

where we recall that $m=\gcd(p,N)$ and we defined

$\displaystyle q=\frac{p}{m}\,,\qquad n=\frac{N}{m}\,.$ (99)

The above expression means that we are gauging a diagonal $SU(N-n)$ subgroup
of the symmetries of the component theories. This gauging then leaves a
$SU(N)\times U(1)$ symmetry from the right theory, which also has
$\gcd(q,qN-n)=\gcd(q,n)=1$. The same is not true for the left theory, which
instead has $\gcd((m-1)q,N-n)=\gcd((m-1)q,(m-1)n)=m-1$. However, iterating
this relation for a total of $m-1$ times, we obtain that the $D_{p}[SU(N)]$
theory can be expressed as a linear sequence of $m-1$ gaugings of
$D_{p_{i}}(SU(N_{i}),[Y_{i}])$ theories with $m_{i}=\gcd(p_{i},N_{i})=1$.
Moreover, the flavor symmetry preserved by these gaugings is exactly the one
predicted in (89). We have just reviewed that the 3d quiver for the component
theories are linear unitary quivers. The observation of Closset:2020afy ;
Giacomelli:2020ryy  is that one can use (98) to express the 3d quivers for the
$D_{p}[SU(N)]$ theories with $m=\gcd(p,N)\neq 1$ as a linear quiver with mixed
unitary and special unitary gauge groups.

Let us consider for example the $D_{4}(SU(6))$ theory. In this case $m=2$ and
we have one exactly marginal deformation. Indeed, following (98) we can
express the theory as a gauging of two theories with no exactly marginal
deformations

$\displaystyle D_{4}(SU(6))=D_{2}(SU(3))\longleftarrow SU(3)\longrightarrow
D_{2}(SU(9))\,.$ (100)

The 3d reduction of $D_{2}(SU(3))$ is $U(1)$ SQED with 3 flavors, while that
of $D_{2}(SU(9))$ is $U(4)$ SQCD with 9 flavors. Gauging a diagonal
combination of the $SU(3)$ symmetry of the former theory and of the one acting
on 3 out of the 9 flavors of the latter theory, we obtain that the 3d
reduction of $D_{4}(SU(6))$ can be described by the quiver

(101)

###  4.2 Comparison between $\mathcal{C}_{p}[SU(N)]$ and 3d quivers for
$D_{p}[SU(N)]$

We will now discuss more in details the relation between the family of 3d
$\mathcal{N}=2$ $\mathcal{C}_{p}[SU(N)]$ quivers and those obtained from the
circle reduction of the $D_{p}[SU(N)]$ theories. For the entirety of this
section, as we did in the rest of the paper, we focus on $p\leq N$.

#####  $\gcd(p,N)=1$.

Let us begin from the case $\gcd(p,N)=1$. As we have just reviewed, the quiver
for the 3d reduction of $D_{p}[SU(N)]$ is given by the
$T^{\sigma}_{\rho}[SU(N)]$ theory with $\sigma=[1^{N}]$ and
$\rho=[(r+1)^{l},r^{p-l}]$, whose matter and gauge content turns out to
coincide with that of the $\mathcal{C}_{p}[SU(N)]$ theory. The only difference
is in the superpotential: the 3d reduction of $D_{p}[SU(N)]$ has
$\mathcal{N}=4$ supersymmetry and so it only has the cubic interaction
$\mathcal{W}_{\text{cub}}$ between the bifundamentals and the adjoint chirals,
while the $\mathcal{C}_{p}[SU(N)]$ theory as we defined it in Section 2 has
also a monopole superpotential which breaks supersymmetry down to
$\mathcal{N}=2$. Moreover, in the same section we have shown that such
monopole deformation of the $\mathcal{N}=4$ theory makes it flow to a WZ
theory, consisting of a chiral in the adjoint of the $SU(N)$ flavor symmetry
with interaction $\mathcal{W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. It
is then natural to wonder whether there exists a deformation of the
$D_{p}[SU(N)]$ theory directly in 4d that breaks supersymmetry from
$\mathcal{N}=2$ to $\mathcal{N}=1$ and that makes it flow to the same WZ
theory. Interestingly, such a deformation has been recently studied in
Maruyoshi:2023mnv  (see also Xie:2021omd ; Kang:2023dsa ; Bajeot:2023gyl  for
analogous previous observations), so let us briefly review their result.

As we reviewed above, all $D_{p}[SU(N)]$ theories with $\gcd(p,N)=1$ have a
relevant chiral CB operator with the minimal scaling dimension
$\Delta(u_{\text{min}})=(p+1)/p$. Turning this on in the superpotential breaks
supersymmetry from $\mathcal{N}=2$ to $\mathcal{N}=1$, since it breaks the
$\mathcal{N}=2$ R-symmetry $SU(2)_{R}\times U(1)_{r}$ to an $\mathcal{N}=1$
R-symmetry $U(1)_{R}$ that is a combination of the Cartan $U(1)_{I_{3}}\subset
SU(2)_{R}$ and of $U(1)_{r}$. The exact combination can be determined from the
knowledge of the scaling dimension of the operator $u_{\text{min}}$ and
recalling that for a 4d $\mathcal{N}=2$ CB operator $r=2\Delta$ and $I_{3}=0$

$\displaystyle
U(1)_{R}=\frac{1}{p+1}\left(2\,U(1)_{I_{3}}+p\,U(1)_{r}\right)\,.$ (102)

Since a moment map operator in a 4d $\mathcal{N}=2$ SCFT has $r=0$ and
$I_{3}=1$, one has that under such a symmetry the moment map operator $\mu$ of
$D_{p}[SU(N)]$ for the $SU(N)$ flavor symmetry has R-charge

$\displaystyle R[\mu]=\frac{2}{p+1}\,.$ (103)

This is mapped to the chiral operator $\Phi$ of the $\mathcal{N}=1$ theory
after the deformation that is in the adjoint of $SU(N)$ and couples as
$\mathcal{W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. Computing the
$\mathcal{N}=1$ $a$ and $c$ central charges Anselmi:1997am  by exploiting the
anomalies (90)-(94)-(96) of $D_{p}[SU(N)]$ and the embedding (102)151515We
also need that
$\mathrm{Tr}\,r=\mathrm{Tr}\,r^{3}=48(a_{\mathcal{N}=2}-c_{\mathcal{N}=2})$.

$\displaystyle a_{\mathcal{N}=1}$
$\displaystyle=\frac{3}{32}\left(3\mathrm{Tr}\,R^{3}-\mathrm{Tr}\,R\right)=-\frac{3(p-1)(p^{2}-4p+1)}{16(p+1)^{3}}(N^{2}-1)\,,$
$\displaystyle c_{\mathcal{N}=1}$
$\displaystyle=\frac{1}{32}\left(9\mathrm{Tr}\,R^{3}-5\mathrm{Tr}\,R\right)=-\frac{(p-1)(p^{2}-7p+1)}{8(p+1)^{3}}(N^{2}-1)\,,$
(104)

one also finds that these coincide with those of a chiral in the adjoint of
$SU(N)$ and with R-charge $2/(p+1)$. An additional check of this claim that
was done in Maruyoshi:2023mnv  was at the level of a limit of the index, which
is related to the Schur limit Gadde:2011uv  of the index of the $D_{p}[SU(N)]$
theory Xie:2016evu ; Song:2017oew .

The confining dualities for the 3d $\mathcal{C}_{p}[SU(N)]$ theories that we
discussed in Section 2 thus represent an additional check for the result of
Maruyoshi:2023mnv  about the 4d $D_{p}[SU(N)]$ theories. When compactifying a
4d $\mathcal{N}=2$ theory on a circle of radius $\beta$, we obtain a 3d
effective theory with KK modes whose CB is obtained from the 4d CB with $2r$
additional compact directions, where we recall that $r$ is the rank of the 4d
theory. Flowing to energies smaller than $\frac{1}{\beta}$ the KK tower is
integrated out and these directions decompactify, so that we obtain a 3d CB
whose complex dimension is twice that of the 4d CB. Since the CB of the 3d
theory is parametrized by monopole operators, it is then natural to expect
that the 4d CB operator $u_{\text{min}}$ descends to a monopole operator in
the 3d $\mathcal{N}=4$ quiver theory for the dimensional reduction of
$D_{p}[SU(N)]$, although determining the precise mapping between the 4d and
the 3d CB operators is difficult due to the fact that the 4d theory is non-
Lagrangian. Moreover, the monopole superpotential of $\mathcal{C}_{p}[SU(N)]$
consists of many terms, while the deformation of $D_{p}[SU(N)]$ that makes it
flow to a WZ model is just given by one operator $u_{\text{min}}$. The
generation of monopole superpotentials in the dimensional reduction of 4d
$\mathcal{N}=1$ theories to 3d $\mathcal{N}=2$ theories is actually quite
common, see e.g. Aharony:2013dha ; Aharony:2013kma , but repeating a similar
analysis here is difficult again due to the non-Lagrangian nature of the
$D_{p}[SU(N)]$ theory. However, we have observed that the role of the monopole
superpotential of $\mathcal{C}_{p}[SU(N)]$ is to set the R-charge of the
$SU(N)$ moment map, consisting of the meson matrix constructed with the
flavors at the end of the tail of the quiver (23), to $2/(p+1)$, which is
exactly what happens also in 4d (103). Moreover, the monopole superpotential
also breaks the topological symmetries except for a residual
$\mathbb{Z}_{p+1}$, which is also the only other symmetry present in 4d on top
of $SU(N)$ and $U(1)_{R}$. Hence, we expect the full monopole superpotential
of $\mathcal{C}_{p}[SU(N)]$ to be present in the dimensional reduction of
$D_{p}[SU(N)]$ deformed by $u_{\text{min}}$ in order to match the symmetries
and the charges between the 4d and the 3d theories.

#####  $\gcd(p,N)\neq 1$ and $N\neq pr$.

Let us move now to the case where $\gcd(p,N)\neq 1$, but excluding the
possibility that $N$ is an integer multiple of $p$, that is $l\neq 0$. The 3d
$\mathcal{C}_{p}[SU(N)]$ theory is still given by the
$T^{\sigma}_{\rho}[SU(N)]$ theory with $\sigma=[1^{N}]$ and
$\rho=[(r+1)^{l},r^{p-l}]$ deformed by the monopole superpotential, but now
the 3d reduction of $D_{p}[SU(N)]$ is a linear quiver with mixed unitary and
special unitary nodes. However, we can find a deformation of the latter quiver
that makes it flow to the $T^{\sigma}_{\rho}[SU(N)]$ theory.

Consider for example the case $p=4$ and $N=6$, or equivalently $r=1$ and
$l=2$. As we have seen above, the quiver for the 3d reduction of
$D_{4}(SU(6))$ is given by the quiver in (101). On the other hand, the 3d
$\mathcal{N}=4$ theory obtained from $\mathcal{C}_{4}[SU(6)]$ by removing the
monopole superpotential is the $T^{[1^{6}]}_{[2^{2},1^{2}]}[SU(6)]$ theory,
whose quiver is

(105)

Nevertheless, we observe that the quiver (105) can be obtained from (101) by
replacing the $SU(3)$ gauge node, which is connected to 5 flavors, with a
$U(2)$ gauge node with the same number of flavors.

This is a general feature that persists for other values of $p$ and $N$: the
3d quiver of $D_{p}[SU(N)]$ generically has $m-1$ special unitary gauge nodes
and the $i$-th node is $SU(N-in)$ with $2(N-in)-1$ flavors (recall that
$m=\gcd(p,N)$ and $n=N/m$), and the 3d $\mathcal{N}=4$ quiver associated to
$\mathcal{C}_{p}[SU(N)]$ by removing the monopole superpotential has such
nodes replaced by $U(N-in-1)$ nodes again with $2(N-in)-1$ flavors. As shown
in Closset:2020afy , there exists in general a deformation that makes us flow
from a $SU(M)$ theory with $2M-1$ flavors to a $U(M-1)$ theory with $2M-1$
flavors. This consists of a vev for the $SU(M)$ adjoint chiral of the form

$\displaystyle\langle\Phi\rangle=\mathrm{diag}(\mathfrak{m},\cdots,\mathfrak{m},-(M-1)\mathfrak{m})\,,$
(106)

which also induces an equal mass for the flavors $Q$, $\tilde{Q}$

$\displaystyle\delta\mathcal{W}=\mathfrak{m}\,\mathrm{Tr}_{U(2M-1)}\mathrm{Tr}_{SU(M)}Q\tilde{Q}\,,$
(107)

where $\mathrm{Tr}_{SU(M)}$ is the trace over the color indices while
$\mathrm{Tr}_{U(2M-1)}$ is over the flavor indices. Notice indeed that such a
vev breaks the $SU(M)$ gauge group down to the expected $U(M-1)$ subgroup.
This deformation can be studied in the mirror dual of the $SU(M)$ SQCD with
$2M-1$ flavors, where it corresponds to a vev for a meson which can be induced
by an FI deformation that can be easily analyzed. The result is precisely the
mirror dual of the $U(M-1)$ SQCD with $2M-1$ flavors Closset:2020afy .

After such a deformation of the 3d quiver of $D_{p}[SU(N)]$, the unitary nodes
that were not directly affected by the deformation might have become ugly, so
they need to be dualized with the effect that their rank is reduced by one and
a free twisted hyper is produced Gaiotto:2008ak . It is only after such
dualizations that we recover the 3d $\mathcal{N}=4$ unitary linear quiver
corresponding to $\mathcal{C}_{p}[SU(N)]$ but with the monopole deformation
removed. This did not happen for the example of the $D_{4}(SU(6))$ theory that
we have seen before, but it is usually the case when $m>2$ for the unitary
nodes that are in between two special unitary nodes in the 3d quiver of
$D_{p}[SU(N)]$.

Consider for example the case $p=6$ and $N=9$, for which $m=\gcd(6,9)=3$. The
quiver for the 3d reduction of $D_{6}(SU(9))$ is

(108)

which reflects the fact that, according to (98), $D_{6}(SU(9))$ can be
understood as the $SU(6)$ gauging of $D_{4}(SU(6))$ and $D_{2}(SU(15))$.
Turning on the deformation (106) at the two special unitary nodes we obtain
the quiver

(109)

As anticipated, the $U(4)$ node that used to be in between the special unitary
nodes that we deformed has 7 flavors and is thus ugly. If we dualize it, we
produce a free twisted hyper and obtain the quiver

(110)

This is precisely the $T^{[1^{9}]}_{[2^{3},1^{3}]}[SU(9)]$ theory that is
related to $\mathcal{C}_{6}[SU(9)]$ upon turning on the monopole
superpotential deformation

Summarizing, the 3d $\mathcal{N}=2$ $\mathcal{C}_{p}[SU(N)]$ theory for
$\gcd(p,N)\neq 1$ and $N\neq pr$ can be obtained from the 4d $\mathcal{N}=2$
$D_{p}[SU(N)]$ SCFT as follows:

  1. 1.

compactify the $D_{p}[SU(N)]$ theory on a circle so to get a 3d
$\mathcal{N}=4$ SCFT that is the IR limit of a 3d $\mathcal{N}=4$ linear
quiver with mixed unitary and special unitary gauge nodes;

  2. 2.

turn on deformations of the form (106) which effectively replace each special
unitary gauge node $SU(M)$ of this quiver with a unitary gauge node $U(M-1)$
of the same rank;

  3. 3.

dualize each ugly node $U(M)$ with $2M-1$ flavors into the good node $U(M-1)$
with $2M-1$ flavors plus a free twisted hyper;

  4. 4.

turn on a suitable monopole superpotential.

In particular, the last deformation breaks supersymmetry from $\mathcal{N}=4$
to $\mathcal{N}=2$ and makes the theory flow to a WZ model consisting of an
adjoint chiral $\Phi$ for the $SU(N)$ flavor symmetry with superpotential
$\mathcal{W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$.

##### Some open questions.

We conclude this section by commenting on some open questions about the
relationship between the 3d $\mathcal{N}=2$ ${\cal C}_{p}[SU(N)]$ theories and
the 4d $\mathcal{N}=2$ $D_{p}[SU(N)]$ SCFTs, which would be interesting to
further investigate in future works.

We have seen that for $\gcd(p,N)=1$ the 3d reduction of $D_{p}[SU(N)]$ flows
to ${\cal C}_{p}[SU(N)]$ upon turning on the monopole superpotential
deformation, which is consistent with the fact that both ${\cal C}_{p}[SU(N)]$
and $D_{p}[SU(N)]$ deformed by the CB operator $u_{\text{min}}$ flow to a WZ
theory. However, for $\gcd(p,N)\neq 1$ and $N\neq pr$ an additional
deformation, corresponding to the steps 2. and 3. listed above, is needed. It
would be interesting to understand whether these operations can be uplifted to
4d, so to find a deformation of $D_{p}[SU(N)]$ that makes it flow to a WZ
model which generalizes the result of Maruyoshi:2023mnv  to the case of
$\gcd(p,N)\neq 1$.

Another open question is the relationship between $\mathcal{C}_{p}[SU(N)]$ and
$D_{p}[SU(N)]$ when $N=pr$. In this case, $\mathcal{C}_{p}[SU(N)]$ is given by
a sequence of balanced gauge nodes $U(jr)$ for $j=1,\ldots,p-1$, while
$D_{p}[SU(N)]$ is fully Lagrangian already in 4d and given by a similar quiver
but where the gauge nodes are special unitary $SU(jr)$ rather than unitary
(see e.g. Cecotti:2013lda ; Giacomelli:2017ckh ). Hence, the two quivers
cannot be related by the deformation (106) as for $N\neq pr$.

Finally, in our discussion we focused on the quivers of these theories and we
neglected possible gauge singlet fields. In fact, the confining duality for
the $\mathcal{C}_{p}[SU(N)]$ theory also predicts the presence of singlet
chirals flipping traces of powers of the adjoint field $\Phi$, which are not
present in the analogous duality for $D_{p}[SU(N)]$. On the other hand, the 3d
reduction of $D_{p}[SU(N)]$ generically also has some decoupled twisted
hypers, which are in number Giacomelli:2020ryy

$H_{\text{free}}=\frac{(N-m)(p-N-m)}{2m}\,,$ (111)

where we recall that $m=\gcd(p,N)$. These are in particular crucial in order
to match the CB dimension between 4d and 3d. Moreover, they encode the fact
that generically on the most generic point of the HB of $D_{p}[SU(N)]$ we do
not have a collection of free hypers but also some non-Higgsable (i.e. with
trivial HB) SCFT, such as the $(A_{1},A_{2})$ AD theory, which in 3d reduces
just to twisted hypers. It would be interesting to understand how to reconcile
the presence of the singlet chirals in the confining duality of ${\cal
C}_{p}[SU(N)]$ and of the twisted hypers in the 3d reduction of
$D_{p}[SU(N)]$.

##  5 The 4d $\mathcal{N}=1$ $\mathsf{C}_{p}[USp(2N)]$ family

In this section we introduce the 4d $\mathcal{N}=1$ $\mathsf{C}_{p}[USp(2N)]$
theories, where $N$ and $p$ are positive integers such that $N>p$. We show
that the $\mathsf{C}_{p}[USp(2N)]$ theories confine to a WZ model of a chiral
$\Phi$ in the traceless antisymmetric representation of $USp(2N)$ with
superpotential ${\cal W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$. Upon a
circle reduction and suitable deformations, they reduce to the ${\cal
C}_{p}[SU(N)]$ theories we introduced in Section 2 (see e.g. Benini:2017dud ;
Amariti:2018wht ; Pasquetti:2019hxf ; Bajeot:2022lah ; Bottini:2021vms ;
Comi:2022aqo ; Bajeot:2023gyl  for similar dimensional reduction limits), and
their respective confining dualities are similarly related.

###  5.1 The $\mathsf{C}_{p}[USp(2N)]$ family

The $\mathsf{C}_{p}[USp(2N)]$ theories are 4d ${\cal N}=1$ quiver theories
with $p-1$ $USp(2n_{j})$ gauge nodes and a $USp(2N)$ flavor
symmetry.161616Unlike what we did in Section 2, here for simplicity we ignore
possible finite symmetries and only focus on the Lie algebras of the global
symmetries. As in Section 2, we distinguish two situations depending on
whether $N$ is a multiple of $p$ or not. It is then useful to define

$\displaystyle N=pr+l\,,\qquad l=0,\ldots,p-1\,.$ (112)

The ranks $n_{j}$ of the gauge groups are determined as in (4).

#####  $l\neq 0$.

In this case the $\mathsf{C}_{p}[USp(2(pr+l))]$ theory can be depicted
as171717In this section and in the following, where we work solely in four
dimensions, we adopt a different notation for quivers than what we did in 3d.
Each node, round or square, denotes a gauge or flavor symplectic group. Lines
are chirals in the fundamental representation of the nodes to whom they are
linked. Square nodes labelled with $1$ do not represent flavor symmetries and
are used only to encode the number of chiral fields. Unless stated otherwise,
arcs are traceful antisymmetric tensors and crosses denotes flipping fields.

(113)

In analogy with Section 2, we call a $USp(2N_{c})$ gauge node _balanced_ if it
sees exactly $2F=4N_{c}$ fundamental chirals, counting only the horizontal
bifundamentals and not chirals in the saw. If instead it sees $2F>4N$ it is
called _overbalanced_. In figure (113) all the gauge nodes are balanced except
the $(p-l)$-th one which is overbalanced with excess number 1, similarly to
3d. We will also refer to the first $p-l-1$ nodes as the _left tail_ and to
the last $l-1$ as the _right tail_.

The $\mathsf{C}_{p}[USp(2N)]$ theory in (113) has also a superpotential
consisting of several terms, which we now illustrate. ${\cal W}_{\text{cub}}$
encodes cubic couplings between each antisymmetric field and the
bifundamentals on its sides. ${\cal W}_{\text{saw}}$ contains all the cubic
superpotential terms for the triangles in the saw of the form
$r_{j}b_{j}l_{j+1}$. Then we have also the $pb_{1}\ldots b_{p-2}s$ term and
two flippers that in the figure appear as the two singlet arcs.

The vanishing of the NSVZ $\beta$-functions and the superpotential constrain
the R-charges as follows:

$\displaystyle R[\text{antisymmetrics}]=2-\frac{2}{p+1}\,,$ $\displaystyle
R[b_{j}]=\frac{1}{p+1}\,,$ (114)

as well as181818To obtain the following relations we can fix the charges of
antisymmetrics and bifundamentals as in (5.1). One then solves the NSVZ
$\beta$-function condition to obtain that $R[r_{j}]+R[l_{j}]$ equals
$\tfrac{p-1}{p+1}$ if the $j$-th node is balanced or $\tfrac{2p-1}{p+1}$ if
the $j$-th node is the unbalanced one. Then one also takes into account the
constraint due to ${\cal W}_{\text{saw}}$ imposing
$R[r_{j}]+R[l_{j+1}]=2-\tfrac{1}{p+1}$. Analogously $R[s]$ and $R[p]$ follow.

$\displaystyle
R[r_{j}]=\begin{cases}R[r_{j-1}]-\frac{3}{p+1}\qquad&\text{if}\quad j\neq
p-l\\\ R[r_{p-l}]+\frac{2p-3}{p+1}\qquad&\text{if}\quad j=p-l\end{cases}\,,$
$\displaystyle R[l_{j}]=2-\frac{1}{p+1}-R[r_{j-1}]\,,$ $\displaystyle
R[p]=\begin{cases}2\frac{2p-1}{p+1}-R[r_{1}]\qquad&\text{if}\quad l=1\\\
2\frac{p-1}{p+1}-R[r_{1}]\qquad&\text{otherwise}\end{cases}\,,$ $\displaystyle
R[s]=\begin{cases}2\frac{2p-1}{p+1}-R[l_{p\\!-\\!1}]\qquad&\text{if}\quad
l=p-1\\\
2\frac{p-1}{p+1}-R[l_{p\\!-\\!1}]\qquad&\text{otherwise}\end{cases}\,.$ (115)

All the R-charges are fixed from the choice of $R[r_{1}]$. This implies that
there is a non-anomalous $U(1)_{C}$ global symmetry that can mix with the
R-symmetry. We will comment on this symmetry later.

#####  $l=0$.

In this case the $\mathsf{C}_{p}[USp(2pr)]$ theory is given by the following
quiver theory:

(116)

Notice that in this case all nodes are balanced, similarly to 3d.

The superpotential contains, other than the cubic and saw terms, a term linear
in the meson $pb_{1}\ldots b_{p\\!-\\!2}ar_{p\\!-\\!1}$191919The antisymmetric
$a$ used to dress the meson can be taken to be any of the antisymmetrics. All
the possible choices are identified in the chiral ring of the theory. For
simplicity we assume that the antisymmetric $a$ is the one charged under the
$USp(2(p-1)r)$ gauge group. and also a flipping term for the similar meson
$pb_{1}\ldots b_{p\\!-\\!2}r_{p\\!-\\!1}$. These are the analogues of the
dressed monopole superpotential and the flip of the undressed monopole that we
had in 3d.

The R-charges of the bifundamental and antisymmetric chirals are as in (5.1),
while the R-charge of the rest of the fields can again be obtained imposing
the vanishing of the NSVZ $\beta$-function and the superpotential constraints,
finding relations similar to those in (5.1).

##### Global symmetries and gauge invariant operators.

The $\mathsf{C}_{p}[USp(2N)]$ theory is characterized by a $USp(2N)$ flavor
symmetry. There is also an extra $U(1)_{C}$ symmetry that is compatible with
the superpotential in figure (113) and (116). The charge assignment for this
symmetry is:

$\displaystyle C[b_{j}]=C[\text{antisymmetrics}]=0\,,$ $\displaystyle
C[r_{j}]=C[s]=+1\,,$ $\displaystyle C[l_{j}]=C[r]=-1\,.$ (117)

From which it follows that also the flippers $F[sb_{p\\!-\\!1}]$ and
$F[l_{p\\!-\\!l\\!-\\!1}b_{p\\!-\\!l\\!+\\!1}\ldots b_{p\\!-\\!1}]$ have
$C$-charge $+1$ and $-1$ respectively. We claim that the $U(1)_{C}$ symmetry
is actually unfaithful, i.e. it trivializes in the low energy theory.
Therefore all the gauge invariant operators that can be constructed with a
non-zero charge under $U(1)_{C}$ are expected to vanish in the chiral ring.
This claim is supported by the fact that all the anomalies involving
$U(1)_{C}$ vanish and by the validity of the confining duality, since on the
dual side this symmetry is not present.

Below we discuss some interesting gauge invariant operators of the
$\mathsf{C}_{p}[USp(2N)]$ theory which we will later map to the singlets in
the confined dual frame (122). There is a traceless antisymmetric operator
$\mu=b_{p\\!-\\!1}^{2}$, with R-charge

$\displaystyle R[\mu]=\frac{2}{p+1}\,.$ (118)

Sometimes we will, by abuse of notation, refer to this operator as “moment
map”, in analogy with the 3d story in Section 2.

In addition we have also other gauge invariant operators that are singlets of
the $USp(2N)$ global symmetry. In the theories (113) and (116) we can
construct mesons using the fields in the saw of the quiver

$\displaystyle\mathsf{M}^{(k)}=\begin{cases}l_{j}b_{j}\ldots
b_{j-1+k}r_{j+k}\\\ pb_{1}\ldots b_{k-1}r_{k}\\\ l_{p-k}b_{p-k}\ldots
b_{p-2}s\end{cases}\qquad\text{for}\quad k\geq 1\,,$ (119)

such that the nodes $j$ and $j-1+k$ nodes belong to either the left or right
balanced tails. These operators have R-charge

$\displaystyle R[\mathsf{M}^{(k)}]=2-2\frac{k+1}{p+1}\qquad\text{for}\quad
k\geq 1\,.$ (120)

In fact, for any fixed value $k$ there are many such operators, however we
claim that the mesons of the same length $k$ constructed from the same
balanced tail are degenerate. We are then left with two sets of operators,
those constructed from the left tail and those from the right one, which we
name $\mathsf{M}_{L/R}^{(k)}$. By looking at the quivers in (113) and (116)
one finds that the operators are

$\displaystyle l\neq 0\quad:\quad\begin{cases}\mathsf{M}^{(k)}_{L}&\quad
k=1,\ldots,p-l-2\qquad\text{from the left tail}\,,\\\
\mathsf{M}^{(k)}_{R}&\quad k=1,\ldots,l-2\qquad\text{from the right
tail}\,,\\\ \end{cases}$ $\displaystyle l=0\quad:\quad\mathsf{M}^{(k)}\quad
k=1,\ldots,p-3\qquad\text{from the (only) tail}\,.$ (121)

These operators are not charged under the $USp(2N)$ global symmetry, therefore
we can construct exactly marginal deformations for the
$\mathsf{C}_{p}[USp(2N)]$ theory given by the product of
$\mathsf{M}^{(k)}_{L/R}$ operators that have total R-charge 2. For example,
considering only the product of two $\mathsf{M}^{(k)}_{L/R}$ operators we have
that $\mathsf{M}^{(k)}_{L/R}\mathsf{M}^{(p-j-1)}_{L/R}$ always have R-charge
2, provided that the operators exists in the given $\mathsf{C}^{p}[USp(2N)]$
theory.

By simply looking at the quivers in figures (113) and (116), one may notice
that there are many more gauge invariant operators that we have not
considered. However, we claim that all the gauge invariant operators carrying
a non-zero charge under the $U(1)_{C}$ unfaithful symmetry are set to zero in
the chiral ring, as we already mentioned previously.

####  5.1.1 Statement of the confining duality

We claim that the $\mathsf{C}_{p}[USp(2N)]$ theories confine to a traceless
antisymmetric field with ${\cal W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{p+1})$
superpotential plus flippers

(122)

where in the figure we depict the $\mathsf{C}_{p}[USp(2N)]$ theory in short as
the label $\mathsf{C}_{p}$ embedded in an arc.
On the r.h.s. of figure (122) we have flipping fields for the traces of $\Phi$
from power 2 up to $k(l)$, with

$\displaystyle k(l)=\min(l,p-l)\,.$ (123)

The above formula enjoys the property $k(l)=k(p-l)$. This will turn out to be
important when we will use this duality to prove the IKS duality, which maps
the gauge ranks from $N=pr+l$ to $N^{\prime}=pr^{\prime}+(p-l)$.
The superpotential on the l.h.s. of the duality in (122) labelled as ${\cal
W}_{\text{multi-meson}}$ contains all the possible exactly marginal
deformations given as the product of two or more $\mathsf{M}^{(k)}_{L/R}$
operators. Correspondingly on the r.h.s. we have the superpotential ${\cal
W}_{\text{multi-trace}}$ which contains all the exactly marginal double trace
operators. The reason why this duality is considered at a generic point of the
conformal manifold is the same as in the 3d confining duality for the ${\cal
C}^{p}[SU(N)]$ theory, discussed below figure (23).

As a first simple check of the confining duality in (122) we checked that the
anomalies $\mathop{\text{Tr}}\nolimits(U(1)_{R})$,
$\mathop{\text{Tr}}\nolimits(U(1)_{R}^{3})$ and
$\mathop{\text{Tr}}\nolimits(U(1)_{R}USp(2N)^{2})$ of the
$\mathsf{C}_{p}[USp(2N)]$ theory check with those of the WZ model. In the
following section we will provide a derivation of this duality.

####  5.1.2 The mirror theories $\check{\mathsf{C}}_{p}[USp(2N)]$

It is possible to construct a _mirror-like_ dual of the 4d ${\cal N}=1$
theories $\check{\mathsf{C}}_{p}[USp(2N)]$ following the strategies initiated
in Hwang:2020wpd  (see also Bottini:2021vms ; Hwang:2021ulb ; Comi:2022aqo ).
The mirror duals are different for $l\neq 0$ and $l=0$

In the theories in (LABEL:fig:4dCpmir) the superpotential contains ${\cal
W}_{\text{cub}}$, encoding all the cubic couplings between horizontal
bifundamentals and the antisymmetric fields on their sides, and ${\cal
W}_{\text{saw}}$, encoding the cubic couplings for each tooth composing the
saw. On top of that we turn on dressed mesonic operators constructed from the
fundamental flavors in the midlle of the quiver and flipping fields.

The R-charges of all the fields can be fixed from the superpotential and
vanishing $\beta$-function constraints. We get that all the horizontal
bifundamentals and the antisymmetric fields have R-charges (for both $l\neq 0$
and $l=0$)

$\displaystyle R[\text{bif}]=\frac{1}{2}-\frac{1}{2(p+1)}\,,$ $\displaystyle
R[\text{antisymmetrics}]=\frac{2}{p+1}\,.$ (125)

The R-charges of the $p,q$ fields for $l\neq 0$ and of the $s$ field for $l=0$
are fixed to be

$\displaystyle l\neq 0\quad:\quad\begin{cases}&R[p]=1-\frac{l}{p+1}\\\
&R[q]=1-\frac{p-l}{p+1}\end{cases}\,,$ $\displaystyle l=0\quad:\quad
R[s]=1-\frac{p}{p+1}\,.$ (126)

The R-charge of the fields in the saw are

$\displaystyle R[l_{j}]=1-\frac{1}{p+1}+R[l_{j-1}]\,,$ $\displaystyle
R[r_{j}]=1+\frac{1}{p+1}-R[l_{j+1}]\,.$ (127)

Notice that all the R-charges are fixed from the assignment of $R[l_{1}]$,
indicating again the existance of a non-anomalous abelian $U(1)_{C^{\prime}}$
symmetry. The above formula holds both for $l\neq 0$ and for $l=0$.

##### Global symmetries and gauge invariant operators.

The UV global symmetry of the quiver theories in (LABEL:fig:4dCpmir) is:

$\displaystyle SU(2)^{N}\times SU(2)_{F}\times U(1)_{C^{\prime}}\,,$ (128)

where $SU(2)^{N}$ are the $N$ symmetries rotating the chirals in the saw,
$SU(2)_{F}$ is the symmetry acting on the fundamental flavors in the middle of
the quiver, and $U(1)_{C^{\prime}}$ is a non-anomalous abelian global symmetry
consistent with the superpotential as explained at the end of the previous
section. In the IR the $SU(2)^{N}$ symmetries enhance to form a $USp(2N)$
group, which is mapped to the $USp(2N)$ global symmetry of the
$\mathsf{C}_{p}[USp(2N)]$ theory. The remaining $SU(2)_{F}\times
U(1)_{C^{\prime}}$ symmetries are unfaithful and thus we do not map them to
some symmetry of the $\mathsf{C}_{p}[USp(2N)]$ theory. All the gauge invariant
operators that can be constructed and that are charged under these symmetries
thus vanish in the chiral ring. Moreover, we checked that all the anomalies
involving $SU(2)_{F}$ and $U(1)_{C^{\prime}}$ are exactly zero.

We can construct a gauge invariant operator which is the traceless
antisymmetric representation of the emergent $USp(2N)$ symmetry. This operator
is built by casting together mesonic operators in a $N\times N$ matrix. The
$N-1$ elements on the diagonal are given as the traces of the $N-1$
antisymmetric fields. The off-diagonal entries are filled with mesons built
from the saw as $l_{j}b_{j}\ldots b_{k-1}r_{k}$. The resulting operator has
R-charge $\tfrac{2}{p+1}$ and is naturally mapped to the moment map $\mu$ of
the $\mathsf{C}_{p}[USp(2N)]$ theory.

There are also gauge invariant operators that are singlets under the $USp(2N)$
global symmetry. These are constructed by taking the square meson built from
central flavors and dressing them with powers of the antisymmetric. Their
R-charges are

$\displaystyle l\neq
0\quad:\quad\begin{cases}R[p^{2}A_{L}^{j}]=2-2\frac{l-j}{p+1}\qquad&j=0,\ldots,l-2\\\
R[q^{2}A_{R}^{j}]=2-2\frac{p-l-j}{p+1}\qquad&j=0,\ldots,p-l-2\end{cases}\,,$
$\displaystyle l=0\quad:\quad R[s^{2}A^{j}]=2\frac{j+1}{p+1}\qquad
j=0,\ldots,p-2\,.$ (129)

These operators for $l\neq 0$ are mapped as

$\displaystyle p^{2}A_{L}^{j}\quad$
$\displaystyle\leftrightarrow\quad\mathsf{M}_{L}^{(l-1-j)}\qquad
j=0,\ldots,l-2\,,$ $\displaystyle q^{2}A_{R}^{j}\quad$
$\displaystyle\leftrightarrow\quad\mathsf{M}_{R}^{(p-l-1-j)}\qquad
j=0,\ldots,p-l-2\,.$ (130)

For $l=0$ they are mapped as

$\displaystyle s^{2}A^{j}\quad\leftrightarrow\quad\mathsf{M}^{(p-1-j)}\qquad
j=0,\ldots,p-l-1\,.$ (131)

#### Deriving the mirror theories

The strategy to derive these dualities is very similar to the 3d case in
Section 2. We start from a pair of 4d ${\cal N}=1$ theories that enjoy a
mirror-like duality. These theories have been introduced in Hwang:2020wpd
(see also Hwang:2021ulb ; Comi:2022aqo ) and are called
$E_{\rho}^{\sigma}[USp(2N)]$. These theories are inspired by and, under a
suitable dimensional reduction, they reduce to the 3d ${\cal N}=4$
$T_{\rho}^{\sigma}[SU(N)]$ theories., where $\rho$ and $\sigma$ labels are
partitions of $N$. $\sigma$ encodes the subgroup of $USp(2N)$ which is the
manifest global symmetry of the theory (analogous to the 3d Higgs branch
symmetry), while $\rho$ defines a subgroup of $USp(2N)$ which is global
symmetry that emerges in the IR from the enhancement of a collection of
$SU(2)$ UV symmetries (analogous to the 3d Coulomb branch symmetry). For the
case of interest, the $\mathsf{C}_{p}[USp(2N=2(pr+l))]$ theories are obtained
starting from a $E_{\rho}^{\sigma}[USp(2(pr+l)]$ theory with partitions

$\displaystyle\sigma=[1^{N}]\,,\qquad\rho=[(r+1)^{l},r^{p-l}]\,.$

Notice that these are the same partitions in (10) that defined the ${\cal
C}_{p}[SU(N)]$ theories. In particular for $l=0$ we have $\rho=[r^{p}]$. From
the partition $\sigma$ we see that the theory has a $USp(2N)$ manifest global
symmetry while from the partition $\rho$ we observe a $USp(2l)\times
USp(2(p-l))$ enhanced global symmetry. Similarly to 3d, the mirrors are
obtained by swapping $\rho\leftrightarrow\sigma$, meaning that the manifest
and emergent global symmetries are swapped.

Starting from this mirror-like pair we want to deform the theories in order to
break completely the $USp(2l)\times USp(2(p-l))$ symmetry, leaving a theory
with only a $USp(2N)$ global symmetry group which is that of the
$\mathsf{C}_{p}[USp(2N)]$ theory. The precise step followed in order to deform
the mirror-like pair are different for $l\neq 0$ and $l=0$. Let us then
discuss separately the two cases in the following paragraphs.

#####  $l\neq 0$.

In this case the strategy to derive the $\check{\mathsf{C}}_{p}[USp(2N)]$
mirror theory is schematically depicted below. On the l.h.s. of the figure we
depict the steps to obtain $\mathsf{C}_{p}[USp(2N)]$ from the relevant
$E_{\rho}^{\sigma}[USp(2N)]$ theory, while on the r.h.s. we depict the mirror
dual procedure. Notice that we do not write all the singlets and all the
superpotential terms at each step to make the discussion simpler. Indeed, one
can keep track of all the details to find precisely the stated duality between
the $\mathsf{C}_{p}[USp(2N)]$ theories and their mirror
$\check{\mathsf{C}}_{p}[USp(2N)]$.

(133)

We start from the 4d ${\cal N}=1$ mirror pair in the first line of the figure
above. The two starting theories have a $USp(2(pr+l))\times USp(2l)\times
USp(2p-2l)$ global symmetry. On the electric side the first group is manifest
while the remaining two are emergent in the IR and come from the enhancement
of the $SU(2)^{p}=SU(2)^{l}\times SU(2)^{p-l}$ groups. Similarly, on the
magnetic side the $USp(2(pr+l))$ group is emergent while the remaining two
groups are manifest.202020To actually see the claimed enhancement it is very
important to consider the correct singlets in the theory. However, these are
not very important in the following discussion and therefore we do not
consider them. For more details one can see Hwang:2020wpd ; Comi:2022aqo
These theories can be deformed by an analogues of the nilpotent deformations
that are usually performed in 3d, as discussed extensively in Hwang:2020wpd .
Even though these are not nilpotent in 4d, we will still refer to them as
“nilpotent” by abuse of notation. In the first step, leading to the second
line in figure (133), we turn on a next-to-maximal nilpotent masses for the
antisymmetric operators charged under $USp(2l)$ and $USp(2p-2l)$. On the
electric side this deformation has the effect of integrating out part of the
fields in the saw, while on the magnetic side the mass has the effect of
integrating out many of the fundamental flavors and breaking $USp(2p)\to
SU(2)_{V_{1}}$ and $USp(2p-2l)\to SU(2)_{V_{2}}$. In the second step, leading
to the third line of figure (133), on the magnetic side we turn on the
superpotential corresponding to $bpq$, with the effect of breaking the
$SU(2)_{V_{1}}\times SU(2)_{V_{2}}$ symmetry to a $SU(2)_{F}$ diagonal
subgroup. On the electric side, the superpotential dual to $pbq$, has the
effect of integrating two out of the four chirals seen by the $USp(2(p-l)r)$
gauge node and also turns on the ${\cal W}_{\text{triangle}}$ superpotential.
All in all, we land on the duality in the third line of figure (133), which is
precisely the stated duality between the theory $\mathsf{C}_{p}[USp(2N)]$
theory in (113) and its mirror in the first line of (LABEL:fig:4dCpmir). Again
we point out that in the presentation we did not keep track of the singlet
fields for simplicity, but this can be done explicitly.

#####  $l=0$.

In this case the strategy is schematically depicted in the figure below.
Notice that we depict only some singlets and we do not write all the
superpotential terms at each step, to make the discussion simpler. Indeed, one
can keep track of all the details to find precisely the stated duality between
the $\mathsf{C}_{p}[USp(2N)]$ theories and their mirror
$\check{\mathsf{C}}_{p}[USp(2N)]$.

(134)

We start from the 4d ${\cal N}=1$ duality in the first line of the figure
above. The global symmetry is $USp(2pr)\times USp(2p)$, where the first group
is emergent on the r.h.s. while the second is emergent on the l.h.s. In the
first step, leading to the second line of figure (134), we perform a nilpotent
mass for the antisymmetric operator of $USp(2p)$ with the effect of breaking
$USp(2p)\to SU(2)$. On the electric side the mass has the effect of
integrating out half of the saw, so that we are left with a single $U(1)$
symmetry acting on the fields of the saw which is the Cartan of the $SU(2)$.
On the magnetic side the effect is that we are left with a single flavor in a
$s^{2}A^{p}$ superpotential. In the second step, leading to the third line of
figure (134), we turn on $q^{2}b^{2}$ on the magnetic side. The dual
deformation on the electric side is the long triangle superpotential. After
this deformation the $SU(2)$ symmetry becomes unfaithful. All in all we are
left with the duality in the third line of figure (134), which is precisely
the duality between the $\mathsf{C}_{p}[USp(2N)]$ theory is (116) and its
mirror in the second line of (LABEL:fig:4dCpmir).

###  5.2 Derivation of the confining duality

The confining duality in (122) can be derived iteratively with the same
strategy as in the 3d case decribed in Section 2.2. We will not describe the
iterations explicitly since these are identical to those in 3d, but we just
present the inductive step and the inductive basis that are needed.

##### Inductive step.

If $N=pr+l>p+1$, the $\mathsf{C}_{p}[USp(2N)]$ theory ejoys a duality with a
$\mathsf{C}_{p}[USp(2(N-1))]$ theory plus singlets

(135)

where the operators of the $\mathsf{C}_{p}[USp(2(N-1))]$ theory are primed in
order to distinguish them from those of the $\mathsf{C}_{p}[USp(2N)]$. On the
r.h.s. the superpotential includes all the possible terms between
$\mu^{\prime},b,a$ of degree $p+1$, such that $b$ appears only with even
powers. There is also a flipping term for the mesons $\mathsf{M}_{L}^{(p-l)}$.
Notice that for $l=0,1$ this operator does not exist, therefore we do not have
any flipping terms and thus there is no extra singlet.

For $l\neq 0$ the operator map of the duality (135) is For $l\neq 0$ the map
of the operators in the duality (49) is

$\displaystyle\mu$
$\displaystyle\quad\longleftrightarrow\quad\\{\mu^{\prime},b,a\\}\,\,,$
$\displaystyle\mathsf{M}_{L}^{(j)}$
$\displaystyle\quad\longleftrightarrow\quad{\mathsf{M}^{\prime}}_{L}^{(j)}\,,$
$\displaystyle\mathsf{M}_{R}^{(j)}$
$\displaystyle\quad\longleftrightarrow\quad\\{{\mathsf{M}^{\prime}}_{R}^{(j)},F[{\mathsf{M}^{\prime}}_{L}^{(p-l)}]\\}\,.$
(136)

For $l=0$ the situation is simpler, since on the electric side we have no
$\mathsf{M}_{R}^{j}$ operators and in the dual we have no
${\mathsf{M}^{\prime}}_{L}^{j}$. The operator map is then simply

$\displaystyle\mu$
$\displaystyle\quad\longleftrightarrow\quad\\{\mu^{\prime},b,a\\}\,\,,$
$\displaystyle\mathsf{M}^{(j)}$
$\displaystyle\quad\longleftrightarrow\quad{\mathsf{M}^{\prime}}_{R}^{(j)}\,.$
(137)

Notice that on the r.h.s. we have $l=p-1$ and looking at (5.1) this implies
that the set ${\mathsf{M}^{\prime}}_{L}^{(j)}$ is empty.

##### Inductive basis.

The basis of the inductive derivation consists in the confining duality (122)
for the $\mathsf{C}_{p}[USp(2(p+1))]$ theory, that is $N=p+1$ or equivalently
$r=1$, $l=1$

(138)

We do not provide a complete derivation of the two above dualities (135) and
(138). The derivation can be done by repeating the same strategy used in 3d in
Section 2.2, with the only difference being that we now employ the new 4d
mirror theories $\check{\mathsf{C}}_{p}[USp(2N)]$ and the 4d
Intriligator–Pouliot duality Intriligator:1995ne  instead of the Aharony
duality.

##  6 Derivation of the 4d ${\cal N}=1$ $USp(2N)$ Intriligator duality

In this section we use the confining duality (122) for the 4d ${\cal N}=1$
$\mathsf{C}_{p}[USp(2N)]$ theories to provide a derivation of the version with
symplectic gauge group of the Kutasov–Schwimmer duality (IKS), which was
proposed by Intriligator in Intriligator:1995ff . The derivation only assumes
the validity of the confinement property of the $\mathsf{C}_{p}[USp(2N)]$
theories which, as we discussed in the previous section, is a consequence of
the Intriligator–Pouliot (IP) duality. This implies that the IKS duality can
be derived only assuming the validity of IP duality.

The IKS duality relates two gauge theories with symplectic gauge group, $2F$
fundamental chirals and an antisymmetric field as

(139)

with $\tilde{N}=p(F-2)-N$. The mapping of the chiral ring generators is

$\left\\{\begin{tabular}[]{c}$Q\Phi^{j}Q$\\\
$\mathop{\text{Tr}}\nolimits(\Phi^{j})$\end{tabular}\right\\}\qquad\longleftrightarrow\qquad\left\\{\begin{tabular}[]{c}$F[q\phi^{p-1-j}q]$\\\
$\mathop{\text{Tr}}\nolimits(\phi^{j})$\end{tabular}\right\\}\qquad\begin{tabular}[]{c}$j=0,1,\ldots,p-1$
\,,\\\ $j=2,3,\ldots,p-1$ \,.\end{tabular}$ (140)

Notice that for $p=1$ the antisymmetric chiral is massive and we recover the
Intriligator–Pouliot (IP) duality Intriligator:1995ne .

The derivation of the IKS duality is very similar to that of the 3d KP duality
described in Section 3. In short, we first deconfine the antisymmetric chiral
with a suitable $\mathsf{C}_{p}[USp(2N)]$ theory, we then apply iteratively
the basic IP duality along the quiver and finally we reconfine the resulting
$\mathsf{C}_{p}[USp(2\tilde{N})]$ theory to land on the expected IKS dual
frame. Let us next describe this procedure a bit more in detail, focusing on
the case $l\neq 0$ for concreteness. The derivation for $l=0$ works similarly
with just some minor differences.

We first consider a different version of the confining duality (122) in which
we flip the flipper $F[l_{p\\!-\\!l\\!-\\!1}b_{p\\!-\\!l\\!+\\!1}\ldots
b_{p\\!-\\!1}]$ of the $\mathsf{C}_{p}[USp(2N)]$ theory in figure
(113),212121Similarly, for $l=0$ one flips the singlet $r$ in figure (116).
which is in the fundamental representation of the global $USp(2N)$ symmetry.
The resulting theory will then confine to a $USp(2N)$ antisymmetric plus a
singlet. We name the flipped theory as $\mathsf{C}^{\text{F}}_{p}[USp(2N)]$
and the new confining duality can be depicted as:

(141)

Notice that the extra fundamental chiral $q$ on the r.h.s. does not enter in
the superpotential.

To derive the IKS duality we start from the l.h.s. of figure (139) and use the
modified confining duality in (141) to deconfine the antisymmetric with
superpotential ${\cal W}\sim\mathop{\text{Tr}}\nolimits(\Phi^{j})$ and a
fundamental chiral. Writing explicitly the quiver of the
$\mathsf{C}^{\text{F}}_{p}[USp(2N)]$ theory, we have

(142)

where on the l.h.s. we added for convenience a set of flippers for the traces
of $\Phi$ from $2$ up to $k(l)$, where we recall that

$\displaystyle k(l)=\min(l,p-l)\,.$ (143)

We also added ${\cal W}_{\text{multi-trace}}$ in order to use the deconfining
duality in (141). On the r.h.s. of (142) we are not keeping track of all the
exactly marginal deformations corresponding to ${\cal W}_{\text{multi-meson}}$
to make the discussion simpler. We will comment about them at the end of the
subsection.

At this point, we notice that on the r.h.s. (142) the rightmost gauge node
does not have an antisymmetric field, therefore we can apply the IP duality.
Also, some new fields and superpotential terms are created. We do not give all
the details, which can be recovered by studying in detail the precise effect
of IP duality. The result is the following theory: The main effect of the
dualization is that the antisymmetric and one fundamental of the adjacent node
are removed. We then apply the IP duality on the second-to-last gauge node to
obtain

We keep iterating the IP duality to reach finally

(146)

Where we have defined $s=F-r-3$. We can now recognize the tail to be the
modified $\mathsf{C}^{\text{F}}_{p}[USp(2\tilde{N})]$ theory with
$N=p(F-r-3)+(p-l)$. Confining it using the duality in (141) we obtain

(147)

By keeping track of all the superpotential terms involving the
$\mathsf{M}_{L/R}^{(j)}$ operators, we obtain the theory in (146) with all the
exactly marginal deformations turned on. Therefore, the confinement generates
precisely the theory in (147) including the exactly marginal interactions
$\mathcal{W}_{\text{multi-trace}}$. Moreover, the flippers produced at this
last step map directly to those we introduced in (142), so we can just remove
them from both sides to recover the IKS duality (139) on a generic point of
the conformal manifold.

##  7 Conclusions and outlook

In this paper we provided a derivation of some Kutasov–Schwimmer-like (KS-
like) dualities which involve a rank-2 field and a degree-$(p+1)$
superpotential, specifically the 3d $\mathcal{N}=2$ Kim–Park duality for
unitary gauge group and adjoint matter, and the 4d $\mathcal{N}=1$
Intriligator duality for symplectic gauge group and antisymmetric matter. The
derivation uses the technique of the deconfinement, which consists of trading
the rank-2 field for an auxiliary quiver gauge theory that can then be further
dualized to the desired dual frame.

We proceeded in two steps. We first introduced two new classes of linear
quivers, the 3d $\mathcal{N}=2$ ${\cal C}_{p}[SU(N)]$ and the 4d
$\mathcal{N}=1$ $\mathsf{C}_{p}[USp(2N)]$ theories, and showed that they
respectively confine into an $SU(N)$ adjoint and a $USp(2N)$ antisymmetric
chiral with degree-$(p+1)$ superpotential. We then used these confining
dualities to deconfine the rank-$2$ fields in the KS-like dualities. This
allowed us to obtain quiver gauge theories that we could further dualize with
known Seiberg-like dualities, which only involve fields in the fundamental
representation, so to obtain the expected dual gauge group connected to an
additional linear quiver. Finally, such dual quiver could be traded back for
the rank-2 field using again our new confining dualities, so to find the
desired KS-like dual with the correct superpotential and flipping fields.

All the steps in our derivation assume the validity of only some fundamental
Seiberg-like duality, which are the Intriligator–Pouliot duality in 4d and the
Aharony duality in 3d (since the variants with monopole superpotential can be
obtained as deformations of it). As a byproduct, we are thus able to provide a
proof, which was previously missing, of the integral identities of partition
functions and supersymmetric indices of the KS-like dualities by only assuming
the validity of those for the Seiberg-like dualities, which instead have been
already proven in the math literature.

Our ${\cal C}_{p}[SU(N)]$ theories and their confining duality, which played a
crucial role in the derivation of the KS-like dualities, are deeply connected
to the 4d $\mathcal{N}=2$ $D_{p}[SU(N)]$ SCFTs. Indeed, for $\gcd(p,N)=1$ we
showed that the quiver of the former can be obtained as a monopole
superpotential deformation of the 3d $\mathcal{N}=4$ quiver for the circle
compactification of the latter. Such monopole deformation is expected to be